**Meet the editor**

Dr. V. N. Stavrou is currently an adjunct member at the Hellenic Naval Academy, Piraeus, Greece. He received his MSc degree and PhD degree in Theoretical Solid State Physics from the University of Essex in England, in 1995 and in 1999, respectively. He has held postdoctoral positions at the following research institutions: (a) Deutsche Forschungsanstalt fuer Luft und Raumfahrt e.V (German Aerospace Research Center) in Germany; (b) Helsinki University of Technology; (c) State University of New York (SUNY) at Buffalo, USA; and (d) the University of Iowa, USA. He is specifically interested in researching on the electronic, optical, and lattice properties of semiconducting low-dimensional structures (quantum dots and quantum wells, among others). These properties are of special importance for quantum computing architecture and laser technology. He has published his research work in reputable journals related, among others, to quantum computing, spintronics, decoherence in quantum dots, diluted magnetic semiconductors, and phonon models in low-dimensional structures.

Contents

**Preface VII**

Er'el Granot

**Section 1 Nonmagnetic Quantum Dots 1**

**Quantum Dots 21**

**Realization 43** Ákos Nemcsics

**Rolling Bearing 77** Ke Yan and Bei Yan

Jung Y. Huang

Wei Ma

Chapter 1 **Exact Model for Single Atom Transistor 3**

Ciann-Dong Yang and Shih-Ming Huang

Chapter 3 **Droplet Epitaxy as a Tool for the QD-Based Circuit**

Chapter 5 **CdTe Quantum Dot Fluorescence Thermometry of**

Zequn Chen, Chuli Sun, Wei Guo and Zhuo Chen

Chapter 6 **Quantum Dots-Based Nano-Coatings for Inhibition of Microbial**

Chapter 7 **Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells 99**

Chapter 8 **Redox-Mediated Quantum Dots as Fluorescence Probe and**

Eepsita Priyadarshini, Kamla Rawat and Himadri Bihari Bohidar

Chapter 4 **Colloidal III–V Nitride Quantum Dots 61**

**Biofilms: A Mini Review 87**

**Their Biological Application 133**

Chapter 2 **A Quantum Trajectory Interpretation of Magnetic Resistance in**

## Contents

#### **Preface XI**


#### Chapter 9 **Enhancement of Photosynthetic Productivity by Quantum Dots Application 147**

Angela Janet Murray, John Love, Mark D. Redwood, Rafael L. Orozco, Richard K. Tennant, Frankie Woodhall, Alex Goodridge and Lynne Elaine Macaskie

Preface

Quantum dots are mainly made with semiconducting materials in which the carriers are confined in all three dimensions. The carrier confinement makes the quantum dots signifi‐ cant in technology due to highly tunable electronic and optical properties. Their optoelec‐ tronic properties depend on the size and the shape of the dots, among other external parameters. Potential applications of these nanostructures include semiconductor lasers, quantum computing, transistors, and sensors, among others. Due to worldwide interest in the domain of technology, several fabrication techniques have been used to control the growth of quantum dots and the quality of the samples. Stranski-Krastanov (SK) random growth in molecular beam epitaxy (MBE) and metalorganic chemical vapor deposition (MOCVD) are the most commonly used growth methods to create high-quality quantum dot samples, in both size and shape. Several advanced computational methods like Monte-Carlo simulations, *k · p* theory, ab initio calculations, and density functional theory, among

others, have been developed to calculate the electronic structure of quantum dots.

biology, and medicine are included in Section 2.

In the current book, there is a collection of interesting topics related to quantum dots made with magnetic and nonmagnetic semiconductor materials and their applications. This book is divided into two sections. In Section 1, the chapters are related to nonmagnetic quantum dots and their applications. More specifically, exact models and numerical methods have been presented to describe the analytical solution of the carrier wave functions, the quantum mechanical aspects of quantum dots, and the comparison of the latter to experimental data. Furthermore, methods to produce quantum dots, synthesis techniques of colloidal quantum dots, and applications on sensors and biology, among others, are included in this section. In Section 2, a few topics of magnetic quantum dots and their applications are presented. The section starts with a theoretical model to describe the magnetization dynamics in magnetic quantum dot array and the description of dilute magnetic semiconducting quantum dots and their applications. Additionally, a few applications of magnetic quantum dot in sensors,

As an editor of this book, I would like to thank all the authors for their contribution through the up-to-date research of their high-quality work. Lastly, I would like to express my thanks and gratitude to the InTechOpen team for their support during the preparation of this book.

> **Dr. Vasilios N. Stavrou** Hellenic Naval Academy

> > Piraeus, Greece

#### **Section 2 Magnetic Quantum Dots 175**


## Preface

Chapter 9 **Enhancement of Photosynthetic Productivity by Quantum Dots**

Chapter 10 **Magnetization Dynamics in Arrays of Quantum Dots 177**

Chapter 11 **Dilute Magnetic Semiconducting Quantum Dots: Smart**

Jejiron Maheswari Baruah and Jyoti Narayan

Kannaiyan Pandian and Oluwatobi S. Oluwafemi

Chapter 13 **Quantum Dots and Fluorescent and Magnetic Nanocomposites: Recent Investigations and Applications in Biology and**

**Materials for Spintronics 187**

Chapter 12 **Mn-Doped ZnSe Quantum Dots as Fluorimetric**

**Mercury Sensor 201**

**Medicine 221** Anca Armăşelu

Pablo F. Zubieta Rico, Daniel Olguín and Yuri V. Vorobiev

Sundararajan Parani, Ncediwe Tsolekile, Bambesiwe M.M. May,

Angela Janet Murray, John Love, Mark D. Redwood, Rafael L. Orozco, Richard K. Tennant, Frankie Woodhall, Alex Goodridge and

**Application 147**

**VI** Contents

Lynne Elaine Macaskie

**Section 2 Magnetic Quantum Dots 175**

Quantum dots are mainly made with semiconducting materials in which the carriers are confined in all three dimensions. The carrier confinement makes the quantum dots signifi‐ cant in technology due to highly tunable electronic and optical properties. Their optoelec‐ tronic properties depend on the size and the shape of the dots, among other external parameters. Potential applications of these nanostructures include semiconductor lasers, quantum computing, transistors, and sensors, among others. Due to worldwide interest in the domain of technology, several fabrication techniques have been used to control the growth of quantum dots and the quality of the samples. Stranski-Krastanov (SK) random growth in molecular beam epitaxy (MBE) and metalorganic chemical vapor deposition (MOCVD) are the most commonly used growth methods to create high-quality quantum dot samples, in both size and shape. Several advanced computational methods like Monte-Carlo simulations, *k · p* theory, ab initio calculations, and density functional theory, among others, have been developed to calculate the electronic structure of quantum dots.

In the current book, there is a collection of interesting topics related to quantum dots made with magnetic and nonmagnetic semiconductor materials and their applications. This book is divided into two sections. In Section 1, the chapters are related to nonmagnetic quantum dots and their applications. More specifically, exact models and numerical methods have been presented to describe the analytical solution of the carrier wave functions, the quantum mechanical aspects of quantum dots, and the comparison of the latter to experimental data. Furthermore, methods to produce quantum dots, synthesis techniques of colloidal quantum dots, and applications on sensors and biology, among others, are included in this section. In Section 2, a few topics of magnetic quantum dots and their applications are presented. The section starts with a theoretical model to describe the magnetization dynamics in magnetic quantum dot array and the description of dilute magnetic semiconducting quantum dots and their applications. Additionally, a few applications of magnetic quantum dot in sensors, biology, and medicine are included in Section 2.

As an editor of this book, I would like to thank all the authors for their contribution through the up-to-date research of their high-quality work. Lastly, I would like to express my thanks and gratitude to the InTechOpen team for their support during the preparation of this book.

> **Dr. Vasilios N. Stavrou** Hellenic Naval Academy Piraeus, Greece

**Section 1**

**Nonmagnetic Quantum Dots**

**Nonmagnetic Quantum Dots**

**Chapter 1**

Provisional chapter

**Exact Model for Single Atom Transistor**

Exact Model for Single Atom Transistor

An exact model for a single atom transistor was developed. Using two simplifying assumptions (1) that the device is restricted to a narrow conducting wire and (2) that the atom can be simulated by a point impurity potential, the model can be simplified considerably and an exact analytical solution can be derived. Thus, analytical solution is approximated to a close-form solution in three important regimes: at the vicinity of the resonance energy (near the maximum peak), at the vicinity of the inverse resonance, i.e., Fano resonance (near the minimum), and at the threshold energy where a universal transmission pattern appears. Finally, physical values are applied to demonstrate that this device can operate as a transistor, when it is calibrated to work at the vicinity of its

DOI: 10.5772/intechopen.70445

Keywords: quantum dots, quantum point defect, point impurity, quantum transistor, single

In accordance with the rapid growth of calculation power, the transistor dimensions shrink exponentially. Surprisingly, more than 50 years after Gordon Moore made his observation in 1965 (or, more accurately, its revised form a decade later), that the number of transistors on a single chip doubles every couple of years, this observation is still valid [1, 2]. The number of transistors in a chip keeps growing despite the fact that the chip clock speed and its power

To meet the demands of the current trend, the average transistor size should decrease to the dimensions of a single atom, which is the smallest quantum dot, within about a decade.

The ability to move and manipulate single Xenon atoms (in Eigler and Schweizer lab at IBM's

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Almaden Research Center) in the early 1990s was a great leap in that direction [3].

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70445

maximum and minimum points.

Er'el Granot

Er'el Granot

Abstract

atom transistor

consumption seem to be stagnated.

1. Introduction

Provisional chapter

## **Exact Model for Single Atom Transistor** Exact Model for Single Atom Transistor

## Er'el Granot

Additional information is available at the end of the chapter Er'el Granot Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70445

Abstract

An exact model for a single atom transistor was developed. Using two simplifying assumptions (1) that the device is restricted to a narrow conducting wire and (2) that the atom can be simulated by a point impurity potential, the model can be simplified considerably and an exact analytical solution can be derived. Thus, analytical solution is approximated to a close-form solution in three important regimes: at the vicinity of the resonance energy (near the maximum peak), at the vicinity of the inverse resonance, i.e., Fano resonance (near the minimum), and at the threshold energy where a universal transmission pattern appears. Finally, physical values are applied to demonstrate that this device can operate as a transistor, when it is calibrated to work at the vicinity of its maximum and minimum points.

DOI: 10.5772/intechopen.70445

Keywords: quantum dots, quantum point defect, point impurity, quantum transistor, single atom transistor

## 1. Introduction

In accordance with the rapid growth of calculation power, the transistor dimensions shrink exponentially. Surprisingly, more than 50 years after Gordon Moore made his observation in 1965 (or, more accurately, its revised form a decade later), that the number of transistors on a single chip doubles every couple of years, this observation is still valid [1, 2]. The number of transistors in a chip keeps growing despite the fact that the chip clock speed and its power consumption seem to be stagnated.

To meet the demands of the current trend, the average transistor size should decrease to the dimensions of a single atom, which is the smallest quantum dot, within about a decade.

The ability to move and manipulate single Xenon atoms (in Eigler and Schweizer lab at IBM's Almaden Research Center) in the early 1990s was a great leap in that direction [3].

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

In the attempts to meet this requirement, scientists already demonstrated the operation on several atoms and dopants [4–9] and even on single atoms [10–12]. The atom can be utilized as a stationary gate [10] or as a dynamic switch [13].

Such a device indeed consists of a single atom, but its conductor leads are of mesoscopic dimensions. Consequently, this is a complicated device to simulate and requires heavy software.

However, since the process is dominated by resonant tunneling, the model can be simplified considerably. In this case, only a single energy level of the quantum dot is relevant to the process, and therefore, it can be simulated by a point defect potential.

A point defect potential has a single eigen bound state, and therefore, it can simulate a quantum dot or a small atom in a relatively narrow spectral domain.

While a delta function can simulate a point defect in one-dimensional (1D) systems, a twodimensional (2D) delta function cannot scatter and therefore cannot simulate a quantum dot. Azbel suggested to use an Impurity-D-Function (IDF) to simulate point defects in 2D quantum systems [14, 15] (for a comprehensive discussion and derivations, see Ref. [16]).

Therefore, the system can be described by the following stationary Schrödinger equation

in which normalized units (where Planck constant is ħ = 1, and the electron's mass is m = 1/2)

*x*/*L*


Single Atom

A Single Atom

Conductor Conductor

Barrier/Insulator

U yð Þ¼ 0 0 <sup>&</sup>lt; <sup>y</sup> <sup>&</sup>lt; <sup>w</sup> <sup>∞</sup> else �

V xð Þ¼ V xj j <sup>&</sup>lt; <sup>L</sup> <sup>0</sup> else �

and for the point impurity potential, we use an asymmetric Impurity D Functions (see Refs.

where <sup>r</sup> <sup>¼</sup> <sup>b</sup>xx <sup>þ</sup> <sup>b</sup>yy, and the location of the point impurity is <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>b</sup>xx<sup>0</sup> <sup>þ</sup> <sup>b</sup>yy0. <sup>ρ</sup><sup>0</sup> is the de-Broglie wavelength of the impurity's bound eigenstate. The eigenenergy of the bound state of this

<sup>π</sup> <sup>p</sup> exp �y<sup>2</sup>=ρ<sup>2</sup> � �

ffi �

8:98 ρ2 0

<sup>0</sup> ½ � ð Þ � r<sup>0</sup> Ψð Þ¼ x; y EΨð Þ x; y (1)

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445 5

<sup>ρ</sup>ln <sup>ρ</sup>0=<sup>ρ</sup> � � <sup>δ</sup>ð Þ<sup>x</sup> ; (4)

(2)

(3)

(5)

Ψð Þþ x; y V xð Þþ U yð Þ� D r

is the boundaries' potential, which confines the dynamics to the wire geometry.

Dð Þ¼ r lim ρ!0

The potential of the gap between the wires is represented by the finite potential barrier,

2 ffiffiffi

<sup>E</sup><sup>0</sup> ¼ � 16expð Þ �<sup>γ</sup> ρ2 0

�∇<sup>2</sup>

*y*/*w*


0

0.5

1

1.5

2

were used. In this equation,

Figure 1. Model schematic.

[14, 15])

impurity is

where γ ffi 0.577 is Euler constant [22].

Several years later, the IDF was utilized in simulations of resonant tunneling through an opaque quantum barrier via a point defect in the presence [17] and absence [18] of a magnetic field. However, in these models, it was taken that there is a degeneracy in the y-direction, i.e., it was assumed that the barrier's transverse dimension is infinite and therefore cannot be applied in a system, where the current is carried by narrow wires (as in modern single atom transistor's devices).

On the other hand, conductance of nanowires with defects, but without a barrier, received lots of attention in the literature, exhibiting a wealth of physical phenomena [4–9, 19–21].

It is the purpose of this chapter to integrate the two, i.e., to formulate a model, which incorporates resonant tunneling via a point defect and wire conductance. That is, both the potential barrier and the impurity are located in the nanowire.

## 2. The model

The system is illustrated in Figure 1. It consists of two semi-infinite conducting wires, which are separated by an insulating gap. Within this gap, there is a quantum dot, which characterizes the resonance atom. To simplify the analysis, it is assumed that the wire boundaries in the y direction are totally reflecting, i.e., the wire is bounded by an infinitely large potential. Moreover, the single atom is modeled by a point defect potential.

It is also taken that this is a 2D model, i.e., there are no variations in the third dimension. This is a good approximation provided the wire is narrower in the z-dimension. Another advantage in constructing the model in 2D is that point impurities potential cannot exist in higher than two dimensions (see Ref. [16]).

#### Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445 5

Figure 1. Model schematic.

In the attempts to meet this requirement, scientists already demonstrated the operation on several atoms and dopants [4–9] and even on single atoms [10–12]. The atom can be utilized as

Such a device indeed consists of a single atom, but its conductor leads are of mesoscopic dimensions. Consequently, this is a complicated device to simulate and requires heavy

However, since the process is dominated by resonant tunneling, the model can be simplified considerably. In this case, only a single energy level of the quantum dot is relevant to the

A point defect potential has a single eigen bound state, and therefore, it can simulate a

While a delta function can simulate a point defect in one-dimensional (1D) systems, a twodimensional (2D) delta function cannot scatter and therefore cannot simulate a quantum dot. Azbel suggested to use an Impurity-D-Function (IDF) to simulate point defects in 2D quantum

Several years later, the IDF was utilized in simulations of resonant tunneling through an opaque quantum barrier via a point defect in the presence [17] and absence [18] of a magnetic field. However, in these models, it was taken that there is a degeneracy in the y-direction, i.e., it was assumed that the barrier's transverse dimension is infinite and therefore cannot be applied in a system, where the current is carried by narrow wires (as in modern single atom transis-

On the other hand, conductance of nanowires with defects, but without a barrier, received lots

It is the purpose of this chapter to integrate the two, i.e., to formulate a model, which incorporates resonant tunneling via a point defect and wire conductance. That is, both the potential

The system is illustrated in Figure 1. It consists of two semi-infinite conducting wires, which are separated by an insulating gap. Within this gap, there is a quantum dot, which characterizes the resonance atom. To simplify the analysis, it is assumed that the wire boundaries in the y direction are totally reflecting, i.e., the wire is bounded by an infinitely large potential.

It is also taken that this is a 2D model, i.e., there are no variations in the third dimension. This is a good approximation provided the wire is narrower in the z-dimension. Another advantage in constructing the model in 2D is that point impurities potential cannot exist in higher than

of attention in the literature, exhibiting a wealth of physical phenomena [4–9, 19–21].

barrier and the impurity are located in the nanowire.

Moreover, the single atom is modeled by a point defect potential.

process, and therefore, it can be simulated by a point defect potential.

quantum dot or a small atom in a relatively narrow spectral domain.

systems [14, 15] (for a comprehensive discussion and derivations, see Ref. [16]).

a stationary gate [10] or as a dynamic switch [13].

4 Nonmagnetic and Magnetic Quantum Dots

software.

tor's devices).

2. The model

two dimensions (see Ref. [16]).

Therefore, the system can be described by the following stationary Schrödinger equation

$$-\nabla^2 \Psi(\mathbf{x}, y) + [V(\mathbf{x}) + \mathcal{U}(y) - D(\mathbf{r}' - \mathbf{r}\_0)] \Psi(\mathbf{x}, y) = E \Psi(\mathbf{x}, y) \tag{1}$$

in which normalized units (where Planck constant is ħ = 1, and the electron's mass is m = 1/2) were used. In this equation,

$$\mathcal{U}I(y) = \begin{cases} 0 & 0 < y < w \\ \infty & \text{else} \end{cases} \tag{2}$$

is the boundaries' potential, which confines the dynamics to the wire geometry.

The potential of the gap between the wires is represented by the finite potential barrier,

$$V(\mathbf{x}) = \begin{cases} V & |\mathbf{x}| < L \\ 0 & \text{else} \end{cases} \tag{3}$$

and for the point impurity potential, we use an asymmetric Impurity D Functions (see Refs. [14, 15])

$$D(\mathbf{r}) = \lim\_{\rho \to 0} \frac{2\sqrt{\pi} \exp\left(-y^2/\rho^2\right)}{\rho \ln(\rho\_0/\rho)} \delta(\mathbf{x}),\tag{4}$$

where <sup>r</sup> <sup>¼</sup> <sup>b</sup>xx <sup>þ</sup> <sup>b</sup>yy, and the location of the point impurity is <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>b</sup>xx<sup>0</sup> <sup>þ</sup> <sup>b</sup>yy0. <sup>ρ</sup><sup>0</sup> is the de-Broglie wavelength of the impurity's bound eigenstate. The eigenenergy of the bound state of this impurity is

$$E\_0 = -\frac{16\exp(-\gamma)}{\rho\_0^2} \cong -\frac{8.98}{\rho\_0^2} \tag{5}$$

where γ ffi 0.577 is Euler constant [22].

It should be stressed that this point impurity potential is an excellent approximation to a small quantum dot defect, i.e., a finite but small impurity, with a radius a and potential V<sup>0</sup> provided

$$\rho\_0 = 2a \exp\left(\frac{2}{V\_0 a^2} + \frac{\mathcal{V}}{2}\right). \tag{6}$$

where Em � E � (mπ/w)

with the boundary condition

G<sup>þ</sup>

Therefore,

where

and

km � ffiffiffiffiffiffi Em <sup>p</sup> <sup>¼</sup> � ∂2 <sup>∂</sup>x<sup>2</sup> <sup>G</sup><sup>þ</sup>

> ∂ ∂x G<sup>þ</sup>

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ¼ ; Em

where the tags stand for spatial derivatives.

χ<sup>þ</sup> k,nð Þ¼ x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E � ð Þ mπ=w

q

The general Green function is then

<sup>2</sup> and G<sup>þ</sup>

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ ; E solves the equation

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ¼� ; Em δ x � x<sup>0</sup> ð Þ (13)

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445

x > x<sup>0</sup>

x < x<sup>0</sup>

(15)

7

(16)

(17)

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ¼ ; Em 0 forx ! �∞: (14)

E,mð Þ �x<sup>0</sup>

E,mð Þ �x<sup>0</sup>

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þþ ; Em ½ � V xð Þþ Em G<sup>þ</sup>

Em p G<sup>þ</sup>

χ�

χ�

In the case of a rectangular barrier (in a slightly different writing, see Ref. [23])

and Km � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tn <sup>¼</sup> expð Þ �2iknL

ffi <sup>2</sup> expð Þ �2iknL � <sup>2</sup>KnL 1 þ i Kð Þ <sup>n</sup>=kn � kn=Kn =2

> <sup>2</sup> <sup>1</sup> � ikn Kn

> > ikn Kn

Kn kn þ kn Kn

<sup>2</sup> <sup>1</sup> <sup>þ</sup>

Cn <sup>¼</sup> <sup>1</sup>

Dn <sup>¼</sup> <sup>1</sup>

Rn ¼ � <sup>i</sup> 2

V � Em <sup>p</sup> <sup>¼</sup>

E,mð Þx =χ�

E,mð Þ� x<sup>0</sup> χ�

E,mð Þ� x<sup>0</sup> χ�

E,mð Þ �x =χ�

E,mð Þ x<sup>0</sup>

E,mð Þ �x<sup>0</sup>

E,m 0 ð Þ �x =χ�

E,m 0 ð Þ �x =χ�

expð Þþ iknx tnRnexpð Þ �iknx x < �L tnCnexpð Þþ �Knx tnDnexpð Þ Knx j j x < L

q

;

cosh 2ð Þþ KnL i Kð Þ <sup>n</sup>=kn � kn=Kn sinh 2ð Þ KnL =2

tnexpð Þ iknx x > L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V � E þ ð Þ mπ=w

� �expð Þ KnL <sup>þ</sup> iknL ; (18)

� �expð Þ �KnL <sup>þ</sup> iknL (19)

� �sinh 2ð Þ KnL : (20)

2

;

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ ; Em <sup>∓</sup><sup>i</sup> ffiffiffiffiffiffi

χ� E,m 0 ð Þx =χ�

8 >>>><

>>>>:

8 ><

>:

2

χ� E,m 0 ð Þx =χ�

#### 3. Derivation of the exact analytical solution

The solution of Eq. (1) reads (see Refs. [16, 19])

$$
\Psi(\mathbf{r}) = \Psi\_{inc}(\mathbf{r}) - \frac{G^+(\mathbf{r}, \mathbf{r}\_0) \Psi\_{inc}(\mathbf{r}\_0)}{1 + \int d\mathbf{r}' G^+(\mathbf{r}', \mathbf{r}\_0) D(\mathbf{r}' - \mathbf{r}\_0)} \int d\mathbf{r}' D(\mathbf{r}' - \mathbf{r}\_0) \tag{7}
$$

where Ψinc(r) is the incoming wavefunction, G<sup>+</sup> (r, r0) is the outgoing 2D Green function, i.e., G+ (r, r0) is the solution of the partial differential equation

$$-\nabla^2 G^+(\mathbf{r}, \mathbf{r}\_0) + [V(\mathbf{x}) + \mathcal{U}(\mathcal{y}) - E]G^+(\mathbf{r}, \mathbf{r}\_0) = -\delta(\mathbf{r} - \mathbf{r}\_0).\tag{8}$$

Both the incoming wavefunction ψinc(r) and the Green function can be written as a superposition of the homogenous solution of Eq. (1) φ<sup>þ</sup> m,Eð Þr , i.e., solution of the equation where the impurity is absent. These solutions are characterized by two quantum parameters: the energy E and the mode number m, namely

$$-\nabla^2 \phi\_{m,E}^{\pm}(\mathbf{r}) + [V(\mathbf{x}) + \mathcal{U}(\mathbf{y}) - E] \phi\_{m,E}^{\pm}(\mathbf{r}) = 0\tag{9}$$

where

$$\phi\_{m,E}^{\pm}(\mathbf{r}) = \sqrt{\frac{2}{w}} \sin \left( m \pi y / w \right) \chi\_{E,m}^{\pm}(\mathbf{x}) \tag{10}$$

and χ� E,mð Þx are the homogeneous solutions of the 1D equation

$$-\frac{\partial^2}{\partial \mathbf{x}^2} \chi\_{E,m}^{\pm}(\mathbf{x}) + \left[ V(\mathbf{x}) + (m\pi/w)^2 - E \right] \chi\_{E,m}^{\pm}(\mathbf{x}) = \mathbf{0},\tag{11}$$

where the superscript "+" and "�" stand for propagation to the right and to the left respectively.

Similarly, it is convenient to formulate the 2D Green function in terms of the 1D one [G<sup>þ</sup> <sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ ; E ]:

$$G^{+}(\mathbf{r}, \mathbf{r}') = \frac{2}{w} \sum\_{n=1}^{\ast} \sin \left( n \pi y / w \right) \sin \left( n \pi y' / w \right) \mathbb{G}\_{1D}^{+}(\mathbf{x}, \mathbf{x}'; \mathbf{E}\_n) \tag{12}$$

where Em � E � (mπ/w) <sup>2</sup> and G<sup>þ</sup> <sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ ; E solves the equation

$$-\frac{\partial^2}{\partial \mathbf{x}^2} \mathcal{G}\_{\mathrm{1D}}^+(\mathbf{x}, \mathbf{x}'; E\_m) + [V(\mathbf{x}) + E\_m] \mathcal{G}\_{\mathrm{1D}}^+(\mathbf{x}, \mathbf{x}'; E\_m) = -\delta(\mathbf{x} - \mathbf{x}') \tag{13}$$

with the boundary condition

$$\frac{\partial}{\partial \mathbf{x}} G\_{1D}^{+}(\mathbf{x}, \mathbf{x}'; E\_m) \mp i \sqrt{E\_m} G\_{1D}^{+}(\mathbf{x}, \mathbf{x}'; E\_m) = 0 \text{ for } \mathbf{x} \to \pm \infty. \tag{14}$$

Therefore,

It should be stressed that this point impurity potential is an excellent approximation to a small quantum dot defect, i.e., a finite but small impurity, with a radius a and potential V<sup>0</sup> provided

> 2 <sup>V</sup>0a<sup>2</sup> <sup>þ</sup> <sup>γ</sup> 2

Gþð Þ r;r<sup>0</sup> Ψincð Þ r<sup>0</sup>

Both the incoming wavefunction ψinc(r) and the Green function can be written as a superposi-

impurity is absent. These solutions are characterized by two quantum parameters: the energy

m,Eð Þþ r ½ � V xð Þþ U yð Þ� E ϕ�

sin ð Þ mπy=w χ�

h i

where the superscript "+" and "�" stand for propagation to the right and to the left respec-

Similarly, it is convenient to formulate the 2D Green function in terms of the 1D one

sin ð Þ nπy=w sin nπy<sup>0</sup> ð Þ =w G<sup>þ</sup>

<sup>2</sup> � <sup>E</sup>

χ�

ffiffiffiffi 2 w r

E,mð Þþ x V xð Þþ ð Þ mπ=w

<sup>0</sup> ð Þ ,r<sup>0</sup> D r

<sup>0</sup> ð Þ � r<sup>0</sup>

Gþð Þþ r;r<sup>0</sup> ½ � V xð Þþ U yð Þ� E Gþð Þ¼� r;r<sup>0</sup> δð Þ r � r<sup>0</sup> : (8)

ð dr 0 D r

(r, r0) is the outgoing 2D Green function, i.e.,

m,Eð Þr , i.e., solution of the equation where the

m,Eð Þ¼ r 0 (9)

E,mð Þx (10)

E,mð Þ¼ x 0; (11)

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ ; En (12)

� �

: (6)

<sup>0</sup> ð Þ � r<sup>0</sup> (7)

ρ<sup>0</sup> ¼ 2a exp

1 þ ð dr 0 G<sup>þ</sup> r

3. Derivation of the exact analytical solution

Ψð Þ¼ r Ψincð Þ� r

(r, r0) is the solution of the partial differential equation

The solution of Eq. (1) reads (see Refs. [16, 19])

6 Nonmagnetic and Magnetic Quantum Dots

where Ψinc(r) is the incoming wavefunction, G<sup>+</sup>

tion of the homogenous solution of Eq. (1) φ<sup>þ</sup>

� ∂2 <sup>∂</sup>x<sup>2</sup> <sup>χ</sup>�

G<sup>þ</sup> r,r <sup>0</sup> ð Þ¼ <sup>2</sup> w X∞ n¼1

�∇<sup>2</sup> ϕ�

> ϕ� m,Eð Þ¼ r

E,mð Þx are the homogeneous solutions of the 1D equation

�∇<sup>2</sup>

E and the mode number m, namely

G+

where

and χ�

tively.

[G<sup>þ</sup>

<sup>1</sup><sup>D</sup> x, x<sup>0</sup> ð Þ ; E ]:

$$\mathbf{G}\_{\rm{ID}}^{+}(\mathbf{x},\mathbf{x}';\mathbf{E}\_{m}) = \begin{cases} \frac{\chi\_{\rm{E},m}^{\pm}(\mathbf{x})/\chi\_{\rm{E},m}^{\pm}(\mathbf{x}\_{0})}{\chi\_{\rm{E},m}^{\pm}(\mathbf{x})/\chi\_{\rm{E},m}^{\pm}(\mathbf{x}\_{0}) - \chi\_{\rm{E},m}^{\pm}(-\mathbf{x})/\chi\_{\rm{E},m}^{\pm}(-\mathbf{x}\_{0})} & \mathbf{x} > \mathbf{x}\_{0} \\\\ \frac{\chi\_{\rm{E},m}^{\pm}(-\mathbf{x})/\chi\_{\rm{E},m}^{\pm}(-\mathbf{x}\_{0})}{\chi\_{\rm{E},m}^{\pm}(\mathbf{x})/\chi\_{\rm{E},m}^{\pm}(\mathbf{x}\_{0}) - \chi\_{\rm{E},m}^{\pm}(-\mathbf{x})/\chi\_{\rm{E},m}^{\pm}(-\mathbf{x}\_{0})} & \mathbf{x} < \mathbf{x}\_{0} \end{cases} \tag{15}$$

where the tags stand for spatial derivatives.

In the case of a rectangular barrier (in a slightly different writing, see Ref. [23])

$$\chi\_{k,n}^{+}(\mathbf{x}) = \begin{cases} \exp(i\mathbf{k}\_{n}\mathbf{x}) + t\_{n}R\_{n}\exp(-i\mathbf{k}\_{n}\mathbf{x}) & \mathbf{x} < -L \\\ t\_{n}\mathbb{C}\_{n}\exp(-\mathbf{K}\_{n}\mathbf{x}) + t\_{n}D\_{n}\exp(\mathbf{K}\_{n}\mathbf{x}) & |\mathbf{x}| < L \\\ t\_{n}\exp(i\mathbf{k}\_{n}\mathbf{x}) & \mathbf{x} > L \end{cases} \tag{16}$$

where

$$k\_{n} \equiv \sqrt{E\_{n}} = \sqrt{E - (m\pi/w)^{2}} \text{ and } K\_{n} \equiv \sqrt{V - E\_{n}} = \sqrt{V - E + (m\pi/w)^{2}};$$

$$t\_{n} = \frac{\exp(-2ik\_{n}L)}{\cosh(2K\_{n}L) + i(K\_{n}/k\_{n} - k\_{n}/K\_{n})\sinh(2K\_{n}L)/2} \tag{17}$$

$$\cong 2\frac{\exp(-2ik\_{n}L - 2K\_{n}L)}{1 + i(K\_{n}/k\_{n} - k\_{n}/K\_{n})/2},$$

$$C\_{n} = \frac{1}{2}\left(1 - \frac{ik\_{n}}{K\_{n}}\right)\exp(K\_{n}L + ik\_{n}L),\tag{18}$$

$$D\_n = \frac{1}{2} \left( 1 + \frac{i k\_n}{K\_n} \right) \exp\left( -K\_n L + i k\_n L \right) \tag{19}$$

and

$$R\_n = -\frac{\mathrm{i}}{2} \left( \frac{\mathrm{K}\_n}{\mathrm{k}\_n} + \frac{\mathrm{k}\_n}{\mathrm{K}\_n} \right) \sinh(2\mathrm{K}\_n \mathrm{L}).\tag{20}$$

The general Green function is then

$$\mathbf{G}\_{1D}^{+}(\mathbf{x},\mathbf{x}\_{0};\mathcal{E}\_{m}) \cong \frac{1}{\mathcal{M}\_{n}} \begin{cases} \frac{\exp(i\mathbf{k}\_{n}\mathbf{x})}{\mathcal{C}\_{n}\exp(-\mathcal{K}\_{n}\mathbf{x}\_{0}) + \mathcal{D}\_{n}\exp(\mathcal{K}\_{n}\mathbf{x}\_{0})} & L < \mathbf{x} \\\\ \frac{\mathcal{C}\_{n}\exp(-\mathcal{K}\_{n}\mathbf{x}) + \mathcal{D}\_{n}\exp(\mathcal{K}\_{n}\mathbf{x})}{\mathcal{C}\_{n}\exp(-\mathcal{K}\_{n}\mathbf{x}\_{0}) + \mathcal{D}\_{n}\exp(\mathcal{K}\_{n}\mathbf{x}\_{0})} & \mathbf{x}\_{0} < \mathbf{x} < L \\\\ \frac{\mathcal{C}\_{n}\exp(\mathcal{K}\_{n}\mathbf{x}) + \mathcal{D}\_{n}\exp(-\mathcal{K}\_{n}\mathbf{x})}{\mathcal{C}\_{n}\exp(\mathcal{K}\_{n}\mathbf{x}\_{0}) + \mathcal{D}\_{n}\exp(-\mathcal{K}\_{n}\mathbf{x}\_{0})} & -L < \mathbf{x} < \mathbf{x}\_{0} \\\\ \frac{\exp(-i\mathcal{K}\_{n}\mathbf{x})}{\mathcal{C}\_{n}\exp(\mathcal{K}\_{n}\mathbf{x}\_{0}) + \mathcal{D}\_{n}\exp(-\mathcal{K}\_{n}\mathbf{x}\_{0})} & \mathbf{x} < -L \end{cases} \tag{21}$$

where

$$M\_n = -K\_n[\tanh[K\_n(L - \mathbf{x}\_0) + i\theta(k\_n)] + \tanh[K\_n(L + \mathbf{x}\_0) + i\theta(k\_n)]],$$

and then

$$\mathcal{G}\_{1D}^{+}(\mathbf{x}, \mathbf{x}\_{0}; E\_{m}) = \frac{\frac{\tanh[K\_{m}(L-\mathbf{x}) + i\theta(k\_{m})]}{\tanh[K\_{m}(L-\mathbf{x}\_{0}) + i\theta(k\_{m})]}}{-K\_{m}[\tanh[K\_{m}(L-\mathbf{x}\_{0}) + i\theta(k\_{m})] + \tanh[K\_{m}(L+\mathbf{x}\_{0}) + i\theta(k\_{m})])} \text{ for } |\mathbf{x}| < L \quad \text{(22)}$$

using

$$
\tan \theta(k) = -k/K. \tag{23}
$$

which can be written as

p

8 >>>><

>>>>:

� δð Þ� p � m

sin <sup>p</sup>π<sup>y</sup> w � �χ<sup>þ</sup>

E,pð Þx

1 <sup>2</sup><sup>π</sup> ln <sup>ρ</sup><sup>0</sup> ρ � � þ 2 w X∞ n¼1

In the case where the incoming particle's energy satisfies

<sup>t</sup><sup>11</sup> � <sup>1</sup> � sin <sup>2</sup> <sup>π</sup>y<sup>0</sup>

then only a single mode propagates, in which case

1 <sup>2</sup><sup>π</sup> ln <sup>ρ</sup><sup>0</sup> ρ � � <sup>þ</sup> <sup>2</sup> w X∞ n¼1

sin <sup>m</sup>πy<sup>0</sup> w

ð Þ π=w

<sup>Ψ</sup>ð Þ¼ <sup>x</sup> ! <sup>∞</sup>, y sin <sup>π</sup><sup>y</sup>

where t<sup>11</sup> is the transmission coefficient to remain at x ! ∞ in the first mode, which is

w � � <sup>2</sup> <sup>w</sup> G<sup>þ</sup>

A plot of T<sup>11</sup> = |t11|2 as a function of the incoming particle's energy is presented in Figure 2.

Clearly, a resonance occurs when the real part of the denominator of Eq. (29) vanishes, i.e.

In general, it is a complex transcendental equation; however, in case of an opaque barrier,

1 2Kn

¼ �2

G1Dð Þffi� x0; x0; En

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � �ℜG<sup>þ</sup>

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � �G<sup>þ</sup>

� � sin <sup>p</sup>πy<sup>0</sup>

w � � χ<sup>þ</sup>

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � �G<sup>þ</sup>

<sup>2</sup> <sup>&</sup>lt; <sup>E</sup> <sup>&</sup>lt; ð Þ <sup>2</sup>π=<sup>w</sup>

w � �χ<sup>þ</sup>

E,mð Þ x<sup>0</sup> χ<sup>þ</sup> E,pð Þ x<sup>0</sup>

2

<sup>1</sup>Dð Þ x0; x0; E<sup>1</sup>

<sup>1</sup>Dð Þ <sup>x</sup>0; <sup>x</sup>0; En exp � <sup>n</sup>πρ

<sup>1</sup>Dð Þ <sup>x</sup>0; <sup>x</sup>0; En exp � <sup>n</sup>πρ

2 <sup>w</sup> <sup>G</sup><sup>þ</sup>

<sup>1</sup><sup>D</sup> x0; x0; Ep � �

E, <sup>1</sup>ð Þx t<sup>11</sup> (28)

2w

2w

exp½ � �2KnL kncosh 2ð Þ Knx<sup>0</sup>

� �<sup>2</sup> � � (29)

� �<sup>2</sup> � � <sup>¼</sup> <sup>0</sup>: (30)

þ iε<sup>n</sup> (31)

<sup>V</sup> ; (32)

2w � �<sup>2</sup> � �

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445

> 9 >>>>=

(27)

9

>>>>; :

<sup>1</sup>Dð Þ <sup>x</sup>0; <sup>x</sup>0; En exp � <sup>n</sup>πρ

<sup>Ψ</sup>ð Þ¼ <sup>x</sup> <sup>&</sup>gt; L, y <sup>X</sup>

when

when

1 2π ln <sup>ρ</sup><sup>0</sup> ρ � �

Eq. (24) can be further simplified to

þ 2 w X∞ n¼1

<sup>ε</sup><sup>n</sup> � exp½ � �2KnL sin 2½ � <sup>θ</sup>ð Þ kn cosh 2ð Þ Knx<sup>0</sup> Kn

Then

$$\begin{split}G\_{1D}^{+}(\mathbf{x}\_{0},\mathbf{x}\_{0};\mathbf{E}\_{m})&=\frac{1}{-K\_{m}[\tanh[K\_{m}(L-\mathbf{x}\_{0})+i\theta(k\_{m})]+\tanh[K\_{m}(L+\mathbf{x}\_{0})+i\theta(k\_{m})]]}\\&\cong\frac{\operatorname{2}{-2K\_{m}[1-2\exp[-2K\_{m}L-2i\theta(k\_{m})]\cosh[2K\_{m}\mathbf{x}\_{0}]]}\end{split} \tag{24}$$

where the last term is an approximation in the limit of opaque barriers.

When the incoming wavefunction is the mth mode

$$
\Psi\_{\rm inc}(\mathbf{r}) = \sin\left(\frac{m\pi y}{w}\right) \chi^{+}\_{E,m}(\mathbf{x})\tag{25}
$$

then, the solution (in all space) reads

$$\begin{split} \Psi(\mathbf{r}) &= \sin\left(\frac{m\pi y}{\varpi}\right) \chi\_{E,m}^{+}(\mathbf{x}) \\ &+ \frac{\sin\left(\frac{m\pi y\_0}{\varpi}\right) \chi\_{E,m}^{+}(\mathbf{x}\_0) \frac{2}{\varpi} \sum\_{n=1}^{\infty} \sin\left(\frac{n\pi y}{\varpi}\right) \sin\left(\frac{n\pi y\_0}{\varpi}\right) G\_{1D}^{+}(\mathbf{x}, \mathbf{x}\_0; E\_n) \\ &+ \frac{1}{2\pi} \ln\left(\frac{\rho\_0}{\rho}\right) + \frac{2}{\varpi} \sum\_{n=1}^{\infty} \sin^2\left(\frac{n\pi y\_0}{\varpi}\right) G\_{1D}^{+}(\mathbf{x}\_0, \mathbf{x}\_0; E\_n) \exp\left(-\left(\frac{n\pi \rho}{2\varpi}\right)^2\right) \end{split} \tag{26}$$

which can be written as

G<sup>þ</sup>

8 Nonmagnetic and Magnetic Quantum Dots

where

and then

G<sup>þ</sup>

using

Then

<sup>1</sup>Dð Þ¼ x; x0; Em

G<sup>þ</sup>

<sup>1</sup>Dð Þ¼ x0; x0; Em

then, the solution (in all space) reads

þ

<sup>Ψ</sup>ð Þ¼ <sup>r</sup> sin <sup>m</sup>π<sup>y</sup>

sin

1 2π ffi

When the incoming wavefunction is the mth mode

w � �

mπy<sup>0</sup> w � �

ln <sup>ρ</sup><sup>0</sup> ρ � � χ<sup>þ</sup> E,mð Þx

χ<sup>þ</sup> E,mð Þ x<sup>0</sup>

þ 2 w X∞ n¼1

where the last term is an approximation in the limit of opaque barriers.

<sup>Ψ</sup>incð Þ¼ <sup>r</sup> sin <sup>m</sup>π<sup>y</sup>

2 w X∞ n¼1

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � �

<sup>1</sup>Dð Þffi x, x0; Em

1 Mn 8

>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

expð Þ iknx Cnexpð Þþ �Knx<sup>0</sup> Dnexpð Þ Knx<sup>0</sup>

Cnexpð Þþ �Knx Dnexpð Þ Knx Cnexpð Þþ �Knx<sup>0</sup> Dnexpð Þ Knx<sup>0</sup>

Cnexpð Þþ Knx Dnexpð Þ �Knx Cnexpð Þþ Knx<sup>0</sup> Dnexpð Þ �Knx<sup>0</sup>

expð Þ �iknx Cnexpð Þþ Knx<sup>0</sup> Dnexpð Þ �Knx<sup>0</sup>

Mn ¼ �Kn½ � tanh½Knð Þþ L � x<sup>0</sup> iθð Þ kn � þ tanh½ � Knð Þþ L þ x<sup>0</sup> iθð Þ kn

tanh½ � Kmð Þþ L�x iθð Þ km tanh½ � Kmð Þþ L�x<sup>0</sup> iθð Þ km �Km½ � tanh½Kmð Þþ L � x<sup>0</sup> iθð Þ km � þ tanh½ � Kmð Þþ L þ x<sup>0</sup> iθð Þ km

L < x

x<sup>0</sup> < x < L

(21)

forj j x < L (22)

(24)

(26)

�L < x < x<sup>0</sup>

x < �L

tan θð Þ¼� k k=K: (23)

E,mð Þx (25)

<sup>1</sup>Dð Þ x; x0; En

2w � �<sup>2</sup> � �

1 �Km½ � tanh½Kmð Þþ L � x<sup>0</sup> iθð Þ km � þ tanh½ � Kmð Þþ L þ x<sup>0</sup> iθð Þ km

1 �2Km 1 � 2exp½ � �2KmL � 2iθð Þ km cosh 2½ � Kmx<sup>0</sup> � �

> w � �

sin <sup>n</sup>π<sup>y</sup> w � �

G<sup>þ</sup>

χ<sup>þ</sup>

sin

nπy<sup>0</sup> w � �

<sup>1</sup>Dð Þ <sup>x</sup>0; <sup>x</sup>0; En exp � <sup>n</sup>πρ

G<sup>þ</sup>

$$\begin{split} \Psi(\mathbf{x} > L, y) &= \sum\_{p} \sin\left(\frac{p\pi y}{w}\right) \chi\_{\mathbf{E}, p}^{+}(\mathbf{x}) \\ &\times \left\{ \delta(p - m) - \frac{\sin\left(\frac{m\pi y\_{0}}{w}\right) \sin\left(\frac{p\pi y\_{0}}{w}\right) \frac{\chi\_{\mathbf{E}, m}^{+}(\mathbf{x}\_{0})}{\chi\_{\mathbf{E}, p}^{+}(\mathbf{x})} \frac{2}{w} G\_{10}^{+}(\mathbf{x}\_{0}, \mathbf{x}\_{0}; E\_{p}) \right. \\ &\left. - \frac{1}{2\pi} \ln\left(\frac{\rho\_{0}}{\rho}\right) + \frac{2}{w} \sum\_{n=1}^{\infty} \sin^{2}\left(\frac{n\pi y\_{0}}{w}\right) G\_{10}^{+}(\mathbf{x}\_{0}, \mathbf{x}\_{0}; E\_{n}) \exp\left(-\left(\frac{m\pi \rho}{2w}\right)^{2}\right) \right\}. \end{split} \tag{27}$$

In the case where the incoming particle's energy satisfies

$$\left(\pi/w\right)^{2} < E < \left(2\pi/w\right)^{2}$$

then only a single mode propagates, in which case

$$\Psi(\mathbf{x}\rightarrow\ast,\mathbf{y}) = \sin\left(\frac{\pi y}{w}\right) \chi\_{\mathbf{E},1}^{+}(\mathbf{x}) t\_{11} \tag{28}$$

where t<sup>11</sup> is the transmission coefficient to remain at x ! ∞ in the first mode, which is

$$t\_{11} \equiv 1 - \frac{\sin^2\left(\frac{\pi y\_0}{w}\right) \frac{2}{w} G\_{1D}^+(\mathbf{x}\_0, \mathbf{x}\_0; \mathbf{E}\_1)}{\frac{1}{2\pi} \ln\left(\frac{\rho\_0}{\rho}\right) + \frac{2}{w} \sum\_{n=1}^{\prime\prime} \sin^2\left(\frac{n\pi y\_0}{w}\right) G\_{1D}^+(\mathbf{x}\_0, \mathbf{x}\_0; \mathbf{E}\_n) \exp\left(-\left(\frac{n\pi \rho}{2w}\right)^2\right)}\tag{29}$$

A plot of T<sup>11</sup> = |t11|2 as a function of the incoming particle's energy is presented in Figure 2.

Clearly, a resonance occurs when the real part of the denominator of Eq. (29) vanishes, i.e. when

$$\frac{1}{2\pi}\ln\left(\frac{\rho\_0}{\rho}\right) + \frac{2}{w}\sum\_{n=1}^{\infty} \sin^2\left(\frac{n\pi y\_0}{w}\right) \Re G\_{1D}^+(\mathbf{x}\_0, \mathbf{x}\_0; E\_n) \exp\left(-\left(\frac{n\pi \rho}{2w}\right)^2\right) = 0. \tag{30}$$

In general, it is a complex transcendental equation; however, in case of an opaque barrier, Eq. (24) can be further simplified to

$$G\_{1D}(\mathbf{x}\_0, \mathbf{x}\_0; E\_n) \cong -\frac{1}{2\mathcal{K}\_n} + i\varepsilon\_n \tag{31}$$

when

$$\varepsilon\_{n} \equiv \frac{\exp[-2K\_{n}L]\sin\left[2\theta(k\_{n})\right]\cosh(2K\_{n}\mathbf{x}\_{0})}{K\_{n}} = -2\frac{\exp[-2K\_{n}L]k\_{n}\cosh(2K\_{n}\mathbf{x}\_{0})}{V},\tag{32}$$

Figure 2. Plot of the T<sup>11</sup> = |t11|2 , i.e., the probability to remain in the first mode of propagation as a function of the normalized energy. The barrier parameters were L = 2w and V = 2/w<sup>2</sup> , and the defect parameters were ρ<sup>0</sup> = 300w, x<sup>0</sup> = 0, and y<sup>0</sup> = w/2. The dotted line represents the barrier's energy Eb = V + π<sup>2</sup> /w<sup>2</sup> , and the dashed line represents the resonance energy Eres.

$$\text{rand } \sin\left[2\theta(k\_1)\right] = \frac{-2kK}{K^2 + k^2} = \frac{-2kK}{V} \,. \tag{33}$$

which has a solution provided 4ρ<sup>0</sup> > w, otherwise the impurity can be regarded as a perturba-

w � � 1

2

� �<sup>=</sup> sin <sup>2</sup> <sup>π</sup>y<sup>0</sup>

ln <sup>4</sup>ρ<sup>0</sup> w

1

w

w

� � � � �<sup>2</sup> ( ) (39)

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445

� � � ð Þ <sup>1</sup> <sup>þ</sup> <sup>i</sup>2K1ε<sup>n</sup>

V xð Þ¼�λδð Þx (41)

� �: (42)

(40)

11

(43)

When the resonant level exits, then the resonance energy ER is approximately

<sup>w</sup><sup>2</sup> <sup>1</sup> � sin <sup>4</sup> <sup>π</sup>y<sup>0</sup>

K<sup>1</sup> <sup>w</sup> <sup>2</sup><sup>π</sup> ln <sup>4</sup>ρ<sup>0</sup> w

Since in this regime only, one transverse mode is propagating, the system in practice reduces to a 1D problem, where the 2D impurity can be replaced by a 1D delta function

<sup>w</sup>ln 4ρ0=<sup>w</sup> � � sin <sup>2</sup> <sup>π</sup>y<sup>0</sup>

Therefore, in the 1D analogy the point potential depends not only on the impurity's de-Broglie

In this case, the barrier's transmission can be as high as 1. It depends on the location of the

However, there is a point where a minimum occurs. When the incoming particle's energy

� �expð Þ ik1<sup>x</sup> expð Þ �2ik1<sup>L</sup> <sup>þ</sup> <sup>i</sup><sup>Ξ</sup>

cosh 2ð Þ K1x<sup>0</sup>

<sup>1</sup>Dð Þ <sup>x</sup>0; <sup>x</sup>0; En exp � <sup>n</sup>πρ

tan Ξ ¼ �ð Þ K=k � k=K =2: (44)

2w

� �<sup>2</sup> � � <sup>¼</sup> 0 (45)

π2

E,1ð Þx

<sup>λ</sup> <sup>¼</sup> <sup>4</sup><sup>π</sup>

wavelength in free space, but on its location (y0) and the wire's width as well.

w

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � �ℜG<sup>þ</sup>

which at the vicinity of the second mode threshold can be approximated by

point defect in the horizontal dimension, namely, at the resonance energy

<sup>Ψ</sup>ð Þ¼ <sup>r</sup> <sup>i</sup> sin <sup>π</sup><sup>y</sup>

tion and does not carry a resonant level.

In this approximation,

potential

where

where

satisfies

1 2π ln <sup>ρ</sup><sup>0</sup> ρ � � þ 2 w X∞ n¼2

ER ffi V þ

<sup>Ψ</sup>ð Þffi <sup>r</sup> sin <sup>π</sup><sup>y</sup>

In Figure 2, the resonance energy is presented by a dashed line.

w � �χ<sup>þ</sup>

Then, Eq. (30) can be approximated as

$$\frac{1}{2\pi}\ln\left(\frac{\rho\_0}{\rho}\right) - \frac{2}{w}\sum\_{n=1}^{\circ}\sin^2\left(\frac{n\pi y\_0}{w}\right)\frac{1}{2K\_n}\exp\left(-\left(\frac{n\pi\rho}{2w}\right)^2\right) = 0\tag{34}$$

In the case where the conducting wires is very narrow or the barrier is very high, i.e.,

$$\left(\pi/w\right)^{2} + V >> E\_{0} \tag{35}$$

then

$$\frac{1}{2\pi}\ln\left(\frac{\rho\_0}{\rho}\right) - \frac{1}{w}\sin^2\left(\frac{\pi y\_0}{w}\right)\frac{1}{\sqrt{V - E + \left(\pi/w\right)^2}} - \frac{1}{\pi}\sum\_{n=2}^{\infty}\sin^2\left(\frac{n\pi y\_0}{w}\right)\frac{1}{\sqrt{n^2 - 1}}\exp\left(-\left(\frac{n\pi\rho}{2w}\right)^2\right) = 0 \tag{36}$$

since

$$\sum\_{n=2}^{\infty} \sin^2 \left(\frac{n\pi y\_0}{w}\right) \frac{1}{\pi\sqrt{n^2 - 1}} \exp\left(-\left(\frac{n\pi\rho}{2w}\right)^2\right) \cong -\ln\left(4\rho/w\right)/2\pi\tag{37}$$

then

$$\frac{1}{2\pi}\ln\left(\frac{4\rho\_0}{w}\right) = \frac{1}{w}\sin^2\left(\frac{\pi y\_0}{w}\right)\frac{1}{\sqrt{V - E + \left(\pi/w\right)^2}}\tag{38}$$

which has a solution provided 4ρ<sup>0</sup> > w, otherwise the impurity can be regarded as a perturbation and does not carry a resonant level.

When the resonant level exits, then the resonance energy ER is approximately

$$E\_R \cong V + \frac{\pi^2}{w^2} \left\{ 1 - \sin^4 \left( \frac{\pi y\_0}{w} \right) \left[ \frac{1}{2} \ln \left( \frac{4 \rho\_0}{w} \right) \right]^{-2} \right\} \tag{39}$$

In Figure 2, the resonance energy is presented by a dashed line.

In this approximation,

$$\Psi(\mathbf{r}) \cong \sin\left(\frac{\pi y}{w}\right) \chi\_{E,1}^{+}(\mathbf{x}) \frac{1}{K\_1 \frac{w}{2\pi} \ln\left(\frac{4\rho\_0}{w}\right) / \sin^2\left(\frac{\pi y\_0}{w}\right) - (1 + i2K\_1\varepsilon\_n)}\tag{40}$$

Since in this regime only, one transverse mode is propagating, the system in practice reduces to a 1D problem, where the 2D impurity can be replaced by a 1D delta function potential

$$V(\mathbf{x}) = -\lambda \delta(\mathbf{x})\tag{41}$$

where

and sin 2½ �¼ θð Þ k<sup>1</sup>

/w<sup>2</sup>

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � � 1

In the case where the conducting wires is very narrow or the barrier is very high, i.e.,

π X∞ n¼2

<sup>n</sup><sup>2</sup> � <sup>1</sup> <sup>p</sup> exp � <sup>n</sup>πρ

sin <sup>2</sup> <sup>π</sup>y<sup>0</sup> w

ð Þ π=w

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>V</sup> � <sup>E</sup> <sup>þ</sup> ð Þ <sup>π</sup>=<sup>w</sup> <sup>2</sup> <sup>q</sup> � <sup>1</sup>

π ffiffiffiffiffiffiffiffiffiffiffiffiffi

¼ 1 w

Then, Eq. (30) can be approximated as

Figure 2. Plot of the T<sup>11</sup> = |t11|2

10 Nonmagnetic and Magnetic Quantum Dots

then

1 <sup>2</sup><sup>π</sup> ln <sup>ρ</sup><sup>0</sup> ρ � �

since

then

� 1

<sup>w</sup> sin <sup>2</sup> <sup>π</sup>y<sup>0</sup> w

> X∞ n¼2

1 2π

energy. The barrier parameters were L = 2w and V = 2/w<sup>2</sup>

dotted line represents the barrier's energy Eb = V + π<sup>2</sup>

ln <sup>ρ</sup><sup>0</sup> ρ � �

� � 1

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � � 1

> 1 2π

ln <sup>4</sup>ρ<sup>0</sup> w � �

� 2 w X∞ n¼1

�2kK <sup>K</sup><sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>¼</sup> �2kK

2Kn

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � � 1

2w � �<sup>2</sup> � �

� � 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V � E þ ð Þ π=w

2 <sup>q</sup> (38)

exp � <sup>n</sup>πρ 2w � �<sup>2</sup> � �

, i.e., the probability to remain in the first mode of propagation as a function of the normalized

<sup>2</sup> <sup>þ</sup> <sup>V</sup> >> <sup>E</sup><sup>0</sup> (35)

, and the defect parameters were ρ<sup>0</sup> = 300w, x<sup>0</sup> = 0, and y<sup>0</sup> = w/2. The

, and the dashed line represents the resonance energy Eres.

ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>n</sup><sup>2</sup> � <sup>1</sup> <sup>p</sup> exp � <sup>n</sup>πρ

2w � �<sup>2</sup> � �

ffi �ln 4ρ=<sup>w</sup> � �=2<sup>π</sup> (37)

¼ 0 (36)

<sup>V</sup> : (33)

¼ 0 (34)

$$
\lambda = \frac{4\pi}{w \ln\left(4\rho\_0/w\right)} \sin^2\left(\frac{\pi y\_0}{w}\right). \tag{42}
$$

Therefore, in the 1D analogy the point potential depends not only on the impurity's de-Broglie wavelength in free space, but on its location (y0) and the wire's width as well.

In this case, the barrier's transmission can be as high as 1. It depends on the location of the point defect in the horizontal dimension, namely, at the resonance energy

$$\Psi(\mathbf{r}) = i \sin\left(\frac{\pi y}{w}\right) \exp(ik\_1 \mathbf{x}) \frac{\exp(-2ik\_1 L + i\Xi)}{\cosh(2K\_1 \mathbf{x}\_0)}\tag{43}$$

where

$$
\tan \Xi = -(\mathcal{K}/\mathcal{k} - \mathcal{k}/\mathcal{K})/2. \tag{44}
$$

However, there is a point where a minimum occurs. When the incoming particle's energy satisfies

$$\frac{1}{2\pi}\ln\left(\frac{\rho\_0}{\rho}\right) + \frac{2}{w}\sum\_{n=2}^{\infty} \sin^2\left(\frac{n\pi y\_0}{w}\right) \Re G\_{1D}^+(\mathbf{x}\_0, \mathbf{x}\_0; E\_n) \exp\left(-\left(\frac{n\pi \rho}{2w}\right)^2\right) = 0\tag{45}$$

which at the vicinity of the second mode threshold can be approximated by

$$\frac{1}{2\pi}\ln\left(\frac{\rho\_0}{\rho}\right) - \frac{1}{w}\sin^2\left(\frac{2\pi y\_0}{w}\right)\frac{1}{\sqrt{V - E + \left(2\pi/w\right)^2}} - \frac{1}{\pi}\sum\_{n=3}^{\infty}\sin^2\left(\frac{n\pi y\_0}{w}\right)\frac{1}{\sqrt{n^2 - 4}}\exp\left(-\left(\frac{n\pi\rho}{2w}\right)^2\right) = 0 \tag{46}$$

or

$$\frac{1}{2\pi}\ln\left(\frac{3.8\rho\_0}{w}\right) = \frac{1}{w}\sin^2\left(\frac{2\pi y\_0}{w}\right)\frac{1}{\sqrt{V - E + \left(2\pi/w\right)^2}}.\tag{47}$$

which is an exponentially small value. This result agrees with Ref. [24].

In Figures 4–9, a 2D probability density plots (of |ψ(x, y)|2

the interplay between the waveguide and the point defect.

Another important working point is when Kp = 0, i.e., V � E + (pπ/w)

in the 1D approximation.

is high.

all of them are reflected from the barrier.

than the barrier's height: E = 10.5w� <sup>2</sup> < (π/w)

It should be stressed, however, that this is a pure 2D phenomenon, which is a consequence of the interaction between the point defect and the wire, and therefore, this minimum disappears

In Figure 4, the energy is too low for the particles to penetrate the barrier, and therefore, almost

In Figure 5, the particle's energy is close to the resonance energy, and therefore, a quasibound state is generated at the vicinity of the defect, and the transmission probability

In Figure 8, the particle's energy is close to the minimum (Eq. (48)), which was generated by

Figure 4. A false colors presentation of the probability density |Ψ(x, y)|2 when the incoming particle's energy is lower

<sup>2</sup> <sup>+</sup> <sup>V</sup> ffi 11.87w� <sup>2</sup>

represent the barrier's boundaries, and the cross at the center of the circle represents the impurity's location.

Figures 6 and 7 are examples for local minimum and local maximum respectively.

At this energy, a universal behavior appears. The scattered wavefunction reads

) for various energies are presented.

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445 13

<sup>2</sup> = 0 and kp <sup>¼</sup> ffiffiffiffi

. The parameters are same as in Figure 3. The dashed lines

V <sup>p</sup> .

Again, we see that this equation does not always have a solution. It is required that 3.8ρ<sup>0</sup> > w, in which case

$$E\_{\rm min} = V + \left(\frac{2\pi}{w}\right)^2 \left[1 - \sin^4\left(\frac{2\pi y\_0}{w}\right) \left[\ln\left(\frac{3.8\rho\_0}{w}\right)\right]^{-2}\right] \tag{48}$$

This minimum is presented in Figure 3 by a dotted line.

In which case, the denominator of Eq. (29) is exactly sin <sup>2</sup> <sup>π</sup>y<sup>0</sup> w � � <sup>2</sup> <sup>w</sup> G<sup>þ</sup> <sup>1</sup>Dð Þ x0; x0; E<sup>1</sup> , and therefore at this point, the transmission is exponentially small, and not zero as in the zero potential case, i.e.,

$$\begin{split} \Psi\_{\min}(\mathbf{r}) &= \sin\left(\frac{\pi y}{w}\right) \chi\_{\mathbf{E},1}^{+}(\mathbf{x}) i \frac{\sin^{2}\left(2\pi y\_{0}/w\right)}{\sin^{2}\left(\pi y\_{0}/w\right)} \frac{\mathfrak{S}\mathbf{G}\_{1D}^{+}(\mathbf{x}\_{0},\mathbf{x}\_{0};\mathbf{E}\_{2})}{\mathfrak{R}\mathbf{G}\_{1D}^{+}(\mathbf{x}\_{0},\mathbf{x}\_{0};\mathbf{E}\_{1})} \\ &= -\sin\left(\frac{\pi y}{w}\right) \chi\_{\mathbf{E},1}^{+}(\mathbf{x}) i \frac{\sin^{2}\left(2\pi y\_{0}/w\right)}{\sin^{2}\left(\pi y\_{0}/w\right)} 2\mathbf{K}\_{1}\varepsilon\_{2} \end{split} \tag{49}$$

Figure 3. Plot of T<sup>11</sup> = |t11|2 , i.e., the probability to remain in the base (1) mode of propagation as a function of the normalized energy. The barrier parameters were L = 2w and V = 2/w<sup>2</sup> , and the defect parameters were ρ<sup>0</sup> = 30w, x<sup>0</sup> = 0, and y<sup>0</sup> = 0.2w. The dotted line represents the minimum transmission point Emin, and the dashed line represents the resonance energy Eres.

which is an exponentially small value. This result agrees with Ref. [24].

1 <sup>2</sup><sup>π</sup> ln <sup>ρ</sup><sup>0</sup> ρ � �

or

in which case

Figure 3. Plot of T<sup>11</sup> = |t11|2

energy Eres.

� 1

12 Nonmagnetic and Magnetic Quantum Dots

<sup>w</sup> sin <sup>2</sup> <sup>2</sup>πy<sup>0</sup> w

> 1 2π

� � 1

ln <sup>3</sup>:8ρ<sup>0</sup> w � �

Emin ¼ V þ

This minimum is presented in Figure 3 by a dotted line.

<sup>Ψ</sup>minð Þ¼ <sup>r</sup> sin <sup>π</sup><sup>y</sup>

normalized energy. The barrier parameters were L = 2w and V = 2/w<sup>2</sup>

In which case, the denominator of Eq. (29) is exactly sin <sup>2</sup> <sup>π</sup>y<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>V</sup> � <sup>E</sup> <sup>þ</sup> ð Þ <sup>2</sup>π=<sup>w</sup> <sup>2</sup> <sup>q</sup> � <sup>1</sup>

> ¼ 1 w

2π w � �<sup>2</sup>

w � � χ<sup>þ</sup> E,1ð Þx i

> w � � χ<sup>þ</sup> E,1ð Þx i

¼ � sin <sup>π</sup><sup>y</sup>

π X∞ n¼3

sin <sup>2</sup> <sup>2</sup>πy<sup>0</sup> w

Again, we see that this equation does not always have a solution. It is required that 3.8ρ<sup>0</sup> > w,

<sup>1</sup> � sin <sup>4</sup> <sup>2</sup>πy<sup>0</sup>

this point, the transmission is exponentially small, and not zero as in the zero potential case, i.e.,

sin <sup>2</sup> <sup>n</sup>πy<sup>0</sup> w � � 1

� � 1

w � �

sin <sup>2</sup> <sup>2</sup>πy0=<sup>w</sup> � � sin <sup>2</sup> <sup>π</sup>y0=<sup>w</sup> � �

> sin <sup>2</sup> <sup>2</sup>πy0=<sup>w</sup> � � sin <sup>2</sup> <sup>π</sup>y0=<sup>w</sup> � � <sup>2</sup>K1ε<sup>2</sup>

� � � � �<sup>2</sup> " #

w � � <sup>2</sup> <sup>w</sup> G<sup>þ</sup>

, i.e., the probability to remain in the base (1) mode of propagation as a function of the

y<sup>0</sup> = 0.2w. The dotted line represents the minimum transmission point Emin, and the dashed line represents the resonance

, and the defect parameters were ρ<sup>0</sup> = 30w, x<sup>0</sup> = 0, and

ℑG<sup>þ</sup>

ℜG<sup>þ</sup>

<sup>1</sup>Dð Þ x0; x0; E<sup>2</sup>

<sup>1</sup>Dð Þ x0; x0; E<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V � E þ ð Þ 2π=w

> ln <sup>3</sup>:8ρ<sup>0</sup> w

<sup>n</sup><sup>2</sup> � <sup>4</sup> <sup>p</sup> exp � <sup>n</sup>πρ

2 <sup>q</sup> : (47)

2w � �<sup>2</sup> � �

<sup>1</sup>Dð Þ x0; x0; E<sup>1</sup> , and therefore at

¼ 0 (46)

(48)

(49)

It should be stressed, however, that this is a pure 2D phenomenon, which is a consequence of the interaction between the point defect and the wire, and therefore, this minimum disappears in the 1D approximation.

In Figures 4–9, a 2D probability density plots (of |ψ(x, y)|2 ) for various energies are presented.

In Figure 4, the energy is too low for the particles to penetrate the barrier, and therefore, almost all of them are reflected from the barrier.

In Figure 5, the particle's energy is close to the resonance energy, and therefore, a quasibound state is generated at the vicinity of the defect, and the transmission probability is high.

Figures 6 and 7 are examples for local minimum and local maximum respectively.

In Figure 8, the particle's energy is close to the minimum (Eq. (48)), which was generated by the interplay between the waveguide and the point defect.

Another important working point is when Kp = 0, i.e., V � E + (pπ/w) <sup>2</sup> = 0 and kp <sup>¼</sup> ffiffiffiffi V <sup>p</sup> .

At this energy, a universal behavior appears. The scattered wavefunction reads

Figure 4. A false colors presentation of the probability density |Ψ(x, y)|2 when the incoming particle's energy is lower than the barrier's height: E = 10.5w� <sup>2</sup> < (π/w) <sup>2</sup> <sup>+</sup> <sup>V</sup> ffi 11.87w� <sup>2</sup> . The parameters are same as in Figure 3. The dashed lines represent the barrier's boundaries, and the cross at the center of the circle represents the impurity's location.

Figure 5. Same as Figure 4 but when the income particle's energy is close to the resonance energy, i.e. <sup>E</sup> = 11.69w� <sup>2</sup> ffi Eres.

Figure 7. Same as Figure 4 but when the income particle's energy is close to a local maximum at E = 15.4w <sup>2</sup>

Figure 8. Same as Figure 4 but when the income particle's energy is close to a local minima at E = 40.02w <sup>2</sup>

.

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445 15

.

Figure 6. Same as Figure 4 but when the income particle's energy is close to a local minimum at E = 12.45w� <sup>2</sup> .

Figure 7. Same as Figure 4 but when the income particle's energy is close to a local maximum at E = 15.4w <sup>2</sup> .

Figure 5. Same as Figure 4 but when the income particle's energy is close to the resonance energy, i.e. <sup>E</sup> = 11.69w� <sup>2</sup> ffi Eres.

14 Nonmagnetic and Magnetic Quantum Dots

Figure 6. Same as Figure 4 but when the income particle's energy is close to a local minimum at E = 12.45w� <sup>2</sup>

.

Figure 8. Same as Figure 4 but when the income particle's energy is close to a local minima at E = 40.02w <sup>2</sup> .

$$\Psi(\mathbf{r}) = \sin\left(\frac{m\pi y}{w}\right)\chi\_{E,m}^{+}(\mathbf{x}) + \frac{\sin\left(\frac{m\pi y\_0}{w}\right)\chi\_{E,m}^{+}(\mathbf{x}\_0)\sin\left(\frac{m\pi y}{w}\right)G\_{1D}^{+}(\mathbf{x},\mathbf{x}\_0;E\_p)}{\sin\left(\frac{m\pi y\_0}{w}\right)G\_{1D}^{+}(\mathbf{x}\_0,\mathbf{x}\_0;E\_p)}.\tag{50}$$

The fact that the second part is independent of x is also in agreement with Ref. [25].

E,mð Þþ x

2

reads

resonance.

transistor.

the potential barrier V = 0.15 eV.

<sup>Ψ</sup>ð Þ¼ j j <sup>x</sup> <sup>&</sup>gt; L, y sin <sup>m</sup>π<sup>y</sup>

the device's conductivity in units of e

w � �χ<sup>þ</sup>

4. Physical realization and implementation

But unlike Ref. [25], due the barrier, the second mode does propagate, but the expression is still generic (in the sense that it is independent of the impurity's parameter), beyond the barrier it

> m p χ<sup>þ</sup>

This special universal case is illustrated in Figure 9, and it is a manifestation of the effect of Ref. [25], where the footprints of the defect are clearly seen but without any fingerprints. That is, the defect is clearly there, but the scattering is independent of its strength (its eigenenergy).

Let us apply this model to a 1.5-nm wide silicon wire, which is contaminated by a single phosphorous atom. In this case w = 1.5 nm, the phosphorous atom radius is a = 0.098 nm, the effective electron mass in silicon is me ffi 0.2m. Then the wire transmission (proportional to

Eq. (6) (which is proportional to the transistor gate voltage) is plotted in Figure 10 for two scenarios. In the first scenario, the electron's energy, i.e., the Fermi energy, is E = 0.9 eV and in the second, it is equal to E = 3 eV. In the former scenario, the device works at the vicinity of the quantum dot's resonance, and in the latter, it works at the vicinity of the fano-(anti)

In both scenarios, a change of about a volt in the gate voltage can change drastically the wire's current. Therefore, it can be implemented as a simplified but rich model for a single atom

Figure 10. The wire's transmission as a function of the potential on the atom. In the left plot, the electron energy is E = 0.9 eV and in the right plot, E = 3 eV. The other parameters are w = 1.5 nm, me ffi 0.2m, a = 0.098 nm, L = 2w = 3 nm, and

E,mð Þ <sup>x</sup><sup>0</sup> sin <sup>p</sup>π<sup>y</sup>

w

� �exp <sup>i</sup> ffiffiffiffi

/h) as a function of the potential at the atom's center V0,

V <sup>p</sup> ð Þ j j <sup>x</sup> � <sup>L</sup> � �: (54)

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445 17

This expression is universal in the sense that it is independent of the point defect potential. It depends only on its location. In case this is a surface defect, i.e., y0/w < < 1 then even the dependence on the vertical location vanishes

$$\Psi(\mathbf{r}) = \sin\left(\frac{m\pi y}{w}\right)\chi\_{E,m}^{+}(\mathbf{x}) + \frac{m}{p}\chi\_{E,m}^{+}(\mathbf{x}\_{0})\sin\left(\frac{p\pi y}{w}\right)\frac{G\_{1D}^{+}(\mathbf{x},\mathbf{x}\_{0};E\_{p})}{G\_{1D}^{+}(\mathbf{x}\_{0},\mathbf{x}\_{0};E\_{p})}.\tag{51}$$

This universality agree with Ref. [25].

For |x| < L Eq. (51) reduces to the simple form

$$\Psi(|\mathbf{x}| < L, \mathbf{y}) = \sin\left(\frac{m\pi y}{w}\right) \chi^{+}\_{\bar{E}, m}(\mathbf{x}) + \frac{\sin\left(\frac{m\pi y\_0}{w}\right)}{\sin\left(\frac{m\pi y\_0}{w}\right)} \chi^{+}\_{\bar{E}, m}(\mathbf{x}\_0) \sin\left(\frac{p\pi y}{w}\right) \tag{52}$$

and in the case of a surface defect, it reduces to even a simpler expression

$$\Psi(|\mathbf{x}| < L, y) = \sin\left(\frac{m\pi y}{w}\right) \chi^+\_{E,m}(\mathbf{x}) + \frac{m}{p} \chi^+\_{E,m}(\mathbf{x}\_0) \sin\left(\frac{p\pi y}{w}\right) \tag{53}$$

Figure 9. At the transition level E = V + (π/w) 2 , a universal pattern appears.

The fact that the second part is independent of x is also in agreement with Ref. [25].

But unlike Ref. [25], due the barrier, the second mode does propagate, but the expression is still generic (in the sense that it is independent of the impurity's parameter), beyond the barrier it reads

$$\Psi(|\mathbf{x}|>L,\mathbf{y}) = \sin\left(\frac{m\pi y}{w}\right)\chi^+\_{E,m}(\mathbf{x}) + \frac{m}{p}\chi^+\_{E,m}(\mathbf{x}\_0)\sin\left(\frac{p\pi y}{w}\right)\exp\left(i\sqrt{V}(|\mathbf{x}|-L)\right). \tag{54}$$

This special universal case is illustrated in Figure 9, and it is a manifestation of the effect of Ref. [25], where the footprints of the defect are clearly seen but without any fingerprints. That is, the defect is clearly there, but the scattering is independent of its strength (its eigenenergy).

#### 4. Physical realization and implementation

<sup>Ψ</sup>ð Þ¼ <sup>r</sup> sin <sup>m</sup>π<sup>y</sup>

16 Nonmagnetic and Magnetic Quantum Dots

dependence on the vertical location vanishes

<sup>Ψ</sup>ð Þ¼ <sup>r</sup> sin <sup>m</sup>π<sup>y</sup>

For |x| < L Eq. (51) reduces to the simple form

<sup>Ψ</sup>ð Þ¼ j j <sup>x</sup> <sup>&</sup>lt; L, y sin <sup>m</sup>π<sup>y</sup>

This universality agree with Ref. [25].

Figure 9. At the transition level E = V + (π/w)

w 

> w

χ<sup>þ</sup> E,mð Þþ x

w 

and in the case of a surface defect, it reduces to even a simpler expression

2

, a universal pattern appears.

<sup>Ψ</sup>ð Þ¼ j j <sup>x</sup> <sup>&</sup>lt; L, y sin <sup>m</sup>π<sup>y</sup>

χ<sup>þ</sup> E,mð Þþ x

w 

χ<sup>þ</sup> E,mð Þþ x

χ<sup>þ</sup> E,mð Þþ x sin <sup>m</sup>πy<sup>0</sup> w χ<sup>þ</sup>

This expression is universal in the sense that it is independent of the point defect potential. It depends only on its location. In case this is a surface defect, i.e., y0/w < < 1 then even the

> m p χ<sup>þ</sup>

E,mð Þ <sup>x</sup><sup>0</sup> sin <sup>p</sup>π<sup>y</sup>

sin <sup>p</sup>πy<sup>0</sup> w G<sup>þ</sup>

E,mð Þ <sup>x</sup><sup>0</sup> sin <sup>p</sup>π<sup>y</sup>

sin <sup>m</sup>πy<sup>0</sup> w sin <sup>p</sup>πy<sup>0</sup> w <sup>χ</sup><sup>þ</sup>

> m p χ<sup>þ</sup>

w G<sup>þ</sup>

w G<sup>þ</sup>

<sup>1</sup><sup>D</sup> x0; x0; Ep

G<sup>þ</sup>

<sup>1</sup><sup>D</sup> x; x0; Ep 

<sup>1</sup><sup>D</sup> x; x0; Ep 

<sup>1</sup><sup>D</sup> x0; x0; Ep

E,mð Þ <sup>x</sup><sup>0</sup> sin <sup>p</sup>π<sup>y</sup>

w 

E,mð Þ <sup>x</sup><sup>0</sup> sin <sup>p</sup>π<sup>y</sup>

w 

: (50)

: (51)

(52)

(53)

Let us apply this model to a 1.5-nm wide silicon wire, which is contaminated by a single phosphorous atom. In this case w = 1.5 nm, the phosphorous atom radius is a = 0.098 nm, the effective electron mass in silicon is me ffi 0.2m. Then the wire transmission (proportional to the device's conductivity in units of e 2 /h) as a function of the potential at the atom's center V0, Eq. (6) (which is proportional to the transistor gate voltage) is plotted in Figure 10 for two scenarios. In the first scenario, the electron's energy, i.e., the Fermi energy, is E = 0.9 eV and in the second, it is equal to E = 3 eV. In the former scenario, the device works at the vicinity of the quantum dot's resonance, and in the latter, it works at the vicinity of the fano-(anti) resonance.

In both scenarios, a change of about a volt in the gate voltage can change drastically the wire's current. Therefore, it can be implemented as a simplified but rich model for a single atom transistor.

Figure 10. The wire's transmission as a function of the potential on the atom. In the left plot, the electron energy is E = 0.9 eV and in the right plot, E = 3 eV. The other parameters are w = 1.5 nm, me ffi 0.2m, a = 0.098 nm, L = 2w = 3 nm, and the potential barrier V = 0.15 eV.

## Author details

#### Er'el Granot

Address all correspondence to: erel@ariel.ac.il

Department of Electrical and Electronics Engineering, Ariel University, Ariel, Israel

### References

[1] Moore G. Chapter 7: Moore's law at 40. In: Brock D, editor. Understanding Moore's Law: Four Decades of Innovation. Philadelphia, PA: Chemical Heritage Foundation; 2006. pp. 67-84

[13] Obermair Ch, Xie F-Q, Schimmel Th. The single-atom transistor: Perspectives for quan-

Exact Model for Single Atom Transistor http://dx.doi.org/10.5772/intechopen.70445 19

[14] Azbel MY. Variable-range-hopping magnetoresistance. Physical Review B. 1991;43:2435 [15] Azbel MY. Quantum particle in a random potential: Implications of an exact solution.

[16] Granot E. Point scatterers and resonances in low number of dimensions. Physica E.

[17] Granot E, Azbel MY. Resonant angular dependence in a weak magnetic field. Journal of

[18] Granot E, Azbel MY. Resonant tunneling in two dimensions via an impurity. Physical

[19] Granot E. Near-threshold-energy conductance of a thin wire. Physical Review B.

[20] Granot E. Symmetry breaking and current patterns due to a weak imperfection. Physical

[21] Weber B, Mahapatra S, Ryu H, Lee S, Fuhrer A, Reusch TCG, Thompson DL, Lee WCT, Klimeck G, Hollenberg LCL, Simmons MY. Ohm's law survives to the atomic scale.

[22] Abramowitz M, Stegun IA. Handbook of Mathematical Functions. New York: Dover

[24] Granot E. Universal conductance reduction in a quantum wire. Europhysics Letters.

[25] Granot E. Transmission coefficient for a point scatterer at specific energies is affected by the presence of the scatterer but independent of the scatterer's characteristics. Physical

[23] Merzbacher E. Quantum Mechanics. Hoboken, NJ: Wiley; 1970

tum electronics on the atomic-scale. Europhysics News. 2010;41:25-28

Physical Review Letters. 1991;67:1787

Physics: Condensed Matter. 1999;11:4031

2006;31:13-16

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Review B. 1994;50:8868

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[13] Obermair Ch, Xie F-Q, Schimmel Th. The single-atom transistor: Perspectives for quantum electronics on the atomic-scale. Europhysics News. 2010;41:25-28

Author details

18 Nonmagnetic and Magnetic Quantum Dots

Address all correspondence to: erel@ariel.ac.il

scope. Nature. 1990;344:524-526

Physical Review B. 2008;78:195309

Letters. 2003;91:136104

transistor. Nano Letters. 2010;10:11-15

conductors. Physical Review B. 2004;69:113301

Society of New South Wales. 2012;145(443 & 444):66-74

Department of Electrical and Electronics Engineering, Ariel University, Ariel, Israel

[1] Moore G. Chapter 7: Moore's law at 40. In: Brock D, editor. Understanding Moore's Law: Four Decades of Innovation. Philadelphia, PA: Chemical Heritage Foundation; 2006. pp.

[2] Takahashi D. Forty Years of Moore's Law. San Jose, CA: Seattle Times; April 18, 2005

[3] Eigler DM, Schweizer EK. Positioning single atoms with a scanning tunnelling micro-

[4] Koenraad PM, Flatté ME. Single dopants in semiconductors. Nature Materials. 2011;10:

[5] Lansbergen GP, et al. Gate-induced quantum-confinement transition of a single dopant

[6] Calvet LE, Snyder JP, Wernsdorfer W. Excited-state spectroscopy of single Pt atoms in Si.

[7] Tan KY, et al. Transport spectroscopy of single phosphorus donors in a silicon nanoscale

[8] Hollenberg LCL, et al. Charge-based quantum computing using single donors in semi-

[9] Schofield SR, et al. Atomically precise placement of single dopants in Si. Physical Review

[10] Fuechsle M, Miwa JA, Mahapatra S, Ryu H, Lee S, Warschkow O, Hollenberg LCL, Klimeck G, Simmons MY. A single-atom transistor. Nature Nanotechnology. 2012;7:242-

[11] Xie F-Q, Maul R, Wenzel W, Schn G, Obermair Ch, Schimmel Th. Single-atom transistors: Atomic-scale electronic devices in experiment and simulation. In: International Beilstein Symposium on Functional Nanoscience; Frankfurt am Main, May 2010. pp. 213-228 [12] Fuechsle M, Miwa JA, Mahapatra S, Warschkow O, Hollenberg LCL, Simmons MY. Realisation of a single-atom transistor in silicon. Journal and Proceedings of the Royal

atom in a silicon FinFET. Nature Physics. 2008;4:656-661

Er'el Granot

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246


**Chapter 2**

Provisional chapter

**A Quantum Trajectory Interpretation of Magnetic**

DOI: 10.5772/intechopen.74409

For a complete description of the electronic motion in a quantum dot, we need a method that can describe not only the trajectory behavior of the electron but also its probabilistic wave behavior. Quantum Hamilton mechanics, which possesses the desired ability of manifesting the wave-particle duality of electrons moving in a quantum dot, is introduced in this chapter to recover the quantum-mechanical meanings of the classical terms such as backscattering and commensurability and to give a quantum-mechanical interpretation of the observed oscillation in the magneto-resistance curve. Solutions of quantum Hamilton equations reveal the existence of electronic standing waves in a quantum dot, whose occurrence is found to be accompanied by a jump in the electronic resistance. The comparison with the experimental data shows that the predicted locations of the resistance

jump match closely with the peaks of the measured magneto-resistance.

Keywords: quantum dots, quantum Hamilton mechanics, standing waves, quantum

As the size of electronic devices is narrowed down to the nanoscale, quantum effects become so prominent that classical mechanics is no longer able to provide an accurate description for electrons moving in nanostructures. However, due to the lack of the sense of trajectory in quantum mechanics, classical or semi-classical mechanics so far has been the sole tool in determining ballistic orbits in quantum dots. Classical orbits satisfying commensurability conditions of geometrical resonances were derived in the literature to determine the magnetotransport behavior of periodic quantum systems. It was reported that the observed regular peaks in the magneto-resistance corresponded to backscattering of commensurate orbits [1],

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

> © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A Quantum Trajectory Interpretation of Magnetic

**Resistance in Quantum Dots**

Resistance in Quantum Dots

Ciann-Dong Yang and Shih-Ming Huang

Ciann-Dong Yang and Shih-Ming Huang

http://dx.doi.org/10.5772/intechopen.74409

trajectory, magneto-resistance

Abstract

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots** A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

DOI: 10.5772/intechopen.74409

Ciann-Dong Yang and Shih-Ming Huang Ciann-Dong Yang and Shih-Ming Huang

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74409

#### Abstract

For a complete description of the electronic motion in a quantum dot, we need a method that can describe not only the trajectory behavior of the electron but also its probabilistic wave behavior. Quantum Hamilton mechanics, which possesses the desired ability of manifesting the wave-particle duality of electrons moving in a quantum dot, is introduced in this chapter to recover the quantum-mechanical meanings of the classical terms such as backscattering and commensurability and to give a quantum-mechanical interpretation of the observed oscillation in the magneto-resistance curve. Solutions of quantum Hamilton equations reveal the existence of electronic standing waves in a quantum dot, whose occurrence is found to be accompanied by a jump in the electronic resistance. The comparison with the experimental data shows that the predicted locations of the resistance jump match closely with the peaks of the measured magneto-resistance.

Keywords: quantum dots, quantum Hamilton mechanics, standing waves, quantum trajectory, magneto-resistance

## 1. Introduction

As the size of electronic devices is narrowed down to the nanoscale, quantum effects become so prominent that classical mechanics is no longer able to provide an accurate description for electrons moving in nanostructures. However, due to the lack of the sense of trajectory in quantum mechanics, classical or semi-classical mechanics so far has been the sole tool in determining ballistic orbits in quantum dots. Classical orbits satisfying commensurability conditions of geometrical resonances were derived in the literature to determine the magnetotransport behavior of periodic quantum systems. It was reported that the observed regular peaks in the magneto-resistance corresponded to backscattering of commensurate orbits [1],

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and the critical magnetic fields determined from the backscattering orbits showed an excellent agreement with the observed peak positions in the magneto-resistance curves [2]. A recent study showed that the ballistic motion of electrons within quantum dots can be controlled by an externally applied magnetic field so that the resulting conductance images resemble the classical transmitted and backscattered trajectories [3].

by an applied magnetic field in such a way that the electron's cyclotron angular velocity is exactly counterbalanced by its quantum angular velocity. We point out that magnetic stagnation is a degenerate case from the electronic standing-wave motion as the wave number N approaches to infinity. The magnetic fields yielding the phenomenon of magnetic stagnation can be determined by the quantum Hamilton equations derived here. Knowing these critical

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

http://dx.doi.org/10.5772/intechopen.74409

23

In the following sections, we first introduce quantum Hamilton mechanics and apply it to derive Hamilton equations, which are then used in Section 2 to describe the electronic quantum motions in a quantum dot. By solving the Hamilton equations of motion, Section 3 demonstrates electronic standing-wave motions in various quantum states and characterizes the magnetic field leading to the phenomenon of magnetic stagnation. In Section 4, we show that the magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field by comparing the theoretical predictions obtained from Section 3 with the

To probe the quantum to classical transition, which involves both classical and quantum features, quantum dots are the most natural systems [14]. Analyzing such systems, we need an approach that can provide both classical and quantum descriptions. Quantum Hamilton mechanics is one of the candidates satisfying this requirement. This chapter will apply quantum Hamilton mechanics to an open quantum dot with circular shape, which is connected to reservoirs with strong coupling. The electronic transport through an open quantum dot can be realized by nano-fabrication techniques as a two-dimensional electron gases system (2DES) at

Figure 1. Schematic illustration of a semiconductor heterostructure with a circular dot between the two tunnel barriers.

magnetic fields allows us to control the magneto-resistance precisely.

experimental results of the magneto-resistance curve [4, 13].

an AlGaAs/GaAs heterostructure, as depicted in Figure 1.

2. Quantum Hamilton dynamics in a 2D quantum dot

The use of an anisotropic harmonic function, instead of an abrupt hard potential, to describe the confining potential in a quantum dot was shown to be helpful to improve the accuracy of predicting magneto-resistance peaks based on backscattering orbits [4]. Nowadays, the confinement potential forming an electron billiard can be practically patterned to almost arbitrary profile, through which ballistic orbits with chaotic dynamics can be generated to characterize magneto transport [5]. However, the chaotic behavior and its change with magnetic field could not be described in the usual quantum-mechanical picture due to the lack of a trajectory interpretation. Regarding this aspect, the classical description becomes a valued tool for detailed understanding of the transition from low to high magnetic fields in quantum dot arrays [6]. On the other hand, quantum mechanical model for electron billiards was known as quantum billiards [7], in which moving point particles are replaced by waves. Quantum billiards are most convenient for illustrating the phenomenon of Fano interference [8] and its interplay with Aharonov-Bohm interference [9], which otherwise cannot be described by classical methods.

From the existing researches, we have an observation that the ballistic motion in electron billiards was solely described by classical mechanics, while the wave motion in quantum billiards could only be described by quantum mechanics. The aim of this chapter is to give a unified treatment of electron billiards and quantum billiards. We point out that quantum Hamilton mechanics [10, 11] can describe both ballistic motion and wave motion of electrons in a quantum dot to provide us with a quantum commensurability condition to determine backscattering orbits as well as with the wave behavior to characterize the magneto-resistance in a quantum dot.

Quantum Hamilton mechanics is a dynamical realization of quantum mechanics in the complex space [12], under which each quantum operator is realized as a complex function and each wavefunction is represented by a set of complex-valued Hamilton equations of motion. With quantum Hamilton mechanics, we can recover the quantum-mechanical meanings of the classical commensurability condition by showing that there are integral numbers of oscillation in the radial direction, as an electron undergoes a complete angular oscillation around a quantum dot. When the radial and angular dynamics are commensurable, the shape of electronic quantum orbits is found to be stationary like a standing wave. Furthermore, the wave number N, distributed on the circumference of the quantum dot, can be controlled by the applied magnetic field. It will be shown that the classical backscattering orbits discovered in the literature resemble the shape of the quantum standing waves derived here with their wave numbers equal to the numbers of electron's bounces within the quantum dot.

The electronic standing-wave motions considered in this chapter will reveal that a jump of the magneto-resistance in quantum dots is accompanied by a phenomenon of magnetic stagnation, which is a quantum effect that an electron is stagnated or trapped within a quantum dot by an applied magnetic field in such a way that the electron's cyclotron angular velocity is exactly counterbalanced by its quantum angular velocity. We point out that magnetic stagnation is a degenerate case from the electronic standing-wave motion as the wave number N approaches to infinity. The magnetic fields yielding the phenomenon of magnetic stagnation can be determined by the quantum Hamilton equations derived here. Knowing these critical magnetic fields allows us to control the magneto-resistance precisely.

In the following sections, we first introduce quantum Hamilton mechanics and apply it to derive Hamilton equations, which are then used in Section 2 to describe the electronic quantum motions in a quantum dot. By solving the Hamilton equations of motion, Section 3 demonstrates electronic standing-wave motions in various quantum states and characterizes the magnetic field leading to the phenomenon of magnetic stagnation. In Section 4, we show that the magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field by comparing the theoretical predictions obtained from Section 3 with the experimental results of the magneto-resistance curve [4, 13].

## 2. Quantum Hamilton dynamics in a 2D quantum dot

and the critical magnetic fields determined from the backscattering orbits showed an excellent agreement with the observed peak positions in the magneto-resistance curves [2]. A recent study showed that the ballistic motion of electrons within quantum dots can be controlled by an externally applied magnetic field so that the resulting conductance images resemble the

The use of an anisotropic harmonic function, instead of an abrupt hard potential, to describe the confining potential in a quantum dot was shown to be helpful to improve the accuracy of predicting magneto-resistance peaks based on backscattering orbits [4]. Nowadays, the confinement potential forming an electron billiard can be practically patterned to almost arbitrary profile, through which ballistic orbits with chaotic dynamics can be generated to characterize magneto transport [5]. However, the chaotic behavior and its change with magnetic field could not be described in the usual quantum-mechanical picture due to the lack of a trajectory interpretation. Regarding this aspect, the classical description becomes a valued tool for detailed understanding of the transition from low to high magnetic fields in quantum dot arrays [6]. On the other hand, quantum mechanical model for electron billiards was known as quantum billiards [7], in which moving point particles are replaced by waves. Quantum billiards are most convenient for illustrating the phenomenon of Fano interference [8] and its interplay with Aharonov-Bohm interference [9], which otherwise cannot be described by

From the existing researches, we have an observation that the ballistic motion in electron billiards was solely described by classical mechanics, while the wave motion in quantum billiards could only be described by quantum mechanics. The aim of this chapter is to give a unified treatment of electron billiards and quantum billiards. We point out that quantum Hamilton mechanics [10, 11] can describe both ballistic motion and wave motion of electrons in a quantum dot to provide us with a quantum commensurability condition to determine backscattering orbits as well as with the wave behavior to characterize the magneto-resistance

Quantum Hamilton mechanics is a dynamical realization of quantum mechanics in the complex space [12], under which each quantum operator is realized as a complex function and each wavefunction is represented by a set of complex-valued Hamilton equations of motion. With quantum Hamilton mechanics, we can recover the quantum-mechanical meanings of the classical commensurability condition by showing that there are integral numbers of oscillation in the radial direction, as an electron undergoes a complete angular oscillation around a quantum dot. When the radial and angular dynamics are commensurable, the shape of electronic quantum orbits is found to be stationary like a standing wave. Furthermore, the wave number N, distributed on the circumference of the quantum dot, can be controlled by the applied magnetic field. It will be shown that the classical backscattering orbits discovered in the literature resemble the shape of the quantum standing waves derived here with their wave

The electronic standing-wave motions considered in this chapter will reveal that a jump of the magneto-resistance in quantum dots is accompanied by a phenomenon of magnetic stagnation, which is a quantum effect that an electron is stagnated or trapped within a quantum dot

numbers equal to the numbers of electron's bounces within the quantum dot.

classical transmitted and backscattered trajectories [3].

22 Nonmagnetic and Magnetic Quantum Dots

classical methods.

in a quantum dot.

To probe the quantum to classical transition, which involves both classical and quantum features, quantum dots are the most natural systems [14]. Analyzing such systems, we need an approach that can provide both classical and quantum descriptions. Quantum Hamilton mechanics is one of the candidates satisfying this requirement. This chapter will apply quantum Hamilton mechanics to an open quantum dot with circular shape, which is connected to reservoirs with strong coupling. The electronic transport through an open quantum dot can be realized by nano-fabrication techniques as a two-dimensional electron gases system (2DES) at an AlGaAs/GaAs heterostructure, as depicted in Figure 1.

Figure 1. Schematic illustration of a semiconductor heterostructure with a circular dot between the two tunnel barriers.

Under the framework of quantum Hamilton mechanics [10, 12], the equivalent mathematical model of a quantum dot is described as an electron moving in an electromagnetic field with scalar potential V and vector potential A. The related Hamiltonian operator Hb can be realized as the following complex Hamiltonian function,

$$H(t, \mathbf{q}, \mathbf{p}) = \frac{1}{2m} \left(\mathbf{p} + \frac{c}{\varepsilon}\mathbf{A}\right) \cdot \left(\mathbf{p} + \frac{c}{\varepsilon}\mathbf{A}\right) + V(t, \mathbf{q}) + \frac{\hbar}{2im} \nabla \cdot \mathbf{p}.\tag{1}$$

S ¼ �iℏln Ψ, (7)

http://dx.doi.org/10.5772/intechopen.74409

�iEt=<sup>ℏ</sup>ψð Þ <sup>r</sup>; <sup>θ</sup> , (9)

Hb ψ ¼ Hψ, (11)

1 r2 ℏ i ∂2 ln ψ ∂θ<sup>2</sup>

� �: (12)

, (13)

<sup>∂</sup><sup>θ</sup> <sup>þ</sup> <sup>ω</sup>L: (14)

Ψ: (8)

25

� iℏω<sup>L</sup>

∂Ψ ∂θ þ 1 2 m<sup>∗</sup>ω<sup>2</sup> r 2

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

to Eq. (6) to produce the expected Schrodinger equation:

∂2 Ψ ∂r<sup>2</sup> þ

where ψð Þ r; θ satisfies the time-independent Schrodinger equation

∂2 ∂r<sup>2</sup> þ 1 r ∂ ∂r þ 1 r2 ∂2 ∂θ<sup>2</sup>

1 r ∂Ψ ∂r þ 1 r2 ∂2 Ψ ∂θ<sup>2</sup>

� �

Ψð Þ¼ t;r; θ e

On the other hand, Eq. (3) can be rewritten by using the substitutions Eqs. (5) and (7) as

where H and Hb are defined, respectively, by Eqs. (3) and (10). This is a direct proof of the fact that the complex Hamiltonian H is a functional realization of the Hamiltonian operator Hb in a complex space. Indeed, it can be shown [10] that every quantum operator Ab can be realized as a complex function A via the relation Abψ ¼ Aψ. The combination of Eqs. (10) and (11) reveals the energy conservation law H ¼ E, which is a natural result of Hamilton mechanics by noting that the Hamiltonian H in Eq. (3) does not contain time t explicitly and must be a motion

Upon performing the differentiations ∂pr=∂r and ∂pθ=∂θ involved in Eq. (3), we have to specify in advance the action function S or equivalently the wavefunction ψ via the relation Eq. (7). This requirement makes the complex Hamiltonian H state-dependent. For a given quantum

Apart from deriving the Schrodinger equation, the above complex Hamiltonian also gives electronic quantum motions in the state ψ in terms of the Hamilton equations of motion,

> ℏ 2im<sup>∗</sup> 1 <sup>r</sup> <sup>¼</sup> <sup>ℏ</sup> im<sup>∗</sup>

<sup>m</sup><sup>∗</sup>r<sup>2</sup> <sup>p</sup><sup>θ</sup> <sup>þ</sup> <sup>ω</sup><sup>L</sup> <sup>¼</sup> <sup>ℏ</sup>

1 r pr þ ℏ i ∂2 ln ψ ∂r<sup>2</sup> þ

∂lnψ ∂r þ ℏ 2im<sup>∗</sup> 1 r

∂lnψ

im<sup>∗</sup> r2

state described by ψ, the complex Hamiltonian H can be expressed explicitly as:

1 2 m<sup>∗</sup>ω<sup>2</sup> r 2 þ ℏ 2im<sup>∗</sup>

þ ωLp<sup>θ</sup> þ

¼ 1 <sup>m</sup><sup>∗</sup> pr <sup>þ</sup>

<sup>¼</sup> <sup>1</sup>

� �

Due to the time-independent nature of the applied potentials A and V, the wavefunction Ψ in

� iℏω<sup>L</sup>

∂ ∂θ þ 1 2 m<sup>∗</sup>ω<sup>2</sup> r 2

� �<sup>ψ</sup> <sup>¼</sup> <sup>E</sup>ψ: (10)

<sup>i</sup><sup>ℏ</sup> <sup>∂</sup><sup>Ψ</sup>

<sup>∂</sup><sup>t</sup> ¼ � <sup>ℏ</sup><sup>2</sup> 2m<sup>∗</sup>

Eq. (8) assumes the following form of solution,

<sup>H</sup><sup>b</sup> <sup>ψ</sup><sup>≜</sup> � <sup>ℏ</sup><sup>2</sup>

constant equal to the system's total energy E.

� �

dr dt <sup>¼</sup> <sup>∂</sup><sup>H</sup> ∂pr

> dθ dt <sup>¼</sup> <sup>∂</sup><sup>H</sup> ∂p<sup>θ</sup>

<sup>H</sup> <sup>¼</sup> <sup>1</sup>

<sup>2</sup>m<sup>∗</sup> <sup>p</sup><sup>2</sup> <sup>r</sup> þ 1 r2 p2 θ

2m<sup>∗</sup>

We adopt polar coordinates q ¼ ð Þ r; θ and momentum p ¼ pr; p<sup>θ</sup> � � in the above equation to describe the electronic quantum motion in a 2D circular quantum dot. The resulting complex Hamiltonian Eq. (1) becomes

$$H = \frac{1}{2m^\*} \left[ \left( p\_r + \frac{e}{c} A\_r \right)^2 + \frac{1}{r^2} \left( p\_\theta + \frac{e}{c} A\_\theta \right)^2 \right] + V(r, \theta) + \frac{\hbar}{2im^\*} \left( \frac{1}{r} p\_r + \frac{\partial p\_r}{\partial r} + \frac{1}{r^2} \frac{\partial p\_\theta}{\partial \theta} \right), \tag{2}$$

where <sup>m</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:067me is the electron's effective mass in AlGaAs/GaAs heterostructure. The scalar potential V rð Þ ; θ acts as a confinement potential in the dot, and is modeled by the parabolic function <sup>V</sup> <sup>¼</sup> kr<sup>2</sup> <sup>=</sup><sup>2</sup> <sup>¼</sup> <sup>m</sup><sup>∗</sup>ω<sup>2</sup> <sup>0</sup>r<sup>2</sup>=2 to simulate a soft-wall potential. The vector potential A is determined from the applied magnetic field B via the relation B ¼ ∇ � A. Here, we consider a constant <sup>B</sup> along the <sup>z</sup> direction, which amounts to Ar <sup>¼</sup> 0 and <sup>A</sup><sup>θ</sup> <sup>¼</sup> Br<sup>2</sup> =2. Substituting the above assignments of V and A into the complex Hamiltonian Eq. (2), we obtain

$$H = \frac{1}{2m^\*} \left( p\_r^2 + \frac{1}{r^2} p\_\theta^2 \right) + \omega\_L p\_\theta + \frac{1}{2} m^\* \omega^2 r^2 + \frac{\hbar}{2im^\*} \left( \frac{1}{r} p\_r + \frac{\partial p\_r}{\partial r} + \frac{1}{r^2} \frac{\partial p\_\theta}{\partial \theta} \right) \tag{3}$$

where <sup>ω</sup><sup>L</sup> <sup>¼</sup> eB<sup>=</sup> <sup>2</sup>m<sup>∗</sup> ð Þ<sup>c</sup> is the Larmor frequency and <sup>ω</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 <sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> L q is the composite frequency. The quantum Hamilton-Jacobi equation associated with the Hamiltonian H defined in Eq. (1) reads

$$\left. \frac{\partial \mathcal{S}}{\partial t} + H(t, \mathbf{q}, \mathbf{p}) \right|\_{p\_i = \partial \mathcal{S}/\partial q\_i} = 0,\tag{4}$$

where S is the quantum action function to be determined. By making use of the substitution

$$p\_r = \frac{\partial S}{\partial r}, \ p\_\theta = \frac{\partial S}{\partial \theta'} \tag{5}$$

the quantum Hamilton-Jacobi Eq. (4) associated with the Hamiltonian in Eq. (3) turns out to be

$$\frac{\partial \mathbf{S}}{\partial t} + \frac{1}{2m^\*} \left[ \left( \frac{\partial \mathbf{S}}{\partial r} \right)^2 + \frac{1}{r^2} \left( \frac{\partial \mathbf{S}}{\partial \theta} \right)^2 \right] + \omega\_L \frac{\partial \mathbf{S}}{\partial \theta} + \frac{1}{2} m^\* \omega^2 r^2 - \frac{i\hbar}{2m^\*} \left( \frac{1}{r} \frac{\partial \mathbf{S}}{\partial r} + \frac{\partial^2 \mathbf{S}}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 \mathbf{S}}{\partial \theta^2} \right) = \mathbf{0}. \tag{6}$$

The recognition of the complex Hamiltonian H in Eqs. (1) and (3) as a complex realization of the Hamiltonian operator Hb is confirmed by the fact that the quantum Hamilton-Jacobi equation in Eqs. (4) and (6) yields the same Schrodinger equation as constructed from Hb . This can be seen by applying the following transformation

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots http://dx.doi.org/10.5772/intechopen.74409 25

$$S = -i\hbar \ln \Psi\_{\prime} \tag{7}$$

to Eq. (6) to produce the expected Schrodinger equation:

Under the framework of quantum Hamilton mechanics [10, 12], the equivalent mathematical model of a quantum dot is described as an electron moving in an electromagnetic field with scalar potential V and vector potential A. The related Hamiltonian operator Hb can be realized

> � p þ c e A � �

describe the electronic quantum motion in a 2D circular quantum dot. The resulting complex

where <sup>m</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:067me is the electron's effective mass in AlGaAs/GaAs heterostructure. The scalar potential V rð Þ ; θ acts as a confinement potential in the dot, and is modeled by the parabolic

mined from the applied magnetic field B via the relation B ¼ ∇ � A. Here, we consider a

1 2 m<sup>∗</sup>ω<sup>2</sup> r 2 þ ℏ 2im<sup>∗</sup>

The quantum Hamilton-Jacobi equation associated with the Hamiltonian H defined in Eq. (1) reads

where S is the quantum action function to be determined. By making use of the substitution

the quantum Hamilton-Jacobi Eq. (4) associated with the Hamiltonian in Eq. (3) turns out to be

The recognition of the complex Hamiltonian H in Eqs. (1) and (3) as a complex realization of the Hamiltonian operator Hb is confirmed by the fact that the quantum Hamilton-Jacobi equation in Eqs. (4) and (6) yields the same Schrodinger equation as constructed from Hb . This can be seen by

, p<sup>θ</sup> <sup>¼</sup> <sup>∂</sup><sup>S</sup>

� � � � pi ¼∂S=∂qi

þ H tð Þ ; q; p

pr <sup>¼</sup> <sup>∂</sup><sup>S</sup> ∂r

þ ω<sup>L</sup> ∂S ∂θ þ 1 2 m<sup>∗</sup>ω<sup>2</sup> r <sup>2</sup> � <sup>i</sup><sup>ℏ</sup> 2m<sup>∗</sup>

þ V tð Þþ ; q

2im<sup>∗</sup>

<sup>0</sup>r<sup>2</sup>=2 to simulate a soft-wall potential. The vector potential A is deter-

1 r pr þ ∂pr ∂r þ 1 r2 ∂p<sup>θ</sup> ∂θ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 <sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> L

> 1 r ∂S ∂r þ ∂2 S ∂r<sup>2</sup> þ

q

� �

1 r pr þ ∂pr ∂r þ 1 r2 ∂p<sup>θ</sup> ∂θ

<sup>þ</sup> V rð Þþ ; <sup>θ</sup> <sup>ℏ</sup>

ℏ

<sup>2</sup>im <sup>∇</sup> � <sup>p</sup>: (1)

, (2)

(3)

=2. Substituting the

is the composite frequency.

¼ 0, (4)

<sup>∂</sup><sup>θ</sup> , (5)

� �

1 r2 ∂2 S ∂θ<sup>2</sup>

¼ 0: (6)

� � in the above equation to

� �

as the following complex Hamiltonian function,

1 <sup>2</sup><sup>m</sup> <sup>p</sup> <sup>þ</sup>

þ 1 <sup>r</sup><sup>2</sup> <sup>p</sup><sup>θ</sup> <sup>þ</sup>

� �<sup>2</sup> � �

We adopt polar coordinates q ¼ ð Þ r; θ and momentum p ¼ pr; p<sup>θ</sup>

c e A � �

> e c A<sup>θ</sup>

constant <sup>B</sup> along the <sup>z</sup> direction, which amounts to Ar <sup>¼</sup> 0 and <sup>A</sup><sup>θ</sup> <sup>¼</sup> Br<sup>2</sup>

∂S ∂t

above assignments of V and A into the complex Hamiltonian Eq. (2), we obtain

þ ωLp<sup>θ</sup> þ

H tð Þ¼ ; q; p

e c Ar � �<sup>2</sup>

<sup>=</sup><sup>2</sup> <sup>¼</sup> <sup>m</sup><sup>∗</sup>ω<sup>2</sup>

<sup>2</sup>m<sup>∗</sup> <sup>p</sup><sup>2</sup> <sup>r</sup> þ 1 r2 p2 θ

� �

where <sup>ω</sup><sup>L</sup> <sup>¼</sup> eB<sup>=</sup> <sup>2</sup>m<sup>∗</sup> ð Þ<sup>c</sup> is the Larmor frequency and <sup>ω</sup> <sup>¼</sup>

Hamiltonian Eq. (1) becomes

24 Nonmagnetic and Magnetic Quantum Dots

<sup>2</sup>m<sup>∗</sup> pr <sup>þ</sup>

<sup>H</sup> <sup>¼</sup> <sup>1</sup>

∂S ∂r � �<sup>2</sup>

applying the following transformation

þ 1 r2

� �<sup>2</sup> " #

∂S ∂θ

<sup>H</sup> <sup>¼</sup> <sup>1</sup>

function <sup>V</sup> <sup>¼</sup> kr<sup>2</sup>

∂S ∂t þ 1 2m<sup>∗</sup>

$$i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m^\*}\left(\frac{\partial^2\Psi}{\partial r^2} + \frac{1}{r}\frac{\partial\Psi}{\partial r} + \frac{1}{r^2}\frac{\partial^2\Psi}{\partial\theta^2}\right) - i\hbar\omega\_L\frac{\partial\Psi}{\partial\theta} + \frac{1}{2}m^\*\omega^2r^2\Psi. \tag{8}$$

Due to the time-independent nature of the applied potentials A and V, the wavefunction Ψ in Eq. (8) assumes the following form of solution,

$$\Psi'(t, r, \theta) = e^{-iEt/\hbar} \psi(r, \theta),\tag{9}$$

where ψð Þ r; θ satisfies the time-independent Schrodinger equation

$$
\hat{H}\psi \triangleq \left[ -\frac{\hbar^2}{2m^\*} \left( \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2} \right) - i\hbar\omega\_L \frac{\partial}{\partial \theta} + \frac{1}{2}m^\*\omega^2 r^2 \right] \psi = E\psi. \tag{10}
$$

On the other hand, Eq. (3) can be rewritten by using the substitutions Eqs. (5) and (7) as

$$
\dot{H}\psi = H\psi.\tag{11}
$$

where H and Hb are defined, respectively, by Eqs. (3) and (10). This is a direct proof of the fact that the complex Hamiltonian H is a functional realization of the Hamiltonian operator Hb in a complex space. Indeed, it can be shown [10] that every quantum operator Ab can be realized as a complex function A via the relation Abψ ¼ Aψ. The combination of Eqs. (10) and (11) reveals the energy conservation law H ¼ E, which is a natural result of Hamilton mechanics by noting that the Hamiltonian H in Eq. (3) does not contain time t explicitly and must be a motion constant equal to the system's total energy E.

Upon performing the differentiations ∂pr=∂r and ∂pθ=∂θ involved in Eq. (3), we have to specify in advance the action function S or equivalently the wavefunction ψ via the relation Eq. (7). This requirement makes the complex Hamiltonian H state-dependent. For a given quantum state described by ψ, the complex Hamiltonian H can be expressed explicitly as:

$$H = \frac{1}{2m^\*} \left( p\_r^2 + \frac{1}{r^2} p\_\theta^2 \right) + \omega \iota\_r p\_\theta + \frac{1}{2} m^\* \omega^2 r^2 + \frac{\hbar}{2im^\*} \left( \frac{1}{r} p\_r + \frac{\hbar}{i} \frac{\partial^2 \ln \psi}{\partial r^2} + \frac{1}{r^2} \frac{\hbar}{i} \frac{\partial^2 \ln \psi}{\partial \theta^2} \right). \tag{12}$$

Apart from deriving the Schrodinger equation, the above complex Hamiltonian also gives electronic quantum motions in the state ψ in terms of the Hamilton equations of motion,

$$\frac{dr}{dt} = \frac{\partial H}{\partial p\_r} = \frac{1}{m^\*} p\_r + \frac{\hbar}{2im^\*} \frac{1}{r} = \frac{\hbar}{im^\*} \frac{\partial \ln \psi}{\partial r} + \frac{\hbar}{2im^\*} \frac{1}{r'} \tag{13}$$

$$\frac{d\varOmega}{dt} = \frac{\varOmega H}{\varOmega p\_{\varOmega}} = \frac{1}{m^\* r^2} p\_{\varOmega} + \varomega\_L = \frac{\hbar}{\varOmega r^\* r^2} \frac{\varOmega \ln \psi}{\varOmega \theta} + \varomega\_L. \tag{14}$$

The appearance of the imaginary number <sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> on the right-hand side of the above equations indicates that the quantum trajectory ð Þ r tð Þ; θð Þt has to be defined in the complex space as ð Þ¼ r tð Þ; θð Þt rRð Þþ t irIð Þt ; θRð Þþ t iθ<sup>I</sup> ð Þ ð Þt to guarantee the solvability of Eqs. (13) and (14). It is just the coupling connection between the real and imaginary parts that gives rise to the quantum phenomena, as we have observed in the real world, such as wave-particle duality [15], tunneling [16], and Heisenberg uncertainty principle [17]. For a given 1D wavefunction Ψð Þ t; x expressed in Cartesian coordinates, the complex Hamiltonian Eq. (1) has a simple form:

$$H(t, \mathbf{x}, p) = \frac{1}{2m}p^2 + V(t, \mathbf{x}) + \frac{\hbar}{2im} \frac{\partial p}{\partial \mathbf{x}} = \frac{1}{2m}p^2 + V(t, \mathbf{x}) - \frac{\hbar^2}{2m} \frac{\partial^2 \ln \Psi(t, \mathbf{x})}{\partial \mathbf{x}^2}.\tag{15}$$

The Hamilton equation for x turns out to be

$$\dot{\mathbf{x}} = \frac{\partial H}{\partial p} = \frac{p}{m} = \frac{1}{m} \frac{\partial \mathbf{S}}{\partial \mathbf{x}} = \frac{\hbar}{im} \frac{\partial \Psi}{\partial \mathbf{x}}, \quad \mathbf{x} \in \mathbb{C} \tag{16}$$

resistance. In the following sections, we will characterize the stagnation magnetic field from the equations of motion Eqs. (19) and (20) and verify the consistency between this theoretical

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

http://dx.doi.org/10.5772/intechopen.74409

27

The conductivity of a quantum dot depends on how electrons move under the confinement potential within the quantum dot. Eqs. (19) and (20) provides us with all the required information to describe the underlying electronic quantum motion. The radial motion rð Þτ described by Eq. (19) and the angular motion θ τð Þ described by Eq. (20) are, individually, periodic time functions, whose periods, T<sup>r</sup> and Tθ, can be computed by using the residue theorem. In case that the radial and angular motions are not commensurable, i.e., Tr=Tθ∉Q, the overall motion is not periodic and the electron's orbit precesses continuously around the periphery of the quantum dot, as shown in Figure 2a. By way of this precession orbit, an electron can pass through the quantum dot from the entrance to the exit and contribute to the

On the other hand, if Tr=T<sup>θ</sup> is a rational number, the shape of the electron's orbit is stationary like a standing wave, as shown in Figure 2b. Except that the orientation of the standing wave happens to align with the direction from the entrance to the exit, as shown in Figure 2c, passage through the quantum dot is prohibited, when a standing-wave motion emerges. A

where N is a positive integer. This condition ensures that when the electron undergoes a complete oscillation in the θ direction, there are integral numbers of oscillation in the r direction. Once electronic standing waves emerge in a quantum dot, the electron after a complete θ revolution will return to the entrance to the quantum dot and consequently contribute to the

As shown in Figure 2d, the standing-wave motion degenerates into a confined motion such that the electron is trapped into a closed trajectory, in the extreme case N ! ∞. When the electron is trapped or stagnated, it is in no way to pass through the quantum dot and causes a remarkable increase in resistance. The special magnetic field corresponding to N ! ∞ plays

The pattern and the orientation of the standing waves can be controlled by the applied magnetic field B via the relation Eq. (20), which indicates that the angular motion depends on

> <sup>q</sup> <sup>¼</sup> eB<sup>=</sup> <sup>2</sup>m<sup>∗</sup> ð Þ<sup>c</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 <sup>0</sup> <sup>þ</sup> e2B<sup>2</sup>

= 2m<sup>∗</sup> ð Þc 2 <sup>q</sup> , (22)

the major role in the magneto-resistance and is to be derived below.

<sup>ω</sup> <sup>¼</sup> <sup>ω</sup><sup>L</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 <sup>0</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> L

ω<sup>L</sup>

T<sup>θ</sup> ¼ NT<sup>r</sup> (21)

standing-wave (non-precessing) orbit has to satisfy the commensurability condition,

prediction and the experimental measurement of resistance.

3. Standing waves and critical magnetic field

conductance.

resistance of the quantum dot.

the parameter,

which can be conceived of as a complex-valued version of Bohmian mechanics [18, 19]. The complex quantum trajectory method based on Eq. (16) has been recently developed into a potential computational tool to analyze wave-packet interference [20] and wave-packet scattering [21].

The wavefunction ψ has to be solved in advance from the Schrodinger Eq. (10), before we determine the electron's quantum trajectory ð Þ r tð Þ; θð Þt from Eqs. (13) and (14). In terms of the dimensionless radial distance <sup>r</sup> <sup>¼</sup> ð Þ <sup>ℏ</sup>=m<sup>ω</sup> �1=<sup>2</sup> r, the eigenvalues En,l and the related eigenfunction ψn,l can be solved analytically as [22].

$$E\_{n,l} = (2\mathfrak{n} + |l| + 1)\hbar\omega + l\hbar\omega\_{L^\ast} \tag{17}$$

$$\psi\_{n,l}(\rho,\theta) = \mathcal{R}\_{n,l}(\rho)\Theta\_l(\theta) = \mathcal{C}\_{n,l}e^{-\rho^2/2}\rho^{\parallel l}L\_n^{\parallel l}(\rho^2)e^{il\theta},\tag{18}$$

where n ¼ 0, 1, 2, ⋯ is the radial quantum number, l ¼ 0, � 1, � 2, ⋯ is the angular quantum number, and Cn,l is a normalization factor. The electronic motion in the eigenstate ψn,l now can be established by integrating Eqs. (13) and (14) with ψ given by Eq. (18):

$$\frac{d\rho}{d\tau} = \frac{1}{i} \left( \frac{d \ln R\_{n,l}(\rho)}{d\rho} + \frac{1}{2\rho} \right) \triangleq f\_{n,l}(\rho), \tag{19}$$

$$\frac{d\theta}{d\tau} = \frac{l}{\rho^2} + \frac{\omega\_L}{\omega}.\tag{20}$$

where the dimensionless time is expressed by τ ¼ tω. Eq. (20) indicates that the angular dynamics θ τð Þ is influenced by the magnetic field <sup>B</sup> via the relation <sup>ω</sup><sup>L</sup> <sup>¼</sup> eB<sup>=</sup> <sup>2</sup>m<sup>∗</sup> ð Þ<sup>c</sup> and reveals the existence of a critical B such that the Larmor angular velocity ωL=ω counterbalances the quantum angular velocity <sup>l</sup>=r<sup>2</sup> to yield <sup>d</sup>θ=d<sup>τ</sup> <sup>¼</sup> 0. The stagnation magnetic field denotes the critical B that stagnates the electron with zero angular displacement within a quantum dot. The occurrence of magnetic stagnation retards the electronic transport and causes a jump in resistance. In the following sections, we will characterize the stagnation magnetic field from the equations of motion Eqs. (19) and (20) and verify the consistency between this theoretical prediction and the experimental measurement of resistance.

## 3. Standing waves and critical magnetic field

The appearance of the imaginary number <sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffi

1

The Hamilton equation for x turns out to be

dimensionless radial distance <sup>r</sup> <sup>¼</sup> ð Þ <sup>ℏ</sup>=m<sup>ω</sup> �1=<sup>2</sup>

ψn,l

ψn,l can be solved analytically as [22].

<sup>2</sup><sup>m</sup> <sup>p</sup><sup>2</sup> <sup>þ</sup> V tð Þþ ; <sup>x</sup>

<sup>x</sup>\_ <sup>¼</sup> <sup>∂</sup><sup>H</sup> <sup>∂</sup><sup>p</sup> <sup>¼</sup> <sup>p</sup>

H tð Þ¼ ; x; p

26 Nonmagnetic and Magnetic Quantum Dots

tering [21].

tions indicates that the quantum trajectory ð Þ r tð Þ; θð Þt has to be defined in the complex space as ð Þ¼ r tð Þ; θð Þt rRð Þþ t irIð Þt ; θRð Þþ t iθ<sup>I</sup> ð Þ ð Þt to guarantee the solvability of Eqs. (13) and (14). It is just the coupling connection between the real and imaginary parts that gives rise to the quantum phenomena, as we have observed in the real world, such as wave-particle duality [15], tunneling [16], and Heisenberg uncertainty principle [17]. For a given 1D wavefunction Ψð Þ t; x expressed in Cartesian coordinates, the complex Hamiltonian Eq. (1) has a simple form:

<sup>2</sup><sup>m</sup> <sup>p</sup><sup>2</sup> <sup>þ</sup> V tð Þ� ; <sup>x</sup>

ℏ 2im ∂p <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>

<sup>m</sup> <sup>¼</sup> <sup>1</sup> m ∂S <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>ℏ</sup> im ∂Ψ ∂x

ð Þ¼ r; θ Rn,lð Þ r Θlð Þ¼ θ Cn,le

be established by integrating Eqs. (13) and (14) with ψ given by Eq. (18):

dr <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>1</sup> i

which can be conceived of as a complex-valued version of Bohmian mechanics [18, 19]. The complex quantum trajectory method based on Eq. (16) has been recently developed into a potential computational tool to analyze wave-packet interference [20] and wave-packet scat-

The wavefunction ψ has to be solved in advance from the Schrodinger Eq. (10), before we determine the electron's quantum trajectory ð Þ r tð Þ; θð Þt from Eqs. (13) and (14). In terms of the

where n ¼ 0, 1, 2, ⋯ is the radial quantum number, l ¼ 0, � 1, � 2, ⋯ is the angular quantum number, and Cn,l is a normalization factor. The electronic motion in the eigenstate ψn,l now can

> þ 1 2r

dlnRn,lð Þ r dr

> dθ <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>l</sup> r<sup>2</sup> þ ω<sup>L</sup>

� �

where the dimensionless time is expressed by τ ¼ tω. Eq. (20) indicates that the angular dynamics θ τð Þ is influenced by the magnetic field <sup>B</sup> via the relation <sup>ω</sup><sup>L</sup> <sup>¼</sup> eB<sup>=</sup> <sup>2</sup>m<sup>∗</sup> ð Þ<sup>c</sup> and reveals the existence of a critical B such that the Larmor angular velocity ωL=ω counterbalances the quantum angular velocity <sup>l</sup>=r<sup>2</sup> to yield <sup>d</sup>θ=d<sup>τ</sup> <sup>¼</sup> 0. The stagnation magnetic field denotes the critical B that stagnates the electron with zero angular displacement within a quantum dot. The occurrence of magnetic stagnation retards the electronic transport and causes a jump in

�<sup>1</sup> <sup>p</sup> on the right-hand side of the above equa-

ℏ2 2m ∂2

r, the eigenvalues En,l and the related eigenfunction

En,l ¼ ð Þ 2n þ jlj þ 1 ℏω þ lℏωL, (17)

�r2=<sup>2</sup> r∣l∣ L∣l∣ <sup>n</sup> <sup>r</sup><sup>2</sup> � �<sup>e</sup>

≜f n,l

ln Ψð Þ t; x

, x ∈ C (16)

<sup>∂</sup>x<sup>2</sup> : (15)

il<sup>θ</sup>, (18)

ð Þ r , (19)

<sup>ω</sup> : (20)

The conductivity of a quantum dot depends on how electrons move under the confinement potential within the quantum dot. Eqs. (19) and (20) provides us with all the required information to describe the underlying electronic quantum motion. The radial motion rð Þτ described by Eq. (19) and the angular motion θ τð Þ described by Eq. (20) are, individually, periodic time functions, whose periods, T<sup>r</sup> and Tθ, can be computed by using the residue theorem. In case that the radial and angular motions are not commensurable, i.e., Tr=Tθ∉Q, the overall motion is not periodic and the electron's orbit precesses continuously around the periphery of the quantum dot, as shown in Figure 2a. By way of this precession orbit, an electron can pass through the quantum dot from the entrance to the exit and contribute to the conductance.

On the other hand, if Tr=T<sup>θ</sup> is a rational number, the shape of the electron's orbit is stationary like a standing wave, as shown in Figure 2b. Except that the orientation of the standing wave happens to align with the direction from the entrance to the exit, as shown in Figure 2c, passage through the quantum dot is prohibited, when a standing-wave motion emerges. A standing-wave (non-precessing) orbit has to satisfy the commensurability condition,

$$T\_{\theta} = NT\_{\rho} \tag{21}$$

where N is a positive integer. This condition ensures that when the electron undergoes a complete oscillation in the θ direction, there are integral numbers of oscillation in the r direction. Once electronic standing waves emerge in a quantum dot, the electron after a complete θ revolution will return to the entrance to the quantum dot and consequently contribute to the resistance of the quantum dot.

As shown in Figure 2d, the standing-wave motion degenerates into a confined motion such that the electron is trapped into a closed trajectory, in the extreme case N ! ∞. When the electron is trapped or stagnated, it is in no way to pass through the quantum dot and causes a remarkable increase in resistance. The special magnetic field corresponding to N ! ∞ plays the major role in the magneto-resistance and is to be derived below.

The pattern and the orientation of the standing waves can be controlled by the applied magnetic field B via the relation Eq. (20), which indicates that the angular motion depends on the parameter,

$$\frac{\omega\_{\rm L}}{\omega} = \frac{\omega\_{\rm L}}{\sqrt{\omega\_{0}^{2} + \omega\_{\rm L}^{2}}} = \frac{eB/(2m^{\*}c)}{\sqrt{\omega\_{0}^{2} + c^{2}B^{2}/(2m^{\*}c)}^{2}},\tag{22}$$

According to the residue theorem, the contour integral in Eq. (23) is equal to 2πi times the sum

dr f n,l

T<sup>r</sup> ¼ ∮ <sup>c</sup><sup>r</sup>

ð Þ r evaluated at its poles within the contour cr, i.e.,

ð Þ <sup>r</sup> <sup>¼</sup> <sup>2</sup>π<sup>i</sup>

integrals along the contours belonging to the same set Ω<sup>k</sup> have the same contour integral,

contour sets, Ω1, Ω2, …, ΩM, with each contour set corresponding to one particular way of pole encirclement. Along all the possible contours, the period T<sup>r</sup> defined by Eq. (23) can only have

In case of l ¼ 0, the radial dynamics and azimuth dynamics are decoupled according to Eqs. (19) and (20). A look on the ground state ð Þ¼ n; l ð Þ 0; 0 is helpful to understand some common features in the states with l ¼ 0. The related wavefunction is given by Eq. (18) as <sup>R</sup>0,0ð Þ¼ <sup>r</sup> <sup>e</sup>�r2=<sup>2</sup> and <sup>Θ</sup>0ð Þ¼ <sup>θ</sup> 1. Substituting this wavefunction into Eqs. (19) and (20) yields

> <sup>2</sup>r<sup>2</sup> � <sup>1</sup> <sup>2</sup><sup>r</sup> , <sup>d</sup><sup>θ</sup>

It appears that that the ground-state electron rotates with a constant angular velocity

T<sup>θ</sup> is simply 2π=ð Þ ωL=ω , and the radial period T<sup>r</sup> can be computed from Eqs. (24) and

2r 2r<sup>2</sup> � 1

The commensurability condition Eq. (21) with the calculated T<sup>r</sup> and T<sup>θ</sup> for the ground state

2

<sup>r</sup> , defined by

ð Þ <sup>r</sup> <sup>¼</sup> <sup>T</sup>ð Þ<sup>k</sup>

X k

<sup>r</sup> . If the number of different ways of pole encirclement is M, we can define M

n o then constitutes a set of quantization levels for the period <sup>T</sup><sup>r</sup>

<sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>ω</sup><sup>L</sup>

2

ð Þ <sup>r</sup> evaluated at its <sup>k</sup>th pole. Let <sup>Ω</sup><sup>k</sup> be the set containing all of

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

Rk, (24)

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29

<sup>ω</sup> : (26)

<sup>p</sup> <sup>=</sup>2. Therefore, the azimuth period

dr ¼ π, (27)

<sup>p</sup> <sup>=</sup>2 on the complex plane of <sup>r</sup>.

ð Þ r . According to the residue theorem, the

<sup>r</sup> , ∀c<sup>r</sup> ∈ Ωk, k ¼ 1, 2, ⋯, M: (25)

of the residues of 1=f n,l

denoted by Tð Þ<sup>k</sup>

M discrete values, Tð Þ<sup>1</sup>

The sequence Tð Þ<sup>1</sup>

(26) as

turns out to be

in the quantum state ψn,l

(A) Standing Wave with l¼0

where Rk is the residue of 1=f n,l

contours which enclose the same poles of 1=f n,l

<sup>r</sup> , Tð Þ<sup>2</sup>

<sup>r</sup> ; Tð Þ<sup>2</sup>

T<sup>r</sup> ¼ ∮ <sup>c</sup><sup>r</sup>

<sup>r</sup> ; ⋯; Tð Þ <sup>M</sup> r

ð Þ r; θ .

the equations of motion for the ground-state electron,

<sup>ω</sup>L=<sup>ω</sup> around its equilibrium radial position <sup>r</sup>eq <sup>¼</sup> ffiffiffi

where <sup>c</sup><sup>r</sup> is any contour enclosing the pole <sup>r</sup>eq <sup>¼</sup> ffiffiffi

dr <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>i</sup>

<sup>T</sup><sup>r</sup> <sup>¼</sup> <sup>1</sup> i ∮ cr

<sup>r</sup> , …, Tð Þ <sup>M</sup>

dr f n,l

Figure 2. Four types of electronic quantum trajectory in a quantum dot. (a) a magnetic field (Bc ¼ 0:4T) not satisfying the commensurability condition Eq. (21) yields precessing trajectories. (b) A critical magnetic field Bc ¼ 0:204T yields standing-wave like quantum trajectory, whose five wavelengths on the circumference do not contact the exit of the quantum dot. (c) A magnetic field Bc ¼ 0:26T yields similar standing-wave trajectory as that in part (b) but with six wavelengths which contact both the entrance and exit of the quantum. (d) At Bc ¼ 0:65T, a stagnation magnetic field, the electron is stagnated within an isolated region of θ.

which, in turn, is solely determined by the magnetic field B. We are able to control the resistance of the quantum dot by varying the magnitude of B to satisfy the commensurability condition Eq. (21). Our next issue is to characterize the critical magnetic field Bc that satisfies the commensurability condition Eq. (21). The period T<sup>r</sup> in Eq. (21) can be evaluated by the radial dynamics Eq. (19). The radial motion rð Þt is a periodic time function whose trace on the complex r plane forms a closed path cr, along which the period T<sup>r</sup> can be computed as a contour integral as

$$T\_{\rho} = \int d\tau = \oint\_{c\_{\rho}} \frac{d\rho}{f\_{n,l}(\rho)}. \tag{23}$$

According to the residue theorem, the contour integral in Eq. (23) is equal to 2πi times the sum of the residues of 1=f n,l ð Þ r evaluated at its poles within the contour cr, i.e.,

$$T\_{\rho} = \oint\_{c\_{\rho}} \frac{d\rho}{f\_{n,l}(\rho)} = 2\pi i \sum\_{k} R\_{k\prime} \tag{24}$$

where Rk is the residue of 1=f n,l ð Þ <sup>r</sup> evaluated at its <sup>k</sup>th pole. Let <sup>Ω</sup><sup>k</sup> be the set containing all of contours which enclose the same poles of 1=f n,l ð Þ r . According to the residue theorem, the integrals along the contours belonging to the same set Ω<sup>k</sup> have the same contour integral, denoted by Tð Þ<sup>k</sup> <sup>r</sup> . If the number of different ways of pole encirclement is M, we can define M contour sets, Ω1, Ω2, …, ΩM, with each contour set corresponding to one particular way of pole encirclement. Along all the possible contours, the period T<sup>r</sup> defined by Eq. (23) can only have M discrete values, Tð Þ<sup>1</sup> <sup>r</sup> , Tð Þ<sup>2</sup> <sup>r</sup> , …, Tð Þ <sup>M</sup> <sup>r</sup> , defined by

$$T\_{\rho} = \oint\_{c\_{\rho}} \frac{d\rho}{f\_{n,l}(\rho)} = T\_{\rho}^{(k)}, \quad \forall c\_{\rho} \in \Omega\_{k}, \quad k = 1, 2, \cdots, M. \tag{25}$$

The sequence Tð Þ<sup>1</sup> <sup>r</sup> ; Tð Þ<sup>2</sup> <sup>r</sup> ; ⋯; Tð Þ <sup>M</sup> r n o then constitutes a set of quantization levels for the period <sup>T</sup><sup>r</sup> in the quantum state ψn,l ð Þ r; θ .

## (A) Standing Wave with l¼0

which, in turn, is solely determined by the magnetic field B. We are able to control the resistance of the quantum dot by varying the magnitude of B to satisfy the commensurability condition Eq. (21). Our next issue is to characterize the critical magnetic field Bc that satisfies the commensurability condition Eq. (21). The period T<sup>r</sup> in Eq. (21) can be evaluated by the radial dynamics Eq. (19). The radial motion rð Þt is a periodic time function whose trace on the complex r plane forms a closed path cr, along which the period T<sup>r</sup> can be computed as a contour integral as

Figure 2. Four types of electronic quantum trajectory in a quantum dot. (a) a magnetic field (Bc ¼ 0:4T) not satisfying the commensurability condition Eq. (21) yields precessing trajectories. (b) A critical magnetic field Bc ¼ 0:204T yields standing-wave like quantum trajectory, whose five wavelengths on the circumference do not contact the exit of the quantum dot. (c) A magnetic field Bc ¼ 0:26T yields similar standing-wave trajectory as that in part (b) but with six wavelengths which contact both the entrance and exit of the quantum. (d) At Bc ¼ 0:65T, a stagnation magnetic field, the

dτ ¼ ∮ <sup>c</sup><sup>r</sup>

dr f n,l

ð Þ <sup>r</sup> : (23)

T<sup>r</sup> ¼ ð

electron is stagnated within an isolated region of θ.

28 Nonmagnetic and Magnetic Quantum Dots

In case of l ¼ 0, the radial dynamics and azimuth dynamics are decoupled according to Eqs. (19) and (20). A look on the ground state ð Þ¼ n; l ð Þ 0; 0 is helpful to understand some common features in the states with l ¼ 0. The related wavefunction is given by Eq. (18) as <sup>R</sup>0,0ð Þ¼ <sup>r</sup> <sup>e</sup>�r2=<sup>2</sup> and <sup>Θ</sup>0ð Þ¼ <sup>θ</sup> 1. Substituting this wavefunction into Eqs. (19) and (20) yields the equations of motion for the ground-state electron,

$$\frac{d\rho}{d\tau} = \mathrm{i}\,\frac{2\rho^2 - 1}{2\rho}, \quad \frac{d\theta}{d\tau} = \frac{\omega\_L}{\omega}.\tag{26}$$

It appears that that the ground-state electron rotates with a constant angular velocity <sup>ω</sup>L=<sup>ω</sup> around its equilibrium radial position <sup>r</sup>eq <sup>¼</sup> ffiffiffi 2 <sup>p</sup> <sup>=</sup>2. Therefore, the azimuth period T<sup>θ</sup> is simply 2π=ð Þ ωL=ω , and the radial period T<sup>r</sup> can be computed from Eqs. (24) and (26) as

$$T\_{\rho} = \frac{1}{\mathrm{i}} \oint\_{c\_{\rho}} \frac{2\rho}{2\rho^2 - 1} d\rho = \pi,\tag{27}$$

where <sup>c</sup><sup>r</sup> is any contour enclosing the pole <sup>r</sup>eq <sup>¼</sup> ffiffiffi 2 <sup>p</sup> <sup>=</sup>2 on the complex plane of <sup>r</sup>.

The commensurability condition Eq. (21) with the calculated T<sup>r</sup> and T<sup>θ</sup> for the ground state turns out to be

$$\frac{\omega\_{\rm L}}{\omega} = \frac{2}{N}, \quad N = 3, 4, 5, \cdots \tag{28}$$

where we note ωL=ω < 1 from its definition in Eq. (22). The critical magnetic field Bc now can be solved from Eq. (28) as

$$B\_{\varepsilon} = \frac{B\_0}{\sqrt{N^2/4 - 1}}, \quad N = 3, 4, 5, \cdots \tag{29}$$

where B<sup>0</sup> is the magnetic field whose Larmor frequency ω<sup>L</sup> is equal to the natural frequency ω<sup>0</sup> of the harmonic oscillator, i.e., <sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>m<sup>∗</sup> ð Þ <sup>c</sup>=<sup>e</sup> <sup>ω</sup>0. The relation expressed by Eq. (29) characterizes all the magnetic fields that force the electron to behave like a standing wave in the ground state of a quantum dot.

Regarding excited states, there are multiple periods in the radial motion rð Þτ as indicated by Eq. (25). Taking first excited state ð Þ¼ n; l ð Þ 1; 0 as an illustrating example, the quantum dynamics is described by

$$i\frac{d\rho}{d\tau} = i\frac{2\rho^4 - 11\rho^2 + 6}{2\rho(\rho^2 - 2)} \quad \frac{d\theta}{d\tau} = \frac{\omega\_L}{\omega} \,\tag{30}$$

which has four equilibrium points at

$$
\rho\_{eq} = \pm \frac{\sqrt{11 \pm \sqrt{73}}}{2} \tag{31}
$$

According to different encirclements of equilibrium points, four sets of complex trajectories rð Þτ can be identified as shown in Figure 3a, where Ω<sup>1</sup> and Ω<sup>2</sup> denote the sets of all trajectories enclosing only one equilibrium point, Ω<sup>3</sup> denotes the set enclosing two equilibrium points, and Ω<sup>4</sup> denotes the set enclosing all the four equilibrium points.

Corresponding to the four different ways of encirclement, the four quantization levels of T<sup>r</sup> can be computed from Eq. (24) as

$$T\_{\rho} = \frac{2\pi (73 \pm 3\sqrt{73})}{292}, \pi, 2\pi. \tag{32}$$

The related critical magnetic field Bc can be determined by substituting Eq. (33) into Eq. (22). Comparing Eq. (28) with Eq. (33), we can see that the critical Bc, which raises standing waves in the ground state, also raises standing waves in the first excited state. The peaks of the magnetoresistance just concentrate on the dominant critical magnetic field that concurrently produces

Figure 2b. (c) Typical time response of an oscillatory Reð Þ θ corresponds to the trajectory shown in Figure 2d.

Figure 3. (a) Four sets of complex trajectories rð Þτ are identified according to different encirclements of equilibrium points in the state ð Þ¼ n; l ð Þ 1; 0 . (b) Typical time response of an increasing Reð Þ θ corresponds to the trajectory shown in

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

http://dx.doi.org/10.5772/intechopen.74409

31

In the case of l > 0, the cyclotron angular velocity ωL=ω and the quantum angular velocity l=r<sup>2</sup> are in the same direction so as to give an intensified resultant <sup>θ</sup>\_ <sup>¼</sup> <sup>l</sup>=r<sup>2</sup> <sup>þ</sup> <sup>ω</sup>L=ω. The coupling between the azimuth motion θ τð Þ and the radial dynamics rð Þτ makes the evaluation of T<sup>θ</sup> more difficult; however, because <sup>r</sup>ð Þ<sup>τ</sup> is a periodic function, we can evaluate <sup>θ</sup>\_ in Eq. (20) by

ave, if only the period of θ τð Þ is concerned,

standing waves in different states.

simply replacing l=r<sup>2</sup> with its average value l=r<sup>2</sup>

(B) Standing Wave with l > 0.

The commensurability condition for the occurrence of standing wave in the four contour sets now can be derived from Eq. (21) as

$$\frac{\frac{\omega\_{\rm L}}{\omega}}{\frac{\omega}{\omega}}=\begin{cases}\frac{73+3\sqrt{73}}{16N}, N \ge 7, & \rho(\tau) \in \Omega\_{1} \\\\ \frac{73-3\sqrt{73}}{16N}, N \ge 3, & \rho(\tau) \in \Omega\_{2} \\\frac{2}{N}, & N \ge 3, & \rho(\tau) \in \Omega\_{3} \\\frac{1}{N}, & N \ge 2, & \rho(\tau) \in \Omega\_{4} \end{cases}\tag{33}$$

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots http://dx.doi.org/10.5772/intechopen.74409 31

Figure 3. (a) Four sets of complex trajectories rð Þτ are identified according to different encirclements of equilibrium points in the state ð Þ¼ n; l ð Þ 1; 0 . (b) Typical time response of an increasing Reð Þ θ corresponds to the trajectory shown in Figure 2b. (c) Typical time response of an oscillatory Reð Þ θ corresponds to the trajectory shown in Figure 2d.

The related critical magnetic field Bc can be determined by substituting Eq. (33) into Eq. (22). Comparing Eq. (28) with Eq. (33), we can see that the critical Bc, which raises standing waves in the ground state, also raises standing waves in the first excited state. The peaks of the magnetoresistance just concentrate on the dominant critical magnetic field that concurrently produces standing waves in different states.

#### (B) Standing Wave with l > 0.

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>2</sup>

Bc <sup>¼</sup> <sup>B</sup><sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N2 =4 � 1

dr <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>i</sup>

Ω<sup>4</sup> denotes the set enclosing all the four equilibrium points.

ω<sup>L</sup> ω ¼

be solved from Eq. (28) as

30 Nonmagnetic and Magnetic Quantum Dots

state of a quantum dot.

which has four equilibrium points at

be computed from Eq. (24) as

now can be derived from Eq. (21) as

ics is described by

where we note ωL=ω < 1 from its definition in Eq. (22). The critical magnetic field Bc now can

where B<sup>0</sup> is the magnetic field whose Larmor frequency ω<sup>L</sup> is equal to the natural frequency ω<sup>0</sup> of the harmonic oscillator, i.e., <sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>m<sup>∗</sup> ð Þ <sup>c</sup>=<sup>e</sup> <sup>ω</sup>0. The relation expressed by Eq. (29) characterizes all the magnetic fields that force the electron to behave like a standing wave in the ground

Regarding excited states, there are multiple periods in the radial motion rð Þτ as indicated by Eq. (25). Taking first excited state ð Þ¼ n; l ð Þ 1; 0 as an illustrating example, the quantum dynam-

> dθ <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>ω</sup><sup>L</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>11</sup> � ffiffiffiffiffi 73

p p

According to different encirclements of equilibrium points, four sets of complex trajectories rð Þτ can be identified as shown in Figure 3a, where Ω<sup>1</sup> and Ω<sup>2</sup> denote the sets of all trajectories enclosing only one equilibrium point, Ω<sup>3</sup> denotes the set enclosing two equilibrium points, and

Corresponding to the four different ways of encirclement, the four quantization levels of T<sup>r</sup> can

The commensurability condition for the occurrence of standing wave in the four contour sets

<sup>16</sup><sup>N</sup> , N <sup>≥</sup> <sup>7</sup>, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>1</sup>

<sup>16</sup><sup>N</sup> , N <sup>≥</sup> <sup>3</sup>, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>2</sup>

<sup>N</sup> , N <sup>≥</sup> <sup>3</sup>, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>3</sup>

<sup>N</sup> , N <sup>≥</sup> <sup>2</sup>, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>4</sup>

<sup>T</sup><sup>r</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>73</sup> � <sup>3</sup> ffiffiffiffiffi <sup>73</sup> � � <sup>p</sup>

> <sup>73</sup> <sup>þ</sup> <sup>3</sup> ffiffiffiffiffi 73 p

> <sup>73</sup> � <sup>3</sup> ffiffiffiffiffi 73 p

2

8

>>>>>>>>>>><

>>>>>>>>>>>:

1

<sup>2</sup>r<sup>4</sup> � <sup>11</sup>r<sup>2</sup> <sup>þ</sup> <sup>6</sup> 2r rð Þ <sup>2</sup> � 2

req ¼ �

<sup>N</sup> , N <sup>¼</sup> <sup>3</sup>, <sup>4</sup>, <sup>5</sup>, <sup>⋯</sup>, (28)

<sup>q</sup> , N <sup>¼</sup> <sup>3</sup>, <sup>4</sup>, <sup>5</sup>, <sup>⋯</sup>, (29)

<sup>ω</sup> , (30)

<sup>2</sup> (31)

<sup>292</sup> ,π, <sup>2</sup>π: (32)

9

>>>>>>>>>>>=

(33)

>>>>>>>>>>>;

In the case of l > 0, the cyclotron angular velocity ωL=ω and the quantum angular velocity l=r<sup>2</sup> are in the same direction so as to give an intensified resultant <sup>θ</sup>\_ <sup>¼</sup> <sup>l</sup>=r<sup>2</sup> <sup>þ</sup> <sup>ω</sup>L=ω. The coupling between the azimuth motion θ τð Þ and the radial dynamics rð Þτ makes the evaluation of T<sup>θ</sup> more difficult; however, because <sup>r</sup>ð Þ<sup>τ</sup> is a periodic function, we can evaluate <sup>θ</sup>\_ in Eq. (20) by simply replacing l=r<sup>2</sup> with its average value l=r<sup>2</sup> ave, if only the period of θ τð Þ is concerned,

$$T\_{\theta} = \frac{2\pi}{\overline{\theta}\_{\text{ave}}} = \frac{2\pi}{(l/\rho^2)\_{\text{ave}} + \omega\_{\text{L}}/\omega} = NT\_{\rho}. \tag{34}$$

The comparison between Eqs. (33) and (40) leads to the observation that the number of the allowed integer N decreases dramatically when l increases from 0 to 1. Since the total different number of N accounts for the number of different ways by which standing wave can be formed, the possibility for the occurrence of standing-wave motion and thus the electronic resistance decreases with increasing angular quantum number l. The main reason is that the increment of the angular velocity <sup>θ</sup>\_ <sup>¼</sup> <sup>l</sup>=r<sup>2</sup> <sup>þ</sup> <sup>ω</sup>L=<sup>ω</sup> with large <sup>l</sup> accelerates the electron's angular motion around the quantum dot and thus improves the conductance of the quantum dot.

In this case, the cyclotron angular velocity ωL=ω and the quantum angular velocity l=r<sup>2</sup> are in opposite directions so as to give a weakened resultant <sup>θ</sup>\_ <sup>¼</sup> <sup>l</sup>=r<sup>2</sup> <sup>þ</sup> <sup>ω</sup>L=ω. The resultant angu-

The cases of <sup>l</sup> <sup>¼</sup> 0 and <sup>l</sup> <sup>&</sup>gt; 0 considered previously belong to category (1) with <sup>θ</sup>\_

where the admissible integer N for the three categories is summarized in Table 1.

dot, as shown in Figure 2b and <sup>c</sup>. For an angular quantum number with �<sup>1</sup> <sup>&</sup>lt; <sup>l</sup>=r<sup>2</sup>

ave to yield

Range of l Critical ωL=ω Range of integer N

<sup>θ</sup>\_ ave <sup>¼</sup> <sup>l</sup>=r<sup>2</sup>

<sup>ω</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> NT<sup>r</sup> � <sup>l</sup> r2 ave

<sup>ω</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> NT<sup>r</sup> � <sup>l</sup> r2 ave

<sup>ω</sup> <sup>¼</sup> �2<sup>π</sup> NT<sup>r</sup> � <sup>l</sup> r2 ave

ω<sup>L</sup> <sup>ω</sup> ¼ � <sup>l</sup> r2 ave

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> �2<sup>π</sup> NT<sup>r</sup> � <sup>l</sup> r2 ave

Table 1. The relation between critical Larmor frequency and angular quantum number l.

ω<sup>L</sup> ω ¼ �

The critical magnetic field Bc given by Eq. (41) with θ\_ ave > 0 and θ\_

the case of l < 0 belongs to categories (2) and (3). Taking into account the motion with θ\_

2π NT<sup>r</sup>

wave motions oscillating, respectively, counterclockwise and clockwise around the quantum

category (2), there exists a special Larmor angular velocity ωL=ω such that it counterbalances

� <sup>l</sup> r2 

ave

ave may be positive, negative or zero, depending on the magnitude of l=r<sup>2</sup>

ave <sup>≥</sup> 0, (2) �<sup>1</sup> <sup>&</sup>lt; <sup>l</sup>=r<sup>2</sup>

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

http://dx.doi.org/10.5772/intechopen.74409

, (41)

ave þ ωL=ω ¼ 0: (42)

2π=T<sup>r</sup> 1þ l=r<sup>2</sup> ð Þave

N ! ∞

N ≥ <sup>2</sup>π=T<sup>r</sup> � l=r<sup>2</sup> ð Þave

�2π=T<sup>r</sup> l=r<sup>2</sup> ð Þave

N > <sup>2</sup>π=T<sup>r</sup> 1þ l=r<sup>2</sup> ð Þave

ave < 0 produces standing-

< N ≤ <sup>2</sup>π=T<sup>r</sup> l=r<sup>2</sup> ð Þave

≤ N < �2π=T<sup>r</sup> 1þ l=r<sup>2</sup> ð Þave ave,

33

ave < 0, and (3)

ave > 0, while

ave < 0,

ave < 0 in

(C) Standing Waves with l < 0

which can be classified into three categories: (1) l=r<sup>2</sup>

ave ≤ � 1, as listed in Table 1.

the quantum angular velocity l=r<sup>2</sup>

ave ≥ 0 <sup>ω</sup><sup>L</sup>

ave ≤ � 1 <sup>ω</sup><sup>L</sup>

ave < 0 <sup>ω</sup><sup>L</sup>

lar velocity θ\_

Eq. (36) now becomes

l=r<sup>2</sup>

l=r<sup>2</sup>

l=r<sup>2</sup>

�<sup>1</sup> <sup>&</sup>lt; <sup>l</sup>=r<sup>2</sup>

The time average l=r<sup>2</sup> � � ave is computed over one period of rð Þτ and can be converted into a contour integral along the contour c<sup>r</sup> traced by rð Þτ on the complex plane:

$$\left(\frac{l}{\rho^2}\right)\_{ave} = \frac{l}{T\_\rho}\int\_0^{T\_\rho} \frac{d\tau}{\rho^2(\tau)} = \frac{l}{T\_\rho}\oint\_{c\_\rho} \frac{d\rho}{\rho^2 f\_{n,l}(\rho)}.\tag{35}$$

Substituting the above l=r<sup>2</sup> � � ave into Eq. (34), we obtain the critical value of ωL=ω as

$$\frac{\omega\_L}{\omega} = \frac{2\pi}{NT\_\rho} - \left(\frac{l}{\rho^2}\right)\_{\text{ave}}.\tag{36}$$

Due to the constraint 0 ≤ ωL=ω ≤ 1, the admissible integer N lies in the interval

$$\frac{2\pi/T\_{\rho}}{1+(l/\rho^2)\_{\text{ave}}} < N \le \frac{2\pi/T\_{\rho}}{(l/\rho^2)\_{\text{ave}}},\tag{37}$$

where T<sup>r</sup> and l=r<sup>2</sup> � � ave are given by Eqs. (24) and (35), respectively. The admissible range of N is narrowed by increasing angular quantum number l, as can be seen from inequality Eq. (37). There is a maximum allowable l beyond which inequality Eq. (37) has no integer solution and standing-wave motion within the quantum dot disappears. To compare with the quantum state ð Þ¼ n; l ð Þ 1; 0 considered previously, let us study the state ð Þ¼ n; l ð Þ 1; 1 whose quantum motion is described by

$$i\frac{d\rho}{d\tau} = i\frac{2\rho^4 - 11\rho^2 + 6}{2\rho(\rho^2 - 2)},\\\frac{d\theta}{d\tau} = \frac{1}{\rho^2} + \frac{\omega\_L}{\omega}.\tag{38}$$

The period T<sup>r</sup> is the same as that derived in Eq. (32), and the period T<sup>θ</sup> can be computed by Eq. (34) with l=r<sup>2</sup> � � ave evaluated by the contour integral Eq. (35) as

$$\begin{pmatrix} 1\\ \left(\frac{1}{\rho^2}\right)\_{\text{ave}} = \begin{cases} l\left(11 + \sqrt{73}\right)/12, & \rho(\tau) \in \Omega\_1\\ l\left(11 - \sqrt{73}\right)/12, & \rho(\tau) \in \Omega\_2\\ 2l/3, & \rho(\tau) \in \Omega\_3 \cup \Omega\_4 \end{cases} \end{pmatrix} \tag{39}$$

Using T<sup>r</sup> and l=r<sup>2</sup> � � ave in Eqs. (36) and (37), the critical value of ωL=ω in the state ð Þ¼ n; l ð Þ 1; 1 becomes

$$\frac{\omega\_L}{\omega} \begin{cases} \frac{73 + 3\sqrt{73}}{16N} - \frac{11 + \sqrt{73}}{12}, N = 3, & \rho(\tau) \in \Omega\_1 \\ \frac{73 - 3\sqrt{73}}{16N} - \frac{11 - \sqrt{73}}{12}, 3 \le N \le 14, & \rho(\tau) \in \Omega\_2 \\ \frac{2/N - 2/3, N = 2, 3, & \rho(\tau) \in \Omega\_3 \\ 1/N - 2/3, N = 1, & \rho(\tau) \in \Omega\_4 \end{cases} \tag{40}$$

The comparison between Eqs. (33) and (40) leads to the observation that the number of the allowed integer N decreases dramatically when l increases from 0 to 1. Since the total different number of N accounts for the number of different ways by which standing wave can be formed, the possibility for the occurrence of standing-wave motion and thus the electronic resistance decreases with increasing angular quantum number l. The main reason is that the increment of the angular velocity <sup>θ</sup>\_ <sup>¼</sup> <sup>l</sup>=r<sup>2</sup> <sup>þ</sup> <sup>ω</sup>L=<sup>ω</sup> with large <sup>l</sup> accelerates the electron's angular motion around the quantum dot and thus improves the conductance of the quantum dot.

#### (C) Standing Waves with l < 0

<sup>T</sup><sup>θ</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> θ\_ ave

> ave ¼ l Tr ð<sup>T</sup><sup>r</sup> 0

l r2 � �

> dr <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>i</sup>

1 r2 � �

ω<sup>L</sup> ω

8 >>>>>>><

>>>>>>>:

ave ¼

<sup>73</sup> <sup>þ</sup> <sup>3</sup> ffiffiffiffiffi 73 p

<sup>73</sup> � <sup>3</sup> ffiffiffiffiffi 73 p

8 ><

>:

<sup>16</sup><sup>N</sup> � <sup>11</sup> <sup>þ</sup> ffiffiffiffiffi

<sup>16</sup><sup>N</sup> � <sup>11</sup> � ffiffiffiffiffi

contour integral along the contour c<sup>r</sup> traced by rð Þτ on the complex plane:

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> NT<sup>r</sup>

Due to the constraint 0 ≤ ωL=ω ≤ 1, the admissible integer N lies in the interval

2π=T<sup>r</sup> 1 þ l=r<sup>2</sup> ð Þave

The time average l=r<sup>2</sup> � �

32 Nonmagnetic and Magnetic Quantum Dots

Substituting the above l=r<sup>2</sup> � �

where T<sup>r</sup> and l=r<sup>2</sup> � �

motion is described by

Eq. (34) with l=r<sup>2</sup> � �

Using T<sup>r</sup> and l=r<sup>2</sup> � �

becomes

<sup>¼</sup> <sup>2</sup><sup>π</sup>

dτ <sup>r</sup><sup>2</sup>ð Þ<sup>τ</sup> <sup>¼</sup> <sup>l</sup> Tr ∮ cr

> � <sup>l</sup> r2 � �

< N ≤

is narrowed by increasing angular quantum number l, as can be seen from inequality Eq. (37). There is a maximum allowable l beyond which inequality Eq. (37) has no integer solution and standing-wave motion within the quantum dot disappears. To compare with the quantum state ð Þ¼ n; l ð Þ 1; 0 considered previously, let us study the state ð Þ¼ n; l ð Þ 1; 1 whose quantum

The period T<sup>r</sup> is the same as that derived in Eq. (32), and the period T<sup>θ</sup> can be computed by

73 p

73 p

2=N � 2=3, N ¼ 2, 3, rð Þτ ∈ Ω<sup>3</sup> 1=N � 2=3, N ¼ 1, rð Þτ ∈ Ω<sup>4</sup>

<sup>73</sup> � � <sup>p</sup> <sup>=</sup>12, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>1</sup>

<sup>73</sup> � � <sup>p</sup> <sup>=</sup>12, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>2</sup> 2l=3, rð Þτ ∈ Ω3∪Ω<sup>4</sup>

ave in Eqs. (36) and (37), the critical value of ωL=ω in the state ð Þ¼ n; l ð Þ 1; 1

<sup>12</sup> , N <sup>¼</sup> <sup>3</sup>, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>1</sup>

<sup>12</sup> , <sup>3</sup> <sup>≤</sup> <sup>N</sup> <sup>≤</sup> <sup>14</sup>, <sup>r</sup>ð Þ<sup>τ</sup> <sup>∈</sup> <sup>Ω</sup><sup>2</sup>

<sup>2</sup>r<sup>4</sup> � <sup>11</sup>r<sup>2</sup> <sup>þ</sup> <sup>6</sup> <sup>2</sup>r rð Þ <sup>2</sup> � <sup>2</sup> ,

ave evaluated by the contour integral Eq. (35) as

<sup>l</sup> <sup>11</sup> <sup>þ</sup> ffiffiffiffiffi

<sup>l</sup> <sup>11</sup> � ffiffiffiffiffi

<sup>l</sup>=r<sup>2</sup> ð Þave <sup>þ</sup> <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> NTr: (34)

ð Þ <sup>r</sup> : (35)

: (36)

, (37)

<sup>ω</sup> : (38)

(39)

(40)

9 >=

>;

9 >>>>>>>=

>>>>>>>;

ave is computed over one period of rð Þτ and can be converted into a

ave into Eq. (34), we obtain the critical value of ωL=ω as

ave

2π=T<sup>r</sup> l=r<sup>2</sup> ð Þave

ave are given by Eqs. (24) and (35), respectively. The admissible range of N

dθ <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>1</sup> r<sup>2</sup> þ ω<sup>L</sup>

dr r<sup>2</sup>f n,l

> In this case, the cyclotron angular velocity ωL=ω and the quantum angular velocity l=r<sup>2</sup> are in opposite directions so as to give a weakened resultant <sup>θ</sup>\_ <sup>¼</sup> <sup>l</sup>=r<sup>2</sup> <sup>þ</sup> <sup>ω</sup>L=ω. The resultant angular velocity θ\_ ave may be positive, negative or zero, depending on the magnitude of l=r<sup>2</sup> ave, which can be classified into three categories: (1) l=r<sup>2</sup> ave <sup>≥</sup> 0, (2) �<sup>1</sup> <sup>&</sup>lt; <sup>l</sup>=r<sup>2</sup> ave < 0, and (3) l=r<sup>2</sup> ave ≤ � 1, as listed in Table 1.

> The cases of <sup>l</sup> <sup>¼</sup> 0 and <sup>l</sup> <sup>&</sup>gt; 0 considered previously belong to category (1) with <sup>θ</sup>\_ ave > 0, while the case of l < 0 belongs to categories (2) and (3). Taking into account the motion with θ\_ ave < 0, Eq. (36) now becomes

$$\frac{\omega\_L}{\omega} = \pm \frac{2\pi}{NT\_\rho} - \left(\frac{l}{\rho^2}\right)\_{\text{ave}}\tag{41}$$

where the admissible integer N for the three categories is summarized in Table 1.

The critical magnetic field Bc given by Eq. (41) with θ\_ ave > 0 and θ\_ ave < 0 produces standingwave motions oscillating, respectively, counterclockwise and clockwise around the quantum dot, as shown in Figure 2b and <sup>c</sup>. For an angular quantum number with �<sup>1</sup> <sup>&</sup>lt; <sup>l</sup>=r<sup>2</sup> ave < 0 in category (2), there exists a special Larmor angular velocity ωL=ω such that it counterbalances the quantum angular velocity l=r<sup>2</sup> ave to yield

$$
\dot{\theta}\_{\text{ave}} = \left( \mathbb{I} / \rho^2 \right)\_{\text{ave}} + \omega\_{\text{L}} / \omega = \mathbf{0}. \tag{42}
$$


Table 1. The relation between critical Larmor frequency and angular quantum number l.

The critical Bc satisfying θ\_ ave ¼ 0 produces isolated standing waves that form closed trajectories as shown in Figure 2d. This critical Bc nullifies the electron's net angular displacement and is called stagnation magnetic field. Because a passage through the quantum dot requires a net angular displacement equal to π, an electron with θ\_ ave ¼ 0 is unable to pass the quantum dot and makes no contribution to the conductivity.

In a case study of l < 0, we consider the state of ð Þ¼ n; l ð Þ 1; �1 , whose quantum equations of motion read

$$i\frac{d\rho}{d\tau} = i\frac{2\rho^4 - 11\rho^2 + 6}{2\rho(\rho^2 - 2)}, \quad \frac{d\theta}{d\tau} = \frac{-1}{\rho^2} + \frac{\omega\_L}{\omega}.\tag{43}$$

the quantum angular velocity l=r<sup>2</sup> � �

Apart from the consequence of θ\_

θ\_

wave number N. From Eq. (21), the relation between θ\_

ave such that the electron's net angular displacement Δθave

http://dx.doi.org/10.5772/intechopen.74409

35

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

ave ¼ 0, the effect of magnetic stagnation is also reflected in the

ave and N can be expressed by

<sup>l</sup>=r<sup>2</sup> ð Þave <sup>þ</sup> <sup>ω</sup>L=<sup>ω</sup> (44)

<sup>73</sup> � � <sup>p</sup> <sup>=</sup><sup>12</sup>

is zero and the electron is stagnated within the quantum dot. The instantaneous dynamics of rð Þτ and θ τð Þ are solved from Eq. (43) at the stagnation frequency ωL=ω ¼ 2=3 and the results are shown in Figure 3c. As expected, the net change of θ τð Þ is zero after a period of oscillation. The projection of the computed complex trajectory on the real x � y plane is a closed path as illustrated in Figure 2d. This closed path produced by magnetic stagnation isolates the electron

<sup>¼</sup> <sup>2</sup>π=T<sup>r</sup>

There are infinitely many wavelengths distributed on the circumference of the quantum dot, as

and 2=3, at which the wave number N approaches to infinity. These two stagnation frequencies

Figure 4. Typical standing-wave motions in the state ð Þ¼ n; l ð Þ 1; �1 with wave number N ¼ 5,7 and 9. The trajectory sets

Ω1, Ω2, and Ω<sup>3</sup> refer to the three sets of radial trajectory defined in Figure 3a.

ave approaches to zero. The variation of the wave number N with respect to the critical Larmor frequency ωL=ω for the quantum state ð Þ¼ n; l ð Þ 1; �1 is demonstrated in Figure 5a. A prominent change of <sup>N</sup> appears in the vicinity of the two stagnation frequencies <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi

from the exit of the quantum dot and is the main cause of electronic resistance.

<sup>N</sup> <sup>¼</sup> <sup>2</sup>π=T<sup>r</sup> θave

The radial trajectories rð Þτ are the same as those depicted in Figure 3a. Along different sets of radial trajectory, different modes of standing-wave motion are excited. According to the value of l=r<sup>2</sup> � � ave ¼ � <sup>1</sup>=r<sup>2</sup> � � ave given by Eq. (39), it is found that the trajectory set Ω<sup>1</sup> belongs to category (3), while the sets Ω2, Ω<sup>3</sup> and Ω<sup>4</sup> belong to category (2), as tabulated in Table 2.

Typical standing waves in Ω1, Ω<sup>2</sup> and Ω<sup>3</sup> are shown in Figure 4 for N ¼ 5, 7 and 9. We can see that the geometrical meaning of the integer N defined in Eq. (21) is just the number of electronic waves distributed on the circumference of the quantum dot. Due to θ\_ ave < 0 in Ω<sup>1</sup> trajectory set, as indicated in Table 2, the mean rotation direction of the electron in Ω<sup>1</sup> is clockwise. Because θ\_ ave merely denotes the mean angular velocity, locally we may have θ\_ > 0 during some short periods in which the electron rotates in an opposite direction as shown in Figure 3b.

In the state <sup>ψ</sup>, we have two stagnation frequencies at <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi <sup>73</sup> � � <sup>p</sup> <sup>=</sup>12 and <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>2</sup>=3. In the presence of magnetic stagnation, Larmor angular velocity ωL=ω is counterbalanced by


Table 2. Distribution of the critical frequencies in the state ð Þ¼ n; l ð Þ 1; �1 .

the quantum angular velocity l=r<sup>2</sup> � � ave such that the electron's net angular displacement Δθave is zero and the electron is stagnated within the quantum dot. The instantaneous dynamics of rð Þτ and θ τð Þ are solved from Eq. (43) at the stagnation frequency ωL=ω ¼ 2=3 and the results are shown in Figure 3c. As expected, the net change of θ τð Þ is zero after a period of oscillation. The projection of the computed complex trajectory on the real x � y plane is a closed path as illustrated in Figure 2d. This closed path produced by magnetic stagnation isolates the electron from the exit of the quantum dot and is the main cause of electronic resistance.

The critical Bc satisfying θ\_

34 Nonmagnetic and Magnetic Quantum Dots

motion read

in Table 2.

θ\_

to the value of l=r<sup>2</sup> � �

Ω<sup>1</sup> 0 ≤ <sup>ω</sup><sup>L</sup>

Ω<sup>2</sup> <sup>11</sup>� ffiffiffiffi

Ω<sup>3</sup> <sup>ω</sup><sup>L</sup>

Ω<sup>4</sup> <sup>ω</sup><sup>L</sup>

73 p <sup>12</sup> <sup>≤</sup> <sup>ω</sup><sup>L</sup>

> <sup>ω</sup> <sup>&</sup>lt; <sup>11</sup>� ffiffiffiffi 73 p 12

Table 2. Distribution of the critical frequencies in the state ð Þ¼ n; l ð Þ 1; �1 .

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>11</sup>� ffiffiffiffi 73 p 12

0 ≤ <sup>ω</sup><sup>L</sup>

<sup>ω</sup> <sup>&</sup>gt; <sup>2</sup> 3

0 ≤ <sup>ω</sup><sup>L</sup> <sup>ω</sup> <sup>&</sup>lt; <sup>2</sup> 3

<sup>ω</sup> <sup>&</sup>gt; <sup>2</sup> 3

0 ≤ <sup>ω</sup><sup>L</sup> <sup>ω</sup> < <sup>2</sup> 3

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>2</sup> 3

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>2</sup> 3

angular displacement equal to π, an electron with θ\_

dr <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>i</sup>

ave ¼ � <sup>1</sup>=r<sup>2</sup> � �

waves distributed on the circumference of the quantum dot. Due to θ\_

In the state <sup>ψ</sup>, we have two stagnation frequencies at <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi

<sup>ω</sup> < 1 <sup>ω</sup><sup>L</sup>

<sup>ω</sup> <sup>&</sup>lt; <sup>1</sup> <sup>ω</sup><sup>L</sup>

and makes no contribution to the conductivity.

ave ¼ 0 produces isolated standing waves that form closed trajecto-

<sup>d</sup><sup>τ</sup> <sup>¼</sup> �<sup>1</sup> r<sup>2</sup> þ ω<sup>L</sup>

ave given by Eq. (39), it is found that the trajectory set Ω<sup>1</sup>

ave ¼ 0 is unable to pass the quantum dot

<sup>ω</sup> : (43)

ave < 0 in Ω<sup>1</sup> trajectory set,

<sup>73</sup> � � <sup>p</sup> <sup>=</sup>12 and <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>2</sup>=3.

<sup>12</sup> <sup>4</sup> <sup>≤</sup> <sup>N</sup> <sup>≤</sup> <sup>9</sup>

<sup>12</sup> <sup>N</sup> <sup>≥</sup> <sup>4</sup>

<sup>12</sup> <sup>N</sup> <sup>≥</sup> <sup>15</sup>

<sup>12</sup> <sup>N</sup> ! <sup>∞</sup>

<sup>3</sup> N ≥ 7

<sup>3</sup> N ! ∞

<sup>3</sup> N ≥ 3

<sup>3</sup> N ≥ 4

<sup>3</sup> N ! ∞

<sup>3</sup> N ≥ 2

ries as shown in Figure 2d. This critical Bc nullifies the electron's net angular displacement and is called stagnation magnetic field. Because a passage through the quantum dot requires a net

In a case study of l < 0, we consider the state of ð Þ¼ n; l ð Þ 1; �1 , whose quantum equations of

The radial trajectories rð Þτ are the same as those depicted in Figure 3a. Along different sets of radial trajectory, different modes of standing-wave motion are excited. According

belongs to category (3), while the sets Ω2, Ω<sup>3</sup> and Ω<sup>4</sup> belong to category (2), as tabulated

Typical standing waves in Ω1, Ω<sup>2</sup> and Ω<sup>3</sup> are shown in Figure 4 for N ¼ 5, 7 and 9. We can see that the geometrical meaning of the integer N defined in Eq. (21) is just the number of electronic

as indicated in Table 2, the mean rotation direction of the electron in Ω<sup>1</sup> is clockwise. Because

In the presence of magnetic stagnation, Larmor angular velocity ωL=ω is counterbalanced by

<sup>ω</sup> ¼ � <sup>73</sup>þ<sup>3</sup> ffiffiffiffi 73 p <sup>16</sup><sup>N</sup> <sup>þ</sup> <sup>11</sup><sup>þ</sup> ffiffiffiffi 73 p

<sup>ω</sup> <sup>¼</sup> <sup>73</sup>�<sup>3</sup> ffiffiffiffi 73 p <sup>16</sup><sup>N</sup> <sup>þ</sup> <sup>11</sup>� ffiffiffiffi 73 p

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>11</sup>� ffiffiffiffi 73 p

ω<sup>L</sup> <sup>ω</sup> ¼ � <sup>73</sup>�<sup>3</sup> ffiffiffiffi 73 p <sup>16</sup><sup>N</sup> <sup>þ</sup> <sup>11</sup>� ffiffiffiffi 73 p

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>2</sup> <sup>N</sup> <sup>þ</sup> <sup>2</sup>

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>2</sup>

ω<sup>L</sup> <sup>ω</sup> ¼ � <sup>2</sup> <sup>N</sup> <sup>þ</sup> <sup>2</sup>

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>1</sup> <sup>N</sup> <sup>þ</sup> <sup>2</sup>

ω<sup>L</sup> <sup>ω</sup> <sup>¼</sup> <sup>2</sup>

ω<sup>L</sup> <sup>ω</sup> ¼ � <sup>1</sup> <sup>N</sup> <sup>þ</sup> <sup>2</sup>

Set Frequency range Critical frequency Integer N

periods in which the electron rotates in an opposite direction as shown in Figure 3b.

ave merely denotes the mean angular velocity, locally we may have θ\_ > 0 during some short

<sup>2</sup>r<sup>4</sup> � <sup>11</sup>r<sup>2</sup> <sup>þ</sup> <sup>6</sup> <sup>2</sup>r rð Þ <sup>2</sup> � <sup>2</sup> , <sup>d</sup><sup>θ</sup> Apart from the consequence of θ\_ ave ¼ 0, the effect of magnetic stagnation is also reflected in the wave number N. From Eq. (21), the relation between θ\_ ave and N can be expressed by

$$N = \frac{2\pi/T\_{\rho}}{\Theta\_{\text{ave}}} = \frac{2\pi/T\_{\rho}}{(l/\rho^2)\_{\text{ave}} + \omega\_{\text{L}}/\omega} \tag{44}$$

There are infinitely many wavelengths distributed on the circumference of the quantum dot, as θ\_ ave approaches to zero. The variation of the wave number N with respect to the critical Larmor frequency ωL=ω for the quantum state ð Þ¼ n; l ð Þ 1; �1 is demonstrated in Figure 5a. A prominent change of <sup>N</sup> appears in the vicinity of the two stagnation frequencies <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi <sup>73</sup> � � <sup>p</sup> <sup>=</sup><sup>12</sup> and 2=3, at which the wave number N approaches to infinity. These two stagnation frequencies

Figure 4. Typical standing-wave motions in the state ð Þ¼ n; l ð Þ 1; �1 with wave number N ¼ 5,7 and 9. The trajectory sets Ω1, Ω2, and Ω<sup>3</sup> refer to the three sets of radial trajectory defined in Figure 3a.

Figure 5. (a) The variation of wave number N with respect to the Larmor frequency ωL=ω in the quantum state ð Þ¼ <sup>n</sup>; <sup>l</sup> ð Þ <sup>1</sup>; �<sup>1</sup> . (b) The two stagnation frequencies, <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi <sup>73</sup> � � <sup>p</sup> <sup>=</sup>12 and 2=3, coincide with the two peaks of the experimental curve of resistance.

coincide with the locations of the resistance peaks by comparing with the experimental results as shown in Figure 5b.

#### 4. Experimental verification

This section will compare the above theoretical predictions with the existing experimental data [4, 13] to confirm the fact that the effect of magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field. The experiment was performed in an AlGaAs/GaAs heterostructure with a carrier concentration ne <sup>¼</sup> <sup>2</sup>:<sup>5</sup> � 1011cm�2. Resistance was measured at temperature T ¼ 1:4K using a sensitive lock-in amplifier at currents of typically 1 nA and a frequency of 12 Hz. The resulting resistance measurement in the range of low magnetic field B ≤ 1:3 is depicted in Figure 5b showing a strong peak located around B ¼ 0:22T and three weak peaks at B ¼ 0:65T, 0:97T, and 1:21T.

Thus far, our analysis on quantum trajectory focuses on some specific states. In order to know the influence of the applied magnetic field on the resistance, we have to consider all the possible quantum states occupied in the device. At temperature T ¼ 1:4K, where the resistance is measured, the possible states to be occupied can be estimated by the Fermi-Dirac distribution,

$$f(E) = \frac{1}{1 + e^{\left(E\_{\pi\_l l} - E\_F\right)/k\_B T}},\tag{45}$$

An incident electron subjected to an applied magnetic field B may enter any one of the occupied states listed in Table 3. The electronic resistance induced by B depends on the global transportation behavior across the quantum dot through all the allowable states. Magnetic stagnation slows down the electron's angular rate and retards the passage of the electron. The angular motion is fully retarded and the electron is trapped in the quantum dot without

01 2 3 4

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

http://dx.doi.org/10.5772/intechopen.74409

37

�1 2/3 2/3, 0.205 2/3, 0.373, 0.119 2/3, 0.52, 0.19 2/3, 0.252, 0.124 �2 4/5 4/5, 0.316 4/5, 0.543, 0.2 4/5, 0.73, 0.31, 0.146 0.4, 0.213, 0.115

l 000 0 0 0

�3 6/7 6/7, 0.391 6/7, 0.64, 0.26 0.39, 0.196 <sup>∗</sup> �4 8/9 8/9, 0.445 0.7, 0.3, 8/9 0.454, 0.237 <sup>∗</sup> �<sup>5</sup> <sup>∗</sup> 0.486 0.75, 0.347 ∗ ∗ �<sup>6</sup> <sup>∗</sup> 0.52 0.78, 0.38 ∗ ∗ �<sup>7</sup> <sup>∗</sup> 0.546 ∗∗ ∗ �<sup>8</sup> <sup>∗</sup> 0.57 ∗∗ ∗

> � ¼ �<sup>X</sup> n, l

where the summation is taken over all the states listed in Table 3. The expression of ωL=ω as a function of B has already been given by Eq. (22). Upon comparing the prediction of Eq. (46) with the experimental results, we evaluate the constants in ωL=ω according to the experimental

<sup>ω</sup> <sup>¼</sup> <sup>0</sup>:88<sup>B</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

which in turn is substituted into Eq. (46) to express the magneto-stagnation function S Bð Þ as an

The electron's total angular velocity via all admissible quantum states at T ¼ 1:4K can be esti-

there is a high resistance to the electron's angular movability. Accordingly, S Bð Þ can be reasonably treated as an alternative description of electronic resistance. Figure 5a and b illustrates the first

ln <sup>ω</sup><sup>L</sup> ω þ l r2 � �

� � � �

ave ! 0. To quantify the effect of magnetic stagnation,

<sup>0</sup>:<sup>4096</sup> <sup>þ</sup> <sup>0</sup>:7744B<sup>2</sup> <sup>p</sup> (47)

ave ! 0, a large value of S Bð Þ implies that

ave

� � � � n,l

<sup>k</sup>=<sup>m</sup> <sup>p</sup> <sup>¼</sup> <sup>0</sup>:64meV. Using

(46)

contribution to the conductance, as θ\_

ωL=ω n

these data in Eq. (22) yields

explicit function of B.

we define a magneto-stagnation function as following

S Bð Þ¼�<sup>X</sup>

mated by the function S Bð Þ. Because of S Bð Þ! <sup>∞</sup> as <sup>θ</sup>\_

n, l

ln θ\_

Table 3. Stagnation frequencies ωL=ω evaluated in the quantum states ð Þ n; l at T ¼ 1:4 K.

setup [4, 13], which gave <sup>ℏ</sup>ω<sup>c</sup> <sup>¼</sup> <sup>ℏ</sup>eB=<sup>m</sup> <sup>¼</sup> <sup>1</sup>:76B meVand <sup>ℏ</sup>ω<sup>0</sup> <sup>¼</sup> <sup>ℏ</sup> ffiffiffiffiffiffiffiffi

ω<sup>L</sup>

aveð Þ <sup>n</sup>; <sup>l</sup>; <sup>B</sup> � � �

where En,l is the energy level given by Eq. (17), and EF ¼ 8:5 meV is the Fermi energy of the AlGaAs/GaAs heterostructure. All the possibly occupied states and their associated stagnation frequencies are listed in Table 3.


Table 3. Stagnation frequencies ωL=ω evaluated in the quantum states ð Þ n; l at T ¼ 1:4 K.

coincide with the locations of the resistance peaks by comparing with the experimental results as

Figure 5. (a) The variation of wave number N with respect to the Larmor frequency ωL=ω in the quantum state

This section will compare the above theoretical predictions with the existing experimental data [4, 13] to confirm the fact that the effect of magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field. The experiment was performed in an AlGaAs/GaAs heterostructure with a carrier concentration ne <sup>¼</sup> <sup>2</sup>:<sup>5</sup> � 1011cm�2. Resistance was measured at temperature T ¼ 1:4K using a sensitive lock-in amplifier at currents of typically 1 nA and a frequency of 12 Hz. The resulting resistance measurement in the range of low magnetic field B ≤ 1:3 is depicted in Figure 5b showing a strong peak located around

Thus far, our analysis on quantum trajectory focuses on some specific states. In order to know the influence of the applied magnetic field on the resistance, we have to consider all the possible quantum states occupied in the device. At temperature T ¼ 1:4K, where the resistance is measured, the possible states to be occupied can be estimated by the Fermi-Dirac distribution,

where En,l is the energy level given by Eq. (17), and EF ¼ 8:5 meV is the Fermi energy of the AlGaAs/GaAs heterostructure. All the possibly occupied states and their associated stagnation

<sup>1</sup> <sup>þ</sup> <sup>e</sup>ð Þ En,l�EF <sup>=</sup>kBT , (45)

<sup>73</sup> � � <sup>p</sup> <sup>=</sup>12 and 2=3, coincide with the two peaks of the

f Eð Þ¼ <sup>1</sup>

B ¼ 0:22T and three weak peaks at B ¼ 0:65T, 0:97T, and 1:21T.

ð Þ¼ <sup>n</sup>; <sup>l</sup> ð Þ <sup>1</sup>; �<sup>1</sup> . (b) The two stagnation frequencies, <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi

shown in Figure 5b.

experimental curve of resistance.

36 Nonmagnetic and Magnetic Quantum Dots

4. Experimental verification

frequencies are listed in Table 3.

An incident electron subjected to an applied magnetic field B may enter any one of the occupied states listed in Table 3. The electronic resistance induced by B depends on the global transportation behavior across the quantum dot through all the allowable states. Magnetic stagnation slows down the electron's angular rate and retards the passage of the electron. The angular motion is fully retarded and the electron is trapped in the quantum dot without contribution to the conductance, as θ\_ ave ! 0. To quantify the effect of magnetic stagnation, we define a magneto-stagnation function as following

$$S(B) = -\sum\_{n\_\ell} \ln \left| \dot{\theta}\_{\text{ave}}(n, l, B) \right| = -\sum\_{n\_\ell} \ln \left| \frac{\omega\_L}{\omega} + \left( \frac{l}{\rho^2} \right)\_{\text{ave}} \right|\_{n, l} \tag{46}$$

where the summation is taken over all the states listed in Table 3. The expression of ωL=ω as a function of B has already been given by Eq. (22). Upon comparing the prediction of Eq. (46) with the experimental results, we evaluate the constants in ωL=ω according to the experimental setup [4, 13], which gave <sup>ℏ</sup>ω<sup>c</sup> <sup>¼</sup> <sup>ℏ</sup>eB=<sup>m</sup> <sup>¼</sup> <sup>1</sup>:76B meVand <sup>ℏ</sup>ω<sup>0</sup> <sup>¼</sup> <sup>ℏ</sup> ffiffiffiffiffiffiffiffi <sup>k</sup>=<sup>m</sup> <sup>p</sup> <sup>¼</sup> <sup>0</sup>:64meV. Using these data in Eq. (22) yields

$$\frac{\omega\_L}{\omega} = \frac{0.88B}{\sqrt{0.4096 + 0.7744B^2}}\tag{47}$$

which in turn is substituted into Eq. (46) to express the magneto-stagnation function S Bð Þ as an explicit function of B.

The electron's total angular velocity via all admissible quantum states at T ¼ 1:4K can be estimated by the function S Bð Þ. Because of S Bð Þ! <sup>∞</sup> as <sup>θ</sup>\_ ave ! 0, a large value of S Bð Þ implies that there is a high resistance to the electron's angular movability. Accordingly, S Bð Þ can be reasonably treated as an alternative description of electronic resistance. Figure 5a and b illustrates the first evidence of this correspondence. At the two stagnation frequencies <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi <sup>73</sup> � � <sup>p</sup> <sup>=</sup>12 and 2=3, corresponding to the two peaks of the resistance curve around B ¼ 0:22T and B ¼ 0:65T, S Bð Þ approaches to infinity, even though only the state ð Þ¼ n; l ð Þ 1; �1 is considered in Figure 5a.

at B ¼ 0:22 T from Eq. (47). The agreement between the experimental data of magnetoresistance and the magneto-stagnation function S Bð Þ constructed from the quantum Hamilton dynamics Eqs. (19) and (20) is not surprising, if we recall that Eqs. (19) and (20) is fully

Parallel to the existing probabilistic description for a quantum dot by a probability density function ψ<sup>∗</sup>ψ, this chapter considered an alternative trajectory description according to a dynamic representation of ψ constructed from quantum Hamilton mechanics. The equivalence between a given wavefunction ψð Þx and its dynamic representation x\_ ¼ f xð Þ ensures that the various quantum properties possessed by ψ also manifest in its dynamic representation. The established Hamilton dynamics for a quantum dot predicts that there are special magnetic fields, which can trap electrons within the quantum dot and cause a significant raise in the resistance. The comparison with experimental data validates this theoretical prediction. Apart from the magneto-transport considered in this chapter, many other features of a quantum dot, which were studied previously from a probabilistic perspective based on ψ, now can be reexamined

from a trajectory viewpoint based on the dynamic representation of ψ proposed here.

Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan,

[1] Ochiai Y, Widjaja AW, Sasaki N, Yamamoto K, Akis R, Ferry DK, Bird JP, Ishibashi K, Aoyagi Y, Sugano T. Backscattering of ballistic electrons in a corrugated-gate quantum

[2] Lin LH, Aoki N, Nakao K, Ishibashi K, Aoyagi Y, Sugano T, Holmberg N, Vasileska D, Akis R, Bird JP, Ferry DK, Ochiai Y. Magneto-transport in corrugated quantum wires.

[3] Aoki N, Brunner R, Burke AM, Akis R, Meisels R, Ferry DK, Ochiai Y. Direct imaging of electron states in open quantum dots. Physical Review Letters. 2012;108:136804

Physica E: Low-dimensional Systems and Nanostructures. 2000;7:750-755

ð Þ r; θ , which is responsible for the observed magneto-

http://dx.doi.org/10.5772/intechopen.74409

39

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

determined by the wavefunction ψn,l

resistance in quantum dots.

5. Conclusions

Author details

Taiwan

References

Ciann-Dong Yang\* and Shih-Ming Huang

\*Address all correspondence to: cdyang@mail.ncku.edu.tw

wire. Physical Review B. 1997;56:1073-1076

If magnetic stagnation takes place simultaneously in many states, its effect will be amplified. Stagnation frequencies such as ωL=ω ¼ 2=3, 4=5, and 6=7 appear concurrently in different quantum states, as can be seen from Table 3. Because the stagnation function considers the superposition of ln θ\_ aveð Þ <sup>n</sup>; <sup>l</sup>; <sup>B</sup> � � � � coming from all the allowable states, the value of S Bð Þ is intensified at such stagnation frequency concurring in different states. According to the conversion formula Eq. (47), the magnetic field relating to the stagnation frequencies ωL=ω ¼ 2=3, 4=5, and 6=7 is found to be B ¼ 0:65T, 0:97T, and 1:21T, respectively, which are just the locations of the three weak peaks of the resistance curve as shown in Figure 5b.

Figure 6 demonstrates the strong correspondence between the stagnation function S Bð Þ and the resistance curve, where the resistance curve is superposed on the gray-level plot of the stagnation function S Bð Þ with the intensity of darkness representing the magnitude of S Bð Þ. As can be seen, the gray-level distribution matches closely with the resistance curve and in that the dark bands of S Bð Þ correctly locate the peaks of the resistance. The gray-level plot of S Bð Þ has several narrow dark bands and one broad dark band. The narrow dark bands come from the isolated stagnation frequencies at ωL=ω ¼ 2=3, 4=5, and 6=7, and their locations coincide with the three weak peaks of the resistance curve. The broad dark band of S Bð Þ covers the neighborhood of the strong peak of the resistance curve, which is formed by a series of closely distributed stagnation frequencies centered at ωL=ω ¼ 0:29, or equivalently,

Figure 6. A gray-level plot of the stagnation function S Bð Þ with the darkness intensity representing the value of S Bð Þ is compared with the resistance curve. The resistance curve [4, 13] has a strong peak located around B ¼ 0:22 T and three weak peaks at B ¼ 0:65 T, 0:97 T, and 1:21 T. It appears that the locations of the three narrow dark bands coincide with the three weak peaks of the resistance curve, while the broad dark band covers the neighborhood of the strong peak of the resistance curve.

at B ¼ 0:22 T from Eq. (47). The agreement between the experimental data of magnetoresistance and the magneto-stagnation function S Bð Þ constructed from the quantum Hamilton dynamics Eqs. (19) and (20) is not surprising, if we recall that Eqs. (19) and (20) is fully determined by the wavefunction ψn,l ð Þ r; θ , which is responsible for the observed magnetoresistance in quantum dots.

### 5. Conclusions

evidence of this correspondence. At the two stagnation frequencies <sup>ω</sup>L=<sup>ω</sup> <sup>¼</sup> <sup>11</sup> � ffiffiffiffiffi

superposition of ln θ\_

38 Nonmagnetic and Magnetic Quantum Dots

resistance curve.

aveð Þ <sup>n</sup>; <sup>l</sup>; <sup>B</sup> � � �

2=3, corresponding to the two peaks of the resistance curve around B ¼ 0:22T and B ¼ 0:65T, S Bð Þ approaches to infinity, even though only the state ð Þ¼ n; l ð Þ 1; �1 is considered in Figure 5a. If magnetic stagnation takes place simultaneously in many states, its effect will be amplified. Stagnation frequencies such as ωL=ω ¼ 2=3, 4=5, and 6=7 appear concurrently in different quantum states, as can be seen from Table 3. Because the stagnation function considers the

intensified at such stagnation frequency concurring in different states. According to the conversion formula Eq. (47), the magnetic field relating to the stagnation frequencies ωL=ω ¼ 2=3, 4=5, and 6=7 is found to be B ¼ 0:65T, 0:97T, and 1:21T, respectively, which are just the

Figure 6 demonstrates the strong correspondence between the stagnation function S Bð Þ and the resistance curve, where the resistance curve is superposed on the gray-level plot of the stagnation function S Bð Þ with the intensity of darkness representing the magnitude of S Bð Þ. As can be seen, the gray-level distribution matches closely with the resistance curve and in that the dark bands of S Bð Þ correctly locate the peaks of the resistance. The gray-level plot of S Bð Þ has several narrow dark bands and one broad dark band. The narrow dark bands come from the isolated stagnation frequencies at ωL=ω ¼ 2=3, 4=5, and 6=7, and their locations coincide with the three weak peaks of the resistance curve. The broad dark band of S Bð Þ covers the neighborhood of the strong peak of the resistance curve, which is formed by a series of closely distributed stagnation frequencies centered at ωL=ω ¼ 0:29, or equivalently,

Figure 6. A gray-level plot of the stagnation function S Bð Þ with the darkness intensity representing the value of S Bð Þ is compared with the resistance curve. The resistance curve [4, 13] has a strong peak located around B ¼ 0:22 T and three weak peaks at B ¼ 0:65 T, 0:97 T, and 1:21 T. It appears that the locations of the three narrow dark bands coincide with the three weak peaks of the resistance curve, while the broad dark band covers the neighborhood of the strong peak of the

locations of the three weak peaks of the resistance curve as shown in Figure 5b.

� coming from all the allowable states, the value of S Bð Þ is

<sup>73</sup> � � <sup>p</sup> <sup>=</sup>12 and

Parallel to the existing probabilistic description for a quantum dot by a probability density function ψ<sup>∗</sup>ψ, this chapter considered an alternative trajectory description according to a dynamic representation of ψ constructed from quantum Hamilton mechanics. The equivalence between a given wavefunction ψð Þx and its dynamic representation x\_ ¼ f xð Þ ensures that the various quantum properties possessed by ψ also manifest in its dynamic representation. The established Hamilton dynamics for a quantum dot predicts that there are special magnetic fields, which can trap electrons within the quantum dot and cause a significant raise in the resistance. The comparison with experimental data validates this theoretical prediction. Apart from the magneto-transport considered in this chapter, many other features of a quantum dot, which were studied previously from a probabilistic perspective based on ψ, now can be reexamined from a trajectory viewpoint based on the dynamic representation of ψ proposed here.

## Author details

Ciann-Dong Yang\* and Shih-Ming Huang

\*Address all correspondence to: cdyang@mail.ncku.edu.tw

Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan

## References


[4] Brunner R, Meisels R, Kuchar F, ElHassan M, Bird JP, Ishibashi K. Investigations of backscattering peaks and of the nature of the confining potential in open quantum dots. Physica E: Low-dimensional Systems and Nanostructures. 2004;21:491-495

[20] Chou CC, Sanz AS, Miret-Artes S, Wyatt RE. Hydrodynamic view of wave-packet inter-

A Quantum Trajectory Interpretation of Magnetic Resistance in Quantum Dots

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41

[21] Wyatt RE, Rowland BA. Computational investigation of wave packet scattering in the complex plane: Propagation on a grid. Journal of Chemical Theory and Computation.

[22] Darwin CG. The diamagnetism of the free electron. Mathematical Proceedings of the

ference: Quantum caves. Physical Review Letters. 2009;102:250401

Cambridge Philosophical Society. 1931;27:86-90

2009;5:443-451


[20] Chou CC, Sanz AS, Miret-Artes S, Wyatt RE. Hydrodynamic view of wave-packet interference: Quantum caves. Physical Review Letters. 2009;102:250401

[4] Brunner R, Meisels R, Kuchar F, ElHassan M, Bird JP, Ishibashi K. Investigations of backscattering peaks and of the nature of the confining potential in open quantum dots.

[5] Brunner R, Meisels R, Kuchar F, Akis R, Ferry DK, Bird JP. Classical and quantum dynamics in an array of electron billiards. Physica E: Low-dimensional Systems and Nanostructures.

[6] Brunner R, Meisels R, Kuchar F, Akis A, Ferry DK, Bird JP. Magneto-transport in open quantum dot arrays at the transition from low to high magnetic field: Regularity and

[7] Morfonios CV, Schmelcher P. Control of Magnetotransport in Qunatum Billiards. Swit-

[8] Fransson J, Kang M, Yoon Y, Xiao S, Ochiai Y, Reno J, Aoki N, Bird JP. Tuning the Fano

[9] Poniedziałek MR, Szafran B. Multisubband transport and magnetic deflection of Fermi electron trajectories in three terminal junctions and rings. Journal of Physics. Condensed

[10] Yang CD. Quantum Hamilton mechanics Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom. Annals of Physics. 2006;321:

[11] Yang CD. Complex mechanics. In: Progress in Nonlinear Science. Vol. 1. Hong Kong:

[12] Yang CD. Modeling quantum harmonic oscillator in complex domain. Chaos, Solitons, &

[13] Elhassan M, Akis R, Bird JP, Ferry DK, Ida T, Ishibashi K. Magnetically induced Bragg scattering of electrons in quantum-dot crystals. Physical Review B. 2004;70:205341 [14] Ferry DK, Burke AM, Akis R, Brunner R, Day TE, Meisels R, Kuchar F, Bird JP, Bennett BR. Open quantum dots-probing the quantum to classical transition. Semiconductor Science

[15] Yang CD. Wave-particle duality in complex space. Ann. Physics. 2005;319:444-470

complex Bohmian mechanics. Physics Letters A. 2008;372:6240-6253

[16] Yang CD. Complex tunneling dynamics. Chaos,Solitons and Fractals. 2007;32:312-345

[17] Yang CD. Trajectory interpretation of the uncertainty principle in 1D systems using

[18] Bohm D. A suggested interpretation of the quantum theory in terms of hidden variables.

[19] Holland PR. The Quantum Theory of Motion. Cambridge: Cambridge University Press; 1993

Physica E: Low-dimensional Systems and Nanostructures. 2004;21:491-495

chaos. International Journal of Modern Physics B. 2007;21:1288-1296

resonance with an intruder continuum. Nano Letters. 2014;14:788-793

zerland: Springer International Publishing; 2017

2008;40:1315-1318

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Matter. 2012;24:085801

Asian Academic Publisher; 2010

and Technology. 2011;26:043001

Physical Review. 1952;85:166-193

Fractals. 2006;30:342-362

2876-2926


**Chapter 3**

**Provisional chapter**

**Droplet Epitaxy as a Tool for the QD-Based Circuit**

**Droplet Epitaxy as a Tool for the QD-Based Circuit** 

DOI: 10.5772/intechopen.70613

The chapter describes a novel technology, called droplet epitaxy, in the view point of quantum-circuit realization. This technology is useful when quantum dots are to be produced, of different shape and size in various densities. There are self-assembling methods to achieve spatial ordering or spatial positioning. Out of some of the possible applications

**Keywords:** droplet epitaxy, quantum dot, self-assembling, lateral alignment, vertical stacking

The most frequently quoted integration tendency in microelectronics is covered by the socalled Moor's law, which predicts the growth of the component concentration on microchips, doubling in every 2 years and forecasting further miniaturization. The CMOS technology itself is approaching its theoretical limits. Further limitations are also caused by quantum effects and certain anomalies in the technology in materials science. The spread in size, when CMOS technology is approaching the nano-region, represents further problems in microchip design. These difficulties make us wander about the next step in microchip technology, which would follow the present CMOS technology. The answer is hidden either in the promising state of spinotronics, an electronics based on graphene, or in circuits based on Josephson junction [1–4]. Quantum dots (QDs) or groups of QDs are also possible candidates of a new technology for the creation of electronic circuitry (**Figure 1A**). Nanotechnology based on GaAs and related compounds are also the most likely candidates for the development of new

as an example, the register and cellular automata circuit will be described.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Realization**

**Realization**

Ákos Nemcsics

**Abstract**

**1. Introduction**

technology.

Ákos Nemcsics

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70613

**Provisional chapter**

## **Droplet Epitaxy as a Tool for the QD-Based Circuit Realization Droplet Epitaxy as a Tool for the QD-Based Circuit Realization**

DOI: 10.5772/intechopen.70613

## Ákos Nemcsics Ákos Nemcsics

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70613

#### **Abstract**

The chapter describes a novel technology, called droplet epitaxy, in the view point of quantum-circuit realization. This technology is useful when quantum dots are to be produced, of different shape and size in various densities. There are self-assembling methods to achieve spatial ordering or spatial positioning. Out of some of the possible applications as an example, the register and cellular automata circuit will be described.

**Keywords:** droplet epitaxy, quantum dot, self-assembling, lateral alignment, vertical stacking

## **1. Introduction**

The most frequently quoted integration tendency in microelectronics is covered by the socalled Moor's law, which predicts the growth of the component concentration on microchips, doubling in every 2 years and forecasting further miniaturization. The CMOS technology itself is approaching its theoretical limits. Further limitations are also caused by quantum effects and certain anomalies in the technology in materials science. The spread in size, when CMOS technology is approaching the nano-region, represents further problems in microchip design. These difficulties make us wander about the next step in microchip technology, which would follow the present CMOS technology. The answer is hidden either in the promising state of spinotronics, an electronics based on graphene, or in circuits based on Josephson junction [1–4]. Quantum dots (QDs) or groups of QDs are also possible candidates of a new technology for the creation of electronic circuitry (**Figure 1A**). Nanotechnology based on GaAs and related compounds are also the most likely candidates for the development of new technology.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

shape and size in various densities. This technology is used already, for boosting efficiency, in device technology, such as lasers, LEDs, and solar cells. Their accurate positioning in complex structures like nano-sized circuits is very important. The lithographic direct processing used in microelectronics cannot be applied anymore; instead, the material's self-assembling prop-

Droplet Epitaxy as a Tool for the QD-Based Circuit Realization

http://dx.doi.org/10.5772/intechopen.70613

45

There are various self-assembling methods to achieve spatial arranging or positioning. The nucleation of the nano-structure, which can be induced by local stress field, is to be used for locating laterally or vertically some nano-objects. This method, called controlled self-assembling, has three different forms. The first forms self-contained objects like QDs, QD pairs, or QD clovers (four-coupled QDs) by manipulating the technological parameters. The second uses the natural features (steps of monolayers (MLs), or dislocations, etc.) to induce the required ordering by self-assembling effect. The third induces the required order by applying artificial influence on the process, like for instance focused ion beam (FIB) or creation of nanoholes (NHs). The combination of these methods can provide possibility to create complex nano-structures. This is called hierarchical self-organization, which provides potentional cre-

The chapter is organized in the following way. In the first parts, following "Introduction" section, we briefly describe the technique of DE. The following part describes the recent opportunities of the self-organizing-based creation of DE nanostructures. The last part describes two possibilities of applications: as an example, the QD register and circuit of quantum cellular automata will be discussed. The purpose of this paper is to speak to people engaged in circuit research with the aim of bringing together material scientists and circuit designers onto a common platform in order to overcome the problem of the present restrictions in further

For the fabrication of QDs and other zero-dimensional nano-structures, various techniques have been developed. The molecular beam epitaxy (MBE) is the most advanced technology in this area for nano-structure preparation. For a long time, the only known method for the production of epitaxially grown zero-dimensional structures was the strain-induced method, based on lattice mismatch in Stranski-Krastanov growth mode [8–12]. InAs-based QDs on GaAs surface are the archetypal system. The driving force of the self-organized QD formation is the strain energy induced by the lattice mismatch, which in approximately 7% of the case the conditions restrict the material choice. Two groups of shape formations, like pyramids

The DE is a viable alternative technology to the production of strain-driven QDs [13–20]. Here, the material choice is not restricted by the lattice mismatch condition, which is a further advantage to a process, based on the strain-induced growth mode. DE also makes possible the fabrication of strain-free QDs and other nano-structures. This shape diversity of the produced nano-structures makes it advantageous in applications. The technology used for the growth

erties is to be used, which is an inherent feature of every substance used.

**2. Fundamentals of droplet-epitaxial technology**

and domes, can be created with defect-free QD transformations.

ation of quantum circuits.

miniaturization.

**Figure 1.** (A) Quantum circuit is a possible solution beyond CMOS technology; (B) realization of quantum circuits such as register or quantum dot cellular automata (QCA). The quantum dot-based realization can bridge the bit- and qubitbased circuits.

The work of quantum circuits (like quantum computing) is based on quantum mechanical phenomena, so the realization must be in a nanometer scale. In this field, one of the promising candidates is the QD-based technology. The QD-based computing technology fundamentally differs from earlier systems. Conventional digital computing technology uses voltage values to represent binary states. By contrast, QD-based computing system uses the position of electrons in QDs to represent binary states. Here, we can distinguish two main types according to the interactions. One of them utilizes superposition and entanglement, and another one utilizes electrostatic interaction and tunneling. For the computation, the first one uses the so-called qbits.

The quantum computer uses the quantum states to encode and process information. The unit of quantum information is the qubit, which can be shown as a two-stage system such as a QD. Opposite to some classical object, a quantum system can exist not only in the ground state |0 > or the excited state |1>, but in some linear superposition of these two stages. The possibility of the handling of these stages provides the main advantage of quantum computing [5]. One type uses the charge of an electron to form a qubit. The qubit realization is possible by single or two electron QDs. In close neighbor, two semiconducting QDs can be coupled with each other. They spatially confine an individual charge carrier in a discrete energy level, interact quantum mechanically with each other. The ordered QD pair ensemble system offers the potential of implementing tunable qubit arrays. The utilization of the ordering of charge-coupled QDs enables to realize also quantum circuits with utilization of classical bit. One of them is called as computational register and the other one is memory register, respectively [6, 7]. One of the main tasks of the quantum computer is the encoding of the qubit. The QD-based circuits can bridge the qubit- and bit-based circuits.

In this chapter, a very novel technology, called droplet epitaxy (DE), will be discussed in the applicational view point. What kind of possibilities can be served by DE for the technology of quantum circuitry? This method is useful when QDs are to be produced, of different shape and size in various densities. This technology is used already, for boosting efficiency, in device technology, such as lasers, LEDs, and solar cells. Their accurate positioning in complex structures like nano-sized circuits is very important. The lithographic direct processing used in microelectronics cannot be applied anymore; instead, the material's self-assembling properties is to be used, which is an inherent feature of every substance used.

There are various self-assembling methods to achieve spatial arranging or positioning. The nucleation of the nano-structure, which can be induced by local stress field, is to be used for locating laterally or vertically some nano-objects. This method, called controlled self-assembling, has three different forms. The first forms self-contained objects like QDs, QD pairs, or QD clovers (four-coupled QDs) by manipulating the technological parameters. The second uses the natural features (steps of monolayers (MLs), or dislocations, etc.) to induce the required ordering by self-assembling effect. The third induces the required order by applying artificial influence on the process, like for instance focused ion beam (FIB) or creation of nanoholes (NHs). The combination of these methods can provide possibility to create complex nano-structures. This is called hierarchical self-organization, which provides potentional creation of quantum circuits.

The chapter is organized in the following way. In the first parts, following "Introduction" section, we briefly describe the technique of DE. The following part describes the recent opportunities of the self-organizing-based creation of DE nanostructures. The last part describes two possibilities of applications: as an example, the QD register and circuit of quantum cellular automata will be discussed. The purpose of this paper is to speak to people engaged in circuit research with the aim of bringing together material scientists and circuit designers onto a common platform in order to overcome the problem of the present restrictions in further miniaturization.

## **2. Fundamentals of droplet-epitaxial technology**

The work of quantum circuits (like quantum computing) is based on quantum mechanical phenomena, so the realization must be in a nanometer scale. In this field, one of the promising candidates is the QD-based technology. The QD-based computing technology fundamentally differs from earlier systems. Conventional digital computing technology uses voltage values to represent binary states. By contrast, QD-based computing system uses the position of electrons in QDs to represent binary states. Here, we can distinguish two main types according to the interactions. One of them utilizes superposition and entanglement, and another one utilizes electrostatic interaction and tunneling. For the computation, the first one uses the

**Figure 1.** (A) Quantum circuit is a possible solution beyond CMOS technology; (B) realization of quantum circuits such as register or quantum dot cellular automata (QCA). The quantum dot-based realization can bridge the bit- and qubit-

The quantum computer uses the quantum states to encode and process information. The unit of quantum information is the qubit, which can be shown as a two-stage system such as a QD. Opposite to some classical object, a quantum system can exist not only in the ground state |0 > or the excited state |1>, but in some linear superposition of these two stages. The possibility of the handling of these stages provides the main advantage of quantum computing [5]. One type uses the charge of an electron to form a qubit. The qubit realization is possible by single or two electron QDs. In close neighbor, two semiconducting QDs can be coupled with each other. They spatially confine an individual charge carrier in a discrete energy level, interact quantum mechanically with each other. The ordered QD pair ensemble system offers the potential of implementing tunable qubit arrays. The utilization of the ordering of charge-coupled QDs enables to realize also quantum circuits with utilization of classical bit. One of them is called as computational register and the other one is memory register, respectively [6, 7]. One of the main tasks of the quantum computer is the encoding of the qubit. The QD-based circuits can

In this chapter, a very novel technology, called droplet epitaxy (DE), will be discussed in the applicational view point. What kind of possibilities can be served by DE for the technology of quantum circuitry? This method is useful when QDs are to be produced, of different

so-called qbits.

based circuits.

44 Nonmagnetic and Magnetic Quantum Dots

bridge the qubit- and bit-based circuits.

For the fabrication of QDs and other zero-dimensional nano-structures, various techniques have been developed. The molecular beam epitaxy (MBE) is the most advanced technology in this area for nano-structure preparation. For a long time, the only known method for the production of epitaxially grown zero-dimensional structures was the strain-induced method, based on lattice mismatch in Stranski-Krastanov growth mode [8–12]. InAs-based QDs on GaAs surface are the archetypal system. The driving force of the self-organized QD formation is the strain energy induced by the lattice mismatch, which in approximately 7% of the case the conditions restrict the material choice. Two groups of shape formations, like pyramids and domes, can be created with defect-free QD transformations.

The DE is a viable alternative technology to the production of strain-driven QDs [13–20]. Here, the material choice is not restricted by the lattice mismatch condition, which is a further advantage to a process, based on the strain-induced growth mode. DE also makes possible the fabrication of strain-free QDs and other nano-structures. This shape diversity of the produced nano-structures makes it advantageous in applications. The technology used for the growth governs the size, shape, and the elementary distribution of the developed structures. These physical parameters are very important in applications.

DE formation of ring-like QDs is similar as the previous description earlier, but the technological parameters are somewhat different; however, the AlGaAs layer preparation process is the same [22]. After that, the sample is cooled to 300°C. On the surface, Ga is deposited as described before. The same Ga is deposited with the flux of 0.19 ML/s without any arsenic flux. During the annealing, the temperature remained the same (300°C), but the arsenic pressure changed to 4 × 10−6 Torr. During the nano-structure formation, diffusion of the constitu-

Droplet Epitaxy as a Tool for the QD-Based Circuit Realization

http://dx.doi.org/10.5772/intechopen.70613

47

A further recent method for the fabrication of strain-free QDs is the filling of nano-holes [23]. The nano-hole is created by localized thermal etching by liquid metallic droplet, and the created nano-hole is filled subsequently. A localized thermal etching takes place at conventional MBE growth temperatures, and we expect only very low level of crystal defects. The nano-holes are created in a self-organized fashion by local material removal. For inverted QD fabrication, nano-holes are generated by using Al droplets on AlAs surface. Following that, the holes are filled with GaAs to form QDs of controllable height. The nano-hole filling is carried out with GaAs in pulsed mode. The creation of QDs occurs with an inverted

QD pairs can be prepared on AlGaAs surface by using the anisotropy of the (001)-oriented surface [25]. There are two known preparational processes. One of them is carried out under lower temperature, with a fewer amounts of deposited MLs. The other one is prepared under higher temperature at a higher amount of deposited Ga. In the first case, AlGaAs with an Al content of 0.27% is grown on GaAs (001) surface. Following that, Ga droplets are created at 330°C temperature on the substrate. The crystallization occurs at 200°C, under strict control of the arsenic flux. The resulting structure basically consists of two QDs aligned in the [0\_

tallographic direction. In the other technology also, AlGaAs surface is being used. At 550°C substrate temperature, a large amount of Ga is deposited, to create droplets on the surface.

**Figure 3.** According to the technological parameters, the initial metallic droplet can lead to various zero-dimensional semiconductor nano-structures (where QR is quantum ring, NH is nano-hole, inv.QD is QD produced by nano-hole

filling: inverted QD technology) (the AFM and TEM pictures originate from Refs. [22, 24], respectively).

11] crys-

ents has an important role.

technology (**Figure 3**).

In DE applications, GaAs and related substances will be used as sample materials. That case, the clustering on the surface is carried out with the help of Volmer-Weber growth mode. This is a common idea, based on the splitting of the III- and V-column material supply, during the MBE growth (**Figure 2A**). The QD preparation consists of two main parts such as the formation of metallic nano-sized droplet on the surface and its crystallization. Here, the QD preparation consists of two main parts such as the formation of metallic nano-sized droplet on the surface and its crystallization with the help of the non-metallic component of the compound semiconductor [20]. In this way, not only conventional-shaped QDs but ring-like or double-ring-like zero-dimensional nano-structures can be created. Further possible nano-structures are the filled nano-hole and QD pairs or other ensembles, depending on growth parameters (**Figure 2B**). It must be noted that this DE technique is entirely compatible with the MBE technology. This attribute makes possible to combine the DE method with the other conventional MBE processes.

A typical QD preparation is illustrated in the following [21]: at first, on GaAs (001) wafer, an Al0.3Ga0.7As layer is grown. After the layer preparation, the sample is cooled to 200°C. Following this, Ga (*θ* = 3.75 ML) is deposited with the flux of 0.75 ML/s without any arsenic flux. After the Ga deposition, a 60-s waiting time comes. The annealing is carried out at a temperature of 350°C and at an As pressure of 5 × 10−5 Torr. The process of GaAs crystallization starts at the edge of the droplet, initialized by the three-phase line at this point, serving as discontinuity for the crystal seeding. Although, in principle, interaction can take place at any point of the droplet, due to the thermal movement, the atoms, arriving to the edge, will start the seeding of the crystallization process.

**Figure 2.** (A) The droplet epitaxial nano-structure production consists of two basic growth sequences; (B) versatile shaped nano-object can be created depending on the technological parameters (where QR is quantum ring, DQR is double quantum ring, and NH is nano-hole).

DE formation of ring-like QDs is similar as the previous description earlier, but the technological parameters are somewhat different; however, the AlGaAs layer preparation process is the same [22]. After that, the sample is cooled to 300°C. On the surface, Ga is deposited as described before. The same Ga is deposited with the flux of 0.19 ML/s without any arsenic flux. During the annealing, the temperature remained the same (300°C), but the arsenic pressure changed to 4 × 10−6 Torr. During the nano-structure formation, diffusion of the constituents has an important role.

governs the size, shape, and the elementary distribution of the developed structures. These

In DE applications, GaAs and related substances will be used as sample materials. That case, the clustering on the surface is carried out with the help of Volmer-Weber growth mode. This is a common idea, based on the splitting of the III- and V-column material supply, during the MBE growth (**Figure 2A**). The QD preparation consists of two main parts such as the formation of metallic nano-sized droplet on the surface and its crystallization. Here, the QD preparation consists of two main parts such as the formation of metallic nano-sized droplet on the surface and its crystallization with the help of the non-metallic component of the compound semiconductor [20]. In this way, not only conventional-shaped QDs but ring-like or double-ring-like zero-dimensional nano-structures can be created. Further possible nano-structures are the filled nano-hole and QD pairs or other ensembles, depending on growth parameters (**Figure 2B**). It must be noted that this DE technique is entirely compatible with the MBE technology. This attribute makes possible to combine the DE method with the other conventional MBE processes.

A typical QD preparation is illustrated in the following [21]: at first, on GaAs (001) wafer, an Al0.3Ga0.7As layer is grown. After the layer preparation, the sample is cooled to 200°C. Following this, Ga (*θ* = 3.75 ML) is deposited with the flux of 0.75 ML/s without any arsenic flux. After the Ga deposition, a 60-s waiting time comes. The annealing is carried out at a temperature of 350°C and at an As pressure of 5 × 10−5 Torr. The process of GaAs crystallization starts at the edge of the droplet, initialized by the three-phase line at this point, serving as discontinuity for the crystal seeding. Although, in principle, interaction can take place at any point of the droplet, due to the thermal movement, the atoms, arriving to the edge, will start the seeding

**Figure 2.** (A) The droplet epitaxial nano-structure production consists of two basic growth sequences; (B) versatile shaped nano-object can be created depending on the technological parameters (where QR is quantum ring, DQR is

physical parameters are very important in applications.

46 Nonmagnetic and Magnetic Quantum Dots

of the crystallization process.

double quantum ring, and NH is nano-hole).

A further recent method for the fabrication of strain-free QDs is the filling of nano-holes [23]. The nano-hole is created by localized thermal etching by liquid metallic droplet, and the created nano-hole is filled subsequently. A localized thermal etching takes place at conventional MBE growth temperatures, and we expect only very low level of crystal defects. The nano-holes are created in a self-organized fashion by local material removal. For inverted QD fabrication, nano-holes are generated by using Al droplets on AlAs surface. Following that, the holes are filled with GaAs to form QDs of controllable height. The nano-hole filling is carried out with GaAs in pulsed mode. The creation of QDs occurs with an inverted technology (**Figure 3**).

QD pairs can be prepared on AlGaAs surface by using the anisotropy of the (001)-oriented surface [25]. There are two known preparational processes. One of them is carried out under lower temperature, with a fewer amounts of deposited MLs. The other one is prepared under higher temperature at a higher amount of deposited Ga. In the first case, AlGaAs with an Al content of 0.27% is grown on GaAs (001) surface. Following that, Ga droplets are created at 330°C temperature on the substrate. The crystallization occurs at 200°C, under strict control of the arsenic flux. The resulting structure basically consists of two QDs aligned in the [0\_ 11] crystallographic direction. In the other technology also, AlGaAs surface is being used. At 550°C substrate temperature, a large amount of Ga is deposited, to create droplets on the surface.

**Figure 3.** According to the technological parameters, the initial metallic droplet can lead to various zero-dimensional semiconductor nano-structures (where QR is quantum ring, NH is nano-hole, inv.QD is QD produced by nano-hole filling: inverted QD technology) (the AFM and TEM pictures originate from Refs. [22, 24], respectively).

can be carried out by using the naturally occurring anomalies on the crystalline surface or can be made artificially by external influences. There are three kinds of linear alignmentation methods (**Figure 5**). One of them is the surface cross-hatch-induced mode, and the other kind is the alignmentation created by ML step. These are utilization of naturally formed surface effects. The third one is a fully artificial method, where the alignmentation is induced by ion

The dislocations, generated at the substrate/layer junction, show themselves on the surface as ridges and troughs. At a sufficiently high density of dislocations, the development of misfit dislocation network shows up at this junction. Such a network, consisting of two arrays of single dislocations with alternating glide planes, will result in a quadratic surface structure. The dislocation network shows itself at the surface, which is called as a cross-hatch pattern. This pattern coexists with the crystallites, giving the possibility of using the interplay

The cross-hatch pattern creation has been already demonstrated in different material sys-

misfit dislocations and glides. Its production is as follows. Self-assembled InAs QDs are grown on cross-hatched surface, consisting of 50 nm In0.15Ga0.85As layers on GaAs (0 0 1) substrate. The lattice-mismatched In0.15Ga0.85As layer is left growing well beyond the critical layer thickness for the formation of misfit dislocation in order to form long orthogonal

On top of the cross-hatched surface, InAs layer growth at a low growth rate of 0.01 ML/s and at a thickness of 0.8 ML originates spontaneous QD formation. It was found that the

**Figure 5.** Linearly aligned QDs; (A) the QD alignment is induced by cross-hatch (B) and by monolayer (ML) steps, (C)

and by ion-induced surface damage (the AFM and SEM pictures originate from Refs. [31, 39], respectively).

As/InP [29], and Si1−*<sup>x</sup>*

Gex

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11] crystalline directions (**Figure 5A**).

/Si [30], attributed to

Ga1−*<sup>x</sup>*

between the two strain-relief mechanisms for self-ordering of QDs.

As/GaAs [28], Inx

cross-hatch pattern oriented along the [1 <sup>1</sup> 0] and [0\_

beam-created surface damage.

Ga1−*<sup>x</sup>*

tems such as Inx

**Figure 4.** (A) and (B) AFM picture of QD pairs; (C) and (D) QD clovers (the AFM pictures originate from Refs. [24–26], respectively).

The structure is crystallized by an accurate control of the flux. The resulting dots are rather large. The individual pairs have an interdot distance of about 130 nm and are aligned along the [0\_ 11] direction. QD pairs are shown in **Figure 4A** and **B**.

Nano-objects consisting of four parts can also be grown by DE. The structure is a split-ring formation. A typical technological process, when the samples are grown on GaAs (001) substrates, is as follows. First, In0.15Ga0.85 of 20 ML is deposited with a rate of 1 ML/s at 360°C. Then, the formed droplets are exposed to arsenic beam for 5 min at a temperature of 200°C to crystallize the nano-droplets. Following this, the substrate temperature is raised to 450°C for the regrowth process with a growth rate of 0.05 ML/s. The structures are shown in **Figure 4C** and **D** [26, 27].

## **3. Ordered nano-structures**

The self-assembling ordered QDs can be linearly, circularly, and also vertically alignmented. The most promising method for achieving long-range laterally ordered self-assembled QDs is the combination of substrate pre-patterning and self-assembled growth. The pre-patterning can be carried out by using the naturally occurring anomalies on the crystalline surface or can be made artificially by external influences. There are three kinds of linear alignmentation methods (**Figure 5**). One of them is the surface cross-hatch-induced mode, and the other kind is the alignmentation created by ML step. These are utilization of naturally formed surface effects. The third one is a fully artificial method, where the alignmentation is induced by ion beam-created surface damage.

The dislocations, generated at the substrate/layer junction, show themselves on the surface as ridges and troughs. At a sufficiently high density of dislocations, the development of misfit dislocation network shows up at this junction. Such a network, consisting of two arrays of single dislocations with alternating glide planes, will result in a quadratic surface structure. The dislocation network shows itself at the surface, which is called as a cross-hatch pattern. This pattern coexists with the crystallites, giving the possibility of using the interplay between the two strain-relief mechanisms for self-ordering of QDs.

The cross-hatch pattern creation has been already demonstrated in different material systems such as Inx Ga1−*<sup>x</sup>* As/GaAs [28], Inx Ga1−*<sup>x</sup>* As/InP [29], and Si1−*<sup>x</sup>* Gex /Si [30], attributed to misfit dislocations and glides. Its production is as follows. Self-assembled InAs QDs are grown on cross-hatched surface, consisting of 50 nm In0.15Ga0.85As layers on GaAs (0 0 1) substrate. The lattice-mismatched In0.15Ga0.85As layer is left growing well beyond the critical layer thickness for the formation of misfit dislocation in order to form long orthogonal cross-hatch pattern oriented along the [1 <sup>1</sup> 0] and [0\_ 11] crystalline directions (**Figure 5A**). On top of the cross-hatched surface, InAs layer growth at a low growth rate of 0.01 ML/s and at a thickness of 0.8 ML originates spontaneous QD formation. It was found that the

The structure is crystallized by an accurate control of the flux. The resulting dots are rather large. The individual pairs have an interdot distance of about 130 nm and are aligned along

**Figure 4.** (A) and (B) AFM picture of QD pairs; (C) and (D) QD clovers (the AFM pictures originate from Refs. [24–26],

Nano-objects consisting of four parts can also be grown by DE. The structure is a split-ring formation. A typical technological process, when the samples are grown on GaAs (001) substrates, is as follows. First, In0.15Ga0.85 of 20 ML is deposited with a rate of 1 ML/s at 360°C. Then, the formed droplets are exposed to arsenic beam for 5 min at a temperature of 200°C to crystallize the nano-droplets. Following this, the substrate temperature is raised to 450°C for the regrowth process with a growth rate of 0.05 ML/s. The structures are shown in

The self-assembling ordered QDs can be linearly, circularly, and also vertically alignmented. The most promising method for achieving long-range laterally ordered self-assembled QDs is the combination of substrate pre-patterning and self-assembled growth. The pre-patterning

11] direction. QD pairs are shown in **Figure 4A** and **B**.

the [0\_

respectively).

**Figure 4C** and **D** [26, 27].

48 Nonmagnetic and Magnetic Quantum Dots

**3. Ordered nano-structures**

**Figure 5.** Linearly aligned QDs; (A) the QD alignment is induced by cross-hatch (B) and by monolayer (ML) steps, (C) and by ion-induced surface damage (the AFM and SEM pictures originate from Refs. [31, 39], respectively).

substrate temperature reduction immediately after the QD formation will result in a majority of QDs alignment on the cross-hatch pattern. With a short growth interruption, duration of 30 s, before reducing the substrate temperature, the QDs will form, almost exclusively on the cross-hatches and the surface formation is named QD hatches. Exceeding the optimum interruption time will result in inhomogeneous, sparsely connected QD hatches, possibly due to desorption of In atoms [31].

The second possibility to self-aligned QD ordering in a crystalline layer uses step bunching of preexisting ML steps on the miscut (0 0 1) substrate (**Figure 5B**). Crystallite ordering on vicinal-oriented (0 0 1) surface is guided by spontaneously formed step-bunched ripple patterns. The ripple distance and orientation can be engineered by varying the polar and azimuthal miscut directions of the substrate [32].

The focused ion beam bombardment is a widely used technique for surface preparation and nano-patterning for the fabrication of self-assembling nanostructures such as nano-ripples, nano-needles, nano-holes, and also QDs. FIB-induced self-assembly of ordered nano-structures has been reported on metals, semiconductors, and insulators as well [32–38].

Ordered Ga nano-droplets can be self-assembled under ion beam bombardment at off-normal incidence [39]. The homogeneity, size, and density of Ga nano-droplets can be controlled by the incident ion beam angle. The beam current also plays a crucial role in the self-ordering of Ga nano-droplets. It has been found that the droplets exhibit a similar droplet size but higher density and better homogeneity with an increased current of ion beam. Compared to the destructive formation of nano-droplets by direct ion beam bombardment, the controllable assembly of nano-droplets on intact surfaces can be used as templates for DE fabrication of arranged semiconductor nano-structures (**Figure 5C**).

**Figure 6.** Circularly aligned QDs (QD molecule); (A) the QD nucleation is at droplet edge (B) and at the rim of hole opening. The arrows indicate the seeding places (the AFM pictures originate from Refs. [46, 47], respectively).

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**Figure 7.** The vertical alignment; (A) the vertical stacking is induced by strain, (B) vertically coupled QDs by nano-hole

(NH) filling (the TEM pictures originate from Refs. [24, 48], respectively).

The start of circularly aligned QD molecule can be initialized by a droplet edge (**Figure 6A**) or by a rim of nano-holes (**Figure 6B**). It is a simple method of preparing ring-shape InP nano-structures on In0.49Ga0.51P by using DE. The surface morphology of the structure depends strongly on the ML of In. For instance, the ring-shaped nano-structure is formed at 1.6-ML In thickness. The ring-shaped QD molecule is formed when the deposited ML of In is less than 3.2 ML. It has been found that the density, height, and average number of QD per molecule are dependent on the In MLs and on its deposition rate [40].

A relatively simple way to fabricate vertical QD molecules is to grow stacks of QDs (**Figure 7A**). It is known that the surface strain field modulation from a buried island layer influences the island nucleation in the next layer and this leads to a spontaneous vertical alignment [41, 42]. The electronic coupling between vertically aligned QDs has been demonstrated earlier [40, 43–48]. Further vertically stacked QD ensembles can be created by sequentially filled nano-hole. First, Al or Ga droplets are created on the AlGaAs surface. After them an annealing appears, where the substrate temperature is ranged between 550 and 650°C, the arsenic pressure is under 10−7 Torr (**Figure 7B**). During this annealing, the initial droplet transforms into a nano-hole surrounded by a protrusion. The nano-hole is filled by pulsed mode. The filling consists of 0.5-s GaAs deposition followed by a 30-s pause. The stacked QDs are separated by an AlGaAs barrier layer deposition [49].

substrate temperature reduction immediately after the QD formation will result in a majority of QDs alignment on the cross-hatch pattern. With a short growth interruption, duration of 30 s, before reducing the substrate temperature, the QDs will form, almost exclusively on the cross-hatches and the surface formation is named QD hatches. Exceeding the optimum interruption time will result in inhomogeneous, sparsely connected QD hatches, possibly

The second possibility to self-aligned QD ordering in a crystalline layer uses step bunching of preexisting ML steps on the miscut (0 0 1) substrate (**Figure 5B**). Crystallite ordering on vicinal-oriented (0 0 1) surface is guided by spontaneously formed step-bunched ripple patterns. The ripple distance and orientation can be engineered by varying the polar and azimuthal

The focused ion beam bombardment is a widely used technique for surface preparation and nano-patterning for the fabrication of self-assembling nanostructures such as nano-ripples, nano-needles, nano-holes, and also QDs. FIB-induced self-assembly of ordered nano-structures

Ordered Ga nano-droplets can be self-assembled under ion beam bombardment at off-normal incidence [39]. The homogeneity, size, and density of Ga nano-droplets can be controlled by the incident ion beam angle. The beam current also plays a crucial role in the self-ordering of Ga nano-droplets. It has been found that the droplets exhibit a similar droplet size but higher density and better homogeneity with an increased current of ion beam. Compared to the destructive formation of nano-droplets by direct ion beam bombardment, the controllable assembly of nano-droplets on intact surfaces can be used as templates for DE fabrication of

The start of circularly aligned QD molecule can be initialized by a droplet edge (**Figure 6A**) or by a rim of nano-holes (**Figure 6B**). It is a simple method of preparing ring-shape InP nano-structures on In0.49Ga0.51P by using DE. The surface morphology of the structure depends strongly on the ML of In. For instance, the ring-shaped nano-structure is formed at 1.6-ML In thickness. The ring-shaped QD molecule is formed when the deposited ML of In is less than 3.2 ML. It has been found that the density, height, and average number of QD per molecule

A relatively simple way to fabricate vertical QD molecules is to grow stacks of QDs (**Figure 7A**). It is known that the surface strain field modulation from a buried island layer influences the island nucleation in the next layer and this leads to a spontaneous vertical alignment [41, 42]. The electronic coupling between vertically aligned QDs has been demonstrated earlier [40, 43–48]. Further vertically stacked QD ensembles can be created by sequentially filled nano-hole. First, Al or Ga droplets are created on the AlGaAs surface. After them an annealing appears, where the substrate temperature is ranged between 550 and 650°C, the arsenic pressure is under 10−7 Torr (**Figure 7B**). During this annealing, the initial droplet transforms into a nano-hole surrounded by a protrusion. The nano-hole is filled by pulsed mode. The filling consists of 0.5-s GaAs deposition followed by a 30-s pause. The stacked

has been reported on metals, semiconductors, and insulators as well [32–38].

due to desorption of In atoms [31].

50 Nonmagnetic and Magnetic Quantum Dots

miscut directions of the substrate [32].

arranged semiconductor nano-structures (**Figure 5C**).

are dependent on the In MLs and on its deposition rate [40].

QDs are separated by an AlGaAs barrier layer deposition [49].

**Figure 6.** Circularly aligned QDs (QD molecule); (A) the QD nucleation is at droplet edge (B) and at the rim of hole opening. The arrows indicate the seeding places (the AFM pictures originate from Refs. [46, 47], respectively).

**Figure 7.** The vertical alignment; (A) the vertical stacking is induced by strain, (B) vertically coupled QDs by nano-hole (NH) filling (the TEM pictures originate from Refs. [24, 48], respectively).

## **4. Applications in quantum circuitry**

In this chapter, we discuss two types of circuits composed from aligned QDs. One of them is the linearly aligned register. A QD register for quantum computing can be realized by uniformly aligned QDs or by QD pairs with the help of directed DE assembly [50]. A possible realization can be the following. The linearly aligned GaAs QDs is created on an AlGaAs surface. This structure is embedded by a barrier material of AlGaAs. When the cover layer few MLs only then the subsequently deposited metallic droplets are positioned most likely by the QD sites below (**Figure 8**).

electrons are injected into the cell (**Figure 9B**). Due to Coulombic repulsion, the two electrons reside in opposite corners representing two polarizations. Some basic elements for QD cellu-

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The DE-grown QDs as building elements for quantum computing were first proposed in 2009 [61, 62]. The alignmentation of the QD clovers can be realized like a single dot, which can lead to the wire implementation. Here, the linear inhomogeneity of the surface can be utilized. For the gate realization, these inhomogeneities for directed assembly must be generated artificially. The QD cellular automata can be realized in more levels (**Figure 9C** and **D**). The couplings between the circuits on the adjacent levels can be carried out with vertical

For any required operation, it is very important to determine the optimal size of the QDs and their distances from each other. Not only the size but the working temperature is also important. At the realization, it is important to take into consideration that the switching fidelity increases with decreasing temperature [63]. It is predicted that the density of the QD-based circuits could exceed the device density of 1012 cm−2 and the operating speed could reach the frequency of THz region [64]. The clocking in THz region can also be realized with the help of DE. One of the effective ways to generate THz pulses is realized by near-infrared femtosecond laser irradiation on semiconductor/metal surfaces with the help of plasmon enhancement [65–67]. The DE is an appropriate technology to create such positioned semiconductor and metallic nano-particles. The structure can be realized by DE-grown self-alignment of QD molecules and metallic nano-particle [25, 68]. Promising perspective is provided with a recent result to the realization of the nano-positioned metallic nano-particle on QD molecule, which

**Figure 9.** Realization of QD clover-based QD cellular automata; (A) the basic unit and the majority gate and inverter gate

of the QD cellular automata; (B) the realization of QD cellular automata by QD.

lar automata logic implementation are wire, inverter, and majority voter [52–60].

can be useful not only at the THz clocking but also at QD register, too [69].

alignment.

The second discussed structure is the QD cellular automata, which was firstly proposed in the beginning of 1990s [51]. The QD-based cellular automaton is one of the most promising device structures in the future [52–55]. The circuit consists of coupled QD array to realize Boolean logic functions [9] and to perform useful computations. Two main advantages of QD cellular automata are the exceptionally high logic integration derived from the small QD size, and the low power consumption. QD cellular automata can be used to implement complex digital circuits by properly arranged QD clovers. Such circuits are, for example, full adder, multiplexer, programmable logic array, multivibrator or can be also designed memory circuits, such as quantum dot cellular automatic random access memory and serial memory. The basic building block of QD cellular automata device named cell is presented in **Figure 9A**. QD cellular automata unit cell consists of four QDs in a square array coupled by tunnel barriers, and two

**Figure 8.** Realization of QD register: (A) cross section of literally aligned QD series with vertically positioned selfassembling metallic clusters; (B) along cross-hatch-aligned QDs; (C) the alignmentation of QD pairs is also possible.

electrons are injected into the cell (**Figure 9B**). Due to Coulombic repulsion, the two electrons reside in opposite corners representing two polarizations. Some basic elements for QD cellular automata logic implementation are wire, inverter, and majority voter [52–60].

**4. Applications in quantum circuitry**

QD sites below (**Figure 8**).

52 Nonmagnetic and Magnetic Quantum Dots

In this chapter, we discuss two types of circuits composed from aligned QDs. One of them is the linearly aligned register. A QD register for quantum computing can be realized by uniformly aligned QDs or by QD pairs with the help of directed DE assembly [50]. A possible realization can be the following. The linearly aligned GaAs QDs is created on an AlGaAs surface. This structure is embedded by a barrier material of AlGaAs. When the cover layer few MLs only then the subsequently deposited metallic droplets are positioned most likely by the

The second discussed structure is the QD cellular automata, which was firstly proposed in the beginning of 1990s [51]. The QD-based cellular automaton is one of the most promising device structures in the future [52–55]. The circuit consists of coupled QD array to realize Boolean logic functions [9] and to perform useful computations. Two main advantages of QD cellular automata are the exceptionally high logic integration derived from the small QD size, and the low power consumption. QD cellular automata can be used to implement complex digital circuits by properly arranged QD clovers. Such circuits are, for example, full adder, multiplexer, programmable logic array, multivibrator or can be also designed memory circuits, such as quantum dot cellular automatic random access memory and serial memory. The basic building block of QD cellular automata device named cell is presented in **Figure 9A**. QD cellular automata unit cell consists of four QDs in a square array coupled by tunnel barriers, and two

**Figure 8.** Realization of QD register: (A) cross section of literally aligned QD series with vertically positioned selfassembling metallic clusters; (B) along cross-hatch-aligned QDs; (C) the alignmentation of QD pairs is also possible.

The DE-grown QDs as building elements for quantum computing were first proposed in 2009 [61, 62]. The alignmentation of the QD clovers can be realized like a single dot, which can lead to the wire implementation. Here, the linear inhomogeneity of the surface can be utilized. For the gate realization, these inhomogeneities for directed assembly must be generated artificially. The QD cellular automata can be realized in more levels (**Figure 9C** and **D**). The couplings between the circuits on the adjacent levels can be carried out with vertical alignment.

For any required operation, it is very important to determine the optimal size of the QDs and their distances from each other. Not only the size but the working temperature is also important. At the realization, it is important to take into consideration that the switching fidelity increases with decreasing temperature [63]. It is predicted that the density of the QD-based circuits could exceed the device density of 1012 cm−2 and the operating speed could reach the frequency of THz region [64]. The clocking in THz region can also be realized with the help of DE. One of the effective ways to generate THz pulses is realized by near-infrared femtosecond laser irradiation on semiconductor/metal surfaces with the help of plasmon enhancement [65–67]. The DE is an appropriate technology to create such positioned semiconductor and metallic nano-particles. The structure can be realized by DE-grown self-alignment of QD molecules and metallic nano-particle [25, 68]. Promising perspective is provided with a recent result to the realization of the nano-positioned metallic nano-particle on QD molecule, which can be useful not only at the THz clocking but also at QD register, too [69].

**Figure 9.** Realization of QD clover-based QD cellular automata; (A) the basic unit and the majority gate and inverter gate of the QD cellular automata; (B) the realization of QD cellular automata by QD.

## **5. Conclusion**

There are still a number of open scientific problems awaiting a solution. For the perfect operation of the circuits, the optimal QDs and their distances from each other must be determined. It is a fact that the sizes and the shape of the QD are not independent from their elementary density, which finally determines the distances among the QDs. The task is rather complex. If we can understand the details of the evolution mechanism of the DE-grown nano-structures, we can approach the technological solution of the circuit formation. It is another possibility to take into account the technological capability at the circuit design, which increases the importance of the mutually common thinking among different professionals. Lately, the number of published papers in this area has increased drastically, which is an encouraging sign for the possible technological solution.

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## **Acknowledgements**

This work was supported party by NKFI-OTKA-114457(FemtoTera) and partly by OE(KVK and ADTI) research grants, which are acknowledged.

## **Author details**

Ákos Nemcsics

Address all correspondence to: nemcsics.akos@kvk.uni-obuda.hu

Institute for Microelectronics and Technology, Obuda University, Budapest, Hungary

## **References**


[5] Deutsch D. Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society of London A. 1985;**97**:400

**5. Conclusion**

54 Nonmagnetic and Magnetic Quantum Dots

possible technological solution.

and ADTI) research grants, which are acknowledged.

Address all correspondence to: nemcsics.akos@kvk.uni-obuda.hu

and Design; 28-31 August 2001; Espoo, Finland; 2001. p.I-9

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**Acknowledgements**

**Author details**

Ákos Nemcsics

**References**

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There are still a number of open scientific problems awaiting a solution. For the perfect operation of the circuits, the optimal QDs and their distances from each other must be determined. It is a fact that the sizes and the shape of the QD are not independent from their elementary density, which finally determines the distances among the QDs. The task is rather complex. If we can understand the details of the evolution mechanism of the DE-grown nano-structures, we can approach the technological solution of the circuit formation. It is another possibility to take into account the technological capability at the circuit design, which increases the importance of the mutually common thinking among different professionals. Lately, the number of published papers in this area has increased drastically, which is an encouraging sign for the

This work was supported party by NKFI-OTKA-114457(FemtoTera) and partly by OE(KVK

Institute for Microelectronics and Technology, Obuda University, Budapest, Hungary

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**Chapter 4**

Provisional chapter

**Colloidal III–V Nitride Quantum Dots**

Colloidal III–V Nitride Quantum Dots

Zequn Chen, Chuli Sun, Wei Guo and Zhuo Chen

Colloidal quantum dots (QDs) have attracted intense attention in both fundamental studies and practical applications. To date, the size, morphology, and composition-controlled syntheses have been successfully achieved in II–VI semiconductor nanocrystals. Recently, III-nitride semiconductor quantum dots have begun to draw significant interest due to their promising applications in solid-state lighting, lasing technologies, and optoelectronic devices. The quality of nitride nanocrystals is, however, dramatically lower than that of II– VI semiconductor nanocrystals. In this review, the recent development in the synthesis techniques and properties of colloidal III–V nitride quantum dots as well as their applica-

DOI: 10.5772/intechopen.70844

Due to the uniquely tunable electronic structure and low-cost synthesis in a controllable way, colloidal quantum dots (QDs) have attracted intense attention in both fundamental studies and practical applications [1–5], such as solar cell, quantum dot light-emitting diode, and spectrometer. Usually, semiconductor quantum dot properties can be varied by their size, composition, morphology, and phase structure. To date, with the rapid development of synthesis technique, the size, morphology, and composition-controlled syntheses of colloidal

III–V semiconductors are crystalline binary compounds formed by combining metallic elements from group III and nonmetallic elements from group V of the periodic table [10]. In the III–V nitride, the wurtzite phase is the stable form and they have direct bandgaps ranging from 0.7 eV for InN, to 3.4 eV for GaN, and to 6.2 eV for AlN. They can combine with each other to form alloys with bandgaps value from 0.7 to 6.2 eV, covering a wide range of spectra from

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Keywords: colloidal synthesis, III–V nitride, quantum dots, semiconductor,

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70844

Zhuo Chen

Abstract

tions are introduced.

1. Introduction

optoelectronic properties

quantum dots have been successfully achieved [6–9].

Zequn Chen, Chuli Sun, Wei Guo and

#### **Colloidal III–V Nitride Quantum Dots** Colloidal III–V Nitride Quantum Dots

Zequn Chen, Chuli Sun, Wei Guo and Zhuo Chen Zequn Chen, Chuli Sun, Wei Guo and

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70844

#### Abstract

Zhuo Chen

Colloidal quantum dots (QDs) have attracted intense attention in both fundamental studies and practical applications. To date, the size, morphology, and composition-controlled syntheses have been successfully achieved in II–VI semiconductor nanocrystals. Recently, III-nitride semiconductor quantum dots have begun to draw significant interest due to their promising applications in solid-state lighting, lasing technologies, and optoelectronic devices. The quality of nitride nanocrystals is, however, dramatically lower than that of II– VI semiconductor nanocrystals. In this review, the recent development in the synthesis techniques and properties of colloidal III–V nitride quantum dots as well as their applications are introduced.

DOI: 10.5772/intechopen.70844

Keywords: colloidal synthesis, III–V nitride, quantum dots, semiconductor, optoelectronic properties

## 1. Introduction

Due to the uniquely tunable electronic structure and low-cost synthesis in a controllable way, colloidal quantum dots (QDs) have attracted intense attention in both fundamental studies and practical applications [1–5], such as solar cell, quantum dot light-emitting diode, and spectrometer. Usually, semiconductor quantum dot properties can be varied by their size, composition, morphology, and phase structure. To date, with the rapid development of synthesis technique, the size, morphology, and composition-controlled syntheses of colloidal quantum dots have been successfully achieved [6–9].

III–V semiconductors are crystalline binary compounds formed by combining metallic elements from group III and nonmetallic elements from group V of the periodic table [10]. In the III–V nitride, the wurtzite phase is the stable form and they have direct bandgaps ranging from 0.7 eV for InN, to 3.4 eV for GaN, and to 6.2 eV for AlN. They can combine with each other to form alloys with bandgaps value from 0.7 to 6.2 eV, covering a wide range of spectra from

distribution, and eproduction in any medium, provided the original work is properly cited.

ultraviolet (UV) to infrared region, exhibiting large potential applications for electronic and optoelectronic devices.

increasing attention. It has a good property on electron transport, making it face the huge application on high-speed electronic device due to its small-effective mass [18]. InN is also considered as an excellent material for low-cost, high-efficiency solar cell, photomask, lightemitting diodes, laser diodes, sensors, and THz radiation [19–22]. In addition, InN is considered as a promising candidate for biological imaging and in vivo medical applications because of its nontoxicity and its infrared emission in the optically transparent region of water and

Colloidal III–V Nitride Quantum Dots http://dx.doi.org/10.5772/intechopen.70844 63

With the increasing importance on InN, how to obtain the InN with high quality is also attracting much attention. Although several synthesis methods, such as solvothermal methods [24, 25], sputtering [26], and molecular beam epitaxy (MBE) [27], were developed to prepare 0D InN QDs, more effort needs to be devoted to improve the quality of InN QDs with a

Group III nitrides have a high light-emitting efficiency due to its direct bandgap as well as high radiative transition rate. GaN and InN can form component continuous solid solution and superlattice, like InGaN. The alloy's bandgap can be tuned by controlling the ratio of GaN/InN and with the increasing composition of In, the VBM of the InGaN increases in energy almost linearly [28]. In addition, due to the quantum size effect, the bandgap of the InGaN can be further tuned by changing the size and shape of the QDs, so that these semiconductors can be

Unlike the synthesis methods of GaN and InN, the way to prepare InGaN quantum dots are mainly top-down approach, such as plasma-assisted molecular beam epitaxy (PA-MBE) [30], metal-organic vapour phase epitaxy (MOVPE) [31], and metal-organic chemical vapour depo-

There are abundant papers and books to review the development of 1D and 2D III–V nitride and other nanostructures. However, there are few reviews on the research status of the colloidal III–V nitride quantum dots. This research field is important for fundamental science and technology and is growing fast. In this review, we focus on recent progresses in the synthesis, crystal

There are three common crystal structures shared by the group III nitrides, namely, the wurtzite, zinc blende, and rocksalt structures. Usually, the thermodynamically stable structures are wurtzite for bulk AlN, GaN, and InN. The large difference in electronegativity between the group III and group V elements (Al = 1.18, Ga = 1.13, In = 0.99, N = 3.1) results in very strong chemical bonds within the III-nitride material system, which not only is at the origin of most of the exceptional III-nitride physical properties (listed in Table 1), but also greatly hinder the low-temperature solution synthesis of III–V nitrides [33]. For the growth of

III–V nitride QDs, vapour deposition based on substrate is a more popular approach.

structure, and optoelectronic properties of colloidal III–V nitride quantum dots.

blood [23].

controlled way.

sition (MOCVD) [32].

2. Crystal structure

1.3. Indium gallium nitride quantum dots

used for red to ultraviolet emitting devices [29].

Recently, III-nitride semiconductor (such as GaN, InN, and AlN) quantum dots (QDs) have begun to draw significant interest due to their promising applications in solid-state lighting, lasing technologies, and optoelectronic devices. The quality of nitride nanocrystals is, however, dramatically lower than that of II–VI semiconductor nanocrystals. For synthesis of III–V nitride quantum dots with uniform distribution, it is important to have a very fast nucleation and relatively slow growth process, which requires the growth unit concentration to reach high super-saturation level. However, such condition is very difficult to achieve for nitride quantum dots due to their strong covalent bonding and lack of suitable precursors. Although during the past few decades, various methods and precursors have been studied, the effective reaction with a control over group III elements and nitrogen in the solution has still remained difficult. Herein, we review the recent development in the synthesis techniques and properties of colloidal III–V nitride quantum dots as well as their applications. Meanwhile, the overview will partially involve the development and improvement of III-nitride QDs grown by vapourphase methods. More detailed introduction of colloidal III–V nitride quantum dots, that is, GaN, InN, and their alloys, are presented in the following discussion.

#### 1.1. Gallium nitride quantum dots

GaN is a technologically important direct semiconductor for development of short-wavelength optoelectronic devices, high-speed microwave device, and high-density integrated circuit [11– 13]. Its bandgap is 3.4 eV. GaN also have chemical and radiation resistance, and is therefore being considered as a stable photocatalyst in photoelectrochemical (PEC) cells for the production of fuels [14]. Colloidal QDs made from this material are expected to comprise good thermal, chemical, and radiation stability with the excellent optical properties. Therefore, since Xie [15] group succeeded in preparing GaN nanoparticles by simple inorganic reactions at 300C, considerable efforts have been made towards the solution-based synthesis of GaN QDs at low temperature and the understanding of optical and electronic properties. In the past 20 years, many researchers have prepared zero-dimensional (0D) GaN nanostructures by topdown approach. Various methods based on a bottom-up approach, like solvothermal methods, thermal decomposition, and so on, have been used to synthesize 0D GaN QDs. However, for the wet-chemical approach, controlling the size of 0D GaN QDs and even their optical and electrical properties remains a significant challenge.

#### 1.2. Indium nitride quantum dots

Among III–V nitrides, indium nitride is of relatively low-thermal stability. For example, the InN thin film conducts thermal decomposition under dinitrogen desorption at 500–550C [16]. InN is one of the least studied materials in the III–V compounds. Previously, it was believed that the fundamental bandgap of InN was 1.9 eV, until much more recent studies on higher quality films of InN have clearly shown that the true value of its direct bandgap is at 0.7 eV [17], making it a very promising compound for optoelectronic applications. Recently, with the further study of the group III nitride, the semiconductor properties of InN have attracted increasing attention. It has a good property on electron transport, making it face the huge application on high-speed electronic device due to its small-effective mass [18]. InN is also considered as an excellent material for low-cost, high-efficiency solar cell, photomask, lightemitting diodes, laser diodes, sensors, and THz radiation [19–22]. In addition, InN is considered as a promising candidate for biological imaging and in vivo medical applications because of its nontoxicity and its infrared emission in the optically transparent region of water and blood [23].

With the increasing importance on InN, how to obtain the InN with high quality is also attracting much attention. Although several synthesis methods, such as solvothermal methods [24, 25], sputtering [26], and molecular beam epitaxy (MBE) [27], were developed to prepare 0D InN QDs, more effort needs to be devoted to improve the quality of InN QDs with a controlled way.

#### 1.3. Indium gallium nitride quantum dots

ultraviolet (UV) to infrared region, exhibiting large potential applications for electronic and

Recently, III-nitride semiconductor (such as GaN, InN, and AlN) quantum dots (QDs) have begun to draw significant interest due to their promising applications in solid-state lighting, lasing technologies, and optoelectronic devices. The quality of nitride nanocrystals is, however, dramatically lower than that of II–VI semiconductor nanocrystals. For synthesis of III–V nitride quantum dots with uniform distribution, it is important to have a very fast nucleation and relatively slow growth process, which requires the growth unit concentration to reach high super-saturation level. However, such condition is very difficult to achieve for nitride quantum dots due to their strong covalent bonding and lack of suitable precursors. Although during the past few decades, various methods and precursors have been studied, the effective reaction with a control over group III elements and nitrogen in the solution has still remained difficult. Herein, we review the recent development in the synthesis techniques and properties of colloidal III–V nitride quantum dots as well as their applications. Meanwhile, the overview will partially involve the development and improvement of III-nitride QDs grown by vapourphase methods. More detailed introduction of colloidal III–V nitride quantum dots, that is,

GaN is a technologically important direct semiconductor for development of short-wavelength optoelectronic devices, high-speed microwave device, and high-density integrated circuit [11– 13]. Its bandgap is 3.4 eV. GaN also have chemical and radiation resistance, and is therefore being considered as a stable photocatalyst in photoelectrochemical (PEC) cells for the production of fuels [14]. Colloidal QDs made from this material are expected to comprise good thermal, chemical, and radiation stability with the excellent optical properties. Therefore, since Xie [15] group succeeded in preparing GaN nanoparticles by simple inorganic reactions at 300C, considerable efforts have been made towards the solution-based synthesis of GaN QDs at low temperature and the understanding of optical and electronic properties. In the past 20 years, many researchers have prepared zero-dimensional (0D) GaN nanostructures by topdown approach. Various methods based on a bottom-up approach, like solvothermal methods, thermal decomposition, and so on, have been used to synthesize 0D GaN QDs. However, for the wet-chemical approach, controlling the size of 0D GaN QDs and even their optical and

Among III–V nitrides, indium nitride is of relatively low-thermal stability. For example, the InN thin film conducts thermal decomposition under dinitrogen desorption at 500–550C [16]. InN is one of the least studied materials in the III–V compounds. Previously, it was believed that the fundamental bandgap of InN was 1.9 eV, until much more recent studies on higher quality films of InN have clearly shown that the true value of its direct bandgap is at 0.7 eV [17], making it a very promising compound for optoelectronic applications. Recently, with the further study of the group III nitride, the semiconductor properties of InN have attracted

GaN, InN, and their alloys, are presented in the following discussion.

optoelectronic devices.

62 Nonmagnetic and Magnetic Quantum Dots

1.1. Gallium nitride quantum dots

electrical properties remains a significant challenge.

1.2. Indium nitride quantum dots

Group III nitrides have a high light-emitting efficiency due to its direct bandgap as well as high radiative transition rate. GaN and InN can form component continuous solid solution and superlattice, like InGaN. The alloy's bandgap can be tuned by controlling the ratio of GaN/InN and with the increasing composition of In, the VBM of the InGaN increases in energy almost linearly [28]. In addition, due to the quantum size effect, the bandgap of the InGaN can be further tuned by changing the size and shape of the QDs, so that these semiconductors can be used for red to ultraviolet emitting devices [29].

Unlike the synthesis methods of GaN and InN, the way to prepare InGaN quantum dots are mainly top-down approach, such as plasma-assisted molecular beam epitaxy (PA-MBE) [30], metal-organic vapour phase epitaxy (MOVPE) [31], and metal-organic chemical vapour deposition (MOCVD) [32].

There are abundant papers and books to review the development of 1D and 2D III–V nitride and other nanostructures. However, there are few reviews on the research status of the colloidal III–V nitride quantum dots. This research field is important for fundamental science and technology and is growing fast. In this review, we focus on recent progresses in the synthesis, crystal structure, and optoelectronic properties of colloidal III–V nitride quantum dots.

## 2. Crystal structure

There are three common crystal structures shared by the group III nitrides, namely, the wurtzite, zinc blende, and rocksalt structures. Usually, the thermodynamically stable structures are wurtzite for bulk AlN, GaN, and InN. The large difference in electronegativity between the group III and group V elements (Al = 1.18, Ga = 1.13, In = 0.99, N = 3.1) results in very strong chemical bonds within the III-nitride material system, which not only is at the origin of most of the exceptional III-nitride physical properties (listed in Table 1), but also greatly hinder the low-temperature solution synthesis of III–V nitrides [33]. For the growth of III–V nitride QDs, vapour deposition based on substrate is a more popular approach.


nucleation can reduce saturation. As long as the rate of the consuming concentration for nanocrystal growth reaction is less than the rate of precursor injection, there are no new nanocrystals formed. Therefore, the size of the particle distributions mostly depends on the time from nucleation to growth. This method has some advantages that cannot be replaced by other methods: (1) uniform morphology; (2) narrow size distribution; and (3) high

Colloidal III–V Nitride Quantum Dots http://dx.doi.org/10.5772/intechopen.70844 65

The preparation of colloidal GaN quantum dots via thermal decomposition commonly requires the suitable Ga and N resources as the precursors. As the growth temperature increases, the precursors will be decomposed rapidly, react, and then generate lots of small nanometal clusters. Finally, the GaN nanoparticles are formed after the further growth and the size of the particles depends on the reaction time and temperature. The precursor contains polymeric gallium imide [40, 41] and gallium cupferron with hexamethyldisilazane, [42] etc. The selection of the precursor is very important for successful growth of colloidal GaN quantum dots. Janik and Wells [41] prepared powders of mixed hexagonal/cubic nanocrystals of GaN by deamination of polymeric gallium imide ({Ga(NH)3/2}n) at 210C. Their work showed that nanosize GaN could be prepared by polymeric gallium imide ({Ga(NH)3/2}n) at high temperature, due to the lack of any organic substituents in the precursor, which made {Ga (NH)3/2}<sup>n</sup> a good candidate for the generation of carbon-free GaN. However, it is unfortunate that these methods for the size control were limited and did not allow the nanocrystals to be dispersed in solvents to form transparent solutions of QDs suitable for optical measurements. One important factor in the synthesis of GaN is the purity of the final product. Carbon is usually left on QD surfaces after pyrolysis and it is difficult to remove. The elemental analyses of the GaN QDs showed 2.49–3.65% carbon content. Although the GaN QDs obtained by the above methods were of poor quality, it opened a window for researchers to find better ways to

According to the previous reports, Mićić et al. [40] still used {Ga(NH)3/2}n to prepare the GaN but they added trioctylamine (TOA) and hexadecylamine (HDA) during the heating process. The HDA could improve hydrophobicity of the GaN surface because of its less sterical hinderance and much dense surface cap. Mićić also showed that TOA/HDA decreased carbon adsorption on the QD particles and after purification yields a white colloidal solution. A transmission electron microscopy (TEM) image of the GaN quantum dots is given in Figure 1, which shows the spherical GaN QDs with diameter ranging from 23 to 45 Å. High-resolution micrographs show <111> lattice fringes in some particles have the proper orientation for observing fringes (Figure 1, bottom right panels). The particle size was estimated by simply counting the lattice fringes for each particle (interplanar spacing for <111> GaN = 2.52 Å); and the average diameter is 30 Å 40%. The bottom left panel in Figure 1 shows the electron

The polymer is not a good solvent for controlling the size and the high-yield production of QDs. In the polymer solvent, during the nucleation process, due to the melt in the solvent, dispersibility is not good, the nanoclusters may be happened to aggregate. To solve this

crystallinity due to relatively high reaction temperature.

3.1.1. Gallium nitride

obtain the high-quality GaN QDs.

diffraction pattern of the GaN nanocrystals.

Table 1. Physical properties of III–V nitride semiconductors [33].

## 3. Colloidal III–V nitride quantum dots

#### 3.1. Syntheses of colloidal nitride quantum dots

So far, many efforts have been devoted to synthesize colloidal nitride quantum dots by the solution-phase routes. Over the past 20 years, many groups have prepared the colloidal III– V nitride quantum dots by various solution-based methods. These methods can be classified into the following approaches, namely solvothermal [24, 25, 34], hydrothermal [35], and thermal decomposition of single [36, 37] and two precursors [38]. The hydrothermal process refers to the reaction of reactants and water in a pressurized reaction environment to form nanoparticles. The process of forming of nanocrystals undergoes two stages, dissolution and crystallization. In the primary reaction, the aggregation and binding of the precursor particles are destroyed, making the particles dissolute in the hydrothermal solvent, transporting into the solution in the form of ions or ionic groups, and crystallizing the crystalline grain after nucleation. Solvothermal method was developed on the basis of hydrothermal method, which uses the organic solvent as solution, replacing water. In this way, some compounds sensitive to water (reacting with water, hydrolysis, resolution, or instability) like III–V group semiconductors, carbides, fluorides, and so on, can be prepared. LaMer and Dinegra thought that the preparation of monodisperse nanocluster needed a transient and discrete nucleation process, and then controlled the crystal nucleus growth slowly [39]. Putting the reactants rapidly into the container makes the precursor concentration higher than the nucleation threshold value. A short period of sudden

nucleation can reduce saturation. As long as the rate of the consuming concentration for nanocrystal growth reaction is less than the rate of precursor injection, there are no new nanocrystals formed. Therefore, the size of the particle distributions mostly depends on the time from nucleation to growth. This method has some advantages that cannot be replaced by other methods: (1) uniform morphology; (2) narrow size distribution; and (3) high crystallinity due to relatively high reaction temperature.

#### 3.1.1. Gallium nitride

3. Colloidal III–V nitride quantum dots

Table 1. Physical properties of III–V nitride semiconductors [33].

Refractive index, n 2.2 (0.60 μm)

ɛ (0) 9.14 10.4 (Ekc)

ɛ (∞) 4.84 5.8 (Ekc)

Thermal conductivity, (κher/cm K) 2.0 1.7–1.8

Thermal expansion coefficient α<sup>a</sup> (10�<sup>6</sup> K�<sup>1</sup>

64 Nonmagnetic and Magnetic Quantum Dots

Thermal expansion coefficient α<sup>c</sup> (10�<sup>6</sup> K�<sup>1</sup>

3.1. Syntheses of colloidal nitride quantum dots

So far, many efforts have been devoted to synthesize colloidal nitride quantum dots by the solution-phase routes. Over the past 20 years, many groups have prepared the colloidal III– V nitride quantum dots by various solution-based methods. These methods can be classified into the following approaches, namely solvothermal [24, 25, 34], hydrothermal [35], and thermal decomposition of single [36, 37] and two precursors [38]. The hydrothermal process refers to the reaction of reactants and water in a pressurized reaction environment to form nanoparticles. The process of forming of nanocrystals undergoes two stages, dissolution and crystallization. In the primary reaction, the aggregation and binding of the precursor particles are destroyed, making the particles dissolute in the hydrothermal solvent, transporting into the solution in the form of ions or ionic groups, and crystallizing the crystalline grain after nucleation. Solvothermal method was developed on the basis of hydrothermal method, which uses the organic solvent as solution, replacing water. In this way, some compounds sensitive to water (reacting with water, hydrolysis, resolution, or instability) like III–V group semiconductors, carbides, fluorides, and so on, can be prepared. LaMer and Dinegra thought that the preparation of monodisperse nanocluster needed a transient and discrete nucleation process, and then controlled the crystal nucleus growth slowly [39]. Putting the reactants rapidly into the container makes the precursor concentration higher than the nucleation threshold value. A short period of sudden

Lattice constant, a (Å) 3.112 3.189 3.545 Lattice constant, c (Å) 4.982 5.186 5.703

Electron effective mass, me (m0) 0.2 0.11 Hole effective mass, mh (m0) 0.8 0.5 (mhh) 0.17 (mlh)

Melting point (�C) 2000 >1700 1100 <sup>Δ</sup>G0 (kcal/mol) �68.2 �33.0 �23.0 Heat capacity, Cp (cal/mol K) 7.6 9.7 10.0

2.5 (0.23 μm)

AlN GaN InN

) 5.27 (20–800�C) 4.3 (17–477�C) 5.6 (280�C)

) 4.15 (20–800�C) 4.0 (20–800�C) 3.8 (280�C)

2.35 (1.0 μm) 2.60 (0.38 μm)

9.5 (E⊥c)

5.4 (E⊥c)

2.56 (1.0 μm) 3.12 (0.66 μm)

9.3

The preparation of colloidal GaN quantum dots via thermal decomposition commonly requires the suitable Ga and N resources as the precursors. As the growth temperature increases, the precursors will be decomposed rapidly, react, and then generate lots of small nanometal clusters. Finally, the GaN nanoparticles are formed after the further growth and the size of the particles depends on the reaction time and temperature. The precursor contains polymeric gallium imide [40, 41] and gallium cupferron with hexamethyldisilazane, [42] etc. The selection of the precursor is very important for successful growth of colloidal GaN quantum dots. Janik and Wells [41] prepared powders of mixed hexagonal/cubic nanocrystals of GaN by deamination of polymeric gallium imide ({Ga(NH)3/2}n) at 210C. Their work showed that nanosize GaN could be prepared by polymeric gallium imide ({Ga(NH)3/2}n) at high temperature, due to the lack of any organic substituents in the precursor, which made {Ga (NH)3/2}<sup>n</sup> a good candidate for the generation of carbon-free GaN. However, it is unfortunate that these methods for the size control were limited and did not allow the nanocrystals to be dispersed in solvents to form transparent solutions of QDs suitable for optical measurements. One important factor in the synthesis of GaN is the purity of the final product. Carbon is usually left on QD surfaces after pyrolysis and it is difficult to remove. The elemental analyses of the GaN QDs showed 2.49–3.65% carbon content. Although the GaN QDs obtained by the above methods were of poor quality, it opened a window for researchers to find better ways to obtain the high-quality GaN QDs.

According to the previous reports, Mićić et al. [40] still used {Ga(NH)3/2}n to prepare the GaN but they added trioctylamine (TOA) and hexadecylamine (HDA) during the heating process. The HDA could improve hydrophobicity of the GaN surface because of its less sterical hinderance and much dense surface cap. Mićić also showed that TOA/HDA decreased carbon adsorption on the QD particles and after purification yields a white colloidal solution. A transmission electron microscopy (TEM) image of the GaN quantum dots is given in Figure 1, which shows the spherical GaN QDs with diameter ranging from 23 to 45 Å. High-resolution micrographs show <111> lattice fringes in some particles have the proper orientation for observing fringes (Figure 1, bottom right panels). The particle size was estimated by simply counting the lattice fringes for each particle (interplanar spacing for <111> GaN = 2.52 Å); and the average diameter is 30 Å 40%. The bottom left panel in Figure 1 shows the electron diffraction pattern of the GaN nanocrystals.

The polymer is not a good solvent for controlling the size and the high-yield production of QDs. In the polymer solvent, during the nucleation process, due to the melt in the solvent, dispersibility is not good, the nanoclusters may be happened to aggregate. To solve this

Figure 1. TEM image of GaN QDs taken in bright field. Top panel shows low magnification of QDs and some linear alignment. Bottom two right panels show high magnification and lattice fringes of QD oriented with the <111> axis in the plane of the micrograph. Bottom left panel shows electron diffraction pattern of GaN QDs indicating zinc-blende structure [40].

no GaN was produced. Elemental analysis on the GaN product prepared in the presence of HDA revealed a Ga/N mass ratio of 4.88:1 (theoretical 4.98:1), indicative of nearly stoichiomet-

Figure 2. TEM image of GaN nanoparticles obtained from pyrolysis of Ga2[N(CH3)2]6. Scale bar is 10 nm [43].

The sample also had a high-carbon content (C/N) of 2.27:1, indicating that carbon was incorporated into the particles. Certainly, it was consistent with capping of nanosize particles by

Generally, to get the GaN, the post-treatment temperature is at least 500�C. Nitrides of lanthanide and transition metals (M) can be prepared by solid reaction at the temperature ranging

Xie et al. [15] reported that the crystalline GaN particles could be synthesized by simple inorganic reactions at temperature of 300�C in the autoclave (no capping ligand in the whole preparation process). They used GaCl3 and Li3N as gallium and nitrogen precursors in the

These crystallites of GaN have an average size of 32 nm and display a uniform shape (Figure 3A). The images of GaN particles were observed by High-resolution electron microscopy (HREM). In

MCln þ Li3N ! MN þ LiCl (1)

Colloidal III–V Nitride Quantum Dots http://dx.doi.org/10.5772/intechopen.70844 67

GaCl3 þ Li3N ! GaN þ 3LiCl (2)

capping ligands, such as HDA or TOA that contains long aliphatic chains.

from 600 to 1000�C. The chemical equation is as follows:

liquid, respectively, and the reaction equation is:

ric GaN.

problem, Pan et al. [43] found that GaN could be obtained by dimeric amidogallium precursor (Ga2[N(CH3)2]6) through pyrolysis without the need for the polymeric intermediate. In this way, not only it produced colloidal GaN quantum dots, but also offered the possibilities of controlling the dots' size. In addition, the gaseous ammonia needed in the nucleation process was cancelled. Colloidal GaN QDs was got by transmission electron microscopy (TEM) imaging. Figure 2 shows a TEM image of the GaN nanoparticles. Several spherical particles with diameters of 2–4 nm are shown in the Figure 2. Although the particle size distribution obtained here is not comparable with those obtained in the highly optimized II–VI group systems, it is believed that this reaction will be improved through optimizing the reaction conditions. However, the as-prepared samples' crystallinity is poor, which can be confirmed by the TEM images. No lattice fringes were observed in HRTEM images. HAD may has a contribution for the pyrolysis reaction. When HDA was eliminated from the reaction mixture,

Figure 2. TEM image of GaN nanoparticles obtained from pyrolysis of Ga2[N(CH3)2]6. Scale bar is 10 nm [43].

no GaN was produced. Elemental analysis on the GaN product prepared in the presence of HDA revealed a Ga/N mass ratio of 4.88:1 (theoretical 4.98:1), indicative of nearly stoichiometric GaN.

The sample also had a high-carbon content (C/N) of 2.27:1, indicating that carbon was incorporated into the particles. Certainly, it was consistent with capping of nanosize particles by capping ligands, such as HDA or TOA that contains long aliphatic chains.

Generally, to get the GaN, the post-treatment temperature is at least 500�C. Nitrides of lanthanide and transition metals (M) can be prepared by solid reaction at the temperature ranging from 600 to 1000�C. The chemical equation is as follows:

problem, Pan et al. [43] found that GaN could be obtained by dimeric amidogallium precursor (Ga2[N(CH3)2]6) through pyrolysis without the need for the polymeric intermediate. In this way, not only it produced colloidal GaN quantum dots, but also offered the possibilities of controlling the dots' size. In addition, the gaseous ammonia needed in the nucleation process was cancelled. Colloidal GaN QDs was got by transmission electron microscopy (TEM) imaging. Figure 2 shows a TEM image of the GaN nanoparticles. Several spherical particles with diameters of 2–4 nm are shown in the Figure 2. Although the particle size distribution obtained here is not comparable with those obtained in the highly optimized II–VI group systems, it is believed that this reaction will be improved through optimizing the reaction conditions. However, the as-prepared samples' crystallinity is poor, which can be confirmed by the TEM images. No lattice fringes were observed in HRTEM images. HAD may has a contribution for the pyrolysis reaction. When HDA was eliminated from the reaction mixture,

structure [40].

66 Nonmagnetic and Magnetic Quantum Dots

Figure 1. TEM image of GaN QDs taken in bright field. Top panel shows low magnification of QDs and some linear alignment. Bottom two right panels show high magnification and lattice fringes of QD oriented with the <111> axis in the plane of the micrograph. Bottom left panel shows electron diffraction pattern of GaN QDs indicating zinc-blende

$$\text{MCl}\_{\text{n}} + \text{Li}\_{3}\text{N} \to \text{MN} + \text{LiCl} \tag{1}$$

Xie et al. [15] reported that the crystalline GaN particles could be synthesized by simple inorganic reactions at temperature of 300�C in the autoclave (no capping ligand in the whole preparation process). They used GaCl3 and Li3N as gallium and nitrogen precursors in the liquid, respectively, and the reaction equation is:

$$\text{GaCl}\_3 + \text{Li}\_3\text{N} \to \text{GaN} + \text{3LiCl} \tag{2}$$

These crystallites of GaN have an average size of 32 nm and display a uniform shape (Figure 3A). The images of GaN particles were observed by High-resolution electron microscopy (HREM). In

morphology due to poor control in the nucleation process. Moreover, the vapour-phase methods often encounter the aggregation problem due to the large surface energy of nanocrystals. It is concluded that each of the aforementioned methods for synthesis of monodisperse InN nanocrystals faces significant challenges in achieving well-defined size and shape. Our group [49] had addressed these critical issues by exploiting a new synthesis approach that resulted in monodisperse InN nanocrystals with uniform size and morphology and superior optical quality, and first successfully prepared the cubic InN nanocrystals with aforementioned advantages, by combining solution- and vapour-phase methods under silica shell confinement (SVSC), as

In this method, the In2O3 nanocrystals with well-defined size and morphology were first synthesized by a solution-based method. The In2O3 nanocrystals were coated by silica shell before nitridation, this is because silica is inert and can be easily removed by HF acid. The obtained In2O3@SiO2 nanopowders were put into a tube furnace. After being purged with NH3

Finally, large-scale InN@SiO2nanocrystals with uniform size and morphology were obtained through the SVSC route. After removing the silica shell, InN NCs can be dispersed into DI water and then transferred to various nonpolar organic solvents by phase transfer, as shown in

Figure 5(a) and (b) shows transmission electron microscopy (TEM) and high-resolution transmission electron microscopy (HRTEM) images of the InN nanocrystals. The indium nitride nanocrystals with nearly monodisperse spherical shape can be observed from the TEM images. The diameter distribution of the synthesized indium nitride nanocrystals can be revealed from the Figure 5(c), revealing a fairly uniform size distribution of the InN nanocrystals from 5.0 to

Figure 4. (a) Schematic of the SVSC method for InN nanocrystals. (b) The upper layer is hexane and the bottom layer is distilled water. The left bottle contains InN nanocrystals in hexane and the right one contains InN nanocrystals in water.

(c) Large-scale InN@SiO2 nanopowders (0.47 g) synthesized by the SVSC method [49].

C and kept for 5 h under NH3 flow at 300 ml/min.

Colloidal III–V Nitride Quantum Dots http://dx.doi.org/10.5772/intechopen.70844 69

schematically shown in Figure 4(a).

Figures 4(b) and (c).

gas for 20 min, the furnace was heated to 500–700o

Figure 3. (A) A TEM micrograph of nanocrystalline GaN. (B and C) HREM images of nanocrystalline GaN: (B) lattice fringes of (001) plane in GaN with a wurtzite structure and (C) lattice fringes of (100) and (110) planes in GaN (marked A and B, respectively) with a rocksalt structure [15].

Figure 3B, the (001) lattice fringes of GaN in the wurtzite structure appear frequently, indicating the preferential orientation of the plate-like GaN particles. The areas marked by arrowheads (A and B) in Figure 3C represent a typical structural image of [100] and [110] orientations, respectively, of GaN in the rocksalt structure.

#### 3.1.2. Indium nitride

Among the nitride, InN is the most unstable, which decomposes above 500�C [44]. Hence, it is difficult to prepare InN crystalline. Historically, polycrystalline indium nitride was synthesized by radio frequency sputtering [28], which results in high free-electron concentration, significant oxygen contamination, and an absorption edge at about 1.9 eV [45–47]. Recently, high-quality crystalline InN has been grown by molecular beam epitaxy (MBE) [27]. The typically observed bandgap of high-quality wurtzite-InN grown by MBE is around 0.65–0.7 eV [34, 48]. However, the quality of indium nitride samples grown by low-cost solution or other vapour methods has still remained challenging. Therefore, there still exists a huge challenge to synthesize high-quality InN nanocrystals using low-cost solution or vapour methods. So far, Xiao et al. [24] adopted the solvothermal using NaNH2 and In2S3 as novel nitrogen and indium sources to prepare the indium nitride at 180–200 �C with the particle size ranging from 10 to 30 nm, and the reaction equation is:

$$\text{In}\_2\text{S}\_3 + 6\text{NaNH}\_2 \to 2\text{InN} + 3\text{Na}\_2\text{S} + 4\text{NH}\_3\tag{3}$$

Hsieh [25] also used solvothermal to prepare the InN NCs with an average diameter of 6.2 � 2.0 nm utilizing InBr3 and NaNH2 in a low temperature, ambient pressure, and liquidphase condition.

Generally, ammonia is used as nitrogen source in the vapour-phase growth process. The relative high growth temperature in vapour-phase methods can help nucleation overcome the reaction difficulty encountered by solution methods. However, unlike the solution-based methods (where the nucleation and growth process could be separated by choosing appropriate ligands and solvents), the vapour-phase methods usually trigger off nonuniform nanocrystal

morphology due to poor control in the nucleation process. Moreover, the vapour-phase methods often encounter the aggregation problem due to the large surface energy of nanocrystals. It is concluded that each of the aforementioned methods for synthesis of monodisperse InN nanocrystals faces significant challenges in achieving well-defined size and shape. Our group [49] had addressed these critical issues by exploiting a new synthesis approach that resulted in monodisperse InN nanocrystals with uniform size and morphology and superior optical quality, and first successfully prepared the cubic InN nanocrystals with aforementioned advantages, by combining solution- and vapour-phase methods under silica shell confinement (SVSC), as schematically shown in Figure 4(a).

In this method, the In2O3 nanocrystals with well-defined size and morphology were first synthesized by a solution-based method. The In2O3 nanocrystals were coated by silica shell before nitridation, this is because silica is inert and can be easily removed by HF acid. The obtained In2O3@SiO2 nanopowders were put into a tube furnace. After being purged with NH3 gas for 20 min, the furnace was heated to 500–700o C and kept for 5 h under NH3 flow at 300 ml/min. Finally, large-scale InN@SiO2nanocrystals with uniform size and morphology were obtained through the SVSC route. After removing the silica shell, InN NCs can be dispersed into DI water and then transferred to various nonpolar organic solvents by phase transfer, as shown in Figures 4(b) and (c).

Figure 3B, the (001) lattice fringes of GaN in the wurtzite structure appear frequently, indicating the preferential orientation of the plate-like GaN particles. The areas marked by arrowheads (A and B) in Figure 3C represent a typical structural image of [100] and [110]

Figure 3. (A) A TEM micrograph of nanocrystalline GaN. (B and C) HREM images of nanocrystalline GaN: (B) lattice fringes of (001) plane in GaN with a wurtzite structure and (C) lattice fringes of (100) and (110) planes in GaN (marked A

Among the nitride, InN is the most unstable, which decomposes above 500�C [44]. Hence, it is difficult to prepare InN crystalline. Historically, polycrystalline indium nitride was synthesized by radio frequency sputtering [28], which results in high free-electron concentration, significant oxygen contamination, and an absorption edge at about 1.9 eV [45–47]. Recently, high-quality crystalline InN has been grown by molecular beam epitaxy (MBE) [27]. The typically observed bandgap of high-quality wurtzite-InN grown by MBE is around 0.65–0.7 eV [34, 48]. However, the quality of indium nitride samples grown by low-cost solution or other vapour methods has still remained challenging. Therefore, there still exists a huge challenge to synthesize high-quality InN nanocrystals using low-cost solution or vapour methods. So far, Xiao et al. [24] adopted the solvothermal using NaNH2 and In2S3 as novel nitrogen and indium sources to prepare the indium nitride at 180–200 �C with the particle size ranging from 10 to 30

Hsieh [25] also used solvothermal to prepare the InN NCs with an average diameter of 6.2 � 2.0 nm utilizing InBr3 and NaNH2 in a low temperature, ambient pressure, and liquid-

Generally, ammonia is used as nitrogen source in the vapour-phase growth process. The relative high growth temperature in vapour-phase methods can help nucleation overcome the reaction difficulty encountered by solution methods. However, unlike the solution-based methods (where the nucleation and growth process could be separated by choosing appropriate ligands and solvents), the vapour-phase methods usually trigger off nonuniform nanocrystal

In2S3 þ 6NaNH2 ! 2InN þ 3Na2S þ 4NH3 (3)

orientations, respectively, of GaN in the rocksalt structure.

and B, respectively) with a rocksalt structure [15].

68 Nonmagnetic and Magnetic Quantum Dots

3.1.2. Indium nitride

nm, and the reaction equation is:

phase condition.

Figure 5(a) and (b) shows transmission electron microscopy (TEM) and high-resolution transmission electron microscopy (HRTEM) images of the InN nanocrystals. The indium nitride nanocrystals with nearly monodisperse spherical shape can be observed from the TEM images. The diameter distribution of the synthesized indium nitride nanocrystals can be revealed from the Figure 5(c), revealing a fairly uniform size distribution of the InN nanocrystals from 5.0 to

Figure 4. (a) Schematic of the SVSC method for InN nanocrystals. (b) The upper layer is hexane and the bottom layer is distilled water. The left bottle contains InN nanocrystals in hexane and the right one contains InN nanocrystals in water. (c) Large-scale InN@SiO2 nanopowders (0.47 g) synthesized by the SVSC method [49].

nanostructure size. Therefore, they used QSC-PEC etching to fabricate InGaN QDs of controlled size starting from the InGaN thin film. They used H2SO4 aqueous solution as the electrolyte and a tunable, relatively narrow band laser source as photoexcitation. The sample consisted of In0.13Ga0.87N films (3–20 nm) grown on c-plane GaN/sapphire. They discussed the influence of solution pH during quantum size-controlled PEC etch process. [51] When the solution pH lies between 5 and 11, both Ga- and In-oxides are formed at the surface. Etching rates are very low and InGaN QDs are not formed. In the dark etching of InGaN at pH above 5, the above situation may also occur. However, when the solution pH is below 3, oxide-free QDs with self-terminated sizes can be successfully realized. In strongly acidic solutions, the oxides are not formed during the PEC etching process, due to all the oxide productions can be dissolved by the electrolyte. Therefore, PEC etching can be used to prepare InGaN QDs. Meanwhile, there are other methods of preparing InGaN quantum dot, such as plasma-assisted molecular beam epitaxy (PA-MBE) [30], metal-organic vapour phase epitaxy (MOVPE) [31],

Colloidal III–V Nitride Quantum Dots http://dx.doi.org/10.5772/intechopen.70844 71

Figure 6. Photoluminescence spectrum of InN@SiO2 nanocrystals at room temperature [49].

The field of colloidal III–V nitride quantum dots has been constantly gaining interest among science and engineering communities during the past decades. In this chapter, we have summarized the research status and progress in this field, including their preparation techniques and optoelectronic properties. Although much progress has already been made in the field of colloidal III–V nitride quantum dots, significant challenges involving the fabrication of quantum dots with uniform morphology and size and control of the electronic and optical properties in terms of composition and structure remain to be solved. Once high-quality colloidal III– V nitride quantum dots are synthesized successfully, more new discoveries and applications

and metal-organic chemical vapour deposition (MOCVD) [32].

4. Summary and future directions

Figure 5. (a) Low-resolution TEM; (b) HRTEM images of the InN nanocrystals; (c) size distribution of the InN nanocrystals; and (d) XRD spectrum of the InN@SiO2nanocrystals obtained at 550C [49].

6.3 nm. The average diameter of the InN nanocrystals is calculated to be 5.7 0.6 nm, after counting about 200 nanocrystals. Figure 5d shows the X-ray diffraction (XRD) pattern of the InN nanocrystals. All the peaks can be matched with the cubic InN (JCPDS No. 88-2365) except the peak at 2θ = 22.5, which corresponds to the amorphous silica. No peaks of In2O3 were observed in the XRD pattern, indicating that all In2O3 nanocrystals had been converted to InN. As illustrated in Figure 6, the InN@SiO2 nanocrystals in the form of powders (not in the solution) exhibited infrared PL at room temperature. The fluctuation of PL signal at 1900 nm could be due to water absorption and might not originate from the sample. The PL spectrum is characterized by the presence of three distinct emission peaks.

#### 3.1.3. Indium gallium nitride

Xiao et al. [50] demonstrated for the first time, a new route that quantum size effect can be used to prepare the epitaxial nanostructures with great significance for the achievement of a broad range of future nanoelectronic and nanophotonic devices. The process is quantum sizecontrolled photoelectrochemical (QSC-PEC) etching. Quantum dots' bandgap depends on the

Figure 6. Photoluminescence spectrum of InN@SiO2 nanocrystals at room temperature [49].

nanostructure size. Therefore, they used QSC-PEC etching to fabricate InGaN QDs of controlled size starting from the InGaN thin film. They used H2SO4 aqueous solution as the electrolyte and a tunable, relatively narrow band laser source as photoexcitation. The sample consisted of In0.13Ga0.87N films (3–20 nm) grown on c-plane GaN/sapphire. They discussed the influence of solution pH during quantum size-controlled PEC etch process. [51] When the solution pH lies between 5 and 11, both Ga- and In-oxides are formed at the surface. Etching rates are very low and InGaN QDs are not formed. In the dark etching of InGaN at pH above 5, the above situation may also occur. However, when the solution pH is below 3, oxide-free QDs with self-terminated sizes can be successfully realized. In strongly acidic solutions, the oxides are not formed during the PEC etching process, due to all the oxide productions can be dissolved by the electrolyte. Therefore, PEC etching can be used to prepare InGaN QDs. Meanwhile, there are other methods of preparing InGaN quantum dot, such as plasma-assisted molecular beam epitaxy (PA-MBE) [30], metal-organic vapour phase epitaxy (MOVPE) [31], and metal-organic chemical vapour deposition (MOCVD) [32].

#### 4. Summary and future directions

6.3 nm. The average diameter of the InN nanocrystals is calculated to be 5.7 0.6 nm, after counting about 200 nanocrystals. Figure 5d shows the X-ray diffraction (XRD) pattern of the InN nanocrystals. All the peaks can be matched with the cubic InN (JCPDS No. 88-2365) except the peak at 2θ = 22.5, which corresponds to the amorphous silica. No peaks of In2O3 were observed in the XRD pattern, indicating that all In2O3 nanocrystals had been converted to InN. As illustrated in Figure 6, the InN@SiO2 nanocrystals in the form of powders (not in the solution) exhibited infrared PL at room temperature. The fluctuation of PL signal at 1900 nm could be due to water absorption and might not originate from the sample. The PL spectrum is

Figure 5. (a) Low-resolution TEM; (b) HRTEM images of the InN nanocrystals; (c) size distribution of the InN

Xiao et al. [50] demonstrated for the first time, a new route that quantum size effect can be used to prepare the epitaxial nanostructures with great significance for the achievement of a broad range of future nanoelectronic and nanophotonic devices. The process is quantum sizecontrolled photoelectrochemical (QSC-PEC) etching. Quantum dots' bandgap depends on the

characterized by the presence of three distinct emission peaks.

nanocrystals; and (d) XRD spectrum of the InN@SiO2nanocrystals obtained at 550C [49].

3.1.3. Indium gallium nitride

70 Nonmagnetic and Magnetic Quantum Dots

The field of colloidal III–V nitride quantum dots has been constantly gaining interest among science and engineering communities during the past decades. In this chapter, we have summarized the research status and progress in this field, including their preparation techniques and optoelectronic properties. Although much progress has already been made in the field of colloidal III–V nitride quantum dots, significant challenges involving the fabrication of quantum dots with uniform morphology and size and control of the electronic and optical properties in terms of composition and structure remain to be solved. Once high-quality colloidal III– V nitride quantum dots are synthesized successfully, more new discoveries and applications will be exploited, such as solid-state lighting, lasing technologies, and optoelectronic devices, as well as the booming quantum photonics technology.

[8] Xia YN, Xiong YJ, Lim B, Skrabalak SE. Shape-controlled synthesis of metal nanocrystals: Simple chemistry meets complex physics. Angewandte Chemie International Edition.

Colloidal III–V Nitride Quantum Dots http://dx.doi.org/10.5772/intechopen.70844 73

[9] Puntes VF, Krishnan KM, Alivisatos AP. Colloidal nanocrystal shape and size control:

[10] Reiss P, Carriere M, Lincheneau C, Vaure L, Tamang S. Synthesis of semiconductor nanocrystals, focusing on nontoxic and earth-abundant materials. Chemical Review.

[11] Nakamura S, Pearton SJ, Fasol G. The Blue Laser Diode: The Complete Story. 2nd ed.

[12] Wu Y, Jacob-mitos M, Moore ML, Heikman S. A 97.8% efficient GaN HEMT boost converter with 300-W output power at 1 MHz. IEEE Electron Device Letters. 2008;29:824-826.

[13] Nakamura S, Faso G. The Blue Laser Diode. The Complete Story. Berlin: Springer; 1999.

[14] Kocha SS, Peterson MW, Arent DJ, Redwing JM, Tischler MA, Turner JA. Electrochemical investigation of the gallium nitride-aqueous electrolyte interface. Journal of the Electro-

[15] Xie Y, Qian Y, Wang W, Zhang S, Zhang Y. A benzene-thermal synthetic route to nanocrystalline GaN. Science. 1996;272:1926-1927. DOI: 10.1126/science.272.5270.1926 [16] Guo Q, Kato O, Yoshida A. Thermal stability of indium nitride single crystal films.

[17] Yasushi N, Yoshiki S, Tomohiro Y. RF-molecular beam epitaxy growth and properties of InN and related alloys. The Japan Society of Applied Physics. 2003;42:2549-2559. DOI:

[18] Faso GS, Nakamura S. The Blue Laser Diode: GaN Based Light Emitters and Lasers.

[19] Starikov E, Shiktorov P, Gruninskis V. Monte Carlo calculations of THz generation in wide gap semiconductors. Physica B. 2002;341:171-175. DOI: 10.1016/S0921-4526(01)

[20] Neff H, Semchinova OK, AMN L, Filimonov A, Holzhueter G. Photovoltaic properties and technological aspectsof In1-xGaxN/Si, Ge (0 < x < 0.6) heterojunction solar cells. Solar Energy Materials & Solar Cells. 2006;90:982-997. DOI: 10.1016/j.solmat.2005.06.002 [21] Nguyen HPT, Chang Y-L, et al. InN p-i-n nanowire solar cells on Si. IEEE Journal of Selected Topics in Quantum Electronics. 2011;17:1062-1069. DOI: 10.1109/JSTQE.2010.

[22] Wu JQ. When group-III nitrides go infrared: New properties and perspectives. Journal of

Applied Physics. 2009;106:011101-011128. DOI: 10.1063/1.3155798

The case of cobalt. Science. 2001;291:1019-1020. DOI: 10.1126/science.1057553

New York: Springer-Verlag; 2000. 104 p. DOI: 10.1007/978-3-662-04156-7

2009;48:60-103. DOI: 10.1002/anie.200802248

DOI: 10.1109/LED.2008.2000921

10.1143/JJAP.42.2549

01374-6

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Berlin, Hong Kong: Springer; 1997. 15 p. DOI: 10.1007/978-3-662-03462-0

## Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant No. 51472031, 51102017 and 21503014).

## Author details

Zequn Chen<sup>1</sup> , Chuli Sun<sup>2</sup> , Wei Guo<sup>2</sup> and Zhuo Chen<sup>1</sup> \*

\*Address all correspondence to: zchen@bit.edu.cn

1 Department of Materials Physics and Chemistry, Beijing Key Laboratory of Construction Tailorable Advanced Functional Materials and Green Applications, School of Materials Science and Engineering, Beijing Institute of Technology Institution, Beijing, P.R. China

2 Department of Physics, Beijing Institute of Technology, Beijing, P.R. China

## References


[8] Xia YN, Xiong YJ, Lim B, Skrabalak SE. Shape-controlled synthesis of metal nanocrystals: Simple chemistry meets complex physics. Angewandte Chemie International Edition. 2009;48:60-103. DOI: 10.1002/anie.200802248

will be exploited, such as solid-state lighting, lasing technologies, and optoelectronic devices,

This work was financially supported by the National Natural Science Foundation of China

1 Department of Materials Physics and Chemistry, Beijing Key Laboratory of Construction Tailorable Advanced Functional Materials and Green Applications, School of Materials Science

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\*

, Wei Guo<sup>2</sup> and Zhuo Chen<sup>1</sup>

and Engineering, Beijing Institute of Technology Institution, Beijing, P.R. China

2 Department of Physics, Beijing Institute of Technology, Beijing, P.R. China

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as well as the booming quantum photonics technology.

(Grant No. 51472031, 51102017 and 21503014).

\*Address all correspondence to: zchen@bit.edu.cn

, Chuli Sun<sup>2</sup>

Acknowledgements

72 Nonmagnetic and Magnetic Quantum Dots

Author details

Zequn Chen<sup>1</sup>

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**Chapter 5**

**Provisional chapter**

**CdTe Quantum Dot Fluorescence Thermometry of**

**CdTe Quantum Dot Fluorescence Thermometry of** 

DOI: 10.5772/intechopen.70866

Temperature is one of the most important parameters affecting the service life and performance of a rolling element bearing component. In this paper, a nonintrusive method is developed to monitor the temperature variation of the inner raceway during bearing operation utilizing CdTe quantum dots as the temperature sensors. The CdTe quantum dots were synthesized and were used in constructing a sensor film by means of layer-bylayer electrostatic self-assembly method on an ultrathin glass slice. The peak wavelength shift of the fluorescence spectrum of the sensor film shows a linear and reversible relationship with temperature, and it is used to sense the temperature of the inner raceway. The resolution of the CdTe optothermal sensor is determined to be 0.14 nm/°C. The temperature measurement of rolling element bearing was conducted on a bearing test rig incorporated with an optical fiber fluorescence spectrum detecting system. To verify the accuracy of the temperature obtained by quantum dots sensor film, a thermocouple was used to test the temperature of the inner raceway right before and after the operation. Results show that the temperature obtained by the CdTe quantum dots film sensor is consistent with that by the thermocouple, with an error typically below 10% or smaller. **Keywords:** high speed rolling bearing, inner ring temperature monitoring, quantum dots

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Rolling bearings are basic mechanical components widely used in machinery for low friction, high rigidity, and reliability. They are required to operate at high speed for long period of time under uneven conditions with minimum maintenance. The operating status of bearings directly affects the performance of rotating machinery. Bearing failure can make machine

**Rolling Bearing**

**Rolling Bearing**

Ke Yan and Bei Yan

**Abstract**

**1. Introduction**

Ke Yan and Bei Yan

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70866

### **CdTe Quantum Dot Fluorescence Thermometry of Rolling Bearing CdTe Quantum Dot Fluorescence Thermometry of Rolling Bearing**

DOI: 10.5772/intechopen.70866

Ke Yan and Bei Yan Ke Yan and Bei Yan

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70866

#### **Abstract**

Temperature is one of the most important parameters affecting the service life and performance of a rolling element bearing component. In this paper, a nonintrusive method is developed to monitor the temperature variation of the inner raceway during bearing operation utilizing CdTe quantum dots as the temperature sensors. The CdTe quantum dots were synthesized and were used in constructing a sensor film by means of layer-bylayer electrostatic self-assembly method on an ultrathin glass slice. The peak wavelength shift of the fluorescence spectrum of the sensor film shows a linear and reversible relationship with temperature, and it is used to sense the temperature of the inner raceway. The resolution of the CdTe optothermal sensor is determined to be 0.14 nm/°C. The temperature measurement of rolling element bearing was conducted on a bearing test rig incorporated with an optical fiber fluorescence spectrum detecting system. To verify the accuracy of the temperature obtained by quantum dots sensor film, a thermocouple was used to test the temperature of the inner raceway right before and after the operation. Results show that the temperature obtained by the CdTe quantum dots film sensor is consistent with that by the thermocouple, with an error typically below 10% or smaller.

**Keywords:** high speed rolling bearing, inner ring temperature monitoring, quantum dots

## **1. Introduction**

Rolling bearings are basic mechanical components widely used in machinery for low friction, high rigidity, and reliability. They are required to operate at high speed for long period of time under uneven conditions with minimum maintenance. The operating status of bearings directly affects the performance of rotating machinery. Bearing failure can make machine

and reproduction in any medium, provided the original work is properly cited.

breakdown, lead to cost increase and even human death [1]. Hence, the development of strategies for monitoring bearing health conditions while in operation has been of significant importance.

was dissolved into 50 ml deionized water in a three-neck flask and 18 μl TGA was added under stirring, then the pH was adjusted to 10.5–11 with 1 M NaOH solution. After that,

proceeded for about 5 min, the flask was attached to a condenser and refluxed at 100°C under open-air condition. By controlling refluxing time, CdTe QDs with desired size and color can

To implement the temperature measurement, QDs sensors were fabricated by the layer-bylayer electrostatic self-assembly (LBL ESA) technique [17, 18]. Quartz slides of 200 μm thickness were placed into Piranha Solution for 30 min for cleaning before the deposition of the QD coatings. Then the Quartz slides were immersed into 50 ml of a solution of 1%wt PDDA with the pH adjusted to 8.0 for the absorption of polycation for 15 min. Next, the substrates were

QDs solution synthesized by ourselves for 10 min to absorb QDs followed by cleaned and dried. Repeating the above steps, a sensitive coating denoted by [PDDA/CdTe]n was formed, where n was chosen to be 15 for this paper. Afterward, the sensor films were cured at 150°C in a vacuum chamber. Thermal treatment yields a more repetitive and stable response when suffered to temperature change. When the fabrication was completed, the sensors were kept

In order to utilize QDs as sensors for the temperature measurement of rolling bearings, the temperature-dependent emission properties of the sensor was first characterized using the experimental setup showing in **Figure 1**. The QDs sensor is placed on a heater cell with a thermocouple to monitor its temperature. Since the QDs have a wide absorption spectrum, a mercury lamp at 365 nm is used as the excitation light source. The light generated by the lamp is reflected by a dichroic mirror and directed to the QDs sensor through a focus lens, which is also used to prevent the excitation signal from masking the fluorescence of the QDs sensor. The fluorescence is collected by the same lens and leaded to an Andor Shamrock SR-303i spectrograph. Finally, an Andor iDus DU420A-BV CCD camera together with a computer is used to analyze the optical response of the QD with respect to changes in the temperature values. By adjusting the heater cell, its temperature varies and changes in wavelength and intensity

To study the properties of the QDs sensor, the heater cell in **Figure 1** was adjusted to run several temperature cycles from room temperature to 70°C and back. **Figure 2(a)** shows the emission spectrum varies as the temperature increases and decreases. As it can be seen from the picture, the photoluminescence intensity decreases with the increase of temperature, while the peak wavelength red shifts and FWHM increases as the temperature increases. It means that there are three features that could be used to detect the variation of temperature. However, as the photoluminescence intensity is affected by the power of the excitation source as well as the distance between the focus lens and the QD sensor, it is not suitable for the temperature monitoring of rolling bearings, where slight or heavy vibration usually occurs. Besides, the average temperature sensitivity of the FWHM is generally small compared with peak

in darkness until the temperature response of emission spectrum was studied.

of the luminescence emission of the quantum dots are registered.

, which was dissolved in 50 ml deionized water was added into the above

was added into the precursor solution. After the reactions

CdTe Quantum Dot Fluorescence Thermometry of Rolling Bearing

http://dx.doi.org/10.5772/intechopen.70866

79

. Then, the slides were removed into the CdTe

0.04 mmol K<sup>2</sup>

be obtained.

**2.2. Sensor calibration**

TeO<sup>3</sup>

solution. Then, 80 mg of NaBH4

cleaned in deionized water and dried by N<sup>2</sup>

The contact friction between the inner component leads to large heat generation and elevated temperature, which could cause thinner lubricant film, higher asperity contact, and reduction of material properties. Thus, the temperature is considered to be one of the most important parameters affecting the service life and performance of a rolling element bearing component. However, because of the complex structure and extreme operating conditions, instrument for real-time, nonintrusive monitoring of bearing temperatures has been limited. This is particularly true for the rolling element of a bearing, whose temperature is often indirectly obtained from the measured temperature of outer raceway. Indirect measurements are known to be error-prone. Thus far, direct measurement of the temperature of the inner bearing components such as the inner raceway and cage has eluded researchers [2–6]. Joshi [2] has developed a battery-powered telemeter and a remotely powered telemeter to measure the cage temperature in a tapered roller bearing. Also, Jia et al. [6] used a remotely powered wireless temperature sensor to monitor the cage temperature in real-time. However, the battery-powered telemeter has an extremely short functional life, and both the wireless ones are easily affected by the electromagnetic environment and are not suitable for high speed situations.

Recently, luminescent semiconductor nanocrystals, quantum dots, have attracted extensive attentions due to its unique optical properties and have been applied in light-emitting diodes, solar cells, and bio-labeling [7–9]. These semiconductor nanoparticles offer several advantages including narrow fluorescence emission, tunable wavelength, relatively high quantum yield, outstanding photo stability as well as flexible photo excitation. It also has been reported that the behavior of the luminescent properties of quantum dots with temperature has suitable characteristics for application as temperature probes [10–15]. The luminescence properties, such as the excited state lifetime, emission intensity, and peak wavelength, have been proven to be good indicators of temperature. The wide range of temperature in which luminescent properties change makes them very suitable for temperature sensing applications.

This paper presents a study on the use of CdTe quantum dots as thermal sensor to measure the temperature of inner raceway of rolling bearing while in operation. The quantum dots sensor film is fabricated by means of layer-by-layer electrostatic self-assembly method on an ultrathin glass slice. The peak wavelength shows a linear and reversible relationship as temperature changes. The factors that would affect the acquired fluorescence signal have been studied. Results show that this method is feasible and effective for the temperature measurement of rolling bearing, especially in very high speed conditions.

### **2. Sensor preparation and calibration**

#### **2.1. Sensor preparation**

Colloidal solutions of CdTe quantum dots stabilized by TGA were prepared according to the method given by the previously reported paper [16]. Typically, 0.2 mmol Cd(CH<sup>3</sup> COO)<sup>2</sup> ·2H<sup>2</sup> O was dissolved into 50 ml deionized water in a three-neck flask and 18 μl TGA was added under stirring, then the pH was adjusted to 10.5–11 with 1 M NaOH solution. After that, 0.04 mmol K<sup>2</sup> TeO<sup>3</sup> , which was dissolved in 50 ml deionized water was added into the above solution. Then, 80 mg of NaBH4 was added into the precursor solution. After the reactions proceeded for about 5 min, the flask was attached to a condenser and refluxed at 100°C under open-air condition. By controlling refluxing time, CdTe QDs with desired size and color can be obtained.

To implement the temperature measurement, QDs sensors were fabricated by the layer-bylayer electrostatic self-assembly (LBL ESA) technique [17, 18]. Quartz slides of 200 μm thickness were placed into Piranha Solution for 30 min for cleaning before the deposition of the QD coatings. Then the Quartz slides were immersed into 50 ml of a solution of 1%wt PDDA with the pH adjusted to 8.0 for the absorption of polycation for 15 min. Next, the substrates were cleaned in deionized water and dried by N<sup>2</sup> . Then, the slides were removed into the CdTe QDs solution synthesized by ourselves for 10 min to absorb QDs followed by cleaned and dried. Repeating the above steps, a sensitive coating denoted by [PDDA/CdTe]n was formed, where n was chosen to be 15 for this paper. Afterward, the sensor films were cured at 150°C in a vacuum chamber. Thermal treatment yields a more repetitive and stable response when suffered to temperature change. When the fabrication was completed, the sensors were kept in darkness until the temperature response of emission spectrum was studied.

#### **2.2. Sensor calibration**

breakdown, lead to cost increase and even human death [1]. Hence, the development of strategies for monitoring bearing health conditions while in operation has been of significant

The contact friction between the inner component leads to large heat generation and elevated temperature, which could cause thinner lubricant film, higher asperity contact, and reduction of material properties. Thus, the temperature is considered to be one of the most important parameters affecting the service life and performance of a rolling element bearing component. However, because of the complex structure and extreme operating conditions, instrument for real-time, nonintrusive monitoring of bearing temperatures has been limited. This is particularly true for the rolling element of a bearing, whose temperature is often indirectly obtained from the measured temperature of outer raceway. Indirect measurements are known to be error-prone. Thus far, direct measurement of the temperature of the inner bearing components such as the inner raceway and cage has eluded researchers [2–6]. Joshi [2] has developed a battery-powered telemeter and a remotely powered telemeter to measure the cage temperature in a tapered roller bearing. Also, Jia et al. [6] used a remotely powered wireless temperature sensor to monitor the cage temperature in real-time. However, the battery-powered telemeter has an extremely short functional life, and both the wireless ones are easily affected

by the electromagnetic environment and are not suitable for high speed situations.

properties change makes them very suitable for temperature sensing applications.

ment of rolling bearing, especially in very high speed conditions.

**2. Sensor preparation and calibration**

**2.1. Sensor preparation**

Recently, luminescent semiconductor nanocrystals, quantum dots, have attracted extensive attentions due to its unique optical properties and have been applied in light-emitting diodes, solar cells, and bio-labeling [7–9]. These semiconductor nanoparticles offer several advantages including narrow fluorescence emission, tunable wavelength, relatively high quantum yield, outstanding photo stability as well as flexible photo excitation. It also has been reported that the behavior of the luminescent properties of quantum dots with temperature has suitable characteristics for application as temperature probes [10–15]. The luminescence properties, such as the excited state lifetime, emission intensity, and peak wavelength, have been proven to be good indicators of temperature. The wide range of temperature in which luminescent

This paper presents a study on the use of CdTe quantum dots as thermal sensor to measure the temperature of inner raceway of rolling bearing while in operation. The quantum dots sensor film is fabricated by means of layer-by-layer electrostatic self-assembly method on an ultrathin glass slice. The peak wavelength shows a linear and reversible relationship as temperature changes. The factors that would affect the acquired fluorescence signal have been studied. Results show that this method is feasible and effective for the temperature measure-

Colloidal solutions of CdTe quantum dots stabilized by TGA were prepared according to the

COO)<sup>2</sup>

·2H<sup>2</sup> O

method given by the previously reported paper [16]. Typically, 0.2 mmol Cd(CH<sup>3</sup>

importance.

78 Nonmagnetic and Magnetic Quantum Dots

In order to utilize QDs as sensors for the temperature measurement of rolling bearings, the temperature-dependent emission properties of the sensor was first characterized using the experimental setup showing in **Figure 1**. The QDs sensor is placed on a heater cell with a thermocouple to monitor its temperature. Since the QDs have a wide absorption spectrum, a mercury lamp at 365 nm is used as the excitation light source. The light generated by the lamp is reflected by a dichroic mirror and directed to the QDs sensor through a focus lens, which is also used to prevent the excitation signal from masking the fluorescence of the QDs sensor. The fluorescence is collected by the same lens and leaded to an Andor Shamrock SR-303i spectrograph. Finally, an Andor iDus DU420A-BV CCD camera together with a computer is used to analyze the optical response of the QD with respect to changes in the temperature values. By adjusting the heater cell, its temperature varies and changes in wavelength and intensity of the luminescence emission of the quantum dots are registered.

To study the properties of the QDs sensor, the heater cell in **Figure 1** was adjusted to run several temperature cycles from room temperature to 70°C and back. **Figure 2(a)** shows the emission spectrum varies as the temperature increases and decreases. As it can be seen from the picture, the photoluminescence intensity decreases with the increase of temperature, while the peak wavelength red shifts and FWHM increases as the temperature increases. It means that there are three features that could be used to detect the variation of temperature. However, as the photoluminescence intensity is affected by the power of the excitation source as well as the distance between the focus lens and the QD sensor, it is not suitable for the temperature monitoring of rolling bearings, where slight or heavy vibration usually occurs. Besides, the average temperature sensitivity of the FWHM is generally small compared with peak

wavelength. Therefore, here we choose the peak wavelength as the parameter for temperature measurement of rolling bearings. The temperature dependence of peak wavelength in three thermal cycles is depicted in **Figure 2(b)**. It is shown that the response of emission spectrum peak wavelength is linear and reversible as the temperature changes. The R square value with respect to the linear approximation is about 0.994 for both cases. And the sensitivity shown by the sensor is around 0.14 nm/°C. The wavelength shift can be explained by the fact that heat expands the crystalline of the quantum dots material and causes a change in the band gap [13],

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For the temperature measurement of rolling bearings, the influence of rotating to the fluorescence signal acquisition was studied first. The same setup shown in **Figure 1** was used with some modification. The QD sensor was mounted to a disk, which was drove by a motor, with its rotating speed detected by a photoelectric tachometer. The excitation light generated by the mercury lamp continuously is illuminated on the disk. What is different from the calibration state is that the QD sensor is excited at intervals when the disk is rotated with a certain speed. And the total amount of fluorescence detected by the CCD changes in time as the QD sensor moves into, though, and out of the focus lens's field of view, as is shown in **Figure 3**. Generally, a specific exposure time is needed to collect fluorescence when the spectrograph is set to acquire the emission spectrum. Assuming that the exposure time is T, the angular velocity of the disk is ω, and the central angle of the QD sensor to the disk is α, the total time the QD sensor excited within the exposure time is: t = Tα/2π. This means that there is no difference whether the QD sensor is stable or in rotation, but the exposure time multiplies a factor of α/2π. And it is proved by the results shown in **Figure 4**. We studied the

**Figure 3.** The rotation causes the QDs sensor to move with respect to the fluorescence collection input aperture.

which is only decided by the properties of the quantum dots.

**3. Rolling bearing temperature measurement**

**Figure 1.** Schematic representation of the experimental setup.

**Figure 2.** (a) Behavior of emission spectrum of QDs with different temperatures in heating and cooling. (b) Dependence of the emission peak wavelength with respect to the temperature in three thermal cycles.

wavelength. Therefore, here we choose the peak wavelength as the parameter for temperature measurement of rolling bearings. The temperature dependence of peak wavelength in three thermal cycles is depicted in **Figure 2(b)**. It is shown that the response of emission spectrum peak wavelength is linear and reversible as the temperature changes. The R square value with respect to the linear approximation is about 0.994 for both cases. And the sensitivity shown by the sensor is around 0.14 nm/°C. The wavelength shift can be explained by the fact that heat expands the crystalline of the quantum dots material and causes a change in the band gap [13], which is only decided by the properties of the quantum dots.

## **3. Rolling bearing temperature measurement**

For the temperature measurement of rolling bearings, the influence of rotating to the fluorescence signal acquisition was studied first. The same setup shown in **Figure 1** was used with some modification. The QD sensor was mounted to a disk, which was drove by a motor, with its rotating speed detected by a photoelectric tachometer. The excitation light generated by the mercury lamp continuously is illuminated on the disk. What is different from the calibration state is that the QD sensor is excited at intervals when the disk is rotated with a certain speed. And the total amount of fluorescence detected by the CCD changes in time as the QD sensor moves into, though, and out of the focus lens's field of view, as is shown in **Figure 3**.

Generally, a specific exposure time is needed to collect fluorescence when the spectrograph is set to acquire the emission spectrum. Assuming that the exposure time is T, the angular velocity of the disk is ω, and the central angle of the QD sensor to the disk is α, the total time the QD sensor excited within the exposure time is: t = Tα/2π. This means that there is no difference whether the QD sensor is stable or in rotation, but the exposure time multiplies a factor of α/2π. And it is proved by the results shown in **Figure 4**. We studied the

**Figure 3.** The rotation causes the QDs sensor to move with respect to the fluorescence collection input aperture.

**Figure 2.** (a) Behavior of emission spectrum of QDs with different temperatures in heating and cooling. (b) Dependence

of the emission peak wavelength with respect to the temperature in three thermal cycles.

**Figure 1.** Schematic representation of the experimental setup.

80 Nonmagnetic and Magnetic Quantum Dots

effect under different speeds: 650, 1525, 2515, and 3012 r/min, with the exposure time of the spectrograph set at 500 ms. There are 20 spectrum lines captured every 1 min for different speeds. As is shown in the picture, the emission intensity acquired at different speed keeps nearly unchanged from 650 to 3012 r/min, which means that the rotating speed has no effect on the fluorescence signal. Therefore, we could take temperature measurement of rolling bearings by a common fluorescence spectrum measurement system, and the utilization of QD as temperature senor for rolling bearing thermometry could be applied in very high speed conditions.

#### **3.1. Experimental setup**

A new optical fiber fluorescence spectrum detecting system was established to measure the temperature of the inner raceway of the ball bearing of a bearing test rig, which is depicted in **Figure 5**. The bearing test rig was built on a rigid platform and one of its bearing was chosen as testing target. The cover of the bearing block was taken away for convenient measuring. The QDs sensor was mounted to the inner raceway by an epoxy binding agent. An optic fiber with a fluorescent probe was used to conduct the excitation light and collect the fluorescence. By setting the bearing test rig operating at different constant speeds, the fluorescence spectrum of the CdTe film sensor was acquired by the spectrograph (QEpro6500) every 1 min from the moment the setup started until running 20 min, which is thus used to obtain information

on the temperature variation of inner raceway. To verify the accuracy of the temperature obtained by quantum dots sensor film, a thermocouple was used to test the temperature of

**Figure 5.** Rolling bearing temperature measurement system. (1) Testing bearing; (2) QDs sensor; (3) optic fiber; (4)

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405 nm laser; (5) QEPro spectrograph; (6) computer. Inset: No excitation light irradiates on QDs sensor.

Temperature measurements were conducted at shaft speed of 1000, 1600, 2200, and 2800 r/ min, respectively without any load. **Figure 6** depicts 20 fluorescence spectrum lines acquired every 1 min from the moment the setup started until running 20 min. As we can see from the picture, for different constant shaft speeds, the variation tendency of the fluorescence spectrum is similar to each other: the photoluminescence intensity decreases and the peak wavelength red shifts as time goes on, indicating that the temperature of the bearing inner raceway rises up in the testing 20 min. Comparing the fluorescence spectrum obtained at different shaft speeds, the red shift of the peak wavelength increases as the shaft speed increases, which means that the heat generation and the temperature rise vary with speed. The variation of bearing inner raceway temperature measured by QDs sensor with time at four different conditions of shaft speed is shown in **Figure 7**. It clearly shows that there are more heat gen-

The accuracy of the temperature obtained by the QDs sensor was studied by comparing with the temperature tested by a thermocouple. The thermocouple was used to test the temperature of the inner raceway at the point near QDs sensor right before and after the operation. Results show that the temperature acquired by these two methods has good consistency. The temperature of the inner raceway at 2800 r/min was about 26.1 and 50.2°C before and after the operation of the test rig by thermocouple, while the first and last obtained spectrum line indicate that the temperature was 28.7 and 51.4°C, respectively. The error of the temperature rise between these two methods is 5.8%. Because of the different measuring time and other affects, the temperature measured by two methods shows little difference, but the temperature rise

the inner raceway right before and after the operation.

eration and large temperature rise at higher speed.

error all blow 10% for different shaft speed.

**3.2. Results and discussion**

**Figure 4.** The influence of rotating speed of fluorescence signal.

**Figure 5.** Rolling bearing temperature measurement system. (1) Testing bearing; (2) QDs sensor; (3) optic fiber; (4) 405 nm laser; (5) QEPro spectrograph; (6) computer. Inset: No excitation light irradiates on QDs sensor.

on the temperature variation of inner raceway. To verify the accuracy of the temperature obtained by quantum dots sensor film, a thermocouple was used to test the temperature of the inner raceway right before and after the operation.

#### **3.2. Results and discussion**

effect under different speeds: 650, 1525, 2515, and 3012 r/min, with the exposure time of the spectrograph set at 500 ms. There are 20 spectrum lines captured every 1 min for different speeds. As is shown in the picture, the emission intensity acquired at different speed keeps nearly unchanged from 650 to 3012 r/min, which means that the rotating speed has no effect on the fluorescence signal. Therefore, we could take temperature measurement of rolling bearings by a common fluorescence spectrum measurement system, and the utilization of QD as temperature senor for rolling bearing thermometry could be applied in very

A new optical fiber fluorescence spectrum detecting system was established to measure the temperature of the inner raceway of the ball bearing of a bearing test rig, which is depicted in **Figure 5**. The bearing test rig was built on a rigid platform and one of its bearing was chosen as testing target. The cover of the bearing block was taken away for convenient measuring. The QDs sensor was mounted to the inner raceway by an epoxy binding agent. An optic fiber with a fluorescent probe was used to conduct the excitation light and collect the fluorescence. By setting the bearing test rig operating at different constant speeds, the fluorescence spectrum of the CdTe film sensor was acquired by the spectrograph (QEpro6500) every 1 min from the moment the setup started until running 20 min, which is thus used to obtain information

high speed conditions.

82 Nonmagnetic and Magnetic Quantum Dots

**3.1. Experimental setup**

**Figure 4.** The influence of rotating speed of fluorescence signal.

Temperature measurements were conducted at shaft speed of 1000, 1600, 2200, and 2800 r/ min, respectively without any load. **Figure 6** depicts 20 fluorescence spectrum lines acquired every 1 min from the moment the setup started until running 20 min. As we can see from the picture, for different constant shaft speeds, the variation tendency of the fluorescence spectrum is similar to each other: the photoluminescence intensity decreases and the peak wavelength red shifts as time goes on, indicating that the temperature of the bearing inner raceway rises up in the testing 20 min. Comparing the fluorescence spectrum obtained at different shaft speeds, the red shift of the peak wavelength increases as the shaft speed increases, which means that the heat generation and the temperature rise vary with speed. The variation of bearing inner raceway temperature measured by QDs sensor with time at four different conditions of shaft speed is shown in **Figure 7**. It clearly shows that there are more heat generation and large temperature rise at higher speed.

The accuracy of the temperature obtained by the QDs sensor was studied by comparing with the temperature tested by a thermocouple. The thermocouple was used to test the temperature of the inner raceway at the point near QDs sensor right before and after the operation. Results show that the temperature acquired by these two methods has good consistency. The temperature of the inner raceway at 2800 r/min was about 26.1 and 50.2°C before and after the operation of the test rig by thermocouple, while the first and last obtained spectrum line indicate that the temperature was 28.7 and 51.4°C, respectively. The error of the temperature rise between these two methods is 5.8%. Because of the different measuring time and other affects, the temperature measured by two methods shows little difference, but the temperature rise error all blow 10% for different shaft speed.

**4. Conclusions**

**Acknowledgements**

**Author details**

\* and Bei Yan<sup>2</sup>

Jiaotong University, Xi'an, China

Ke Yan1

Xi'an, China

**References**

In this paper, we have proposed a quantum dot fluorescence-based thermometry method for rolling element bearings. Temperature sensor has been fabricated by the deposition of quantum dot films on quartz slide by means of layer-by-layer technique. It has been shown that the emission peak wavelength of the QD sensor has a very linear relationship with the temperature, making it applicable of noncontact temperature measurement of rotating surface. We have managed to take temperature measurement of rolling bearings by a common fluorescence spectrum measurement system. The rotating speed shows no effect on the acquired fluorescence signal, which makes the QD fluorescence-based thermometry method, suitable for very high rotating speed temperature measurement. The practical experiment proves that the CdTe quantum dot fluorescence thermometry could be a feasible and accurate tempera-

CdTe Quantum Dot Fluorescence Thermometry of Rolling Bearing

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85

The research work is financially supported by the Youth Project of National Natural Science Foundation of China (Grant No. 51405375) and the China Postdoctoral Science Foundation.

1 Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi'an

2 State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University,

[2] Sadeghi F, Marble S, Joshi A. Bearing cage temperature measurement using radio telemetry. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of

[3] Henao-Sepulveda JA, Toledo-Quiñones M, Jia Y. Contactless monitoring of ball bearing temperature. Proceedings of the Instrumentation and Measurement Technology Conference, IMTC 2005. Proceedings of the IEEE. Institute of Electrical and Electronics

[1] Harris TA, Kotzalas MN. Rolling Bearing Analysis. New York: Wiley; 2001

ture measurement method of bearing inner raceway in operation.

\*Address all correspondence to: yanke@mail.xjtu.edu.cn

Engineering Tribology. 2001;**215**(5):471-481

Engineers Inc,USA: 2005. pp. 1571-1573

**Figure 6.** Fluorescence spectrum of different shaft speed in 20 min.

**Figure 7.** The temperature of inner raceway as a function of time at different speed.

## **4. Conclusions**

In this paper, we have proposed a quantum dot fluorescence-based thermometry method for rolling element bearings. Temperature sensor has been fabricated by the deposition of quantum dot films on quartz slide by means of layer-by-layer technique. It has been shown that the emission peak wavelength of the QD sensor has a very linear relationship with the temperature, making it applicable of noncontact temperature measurement of rotating surface. We have managed to take temperature measurement of rolling bearings by a common fluorescence spectrum measurement system. The rotating speed shows no effect on the acquired fluorescence signal, which makes the QD fluorescence-based thermometry method, suitable for very high rotating speed temperature measurement. The practical experiment proves that the CdTe quantum dot fluorescence thermometry could be a feasible and accurate temperature measurement method of bearing inner raceway in operation.

## **Acknowledgements**

The research work is financially supported by the Youth Project of National Natural Science Foundation of China (Grant No. 51405375) and the China Postdoctoral Science Foundation.

## **Author details**

Ke Yan1 \* and Bei Yan<sup>2</sup>

\*Address all correspondence to: yanke@mail.xjtu.edu.cn

1 Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi'an Jiaotong University, Xi'an, China

2 State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, China

## **References**

**Figure 7.** The temperature of inner raceway as a function of time at different speed.

**Figure 6.** Fluorescence spectrum of different shaft speed in 20 min.

84 Nonmagnetic and Magnetic Quantum Dots


[4] Nickel DA, Sadeghi F. In situ tribocomponent temperature measurement using a radio telemeter. Tribology Transactions. 1997;**40**(3):514-520

**Chapter 6**

**Provisional chapter**

**Quantum Dots-Based Nano-Coatings for Inhibition of**

 Infection of implants by microbial biofilm is chiefly caused by *Staphylococci, Pseudomonas* and *Candida* species. The growth of microbes by forming biofilms offers them protection from antibiotics, drugs and host defense mechanisms. The eradication of biofilms from implants and medical devices is difficult because of the protection by the biofilm forming pathogenic microbes. Hence, researches are focused on development of antibiofilm materials, which are basically constituted of antimicrobial substances or antimicrobial coatings. Nanomaterialbased coatings offer a promising solution in this regard. Quantum dots (QDs) are the group of semiconductor nanoparticles with high photoluminescent properties compared to conventional organic fluorophores. Thus, drug-conjugated QDs can be a promising alternative for biofilm treatment, and these can serve as excellent alternatives for the mitigation of recalcitrant biomaterial-associated infections caused by resistant strains. Furthermore, their use as antibiofilm coating would avoid the dispersion of antimicrobial agents in the surrounding cells and tissues, thereby minimizing the risks of developing microbial resistivity.

**Keywords:** quantum dots, microbial biofilms, fluorescence, infections, antibiofilm materials

Quantum dots (QDs) represent a class of colloidal semiconductor nanocrystals having fluorescent properties that absorb photons at a particular (lower) wavelength and emit at a higher wavelength. These QDs are basically composed of a core and corona layer. The photoluminescence emission wavelength of QDs is directly proportional to its size. The core of the QDs may contain one or more heavy elements such as cadmium, selenium, zinc or tellurium. QDs possess significant superiority over the conventional fluorophores

**Quantum Dots-Based Nano-Coatings for Inhibition of** 

DOI: 10.5772/intechopen.70785

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Microbial Biofilms: A Mini Review**

**Microbial Biofilms: A Mini Review**

Eepsita Priyadarshini, Kamla Rawat and

Eepsita Priyadarshini, Kamla Rawat and

http://dx.doi.org/10.5772/intechopen.70785

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Himadri Bihari Bohidar

**Abstract**

**1. Introduction**

Himadri Bihari Bohidar


**Provisional chapter**

## **Quantum Dots-Based Nano-Coatings for Inhibition of Microbial Biofilms: A Mini Review Microbial Biofilms: A Mini Review**

**Quantum Dots-Based Nano-Coatings for Inhibition of** 

DOI: 10.5772/intechopen.70785

Eepsita Priyadarshini, Kamla Rawat and Himadri Bihari Bohidar Himadri Bihari Bohidar Additional information is available at the end of the chapter

Eepsita Priyadarshini, Kamla Rawat and

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70785

#### **Abstract**

[4] Nickel DA, Sadeghi F. In situ tribocomponent temperature measurement using a radio

[5] Scott S, Sadeghi F, Peroulis D. An inherently-robust 300 C MEMS temperature sensor for wireless health monitoring of ball and rolling element bearings. Proceedings of IEEE Conference on Sensors. Institute of Electrical and Electronics Engineers Inc, USA; 2009.

[6] Liu SC, Jia Y, Henao-Sepulveda J, Toledo-Quinones M. Wireless temperature sensor for bearing health monitoring. Proceedings of SPIE, Bellingham, WA. 2004;**5391**:368-376 [7] Wang XD, Wolfbeis OS, Meier RJ. Luminescent probes and sensors for temperature.

[8] Jorge P, Martins MA, Trindade T, Santos JL, Farahi F. Optical fiber sensing using quan-

[9] Costa-Fernández JM, Pereiro R, Sanz-Medel A. The use of luminescent quantum dots for

[10] Cutolo A, Bravo J, Goicoechea J, Corres JM, Arregui FJ, Matias IR, Culshaw B, López-Higuera JM. Fiber optic temperature sensor depositing quantum dots inside hollow core fibers using the layer by layer technique. Proceedings of SPIE - The International Society

[11] de Bastida G, Arregui FJ, Goicoechea J, Matias IR. Quantum dots-based optical fiber temperature sensors fabricated by layer-by-layer. Sensors Journal, IEEE. 2006;**6**(6):1378-1379

[12] Jorge PAS, Mayeh M, Benrashid R, Caldas P, Santos JL, Farahi F. Quantum dots as self-referenced optical fibre temperature probes for luminescent chemical sensors.

[13] Pugh-Thomas D, Walsh BM, Gupta MC. CdSe(ZnS) nanocomposite luminescent high

[14] Walker GW, Sundar VC, Rudzinski CM, Wun AW, Bawendi MG, Nocera DG. Quantum-

[15] Wang H-l, Yang A-J, Sui C-H. Luminescent high temperature sensor based on the CdSe/

[16] Wu S, Dou J, Zhang J, Zhang S. A simple and economical one-pot method to synthesize high-quality water soluble CdTe QDs. Journal of Materials Chemistry. 2012;**22**(29):14573

[17] Decher G. Fuzzy nanoassemblies: Toward layered polymeric multicomposites. Science.

[18] Crisp MT, Kotov NA. Preparation of nanoparticle coatings on surfaces of complex geom-

dot optical temperature probes. Applied Physics Letters. 2003;**83**(17):3555

ZnS quantum dot thin film. Optoelectronics Letters. 2013;**9**(6):421-424

optical sensing. TrAC Trends in Analytical Chemistry. 2006;**25**(3):207-218

telemeter. Tribology Transactions. 1997;**40**(3):514-520

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pp. 975-978

86 Nonmagnetic and Magnetic Quantum Dots

 Infection of implants by microbial biofilm is chiefly caused by *Staphylococci, Pseudomonas* and *Candida* species. The growth of microbes by forming biofilms offers them protection from antibiotics, drugs and host defense mechanisms. The eradication of biofilms from implants and medical devices is difficult because of the protection by the biofilm forming pathogenic microbes. Hence, researches are focused on development of antibiofilm materials, which are basically constituted of antimicrobial substances or antimicrobial coatings. Nanomaterialbased coatings offer a promising solution in this regard. Quantum dots (QDs) are the group of semiconductor nanoparticles with high photoluminescent properties compared to conventional organic fluorophores. Thus, drug-conjugated QDs can be a promising alternative for biofilm treatment, and these can serve as excellent alternatives for the mitigation of recalcitrant biomaterial-associated infections caused by resistant strains. Furthermore, their use as antibiofilm coating would avoid the dispersion of antimicrobial agents in the surrounding cells and tissues, thereby minimizing the risks of developing microbial resistivity.

**Keywords:** quantum dots, microbial biofilms, fluorescence, infections, antibiofilm materials

### **1. Introduction**

Quantum dots (QDs) represent a class of colloidal semiconductor nanocrystals having fluorescent properties that absorb photons at a particular (lower) wavelength and emit at a higher wavelength. These QDs are basically composed of a core and corona layer. The photoluminescence emission wavelength of QDs is directly proportional to its size. The core of the QDs may contain one or more heavy elements such as cadmium, selenium, zinc or tellurium. QDs possess significant superiority over the conventional fluorophores

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

in terms of physicochemical and fluorescent properties. The distinguishable fluorescent properties, smaller size, photostability, resistivity to metabolic degradation and capability of conjugation to ligands/biomolecules make QDs a superior choice for biological applications compared to conventional fluorophores.

targeted towards exploring novel biocompatible nanomaterials with effective antibiofilm and optical properties. QDs can be suitable alternatives because of their intriguing optical, fluorescence, high quantum yield, photostability and easy conjugation efficiency. QDs easily attach to microbial surface because of their small size and their dispersion stability is basically governed by colloidal theory [17]. These are excellent candidates in biomedical applications such as imaging, diagnosis and sensing and drug discovery. Developing QDs-based nanocomposites as coating materials on implants and catheters can thus combat pathogenic invasion and biofilm formation. QDs could be engineered with coating agents and conjugated with bioactive ligands or biorecognition elements for targeted treatment,

Quantum Dots-Based Nano-Coatings for Inhibition of Microbial Biofilms: A Mini Review

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89

Biofilm can be defined as microbial cells enclosed in an exopolysaccharide matrix and adhered to a cell surface. Formation of biofilms by bacteria and fungus is a defense strategy for protection from environment. Microbes secrete extracellular polymeric substances (EPS) that act as a primary scaffold for attachment to solid substrate [18] and its basic constituents are proteins, polysaccharides, nucleic acids with some lipids and humic substances [19]. Three-dimensional study of the EPS layer suggested that it forms a gel-like network wherein microbes are embedded and it also maintains the attachment of bacteria to the solid substrate [20]. Stability to the 3D structure of EPS is rendered by the hydrophobic interactions as well as van der Waals attraction between amino acids/peptides and cations such as Ca2+ and Mg2+ [21]. Biofilm formation and its structure depend on the environmental conditions to which the bacteria are exposed. When cells are in a nutrient stress condition, an increase in EPS secretion occurs, which promotes hydrophobic interactions to allow attachment to solid substrate [22]. It has been suggested that the presence of a high concentration of EPS negatively affects the diffusion of lipophilic compounds (such as

sanitizers, antibiotics and hydrocarbons), across the microbial cell surface [23, 24].

Among bacteria, *P. aeruginosa* is an opportunistic pathogen that causes a number of infections in humans. It develops resistance to antibiotics by forming biofilm matrices that comprise polysaccharide-EPS. It has been noticed that it forms and regulates biofilm via quorum sensing mechanism and therefore most of the researches have focused on disrupting the quorum sensing pathway [25]. Similarly, biofilm formation in *Staphylococcus epidermidis* has also been analyzed using different methods such as microtiter plate, congo red agar plate test and via molecular detection of the ica locus [26–28]. It was found that production of a slimy substance assisted in forming biofilm and was associated with virulence also. The production and formation of biofilm depend on the media constituents; however, the exact mechanism behind the formation of a mature biofilm is still being investigated. However, on the basis of in vitro experimental models, biofilm formation can be segregated into four different stages: (i) attachment of microbial cells to surface, (ii) formation of multi-layer structure via the accumulation and aggregation of cells, (iii) maturation of biofilm and (iv) detachment of cells from biofilm into planktonic state and initiation of a new biofilm cycle [29, 30]. The initial step of attachment is normally driven by hydrophobic, electrostatic and Lifshitze-van der Waals forces, and hence is nonspecific in

biofilm visualization, and inhibition.

**2. Biofilm formation, its mechanism and transmission**

In the last three decades, several microbes (fungi, yeast and bacteria) have emerged as major human pathogens and have been responsible for causing life threatening diseases especially in immunecompromised individuals and patients with serious medical issues [1]. The widespread and prolonged use of antifungal agents and drugs for treating the infection caused by the pathogens has resulted in increasing incidences of multidrug resistance (MDR). Additionally, several mutant strains have developed that show high resistance to the antifungal drugs being used [1]. For example, *Candida albicans*, a dimorphic opportunistic pathogen, occurs as a normal commensal in humans but becomes pathogenic in immunecompromised individuals. The azole resistive clinical isolates of *C. albicans* result in cross-resistance to several unrelated drugs and this arises because of the phenomenon of multidrug resistance (MDR) [2, 3]. Similarly, *Pseudomonas aeruginosa* and *Staphylococcus aureus* are the two most pathogenic bacteria known to cause severe infection and biofilm formation [4]. Several mechanisms are responsible for development of MDR, some of which involve an overexpression of drug efflux pumps encoding genes such as *CDR1* and *CDR2* belonging to ATP-binding cassette [2, 5, 6]; overexpression of the drug and *MDR1* belonging to the major facilitator superfamily transporters [3, 6] and overexpression of mutations in *ERG11* and encoding the target enzyme of azoles, lanosterol 14α-demethylase [7]. Hence, microbial infection has become a major problem with concerns focusing on those that have become resistant to antibiotics. Around 2 million people are affected annually with antibiotic-resistant bacteria of which approximately 23,000 people die as per the studies of U.S. Center for Disease Control and Prevention [8].

Microbial communities adhere to a solid surface especially in surface/water interference forming biofilms [9]. Microbes attach to the surface by means of extracellular polymeric substances (EPS), and this acts as their survival means against harsh environmental conditions. Biofilm formation is however associated with surface deterioration and corrosion. In addition, pathogenic microbes form biofilms on medical devices and implants, and this has become a great concern in the arena of healthcare. Biofilm also enhances microbial activity and provides protection against harsh environmental conditions such as drugs, antibiotics and common sanitizers. Because of the emerging conditions of MDR, there is a demand for developing new drugs, antimicrobial agents and modifiers capable of inhibiting microbial growth and biofilm formation. With the necessity of developing antimicrobial agents with diverse functionality and ability to kill both strains of bacteria, nanomaterials have been widely investigated in this regard. Silver nanoparticles [10], copper oxide nanoparticles [11–13], metal oxide nanoparticles [12, 13] and even carbon nanomaterials [14] have been reported for their excellent antimicrobial efficiency. Among these, silver nanoparticles have been extensively used as antimicrobial and antibiofilm agents due to their broad spectrum antimicrobial activity, multiple cellular targets and minimum host toxicity. However, high concentration of silver is toxic to humans and its persistent use causes argyrosis and argia [15, 16]. Hence, the demand is for exploring novel nanomaterials with effective antimicrobial and antibiofilm properties along with biocompatibility. Therefore, the requirement must be targeted towards exploring novel biocompatible nanomaterials with effective antibiofilm and optical properties. QDs can be suitable alternatives because of their intriguing optical, fluorescence, high quantum yield, photostability and easy conjugation efficiency. QDs easily attach to microbial surface because of their small size and their dispersion stability is basically governed by colloidal theory [17]. These are excellent candidates in biomedical applications such as imaging, diagnosis and sensing and drug discovery. Developing QDs-based nanocomposites as coating materials on implants and catheters can thus combat pathogenic invasion and biofilm formation. QDs could be engineered with coating agents and conjugated with bioactive ligands or biorecognition elements for targeted treatment, biofilm visualization, and inhibition.

## **2. Biofilm formation, its mechanism and transmission**

in terms of physicochemical and fluorescent properties. The distinguishable fluorescent properties, smaller size, photostability, resistivity to metabolic degradation and capability of conjugation to ligands/biomolecules make QDs a superior choice for biological applica-

In the last three decades, several microbes (fungi, yeast and bacteria) have emerged as major human pathogens and have been responsible for causing life threatening diseases especially in immunecompromised individuals and patients with serious medical issues [1]. The widespread and prolonged use of antifungal agents and drugs for treating the infection caused by the pathogens has resulted in increasing incidences of multidrug resistance (MDR). Additionally, several mutant strains have developed that show high resistance to the antifungal drugs being used [1]. For example, *Candida albicans*, a dimorphic opportunistic pathogen, occurs as a normal commensal in humans but becomes pathogenic in immunecompromised individuals. The azole resistive clinical isolates of *C. albicans* result in cross-resistance to several unrelated drugs and this arises because of the phenomenon of multidrug resistance (MDR) [2, 3]. Similarly, *Pseudomonas aeruginosa* and *Staphylococcus aureus* are the two most pathogenic bacteria known to cause severe infection and biofilm formation [4]. Several mechanisms are responsible for development of MDR, some of which involve an overexpression of drug efflux pumps encoding genes such as *CDR1* and *CDR2* belonging to ATP-binding cassette [2, 5, 6]; overexpression of the drug and *MDR1* belonging to the major facilitator superfamily transporters [3, 6] and overexpression of mutations in *ERG11* and encoding the target enzyme of azoles, lanosterol 14α-demethylase [7]. Hence, microbial infection has become a major problem with concerns focusing on those that have become resistant to antibiotics. Around 2 million people are affected annually with antibiotic-resistant bacteria of which approximately 23,000 people die as per the studies of U.S. Center for Disease Control and Prevention [8].

Microbial communities adhere to a solid surface especially in surface/water interference forming biofilms [9]. Microbes attach to the surface by means of extracellular polymeric substances (EPS), and this acts as their survival means against harsh environmental conditions. Biofilm formation is however associated with surface deterioration and corrosion. In addition, pathogenic microbes form biofilms on medical devices and implants, and this has become a great concern in the arena of healthcare. Biofilm also enhances microbial activity and provides protection against harsh environmental conditions such as drugs, antibiotics and common sanitizers. Because of the emerging conditions of MDR, there is a demand for developing new drugs, antimicrobial agents and modifiers capable of inhibiting microbial growth and biofilm formation. With the necessity of developing antimicrobial agents with diverse functionality and ability to kill both strains of bacteria, nanomaterials have been widely investigated in this regard. Silver nanoparticles [10], copper oxide nanoparticles [11–13], metal oxide nanoparticles [12, 13] and even carbon nanomaterials [14] have been reported for their excellent antimicrobial efficiency. Among these, silver nanoparticles have been extensively used as antimicrobial and antibiofilm agents due to their broad spectrum antimicrobial activity, multiple cellular targets and minimum host toxicity. However, high concentration of silver is toxic to humans and its persistent use causes argyrosis and argia [15, 16]. Hence, the demand is for exploring novel nanomaterials with effective antimicrobial and antibiofilm properties along with biocompatibility. Therefore, the requirement must be

tions compared to conventional fluorophores.

88 Nonmagnetic and Magnetic Quantum Dots

Biofilm can be defined as microbial cells enclosed in an exopolysaccharide matrix and adhered to a cell surface. Formation of biofilms by bacteria and fungus is a defense strategy for protection from environment. Microbes secrete extracellular polymeric substances (EPS) that act as a primary scaffold for attachment to solid substrate [18] and its basic constituents are proteins, polysaccharides, nucleic acids with some lipids and humic substances [19]. Three-dimensional study of the EPS layer suggested that it forms a gel-like network wherein microbes are embedded and it also maintains the attachment of bacteria to the solid substrate [20]. Stability to the 3D structure of EPS is rendered by the hydrophobic interactions as well as van der Waals attraction between amino acids/peptides and cations such as Ca2+ and Mg2+ [21]. Biofilm formation and its structure depend on the environmental conditions to which the bacteria are exposed. When cells are in a nutrient stress condition, an increase in EPS secretion occurs, which promotes hydrophobic interactions to allow attachment to solid substrate [22]. It has been suggested that the presence of a high concentration of EPS negatively affects the diffusion of lipophilic compounds (such as sanitizers, antibiotics and hydrocarbons), across the microbial cell surface [23, 24].

Among bacteria, *P. aeruginosa* is an opportunistic pathogen that causes a number of infections in humans. It develops resistance to antibiotics by forming biofilm matrices that comprise polysaccharide-EPS. It has been noticed that it forms and regulates biofilm via quorum sensing mechanism and therefore most of the researches have focused on disrupting the quorum sensing pathway [25]. Similarly, biofilm formation in *Staphylococcus epidermidis* has also been analyzed using different methods such as microtiter plate, congo red agar plate test and via molecular detection of the ica locus [26–28]. It was found that production of a slimy substance assisted in forming biofilm and was associated with virulence also. The production and formation of biofilm depend on the media constituents; however, the exact mechanism behind the formation of a mature biofilm is still being investigated. However, on the basis of in vitro experimental models, biofilm formation can be segregated into four different stages: (i) attachment of microbial cells to surface, (ii) formation of multi-layer structure via the accumulation and aggregation of cells, (iii) maturation of biofilm and (iv) detachment of cells from biofilm into planktonic state and initiation of a new biofilm cycle [29, 30]. The initial step of attachment is normally driven by hydrophobic, electrostatic and Lifshitze-van der Waals forces, and hence is nonspecific in

forms (silver, copper, gold etc.) have been used as antimicrobial agents, there efficiency is diminishing due to MDR. Investigations on the antibacterial and antifungal property of QDs have been conducted, which suggests that they can serve as excellent candidates for biomedi-

Quantum Dots-Based Nano-Coatings for Inhibition of Microbial Biofilms: A Mini Review

http://dx.doi.org/10.5772/intechopen.70785

Aqueous solubility and compatibility make graphene quantum dots (GQDs) useful in biomedicine. GQDs are reported to be biocompatible at cellular levels investigated via WST-1 assay, LDH production, ROS generation and *in vitro* and *in vivo* distribution [40]. GQDs also possess antibacterial property against *Escherichia coli* and *S. aureus*, and GQDs with low dose

of these particles. The designed band-aids showed a good anti-disinfectant property. They analyzed the effect of formed GQDs on biofilm formation and destruction and observed a reduction in biofilm formation by *S. aureus* at 100 μg/mL and 100 mM of GQDs and H<sup>2</sup>

concentration, respectively. Furthermore, they also observed that GQDs alone also showed antibiofilm properties [41]. The studies thus suggested that appropriately designed GQDs had the ability to breakdown existing biofilms and simultaneously prevented the formation of new ones. Habiba et al. suggested the antimicrobial property of silver-graphene quantum dots against *P. aeruginosa* and *S. aureus*. They observed a synergistic effect between silver nanoparticles and GQDs with 25 and 50 g/ml of silver-graphene quantum dots inhibiting *S. aureus* and *P. aeruginosa* growth, respectively. Thus, the potential applicability of Ag-GQDs as fabrication

Furthermore, the use of semiconductor QDs will allow visualization of biofilm inhibition due to their fluorescent properties. The current methods being used for biofilm analysis are SEM, AFM, MRI and Raman spectroscopy that require lengthy and costly procedures apart from sample modulation, which sometimes provide partial details of the samples concerned [43, 44]. Other than this, conventional fluorescent dyes conjugated with carbohydrate recognition elements are used for biofilm analysis via confocal laser microscopy [45]. However, the use of a synthetic complex is sometimes toxic to cells thereby preventing in situ analysis. Therefore, QDs can be an exceptional solution for this. Moreover, amphiphilic carbon dots (CDs) have been shown to penetrate the EPS layer of *P. aeruginosa*, allowing direct visualization of its architecture, growth and how external agents affect its inhibition. The hydrocarbon side chains of CDs dock to the EPS network resulting in making the EPS scaffold highly fluorescent [46]. In yet another study, QDs with two varied surface chemistry [−COOH and polyethylene glycol (PEG) modified] were analyzed for their mobility and distribution in *P. aeruginosa* PAO1 biofilms. It was inferred that the QDs did not penetrate the bacterial cell but did colocalize with EPS matrix of the biofilm. While surface functionalization and QDs flow rate did not show any distinctive difference, analysis of center of density suggested that QDs with –COOH surface groups diffused easily compared to PEGlyated QDs. Biofilms treated with PEGlyated QDs had rough polysaccharide layers and cell distribution compared to –COOH functionalized QDs. It was thus concluded that treatment with nanomaterials can result in varying the structural parameters of biofilm [47]. The fluorescent property of QDs would thus allow recognition of biofilm formation at different growth stages and environmental conditions. Additionally, spectroscopic analysis can also be performed, which would allow better understanding of the phenomenon of binding of QDs to EPS. Conjugated QDs have also been used for biofilm imaging analysis. In a study, CdTe


O2

91

cal applications because of their solubility and biocompatibility.

and antibacterial coating agents was clearly established [42].

of H2 O2

**Figure 1.** Schematic illustration of the different stages involved in biofilm formation and detachment.

nature. Additionally, certain specific proteins also assist in binding of the microbes to the surfaces [30]. The second step of accumulation is mediated via microbial surface components that recognize adhesive matrix molecules and occurs via an active process. This step involves the establishment of biofilm on the microbial surface. This process is followed by the maturation step. In this step, the characteristic features of the biofilm are formed on basis of specific microbial type. In the final step, where a new phase of invasion is initiated involves the detachment and dispersion of the microbes [31, 32]. **Figure 1** shows the schematic illustration of the different stages in the cycle of biofilm formation and detachment.

*P. aeruginosa*, *Vibrio cholerae* and some *mycobacterial* species are the common human pathogens that form biofilms and hence have the possibility of infecting humans. There are several mechanisms via which pathogenic microbes in the biofilm can initiate an infection. The seeding dispersal of a large number of pathogenic cells is one of the possible mechanisms that can initiate an infection, as the microbes are not sessile in a biofilm and hence can easily detach and initiate an infection. Secondly, the virulent phenotypes present in a biofilm can expand their colony and initiate infection. This is highly possible as biofilm has a huge heterogeneity in its phenotypical constitution [33, 34]. In addition to these, several other mechanisms have been hypothesized that could possibly allow the survival of a pathogenic organism and its transmission. For example, the detachment of pathogenic microbes from the biofilm, quorum sensing [35], co-aggregation and auto-aggregation [35, 36], modification in biosynthesis of EPS and metabolic pathways and genetic mutations [37] are important issues. However, the complete understanding of the mechanism of biofilm formation and virulence requires complete analysis of the pathogen's life cycle, environmental parameters and the different phenotypes.

## **3. Role of QDs in inhibiting biofilm formation**

Biomedical implants are a necessity in modern health care; biofilm formation on these implants and devices is a major cause of their failure. Mostly *S. epidermidis* and *S. aureus* are observed in contaminated biomedical implants and devices [38]. Biofilms formed on implants and medical devices are difficult to remove as they are protected by exopolymeric matrix secreted by the pathogenic microbe [39]. Although a number of metal and their nanosize forms (silver, copper, gold etc.) have been used as antimicrobial agents, there efficiency is diminishing due to MDR. Investigations on the antibacterial and antifungal property of QDs have been conducted, which suggests that they can serve as excellent candidates for biomedical applications because of their solubility and biocompatibility.

Aqueous solubility and compatibility make graphene quantum dots (GQDs) useful in biomedicine. GQDs are reported to be biocompatible at cellular levels investigated via WST-1 assay, LDH production, ROS generation and *in vitro* and *in vivo* distribution [40]. GQDs also possess antibacterial property against *Escherichia coli* and *S. aureus*, and GQDs with low dose of H2 O2 -based band-aids have also been prepared based on the peroxidase-like property of these particles. The designed band-aids showed a good anti-disinfectant property. They analyzed the effect of formed GQDs on biofilm formation and destruction and observed a reduction in biofilm formation by *S. aureus* at 100 μg/mL and 100 mM of GQDs and H<sup>2</sup> O2 concentration, respectively. Furthermore, they also observed that GQDs alone also showed antibiofilm properties [41]. The studies thus suggested that appropriately designed GQDs had the ability to breakdown existing biofilms and simultaneously prevented the formation of new ones. Habiba et al. suggested the antimicrobial property of silver-graphene quantum dots against *P. aeruginosa* and *S. aureus*. They observed a synergistic effect between silver nanoparticles and GQDs with 25 and 50 g/ml of silver-graphene quantum dots inhibiting *S. aureus* and *P. aeruginosa* growth, respectively. Thus, the potential applicability of Ag-GQDs as fabrication and antibacterial coating agents was clearly established [42].

nature. Additionally, certain specific proteins also assist in binding of the microbes to the surfaces [30]. The second step of accumulation is mediated via microbial surface components that recognize adhesive matrix molecules and occurs via an active process. This step involves the establishment of biofilm on the microbial surface. This process is followed by the maturation step. In this step, the characteristic features of the biofilm are formed on basis of specific microbial type. In the final step, where a new phase of invasion is initiated involves the detachment and dispersion of the microbes [31, 32]. **Figure 1** shows the schematic illustration of the different

**Figure 1.** Schematic illustration of the different stages involved in biofilm formation and detachment.

*P. aeruginosa*, *Vibrio cholerae* and some *mycobacterial* species are the common human pathogens that form biofilms and hence have the possibility of infecting humans. There are several mechanisms via which pathogenic microbes in the biofilm can initiate an infection. The seeding dispersal of a large number of pathogenic cells is one of the possible mechanisms that can initiate an infection, as the microbes are not sessile in a biofilm and hence can easily detach and initiate an infection. Secondly, the virulent phenotypes present in a biofilm can expand their colony and initiate infection. This is highly possible as biofilm has a huge heterogeneity in its phenotypical constitution [33, 34]. In addition to these, several other mechanisms have been hypothesized that could possibly allow the survival of a pathogenic organism and its transmission. For example, the detachment of pathogenic microbes from the biofilm, quorum sensing [35], co-aggregation and auto-aggregation [35, 36], modification in biosynthesis of EPS and metabolic pathways and genetic mutations [37] are important issues. However, the complete understanding of the mechanism of biofilm formation and virulence requires complete analysis of the

Biomedical implants are a necessity in modern health care; biofilm formation on these implants and devices is a major cause of their failure. Mostly *S. epidermidis* and *S. aureus* are observed in contaminated biomedical implants and devices [38]. Biofilms formed on implants and medical devices are difficult to remove as they are protected by exopolymeric matrix secreted by the pathogenic microbe [39]. Although a number of metal and their nanosize

pathogen's life cycle, environmental parameters and the different phenotypes.

**3. Role of QDs in inhibiting biofilm formation**

stages in the cycle of biofilm formation and detachment.

90 Nonmagnetic and Magnetic Quantum Dots

Furthermore, the use of semiconductor QDs will allow visualization of biofilm inhibition due to their fluorescent properties. The current methods being used for biofilm analysis are SEM, AFM, MRI and Raman spectroscopy that require lengthy and costly procedures apart from sample modulation, which sometimes provide partial details of the samples concerned [43, 44]. Other than this, conventional fluorescent dyes conjugated with carbohydrate recognition elements are used for biofilm analysis via confocal laser microscopy [45]. However, the use of a synthetic complex is sometimes toxic to cells thereby preventing in situ analysis. Therefore, QDs can be an exceptional solution for this. Moreover, amphiphilic carbon dots (CDs) have been shown to penetrate the EPS layer of *P. aeruginosa*, allowing direct visualization of its architecture, growth and how external agents affect its inhibition. The hydrocarbon side chains of CDs dock to the EPS network resulting in making the EPS scaffold highly fluorescent [46]. In yet another study, QDs with two varied surface chemistry [−COOH and polyethylene glycol (PEG) modified] were analyzed for their mobility and distribution in *P. aeruginosa* PAO1 biofilms. It was inferred that the QDs did not penetrate the bacterial cell but did colocalize with EPS matrix of the biofilm. While surface functionalization and QDs flow rate did not show any distinctive difference, analysis of center of density suggested that QDs with –COOH surface groups diffused easily compared to PEGlyated QDs. Biofilms treated with PEGlyated QDs had rough polysaccharide layers and cell distribution compared to –COOH functionalized QDs. It was thus concluded that treatment with nanomaterials can result in varying the structural parameters of biofilm [47]. The fluorescent property of QDs would thus allow recognition of biofilm formation at different growth stages and environmental conditions. Additionally, spectroscopic analysis can also be performed, which would allow better understanding of the phenomenon of binding of QDs to EPS. Conjugated QDs have also been used for biofilm imaging analysis. In a study, CdTe QDs tagged with Concanavalin A for labeling the saccharide molecules on the surface of *C. albicans* was studied. It relied on the ability of Concanavalin A to specifically bind to α-D mannose and glucose residues of saccharides. They observed that almost 93% of cells were labeled with the modified CdTe particles and were highly specific in activity [48]. Similarly, CdSe/ZnS QDs surface capped by 3-mercaptopropionic acid (MPA) and the amino acids (leucine or phenylalanine) were also used for labeling the biofilm produced by *Shewanella.* Amphiphilic core/shell CdSe/ZnS QDs were used for labeling the hydrophobic microdomains of biofilm produced by *Shewanella oneidensis*, a Gram-negative bacteria. It was inferred that CdSe/ZnS@dihydrolipoic acid-Leu or CdSe/ZnS@dihydrolipoic acid-Phe QDs showed increased hydrophobicity in comparison to CdSe-core QDs capped with 3-mercaptopropionic acid (MPA). Thus, the functional group on QD surface and the ligand density played an integral role in interaction with biofilm matrix. While the hydrophilic MPA-capped QDs were homogeneously associated, DHLA-Leu and DHLA-Phe QDs were specifically confined assisting in identifying the hydrophobic microdomains of biofilm. Hence, appropriate conjugation of surface functional groups can significantly dictate their interaction with biofilm [49]. Quite recently, selenium nanoparticles have been reported for their tremendous potential in biofilm inhibition in *C. albicans*. For the study, selenium nanoparticles were synthesized via laser ablation method and were used to analyze biofilm inhibition. They observed a very good attachment of selenium nanoparticles to the *Candida* surface, which was due to electrostatic attraction between the positively charged surface of *Candida* and negatively charged Se nanoparticles. The particles affect the cellular morphology of the fungus by substitution of sulfur groups of amino acids by the Se particles. This consequently altered the protein structure and damaged Candida morphology. Size and crystallinity of particles had a significant effect on biofilm inhibition [50].

With this, we envision that QD-based antibiofilm coatings can be promising probes in investigating biofilm imaging, treatment and their eradication. Furthermore, their broad spectrum activity and minimal host toxicity are additional advantages in this regard. Hence, the use of semiconductor QDs would not only allow detecting the inhibition process but also favor their

Quantum Dots-Based Nano-Coatings for Inhibition of Microbial Biofilms: A Mini Review

http://dx.doi.org/10.5772/intechopen.70785

93

There is a steady increase in the use of QDs. Despite the several advantages offered by QDs, with some improvements, these can emerge as excellent probes for biological applications. Focus should be towards improved protocols for functionalizing the surface of QDs simultaneously making sure that its properties remain unaltered and secondly, appropriately modifying the surface of QDs so that they do not aggregate in a protein-rich solution or cystol. These methods along with the said advantages would assist in utilizing QDs for biological and biomedical applications. Furthermore, the QDs can be tagged with antimicrobial drugs or drugs can be encapsulated inside the QD core thereby increasing the potency of drugs even at low concentration. Synergistic effect of silver nanoparticles with antibiotics such as penicillin G, amoxicillin, erythromycin, clindamycin and vancomycin is known. Therefore, studies on the synergism between QDs and drug molecules have to be analyzed in detail. This would also assist in providing insights into the molecular mechanism of action of QDs and any kind of cellular changes occurring in the pathogen upon its interaction with pathogenic microbes. Additionally, QDs labeling would allow a high throughput analysis of biofilm inhibition and disruptions that will have significant effect in healthcare sector to identify and combat biofilm

This research was supported by DST-PURSE-II funding. KR acknowledges the receipt of a DST-Inspire Faculty award from Department of Science and Technology, Government of India.

, Kamla Rawat2,3 and Himadri Bihari Bohidar1,2\*

1 School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India

2 Special Centre for Nano Sciences, Jawaharlal Nehru University, New Delhi, India

visible monitoring.

**4. Conclusion and future perspective**

formation and pathogenic infections.

**Acknowledgements**

**Author details**

Eepsita Priyadarshini<sup>1</sup>

\*Address all correspondence to: bohi0700@mail.jnu.ac.in

3 Inter University Accelerator Centre, New Delhi, India

**Figure 2** presents the mode of action of quantum dots. The application of QDs as antibiofilm agents can inhibit microbial biofilm at two stages. It can act at the initial stage, where its presence would hinder further attachment of microbial cells to the solid substrate thereby preventing the progression to mature biofilm stage and EPS secretion. Secondly, QDs can act on the matured biofilm, where its penetration into the cells would result in killing of the microbes and subsequent dispersion of the formed biofilm.

**Figure 2.** Schematic illustration showing the possible mode of action of antimicrobial quantum dots on biofilm.

With this, we envision that QD-based antibiofilm coatings can be promising probes in investigating biofilm imaging, treatment and their eradication. Furthermore, their broad spectrum activity and minimal host toxicity are additional advantages in this regard. Hence, the use of semiconductor QDs would not only allow detecting the inhibition process but also favor their visible monitoring.

## **4. Conclusion and future perspective**

QDs tagged with Concanavalin A for labeling the saccharide molecules on the surface of *C. albicans* was studied. It relied on the ability of Concanavalin A to specifically bind to α-D mannose and glucose residues of saccharides. They observed that almost 93% of cells were labeled with the modified CdTe particles and were highly specific in activity [48]. Similarly, CdSe/ZnS QDs surface capped by 3-mercaptopropionic acid (MPA) and the amino acids (leucine or phenylalanine) were also used for labeling the biofilm produced by *Shewanella.* Amphiphilic core/shell CdSe/ZnS QDs were used for labeling the hydrophobic microdomains of biofilm produced by *Shewanella oneidensis*, a Gram-negative bacteria. It was inferred that CdSe/ZnS@dihydrolipoic acid-Leu or CdSe/ZnS@dihydrolipoic acid-Phe QDs showed increased hydrophobicity in comparison to CdSe-core QDs capped with 3-mercaptopropionic acid (MPA). Thus, the functional group on QD surface and the ligand density played an integral role in interaction with biofilm matrix. While the hydrophilic MPA-capped QDs were homogeneously associated, DHLA-Leu and DHLA-Phe QDs were specifically confined assisting in identifying the hydrophobic microdomains of biofilm. Hence, appropriate conjugation of surface functional groups can significantly dictate their interaction with biofilm [49]. Quite recently, selenium nanoparticles have been reported for their tremendous potential in biofilm inhibition in *C. albicans*. For the study, selenium nanoparticles were synthesized via laser ablation method and were used to analyze biofilm inhibition. They observed a very good attachment of selenium nanoparticles to the *Candida* surface, which was due to electrostatic attraction between the positively charged surface of *Candida* and negatively charged Se nanoparticles. The particles affect the cellular morphology of the fungus by substitution of sulfur groups of amino acids by the Se particles. This consequently altered the protein structure and damaged Candida morphology. Size and

crystallinity of particles had a significant effect on biofilm inhibition [50].

and subsequent dispersion of the formed biofilm.

92 Nonmagnetic and Magnetic Quantum Dots

**Figure 2** presents the mode of action of quantum dots. The application of QDs as antibiofilm agents can inhibit microbial biofilm at two stages. It can act at the initial stage, where its presence would hinder further attachment of microbial cells to the solid substrate thereby preventing the progression to mature biofilm stage and EPS secretion. Secondly, QDs can act on the matured biofilm, where its penetration into the cells would result in killing of the microbes

**Figure 2.** Schematic illustration showing the possible mode of action of antimicrobial quantum dots on biofilm.

There is a steady increase in the use of QDs. Despite the several advantages offered by QDs, with some improvements, these can emerge as excellent probes for biological applications. Focus should be towards improved protocols for functionalizing the surface of QDs simultaneously making sure that its properties remain unaltered and secondly, appropriately modifying the surface of QDs so that they do not aggregate in a protein-rich solution or cystol. These methods along with the said advantages would assist in utilizing QDs for biological and biomedical applications. Furthermore, the QDs can be tagged with antimicrobial drugs or drugs can be encapsulated inside the QD core thereby increasing the potency of drugs even at low concentration. Synergistic effect of silver nanoparticles with antibiotics such as penicillin G, amoxicillin, erythromycin, clindamycin and vancomycin is known. Therefore, studies on the synergism between QDs and drug molecules have to be analyzed in detail. This would also assist in providing insights into the molecular mechanism of action of QDs and any kind of cellular changes occurring in the pathogen upon its interaction with pathogenic microbes. Additionally, QDs labeling would allow a high throughput analysis of biofilm inhibition and disruptions that will have significant effect in healthcare sector to identify and combat biofilm formation and pathogenic infections.

## **Acknowledgements**

This research was supported by DST-PURSE-II funding. KR acknowledges the receipt of a DST-Inspire Faculty award from Department of Science and Technology, Government of India.

## **Author details**

Eepsita Priyadarshini<sup>1</sup> , Kamla Rawat2,3 and Himadri Bihari Bohidar1,2\*

\*Address all correspondence to: bohi0700@mail.jnu.ac.in

1 School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India

2 Special Centre for Nano Sciences, Jawaharlal Nehru University, New Delhi, India

3 Inter University Accelerator Centre, New Delhi, India

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96 Nonmagnetic and Magnetic Quantum Dots


**Chapter 7**

Provisional chapter

**Biomolecule-Conjugated Quantum Dot Nanosensors as**

DOI: 10.5772/intechopen.72858

Biomolecule-Conjugated Quantum Dot Nanosensors as

A single-molecule tracking/imaging technique with semiconductor quantum dot (QD) nanosensors conjugated with appropriate peptides or antibodies is appealing for probing cellular dynamic events in living cells. We developed a 2D analysis of singlemolecule trajectories using normalized variance versus mean square displacement (MSD) to provide high-quality statistics sampled by nanosensors while preserving single-molecule sensitivity. This plot can be more informative than MSD alone to reflect the diffusive dynamics of a protein in its cellular environment. We illustrate the performance of this technique with selected examples, which are designed to expose the functionalities and importance in live cells. Our findings suggest that biomoleculeconjugated QD nanosensors can be used to reveal interactions, stoichiometries, and conformations of proteins, and provide an understanding of the mode of the interaction,

stable states, and dynamical pathways of biomolecules in live cells.

Keywords: quantum dot, single-particle tracking, fluorescence imaging, stochastic thermodynamics, single-molecule trajectory, living cell, plasma membrane, epidermal growth factor receptor, lipid domain, actin filaments, cell-penetrating peptides

Studying the movement of individual biomolecules in live cells and their interactions with the surrounding microenvironment would greatly improve our understanding of how biomolecules behave in their native cellular environment [1, 2]. Deciphering those functions and relevant regulation mechanisms is also important for developing new therapeutic strategies for diseases [2]. The major factors affecting protein mobility include local viscosity, protein– protein interaction, molecular crowding, and dimensionality of accessible space [3]. However, such factors are difficult to reconstitute in vitro using purified constituents. Therefore, there is a compelling demand for a tool to directly access the properties of the molecular assemblies

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Probes for Cellular Dynamic Events in Living Cells**

Probes for Cellular Dynamic Events in Living Cells

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72858

Jung Y. Huang

Jung Y. Huang

Abstract

1. Introduction

#### **Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells** Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

DOI: 10.5772/intechopen.72858

Jung Y. Huang Jung Y. Huang

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72858

#### Abstract

A single-molecule tracking/imaging technique with semiconductor quantum dot (QD) nanosensors conjugated with appropriate peptides or antibodies is appealing for probing cellular dynamic events in living cells. We developed a 2D analysis of singlemolecule trajectories using normalized variance versus mean square displacement (MSD) to provide high-quality statistics sampled by nanosensors while preserving single-molecule sensitivity. This plot can be more informative than MSD alone to reflect the diffusive dynamics of a protein in its cellular environment. We illustrate the performance of this technique with selected examples, which are designed to expose the functionalities and importance in live cells. Our findings suggest that biomoleculeconjugated QD nanosensors can be used to reveal interactions, stoichiometries, and conformations of proteins, and provide an understanding of the mode of the interaction, stable states, and dynamical pathways of biomolecules in live cells.

Keywords: quantum dot, single-particle tracking, fluorescence imaging, stochastic thermodynamics, single-molecule trajectory, living cell, plasma membrane, epidermal growth factor receptor, lipid domain, actin filaments, cell-penetrating peptides

## 1. Introduction

Studying the movement of individual biomolecules in live cells and their interactions with the surrounding microenvironment would greatly improve our understanding of how biomolecules behave in their native cellular environment [1, 2]. Deciphering those functions and relevant regulation mechanisms is also important for developing new therapeutic strategies for diseases [2]. The major factors affecting protein mobility include local viscosity, protein– protein interaction, molecular crowding, and dimensionality of accessible space [3]. However, such factors are difficult to reconstitute in vitro using purified constituents. Therefore, there is a compelling demand for a tool to directly access the properties of the molecular assemblies

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

and kinetics of interaction in live cells. Single-molecule imaging and tracking based on fluorescence microscopy have been developed to meet this challenge [4, 5].

Stochastic thermodynamics is a relatively new subject, which focuses on the description of the individual trajectories [14]. This framework can serve as a solid foundation for single-molecule technique but has not been sufficiently clarified in the literature. In this chapter, we first briefly review some basic concepts of stochastic thermodynamics that are specifically relevant to the analysis of the trajectories from a single-molecule optical imaging and tracking technique. Repetitive measurements of the cellular locations ~x tð Þ of nanosensors provide an entire history of that observation. Provided that the repetitive measurements could scan all of the system degrees of freedom, the time evolution of the system may be encoded into the trace of this nanosensor. Based on this understanding, we illustrated the procedures of implementing such concepts in the analysis of trajectories for receptor proteins on the plasma membranes of living

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

http://dx.doi.org/10.5772/intechopen.72858

101

Note that the separation of time scales can also render the dimensionality of a mesosystem much lower than the 3N–dimensional coordinate space. Thus, the trajectories through 3N– dimensional space are effectively restrained to an intrinsic manifold of much lower dimensionality [15, 16]. To show the usefulness of this concept in single molecule tracking, we took advantage of the brightness and photostability of QDs to investigate the translocation behavior of the human immunodeficiency virus 1 (HIV-1) transactivator of transcription peptide (TatP) conjugated quantum dot (TatP-QD) nanosensors in complex cellular terrains [17]. As TatP-QDs translocate across the plasma membranes of living cells, the particles can be viewed as nanoscale pens [18] to record the influence of the hierarchical structure of the cellular environment on TatP-QD trajectories. Analysis of the resulting three-dimensional (3D) trajectories disclosed the interaction between the TatP-QDs and bioactive groups on the plasma membrane [19, 20]. An understanding of the cellular uptake of TatP is also essential for the development of TatP-

This chapter aims to expose the connections between the framework of stochastic thermodynamics, single-molecule optical tracking, and trajectory analysis. Applications were mentioned to illustrate the type of information that can be deduced from these studies. The chapter is not a complete review on relevant subjects. Many important research studies have not been mentioned or referred. The author simply hopes this article will encourage interested readers

For a mesosystem in contact with a heat bath, the probability of finding it in a specific microstate is given by the Boltzmann factor (i.e., exp ½ � �U xð Þ ; p =kBT [13], where U denotes the total internal energy with x and p being the generalized coordinates and momentums of particles enclosed in the system). Stochastic thermodynamics provides the framework for extending the notions of work, heat and entropy production from classical thermodynamics to individual trajectories of non-equilibrium processes. It brings out the fluctuation-dissipation theorems (FDT) to constrain the probability distributions for work, heat, and entropy production along each trajectory [14]. Some milestone developments in this discipline include:

cells and obtained useful information from this important system.

based delivery strategies for therapeutic applications.

to design new experiments that would fill in the holes of this article.

2. Formalism of single-molecule trajectory analysis

2.1. A brief overview of stochastic thermodynamics

The primary factor controlling the motion of a protein in a living cell is often not the friction in the cellular medium but the interactions with its molecular partners, which often result in a transient stall or transport of molecules [3]. The binding energies between the protein of interest and its interacting partners are also of interest because regulatory processes can be mediated by changes in these binding energies. Biological media are spatially inhomogeneous, which is poorly conveyed by measuring just a few, sparse single-molecule trajectories. Thus, to fully realize the potential of a single-molecule imaging and tracking technique, an efficient and reliable analytical method is required to help extract useful information from the large amounts of trajectory data. This type of analysis usually involves the computing of the mean square displacement (MSD) along the trajectories of the molecules [6, 7].

The key component of the single-molecule imaging and tracking technique is a set of bright fluorophores with different emission wavelengths. Semiconductor quantum dots (QDs) have unique optical properties, such as high emission efficiency, wavelength tunability, and longterm stability, which make them appealing as in vivo and in vitro fluorophores [8, 9]. The ability to make water-soluble QDs that can be targeted to specific biomolecules has led to a variety of applications in cellular sensing and imaging. For example, Zhang et al. developed a QDslabeled silica nanoprobe for the detection of apoptotic cells in response to therapy [10]. Different classes of fluorescent nanoprobes were also developed for the imaging of cellular metal ions [11], which serve as essential cofactors in energy metabolism, signal transduction, and nucleic acid processing. Recently, Jainb et al. developed a synthetic technique of QD immunoconstructs by coupling antibodies (Ab) to QD [12]. The resulting QD-Ab conjugates can maintain a high and stable quantum yield for in vivo environments and acts as an ideal nanosensor to specific antigene. Therefore, a molecular-level of understanding of the cellular functions in the context of their native environments becomes possible.

Labeled biomolecules in their native environments can be considered a mesosystem with a length scale ranging from a few nanometers to <1 μm. Our understanding of thermodynamically equilibrated mesosystems roots solidly in equilibrium statistical mechanics. For small deviations from the equilibrium, researchers can invoke the linear response theory to relate the transport properties caused by the external fields to the equilibrium correlation functions. Beyond this linear response regime, no universally exact results are currently available [13].

Under non-equilibrium conditions, the temperature of a mesosystem in solution remains welldefined, yielding a value that is the same as that of the embedding solution [13]. For a complex biomolecular system comprised of N relatively rigid domains, the configuration can be described by a 3N–dimensional vector of Cartesian coordinates. The interactions among these rigid units introduce cooperative couplings between the units that yield a separation of time scales [14]. The resulting time-scale separation occurs between the observable slow degrees of freedom of the system and the fast ones that are made up by both the system and thermal bath. The collection of the slow degrees of freedom offers a natural approach to define the states of a system. The state changes with time, either due to the external driving or from ever-present thermal fluctuations that trace out a trajectory. The thermodynamic quantities defined along the trajectories follow a distribution with some universal constraints [14].

Stochastic thermodynamics is a relatively new subject, which focuses on the description of the individual trajectories [14]. This framework can serve as a solid foundation for single-molecule technique but has not been sufficiently clarified in the literature. In this chapter, we first briefly review some basic concepts of stochastic thermodynamics that are specifically relevant to the analysis of the trajectories from a single-molecule optical imaging and tracking technique. Repetitive measurements of the cellular locations ~x tð Þ of nanosensors provide an entire history of that observation. Provided that the repetitive measurements could scan all of the system degrees of freedom, the time evolution of the system may be encoded into the trace of this nanosensor. Based on this understanding, we illustrated the procedures of implementing such concepts in the analysis of trajectories for receptor proteins on the plasma membranes of living cells and obtained useful information from this important system.

Note that the separation of time scales can also render the dimensionality of a mesosystem much lower than the 3N–dimensional coordinate space. Thus, the trajectories through 3N– dimensional space are effectively restrained to an intrinsic manifold of much lower dimensionality [15, 16]. To show the usefulness of this concept in single molecule tracking, we took advantage of the brightness and photostability of QDs to investigate the translocation behavior of the human immunodeficiency virus 1 (HIV-1) transactivator of transcription peptide (TatP) conjugated quantum dot (TatP-QD) nanosensors in complex cellular terrains [17]. As TatP-QDs translocate across the plasma membranes of living cells, the particles can be viewed as nanoscale pens [18] to record the influence of the hierarchical structure of the cellular environment on TatP-QD trajectories. Analysis of the resulting three-dimensional (3D) trajectories disclosed the interaction between the TatP-QDs and bioactive groups on the plasma membrane [19, 20]. An understanding of the cellular uptake of TatP is also essential for the development of TatPbased delivery strategies for therapeutic applications.

This chapter aims to expose the connections between the framework of stochastic thermodynamics, single-molecule optical tracking, and trajectory analysis. Applications were mentioned to illustrate the type of information that can be deduced from these studies. The chapter is not a complete review on relevant subjects. Many important research studies have not been mentioned or referred. The author simply hopes this article will encourage interested readers to design new experiments that would fill in the holes of this article.

## 2. Formalism of single-molecule trajectory analysis

#### 2.1. A brief overview of stochastic thermodynamics

and kinetics of interaction in live cells. Single-molecule imaging and tracking based on fluores-

The primary factor controlling the motion of a protein in a living cell is often not the friction in the cellular medium but the interactions with its molecular partners, which often result in a transient stall or transport of molecules [3]. The binding energies between the protein of interest and its interacting partners are also of interest because regulatory processes can be mediated by changes in these binding energies. Biological media are spatially inhomogeneous, which is poorly conveyed by measuring just a few, sparse single-molecule trajectories. Thus, to fully realize the potential of a single-molecule imaging and tracking technique, an efficient and reliable analytical method is required to help extract useful information from the large amounts of trajectory data. This type of analysis usually involves the computing of the mean

The key component of the single-molecule imaging and tracking technique is a set of bright fluorophores with different emission wavelengths. Semiconductor quantum dots (QDs) have unique optical properties, such as high emission efficiency, wavelength tunability, and longterm stability, which make them appealing as in vivo and in vitro fluorophores [8, 9]. The ability to make water-soluble QDs that can be targeted to specific biomolecules has led to a variety of applications in cellular sensing and imaging. For example, Zhang et al. developed a QDslabeled silica nanoprobe for the detection of apoptotic cells in response to therapy [10]. Different classes of fluorescent nanoprobes were also developed for the imaging of cellular metal ions [11], which serve as essential cofactors in energy metabolism, signal transduction, and nucleic acid processing. Recently, Jainb et al. developed a synthetic technique of QD immunoconstructs by coupling antibodies (Ab) to QD [12]. The resulting QD-Ab conjugates can maintain a high and stable quantum yield for in vivo environments and acts as an ideal nanosensor to specific antigene. Therefore, a molecular-level of understanding of the cellular

Labeled biomolecules in their native environments can be considered a mesosystem with a length scale ranging from a few nanometers to <1 μm. Our understanding of thermodynamically equilibrated mesosystems roots solidly in equilibrium statistical mechanics. For small deviations from the equilibrium, researchers can invoke the linear response theory to relate the transport properties caused by the external fields to the equilibrium correlation functions. Beyond this linear response regime, no universally exact results are currently available [13].

Under non-equilibrium conditions, the temperature of a mesosystem in solution remains welldefined, yielding a value that is the same as that of the embedding solution [13]. For a complex biomolecular system comprised of N relatively rigid domains, the configuration can be described by a 3N–dimensional vector of Cartesian coordinates. The interactions among these rigid units introduce cooperative couplings between the units that yield a separation of time scales [14]. The resulting time-scale separation occurs between the observable slow degrees of freedom of the system and the fast ones that are made up by both the system and thermal bath. The collection of the slow degrees of freedom offers a natural approach to define the states of a system. The state changes with time, either due to the external driving or from ever-present thermal fluctuations that trace out a trajectory. The thermodynamic quantities defined along

cence microscopy have been developed to meet this challenge [4, 5].

100 Nonmagnetic and Magnetic Quantum Dots

square displacement (MSD) along the trajectories of the molecules [6, 7].

functions in the context of their native environments becomes possible.

the trajectories follow a distribution with some universal constraints [14].

For a mesosystem in contact with a heat bath, the probability of finding it in a specific microstate is given by the Boltzmann factor (i.e., exp ½ � �U xð Þ ; p =kBT [13], where U denotes the total internal energy with x and p being the generalized coordinates and momentums of particles enclosed in the system). Stochastic thermodynamics provides the framework for extending the notions of work, heat and entropy production from classical thermodynamics to individual trajectories of non-equilibrium processes. It brings out the fluctuation-dissipation theorems (FDT) to constrain the probability distributions for work, heat, and entropy production along each trajectory [14]. Some milestone developments in this discipline include:

1) Both the steady-state and transient FDT were valid for a large class of systems, including chaotic dynamics [21], driven Langevin dynamics [22], and driven diffusive dynamics [23].

V Rk

and therefore smaller V R<sup>2</sup> � �. We proposed a histogram of R<sup>τ</sup>

! 0 � �<sup>T</sup>

Figure 1 (a) Plot of the potential energy surface with U x!� � <sup>¼</sup> <sup>0</sup>:05 0:<sup>1</sup> <sup>x</sup>

<sup>0</sup> ¼ ð Þ 3; 3 , x !

<sup>10</sup> ¼ ð Þ 2; 0 , and x

coded by different colors. (c) the statistics of 500 trajectories were summarized in the V R<sup>2</sup> � �-R<sup>τ</sup>

!

to make the contour plot visible. (b) Twenty-five randomly selected trajectories were displayed with appearing times

!

! ð Þ <sup>20</sup> ��, where <sup>x</sup>

<sup>2</sup> ≃0:27 and 0.4.

� x ! �<sup>x</sup> ! 0 � �=<sup>2</sup> <sup>þ</sup> <sup>P</sup>

contrast, V R<sup>τ</sup>

particle, and V R<sup>τ</sup>

forces involved [31].

þ2e � x !�<sup>x</sup> ! ð Þ <sup>20</sup> T � x !�<sup>x</sup>

V R<sup>2</sup> � �≃0:1, R<sup>τ</sup>

energy surface U x!� � <sup>¼</sup> k x! �<sup>x</sup>

<sup>2</sup> � � <sup>¼</sup> ð Þ <sup>2</sup>γ=dt ð Þ xk

<sup>2</sup> � Var Fkxk ½ �

: (5)

http://dx.doi.org/10.5772/intechopen.72858

<sup>2</sup> and normalized variance V R<sup>2</sup> � �

� x ! �<sup>x</sup> ! i0 � �=wi

� �

2

103

γD þ Fkxk

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

<sup>2</sup> � � will fall below the free diffusion limit (V R<sup>2</sup> � � <sup>¼</sup> 2) when a probing particle

<sup>2</sup> � � revealed the nature (e.g., deterministic or stochastic) of the interactive

<sup>i</sup> α<sup>i</sup> exp � x

! �<sup>x</sup> ! 0 � �<sup>T</sup> � x ! �<sup>x</sup> ! 0 � � � <sup>11</sup> <sup>e</sup>

<sup>20</sup> ¼ ð Þ 0; 2 . The energy surface was shifted upwardly by 1

! ð Þ <sup>10</sup> � �

� x !�<sup>x</sup> ! ð Þ <sup>10</sup> T � x !�<sup>x</sup>

<sup>2</sup> plot, exhibiting a peak at

! �<sup>x</sup> ! i0 � �<sup>T</sup>

We applied this function to display the relative influence on the trajectories by deterministic forces Fkð Þ<sup>t</sup> and by the stochastic force <sup>f</sup> <sup>k</sup>. As particles diffuse under a force field, V R<sup>2</sup> � � increases with force strength [30]. As a particle diffuses near a short-range force field, it can be stalled briefly by the force field, resulting in large variances in the diffusion step size. In

moves in an environment where its surrounding medium can be polarized by the particle either electrically or orientationally [30]. This dressing effect could lead to smaller variances,

in a contour plot. Here, the MSD values were used to quantify the diffusion of a probing

In the following, we presented some simulated results to illustrate the features of this ad hoc data-driven methodology in the framework of stochastic thermodynamics. Using Eq. (1), we first simulated 2D motions of Brownian particles in a force field, which had a potential

(see Figure 1a). The potential energy surface comprised two isotropic Gaussian wells and a long-range harmonic potential to prevent the particle from drifting off to infinity. The central positions of the harmonic potential and two Gaussian wells are located at (3, 3), (2, 0) and (0, 2), respectively. The strength parameters of the harmonic and Gaussian potentials were chosen to yield deterministic forces, which were a factor of 0.14 and 3.8 to that of the stochastic force

2) The Jarzynski relation (JR) was derived [24, 25], which relates the free energy difference between two equilibrium states to the average work done to drive the system from one state to the other along a non-equilibrium process. For non-equilibrium systems driven by timedependent forces, a refinement of the JR became extremely useful to determine the free energy landscapes of biomolecules [26, 27].

3) The exchanged heat and applied work could also be rigorously defined along individual trajectories of the driven Brownian motion. The entropy produced in a medium could thereby be related to the stochastic action, which also serves as the weight of trajectories [28].

#### 2.2. Trajectory analysis of single-molecule stochastic processes

2.2.1. Two-dimensional plot of normalized variance and mean square displacement of single-molecule trajectories

In a living cell, a biomolecule subjected to random influences can explore its possible outcomes and evolve to yield dispersion over state space. This evolution contains contributions from both deterministic and stochastic forces. The time-scale separation mentioned above implies that the dynamics will become Markovian and follow a generalized Langevin equation [29]

$$
\nabla \partial\_t \overrightarrow{\mathbf{x}}\_k(t) = -\nabla\_k \mathcal{U} \left( \overrightarrow{\mathbf{x}}\_k(t) \right) / \gamma \; \; \; + \sqrt{2D} \, d\mathcal{W}\_t = \mathcal{F}\_k(\mathbf{t}) / \gamma \; \; \; + f\_k(\mathbf{t}) / \gamma,\tag{1}
$$

where subindex k represents the k-th particle at the position xk !. The frictional parameter γ is relevant to the diffusion coefficient D with γ:D ¼ kBT. The total deterministic force acting on the diffusive particle is expressed as Fkð Þt . The stochastic force f <sup>k</sup>ð Þt follows the Weiner process with zero mean and a delta correlation of < f <sup>k</sup>ð Þ� t þ τ f <sup>k</sup>ð Þt >¼ 2γkBTδ τð Þ.

By rewriting Eq. (1) as dxk <sup>¼</sup> Fk=<sup>γ</sup> � dt <sup>þ</sup> ffiffiffiffiffiffi <sup>2</sup><sup>D</sup> <sup>p</sup> dWt and invoking the stochastic chain rule, we derived an equation for the square displacement [30]

$$\left(d\mathbf{x}\_k\right)^2 = 2F\_k/\gamma \cdot \mathbf{x}\_k dt + 2Ddt + 2\sqrt{2D}\mathbf{x}\_k dW\_t. \tag{2}$$

From Eq. (2), we further derived the local MSD Rk <sup>2</sup> as

$$
\overline{\text{R}\_k}^2 = \overline{\text{dx}\_k}^2 = 2/\gamma \cdot \overline{F\_k \text{x}\_k} dt + 2\text{D}dt,\tag{3}
$$

and similarly, the variance of the square displacement Var dxk <sup>2</sup> � � <sup>¼</sup> dxk <sup>2</sup> � �<sup>2</sup> � dxk <sup>2</sup> h i<sup>2</sup> as

$$\operatorname{Var}\left(d\mathbf{x}\_k^{\mathcal{Q}}\right) = 8D\left(\overline{\mathbf{x}\_k}\right)^2 dt - \left(4/\gamma\right) \cdot \operatorname{Var}[\mathbf{F}\_k \mathbf{x}\_k] \left(dt\right)^2,\tag{4}$$

where Var[.] denotes the variance operation. A normalized variance of square displacement was defined as V Rk <sup>2</sup> � � <sup>¼</sup> Var Rk <sup>2</sup> � �= Rk <sup>2</sup> � �<sup>2</sup> , which yielded [30]

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells http://dx.doi.org/10.5772/intechopen.72858 103

$$V\left(\mathbf{R}\_k^{\,2}\right) = \frac{\left(2\gamma/dt\ \ \right)\left(\overline{\mathbf{x}\_k}\right)^2 - Var[F\_k \mathbf{x}\_k]}{\gamma D + \overline{F\_k \mathbf{x}\_k}}.\tag{5}$$

We applied this function to display the relative influence on the trajectories by deterministic forces Fkð Þ<sup>t</sup> and by the stochastic force <sup>f</sup> <sup>k</sup>. As particles diffuse under a force field, V R<sup>2</sup> � � increases with force strength [30]. As a particle diffuses near a short-range force field, it can be stalled briefly by the force field, resulting in large variances in the diffusion step size. In contrast, V R<sup>τ</sup> <sup>2</sup> � � will fall below the free diffusion limit (V R<sup>2</sup> � � <sup>¼</sup> 2) when a probing particle moves in an environment where its surrounding medium can be polarized by the particle either electrically or orientationally [30]. This dressing effect could lead to smaller variances, and therefore smaller V R<sup>2</sup> � �. We proposed a histogram of R<sup>τ</sup> <sup>2</sup> and normalized variance V R<sup>2</sup> � � in a contour plot. Here, the MSD values were used to quantify the diffusion of a probing particle, and V R<sup>τ</sup> <sup>2</sup> � � revealed the nature (e.g., deterministic or stochastic) of the interactive forces involved [31].

1) Both the steady-state and transient FDT were valid for a large class of systems, including chaotic dynamics [21], driven Langevin dynamics [22], and driven diffusive dynamics [23].

2) The Jarzynski relation (JR) was derived [24, 25], which relates the free energy difference between two equilibrium states to the average work done to drive the system from one state to the other along a non-equilibrium process. For non-equilibrium systems driven by timedependent forces, a refinement of the JR became extremely useful to determine the free energy

3) The exchanged heat and applied work could also be rigorously defined along individual trajectories of the driven Brownian motion. The entropy produced in a medium could thereby

2.2.1. Two-dimensional plot of normalized variance and mean square displacement of single-molecule

In a living cell, a biomolecule subjected to random influences can explore its possible outcomes and evolve to yield dispersion over state space. This evolution contains contributions from both deterministic and stochastic forces. The time-scale separation mentioned above implies that the dynamics will become Markovian and follow a generalized Langevin equation [29]

<sup>=</sup><sup>γ</sup> <sup>þ</sup> ffiffiffiffiffiffi

relevant to the diffusion coefficient D with γ:D ¼ kBT. The total deterministic force acting on the diffusive particle is expressed as Fkð Þt . The stochastic force f <sup>k</sup>ð Þt follows the Weiner process

<sup>2</sup> <sup>¼</sup> <sup>2</sup>Fk=<sup>γ</sup> � xk dt <sup>þ</sup> <sup>2</sup>Ddt <sup>þ</sup> <sup>2</sup> ffiffiffiffiffiffi

2

where Var[.] denotes the variance operation. A normalized variance of square displacement

, which yielded [30]

<sup>2</sup> as

dt � ð Þ� <sup>4</sup>=<sup>γ</sup> Var Fkxk ½ �ð Þ dt <sup>2</sup>

<sup>2</sup><sup>D</sup> <sup>p</sup> dWt <sup>¼</sup> Fkð Þ<sup>t</sup> <sup>=</sup><sup>γ</sup> <sup>þ</sup> <sup>f</sup> <sup>k</sup>ð Þ<sup>t</sup> <sup>=</sup>γ, (1)

<sup>2</sup><sup>D</sup> <sup>p</sup> dWt and invoking the stochastic chain rule, we

<sup>2</sup> <sup>¼</sup> <sup>2</sup>=<sup>γ</sup> � Fkxk dt <sup>þ</sup> <sup>2</sup>Ddt, (3)

<sup>2</sup> � � <sup>¼</sup> dxk

!. The frictional parameter γ is

<sup>2</sup><sup>D</sup> <sup>p</sup> xkdWt: (2)

<sup>2</sup> � �<sup>2</sup> � dxk

<sup>2</sup> h i<sup>2</sup>

, (4)

as

be related to the stochastic action, which also serves as the weight of trajectories [28].

2.2. Trajectory analysis of single-molecule stochastic processes

landscapes of biomolecules [26, 27].

102 Nonmagnetic and Magnetic Quantum Dots

∂<sup>t</sup> x !

<sup>k</sup>ðÞ¼� <sup>t</sup> <sup>∇</sup><sup>k</sup> U x!

By rewriting Eq. (1) as dxk <sup>¼</sup> Fk=<sup>γ</sup> � dt <sup>þ</sup> ffiffiffiffiffiffi

derived an equation for the square displacement [30]

ð Þ dxk

Rk <sup>2</sup> <sup>¼</sup> dxk

and similarly, the variance of the square displacement Var dxk

<sup>2</sup> � � <sup>¼</sup> <sup>8</sup>Dð Þ xk

<sup>2</sup> � �= Rk

<sup>2</sup> � �<sup>2</sup>

Var dxk

<sup>2</sup> � � <sup>¼</sup> Var Rk

was defined as V Rk

From Eq. (2), we further derived the local MSD Rk

where subindex k represents the k-th particle at the position xk

<sup>k</sup>ð Þt � �

with zero mean and a delta correlation of < f <sup>k</sup>ð Þ� t þ τ f <sup>k</sup>ð Þt >¼ 2γkBTδ τð Þ.

trajectories

In the following, we presented some simulated results to illustrate the features of this ad hoc data-driven methodology in the framework of stochastic thermodynamics. Using Eq. (1), we first simulated 2D motions of Brownian particles in a force field, which had a potential energy surface U x!� � <sup>¼</sup> k x! �<sup>x</sup> ! 0 � �<sup>T</sup> � x ! �<sup>x</sup> ! 0 � �=<sup>2</sup> <sup>þ</sup> <sup>P</sup> <sup>i</sup> α<sup>i</sup> exp � x ! �<sup>x</sup> ! i0 � �<sup>T</sup> � x ! �<sup>x</sup> ! i0 � �=wi 2 � � (see Figure 1a). The potential energy surface comprised two isotropic Gaussian wells and a long-range harmonic potential to prevent the particle from drifting off to infinity. The central positions of the harmonic potential and two Gaussian wells are located at (3, 3), (2, 0) and (0, 2), respectively. The strength parameters of the harmonic and Gaussian potentials were chosen to yield deterministic forces, which were a factor of 0.14 and 3.8 to that of the stochastic force

Figure 1 (a) Plot of the potential energy surface with U x!� � <sup>¼</sup> <sup>0</sup>:05 0:<sup>1</sup> <sup>x</sup> ! �<sup>x</sup> ! 0 � �<sup>T</sup> � x ! �<sup>x</sup> ! 0 � � � <sup>11</sup> <sup>e</sup> � x !�<sup>x</sup> ! ð Þ <sup>10</sup> T � x !�<sup>x</sup> ! ð Þ <sup>10</sup> � � þ2e � x !�<sup>x</sup> ! ð Þ <sup>20</sup> T � x !�<sup>x</sup> ! ð Þ <sup>20</sup> ��, where <sup>x</sup> ! <sup>0</sup> ¼ ð Þ 3; 3 , x ! <sup>10</sup> ¼ ð Þ 2; 0 , and x ! <sup>20</sup> ¼ ð Þ 0; 2 . The energy surface was shifted upwardly by 1 to make the contour plot visible. (b) Twenty-five randomly selected trajectories were displayed with appearing times coded by different colors. (c) the statistics of 500 trajectories were summarized in the V R<sup>2</sup> � �-R<sup>τ</sup> <sup>2</sup> plot, exhibiting a peak at V R<sup>2</sup> � �≃0:1, R<sup>τ</sup> <sup>2</sup> ≃0:27 and 0.4.

( ffiffiffiffiffiffi <sup>2</sup><sup>D</sup> <sup>p</sup> <sup>=</sup><sup>γ</sup> <sup>¼</sup> <sup>0</sup>:35). We updated the locations of the Brownian particles every 0.01 s using the Euler–Maruyama solver, starting at initial position (0, 0). We generated a total of 500 trajectories, each of a 10 s duration. Figure 1b displays 25 randomly selected trajectories with appearing times coded by different colors.

peqð Þ¼ <sup>x</sup> <sup>e</sup>�βU xð Þ=Zeq, where U xð Þ is the potential of mean force (PMF) with <sup>x</sup> denoting the generalized coordinates, and Zeq representing the equilibrium partition function [32]. We can

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

caliber [14, 33]. For non-equilibrium processes, the entropy produced along a trajectory with time

stochastic trajectories to display the degree of dynamic dispersion. For coarser time resolutions (i.e., larger Δt), the transition rates converged to their equilibrium values and the information

Conformational trajectories of a biomolecular system, comprising N relatively rigid domains, can be displayed in a 3N–dimensional phase space. As noted above, cooperative couplings between these rigid units often yield a separation of time scales, which causes the system's slow degrees of freedom to be separated from the fast ones made up by the system and thermal bath. An intrinsic manifold of much lower dimensionality is thus embedded in the high-dimensional configuration trajectories. Unfortunately, the projection of dynamical configurations <sup>f</sup> : <sup>R</sup><sup>3</sup><sup>N</sup> ! M R<sup>m</sup> ð Þ into a reduced dimensional space, which is specified by m collective variables <sup>f</sup><sup>~</sup> <sup>¼</sup> <sup>f</sup>1;f<sup>2</sup> ½ � ;…;f<sup>m</sup> <sup>∈</sup> <sup>M</sup>, is highly nonlinear and unavailable from analytical theory. The first issue encountered in depicting the complex dynamics in a low-dimensional space is how to identify a set of appropriate slow variables f~ . In recent years, a number of machine learning approaches have been developed to infer such mappings by discovering

Recently, Wang and Ferguson successfully applied the generalized Takens Delay Embedding Theorem [34] to retrieve a low-dimensional representation of the free energy landscape from univariate time series of single-molecule physical observable. The authors also determined that the univariate time series could be expanded into a high-dimensional space in which the dynamics were equivalent to those of the molecular motions in real space. Single-molecule optical techniques based on a variety of nanosensors can provide the time series of experimentally accessible observables. By measuring the impact of cellular environments on the trajectory ensemble of those nanosensors, it is possible to reveal the influence of the cellular environments. Figure 3 presents a conceptual drawing to illustrate the translocation process of biomolecule-conjugated quantum dot nanosensors across the cellular plasma membrane.

h i ¼ � <sup>Ð</sup>

i,j πiri!<sup>j</sup>ð Þ Δt ln ri!<sup>j</sup>ð Þ Δt , which required averaging over the

! ð Þ<sup>t</sup> of the nanosensors to be generated by a stochastic

<sup>i</sup>ð Þt because the probing particles could move in n

! ξ � � � �!

! ð Þ<sup>t</sup> on slow degrees of freedom to

; i ¼ 1,

! ð Þ<sup>t</sup> was implicitly dependent on the generalized coordi-

dxpeqð Þx ln peqð Þx

http://dx.doi.org/10.5772/intechopen.72858

h i as a

105

measure the static dispersion of the system using entropy S peqð Þx

2.2.2. Spectral-embedding analysis of single-molecule trajectories

low-dimensional manifolds within high-dimensional trajectories [15].

!

different realizations of the local environment with interaction potentials Ui x

; i ¼ 1, ::, n.

resolution <sup>Δ</sup><sup>t</sup> becomes <sup>S</sup>ð Þ� <sup>Δ</sup><sup>t</sup> <sup>P</sup><sup>N</sup>

We assumed the trajectory ensemble x

nates of the fast degrees of freedom ξ

::, n. In the following, we will describe a projection of x

! ξ � � � �!

process governed by Eq. (1). Here, x

disclose the influences of Ui x

about the dynamics is lost.

As the particles move toward the center of the harmonic potential, they are attracted to the two Gaussian wells. Well 2, centered at (2, 0), had the same width but was deeper than well 1 by a factor of 2. Thus, at the end of the simulation, the particles near (2, 0) were about twice that of those near well 1. As displayed in Figure 1c, the diffusion yielded a dual-peak structure at V R<sup>2</sup> � �≃0:1, R<sup>τ</sup> <sup>2</sup> ≃ 0:27 and 0.4 in the V R<sup>2</sup> � �-R<sup>τ</sup> <sup>2</sup> plot, indicating that as a particle repeatedly visits or stays in a spatial region, the characteristic V R<sup>τ</sup> <sup>2</sup> � � and <sup>R</sup><sup>τ</sup> <sup>2</sup> of the location will be imposed on the trajectories.

Next, we reduced the width of well 2 by a factor 3 while keeping its depth at the same value (see Figure 2a. At the end of the simulation, the particles located near (0, 2) became one third that of those near Well 1 (see red spots in Figure 2b). Figure 2c displays a peak at V R<sup>2</sup> � � ≃0:25 and Rτ <sup>2</sup> ≃0:25. Although the population at well 2 was lower, its influence on the trajectories with a higher V R<sup>2</sup> � � value was visible. For a brief summary of this simulation, we would like to point out an attractive feature of the V R<sup>2</sup> � �-R<sup>τ</sup> <sup>2</sup> plot. When a molecule repeatedly visits or stays in a spatial region, the characteristic V R<sup>τ</sup> <sup>2</sup> � � and <sup>R</sup><sup>τ</sup> <sup>2</sup> of the location will be imposed on the trajectories, which then results in the formation of a peak at the corresponding position on the plot.

We used the hidden Markov model (HMM) to further reveal the dynamics by identifying the underlying state changes and their corresponding occupation probability π<sup>i</sup> and transition rates ri!<sup>j</sup>. Note that in the ergodic limit, the system will reach an equilibrium with a distribution of

Figure 2 (a) Plot of the potential energy surface with U x!� � <sup>¼</sup> <sup>0</sup>:05 0:<sup>1</sup> <sup>x</sup> ! �<sup>x</sup> ! 0 � �<sup>T</sup> � x ! �<sup>x</sup> ! 0 � � � <sup>11</sup> <sup>e</sup> � x !�<sup>x</sup> ! ð Þ <sup>10</sup> T � x !�<sup>x</sup> ! ð Þ <sup>10</sup> � � þ2e �3 x !�<sup>x</sup> ! ð Þ <sup>20</sup> T � x !�<sup>x</sup> ! ð Þ <sup>20</sup> ��, where <sup>x</sup> ! <sup>0</sup> ¼ ð Þ 3; 3 , x ! <sup>10</sup> ¼ ð Þ 2; 0 , and x ! <sup>20</sup> ¼ ð Þ 0; 2 . The energy surface was shifted upwardly by 1 to make the contour plot visible. (b) twenty-five randomly selected trajectories from the simulation were displayed. (c) the statistics of 500 trajectories were summarized in the V R<sup>2</sup> � �-R<sup>τ</sup> <sup>2</sup> plot, exhibiting a peak at V R<sup>2</sup> � �≃0:25, R<sup>τ</sup> <sup>2</sup> ≃0:25.

peqð Þ¼ <sup>x</sup> <sup>e</sup>�βU xð Þ=Zeq, where U xð Þ is the potential of mean force (PMF) with <sup>x</sup> denoting the generalized coordinates, and Zeq representing the equilibrium partition function [32]. We can measure the static dispersion of the system using entropy S peqð Þx h i ¼ � <sup>Ð</sup> dxpeqð Þx ln peqð Þx h i as a caliber [14, 33]. For non-equilibrium processes, the entropy produced along a trajectory with time resolution <sup>Δ</sup><sup>t</sup> becomes <sup>S</sup>ð Þ� <sup>Δ</sup><sup>t</sup> <sup>P</sup><sup>N</sup> i,j πiri!<sup>j</sup>ð Þ Δt ln ri!<sup>j</sup>ð Þ Δt , which required averaging over the stochastic trajectories to display the degree of dynamic dispersion. For coarser time resolutions (i.e., larger Δt), the transition rates converged to their equilibrium values and the information about the dynamics is lost.

#### 2.2.2. Spectral-embedding analysis of single-molecule trajectories

( ffiffiffiffiffiffi

V R<sup>2</sup> � �≃0:1, R<sup>τ</sup>

Rτ

þ2e �3 x !�<sup>x</sup> ! ð Þ <sup>20</sup> T � x !�<sup>x</sup> ! ð Þ <sup>20</sup> ��

times coded by different colors.

104 Nonmagnetic and Magnetic Quantum Dots

imposed on the trajectories.

out an attractive feature of the V R<sup>2</sup> � �-R<sup>τ</sup>

Figure 2 (a) Plot of the potential energy surface with U x!� �

<sup>0</sup> ¼ ð Þ 3; 3 , x !

<sup>10</sup> ¼ ð Þ 2; 0 , and x

!

to make the contour plot visible. (b) twenty-five randomly selected trajectories from the simulation were displayed. (c) the

, where x !

statistics of 500 trajectories were summarized in the V R<sup>2</sup> � �-R<sup>τ</sup>

spatial region, the characteristic V R<sup>τ</sup>

<sup>2</sup><sup>D</sup> <sup>p</sup> <sup>=</sup><sup>γ</sup> <sup>¼</sup> <sup>0</sup>:35). We updated the locations of the Brownian particles every 0.01 s using the Euler–Maruyama solver, starting at initial position (0, 0). We generated a total of 500 trajectories, each of a 10 s duration. Figure 1b displays 25 randomly selected trajectories with appearing

As the particles move toward the center of the harmonic potential, they are attracted to the two Gaussian wells. Well 2, centered at (2, 0), had the same width but was deeper than well 1 by a factor of 2. Thus, at the end of the simulation, the particles near (2, 0) were about twice that of those near well 1. As displayed in Figure 1c, the diffusion yielded a dual-peak structure at

Next, we reduced the width of well 2 by a factor 3 while keeping its depth at the same value (see Figure 2a. At the end of the simulation, the particles located near (0, 2) became one third that of those near Well 1 (see red spots in Figure 2b). Figure 2c displays a peak at V R<sup>2</sup> � � ≃0:25 and

<sup>2</sup> ≃0:25. Although the population at well 2 was lower, its influence on the trajectories with a higher V R<sup>2</sup> � � value was visible. For a brief summary of this simulation, we would like to point

and R<sup>τ</sup>

ries, which then results in the formation of a peak at the corresponding position on the plot.

We used the hidden Markov model (HMM) to further reveal the dynamics by identifying the underlying state changes and their corresponding occupation probability π<sup>i</sup> and transition rates ri!<sup>j</sup>. Note that in the ergodic limit, the system will reach an equilibrium with a distribution of

¼ 0:05 0:1 x

! �<sup>x</sup> ! 0 � �<sup>T</sup>

<sup>2</sup> plot, exhibiting a peak at V R<sup>2</sup> � �≃0:25, R<sup>τ</sup>

� x ! �<sup>x</sup> ! 0 � �

<sup>20</sup> ¼ ð Þ 0; 2 . The energy surface was shifted upwardly by 1

� �

� 11 e � x !�<sup>x</sup> ! ð Þ <sup>10</sup> T � x !�<sup>x</sup> ! ð Þ <sup>10</sup>

<sup>2</sup> ≃0:25.

<sup>2</sup> plot, indicating that as a particle repeatedly

<sup>2</sup> of the location will be

and R<sup>τ</sup>

<sup>2</sup> plot. When a molecule repeatedly visits or stays in a

<sup>2</sup> of the location will be imposed on the trajecto-

<sup>2</sup> � �

<sup>2</sup> ≃ 0:27 and 0.4 in the V R<sup>2</sup> � �-R<sup>τ</sup>

<sup>2</sup> � �

visits or stays in a spatial region, the characteristic V R<sup>τ</sup>

Conformational trajectories of a biomolecular system, comprising N relatively rigid domains, can be displayed in a 3N–dimensional phase space. As noted above, cooperative couplings between these rigid units often yield a separation of time scales, which causes the system's slow degrees of freedom to be separated from the fast ones made up by the system and thermal bath. An intrinsic manifold of much lower dimensionality is thus embedded in the high-dimensional configuration trajectories. Unfortunately, the projection of dynamical configurations <sup>f</sup> : <sup>R</sup><sup>3</sup><sup>N</sup> ! M R<sup>m</sup> ð Þ into a reduced dimensional space, which is specified by m collective variables <sup>f</sup><sup>~</sup> <sup>¼</sup> <sup>f</sup>1;f<sup>2</sup> ½ � ;…;f<sup>m</sup> <sup>∈</sup> <sup>M</sup>, is highly nonlinear and unavailable from analytical theory. The first issue encountered in depicting the complex dynamics in a low-dimensional space is how to identify a set of appropriate slow variables f~ . In recent years, a number of machine learning approaches have been developed to infer such mappings by discovering low-dimensional manifolds within high-dimensional trajectories [15].

Recently, Wang and Ferguson successfully applied the generalized Takens Delay Embedding Theorem [34] to retrieve a low-dimensional representation of the free energy landscape from univariate time series of single-molecule physical observable. The authors also determined that the univariate time series could be expanded into a high-dimensional space in which the dynamics were equivalent to those of the molecular motions in real space. Single-molecule optical techniques based on a variety of nanosensors can provide the time series of experimentally accessible observables. By measuring the impact of cellular environments on the trajectory ensemble of those nanosensors, it is possible to reveal the influence of the cellular environments. Figure 3 presents a conceptual drawing to illustrate the translocation process of biomolecule-conjugated quantum dot nanosensors across the cellular plasma membrane.

We assumed the trajectory ensemble x ! ð Þ<sup>t</sup> of the nanosensors to be generated by a stochastic process governed by Eq. (1). Here, x ! ð Þ<sup>t</sup> was implicitly dependent on the generalized coordinates of the fast degrees of freedom ξ ! <sup>i</sup>ð Þt because the probing particles could move in n different realizations of the local environment with interaction potentials Ui x ! ξ � � � �! ; i ¼ 1, ::, n. In the following, we will describe a projection of x ! ð Þ<sup>t</sup> on slow degrees of freedom to disclose the influences of Ui x ! ξ � � � �! ; i ¼ 1, ::, n.

We first simulated 3D Brownian motion with Eq. (1) under the three conditions: isotropic

Spectral-embedding analysis can be implemented in the diffusion-map framework to enable an efficient construction of good slow observables and thereby can expose the lowdimensional manifolds underlying the high-dimensional datasets [35]. A graph-based method provided a discretized approximation of the manifold for efficiently constructing eigen-

h i and projected the time series onto a low-dimensional mani-

fold by exploiting the spectral-embedding technique [35]. The first eigenvector was trivial because its eigenvalue gave only the data density in a cluster. We then focused on the next two eigenvectors, f<sup>2</sup> and f3, which offered the most critical information on the interactions between the nanosensors and their environments. Figure 4d–f display the f2-f<sup>3</sup> plot of the simulated trajectory data, shown in gray. For comparison, the trajectories presented in Figure 4a–c are also reproduced here with appearing times color-coded. A V-shaped distribution in the f2-f<sup>3</sup> plot developed gradually with force field strength, suggesting the V-shaped

In Figure 5a, the isotropic diffusion produced statistics with a clear peak at V R<sup>2</sup> � �≃0:0023 and

reveal the influences of the cellular environment on nanosensors but from different viewpoints.

<sup>2</sup> plot. As field strength Fk=<sup>f</sup> <sup>k</sup> increased to 0.03, the peak shifted

<sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> histogram from simulated trajectories of particle diffusing (a) isotropically

� � �

� ¼ 0:01 and (c) Fk=f <sup>k</sup> � � �

� ¼ 0:03.

<sup>2</sup> ≃0:59, indicating a larger diffusion stepsize variation under

<sup>2</sup> analysis and spectral-embedding technique can

decomposition of the datasets [36]. We assembled the time-delayed vector x~kðÞ¼ t x

feature was a useful indicator of directed movement under a force field [17].

� <sup>¼</sup> 0, <sup>D</sup> <sup>¼</sup> <sup>0</sup>:05μm<sup>2</sup>=s) or under a unidirectional force field with (b) Fk=<sup>f</sup> <sup>k</sup>

� ¼ 0:01 and 0.03. Starting at the origin we generated an ensemble of 300 trajectories for each case. Every trajectory contained 100 diffusion steps with a time resolution of 0.02 s. We display three typical trajectories in Figure 4a–c, which clearly exhibit increased spread along z

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

� <sup>¼</sup> 0, <sup>D</sup> <sup>¼</sup> <sup>0</sup>:05μm<sup>2</sup>=<sup>s</sup> and under z-unidirectional force fields with

! k ð Þ<sup>i</sup> h i

http://dx.doi.org/10.5772/intechopen.72858

i¼1,s

107

diffusion with Fk=f <sup>k</sup>

Fk=f <sup>k</sup> � � �

¼ x ! <sup>k</sup>ð Þ0 ; x !

Rτ

� � �

<sup>k</sup>ð Þτ ;…; x !

<sup>2</sup> ≃0:86 in the V R<sup>2</sup> � �-R<sup>τ</sup>

Figure 5 2D contour plot of V R<sup>τ</sup>

with Fk=f <sup>k</sup> � � �

upwardly to V R<sup>2</sup> � � ≃0:014 and R<sup>τ</sup>

the force field. Thus, both of the V R<sup>2</sup> � �-R<sup>τ</sup>

due to the action of the unidirectional force field.

<sup>k</sup>ð Þ ð Þ s � 1 τ

Figure 3 A schematic showing the translocation process of biomolecule-conjugated QDs that depict cellular dynamic processes by recording the impact of cellular environments on the trajectory ensemble of the nanosensors.

Figure 4 Simulated trajectories of particle diffusing (a,d) isotropically with Fk=f <sup>k</sup> <sup>¼</sup> <sup>0</sup> <sup>D</sup> <sup>¼</sup> <sup>0</sup>:05μm<sup>2</sup>=s) or under a unidirectional force field with (b,e) Fk=f <sup>k</sup> ¼ 0:01 and (c,f) Fk=f <sup>k</sup> ¼ 0:03 were displayed with appearing times colorcoded. For each case, a total of 300 simulated trajectories (gray in d, e, and f) were shown on a low-dimensional manifold of two principal spectral-embedding eigenvectors. The trajectories shown in (a, b, and c) were reproduced on the manifold with appearing times color-coded.

We first simulated 3D Brownian motion with Eq. (1) under the three conditions: isotropic diffusion with Fk=f <sup>k</sup> � � � � <sup>¼</sup> 0, <sup>D</sup> <sup>¼</sup> <sup>0</sup>:05μm<sup>2</sup>=<sup>s</sup> and under z-unidirectional force fields with Fk=f <sup>k</sup> � � � � ¼ 0:01 and 0.03. Starting at the origin we generated an ensemble of 300 trajectories for each case. Every trajectory contained 100 diffusion steps with a time resolution of 0.02 s. We display three typical trajectories in Figure 4a–c, which clearly exhibit increased spread along z due to the action of the unidirectional force field.

Spectral-embedding analysis can be implemented in the diffusion-map framework to enable an efficient construction of good slow observables and thereby can expose the lowdimensional manifolds underlying the high-dimensional datasets [35]. A graph-based method provided a discretized approximation of the manifold for efficiently constructing eigendecomposition of the datasets [36]. We assembled the time-delayed vector x~kðÞ¼ t x ! k ð Þ<sup>i</sup> h i i¼1,s ! ! ! h i and projected the time series onto a low-dimensional mani-

¼ x <sup>k</sup>ð Þ0 ; x <sup>k</sup>ð Þτ ;…; x <sup>k</sup>ð Þ ð Þ s � 1 τ fold by exploiting the spectral-embedding technique [35]. The first eigenvector was trivial because its eigenvalue gave only the data density in a cluster. We then focused on the next two eigenvectors, f<sup>2</sup> and f3, which offered the most critical information on the interactions between the nanosensors and their environments. Figure 4d–f display the f2-f<sup>3</sup> plot of the simulated trajectory data, shown in gray. For comparison, the trajectories presented in Figure 4a–c are also reproduced here with appearing times color-coded. A V-shaped distribution in the f2-f<sup>3</sup> plot developed gradually with force field strength, suggesting the V-shaped feature was a useful indicator of directed movement under a force field [17].

Figure 3 A schematic showing the translocation process of biomolecule-conjugated QDs that depict cellular dynamic

 

<sup>¼</sup> <sup>0</sup> <sup>D</sup> <sup>¼</sup> <sup>0</sup>:05μm<sup>2</sup>=s) or under a

¼ 0:03 were displayed with appearing times color-

processes by recording the impact of cellular environments on the trajectory ensemble of the nanosensors.

Figure 4 Simulated trajectories of particle diffusing (a,d) isotropically with Fk=f <sup>k</sup>

¼ 0:01 and (c,f) Fk=f <sup>k</sup>

 

coded. For each case, a total of 300 simulated trajectories (gray in d, e, and f) were shown on a low-dimensional manifold of two principal spectral-embedding eigenvectors. The trajectories shown in (a, b, and c) were reproduced on the

 

unidirectional force field with (b,e) Fk=f <sup>k</sup>

106 Nonmagnetic and Magnetic Quantum Dots

manifold with appearing times color-coded.

In Figure 5a, the isotropic diffusion produced statistics with a clear peak at V R<sup>2</sup> � �≃0:0023 and Rτ <sup>2</sup> ≃0:86 in the V R<sup>2</sup> � �-R<sup>τ</sup> <sup>2</sup> plot. As field strength Fk=<sup>f</sup> <sup>k</sup> increased to 0.03, the peak shifted upwardly to V R<sup>2</sup> � � ≃0:014 and R<sup>τ</sup> <sup>2</sup> ≃0:59, indicating a larger diffusion stepsize variation under the force field. Thus, both of the V R<sup>2</sup> � �-R<sup>τ</sup> <sup>2</sup> analysis and spectral-embedding technique can reveal the influences of the cellular environment on nanosensors but from different viewpoints.

Figure 5 2D contour plot of V R<sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> histogram from simulated trajectories of particle diffusing (a) isotropically with Fk=f <sup>k</sup> � � � � <sup>¼</sup> 0, <sup>D</sup> <sup>¼</sup> <sup>0</sup>:05μm<sup>2</sup>=s) or under a unidirectional force field with (b) Fk=<sup>f</sup> <sup>k</sup> � � � � ¼ 0:01 and (c) Fk=f <sup>k</sup> � � � � ¼ 0:03.

## 3. Apparatus and experimental procedures

#### 3.1. Optical setup

The schematic of our single-particle fluorescence microscopy apparatus with light-sheet excitation is shown in Figure 6a. The output beam from a solid-state laser with different wavelengths was shaped into a light sheet of 3 μm thickness at the beam waist, yielding a diffraction-limited beam propagation with a Rayleigh range of 41 μm in the x direction. By using a galvanometer scanner, the light sheet can be positioned at a sample in a range of 34 μm along the y and z directions with an accuracy of 0.5 μm [17].

3.2. Linking localized coordinates of nanosensors for generating 3D trajectories

than 0.01 =μm2, D < 0:1μm<sup>2</sup>=s, and a linking accuracy that could be higher than 98%.

HeLa and A431 cells were cultured in Dulbecco's Modified Eagle's medium (DMEM) without phenol red supplemented with 10% (v/v) fetal bovine serum. MCF12A cells were cultured in a 1:1 mixture of DMEM and Ham's F12 medium containing 20 ng/mL Human EGF, 0.01 mg/mL bovine insulin, 500 ng/mL hydrocortisone, and 5%(v/v) horse serum [39]. Before singlemolecule live-cell imaging was performed, the cells were plated in an eight-well chamber slide. When a 70% confluence was reached, HeLa and A431 cells were deprived of serum for 24 h

To label EGFR, anti-EGFR antibody (10 nM; Thermo Scientific) was conjugated with Qdot525 (from Invitrogen, Carlsbad, CA, USA). Cells were incubated with the EGFR-Ab-Qdot525 for 15 min and washed three times with phosphate buffered saline (PBS). Fluorescent EGF (EGF-Qdot585) was synthesized by conjugating biotin-EGF (from Invitrogen) to Qdot585-streptavidin in PBS. To activate EGFRs, cells were incubated in the presence of 40 ng/mL EGF-Qdot585 [31, 39].

Figure 7 Linking accuracy of localized coordinates as a function of number density of Brownian particles. Trajectories of a group of particles diffusing in a spatial region with different particle densities and diffusion coefficients were simulated with Eq. (1) and then coarse-grained with the same sampling scheme as that used in our scanning light-sheet microscope.

3.3. Cell culture and reagents

and MCF12A cells for 3 h.

Single-particle trajectories were recorded for as long as 100 s, with a frame time of 25 ms. The localized coordinates of the nanosensors were extracted from a set of images acquired by synchronously scanning the light sheet and imaging focal plane. Connecting the acquired location coordinates to generate 3D trajectories was challenging. We first carried out multiple particle tracking by solving a linear-assignment problem [37] to identify the assignment matrix between the measured location coordinates and their predicted positions. A Kalman filter was also implemented to provide an optimal estimate of Brownian motion in the presence of Gaussian noise [38]. To verify the functionality of our linking method for 3D–trajectory generation, we simulated a group of particles diffusing in a spatial region with a different number of densities and diffusion coefficients. The resulting 3D trajectories were coarse-grained to yield time series of location coordinates with the same data-taking procedure as that used in our light-sheet microscope. The simulation results are shown in Figure 7, which indicated a particle density lower

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A 60 1.45 numerical aperture oil immersion objective lens (APON 60XOTIRFM, Olympus) was used to ensure both high spatial resolution and high photon collection efficiency. However, this objective lens had a limited depth of field (500 nm). For accurately resolving the depth of a fluorophore, we exploited the astigmatism created by a cylindrical lens (CL2). The combination of the CL2 (f = 100 cm and an imaging lens (f = 20 cm) with a separation of 5.5 cm generates an effective focal length of 17.5 cm on the sagittal focal plane and 20 cm on the tangential focal plane, which encodes the fluorophore depth as an elliptically shaped point spread function (PSF; see Figure 6a). We also inserted an electrically tunable lens (ETL) at the pupil plane of a 4f optical system (formed by the relay lens and the imaging lens) to enable fast acquisition of images at different depths. The fluorescence images of fluorophores on living cells were recorded with a scientific complementary metal-oxide semiconductor (sCMOS) camera (ORCA-Flash 4.0 V2, Hamamatsu). Fluorescence images of quantum dot nanosensors on a living HeLa cell acquired with this apparatus are displayed without CL2 in Figure 6b and with CL2 in Figure 6c, respectively. Elliptically shaped spots were observable, which were localized within 1 μm of the imaging plane at a lateral accuracy of 27 nm and an axial accuracy of 52 nm [17].

Figure 6 (a) Schematic of the light-sheet microscope used to record 3D trajectories of probing nanoparticles in a living cell. The excitation beam was shaped to form a 3 μm-thick light sheet, giving a Rayleigh range of 41 μm. A twodimensional (2D) scanner was inserted to move the light sheet by 34 μm along the y and z directions at the sample position. The imaging arm was perpendicular to the excitation direction, and an imaging plane in the sample was relayed and imaged to a sCMOS camera. The position of the imaging plane was adjusted using an ETL to yield a set of depthresolved images. An astigmatism was introduced using a CL2 to encode information on the fluorophore depth into an elliptically distorted point spread function. Fluorescence images of Qdot585 for a living HeLa cell acquired with this apparatus without CL2 (b) or with CL2 (c) are shown. (a) has been reproduced from ref. [17].

#### 3.2. Linking localized coordinates of nanosensors for generating 3D trajectories

Single-particle trajectories were recorded for as long as 100 s, with a frame time of 25 ms. The localized coordinates of the nanosensors were extracted from a set of images acquired by synchronously scanning the light sheet and imaging focal plane. Connecting the acquired location coordinates to generate 3D trajectories was challenging. We first carried out multiple particle tracking by solving a linear-assignment problem [37] to identify the assignment matrix between the measured location coordinates and their predicted positions. A Kalman filter was also implemented to provide an optimal estimate of Brownian motion in the presence of Gaussian noise [38]. To verify the functionality of our linking method for 3D–trajectory generation, we simulated a group of particles diffusing in a spatial region with a different number of densities and diffusion coefficients. The resulting 3D trajectories were coarse-grained to yield time series of location coordinates with the same data-taking procedure as that used in our light-sheet microscope. The simulation results are shown in Figure 7, which indicated a particle density lower than 0.01 =μm2, D < 0:1μm<sup>2</sup>=s, and a linking accuracy that could be higher than 98%.

#### 3.3. Cell culture and reagents

3. Apparatus and experimental procedures

along the y and z directions with an accuracy of 0.5 μm [17].

The schematic of our single-particle fluorescence microscopy apparatus with light-sheet excitation is shown in Figure 6a. The output beam from a solid-state laser with different wavelengths was shaped into a light sheet of 3 μm thickness at the beam waist, yielding a diffraction-limited beam propagation with a Rayleigh range of 41 μm in the x direction. By using a galvanometer scanner, the light sheet can be positioned at a sample in a range of 34 μm

A 60 1.45 numerical aperture oil immersion objective lens (APON 60XOTIRFM, Olympus) was used to ensure both high spatial resolution and high photon collection efficiency. However, this objective lens had a limited depth of field (500 nm). For accurately resolving the depth of a fluorophore, we exploited the astigmatism created by a cylindrical lens (CL2). The combination of the CL2 (f = 100 cm and an imaging lens (f = 20 cm) with a separation of 5.5 cm generates an effective focal length of 17.5 cm on the sagittal focal plane and 20 cm on the tangential focal plane, which encodes the fluorophore depth as an elliptically shaped point spread function (PSF; see Figure 6a). We also inserted an electrically tunable lens (ETL) at the pupil plane of a 4f optical system (formed by the relay lens and the imaging lens) to enable fast acquisition of images at different depths. The fluorescence images of fluorophores on living cells were recorded with a scientific complementary metal-oxide semiconductor (sCMOS) camera (ORCA-Flash 4.0 V2, Hamamatsu). Fluorescence images of quantum dot nanosensors on a living HeLa cell acquired with this apparatus are displayed without CL2 in Figure 6b and with CL2 in Figure 6c, respectively. Elliptically shaped spots were observable, which were localized within 1 μm of the imaging plane at a lateral accuracy of 27 nm and an axial accuracy

Figure 6 (a) Schematic of the light-sheet microscope used to record 3D trajectories of probing nanoparticles in a living cell. The excitation beam was shaped to form a 3 μm-thick light sheet, giving a Rayleigh range of 41 μm. A twodimensional (2D) scanner was inserted to move the light sheet by 34 μm along the y and z directions at the sample position. The imaging arm was perpendicular to the excitation direction, and an imaging plane in the sample was relayed and imaged to a sCMOS camera. The position of the imaging plane was adjusted using an ETL to yield a set of depthresolved images. An astigmatism was introduced using a CL2 to encode information on the fluorophore depth into an elliptically distorted point spread function. Fluorescence images of Qdot585 for a living HeLa cell acquired with this

apparatus without CL2 (b) or with CL2 (c) are shown. (a) has been reproduced from ref. [17].

3.1. Optical setup

108 Nonmagnetic and Magnetic Quantum Dots

of 52 nm [17].

HeLa and A431 cells were cultured in Dulbecco's Modified Eagle's medium (DMEM) without phenol red supplemented with 10% (v/v) fetal bovine serum. MCF12A cells were cultured in a 1:1 mixture of DMEM and Ham's F12 medium containing 20 ng/mL Human EGF, 0.01 mg/mL bovine insulin, 500 ng/mL hydrocortisone, and 5%(v/v) horse serum [39]. Before singlemolecule live-cell imaging was performed, the cells were plated in an eight-well chamber slide. When a 70% confluence was reached, HeLa and A431 cells were deprived of serum for 24 h and MCF12A cells for 3 h.

To label EGFR, anti-EGFR antibody (10 nM; Thermo Scientific) was conjugated with Qdot525 (from Invitrogen, Carlsbad, CA, USA). Cells were incubated with the EGFR-Ab-Qdot525 for 15 min and washed three times with phosphate buffered saline (PBS). Fluorescent EGF (EGF-Qdot585) was synthesized by conjugating biotin-EGF (from Invitrogen) to Qdot585-streptavidin in PBS. To activate EGFRs, cells were incubated in the presence of 40 ng/mL EGF-Qdot585 [31, 39].

Figure 7 Linking accuracy of localized coordinates as a function of number density of Brownian particles. Trajectories of a group of particles diffusing in a spatial region with different particle densities and diffusion coefficients were simulated with Eq. (1) and then coarse-grained with the same sampling scheme as that used in our scanning light-sheet microscope.

To sequester cholesterol molecules on the plasma membranes, cells were treated with 10 μg/ml nystatin for 1 h before staining with either the antibody or EGF. To disrupt the actin filaments, the cells were pretreated with 10 μm cytochalasin D (Cyto D) for 1 h.

conjugated QD nanosensors in a live cell study, we will briefly review some previous results of

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

4.1. Ligand binding induced receptor protein translocation in plasma membranes of living

Typical single-molecule tracks of unliganded Qdot525-Ab-EGFR and liganded Qdot585-EGF-EGFR on live cells exhibited confined diffusion interspaced by directed movement [31]. We binned the measured MSD in a histogram to deduce the probability density function of the diffusion coefficient. Figure 8a shows the data taken at a frame rate of τ ¼ 25 ms. For unliganded EGFR at rest, two sets of diffusers were observed with the diffusion coefficient of

/s and the slower one at 0.3 μm2

the population of 0:3μm<sup>2</sup>=s < D < 6 μm<sup>2</sup>=s, whereas it increased the populations of D <sup>D</sup> <sup>≤</sup> <sup>0</sup>:<sup>1</sup> <sup>μ</sup>m<sup>2</sup>=<sup>s</sup> and D <sup>¼</sup> <sup>9</sup> <sup>μ</sup>m<sup>2</sup>=s. It is clearly shown that receptor ligation can affect the diffu-

We used Cytochalasin D to disrupt the cellular actin frameworks. As presented in Figure 8b, the major population of unliganded Qdot525-Ab-EGFR (open symbols) on EGF-activated cells shifted from the slow state (D < 0:1 μm<sup>2</sup>=s) to the fast state (0:1μm<sup>2</sup>=s < D < 2 μm<sup>2</sup>=s) after Cytochalasin D treatment. This influence is even more pronounced on liganded Qdot585-EGF-EGFR (filled symbols). As pointed out previously, the primary effector controlling the motion of a protein in a living cell is often not due to the friction in the cellular medium but the interactions with its molecular partners and microenvironment. To illustrate this subject further, we used the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> technique to analyze the statistics of single-molecule trajectories,

For live HeLa cells at rest, the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of measured tracks of unliganded Qdot525-Ab-EGFR are presented in Figure 9a. In this plot, peak 2 was the most populated and stable state

Figure 8 (a) Histogram of the diffusion coefficient of unliganded EGFR (open red circles) in the cells at rest and the liganded EGFR (solid blue squares) in EGF-activated cells. (b) Histogram of the diffusion coefficient of unliganded EGFR (open symbols) and the singly liganded species (filled symbols) on activated HeLa cells without (circles) or with (squares)

/s. EGF activation suppressed

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applying single-molecule tracking techniques to EGFR studies [31, 39].

sion of EGFR and leads to a population change in the two diffusion states.

specifically focusing on the slow diffusion of EGFR.

cells

the fast species peaking at 9 μm<sup>2</sup>

Cytochalasin D pretreatment.

The N terminals of the Tat peptides (from Invitrogen) were biotinylated. Conjugated TatP-QDs were prepared by incubating 20-nm diameter streptavidin-coated Qdot585 in PBS with excess biotinylated TatP (5 μm TatP:50 nM Qdot585) at room temperature for 30 min. Although streptavidin is a tetramer and each subunit can bind biotin with equal affinity, the covalent attachment of streptavidin to the surface of a quantum dot makes two of the four binding sites inaccessible to the biotinylated TatP. As each QD has approximately 5 to 10 streptavidin molecules on its surface, we estimated that an average of 14 Tat peptides was conjugated to each QD.

## 4. Experimental results

The plasma membrane of a living cell is not merely a sea of lipids and proteins, but is more complex with individual components organized into spatially distinct compartments to yield strategic advantages for protein function and signaling [40]. Organizing proteins and lipids into nanodomains could also shield these assemblies from other proteins to tailor specific interactions, thereby mediating signal transduction to relay cellular messages from the external environment to the nucleus [41, 42].

The first event of cellular signaling occurs at various types of receptor proteins in the plasma membrane. To faithfully sense a signal that varies in space and time, live cells face an optimization problem of placing a set of distributed and mobile receptors by balancing two opposing objectives [43]: 1) the need to locally assemble the nanosensors to reduce the estimation noise; and 2) the need to spread these nanosensors to reduce spatial sensing errors. Receptor signaling dysregulation is attributed to the pathogenesis of several diseases [44, 45]. Therefore, understanding the interactions, molecular processes and relevant structures of such signaling assemblies is imperative. One such receptor protein is the epidermal growth factor receptor (EGFR), which can drive cell growth and survival [44]. There is tremendous interest in unraveling how those transducing proteins diffuse and interact on plasma membranes of living cells. However, it remains a challenge to study these cellular events at the singlemolecule level in a live cell as living cells are highly heterogeneous and stochastically dynamic.

Based on our current knowledge of molecular diffusion in the plasma membrane, there are two types of interactions between a receptor and its local environment [46, 47]. First, the protein can induce a local ordering of the surrounding lipid molecules via a lipid-protein interaction. In addition, the cortical actin framework can induce membrane compartments [30]. To study the diffusing behaviors of EGFRs and the interaction with their cellular environment, we tagged EGFRs with antibody-conjugated quantum dots (Qdot525-Ab) and exploited fluorescent EGF, which was synthesized by conjugating EGF with quantum dots (QD585- EGF), to activate the EGFRs. By using this scheme, we could study the diffusive dynamics of paired EGFRs by selecting a pair of liganded and unliganded EGFR or a pair of liganded EGFRs, and follow their relative motions [31]. To appreciate the potential of biomoleculeconjugated QD nanosensors in a live cell study, we will briefly review some previous results of applying single-molecule tracking techniques to EGFR studies [31, 39].

To sequester cholesterol molecules on the plasma membranes, cells were treated with 10 μg/ml nystatin for 1 h before staining with either the antibody or EGF. To disrupt the actin filaments,

The N terminals of the Tat peptides (from Invitrogen) were biotinylated. Conjugated TatP-QDs were prepared by incubating 20-nm diameter streptavidin-coated Qdot585 in PBS with excess biotinylated TatP (5 μm TatP:50 nM Qdot585) at room temperature for 30 min. Although streptavidin is a tetramer and each subunit can bind biotin with equal affinity, the covalent attachment of streptavidin to the surface of a quantum dot makes two of the four binding sites inaccessible to the biotinylated TatP. As each QD has approximately 5 to 10 streptavidin molecules on its surface, we estimated that an average of 14 Tat peptides was conjugated to each QD.

The plasma membrane of a living cell is not merely a sea of lipids and proteins, but is more complex with individual components organized into spatially distinct compartments to yield strategic advantages for protein function and signaling [40]. Organizing proteins and lipids into nanodomains could also shield these assemblies from other proteins to tailor specific interactions, thereby mediating signal transduction to relay cellular messages from the exter-

The first event of cellular signaling occurs at various types of receptor proteins in the plasma membrane. To faithfully sense a signal that varies in space and time, live cells face an optimization problem of placing a set of distributed and mobile receptors by balancing two opposing objectives [43]: 1) the need to locally assemble the nanosensors to reduce the estimation noise; and 2) the need to spread these nanosensors to reduce spatial sensing errors. Receptor signaling dysregulation is attributed to the pathogenesis of several diseases [44, 45]. Therefore, understanding the interactions, molecular processes and relevant structures of such signaling assemblies is imperative. One such receptor protein is the epidermal growth factor receptor (EGFR), which can drive cell growth and survival [44]. There is tremendous interest in unraveling how those transducing proteins diffuse and interact on plasma membranes of living cells. However, it remains a challenge to study these cellular events at the singlemolecule level in a live cell as living cells are highly heterogeneous and stochastically dynamic. Based on our current knowledge of molecular diffusion in the plasma membrane, there are two types of interactions between a receptor and its local environment [46, 47]. First, the protein can induce a local ordering of the surrounding lipid molecules via a lipid-protein interaction. In addition, the cortical actin framework can induce membrane compartments [30]. To study the diffusing behaviors of EGFRs and the interaction with their cellular environment, we tagged EGFRs with antibody-conjugated quantum dots (Qdot525-Ab) and exploited fluorescent EGF, which was synthesized by conjugating EGF with quantum dots (QD585- EGF), to activate the EGFRs. By using this scheme, we could study the diffusive dynamics of paired EGFRs by selecting a pair of liganded and unliganded EGFR or a pair of liganded EGFRs, and follow their relative motions [31]. To appreciate the potential of biomolecule-

the cells were pretreated with 10 μm cytochalasin D (Cyto D) for 1 h.

4. Experimental results

110 Nonmagnetic and Magnetic Quantum Dots

nal environment to the nucleus [41, 42].

### 4.1. Ligand binding induced receptor protein translocation in plasma membranes of living cells

Typical single-molecule tracks of unliganded Qdot525-Ab-EGFR and liganded Qdot585-EGF-EGFR on live cells exhibited confined diffusion interspaced by directed movement [31]. We binned the measured MSD in a histogram to deduce the probability density function of the diffusion coefficient. Figure 8a shows the data taken at a frame rate of τ ¼ 25 ms. For unliganded EGFR at rest, two sets of diffusers were observed with the diffusion coefficient of the fast species peaking at 9 μm<sup>2</sup> /s and the slower one at 0.3 μm2 /s. EGF activation suppressed the population of 0:3μm<sup>2</sup>=s < D < 6 μm<sup>2</sup>=s, whereas it increased the populations of D <sup>D</sup> <sup>≤</sup> <sup>0</sup>:<sup>1</sup> <sup>μ</sup>m<sup>2</sup>=<sup>s</sup> and D <sup>¼</sup> <sup>9</sup> <sup>μ</sup>m<sup>2</sup>=s. It is clearly shown that receptor ligation can affect the diffusion of EGFR and leads to a population change in the two diffusion states.

We used Cytochalasin D to disrupt the cellular actin frameworks. As presented in Figure 8b, the major population of unliganded Qdot525-Ab-EGFR (open symbols) on EGF-activated cells shifted from the slow state (D < 0:1 μm<sup>2</sup>=s) to the fast state (0:1μm<sup>2</sup>=s < D < 2 μm<sup>2</sup>=s) after Cytochalasin D treatment. This influence is even more pronounced on liganded Qdot585-EGF-EGFR (filled symbols). As pointed out previously, the primary effector controlling the motion of a protein in a living cell is often not due to the friction in the cellular medium but the interactions with its molecular partners and microenvironment. To illustrate this subject further, we used the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> technique to analyze the statistics of single-molecule trajectories, specifically focusing on the slow diffusion of EGFR.

For live HeLa cells at rest, the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of measured tracks of unliganded Qdot525-Ab-EGFR are presented in Figure 9a. In this plot, peak 2 was the most populated and stable state

Figure 8 (a) Histogram of the diffusion coefficient of unliganded EGFR (open red circles) in the cells at rest and the liganded EGFR (solid blue squares) in EGF-activated cells. (b) Histogram of the diffusion coefficient of unliganded EGFR (open symbols) and the singly liganded species (filled symbols) on activated HeLa cells without (circles) or with (squares) Cytochalasin D pretreatment.

peaks at (0.006, 1.24), (0.014, 1.18), and (0.031,1.12), suggesting that lipid raft domains on activated cells play a minor role in restraining the motion of unliganded Qdot525-Ab-EGFR. We proposed the following picture to explain our experimental results: Unliganded EGFRs at rest may locate outside the cholesterol-enriched lipid domains. EGF binding causes the receptors to move into the cholesterol-enriched lipid domains. Pretreatment of cells with nystatin, which can disrupt these lipid domains, results in local environmental changes of the ligandbound EGFR. This interpretation is further supported by the observations shown in Figure 9d and e in which nystatin pretreatment did not alter the peak positions of unliganded EGFR on

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The V R<sup>2</sup> � � peak values of unliganded EGFRs in native cells at rest were near the free diffusion limit (Figure 9a). However, those unliganded species could have higher V R<sup>2</sup> � � peak values in highly EGF-activated cells due to much stronger confinements from EGF-promoted actin polymerization [30]. EGF ligation reduced the V R<sup>2</sup> � � values of Qdot585-EGF-EGFR to below the free diffusion limit (Figure 9b) due to the dressing effect by the lipid domain. These experimental findings were verified in three cell lines; including two cancer cell lines (HeLa and A431) and one non-tumorigenic breast epithelial cell line (MCF12A). These cell lines possess a wide range of EGFR expression levels and concentrations of membrane cholesterol. Therefore, the experimental results may represent a general behavior of unliganded and

Receptor dimerization plays a critical role in initializing a signal cascade [48]. Do nearby receptor proteins move correlatively prior to dimer formation? Imagine when a receptor protein moves in the plasma membrane of a live cell, it may induce order in its surrounding lipid molecules through the protein-lipid interaction. A receptor protein and the induced lipid ordering can be viewed as a lipid-dressed protein. As two nearby proteins move in the plasma

We can simulate the diffusive behaviors of two dressed proteins in proximity using coupled Langevin equations [49]. To display the mutual correlation between the two trajectories quan-

!

P t

The summations were taken over a time mesh along the single-molecule tracks. By using this approach, we simulated the correlated motion of two Brownian-like particles with their spatial separation perturbed by a correlated thermal fluctuation [49]. We carried out the simulations

randomly within a 1-μm radius circle centered at (0,0). The coupled Langevin equations were

<sup>k</sup>ðÞ¼ <sup>t</sup> Akð Þ<sup>t</sup> <sup>e</sup><sup>i</sup>θ<sup>k</sup> ð Þ<sup>t</sup> , and defined the degree of

t A2 2

<sup>q</sup> : (6)

<sup>1</sup> ¼ ð Þ 0; 0 and places the other

<sup>A</sup>1ð Þ<sup>t</sup> <sup>A</sup>2ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> cos ½ � <sup>θ</sup>2ð Þ� <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>θ</sup>1ð Þ<sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P <sup>t</sup>A<sup>1</sup> 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>P</sup>

!

4.2. Correlated motion of receptor proteins in plasma membrane of live cells

membrane, they may interact with each other through the ordered lipid molecules.

� � �

EGF-activated cells.

activated receptors in live cells.

mutual correlation as

Cð Þ¼ τ Re

titatively, we expressed the position vectors as x

P x ! 1 ∗ ð Þ� t x ! <sup>2</sup>ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P x ! <sup>1</sup>ð Þt � � �

� � � <sup>2</sup> r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

from an initial condition that positioned one receptor at x

P x ! <sup>2</sup>ð Þt � � �

<sup>2</sup> r

Figure 9 (a) The plot of V R<sup>2</sup> <sup>R</sup><sup>2</sup> for unliganded Qdot525-Ab-EGFR on live HeLa cells at rest. Simulated curves of the peak positions for receptor molecules under free Brownian motion (red dash line), diffusive motion with the confinement from actin corrals alone (green dash line), or both the actin corrals and lipid raft domains (blue dash line) are included for comparison. The same plot is shown for liganded Qdot585-EGF-EGFR in (b) EGF-activated cells, and (c) activated cells pretreated with nystatin. (d) Unliganded Qdot525-Ab-EGFR on live EGF-activated HeLa cells. (e) the same plot as (d) for unliganded Qdot525-Ab-EGFR on activated cells pretreated with nystatin. (a–c) have been reproduced from ref. [31].

among the three peaks detected, located at the <sup>R</sup><sup>2</sup> ; V R<sup>2</sup> coordinates of (0.01, 1.45), (0.02,

1.39), and (0.04, 1.33), respectively. Three simulated curves were plotted for proteins under free Brownian motion (red dash line), confinement by actin corrals alone (green dash line) or by both actin corrals and lipid domains (blue dash curves) and presented in Figure 9a for comparison. The peak positions of the three states clearly fell on the curve of the actin confinement, indicating that these unliganded receptor molecules were not free diffusers, but instead confined by the actin corrals alone.

With the EGFR at rest as the control, we proceeded to examine the diffusion of liganded Qdot585-EGF-EGFR. Figure 9b shows the V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot of the liganded EGFR on activated HeLa cells. Three peaks were found at (0.01, 0.42), (0.02, 0.47), and (0.04, 0.54). These peak positions agreed better with the model that included the confinement effects of actin corrals and lipid raft domains. We used nystatin on the cells to sequester the membrane cholesterol and disrupt the lipid raft domains. Figure 9c displays the V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot of the liganded EGFR on EGF-activated cells pretreated by nystatin. The three peaks of Figure 9c, at (0.01, 1.39), (0.02, 1.33), and (0.04, 1.28), fell again on the curve for the actin confinement model.

For unliganded Qdot525-Ab-EGFR on EGF-activated cells, the corresponding V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot is presented in Figure 9d. Although peak 1 appeared to be produced by the free diffusion proteins, peaks 2 and 3 agree with the confinement model of the actin corrals alone. Qdot525-Ab-EGFR on the nystatin-pretreated cells (shown in Figure 9e) revealed similar peaks at (0.006, 1.24), (0.014, 1.18), and (0.031,1.12), suggesting that lipid raft domains on activated cells play a minor role in restraining the motion of unliganded Qdot525-Ab-EGFR.

We proposed the following picture to explain our experimental results: Unliganded EGFRs at rest may locate outside the cholesterol-enriched lipid domains. EGF binding causes the receptors to move into the cholesterol-enriched lipid domains. Pretreatment of cells with nystatin, which can disrupt these lipid domains, results in local environmental changes of the ligandbound EGFR. This interpretation is further supported by the observations shown in Figure 9d and e in which nystatin pretreatment did not alter the peak positions of unliganded EGFR on EGF-activated cells.

The V R<sup>2</sup> � � peak values of unliganded EGFRs in native cells at rest were near the free diffusion limit (Figure 9a). However, those unliganded species could have higher V R<sup>2</sup> � � peak values in highly EGF-activated cells due to much stronger confinements from EGF-promoted actin polymerization [30]. EGF ligation reduced the V R<sup>2</sup> � � values of Qdot585-EGF-EGFR to below the free diffusion limit (Figure 9b) due to the dressing effect by the lipid domain. These experimental findings were verified in three cell lines; including two cancer cell lines (HeLa and A431) and one non-tumorigenic breast epithelial cell line (MCF12A). These cell lines possess a wide range of EGFR expression levels and concentrations of membrane cholesterol. Therefore, the experimental results may represent a general behavior of unliganded and activated receptors in live cells.

#### 4.2. Correlated motion of receptor proteins in plasma membrane of live cells

among the three peaks detected, located at the <sup>R</sup><sup>2</sup> ; V R<sup>2</sup>

confined by the actin corrals alone.

112 Nonmagnetic and Magnetic Quantum Dots

1.39), and (0.04, 1.33), respectively. Three simulated curves were plotted for proteins under free Brownian motion (red dash line), confinement by actin corrals alone (green dash line) or by both actin corrals and lipid domains (blue dash curves) and presented in Figure 9a for comparison. The peak positions of the three states clearly fell on the curve of the actin confinement, indicating that these unliganded receptor molecules were not free diffusers, but instead

Figure 9 (a) The plot of V R<sup>2</sup> <sup>R</sup><sup>2</sup> for unliganded Qdot525-Ab-EGFR on live HeLa cells at rest. Simulated curves of the peak positions for receptor molecules under free Brownian motion (red dash line), diffusive motion with the confinement from actin corrals alone (green dash line), or both the actin corrals and lipid raft domains (blue dash line) are included for comparison. The same plot is shown for liganded Qdot585-EGF-EGFR in (b) EGF-activated cells, and (c) activated cells pretreated with nystatin. (d) Unliganded Qdot525-Ab-EGFR on live EGF-activated HeLa cells. (e) the same plot as (d) for unliganded Qdot525-Ab-EGFR on activated cells pretreated with nystatin. (a–c) have been reproduced from ref. [31].

With the EGFR at rest as the control, we proceeded to examine the diffusion of liganded Qdot585-EGF-EGFR. Figure 9b shows the V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot of the liganded EGFR on activated HeLa cells. Three peaks were found at (0.01, 0.42), (0.02, 0.47), and (0.04, 0.54). These peak positions agreed better with the model that included the confinement effects of actin corrals and lipid raft domains. We used nystatin on the cells to sequester the membrane cholesterol and disrupt the lipid raft domains. Figure 9c displays the V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot of the liganded EGFR on EGF-activated cells pretreated by nystatin. The three peaks of Figure 9c, at (0.01, 1.39), (0.02, 1.33), and (0.04, 1.28), fell again on the curve for the actin confinement model.

For unliganded Qdot525-Ab-EGFR on EGF-activated cells, the corresponding V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot is presented in Figure 9d. Although peak 1 appeared to be produced by the free diffusion proteins, peaks 2 and 3 agree with the confinement model of the actin corrals alone. Qdot525-Ab-EGFR on the nystatin-pretreated cells (shown in Figure 9e) revealed similar

coordinates of (0.01, 1.45), (0.02,

Receptor dimerization plays a critical role in initializing a signal cascade [48]. Do nearby receptor proteins move correlatively prior to dimer formation? Imagine when a receptor protein moves in the plasma membrane of a live cell, it may induce order in its surrounding lipid molecules through the protein-lipid interaction. A receptor protein and the induced lipid ordering can be viewed as a lipid-dressed protein. As two nearby proteins move in the plasma membrane, they may interact with each other through the ordered lipid molecules.

We can simulate the diffusive behaviors of two dressed proteins in proximity using coupled Langevin equations [49]. To display the mutual correlation between the two trajectories quantitatively, we expressed the position vectors as x ! <sup>k</sup>ðÞ¼ <sup>t</sup> Akð Þ<sup>t</sup> <sup>e</sup><sup>i</sup>θ<sup>k</sup> ð Þ<sup>t</sup> , and defined the degree of mutual correlation as

$$\mathbf{C}(\tau) = \text{Re}\left[\frac{\sum \overrightarrow{\mathbf{x}}\_1^\*(t) \cdot \overrightarrow{\mathbf{x}}\_2(t+\tau)}{\sqrt{\sum \left|\overrightarrow{\mathbf{x}}\_1(t)\right|^2} \sqrt{\sum \left|\overrightarrow{\mathbf{x}}\_2(t)\right|^2}}\right] = \frac{\sum\_l A\_1(t) A\_2(t+\tau) \cos\left[\theta\_2(t+\tau) - \theta\_1(t)\right]}{\sqrt{\sum\_l A\_1^2} \sqrt{\sum\_l A\_2^2}}.\tag{6}$$

The summations were taken over a time mesh along the single-molecule tracks. By using this approach, we simulated the correlated motion of two Brownian-like particles with their spatial separation perturbed by a correlated thermal fluctuation [49]. We carried out the simulations from an initial condition that positioned one receptor at x ! <sup>1</sup> ¼ ð Þ 0; 0 and places the other randomly within a 1-μm radius circle centered at (0,0). The coupled Langevin equations were then solved to generate a pair of trajectories, and the degree of mutual correlation was calculated using Eq. (6). Figure 10a and b display the histograms of simulated trajectories with mutual correlations of 0 and 0.5, respectively. Figure 10c illustrates the histogram of correlation in experimental single-molecule tracks of paired Qdot525-Ab-EGFR and Qdot585-EGF-EGFR on live HeLa cells. As shown, the tracks clearly exhibited a correlation peak at 0.5.

The V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot of the correlated motion between two liganded EGFRs is presented in Figure 11C. Compared to the plot shown in Figure 11b, the V R<sup>2</sup> values of the two major states were decreased when the unliganded companion of Figure 11b was replaced by the liganded EGFR, perhaps because the lipid raft domains surrounding the two receptor mole-

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

Ligand binding promotes receptor dimerization and leads to a downstream signaling cascade. Researchers have increasingly determined that lipid domains rich in raft sphingolipids (GM1) and cholesterol can facilitate signaling receptors to form a dimer [50–52]. The recent identification of cholesterol-dependent nanoassemblies with biophysical techniques also suggests that a cholesterol-mediated interaction exists between lipid domains to affect the organization, sta-

We selected and analyzed those highly correlated segments from single-molecule trajectories to

plots of paired Qdot585-EGF-EGFRs in three different cell lines. The contour plots are more

which may be attributed to effective receptor-lipid and receptor-receptor interactions in these cells [33, 34]. To inspect the nature of the interactions and their relevance to receptor-induced lipid ordering, we again took advantages of the drug effects of nystatin and MβCD. Figure 12b

Correlated Qdot585-EGF-EGFRs appeared to have a weaker interaction in the nystatin-treated

<sup>2</sup> value of the correlated Qdot585-EGF-EGFRs is considerably lower in A431 cells,

<sup>2</sup> plots for the three cell lines pretreated with nystatin and MβCD.

<sup>2</sup> plots of correlated Qdot585-EGF-EGFRs diffusing in the plasma membrane of (a) native cells, (b)

nystatin-pretreated cells, and (c) MβCD-pretreated cells. Trajectory segments with a degree of correlation exceeding 0.8 were selected for analysis. Data are shown in red for HeLa cells, green for A431 cells, and blue for MCF12A cells. This

<sup>2</sup> value. This observation may be explained by a

<sup>2</sup> -R<sup>τ</sup>

http://dx.doi.org/10.5772/intechopen.72858

2

115

reveal the influence of cholesterol-mediated interactions [39]. Figure 12 displays the V R<sup>τ</sup>

scattered, indicating that these data are indeed highly sensitive to receptor interaction.

cules merged and yielded a larger dressing force during the highly correlated motion.

4.3. Cholesterol-mediated interaction between liganded EGF receptors

bility, and function of membrane receptor proteins [52–54].

<sup>2</sup> -R<sup>τ</sup>

A431 cells as evidenced by an increased V R<sup>τ</sup>

The V R<sup>τ</sup>

Figure 12 V R<sup>τ</sup>

<sup>2</sup> -R<sup>τ</sup>

figure has been reproduced from ref. [39].

and c display the V R<sup>τ</sup>

By using this method, we were able to select those highly correlated segments from the singlemolecule tracks and analyzed the correlated motion [31]. We plotted the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of the motion of the unliganded Qdot525-Ab-EGFR correlated with liganded Qdot585-EGF-EGFR in Figure 11a. As the unliganded Qdot525-Ab-EGFR moved correlatively with a nearby liganded Qdot585-EGF-EGFR, the diffusion motility R<sup>2</sup> of state 1 decreased drastically to near 10�<sup>3</sup> , accompanied by a reduction of V R<sup>2</sup> to 0.1. It is interesting to note that the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of the reverse case (i.e., Qdot585-EGF-EGFR relative to Qdot525-Ab-EGFR) differed in R<sup>2</sup> (Figure 11b). The resident time of the liganded EGFR in state 2 became longer, and the V R<sup>2</sup> of both states 1 and 2 increased to 1, indicating that the liganded and unliganded EGFR resided in different lipid environments.

Figure 10 Histogram of the degree of correlation in simulated trajectories of (a) independent (Cð Þ¼ τ 0) or (b) correlated (Cð Þ¼ τ 0:5) diffusing particles. (c) Histogram of the degree of correlation existing in experimental trajectories of unliganded Qdot525-Ab-EGFR and liganded Qdot585-EGF-EGFR.

Figure 11 V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of correlated motion of dual EGFRs. (a) Unliganded Qdot525-Ab-EGFR correlatively moving with a nearby liganded Qdot585-EGF-EGFR companion, (b) liganded Qdot585-EGF-EGFR correlatively moving with a nearby unliganded Qdot525-Ab-EGFR, and (c) correlated motion of dual liganded EGFRs. This figure has been reproduced from ref. [31].

The V R<sup>2</sup> <sup>R</sup><sup>2</sup> plot of the correlated motion between two liganded EGFRs is presented in Figure 11C. Compared to the plot shown in Figure 11b, the V R<sup>2</sup> values of the two major states were decreased when the unliganded companion of Figure 11b was replaced by the liganded EGFR, perhaps because the lipid raft domains surrounding the two receptor molecules merged and yielded a larger dressing force during the highly correlated motion.

#### 4.3. Cholesterol-mediated interaction between liganded EGF receptors

then solved to generate a pair of trajectories, and the degree of mutual correlation was calculated using Eq. (6). Figure 10a and b display the histograms of simulated trajectories with mutual correlations of 0 and 0.5, respectively. Figure 10c illustrates the histogram of correlation in experimental single-molecule tracks of paired Qdot525-Ab-EGFR and Qdot585-EGF-EGFR on live HeLa cells. As shown, the tracks clearly exhibited a correlation peak at 0.5.

By using this method, we were able to select those highly correlated segments from the singlemolecule tracks and analyzed the correlated motion [31]. We plotted the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of the motion of the unliganded Qdot525-Ab-EGFR correlated with liganded Qdot585-EGF-EGFR in Figure 11a. As the unliganded Qdot525-Ab-EGFR moved correlatively with a nearby liganded Qdot585-EGF-EGFR, the diffusion motility R<sup>2</sup> of state 1 decreased drastically to near 10�<sup>3</sup>

accompanied by a reduction of V R<sup>2</sup> to 0.1. It is interesting to note that the V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of the reverse case (i.e., Qdot585-EGF-EGFR relative to Qdot525-Ab-EGFR) differed in R<sup>2</sup> (Figure 11b). The resident time of the liganded EGFR in state 2 became longer, and the V R<sup>2</sup> of both states 1 and 2 increased to 1, indicating that the liganded and unliganded EGFR

Figure 10 Histogram of the degree of correlation in simulated trajectories of (a) independent (Cð Þ¼ τ 0) or (b) correlated (Cð Þ¼ τ 0:5) diffusing particles. (c) Histogram of the degree of correlation existing in experimental trajectories of

Figure 11 V R<sup>2</sup> � <sup>R</sup><sup>2</sup> plot of correlated motion of dual EGFRs. (a) Unliganded Qdot525-Ab-EGFR correlatively moving with a nearby liganded Qdot585-EGF-EGFR companion, (b) liganded Qdot585-EGF-EGFR correlatively moving with a nearby unliganded Qdot525-Ab-EGFR, and (c) correlated motion of dual liganded EGFRs. This figure has been

resided in different lipid environments.

114 Nonmagnetic and Magnetic Quantum Dots

unliganded Qdot525-Ab-EGFR and liganded Qdot585-EGF-EGFR.

reproduced from ref. [31].

,

Ligand binding promotes receptor dimerization and leads to a downstream signaling cascade. Researchers have increasingly determined that lipid domains rich in raft sphingolipids (GM1) and cholesterol can facilitate signaling receptors to form a dimer [50–52]. The recent identification of cholesterol-dependent nanoassemblies with biophysical techniques also suggests that a cholesterol-mediated interaction exists between lipid domains to affect the organization, stability, and function of membrane receptor proteins [52–54].

We selected and analyzed those highly correlated segments from single-molecule trajectories to reveal the influence of cholesterol-mediated interactions [39]. Figure 12 displays the V R<sup>τ</sup> <sup>2</sup> -R<sup>τ</sup> 2 plots of paired Qdot585-EGF-EGFRs in three different cell lines. The contour plots are more scattered, indicating that these data are indeed highly sensitive to receptor interaction.

The V R<sup>τ</sup> <sup>2</sup> value of the correlated Qdot585-EGF-EGFRs is considerably lower in A431 cells, which may be attributed to effective receptor-lipid and receptor-receptor interactions in these cells [33, 34]. To inspect the nature of the interactions and their relevance to receptor-induced lipid ordering, we again took advantages of the drug effects of nystatin and MβCD. Figure 12b and c display the V R<sup>τ</sup> <sup>2</sup> -R<sup>τ</sup> <sup>2</sup> plots for the three cell lines pretreated with nystatin and MβCD. Correlated Qdot585-EGF-EGFRs appeared to have a weaker interaction in the nystatin-treated A431 cells as evidenced by an increased V R<sup>τ</sup> <sup>2</sup> value. This observation may be explained by a

Figure 12 V R<sup>τ</sup> <sup>2</sup> -R<sup>τ</sup> <sup>2</sup> plots of correlated Qdot585-EGF-EGFRs diffusing in the plasma membrane of (a) native cells, (b) nystatin-pretreated cells, and (c) MβCD-pretreated cells. Trajectory segments with a degree of correlation exceeding 0.8 were selected for analysis. Data are shown in red for HeLa cells, green for A431 cells, and blue for MCF12A cells. This figure has been reproduced from ref. [39].

less stable lipid domain due to a lower amount of cholesterol, which resulted in a larger variance in the diffusing step size of the correlated receptors. In contrast, the interaction became stronger in nystatin-treated MCF-12A cells, suggesting the effect of the cholesterol-mediated interaction was opposite to that of the receptor-lipid interaction. The V R<sup>τ</sup> <sup>2</sup> of correlated Qdot585-EGF-EGFR in A431 increased by two orders of magnitude from 10�<sup>2</sup> for native cells to 1 for MβCD treated cells. Of note, the V R<sup>τ</sup> <sup>2</sup> value could be increased to higher than 10 in MβCD treated HeLa and MCF-12A cells, which suggests that a deterministic interaction dominated because the screening effect from the membrane cholesterol was reduced after membrane cholesterol was depleted. These results identified a vital role for membrane cholesterol in mediating the interaction between liganded receptors in the three cell lines.

#### 4.4. Nonraft lipids and sphingolipids in live plasma membranes segregate into separated nanodomains

Previous data analysis implicitly assumed the coexistence of different lipid phases in plasma membranes. Indeed, lipid–lipid interactions were capable of inducing liquid ordered (Lo) liquid disordered (Ld) phase coexistence in model lipid membranes [55, 56]. It was conjectured that plasma membrane composition is poised for selective and functional raft clustering at physiological temperatures [57]. However, such lipid nanodomains have remained largely unresolved in the plasma membrane of living cells. Researchers recently used a fluorescence correlation technique to successfully distinguish between free and anomalous molecular diffusion in a 30-nm focal spot of a stimulated emission depletion (STED) nanoscope [58]. The observed differences were attributed to transient cholesterol-assisted and cytoskeletondependent binding of sphingolipids to other membrane constituents. However, the optical force acting on the highly excited lipid molecules by the STED spot may not be negligible.

In our current study, we investigated lipid nanodomains in live plasma membranes at a much lower excitation level with light-sheet microscopy. We probed the nonraft lipids in living HeLa cells with carbocyanine dyes 1,1<sup>0</sup> -didodecyl- 3,3,3<sup>0</sup> ,30 -tetramethylindocarbocyanine perchlorate (DiI-C12), which serves as an excellent lipophilic fluorescent probe with a strong partition tendency into the Ld phase [59]. The preference originated from the fact that highly packed lipids in Lo phase usually exclude exogenous molecules. BODIPY FL C5-ganglioside GM1 (BODIPY C5-GM1) from ThermoFisher Scientific was used as a direct indication of lipid rafts. By synchronously adjusting the light-sheet position and the focal plane of the high NA objective lens, we were alble to acquire a 3D image of DiI-C12 and BODIPY C5-GM1 in a living HeLa cell. Figure 13a displays a typical image of the scan with DiI-C12 shown in blue and BODIPY C5-GM1 in orange. The 3D point spread function (PSF) of the light-sheet microscope was deduced (shown in Figure 13b) using a fluorescent bead with a diameter of 100 nm. We exploited this PSF to deconvolve the image [60] of DiI-C12 and BODIPY C5-GM1. The resulting deconvolved image is shown in Figure 13a.

To retrieve information about the lipid clustering process from the measured pair correlation, we simulated lipid clustering dynamics. First, we randomly distributed M clusters in an imaging region with each cluster containing N molecules in a circle with diameter R. Figure 15a illustrates an example of two randomly distributed clusters (M = 2) at a spatial resolution of 2 nm. We then binned the molecules in a cluster to the pixel size of the camera used (see Figure 15b). By using these cluster images, we calculated the auto- and crosscorrelation functions with three different lipid clustering models: 1) a random clustering model, 2) an aggregation model, and 3) a segregation model. The random clustering model assumed that two lipid species were independently and randomly distributed in an area. In contrast, the lipid molecules in the aggregation and segregation models were stochastically distributed in each cluster with a Gaussian distribution. There is a major difference between

Figure 14 The auto-(left) and cross-(right) correlation function of DiI-C12 and BODIPY C5-GM1 calculated from the

Figure 13 (a) 3D fluorescence distribution (left) and its deconvoluted image (right) from DiI-C12 (blue) and BODIPY C5-GM1 (orange) fluorophores in a living HeLa cell. The cross section (z ¼ -2:25 μm) of the distribution was shown below. (b)the point spread function of the light-sheet microscope on the xy- (left) and xz-plane (right) emitted from a

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fluorescent bead with a diameter of 100 nm.

deconvoluted image of Figure 13a.

Following the formalism developed by Veatch et al. [61], we calculated both the auto-correlation function A rð Þ¼ h i <sup>r</sup>ð Þ <sup>R</sup> <sup>r</sup>ð Þ <sup>R</sup> � <sup>r</sup> <sup>=</sup>h i <sup>r</sup>ð Þ<sup>r</sup> <sup>2</sup> and the cross-correlation function C rð Þ¼ <sup>h</sup>r1ð Þ <sup>R</sup> <sup>r</sup>2ðR� rÞi= r<sup>1</sup> h i ð Þr r<sup>2</sup> ð Þ h i ð Þr of DiI-C12 and BODIPY C5-GM1. The results are presented in Figure 14, which indicates that both species may form clusters with an average diameter of less than 150 nm. Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells http://dx.doi.org/10.5772/intechopen.72858 117

less stable lipid domain due to a lower amount of cholesterol, which resulted in a larger variance in the diffusing step size of the correlated receptors. In contrast, the interaction became stronger in nystatin-treated MCF-12A cells, suggesting the effect of the cholesterol-mediated interaction

EGFR in A431 increased by two orders of magnitude from 10�<sup>2</sup> for native cells to 1 for MβCD

HeLa and MCF-12A cells, which suggests that a deterministic interaction dominated because the screening effect from the membrane cholesterol was reduced after membrane cholesterol was depleted. These results identified a vital role for membrane cholesterol in mediating the interac-

4.4. Nonraft lipids and sphingolipids in live plasma membranes segregate into separated

Previous data analysis implicitly assumed the coexistence of different lipid phases in plasma membranes. Indeed, lipid–lipid interactions were capable of inducing liquid ordered (Lo) liquid disordered (Ld) phase coexistence in model lipid membranes [55, 56]. It was conjectured that plasma membrane composition is poised for selective and functional raft clustering at physiological temperatures [57]. However, such lipid nanodomains have remained largely unresolved in the plasma membrane of living cells. Researchers recently used a fluorescence correlation technique to successfully distinguish between free and anomalous molecular diffusion in a 30-nm focal spot of a stimulated emission depletion (STED) nanoscope [58]. The observed differences were attributed to transient cholesterol-assisted and cytoskeletondependent binding of sphingolipids to other membrane constituents. However, the optical force acting on the highly excited lipid molecules by the STED spot may not be negligible.

In our current study, we investigated lipid nanodomains in live plasma membranes at a much lower excitation level with light-sheet microscopy. We probed the nonraft lipids in living HeLa

(DiI-C12), which serves as an excellent lipophilic fluorescent probe with a strong partition tendency into the Ld phase [59]. The preference originated from the fact that highly packed lipids in Lo phase usually exclude exogenous molecules. BODIPY FL C5-ganglioside GM1 (BODIPY C5-GM1) from ThermoFisher Scientific was used as a direct indication of lipid rafts. By synchronously adjusting the light-sheet position and the focal plane of the high NA objective lens, we were alble to acquire a 3D image of DiI-C12 and BODIPY C5-GM1 in a living HeLa cell. Figure 13a displays a typical image of the scan with DiI-C12 shown in blue and BODIPY C5-GM1 in orange. The 3D point spread function (PSF) of the light-sheet microscope was deduced (shown in Figure 13b) using a fluorescent bead with a diameter of 100 nm. We exploited this PSF to deconvolve the image [60] of DiI-C12 and BODIPY C5-GM1. The resulting

Following the formalism developed by Veatch et al. [61], we calculated both the auto-correlation function A rð Þ¼ h i <sup>r</sup>ð Þ <sup>R</sup> <sup>r</sup>ð Þ <sup>R</sup> � <sup>r</sup> <sup>=</sup>h i <sup>r</sup>ð Þ<sup>r</sup> <sup>2</sup> and the cross-correlation function C rð Þ¼ <sup>h</sup>r1ð Þ <sup>R</sup> <sup>r</sup>2ðR� rÞi= r<sup>1</sup> h i ð Þr r<sup>2</sup> ð Þ h i ð Þr of DiI-C12 and BODIPY C5-GM1. The results are presented in Figure 14, which indicates that both species may form clusters with an average diameter of less than 150 nm.

,30



<sup>2</sup>

value could be increased to higher than 10 in MβCD treated

of correlated Qdot585-EGF-

was opposite to that of the receptor-lipid interaction. The V R<sup>τ</sup>

<sup>2</sup>

tion between liganded receptors in the three cell lines.

treated cells. Of note, the V R<sup>τ</sup>

116 Nonmagnetic and Magnetic Quantum Dots

cells with carbocyanine dyes 1,1<sup>0</sup>

deconvolved image is shown in Figure 13a.

nanodomains

Figure 13 (a) 3D fluorescence distribution (left) and its deconvoluted image (right) from DiI-C12 (blue) and BODIPY C5-GM1 (orange) fluorophores in a living HeLa cell. The cross section (z ¼ -2:25 μm) of the distribution was shown below. (b)the point spread function of the light-sheet microscope on the xy- (left) and xz-plane (right) emitted from a fluorescent bead with a diameter of 100 nm.

Figure 14 The auto-(left) and cross-(right) correlation function of DiI-C12 and BODIPY C5-GM1 calculated from the deconvoluted image of Figure 13a.

To retrieve information about the lipid clustering process from the measured pair correlation, we simulated lipid clustering dynamics. First, we randomly distributed M clusters in an imaging region with each cluster containing N molecules in a circle with diameter R. Figure 15a illustrates an example of two randomly distributed clusters (M = 2) at a spatial resolution of 2 nm. We then binned the molecules in a cluster to the pixel size of the camera used (see Figure 15b). By using these cluster images, we calculated the auto- and crosscorrelation functions with three different lipid clustering models: 1) a random clustering model, 2) an aggregation model, and 3) a segregation model. The random clustering model assumed that two lipid species were independently and randomly distributed in an area. In contrast, the lipid molecules in the aggregation and segregation models were stochastically distributed in each cluster with a Gaussian distribution. There is a major difference between

It was discovered that the raft lipid species GM1 can be tightened by pentameric cholera toxinβ (CTxB), which initiates a minimum raft coalescence to form the GM1 nanodomains [62]. The plasma membranes in our study were in their native state without perturbations from either intense laser spot or cross linking reagent. Thus, our evidence for segregation of nonraft lipids and GM1 into separate nanodomains supports the idea that such a phase coexistence in a

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4.5. Probing translocation of HIV-1 tat peptides in living cells with tat-conjugated quantum

Viral infection can initiate at entry points on plasma membranes via lipid domains. Drug delivery may benefit from our understanding of this entry process because upon arriving at target tissues, drug molecules must also cross the plasma membrane to reach the sites of action. It is of particular interest to make drug molecules that cross cellular membranes directly to avoid the complications of vesicle-mediated internalization pathways. Recently, cellpenetrating peptides (CPPs), which are short sequences (8 to 30) of amino acids (aa) with a net positive charge in water [63], were found to exhibit such a membrane-crossing capability. An 11 aa segment in the trans activator of a transcription protein of the human immunodeficiency virus is a prototypical example of a CPP that can effectively penetrate a cell [64, 65]. The interactions involved in the approach to developing a TatP-coated nanoscale probe may determine whether the uptake of the probe succeeds or fails. To illustrate the potential of biomolecule-conjugated QDs as a cellular dynamic probe, in this section we briefly discuss the results of the translocation of TatP-conjugated QDs across the plasma membranes of live

The first step for cellular internalization may involve some form of interaction between the Tat peptides and the surface of the cell. The strong anionic charge present on the glycosaminoglycan (GAG) chains of the proteoglycans (PGs) makes them favorable first-binding sites for the cationic Tat peptides [20, 66, 67]. To verify this scenario, we treated cells with Heparinase III enzyme (HSase) to cleave heparan sulfate (HS) groups from heparan sulfate proteoglycans (HSPGs). We observed a reduction in TatP-QD internalization of 74% at 30 min. Treatment with Cyto D, which can inhibit actin polymerization and thereby disrupt the cellular actin framework [68], resulted in a similar drop in TatP-QD internalization. The results indicate that both HS-mediated binding and the interaction with intracellular actin filaments are crucial for

4.5.1. TatP-QDs approaching cell surface aggregate at selected regions of plasma membrane

For single-particle tracking, we prepared a cell culture medium containing 1 μm free TatPs and 1 nM TatP-QD nanosensors. The major species of free TatPs were used to restructure the environment of the membrane-peptide interaction, whereas TatP-QDs served as nanoscale dynamic pens to depict the landscape of the membrane-peptide interaction. We conducted single-particle trajectory analysis of the TatP-QDs with light-sheet microscopy to reveal the translocation dynamics. A unique affordance of our light-sheet microscope was the ability to track TatP-QDs in parallel, providing a global view of the dynamics of the approaching TatP-QDs. However, due to the

native plasma membrane not only exists but also is a general behavior of living cells.

cells using the single-molecule tracking technique [17].

the rapid intake of TatP-QDs.

dot nanosensors

Figure 15 (a) Two randomly distributed clusters (M = 2) at a spatial resolution of 2 nm was prepared in a simulation area, (b) the two clusters were binned to the pixel size of the camera used. (c) the auto- (left) and cross- (right) correlation function using the parameters: R = 72 nm, N = 100, and M = 100 (2.8 clusters/μm<sup>2</sup> ) were simulated with the random clustering model (filled circles), aggregation model (open triangles), and segregation model (open squares).

the latter two models. In the aggregation model, any two clusters will attract each other to form an overlapping cluster with the same center of mass (but with different radii). In the segregation model, two clusters will experience a repulsive force to yield the minimum separation distance Δ. To simulate the segregation process, we first used a random number generator to produce the center position X<sup>1</sup> of one cluster and then used the multiplicative congruential generation algorithm to position the other cluster to lie in the interval of ½ Þ 0; X<sup>1</sup> � Δ or ð � X<sup>1</sup> þ Δ; 1 . In this way, the second cluster was located randomly but was excluded from the neighborhood of X<sup>1</sup> with a minimum separation Δ. For a fair comparison, we kept the number of clusters, cluster size, and number of molecules in each cluster at the same values. Figure 15 shows the simulation results for the following parameters: the waist size of the cluster was 72 nm, the number of molecules in a cluster was 100, and the number of clusters in an image was 100 (i.e., 2.8 clusters/μm2 ), which yielded the short-distance autocorrelation and agreed well with the experimental result shown in Figure 14.

By comparing Figure 15c with Figure 14, we concluded that our data could not fit to the model of random clustering. From the comparison of cross-correlation shown in Figures 15c and 14, we could remove the aggregation model, and we concluded that our data was better described by the segregation model with a segregation distance <100 nm.

It was discovered that the raft lipid species GM1 can be tightened by pentameric cholera toxinβ (CTxB), which initiates a minimum raft coalescence to form the GM1 nanodomains [62]. The plasma membranes in our study were in their native state without perturbations from either intense laser spot or cross linking reagent. Thus, our evidence for segregation of nonraft lipids and GM1 into separate nanodomains supports the idea that such a phase coexistence in a native plasma membrane not only exists but also is a general behavior of living cells.

#### 4.5. Probing translocation of HIV-1 tat peptides in living cells with tat-conjugated quantum dot nanosensors

Viral infection can initiate at entry points on plasma membranes via lipid domains. Drug delivery may benefit from our understanding of this entry process because upon arriving at target tissues, drug molecules must also cross the plasma membrane to reach the sites of action. It is of particular interest to make drug molecules that cross cellular membranes directly to avoid the complications of vesicle-mediated internalization pathways. Recently, cellpenetrating peptides (CPPs), which are short sequences (8 to 30) of amino acids (aa) with a net positive charge in water [63], were found to exhibit such a membrane-crossing capability. An 11 aa segment in the trans activator of a transcription protein of the human immunodeficiency virus is a prototypical example of a CPP that can effectively penetrate a cell [64, 65]. The interactions involved in the approach to developing a TatP-coated nanoscale probe may determine whether the uptake of the probe succeeds or fails. To illustrate the potential of biomolecule-conjugated QDs as a cellular dynamic probe, in this section we briefly discuss the results of the translocation of TatP-conjugated QDs across the plasma membranes of live cells using the single-molecule tracking technique [17].

The first step for cellular internalization may involve some form of interaction between the Tat peptides and the surface of the cell. The strong anionic charge present on the glycosaminoglycan (GAG) chains of the proteoglycans (PGs) makes them favorable first-binding sites for the cationic Tat peptides [20, 66, 67]. To verify this scenario, we treated cells with Heparinase III enzyme (HSase) to cleave heparan sulfate (HS) groups from heparan sulfate proteoglycans (HSPGs). We observed a reduction in TatP-QD internalization of 74% at 30 min. Treatment with Cyto D, which can inhibit actin polymerization and thereby disrupt the cellular actin framework [68], resulted in a similar drop in TatP-QD internalization. The results indicate that both HS-mediated binding and the interaction with intracellular actin filaments are crucial for the rapid intake of TatP-QDs.

#### 4.5.1. TatP-QDs approaching cell surface aggregate at selected regions of plasma membrane

the latter two models. In the aggregation model, any two clusters will attract each other to form an overlapping cluster with the same center of mass (but with different radii). In the segregation model, two clusters will experience a repulsive force to yield the minimum separation distance Δ. To simulate the segregation process, we first used a random number generator to produce the center position X<sup>1</sup> of one cluster and then used the multiplicative congruential generation algorithm to position the other cluster to lie in the interval of ½ Þ 0; X<sup>1</sup> � Δ or ð � X<sup>1</sup> þ Δ; 1 . In this way, the second cluster was located randomly but was excluded from the neighborhood of X<sup>1</sup> with a minimum separation Δ. For a fair comparison, we kept the number of clusters, cluster size, and number of molecules in each cluster at the same values. Figure 15 shows the simulation results for the following parameters: the waist size of the cluster was 72 nm, the number of molecules in a cluster was 100, and the number of

clustering model (filled circles), aggregation model (open triangles), and segregation model (open squares).

Figure 15 (a) Two randomly distributed clusters (M = 2) at a spatial resolution of 2 nm was prepared in a simulation area, (b) the two clusters were binned to the pixel size of the camera used. (c) the auto- (left) and cross- (right) correlation

By comparing Figure 15c with Figure 14, we concluded that our data could not fit to the model of random clustering. From the comparison of cross-correlation shown in Figures 15c and 14, we could remove the aggregation model, and we concluded that our data was better described

), which yielded the short-distance auto-

) were simulated with the random

clusters in an image was 100 (i.e., 2.8 clusters/μm2

by the segregation model with a segregation distance <100 nm.

function using the parameters: R = 72 nm, N = 100, and M = 100 (2.8 clusters/μm<sup>2</sup>

118 Nonmagnetic and Magnetic Quantum Dots

correlation and agreed well with the experimental result shown in Figure 14.

For single-particle tracking, we prepared a cell culture medium containing 1 μm free TatPs and 1 nM TatP-QD nanosensors. The major species of free TatPs were used to restructure the environment of the membrane-peptide interaction, whereas TatP-QDs served as nanoscale dynamic pens to depict the landscape of the membrane-peptide interaction. We conducted single-particle trajectory analysis of the TatP-QDs with light-sheet microscopy to reveal the translocation dynamics. A unique affordance of our light-sheet microscope was the ability to track TatP-QDs in parallel, providing a global view of the dynamics of the approaching TatP-QDs. However, due to the limited image-taking speed of the camera used, we were only able to track TatP-QDs within a short distance from the cell surface.

4.5.2. Spectral-embedding analysis of trajectory aggregates of TatP-QDs

identify the mechanism underlying the trajectory aggregation of TatP-QDs.

! � <sup>ξ</sup> � �!

!

and their cellular environments.

trajectory r

action potentials Ui r

set of trajectories: r

As TatP-QDs translocate across the plasma membrane of a living cell, the probing particles can record the influences of the cellular environment on their trajectories. We considered the

by its local environment, which may have different realizations with the corresponding inter-

could implicitly record the configurations of the local environment. Recently, Wang and Ferguson generated a reconstruction of single-molecule free-energy surfaces from time-series data of a physical observable by using the generalized Takens Delay Embedding Theorem [34]. Here we focused on retrieving the eigenmodes of the trajectory coordinates of TatP-QDs to

We used the spectral-embedding technique [35] to extract a low-dimensional manifold from a

approximation of the manifold [36] and enabled an efficient construction of the eigendecomposition. The first eigenvector we retrieved was trivial with the corresponding eigenvalue giving only the data density in a cluster. We then focused on the next two eigenvectors, f<sup>2</sup> and f3, which offered the most critical information on the interactions between TatP-QDs

Figure 17 presents two trajectory aggregates of the TatP-QDs: one (left) is near a living cell, and the other (right) is directly on top of the cell surface. All of the coordinates of the trajectory aggregates are shown in gray. The location coordinates of the TatP-QDs associated with the left

Figure 17 Two trajectory clusters (gray) of TatP-QDs near a living HeLa cell (green) are presented on a manifold of

tory segments within 2% variance of the peak shown in Figure 18 are displayed in blue on the far side, red near the z ¼ 0

plane, and yellow for those closest to the cell membrane. This figure has been reproduced from ref. [17].

<sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> coordinates of the trajec-

spectral-embedding eigenvectors (top inset). For each trajectory cluster, the V R<sup>τ</sup>

! ð Þ<sup>t</sup> to be produced by a stochastic process with influences on the probing particle

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

; i ¼ 1, ::, n. Thus, the trajectory coordinates of the nanosensors

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<sup>i</sup>ð Þt ; i ¼ 1, ::, n. A graph-based method provided a useful discretized

Without external interaction, these Tat-QD nanosensors were expected to traverse the extracellular space through a random walk search, attach to the membrane, and then diffuse to find a suitable entrance site. Figure 16a displays three trajectories of TatP-QDs, color-coded to indicate the approaching times. The green surface depicts the cell surface rendered from the phase contrast images taken by scanning the imaging focal plane at different z positions in the cell. The determination of the cell profile was limited by the diffraction effect of the objective lens used, yielding a resolution of 200 nm in the lateral plane and 500 nm in the axial direction. As indicated in the top inset, the initial approaching trajectories of some of the Tat-QDs resembled directed movement under a force field, and the motion became more diffuse as the TatP-QDs come closer to the cell surface. A longer observation period accumulated more approaching events and revealed the trajectory aggregates at selected regions of the plasma membrane (Figure 16b).

The binding affinity of TatP for HSPGs was greater than that for anionic lipids by 2 to 3 orders of magnitude. Given that the anionic HSPG chains on the plasma membrane [20, 69, 70] may be favorable binding sites for cationic CPPs, we hypothesized that the trajectory aggregates were caused by HS groups in the HSPG chains. To verify this hypothesis, we treated the cells with HSase to cleave the HS groups from the HSPGs, which revealed considerably fewer and more randomly positioned spots in the extracellular space. Thus, the observed trajectory aggregation seems to be caused by the binding to HS groups on the membranes and suggests that HSPGs play a critical role in redirecting the TatP entry process toward spatially restricted sites on the plasma membrane.

Figure 16 (a) Three trajectories of TatP-QDs near a living HeLa cell (green) were color-coded to indicate their appearing times. The green profile denotes the cellular surface rendered from optical sectioning phase contrast images; (b) when duration was increased to acquire information on more approaching events, trajectory aggregates were observed at selected regions on a native HeLa cell. This figure has been reproduced from ref. [17].

#### 4.5.2. Spectral-embedding analysis of trajectory aggregates of TatP-QDs

limited image-taking speed of the camera used, we were only able to track TatP-QDs within a

Without external interaction, these Tat-QD nanosensors were expected to traverse the extracellular space through a random walk search, attach to the membrane, and then diffuse to find a suitable entrance site. Figure 16a displays three trajectories of TatP-QDs, color-coded to indicate the approaching times. The green surface depicts the cell surface rendered from the phase contrast images taken by scanning the imaging focal plane at different z positions in the cell. The determination of the cell profile was limited by the diffraction effect of the objective lens used, yielding a resolution of 200 nm in the lateral plane and 500 nm in the axial direction. As indicated in the top inset, the initial approaching trajectories of some of the Tat-QDs resembled directed movement under a force field, and the motion became more diffuse as the TatP-QDs come closer to the cell surface. A longer observation period accumulated more approaching events and revealed the

The binding affinity of TatP for HSPGs was greater than that for anionic lipids by 2 to 3 orders of magnitude. Given that the anionic HSPG chains on the plasma membrane [20, 69, 70] may be favorable binding sites for cationic CPPs, we hypothesized that the trajectory aggregates were caused by HS groups in the HSPG chains. To verify this hypothesis, we treated the cells with HSase to cleave the HS groups from the HSPGs, which revealed considerably fewer and more randomly positioned spots in the extracellular space. Thus, the observed trajectory aggregation seems to be caused by the binding to HS groups on the membranes and suggests that HSPGs play a critical role in redirecting the TatP entry process toward spatially restricted sites on the

Figure 16 (a) Three trajectories of TatP-QDs near a living HeLa cell (green) were color-coded to indicate their appearing times. The green profile denotes the cellular surface rendered from optical sectioning phase contrast images; (b) when duration was increased to acquire information on more approaching events, trajectory aggregates were observed at

selected regions on a native HeLa cell. This figure has been reproduced from ref. [17].

trajectory aggregates at selected regions of the plasma membrane (Figure 16b).

short distance from the cell surface.

120 Nonmagnetic and Magnetic Quantum Dots

plasma membrane.

As TatP-QDs translocate across the plasma membrane of a living cell, the probing particles can record the influences of the cellular environment on their trajectories. We considered the trajectory r ! ð Þ<sup>t</sup> to be produced by a stochastic process with influences on the probing particle by its local environment, which may have different realizations with the corresponding interaction potentials Ui r ! � <sup>ξ</sup> � �! ; i ¼ 1, ::, n. Thus, the trajectory coordinates of the nanosensors could implicitly record the configurations of the local environment. Recently, Wang and Ferguson generated a reconstruction of single-molecule free-energy surfaces from time-series data of a physical observable by using the generalized Takens Delay Embedding Theorem [34]. Here we focused on retrieving the eigenmodes of the trajectory coordinates of TatP-QDs to identify the mechanism underlying the trajectory aggregation of TatP-QDs.

We used the spectral-embedding technique [35] to extract a low-dimensional manifold from a set of trajectories: r ! <sup>i</sup>ð Þt ; i ¼ 1, ::, n. A graph-based method provided a useful discretized approximation of the manifold [36] and enabled an efficient construction of the eigendecomposition. The first eigenvector we retrieved was trivial with the corresponding eigenvalue giving only the data density in a cluster. We then focused on the next two eigenvectors, f<sup>2</sup> and f3, which offered the most critical information on the interactions between TatP-QDs and their cellular environments.

Figure 17 presents two trajectory aggregates of the TatP-QDs: one (left) is near a living cell, and the other (right) is directly on top of the cell surface. All of the coordinates of the trajectory aggregates are shown in gray. The location coordinates of the TatP-QDs associated with the left

Figure 17 Two trajectory clusters (gray) of TatP-QDs near a living HeLa cell (green) are presented on a manifold of spectral-embedding eigenvectors (top inset). For each trajectory cluster, the V R<sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> coordinates of the trajectory segments within 2% variance of the peak shown in Figure 18 are displayed in blue on the far side, red near the z ¼ 0 plane, and yellow for those closest to the cell membrane. This figure has been reproduced from ref. [17].

cluster present a nearly circular distribution on the f2-f<sup>3</sup> plane. The right cluster, however, displays a V-shaped distribution. Our results indicate that the spectrally decomposed structure of the trajectory aggregates provides the information on the interaction of the TatP-QDs with their cellular environments.

We also analyzed each trajectory aggregate by selecting segments that fell within 2% variance

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

trajectories offered insight into the environmental influences on the TatP-QDs. For example in the left trajectory cluster of Figure 17, these special points are shown in blue on the far side and red near the z = 0 plane. The blue dots were uniformly distributed at the rim of the circle on the f2-f<sup>3</sup> plane, and the distribution of the red dots, which were close to the cell membrane, appeared to be denser on the f<sup>2</sup> > 0 side. In the trajectory aggregate directly on top of the cell, the blue dots were located at the right leg (f<sup>2</sup> > 0) and the red points dots were concentrated at the left leg (f<sup>2</sup> < 0) of a V-shaped distribution. Yellow dots, which represent trajectory segments closest to the cell membrane, aggregated at the tip of the V-shaped distribution, suggesting the formation of hot spots of interaction on the cell membrane, which may be

We also applied spectral embedding [17, 70] to classify 23,382 TatP-QD trajectories measured on 30 cells. In Figure 19, the resulting circular or V-shaped distributions on the f2-f<sup>3</sup> plane are displayed in green. For classification, the norm of the residuals, defined as the sum of the squared deviation from the circular distribution of free diffusion, was used as the metric. The

contours were also included for comparison. As shown in Figure 20, the class of circular

Figure 19 Spectral embedding manifold plots (green in insets) of 23,382 trajectories of TatP-QDs measured on 30 living

coordinates of the trajectory segments within 2% variance of the peaks shown in Figure 18 are displayed in blue, with

HeLa cells (up row) and 5112 trajectories measured on Cyto D-treated cells (bottom row). The V R<sup>τ</sup>

associated contour curves revealing the peak profiles. This figure has been reproduced from ref. [17].

<sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> coordinates on the

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123

<sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> peak with the corresponding

<sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup>

<sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> peak. Labeling the resulting <sup>V</sup> <sup>R</sup><sup>τ</sup>

supported by specifically oriented actin filaments.

coordinates (blue) within 2% variance of the V R<sup>τ</sup>

4.5.4. Classification of TatP-QD trajectories

of the V R<sup>τ</sup>

#### 4.5.3. Influence of actin framework on translocation of TatP-QDs

The findings of a recent study indicated that on attachment to a membrane surface, Tat peptides can remodel the actin framework in an actin-encapsulated giant unilamellar vesicles (GUV) [69]. However, it remains unclear whether such multiplexed membrane and cytoskeletal interactions can also occur in a living cell. The trajectories of nanoscale probing particles may provide the answer. To extract relevant stochastic and geometrical structures from the data and gain insights into the mechanism that generated the data, we generated the V R<sup>τ</sup> <sup>2</sup> � �-R<sup>τ</sup> <sup>2</sup> plots of the trajectories in Figure 18, which summarizes the single-particle diffusion statistics from 23,382 TatP-QD trajectories.

A single peak at the coordinates (0.15, 0.21), which is well below the free diffusion limit of V R<sup>τ</sup> <sup>2</sup> � � <sup>¼</sup> 2, is shown in Figure 18a, suggesting that the TatP-QDs did not diffuse freely near a native HeLa cell. The peak split into two and shifted downward to V R<sup>τ</sup> <sup>2</sup> � � <sup>¼</sup> <sup>0</sup>:06 for Cyto D-treated cells (Figure 18b), indicating that TatP-QDs experience a stronger interaction with a strained cellular membrane. This finding is understandable because, without the support of an actin framework, the plasma membrane may develop a higher local curvature as a result of TatP-QD attachment. In native cells, the effect of the interaction between Tat peptides and the cell membrane may be counter balanced by that of the Tat and actin filaments, resulting in a higher V R<sup>τ</sup> <sup>2</sup> � �.

Figure 18 2D contour plot of V R<sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> histogram for TatP-QDs moving in the neighborhood of (a) living HeLa cell and (b) Cyto D-pretreated cell. This figure has been reproduced from ref. [17].

We also analyzed each trajectory aggregate by selecting segments that fell within 2% variance of the V R<sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> peak. Labeling the resulting <sup>V</sup> <sup>R</sup><sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> coordinates on the trajectories offered insight into the environmental influences on the TatP-QDs. For example in the left trajectory cluster of Figure 17, these special points are shown in blue on the far side and red near the z = 0 plane. The blue dots were uniformly distributed at the rim of the circle on the f2-f<sup>3</sup> plane, and the distribution of the red dots, which were close to the cell membrane, appeared to be denser on the f<sup>2</sup> > 0 side. In the trajectory aggregate directly on top of the cell, the blue dots were located at the right leg (f<sup>2</sup> > 0) and the red points dots were concentrated at the left leg (f<sup>2</sup> < 0) of a V-shaped distribution. Yellow dots, which represent trajectory segments closest to the cell membrane, aggregated at the tip of the V-shaped distribution, suggesting the formation of hot spots of interaction on the cell membrane, which may be supported by specifically oriented actin filaments.

#### 4.5.4. Classification of TatP-QD trajectories

cluster present a nearly circular distribution on the f2-f<sup>3</sup> plane. The right cluster, however, displays a V-shaped distribution. Our results indicate that the spectrally decomposed structure of the trajectory aggregates provides the information on the interaction of the TatP-QDs with

The findings of a recent study indicated that on attachment to a membrane surface, Tat peptides can remodel the actin framework in an actin-encapsulated giant unilamellar vesicles (GUV) [69]. However, it remains unclear whether such multiplexed membrane and cytoskeletal interactions can also occur in a living cell. The trajectories of nanoscale probing particles may provide the answer. To extract relevant stochastic and geometrical structures from the data and gain insights into the mechanism that generated the data, we generated the

A single peak at the coordinates (0.15, 0.21), which is well below the free diffusion limit of

D-treated cells (Figure 18b), indicating that TatP-QDs experience a stronger interaction with a strained cellular membrane. This finding is understandable because, without the support of an actin framework, the plasma membrane may develop a higher local curvature as a result of TatP-QD attachment. In native cells, the effect of the interaction between Tat peptides and the cell membrane may be counter balanced by that of the Tat and actin filaments, resulting in a

native HeLa cell. The peak split into two and shifted downward to V R<sup>τ</sup>

<sup>2</sup> plots of the trajectories in Figure 18, which summarizes the single-particle diffu-

¼ 2, is shown in Figure 18a, suggesting that the TatP-QDs did not diffuse freely near a

<sup>2</sup> � �


¼ 0:06 for Cyto

their cellular environments.

122 Nonmagnetic and Magnetic Quantum Dots

V R<sup>τ</sup> <sup>2</sup> � �

V R<sup>τ</sup> <sup>2</sup> � �

higher V R<sup>τ</sup>

<sup>2</sup> � � .

Figure 18 2D contour plot of V R<sup>τ</sup>

<sup>2</sup> h i � �

cell and (b) Cyto D-pretreated cell. This figure has been reproduced from ref. [17].


4.5.3. Influence of actin framework on translocation of TatP-QDs

sion statistics from 23,382 TatP-QD trajectories.

We also applied spectral embedding [17, 70] to classify 23,382 TatP-QD trajectories measured on 30 cells. In Figure 19, the resulting circular or V-shaped distributions on the f2-f<sup>3</sup> plane are displayed in green. For classification, the norm of the residuals, defined as the sum of the squared deviation from the circular distribution of free diffusion, was used as the metric. The coordinates (blue) within 2% variance of the V R<sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> peak with the corresponding contours were also included for comparison. As shown in Figure 20, the class of circular

Figure 19 Spectral embedding manifold plots (green in insets) of 23,382 trajectories of TatP-QDs measured on 30 living HeLa cells (up row) and 5112 trajectories measured on Cyto D-treated cells (bottom row). The V R<sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> coordinates of the trajectory segments within 2% variance of the peaks shown in Figure 18 are displayed in blue, with associated contour curves revealing the peak profiles. This figure has been reproduced from ref. [17].

A biomolecule subjected to random influences can explore its possible outcomes and evolves to yield a dispersion over its state space. The evolution may contain both contributions from deterministic and stochastic forces. To provide high-quality statistics sampled by appropriate probing biomolecules while preserving single-molecule sensitivity, we developed a 2D analysis

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

MSDs were used to quantify the diffusion of a protein in its cellular environment, and the normalized variance discloses the nature of these interactions. Thus, the plot can be more sensitive than MSD alone to reflect the diffusive dynamics of a protein in cellular environments. We applied this 2D analysis technique to the dimerization processes of EGFRs in live cells under varying cellular conditions. Based on this study, we found that unliganded species appear to remain outside the cholesterol-enriched lipid domains. After ligand binding, EGFR molecules may relocate to lipid raft domains. This experimental finding was verified using three cell lines with a wide range of EGFR expression levels and membrane cholesterol concentrations, suggesting that these results may represent a general behavior of unliganded and activated receptors in live cells. Reigada et al. recently applied near-field scanning optical microscopy to fixed monocytes and found that raftophilic proteins did not physically intermix at the nanoscale with CTxB-GM1 nanodomains but converged within a characteristic distance [62]. Our result of

<sup>2</sup> (normalized variance-vs-MSD). Here the

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125

<sup>2</sup> -R<sup>τ</sup>

<sup>2</sup> to

<sup>2</sup> and <sup>R</sup><sup>τ</sup>

unliganded EGFR agrees with this finding but on live cells without CTxB tightening.

Selectively tagging EGFR species with semiconductor quantum dots allowed us to monitor the correlated motions of unliganded and liganded species. Paired liganded receptors, which diffused in proximity on the plasma membrane interacted with each other and caused the receptors to move correlatively. The correlated motions can be caused by the correlated fluctuations in the lipid environment, which occur when the two receptors are closely separated. The correlated motion can be changed by manipulating either the distribution or total quantity of the membrane cholesterol, suggesting that the membrane cholesterol plays a vital role in mediating the interactions between the liganded receptors. Our quantitative 2D analysis method can capture the dynamic receptor interactions at the single-molecule level, providing

We further used the HIV-1 Tat peptide-conjugated QD as a nanosensor to illustrate the translocation dynamics of the Tat peptides in living cells. By using spectral-embedding analysis, we extracted an intrinsic low-dimensional manifold, which was formed by the isotropic diffusion and a fraction of the directed movement, from the measured trajectories. Our result suggest that HSPGs play a significant role in redirecting the TatP-QD entry process toward spatially

determine the underlying cause of the trajectory aggregation. We found that the membrane deformation induced by the Tat-peptide attachment increased with the disruption of the actin framework, which resulted in higher interactions on the TatP-QDs. In native cells, the Tat peptides could remodel the actin framework to reduce their interaction with the local mem-

Semiconductor quantum dots conjugated with appropriate peptides or antibodies are appealing for probing cellular dynamic events in living cells. The nanosensors have the advantages of high emission efficiency, wavelength tunability, and long-term stability, which have led to a variety of applications in cellular sensing and imaging. Biomolecule-conjugated QD nanosensors are also

restricted sites on the plasma membrane. We further applied 2D analysis of V R<sup>τ</sup>

of single-molecule trajectories with V R<sup>τ</sup>

details that are often obscured in other methods.

brane environment.

Figure 20 Classification (center) of spectral embedding manifold plots (green in insets) of 23,382 trajectories of TatP-QDs measured on 30 living HeLa cells. The V R<sup>τ</sup> <sup>2</sup> h i � � -½ � <sup>R</sup>τð Þ<sup>t</sup> <sup>2</sup> coordinates of the trajectory segments within 2% variance of the peaks shown in Figure 18 are displayed in blue, with associated contour curves revealing the peak profiles. This figure has been reproduced from ref. [17].

distribution with a norm of the residuals of 0.91 � 0.34 contained about 66% of the data from the native cells. The moderate (2 to 6) and highly anisotropic (>6) trajectory data occupied 31 and 3%, respectively. We acquired 5112 trajectories for the Cyto D-treated cells. The proportions of the moderate and highly anisotropic trajectories decreased to 27 and 1%, respectively. Treatment with Cyto D reduced the cellular uptake of the TatP-QDs to 25% of that of the native cells, indicating that both the isotropic and moderate anisotropic classes played a minor role in the initial cellular uptake. The trajectories belonging to the highly anisotropic class resulted in 75% uptake. These findings indicate the formation of funnel passages for the TatP-QDs due to the combined effect of HS-binding and actin remodeling.

#### 5. Conclusion

Probing the distribution and mobility of proteins in live cellular environments is crucial for understanding cellular functions and regulatory mechanisms, which also serve as the basis for developing therapeutic strategies. Factors that affect protein mobility are difficult to reconstitute in vitro using purified constituents. Single-molecule imaging and tracking provide direct access to probe the properties of molecular assemblies and the kinetics of the interaction in live cellular environments. However, biological media are spatially inhomogeneous, which is poorly conveyed by measuring just a few, sparse single-molecule trajectories. Finding a way to efficiently and reliably extract useful information from a large amount of trajectory data is an obstacle of this technique.

A biomolecule subjected to random influences can explore its possible outcomes and evolves to yield a dispersion over its state space. The evolution may contain both contributions from deterministic and stochastic forces. To provide high-quality statistics sampled by appropriate probing biomolecules while preserving single-molecule sensitivity, we developed a 2D analysis of single-molecule trajectories with V R<sup>τ</sup> <sup>2</sup> and <sup>R</sup><sup>τ</sup> <sup>2</sup> (normalized variance-vs-MSD). Here the MSDs were used to quantify the diffusion of a protein in its cellular environment, and the normalized variance discloses the nature of these interactions. Thus, the plot can be more sensitive than MSD alone to reflect the diffusive dynamics of a protein in cellular environments. We applied this 2D analysis technique to the dimerization processes of EGFRs in live cells under varying cellular conditions. Based on this study, we found that unliganded species appear to remain outside the cholesterol-enriched lipid domains. After ligand binding, EGFR molecules may relocate to lipid raft domains. This experimental finding was verified using three cell lines with a wide range of EGFR expression levels and membrane cholesterol concentrations, suggesting that these results may represent a general behavior of unliganded and activated receptors in live cells. Reigada et al. recently applied near-field scanning optical microscopy to fixed monocytes and found that raftophilic proteins did not physically intermix at the nanoscale with CTxB-GM1 nanodomains but converged within a characteristic distance [62]. Our result of unliganded EGFR agrees with this finding but on live cells without CTxB tightening.

Selectively tagging EGFR species with semiconductor quantum dots allowed us to monitor the correlated motions of unliganded and liganded species. Paired liganded receptors, which diffused in proximity on the plasma membrane interacted with each other and caused the receptors to move correlatively. The correlated motions can be caused by the correlated fluctuations in the lipid environment, which occur when the two receptors are closely separated. The correlated motion can be changed by manipulating either the distribution or total quantity of the membrane cholesterol, suggesting that the membrane cholesterol plays a vital role in mediating the interactions between the liganded receptors. Our quantitative 2D analysis method can capture the dynamic receptor interactions at the single-molecule level, providing details that are often obscured in other methods.

distribution with a norm of the residuals of 0.91 � 0.34 contained about 66% of the data from the native cells. The moderate (2 to 6) and highly anisotropic (>6) trajectory data occupied 31 and 3%, respectively. We acquired 5112 trajectories for the Cyto D-treated cells. The proportions of the moderate and highly anisotropic trajectories decreased to 27 and 1%, respectively. Treatment with Cyto D reduced the cellular uptake of the TatP-QDs to 25% of that of the native cells, indicating that both the isotropic and moderate anisotropic classes played a minor role in the initial cellular uptake. The trajectories belonging to the highly anisotropic class resulted in 75% uptake. These findings indicate the formation of funnel passages for the TatP-QDs due to

Figure 20 Classification (center) of spectral embedding manifold plots (green in insets) of 23,382 trajectories of TatP-QDs

peaks shown in Figure 18 are displayed in blue, with associated contour curves revealing the peak profiles. This figure


Probing the distribution and mobility of proteins in live cellular environments is crucial for understanding cellular functions and regulatory mechanisms, which also serve as the basis for developing therapeutic strategies. Factors that affect protein mobility are difficult to reconstitute in vitro using purified constituents. Single-molecule imaging and tracking provide direct access to probe the properties of molecular assemblies and the kinetics of the interaction in live cellular environments. However, biological media are spatially inhomogeneous, which is poorly conveyed by measuring just a few, sparse single-molecule trajectories. Finding a way to efficiently and reliably extract useful information from a large amount of trajectory data is an obstacle of

the combined effect of HS-binding and actin remodeling.

<sup>2</sup> h i � �

5. Conclusion

measured on 30 living HeLa cells. The V R<sup>τ</sup>

has been reproduced from ref. [17].

124 Nonmagnetic and Magnetic Quantum Dots

this technique.

We further used the HIV-1 Tat peptide-conjugated QD as a nanosensor to illustrate the translocation dynamics of the Tat peptides in living cells. By using spectral-embedding analysis, we extracted an intrinsic low-dimensional manifold, which was formed by the isotropic diffusion and a fraction of the directed movement, from the measured trajectories. Our result suggest that HSPGs play a significant role in redirecting the TatP-QD entry process toward spatially restricted sites on the plasma membrane. We further applied 2D analysis of V R<sup>τ</sup> <sup>2</sup> -R<sup>τ</sup> <sup>2</sup> to determine the underlying cause of the trajectory aggregation. We found that the membrane deformation induced by the Tat-peptide attachment increased with the disruption of the actin framework, which resulted in higher interactions on the TatP-QDs. In native cells, the Tat peptides could remodel the actin framework to reduce their interaction with the local membrane environment.

Semiconductor quantum dots conjugated with appropriate peptides or antibodies are appealing for probing cellular dynamic events in living cells. The nanosensors have the advantages of high emission efficiency, wavelength tunability, and long-term stability, which have led to a variety of applications in cellular sensing and imaging. Biomolecule-conjugated QD nanosensors are also

useful for studying the interactions, stoichiometries, and conformational changes of proteins in living cells, which provides an understanding of the mode of the interaction and free-energy surfaces, and can reveal the stable states and dynamic pathways of biomolecules in live cells. The application examples presented in this chapter clearly support the use of biomolecule-conjugated QDs as probes for the cellular dynamics in living cells.

sCMOS complementary metal-oxide semiconductor TatP transactivator of transcription (Tat) peptide

Address all correspondence to: jyhuang@faculty.nctu.edu.tw

Nature Reviews Molecular Cell Biology. 2001;2:444-456

Department of Photonics and The T.K.B. Research Center for Photonics, Chiao Tung

[1] Lippincott-Schwartz J, Snapp E, Kenworthy A. Studying protein dynamics in living cells.

Biomolecule-Conjugated Quantum Dot Nanosensors as Probes for Cellular Dynamic Events in Living Cells

http://dx.doi.org/10.5772/intechopen.72858

127

[2] Hung MC, Link W. Protein localization in disease and therapy. Journal of Cell Science.

[3] Masson JB, Dionne P, Salvatico C, Renner M, Specht CG, Triller A, Dahan T. Mapping the energy and diffusion landscapes of membrane proteins at the cell surface using high-density single-molecule imaging and Bayesian inference: Application to the multiscale dynamics of

[4] Kusumi A, Tsunoyama TA, Hirosawa KM, Kasai RS, Fujiwara TK. Tracking single mole-

[5] Liu Z, Lavis LD, Betzig E. Imaging live-cell dynamics and structure at the single-molecule

[6] Serag MF, Abadi M, Habuchi S. Single-molecule diffusion and conformational dynamics by spatial integration of temporal fluctuations. Nature Communications. 2014;5:5123 [7] Michalet X. Mean square displacement analysis of single-particle trajectories with localization error: Brownian motion in an isotropic medium. Physical Review E. 2010;82:041914 [8] Medintz IL, Uyeda HT, Goldman ER, Mattoussi H. Quantum dot bioconjugates for imag-

[9] Jaiswal JK, Simon SM. Imaging live cells using quantum dots. Cold Spring Harbor Pro-

[10] Zhang JJ, Zheng TT, Cheng FF, Zhang JR, Zhu JJ. Toward the early evaluation of therapeutic effects: An electrochemical platform for ultrasensitive detection of apoptotic cells.

glycine receptors in the neuronal membrane. Biophysical Journal. 2014;106:74-83

cules at work in living cells. Biophysical Journal. 2014;106:74-83

ing, labelling and sensing. Nature Materials. 2005;4:435-446

level. Molecular Cell. 2015;58:644-659

tocols. 2015. p. doi:10.1101/pdb.top086322

Analytical Chemistry. 2011;83:7902-7909

TatP-QD TatP-conjugated quantum dot

Author details

University, Hsinchu, Taiwan

2011;124:3381-3392

Jung Y. Huang

References

## Acknowledgements

This research was funded by the Ministry of Science and Technology of the Republic of China (grant number MOST 106-2112-M-009-019-MY3). Parts of this chapter are taken from the authors' former work with permission of the Creative Commons Attribution license.

## Conflicts of interest

The author declares that he has no competing interests.

## Nomenclature



## Author details

Jung Y. Huang

useful for studying the interactions, stoichiometries, and conformational changes of proteins in living cells, which provides an understanding of the mode of the interaction and free-energy surfaces, and can reveal the stable states and dynamic pathways of biomolecules in live cells. The application examples presented in this chapter clearly support the use of biomolecule-conjugated

This research was funded by the Ministry of Science and Technology of the Republic of China (grant number MOST 106-2112-M-009-019-MY3). Parts of this chapter are taken from the

authors' former work with permission of the Creative Commons Attribution license.

QDs as probes for the cellular dynamics in living cells.

The author declares that he has no competing interests.

Acknowledgements

126 Nonmagnetic and Magnetic Quantum Dots

Conflicts of interest

Nomenclature

Ab antibody

CTxB cholera toxin-β cyto D cytochalasin D

CPP cell-penetrating peptide

ETL electrically tunable lens

GUV giant unilamellar vesicles

MβCD methyl β-cyclodextrin

PMF potential of mean force PSF point spread function

HSPG heparan sulfate proteoglycan

MSD mean square displacement

GAG glycosaminoglycan

HS heparan sulfate

PG proteoglycan

QD quantum dot

FDT fluctuation-dissipation theorems

Address all correspondence to: jyhuang@faculty.nctu.edu.tw

Department of Photonics and The T.K.B. Research Center for Photonics, Chiao Tung University, Hsinchu, Taiwan

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**Chapter 8**

Provisional chapter

**Redox-Mediated Quantum Dots as Fluorescence Probe**

DOI: 10.5772/intechopen.70761

Redox-Mediated Quantum Dots as Fluorescence Probe

Semiconductor quantum dots (QDs) as a new class of fluorescent labels have become valuable fluorescent platforms for biological applications due to their unique optical properties. In addition to their well-known size-dependent emission spectra, QDs are extremely sensitive to the presence of additional charges either on their surfaces or in the surrounding environment, which leads to a variety of optical properties and electronic consequences. By using thiols as bridges between QDs and redox-active ligands, the fluorescence effects of functionalized QD conjugates were investigated because QDs are prone to exchange electrons or energy with the attached ligands upon excitation, resulting in their fluorescence change. The recovery/enhancement or quenching of the QD conjugate fluorescence could be reversibly tuned with the transformation with the redox state of surface ligands. Moreover, quenching of the QD emission is highly dependent on the relative position of the oxidation levels of QDs and the redox-active ligand used. Importantly, the utility of these systems could enhance the compatibility of functionalized QDs in biological systems and can be used for monitoring the fluorescence change to trace in vitro and intracellular target analyte sensing. We believe that redox-mediated quantum dots as fluorescence

Keywords: quantum dots, redox-mediation, charge transfer, fluorescence, biosensor

Semiconductor quantum dots (QDs) or nanocrystals with sizes smaller than the so-called Bohr exciton radius (a few nanometers), resulting in an effect called quantum confinement due to the appearance of discrete energy states in both the conduction and valence bands [1, 2]. Optoelectronics of colloidal QDs offer a compelling combination of solution processing and fluorescence tunability through quantum size effects [3, 4]. They, however, are affected by a variety of parameters including defects in the nanocrystal structure and the surface or with the

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**and Their Biological Application**

and Their Biological Application

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

probe are a significant step forward toward biosensing.

http://dx.doi.org/10.5772/intechopen.70761

Wei Ma

Wei Ma

Abstract

1. Introduction

Provisional chapter

## **Redox-Mediated Quantum Dots as Fluorescence Probe and Their Biological Application** Redox-Mediated Quantum Dots as Fluorescence Probe and Their Biological Application

DOI: 10.5772/intechopen.70761

Wei Ma

Additional information is available at the end of the chapter Wei Ma

http://dx.doi.org/10.5772/intechopen.70761 Additional information is available at the end of the chapter

#### Abstract

Semiconductor quantum dots (QDs) as a new class of fluorescent labels have become valuable fluorescent platforms for biological applications due to their unique optical properties. In addition to their well-known size-dependent emission spectra, QDs are extremely sensitive to the presence of additional charges either on their surfaces or in the surrounding environment, which leads to a variety of optical properties and electronic consequences. By using thiols as bridges between QDs and redox-active ligands, the fluorescence effects of functionalized QD conjugates were investigated because QDs are prone to exchange electrons or energy with the attached ligands upon excitation, resulting in their fluorescence change. The recovery/enhancement or quenching of the QD conjugate fluorescence could be reversibly tuned with the transformation with the redox state of surface ligands. Moreover, quenching of the QD emission is highly dependent on the relative position of the oxidation levels of QDs and the redox-active ligand used. Importantly, the utility of these systems could enhance the compatibility of functionalized QDs in biological systems and can be used for monitoring the fluorescence change to trace in vitro and intracellular target analyte sensing. We believe that redox-mediated quantum dots as fluorescence probe are a significant step forward toward biosensing.

Keywords: quantum dots, redox-mediation, charge transfer, fluorescence, biosensor

## 1. Introduction

Semiconductor quantum dots (QDs) or nanocrystals with sizes smaller than the so-called Bohr exciton radius (a few nanometers), resulting in an effect called quantum confinement due to the appearance of discrete energy states in both the conduction and valence bands [1, 2]. Optoelectronics of colloidal QDs offer a compelling combination of solution processing and fluorescence tunability through quantum size effects [3, 4]. They, however, are affected by a variety of parameters including defects in the nanocrystal structure and the surface or with the

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

surrounding medium [5]. QDs, in particular, have a large-area solution processing on their surfaces and are always capped with functional ligands, which provide surface passivation and promote compatibility with the surrounding medium [6, 7]. These ligands along with the surrounding matrix alter the overall optical and electronic properties of QDs as a result of efficient elimination of the surface native defects, often attributed to the saturation of dangling bonds, improved passivation, and higher packing densities [8–10]. So far, the processibility of colloidal QDs is also exploited in a diversity of applications by fine-tuning their surface ligand characteristics of the semiconductor nanoparticles [11–14]. For example, a water-soluble surface ligand is required for biological sensors [15], an electron conductive layer is important for photoelectric devices [16], and a polymerizable surface is needed to make fluorescence polymer composites [17]. Unlike most organic dyes, QDs are also highly sensitive to charge transfer, thus altering their fluorescence properties [18, 19]. Notably, coupling redox-active ligands to the QDs surface can promote transfer of external electrons (and holes) to QD [18, 19]. Due to an efficient Auger recombination, the presence of additional charges can lead to quenching of the QD fluorescence [20]. The quenching degree of QD fluorescence depends on the location of the added charge, with a complete quenching observed for charges existing in the QD core, due to the strong spatial overlap between charge(s) and exciton, whereas partial quenching is observed for charge(s) locating on the QD surface (due to weaker overlap with the exciton) [19, 20]. When electron transfer between QDs and the molecules bound to their surface occurs, the nanocrystal and its attached ligand molecule exist in highly reactive charged forms long enough to interact with the surrounding environment. The redox-active moiety-functionalized QDs may promote the transfer of external electrons and holes to either the QDs core conduction band or the QDs surface states [21]. Therefore, controlling charge transfer of redox-active surface ligands across functionalized QD conjugates has been attracting increasing interests for advanced diagnostics and in vivo imaging as well as ultrasensitive biosensing [22, 23]. Redox-active compounds including metal complexes, ions, and dyes have already been investigated for use in photo-induced electron-transfer QD sensing. Since the development of high-performance QDs and the advent of excellent coupling techniques to modify them with biological systems [23–25], there has been a urgent need to exploit the interactions of QDs with the redox-active ligand for sensing [26, 27]. A few preliminary researches have reported the redox-active ligand-functionalized QDs and their use to monitor specific biological events. Biofunctional QDs enjoy increasing interest in basic and applied science because of the many possible applications of these structures to fields including proteomics, microarray technology, and biosensors. It is expected that these redox-active ligandfunctionalized nanocrystal will be able to perform specific functions, such as biorecognition in the context of an electrical measurement, better than either purely organic or inorganic systems.

brain activity and neurotransmission (i.e., dopamine), blood clotting (i.e., vitamin K), protein post-translational modification (i.e., topaquinone), cellular signaling molecule metabolism (i.e., estrogens and catecholamines), and antioxidant metabolism (i.e., ubiquinone and tocopherol congeners) [30–32]. Redox moiety was introduced into the surface ligands to achieve the redox-switchable fluorescence properties that could be useful for signal multiplexing, since QDs are highly sensitive to the electron-transfer processes. Recently, the research of functionalized QDs fabrication of redox quinone/hydroquinone on the surface of nanocrystals enjoys increasing interest and performs the specific functions, such as biosensing, ultrasensitive

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3. Ubiquinone-quantum dot bioconjugates and their application

In particular, ubiquinones [coenzyme Q, (CoQ)] are composed of the redox-active ubiquinonyl ring with a tail of isoprenoid units in different homolog forms occurring in nature, which are the only lipid-soluble antioxidant and plays a very important role in the cell membrane physiology [33]. CoQ acts as a mobile electron carrier in the energy-transducing membranes of mitochondria, which can be reduced by NAD(P)H-dependent enzymes. The reduced form CoQH2 is a potent radical scavenger and antioxidant that protects membranes and lipoproteins from peroxidations as a potent radical scavenger [34, 35]. The redox state can be determined not only by the extent of oxidation (oxidative stress), but also by that of reduction (enzymatic reaction). As well-known, fluorescence enhancement/quenching in QDs can be switched by electrochemically modulating electron transfer between attached molecules and QDs (Figure 1) [36]. For this purpose, three CoQ disulfide derivatives ([CoQCnS]2) possessing the basic ubiquinone structure of 2,3-dimethoxy-5-methyl-1,4-benzoquinone with different mercaptoalkyl side chain lengths at the 6-position (n = 1, 5, and 10) (Figure 2, left). The emission of functionalized QDs can be reversibly tuned in two directions, enhancement or

Figure 1. Schematic of fluorescence enhancement/quenching characteristics of CoQH2 and CoQ-functionalized CdTe/ZnS

QDs. Adapted with permission from [36]. Copyright 2011 Wiley-VCH Verlag GmbH & Co. KGaA.

detection, and biomimetic research.

#### 2. Quinone/hydroquinone as redox-active surface ligands of QDs

Quinone/hydroquinone is ubiquitous in nature and constitutes an important class of naturally occurring redox molecules [28]. It is well-known that quinone/hydroquinone fulfills a universal and possibly unique function in electron transfer and energy conserving system [29]. Especially, a number of quinones/hydroquinones have the critical biological functions involving brain activity and neurotransmission (i.e., dopamine), blood clotting (i.e., vitamin K), protein post-translational modification (i.e., topaquinone), cellular signaling molecule metabolism (i.e., estrogens and catecholamines), and antioxidant metabolism (i.e., ubiquinone and tocopherol congeners) [30–32]. Redox moiety was introduced into the surface ligands to achieve the redox-switchable fluorescence properties that could be useful for signal multiplexing, since QDs are highly sensitive to the electron-transfer processes. Recently, the research of functionalized QDs fabrication of redox quinone/hydroquinone on the surface of nanocrystals enjoys increasing interest and performs the specific functions, such as biosensing, ultrasensitive detection, and biomimetic research.

## 3. Ubiquinone-quantum dot bioconjugates and their application

surrounding medium [5]. QDs, in particular, have a large-area solution processing on their surfaces and are always capped with functional ligands, which provide surface passivation and promote compatibility with the surrounding medium [6, 7]. These ligands along with the surrounding matrix alter the overall optical and electronic properties of QDs as a result of efficient elimination of the surface native defects, often attributed to the saturation of dangling bonds, improved passivation, and higher packing densities [8–10]. So far, the processibility of colloidal QDs is also exploited in a diversity of applications by fine-tuning their surface ligand characteristics of the semiconductor nanoparticles [11–14]. For example, a water-soluble surface ligand is required for biological sensors [15], an electron conductive layer is important for photoelectric devices [16], and a polymerizable surface is needed to make fluorescence polymer composites [17]. Unlike most organic dyes, QDs are also highly sensitive to charge transfer, thus altering their fluorescence properties [18, 19]. Notably, coupling redox-active ligands to the QDs surface can promote transfer of external electrons (and holes) to QD [18, 19]. Due to an efficient Auger recombination, the presence of additional charges can lead to quenching of the QD fluorescence [20]. The quenching degree of QD fluorescence depends on the location of the added charge, with a complete quenching observed for charges existing in the QD core, due to the strong spatial overlap between charge(s) and exciton, whereas partial quenching is observed for charge(s) locating on the QD surface (due to weaker overlap with the exciton) [19, 20]. When electron transfer between QDs and the molecules bound to their surface occurs, the nanocrystal and its attached ligand molecule exist in highly reactive charged forms long enough to interact with the surrounding environment. The redox-active moiety-functionalized QDs may promote the transfer of external electrons and holes to either the QDs core conduction band or the QDs surface states [21]. Therefore, controlling charge transfer of redox-active surface ligands across functionalized QD conjugates has been attracting increasing interests for advanced diagnostics and in vivo imaging as well as ultrasensitive biosensing [22, 23]. Redox-active compounds including metal complexes, ions, and dyes have already been investigated for use in photo-induced electron-transfer QD sensing. Since the development of high-performance QDs and the advent of excellent coupling techniques to modify them with biological systems [23–25], there has been a urgent need to exploit the interactions of QDs with the redox-active ligand for sensing [26, 27]. A few preliminary researches have reported the redox-active ligand-functionalized QDs and their use to monitor specific biological events. Biofunctional QDs enjoy increasing interest in basic and applied science because of the many possible applications of these structures to fields including proteomics, microarray technology, and biosensors. It is expected that these redox-active ligandfunctionalized nanocrystal will be able to perform specific functions, such as biorecognition in the context of an electrical measurement, better than either purely organic or inorganic systems.

134 Nonmagnetic and Magnetic Quantum Dots

2. Quinone/hydroquinone as redox-active surface ligands of QDs

Quinone/hydroquinone is ubiquitous in nature and constitutes an important class of naturally occurring redox molecules [28]. It is well-known that quinone/hydroquinone fulfills a universal and possibly unique function in electron transfer and energy conserving system [29]. Especially, a number of quinones/hydroquinones have the critical biological functions involving In particular, ubiquinones [coenzyme Q, (CoQ)] are composed of the redox-active ubiquinonyl ring with a tail of isoprenoid units in different homolog forms occurring in nature, which are the only lipid-soluble antioxidant and plays a very important role in the cell membrane physiology [33]. CoQ acts as a mobile electron carrier in the energy-transducing membranes of mitochondria, which can be reduced by NAD(P)H-dependent enzymes. The reduced form CoQH2 is a potent radical scavenger and antioxidant that protects membranes and lipoproteins from peroxidations as a potent radical scavenger [34, 35]. The redox state can be determined not only by the extent of oxidation (oxidative stress), but also by that of reduction (enzymatic reaction). As well-known, fluorescence enhancement/quenching in QDs can be switched by electrochemically modulating electron transfer between attached molecules and QDs (Figure 1) [36]. For this purpose, three CoQ disulfide derivatives ([CoQCnS]2) possessing the basic ubiquinone structure of 2,3-dimethoxy-5-methyl-1,4-benzoquinone with different mercaptoalkyl side chain lengths at the 6-position (n = 1, 5, and 10) (Figure 2, left). The emission of functionalized QDs can be reversibly tuned in two directions, enhancement or

Figure 1. Schematic of fluorescence enhancement/quenching characteristics of CoQH2 and CoQ-functionalized CdTe/ZnS QDs. Adapted with permission from [36]. Copyright 2011 Wiley-VCH Verlag GmbH & Co. KGaA.

quenching, depending on the different redox state of substrates bound to the surface of the QDs (Figure 2, right). Following photoexcitation of functionalized QD bioconjugates, the conductive band electron of QDs is transported to the lowest unoccupied molecular orbital of the oxidized ubiquinone acceptor and the electron is then went back to the valence band of QDs via nonradiative pathways. Thus, ubiquinones play the surface trap states acting as nonradiative deexcitation paths for photo-induced electron carriers, resulting in fluorescence quenching. It is worth noting that reduced ubiquinol ligands on the surface of QDs yield an obvious fluorescence enhancement. In this case, the photo-excited CoQH2-QD bioconjugates decay to the ground state because the ubiquinols serve as poor electron donors. This switching results in recovering a high fluorescence compared to bare QDs. Furthermore, the reduced ubiquinols provide an efficient passivation of the surface trap states to overcome the potential surface defects, leading to a significantly enhanced fluorescence in CoQH2-QD bioconjugates. According to energy band, bandgap of surface-capping ligand ubiquinol is larger than that of CdSe/ZnS QDs and hole trapping is also negligible. Upon photoexcitation, the resulting electrons and holes are confined in the surface regions of the ubiquinol-functionalized QDs, thus increasing the fluorescence. In addition, the fluorescence efficiency and stability of CoQH2-QD bioconjugates against photo-oxidation has shown significant improvement due to the antioxidation effect of ubiquinol. Therefore, there is the remarkable fluorescence difference between CoQ and CoQH2-capped QDs. Notably, the capping layer of reduced ubiquinol ligands enhances the QDs' fluorescence intensity significantly, while a modification using the oxidized ubiquinone ligands presents efficient quenching on fluorescence intensity of QDs under the identical conditions (Figure 2, right). We show fluorescence quenching efficiency to be dependent on alkyl chain spacer length of surface ligands, as more pronounced quenching was observed for C2 spacer-modified QDs. Surface-attached CdTe/ZnS QDs exploiting coenzyme Q derivatives CoQ and CoQH2 can be chemically attached to the surface of the QDs in an effort to mimic the electron transfer in the part of mitochondrial respiratory chain. Our system is extremely sensitive to NADH and superoxide radical (O2•) species, and mimics a biological electron-transfer system in the part of the mitochondrial respiratory chain. In addition, in situ fluorescence spectra-electrochemical results further validate that the reduced state of ubiquinols significantly increase the fluorescence of QD bioconjugates, while the oxidized state of the ubiquinones decrease the fluorescence at varying degrees.

To further enhance the compatibility of ubiquinone-QD bioconjugates in biological system, the ligands Q2NS, Q5NS, and Q10NS were designed and synthesized by a facile click reaction between ubiquinone with terminal alkynes and alkylazide-disulfides via copper(I) tris (benzyltriazolylmethyl) amine catalyzed 1,2,3-triazole formation [37] (Figure 3a). In this system, the quinoid moiety in the QnNS surface ligands was introduced to achieve the redoxswitchable fluorescence properties for signal multiplexing. The 1,2,3-triazole groups can enhance the compatibility of QnNS-QDs in biological system because of the similarity with histidine. Three alkyl spacers (C2, C5, C10) confer various electron-transfer abilities either the core or the surface of QDs. As a final point, the disulfide group facilitates modification of QnNS ligand to the surface of QDs. Using the QnNS-QD bioconjugates, enhancement or quenching of the fluorescence of QD bioconjugates can also be switched by modulating the redox state of

Redox-Mediated Quantum Dots as Fluorescence Probe and Their Biological Application

Interestingly, the emission of QD bioconjugates was enhanced when the surface-attached ubiquinone layer was reduced to ubiquinol in the presence of NADH and complex I in an effort to mimic the initial stages of mitochondrial respiration. The fluorescence intensity of

• is higher, the luminescence of the QDs is quenched to a higher extent, consistent with the formation of a higher coverage of the oxidized ubiquinone-modified QDs. Moreover, these systems provide the general framework for the creation of probes to monitor the reactive oxygen species in living cells according to their redox state, suggesting that this principle can be generalized to many different biological systems and applications. To demonstrate 1,2,3-triazole groups incorporated into the ubiquinone ligands can enhance the compatibility of QD bioconjugates in biological systems, we investigated a time-dependent fluorescence process using ubiquinone-assembled QDs with or without 1,2,3-triazole groups. A significant increase in the incubation time was observed for the same enhancement of fluorescence compared to the triazole-linked ubiquinone-QDs in the presence of NADH and complex I (Figure 3). This is because that the triazole groups behave similarly to histidine ligands and can be used to cap enzymes through proteins- or peptide-affinity coordination of triazole residues, leading to the triazole ubiquinone ligands efficiently improving binding affinity with complex I. The ubiquinone-QD bioconjugate system could be used for monitoring in vitro and intracellular complex I levels by the fluorescence changes of QD. Epidemiological researches show that the activity of complex I of Parkinson patients is impaired. Therefore, this system can be employed as a potent fluorescence probe for early stage Parkinson disease diagnosis and progression monitoring by observing complex I levels

Another novel strategy that uses QDs functionalized with quinonyl ligands was developed [39]. A novel biosensor based on "switch-on" photoluminogenic strategy employing of quinonyl glycosides functionalized QDs for the ingenious and biospecific imaging of human hepatoma Hep-G2 cells that express transmembrane glycoprotein receptors (Figure 4). The closely coupled quinonyl glycoside ligands are envisioned to have dual functions: the quinone part acts as a quencher of QDs and the glycoside part as a ligand for targeting a specific receptor. Moreover, self-assembly of quinonyl glycosides to QDs through a sulfide bond may produce QD bioconjugates that expose the glycosides in a clustering manner, enhancing their binding avidity with the target receptors. We observed that the quenched fluorescence

• was added. As the concentration of

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surface-capping ubiquinone ligands (Figure 3b) [38].

ubiquinol-QDs was decreased gradually when the O2

in human neuroblastoma SH-SY5Y cells.

O2

Figure 2. (Left) Chemical structures of synthesized [CoQCnS]2, n = 1, 5, 10. (Right) Fluorescence spectra of functionalized QDs. (a) CdTe/ZnS QDs, (b) [CoQH2CnS]2, and (c) [CoQCnS]2-functionalized CdTe/ZnS QDs. A: n = 1, B: n = 5, C: n = 10. Adapted with permission from [36]. Copyright 2011 Wiley-VCH Verlag GmbH & Co. KGaA.

To further enhance the compatibility of ubiquinone-QD bioconjugates in biological system, the ligands Q2NS, Q5NS, and Q10NS were designed and synthesized by a facile click reaction between ubiquinone with terminal alkynes and alkylazide-disulfides via copper(I) tris (benzyltriazolylmethyl) amine catalyzed 1,2,3-triazole formation [37] (Figure 3a). In this system, the quinoid moiety in the QnNS surface ligands was introduced to achieve the redoxswitchable fluorescence properties for signal multiplexing. The 1,2,3-triazole groups can enhance the compatibility of QnNS-QDs in biological system because of the similarity with histidine. Three alkyl spacers (C2, C5, C10) confer various electron-transfer abilities either the core or the surface of QDs. As a final point, the disulfide group facilitates modification of QnNS ligand to the surface of QDs. Using the QnNS-QD bioconjugates, enhancement or quenching of the fluorescence of QD bioconjugates can also be switched by modulating the redox state of surface-capping ubiquinone ligands (Figure 3b) [38].

quenching, depending on the different redox state of substrates bound to the surface of the QDs (Figure 2, right). Following photoexcitation of functionalized QD bioconjugates, the conductive band electron of QDs is transported to the lowest unoccupied molecular orbital of the oxidized ubiquinone acceptor and the electron is then went back to the valence band of QDs via nonradiative pathways. Thus, ubiquinones play the surface trap states acting as nonradiative deexcitation paths for photo-induced electron carriers, resulting in fluorescence quenching. It is worth noting that reduced ubiquinol ligands on the surface of QDs yield an obvious fluorescence enhancement. In this case, the photo-excited CoQH2-QD bioconjugates decay to the ground state because the ubiquinols serve as poor electron donors. This switching results in recovering a high fluorescence compared to bare QDs. Furthermore, the reduced ubiquinols provide an efficient passivation of the surface trap states to overcome the potential surface defects, leading to a significantly enhanced fluorescence in CoQH2-QD bioconjugates. According to energy band, bandgap of surface-capping ligand ubiquinol is larger than that of CdSe/ZnS QDs and hole trapping is also negligible. Upon photoexcitation, the resulting electrons and holes are confined in the surface regions of the ubiquinol-functionalized QDs, thus increasing the fluorescence. In addition, the fluorescence efficiency and stability of CoQH2-QD bioconjugates against photo-oxidation has shown significant improvement due to the antioxidation effect of ubiquinol. Therefore, there is the remarkable fluorescence difference between CoQ and CoQH2-capped QDs. Notably, the capping layer of reduced ubiquinol ligands enhances the QDs' fluorescence intensity significantly, while a modification using the oxidized ubiquinone ligands presents efficient quenching on fluorescence intensity of QDs under the identical conditions (Figure 2, right). We show fluorescence quenching efficiency to be dependent on alkyl chain spacer length of surface ligands, as more pronounced quenching was observed for C2 spacer-modified QDs. Surface-attached CdTe/ZnS QDs exploiting coenzyme Q derivatives CoQ and CoQH2 can be chemically attached to the surface of the QDs in an effort to mimic the electron transfer in the part of mitochondrial respiratory chain. Our system is extremely sensitive to NADH and superoxide radical (O2•) species, and mimics a biological electron-transfer system in the part of the mitochondrial respiratory chain. In addition, in situ fluorescence spectra-electrochemical results further validate that the reduced state of ubiquinols significantly increase the fluorescence of QD bioconjugates, while the oxidized state of

136 Nonmagnetic and Magnetic Quantum Dots

the ubiquinones decrease the fluorescence at varying degrees.

Figure 2. (Left) Chemical structures of synthesized [CoQCnS]2, n = 1, 5, 10. (Right) Fluorescence spectra of functionalized QDs. (a) CdTe/ZnS QDs, (b) [CoQH2CnS]2, and (c) [CoQCnS]2-functionalized CdTe/ZnS QDs. A: n = 1, B: n = 5, C: n = 10.

Adapted with permission from [36]. Copyright 2011 Wiley-VCH Verlag GmbH & Co. KGaA.

Interestingly, the emission of QD bioconjugates was enhanced when the surface-attached ubiquinone layer was reduced to ubiquinol in the presence of NADH and complex I in an effort to mimic the initial stages of mitochondrial respiration. The fluorescence intensity of ubiquinol-QDs was decreased gradually when the O2 • was added. As the concentration of O2 • is higher, the luminescence of the QDs is quenched to a higher extent, consistent with the formation of a higher coverage of the oxidized ubiquinone-modified QDs. Moreover, these systems provide the general framework for the creation of probes to monitor the reactive oxygen species in living cells according to their redox state, suggesting that this principle can be generalized to many different biological systems and applications. To demonstrate 1,2,3-triazole groups incorporated into the ubiquinone ligands can enhance the compatibility of QD bioconjugates in biological systems, we investigated a time-dependent fluorescence process using ubiquinone-assembled QDs with or without 1,2,3-triazole groups. A significant increase in the incubation time was observed for the same enhancement of fluorescence compared to the triazole-linked ubiquinone-QDs in the presence of NADH and complex I (Figure 3). This is because that the triazole groups behave similarly to histidine ligands and can be used to cap enzymes through proteins- or peptide-affinity coordination of triazole residues, leading to the triazole ubiquinone ligands efficiently improving binding affinity with complex I. The ubiquinone-QD bioconjugate system could be used for monitoring in vitro and intracellular complex I levels by the fluorescence changes of QD. Epidemiological researches show that the activity of complex I of Parkinson patients is impaired. Therefore, this system can be employed as a potent fluorescence probe for early stage Parkinson disease diagnosis and progression monitoring by observing complex I levels in human neuroblastoma SH-SY5Y cells.

Another novel strategy that uses QDs functionalized with quinonyl ligands was developed [39]. A novel biosensor based on "switch-on" photoluminogenic strategy employing of quinonyl glycosides functionalized QDs for the ingenious and biospecific imaging of human hepatoma Hep-G2 cells that express transmembrane glycoprotein receptors (Figure 4). The closely coupled quinonyl glycoside ligands are envisioned to have dual functions: the quinone part acts as a quencher of QDs and the glycoside part as a ligand for targeting a specific receptor. Moreover, self-assembly of quinonyl glycosides to QDs through a sulfide bond may produce QD bioconjugates that expose the glycosides in a clustering manner, enhancing their binding avidity with the target receptors. We observed that the quenched fluorescence

Figure 3. Schematic of ubiquinone-CdSe/ZnS QDs as redox fluorescence biosensor for Parkinson's disease diagnosis. (a) ubiquinone-terminated disulphides (QnNS) synthesis and self-assembly of QnNS on to CdSe/ZnS QDs. (b) Conceptual visualisation of QnNS-QDs as complex I sensor in vitro. Under oxidized state (QnNS), ubiquinone functions as a favorable electron acceptor, this results in effective QDs' fluorescence quenching. Addition of complex I to QnNS-QDs solution in the presence of NADH, ubiquinone coupled electron transfer and proton translocation from NADH, producing reduced ubiquinol (HQnNS) form on the surface of QDs to mimic the initial stages of the respiratory chain. Ubiquinol when in close proximity to the QDs produces fluorescence enhancement. (c) Energetic diagram of the QDs bioconjugates and possible electron transfer processes: electron transfer from the QDs CB to QnNS LUMO, followed by the back QDs VB. HQnNS only weakly accepts/donates electrons or energy and the excited QDs can return radiatively to the ground state. Under these conditions, the presence of HQnNS results in a significant fluorescence enhancement. (d) Fluorescence spectra of ubiquinone/ubiquinol- functionalised CdSe/ZnS QDs. e, Cyclic voltammetry of QnNS-CdSe/ZnS QDs. (f) Visualisation of QnNS-CdSe/ZnS QDs as an intracellular complex I sensor. The mitochondrial-specific neurotoxin, rotenone, inhibits complex I and leads to Parkinson's-like pathogenesis. Parkinson's disease is characterized by impaired activity of complex I in the electron-transfer chain of mitochondria. Adapted with permission from [38]. Copyright 2013 Nature Publishing Group.

of the functionalized QDs (by quinone) could be recovered by a lectin that selectively binds to the quinonyl glycosides clustering the QDs, but showed insignificant fluctuations toward a panel of nonselective lectins. We further determined that QDs coated with quinonyl galactosides could optically image transmembrane glycoprotein receptors of a hepatoma cell line in a target-specific manner (which they showed much weakened imaging ability toward cells with a reduced receptor level). This unique system, by taking advantage of the effective quenching ability of benzoquinone for QDs and natural ligand-receptor pairing on the cell surface (that recovers the signal), paves the way for the development of highly specific and low-background techniques for bioimaging of cancer cells as well as probing of unknown

Figure 4. Schematic diagram of quinonyl glycosides functionalized QDs as a novel "switch-on" fluorescence probe for specific targeting and imaging transmembrane glycoprotein receptors of human hepatoma Hep-G2 cancer cells [Q-Glc: quinonyl glucoside disulfide; Q-Gal: quinonyl galactoside disulfide; Q: quinonyl disulfide]. Adapted from [39]. Copyright

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cell-surface receptors.

2014 American Chemical Society.

Redox-Mediated Quantum Dots as Fluorescence Probe and Their Biological Application http://dx.doi.org/10.5772/intechopen.70761 139

Figure 4. Schematic diagram of quinonyl glycosides functionalized QDs as a novel "switch-on" fluorescence probe for specific targeting and imaging transmembrane glycoprotein receptors of human hepatoma Hep-G2 cancer cells [Q-Glc: quinonyl glucoside disulfide; Q-Gal: quinonyl galactoside disulfide; Q: quinonyl disulfide]. Adapted from [39]. Copyright 2014 American Chemical Society.

Figure 3. Schematic of ubiquinone-CdSe/ZnS QDs as redox fluorescence biosensor for Parkinson's disease diagnosis. (a) ubiquinone-terminated disulphides (QnNS) synthesis and self-assembly of QnNS on to CdSe/ZnS QDs. (b) Conceptual visualisation of QnNS-QDs as complex I sensor in vitro. Under oxidized state (QnNS), ubiquinone functions as a favorable electron acceptor, this results in effective QDs' fluorescence quenching. Addition of complex I to QnNS-QDs solution in the presence of NADH, ubiquinone coupled electron transfer and proton translocation from NADH, producing reduced ubiquinol (HQnNS) form on the surface of QDs to mimic the initial stages of the respiratory chain. Ubiquinol when in close proximity to the QDs produces fluorescence enhancement. (c) Energetic diagram of the QDs bioconjugates and possible electron transfer processes: electron transfer from the QDs CB to QnNS LUMO, followed by the back QDs VB. HQnNS only weakly accepts/donates electrons or energy and the excited QDs can return radiatively to the ground state. Under these conditions, the presence of HQnNS results in a significant fluorescence enhancement. (d) Fluorescence spectra of ubiquinone/ubiquinol- functionalised CdSe/ZnS QDs. e, Cyclic voltammetry of QnNS-CdSe/ZnS QDs. (f) Visualisation of QnNS-CdSe/ZnS QDs as an intracellular complex I sensor. The mitochondrial-specific neurotoxin, rotenone, inhibits complex I and leads to Parkinson's-like pathogenesis. Parkinson's disease is characterized by impaired activity of complex I in the electron-transfer chain of mitochondria. Adapted with permission from [38]. Copyright 2013

Nature Publishing Group.

138 Nonmagnetic and Magnetic Quantum Dots

of the functionalized QDs (by quinone) could be recovered by a lectin that selectively binds to the quinonyl glycosides clustering the QDs, but showed insignificant fluctuations toward a panel of nonselective lectins. We further determined that QDs coated with quinonyl galactosides could optically image transmembrane glycoprotein receptors of a hepatoma cell line in a target-specific manner (which they showed much weakened imaging ability toward cells with a reduced receptor level). This unique system, by taking advantage of the effective quenching ability of benzoquinone for QDs and natural ligand-receptor pairing on the cell surface (that recovers the signal), paves the way for the development of highly specific and low-background techniques for bioimaging of cancer cells as well as probing of unknown cell-surface receptors.

## 4. Dopamine-functionalized quantum dots and their application

Dopamine (DA) is an essential neurotransmitter in central nervous system and facilitates various functions in brain. DA-induced neurotoxicity has long been known to be triggered by the oxidation of DA and may play a role in pathological processes associated with neurodegeneration. Under oxidative stress, DA could readily oxidize to produce DA quinone catalyzed by tyrosinase in the presence of O2 and contribute to the nucleophilic addition with sulfhydryl groups on free cysteine (Cys), glutathione, or Cys residue contained in protein [40, 41]. The interaction between DA quinone and Cys residue yields the formation of 5-Cys-DA in vitro and in vivo. As well-known, Cys residue is particularly critical for maintaining dynamic redox balance of cell and physiological function, which is directly correlated with the level of cellular stress. However, DA inactivation modification between Cys residue and DA quinone may disturb mitochondrial function, scavenge the thiol protein, inhibit protein function, and possibly lead to cell death [42]. Moreover, this modification decreases in the endogenous level of Cys residues, which is often found at the active site of functional proteins. Recent studies suggest that disturbance of Cys residue homeostasis may either lead to or result from oxidative stress in cell, contributing to mitochondrial dysfunction occurs early, and acts causally in neurodegenerative pathogenesis [43]. Therefore, it is of considerable significance to investigate the nature of this interaction process in physiology and pathology.

were used to monitor the redox process of DAs and the formation of 5-Cys-DAs by mimicking the interaction process that DA oxidizes to form DA quinone, which binds covalently to nucleophilic sulfhydryl groups on Cys residues. The enzymatic process catalyzes the transformation of ligand structure between DA quinone and catechol, leading to the fluorescence change of functionalized QD bioconjugates (Figure 5). Several lines of evidence suggest that disturbance of Cys residue

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Figure 5. Schematic diagram of self-assembly and FL quenching/recovery characteristics of DAs-functionalized CdTe/ ZnS QDs; Inset: schematic of the oxidation of DA and the irreversible interaction between Cys residue and DA quinone.

Figure 6. Schematic representation of redox-mediated indirect fluorescence immunoassay for the detection of biomarkers

using DAs-functionalized CdSe/ZnS QDs. Adapted from [52]. Copyright 2016 American Chemical Society.

Adapted from [51]. Copyright 2015 American Chemical Society.

Due to the superior optical and photophysical properties of QDs, biorecognition or biocatalytic reactions have been followed by fluorescence resonance energy transfer or electron-transfer processes stimulated by redox-active biomolecule-functionalized QDs [44–50]. The DA-functionalized QDs were prepared through the following steps: (1) 596-nm-emitting thiohydracrylic acid capped CdTe/ZnS QDs and a redox-active DA thiol derivative (DAs) as surface-capping ligand were designed and synthesized; (2) the ligand molecule DAs was self-assembled onto the surface of QDs [51]. About 24 DAs molecules per QD were chosen as the optimal ratio from the spectra according to the QDs self-assembled with increasing ratio of DAs. DAs quinone on the surface of QD bioconjugates are generated in the enzymatic oxidation of DAs by tyrosinase/O2, resulting in the fluorescence quenching (Figure 5). With adding a three-fold maximum tyrosinase/O2, the fluorescence intensity of DAs-functionalized QDs was obviously quenched as expected. However, even much more excess tyrosinase/O2 did not greatly affect the fluorescence of bare CdTe/ZnS QDs (≤10% quenching). After DAs-QDs catalyzed by tyrosinase/O2, the resulting product DAs quinone acting as an excellent electron acceptor is efficient for hole trapping of QDs and induces the fluorescence quenching. Fluorescence intensity of DAs quinone-QD bioconjugates recovered gradually upon addition of increasing amounts of Cys. Approximately 96% of the fluorescence was recovered after addition of Cys. It is worth noting that the 5-Cys-DAs containing catechol moiety on the functionalized QDs significantly recovered fluorescence. Here, the photo-excited-functionalized QD bioconjugates decay radiatively to the ground state of QDs because the 5-Cys-DAs ligands could function as poor electron acceptors. This in turn results in a fluorescence recovery due to the transformation from DAs quinone to DAs on the surface of functionalized QDs, blocking the electron transfer from QDs to benzoquinone. Only the presence of Cys residues (Cys or GSH) could induce rapid fluorescence recovery of the DAs quinone-functionalized QDs, confirming the specific coupling of Cys and DAs quinone in this system. In this study, photophysical properties of QDs were used to monitor the redox process of DAs and the formation of 5-Cys-DAs by mimicking the interaction process that DA oxidizes to form DA quinone, which binds covalently to nucleophilic sulfhydryl groups on Cys residues. The enzymatic process catalyzes the transformation of ligand structure between DA quinone and catechol, leading to the fluorescence change of functionalized QD bioconjugates (Figure 5). Several lines of evidence suggest that disturbance of Cys residue

4. Dopamine-functionalized quantum dots and their application

140 Nonmagnetic and Magnetic Quantum Dots

investigate the nature of this interaction process in physiology and pathology.

Due to the superior optical and photophysical properties of QDs, biorecognition or biocatalytic reactions have been followed by fluorescence resonance energy transfer or electron-transfer processes stimulated by redox-active biomolecule-functionalized QDs [44–50]. The DA-functionalized QDs were prepared through the following steps: (1) 596-nm-emitting thiohydracrylic acid capped CdTe/ZnS QDs and a redox-active DA thiol derivative (DAs) as surface-capping ligand were designed and synthesized; (2) the ligand molecule DAs was self-assembled onto the surface of QDs [51]. About 24 DAs molecules per QD were chosen as the optimal ratio from the spectra according to the QDs self-assembled with increasing ratio of DAs. DAs quinone on the surface of QD bioconjugates are generated in the enzymatic oxidation of DAs by tyrosinase/O2, resulting in the fluorescence quenching (Figure 5). With adding a three-fold maximum tyrosinase/O2, the fluorescence intensity of DAs-functionalized QDs was obviously quenched as expected. However, even much more excess tyrosinase/O2 did not greatly affect the fluorescence of bare CdTe/ZnS QDs (≤10% quenching). After DAs-QDs catalyzed by tyrosinase/O2, the resulting product DAs quinone acting as an excellent electron acceptor is efficient for hole trapping of QDs and induces the fluorescence quenching. Fluorescence intensity of DAs quinone-QD bioconjugates recovered gradually upon addition of increasing amounts of Cys. Approximately 96% of the fluorescence was recovered after addition of Cys. It is worth noting that the 5-Cys-DAs containing catechol moiety on the functionalized QDs significantly recovered fluorescence. Here, the photo-excited-functionalized QD bioconjugates decay radiatively to the ground state of QDs because the 5-Cys-DAs ligands could function as poor electron acceptors. This in turn results in a fluorescence recovery due to the transformation from DAs quinone to DAs on the surface of functionalized QDs, blocking the electron transfer from QDs to benzoquinone. Only the presence of Cys residues (Cys or GSH) could induce rapid fluorescence recovery of the DAs quinone-functionalized QDs, confirming the specific coupling of Cys and DAs quinone in this system. In this study, photophysical properties of QDs

Dopamine (DA) is an essential neurotransmitter in central nervous system and facilitates various functions in brain. DA-induced neurotoxicity has long been known to be triggered by the oxidation of DA and may play a role in pathological processes associated with neurodegeneration. Under oxidative stress, DA could readily oxidize to produce DA quinone catalyzed by tyrosinase in the presence of O2 and contribute to the nucleophilic addition with sulfhydryl groups on free cysteine (Cys), glutathione, or Cys residue contained in protein [40, 41]. The interaction between DA quinone and Cys residue yields the formation of 5-Cys-DA in vitro and in vivo. As well-known, Cys residue is particularly critical for maintaining dynamic redox balance of cell and physiological function, which is directly correlated with the level of cellular stress. However, DA inactivation modification between Cys residue and DA quinone may disturb mitochondrial function, scavenge the thiol protein, inhibit protein function, and possibly lead to cell death [42]. Moreover, this modification decreases in the endogenous level of Cys residues, which is often found at the active site of functional proteins. Recent studies suggest that disturbance of Cys residue homeostasis may either lead to or result from oxidative stress in cell, contributing to mitochondrial dysfunction occurs early, and acts causally in neurodegenerative pathogenesis [43]. Therefore, it is of considerable significance to

Figure 5. Schematic diagram of self-assembly and FL quenching/recovery characteristics of DAs-functionalized CdTe/ ZnS QDs; Inset: schematic of the oxidation of DA and the irreversible interaction between Cys residue and DA quinone. Adapted from [51]. Copyright 2015 American Chemical Society.

Figure 6. Schematic representation of redox-mediated indirect fluorescence immunoassay for the detection of biomarkers using DAs-functionalized CdSe/ZnS QDs. Adapted from [52]. Copyright 2016 American Chemical Society.

homeostasis may either lead to or result from oxidative stress in cell. In all major examples of neurodegenerative diseases, there is strong evidence that oxidative stress contribute to mitochondrial dysfunction occurs early and acts causally in disease pathogenesis. Thus, this specific fluorescence changes in our proposed system develop a powerful fluorescence sensor to follow the tracks of the neurotransmitter modification.

[3] Henry CH. Limiting efficiencies of ideal single and multiple energy-gap terrestrialsolar-

Redox-Mediated Quantum Dots as Fluorescence Probe and Their Biological Application

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Inspired by redox-mediated fluorescence strategy, a redox-mediated indirect fluorescence immunoassay was developed for detecting the disease biomarker α-fetoprotein in a model based on DAs-immobilized CdSe/ZnS QDs (Figure 6) [52]. In this system, tyrosinase conjugated with the detection antibody was used as a bridge linking the QD fluorescence signals with the concentration of target disease biomarkers; the tyrosinase could catalyze enzymatic oxidation of DA to DA-quinone, resulting in fluorescence quenching in the presence of the analyte. Using this method, the detection limit for AFP was as low as 10 pM. This work provides a new pathway for the detection of disease biomarkers by RMFIA and has good potential for other applications.

## 5. Conclusion

By using redox-mediated fluorescence strategy, we demonstrated that coupling QDs with redoxactive surface ligand is capable of fluorescence detecting of target analytes with high specificity. Ubiquinone-coupled QDs could be used for quantitative detection of ROS and target-specific imaging of transmembrane receptors in living cells. Dopamine as an electron donor could sensitize QDs through different mechanisms for monitoring dopaminergic neurotoxicity. Moreover, the improvement of QD-dopamine bioconjugates as biosensors was used for clinical diagnostic applications. Cumulatively, these results confirm a critical role for redox molecules, and especially quinone, in charge-transfer interactions with QDs for biological application.

## Author details

Wei Ma

Address all correspondence to: weima@ecust.edu.cn

Key Laboratory for Advanced Materials, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, PR China

## References


homeostasis may either lead to or result from oxidative stress in cell. In all major examples of neurodegenerative diseases, there is strong evidence that oxidative stress contribute to mitochondrial dysfunction occurs early and acts causally in disease pathogenesis. Thus, this specific fluorescence changes in our proposed system develop a powerful fluorescence sensor to follow the tracks

Inspired by redox-mediated fluorescence strategy, a redox-mediated indirect fluorescence immunoassay was developed for detecting the disease biomarker α-fetoprotein in a model based on DAs-immobilized CdSe/ZnS QDs (Figure 6) [52]. In this system, tyrosinase conjugated with the detection antibody was used as a bridge linking the QD fluorescence signals with the concentration of target disease biomarkers; the tyrosinase could catalyze enzymatic oxidation of DA to DA-quinone, resulting in fluorescence quenching in the presence of the analyte. Using this method, the detection limit for AFP was as low as 10 pM. This work provides a new pathway for the detection of disease biomarkers by RMFIA and has good potential for other applications.

By using redox-mediated fluorescence strategy, we demonstrated that coupling QDs with redoxactive surface ligand is capable of fluorescence detecting of target analytes with high specificity. Ubiquinone-coupled QDs could be used for quantitative detection of ROS and target-specific imaging of transmembrane receptors in living cells. Dopamine as an electron donor could sensitize QDs through different mechanisms for monitoring dopaminergic neurotoxicity. Moreover, the improvement of QD-dopamine bioconjugates as biosensors was used for clinical diagnostic applications. Cumulatively, these results confirm a critical role for redox molecules, and

Key Laboratory for Advanced Materials, School of Chemistry and Molecular Engineering, East

[1] Moras JD, Strandberg B, Suc D, et al. Semiconductor clusters, nanocrystals, and quantum

[2] Michalet X, Pinaud FF, Bentolila LA, et al. Quantum dots for live cells, in vivo imaging,

especially quinone, in charge-transfer interactions with QDs for biological application.

of the neurotransmitter modification.

142 Nonmagnetic and Magnetic Quantum Dots

5. Conclusion

Author details

Address all correspondence to: weima@ecust.edu.cn

dots. Science. 1996;271:933

China University of Science and Technology, Shanghai, PR China

and diagnostics. Science. 2005;307(5709):538-544

Wei Ma

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**Chapter 9**

**Provisional chapter**

**Enhancement of Photosynthetic Productivity by**

**Enhancement of Photosynthetic Productivity by** 

DOI: 10.5772/intechopen.74032

The challenge of climate change promotes use of carbon neutral fuels. Biofuels are made via fixing carbon dioxide via photosynthesis which is inefficient. Light trapping pigments use restricted light wavelengths. A study using the microalga *Botryococcus braunii* (which produces bio-oil), the bacterium *Rhodobacter sphaeroides* (which produces hydrogen), and the cyanobacterium *Arthrospira platensis* (for bulk biomass) showed that photosynthetic productivity was increased by up to 2.5-fold by upconverting unused wavelengths of sunlight via using quantum dots. For large scale commercial energy processes, a 100 fold cost reduction was calculated as the break-even point for adoption of classical QD technology into large scale photobioreactors (PBRs). As a potential alternative, zinc sul-

precipitates metals from mine wastewaters. Biogenic ZnS NPs behaved identically to ZnS quantum dots with absorbance and emission maxima of 290 nm (UVB, which is mostly absorbed by the atmosphere) and 410 nm, respectively; the optimal wavelength for chlorophyll a is 430 nm. By using a low concentration of citrate (10 mM) during ZnS synthesis, the excitation wavelength was redshifted to 315 nm (into the UVA, 85% of which reaches the earth's surface) with an emission peak of 425 nm, i.e., appropriate for photosynthesis. The potential for use in large scale photobioreactors is discussed in the light of current PBR designs, with respect to the need for durable UV-transmitting mate-

**Keywords:** photosynthetic enhancement, bioenergy, quantum dots, zinc sulfide, *Botryococcus braunii*, *Arthrospira platensis*, *Rhodobacter sphaeroides*, bio-oil, bio-hydrogen,

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

S derived from another process that

**Quantum Dots Application**

**Quantum Dots Application**

Rafael L. Orozco, Richard K. Tennant, Frankie Woodhall, Alex Goodridge and

http://dx.doi.org/10.5772/intechopen.74032

Additional information is available at the end of the chapter

fide nanoparticles (NPs) were made using waste H2

rials in appropriate QD delivery systems.

Additional information is available at the end of the chapter

Alex Goodridge and Lynne Elaine Macaskie

Lynne Elaine Macaskie

**Abstract**

biomass

Angela Janet Murray, John Love, Mark D. Redwood,

Angela Janet Murray, John Love, Mark D. Redwood, Rafael L. Orozco, Richard K. Tennant, Frankie Woodhall,


#### **Enhancement of Photosynthetic Productivity by Quantum Dots Application Enhancement of Photosynthetic Productivity by Quantum Dots Application**

DOI: 10.5772/intechopen.74032

Angela Janet Murray, John Love, Mark D. Redwood, Rafael L. Orozco, Richard K. Tennant, Frankie Woodhall, Alex Goodridge and Lynne Elaine Macaskie Angela Janet Murray, John Love, Mark D. Redwood, Rafael L. Orozco, Richard K. Tennant, Frankie Woodhall, Alex Goodridge and Lynne Elaine Macaskie

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74032

#### **Abstract**

[49] Ji X, Palui G, Avellini T, Na HB, Yi C, Knappenberger K, Mattoussi H. On the pHdependent quenching of quantum dot photoluminescence by redox active dopamine.

[50] Li DW, Qin LX, Li Y, Nia RP, Long Y-T, Chen H-Y. CdSe/ZnS quantum dot-cytochrome c

[51] Ma W, Liu HT, Long YT. Monitoring dopamine quinone-induced dopaminergic neurotoxicity using dopamine functionalized quantum dots. ACS Applied Materials & Inter-

[52] Zhang WH, Ma W, Long YT. Redox-mediated indirect fluorescence immunoassay for the detection of disease biomarkers using dopamine-functionalized quantum dots. Analyti-

• sensing. Chemical Communications.

Journal of the American Chemical Society. 2012;134:6006-6017

bioconjugates for selective intracellular O2

2011;47:8539-8541

146 Nonmagnetic and Magnetic Quantum Dots

faces. 2015;7(26):14352-14358

cal Chemistry. 2016;88(10):5131-5136

The challenge of climate change promotes use of carbon neutral fuels. Biofuels are made via fixing carbon dioxide via photosynthesis which is inefficient. Light trapping pigments use restricted light wavelengths. A study using the microalga *Botryococcus braunii* (which produces bio-oil), the bacterium *Rhodobacter sphaeroides* (which produces hydrogen), and the cyanobacterium *Arthrospira platensis* (for bulk biomass) showed that photosynthetic productivity was increased by up to 2.5-fold by upconverting unused wavelengths of sunlight via using quantum dots. For large scale commercial energy processes, a 100 fold cost reduction was calculated as the break-even point for adoption of classical QD technology into large scale photobioreactors (PBRs). As a potential alternative, zinc sulfide nanoparticles (NPs) were made using waste H2 S derived from another process that precipitates metals from mine wastewaters. Biogenic ZnS NPs behaved identically to ZnS quantum dots with absorbance and emission maxima of 290 nm (UVB, which is mostly absorbed by the atmosphere) and 410 nm, respectively; the optimal wavelength for chlorophyll a is 430 nm. By using a low concentration of citrate (10 mM) during ZnS synthesis, the excitation wavelength was redshifted to 315 nm (into the UVA, 85% of which reaches the earth's surface) with an emission peak of 425 nm, i.e., appropriate for photosynthesis. The potential for use in large scale photobioreactors is discussed in the light of current PBR designs, with respect to the need for durable UV-transmitting materials in appropriate QD delivery systems.

**Keywords:** photosynthetic enhancement, bioenergy, quantum dots, zinc sulfide, *Botryococcus braunii*, *Arthrospira platensis*, *Rhodobacter sphaeroides*, bio-oil, bio-hydrogen, biomass

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

#### **1.1. Photosynthetic biotechnologies for biomass and fuels**

The term "bioenergy" is used to describe the conversion of materials of biological origin into fuels and also includes the use of living organisms to produce a material that is a fuel or fuel precursor. This chapter focuses on the use of photosynthetic microorganisms: bacteria, cyanobacteria, and algae. These all grow at the expense of sunlight, while at the same time fixing carbon dioxide from the atmosphere or dissolved in water (algae) or converting organic waste into biomass material (bacteria). Traditional photobiotechnologies have used algae which range from seaweeds to small unicellular organisms within the "kingdom" of eukaryotes which also includes all higher forms of life. The "kingdom" of prokaryotes represents a far simpler level of cellular organization and includes single-celled photosynthetic bacteria and also filamentous microorganisms called cyanobacteria (or "blue green algae"). This review will illustrate examples of all three types.

Cyanobacteria (historically misnamed "blue-green algae") are functionally similar to algae but distinct in many ways. The filamentous cyanobacterium *Arthrospira* ("spirulina") *platensis* is grown as a high-value food supplement and also (under less stringent production standards) as animal feed due to its high content of protein and other nutrients [6, 7]. Spirulina production is highly practical, as the alkaline medium it prefers suppresses contaminants that can impact on algae production. Notably, too, it can utilize soluble bicarbonate ion which

The anoxygenic photosynthetic "purple nonsulfur bacteria" (e.g., *Rhodobacter sphaeroides*)

tic precursors; 50–80% w/w) [8] and high-purity hydrogen gas (typically 90% v/v) as part of an integrated biohydrogen refinery, which could exceed the delivered energy densities of mainstream renewable energy systems, such as photovoltaic cells and on-shore wind turbines [9, 10]. Unlike cyanobacteria and higher algae, *R sphaeroides* utilizes organic acids which are almost ubiquitously produced as by-products from various fermentations and wastewater

Lacking complex structures, microbes can achieve much higher productivity than crop plants. The efficiencies of light conversion to fuel are 0.4–0.8% for algal oil and ~1–5% for purple bac-

[12]. Significantly higher photosynthetic productivities are needed to make significant progress toward supplanting fossil fuels. As well as improving the microorganism, the "value" of

Bacterial photobiotechnologies are not yet developed at scale, and in some cases, as for the biohydrogen process noted above, the photobioreactor design can be complex due to the need to exclude air. In contrast, algal biotechnology is relatively well developed [13], even though predict-

Although the basic requirements for algal culture are simple—water, dilute inorganic nutrients

factors limit the basic engineering designs of algal culture platforms which, as a consequence, have changed little in the last 50 years. These limiting factors include mainly light attenuation in water (notably of the photosynthetic, red wavelengths) due to absorbance and scattering,

 dissolution, water temperature, and, often overlooked, the fact that algal cultures typically comprise unicellular organisms that are fundamentally "selfish" and are in a permanent competition with all other individuals in the culture [15]. This latter point means that algae are superbly adapted to acquiring more photons than are actually required for their photosynthetic processes and dissipate the surplus as nonphotosynthetic radiation. In algal cultures, illumination typically follows the Beer-Lambert law, with light intensity decreasing exponentially depending on the biomass concentration [16]. Consequently, in static cultures, cells at the surface of the photic zone experience high intensities of light and temperature, while the majority of the culture is in complete darkness [17–19], the consequence of which is that static

sunlight and its delivery can also be improved, which forms the focus of this chapter.

–106

cultures rapidly become light limited, and overall growth slows or reaches a plateau.

[11], whereas for higher plants, the value is at most 0.16% and normally much less

.

Enhancement of Photosynthetic Productivity by Quantum Dots Application

l), for extended periods, remains problematic [14].

, and light—a number of physical and biological

polyhydroxyalkanoates (bioplas-

http://dx.doi.org/10.5772/intechopen.74032

149


forms in alkaline solution, following the dissolving of gaseous CO2

offer potentially both fuels and chemicals, producing C4

**2. Overview of photobiotechnologies**

able algal culture at industrial scales (105

(nitrate, phosphate and trace elements), CO2

treatment processes.

terial H2

CO2

Photosynthesis is achieved via the use of specialized pigments called chlorophylls that trap light energy for conversion into chemical energy to drive microbial processes and growth. Algae and cyanobacteria contain "chlorophyll a," while algae, like higher plants, also have a second chlorophyll, "chlorophyll b." Photosynthetic bacteria have functionally equivalent pigments called bacteriochlorophylls, and also ancillary pigments involved in light trapping.

Photosynthetic microorganisms are united by the need to maximize solar irradiation onto their light trapping centers. Natural growth occurs in, for example, ponds but, focusing on maximizing productivity, biotechnology has developed various strategies using photobioreactors (PBRs) for process intensification. Typical strategies include various PBR formats for optimal growth and production at scale, molecular engineering of light trapping centers to improve light conversion and strategies to convert the unused portions of sunlight into additional light which forms the focus of this chapter. Examples will be presented as a proof of concept, highlighting some of the barriers towards implementation.

#### **1.2. Examples of photosynthetic biotechnologies: three examples**

By 2030, the global demand for transport fuel is likely to increase significantly, requiring the production of up to approximately 400–500 billion liters of biofuel per year [1, 2]. Biofuel production could rise to 165 billion liters by 2030, if the US, Canada, and Europe adopt a common E15 blending standard [3], but clearly there will be a shortfall. Biofuel production by photosynthetic microbes is an alternative to crop-based biofuels as fertile soil and a hospitable climate are not required, and hence, biofuels could be produced using contaminated land, steeply sloping hillsides, deserts, urban areas, or rooftops. Therefore, unlike crop-based biofuels, microbial biofuels would not necessarily impact upon agricultural food production.

The microscopic alga *Botryococcus braunii* is potentially valuable as it secretes long-chain (C20–40) hydrocarbons which can be processed into "drop-in" liquid fuels [1, 4]. As an alternative approach, algae have been grown as a source of biomass for production of another form of bio-oil. Thermochemical treatment (pyrolysis) produces oil, which is akin to fossil oils when suitably processed via upgrading and refinery processes [5].

Cyanobacteria (historically misnamed "blue-green algae") are functionally similar to algae but distinct in many ways. The filamentous cyanobacterium *Arthrospira* ("spirulina") *platensis* is grown as a high-value food supplement and also (under less stringent production standards) as animal feed due to its high content of protein and other nutrients [6, 7]. Spirulina production is highly practical, as the alkaline medium it prefers suppresses contaminants that can impact on algae production. Notably, too, it can utilize soluble bicarbonate ion which forms in alkaline solution, following the dissolving of gaseous CO2 .

The anoxygenic photosynthetic "purple nonsulfur bacteria" (e.g., *Rhodobacter sphaeroides*) offer potentially both fuels and chemicals, producing C4 -C5 polyhydroxyalkanoates (bioplastic precursors; 50–80% w/w) [8] and high-purity hydrogen gas (typically 90% v/v) as part of an integrated biohydrogen refinery, which could exceed the delivered energy densities of mainstream renewable energy systems, such as photovoltaic cells and on-shore wind turbines [9, 10]. Unlike cyanobacteria and higher algae, *R sphaeroides* utilizes organic acids which are almost ubiquitously produced as by-products from various fermentations and wastewater treatment processes.

Lacking complex structures, microbes can achieve much higher productivity than crop plants. The efficiencies of light conversion to fuel are 0.4–0.8% for algal oil and ~1–5% for purple bacterial H2 [11], whereas for higher plants, the value is at most 0.16% and normally much less [12]. Significantly higher photosynthetic productivities are needed to make significant progress toward supplanting fossil fuels. As well as improving the microorganism, the "value" of sunlight and its delivery can also be improved, which forms the focus of this chapter.

## **2. Overview of photobiotechnologies**

**1. Introduction**

148 Nonmagnetic and Magnetic Quantum Dots

**1.1. Photosynthetic biotechnologies for biomass and fuels**

concept, highlighting some of the barriers towards implementation.

**1.2. Examples of photosynthetic biotechnologies: three examples**

when suitably processed via upgrading and refinery processes [5].

will illustrate examples of all three types.

The term "bioenergy" is used to describe the conversion of materials of biological origin into fuels and also includes the use of living organisms to produce a material that is a fuel or fuel precursor. This chapter focuses on the use of photosynthetic microorganisms: bacteria, cyanobacteria, and algae. These all grow at the expense of sunlight, while at the same time fixing carbon dioxide from the atmosphere or dissolved in water (algae) or converting organic waste into biomass material (bacteria). Traditional photobiotechnologies have used algae which range from seaweeds to small unicellular organisms within the "kingdom" of eukaryotes which also includes all higher forms of life. The "kingdom" of prokaryotes represents a far simpler level of cellular organization and includes single-celled photosynthetic bacteria and also filamentous microorganisms called cyanobacteria (or "blue green algae"). This review

Photosynthesis is achieved via the use of specialized pigments called chlorophylls that trap light energy for conversion into chemical energy to drive microbial processes and growth. Algae and cyanobacteria contain "chlorophyll a," while algae, like higher plants, also have a second chlorophyll, "chlorophyll b." Photosynthetic bacteria have functionally equivalent pigments called bacteriochlorophylls, and also ancillary pigments involved in light trapping. Photosynthetic microorganisms are united by the need to maximize solar irradiation onto their light trapping centers. Natural growth occurs in, for example, ponds but, focusing on maximizing productivity, biotechnology has developed various strategies using photobioreactors (PBRs) for process intensification. Typical strategies include various PBR formats for optimal growth and production at scale, molecular engineering of light trapping centers to improve light conversion and strategies to convert the unused portions of sunlight into additional light which forms the focus of this chapter. Examples will be presented as a proof of

By 2030, the global demand for transport fuel is likely to increase significantly, requiring the production of up to approximately 400–500 billion liters of biofuel per year [1, 2]. Biofuel production could rise to 165 billion liters by 2030, if the US, Canada, and Europe adopt a common E15 blending standard [3], but clearly there will be a shortfall. Biofuel production by photosynthetic microbes is an alternative to crop-based biofuels as fertile soil and a hospitable climate are not required, and hence, biofuels could be produced using contaminated land, steeply sloping hillsides, deserts, urban areas, or rooftops. Therefore, unlike crop-based biofuels, microbial biofuels would not necessarily impact upon agricultural food production. The microscopic alga *Botryococcus braunii* is potentially valuable as it secretes long-chain (C20–40) hydrocarbons which can be processed into "drop-in" liquid fuels [1, 4]. As an alternative approach, algae have been grown as a source of biomass for production of another form of bio-oil. Thermochemical treatment (pyrolysis) produces oil, which is akin to fossil oils Bacterial photobiotechnologies are not yet developed at scale, and in some cases, as for the biohydrogen process noted above, the photobioreactor design can be complex due to the need to exclude air. In contrast, algal biotechnology is relatively well developed [13], even though predictable algal culture at industrial scales (105 –106 l), for extended periods, remains problematic [14].

Although the basic requirements for algal culture are simple—water, dilute inorganic nutrients (nitrate, phosphate and trace elements), CO2 , and light—a number of physical and biological factors limit the basic engineering designs of algal culture platforms which, as a consequence, have changed little in the last 50 years. These limiting factors include mainly light attenuation in water (notably of the photosynthetic, red wavelengths) due to absorbance and scattering, CO2 dissolution, water temperature, and, often overlooked, the fact that algal cultures typically comprise unicellular organisms that are fundamentally "selfish" and are in a permanent competition with all other individuals in the culture [15]. This latter point means that algae are superbly adapted to acquiring more photons than are actually required for their photosynthetic processes and dissipate the surplus as nonphotosynthetic radiation. In algal cultures, illumination typically follows the Beer-Lambert law, with light intensity decreasing exponentially depending on the biomass concentration [16]. Consequently, in static cultures, cells at the surface of the photic zone experience high intensities of light and temperature, while the majority of the culture is in complete darkness [17–19], the consequence of which is that static cultures rapidly become light limited, and overall growth slows or reaches a plateau.

Engineering solutions to these problems typically include: constructing short light paths within the culture system or using high-intensity illumination; mixing cultures using pumps, impellers, paddle-wheels, or bubbles to maintain an overall average illumination experienced by all cells in the culture; controlling temperature; and increasing the concentration of dissolved CO2 . Fundamentally, engineering algal culture systems is a complex problem [20], involving multiple possibilities, and compromises that must be aligned with the final application, as any solution invariably has a cost that will be reflected in that of the product.

Algal culture platforms are conventionally divided into two categories, open or closed systems, each of which has different advantages, uses, and productivities.

Open algal cultures (**Figure 1a** and **b**) are typically shallow ponds, or "raceways", in which mixing is performed by direct displacement of the liquid using impellers or paddle wheels or by bubbles in airlift systems [14]. Raceways are designed to provide predictable, circulatory patterns (**Figure 1c**), enabling a more or less homogenous distribution of nutrients and access to light for all individual algal cells [22]. Although photosynthetically active radiation may not penetrate dense cultures, a combination of shallow ponds (20–50 cm deep) and mixing allows the algae sufficient time in the photic zone to grow (**Figure 1d**). The larger the installation, the more energy is required for mixing, increasing hydrodynamic shear, and the possibility of localized "dead-zones," where mixing is sub-optimal, and resulting in sub-optimal productivities [23]. Moreover, open ponds require large expanses of flat land which, in certain locations, is sought after for other, more lucrative uses, thereby increasing the capital cost of the installation. The addition of CO2 to open systems is also problematic. Finally, open ponds carry the possibility of culture contamination by undesirable organisms such as other algal species or algal predators. While some applications, notably bioremediation, might benefit from a diverse population of different algal species with regard to resilience, stability, and performance [24–26], when the culture of a single algal species is preferred in an open setting, a limited number of extremophiles and rapidly growing algal species are used to minimize contamination [27].

Closed systems (also termed "photobioreactors" or PBRs), in which there is no direct exchange of culture media, gases, and potential contaminants with the environment, offer a number of advantages for algal culture, including better control over culture conditions (light intensity, temperature, pH, oxygen concentration, and CO2 ), higher levels of reproducibility, higher biomass productivity, a lower risk of contamination, enabling culture of a wider variety of species, and, because they are contained, the use of genetically modified algal strains. Several types of PBRs have been devised [28, 29] (**Figure 2**) that can be located either outdoors or, for more accurate temperature control, in greenhouses or in artificially lit chambers. Apart from shaken flasks in an illuminated incubator, the simplest PBR design is a hanging, translucent, or transparent plastic bag or vertical, transparent tube, in which algal cultures are mixed by gas sparging ("bubble columns" or "airlift columns"). Such PBRs have a high surface area to volume ratio suitable for light transmission and satisfactory heat and mass transfer, providing a homogenous culture environment and efficient release of gases. Other advantages include low shear; the lack of moving parts makes bubble columns relatively inexpensive and easy to maintain. Alternatively, algal and media mixing may be achieved by an impeller (so-called "stir-tank" reactors; conceptually similar to an illuminated bacterial fermenter); here, the effectiveness of mixing depends upon the design of the impeller blades, the speed of rotation,

and the depth of liquid. Vertical or horizontal tubular reactors in which media and algae are pumped from a main sump through the structure ("biofence") provide a scale-up capacity to several hundred liters. Shorter light paths are achieved using flat-panel reactor designs.

**Figure 1.** Open algal culture systems. Open-pond systems used for (a) small-scale (≈1000 l) and (b) commercial scale (>100,000 l) algal culture (image courtesy of the South Australian Research Institute (SARDI)), where the medium is displaced by paddle-wheels that are easy to service and cause low hydrodynamic shear. Panel (c) shows a model raceway (200 × 50 and 20 cm water depth) in which currents are driven by impellers. Computational fluid dynamic (CFD) modeling of this system topography (panel c, right) shows the distribution and strength of currents, with regions of low water movement in cold colors and faster water movement represented by warmer colors. The CFD was performed by Robert Rouse and Gavin Tabor (University of Exeter Department of Engineering) using empirical data. The graph in (d) shows the growth of a *Botryococcus braunii* culture (closed squares representing the mean of 3 replicates) in a 20 cm deep raceway and the reduction in photosynthetically active radiation (PAR; open circles representing the mean of readings from 4 sensors placed under the tank and 49 cm intervals) at the bottom of the pond, as the culture grows. Note that after approximately 15 days, PAR is only 10% of the starting level but, due to mixing, the culture continues to grow for

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Illuminating plants with light emitting diodes (LEDs) leads to higher biomass productivity per unit of irradiance [30]. LEDs have several advantages over conventional, incandescent, or fluorescent lights, including small size, durability, long lifetime, cool-emitting temperature, and the option to select specific wavelengths, notably in the photosynthetic red and blue

Despite the concomitant reduction in energy use from LEDs compared to other forms of illumination, and the effectiveness of different PBRs at laboratory scale, the mass production of

wavelengths [31, 32].

a further fortnight. Bars represent the standard error of the mean.

Engineering solutions to these problems typically include: constructing short light paths within the culture system or using high-intensity illumination; mixing cultures using pumps, impellers, paddle-wheels, or bubbles to maintain an overall average illumination experienced by all cells in the culture; controlling temperature; and increasing the concentration of dis-

involving multiple possibilities, and compromises that must be aligned with the final applica-

Algal culture platforms are conventionally divided into two categories, open or closed sys-

Open algal cultures (**Figure 1a** and **b**) are typically shallow ponds, or "raceways", in which mixing is performed by direct displacement of the liquid using impellers or paddle wheels or by bubbles in airlift systems [14]. Raceways are designed to provide predictable, circulatory patterns (**Figure 1c**), enabling a more or less homogenous distribution of nutrients and access to light for all individual algal cells [22]. Although photosynthetically active radiation may not penetrate dense cultures, a combination of shallow ponds (20–50 cm deep) and mixing allows the algae sufficient time in the photic zone to grow (**Figure 1d**). The larger the installation, the more energy is required for mixing, increasing hydrodynamic shear, and the possibility of localized "dead-zones," where mixing is sub-optimal, and resulting in sub-optimal productivities [23]. Moreover, open ponds require large expanses of flat land which, in certain locations, is sought after for other, more lucrative uses, thereby increasing the capital cost of the installation.

ity of culture contamination by undesirable organisms such as other algal species or algal predators. While some applications, notably bioremediation, might benefit from a diverse population of different algal species with regard to resilience, stability, and performance [24–26], when the culture of a single algal species is preferred in an open setting, a limited number of extremo-

Closed systems (also termed "photobioreactors" or PBRs), in which there is no direct exchange of culture media, gases, and potential contaminants with the environment, offer a number of advantages for algal culture, including better control over culture conditions (light intensity,

biomass productivity, a lower risk of contamination, enabling culture of a wider variety of species, and, because they are contained, the use of genetically modified algal strains. Several types of PBRs have been devised [28, 29] (**Figure 2**) that can be located either outdoors or, for more accurate temperature control, in greenhouses or in artificially lit chambers. Apart from shaken flasks in an illuminated incubator, the simplest PBR design is a hanging, translucent, or transparent plastic bag or vertical, transparent tube, in which algal cultures are mixed by gas sparging ("bubble columns" or "airlift columns"). Such PBRs have a high surface area to volume ratio suitable for light transmission and satisfactory heat and mass transfer, providing a homogenous culture environment and efficient release of gases. Other advantages include low shear; the lack of moving parts makes bubble columns relatively inexpensive and easy to maintain. Alternatively, algal and media mixing may be achieved by an impeller (so-called "stir-tank" reactors; conceptually similar to an illuminated bacterial fermenter); here, the effectiveness of mixing depends upon the design of the impeller blades, the speed of rotation,

philes and rapidly growing algal species are used to minimize contamination [27].

temperature, pH, oxygen concentration, and CO2

tion, as any solution invariably has a cost that will be reflected in that of the product.

tems, each of which has different advantages, uses, and productivities.

. Fundamentally, engineering algal culture systems is a complex problem [20],

to open systems is also problematic. Finally, open ponds carry the possibil-

), higher levels of reproducibility, higher

solved CO2

150 Nonmagnetic and Magnetic Quantum Dots

The addition of CO2

**Figure 1.** Open algal culture systems. Open-pond systems used for (a) small-scale (≈1000 l) and (b) commercial scale (>100,000 l) algal culture (image courtesy of the South Australian Research Institute (SARDI)), where the medium is displaced by paddle-wheels that are easy to service and cause low hydrodynamic shear. Panel (c) shows a model raceway (200 × 50 and 20 cm water depth) in which currents are driven by impellers. Computational fluid dynamic (CFD) modeling of this system topography (panel c, right) shows the distribution and strength of currents, with regions of low water movement in cold colors and faster water movement represented by warmer colors. The CFD was performed by Robert Rouse and Gavin Tabor (University of Exeter Department of Engineering) using empirical data. The graph in (d) shows the growth of a *Botryococcus braunii* culture (closed squares representing the mean of 3 replicates) in a 20 cm deep raceway and the reduction in photosynthetically active radiation (PAR; open circles representing the mean of readings from 4 sensors placed under the tank and 49 cm intervals) at the bottom of the pond, as the culture grows. Note that after approximately 15 days, PAR is only 10% of the starting level but, due to mixing, the culture continues to grow for a further fortnight. Bars represent the standard error of the mean.

and the depth of liquid. Vertical or horizontal tubular reactors in which media and algae are pumped from a main sump through the structure ("biofence") provide a scale-up capacity to several hundred liters. Shorter light paths are achieved using flat-panel reactor designs.

Illuminating plants with light emitting diodes (LEDs) leads to higher biomass productivity per unit of irradiance [30]. LEDs have several advantages over conventional, incandescent, or fluorescent lights, including small size, durability, long lifetime, cool-emitting temperature, and the option to select specific wavelengths, notably in the photosynthetic red and blue wavelengths [31, 32].

Despite the concomitant reduction in energy use from LEDs compared to other forms of illumination, and the effectiveness of different PBRs at laboratory scale, the mass production of

**Figure 2.** Examples of closed algal culture systems. (a) Flasks containing 100 ml of algal culture in a shaking, lighted incubator with CO2 -enriched atmosphere. (b) Polyethylene bag containing 100 ml of algal culture, located in a greenhouse with natural and complementary artificial lighting. Mixing is achieved by an air-stone (aquarium) bubbler. (c) Translucent polyethylene, conical bucket for airlift culture, containing 100 l for media. Air is provided by a simple tube at the bottom of the vessel. The conical shape and tap towards the bottom of the container enable simple harvest of the algal suspension. Photograph courtesy of Dr. Mike Allen, Plymouth Marine Laboratory. (d) Airlift column photobioreactor (perspex; 2 m in length; 10 cm in diameter), containing 10 l of culture and lit by a combination of white, blue and red LED's optimized for algal growth. An air inlet at the base of the column provides mixing. (e) Stirred photo-bioreactor with white LED light jacket, containing 2 l of algal culture. (f) Horizontal, tubular photo-bioreactor (or "biofence") containing 600 l of *Phaeodactylum* culture, located at Swansea University (Wales, UK). The culture is pumped through the transparent tubes from a sump enabling control of media composition and temperature. (g) Flatpanel photobioreactor containing 100 ml of algal culture. The light path is 5 mm and the algae are pumped through the reactor from a sump.

microalgae in closed systems remains expensive in terms of construction costs, materials, and energy. Moreover, up-scaling is problematic, as most PBR designs suffer from a number of limiting factors, including poor gas exchange, difficulties in nutrient delivery, heat balance, and, in locations where seasons are marked, available light [33, 34].

## **3. Process intensification: Limitations of light availability**

The problems of light delivery to a photobioreactor are 2-fold. In equatorial regions, the photoperiod (day length) and seasonality are reasonably constant. However, at higher and lower latitudes, the day length and incident light are seasonally variable, and hence, for a proportion of the year, a PBR cannot operate during significant periods of darkness or is impaired by low light intensity. A pilot scale tubular photobioreactor using *R. sphaeroides* to produce hydrogen (**Figure 3a**) was programmed to operate at UK latitude (~55°N) at equinox and maintained on that diurnal cycle for 3 months, with the light intensity varied day to day. Saturation occurred at ~400 W/m2 (**Figure 3c**). **Figure 3d** shows that in spring and autumn, a PBR will not reach light saturation, while in winter, the productivity would reach only ~33% of its potential maximum at midday (**Figure 3c**, **d**). Saturation would only be reached in midsummer (**Figure 3d**); hence, an increase in light intensity of up to 4-fold would be required for

) as typical UK profiles.

Determination of light saturation in terms of biohydrogen productivity. D: Light intensity (W/m2

**Figure 3.** Performance of a programmable photobioreactor. The PBR was operated over 3 months of continuous diurnal operation set to spring and autumn equinox (12 hour days/nights plus dawn/dusk periods). The light intensity was varied from day to day (at random) and parallel rooftop tests confirmed the pilot scale data. A. PBR tubular construction and orange-pigmented *R. sphaeroides*. B. Scale model of full scale PBR constructed on the basis of the pilot data C:

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To attempt to overcome the limitation, the culture biomass intensity/ml can be increased, but this results in significant "self-shading". In illustration, using the spring/autumn illumination profile (**Figure 3c**) of the biohydrogen PBR using low, intermediate and high density cultures gave a hydrogen yield of 11.5, 7.0, and 3.5 ml/min, respectively, i.e., simply introducing more bacteria is

maximum productivity.

**Figure 3.** Performance of a programmable photobioreactor. The PBR was operated over 3 months of continuous diurnal operation set to spring and autumn equinox (12 hour days/nights plus dawn/dusk periods). The light intensity was varied from day to day (at random) and parallel rooftop tests confirmed the pilot scale data. A. PBR tubular construction and orange-pigmented *R. sphaeroides*. B. Scale model of full scale PBR constructed on the basis of the pilot data C: Determination of light saturation in terms of biohydrogen productivity. D: Light intensity (W/m2 ) as typical UK profiles.

microalgae in closed systems remains expensive in terms of construction costs, materials, and energy. Moreover, up-scaling is problematic, as most PBR designs suffer from a number of limiting factors, including poor gas exchange, difficulties in nutrient delivery, heat balance,

**Figure 2.** Examples of closed algal culture systems. (a) Flasks containing 100 ml of algal culture in a shaking, lighted

greenhouse with natural and complementary artificial lighting. Mixing is achieved by an air-stone (aquarium) bubbler. (c) Translucent polyethylene, conical bucket for airlift culture, containing 100 l for media. Air is provided by a simple tube at the bottom of the vessel. The conical shape and tap towards the bottom of the container enable simple harvest of the algal suspension. Photograph courtesy of Dr. Mike Allen, Plymouth Marine Laboratory. (d) Airlift column photobioreactor (perspex; 2 m in length; 10 cm in diameter), containing 10 l of culture and lit by a combination of white, blue and red LED's optimized for algal growth. An air inlet at the base of the column provides mixing. (e) Stirred photo-bioreactor with white LED light jacket, containing 2 l of algal culture. (f) Horizontal, tubular photo-bioreactor (or "biofence") containing 600 l of *Phaeodactylum* culture, located at Swansea University (Wales, UK). The culture is pumped through the transparent tubes from a sump enabling control of media composition and temperature. (g) Flatpanel photobioreactor containing 100 ml of algal culture. The light path is 5 mm and the algae are pumped through the


The problems of light delivery to a photobioreactor are 2-fold. In equatorial regions, the photoperiod (day length) and seasonality are reasonably constant. However, at higher and lower latitudes, the day length and incident light are seasonally variable, and hence, for a proportion of the year, a PBR cannot operate during significant periods of darkness or is impaired by low light intensity. A pilot scale tubular photobioreactor using *R. sphaeroides* to produce hydrogen (**Figure 3a**) was programmed to operate at UK latitude (~55°N) at equinox and maintained on that diurnal cycle for 3 months, with the light intensity varied day to day.

(**Figure 3c**). **Figure 3d** shows that in spring and autumn, a

and, in locations where seasons are marked, available light [33, 34].

Saturation occurred at ~400 W/m2

incubator with CO2

152 Nonmagnetic and Magnetic Quantum Dots

reactor from a sump.

**3. Process intensification: Limitations of light availability**

PBR will not reach light saturation, while in winter, the productivity would reach only ~33% of its potential maximum at midday (**Figure 3c**, **d**). Saturation would only be reached in midsummer (**Figure 3d**); hence, an increase in light intensity of up to 4-fold would be required for maximum productivity.

To attempt to overcome the limitation, the culture biomass intensity/ml can be increased, but this results in significant "self-shading". In illustration, using the spring/autumn illumination profile (**Figure 3c**) of the biohydrogen PBR using low, intermediate and high density cultures gave a hydrogen yield of 11.5, 7.0, and 3.5 ml/min, respectively, i.e., simply introducing more bacteria is counterproductive, and this also increases the running costs with respect to both the make-up feed (trace nutrients) and the final biomass for waste disposal, if the biomass is not used for biofuel.

The second limitation is that the solar spectrum is very wide, yet the wavelengths captured by photosynthetic pigments are quite conservative (**Figure 4**), with algal/cyanobacterial chlorophyll utilizing visible wavelengths, while bacteriochlorophyll utilizes light in the visible/near infrared (NIR) region.

A novel study used a beam splitting approach to supply an algal and a bacterial system (similar to that shown in **Figures 5** and **6**, without quantum dots), taking advantage of their respective preferred wavelengths and giving the potential to operate two parallel PBRs. This enhanced the microbial productivity per incident photon [35], an approach that could be useful in, for example, biohydrogen production, where bacteria and algae both make bio-H2 but by using different pathways [36]. Hence, it may be possible to produce bio-H2 by bacterial and algal reactors side by side, with the additional advantage of providing a "sink" for bacterially produced CO2 into algal biomass.

**Figure 4.** Action spectra and identification of targets for spectral enrichment. Generic action spectra were adapted from [35]. Note that action spectra differ substantially from whole-cell absorption spectra, which show strong wavelengthnonspecific attenuation due to the scattering effect of cells. Above this, small peaks associated with the absorption maxima of chlorophylls can usually be detected. The emission of the desired quantum dots is shown by the dotted line; ideally the emission peak should be narrow and overlap with the absorption maxima of the algal and bacterial chlorophylls.

However, in practice, other than to make a common product in an integrated process, it may be impractical to co-locate both types of PBR due to other requirements; for example, organic acid feedstock for bacterial hydrogen production can be supplied by urban wastewater treatment plants [37], whereas algal biotechnologies typically require a large land area which can be waste or nonarable land. Hence, a generic method is required to "upgrade" solar light by converting unused wavelengths into used wavelengths for a particular process, thereby increasing the usable light and productivity without increasing biomass density and "self-

**Figure 6.** Boosting photosynthetic activity of *Arthrospira* ("spirulina") *platensis.* Growth was as in [35]. Inocula for photonic enhancement experiments were taken 2 days after subculturing (to ensure active growth) and diluted with fresh medium to an OD 660 of 0.364.4 ml was transferred into each vial. Cultures were illuminated by a close-match

was inferred from growth measured after 22 h [35]. To isolate optical effects, QDs were isolated in a glass insert, each

openings in an opaque sheet to allow illumination from beneath with artificial sunlight. Ambient light was excluded by covering the assembly with black velvet and opaque barriers were placed around each mini-reactor. The temperature

with controls of QD-free Na2

**Figure 5.** Boosting photosynthetic activity of *Rhodobacter sphaeroides*. Qdot'792 (to 20 nm) was encapsulated in 2% alginate beads (diam. 2.6 ±0.04 mm). Beads were prepared by mixing QD or blank (50 mM sodium borate buffer, pH 9.0)

washed with deionized water and used immediately. Concentrated cell suspension was diluted to 0.547 gdry weight/l with fresh butyrate medium [35]. Vials (4 ml bacterial suspension, 3 ml beads containing Qdot'792 and ~5 ml headspace) were sealed with gastight stoppers and purged with argon (30 min in darkness) and incubated (30°C, 3d, 10.0W/m2

formation was measured as described previously [35]. Each vial contained 0.056 nmol Qdot'792

through an 18G needle. After curing (60 min), the beads were

) (supplementary material in [35]. Photosynthetic action

. The inserts aligned with 10 mm

SO3

. Optical dividers prevented optical interactions between vials and

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with 2% sodium alginate and dropping into 100 mM CaCl2

solar simulator undergrowth-limiting irradiance (10W/m2

containing 1 ml of Qdot'652 in 100 mM Na2

was controlled by circulating water at 30°C.

ambient light was excluded by covering the assembly with black cloth.

distributed over an illuminated surface of 3.14 cm2

simulated sunlight). H2

shading". This forms a goal for photobiotechnology process intensification.

SO3

counterproductive, and this also increases the running costs with respect to both the make-up feed (trace nutrients) and the final biomass for waste disposal, if the biomass is not used for biofuel.

The second limitation is that the solar spectrum is very wide, yet the wavelengths captured by photosynthetic pigments are quite conservative (**Figure 4**), with algal/cyanobacterial chlorophyll utilizing visible wavelengths, while bacteriochlorophyll utilizes light in the visible/near

A novel study used a beam splitting approach to supply an algal and a bacterial system (similar to that shown in **Figures 5** and **6**, without quantum dots), taking advantage of their respective preferred wavelengths and giving the potential to operate two parallel PBRs. This enhanced the microbial productivity per incident photon [35], an approach that could be useful in, for example, biohydrogen production, where bacteria and algae both make bio-H2

algal reactors side by side, with the additional advantage of providing a "sink" for bacterially

**Figure 4.** Action spectra and identification of targets for spectral enrichment. Generic action spectra were adapted from [35]. Note that action spectra differ substantially from whole-cell absorption spectra, which show strong wavelengthnonspecific attenuation due to the scattering effect of cells. Above this, small peaks associated with the absorption maxima of chlorophylls can usually be detected. The emission of the desired quantum dots is shown by the dotted line; ideally the emission peak should be narrow and overlap with the absorption maxima of the algal and bacterial chlorophylls.

by using different pathways [36]. Hence, it may be possible to produce bio-H2

into algal biomass.

but

by bacterial and

infrared (NIR) region.

154 Nonmagnetic and Magnetic Quantum Dots

produced CO2

**Figure 5.** Boosting photosynthetic activity of *Rhodobacter sphaeroides*. Qdot'792 (to 20 nm) was encapsulated in 2% alginate beads (diam. 2.6 ±0.04 mm). Beads were prepared by mixing QD or blank (50 mM sodium borate buffer, pH 9.0) with 2% sodium alginate and dropping into 100 mM CaCl2 through an 18G needle. After curing (60 min), the beads were washed with deionized water and used immediately. Concentrated cell suspension was diluted to 0.547 gdry weight/l with fresh butyrate medium [35]. Vials (4 ml bacterial suspension, 3 ml beads containing Qdot'792 and ~5 ml headspace) were sealed with gastight stoppers and purged with argon (30 min in darkness) and incubated (30°C, 3d, 10.0W/m2 simulated sunlight). H2 formation was measured as described previously [35]. Each vial contained 0.056 nmol Qdot'792 distributed over an illuminated surface of 3.14 cm2 . Optical dividers prevented optical interactions between vials and ambient light was excluded by covering the assembly with black cloth.

**Figure 6.** Boosting photosynthetic activity of *Arthrospira* ("spirulina") *platensis.* Growth was as in [35]. Inocula for photonic enhancement experiments were taken 2 days after subculturing (to ensure active growth) and diluted with fresh medium to an OD 660 of 0.364.4 ml was transferred into each vial. Cultures were illuminated by a close-match solar simulator undergrowth-limiting irradiance (10W/m2 ) (supplementary material in [35]. Photosynthetic action was inferred from growth measured after 22 h [35]. To isolate optical effects, QDs were isolated in a glass insert, each containing 1 ml of Qdot'652 in 100 mM Na2 SO3 with controls of QD-free Na2 SO3 . The inserts aligned with 10 mm openings in an opaque sheet to allow illumination from beneath with artificial sunlight. Ambient light was excluded by covering the assembly with black velvet and opaque barriers were placed around each mini-reactor. The temperature was controlled by circulating water at 30°C.

However, in practice, other than to make a common product in an integrated process, it may be impractical to co-locate both types of PBR due to other requirements; for example, organic acid feedstock for bacterial hydrogen production can be supplied by urban wastewater treatment plants [37], whereas algal biotechnologies typically require a large land area which can be waste or nonarable land. Hence, a generic method is required to "upgrade" solar light by converting unused wavelengths into used wavelengths for a particular process, thereby increasing the usable light and productivity without increasing biomass density and "selfshading". This forms a goal for photobiotechnology process intensification.

The wavelength dependence of photosynthesis by purple bacteria and microalgae has been known since the early twentieth century and confirmed many times in different species. As shown in **Figure 4**, green algae/cyanobacteria/higher plants show the greatest activity with red light, whereas purple bacteria are most active under near-infrared (NIR) [38–42]. The effect is so powerful that these organisms have developed an apparent "phototaxis" response, accumulating in the optically optimal part of a natural water column [38]. This ability is very important because red light is absorbed strongly by water, and hence, the availability of useful light drops markedly with depth. Blue light has a far greater penetration (see later).

The action spectra of photosynthetic microorganisms have been extensively surveyed. Green microorganisms (and plant chloroplasts) conform to a generic action spectrum, while purple bacteria conform to a distinctly different generic action spectrum (see [35] and **Figure 4**) attributable to the different chlorophylls evolved in the taxanomic groups.

## **4. Quantum dots as a potential means of "upgrading" light**

One method of "upgrading" "waste" light of a particular wavelength is to use the light-emitting properties of quantum dots (QDs). QDs are single crystals of uniform size and shape of ~2–10 nm diameter and usually comprising pairs of semiconductors (e.g., CdSe, PbSe). QDs are replacing fluor dyes in cell biology due to their high brightness and photostability [43]. The properties and potential applications of QDs are described elsewhere in this volume, and indeed, QDs are commercially available in appropriate delivery systems for boosting horticulture and small-scale crop production [44] but have yet to find application in large-scale photobioreactor systems. However, for bioenergy applications and bulk-scale animal feed production, large scale constructions would be required (e.g., see **Figure 3b** and **Table 1**). Hence, a feasibility study was undertaken using the three microorganisms described above to indicate whether photosynthetic boosting via QDs is feasible for algal and bacterial growth systems. The use of LEDs to supply additional lighting at the optimal wavelengths is wellestablished technology [44, 45], and it is assumed to be intrinsically scalable, although a full cost-benefit analysis is required for applications in biofuels production.

and the cyanobacterium/green alga spirulina and *B. braunii,* respectively. The loading densities in

Electricity is taken from the grid (@ €0.107/kWh) and includes "parasitic energy" consumption (pumps for culture and

Projection via use of solar cells, efficient battery technology and LED supplementary illumination. Calculations by

variable cost includes cost of water use, electricity (parasitic energy), labor, fertilizers (N & P) and waste water. V:

**Table 1.** A comparative study [21] on algal (*Chlorella vulgaris*) cultivation technologies which include open pond and closed photobioreactors (PBRs: Tubular and flat panels) and economics of algal biomass production. The high

**kg dry solids/yr**

**Microalgae cost price (€/kg dry solids)**

Enhancement of Photosynthetic Productivity by Quantum Dots Application

1538 36 35

3076 18 50

5127 12.50 68

12,818\* ~ 4.20\* ~30\*

**Ratio electricity\***

**cost# (%)**

http://dx.doi.org/10.5772/intechopen.74032

).

**/variable** 

157

*R. sphaeroides* was used in a test system of mounted vials as shown in **Figure 5**, using simu-

*Arthrospira platensis* (spirulina) was used in a test system (**Figure 6**) using a close-match solar

*B. braunii* was cultured routinely in shake flask cultures. Q dot'652 was added directly into small 25 ml cultures to 10 nm. Cultures were shaken in a temperature-controlled greenhouse

**Figure 8c**). This was a close fit to the increase predicted by the known QD quantum efficiency

.

/s). Photosynthetic action was inferred from

, showed a photonic enhancement of ~10% (**Table 2**;

the tests were 0.0178 (*R. sphaeroides*), 0.0792 (spirulina), and 0.050 (*B.braunii*) nmol/cm2

productivities of flat panels compared to the other systems are reflected in the lower cost price.

**4.2. Experimental test systems**

*4.2.2. Spirulina for biomass production*

lated sunlight.

*4.2.1. Rhodobacter sphaeroides for biohydrogen production*

**PBR type Dimensions; vol, m3 Algae production** 

diam.; 45 m3

spacing; 60 m3

spacing: 60 m3

; 0.03 m water depth; 300 m3

; 0.06 m tube

; 0.03 m plate

; 0.03 m plate

water heating/cooling circulation, centrifuge and blower for the flue gas supplying CO2

Open pond 1000 m2

Tubular 1000 m2

Flat panel 1000 m2

Solar LEDs/flat panel\* 1000 m2

R.L. Orozco (unpublished work).

\*

#

#

culture volume.

simulator (supplementary material in Redwood et al. [35]).

**4.3. Photosynthetic enhancement using commercial quantum dots**

*4.2.3. B. braunii: A single-celled alga for bio-oil production*

(average solar photon flux was 11 μmol/m2

The first test, using *R. sphaeroides* to produce H2

growth at 21 days as estimated by OD600.

#### **4.1. Boosting of three photosystems using quantum dots**

The concept of photonic enhancement is to increase the proportion of the solar spectrum that corresponds to the major peak(s) of the organismal action spectrum (**Figure 4**), at the expense of other irradiance at less active wavelengths. The part of the spectrum to be intensified is referred to as the target band. **Figure 4** (top panel) shows the boundaries of the target band corresponding to the half maximum of the major peak in the organismal action spectrum. Using generic action spectra derived previously [43], the target bands of 640–690 nm and 790–940 nm were determined for algae/cyanobacteria and purple bacteria, respectively. These bands account for 25 and 67% of the total action, respectively.

The study used test quantum dots purchased from Invitrogen: Qdot'792 (ITK carboxyl, no Q21371, lot 834,674; quantum yield (QY) 72%; full width height maximum (FWHM): 82 nm) and QD'652 (ITK carboxyl, no. Q21321MP, lot 891,174; QY 78%; FWHM 26 nm) for cultures of *R. sphaeroides*


\* Electricity is taken from the grid (@ €0.107/kWh) and includes "parasitic energy" consumption (pumps for culture and water heating/cooling circulation, centrifuge and blower for the flue gas supplying CO2 ).
