Contents

#### **Preface XI**



#### Chapter 8 **Surface Functionalization of Graphene with Polymers for Enhanced Properties 207** Wenge Zheng, Bin Shen and Wentao Zhai

## Chapter 9 **Graphene Nanowalls 235**

Mineo Hiramatsu, Hiroki Kondo and Masaru Hori

Preface

Graphene is a one-atom-thick and two-dimensional repetitive hexagonal lattice sp2

ress have been made in this rapid developing arena.

necessarily having to go through the whole book.

dized carbon layer. The extended honeycomb network of graphene is the basic building block of other important allotropes of carbon. 2D graphene can be wrapped to form 0D full‐ erenes, rolled to form 1D carbon nanotubes, and stacked to form 3D graphite. Depending on its unique structure, graphene yields many excellent electrical, thermal, and mechanical properties. It has been interesting to both theoreticians and experimentalists in various fields, such as materials, chemistry, physics, electronics, and biomedicine, and great prog‐

The aim of publishing this book is to present the recent new achievements about graphene research on a variety of topics. And the book is divided into two parts: Part I, from theoreti‐ cal aspect, Graphene tunneling (Chapter 1), Localized states of Fabry-Perot type in graphene nanoribbons (Chapter 2), Electronic properties of deformed graphene nanoribbons (Chapter 3), The Čererenkov effect in graphene-like structures (Chapter 4), and Electronic and vibra‐ tional properties of adsorbed and embedded carbon nanofilms with defects (Chapter 5) are elaborated; Part II, from experimental aspect, Quantum transport in graphene quantum dots (Chapter 6), Advances in resistive switching memories based on graphene oxide (Chapter 7), Surface functionalization of graphene with polymers for enhanced properties (Chapter 8), and Carbon nanowalls: synthesis and applications (Chapter 9) are introduced. Also, indepth discussions ranging from comprehensive understanding to challenges and perspec‐ tives are included for the respective topic. Each chapter is relatively independent of others, and the Table of Contents we hope will help readers quickly find topics of interest without

Last, I appreciate the outstanding contributions from scientists with excellent academic re‐ cords, who are at the top of their fields on the cutting edge of technology, to the book. Research related to graphene updates every day, so it is impossible to embody all the progress in this collection, and hopefully it could be of any help to people who are interested in this field.


**Prof. Jian Ru Gong**

P. R. China

National Center for Nanoscience and Technology, Beijing

## Preface

Chapter 8 **Surface Functionalization of Graphene with Polymers for**

Mineo Hiramatsu, Hiroki Kondo and Masaru Hori

Wenge Zheng, Bin Shen and Wentao Zhai

**Enhanced Properties 207**

Chapter 9 **Graphene Nanowalls 235**

**VI** Contents

Graphene is a one-atom-thick and two-dimensional repetitive hexagonal lattice sp2 -hybri‐ dized carbon layer. The extended honeycomb network of graphene is the basic building block of other important allotropes of carbon. 2D graphene can be wrapped to form 0D full‐ erenes, rolled to form 1D carbon nanotubes, and stacked to form 3D graphite. Depending on its unique structure, graphene yields many excellent electrical, thermal, and mechanical properties. It has been interesting to both theoreticians and experimentalists in various fields, such as materials, chemistry, physics, electronics, and biomedicine, and great prog‐ ress have been made in this rapid developing arena.

The aim of publishing this book is to present the recent new achievements about graphene research on a variety of topics. And the book is divided into two parts: Part I, from theoreti‐ cal aspect, Graphene tunneling (Chapter 1), Localized states of Fabry-Perot type in graphene nanoribbons (Chapter 2), Electronic properties of deformed graphene nanoribbons (Chapter 3), The Čererenkov effect in graphene-like structures (Chapter 4), and Electronic and vibra‐ tional properties of adsorbed and embedded carbon nanofilms with defects (Chapter 5) are elaborated; Part II, from experimental aspect, Quantum transport in graphene quantum dots (Chapter 6), Advances in resistive switching memories based on graphene oxide (Chapter 7), Surface functionalization of graphene with polymers for enhanced properties (Chapter 8), and Carbon nanowalls: synthesis and applications (Chapter 9) are introduced. Also, indepth discussions ranging from comprehensive understanding to challenges and perspec‐ tives are included for the respective topic. Each chapter is relatively independent of others, and the Table of Contents we hope will help readers quickly find topics of interest without necessarily having to go through the whole book.

Last, I appreciate the outstanding contributions from scientists with excellent academic re‐ cords, who are at the top of their fields on the cutting edge of technology, to the book. Research related to graphene updates every day, so it is impossible to embody all the progress in this collection, and hopefully it could be of any help to people who are interested in this field.

> **Prof. Jian Ru Gong** National Center for Nanoscience and Technology, Beijing P. R. China

**Section 1**

**Theoretical Aspect**

**Section 1**

**Theoretical Aspect**

**Chapter 1**

**Provisional chapter**

**Electronic Tunneling in Graphene**

**Electronic Tunneling in Graphene**

Additional information is available at the end of the chapter

In this chapter the transmission of massless and massive Dirac fermions across two-dimensional p-n and n-p-n junctions of graphene which are high enough so that they correspond to 2D potential steps and square barriers, respectively is investigated. It is shown that tunneling without exponential damping occurs when an relativistic particle is incident on a very high barrier. Such an effect has been described by Oskar Klein in 1929 [1] (for an historical review on klein paradox see [2]). He showed that in the limit of a high enough electrostatic potential barrier, it becomes transparent and both reflection and transmission probability remains smaller than one [3]. However, some later authors claimed that the reflection amplitude at the step barrier exceeds unity [4,5], implying that

Throughout this chapter, these negative transmission and higher-than-unity reflection probability is refereed to as the Klein paradox and not to the transparency of the barrier in the limit *<sup>V</sup>*<sup>0</sup> → <sup>∞</sup> (*V*<sup>0</sup> is hight of the barrier). However, by considering the massless electrons tunneling through a potential step which can correspond to a p-n junction of graphene, as the main aim in the first section, it is be clear that the transmission and reflection probability both are positive and the Klein paradox is not then a paradox at all. Thus, one really doesn't need to associate the particle-antiparticle pair creation, which is commonly regarded as an explanation of particle tunneling in the Klein energy interval, to Klein paradox. In fact it will be revealed that the Klein paradox arises because of not considering a *π* phase change of the transmitted wave function of momentum-space which occurs when the energy of the incident electron is smaller than the height of the electrostatic potential step. In the other words, one arrives at negative values for transmission probability merely because of confusing the direction of group velocity with the propagation direction of particle's wave function or equivalently- from a two-dimensional point of view- the propagation angle with the angle that momentum vector under the electrostatic potential step makes with the normal incidence. Then our attentions turn to the tunneling of massless electrons into a barrier with

> ©2012 Jahani, licensee InTech. This is an open access chapter 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. © 2013 Jahani; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Jahani, licensee InTech. This is a paper 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.

Additional information is available at the end of the chapter

transmission probability takes the negative values.

Dariush Jahani

**1. Introduction**

Dariush Jahani

10.5772/51980

http://dx.doi.org/10.5772/51980

## **Chapter 1**

**Provisional chapter**

## **Electronic Tunneling in Graphene**

**Electronic Tunneling in Graphene**

## Dariush Jahani

Dariush Jahani

10.5772/51980

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51980

### **1. Introduction**

In this chapter the transmission of massless and massive Dirac fermions across two-dimensional p-n and n-p-n junctions of graphene which are high enough so that they correspond to 2D potential steps and square barriers, respectively is investigated. It is shown that tunneling without exponential damping occurs when an relativistic particle is incident on a very high barrier. Such an effect has been described by Oskar Klein in 1929 [1] (for an historical review on klein paradox see [2]). He showed that in the limit of a high enough electrostatic potential barrier, it becomes transparent and both reflection and transmission probability remains smaller than one [3]. However, some later authors claimed that the reflection amplitude at the step barrier exceeds unity [4,5], implying that transmission probability takes the negative values.

Throughout this chapter, these negative transmission and higher-than-unity reflection probability is refereed to as the Klein paradox and not to the transparency of the barrier in the limit *<sup>V</sup>*<sup>0</sup> → <sup>∞</sup> (*V*<sup>0</sup> is hight of the barrier). However, by considering the massless electrons tunneling through a potential step which can correspond to a p-n junction of graphene, as the main aim in the first section, it is be clear that the transmission and reflection probability both are positive and the Klein paradox is not then a paradox at all. Thus, one really doesn't need to associate the particle-antiparticle pair creation, which is commonly regarded as an explanation of particle tunneling in the Klein energy interval, to Klein paradox. In fact it will be revealed that the Klein paradox arises because of not considering a *π* phase change of the transmitted wave function of momentum-space which occurs when the energy of the incident electron is smaller than the height of the electrostatic potential step. In the other words, one arrives at negative values for transmission probability merely because of confusing the direction of group velocity with the propagation direction of particle's wave function or equivalently- from a two-dimensional point of view- the propagation angle with the angle that momentum vector under the electrostatic potential step makes with the normal incidence. Then our attentions turn to the tunneling of massless electrons into a barrier with

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. © 2013 Jahani; licensee InTech. This is an open access article 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. © 2013 Jahani, licensee InTech. This is a paper 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.

©2012 Jahani, licensee InTech. This is an open access chapter distributed under the terms of the Creative

the hight *V*<sup>0</sup> and width *D*. It will be found that the probability for an electron (approaching perpendicularly) to penetrate the barrier is equal to one, independent of *V*<sup>0</sup> and *D*. Although this result is very interesting from the point of view of fundamental research, its presence in graphene is unwanted when it comes to applications of graphene to nano-electronics because the pinch-off of the field effect transistors may be very ineffective. One way to overcome these difficulties is by generating a gap in the graphene spectrum. From the point of view of Dirac fermions this is equivalent to the appearing of a mass term in relativistic equation which describes the low-energy excitations of graphene, i.e. 2D the massive Dirac equation:

$$H = -i v\_F \sigma. \nabla \ \pm \Delta \sigma^z \tag{1}$$

10.5772/51980

5

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

(2)

[13,14]. As mentioned it is a real problem for application of graphene into nano-electronics, since for nano-electronics applications of graphene a mass gap in itŠs energy spectrum is needed just like a conventional semiconductor. We also see that, considering the appropriate wave functions in region of electrostatic barrier reveals that transmission is independent of whether the refractive index is negative or positive[15-17]. There is exactly a mistake on this

In the end, throughout a numerical approach the consequences that the extra *π*-shift might

According to classical physics, a particle of energy *E* less than the height *V*<sup>0</sup> of a potential barrier could not penetrate it because the region inside the barrier is classically forbidden, whereas the wave function associated with a free particle must be continuous at the barrier and will show an exponential decay inside it. The wave function must also be continuous on the far side of the barrier, so there is a finite probability that the particle will pass through the barrier( Fig. 1). One important example based on quantum tunnelling is *α*-radioactivity which was proposed by Gamow [20-22] who found the well-known Gamow formula. The story of this discovery is told by Rosenfeld [23] who was one of the leading nuclear physicist

In the following, before proceeding to the case of massless electrons tunneling in graphene, we concern ourselves to evaluation of transmission probability of an electron incident upon

**2.1. Tunneling of an electron with energy lower than the electrostatic potential** For calculating the transmission probability of an electron incident from the left on a potential barrier of hight *V*<sup>0</sup> which is more than the value of energy as indicated in the Figure 1 we

> 

*<sup>ψ</sup><sup>I</sup>* = *<sup>e</sup>*

0 x < 0 *V*<sup>0</sup> 0 < x < *w* 0 x > *w*

*ikx* <sup>+</sup> *re*−*ikx* (3)

*<sup>ψ</sup>I I* <sup>=</sup> *aeiqx* <sup>+</sup> *be*−*iqx* (4)

*<sup>ψ</sup>III* = *teikx* (5)

For regions I, the solution of Schrodinger's equation will be a combination of incident and reflected plane waves while in region II, depending on the energy, the solution will be either

a potential barrier with height much higher than the electron's energy.

*V*(*x*) =

have on the transmission probability and conductance in graphene is discussed [19].

point in the well-known paper "The electronic properties of graphene" [18].

**2. Quantum tunneling**

of the twentieth century.

consider the following potential:

a plane wave or a decaying exponential form.

where ∆ is equal to the half of the induced gap in graphene spectrum and it's positive (negative) sign corresponds to the *<sup>K</sup>* (*K*′ ) point. Then the exact expression for T in gapped graphene is evaluated. Although the presence of massless electrons which is an interesting aspect of graphene is ignored, it"l be seen that how it can save us from doing the calculation once more with zero mass on both sides of the barrier, but non-zero mass inside the barrier. This might be a better model for two pieces of graphene connected by a semiconductor barrier (see fig. 6). Another result that show up is that the expression for T in the former case shows a dependence of transmission on the sign of refractive index, n, while in the latter case it will be revealed that T is independent from the sign of n.

From the above discussion and motivated by mass production of graphene, using 2D massive Dirac-like equation, in the next sections, the scattering of Dirac fermions from a special potential step of height *V*<sup>0</sup> which electrons under it acquire a finite mass, due to the presence of a gap of 2∆ in graphene spectrum is investigated [2], resulting in changing of it's spectrum from the usual linear dispersion to a hyperbolic dispersion and then show that for an electron with energy *E* < *V*<sup>0</sup> incident on such a potential step, the transmission probability turns out to be smaller than one in normal incident, whereas in the case of ∆ → 0, this quantity is found to be unity. In graphene, a p-n junction could correspond to such a potential step if it is sharp enough [6-7].

Here it should be noted that for building up such a potential step, finite gaps are needed to be induced in spatial regions in graphene. One of the methods for inducing these gaps in energy spectra of graphene is to grow it on top of a hexagonal boron nitride with the B-N distance very close to C-C distance of graphene [8,9,10]. One other method is to pattern graphene nanoribbons.[11,12]. In this method graphene planes are patterned such that in several areas of the graphene flake narrow nanoribbons may exist. Here, considering the slabs with *SiO*2-BN interfaces, on top of which a graphene flake is deposit, it is then possible to build up some regions in graphene where the energy spectrum reveals a finite gap, meaning that charge carriers there behave as massive Dirac fermions while there can be still regions where massless Dirac fermions are present. Considering this possibility, therefore, the tunneling of electrons of energy *E* through this type of potential step and also an electrostatic barrier of hight *V*<sup>0</sup> which allows quasi-particles to acquire a finite mass in a region of the width *D* where the dispersion relation of graphene exhibits a parabolic dispersion is investigated. The potential barrier considered here is such that the width of the region of finite mass and the width of the electrostatics barrier is similar. It will be observed that this kind of barrier is not completely transparent for normal incidence contrary to the case of tunneling of massless Dirac fermions in gapless graphene which leads to the total transparency of the barrier

10.5772/51980

[13,14]. As mentioned it is a real problem for application of graphene into nano-electronics, since for nano-electronics applications of graphene a mass gap in itŠs energy spectrum is needed just like a conventional semiconductor. We also see that, considering the appropriate wave functions in region of electrostatic barrier reveals that transmission is independent of whether the refractive index is negative or positive[15-17]. There is exactly a mistake on this point in the well-known paper "The electronic properties of graphene" [18].

In the end, throughout a numerical approach the consequences that the extra *π*-shift might have on the transmission probability and conductance in graphene is discussed [19].

### **2. Quantum tunneling**

2 Graphene - Research and Applications

(negative) sign corresponds to the *<sup>K</sup>* (*K*′

is sharp enough [6-7].

case it will be revealed that T is independent from the sign of n.

the hight *V*<sup>0</sup> and width *D*. It will be found that the probability for an electron (approaching perpendicularly) to penetrate the barrier is equal to one, independent of *V*<sup>0</sup> and *D*. Although this result is very interesting from the point of view of fundamental research, its presence in graphene is unwanted when it comes to applications of graphene to nano-electronics because the pinch-off of the field effect transistors may be very ineffective. One way to overcome these difficulties is by generating a gap in the graphene spectrum. From the point of view of Dirac fermions this is equivalent to the appearing of a mass term in relativistic equation which describes the low-energy excitations of graphene, i.e. 2D the massive Dirac equation:

where ∆ is equal to the half of the induced gap in graphene spectrum and it's positive

graphene is evaluated. Although the presence of massless electrons which is an interesting aspect of graphene is ignored, it"l be seen that how it can save us from doing the calculation once more with zero mass on both sides of the barrier, but non-zero mass inside the barrier. This might be a better model for two pieces of graphene connected by a semiconductor barrier (see fig. 6). Another result that show up is that the expression for T in the former case shows a dependence of transmission on the sign of refractive index, n, while in the latter

From the above discussion and motivated by mass production of graphene, using 2D massive Dirac-like equation, in the next sections, the scattering of Dirac fermions from a special potential step of height *V*<sup>0</sup> which electrons under it acquire a finite mass, due to the presence of a gap of 2∆ in graphene spectrum is investigated [2], resulting in changing of it's spectrum from the usual linear dispersion to a hyperbolic dispersion and then show that for an electron with energy *E* < *V*<sup>0</sup> incident on such a potential step, the transmission probability turns out to be smaller than one in normal incident, whereas in the case of ∆ → 0, this quantity is found to be unity. In graphene, a p-n junction could correspond to such a potential step if it

Here it should be noted that for building up such a potential step, finite gaps are needed to be induced in spatial regions in graphene. One of the methods for inducing these gaps in energy spectra of graphene is to grow it on top of a hexagonal boron nitride with the B-N distance very close to C-C distance of graphene [8,9,10]. One other method is to pattern graphene nanoribbons.[11,12]. In this method graphene planes are patterned such that in several areas of the graphene flake narrow nanoribbons may exist. Here, considering the slabs with *SiO*2-BN interfaces, on top of which a graphene flake is deposit, it is then possible to build up some regions in graphene where the energy spectrum reveals a finite gap, meaning that charge carriers there behave as massive Dirac fermions while there can be still regions where massless Dirac fermions are present. Considering this possibility, therefore, the tunneling of electrons of energy *E* through this type of potential step and also an electrostatic barrier of hight *V*<sup>0</sup> which allows quasi-particles to acquire a finite mass in a region of the width *D* where the dispersion relation of graphene exhibits a parabolic dispersion is investigated. The potential barrier considered here is such that the width of the region of finite mass and the width of the electrostatics barrier is similar. It will be observed that this kind of barrier is not completely transparent for normal incidence contrary to the case of tunneling of massless Dirac fermions in gapless graphene which leads to the total transparency of the barrier

*<sup>H</sup>* = −*ivFσ***.**∇ ± <sup>∆</sup>*σ<sup>z</sup>* (1)

) point. Then the exact expression for T in gapped

According to classical physics, a particle of energy *E* less than the height *V*<sup>0</sup> of a potential barrier could not penetrate it because the region inside the barrier is classically forbidden, whereas the wave function associated with a free particle must be continuous at the barrier and will show an exponential decay inside it. The wave function must also be continuous on the far side of the barrier, so there is a finite probability that the particle will pass through the barrier( Fig. 1). One important example based on quantum tunnelling is *α*-radioactivity which was proposed by Gamow [20-22] who found the well-known Gamow formula. The story of this discovery is told by Rosenfeld [23] who was one of the leading nuclear physicist of the twentieth century.

In the following, before proceeding to the case of massless electrons tunneling in graphene, we concern ourselves to evaluation of transmission probability of an electron incident upon a potential barrier with height much higher than the electron's energy.

#### **2.1. Tunneling of an electron with energy lower than the electrostatic potential**

For calculating the transmission probability of an electron incident from the left on a potential barrier of hight *V*<sup>0</sup> which is more than the value of energy as indicated in the Figure 1 we consider the following potential:

$$V(\mathbf{x}) = \begin{cases} 0 & \mathbf{x} < \mathbf{0} \\ V\_0 \ \mathbf{0} < \mathbf{x} < w \\ \mathbf{0} & \mathbf{x} > w \end{cases} \tag{2}$$

For regions I, the solution of Schrodinger's equation will be a combination of incident and reflected plane waves while in region II, depending on the energy, the solution will be either a plane wave or a decaying exponential form.

$$
\psi\_I = e^{i\mathbf{k}\mathbf{x}} + r e^{-i\mathbf{k}\mathbf{x}} \tag{3}
$$

$$
\psi\_{II} = a e^{iq\chi} + b e^{-iq\chi} \tag{4}
$$

$$
\psi\_{III} = \text{te}^{i\text{kx}} \tag{5}
$$

**Figure 1.** Schematic representation of tunneling in a 2D barrier.

where *a*, *b*,*r*, *t* are probability coefficients that must be determined from applying the boundary conditions. *k* and *q* are the momentum vectors in the regions I an II, respectively:

$$k = \sqrt{\frac{2mE}{\hbar^2}},\tag{6}$$

10.5772/51980

7

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

**Figure 2.** A p-n junction of graphene in which massless electrons incident upon an electrostatic region with no energy gap so

<sup>2</sup> <sup>=</sup> <sup>16</sup>*k*2*q*<sup>2</sup>

For energies lower than *V*0, the wave decays exponentially as it passes through the barrier, since in this case *q* is imaginary. Also note that the perfect transmission happens at *qD* = *nπ* (n an integer). This resonance in transmission occurs physically because of instructive and destructive matching of the transmitted and reflected waves in the potential region. Now that we have got a insight on the quantum tunneling phenomena in non-relativistic limit, the

Here, first a p-n junction of graphene which could be realized with a backgate and could correspond to a potential step of hight *V*<sup>0</sup> on which an massless electron of energy *E* is incident ( see Fig 2) is considered. Two region, therefore, can be considered. The region for which *x* < 0 corresponding to a kinetic energy of *E* and the region corresponding to a kinetic energy of *<sup>E</sup>* − *<sup>V</sup>*0. In order to obtain the transmission and reflection amplitudes, we first need

*<sup>V</sup>*(**r**) = *<sup>V</sup>*<sup>0</sup> <sup>x</sup> <sup>&</sup>gt; <sup>0</sup>

The above Dirac equation for *x* > 0 has the exact solutions which are the same as the free particle solutions except that the energy *E* can be different from the free particle case by the

(*<sup>q</sup>* <sup>+</sup> *<sup>k</sup>*)2*e*−*ikD* <sup>−</sup> (*<sup>q</sup>* <sup>−</sup> *<sup>k</sup>*)2*eikD* . (10)

*<sup>H</sup>* = *vFσ***.***<sup>p</sup>* + *<sup>V</sup>*(**r**), (11)

0 x <sup>&</sup>lt; <sup>0</sup> (12)

that electrons in tunneling process have an effective mass equal to zero.

which from it the transmission probability T can be evaluated as:

*T* = |*t*|

next step is to extent our attentions to the relativistic case.

to write down the following equation:

where

**3. Massless electrons tunneling into potential step**

$$q = \sqrt{\frac{2m(E - V\_0)}{\hbar^2}}.\tag{7}$$

We know that the wave functions and also their first spatial derivatives must be continuous across the boundaries. Imposing these conditions yields:

$$\begin{cases} 1 + r = a + b \\ ik(1 - r) = iq(a - b) \\ ae^{iqD} + be^{-iqD} = te^{ikD} \\ iq(ae^{iqD} - be^{-iqD})a = ikt e^{ikD} \end{cases} \tag{8}$$

The transmission amplitude, t is easily obtained:

$$t = \frac{4e^{-ikD}kq}{(q+k)^2e^{-ikD} - (q-k)^2e^{ikD}} \,\tag{9}$$

**Figure 2.** A p-n junction of graphene in which massless electrons incident upon an electrostatic region with no energy gap so that electrons in tunneling process have an effective mass equal to zero.

which from it the transmission probability T can be evaluated as:

$$T = \left| t \right|^2 = \frac{16k^2q^2}{(q+k)^2e^{-ikD} - (q-k)^2e^{ikD}}.\tag{10}$$

For energies lower than *V*0, the wave decays exponentially as it passes through the barrier, since in this case *q* is imaginary. Also note that the perfect transmission happens at *qD* = *nπ* (n an integer). This resonance in transmission occurs physically because of instructive and destructive matching of the transmitted and reflected waves in the potential region. Now that we have got a insight on the quantum tunneling phenomena in non-relativistic limit, the next step is to extent our attentions to the relativistic case.

#### **3. Massless electrons tunneling into potential step**

Here, first a p-n junction of graphene which could be realized with a backgate and could correspond to a potential step of hight *V*<sup>0</sup> on which an massless electron of energy *E* is incident ( see Fig 2) is considered. Two region, therefore, can be considered. The region for which *x* < 0 corresponding to a kinetic energy of *E* and the region corresponding to a kinetic energy of *<sup>E</sup>* − *<sup>V</sup>*0. In order to obtain the transmission and reflection amplitudes, we first need to write down the following equation:

$$H = v\_{\mathbf{F}} \sigma . p \, + \, V(\mathbf{r}) , \tag{11}$$

where

4 Graphene - Research and Applications

**Figure 1.** Schematic representation of tunneling in a 2D barrier.

where *a*, *b*,*r*, *t* are probability coefficients that must be determined from applying the boundary conditions. *k* and *q* are the momentum vectors in the regions I an II, respectively:

<sup>2</sup>*m*(*<sup>E</sup>* − *<sup>V</sup>*0)

We know that the wave functions and also their first spatial derivatives must be continuous

*<sup>h</sup>*¯ <sup>2</sup> , (6)

*<sup>h</sup>*¯ <sup>2</sup> . (7)

(*<sup>q</sup>* <sup>+</sup> *<sup>k</sup>*)2*e*−*ikD* <sup>−</sup> (*<sup>q</sup>* <sup>−</sup> *<sup>k</sup>*)2*eikD* , (9)

(8)

*k* = � 2*mE*

*q* = �

1 + *r* = *a* + *b ik*(1 − *r*) = *iq*(*a* − *b*) *aeiqD* + *be*−*iqD* = *teikD iq*(*aeiqD* − *be*−*iqD*)*<sup>a</sup>* = *ikteikD*

*<sup>t</sup>* <sup>=</sup> <sup>4</sup>*e*<sup>−</sup>*ikDkq*

across the boundaries. Imposing these conditions yields:

The transmission amplitude, t is easily obtained:

 

$$V(\mathbf{r}) = \begin{cases} V\_0 \ge > 0 \\ 0 \quad \text{x} < 0 \end{cases} \tag{12}$$

The above Dirac equation for *x* > 0 has the exact solutions which are the same as the free particle solutions except that the energy *E* can be different from the free particle case by the addition of the constant potential *V*0. Thus, in the region II, the energy of the Dirac fermions is given by:

$$E = v\_F \sqrt{q\_x^2 + k\_y^2} + V\_{0\prime} \tag{13}$$

10.5772/51980

9

. (21)

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

*R* + *T* = 1. (22)

*<sup>j</sup>* = *vFψ*†*σψ*, (24)

∇**.j**(*x*, *y*) = 0, (25)

*jx*(*x*) = *constant*. (26)

*<sup>x</sup>*, (27)

sin *<sup>φ</sup>* (28)

2, (23)

Here it should be noted that the transmission probability, *T*, as we see later, is not simply

*cos*(*φ* − *θ*)

1 + *λλ*′*cos*(*φ* + *θ*)

given by *tt*<sup>∗</sup> unlike to the refraction probability, *<sup>R</sup>*, which is always equal to *rr*∗:

*<sup>R</sup>* <sup>=</sup> *rr*<sup>∗</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *λλ*′

Physically the reason that T is not given by *tt*<sup>∗</sup> is because in the conservation law:

∇.**j** +

*∂ ∂t* |*ψ*|

it is the probability current, **j**(*x*, *y*), that matters, which is not simply given by probability

changes between the incoming wave and the transmitted wave, T is not, therefore, given

transmission, since the system is translational invariant along the *y*-direction, we get

*j i <sup>x</sup>* <sup>+</sup> *<sup>j</sup> r <sup>x</sup>* <sup>=</sup> *<sup>j</sup> t*

1 = |*r*|

One can then obtain the transmission probability from the relation (R+T=1) as:

<sup>2</sup> + |*t*|

2, however there is the ratio of the two velocities entering. Here, in order to find the

2. The probability current also contains the velocity which means that if velocity

*<sup>x</sup>* denote the incident, reflected and transmission currents, respectively.

<sup>2</sup> *λλ*′ cos *<sup>θ</sup>*

The reader can easily check that using the relation:

which gives for the probability current

Hence one can write the following relation:

From this equation it is obvious that:

density |*ψ*|

which implies that:

by |*t*|

where *j i <sup>x</sup>*, *j r <sup>x</sup>* and *j i*

where **q** is the momentum in the region of electrostaic potential. The wave functions in the two regions can be written as:

$$\psi\_I = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i\phi} \end{pmatrix} e^{i(k\_x x + k\_y y)} + \frac{r}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i(\pi - \phi)} \end{pmatrix} e^{i(-k\_x x + k\_y y)} \tag{14}$$

and

$$\psi\_{II} = \frac{t}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda' e^{i(\theta + \pi)} \end{pmatrix} e^{i(q\_{\text{x}}\mathbf{x} + k\_{\text{y}}\mathbf{y})} \tag{15}$$

where *<sup>r</sup>* and *<sup>t</sup>* are reflected and transmitted amplitudes, respectively, *<sup>λ</sup>*′ <sup>=</sup> *sgn*(*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*0) is the band index of the wave function corresponding to the second region (*x* > 0) and *φ* = arctan( *ky kx* ) is the angle of propagation of the incident electron wave and *θ* = arctan( *ky qx* ) with

$$q\_X = \pm \sqrt{[\frac{(V\_0 - E)^2}{v\_F^2}] - k\_{y\_{\prime}}^2} \tag{16}$$

is the angle of the propagation of the transmitted electron wave<sup>1</sup> and not, as it should be, the angle that momentum vector **q** makes with the x-axis. The reason will be clear later.

The following set of equations are obtained, if one applies the continuity condition of the wave functions at the interface *x* = 0:

$$1 + r = t \tag{17}$$

$$
\lambda e^{i\phi} - r\lambda e^{-i\phi} = \lambda' t e^{i\theta} \,\,\,\,\,\tag{18}
$$

which gives the transmission amplitude, t, as follows:

$$t = \frac{2\lambda\cos\phi}{\lambda'\varepsilon^{i\theta} + \lambda\varepsilon^{-i\phi}}.\tag{19}$$

Multiplying t by it's complex conjugate yields:

$$tt^\* = \frac{2\cos^2\phi}{1 + \lambda\lambda'\cos(\phi + \theta)}.\tag{20}$$

<sup>1</sup> By this definition *θ* falls in the range − *<sup>π</sup>* <sup>2</sup> <sup>&</sup>lt; *<sup>θ</sup>* <sup>&</sup>lt; <sup>−</sup> *<sup>π</sup>* 2 .

Here it should be noted that the transmission probability, *T*, as we see later, is not simply given by *tt*<sup>∗</sup> unlike to the refraction probability, *<sup>R</sup>*, which is always equal to *rr*∗:

$$R = r r^\* = \frac{1 - \lambda \lambda' \cos(\phi - \theta)}{1 + \lambda \lambda' \cos(\phi + \theta)}.\tag{21}$$

The reader can easily check that using the relation:

$$R + T = 1.\tag{22}$$

Physically the reason that T is not given by *tt*<sup>∗</sup> is because in the conservation law:

$$
\nabla \cdot \mathbf{j} + \frac{\partial}{\partial t} |\psi|^2 \,, \tag{23}
$$

which gives for the probability current

$$j = v\_F \psi^\dagger \sigma \psi\_{\prime} \tag{24}$$

it is the probability current, **j**(*x*, *y*), that matters, which is not simply given by probability density |*ψ*| 2. The probability current also contains the velocity which means that if velocity changes between the incoming wave and the transmitted wave, T is not, therefore, given by |*t*| 2, however there is the ratio of the two velocities entering. Here, in order to find the transmission, since the system is translational invariant along the *y*-direction, we get

$$\nabla \mathbf{j}(\mathbf{x}, \mathbf{y}) = \mathbf{0},\tag{25}$$

which implies that:

6 Graphene - Research and Applications

two regions can be written as:

*<sup>ψ</sup><sup>I</sup>* <sup>=</sup> <sup>1</sup> √2

wave functions at the interface *x* = 0:

  1

 *e*

*<sup>ψ</sup>I I* <sup>=</sup> *<sup>t</sup>* √2

*qx* = ±

*λe*

which gives the transmission amplitude, t, as follows:

Multiplying t by it's complex conjugate yields:

<sup>1</sup> By this definition *θ* falls in the range − *<sup>π</sup>*

� [

angle that momentum vector **q** makes with the x-axis. The reason will be clear later.

*<sup>i</sup><sup>φ</sup>* − *<sup>r</sup>λ<sup>e</sup>*

*<sup>t</sup>* <sup>=</sup> <sup>2</sup>*<sup>λ</sup>* cos *<sup>φ</sup>*

*tt*<sup>∗</sup> <sup>=</sup> 2 cos2 *<sup>φ</sup>*

2 .

<sup>2</sup> <sup>&</sup>lt; *<sup>θ</sup>* <sup>&</sup>lt; <sup>−</sup> *<sup>π</sup>*

1 + *λλ*′*cos*(*φ* + *θ*)

*λei<sup>φ</sup>*

is given by:

and

arctan( *ky kx*

addition of the constant potential *V*0. Thus, in the region II, the energy of the Dirac fermions

where **q** is the momentum in the region of electrostaic potential. The wave functions in the

1

where *<sup>r</sup>* and *<sup>t</sup>* are reflected and transmitted amplitudes, respectively, *<sup>λ</sup>*′ <sup>=</sup> *sgn*(*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*0) is the band index of the wave function corresponding to the second region (*x* > 0) and *φ* =

) is the angle of propagation of the incident electron wave and *θ* = arctan(

is the angle of the propagation of the transmitted electron wave<sup>1</sup> and not, as it should be, the

The following set of equations are obtained, if one applies the continuity condition of the

<sup>−</sup>*i<sup>φ</sup>* = *λ*′

(*V*<sup>0</sup> − *<sup>E</sup>*)<sup>2</sup> *v*2 *F*

*r* √2  

 *e* 1

 *e*

*<sup>λ</sup>ei*(*π*−*φ*)

*i*(*qx x*+*ky y*)

] − *k*<sup>2</sup>

*<sup>y</sup>* + *<sup>V</sup>*0, (13)

*i*(−*kx x*+*ky y*)

, (14)

*ky qx* ) with

, (15)

*<sup>y</sup>*, (16)

1 + *r* = *t* (17)

*<sup>λ</sup>*′*ei<sup>θ</sup>* <sup>+</sup> *<sup>λ</sup>e*−*i<sup>φ</sup>* . (19)

*tei<sup>θ</sup>* , (18)

. (20)

� *q*2 *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup>

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) +

 

*λ*′ *<sup>e</sup>i*(*θ*+*π*)

*<sup>E</sup>* = *vF*

$$j\_\mathbf{x}(\mathbf{x}) = \text{constant.} \tag{26}$$

Hence one can write the following relation:

$$\mathbf{j\_{x}^{i}} + \mathbf{j\_{x}^{r}} = \mathbf{j\_{x}^{t}} \tag{27}$$

where *j i <sup>x</sup>*, *j r <sup>x</sup>* and *j i <sup>x</sup>* denote the incident, reflected and transmission currents, respectively. From this equation it is obvious that:

$$1 = |r|^2 + |t|^2 \frac{\lambda \lambda' \cos \theta}{\sin \phi} \tag{28}$$

One can then obtain the transmission probability from the relation (R+T=1) as:

$$T = \frac{2\lambda\lambda'\cos\theta\cos\phi}{1 + \lambda\lambda'\cos(\phi + \theta)}.\tag{29}$$

10.5772/51980

11

. (33)

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

<sup>1</sup> <sup>+</sup> cos *<sup>φ</sup>*, (34)

<sup>1</sup> <sup>+</sup> cos *<sup>φ</sup>*. (35)

*<sup>R</sup>* == <sup>1</sup> <sup>+</sup> *λλ*′ cos(*<sup>φ</sup>* <sup>−</sup> *<sup>θ</sup>*) 1 − *λλ*′ cos(*φ* + *θ*)

These expressions now reveal that both transmission and reflection probability are positive and less than unity. It also shows that if electron arrives perpendicularly upon the step, the probability to go through it is one which is is related to the well-known "absence of backscattering" [24] and is a consequence of the chirality of the massless Dirac electrons [25]. Notice that in the limit *<sup>V</sup>*<sup>0</sup> >> *<sup>E</sup>*, since in this case *qx* → <sup>∞</sup> and therefore *<sup>θ</sup>* → 0, transmission

*<sup>T</sup>*(*φ*) = 2 cos *<sup>φ</sup>*

*<sup>R</sup>*(*φ*) = <sup>1</sup> <sup>−</sup> cos *<sup>φ</sup>*

As it is clear in the case of normal incident the p-n junction become totally transparent, i.e.

In this section the scattering of massless electrons of energy *E* by a n-p-n junction of graphene which can correspond to a square barrier if it is sharp enough I address as depicted in figure

> *r* √2

> > *b* √2

 *e*  

*λ*′ *<sup>e</sup>i*(*π*−*θt*)

*i*(*kx x*+*ky y*)

  1

 *e*

> *e*

*i*(−*kx x*+*ky y*)

*i*(*qx x*+*ky y*)

, (38)

, (36)

, (37)

*<sup>λ</sup>ei*(*π*−*φ*)

1

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) +

*<sup>i</sup>*(*qx <sup>x</sup>*+*ky <sup>y</sup>*) +

  1

*λei<sup>φ</sup>*

we'll be able to calculate *T* only by imposing the continuous condition of wave function at the boundaries and not it's derivative. Note that, in the case of *<sup>E</sup>* < *<sup>V</sup>*0, *<sup>θ</sup><sup>t</sup>* = *<sup>θ</sup>* + *<sup>π</sup>* is the angle of momentum vector **q**, measured from the x-axis while *θ* is the angle of propagation of the wave packed and, therefore, shows the angle that group velocity, *vg*, makes with the x-axis2. <sup>2</sup> Notice that if one consider the case *E* > *V*0, one then see that *θ<sup>t</sup>* = *θ*, implying that momentum and group velocity

**4. Ultra-relativistic tunneling into a potential barrier**

3. By writing the wave functions in the three regions as:

1

 *e*

*λei<sup>φ</sup>*

1

 *e*

*<sup>ψ</sup>III* <sup>=</sup> *<sup>t</sup>* √2

*λ*′ *eiθt*

 

 

*<sup>ψ</sup><sup>I</sup>* <sup>=</sup> <sup>1</sup> √2

*<sup>ψ</sup>I I* <sup>=</sup> *<sup>a</sup>* √2

and reflection probability are:

and

*T*(0) = 1.

are parallel.

This equation shows that for an electron of energy *E* > *V*0, the probability is positive and also less than unity, whereas for an electron of energy *E* < *V*0, as in this case we have *<sup>λ</sup>* <sup>=</sup> 1 and *<sup>λ</sup>*′ <sup>=</sup> *sgn*(*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*0) = <sup>−</sup>1, we find that the probability is negative and therefore the reflection probability, *R*, exceeds unity as it is clear from (21). In fact the assumption of particle-antiparticle (in this case electron-hole) pair production at the interface was considered as an explanation of these higher-than-unity reflection probability and negative transmission and has been so often interpreted as the meaning of the Klein paradox. In particular, throughout this chapter, these features are refereed to as the Klein paradox.

Another odd result will be revealed, if we consider the normal incident of electrons upon the interface of the potential step. Assuming an electron propagating with propagation angle *φ* = 0 on the potential step, we see that both *R* and *T*, in this case, become infinite which does not make sense at all because it would imply the existence of a hypothetical current source corresponding to the electron-hole pair creation at interface of the step. In other words no known physical mechanism can be associated to this results.

As it will be clear in what follows the negative T and higher than one reflection probability that equations (29) and (21) imply, arises from the wrong considered direction of the momentum vector, **q**, of the wave function in the region II. In fact, in the case of *E* < *V*0, momentum and group velocity *vg* which is evaluated as:

$$\mathbf{v}\_{\mathcal{S}} = \frac{\partial E}{\partial q\_{\mathcal{X}}} = \frac{q\_{\mathcal{X}}}{E - V\_0} \, \text{} \tag{30}$$

have opposite directions because we assumed that the transmitted electron moves from left to right and therefore *vg* must be positive implying that *qx* has to assign it's negative value, meaning that the direction of momentum in the region II differs by 180 degree from the direction of which the wave packed propagates. In the other words in the case of *E* < *V*0, the phase of the transmitted wave function in momentum-space undergoes a *π* change in transmitting from the region I to region II. Thus, the appropriate wave functions in the momentum space, *ψI I*, is:

$$
\psi\_{II} = \frac{t}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda' e^{i(\theta + \pi)} \end{pmatrix} \tag{31}
$$

which from them *T* and *R* are given by:

$$T = -\frac{2\lambda\lambda'\cos\theta\cos\phi}{1 + \lambda\lambda'\cos(\phi + \theta)}.\tag{32}$$

$$R = = \frac{1 + \lambda \lambda' \cos(\phi - \theta)}{1 - \lambda \lambda' \cos(\phi + \theta)}.\tag{33}$$

These expressions now reveal that both transmission and reflection probability are positive and less than unity. It also shows that if electron arrives perpendicularly upon the step, the probability to go through it is one which is is related to the well-known "absence of backscattering" [24] and is a consequence of the chirality of the massless Dirac electrons [25]. Notice that in the limit *<sup>V</sup>*<sup>0</sup> >> *<sup>E</sup>*, since in this case *qx* → <sup>∞</sup> and therefore *<sup>θ</sup>* → 0, transmission and reflection probability are:

$$T(\phi) = \frac{2\cos\phi}{1 + \cos\phi},\tag{34}$$

and

8 Graphene - Research and Applications

momentum space, *ψI I*, is:

which from them *T* and *R* are given by:

*<sup>T</sup>* <sup>=</sup> <sup>2</sup>*λλ*′ cos *<sup>θ</sup>* cos *<sup>φ</sup>* 1 + *λλ*′*cos*(*φ* + *θ*)

This equation shows that for an electron of energy *E* > *V*0, the probability is positive and also less than unity, whereas for an electron of energy *E* < *V*0, as in this case we have *<sup>λ</sup>* <sup>=</sup> 1 and *<sup>λ</sup>*′ <sup>=</sup> *sgn*(*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*0) = <sup>−</sup>1, we find that the probability is negative and therefore the reflection probability, *R*, exceeds unity as it is clear from (21). In fact the assumption of particle-antiparticle (in this case electron-hole) pair production at the interface was considered as an explanation of these higher-than-unity reflection probability and negative transmission and has been so often interpreted as the meaning of the Klein paradox. In particular, throughout this chapter, these features are refereed to as the Klein paradox.

Another odd result will be revealed, if we consider the normal incident of electrons upon the interface of the potential step. Assuming an electron propagating with propagation angle *φ* = 0 on the potential step, we see that both *R* and *T*, in this case, become infinite which does not make sense at all because it would imply the existence of a hypothetical current source corresponding to the electron-hole pair creation at interface of the step. In other

As it will be clear in what follows the negative T and higher than one reflection probability that equations (29) and (21) imply, arises from the wrong considered direction of the momentum vector, **q**, of the wave function in the region II. In fact, in the case of *E* < *V*0,

> = *qx <sup>E</sup>* − *<sup>V</sup>*<sup>0</sup>

have opposite directions because we assumed that the transmitted electron moves from left to right and therefore *vg* must be positive implying that *qx* has to assign it's negative value, meaning that the direction of momentum in the region II differs by 180 degree from the direction of which the wave packed propagates. In the other words in the case of *E* < *V*0, the phase of the transmitted wave function in momentum-space undergoes a *π* change in transmitting from the region I to region II. Thus, the appropriate wave functions in the

> 

*<sup>T</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>*λλ*′ cos *<sup>θ</sup>* cos *<sup>φ</sup>*

1 + *λλ*′ cos(*φ* + *θ*)

*λ*′ *<sup>e</sup>i*(*θ*+*π*)

1

**<sup>v</sup>***<sup>g</sup>* <sup>=</sup> *<sup>∂</sup><sup>E</sup> ∂qx*

*<sup>ψ</sup>I I* <sup>=</sup> *<sup>t</sup>* √2

words no known physical mechanism can be associated to this results.

momentum and group velocity *vg* which is evaluated as:

. (29)

, (30)

, (31)

. (32)

$$R(\phi) = \frac{1 - \cos\phi}{1 + \cos\phi}.\tag{35}$$

As it is clear in the case of normal incident the p-n junction become totally transparent, i.e. *T*(0) = 1.

#### **4. Ultra-relativistic tunneling into a potential barrier**

In this section the scattering of massless electrons of energy *E* by a n-p-n junction of graphene which can correspond to a square barrier if it is sharp enough I address as depicted in figure 3. By writing the wave functions in the three regions as:

$$\psi\_I = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i\phi} \end{pmatrix} e^{i(k\_x x + k\_y y)} + \frac{r}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i(\pi - \phi)} \end{pmatrix} e^{i(-k\_x x + k\_y y)} \tag{36}$$

$$\psi\_{II} = \frac{a}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda' e^{i\theta\_l} \end{pmatrix} e^{i(q\_x x + k\_y y)} + \frac{b}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda' e^{i(\pi - \theta\_l)} \end{pmatrix} e^{i(q\_x x + k\_y y)} \tag{37}$$

$$
\psi\_{III} = \frac{t}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i\phi} \end{pmatrix} e^{i(k\_{\text{xx}} + k\_{\text{yy}} y)} \,, \tag{38}
$$

we'll be able to calculate *T* only by imposing the continuous condition of wave function at the boundaries and not it's derivative. Note that, in the case of *<sup>E</sup>* < *<sup>V</sup>*0, *<sup>θ</sup><sup>t</sup>* = *<sup>θ</sup>* + *<sup>π</sup>* is the angle of momentum vector **q**, measured from the x-axis while *θ* is the angle of propagation of the wave packed and, therefore, shows the angle that group velocity, *vg*, makes with the x-axis2.

<sup>2</sup> Notice that if one consider the case *E* > *V*0, one then see that *θ<sup>t</sup>* = *θ*, implying that momentum and group velocity are parallel.

**Figure 3.** an one dimensional schematic view of a n-p-n junction of gapless graphene. In all three zones the energy bands are linear in momentum and therefore we have massless electrons passing through the barrier.

By applying the continuity conditions of the wave functions at the two discontinuities of the barrier (*x* = 0 and *x* = *D*), the following set of equations is obtained:

$$1 + r = a + b \tag{39}$$

10.5772/51980

13

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

Thus, by plugging *a* and *b* into this equation, after some algebraical manipulation t can be

Up to now, we have only obtained the transmission amplitude and not transmission probability. One can multiply t, by itŠs complex conjugation and get the exact expression

It is evident that *T*(*φ*) = *T*(−*φ*) and for values of *qxD* satisfying the relation *qxD* = *nπ*, with n an integer, the barrier becomes totally transparent, as in this case we have *T*(*φ*) = 1. Another interesting result will be obtained when we consider the scattering of an electron incident on the barrier with propagation angle *φ* = 0 (*φ* → 0 leading to *θ<sup>t</sup>* → 0 and *π* for the case of *E* > *V*<sup>0</sup> and *E* < *V*0, respectively) which imply that, no matter what the value of *qxD* is, the barrier becomes completely transparent, i.e. T(0) = 1. However for applications of graphene in nano-electronic devices such as a graphene-based transistors this transparency of the barrier is unwanted, since the transistor can not be pinched off in this case, however, in the next section by evaluating the transmission probability of a n-p-n junction of graphene which quasi-particles can acquire a finite mass there, it will be clear that transmission is smaller than one and therefore suitable for applications purposes. Turning our attention back to expression (47), it is clear that if one considers the cases *E* > *V*<sup>0</sup> and *E* < *V*<sup>0</sup> with the same magnitude for x-component of momentum vector **q**, corresponding to same values for |*V*<sup>0</sup> − *E*|, would arrive at the same results for transmission probability, irrespective of whether the energy of incident electron is higher or smaller than the hight of the barrier3. This is a very interesting result because it shows that transmission is independent of the sign of refractive index n of graphene, since for the case of *E* < *V*<sup>0</sup> group velocity and the momentum vector in the region II have opposite directions and graphene, therefore, meets the negative refractive index. There is a mistake exactly on this point in [18]. In this paper the angle that momentum vector **q** makes with the x-axis have been confused with the propagation angle *θ*. In fact the negative sign of *qx* have not been considered there and

*<sup>e</sup>iqxD*[<sup>2</sup> <sup>−</sup> <sup>2</sup>*λλ*′ cos(*<sup>φ</sup>* <sup>−</sup> *<sup>θ</sup>t*)] <sup>−</sup> *<sup>e</sup>*−*iqxD*[<sup>2</sup> <sup>+</sup> <sup>2</sup>*λλ*′ cos(*<sup>φ</sup>* <sup>+</sup> *<sup>θ</sup>t*)] (46)

(cos *<sup>φ</sup>* cos *<sup>θ</sup><sup>t</sup>* cos(*qxD*))<sup>2</sup> <sup>+</sup> sin2(*qxD*)(<sup>1</sup> <sup>−</sup> *λλ*′ sin *<sup>φ</sup>* sin *<sup>θ</sup>t*)<sup>2</sup> (47)

(cos *<sup>φ</sup>* cos *<sup>θ</sup>* cos(*qxD*))<sup>2</sup> <sup>+</sup> sin2(*qxD*)(<sup>1</sup> <sup>−</sup> *λλ*′ sin *<sup>φ</sup>* sin *<sup>θ</sup>*)<sup>2</sup> , (48)

results in different values for probability when |*E* − *V*0| is the same for both cases of *E* > *V*<sup>0</sup> and *E* < *V*0. In other words, the *π* phase change of the transmitted wave function

<sup>3</sup> Because if we assume that energy of incident electron is smaller than height of the barrier, the band index *λ*′ assigns it's negative value, meaning that the transmission angle *θ<sup>t</sup>* is *θ<sup>t</sup>* = *θ* + *π* and therefore we get sin *θ<sup>t</sup>* = − sin *θ*.

<sup>−</sup>*ikxD* <sup>4</sup>*λλ*′ cos *<sup>φ</sup>* cos *<sup>θ</sup><sup>t</sup>*

for the transmission probability of massless electrons as:

therefore expression for *T* which is written there as

*<sup>T</sup>*(*φ*) = cos2 *<sup>φ</sup>* cos<sup>2</sup> *<sup>θ</sup>*

*<sup>T</sup>*(*φ*) = cos2 *<sup>φ</sup>* cos<sup>2</sup> *<sup>θ</sup><sup>t</sup>*

determined as:

*t* = −*e*

$$
\lambda e^{i\phi} - \lambda r e^{-i\phi} = \lambda' a e^{i\theta\_l} - \lambda' b e^{-i\theta\_l} \tag{40}
$$

$$ae^{iq\_\mathbb{z}D} + be^{-iq\_\mathbb{z}D} = te^{i\mathbb{k}\_\mathbb{z}D} \tag{41}$$

$$
\lambda' a e^{i\theta\_l + iq\_\ge D} - \lambda' b e^{-i\theta\_l - iq\_\ge D} = \lambda t e^{i\phi + ik\_\ge D}. \tag{42}
$$

Here, as previous sections, the transmission amplitude in the first region (incoming wave) is set to 1. For solving the above system of equations with respect to transmission amplitude, t, we first determine *a* from (41) which turns out to be:

$$a = te^{-iq\_\ge D + ik\_\ge D} - be^{-2iq\_\ge D}\_{\text{ } \prime} \tag{43}$$

and then substituting it in equation (42), b can be evaluated as:

$$b = \frac{t e^{i\eta\_t D + ik\_x D} (\lambda' e^{i\theta\_l} - \lambda e^{i\phi})}{2\lambda' \cos \theta\_l} \tag{44}$$

Now equation (40) by the use of relation (39) could be rewritten as follows:

$$2\lambda\cos\phi = a(\lambda'e^{i\theta\_l} + \lambda e^{-i\phi}) - b(\lambda'e^{-i\theta\_l} - \lambda e^{-i\phi}).\tag{45}$$

10.5772/51980

Thus, by plugging *a* and *b* into this equation, after some algebraical manipulation t can be determined as:

10 Graphene - Research and Applications

**Figure 3.** an one dimensional schematic view of a n-p-n junction of gapless graphene. In all three zones the energy bands are

By applying the continuity conditions of the wave functions at the two discontinuities of the

Here, as previous sections, the transmission amplitude in the first region (incoming wave) is set to 1. For solving the above system of equations with respect to transmission amplitude,

*aeiθ<sup>t</sup>* <sup>−</sup> *<sup>λ</sup>*′

1 + *r* = *a* + *b* (39)

*aeiqxD* <sup>+</sup> *be*−*iqxD* <sup>=</sup> *teikxD* (41)

*<sup>a</sup>* <sup>=</sup> *te*−*iqxD*+*ikxD* <sup>−</sup> *be*−2*iqxD*, (43)

*eiθ<sup>t</sup>* − *λeiφ*)

*e*

<sup>−</sup>*iθ<sup>t</sup>* <sup>−</sup> *<sup>λ</sup><sup>e</sup>*

<sup>2</sup>*λ*′ cos *<sup>θ</sup><sup>t</sup>*

<sup>−</sup>*iφ*) <sup>−</sup> *<sup>b</sup>*(*λ*′

*be*−*iθt*−*iqxD* <sup>=</sup> *<sup>λ</sup>teiφ*+*ikxD*. (42)

*be*−*iθ<sup>t</sup>* (40)

(44)

<sup>−</sup>*iφ*). (45)

linear in momentum and therefore we have massless electrons passing through the barrier.

barrier (*x* = 0 and *x* = *D*), the following set of equations is obtained:

*aeiθt*+*iqxD* <sup>−</sup> *<sup>λ</sup>*′

*<sup>i</sup><sup>φ</sup>* <sup>−</sup> *<sup>λ</sup>re*−*i<sup>φ</sup>* <sup>=</sup> *<sup>λ</sup>*′

*<sup>b</sup>* <sup>=</sup> *teiqxD*+*ikxD*(*λ*′

Now equation (40) by the use of relation (39) could be rewritten as follows:

*e <sup>i</sup>θ<sup>t</sup>* + *<sup>λ</sup><sup>e</sup>*

*λe*

*λ*′

t, we first determine *a* from (41) which turns out to be:

and then substituting it in equation (42), b can be evaluated as:

<sup>2</sup>*<sup>λ</sup>* cos *<sup>φ</sup>* <sup>=</sup> *<sup>a</sup>*(*λ*′

$$t = -e^{-ik\_lD} \frac{4\lambda\lambda'\cos\phi\cos\theta\_l}{e^{iq\_lD}[2 - 2\lambda\lambda'\cos(\phi - \theta\_l)] - e^{-iq\_lD}[2 + 2\lambda\lambda'\cos(\phi + \theta\_l)]}\tag{46}$$

Up to now, we have only obtained the transmission amplitude and not transmission probability. One can multiply t, by itŠs complex conjugation and get the exact expression for the transmission probability of massless electrons as:

$$T(\phi) = \frac{\cos^2 \phi \cos^2 \theta\_l}{(\cos \phi \cos \theta\_l \cos (q\_\mathbf{r} D))^2 + \sin^2 (q\_\mathbf{r} D)(1 - \lambda \lambda' \sin \phi \sin \theta\_l)^2} \tag{47}$$

It is evident that *T*(*φ*) = *T*(−*φ*) and for values of *qxD* satisfying the relation *qxD* = *nπ*, with n an integer, the barrier becomes totally transparent, as in this case we have *T*(*φ*) = 1. Another interesting result will be obtained when we consider the scattering of an electron incident on the barrier with propagation angle *φ* = 0 (*φ* → 0 leading to *θ<sup>t</sup>* → 0 and *π* for the case of *E* > *V*<sup>0</sup> and *E* < *V*0, respectively) which imply that, no matter what the value of *qxD* is, the barrier becomes completely transparent, i.e. T(0) = 1. However for applications of graphene in nano-electronic devices such as a graphene-based transistors this transparency of the barrier is unwanted, since the transistor can not be pinched off in this case, however, in the next section by evaluating the transmission probability of a n-p-n junction of graphene which quasi-particles can acquire a finite mass there, it will be clear that transmission is smaller than one and therefore suitable for applications purposes. Turning our attention back to expression (47), it is clear that if one considers the cases *E* > *V*<sup>0</sup> and *E* < *V*<sup>0</sup> with the same magnitude for x-component of momentum vector **q**, corresponding to same values for |*V*<sup>0</sup> − *E*|, would arrive at the same results for transmission probability, irrespective of whether the energy of incident electron is higher or smaller than the hight of the barrier3. This is a very interesting result because it shows that transmission is independent of the sign of refractive index n of graphene, since for the case of *E* < *V*<sup>0</sup> group velocity and the momentum vector in the region II have opposite directions and graphene, therefore, meets the negative refractive index. There is a mistake exactly on this point in [18]. In this paper the angle that momentum vector **q** makes with the x-axis have been confused with the propagation angle *θ*. In fact the negative sign of *qx* have not been considered there and therefore expression for *T* which is written there as

$$T(\phi) = \frac{\cos^2 \phi \cos^2 \theta}{(\cos \phi \cos \theta \cos (q\_\text{J} D))^2 + \sin^2 (q\_\text{J} D)(1 - \lambda \lambda' \sin \phi \sin \theta)^2},\tag{48}$$

results in different values for probability when |*E* − *V*0| is the same for both cases of *E* > *V*<sup>0</sup> and *E* < *V*0. In other words, the *π* phase change of the transmitted wave function

<sup>3</sup> Because if we assume that energy of incident electron is smaller than height of the barrier, the band index *λ*′ assigns it's negative value, meaning that the transmission angle *θ<sup>t</sup>* is *θ<sup>t</sup>* = *θ* + *π* and therefore we get sin *θ<sup>t</sup>* = − sin *θ*.

in momentum-space in the latter case is not counted in. It is worth noticing that both expressions for normal incident lead to same result *T*(0) = 1.

For a very high potential barrier (*V*<sup>0</sup> → <sup>∞</sup>), we have *<sup>θ</sup>* → 0 , *<sup>π</sup>*, and, therefore, we arrive at the following result for T:

$$T(\phi) = \frac{\cos^2 \phi}{\cos^2 \phi \cos^2(q\_\text{x}D) + \sin^2(q\_\text{x}D)} = \frac{\cos^2 \phi}{1 - \cos^2(q\_\text{x}D)\sin^2 \phi'} \tag{49}$$

which reveals that for perpendicular incidence the barrier is again totally transparent.

#### **5. Tunnelling of massive electrons into a p-n junction**

In the two previous sections the tunneling of massless Dirac fermions across p-n and n-p-n junctions was covered. In this section the massive electrons tunneling into a two dimensional potential step (n-p junction) of a gapped graphene which shows a hyperbolic energy spectrum unlike to the linear dispersion relation of a gapless graphene is discussed (see Fig. 4). The low energy excitations, therefore, are governed by the two dimensional massive Dirac equation. Thus, in order to calculate the transmission probability, we first need to obtain the eigenfunctions of the following Dirac equation which describes the massive Dirac fermions in gapped graphene so that we'll be able to write down the wave functions in different regions:

$$H = v\_F \sigma.p \, + \Delta \sigma^z,\tag{50}$$

10.5772/51980

15

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

**Figure 4.** Massive Dirac electron tunneling into a step potential of graphene. As it is clear an opening gap in graphene

where *λ* = ± correspond to the positive and negative energy states, respectively. Now in

*vλ*

where we've used units such that ¯*h* = 1. Plugging the above spinors into the corresponding

, *v<sup>λ</sup>* = *λ*

*λ*∆ ∆2+*v*<sup>2</sup> *F k*2

*λ*∆ ∆2+*v*<sup>2</sup>

*<sup>F</sup> <sup>k</sup>*<sup>2</sup> *<sup>e</sup>iϕ***<sup>k</sup>**

 *e*

 *uλ*

order to obtain the eigenfunctions, one can make the following ansatz:

*ψλ*,**<sup>k</sup>** <sup>=</sup> <sup>1</sup> √2

*λ*∆

∆<sup>2</sup> + *v*<sup>2</sup> *Fk*2

� 1 + √

> 

1

 *e*

*λeiϕ***<sup>k</sup>**

*λ* � 1 − √

It is clear that in the limit ∆ → 0, one arrives at the same eigenfunctions

*ψλ*,**<sup>k</sup>** <sup>=</sup> <sup>1</sup> √2

�

*ψλ*,**<sup>k</sup>** <sup>=</sup> <sup>1</sup> √2 *<sup>F</sup>* in both regions

*i*(*kx x*+*ky y*)

����

<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*<sup>∆</sup> �

 *e*

*i*(*kx x*+*ky y*)

∆<sup>2</sup> + *v*<sup>2</sup> *Fk*2 *e*

*i*(*kx x*+*ky y*)

, (55)

*<sup>i</sup>ϕ<sup>k</sup>* . (56)

. (57)

, (58)

spectrum makes electrons to acquire an effective mass of ∆/2*v*<sup>2</sup>

eigenvalue equation then gives:

*u<sup>λ</sup>* = ���� 1 +

The wave functions, therefore are given by:

where 2∆ is the induced gap in graphene spectrum and *σ* = (*σx*, *σy*) with

$$
\sigma^x = \begin{pmatrix} 0 \ 1 \\ 1 \ 0 \end{pmatrix}, \quad \sigma^y = \begin{pmatrix} 0 \ -i \\ i \ 0 \end{pmatrix}, \quad \sigma^z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \tag{51}
$$

the i=x,x,z, Pauli matrix. Now for obtaining the eigenfunctions one may rewrite the Hamiltonian as:

$$H = \begin{pmatrix} \Delta & \upsilon\_F | \mathbf{p} | e^{-i\rho\_\mathbf{p}} \\ \upsilon\_F | \mathbf{p} | e^{i\rho\_\mathbf{p}} & \Delta \end{pmatrix} \tag{52}$$

where

$$
\varphi\_{\mathbf{P}} = \arctan(p\_{\mathcal{Y}}/p\_{\mathcal{X}}).\tag{53}
$$

As one can easily see the corresponding eigenvalues are given by:

$$E = \lambda \sqrt{\Delta^2 + v\_F^2 \mathbf{P}^2} \,\,\,\,\tag{54}$$

10.5772/51980

**Figure 4.** Massive Dirac electron tunneling into a step potential of graphene. As it is clear an opening gap in graphene spectrum makes electrons to acquire an effective mass of ∆/2*v*<sup>2</sup> *<sup>F</sup>* in both regions

where *λ* = ± correspond to the positive and negative energy states, respectively. Now in order to obtain the eigenfunctions, one can make the following ansatz:

$$
\psi\_{\lambda, \mathbf{k}} = \frac{1}{\sqrt{2}} \begin{pmatrix} u\_{\lambda} \\\\ v\_{\lambda} \end{pmatrix} e^{i(k\_{\lambda}\mathbf{x} + k\_{\lambda}\mathbf{y})} \,, \tag{55}
$$

where we've used units such that ¯*h* = 1. Plugging the above spinors into the corresponding eigenvalue equation then gives:

$$u\_{\lambda} = \sqrt{1 + \frac{\lambda \Delta}{\sqrt{\Delta^2 + v\_F^2 k^2}}}, \quad v\_{\lambda} = \lambda \sqrt{1 - \frac{\lambda \Delta}{\sqrt{\Delta^2 + v\_F^2 k^2}}} e^{i q\_{\lambda}}.\tag{56}$$

The wave functions, therefore are given by:

12 Graphene - Research and Applications

the following result for T:

in different regions:

Hamiltonian as:

where

in momentum-space in the latter case is not counted in. It is worth noticing that both

For a very high potential barrier (*V*<sup>0</sup> → <sup>∞</sup>), we have *<sup>θ</sup>* → 0 , *<sup>π</sup>*, and, therefore, we arrive at

cos2 *<sup>φ</sup>* cos2(*qxD*) + sin2(*qxD*) <sup>=</sup> cos<sup>2</sup> *<sup>φ</sup>*

In the two previous sections the tunneling of massless Dirac fermions across p-n and n-p-n junctions was covered. In this section the massive electrons tunneling into a two dimensional potential step (n-p junction) of a gapped graphene which shows a hyperbolic energy spectrum unlike to the linear dispersion relation of a gapless graphene is discussed (see Fig. 4). The low energy excitations, therefore, are governed by the two dimensional massive Dirac equation. Thus, in order to calculate the transmission probability, we first need to obtain the eigenfunctions of the following Dirac equation which describes the massive Dirac fermions in gapped graphene so that we'll be able to write down the wave functions

*<sup>H</sup>* <sup>=</sup> *vFσ***.***<sup>p</sup>* <sup>+</sup> <sup>∆</sup>*σ<sup>z</sup>*

 0 −*i i* 0 

the i=x,x,z, Pauli matrix. Now for obtaining the eigenfunctions one may rewrite the

 <sup>∆</sup> *vF*|**p**|*e*−*iϕ***<sup>p</sup>** *vF*|**p**|*eiϕ***<sup>p</sup>** ∆

∆2 + *v*2

, *<sup>σ</sup><sup>z</sup>* =

 1 0 0 −1 

*ϕ***<sup>p</sup>** = arctan(*py*/*px*). (53)

where 2∆ is the induced gap in graphene spectrum and *σ* = (*σx*, *σy*) with

, *<sup>σ</sup><sup>y</sup>* =

*H* =

As one can easily see the corresponding eigenvalues are given by:

*E* = *λ* 

*σ<sup>x</sup>* =

 0 1 1 0 

which reveals that for perpendicular incidence the barrier is again totally transparent.

1 − cos2(*qxD*) sin2 *φ*

, (50)

, (52)

*<sup>F</sup>***P**2, (54)

, (51)

, (49)

expressions for normal incident lead to same result *T*(0) = 1.

*<sup>T</sup>*(*φ*) = cos<sup>2</sup> *<sup>φ</sup>*

**5. Tunnelling of massive electrons into a p-n junction**

$$\psi\_{\lambda,\mathbf{k}} = \frac{1}{\sqrt{2}} \begin{pmatrix} \sqrt{1 + \frac{\lambda\Delta}{\sqrt{\Delta^2 + v\_r^2 k^2}}} \\\\ \lambda \sqrt{1 - \frac{\lambda\Delta}{\sqrt{\Delta^2 + v\_r^2 k^2}}} e^{i\varphi\_\mathbf{k}} \end{pmatrix} e^{i(k\_x x + k\_y y)}. \tag{57}$$

It is clear that in the limit ∆ → 0, one arrives at the same eigenfunctions

$$
\psi\_{\lambda, \mathbf{k}} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i\varphi\_{\mathbf{k}}} \end{pmatrix} e^{i(k\_{\mathbf{x}}\mathbf{x} + k\_{\mathbf{y}}\mathbf{y})} \,, \tag{58}
$$

as those of massless Dirac fermions in graphene.

Now that we have found the corresponding eigenfunctions of Hamiltonian (4.52), assuming an electron incident upon a step of height *V*0, we can write the single valley Hamiltonian as:

$$H = v\_F \sigma. \mathcal{p} \, : + \Delta \sigma^z + V(\mathbf{r}), \tag{59}$$

and

where

and

*<sup>t</sup>* <sup>=</sup> <sup>2</sup>*<sup>λ</sup>* cos *<sup>φ</sup> η*

*<sup>R</sup>* <sup>=</sup> *Nr* <sup>−</sup> <sup>2</sup>*λλ*′

<sup>=</sup> <sup>2</sup> *<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′

<sup>=</sup> <sup>2</sup> *<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′

<sup>=</sup> *<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′

<sup>=</sup> (|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*<sup>|</sup> <sup>+</sup> *<sup>λ</sup>*′

is to calculate the transmission probability. So, considering equation (67) and:

*r*

<sup>2</sup> *λλ*′

<sup>=</sup> <sup>4</sup>*λλ*′

*T* = |*t*|

*Nr* <sup>=</sup> *<sup>β</sup>*2*γ*<sup>2</sup> <sup>+</sup> *<sup>α</sup>*2*η*<sup>2</sup> *β*2*γ*2

*Sr* <sup>=</sup> *αη βγ*

*j in*

T is found to be:

*<sup>x</sup>* <sup>=</sup> *λαγ* cos *<sup>φ</sup>*, *<sup>j</sup>*

From (1.66) it is straightforward to show that R is:

*<sup>γ</sup> <sup>λ</sup>*′*eiθ<sup>t</sup>* <sup>+</sup> *<sup>β</sup>*

*Sr* cos(*φ* − *θt*)

∆2

(|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*<sup>|</sup> <sup>+</sup> *<sup>λ</sup>*′∆)(*<sup>E</sup>* <sup>−</sup> *<sup>λ</sup>*∆) (69)

*<sup>E</sup>*<sup>∆</sup> − *<sup>λ</sup>*|*V*<sup>0</sup> − *<sup>E</sup>*|<sup>∆</sup>

(|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *<sup>λ</sup>*′∆)(*<sup>E</sup>* <sup>+</sup> *<sup>λ</sup>*∆) (70)

*t <sup>x</sup>* <sup>=</sup> *<sup>λ</sup>*′

*Nr* + <sup>2</sup>*λλ*′*Sr* cos(*<sup>φ</sup>* + *<sup>θ</sup>t*)

*<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′∆<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*|<sup>∆</sup> <sup>+</sup> *<sup>λ</sup>*′*E*<sup>∆</sup>

∆2

∆<sup>2</sup> + *λ*′

*<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′∆<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*′*E*<sup>∆</sup> <sup>+</sup> *<sup>λ</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*|<sup>∆</sup>

∆)(*E* − *λ*∆)

In the limit ∆ → 0 we get the same reflection as that of massless case. In the limit of no electrostatic potential we arrive at the logical result *R* = 0. This is important because we see later that for a special potential step in this limit R is not zero. Now one remaining problem

*<sup>x</sup>* <sup>=</sup> <sup>−</sup>*λαγ* cos *<sup>φ</sup>*, *<sup>j</sup>*

*ηβ* cos *θ<sup>t</sup> αγ* cos *φ*

*St* cos *φ* cos *θ<sup>t</sup> Nt* + <sup>2</sup>*Stλλ*′ cos(*<sup>φ</sup>* + *<sup>θ</sup>t*) 10.5772/51980

17

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

*<sup>α</sup> <sup>λ</sup>e*−*i<sup>φ</sup>* . (67)

, (68)

*ηβ* cos *θ<sup>t</sup>* (71)

, (72)

where *V*(**r**) = 0 for region I (*x* < 0) and for the region II (*x* > 0), massive Dirac fermions feel a electrostatic potential of hight *<sup>V</sup>*<sup>0</sup> with the kinetic energy *<sup>E</sup>* − *<sup>V</sup>*0. The wave functions in the two regions then are:

$$\psi\_I = \frac{1}{\sqrt{2}} \begin{pmatrix} \alpha \\\\ \gamma \lambda e^{i\phi} \end{pmatrix} e^{i(k\_x x + k\_y y)} + \frac{r}{\sqrt{2}} \begin{pmatrix} \alpha \\\\ \gamma \lambda e^{i(\pi - \phi)} \end{pmatrix} e^{i(-k\_x x + k\_y y)} \tag{60}$$

and

$$
\psi\_{II} = \frac{t}{\sqrt{2}} \begin{pmatrix} \beta \\\\ \lambda' \eta e^{i\theta\_l} \end{pmatrix} e^{i(q\_x x + k\_y y)} \,, \tag{61}
$$

where in order to make things more simple, the following abbreviations is introduced:

$$\alpha = \sqrt{1 + \frac{\lambda \Delta}{\sqrt{\Delta^2 + v\_F^2 (k\_x^2 + k\_y^2)}}}, \quad \gamma = \sqrt{1 - \frac{\lambda \Delta}{\sqrt{\Delta^2 + v\_F^2 (k\_x^2 + k\_y^2)}}},\tag{62}$$

$$\beta = \sqrt{1 + \frac{\lambda' \Delta}{\sqrt{\Delta^2 + v\_F^2 (q\_x^2 + k\_y^2)}}}, \quad \eta = \sqrt{1 - \frac{\lambda' \Delta}{\sqrt{\Delta^2 + v\_F^2 (q\_x^2 + k\_y^2)}}}. \tag{63}$$

Imposing the continuity conditions of *ψ<sup>I</sup>* and *ψI I* at the interface leads to the following system of equations:

$$
\mathfrak{a} + \mathfrak{a}r = \mathfrak{F}t,\tag{64}
$$

$$
\lambda \gamma e^{i\phi} - \lambda \gamma r e^{-i\phi} = \lambda' \eta t e^{i\theta\_l},
\tag{65}
$$

which solving them with respect to r and t gives

$$r = \frac{\lambda \varepsilon^{i\phi} - \lambda' \frac{a\eta}{\beta^\gamma \gamma} e^{i\theta\_l}}{\lambda' \frac{a\eta}{\beta^\gamma \gamma} e^{i\theta\_l} + \lambda e^{-i\phi}} \,\,\,\,\tag{66}$$

and

14 Graphene - Research and Applications

two regions then are:

and

*<sup>ψ</sup><sup>I</sup>* <sup>=</sup> <sup>1</sup> √2

*α* = ���� 1 +

*β* = ���� 1 +

system of equations:

 

�

�

which solving them with respect to r and t gives

*α*

 *e*

*<sup>ψ</sup>I I* <sup>=</sup> *<sup>t</sup>* √2

*λ*∆

*λ*′∆

*F*(*q*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup> *y*)

*λγe*

*F*(*k*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup> *y*)

∆<sup>2</sup> + *v*<sup>2</sup>

∆<sup>2</sup> + *v*<sup>2</sup>

*γλei<sup>φ</sup>*

as those of massless Dirac fermions in graphene.

Now that we have found the corresponding eigenfunctions of Hamiltonian (4.52), assuming an electron incident upon a step of height *V*0, we can write the single valley Hamiltonian as:

where *V*(**r**) = 0 for region I (*x* < 0) and for the region II (*x* > 0), massive Dirac fermions feel a electrostatic potential of hight *<sup>V</sup>*<sup>0</sup> with the kinetic energy *<sup>E</sup>* − *<sup>V</sup>*0. The wave functions in the

> *r* √2

  *α*

*γλei*(*π*−*φ*)

*i*(*qx x*+*ky y*)

<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*<sup>∆</sup> �

<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*′<sup>∆</sup> �

∆<sup>2</sup> + *v*<sup>2</sup>

∆<sup>2</sup> + *v*<sup>2</sup>

*F*(*k*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup> *y*)

*F*(*q*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup> *y*)

*α* + *αr* = *βt*, (64)

*βγ <sup>e</sup>iθ<sup>t</sup>* <sup>+</sup> *<sup>λ</sup>e*−*i<sup>φ</sup>* , (66)

*<sup>η</sup>teiθ<sup>t</sup>* , (65)

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) +

  *β*

, *γ* =

, *η* =

Imposing the continuity conditions of *ψ<sup>I</sup>* and *ψI I* at the interface leads to the following

*<sup>i</sup><sup>φ</sup>* <sup>−</sup> *λγre*−*i<sup>φ</sup>* <sup>=</sup> *<sup>λ</sup>*′

*<sup>r</sup>* <sup>=</sup> *<sup>λ</sup>ei<sup>φ</sup>* <sup>−</sup> *<sup>λ</sup>*′ *αη*

*λ*′ *αη*

����

����

*βγ <sup>e</sup>iθ<sup>t</sup>*

 *e*

*λ*′ *ηeiθ<sup>t</sup>*

where in order to make things more simple, the following abbreviations is introduced:

*<sup>H</sup>* = *vFσ***.***<sup>p</sup>* + <sup>∆</sup>*σ<sup>z</sup>* + *<sup>V</sup>*(**r**), (59)

 *e*

*<sup>i</sup>*(−*kx <sup>x</sup>*+*ky <sup>y</sup>*) (60)

, (61)

, (62)

. (63)

$$t = \frac{2\lambda\cos\phi}{\frac{\eta}{\gamma}\lambda'e^{i\theta\_l} + \frac{\beta}{\alpha}\lambda e^{-i\phi}}.\tag{67}$$

From (1.66) it is straightforward to show that R is:

$$R = \frac{N\_r - 2\lambda\lambda' S\_r \cos(\phi - \theta\_l)}{N\_r + 2\lambda\lambda' S\_r \cos(\phi + \theta\_l)},\tag{68}$$

where

$$\begin{split} N\_{r} &= \frac{\beta^{2}\gamma^{2} + a^{2}\eta^{2}}{\beta^{2}\gamma^{2}} \\ &= 2 \frac{E|V\_{0} - E| - \lambda\lambda'\Delta^{2}}{E|V\_{0} - E| - \lambda\lambda'\Delta^{2} - \lambda|V\_{0} - E|\Delta + \lambda'E\Delta} \\ &= 2 \frac{E|V\_{0} - E| - \lambda\lambda'\Delta^{2}}{(|V\_{0} - E| + \lambda'\Delta)(E - \lambda\Delta)} \end{split} \tag{69}$$

and

$$\begin{split} S\_r &= \frac{a\eta}{\beta\gamma} \\ &= \frac{E|V\_0 - E| - \lambda\lambda'\Delta^2 + \lambda'E\Delta - \lambda|V\_0 - E|\Delta}{E|V\_0 - E| - \lambda\lambda'\Delta^2 - \lambda'E\Delta + \lambda|V\_0 - E|\Delta} \\ &= \frac{(|V\_0 - E| + \lambda'\Delta)(E - \lambda\Delta)}{(|V\_0 - E| - \lambda'\Delta)(E + \lambda\Delta)} \end{split} \tag{70}$$

In the limit ∆ → 0 we get the same reflection as that of massless case. In the limit of no electrostatic potential we arrive at the logical result *R* = 0. This is important because we see later that for a special potential step in this limit R is not zero. Now one remaining problem is to calculate the transmission probability. So, considering equation (67) and:

$$j\_{\mathbf{x}}^{\mathrm{in}} = \lambda a \gamma \cos \phi\_{\prime} \quad j\_{\mathbf{x}}^{\prime} = -\lambda a \gamma \cos \phi\_{\prime} \quad j\_{\mathbf{x}}^{\dagger} = \lambda^{\prime} \eta \beta \cos \theta\_{\prime} \tag{71}$$

T is found to be:

$$\begin{split} T &= |t|^2 \frac{\lambda \lambda' \eta \beta \cos \theta\_t}{\mathfrak{a} \gamma \cos \phi} \\ &= \frac{4 \lambda \lambda' S\_l \cos \phi \cos \theta\_l}{N\_l + 2S\_l \lambda \lambda' \cos(\phi + \theta\_t)} \end{split} \tag{72}$$

where the following abbreviations is defined:

$$S\_{l} = \frac{\eta \beta}{a\gamma} = \left[\frac{v\_F^2 q^2}{\Delta^2 + v\_F^2 q^2} \frac{\Delta^2 + v\_F^2 k^2}{v\_F^2 k^2}\right]^{\frac{1}{2}}$$

$$= \frac{q}{k} \frac{E}{|V\_0 - E|} \tag{73}$$

10.5772/51980

19

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

**Figure 5.** An massive electron of energy *E* incident on a potential barrier of hight *V*<sup>0</sup> and thickness of about 50 *nm*. The opening gap in the all three zones are of the same value and therefore the tunneling phenomenon occurs in a symmetric

As depicted in the figure 5 there are three regions. The first is for *x* < 0 where the potential is equal to zero. The second region is for 0 < *x* < *D* where there is a electrostatic potential of hight *V*<sup>0</sup> and finally, the third region is defined for *x* > 0 and as well as the first region we have *<sup>V</sup>*<sup>0</sup> = 0. At this point, using equations of previous sections, we are able to write the wave functions in these three different regions in terms of incident and reflected waves. The

> *r* √2

> > *b* √2

> > > *e*

*aeiθ<sup>t</sup>* <sup>−</sup> *ηλ*′

In the third region we have only a transmitted wave and therefore the wave function in this

With the continuity of the spinors at the discontinuities, we arrive at the following set of

 *α λγei<sup>φ</sup>*

*<sup>i</sup><sup>φ</sup>* <sup>−</sup> *λγre*−*i<sup>φ</sup>* <sup>=</sup> *ηλ*′

 *α λγei*(*π*−*φ*)

 *β λ*′

*<sup>η</sup>ei*(*π*−*θt*)

 *e*

> *e*

*α* + *αr* = *βa* + *βb* (80)

*i*(−*kx x*+*ky y*)

*i*(−*qx x*+*ky y*)

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) (79)

*be*−*iθ<sup>t</sup>* (81)

. (77)

. (78)

barrier.

region is:

equations:

wave function in region I is then given by:

 *α λγei<sup>φ</sup>*

 *β λ*′ *ηeiθ<sup>t</sup> e*

*λγe*

 *e*

*<sup>ψ</sup>III* <sup>=</sup> *<sup>t</sup>* √2

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) +

*<sup>i</sup>*(*qx <sup>x</sup>*+*ky <sup>y</sup>*) +

*<sup>ψ</sup><sup>I</sup>* <sup>=</sup> <sup>1</sup> √2

*<sup>ψ</sup>I I* <sup>=</sup> *<sup>a</sup>* √2

In the second region we have:

and

$$\begin{split} N\_{l} &= \frac{\eta^{2}\alpha^{2} + \beta^{2}\gamma^{2}}{a^{2}\gamma^{2}} \\ &= 2\frac{E(E|V\_{0} - E| - \lambda\lambda'\Delta^{2})}{v\_{F}^{2}k^{2}|V\_{0} - E|}. \end{split} \tag{74}$$

At this point one can obtain *T*(0) as follows:

$$T(0) = 2\frac{v\_F^2|k\_\mathcal{X}||q\_\mathcal{X}|}{E|V\_0 - E| - \lambda\lambda'\Delta^2 + v\_F^2|k\_\mathcal{X}||q\_\mathcal{X}|}. \tag{75}$$

Note that *St* and *Nt* are positive. It is clear that in the case of *<sup>V</sup>*<sup>0</sup> → 0 and *<sup>V</sup>*<sup>0</sup> → <sup>∞</sup> T is one. Also note that in the limit of ∆ → 0, as:

$$E|V\_0 - E| = \upsilon\_F^2 |k\_x||q\_x|\_\prime \tag{76}$$

we see that probability is unity in agreement with result obtained for massless case. Another interesting result that expression for T shows is that probability is not independent of the band index contrary to the a gapless step that leaded to no independency to band index, *λ* and *<sup>λ</sup>*′ .

#### **6. The barrier case**

Opening nano-electronic opportunities for graphene requires a mass gap in it's energy spectrum just like a conventional semiconductor. In fact the lack of a bandgap on graphene, can limit graphene's uses in electronics because if there is no gaps in graphene spectrum one can't turn off a graphene-made transistor . In this section, motivated by mass production of graphene, we obtain the exact expression for transmission probability of massive Dirac fermions through a two dimensional potential barrier which can correspond to a n-p-n junction of graphene, and show that contrary to the case of massless Dirac fermions which results in complete transparency of the potential barrier for normal incidence, the probability transmission, T, in this case, apart from some resonance conditions that lead to the total transparency of the barrier, is smaller than one. An interesting result is that in the case of *qx* satisfy the relation *qxD* = *nπ*, where n is an integer, we again see that tunneling is easier for a barrier than a potential step, i.e the resonance tunneling is occurred.

10.5772/51980

**Figure 5.** An massive electron of energy *E* incident on a potential barrier of hight *V*<sup>0</sup> and thickness of about 50 *nm*. The opening gap in the all three zones are of the same value and therefore the tunneling phenomenon occurs in a symmetric barrier.

As depicted in the figure 5 there are three regions. The first is for *x* < 0 where the potential is equal to zero. The second region is for 0 < *x* < *D* where there is a electrostatic potential of hight *V*<sup>0</sup> and finally, the third region is defined for *x* > 0 and as well as the first region we have *<sup>V</sup>*<sup>0</sup> = 0. At this point, using equations of previous sections, we are able to write the wave functions in these three different regions in terms of incident and reflected waves. The wave function in region I is then given by:

$$\psi\_I = \frac{1}{\sqrt{2}} \begin{pmatrix} \mathfrak{a} \\ \lambda \gamma e^{i\phi} \end{pmatrix} e^{i(k\_{\bar{x}}x + k\_{\bar{y}}y)} + \frac{r}{\sqrt{2}} \begin{pmatrix} \mathfrak{a} \\ \lambda \gamma e^{i(\pi - \phi)} \end{pmatrix} e^{i(-k\_{\bar{x}}x + k\_{\bar{y}}y)}.\tag{77}$$

In the second region we have:

16 Graphene - Research and Applications

and

and *<sup>λ</sup>*′ .

**6. The barrier case**

where the following abbreviations is defined:

At this point one can obtain *T*(0) as follows:

Also note that in the limit of ∆ → 0, as:

*St* <sup>=</sup> *ηβ αγ* <sup>=</sup>

> = *q k*

 *v*<sup>2</sup> *Fq*2 ∆<sup>2</sup> + *v*<sup>2</sup> *Fq*2

*E* |*V*<sup>0</sup> − *<sup>E</sup>*|

*Nt* <sup>=</sup> *<sup>η</sup>*2*α*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*2*γ*<sup>2</sup> *α*2*γ*2

*<sup>E</sup>*(*E*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′

*v*2 *<sup>F</sup>*|*kx*||*qx*<sup>|</sup>

*<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′∆<sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup>

Note that *St* and *Nt* are positive. It is clear that in the case of *<sup>V</sup>*<sup>0</sup> → 0 and *<sup>V</sup>*<sup>0</sup> → <sup>∞</sup> T is one.

we see that probability is unity in agreement with result obtained for massless case. Another interesting result that expression for T shows is that probability is not independent of the band index contrary to the a gapless step that leaded to no independency to band index, *λ*

Opening nano-electronic opportunities for graphene requires a mass gap in it's energy spectrum just like a conventional semiconductor. In fact the lack of a bandgap on graphene, can limit graphene's uses in electronics because if there is no gaps in graphene spectrum one can't turn off a graphene-made transistor . In this section, motivated by mass production of graphene, we obtain the exact expression for transmission probability of massive Dirac fermions through a two dimensional potential barrier which can correspond to a n-p-n junction of graphene, and show that contrary to the case of massless Dirac fermions which results in complete transparency of the potential barrier for normal incidence, the probability transmission, T, in this case, apart from some resonance conditions that lead to the total transparency of the barrier, is smaller than one. An interesting result is that in the case of *qx* satisfy the relation *qxD* = *nπ*, where n is an integer, we again see that tunneling is easier for

a barrier than a potential step, i.e the resonance tunneling is occurred.

*<sup>E</sup>*|*V*<sup>0</sup> − *<sup>E</sup>*| = *<sup>v</sup>*<sup>2</sup>

*v*2

= 2

*T*(0) = 2

∆<sup>2</sup> + *v*<sup>2</sup> *Fk*2  1 2

, (73)

*<sup>F</sup>k*2|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*<sup>|</sup> . (74)

*<sup>F</sup>*|*kx*||*qx*|, (76)

. (75)

*v*2 *Fk*2

∆2)

*<sup>F</sup>*|*kx*||*qx*<sup>|</sup>

$$\psi\_{II} = \frac{a}{\sqrt{2}} \begin{pmatrix} \beta \\ \lambda' \eta e^{i\theta\_l} \end{pmatrix} e^{i(q\_1 \mathbf{x} + k\_y y)} + \frac{b}{\sqrt{2}} \begin{pmatrix} \beta \\ \lambda' \eta e^{i(\pi - \theta\_l)} \end{pmatrix} e^{i(-q\_1 \mathbf{x} + k\_y y)}.\tag{78}$$

In the third region we have only a transmitted wave and therefore the wave function in this region is:

$$
\psi\_{III} = \frac{t}{\sqrt{2}} \begin{pmatrix} a \\ \lambda \gamma e^{i\phi} \end{pmatrix} e^{i(k\_x x + k\_y y)} \tag{79}
$$

With the continuity of the spinors at the discontinuities, we arrive at the following set of equations:

$$
\alpha + \alpha r = \beta a + \beta b \tag{80}
$$

$$
\lambda \gamma e^{i\phi} - \lambda \gamma re^{-i\phi} = \eta \lambda' a e^{i\theta\_l} - \eta \lambda' b e^{-i\theta\_l} \tag{81}
$$

$$
\beta a e^{i q\_\ge D} + \beta b e^{-i q\_\ge D} = \alpha t e^{i k\_\ge D} \tag{82}
$$

10.5772/51980

21

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

. So in the normal

<sup>2</sup> <sup>−</sup> *λλ*′ sin *<sup>φ</sup>* sin *<sup>θ</sup>*)<sup>2</sup> . (94)

<sup>2</sup> + (*<sup>N</sup>* <sup>−</sup> <sup>2</sup>) sin2(*qxD*) (95)

<sup>2</sup> , (96)

*<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′∆<sup>2</sup> (97)

<sup>1</sup> <sup>−</sup> sin<sup>2</sup> *<sup>φ</sup>* cos2(*qxD*) (98)

Finally by multiplying t by it's complex conjugation, one can obtain the exact expression for

It is clear that in the Klein energy interval (0 <sup>&</sup>lt; *<sup>E</sup>* <sup>&</sup>lt; *<sup>V</sup>*0), *<sup>λ</sup>* and *<sup>λ</sup>*′ has opposite signs so that the term *N*/2 in the above expression is bigger than one and, therefore, we see that unlike to the case of massless Dirac fermions which results in complete transparency of the potential barrier for normal incidence, the transmission T for massive quasi-particles in gapped graphene is smaller than one something that is of interest in a graphene transistor. It is obvious that substituting ∆ with −∆ does not change the T, and hence the result for the

Now considering an electron incident on the barrier with propagation angle *φ* = 0, we know

*qxD* = *n*

*<sup>N</sup>* <sup>=</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>T</sup>*(*φ*) <sup>≃</sup> cos<sup>2</sup> *<sup>φ</sup>*

At this point it is so clear that the transmission depends on the sign of *λλ*′ = ±. In the other words, this equation for the same values of |*V*<sup>0</sup> − *<sup>E</sup>*|, depending on whether *<sup>E</sup>* is higher or smaller than *V*0, results in different values for *T*. The result that have not been revealed before. In the limit |*V*0| >> |*E*|, the exact expression obtained for transmission would be

which reveals that in this limit, *T*(0) is again smaller than one while in the case of *qxD* satisfying the condition *qxD* = *nπ*, with *n* an integer, we still have complete transparency. Furthermore from equations (90) to (93) it is clear that in the limit ∆ → 0, we get *N*/2 = 1 and, therefore, one arrives at the same expressions for *T*(*φ*) corresponding to the case of massless Dirac fermions i.e. equations (48) and (49). Notice that there is transmission resonances just like other barriers studied earlier. It is important to know that resonances occur when a p-n interface is in series with an n-p interface, forming a p-n-p or n-p-n junction.

*π*

*<sup>F</sup>*|*kx*||*qx*<sup>|</sup>

the probability transmission of massive electrons, T, as:

both Dirac points is the same, as it should be.

Now if the following condition is satisfied:

the equation for probability results in:

simplified to:

incidence probability reads:

*<sup>T</sup>*(*φ*) = cos2 *<sup>φ</sup>* cos<sup>2</sup> *<sup>θ</sup>*

that *<sup>θ</sup><sup>t</sup>* becomes 0 (*π*), depending on the positive (negative) sign of *<sup>λ</sup>*′

*<sup>T</sup>*(0) = <sup>2</sup>

*<sup>T</sup>*(0) = <sup>2</sup>

(cos *φ* cos *θ* cos(*qxD*))<sup>2</sup> + sin2(*qxD*)( *<sup>N</sup>*

$$
\eta \lambda' a e^{i\theta\_l + i q\_l D} - \eta \lambda' b e^{-i\theta\_l - i q\_l D} = \gamma \lambda t e^{i\phi + i k\_l D} \tag{83}
$$

Here in order to obtain the transmission *T* we first solve the above set of equations with respect to transmission amplitude t. So we first need to calculate the coefficients *r*,*a*, and *b*. From (82), *a* can be written as follows:

$$a = \frac{\alpha}{\beta} t e^{-iq\_\mathrm{i}D + ik\_\mathrm{i}D} - b e^{-2iq\_\mathrm{i}D},\tag{84}$$

which writing it with respect to transmission amplitude requires to plug b which one can obtain it using the equation (83) as:

$$b = \frac{te^{iq\_xD + ik\_xD}(\lambda' \frac{\alpha \eta}{\beta} e^{i\theta\_l} - \lambda \gamma e^{i\phi})}{2\lambda' \eta \cos \theta\_l},\tag{85}$$

into the corresponding equation for a. Rewriting (81) by the use of relation *α* + *αr* = *βa* + *βb* as:

$$2\lambda\cos\phi = a(\lambda'\frac{\eta}{\gamma}e^{i\theta\_l} + \lambda\frac{\beta}{\alpha}e^{-i\phi}) - b(\lambda'\frac{\eta}{\gamma}e^{-i\theta\_l} - \lambda\frac{\beta}{\alpha}e^{-i\phi}),\tag{86}$$

and then using the equations (85) and (86), the expression for transmission amplitude yields:

$$t = \frac{-4e^{-ik\_\lambda D} \lambda \lambda' \cos \phi \cos \theta}{\left[e^{iq\_\sharp D} (N - 2\lambda \lambda' \cos(\phi - \theta)) - e^{-iq\_\flat D} (N + 2\lambda \lambda' \cos(\phi + \theta))\right]'} \tag{87}$$

where

$$N = \frac{\eta \alpha}{\beta \gamma} + \frac{\beta \gamma}{\eta \alpha}. \tag{88}$$

It is straightforward to show that:

$$N = 2\frac{E|V\_0 - E| - \lambda\lambda'\Delta^2}{v\_F^2 kq},\tag{89}$$

where

$$E = \sqrt{\Delta^2 + v\_F^2 (k\_x^2 + k\_y^2)}\tag{90}$$

$$|V\_0 - E| = \sqrt{\Delta^2 + v\_F^2(q\_x^2 + k\_y^2)}\tag{91}$$

$$k = \sqrt{k\_x^2 + k\_y^2} \tag{92}$$

$$
\eta = \sqrt{q\_x^2 + k\_y^2}.\tag{93}
$$

Finally by multiplying t by it's complex conjugation, one can obtain the exact expression for the probability transmission of massive electrons, T, as:

$$T(\phi) = \frac{\cos^2 \phi \cos^2 \theta}{(\cos \phi \cos \theta \cos (q\_\mathbf{x} D))^2 + \sin^2 (q\_\mathbf{x} D)(\frac{N}{2} - \lambda \lambda' \sin \phi \sin \theta)^2}. \tag{94}$$

It is clear that in the Klein energy interval (0 <sup>&</sup>lt; *<sup>E</sup>* <sup>&</sup>lt; *<sup>V</sup>*0), *<sup>λ</sup>* and *<sup>λ</sup>*′ has opposite signs so that the term *N*/2 in the above expression is bigger than one and, therefore, we see that unlike to the case of massless Dirac fermions which results in complete transparency of the potential barrier for normal incidence, the transmission T for massive quasi-particles in gapped graphene is smaller than one something that is of interest in a graphene transistor. It is obvious that substituting ∆ with −∆ does not change the T, and hence the result for the both Dirac points is the same, as it should be.

Now considering an electron incident on the barrier with propagation angle *φ* = 0, we know that *<sup>θ</sup><sup>t</sup>* becomes 0 (*π*), depending on the positive (negative) sign of *<sup>λ</sup>*′ . So in the normal incidence probability reads:

$$T(0) = \frac{2}{2 + (N - 2)\sin^2(q\_\mathrm{x}D)}\tag{95}$$

Now if the following condition is satisfied:

18 Graphene - Research and Applications

*ηλ*′

<sup>2</sup>*<sup>λ</sup>* cos *<sup>φ</sup>* <sup>=</sup> *<sup>a</sup>*(*λ*′ *<sup>η</sup>*

From (82), *a* can be written as follows:

obtain it using the equation (83) as:

It is straightforward to show that:

as:

where

where

*aeiθt*+*iqxD* <sup>−</sup> *ηλ*′

*<sup>a</sup>* <sup>=</sup> *<sup>α</sup>*

*γ e <sup>i</sup>θ<sup>t</sup>* <sup>+</sup> *<sup>λ</sup> <sup>β</sup> α e*

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>4*e*−*ikxDλλ*′ cos *<sup>φ</sup>* cos *<sup>θ</sup>*

*N* = 2


*E* = 

*k* = *k*2 *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup>

*q* = *q*2 *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup>

Here in order to obtain the transmission *T* we first solve the above set of equations with respect to transmission amplitude t. So we first need to calculate the coefficients *r*,*a*, and *b*.

which writing it with respect to transmission amplitude requires to plug b which one can

<sup>2</sup>*λ*′*<sup>η</sup>* cos *<sup>θ</sup><sup>t</sup>*

<sup>−</sup>*iφ*) <sup>−</sup> *<sup>b</sup>*(*λ*′ *<sup>η</sup>*

into the corresponding equation for a. Rewriting (81) by the use of relation *α* + *αr* = *βa* + *βb*

and then using the equations (85) and (86), the expression for transmission amplitude yields:

*<sup>N</sup>* <sup>=</sup> *ηα*

∆<sup>2</sup> + *v*<sup>2</sup>

*βγ* <sup>+</sup> *βγ*

*<sup>E</sup>*|*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*| − *λλ*′

*v*2

*F*(*k*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup>

∆<sup>2</sup> + *v*<sup>2</sup>

*F*(*q*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup>

*<sup>β</sup> <sup>e</sup>iθ<sup>t</sup>* <sup>−</sup> *λγeiφ*)

*γ e*

[*eiqxD*(*<sup>N</sup>* <sup>−</sup> <sup>2</sup>*λλ*′ cos(*<sup>φ</sup>* <sup>−</sup> *<sup>θ</sup>*)) <sup>−</sup> *<sup>e</sup>*−*iqxD*(*<sup>N</sup>* <sup>+</sup> <sup>2</sup>*λλ*′ cos(*<sup>φ</sup>* <sup>+</sup> *<sup>θ</sup>*))] , (87)

∆2

<sup>−</sup>*iθ<sup>t</sup>* <sup>−</sup> *<sup>λ</sup> <sup>β</sup> α e*

*ηα* . (88)

*<sup>F</sup>kq* , (89)

*<sup>y</sup>*) (90)

*<sup>y</sup>* (92)

*<sup>y</sup>*. (93)

*<sup>y</sup>*) (91)

*<sup>b</sup>* <sup>=</sup> *teiqxD*+*ikxD*(*λ*′ *αη*

*<sup>β</sup>aeiqxD* <sup>+</sup> *<sup>β</sup>be*−*iqxD* <sup>=</sup> *<sup>α</sup>teikxD* (82)

*<sup>β</sup> te*−*iqxD*+*ikxD* <sup>−</sup> *be*−2*iqxD*, (84)

*be*−*iθt*−*iqxD* <sup>=</sup> *γλteiφ*+*ikxD* (83)

, (85)

<sup>−</sup>*iφ*), (86)

$$q\_{\chi}D = n\frac{\pi}{2},\tag{96}$$

the equation for probability results in:

$$T(0) = \frac{2}{N} = \frac{v\_F^2 |k\_x||q\_x|}{E|V\_0 - E| - \lambda \lambda' \Delta^2} \tag{97}$$

At this point it is so clear that the transmission depends on the sign of *λλ*′ = ±. In the other words, this equation for the same values of |*V*<sup>0</sup> − *<sup>E</sup>*|, depending on whether *<sup>E</sup>* is higher or smaller than *V*0, results in different values for *T*. The result that have not been revealed before. In the limit |*V*0| >> |*E*|, the exact expression obtained for transmission would be simplified to:

$$T(\phi) \simeq \frac{\cos^2 \phi}{1 - \sin^2 \phi \cos^2(q\_\chi D)}\tag{98}$$

which reveals that in this limit, *T*(0) is again smaller than one while in the case of *qxD* satisfying the condition *qxD* = *nπ*, with *n* an integer, we still have complete transparency. Furthermore from equations (90) to (93) it is clear that in the limit ∆ → 0, we get *N*/2 = 1 and, therefore, one arrives at the same expressions for *T*(*φ*) corresponding to the case of massless Dirac fermions i.e. equations (48) and (49). Notice that there is transmission resonances just like other barriers studied earlier. It is important to know that resonances occur when a p-n interface is in series with an n-p interface, forming a p-n-p or n-p-n junction.

#### **7. Transmission into spatial regions of finite mass**

In this section the transmission of massless electrons into some regions where the corresponding energy dispersion relation is not linear any more and exhibits a finite gap of ∆ is discussed. Thus, the mass of electrons there can be obtained from the relation *mv*<sup>2</sup> *<sup>F</sup>* <sup>=</sup> <sup>∆</sup>. Starting by looking at a two demential square potential step and after obtaining the probability of penetration of step by electrons, transmission of massless electrons into a region of finite mass is investigated and then see how it turns out to be applicable in a transistor composed of two pieces of graphene connected by a conventional semiconductor or linked by a nanotube.

#### **7.1. Tunnelling through a composed p-n junction**

In this section the scattering of an electron of energy *E* from a potential step of hight *V*<sup>0</sup> which allows massless electrons to acquire a finite mass in the region of the electrostatic potential is investigated(see Fig. 6). The electrostatic potential under the region of finite mass is:

$$V(\mathbf{r}) = \begin{cases} 0 & \mathbf{x} < 0 \\ V\_0 \ 0 < \mathbf{x} < D \\ 0 & \mathbf{x} > D \end{cases} \tag{99}$$

10.5772/51980

23

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

<sup>2</sup> and *R*:

<sup>1</sup> <sup>+</sup> *λλ*′*ηβcos*(*<sup>φ</sup>* <sup>+</sup> *<sup>θ</sup>t*) (106)

*<sup>x</sup>* <sup>=</sup> *<sup>λ</sup>* cos *<sup>φ</sup>* (108)

, (105)

<sup>|</sup>*V*<sup>0</sup> <sup>−</sup> *<sup>E</sup>*<sup>|</sup> (107)

<sup>2</sup> (109)

2. (110)

2, (111)

**Figure 6.** A special potential step of height *V*<sup>0</sup> and width *D* which massless electrons of energy E under it acquire a finite mass.

<sup>2</sup> <sup>=</sup> 2 cos2 *<sup>φ</sup>*

*<sup>R</sup>* <sup>=</sup> *rr*<sup>∗</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *λλ*′

*j in*

*j r*

*j t <sup>x</sup>* <sup>=</sup> *<sup>λ</sup>*′

regions. So by the use of relation (27) the following equation come outs:

1 − |*r*|

 *v*<sup>2</sup> *F*(*q*<sup>2</sup> *<sup>x</sup>* <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *y*)

> *v*2 *F*(*q*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup>

<sup>1</sup> + *λλ*′*ηβcos*(*<sup>φ</sup>* + *<sup>θ</sup>t*)

*<sup>y</sup>*) + <sup>∆</sup><sup>2</sup>

For obtaining the transmission probability we need to evaluate the *x*-component of

*<sup>x</sup>* <sup>=</sup> <sup>−</sup>*<sup>λ</sup>* cos *<sup>φ</sup>*|*r*<sup>|</sup>

Here notice that, using the probability conservation law and the fact that our problem is time independent and invariant along the *y*-direction, *jx*, then has the same values in the two

<sup>2</sup> <sup>=</sup> *λλ*′

*ηβ* cos *θt*|*t*|

*ηβ* cos *θ<sup>t</sup>* cos *<sup>φ</sup>* <sup>|</sup>*t*<sup>|</sup>

*ηβcos*(*φ* − *θt*)

= *vFq*

 1 2

Solving the above equations gives us the following expression for |*t*|


*ηβ* =

probability current in two regions. Using equation (24) we get:

and

where

Assuming an electron of energy E, propagating from the left, the wave functions then in the two zones can be written as:

$$\psi\_I = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i\phi} \end{pmatrix} e^{i(k\_x x + k\_y y)} + \frac{r}{\sqrt{2}} \begin{pmatrix} 1 \\ \\ \lambda e^{i(\pi - \phi)} \end{pmatrix} e^{i(-k\_x x + k\_y y)} \tag{100}$$

$$
\psi\_{II} = \frac{t}{\sqrt{2}} \begin{pmatrix} \beta \\\\ \lambda' \eta e^{i\theta\_l} \end{pmatrix} e^{i(q\_x x + k\_y y)} \tag{101}
$$

where

$$\beta = \sqrt{1 + \frac{\lambda' \Delta}{\sqrt{\Delta^2 + v\_F^2 (q\_x^2 + k\_y^2)}}}, \quad \eta = \sqrt{1 - \frac{\lambda' \Delta}{\sqrt{\Delta^2 + v\_F^2 (q\_x^2 + k\_y^2)}}},\tag{102}$$

and *r* and *t* are reflected and transmitted amplitudes, respectively. Applying the continuity conditions of the wave functions at *x* = 0 yields:

$$1 + r = \beta t \tag{103}$$

$$
\lambda e^{i\phi} - r\lambda e^{-i\phi} = \lambda' \eta t e^{i\theta\_l} \tag{104}
$$

**Figure 6.** A special potential step of height *V*<sup>0</sup> and width *D* which massless electrons of energy E under it acquire a finite mass.

Solving the above equations gives us the following expression for |*t*| <sup>2</sup> and *R*:

$$\left|t\right|^2 = \frac{2\cos^2\phi}{1 + \lambda\lambda'\eta\beta\cos(\phi + \theta\_l)},\tag{105}$$

and

(99)

*<sup>i</sup>*(−*kx <sup>x</sup>*+*ky <sup>y</sup>*) (100)

, (102)

*<sup>i</sup>*(*qx <sup>x</sup>*+*ky <sup>y</sup>*) (101)

20 Graphene - Research and Applications

or linked by a nanotube.

two zones can be written as:

*β* = ���� 1 +

where

*<sup>ψ</sup><sup>I</sup>* <sup>=</sup> <sup>1</sup> √2   1

 *e*

*<sup>ψ</sup>I I* <sup>=</sup> *<sup>t</sup>* √2

*λ*′∆

*F*(*q*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup> *y*)

*λe*

*<sup>i</sup><sup>φ</sup>* − *<sup>r</sup>λ<sup>e</sup>*

∆<sup>2</sup> + *v*<sup>2</sup>

�

conditions of the wave functions at *x* = 0 yields:

*λei<sup>φ</sup>*

*mv*<sup>2</sup>

**7. Transmission into spatial regions of finite mass**

**7.1. Tunnelling through a composed p-n junction**

In this section the transmission of massless electrons into some regions where the corresponding energy dispersion relation is not linear any more and exhibits a finite gap of ∆ is discussed. Thus, the mass of electrons there can be obtained from the relation

*<sup>F</sup>* <sup>=</sup> <sup>∆</sup>. Starting by looking at a two demential square potential step and after obtaining the probability of penetration of step by electrons, transmission of massless electrons into a region of finite mass is investigated and then see how it turns out to be applicable in a transistor composed of two pieces of graphene connected by a conventional semiconductor

In this section the scattering of an electron of energy *E* from a potential step of hight *V*<sup>0</sup> which allows massless electrons to acquire a finite mass in the region of the electrostatic potential is investigated(see Fig. 6). The electrostatic potential under the region of finite mass is:

> 0 x < 0 *V*<sup>0</sup> 0 < x < *D* 0 x > *D*

Assuming an electron of energy E, propagating from the left, the wave functions then in the

*r* √2

> *e*

  1

 *e*

*<sup>λ</sup>ei*(*π*−*φ*)

<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*′<sup>∆</sup> �

∆<sup>2</sup> + *v*<sup>2</sup>

*F*(*q*<sup>2</sup> *<sup>x</sup>* + *<sup>k</sup>*<sup>2</sup> *y*)

1 + *r* = *βt* (103)

*<sup>η</sup>teiθ<sup>t</sup>* (104)

*V*(**r**) =

 

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) +

  *β*

, *η* =

and *r* and *t* are reflected and transmitted amplitudes, respectively. Applying the continuity

<sup>−</sup>*i<sup>φ</sup>* = *λ*′

����

*λ*′ *ηeiθ<sup>t</sup>*

$$R = rr^\* = \frac{1 - \lambda \lambda' \eta \beta \cos(\phi - \theta\_t)}{1 + \lambda \lambda' \eta \beta \cos(\phi + \theta\_t)}\tag{106}$$

where

$$\eta \beta = \left[ \frac{v\_F^2 (q\_\chi^2 + k\_y^2)}{v\_F^2 (q\_\chi^2 + k\_y^2) + \Delta^2} \right]^{\frac{1}{2}} = \frac{v\_F \eta}{|V\_0 - E|} \tag{107}$$

For obtaining the transmission probability we need to evaluate the *x*-component of probability current in two regions. Using equation (24) we get:

$$j\_{\rm x}^{\dot{m}} = \lambda \cos \phi \tag{108}$$

$$f\_{\mathbf{x}}^{r} = -\lambda \cos \phi |r|^2 \tag{109}$$

$$j\_\mathbf{x}^t = \lambda^t \eta \beta \cos \theta\_t |t|^2. \tag{110}$$

Here notice that, using the probability conservation law and the fact that our problem is time independent and invariant along the *y*-direction, *jx*, then has the same values in the two regions. So by the use of relation (27) the following equation come outs:

$$|1 - |r|^2 = \frac{\lambda \lambda' \eta \beta \cos \theta\_t}{\cos \phi} |t|^2 \,\text{\AA} \tag{111}$$

which once again shows that the probability, T, is not given by |*t*| <sup>2</sup> and instead is:

$$T = \frac{\lambda \lambda' \eta \beta \cos \theta\_t}{\cos \phi} |t|^2. \tag{112}$$

10.5772/51980

25

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

, (119)

<sup>1</sup> <sup>+</sup> *cos<sup>φ</sup>* (120)

**Figure 7.** An massless electron of energy E incident (from the left) on a potential barrier of height *V*<sup>0</sup> and width D, which acquires a finite mass under the electrostatic potential, due to the presence of a gap of 2∆ in the region II. The effective mass

which shows that probability always remains smaller than one, as there is no way for k and q to be equal4. Turning our attention back to equation (113), we see that in the limit ∆ → 0

> *<sup>T</sup>* <sup>=</sup> <sup>2</sup>*λλ*′ cos *<sup>θ</sup><sup>t</sup>* cos *<sup>φ</sup>* 1 + *λλ*′*cos*(*φ* + *θ*)

which is just the transmission of massless Dirac fermions through a p-n junction in gapless graphene. This expression now reveals in the limit *<sup>V</sup>*<sup>0</sup> >> *<sup>E</sup>* ≈ <sup>∆</sup> it can be simplified to the

<sup>1</sup> <sup>+</sup> *cos<sup>φ</sup>* , *<sup>R</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> cos *<sup>φ</sup>*

which show that for normal incidence the transmission and reflection probability are unity

Here, before proceeding to some numerical calculations in order to depict consequences that the *π* phase change might have on the probability, I attract the reader's attention to this fact that, the phase change of the wave function in momentum space is equivalent to the rotation of momentum vector, **q** by 180 degree, meaning that the direction of momentum and group velocity is antiparallel which itself lead to negative refraction in graphene reported by Cheianov [26,27]. As it clear for imaginary values of *qx* an evanescent wave is created in the

Now, before ending, in order to emphasize on the importance of the *π*-phase change mentioned earlier some numerical calculations depicting the transmission probability is shown in Fig. 8 which reveal a perceptible difference between result obtained based on considering the *π* − *shi f t* and those obtained if one ignores it. As it is clear for an electron of energy *<sup>E</sup>* = <sup>85</sup>*meV*, barrier thickness of 100*nm* and height of *<sup>V</sup>*<sup>0</sup> = <sup>200</sup>*meV* the probability gets smaller values if the extra phase is not considered. This means that considering the

of electron in this region is then *m* = ∆/*v*<sup>2</sup>

following equation

and zero, respectively.

one arrives at the following solution for T:

zone *I* and a total reflection is observed.

*F*

*<sup>T</sup>* <sup>=</sup> 2 cos *<sup>φ</sup>*

<sup>4</sup> There is no need to say that when there is no electrostatic potential *qx* is positive

The probability, therefore, is given by:

$$T(\phi) = \frac{2\lambda\lambda^{\prime}\eta\beta\cos\theta\_{t}\cos\phi}{1 + \lambda\lambda^{\prime}\eta\beta\cos(\phi + \theta\_{t})}.\tag{113}$$

This result shows that the relation *T*(*φ*) = *T*(−*φ*). Thus, the induced gap in graphene spectrum has nothing to do with relation this relation. We now turn our attention to the case in which an electron is incident perpendicularly upon the step. The probability for this electron to penetrate the step is:

$$T(0) = \frac{2\eta\beta}{1 + \eta\beta}$$

$$\eta = \frac{2v\_F|q\_x|}{|V\_0 - E| + v\_F|q\_x|} \,\prime \tag{114}$$

which shows there is no way for the electron to pass into the step with probability equal to one. However if we consider a potential step which is high enough so that we'll be able to write

$$|V\_0 - E| = \sqrt{v\_F^2 q\_x^2 + \Delta^2} \approx v\_F |q\_x|\_{\prime} \tag{115}$$

we see the step becomes transparent. So by increasing the potential's hight, more electrons can pass through the step. Notice that probability is independent of *λλ*′ unlike to the result (72) [19]. Also note that in the limit ∆ → 0, *qx* we can write:

$$v\_F|q\_x| = |V\_0 - E|\tag{116}$$

which immediately gives *T*(0) as:

$$T(0) = 1,\tag{117}$$

Also note that since for normal incidence we have *<sup>E</sup>* = *vFkx*, from the equation (114) it is evident that in the case of no electrostatic potential (*V*<sup>0</sup> = 0) we get:

$$T = \frac{2q\_{\text{x}}}{k\_{\text{x}} + q\_{\text{x}}} \quad , \quad R = \frac{k\_{\text{x}} - q\_{\text{x}}}{k\_{\text{x}} + q\_{\text{x}}} \, , \tag{118}$$

10.5772/51980

22 Graphene - Research and Applications

The probability, therefore, is given by:

electron to penetrate the step is:

which immediately gives *T*(0) as:

write

which once again shows that the probability, T, is not given by |*t*|

*<sup>T</sup>* <sup>=</sup> *λλ*′

*<sup>T</sup>*(*φ*) = <sup>2</sup>*λλ*′

*ηβ* cos *θ<sup>t</sup>* cos *<sup>φ</sup>* <sup>|</sup>*t*<sup>|</sup>

This result shows that the relation *T*(*φ*) = *T*(−*φ*). Thus, the induced gap in graphene spectrum has nothing to do with relation this relation. We now turn our attention to the case in which an electron is incident perpendicularly upon the step. The probability for this

1 + *ηβ*

which shows there is no way for the electron to pass into the step with probability equal to one. However if we consider a potential step which is high enough so that we'll be able to

we see the step becomes transparent. So by increasing the potential's hight, more electrons can pass through the step. Notice that probability is independent of *λλ*′ unlike to the result

Also note that since for normal incidence we have *<sup>E</sup>* = *vFkx*, from the equation (114) it is

, *<sup>R</sup>* <sup>=</sup> *kx* <sup>−</sup> *qx*

*kx* + *qx*

*<sup>T</sup>*(0) = <sup>2</sup>*ηβ*

 *v*2 *Fq*2


evident that in the case of no electrostatic potential (*V*<sup>0</sup> = 0) we get:

*<sup>T</sup>* <sup>=</sup> <sup>2</sup>*qx kx* + *qx*

(72) [19]. Also note that in the limit ∆ → 0, *qx* we can write:

<sup>=</sup> <sup>2</sup>*vF*|*qx*<sup>|</sup> |*V*<sup>0</sup> − *<sup>E</sup>*| + *vF*|*qx*|

*ηβ* cos *θ<sup>t</sup>* cos *φ* <sup>1</sup> + *λλ*′*ηβcos*(*<sup>φ</sup>* + *<sup>θ</sup>t*) <sup>2</sup> and instead is:

2. (112)

. (113)

, (114)

*<sup>x</sup>* + <sup>∆</sup><sup>2</sup> ≈ *vF*|*qx*|, (115)

*vF*|*qx*| = |*V*<sup>0</sup> − *<sup>E</sup>*| (116)

*T*(0) = 1, (117)

, (118)

**Figure 7.** An massless electron of energy E incident (from the left) on a potential barrier of height *V*<sup>0</sup> and width D, which acquires a finite mass under the electrostatic potential, due to the presence of a gap of 2∆ in the region II. The effective mass of electron in this region is then *m* = ∆/*v*<sup>2</sup> *F*

which shows that probability always remains smaller than one, as there is no way for k and q to be equal4. Turning our attention back to equation (113), we see that in the limit ∆ → 0 one arrives at the following solution for T:

$$T = \frac{2\lambda\lambda^{\prime}\cos\theta\_{\ell}\cos\phi}{1 + \lambda\lambda^{\prime}\cos(\phi + \theta)^{\prime}}\tag{119}$$

which is just the transmission of massless Dirac fermions through a p-n junction in gapless graphene. This expression now reveals in the limit *<sup>V</sup>*<sup>0</sup> >> *<sup>E</sup>* ≈ <sup>∆</sup> it can be simplified to the following equation

$$T = \frac{2\cos\phi}{1 + \cos\phi} \quad , \quad R = \frac{1 - \cos\phi}{1 + \cos\phi} \tag{120}$$

which show that for normal incidence the transmission and reflection probability are unity and zero, respectively.

Here, before proceeding to some numerical calculations in order to depict consequences that the *π* phase change might have on the probability, I attract the reader's attention to this fact that, the phase change of the wave function in momentum space is equivalent to the rotation of momentum vector, **q** by 180 degree, meaning that the direction of momentum and group velocity is antiparallel which itself lead to negative refraction in graphene reported by Cheianov [26,27]. As it clear for imaginary values of *qx* an evanescent wave is created in the zone *I* and a total reflection is observed.

Now, before ending, in order to emphasize on the importance of the *π*-phase change mentioned earlier some numerical calculations depicting the transmission probability is shown in Fig. 8 which reveal a perceptible difference between result obtained based on considering the *π* − *shi f t* and those obtained if one ignores it. As it is clear for an electron of energy *<sup>E</sup>* = <sup>85</sup>*meV*, barrier thickness of 100*nm* and height of *<sup>V</sup>*<sup>0</sup> = <sup>200</sup>*meV* the probability gets smaller values if the extra phase is not considered. This means that considering the

<sup>4</sup> There is no need to say that when there is no electrostatic potential *qx* is positive

10.5772/51980

27

Electronic Tunneling in Graphene http://dx.doi.org/10.5772/51980

[3] D. Dragoman, ArXiv quant ph/0701083.

[4] A. Hansen and F. ravndal, Phys. scr. 23, 1036-1402 (1981)

[6] B. Huard et al., Phys. Rev. Lett. 98, 236803 (2007).

den Brink, Phys. Rev. B 76, 73103 (2007).

[9] J. Zupan, Phys. Rev. B 6, 2477 (1972).

(2006).

London (1931).

Lett. (2008) .

[7] N. Stander et al., Phys. Rev. Lett. 102, 026807 (2009).

and Wenhui Duan, NanoLetters 6, 1469 (2007).

[16] C. Millot, A. J. Stone, Mol. Phys. 77, 439 (1992).

[17] E. M. Mas et al., J. Chem. Phys. 113, 6687 (2000).

[18] A. Castro-Neto et al., Rev. Mod. Phys. 81, 109 (2009).

(Colorado Associated University Press, (1972).

[25] A. Shytov et al. , Phys Rev Lett, 10, 101 (2008).

[5] P. Krekora, Q. su, and R. Grobe, Phys. Rev. Lett. 92,040406-4 (2004).

[8] Gianluca Giovannetti, Petr A. Khomyakov, Geert Brocks, Paul J. Kelly, and Jeroen van

[11] Young-Woo Son, Marvin L. Cohen, and Steven G. Louie, Phys. Rev. Lett. 97, 216803

[12] Qimin Yan, Bing Huang, Jie Yu, Fawei Zheng, Ji Zang, Jian Wu, Bing-Lin Gu, Feng Liu,

[13] Katsnelson, M. I., K. S. Novoselov, and A. K. Geim, Nature Physics 2, 620 (2006).

[14] Katsnelson, M. I., and K. S. Novoselov, Sol. Stat. Comm. 143, 3 (2007).

[19] Setare M R and Jahani D 2010 J. Phys.: Condens. Matter 22 245503

[20] G. Gamow, Quantum theory of atomic nucleus , Z. f. Phys. 51, 204 (1928).

[21] G. Gamow, Quantum theory of nuclear disintegration , Nature 122. 805 (1928).

[22] G. Gamow, Constitution of Atomic Nuclei and Radioactivity (Oxford University Press,

[23] Leon Rosenfeld in Cosmology, Fusion and Other Matters, Edited by F. Reines

[24] A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C. N. Lau, Nano

[15] M. Allesch, E. Schwegler, F. Gygi, G. Galli, J. Chem. Phys. 120, 5192 (2004).

[10] J. Viana Gomes and N. M. R. Peres, J. Phys.: Condens. Matter 20, 325221 (2008).

**Figure 8.** left: Transmission probability as a functions of incident angle for an electron of energy *E* = 85*meV*, *D* = 100*nm* and *V*<sup>0</sup> = 200*meV*. Right: Transmission in gapped graphene for gap value of 20*meV* as a functions of incident angle for an electron of energy *E* = 85*meV*, *D* = 100*nm* and *V*<sup>0</sup> = 200*meV*.

*Buttiker* ¨ formula [28] for conductivity lower conductance is predicted in absence of the extra phase. As it is clear the chance for an electron to penetrate the barrier increases if one chooses the appropriate wave function in the barrier.

The potential application of the theory of extra *π* phase consideration introduced in the previous sections [19] is that we can have higher conductivity in graphene-based electronic devices and also the results of this work is important in combinations of graphene flakes attached with different energy bands in order to get different kind of n-p-n junctions for different uses. Notice that for nanoelectronic application of graphene the existence of a mass gap in graphene's spectrum is essential because it leads to smaller than one transmission which is of most important for devices such as transistors and therefore the results derived in this work concerning gapped graphene could be applicable in nanoelectronic applications of graphene.

In the end of this chapter I would like to remind that one important result that obtained is that Klein paradox is not a paradox at all. More precisely, it was demonstrated theoretically that the reflection and transmission coefficients of a step barrier are both positive and less than unity, and that the hypothesis of particle-antiparticle pair production at the potential step is not necessary as the experimental evidences confirm this conclusion [29].

## **Author details**

#### Dariush Jahani

Young Researchers Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

## **8. References**


[3] D. Dragoman, ArXiv quant ph/0701083.

24 Graphene - Research and Applications

of energy *E* = 85*meV*, *D* = 100*nm* and *V*<sup>0</sup> = 200*meV*.

the appropriate wave function in the barrier.

of graphene.

**Author details** Dariush Jahani

**8. References**

[1] O. Klein. Z. Phys. 53. 157 (1929).

**Figure 8.** left: Transmission probability as a functions of incident angle for an electron of energy *E* = 85*meV*, *D* = 100*nm* and *V*<sup>0</sup> = 200*meV*. Right: Transmission in gapped graphene for gap value of 20*meV* as a functions of incident angle for an electron

*Buttiker* ¨ formula [28] for conductivity lower conductance is predicted in absence of the extra phase. As it is clear the chance for an electron to penetrate the barrier increases if one chooses

The potential application of the theory of extra *π* phase consideration introduced in the previous sections [19] is that we can have higher conductivity in graphene-based electronic devices and also the results of this work is important in combinations of graphene flakes attached with different energy bands in order to get different kind of n-p-n junctions for different uses. Notice that for nanoelectronic application of graphene the existence of a mass gap in graphene's spectrum is essential because it leads to smaller than one transmission which is of most important for devices such as transistors and therefore the results derived in this work concerning gapped graphene could be applicable in nanoelectronic applications

In the end of this chapter I would like to remind that one important result that obtained is that Klein paradox is not a paradox at all. More precisely, it was demonstrated theoretically that the reflection and transmission coefficients of a step barrier are both positive and less than unity, and that the hypothesis of particle-antiparticle pair production at the potential

Young Researchers Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

step is not necessary as the experimental evidences confirm this conclusion [29].

[2] A. Calogeracos and N. Dombey, Contemp. Phys. 40, 313 (1999).


**Chapter 2**

**Provisional chapter**

**Localised States of Fabry-Perot Type in Graphene**

**Localised States of Fabry-Perot Type in Graphene**

Graphene has been spoken of as a "'wonder material"' and described as paradigm shifting in the field of condensed matter physics [1]. The exceptional behavior of single layer graphene is down to its charge carriers being massless, relativistic particles. The anomalous behavior of graphene and its low energy excitation spectrum, implies the emergence of novel electronic characteristics. For example, in graphene-superconductor-graphene junctions specular Andreev reflections occur [1] and in graphene p-n junctions a Veselago lens for electrons has been outlined [2]. It is clear that by incorporating graphene into new and old designs that new physics and applications almost always emerges. Here we investigate Fabry-Perot like localized states in graphene mono and bi-layer graphene. As one will no doubt appreciate, there are many overlaps in the analysis of graphene with the studies of electron transport and light propagation. When we examine the ballistic regime we see that the scattering of electrons by potential barriers is also described in terms of transmission, reflection and refraction profiles; in analogy to any wave phenomenon. Except that there is no counterpart in normal materials to the exceptional quality at which these occur, with electrons capable of tunneling through a potential barrier of height larger than its energy with a probability of one - Klein tunneling. So, normally incident electrons in graphene are perfectly transmitted in analogy to the Klein paradox of relativistic quantum mechanics. A tunable graphene barrier is described in [3] where a local back-gate and a top-gate controlled the carrier density in the bulk of the graphene sheet. The graphene flake was covered in poly-methyl-methacrylate (PMMA) and the top-gate induced the potential barrier. In this work they describe junction configurations associated with the carrier types (p, for holes and n for electrons) and found sharp steps in resistance as the boundaries between n-n-n and n-p-n or p-n-p configurations were crossed. Ballistic transport was examined in the

> ©2012 Zalipaev et al., licensee InTech. This is an open access chapter 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. © 2013 Zalipaev et al.; licensee InTech. This is an open access article 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.

© 2013 Zalipaev et al., licensee InTech. This is a paper 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.

V. V. Zalipaev, D. M. Forrester, C. M. Linton and

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Nano-Ribbons**

**Nano-Ribbons**

F. V. Kusmartsev

**1. Introduction**

10.5772/52267

http://dx.doi.org/10.5772/52267

V.V. Zalipaev, D.M. Forrester, C.M. Linton and F.V. Kusmartsev

[29] J.R. Williams, L. DiCarlo, C.M. Marcus, "Quantum Hall effect in a gate-controlled p-n junction of graphene", Science 317, 638-641 (2007).

**Provisional chapter**

## **Localised States of Fabry-Perot Type in Graphene Nano-Ribbons Localised States of Fabry-Perot Type in Graphene Nano-Ribbons**

V. V. Zalipaev, D. M. Forrester, C. M. Linton and F. V. Kusmartsev V.V. Zalipaev, D.M. Forrester, C.M. Linton and F.V. Kusmartsev

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52267 10.5772/52267

## **1. Introduction**

26 Graphene - Research and Applications

28 New Progress on Graphene Research

University Press)

[26] V.V. Cheianov, V. Falko, B.L. Altshuler, Science 315, 1252 (2007).

[28] Datta S 1995 Electronic Transport in Mesoscopic Systems (Cambridge: Cambridge

[29] J.R. Williams, L. DiCarlo, C.M. Marcus, "Quantum Hall effect in a gate-controlled p-n

[27] V.V. Cheianov, V. Falko, Phys. Rev. B 74, 041403 (2006).

junction of graphene", Science 317, 638-641 (2007).

Graphene has been spoken of as a "'wonder material"' and described as paradigm shifting in the field of condensed matter physics [1]. The exceptional behavior of single layer graphene is down to its charge carriers being massless, relativistic particles. The anomalous behavior of graphene and its low energy excitation spectrum, implies the emergence of novel electronic characteristics. For example, in graphene-superconductor-graphene junctions specular Andreev reflections occur [1] and in graphene p-n junctions a Veselago lens for electrons has been outlined [2]. It is clear that by incorporating graphene into new and old designs that new physics and applications almost always emerges. Here we investigate Fabry-Perot like localized states in graphene mono and bi-layer graphene. As one will no doubt appreciate, there are many overlaps in the analysis of graphene with the studies of electron transport and light propagation. When we examine the ballistic regime we see that the scattering of electrons by potential barriers is also described in terms of transmission, reflection and refraction profiles; in analogy to any wave phenomenon. Except that there is no counterpart in normal materials to the exceptional quality at which these occur, with electrons capable of tunneling through a potential barrier of height larger than its energy with a probability of one - Klein tunneling. So, normally incident electrons in graphene are perfectly transmitted in analogy to the Klein paradox of relativistic quantum mechanics. A tunable graphene barrier is described in [3] where a local back-gate and a top-gate controlled the carrier density in the bulk of the graphene sheet. The graphene flake was covered in poly-methyl-methacrylate (PMMA) and the top-gate induced the potential barrier. In this work they describe junction configurations associated with the carrier types (p, for holes and n for electrons) and found sharp steps in resistance as the boundaries between n-n-n and n-p-n or p-n-p configurations were crossed. Ballistic transport was examined in 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. © 2013 Zalipaev et al.; licensee InTech. This is an open access article 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. © 2013 Zalipaev et al., licensee InTech. This is a paper 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.

©2012 Zalipaev et al., licensee InTech. This is an open access chapter distributed under the terms of the

limits of sharp and smooth potential steps. The PMMA is a transparent thermoplastic that has also been used to great effect in proving that graphene retains its 2D properties when embedded in a polymer heterostructure [4]. The polymers can be made to be sensitive to a specific stimulus that leads to a change in the conductance of the underlying graphene [4] and it is entirely likely that graphene based devices of the future will be hybrids including polymers that can control the carrier charge density. In [5] an experiment was performed to create a n-p-n junction to examine the ballistic regime. Oscillations in the conductance showed up as interferences between the two p-n interfaces and a Fabry-Perot resonator in graphene was created. When there was no magnetic field applied, two consecutive reflections on the p-n interfaces occurred with opposite angles, whereas for a small magnetic field the electronic trajectories bent. Above about 0.3 Tesla the trajectories bent sufficiently to lead to the occurrence of two consecutive reflections with the same incident angle and a *π*-shift in the phase of the electron. Thus, a half period shift in the interference fringes was witnessed and evidence of perfect tunneling at normal incidence accrued.

10.5772/52267

31

http://dx.doi.org/10.5772/52267

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

The oscillations arise due to an interference of an electron ballistic transport in the p-n-p junction, i.e. from Fabry-Perot interference of the electron and hole wave functions comprised between the two p-n interfaces. Thus, the holes or electrons in the top-gated region are multiply reflected between the two interfaces, interfering to give rise to standing waves, similar to those observed in carbon nanotubes [12] or standard graphene devices [13]. Modulations in the charge density distribution change the Fermi wavelength of the charge carriers, which in turn is altering the interference patterns and giving rise to the resistance

In the present work we consider a simplest model of the Fabry-Perot interferometer, which is in fact the p-n-p or n-p-n junction formed by a one dimensional potential. We develop an exact quasi-classical theory of such a system and study the associated Fabry-Perot

Although graphene is commonly referred to as the "'carbon flatland"' there has been a feeling of discontent amongst some that the Mermin-Wagner theorem appeared to be contradicted. However, recent work shows that the buckling of the lattice can give rise to a stable 3D structure that is consistent with this theorem [14]. In what follows we present the general methodology for analysis of graphene nanoribbons using semiclassical techniques that maintain the assumption of a flat lattice. However, it should be mentioned that the effects found from these techniques are powerful in aiding our understanding of potential barriers and are an essential tool for the developing area of graphene barrier engineering. The natural state of graphene to accommodate defects or charged impurities is important for applications. The p-n interfaces described above may be capable of guiding plasmons and to create the electrical analogues of optical devices to produce controllable indices of refraction

In Part *I* of this chapter we investigate the use of powerful semiclassical methods to analyze the relativistic electron and hole tunneling in graphene through a smooth potential barrier. We make comparison to the rectangular barrier. In both cases the barrier is generated as a result of an electrostatic potential in the ballistic regime. The transfer matrix method is employed in complement to the adiabatic WKB approximation for the Dirac system. Crucial to this method of approximation for the smooth barrier problem, when there is a skew electron incidence, is careful consideration of four turning points. These are denoted by *xi*, *i* = 1, 2, 3, 4 and lie in the domain of the barrier. The incident electron energy in this scattering problem belongs to the middle part of the segment [0, *U*0], where *U*<sup>0</sup> is the height of the barrier, and essentially the incident parameter *py* should be large enough to allow

Therefore, between the first two turning points, *x*<sup>1</sup> and *x*2, and also between the next two, *x*<sup>3</sup> and *x*<sup>4</sup> there is no coalescence. Two columns of total internal reflection occur which have solutions that grow and decay exponentially. Looking away from the close vicinity of the asymptotically small boundary layers of *xi*, there exists five domains with WKB type solutions (See Fig. 2): three with oscillatory behavior and two exhibiting asymptotics that are exponentially growing and decaying. Combining these five solutions is done through applying matched asymptotics techniques (see [16]) to the so-called effective Schrödinger equation that is equivalent to the Dirac system (see [17], [18]). This combinatorial procedure generates the WKB formulas that give the elements of the transfer matrix. This transfer

oscillations.

[15].

interference in the electron or hole transport.

normal and quasi-normal incidence.

Quantum interference effects are one of the most pronounced displays of the power of wave quantum mechanics. As an example, the wave nature of light is usually clearly demonstrated with the Fabry-Perot interferometers. Similar interferometers may be used in quantum mechanics to demonstrate the wave nature of electrons and other quantum mechanical particles. For electrons they were first demonstrated in graphene hetero-junctions formed by the application of a top gate voltage [6]. These were simple devices consisting mainly of the resonant cavity, and with transport channels attached. These devices exhibited quantum interference in the regular resistance oscillations that arose when the gate voltage changed.

Within the conventional Fabry-Perot model [7, 8], the resistance peaks correspond to minima in the overall transmission coefficient. The peak separation can be approximated by the condition 2*kF L* = 2*πn*. The charge accumulates a phase shift of 2*π* after completing a single lap (the round-trip) 2*L* in the resonant cavity, where *kF* is the Fermi wave vector of the charges, and *L* is the length of the Fabry-Perot cavity. This is the Fabry-Perot-like resonance condition: the fundamental resonance occurs when half the wavelength of the electron mode fits inside the p-n-p junction representing the Fabry-Perot cavity.

The simplest electron cavity, but still very effective, for the Fabry-Perot resonator may be formed by two parallel metallic wire-like contacts deposited on graphene [9]. There in a simple two terminal graphene structure there are clearly resolved Fabry-Perot oscillations. These have been observed in sub-100 nm devices. With a decrease of the size of the graphene region in these devices, the characteristics of the electron transport changes. Then the channel-dominated diffusive regime is transferred into the contact-dominated ballistic regime. This normally indicates that when the size of the cavity is about 100 nm or less the Fabry-Perot interference may be clearly resolved. The similar Fabry-Perot interferometer for Dirac electrons has been recently developed from carbon nanotubes [10].

Earlier work on the resistance oscillations as a function of the applied gate voltage led to their observation in the p-n-p junctions [6, 11]. It was first reported by Young and Kim [6], but the more pronounced observations of the Fabry-Perot oscillations have been made in the Ref. [11]. There high-quality n-p-n junctions with suspended top gates have been fabricated. They indeed display clear Fabry-Perot resistance oscillations within a small cavity formed by the p-n interfaces.

The oscillations arise due to an interference of an electron ballistic transport in the p-n-p junction, i.e. from Fabry-Perot interference of the electron and hole wave functions comprised between the two p-n interfaces. Thus, the holes or electrons in the top-gated region are multiply reflected between the two interfaces, interfering to give rise to standing waves, similar to those observed in carbon nanotubes [12] or standard graphene devices [13]. Modulations in the charge density distribution change the Fermi wavelength of the charge carriers, which in turn is altering the interference patterns and giving rise to the resistance oscillations.

2 Graphene - Research and Applications

limits of sharp and smooth potential steps. The PMMA is a transparent thermoplastic that has also been used to great effect in proving that graphene retains its 2D properties when embedded in a polymer heterostructure [4]. The polymers can be made to be sensitive to a specific stimulus that leads to a change in the conductance of the underlying graphene [4] and it is entirely likely that graphene based devices of the future will be hybrids including polymers that can control the carrier charge density. In [5] an experiment was performed to create a n-p-n junction to examine the ballistic regime. Oscillations in the conductance showed up as interferences between the two p-n interfaces and a Fabry-Perot resonator in graphene was created. When there was no magnetic field applied, two consecutive reflections on the p-n interfaces occurred with opposite angles, whereas for a small magnetic field the electronic trajectories bent. Above about 0.3 Tesla the trajectories bent sufficiently to lead to the occurrence of two consecutive reflections with the same incident angle and a *π*-shift in the phase of the electron. Thus, a half period shift in the interference fringes was witnessed

Quantum interference effects are one of the most pronounced displays of the power of wave quantum mechanics. As an example, the wave nature of light is usually clearly demonstrated with the Fabry-Perot interferometers. Similar interferometers may be used in quantum mechanics to demonstrate the wave nature of electrons and other quantum mechanical particles. For electrons they were first demonstrated in graphene hetero-junctions formed by the application of a top gate voltage [6]. These were simple devices consisting mainly of the resonant cavity, and with transport channels attached. These devices exhibited quantum interference in the regular resistance oscillations that arose when the gate voltage changed. Within the conventional Fabry-Perot model [7, 8], the resistance peaks correspond to minima in the overall transmission coefficient. The peak separation can be approximated by the condition 2*kF L* = 2*πn*. The charge accumulates a phase shift of 2*π* after completing a single lap (the round-trip) 2*L* in the resonant cavity, where *kF* is the Fermi wave vector of the charges, and *L* is the length of the Fabry-Perot cavity. This is the Fabry-Perot-like resonance condition: the fundamental resonance occurs when half the wavelength of the electron mode

The simplest electron cavity, but still very effective, for the Fabry-Perot resonator may be formed by two parallel metallic wire-like contacts deposited on graphene [9]. There in a simple two terminal graphene structure there are clearly resolved Fabry-Perot oscillations. These have been observed in sub-100 nm devices. With a decrease of the size of the graphene region in these devices, the characteristics of the electron transport changes. Then the channel-dominated diffusive regime is transferred into the contact-dominated ballistic regime. This normally indicates that when the size of the cavity is about 100 nm or less the Fabry-Perot interference may be clearly resolved. The similar Fabry-Perot interferometer for

Earlier work on the resistance oscillations as a function of the applied gate voltage led to their observation in the p-n-p junctions [6, 11]. It was first reported by Young and Kim [6], but the more pronounced observations of the Fabry-Perot oscillations have been made in the Ref. [11]. There high-quality n-p-n junctions with suspended top gates have been fabricated. They indeed display clear Fabry-Perot resistance oscillations within a small cavity formed by

and evidence of perfect tunneling at normal incidence accrued.

fits inside the p-n-p junction representing the Fabry-Perot cavity.

Dirac electrons has been recently developed from carbon nanotubes [10].

the p-n interfaces.

In the present work we consider a simplest model of the Fabry-Perot interferometer, which is in fact the p-n-p or n-p-n junction formed by a one dimensional potential. We develop an exact quasi-classical theory of such a system and study the associated Fabry-Perot interference in the electron or hole transport.

Although graphene is commonly referred to as the "'carbon flatland"' there has been a feeling of discontent amongst some that the Mermin-Wagner theorem appeared to be contradicted. However, recent work shows that the buckling of the lattice can give rise to a stable 3D structure that is consistent with this theorem [14]. In what follows we present the general methodology for analysis of graphene nanoribbons using semiclassical techniques that maintain the assumption of a flat lattice. However, it should be mentioned that the effects found from these techniques are powerful in aiding our understanding of potential barriers and are an essential tool for the developing area of graphene barrier engineering. The natural state of graphene to accommodate defects or charged impurities is important for applications. The p-n interfaces described above may be capable of guiding plasmons and to create the electrical analogues of optical devices to produce controllable indices of refraction [15].

In Part *I* of this chapter we investigate the use of powerful semiclassical methods to analyze the relativistic electron and hole tunneling in graphene through a smooth potential barrier. We make comparison to the rectangular barrier. In both cases the barrier is generated as a result of an electrostatic potential in the ballistic regime. The transfer matrix method is employed in complement to the adiabatic WKB approximation for the Dirac system. Crucial to this method of approximation for the smooth barrier problem, when there is a skew electron incidence, is careful consideration of four turning points. These are denoted by *xi*, *i* = 1, 2, 3, 4 and lie in the domain of the barrier. The incident electron energy in this scattering problem belongs to the middle part of the segment [0, *U*0], where *U*<sup>0</sup> is the height of the barrier, and essentially the incident parameter *py* should be large enough to allow normal and quasi-normal incidence.

Therefore, between the first two turning points, *x*<sup>1</sup> and *x*2, and also between the next two, *x*<sup>3</sup> and *x*<sup>4</sup> there is no coalescence. Two columns of total internal reflection occur which have solutions that grow and decay exponentially. Looking away from the close vicinity of the asymptotically small boundary layers of *xi*, there exists five domains with WKB type solutions (See Fig. 2): three with oscillatory behavior and two exhibiting asymptotics that are exponentially growing and decaying. Combining these five solutions is done through applying matched asymptotics techniques (see [16]) to the so-called effective Schrödinger equation that is equivalent to the Dirac system (see [17], [18]). This combinatorial procedure generates the WKB formulas that give the elements of the transfer matrix. This transfer matrix defines all the transmission and reflection coefficients in the scattering problems discussed here.

10.5772/52267

33

http://dx.doi.org/10.5772/52267

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

[26], [27], and papers [28], [29], [30], for example). The excitation of localized eigenmodes inside a quantum electronic waveguide has a massive effect on the conductivity because these modes could create an internal resonator inside the waveguide. This is a very good reason to research the role of localised eigenmodes for quantum resonator systems and 2-D electronic transport in quantum waveguides. Excitation of some modes could result in the the emergence of stop bands for electronic wave propagation in the dispersion characteristics of the system, whereby propagation through the waveguide is blocked entirely. Other modes

In this review, the semiclassical analysis of resonator eigenstates that are localized near periodic orbits is developed for a resonator of Fabry-Perot type. These are examined inside graphene monolayer nanoribbons in static magnetic fields and electrostatic potentials. The

Graphene has generated a fervor throughout the scientific world and especially in the condensed matter physics community, with its unusual electronic properties in tunneling, charge carrier confinement and the appearance of the integer quantum Hall effect (see [31], [33], [34]), [35], [17])). Its low energy excitations are massless chiral Dirac fermion quasi-particles. The Dirac spectrum, that is valid only at low energies when the chemical potential crosses exactly at the Dirac point (see [31]), describes the physics of quantum electrodynamics for massless fermions, except that in graphene the Dirac electrons move with a Fermi velocity of *vF* = 106*m*/*s*. This is 300 times smaller than the speed of light. Graphene is a material that is easy to work with, it has a high degree of flexibility and agreeable characteristics for lithography. The unusual electronic properties of graphene and its gapless spectrum provide us with the ideal system for investigation of many new and peculiar charge carrier dynamical effects. It is also conceivable, if its promise is fulfilled, that a new form of carbon economy could emerge based upon exploitation of graphenes novel characteristics. The enhancements in devices are not just being found at the nano and micron-sized levels, though these hold the most potential (e.g. the graphene transistor, metamaterials etc), but in composites [36], electrical storage [37], solar harvesting [38] and many more applications. Following this train of thought, graphene is also a viable alternative to the materials normally used in plasmonics and nanophotonics. It absorbs light over the whole electromagnetic spectrum, including UV, visible and far-infrared wavelengths and as we have mentioned, it is capable of confining light and charge carriers into incredibly small volumes. Thus, there are a range of applications where band gap engineering is not required and it is satisfactory to directly use nanoribbons of graphene as optical-electronic devices. In the analysis of graphene one also expects unusual Dirac charge carrier properties in the eigenstates of a Fabry-Perot resonator in a magnetic field. For example, two parts of the semiclassical Maslov spectral series with positive and negative energies, for electrons and holes, correspondingly, with two different Hamiltonian dynamics and families of classical trajectories are apparent. Semiclassical analysis can provide insight into the aforementioned physical systems and good quantitative predictions on quantum observables using classical insights. Application of semiclassical analysis in studying the quantum mechanical behavior of electrons has been demonstrated in descriptions of different nano-structures, electronic transport mechanisms in mesoscopic systems and, as another example, the quantum chaotic dynamics of electronic resonators [25], [26], [27], [39], [40], [41], [42] and many others.

first results for bilayer graphene are also presented in parallel to this.

will result in total transmission.

When the energies are positive around potential height 0.5*U*0, electronic incident, reflected and transmitted states occur outside the barrier. Underneath the barrier a hole state exists (n-p-n junction). The symmetrical nature of the barrier means that we see incident, reflected and transmitted hole states outside the barrier when the energies are negative and close to one-half of the potential height *U*<sup>0</sup> < 0. Thus, underneath the barrier there are electronic states (a p-n-p junction).

Incorporated into the semiclassical method is the assumption that all four turning points are spatially separated. Consequently, the transverse component of the momentum *py* is finite and there is a finite width to the total internal reflection zone. This results in a 1-D Fabry-Perot resonator, which is of great physical importance and may aid understanding in creating plasmonic devices that operate in the range of terahertz to infrared frequencies [19]. Quantum confinement effects are crucial at the nano-scale and plasmon waves can potentially be squeezed into much smaller volumes than noble metals. The basic description of propagating plasma modes is essentially the same in the 2-D electron gas as in graphene, with the notable exception of the linear electronic dispersion and zero band-gap in graphene [20]. Thus, we predicate that the methods applied here are also applicable to systems of 2-D electron gases, such as semiconductor superlattices. Due to the broad absorption range of graphene, nanoribbons as described here, or graphene islands of various geometries may also be incorporated in opto-electronic structures.

In our analysis, if *py* → 0 then we have a quasi-normal incidence whereby and first two, *x*<sup>1</sup> and *x*2, and the second two, *x*<sup>3</sup> and *x*4, turning points coalesce. In the case of normal incidence, there is always total transmission through the barrier. The vital discovery in this form of analysis is that of the existence of modes that are localized in the bulk of the barrier. These modes decay exponentially as the proximity to the barrier decreases. These modes are two discrete, complex and real sets of energy eigen-levels that are determined by the Bohr-Sommerfeld quantization condition, above and below the cut-off energy, respectively. It is shown that the total transmission through the barrier takes place when the energy of an incident electron, which is above the cut-off energy, coincides with the real part of the complex energy eigen-level of one among the first set of modes localized within the barrier. These facts have been confirmed by numerical simulations for the reflection and transmission coefficients using finite elements methods (Comsol package).

In Part II we examine the high energy localized eigenstates in graphene monolayers and double layers. One of the most fundamental prerequisites for understanding electronic transport in quantum waveguide resonators is to be able to explain the nature of the conductance oscillations (see [25], [26], [27]). The inelastic scattering length of charge carriers is much larger than the size of modern electronic devices and consequently electronic motion is ballistic and resistance occurs due to scattering off geometric obstacles or features (e.g. the shape of a resonator micro or nano-cavity or the potential formed by a defect). It is an interesting area of development whereby defects are engineered deliberately into devices to generate a sought effect. In graphene, defects such as missing carbon atoms or the addition of adatoms can lead to interesting and novel effects, e.g magnetism or proximity effects. In the ballistic regime, conductance is analyzed by the total transmission coefficient and the Landauer formula for the zero temperature conductance of a structure (see monographs [25], [26], [27], and papers [28], [29], [30], for example). The excitation of localized eigenmodes inside a quantum electronic waveguide has a massive effect on the conductivity because these modes could create an internal resonator inside the waveguide. This is a very good reason to research the role of localised eigenmodes for quantum resonator systems and 2-D electronic transport in quantum waveguides. Excitation of some modes could result in the the emergence of stop bands for electronic wave propagation in the dispersion characteristics of the system, whereby propagation through the waveguide is blocked entirely. Other modes will result in total transmission.

4 Graphene - Research and Applications

states (a p-n-p junction).

also be incorporated in opto-electronic structures.

coefficients using finite elements methods (Comsol package).

discussed here.

matrix defines all the transmission and reflection coefficients in the scattering problems

When the energies are positive around potential height 0.5*U*0, electronic incident, reflected and transmitted states occur outside the barrier. Underneath the barrier a hole state exists (n-p-n junction). The symmetrical nature of the barrier means that we see incident, reflected and transmitted hole states outside the barrier when the energies are negative and close to one-half of the potential height *U*<sup>0</sup> < 0. Thus, underneath the barrier there are electronic

Incorporated into the semiclassical method is the assumption that all four turning points are spatially separated. Consequently, the transverse component of the momentum *py* is finite and there is a finite width to the total internal reflection zone. This results in a 1-D Fabry-Perot resonator, which is of great physical importance and may aid understanding in creating plasmonic devices that operate in the range of terahertz to infrared frequencies [19]. Quantum confinement effects are crucial at the nano-scale and plasmon waves can potentially be squeezed into much smaller volumes than noble metals. The basic description of propagating plasma modes is essentially the same in the 2-D electron gas as in graphene, with the notable exception of the linear electronic dispersion and zero band-gap in graphene [20]. Thus, we predicate that the methods applied here are also applicable to systems of 2-D electron gases, such as semiconductor superlattices. Due to the broad absorption range of graphene, nanoribbons as described here, or graphene islands of various geometries may

In our analysis, if *py* → 0 then we have a quasi-normal incidence whereby and first two, *x*<sup>1</sup> and *x*2, and the second two, *x*<sup>3</sup> and *x*4, turning points coalesce. In the case of normal incidence, there is always total transmission through the barrier. The vital discovery in this form of analysis is that of the existence of modes that are localized in the bulk of the barrier. These modes decay exponentially as the proximity to the barrier decreases. These modes are two discrete, complex and real sets of energy eigen-levels that are determined by the Bohr-Sommerfeld quantization condition, above and below the cut-off energy, respectively. It is shown that the total transmission through the barrier takes place when the energy of an incident electron, which is above the cut-off energy, coincides with the real part of the complex energy eigen-level of one among the first set of modes localized within the barrier. These facts have been confirmed by numerical simulations for the reflection and transmission

In Part II we examine the high energy localized eigenstates in graphene monolayers and double layers. One of the most fundamental prerequisites for understanding electronic transport in quantum waveguide resonators is to be able to explain the nature of the conductance oscillations (see [25], [26], [27]). The inelastic scattering length of charge carriers is much larger than the size of modern electronic devices and consequently electronic motion is ballistic and resistance occurs due to scattering off geometric obstacles or features (e.g. the shape of a resonator micro or nano-cavity or the potential formed by a defect). It is an interesting area of development whereby defects are engineered deliberately into devices to generate a sought effect. In graphene, defects such as missing carbon atoms or the addition of adatoms can lead to interesting and novel effects, e.g magnetism or proximity effects. In the ballistic regime, conductance is analyzed by the total transmission coefficient and the Landauer formula for the zero temperature conductance of a structure (see monographs [25], In this review, the semiclassical analysis of resonator eigenstates that are localized near periodic orbits is developed for a resonator of Fabry-Perot type. These are examined inside graphene monolayer nanoribbons in static magnetic fields and electrostatic potentials. The first results for bilayer graphene are also presented in parallel to this.

Graphene has generated a fervor throughout the scientific world and especially in the condensed matter physics community, with its unusual electronic properties in tunneling, charge carrier confinement and the appearance of the integer quantum Hall effect (see [31], [33], [34]), [35], [17])). Its low energy excitations are massless chiral Dirac fermion quasi-particles. The Dirac spectrum, that is valid only at low energies when the chemical potential crosses exactly at the Dirac point (see [31]), describes the physics of quantum electrodynamics for massless fermions, except that in graphene the Dirac electrons move with a Fermi velocity of *vF* = 106*m*/*s*. This is 300 times smaller than the speed of light. Graphene is a material that is easy to work with, it has a high degree of flexibility and agreeable characteristics for lithography. The unusual electronic properties of graphene and its gapless spectrum provide us with the ideal system for investigation of many new and peculiar charge carrier dynamical effects. It is also conceivable, if its promise is fulfilled, that a new form of carbon economy could emerge based upon exploitation of graphenes novel characteristics. The enhancements in devices are not just being found at the nano and micron-sized levels, though these hold the most potential (e.g. the graphene transistor, metamaterials etc), but in composites [36], electrical storage [37], solar harvesting [38] and many more applications. Following this train of thought, graphene is also a viable alternative to the materials normally used in plasmonics and nanophotonics. It absorbs light over the whole electromagnetic spectrum, including UV, visible and far-infrared wavelengths and as we have mentioned, it is capable of confining light and charge carriers into incredibly small volumes. Thus, there are a range of applications where band gap engineering is not required and it is satisfactory to directly use nanoribbons of graphene as optical-electronic devices.

In the analysis of graphene one also expects unusual Dirac charge carrier properties in the eigenstates of a Fabry-Perot resonator in a magnetic field. For example, two parts of the semiclassical Maslov spectral series with positive and negative energies, for electrons and holes, correspondingly, with two different Hamiltonian dynamics and families of classical trajectories are apparent. Semiclassical analysis can provide insight into the aforementioned physical systems and good quantitative predictions on quantum observables using classical insights. Application of semiclassical analysis in studying the quantum mechanical behavior of electrons has been demonstrated in descriptions of different nano-structures, electronic transport mechanisms in mesoscopic systems and, as another example, the quantum chaotic dynamics of electronic resonators [25], [26], [27], [39], [40], [41], [42] and many others.

However, it is important to state that the first semiclassical study on two-dimensional graphene systems only recently appeared in [43], [44], [45]. In [43] a semiclassical approximation for the Green's function in graphene monolayer and bilayers was discussed. In [44] and [45] bound states in inhomogeneous magnetic fields in graphene and graphene-based Andreev billiards were studied by semiclassical analysis, accordingly. This was carried out with one-dimensional WKB quantization due to total separation of variables. 10.5772/52267

35

http://dx.doi.org/10.5772/52267

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

element method using COMSOL. For a selectively chosen range of energy eigenvalues and eigenfunctions, agreement between the numerical results and those computed semiclassically is very good. In the graphene Fabry-Perot resonator, the electrostatic potential does not play a role of confinement, it behaves more like an inhomogeneity, but in some cases an electrostatic

In this chapter, we describe the tunneling through smooth potential barriers and the asymptotic solutions for a Dirac system in a classically allowed domain. This is done using WKB methods. We then go on to investigate the classically disallowed domain and tunneling through the smooth barrier. The asymptotic WKB solutions are presented for scattering and for quasi-bound states localized within the smooth barrier. The second part of the chapter, goes into detail about high energy localized eigenstates in monolayer and bilayer graphene. The graphene resonator is described when it is subjected to a magnetic field and ray asymptotic solutions outlined. Finally, the construction of periodic orbits, stability analyses and quantization conditions are thoroughly examined. A numerical analysis is given that compares the analytical techniques and results with those found using finite

In a conventional metal or semiconductor there are no propagating states connecting regions either side of the barrier (regions *I* and *III*). To get through the barrier an electron has to tunnel through the classically forbidden region and the tunneling amplitude depreciates exponentially as a function of the barrier width. Thus, transport between *I* and *III* is strongly suppressed. However, in each of the three regions of a barrier in a graphene system, the valence and conduction band touches, meaning that there are propagating states connecting *I* and *III* at all energies. There is no such suppression of the transport at energies incident

Potential barriers for single quasiparticle tunneling in graphene can be introduced by designing a suitable underlying gate voltage or even as a result of local uniaxial strain [68]. In the following we denote the angle of incidence with respect to the barrier to be *θ*1. We are interested in the dependence of the tunneling transmission on this incidence angle. To illustrate quantum mechanical tunneling one must extract the transmission coefficient from the solution to the graphene barrier problem. The transmission coefficient is the ratio of the flux of the particles that penetrate the potential barrier to the flux of particles incident on the barrier. We demonstrate a rectangular barrier as described in detail in the Reviews by Castro Neto et al [60] and Pereira Jr et al [69]. The problem can be described by the following 2D

*vF*(*σ*¯, <sup>−</sup>*ih*¯ <sup>∇</sup>)*ψ*(**x**) + *<sup>U</sup>*(**x**)*ψ*(**x**) = *<sup>E</sup>ψ*(**x**), *<sup>ψ</sup>*(**x**) = *<sup>u</sup>*

where **x** = (*x*, *y*) and *u*, *v* are the components of the spinor wave function describing electron localization on the sites of sublattice A or B of the honeycomb graphene structure, *vF* is the

*v* 

, (1)

and below the barrier. At normal incidence transmission is always perfect.

potential helps to make a family of POs stable.

**PART I: Tunneling through graphene barriers**

element methods.

**2. The rectangular barrier**

Dirac system (see, for example, [60])

In the second half of this review, the semiclassical Maslov spectral series of the proliferation of high-energy eigenstates (see [48], [49] [50]) of the electrons and holes for a resonator formed inside graphene mono and bilayer nanoribbons with zigzag boundary conditions, is specified. These states are localized around a stable periodic orbit (PO) under the influence of a homogeneous magnetic field and electrostatic potential. The boundaries of the nanoribbon act with perfect reflection to confine the periodic orbit to isolation. This system is a quantum electron-hole Fabry-Perot resonator of a type analogous to the "bouncing ball" high-frequency optical resonators found in studies of electromagnetics and acoustics. The asymptotic analysis of the high-energy localized eigenstates presented here is similar to ones used for optical resonators (see[50], [51], [54], and [55]). In this review, the semiclassical methods presented focus upon the stability of POs and electron and hole eigenstates that depend on the applied magnetic field.

We construct a solitary localized asymptotic solution to the Dirac system in the neighborhood of a classical trajectory called an electronic Gaussian beam (Gaussian wave package). In PO theory there are similarities between the asymptotic techniques used here and those used in the semiclassical analysis (see, for example, [27] (chapters 7, 8) or [39] and cited references). Further, the stability of a continuous family of closed trajectories in asymptotic proximity to a PO, confined between two reflecting interfaces, is studied. The classical theory of linear Hamiltonian systems with periodic coefficients gives the basis to study the stability using monodromy matrix analysis. The asymptotic eigenfunctions for electrons and holes are constructed only for the stable PO as a superposition of two Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbit. A generalized Bohr-Sommerfeld quantization condition gives the asymptotic energy spectral series (see [46] and [47], [48], [49], [50], [51] and [55]). This work highlights that the single quantization condition derived herein for the quantum electron-hole graphene resonator fully agrees with the asymptotic quantization formula of a quite general type spectral problem in [51]. It is worth drawing attention to the fact that in a semiclassical approximation for the Green's function in a graphene monolayer and bilayer, the relationship between the semiclassical phase and the adiabatic Berry phase was discussed in the paper [43]. Our asymptotic solutions, for rays and Gaussian beams, possess the adiabatic phase introduced by Berry [64]. The importance of Berry-like and non-Berry-like phases in the WKB asymptotic theory of coupled differential equations and their roles in semiclassical quantization were discussed in [57], [58], [59].

Our results are a special class of POs that occur for graphene zigzag nanoribbons in a homogeneous magnetic field and piece-wise electrostatic potential that is embedded inside the nanoribbon. They are found by giving, to the leading order, a description of the general form of asymptotic solution of Gaussian beams in a graphene monolayer or bilayer. The key point in the asymptotic analysis is the quantization of the continuous one-parameter (energy) family of POs. For one subclass of lens-shaped POs, these localized eigenstates were evaluated against eigenvalues and eigenfunctions that have been computed by the finite element method using COMSOL. For a selectively chosen range of energy eigenvalues and eigenfunctions, agreement between the numerical results and those computed semiclassically is very good. In the graphene Fabry-Perot resonator, the electrostatic potential does not play a role of confinement, it behaves more like an inhomogeneity, but in some cases an electrostatic potential helps to make a family of POs stable.

In this chapter, we describe the tunneling through smooth potential barriers and the asymptotic solutions for a Dirac system in a classically allowed domain. This is done using WKB methods. We then go on to investigate the classically disallowed domain and tunneling through the smooth barrier. The asymptotic WKB solutions are presented for scattering and for quasi-bound states localized within the smooth barrier. The second part of the chapter, goes into detail about high energy localized eigenstates in monolayer and bilayer graphene. The graphene resonator is described when it is subjected to a magnetic field and ray asymptotic solutions outlined. Finally, the construction of periodic orbits, stability analyses and quantization conditions are thoroughly examined. A numerical analysis is given that compares the analytical techniques and results with those found using finite element methods.

## **PART I: Tunneling through graphene barriers**

## **2. The rectangular barrier**

6 Graphene - Research and Applications

depend on the applied magnetic field.

in [57], [58], [59].

However, it is important to state that the first semiclassical study on two-dimensional graphene systems only recently appeared in [43], [44], [45]. In [43] a semiclassical approximation for the Green's function in graphene monolayer and bilayers was discussed. In [44] and [45] bound states in inhomogeneous magnetic fields in graphene and graphene-based Andreev billiards were studied by semiclassical analysis, accordingly. This was carried out with one-dimensional WKB quantization due to total separation of variables. In the second half of this review, the semiclassical Maslov spectral series of the proliferation of high-energy eigenstates (see [48], [49] [50]) of the electrons and holes for a resonator formed inside graphene mono and bilayer nanoribbons with zigzag boundary conditions, is specified. These states are localized around a stable periodic orbit (PO) under the influence of a homogeneous magnetic field and electrostatic potential. The boundaries of the nanoribbon act with perfect reflection to confine the periodic orbit to isolation. This system is a quantum electron-hole Fabry-Perot resonator of a type analogous to the "bouncing ball" high-frequency optical resonators found in studies of electromagnetics and acoustics. The asymptotic analysis of the high-energy localized eigenstates presented here is similar to ones used for optical resonators (see[50], [51], [54], and [55]). In this review, the semiclassical methods presented focus upon the stability of POs and electron and hole eigenstates that

We construct a solitary localized asymptotic solution to the Dirac system in the neighborhood of a classical trajectory called an electronic Gaussian beam (Gaussian wave package). In PO theory there are similarities between the asymptotic techniques used here and those used in the semiclassical analysis (see, for example, [27] (chapters 7, 8) or [39] and cited references). Further, the stability of a continuous family of closed trajectories in asymptotic proximity to a PO, confined between two reflecting interfaces, is studied. The classical theory of linear Hamiltonian systems with periodic coefficients gives the basis to study the stability using monodromy matrix analysis. The asymptotic eigenfunctions for electrons and holes are constructed only for the stable PO as a superposition of two Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbit. A generalized Bohr-Sommerfeld quantization condition gives the asymptotic energy spectral series (see [46] and [47], [48], [49], [50], [51] and [55]). This work highlights that the single quantization condition derived herein for the quantum electron-hole graphene resonator fully agrees with the asymptotic quantization formula of a quite general type spectral problem in [51]. It is worth drawing attention to the fact that in a semiclassical approximation for the Green's function in a graphene monolayer and bilayer, the relationship between the semiclassical phase and the adiabatic Berry phase was discussed in the paper [43]. Our asymptotic solutions, for rays and Gaussian beams, possess the adiabatic phase introduced by Berry [64]. The importance of Berry-like and non-Berry-like phases in the WKB asymptotic theory of coupled differential equations and their roles in semiclassical quantization were discussed

Our results are a special class of POs that occur for graphene zigzag nanoribbons in a homogeneous magnetic field and piece-wise electrostatic potential that is embedded inside the nanoribbon. They are found by giving, to the leading order, a description of the general form of asymptotic solution of Gaussian beams in a graphene monolayer or bilayer. The key point in the asymptotic analysis is the quantization of the continuous one-parameter (energy) family of POs. For one subclass of lens-shaped POs, these localized eigenstates were evaluated against eigenvalues and eigenfunctions that have been computed by the finite In a conventional metal or semiconductor there are no propagating states connecting regions either side of the barrier (regions *I* and *III*). To get through the barrier an electron has to tunnel through the classically forbidden region and the tunneling amplitude depreciates exponentially as a function of the barrier width. Thus, transport between *I* and *III* is strongly suppressed. However, in each of the three regions of a barrier in a graphene system, the valence and conduction band touches, meaning that there are propagating states connecting *I* and *III* at all energies. There is no such suppression of the transport at energies incident and below the barrier. At normal incidence transmission is always perfect.

Potential barriers for single quasiparticle tunneling in graphene can be introduced by designing a suitable underlying gate voltage or even as a result of local uniaxial strain [68]. In the following we denote the angle of incidence with respect to the barrier to be *θ*1. We are interested in the dependence of the tunneling transmission on this incidence angle. To illustrate quantum mechanical tunneling one must extract the transmission coefficient from the solution to the graphene barrier problem. The transmission coefficient is the ratio of the flux of the particles that penetrate the potential barrier to the flux of particles incident on the barrier. We demonstrate a rectangular barrier as described in detail in the Reviews by Castro Neto et al [60] and Pereira Jr et al [69]. The problem can be described by the following 2D Dirac system (see, for example, [60])

$$
v\_F(\bar{\sigma}\_\prime - i\hbar \nabla)\psi(\mathbf{x}) + \mathcal{U}(\mathbf{x})\psi(\mathbf{x}) = E\psi(\mathbf{x}), \quad \psi(\mathbf{x}) = \begin{pmatrix} u \\ v \end{pmatrix},\tag{1}
$$

where **x** = (*x*, *y*) and *u*, *v* are the components of the spinor wave function describing electron localization on the sites of sublattice A or B of the honeycomb graphene structure, *vF* is the Fermi velocity, the symbol (,) means scalar product, ¯*<sup>h</sup>* is the Planck constant and *<sup>σ</sup>*¯ = (*σ*1, *<sup>σ</sup>*2) with Pauli matrices

$$
\sigma\_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma\_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.
$$

If we assume that the potential representing the barrier does not depend on *y*, i.e. *U* = *U*(*x*), then we can look for a solution in the form

$$
\psi(\mathbf{x}) = e^{i\frac{py}{\hbar}y} \begin{pmatrix} \tilde{u}(\mathbf{x}) \\ \tilde{v}(\mathbf{x}) \end{pmatrix}.
$$

where *py* means value of the transverse momentum component describing the angle of incidence. Then, we obtain the Dirac system of two ODEs

$$
\begin{pmatrix}
\mathcal{U}(\mathbf{x}) - E & v\_F[-i\hbar \partial\_{\mathbf{x}} - ip\_y] \\
v\_F[-i\hbar \partial\_{\mathbf{x}} + ip\_y] & \mathcal{U}(\mathbf{x}) - E
\end{pmatrix}
\begin{pmatrix}
\mathfrak{i} \\
\mathfrak{v}
\end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.
\tag{2}
$$

10.5772/52267

37

http://dx.doi.org/10.5772/52267

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

**Figure 1.** The angles related to the propagation of an electron through the rectangular barrier in the xy plane for a barrier of

through the barrier. The wave inside the barrier is multiply reflected between −*L* and *L*. The

(*E* − *U*0)<sup>2</sup> *υ*2 *F*

− *p*<sup>2</sup> *y*

− *p*<sup>2</sup> *y*

<sup>2</sup> − *e*

<sup>2</sup> − |*a*2|

2 

<sup>−</sup>2*iqxL*−*ikxL* |*β*|

<sup>−</sup>2*iqxL*+*ikxLαβ*¯

<sup>2</sup> = |*c*1|

<sup>2</sup> − |*c*2| 2 . The

, (6)

, (7)

*qx* <sup>=</sup> <sup>1</sup> *h*¯ 

*kx* <sup>=</sup> *px*

*<sup>h</sup>*¯ <sup>=</sup> <sup>1</sup> *h*¯ *E*2 *υ*2 *F*

The wave functions in regions *I* and *I I* are matched at *x* = −*L*. Likewise, the wavefunctions between regions *I I* and *III* are matched at *x* = *L*. It is not necessary to match the derivatives, as is done in an analysis using the Schrödinger equation. One requires the wave functions to be continuous at the boundary of each region to generate relationships between the

<sup>2</sup>*iqxL*−*ikxL* |*α*|

<sup>2</sup>*iqxL*+*ikxLβα*¯ <sup>−</sup> *<sup>e</sup>*

parallel wave vector inside the barrier is given by,

and the wave vectors outside the barrier are defined as,

coefficients, *a*1,2, *b*1,2 and *c*1,2. We seek solutions such that |*a*1|

4*cosθ*1*cosθ*<sup>2</sup>

4*cosθ*1*cosθ*<sup>2</sup>

*<sup>T</sup>*<sup>11</sup> <sup>=</sup> *<sup>e</sup>*−*ikxL*

*<sup>T</sup>*<sup>12</sup> <sup>=</sup> *<sup>e</sup>*−*ikxL*

elements of the transfer matrix for the rectangular barrier are found to be,

 *e*

> *e*

width *W*.

The particle incident with energy *E* < *U*<sup>0</sup> from the left of the barrier has wavevectors *k*1, *q*, and *k*<sup>2</sup> to the left, in the barrier and to the right of the barier, respectively. These regions are denoted *<sup>I</sup>*, *I I* and *III*, respectively. In the symmetric barrier *<sup>k</sup>*<sup>1</sup> = *<sup>k</sup>*<sup>2</sup> = *<sup>k</sup>*. Region *I I* lies between −*L* and *L*, where ±*L* defines the width of the barrier. The wave functions are defined for each of the three regions below:

$$\psi\_{I} = \frac{a\_{1}}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{i\theta\_{1}} \end{pmatrix} e^{i\left(k\_{\text{x}}\mathbf{x} + k\_{\text{y}}\mathbf{y}\right)} + \frac{a\_{2}}{\sqrt{2}} \begin{pmatrix} 1 \\ -e^{-i\theta\_{1}} \end{pmatrix} e^{i\left(-k\_{\text{x}}\mathbf{x} + k\_{\text{y}}\mathbf{y}\right)} \tag{3}$$

$$\psi\_{II} = \frac{b\_1}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{i\theta\_2} \end{pmatrix} e^{i\left(q\_\text{tr} \mathbf{x} + k\_y y\right)} + \frac{b\_2}{\sqrt{2}} \begin{pmatrix} 1 \\ -e^{-i\theta\_2} \end{pmatrix} e^{i\left(-q\_\text{tr} \mathbf{x} + k\_y y\right)} \tag{4}$$

$$\psi\_{III} = \frac{c\_1}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{i\theta\_1} \end{pmatrix} e^{i\left(k\_x x + k\_y y\right)} + \frac{c\_2}{\sqrt{2}} \begin{pmatrix} 1 \\ -e^{-i\theta\_1} \end{pmatrix} e^{i\left(-k\_x x + k\_y y\right)} \tag{5}$$

where we have introduced the wave function, as is done in [31]. The coefficients *c*1, *c*<sup>2</sup> and *<sup>a</sup>*1, *<sup>a</sup>*<sup>2</sup> are related by means of the transfer matrix, *<sup>c</sup>* = *Ta*. The transfer matrix has unique properties, which are demonstrated in Appendix B. In regions *I* and *III* the angle of incidence in momentum space is given by, *<sup>θ</sup>*<sup>1</sup> = arctan *ky*/*kx* and in region *I I*, *<sup>θ</sup>*<sup>2</sup> = arctan *ky*/*qx* . In regions *I*-*III* the valence and conduction bands touch. This allows propagating states to connect the regions at all energies and there is no suppression of transport at the energies below the height of the barrier. There is also perfect transmission at normal incidence. The graphene rectangular barrier can be thought of as a medium with a different refractive index to its surroundings. In an optical analogy, the refractive index of the barrier is 1 − *<sup>U</sup>*0/*<sup>E</sup>* [8]. At the interface of the barrier the incidence angle splits into transmitted and reflected waves with the transmitted wave propagating with angle *θ*<sup>2</sup>

**Figure 1.** The angles related to the propagation of an electron through the rectangular barrier in the xy plane for a barrier of width *W*.

through the barrier. The wave inside the barrier is multiply reflected between −*L* and *L*. The parallel wave vector inside the barrier is given by,

$$q\_{\chi} = \frac{1}{\hbar} \sqrt{\frac{(E - \mathcal{U}\_0)^2}{\upsilon\_F^2} - p\_y^2}$$

and the wave vectors outside the barrier are defined as,

8 Graphene - Research and Applications

with Pauli matrices

Fermi velocity, the symbol (,) means scalar product, ¯*<sup>h</sup>* is the Planck constant and *<sup>σ</sup>*¯ = (*σ*1, *<sup>σ</sup>*2)

, *<sup>σ</sup>*<sup>2</sup> =

 *u*˜(*x*) *v*˜(*x*) ,

If we assume that the potential representing the barrier does not depend on *y*, i.e. *U* = *U*(*x*),

where *py* means value of the transverse momentum component describing the angle of

The particle incident with energy *E* < *U*<sup>0</sup> from the left of the barrier has wavevectors *k*1, *q*, and *k*<sup>2</sup> to the left, in the barrier and to the right of the barier, respectively. These regions are denoted *<sup>I</sup>*, *I I* and *III*, respectively. In the symmetric barrier *<sup>k</sup>*<sup>1</sup> = *<sup>k</sup>*<sup>2</sup> = *<sup>k</sup>*. Region *I I* lies between −*L* and *L*, where ±*L* defines the width of the barrier. The wave functions are

> *a*2 √2

> > *b*2 √2

> > > *c*2 √2

 1 −*e*−*iθ*<sup>1</sup>

 1 −*e*−*iθ*<sup>2</sup>

 1 −*e*−*iθ*<sup>1</sup>

. In regions *I*-*III* the valence and conduction bands touch. This

 *e*

> *e*

> > *e*

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) +

*<sup>i</sup>*(*qx <sup>x</sup>*+*ky <sup>y</sup>*) +

*<sup>i</sup>*(*kx <sup>x</sup>*+*ky <sup>y</sup>*) +

where we have introduced the wave function, as is done in [31]. The coefficients *c*1, *c*<sup>2</sup> and *<sup>a</sup>*1, *<sup>a</sup>*<sup>2</sup> are related by means of the transfer matrix, *<sup>c</sup>* = *Ta*. The transfer matrix has unique properties, which are demonstrated in Appendix B. In regions *I* and *III* the

allows propagating states to connect the regions at all energies and there is no suppression of transport at the energies below the height of the barrier. There is also perfect transmission at normal incidence. The graphene rectangular barrier can be thought of as a medium with a different refractive index to its surroundings. In an optical analogy, the refractive index of the barrier is 1 − *<sup>U</sup>*0/*<sup>E</sup>* [8]. At the interface of the barrier the incidence angle splits into transmitted and reflected waves with the transmitted wave propagating with angle *θ*<sup>2</sup>

 0 − *i i* 0

> *u*˜ *v*˜ = 0 0

. (2)

*<sup>i</sup>*(−*kx <sup>x</sup>*+*ky <sup>y</sup>*) (3)

*<sup>i</sup>*(−*qx <sup>x</sup>*+*ky <sup>y</sup>*) (4)

*<sup>i</sup>*(−*kx <sup>x</sup>*+*ky <sup>y</sup>*) (5)

and in region

*ky*/*kx*

 .

*<sup>σ</sup>*<sup>1</sup> =

incidence. Then, we obtain the Dirac system of two ODEs

 1 *eiθ*1 *e*

 1 *eiθ*2 *e*

 1 *eiθ*1 *e*

angle of incidence in momentum space is given by, *<sup>θ</sup>*<sup>1</sup> = arctan

then we can look for a solution in the form

defined for each of the three regions below:

*<sup>ψ</sup><sup>I</sup>* <sup>=</sup> *<sup>a</sup>*<sup>1</sup> √2

*<sup>ψ</sup>I I* <sup>=</sup> *<sup>b</sup>*<sup>1</sup> √2

*<sup>ψ</sup>III* <sup>=</sup> *<sup>c</sup>*<sup>1</sup> √2

*ky*/*qx* 

*I I*, *<sup>θ</sup>*<sup>2</sup> = arctan

 0 1 1 0 

*ψ*(**x**) = *e i py <sup>h</sup>*¯ *y*

 *<sup>U</sup>*(*x*) − *E vF*[−*ih*¯ *<sup>∂</sup><sup>x</sup>* − *ipy*] *vF*[−*ih*¯ *<sup>∂</sup><sup>x</sup>* + *ipy*] *<sup>U</sup>*(*x*) − *<sup>E</sup>*

$$k\_{\mathbf{x}} = \frac{p\_{\mathbf{x}}}{\hbar} = \frac{1}{\hbar} \sqrt{\frac{E^2}{\nu\_F^2} - p\_{\mathbf{y}}^2}$$

The wave functions in regions *I* and *I I* are matched at *x* = −*L*. Likewise, the wavefunctions between regions *I I* and *III* are matched at *x* = *L*. It is not necessary to match the derivatives, as is done in an analysis using the Schrödinger equation. One requires the wave functions to be continuous at the boundary of each region to generate relationships between the coefficients, *a*1,2, *b*1,2 and *c*1,2. We seek solutions such that |*a*1| <sup>2</sup> − |*a*2| <sup>2</sup> = |*c*1| <sup>2</sup> − |*c*2| 2 . The elements of the transfer matrix for the rectangular barrier are found to be,

$$T\_{11} = \frac{e^{-i\mathbf{k}\_z \mathbf{L}}}{4\cos\theta\_1 \cos\theta\_2} \left( e^{2i\mathbf{q}\_\parallel \mathbf{L} - i\mathbf{k}\_z \mathbf{L}} \left| \mathbf{a} \right|^2 - e^{-2i\mathbf{q}\_\parallel \mathbf{L} - i\mathbf{k}\_z \mathbf{L}} \left| \beta \right|^2 \right), \tag{6}$$

$$T\_{12} = \frac{e^{-ik\_xL}}{4\cos\theta\_1\cos\theta\_2} \left( e^{2i\mathbf{q}\_xL + ik\_xL}\beta\overline{\alpha} - e^{-2i\mathbf{q}\_xL + ik\_xL}\overline{\alpha}\beta \right),\tag{7}$$

$$T\_{21} = \frac{e^{ik\_x L}}{4\cos\theta\_1 \cos\theta\_2} \left( e^{-2iq\_x L - ik\_x L} \vec{\beta} a - e^{2iq\_x L - ik\_x L} a \vec{\beta} \right), \tag{8}$$

10.5772/52267

39

http://dx.doi.org/10.5772/52267

*x*

, (10)

*1*

*2*

*3*

*4*

*5*

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

suitably changing the underlying gate voltage. In the next section we investigate the smooth barrier and expect that there will be similar scattering behavior as through the rectangular

Consider a scattering problem for the Dirac operator describing an electron-hole in the presence of a scalar potential representing a smooth localized barrier with the height *U*<sup>0</sup>

*U*

*U0*

*electron electron*

*hole*

**Figure 2.** The generalization of a smooth potential barrier with Gaussian shape (we assume that *py* > 0). The Dirac electron and hole states arising in resonance tunneling are shown. The quasibound states are to be found above the green strip, |*E*| < *py*, where bound states are located. Quasibound (metastable) states are confined by two tunneling strips between *x*1,

in which we omitted the sign of tilde for brevity. In physical dimensions the energy is *<sup>U</sup>*0*E*, the potential is *<sup>U</sup>*0*U*(*x*), the *<sup>y</sup>*-component of the momentum is *pyU*0/*vF*, and the dimensionless Planck constant (small WKB parameter) is given by *<sup>h</sup>* = *hv*¯ *<sup>F</sup>*/*U*0*D*, where *<sup>U</sup>*<sup>0</sup> is the height of the potential barrier (|*U*(*x*)| < 1 for *<sup>x</sup>* ∈ **<sup>R</sup>**) and *<sup>D</sup>* is a characteristic scale of the potential barrier with respect to the *x*-coordinate. Typical values of *U*<sup>0</sup> and *D* are within the ranges 10-100meV and 100-500nm. For example, for *<sup>U</sup>*0=100meV, *<sup>D</sup>* = <sup>264</sup>*nm*, we

 *u v* = 0 0 

 *U*(*x*) − *E* −*ih∂<sup>x</sup>* − *ipy* −*ih∂<sup>x</sup>* + *ipy U*(*x*) − *E*

*x x <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> a b*

*x x*

barrier. We seek to explore the similarities and the differences between the two.

(see Fig.2). It is convenient to use the dimensionless system

*E*

y

y

**3. The smooth barrier**

*0*

*U +p*

*<sup>0</sup> U −p*

*p*

y

*−p*

y

*x*<sup>2</sup> and *x*3, *x*4, whereas the bound states are located between *x*<sup>2</sup> and *x*3.

have *h* = 0.025 and also we assume that *py* > 0.

$$T\_{22} = \frac{e^{\mathbf{k}\_{\mathbf{x}} \mathbf{L}}}{4 \cos \theta\_1 \cos \theta\_2} \left( e^{-2 \mathbf{i} \mathbf{q}\_x \mathbf{L} + \mathbf{i} \mathbf{k}\_x \mathbf{L}} \left| \mathbf{a} \right|^2 - e^{2 \mathbf{i} \mathbf{q}\_x \mathbf{L} + \mathbf{i} \mathbf{k}\_x \mathbf{L}} \left| \boldsymbol{\beta} \right|^2 \right) . \tag{9}$$

where we make the substitutions *<sup>α</sup>* = *<sup>e</sup>iθ*<sup>1</sup> + *<sup>e</sup>*−*iθ*<sup>2</sup> and *<sup>β</sup>* = *<sup>e</sup>*−*iθ*<sup>2</sup> − *<sup>e</sup>*−*iθ*<sup>1</sup> and their complex conjugate forms are denoted by *<sup>α</sup>*¯ = *<sup>e</sup>iθ*<sup>2</sup> + *<sup>e</sup>*−*iθ*<sup>1</sup> and *<sup>β</sup>*¯ = *<sup>e</sup>iθ*<sup>2</sup> − *<sup>e</sup>*−*iθ*<sup>1</sup> . If we assume that the incident wave approaches from the left, then *<sup>a</sup>*<sup>1</sup> = 1, *<sup>a</sup>*<sup>2</sup> = *<sup>r</sup>*<sup>1</sup> and *<sup>c</sup>*<sup>1</sup> = *<sup>t</sup>*1, where *<sup>r</sup>*<sup>1</sup> is the reflection coefficient and *t*<sup>1</sup> is the transmission coefficient. If the incident wave approaches from the right then *<sup>c</sup>*<sup>1</sup> = *<sup>r</sup>*2, *<sup>c</sup>*<sup>2</sup> = 1 and *<sup>a</sup>*<sup>2</sup> = *<sup>t</sup>*2. We find that *<sup>t</sup>*<sup>1</sup> = *<sup>t</sup>*<sup>2</sup> = *<sup>t</sup>* and the transmission coefficient is *<sup>t</sup>* = 1/*T*22. The reflection coefficients are determined as *<sup>r</sup>*<sup>1</sup> = −*T*21/*T*<sup>22</sup> and *<sup>r</sup>*<sup>2</sup> = *<sup>T</sup>*12/*T*22. The transmission probability is as usual given by |*t*1| <sup>2</sup> with the definition |*t*1| <sup>2</sup> + |*r*1| <sup>2</sup> = 1. At normal incidence the carriers in graphene are transmitted completely through the barrier (Klein tunneling). However, the carriers can be reflected by a potential step when the angle of incidence increases and a non-zero momentum component parallel to the barrier ensues. Thus, the transmission of charge carriers through the potential barrier is anisotropic. When a beam of electrons is fired at an angle into the barrier, it splits up into transmitted and reflected beams, with multiple reflections occurring at the edges of the barrier. As is usual in quantum mechanics, the transmission is found by stipulating that there must be continuity between the wavefunctions. In the above this demand for continuity at the extremities of the barrier allowed us to find the coefficients of the wavefunctions. Thus, using these results and following the work of Castro Neto et al [60], the total transmission as a function of the incident angle is given by *<sup>T</sup>*(*θ*1) = *tt*∗:

$$T = \frac{16\cos^2\theta\_1 \cos^2\theta\_2}{|\alpha|^4 + |\beta|^4 - 2\left|\alpha|^2\left|\beta\right|^2\cos(4q\_xL)}$$

When the tunneling resonance condition 2*Lqx* = *nπ* is met, where n is an integer, *T* = 1. This statement means that a half-integer amount of wavelengths will fit into the length of the potential barrier. The absolute transmission is the manifestation of Klein tunneling, which is unique for relativistic electrons, and it should occur when an incoming electron starts penetrating through a potential barrier of height, *U*<sup>0</sup> (which is far in excess of the electrons rest energy). The transport mechanism in a graphene tunneling structure is unique. This perfect transmission at incidence normal to the barrier is due to the pseudo-spin conservation that gives no backscattering. In order to attain an interference effect between the two interfaces an oblique incidence angle is required and it is under this prerequisite that multiple interference effects emerge. Thus, the potential barrier is analogous to two interfaces at −*L* and *L* and also a Fabry-Perot interferometer [5]. The analogy of the graphene rectangular barrier to the Fabry-Perot resonator when *<sup>θ</sup>*<sup>1</sup> �= 0 extends to the potential barrier operating like an optical cavity. In region *I I* the incoming wave can interfere with itself and with constructive interference, resonances will occur where *<sup>T</sup>*(*θ*<sup>1</sup> �= <sup>0</sup>) = 1 [5]. The potential barriers for single quasiparticle tunneling in graphene are usually created by suitably changing the underlying gate voltage. In the next section we investigate the smooth barrier and expect that there will be similar scattering behavior as through the rectangular barrier. We seek to explore the similarities and the differences between the two.

## **3. The smooth barrier**

10 Graphene - Research and Applications


<sup>2</sup> + |*r*1|

*<sup>T</sup>*<sup>21</sup> <sup>=</sup> *<sup>e</sup>ikxL*

*<sup>T</sup>*<sup>22</sup> <sup>=</sup> *<sup>e</sup>ikxL*

a function of the incident angle is given by *<sup>T</sup>*(*θ*1) = *tt*∗:


4*cosθ*1*cosθ*<sup>2</sup>

4*cosθ*1*cosθ*<sup>2</sup>

 *e*

 *e*

*<sup>r</sup>*<sup>2</sup> = *<sup>T</sup>*12/*T*22. The transmission probability is as usual given by |*t*1|

<sup>−</sup>2*iqxL*−*ikxLβα*¯ <sup>−</sup> *<sup>e</sup>*

<sup>2</sup> − *e*

<sup>−</sup>2*iqxL*+*ikxL* |*α*|

where we make the substitutions *<sup>α</sup>* = *<sup>e</sup>iθ*<sup>1</sup> + *<sup>e</sup>*−*iθ*<sup>2</sup> and *<sup>β</sup>* = *<sup>e</sup>*−*iθ*<sup>2</sup> − *<sup>e</sup>*−*iθ*<sup>1</sup> and their complex conjugate forms are denoted by *<sup>α</sup>*¯ = *<sup>e</sup>iθ*<sup>2</sup> + *<sup>e</sup>*−*iθ*<sup>1</sup> and *<sup>β</sup>*¯ = *<sup>e</sup>iθ*<sup>2</sup> − *<sup>e</sup>*−*iθ*<sup>1</sup> . If we assume that the incident wave approaches from the left, then *<sup>a</sup>*<sup>1</sup> = 1, *<sup>a</sup>*<sup>2</sup> = *<sup>r</sup>*<sup>1</sup> and *<sup>c</sup>*<sup>1</sup> = *<sup>t</sup>*1, where *<sup>r</sup>*<sup>1</sup> is the reflection coefficient and *t*<sup>1</sup> is the transmission coefficient. If the incident wave approaches from the right then *<sup>c</sup>*<sup>1</sup> = *<sup>r</sup>*2, *<sup>c</sup>*<sup>2</sup> = 1 and *<sup>a</sup>*<sup>2</sup> = *<sup>t</sup>*2. We find that *<sup>t</sup>*<sup>1</sup> = *<sup>t</sup>*<sup>2</sup> = *<sup>t</sup>* and the transmission coefficient is *<sup>t</sup>* = 1/*T*22. The reflection coefficients are determined as *<sup>r</sup>*<sup>1</sup> = −*T*21/*T*<sup>22</sup> and

through the barrier (Klein tunneling). However, the carriers can be reflected by a potential step when the angle of incidence increases and a non-zero momentum component parallel to the barrier ensues. Thus, the transmission of charge carriers through the potential barrier is anisotropic. When a beam of electrons is fired at an angle into the barrier, it splits up into transmitted and reflected beams, with multiple reflections occurring at the edges of the barrier. As is usual in quantum mechanics, the transmission is found by stipulating that there must be continuity between the wavefunctions. In the above this demand for continuity at the extremities of the barrier allowed us to find the coefficients of the wavefunctions. Thus, using these results and following the work of Castro Neto et al [60], the total transmission as

*<sup>T</sup>* <sup>=</sup> <sup>16</sup>*cos*2*θ*1*cos*2*θ*<sup>2</sup>

<sup>4</sup> − 2 |*α*|

When the tunneling resonance condition 2*Lqx* = *nπ* is met, where n is an integer, *T* = 1. This statement means that a half-integer amount of wavelengths will fit into the length of the potential barrier. The absolute transmission is the manifestation of Klein tunneling, which is unique for relativistic electrons, and it should occur when an incoming electron starts penetrating through a potential barrier of height, *U*<sup>0</sup> (which is far in excess of the electrons rest energy). The transport mechanism in a graphene tunneling structure is unique. This perfect transmission at incidence normal to the barrier is due to the pseudo-spin conservation that gives no backscattering. In order to attain an interference effect between the two interfaces an oblique incidence angle is required and it is under this prerequisite that multiple interference effects emerge. Thus, the potential barrier is analogous to two interfaces at −*L* and *L* and also a Fabry-Perot interferometer [5]. The analogy of the graphene rectangular barrier to the Fabry-Perot resonator when *<sup>θ</sup>*<sup>1</sup> �= 0 extends to the potential barrier operating like an optical cavity. In region *I I* the incoming wave can interfere with itself and with constructive interference, resonances will occur where *<sup>T</sup>*(*θ*<sup>1</sup> �= <sup>0</sup>) = 1 [5]. The potential barriers for single quasiparticle tunneling in graphene are usually created by

2 |*β*|

<sup>2</sup> *cos*(4*qxL*)

<sup>2</sup> = 1. At normal incidence the carriers in graphene are transmitted completely

<sup>2</sup>*iqxL*−*ikxLαβ*¯

<sup>2</sup>*iqxL*+*ikxL* |*β*|

2 

, (8)

. (9)

<sup>2</sup> with the definition

Consider a scattering problem for the Dirac operator describing an electron-hole in the presence of a scalar potential representing a smooth localized barrier with the height *U*<sup>0</sup> (see Fig.2). It is convenient to use the dimensionless system

**Figure 2.** The generalization of a smooth potential barrier with Gaussian shape (we assume that *py* > 0). The Dirac electron and hole states arising in resonance tunneling are shown. The quasibound states are to be found above the green strip, |*E*| < *py*, where bound states are located. Quasibound (metastable) states are confined by two tunneling strips between *x*1, *x*<sup>2</sup> and *x*3, *x*4, whereas the bound states are located between *x*<sup>2</sup> and *x*3.

$$
\begin{pmatrix}
\mathcal{U}(\mathbf{x}) - E & -i\hbar \partial\_{\mathbf{x}} - ip\_{y} \\
\end{pmatrix}
\begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix},\tag{10}
$$

in which we omitted the sign of tilde for brevity. In physical dimensions the energy is *<sup>U</sup>*0*E*, the potential is *<sup>U</sup>*0*U*(*x*), the *<sup>y</sup>*-component of the momentum is *pyU*0/*vF*, and the dimensionless Planck constant (small WKB parameter) is given by *<sup>h</sup>* = *hv*¯ *<sup>F</sup>*/*U*0*D*, where *<sup>U</sup>*<sup>0</sup> is the height of the potential barrier (|*U*(*x*)| < 1 for *<sup>x</sup>* ∈ **<sup>R</sup>**) and *<sup>D</sup>* is a characteristic scale of the potential barrier with respect to the *x*-coordinate. Typical values of *U*<sup>0</sup> and *D* are within the ranges 10-100meV and 100-500nm. For example, for *<sup>U</sup>*0=100meV, *<sup>D</sup>* = <sup>264</sup>*nm*, we have *h* = 0.025 and also we assume that *py* > 0.

10.5772/52267

41

http://dx.doi.org/10.5772/52267

*<sup>ψ</sup>j*(*x*). (11)

, (13)

, (14)

only *x*<sup>2</sup> and *x*3. When we move down from zone 2 to zone 1, the turning points *x*<sup>2</sup> and *x*<sup>3</sup> disappear. When *E* moves up from zone 3 to zone 4, the turning points *x*<sup>2</sup> and *x*<sup>3</sup> coalesce and disappear such that inside zone 4 we have only *x*<sup>1</sup> and *x*4. When we move up from zone

The WKB oscillatory asymptotic solution to the Dirac system in the classically allowed

 *uj vj* = *e i <sup>h</sup> <sup>S</sup>*(*x*) +∞ ∑ *j*=0

Substituting this series into the Dirac system, and equating to zero corresponding coefficients of successive degrees of the small parameter *h*, we obtain a recurrent system of equations

> *U*(*x*) *px* − *ipy px* + *ipy U*(*x*)

> > 0 *∂<sup>x</sup> ∂<sup>x</sup>* 0

(*<sup>H</sup>* − *EI*)*ψ*<sup>0</sup> = 0, (*<sup>H</sup>* − *EI*)*ψ<sup>j</sup>* = −*Rψj*<sup>−</sup>1, *<sup>j</sup>* > 0, (12)

*<sup>y</sup>* ≡ *<sup>U</sup>*(*x*) ± *<sup>p</sup>*

*y*.

(−*ih*)*<sup>j</sup>*

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

**4. WKB asymptotic solution for Dirac system in classically allowed**

(−*ih*)*<sup>j</sup>*

which determines the unknown *<sup>S</sup>*(*x*) (classical action) and *<sup>ψ</sup>j*(*x*), namely,

*H* =

*<sup>h</sup>*1,2 = *<sup>U</sup>*(*x*) ±

*<sup>e</sup>*1,2 <sup>=</sup> <sup>1</sup> √2

*px* = ±

*R* =

where *<sup>I</sup>* is the identity matrix and *<sup>S</sup>*′ <sup>=</sup> *px*. The Hamiltonian *<sup>H</sup>* has two eigenvalues

 *p*2 *<sup>x</sup>* + *<sup>p</sup>*<sup>2</sup>

> 1 ± *px*+*ipy p*

(*E* − *U*(*x*))<sup>2</sup> − *p*<sup>2</sup>

4 to zone 5, the turning points *x*<sup>1</sup> and *x*<sup>4</sup> coalesce and disappear.

domains is to be sought in the form (see [16]) with real *S*(*x*)

*ψ* = *u v* = *e i <sup>h</sup> <sup>S</sup>*(*x*) +∞ ∑ *j*=0

**domain**

and

with

**Figure 3.** (a) The six different scattering regimes for smooth barrier tunneling. The six zones are separated by the four red diagonal lines, *E* = ±*py* and *E* = ±*py* + *U*0. We will now discuss the right hand side of this diagram. In zone 1 (orange shading), *E* < −*py*, there is total transmission and exponentially small reflection. The asymptotic solutions are of an oscillatory nature everywhere in this zone. In zone 2 (blue), −*py* < *E* < *py* with the cut-off energy at *E* = ±*py*. In zone 2 there is no propagation outside the barrier. However, there are oscillatory solutions within the barrier. Zone 3 (green) is the zone of Klein tunneling. Here, *py* < *E* < *U*<sup>0</sup> − *py* and there are oscillatory solutions everywhere. Zone 4 (hexagons), *U*<sup>0</sup> − *py* < *E* < *U*<sup>0</sup> + *py*, is devoid of propagation through the barrier. There is total reflection and exponentially small transmission in this zone. The fifth zone (sand color) is limited to *E* > *U*<sup>0</sup> + *py* and is characterized by total transmission, exponentially small reflection and the asymptotic solutions are oscillatory everywhere. The sixth zone is one of no propagation and exponentially decaying or growing asymptotic solutions, *U*<sup>0</sup> < *E* < *py*.

In Fig. 3, six zones (horizontal strips in Fig. 3b) are shown that illustrate different scattering regimes for the smooth barrier scattering problem. These six zones are exactly the same as for the rectangular barrier with the height *<sup>U</sup>*0. In zone 1 *<sup>E</sup>* < −*py*, we have total transmission and exponentially small reflection, asymptotic solutions are of oscillatory type everywhere. In zone 2, −*py* < *E* < *py* (*E* = ±*py* is the cut-off energy), there is no propagation outside the barrier, however there are oscillatory solutions within the barrier. In the zone 3, *py* < *E* < *<sup>U</sup>*<sup>0</sup> − *py*, there are oscillatory solutions everywhere (zone of the Klein tunneling). In zone 4, *<sup>U</sup>*<sup>0</sup> − *py* < *<sup>E</sup>* < *<sup>U</sup>*<sup>0</sup> + *py*, there is no propagation through the barrier, we have total reflection and exponentially small transmission. In zone 5 *<sup>E</sup>* > *<sup>U</sup>*<sup>0</sup> + *py*, we have total transmission and exponentially small reflection, asymptotic solutions are of oscillatory type everywhere. Finally, in the zone 6, *<sup>U</sup>*<sup>0</sup> − *py* < *<sup>E</sup>* < *py*, there is no propagation, everywhere asymptotic solutions are of exponential type, decaying or growing.

Firstly, we study the scattering problem for zone 3 (see Fig.2). In this case, there are 5 domains with different WKB asymptotic solutions: <sup>Ω</sup><sup>1</sup> = {*<sup>x</sup>* : −<sup>∞</sup> < *<sup>x</sup>* < *<sup>x</sup>*1}, <sup>Ω</sup><sup>2</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>1</sup> < *<sup>x</sup>* < *<sup>x</sup>*2}, <sup>Ω</sup><sup>3</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>2</sup> < *<sup>x</sup>* < *<sup>x</sup>*3}, <sup>Ω</sup><sup>4</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>3</sup> < *<sup>x</sup>* < *<sup>x</sup>*4} and <sup>Ω</sup><sup>5</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>4</sup> < *<sup>x</sup>* < +∞} and the turning points are *xi* with *i* = 1, 2, 3, 4, are the roots of the equation (*E* − *U*(*x*))<sup>2</sup> − *p*2 *<sup>y</sup>* = 0. The regions <sup>Ω</sup>1, <sup>Ω</sup><sup>3</sup> and <sup>Ω</sup>5, in which (*<sup>E</sup>* − *<sup>U</sup>*(*x*))<sup>2</sup> − *<sup>p</sup>*<sup>2</sup> *<sup>y</sup>* > 0, will be referred to as classically allowed domains, whereas <sup>Ω</sup><sup>2</sup> and <sup>Ω</sup>4, in which (*<sup>E</sup>* − *<sup>U</sup>*(*x*))<sup>2</sup> − *<sup>p</sup>*<sup>2</sup> *<sup>y</sup>* < 0, are classically disallowed domains. Note that as *py* → 0 for fixed value of *E*, the turning points coalesce. We exclude this possibility in our analysis.

It is worth to remark that for fixed *py*, when *E* moves down from zone 3 to zone 2, the turning points *<sup>x</sup>*<sup>1</sup> and *<sup>x</sup>*<sup>4</sup> disappear (*x*<sup>1</sup> → −∞, *<sup>x</sup>*<sup>4</sup> → +∞) such that inside zone 2 we have only *x*<sup>2</sup> and *x*3. When we move down from zone 2 to zone 1, the turning points *x*<sup>2</sup> and *x*<sup>3</sup> disappear. When *E* moves up from zone 3 to zone 4, the turning points *x*<sup>2</sup> and *x*<sup>3</sup> coalesce and disappear such that inside zone 4 we have only *x*<sup>1</sup> and *x*4. When we move up from zone 4 to zone 5, the turning points *x*<sup>1</sup> and *x*<sup>4</sup> coalesce and disappear.

## **4. WKB asymptotic solution for Dirac system in classically allowed domain**

The WKB oscillatory asymptotic solution to the Dirac system in the classically allowed domains is to be sought in the form (see [16]) with real *S*(*x*)

$$\psi = \begin{pmatrix} \mathfrak{u} \\ \upsilon \end{pmatrix} = e^{\frac{i}{\hbar}S(x)} \sum\_{j=0}^{+\infty} (-ih)^j \begin{pmatrix} \mathfrak{u}\_j \\ \upsilon\_j \end{pmatrix} = e^{\frac{i}{\hbar}S(x)} \sum\_{j=0}^{+\infty} (-ih)^j \psi\_j(x). \tag{11}$$

Substituting this series into the Dirac system, and equating to zero corresponding coefficients of successive degrees of the small parameter *h*, we obtain a recurrent system of equations which determines the unknown *<sup>S</sup>*(*x*) (classical action) and *<sup>ψ</sup>j*(*x*), namely,

$$(H - EI)\psi\_0 = 0, \quad (H - EI)\psi\_j = -R\psi\_{j-1}, \quad j > 0,\tag{12}$$

$$H = \begin{pmatrix} \mathcal{U}(\mathbf{x}) & p\_{\mathcal{X}} - ip\_{\mathcal{Y}} \\ p\_{\mathcal{X}} + ip\_{\mathcal{Y}} & \mathcal{U}(\mathbf{x}) \end{pmatrix},\tag{13}$$

$$
\widehat{R} = \begin{pmatrix} 0 & \partial\_{\chi} \\ \partial\_{\chi} & 0 \end{pmatrix}' \tag{14}
$$

where *<sup>I</sup>* is the identity matrix and *<sup>S</sup>*′ <sup>=</sup> *px*. The Hamiltonian *<sup>H</sup>* has two eigenvalues

$$h\_{1,2} = \mathcal{U}(\mathfrak{x}) \pm \sqrt{p\_{\mathfrak{x}}^2 + p\_{\mathfrak{y}}^2} \equiv \mathcal{U}(\mathfrak{x}) \pm p\_{\mathfrak{y}}^2$$

and

12 Graphene - Research and Applications

growing asymptotic solutions, *U*<sup>0</sup> < *E* < *py*.

*p*2

solutions are of exponential type, decaying or growing.

coalesce. We exclude this possibility in our analysis.

*<sup>y</sup>* = 0. The regions <sup>Ω</sup>1, <sup>Ω</sup><sup>3</sup> and <sup>Ω</sup>5, in which (*<sup>E</sup>* − *<sup>U</sup>*(*x*))<sup>2</sup> − *<sup>p</sup>*<sup>2</sup>

as classically allowed domains, whereas <sup>Ω</sup><sup>2</sup> and <sup>Ω</sup>4, in which (*<sup>E</sup>* − *<sup>U</sup>*(*x*))<sup>2</sup> − *<sup>p</sup>*<sup>2</sup>

**Figure 3.** (a) The six different scattering regimes for smooth barrier tunneling. The six zones are separated by the four red diagonal lines, *E* = ±*py* and *E* = ±*py* + *U*0. We will now discuss the right hand side of this diagram. In zone 1 (orange shading), *E* < −*py*, there is total transmission and exponentially small reflection. The asymptotic solutions are of an oscillatory nature everywhere in this zone. In zone 2 (blue), −*py* < *E* < *py* with the cut-off energy at *E* = ±*py*. In zone 2 there is no propagation outside the barrier. However, there are oscillatory solutions within the barrier. Zone 3 (green) is the zone of Klein tunneling. Here, *py* < *E* < *U*<sup>0</sup> − *py* and there are oscillatory solutions everywhere. Zone 4 (hexagons), *U*<sup>0</sup> − *py* < *E* < *U*<sup>0</sup> + *py*, is devoid of propagation through the barrier. There is total reflection and exponentially small transmission in this zone. The fifth zone (sand color) is limited to *E* > *U*<sup>0</sup> + *py* and is characterized by total transmission, exponentially small reflection and the asymptotic solutions are oscillatory everywhere. The sixth zone is one of no propagation and exponentially decaying or

In Fig. 3, six zones (horizontal strips in Fig. 3b) are shown that illustrate different scattering regimes for the smooth barrier scattering problem. These six zones are exactly the same as for the rectangular barrier with the height *<sup>U</sup>*0. In zone 1 *<sup>E</sup>* < −*py*, we have total transmission and exponentially small reflection, asymptotic solutions are of oscillatory type everywhere. In zone 2, −*py* < *E* < *py* (*E* = ±*py* is the cut-off energy), there is no propagation outside the barrier, however there are oscillatory solutions within the barrier. In the zone 3, *py* < *E* < *<sup>U</sup>*<sup>0</sup> − *py*, there are oscillatory solutions everywhere (zone of the Klein tunneling). In zone 4, *<sup>U</sup>*<sup>0</sup> − *py* < *<sup>E</sup>* < *<sup>U</sup>*<sup>0</sup> + *py*, there is no propagation through the barrier, we have total reflection and exponentially small transmission. In zone 5 *<sup>E</sup>* > *<sup>U</sup>*<sup>0</sup> + *py*, we have total transmission and exponentially small reflection, asymptotic solutions are of oscillatory type everywhere. Finally, in the zone 6, *<sup>U</sup>*<sup>0</sup> − *py* < *<sup>E</sup>* < *py*, there is no propagation, everywhere asymptotic

Firstly, we study the scattering problem for zone 3 (see Fig.2). In this case, there are 5 domains with different WKB asymptotic solutions: <sup>Ω</sup><sup>1</sup> = {*<sup>x</sup>* : −<sup>∞</sup> < *<sup>x</sup>* < *<sup>x</sup>*1}, <sup>Ω</sup><sup>2</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>1</sup> < *<sup>x</sup>* < *<sup>x</sup>*2}, <sup>Ω</sup><sup>3</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>2</sup> < *<sup>x</sup>* < *<sup>x</sup>*3}, <sup>Ω</sup><sup>4</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>3</sup> < *<sup>x</sup>* < *<sup>x</sup>*4} and <sup>Ω</sup><sup>5</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>4</sup> < *<sup>x</sup>* < +∞} and the turning points are *xi* with *i* = 1, 2, 3, 4, are the roots of the equation (*E* − *U*(*x*))<sup>2</sup> −

classically disallowed domains. Note that as *py* → 0 for fixed value of *E*, the turning points

It is worth to remark that for fixed *py*, when *E* moves down from zone 3 to zone 2, the turning points *<sup>x</sup>*<sup>1</sup> and *<sup>x</sup>*<sup>4</sup> disappear (*x*<sup>1</sup> → −∞, *<sup>x</sup>*<sup>4</sup> → +∞) such that inside zone 2 we have

*<sup>y</sup>* > 0, will be referred to

*<sup>y</sup>* < 0, are

$$e\_{1,2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \pm \frac{p\_x + ip\_y}{p} \end{pmatrix}$$

with

$$p\_{\mathfrak{x}} = \pm \sqrt{(E - U(\mathfrak{x}))^2 - p\_{\mathfrak{y}}^2}.$$

From now on we will omit the dependence on *x* of *U*, *S*, and quantities derived from them. It turns out to be convenient to use different *e*1,2 instead with

$$e\_{1,2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \pm e^{i\theta} \end{pmatrix}, \quad e^{i\theta} = \frac{p\_x + ip\_y}{E - U}.$$

In this way we will be able to solve problems of electron and hole incidence on the barrier simultaneously. Note that, irrespective of whether *E* > *U* or *E* < *U*,

$$He\_1 = Ee\_1, \quad He\_2 = (2\mathcal{U} - E)e\_2. \tag{15}$$

10.5772/52267

43

http://dx.doi.org/10.5772/52267

<sup>2</sup> *e*2, (19)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

<sup>2</sup> *<sup>e</sup>*2) <sup>&</sup>gt; *dx*

 1 *eiθ*± ,

,

<sup>1</sup> (<sup>1</sup> <sup>+</sup> *<sup>O</sup>*(*h*)), (20)

For higher-order terms,

one can seek a solution to

(*j* > 0) in the form

where from (15), *<sup>σ</sup>*(*j*)

one obtains

<sup>2</sup> is given by

Then, from the orthogonality condition,

*σ*(*j*)

<sup>1</sup> <sup>=</sup> *<sup>e</sup>*−*iθ*/2 √2 cos *θ*

*Sp*(*x*, *xi*) = *<sup>x</sup>*

positive *x*-direction. Thus, to the leading order we have

*ψ* = *u v* <sup>=</sup> *<sup>e</sup>*<sup>±</sup> *<sup>i</sup>*

*xi*

*pxdt*, *J*

*e iθ*<sup>±</sup>

± *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>e</sup>*

asymptotic solution, derived in this section, is valid for the domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 3, 5.

(*<sup>H</sup>* <sup>−</sup> *EI*)*ψ<sup>j</sup>* <sup>=</sup> <sup>−</sup>*<sup>R</sup><sup>ψ</sup>j*<sup>−</sup>1,

(*<sup>H</sup>* <sup>−</sup> *EI*)*ψ<sup>j</sup>* <sup>=</sup> <sup>−</sup>*<sup>R</sup><sup>ψ</sup>j*−<sup>1</sup>

<sup>1</sup> *<sup>e</sup>*<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*(*j*)

<sup>2</sup> <sup>=</sup> <sup>&</sup>lt; *<sup>e</sup>*2, *<sup>R</sup>*(*ψj*−1) <sup>&</sup>gt;

<sup>1</sup> **<sup>e</sup>**<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*(*j*)

where *cj* = *const*. Below we assume that *px* > 0, corresponding to a wave traveling in the

*<sup>h</sup> Sp* (*x*,*xi*) *J* ± *p*

*c*0*e* ±

2*iθ*<sup>±</sup> , *e* ± <sup>1</sup> <sup>=</sup> <sup>1</sup> √2

<sup>=</sup> <sup>±</sup>*px* <sup>+</sup> *ipy <sup>E</sup>* <sup>−</sup> *<sup>U</sup>* .

This asymptotic approximation is not valid near turning points where *<sup>S</sup>*′ = 0 (see Fig. 1) at *<sup>x</sup>* <sup>=</sup> *xj*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3, 4 where *<sup>e</sup>i<sup>θ</sup>* <sup>=</sup> <sup>±</sup>*<sup>i</sup>* and cos *<sup>θ</sup>* <sup>=</sup> 0, while at *<sup>x</sup>* <sup>=</sup> *<sup>a</sup>*, *<sup>b</sup>* we have *<sup>E</sup>* <sup>=</sup> *<sup>U</sup>*. The WKB

<sup>2</sup>(*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*) .

<sup>2</sup> *<sup>e</sup>*2) <sup>&</sup>gt;<sup>=</sup> 0,

2 cos *<sup>θ</sup>* <sup>&</sup>lt; *<sup>e</sup>*1, *<sup>R</sup>*(*σ*(*j*)

*<sup>ψ</sup><sup>j</sup>* <sup>=</sup> *<sup>σ</sup>*(*j*)

*σ*(*j*)

<sup>&</sup>lt; *<sup>e</sup>*1, *<sup>R</sup>*(*σ*(*j*)

 *cj* − *e <sup>i</sup>θ*/2<sup>√</sup>

The classical action *S*(*x*) is given by

$$S = \int p\_{\mathbf{x}} d\mathbf{x} = \pm \int \sqrt{(\mathbf{E} - \mathbf{U})^2 - p\_y^2} \, d\mathbf{x},\tag{16}$$

the sign indicating the direction of the wave, with + corresponding to a wave traveling to the right.

For electrons and holes one can seek a solution to the Dirac system zero-order problem in the form

$$
\psi\_0 = \sigma^{(0)}(\mathfrak{x}) \mathfrak{e}\_1 \tag{17}
$$

with unknown amplitude *<sup>σ</sup>*(0). The solvability of the problem

$$(H - EI)\psi\_1 = -\widehat{R}\psi\_0$$

requires that the orthogonality condition

$$ = 0$$

must hold, written as a scalar product implied with complex conjugation, and from this one obtains the transport equation for *<sup>σ</sup>*(0):

$$\frac{d\sigma^{(0)}}{d\mathbf{x}}(e^{i\theta} + e^{-i\theta}) + \sigma^{(0)}\frac{de^{i\theta}}{d\mathbf{x}} = 0.\tag{18}$$

It has a solution

$$
\sigma^{(0)} = \left(\frac{c\_0}{\sqrt{2\cos\theta}}\right) e^{-i\frac{\theta}{2}}
$$

with *<sup>c</sup>*<sup>0</sup> <sup>=</sup> *const*, where a branch of the analytic function <sup>√</sup>*<sup>z</sup>* is taken that satisfies the condition

$$\operatorname{Im}(\sqrt{z}) \ge 0, \ z \in \mathbb{C}.$$

For higher-order terms,

14 Graphene - Research and Applications

The classical action *S*(*x*) is given by

requires that the orthogonality condition

obtains the transport equation for *<sup>σ</sup>*(0):

the right.

the form

It has a solution

condition

From now on we will omit the dependence on *x* of *U*, *S*, and quantities derived from them.

In this way we will be able to solve problems of electron and hole incidence on the barrier

the sign indicating the direction of the wave, with + corresponding to a wave traveling to

For electrons and holes one can seek a solution to the Dirac system zero-order problem in

(*<sup>H</sup>* <sup>−</sup> *EI*)*ψ*<sup>1</sup> <sup>=</sup> <sup>−</sup>*Rψ*<sup>0</sup>

must hold, written as a scalar product implied with complex conjugation, and from this one

 *c*<sup>0</sup> √2 cos *θ*

with *<sup>c</sup>*<sup>0</sup> <sup>=</sup> *const*, where a branch of the analytic function <sup>√</sup>*<sup>z</sup>* is taken that satisfies the

*z*) ≥ 0, *z* ∈ C.

*<sup>e</sup>*1) >= <sup>0</sup>

<sup>−</sup>*i<sup>θ</sup>* ) + *<sup>σ</sup>*(0) *dei<sup>θ</sup>*

 *e* −*i <sup>θ</sup>* 2

*<sup>ψ</sup>*<sup>0</sup> <sup>=</sup> *<sup>σ</sup>*(0)

<sup>&</sup>lt; *<sup>e</sup>*1, *<sup>R</sup>*(*σ*(0)

*<sup>i</sup><sup>θ</sup>* + *<sup>e</sup>*

*σ*(0) =

*Im*( √ *<sup>i</sup><sup>θ</sup>* <sup>=</sup> *px* <sup>+</sup> *ipy <sup>E</sup>* <sup>−</sup> *<sup>U</sup>* .

*He*<sup>1</sup> = *Ee*1, *He*<sup>2</sup> = (2*<sup>U</sup>* − *<sup>E</sup>*)*e*2. (15)

*<sup>y</sup> dx*, (16)

(*x*)*e*<sup>1</sup> (17)

*dx* <sup>=</sup> 0. (18)

(*E* − *U*)<sup>2</sup> − *p*<sup>2</sup>

 1 ±*ei<sup>θ</sup>* , *e*

*pxdx* = ±

It turns out to be convenient to use different *e*1,2 instead with

*<sup>e</sup>*1,2 <sup>=</sup> <sup>1</sup> √2

*S* = 

with unknown amplitude *<sup>σ</sup>*(0). The solvability of the problem

*<sup>d</sup>σ*(0) *dx* (*<sup>e</sup>*

simultaneously. Note that, irrespective of whether *E* > *U* or *E* < *U*,

$$(H - EI)\psi\_{\hat{l}} = -\widehat{R}\psi\_{\hat{l}-1\nu}$$

one can seek a solution to

$$(H - EI)\psi\_{\hat{\jmath}} = -\hat{R}\psi\_{\hat{\jmath}-1}$$

(*j* > 0) in the form

$$
\psi\_{\dot{j}} = \sigma\_1^{(j)} e\_1 + \sigma\_2^{(j)} e\_{2\prime} \tag{19}
$$

where from (15), *<sup>σ</sup>*(*j*) <sup>2</sup> is given by

$$
\sigma\_2^{(j)} = \frac{}{\mathcal{Q}(E-U)}.
$$

Then, from the orthogonality condition,

$$ = \mathbf{0}\_\prime$$

one obtains

$$
\sigma\_1^{(j)} = \frac{e^{-i\theta/2}}{\sqrt{2\cos\theta}} \left( c\_j - \int e^{i\theta/2} \sqrt{2\cos\theta} < e\_1, \mathbb{R}(\sigma\_2^{(j)} e\_2) > d\mathfrak{x} \right),
$$

where *cj* = *const*. Below we assume that *px* > 0, corresponding to a wave traveling in the positive *x*-direction. Thus, to the leading order we have

$$\psi = \begin{pmatrix} u \\ v \end{pmatrix} = \frac{e^{\pm \frac{l}{h} S\_p(\mathbf{x}, \mathbf{x}\_l)}}{\sqrt{J\_p^{\pm}}} c\_0 e\_1^{\pm} (1 + O(h)), \tag{20}$$

$$S\_p(\mathbf{x}, \mathbf{x}\_i) = \int\_{\mathbf{x}\_i}^{\mathbf{x}} p\_{\mathbf{x}} dt\_{\mathbf{x}} \quad J\_p^{\pm} = 1 + e^{2i\theta^{\pm}}, \quad e\_1^{\pm} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{i\theta^{\pm}} \end{pmatrix},$$
 
$$\dots$$

$$e^{i\theta^{\pm}} = \frac{\pm p\_x + ip\_y}{E - U}.$$

This asymptotic approximation is not valid near turning points where *<sup>S</sup>*′ = 0 (see Fig. 1) at *<sup>x</sup>* <sup>=</sup> *xj*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3, 4 where *<sup>e</sup>i<sup>θ</sup>* <sup>=</sup> <sup>±</sup>*<sup>i</sup>* and cos *<sup>θ</sup>* <sup>=</sup> 0, while at *<sup>x</sup>* <sup>=</sup> *<sup>a</sup>*, *<sup>b</sup>* we have *<sup>E</sup>* <sup>=</sup> *<sup>U</sup>*. The WKB asymptotic solution, derived in this section, is valid for the domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 3, 5.

## **5. WKB asymptotic solution for Dirac system in classically disallowed domain**

The WKB asymptotic solution to the Dirac system in the classically disallowed domain is to be sought in the form

$$\psi = \begin{pmatrix} u \\ v \end{pmatrix} = e^{-\frac{1}{\hbar}S(x)} \sum\_{j=0}^{+\infty} (-ih)^j \begin{pmatrix} u\_j \\ v\_j \end{pmatrix} = e^{-\frac{1}{\hbar}S(x)} \sum\_{j=0}^{+\infty} (-ih)^j \psi\_j(x),\tag{21}$$

with *S*(*x*) real. As in section 4, we obtain a recurrent system of equations which determines the unknown *<sup>S</sup>*(*x*) and *<sup>ψ</sup>j*(*x*), namely,

$$(H - EI)\psi\_0 = 0, \quad (H - EI)\psi\_j = -R\psi\_{j-1}, \quad j > 0,\tag{22}$$

$$H = \begin{pmatrix} \mathcal{U} & i(q\_x - p\_y) \\ i(q\_x + p\_y) & \mathcal{U} \end{pmatrix},\tag{23}$$

10.5772/52267

45

, (25)

http://dx.doi.org/10.5772/52267

Again, for the sake of simplicity, we shall use different *l*1,2

where

the form

where

It has a solution

form

where *<sup>σ</sup>*(*j*)

<sup>2</sup> is given by

*<sup>l</sup>*1,2 <sup>=</sup> <sup>1</sup> √ 1 + *κ*<sup>2</sup>

*<sup>κ</sup>* <sup>=</sup> *qx* <sup>+</sup> *py*

with unknown amplitude *<sup>σ</sup>*(0). Solvability of the problem

*l* ∗ <sup>1</sup> <sup>=</sup> <sup>1</sup> √ 1 + *κ*<sup>2</sup>

orthogonality condition one obtains the transport equation for *<sup>σ</sup>*(0)

*<sup>d</sup>σ*(0)

*c*0 <sup>−</sup> cos 2*<sup>φ</sup>* <sup>=</sup> *<sup>c</sup>*<sup>0</sup>

> *σ*(*j*) <sup>2</sup> <sup>=</sup> <sup>&</sup>lt; *<sup>l</sup>*

requires that the orthogonality condition must hold

The vector *l*<sup>1</sup> is the eigenvector of *H*, whereas *l*

*<sup>σ</sup>*(0) <sup>=</sup>

 1 ±*iκ* =

*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>* , *<sup>κ</sup>* <sup>=</sup> tan *<sup>φ</sup>*, <sup>−</sup>*<sup>π</sup>*

*<sup>ψ</sup>*<sup>0</sup> <sup>=</sup> *<sup>σ</sup>*(0)

For electrons and holes one can seek a solution to the Dirac system zero-order problem in

(*<sup>H</sup>* <sup>−</sup> *EI*)*ψ*<sup>1</sup> <sup>=</sup> <sup>−</sup>*<sup>R</sup><sup>ψ</sup>*<sup>0</sup>

 *κ i* =

*dx* <sup>−</sup> *<sup>σ</sup>*(0) tan 2*<sup>φ</sup>*

 *κ*<sup>2</sup> + 1 *κ*<sup>2</sup> − 1

For higher-order terms, we have (*<sup>H</sup>* <sup>−</sup> *<sup>h</sup>*<sup>1</sup> *<sup>I</sup>*)*ψ<sup>j</sup>* <sup>=</sup> <sup>−</sup>*<sup>R</sup><sup>ψ</sup>j*−<sup>1</sup> and one should seek solution in the

∗

<sup>1</sup> *<sup>l</sup>*<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*(*j*)

<sup>1</sup> , *<sup>R</sup>*(*ψj*−1) <sup>&</sup>gt;

*<sup>ψ</sup><sup>j</sup>* <sup>=</sup> *<sup>σ</sup>*(*j*)

*<sup>l</sup>*1) >= 0,

∗

*dφ*

 sin *φ i* cos *φ*

 .

< *l* ∗ <sup>1</sup> , *<sup>R</sup>*(*σ*(0)  cos *φ* ±*i* sin *φ*

<sup>2</sup> <sup>&</sup>lt; *<sup>φ</sup>* <sup>&</sup>lt;

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

*π* 2 .

(*x*)*l*<sup>1</sup> (26)

<sup>1</sup> is the eigenvector of *<sup>H</sup>*∗. From the

*dx* <sup>=</sup> 0. (27)

, *<sup>c</sup>*<sup>0</sup> = *const*. (28)

<sup>2</sup> *l*2, (29)

<sup>2</sup>(*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*) . (30)

where *<sup>S</sup>*′ <sup>=</sup> *qx*, and the matrix *<sup>R</sup>* is as in (14). The Hamiltonian *<sup>H</sup>* is not Hermitian. It has two eigenvalues and not orthogonal eigenvectors *Hl*1,2 = *<sup>h</sup>*1,2*l*1,2, where

$$\begin{aligned} h\_{1,2} &= \mathcal{U}(\mathbf{x}) \pm \sqrt{p\_y^2 - q\_{\mathbf{x}}^2} \\\\ l\_{1,2} &= \begin{pmatrix} 1 \\ \pm i \sqrt{\frac{q\_x + p\_y}{p\_y - q\_{\mathbf{x}}}} \end{pmatrix} \end{aligned}$$

as we have

$$i\frac{q\_x + p\_y}{E - U} = \pm i\sqrt{\frac{q\_x + p\_y}{p\_y - q\_x}}$$

where

$$q\_x = \pm \sqrt{p\_y^2 - (E - \mathcal{U})^2}, \quad |q\_x| < p\_y.$$

Thus, the function *S*(*x*) in a classically disallowed domain is given by

$$\mathbf{S} = \int q\_{\mathbf{x}} d\mathbf{x} = \pm \int \sqrt{p\_y^2 - (\mathbf{E} - \mathbf{U})^2} d\mathbf{x}.\tag{24}$$

Again, for the sake of simplicity, we shall use different *l*1,2

$$d\_{1,2} = \frac{1}{\sqrt{1+\kappa^2}} \begin{pmatrix} 1 \\ \pm i\kappa \end{pmatrix} = \begin{pmatrix} \cos\phi \\ \pm i\sin\phi \end{pmatrix} ,\tag{25}$$

where

16 Graphene - Research and Applications

be sought in the form

*ψ* = *u v* = *e* − <sup>1</sup> *<sup>h</sup> <sup>S</sup>*(*x*) +∞ ∑ *j*=0

the unknown *<sup>S</sup>*(*x*) and *<sup>ψ</sup>j*(*x*), namely,

**domain**

as we have

where

**5. WKB asymptotic solution for Dirac system in classically disallowed**

(−*ih*)*<sup>j</sup>*

*H* =

two eigenvalues and not orthogonal eigenvectors *Hl*1,2 = *<sup>h</sup>*1,2*l*1,2, where

*<sup>h</sup>*1,2 = *<sup>U</sup>*(*x*) ±

*<sup>l</sup>*1,2 =

*i qx* + *py <sup>E</sup>* <sup>−</sup> *<sup>U</sup>* <sup>=</sup> <sup>±</sup>*<sup>i</sup>*

*qx* = ±

*S* =   *p*2

*qxdx* = ±

 *p*2

Thus, the function *S*(*x*) in a classically disallowed domain is given by

The WKB asymptotic solution to the Dirac system in the classically disallowed domain is to

 *uj vj* = *e* − <sup>1</sup> *<sup>h</sup> <sup>S</sup>*(*x*) +∞ ∑ *j*=0

with *S*(*x*) real. As in section 4, we obtain a recurrent system of equations which determines

 *U i*(*qx* − *py*) *i*(*qx* + *py*) *U*

where *<sup>S</sup>*′ <sup>=</sup> *qx*, and the matrix *<sup>R</sup>* is as in (14). The Hamiltonian *<sup>H</sup>* is not Hermitian. It has

 1 ±*i*

 *p*2 *<sup>y</sup>* − *<sup>q</sup>*<sup>2</sup> *x*,

*qx*+*py py*−*qx*  ,

*qx* + *py py* − *qx*

*<sup>y</sup>* − (*<sup>E</sup>* − *<sup>U</sup>*)2, |*qx*| < *py*.

(*<sup>H</sup>* − *EI*)*ψ*<sup>0</sup> = 0, (*<sup>H</sup>* − *EI*)*ψ<sup>j</sup>* = −*Rψj*−1, *<sup>j</sup>* > 0, (22)

(−*ih*)*<sup>j</sup>*

*<sup>ψ</sup>j*(*x*), (21)

, (23)

*<sup>y</sup>* − (*<sup>E</sup>* − *<sup>U</sup>*)2*dx*. (24)

$$\kappa = \frac{q\_\times + p\_y}{E - \mathcal{U}}, \quad \kappa = \tan \phi, \quad -\frac{\pi}{2} < \phi < \frac{\pi}{2}.$$

For electrons and holes one can seek a solution to the Dirac system zero-order problem in the form

$$
\psi\_0 = \sigma^{(0)}(\mathfrak{x}) l\_1 \tag{26}
$$

with unknown amplitude *<sup>σ</sup>*(0). Solvability of the problem

$$(H - EI)\psi\_1 = -\widehat{R}\psi\_0$$

requires that the orthogonality condition must hold

$$ = 0,$$

where

$$l\_1^\* = \frac{1}{\sqrt{1+\kappa^2}} \begin{pmatrix} \kappa \\ i \end{pmatrix} = \begin{pmatrix} \sin\phi \\ i\cos\phi \end{pmatrix}.$$

The vector *l*<sup>1</sup> is the eigenvector of *H*, whereas *l* ∗ <sup>1</sup> is the eigenvector of *<sup>H</sup>*∗. From the orthogonality condition one obtains the transport equation for *<sup>σ</sup>*(0)

$$\frac{d\sigma^{(0)}}{d\mathbf{x}} - \sigma^{(0)}\tan 2\phi \frac{d\phi}{d\mathbf{x}} = 0. \tag{27}$$

It has a solution

$$
\sigma^{(0)} = \frac{c\_0}{\sqrt{-\cos 2\phi}} = c\_0 \sqrt{\frac{\kappa^2 + 1}{\kappa^2 - 1}}, \quad c\_0 = \text{const.} \tag{28}
$$

For higher-order terms, we have (*<sup>H</sup>* <sup>−</sup> *<sup>h</sup>*<sup>1</sup> *<sup>I</sup>*)*ψ<sup>j</sup>* <sup>=</sup> <sup>−</sup>*<sup>R</sup><sup>ψ</sup>j*−<sup>1</sup> and one should seek solution in the form

$$
\psi\_j = \sigma\_1^{(j)} l\_1 + \sigma\_2^{(j)} l\_{2'} \tag{29}
$$

where *<sup>σ</sup>*(*j*) <sup>2</sup> is given by

$$
\sigma\_2^{(j)} = \frac{}{\mathbf{2}(E-ll)}.\tag{30}
$$

Then, from the orthogonality condition, < *l* ∗ <sup>1</sup> , *<sup>R</sup>*(*σ*(*j*) <sup>1</sup> *<sup>l</sup>*<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*(*j*) <sup>2</sup> *<sup>l</sup>*2) <sup>&</sup>gt;<sup>=</sup> 0 we obtain

$$\sigma\_1^{(j)} = \frac{1}{\sqrt{-\cos 2\phi}} \left( c\_j - \int \sqrt{-\cos 2\phi} < l\_1^\*, R(\sigma\_2^{(j)} l\_2) > d\mathbf{x} \right), \quad c\_j = \text{const.} \tag{31}$$

Below we assume that *qx* > 0. Thus, to the leading order in classically disallowed domains we have

$$\psi = \frac{e^{\mp \frac{1}{h} \mathcal{S}\_q(\mathbf{x}, \mathbf{x}\_l)}}{\sqrt{I\_q^{\pm}}} l\_1^{\pm} (1 + O(h)), \tag{32}$$

10.5772/52267

47

http://dx.doi.org/10.5772/52267

*TL*, (35)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

*<sup>h</sup>* ], (36)

*<sup>h</sup>* ], (37)

], (38)

], (39)

The barrier is represented by the combination of the left and right slopes. The total transfer

with *T<sup>R</sup>* and *T<sup>L</sup>* the transfer matrices of the right and left slopes (see formulas (137), (143) in

*U*(*x*) − *E*

<sup>2</sup> − *p*<sup>2</sup> *ydx*.

> *<sup>h</sup>* ) <sup>+</sup> *<sup>e</sup>* −*i <sup>P</sup>*

> > *<sup>h</sup>* ) <sup>+</sup> *<sup>e</sup> i P*

> > > −*i*(*θ*1+ *<sup>P</sup> h* )

> > > > *i*(*θ*1+ *<sup>P</sup> h* )

*<sup>h</sup>* ) <sup>+</sup> *<sup>s</sup>*1*<sup>e</sup>*

*<sup>h</sup>* ) <sup>+</sup> *<sup>s</sup>*1*<sup>e</sup>*

<sup>12</sup>, det *<sup>T</sup>* <sup>=</sup> 1,

*i*(*θ*1+*θ*2+ *<sup>P</sup>*

−*i*(*θ*1+*θ*2+ *<sup>P</sup>*

*i*(*θ*2+ *<sup>P</sup>*

−*i*(*θ*2+ *<sup>P</sup>*

1 − *e*−2*Qi*/*h*, *i* = 1, 2. They satisfy the classical properties of the transfer

*T* = *T<sup>R</sup>*

*P* = *x*3

*<sup>T</sup>*<sup>11</sup> = *<sup>e</sup>*

*<sup>T</sup>*<sup>22</sup> = *<sup>e</sup>*

*<sup>T</sup>*<sup>12</sup> = −sgn(*py*)*<sup>e</sup>*

*<sup>T</sup>*<sup>21</sup> = −sgn(*py*)*<sup>e</sup>*

*<sup>T</sup>*<sup>22</sup> <sup>=</sup> *<sup>T</sup>*<sup>∗</sup>

and if *<sup>a</sup>*<sup>1</sup> = 1, *<sup>a</sup>*<sup>2</sup> = *<sup>r</sup>*1, *<sup>d</sup>*<sup>1</sup> = *<sup>t</sup>*1, *<sup>d</sup>*<sup>2</sup> = 0, then

If *<sup>a</sup>*<sup>1</sup> = 0, *<sup>a</sup>*<sup>2</sup> = *<sup>t</sup>*2, *<sup>d</sup>*<sup>1</sup> = *<sup>r</sup>*2, *<sup>d</sup>*<sup>2</sup> = 1, then

*x*2

*Q*1 *<sup>h</sup>* <sup>+</sup> *<sup>Q</sup>*<sup>2</sup>

*Q*1 *<sup>h</sup>* <sup>+</sup> *<sup>Q</sup>*<sup>2</sup>

The entries of the matrix *T* read (see formulas (121), (134), (144) in Appendix C)

*<sup>h</sup>* [*s*1*s*2*<sup>e</sup>*

*<sup>h</sup>* [*s*1*s*2*<sup>e</sup>*

*Q*1 *<sup>h</sup>* <sup>+</sup> *<sup>Q</sup>*<sup>2</sup> *<sup>h</sup>* [*s*2*<sup>e</sup>*

*Q*1 *<sup>h</sup>* <sup>+</sup> *<sup>Q</sup>*<sup>2</sup> *<sup>h</sup>* [*s*2*<sup>e</sup>*

<sup>11</sup>, *<sup>T</sup>*<sup>21</sup> <sup>=</sup> *<sup>T</sup>*<sup>∗</sup>

*<sup>t</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> *T*<sup>22</sup> ,

*<sup>r</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>T</sup>*<sup>21</sup> *T*<sup>22</sup> ,

<sup>2</sup> + |*r*1|

*<sup>t</sup>*<sup>2</sup> = *<sup>t</sup>*<sup>1</sup> = *<sup>t</sup>*,

<sup>2</sup> = 1.


 *e i <sup>h</sup> <sup>P</sup>* 0 <sup>0</sup> *<sup>e</sup>*<sup>−</sup> *<sup>i</sup> h P* 

matrix *T*, that is *d* = *Ta*, is given by

Appendix C), respectively, and

where *si* <sup>=</sup> <sup>√</sup>

matrix

where

$$\begin{aligned} S\_q(\mathbf{x}, \mathbf{x}\_i) &= \int\_{\mathbf{x}\_i}^{\mathbf{x}} q\_{\mathbf{x}} dt\_\prime \quad J\_q^{\pm} = \pm ((\kappa^{\pm})^2 - 1)\mu \\\\ l\_1^{\pm} &= \begin{pmatrix} 1 \\ i\kappa^{\pm} \end{pmatrix} \end{aligned}$$

and

$$
\kappa^{\pm} = \frac{\pm q\_{\mathfrak{x}} + p\_{\mathfrak{y}}}{E - U}.
$$

This asymptotic approximation is not valid near turning points *qx* = 0. The WKB asymptotic solution, derived in this section, is valid for the domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 2, 4.

#### **6. WKB asymptotic solution for scattering through the smooth barrier**

Consider a problem of scattering through the smooth barrier (see Fig. 2) under the assumption that |*E*| > |*py*| and all four turning points *xi*, *<sup>i</sup>* = 1, 2, 3, 4 are separated. In this case we have again 5 domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 2, ..., 5 to describe 5 WKB forms of solution to the leading order. In considering a graphene system, if *E* > 0 we observe incident, reflected and transmitted electronic states at *x* < *a* and *x* > *b*, whereas under the barrier *a* < *x* < *b* we have a hole state (n-p-n junction, see Fig. 2).

To formulate the scattering problem for transfer matrix *T*, here we present the WKB solutions in the domains 1 and 5

$$\psi\_1 = \frac{e^{\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^+}} a\_1 e\_1^+ + \frac{e^{-\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^-}} a\_2 e\_1^- \,\mathrm{}\,\tag{33}$$

$$\psi\_{\sf S} = \frac{e^{\frac{i}{\hbar}S\_{\sf p}(\mathbf{x}, \mathbf{x}\_{4})}}{\sqrt{I\_{\sf p}^{+}}} d\_{1}e\_{1}^{+} + \frac{e^{-\frac{i}{\hbar}S\_{\sf p}(\mathbf{x}, \mathbf{x}\_{4})}}{\sqrt{I\_{\sf p}^{-}}} d\_{2}e\_{1}^{-}. \tag{34}$$

The barrier is represented by the combination of the left and right slopes. The total transfer matrix *T*, that is *d* = *Ta*, is given by

$$T = T^R \begin{pmatrix} e^{\frac{i}{\hbar}P} & 0 \\ 0 & e^{-\frac{i}{\hbar}P} \end{pmatrix} T^L \tag{35}$$

with *T<sup>R</sup>* and *T<sup>L</sup>* the transfer matrices of the right and left slopes (see formulas (137), (143) in Appendix C), respectively, and

$$P = \int\_{x\_2}^{x\_3} \sqrt{\left(U(x) - E\right)^2 - p\_y^2} d\mu.$$

The entries of the matrix *T* read (see formulas (121), (134), (144) in Appendix C)

$$T\_{11} = e^{\frac{Q\_1}{\hbar} + \frac{Q\_2}{\hbar}} \left[ s\_1 s\_2 e^{i(\theta\_1 + \theta\_2 + \frac{P}{\hbar})} + e^{-i\frac{P}{\hbar}} \right] \tag{36}$$

$$T\_{22} = e^{\frac{Q\_1}{\hbar} + \frac{Q\_2}{\hbar}} \left[ s\_1 s\_2 e^{-i(\theta\_1 + \theta\_2 + \frac{P}{\hbar})} + e^{i\frac{P}{\hbar}} \right] \tag{37}$$

$$T\_{12} = -\text{sgn}(p\_y)e^{\frac{Q\_1}{\hbar} + \frac{Q\_2}{\hbar}} \left[ s\_2 e^{i(\theta\_2 + \frac{P}{\hbar})} + s\_1 e^{-i(\theta\_1 + \frac{P}{\hbar})} \right] \tag{38}$$

$$T\_{21} = -\text{sgn}(p\_y)e^{\frac{Q\_1}{\hbar} + \frac{Q\_2}{\hbar}} \left[ s\_2 e^{-i(\theta\_2 + \frac{P}{\hbar})} + s\_1 e^{i(\theta\_1 + \frac{P}{\hbar})} \right] \tag{39}$$

where *si* <sup>=</sup> <sup>√</sup> 1 − *e*−2*Qi*/*h*, *i* = 1, 2. They satisfy the classical properties of the transfer matrix

$$T\_{22} = T\_{11\prime}^\* \qquad T\_{21} = T\_{12\prime}^\* \qquad \det T = 1\_{\prime\prime}$$

and if *<sup>a</sup>*<sup>1</sup> = 1, *<sup>a</sup>*<sup>2</sup> = *<sup>r</sup>*1, *<sup>d</sup>*<sup>1</sup> = *<sup>t</sup>*1, *<sup>d</sup>*<sup>2</sup> = 0, then

18 Graphene - Research and Applications

*σ*(*j*)

we have

where

and

<sup>1</sup> <sup>=</sup> <sup>1</sup>

Then, from the orthogonality condition, < *l*

− cos 2*φ*

have a hole state (n-p-n junction, see Fig. 2).

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup> i <sup>h</sup> Sp* (*x*,*x*1) *J* + *p*

*<sup>ψ</sup>*<sup>5</sup> <sup>=</sup> *<sup>e</sup> i <sup>h</sup> Sp* (*x*,*x*4) *J* + *p*

in the domains 1 and 5

 *cj* −

*Sq*(*x*, *xi*) =

∗ <sup>1</sup> , *<sup>R</sup>*(*σ*(*j*)

<sup>−</sup> cos 2*<sup>φ</sup>* <sup>&</sup>lt; *<sup>l</sup>*

*<sup>h</sup> Sq* (*x*,*xi*) *J* ± *q*

*qxdt*, *J*

*<sup>ψ</sup>* <sup>=</sup> *<sup>e</sup>*<sup>∓</sup> <sup>1</sup>

 *<sup>x</sup> xi*

> *l* ± <sup>1</sup> <sup>=</sup>

solution, derived in this section, is valid for the domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 2, 4.

Below we assume that *qx* > 0. Thus, to the leading order in classically disallowed domains

*l* ±

±

 1 *<sup>i</sup>κ*<sup>±</sup> 

*<sup>κ</sup>*<sup>±</sup> <sup>=</sup> <sup>±</sup>*qx* <sup>+</sup> *py*

This asymptotic approximation is not valid near turning points *qx* = 0. The WKB asymptotic

Consider a problem of scattering through the smooth barrier (see Fig. 2) under the assumption that |*E*| > |*py*| and all four turning points *xi*, *<sup>i</sup>* = 1, 2, 3, 4 are separated. In this case we have again 5 domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 2, ..., 5 to describe 5 WKB forms of solution to the leading order. In considering a graphene system, if *E* > 0 we observe incident, reflected and transmitted electronic states at *x* < *a* and *x* > *b*, whereas under the barrier *a* < *x* < *b* we

To formulate the scattering problem for transfer matrix *T*, here we present the WKB solutions

*e*− *i*

*e*− *i*

*<sup>h</sup> Sp* (*x*,*x*1) *J* − *p*

*<sup>h</sup> Sp* (*x*,*x*4) *J* − *p*

*a*2*e* −

*d*2*e* −

<sup>1</sup> , (33)

<sup>1</sup> . (34)

*a*1*e* + <sup>1</sup> <sup>+</sup>

*d*1*e* + <sup>1</sup> <sup>+</sup>

**6. WKB asymptotic solution for scattering through the smooth barrier**

*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>* .

*<sup>q</sup>* <sup>=</sup> <sup>±</sup>((*κ*±)<sup>2</sup> <sup>−</sup> <sup>1</sup>),

<sup>1</sup> *<sup>l</sup>*<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*(*j*)

∗ <sup>1</sup> , *<sup>R</sup>*(*σ*(*j*)

<sup>2</sup> *<sup>l</sup>*2) <sup>&</sup>gt;<sup>=</sup> 0 we obtain

<sup>1</sup> (<sup>1</sup> <sup>+</sup> *<sup>O</sup>*(*h*)), (32)

, *cj* = *const*. (31)

<sup>2</sup> *<sup>l</sup>*2) <sup>&</sup>gt; *dx*

$$t\_1 = \frac{1}{T\_{22}} \prime$$

$$r\_1 = -\frac{T\_{21}}{T\_{22}}\prime$$

$$|t\_1|^2 + |r\_1|^2 = 1.$$

If *<sup>a</sup>*<sup>1</sup> = 0, *<sup>a</sup>*<sup>2</sup> = *<sup>t</sup>*2, *<sup>d</sup>*<sup>1</sup> = *<sup>r</sup>*2, *<sup>d</sup>*<sup>2</sup> = 1, then

$$t\_2 = t\_1 = t\_\prime$$

$$r\_2(p\_y) = \frac{T\_{12}}{T\_{22}}\prime$$

10.5772/52267

49

http://dx.doi.org/10.5772/52267

*<sup>T</sup>*22(*E*) = 0, (44)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

, *<sup>n</sup>* = 0, 1, 2, ..., *<sup>N</sup>*<sup>1</sup> (45)

(*En*). (46)

and as a result we obtain Bohr-Sommerfeld quantization condition for complex energy

<sup>2</sup> log (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*

for |*py*| < *<sup>E</sup>* < *<sup>U</sup>*0. Solutions to this equation are complex resonances *En* = *Re*(*En*) − *<sup>i</sup>*Γ*n*, where <sup>Γ</sup>−<sup>1</sup> *<sup>n</sup>* is the lifetime of the localized resonance. What is important is that the real part of these complex positive resonances is decreasing with *n*, thus showing off the anti-particle hole-like character of the localized modes. For these resonances we have Γ*<sup>n</sup>* > 0. From (45),

that is the equivalent of the formula (14) in [35]. Namely, *w* is the transmission probability through the tunneling strip, ∆*t* is the time interval between the turning points *x*<sup>2</sup> and *x*3, and *<sup>P</sup>*′ is the first derivative of *<sup>P</sup>* with respect to energy. If *py* <sup>→</sup> 0, then *<sup>Q</sup>* <sup>→</sup> 0, and <sup>Γ</sup>*<sup>n</sup>* <sup>→</sup> <sup>+</sup>∞, that is opposite to [35] (to be exact, the estimate for Γ*<sup>n</sup>* in [35] works only for a linear potential

**Figure 4.** (a) The dispersion of energy levels *En*(*py* ) for complex and real bound states for *n* = 0, 1, ....., 15 are shown for *h* = 0.1 and *U* = 1/*coshx*. (b) As in (a), except that *n* = 0, 1, ....., 9 and *U* = 1/*cosh*2*x*. For complex resonant bound states the real part was taken. The energies *E* = ±*py* and *E* = *U*<sup>0</sup> − *py* are shown with thick black lines. Semiclassical solutions are shown by the lines in zones 2 and 3. The upper and lower bounds for the dispersion branches are shown by the boundaries between zones 1, 4 and 6 with zones 2 and 3. The black line *py* = *E*, running between zones 2 and 3 is the upper bound for

−2*Q <sup>h</sup>* ) 

−2*Q*

*<sup>h</sup>* ), <sup>∆</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*P*′

) − *<sup>θ</sup>* − *<sup>i</sup>*

, *w* = − log (1 − *e*

If *<sup>a</sup>*<sup>1</sup> = 0, *<sup>d</sup>*<sup>2</sup> = 0, then

eigen-levels (quasi-discrete)

*P*(*E*) = *h*

we obtain the important estimate

when *py* is not small).

the bound states Γ*<sup>n</sup>* = 0.

 *π*(*n* + 1 2

<sup>Γ</sup>*<sup>n</sup>* <sup>=</sup> *hw* 2∆*t*

$$|t\_2|^2 + |r\_2|^2 = 1$$

(see appendix B). Correspondingly, the unitary scattering matrix connecting

$$
\begin{pmatrix} a\_2 \\ d\_1 \end{pmatrix} = \mathcal{S} \begin{pmatrix} a\_1 \\ d\_2 \end{pmatrix},
$$

may be written as follows

$$
\hat{S} = \begin{pmatrix} r\_1 & t \\ t & r\_2 \end{pmatrix}.
$$

The transmission coefficient *<sup>t</sup>* = 1/*T*22, looks exactly like the formula (131) in [18]. Total transmission takes place only for a symmetric barrier when *<sup>Q</sup>*<sup>2</sup> = *<sup>Q</sup>*<sup>1</sup> = *<sup>Q</sup>* (*θ*<sup>2</sup> = *<sup>θ</sup>*<sup>1</sup> = *<sup>θ</sup>*). Then

$$t = e^{i\theta} \left( \cos\left(\frac{P}{h} + \theta\right) (2e^{\frac{2Q}{h}} - 1) + i\sin\left(\frac{P}{h} + \theta\right) \right)^{-1},\tag{40}$$

$$r\_1(p\_y) = \frac{2\text{sgn}(p\_y)\cos\left(\frac{P}{\hbar} + \theta\right)e^{\frac{2Q}{\hbar} + i\theta}\sqrt{1 - e^{-2Q/\hbar}}}{\cos\left(\frac{P}{\hbar} + \theta\right)(2e^{\frac{2Q}{\hbar}} - 1) + i\sin\left(\frac{P}{\hbar} + \theta\right)} = -r\_2(p\_y). \tag{41}$$

However, it is worth noting that *<sup>r</sup>*1(*py*) = *<sup>r</sup>*2(−*py*). It is clear that if

$$P(E) + h\theta = h\pi(n + \frac{1}{2}), \quad n = 0, 1, 2, \dots \tag{42}$$

then we have total transmission |*t*1| = 1.

#### **7. WKB asymptotic solution for complex resonant (quasibound) states localized within the smooth barrier**

Consider a problem of resonant states localized within the smooth barrier (see Fig. 2). In the first case when the energy of the electron-hole is greater than the cut-off energy (*E* > *Ec* = |*py*|), we have five domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 2, ..., 5 and five WKB forms of solution to the leading order. To determine the correct radiation conditions that are necessary for the localization, we present WKB solutions in the domains 1 and 5

$$\psi\_1 = \frac{e^{-\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^-}} a\_2 e\_1^{-} \qquad \psi\_5 = \frac{e^{\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_4)}}{\sqrt{J\_p^+}} d\_1 e\_1^{+} . \tag{43}$$

If *<sup>a</sup>*<sup>1</sup> = 0, *<sup>d</sup>*<sup>2</sup> = 0, then

20 Graphene - Research and Applications

may be written as follows

*t* = *e iθ* cos ( *P <sup>h</sup>* <sup>+</sup> *<sup>θ</sup>*)(2*<sup>e</sup>*

then we have total transmission |*t*1| = 1.

**localized within the smooth barrier**

we present WKB solutions in the domains 1 and 5

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> *<sup>i</sup>*

*<sup>h</sup> Sp* (*x*,*x*1) *J* − *p*

*a*2*e* −

*<sup>r</sup>*1(*py*) = 2sgn(*py*) cos ( *<sup>P</sup>*

cos ( *<sup>P</sup>*

*<sup>h</sup>* <sup>+</sup> *<sup>θ</sup>*)(2*<sup>e</sup>*

*P*(*E*) + *hθ* = *hπ*(*n* +

However, it is worth noting that *<sup>r</sup>*1(*py*) = *<sup>r</sup>*2(−*py*). It is clear that if

Then

*<sup>r</sup>*2(*py*) = *<sup>T</sup>*<sup>12</sup>

<sup>2</sup> + |*r*2|


(see appendix B). Correspondingly, the unitary scattering matrix connecting

 *a*<sup>2</sup> *d*1 = *S*ˆ *a*<sup>1</sup> *d*2 

*S*ˆ =

 *r*<sup>1</sup> *t t r*<sup>2</sup>

The transmission coefficient *<sup>t</sup>* = 1/*T*22, looks exactly like the formula (131) in [18]. Total transmission takes place only for a symmetric barrier when *<sup>Q</sup>*<sup>2</sup> = *<sup>Q</sup>*<sup>1</sup> = *<sup>Q</sup>* (*θ*<sup>2</sup> = *<sup>θ</sup>*<sup>1</sup> = *<sup>θ</sup>*).

2*Q*

2*Q <sup>h</sup>* <sup>+</sup>*i<sup>θ</sup>* √

1 2

*<sup>h</sup>* − 1) + *i* sin ( *<sup>P</sup>*

*<sup>h</sup>* <sup>+</sup> *<sup>θ</sup>*)*<sup>e</sup>*

**7. WKB asymptotic solution for complex resonant (quasibound) states**

Consider a problem of resonant states localized within the smooth barrier (see Fig. 2). In the first case when the energy of the electron-hole is greater than the cut-off energy (*E* > *Ec* = |*py*|), we have five domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 2, ..., 5 and five WKB forms of solution to the leading order. To determine the correct radiation conditions that are necessary for the localization,

<sup>1</sup> , *<sup>ψ</sup>*<sup>5</sup> <sup>=</sup> *<sup>e</sup>*

*i <sup>h</sup> Sp* (*x*,*x*4) *J* + *p*

*d*1*e* +

<sup>1</sup> . (43)

2*Q*

 .

*<sup>h</sup>* − 1) + *i* sin (

*P <sup>h</sup>* <sup>+</sup> *<sup>θ</sup>*)

1 − *e*−2*Q*/*<sup>h</sup>*

<sup>−</sup><sup>1</sup>

*<sup>h</sup>* <sup>+</sup> *<sup>θ</sup>*) <sup>=</sup> <sup>−</sup>*r*2(*py*). (41)

), *n* = 0, 1, 2, ... , (42)

, (40)

*T*<sup>22</sup> ,

<sup>2</sup> = 1

$$T\_{22}(E) = 0,\tag{44}$$

and as a result we obtain Bohr-Sommerfeld quantization condition for complex energy eigen-levels (quasi-discrete)

$$P(E) = h\left(\pi(n + \frac{1}{2}) - \theta - \frac{i}{2}\log\left(1 - e^{\frac{-2Q}{k}}\right)\right), \quad n = 0, 1, 2, \dots, N\_1 \tag{45}$$

for |*py*| < *<sup>E</sup>* < *<sup>U</sup>*0. Solutions to this equation are complex resonances *En* = *Re*(*En*) − *<sup>i</sup>*Γ*n*, where <sup>Γ</sup>−<sup>1</sup> *<sup>n</sup>* is the lifetime of the localized resonance. What is important is that the real part of these complex positive resonances is decreasing with *n*, thus showing off the anti-particle hole-like character of the localized modes. For these resonances we have Γ*<sup>n</sup>* > 0. From (45), we obtain the important estimate

$$
\Gamma\_{\rm \mu} = \frac{hw}{2\Delta t}, \quad w = -\log\left(1 - e^{\frac{-2Q}{h}}\right), \quad \Delta t = -P'(E\_n). \tag{46}
$$

that is the equivalent of the formula (14) in [35]. Namely, *w* is the transmission probability through the tunneling strip, ∆*t* is the time interval between the turning points *x*<sup>2</sup> and *x*3, and *<sup>P</sup>*′ is the first derivative of *<sup>P</sup>* with respect to energy. If *py* <sup>→</sup> 0, then *<sup>Q</sup>* <sup>→</sup> 0, and <sup>Γ</sup>*<sup>n</sup>* <sup>→</sup> <sup>+</sup>∞, that is opposite to [35] (to be exact, the estimate for Γ*<sup>n</sup>* in [35] works only for a linear potential when *py* is not small).

**Figure 4.** (a) The dispersion of energy levels *En*(*py* ) for complex and real bound states for *n* = 0, 1, ....., 15 are shown for *h* = 0.1 and *U* = 1/*coshx*. (b) As in (a), except that *n* = 0, 1, ....., 9 and *U* = 1/*cosh*2*x*. For complex resonant bound states the real part was taken. The energies *E* = ±*py* and *E* = *U*<sup>0</sup> − *py* are shown with thick black lines. Semiclassical solutions are shown by the lines in zones 2 and 3. The upper and lower bounds for the dispersion branches are shown by the boundaries between zones 1, 4 and 6 with zones 2 and 3. The black line *py* = *E*, running between zones 2 and 3 is the upper bound for the bound states Γ*<sup>n</sup>* = 0.

10.5772/52267

51

<sup>1</sup> , *<sup>x</sup>* <sup>&</sup>gt; *<sup>x</sup>*3. (49)

http://dx.doi.org/10.5772/52267

and outside decaying solutions,

obtain the homogeneous system

**8. Numerical results**

present the transmission probabilities |*t*|

Fabry-Perot multiple interference effects inside the barrier.

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup>* 1 *<sup>h</sup> Sq* (*x*,*x*2) *J* − *q*

*c*¯2*l* −

(bound states) inside the cut-off energy strip for 0 < *E* < |*py*|.

*P*(*E*) = *hπ*(*n* +

<sup>1</sup> , *<sup>x</sup>* <sup>&</sup>lt; *<sup>x</sup>*2, *<sup>ψ</sup>*<sup>3</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>1</sup>

By gluing these WKB solutions together through the two boundary layers near *x*<sup>2</sup> and *x*3, using the results in sections 5.1, 5.2 and the Appendix C, we eliminate *a*¯1,2 and ¯*d*1,2 and

> *i <sup>h</sup> <sup>P</sup>* = 0,

−*i <sup>h</sup> <sup>P</sup>* = 0.

Thus, we derive the Bohr-Sommerfeld quantization condition for real energy eigen-levels

Based upon the analytical descriptions in the preceding sections for the smooth barrier, we present the results for the energy eigenvalues and eigenfunctions. These are shown in Fig's. 4-6 and compare favorably with those obtained through finite difference methods, as detailed in [71]. The energy dispersion curves, *En*(*py*), are shown for the complex resonant and real bound states for *h* = 0.1 and potentials of different widths. In Fig. 4(a), the energy levels are illustrated for the potential, *U* = 1/*coshx*, with *n* = 0, 1, ...., 15. For complex resonant states the real parts are shown. It must be emphasized that in zone "3", which is restricted by *<sup>E</sup>* < *<sup>U</sup>*<sup>0</sup> − *py* and *<sup>E</sup>* = *py* with *py* > 0, the complex quasibound states reside. The bound states are located in zone "2", which lies between *<sup>E</sup>* = ±*py* and *<sup>E</sup>* = *<sup>U</sup>*<sup>0</sup> − *py*. In zone "3" there are nine complex resonances. In Fig. 4(b), the results for a narrower potential of *U* = 1/*cosh*2*x* can be seen (all other parameters being the same as in Fig. 4 (a)). In this case, there are four complex resonances in zone "3" and *n* = 0, 1, ...., 9. The lifetimes of the local resonances, Γ*n*, are shown in Fig. 5 (a) and (b) for the same two potentials as described in Fig. 4. The complex quasi-bound states that are witnessed in zone "3" in Fig. 4 are shown in Fig. 5 for Γ*n*. The thinner potential allows less complex bound states. The bound states have infinite lifetimes. Both types of states are confined within the barrier in the x-direction, while the motion in the y-direction is controlled by the dispersion relations. In Fig. 6 we

resonances, i.e. complex quasi-bound states within potential barrier defined as *U* = 1/*coshx* that correlate with those shown in Fig. 4 in zone "3". Likewise, for the thinner barrier there are four complex quasi-bound states. These resonance states are a clear indication of the

*ic*¯1 + *<sup>c</sup>*¯2*<sup>e</sup>*

*ic*¯1 − *<sup>c</sup>*¯2*<sup>e</sup>*

1 2

*<sup>h</sup> Sq* (*x*,*x*3) *J* + *q*

*c*¯1*l* +

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

), *<sup>n</sup>* = *<sup>N</sup>*<sup>1</sup> + 1, ...*N*2. (50)

<sup>2</sup> for the two potentials. There are nine tunneling

**Figure 5.** (a) The imaginary part Γ*<sup>n</sup>* of the first nine quasi-bound eigenvalues. The semiclassical solutions are shown by the blue lines and the shape of the potential is shown in the inset (*U* = 1/*coshx*). (b) There are four quasibound states for Γ*<sup>n</sup>* associated with a *U* = 1/*cosh*2*x* potential. The narrower potential allows less complex bound states.

**Figure 6.** The transmission probability |*t*| <sup>2</sup> is shown in these colorbar diagrams with respect to dimensionless *py* and *px* = *E*<sup>2</sup> − *p*<sup>2</sup> *<sup>y</sup>* for the barriers (a) *<sup>U</sup>* <sup>=</sup> 1/ cosh *<sup>x</sup>* and (b) *<sup>U</sup>* <sup>=</sup> 1/ cosh 2*x*.

For the second set of real resonances, when the energy of the electron-hole is smaller than the cut-off energy (*E* < |*py*|), we have 2 turning points *x*<sup>2</sup> and *x*3. Between them there are oscillatory WKB solutions

$$\psi\_1 = \frac{e^{\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^+}} \bar{d}\_1 e\_1^+ + \frac{e^{-\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^-}} \bar{d}\_2 e\_1^- \tag{47}$$

or

$$\psi\_1 = \frac{e^{\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_3)}}{\sqrt{J\_p^+}} \overline{a}\_1 e\_1^+ + \frac{e^{-\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_3)}}{\sqrt{J\_p^-}} \overline{a}\_2 e\_1^- \tag{48}$$

and outside decaying solutions,

22 Graphene - Research and Applications

**Figure 6.** The transmission probability |*t*|

oscillatory WKB solutions

 *E*<sup>2</sup> − *p*<sup>2</sup>

or

**Figure 5.** (a) The imaginary part Γ*<sup>n</sup>* of the first nine quasi-bound eigenvalues. The semiclassical solutions are shown by the blue lines and the shape of the potential is shown in the inset (*U* = 1/*coshx*). (b) There are four quasibound states for Γ*<sup>n</sup>* associated

For the second set of real resonances, when the energy of the electron-hole is smaller than the cut-off energy (*E* < |*py*|), we have 2 turning points *x*<sup>2</sup> and *x*3. Between them there are

*e*− *i*

*e*− *i*

*<sup>h</sup> Sp* (*x*,*x*2) *J* − *p*

*<sup>h</sup> Sp* (*x*,*x*3) *J* − *p*

¯*d*1*e* + <sup>1</sup> <sup>+</sup>

*a*¯1*e* + <sup>1</sup> <sup>+</sup>

<sup>2</sup> is shown in these colorbar diagrams with respect to dimensionless *py* and *px* =

¯*d*2*e* −

*a*¯2*e* −

<sup>1</sup> , (47)

<sup>1</sup> , (48)

with a *U* = 1/*cosh*2*x* potential. The narrower potential allows less complex bound states.

*<sup>y</sup>* for the barriers (a) *<sup>U</sup>* <sup>=</sup> 1/ cosh *<sup>x</sup>* and (b) *<sup>U</sup>* <sup>=</sup> 1/ cosh 2*x*.

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup> i <sup>h</sup> Sp* (*x*,*x*2) *J* + *p*

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup> i <sup>h</sup> Sp* (*x*,*x*3) *J* + *p*

$$\psi\_1 = \frac{e^{\frac{1}{\hbar}S\_4(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_q^-}} \overline{\varepsilon}\_2 l\_1^- \quad \mathbf{x} < \mathbf{x}\_2, \quad \psi\_3 = \frac{e^{-\frac{1}{\hbar}S\_4(\mathbf{x}, \mathbf{x}\_3)}}{\sqrt{J\_q^+}} \overline{\varepsilon}\_1 l\_1^+ \quad \mathbf{x} > \mathbf{x}\_3. \tag{49}$$

By gluing these WKB solutions together through the two boundary layers near *x*<sup>2</sup> and *x*3, using the results in sections 5.1, 5.2 and the Appendix C, we eliminate *a*¯1,2 and ¯*d*1,2 and obtain the homogeneous system

$$\begin{aligned} \vec{a}\vec{c}\_1 + \vec{c}\_2 e^{\frac{i}{\hbar}P} &= 0, \\\\ \vec{a}\vec{c}\_1 - \vec{c}\_2 e^{\frac{-i}{\hbar}P} &= 0. \end{aligned}$$

Thus, we derive the Bohr-Sommerfeld quantization condition for real energy eigen-levels (bound states) inside the cut-off energy strip for 0 < *E* < |*py*|.

$$P(E) = h\pi(n + \frac{1}{2}), \quad n = N\_1 + 1, \ldots \\ N\_2. \tag{50}$$

#### **8. Numerical results**

Based upon the analytical descriptions in the preceding sections for the smooth barrier, we present the results for the energy eigenvalues and eigenfunctions. These are shown in Fig's. 4-6 and compare favorably with those obtained through finite difference methods, as detailed in [71]. The energy dispersion curves, *En*(*py*), are shown for the complex resonant and real bound states for *h* = 0.1 and potentials of different widths. In Fig. 4(a), the energy levels are illustrated for the potential, *U* = 1/*coshx*, with *n* = 0, 1, ...., 15. For complex resonant states the real parts are shown. It must be emphasized that in zone "3", which is restricted by *<sup>E</sup>* < *<sup>U</sup>*<sup>0</sup> − *py* and *<sup>E</sup>* = *py* with *py* > 0, the complex quasibound states reside. The bound states are located in zone "2", which lies between *<sup>E</sup>* = ±*py* and *<sup>E</sup>* = *<sup>U</sup>*<sup>0</sup> − *py*. In zone "3" there are nine complex resonances. In Fig. 4(b), the results for a narrower potential of *U* = 1/*cosh*2*x* can be seen (all other parameters being the same as in Fig. 4 (a)). In this case, there are four complex resonances in zone "3" and *n* = 0, 1, ...., 9. The lifetimes of the local resonances, Γ*n*, are shown in Fig. 5 (a) and (b) for the same two potentials as described in Fig. 4. The complex quasi-bound states that are witnessed in zone "3" in Fig. 4 are shown in Fig. 5 for Γ*n*. The thinner potential allows less complex bound states. The bound states have infinite lifetimes. Both types of states are confined within the barrier in the x-direction, while the motion in the y-direction is controlled by the dispersion relations. In Fig. 6 we present the transmission probabilities |*t*| <sup>2</sup> for the two potentials. There are nine tunneling resonances, i.e. complex quasi-bound states within potential barrier defined as *U* = 1/*coshx* that correlate with those shown in Fig. 4 in zone "3". Likewise, for the thinner barrier there are four complex quasi-bound states. These resonance states are a clear indication of the Fabry-Perot multiple interference effects inside the barrier.

#### **PART II: High energy localized eigenstates in graphene monolayers and double layers**

#### **9. Graphene resonator in a magnetic field**

We consider a spectral problem for the Dirac operator describing the electron-hole quantum dynamics in a graphene monolayer without a gap, in the presence of a homogeneous magnetic field **A** and arbitrary scalar potential *U*(**x**) (see [31])

$$
\sigma\_F < \sigma\_\prime - ih\nabla + \frac{e}{c} \mathbf{A} > \psi(\mathbf{x}) + \mathcal{U}(\mathbf{x})\psi(\mathbf{x}) = E\psi(\mathbf{x}), \quad \psi(\mathbf{x}) = \begin{pmatrix} \mu \\ \upsilon \end{pmatrix} / \tag{51}
$$

10.5772/52267

53

http://dx.doi.org/10.5772/52267

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

was noted earlier for the Schrödinger operator (see [55]), if high-energy localised eigenstates are sought, which decay exponentially away from the resonator axis *AB*, the separation of variables will not help construct an exact solution due to the difficulty of satisfying the

The WKB ray asymptotic solution to the Dirac equation is sought through consideration of the eigenvalue problem associated with *Hφ* = *Eφ*. The magnetic vector potential **A** = *<sup>B</sup>*/2(−*x*2, *<sup>x</sup>*1, 0) is given in terms of the axial gauge for a magnetic field. The Hamiltonian of

**Figure 8.** Bilayer graphene consists of two coupled graphene monolayers on top of one another. On the left-hand side, the planes of graphene have perfect Bernal stacking. The *A*<sup>1</sup> atoms of the sublattice of the top sheet overlap the *B*<sup>2</sup> atoms of the sublattice of the bottom sheet as is indicated. A triangular structure is seen when looking directly down upon the bilayers, as is schematically shown, and the *A*<sup>2</sup> and *B*<sup>1</sup> atoms are over the centers of the hexagonal structure of their opposite sheet. The group of four pictures on the right illustrate what happens if a slight shift of one of the graphene planes occurs. Going clockwise around this group, the first is when one graphene layer moves slightly from the ideal Bernal orientation along the *x*<sup>1</sup> direction. The second is with a 5*<sup>o</sup>* tilt and the third and fourth are with a tilt of the plane of 45*<sup>o</sup>* and 90*<sup>o</sup>* , respectively.

� *<sup>U</sup>*(**x**) *vF*[*h*¯(−*i∂x*<sup>1</sup> <sup>−</sup> *<sup>∂</sup>x*<sup>2</sup> ) <sup>−</sup> *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

<sup>2</sup> ] *<sup>U</sup>*(**x**)

*g px* − *ipy* 0 *ζ px* + *ipy g* 0 0 0 0 *g px* − *ipy ζ* 0 *px* + *ipy g*

where *<sup>g</sup>* ≈ 0.4*eV*/*υ<sup>F</sup>* is the interlayer coupling. We consider the case when bilayer graphene has Bernal stacking as shown in Fig. 6. Bernal-stacked bilayer graphene occurs with half

<sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup>

*vF*[*h*¯(−*i∂x*<sup>1</sup> <sup>+</sup> *<sup>∂</sup>x*<sup>2</sup> ) + *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

*<sup>H</sup>* = *<sup>υ</sup><sup>F</sup>*

In contrast, for bilayer graphene the Hamiltonian takes the form,

<sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> 2 ]

,

� ,

the Dirac system (see equations (2) and (10)) takes the form for monolayer graphene,

boundary conditions.

*H* =

**10. Ray asymptotic solution**

where **<sup>x</sup>** = (*x*1, *<sup>x</sup>*2), and *<sup>u</sup>*, *<sup>v</sup>* are the components of the spinor wave function that describes electron localization on the sites of sublattice A or B of a honeycomb graphene structure. Here, *e* is the electron charge, *c* is the speed of light, **A** is magnetic potential in axial **A** = *<sup>B</sup>*/2(−*x*2, *<sup>x</sup>*1, 0) or Landau gauge **<sup>A</sup>** = *<sup>B</sup>*(−*x*2, 0, 0) (magnetic field is directed along the *<sup>x</sup>*<sup>3</sup> axis), and *vF* is the Fermi velocity. The symbol <, > means a scalar product, and ¯*h* is the Planck constant (which is a small parameter (¯*h* → 0) in semiclassical analysis). The vector *<sup>σ</sup>*¯ = (*σ*1, *<sup>σ</sup>*2) with Pauli matrices corresponds to the *<sup>K</sup>* Dirac point of the first Brillouin zone (see [31]). The case of the second *<sup>K</sup>*′ Dirac point with *<sup>σ</sup>*¯ <sup>∗</sup> = (*σ*1, <sup>−</sup>*σ*2) is treated similarly and is not considered here.

**Figure 7.** A periodic orbit inside the graphene nanoribbon resonator with magnetic field and electrostatic potential (electronic trajectory). Magnetic field is directed along the *x*<sup>3</sup> axis, the electrostatic field is piece-wise linear *U*(*x*2) = *β*|*x*2|.

We study the high energy spectral problem, using the semiclassical approximation, for a vertical graphene nanoribbon confined between two flat reflecting interfaces *L*1,2 (see Fig.7). It is assumed that the spinor wave function satisfies zigzag boundary conditions on the interfaces *<sup>L</sup>*1,2: *<sup>u</sup>*|*L*<sup>1</sup> <sup>=</sup> 0, *<sup>v</sup>*|*L*<sup>2</sup> <sup>=</sup> 0. It will be discussed later that the electrostatic field *<sup>U</sup>*(*x*2) = *<sup>β</sup>*|*x*2| makes the orbit shown in Fig. 7 periodic and stable. In the gener al case, as it was noted earlier for the Schrödinger operator (see [55]), if high-energy localised eigenstates are sought, which decay exponentially away from the resonator axis *AB*, the separation of variables will not help construct an exact solution due to the difficulty of satisfying the boundary conditions.

#### **10. Ray asymptotic solution**

24 Graphene - Research and Applications

**9. Graphene resonator in a magnetic field**

*vF* < *σ*¯, −*ih*∇ +

magnetic field **A** and arbitrary scalar potential *U*(**x**) (see [31])

*e c*

**double layers**

is not considered here.

**PART II: High energy localized eigenstates in graphene monolayers and**

We consider a spectral problem for the Dirac operator describing the electron-hole quantum dynamics in a graphene monolayer without a gap, in the presence of a homogeneous

where **<sup>x</sup>** = (*x*1, *<sup>x</sup>*2), and *<sup>u</sup>*, *<sup>v</sup>* are the components of the spinor wave function that describes electron localization on the sites of sublattice A or B of a honeycomb graphene structure. Here, *e* is the electron charge, *c* is the speed of light, **A** is magnetic potential in axial **A** = *<sup>B</sup>*/2(−*x*2, *<sup>x</sup>*1, 0) or Landau gauge **<sup>A</sup>** = *<sup>B</sup>*(−*x*2, 0, 0) (magnetic field is directed along the *<sup>x</sup>*<sup>3</sup> axis), and *vF* is the Fermi velocity. The symbol <, > means a scalar product, and ¯*h* is the Planck constant (which is a small parameter (¯*h* → 0) in semiclassical analysis). The vector *<sup>σ</sup>*¯ = (*σ*1, *<sup>σ</sup>*2) with Pauli matrices corresponds to the *<sup>K</sup>* Dirac point of the first Brillouin zone (see [31]). The case of the second *<sup>K</sup>*′ Dirac point with *<sup>σ</sup>*¯ <sup>∗</sup> = (*σ*1, <sup>−</sup>*σ*2) is treated similarly and

**Figure 7.** A periodic orbit inside the graphene nanoribbon resonator with magnetic field and electrostatic potential (electronic

We study the high energy spectral problem, using the semiclassical approximation, for a vertical graphene nanoribbon confined between two flat reflecting interfaces *L*1,2 (see Fig.7). It is assumed that the spinor wave function satisfies zigzag boundary conditions on the interfaces *<sup>L</sup>*1,2: *<sup>u</sup>*|*L*<sup>1</sup> <sup>=</sup> 0, *<sup>v</sup>*|*L*<sup>2</sup> <sup>=</sup> 0. It will be discussed later that the electrostatic field *<sup>U</sup>*(*x*2) = *<sup>β</sup>*|*x*2| makes the orbit shown in Fig. 7 periodic and stable. In the gener al case, as it

trajectory). Magnetic field is directed along the *x*<sup>3</sup> axis, the electrostatic field is piece-wise linear *U*(*x*2) = *β*|*x*2|.

**A** > *ψ*(**x**) + *U*(**x**)*ψ*(**x**) = *Eψ*(**x**), *ψ*(**x**) =

 *u v* 

, (51)

The WKB ray asymptotic solution to the Dirac equation is sought through consideration of the eigenvalue problem associated with *Hφ* = *Eφ*. The magnetic vector potential **A** = *<sup>B</sup>*/2(−*x*2, *<sup>x</sup>*1, 0) is given in terms of the axial gauge for a magnetic field. The Hamiltonian of the Dirac system (see equations (2) and (10)) takes the form for monolayer graphene,

**Figure 8.** Bilayer graphene consists of two coupled graphene monolayers on top of one another. On the left-hand side, the planes of graphene have perfect Bernal stacking. The *A*<sup>1</sup> atoms of the sublattice of the top sheet overlap the *B*<sup>2</sup> atoms of the sublattice of the bottom sheet as is indicated. A triangular structure is seen when looking directly down upon the bilayers, as is schematically shown, and the *A*<sup>2</sup> and *B*<sup>1</sup> atoms are over the centers of the hexagonal structure of their opposite sheet. The group of four pictures on the right illustrate what happens if a slight shift of one of the graphene planes occurs. Going clockwise around this group, the first is when one graphene layer moves slightly from the ideal Bernal orientation along the *x*<sup>1</sup> direction. The second is with a 5*<sup>o</sup>* tilt and the third and fourth are with a tilt of the plane of 45*<sup>o</sup>* and 90*<sup>o</sup>* , respectively.

$$H = \begin{pmatrix} \mathcal{U}(\mathbf{x}) & v\_F[\hbar(-i\partial\_{\mathbf{x}\_1} - \frac{\mathbf{a}\mathbf{x}\_1}{2}) - i\frac{\mathbf{a}\mathbf{x}\_1}{2} - \frac{\mathbf{a}\mathbf{x}\_2}{2}] \\ v\_F[\hbar(-i\partial\_{\mathbf{x}\_1} + \partial\_{\mathbf{x}\_2}) + i\frac{\mathbf{a}\mathbf{x}\_1}{2} - \frac{\mathbf{a}\mathbf{x}\_2}{2}] & \mathcal{U}(\mathbf{x}) \end{pmatrix} \prime$$

In contrast, for bilayer graphene the Hamiltonian takes the form,

$$H = \upsilon\_F \begin{pmatrix} g & p\_X - ip\_Y & 0 & \mathcal{J} \\ p\_X + ip\_Y & g & 0 & 0 \\ 0 & 0 & g & p\_X - ip\_Y \\ \zeta & 0 & p\_X + ip\_Y & g \end{pmatrix} / $$

where *<sup>g</sup>* ≈ 0.4*eV*/*υ<sup>F</sup>* is the interlayer coupling. We consider the case when bilayer graphene has Bernal stacking as shown in Fig. 6. Bernal-stacked bilayer graphene occurs with half of the carbon atoms in the second layer sitting on top of the empty centers of hexagons in the first layer. An external electric field can tune its bandgap by up to 250*meV* [32]. This form of structure of bilayer graphene can be experimentally created using chemical vapor deposition [32]. Considering the low energy states of electrons, we can reduce the 4 × 4 matrix describing the bilayer graphene to a form similar to that of monolayer graphene [56]. The only difference from the monolayer form is the squaring of the off-diagonal entries and the inclusion of a band mass for bilayer electrons.

10.5772/52267

55

http://dx.doi.org/10.5772/52267

<sup>2</sup> )2] <sup>−</sup> (*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*(**x**))<sup>2</sup> <sup>=</sup> 0. (54)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

− (*E* − *U*(**x**))

+ (*E* − *U*(**x**))

<sup>2</sup> − (*E* − *U*(**x**))<sup>2</sup> = 0. (55)

**A***d***x** = (56)

**A***d***x** = (57)

on the level set *<sup>H</sup><sup>h</sup>* = 0. The Hamiltonian dynamics with *<sup>h</sup>*1,2 or with *<sup>H</sup>e*,*<sup>h</sup>* are equivalent (see [48]).Classical action *S*(*x*) satisfies the Hamilton-Jacobi equation in the monolayer case to be

*αx*<sup>1</sup>

*αx*<sup>1</sup> 2 )2 2

*αx*<sup>1</sup> 2 )2 2

<sup>2</sup> )<sup>2</sup> + (*p*<sup>2</sup> <sup>+</sup>

<sup>2</sup> )<sup>2</sup> + (*p*<sup>2</sup> <sup>+</sup>

*αx*<sup>1</sup> <sup>2</sup> )2]

The solutions of the Hamiltonian-Jacobi equations, for monolayer and bilayers, electrons and holes, are given by the following curvilinear integrals over classical trajectories connecting

[*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*(**x**(*s*))]*ds* <sup>−</sup> *<sup>e</sup>*

*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*(**x**(*s*))*ds* <sup>−</sup> *<sup>e</sup>*

*x*˙ 2 <sup>1</sup> <sup>+</sup> *<sup>x</sup>*˙ 2 <sup>2</sup> <sup>−</sup> *<sup>α</sup>* 2

*x*˙ 2 <sup>1</sup> <sup>+</sup> *<sup>x</sup>*˙ 2 <sup>2</sup> <sup>−</sup> *<sup>α</sup>* 2

for mono and bi-layer, respectfully, where *s* is the arc length. This representation is correct in the neighbourhood of a regular family of classical trajectories emanating from *<sup>M</sup>*(0). For electrons and holes one can seek solution to the Dirac system zero-order problem in the form

*<sup>ψ</sup>*<sup>0</sup> <sup>=</sup> *<sup>σ</sup>*(0)

*c M*

> *c M*

*M*(0)

(−*x*2*x*˙1 + *<sup>x</sup>*1*x*˙2)

*M*(0)

(−*x*2*x*˙1 + *<sup>x</sup>*1*x*˙2)

 *dt*,

> *dt*,

(**x**)**e**<sup>1</sup> (58)

The Hamiliton-Jacobi equation is satisfied in the two-layers of graphene case by,

<sup>2</sup> )<sup>2</sup> + (*Sx*<sup>2</sup> <sup>+</sup>

*M*

*M*(0)

[*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*(**x**(*t*))]

*M*

*M*(0)

*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*(**x**(*t*))

[(*Sx*<sup>1</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup>

*H<sup>e</sup>* = 

*H<sup>h</sup>* = 

[(*Sx*<sup>1</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup>

Likewise, in the case of bilayers,

and

<sup>2</sup> )+(*Sx*<sup>2</sup> <sup>+</sup>

(*p*<sup>1</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup>

(*p*<sup>1</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup>

points *<sup>M</sup>*(0) and *<sup>M</sup>* = (*x*1, *<sup>x</sup>*2) (*M*(0) is fixed and *<sup>M</sup>* is variable)

*<sup>S</sup>*(*M*) =

*<sup>S</sup>*(*M*) =

 *M*

*M*(0)

 *M*

*M*(0)

$$H = \begin{pmatrix} \mathcal{U}(\mathbf{x}) & \frac{1}{2m} [\hbar(-i\partial\_{\mathbf{x}\_1} + \partial\_{\mathbf{x}\_2}) + i\frac{\mathbf{x}\mathbf{x}\_1}{2} - \frac{\mathbf{a}\mathbf{x}\_2}{2}]^2\\ \frac{1}{2m} [\hbar(-i\partial\_{\mathbf{x}\_1} + \partial\_{\mathbf{x}\_2}) + i\frac{\mathbf{a}\mathbf{x}\_1}{2} - \frac{\mathbf{a}\mathbf{x}\_2}{2}]^2 \end{pmatrix}.$$

It is now convenient to introduce some dimensionless variables. The coordinate system *x* ⇒ *xD*, where *D* is a characteristic scale associated with a change in the potential (and correspondingly *<sup>U</sup>*(*x*) <sup>⇒</sup> *<sup>U</sup>*(*xD*)). Then, we write *<sup>U</sup>* <sup>=</sup> *<sup>U</sup>*(*xD*)/*E*0; where we define the characteristic energy scale as *<sup>E</sup>* ⇒ *<sup>E</sup>*0*E*. For single layers of graphene the small parameter, *<sup>h</sup>* << 1, is *<sup>h</sup>* = *<sup>υ</sup>Fh*¯ /*U*0*D*. In double layer graphene it is slightly different; *h* = *αD*/ √2*mU*0, with the magnetic length, as a function of the applied magnetic field, given to be *<sup>α</sup>* <sup>=</sup> *<sup>α</sup>D*/ √<sup>2</sup>*mE*<sup>0</sup> with *<sup>α</sup>* = *eB*/*c*. We now write the dimensionless forms of the one and two layer graphene systems as (with the tildes omitted for brevity),

$$H = \begin{pmatrix} U(\mathbf{x}) & h(-i\partial\_{\mathbf{x}\_1} - \frac{\partial}{\partial x\_1}) - i\frac{\alpha\mathbf{x}\_1}{2} - \frac{\alpha\mathbf{x}\_2}{2} \\ h(-i\partial\_{\mathbf{x}\_1} + \partial\_{\mathbf{x}\_2}) + i\frac{\alpha\mathbf{x}\_1}{2} - \frac{\alpha\mathbf{x}\_2}{2} & U(\mathbf{x}) \end{pmatrix} \tag{52}$$

and

$$H = \begin{pmatrix} U(\mathbf{x}) & (h(-i\partial\_{\mathbf{x}\_1} - \frac{\mathbf{a}\mathbf{x}\_1}{2}) - i\frac{\mathbf{a}\mathbf{x}\_1}{2} - \frac{\mathbf{a}\mathbf{x}\_2}{2})^2\\ (h(-i\partial\_{\mathbf{x}\_1} + \partial\_{\mathbf{x}\_2}) + i\frac{\mathbf{a}\mathbf{x}\_1}{2} - \frac{\mathbf{a}\mathbf{x}\_2}{2})^2 & U(\mathbf{x}) \end{pmatrix} \tag{53}$$

The solution for monolayer graphene is sought in the same form as equation (11). Substituting this series into the Dirac system, and equating to zero the corresponding coefficients of successive degrees of the small parameter *h*, we obtain a recurrent system of equations which determines the unknown *<sup>S</sup>*(**x**) and *<sup>ψ</sup>j*(**x**).

The Hamiltonian *H* has two eigenvalues. In the domain Ω*<sup>e</sup>* = {**x** : *E* > *U*(**x**)}, the Hamiltonian eigenvalue *<sup>h</sup>*<sup>1</sup> = *<sup>U</sup>*(**x**) + *<sup>p</sup>* on the level set *<sup>h</sup>*<sup>1</sup> = *<sup>E</sup>* describes the dynamics of electrons. The corresponding classical trajectories can be obtained from the Hamiltonian system *x*˙ = *H<sup>e</sup> <sup>p</sup>*, *<sup>p</sup>*˙ <sup>=</sup> <sup>−</sup>*H<sup>e</sup> <sup>x</sup>*, **<sup>x</sup>** = (*x*1, *<sup>x</sup>*2), **<sup>p</sup>** = (*p*1, *<sup>p</sup>*2), with an equivalent Hamiltonian (see [48])

$$H^\varepsilon = \frac{1}{2}\left( (p\_1 - \frac{a\mathbf{x}\_2}{2})^2 + (p\_2 + \frac{a\mathbf{x}\_1}{2})^2 - \left(E - \mathcal{U}(\mathbf{x})\right)^2 \right)$$

on the level set *<sup>H</sup><sup>e</sup>* = 0 with *<sup>p</sup>*<sup>1</sup> = *Sx*<sup>1</sup> and *<sup>p</sup>*<sup>2</sup> = *Sx*<sup>2</sup> . Opposite to this case, in the domain <sup>Ω</sup>*<sup>h</sup>* = {**<sup>x</sup>** : *<sup>E</sup>* < *<sup>U</sup>*(**x**)}, the Hamiltonian eigenvalue *<sup>h</sup>*<sup>2</sup> = *<sup>U</sup>*(**x**) − *<sup>p</sup>* on the level set *<sup>h</sup>*<sup>2</sup> = *<sup>E</sup>* describes the dynamics of holes. The corresponding classical trajectories can be obtained from the Hamiltonian system with Hamiltonian

$$H^\hbar = \frac{1}{2}\left(-\left(p\_1 - \frac{a\mathbf{x}\_2}{2}\right)^2 - \left(p\_2 + \frac{a\mathbf{x}\_1}{2}\right)^2 + \left(E - U(\mathbf{x})\right)^2\right)^{\frac{1}{2}}$$

on the level set *<sup>H</sup><sup>h</sup>* = 0. The Hamiltonian dynamics with *<sup>h</sup>*1,2 or with *<sup>H</sup>e*,*<sup>h</sup>* are equivalent (see [48]).Classical action *S*(*x*) satisfies the Hamilton-Jacobi equation in the monolayer case to be

$$[(\mathcal{S}\_{\mathbf{x}\_1} - \frac{a\mathbf{x}\_2}{2}) + (\mathcal{S}\_{\mathbf{x}\_2} + \frac{a\mathbf{x}\_1}{2})^2] - (\mathcal{E} - \mathcal{U}(\mathbf{x}))^2 = \mathbf{0}.\tag{54}$$

Likewise, in the case of bilayers,

$$H^\ell = \left[ (p\_1 - \frac{\mathfrak{a}\mathfrak{x}\_2}{2})^2 + (p\_2 + \frac{\mathfrak{a}\mathfrak{x}\_1}{2})^2 \right]^2 - (E - \mathcal{U}(\mathbf{x}))^2$$

and

26 Graphene - Research and Applications

*H* =

√

*h* = *αD*/

and

to be *<sup>α</sup>* <sup>=</sup> *<sup>α</sup>D*/

system *x*˙ = *H<sup>e</sup>*

(see [48])

1

√

*H* =

*H* =

the inclusion of a band mass for bilayer electrons.

*U*(**x**) <sup>1</sup>

two layer graphene systems as (with the tildes omitted for brevity),

*<sup>h</sup>*(−*i∂x*<sup>1</sup> <sup>+</sup> *<sup>∂</sup>x*<sup>2</sup> ) + *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

(*h*(−*i∂x*<sup>1</sup> <sup>+</sup> *<sup>∂</sup>x*<sup>2</sup> ) + *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

of equations which determines the unknown *<sup>S</sup>*(**x**) and *<sup>ψ</sup>j*(**x**).

*<sup>p</sup>*<sup>1</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> 2

> *<sup>p</sup>*<sup>1</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> 2

*<sup>p</sup>*, *<sup>p</sup>*˙ <sup>=</sup> <sup>−</sup>*H<sup>e</sup>*

*<sup>H</sup><sup>e</sup>* <sup>=</sup> <sup>1</sup> 2 

from the Hamiltonian system with Hamiltonian

*<sup>H</sup><sup>h</sup>* <sup>=</sup> <sup>1</sup> 2 −

<sup>2</sup>*<sup>m</sup>* [*h*¯(−*i∂x*<sup>1</sup> <sup>+</sup> *<sup>∂</sup>x*<sup>2</sup> ) + *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

of the carbon atoms in the second layer sitting on top of the empty centers of hexagons in the first layer. An external electric field can tune its bandgap by up to 250*meV* [32]. This form of structure of bilayer graphene can be experimentally created using chemical vapor deposition [32]. Considering the low energy states of electrons, we can reduce the 4 × 4 matrix describing the bilayer graphene to a form similar to that of monolayer graphene [56]. The only difference from the monolayer form is the squaring of the off-diagonal entries and

> <sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> 2 ]

It is now convenient to introduce some dimensionless variables. The coordinate system *x* ⇒ *xD*, where *D* is a characteristic scale associated with a change in the potential (and correspondingly *<sup>U</sup>*(*x*) <sup>⇒</sup> *<sup>U</sup>*(*xD*)). Then, we write *<sup>U</sup>* <sup>=</sup> *<sup>U</sup>*(*xD*)/*E*0; where we define the characteristic energy scale as *<sup>E</sup>* ⇒ *<sup>E</sup>*0*E*. For single layers of graphene the small parameter, *<sup>h</sup>* << 1, is *<sup>h</sup>* = *<sup>υ</sup>Fh*¯ /*U*0*D*. In double layer graphene it is slightly different;

*<sup>U</sup>*(**x**) *<sup>h</sup>*(−*i∂x*<sup>1</sup> <sup>−</sup> *<sup>∂</sup>x*<sup>2</sup> ) <sup>−</sup> *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

<sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup>

*<sup>U</sup>*(**x**) (*h*(−*i∂x*<sup>1</sup> <sup>−</sup> *<sup>∂</sup>x*<sup>2</sup> ) <sup>−</sup> *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

The solution for monolayer graphene is sought in the same form as equation (11). Substituting this series into the Dirac system, and equating to zero the corresponding coefficients of successive degrees of the small parameter *h*, we obtain a recurrent system

The Hamiltonian *H* has two eigenvalues. In the domain Ω*<sup>e</sup>* = {**x** : *E* > *U*(**x**)}, the Hamiltonian eigenvalue *<sup>h</sup>*<sup>1</sup> = *<sup>U</sup>*(**x**) + *<sup>p</sup>* on the level set *<sup>h</sup>*<sup>1</sup> = *<sup>E</sup>* describes the dynamics of electrons. The corresponding classical trajectories can be obtained from the Hamiltonian

*<sup>p</sup>*<sup>2</sup> +

on the level set *<sup>H</sup><sup>e</sup>* = 0 with *<sup>p</sup>*<sup>1</sup> = *Sx*<sup>1</sup> and *<sup>p</sup>*<sup>2</sup> = *Sx*<sup>2</sup> . Opposite to this case, in the domain <sup>Ω</sup>*<sup>h</sup>* = {**<sup>x</sup>** : *<sup>E</sup>* < *<sup>U</sup>*(**x**)}, the Hamiltonian eigenvalue *<sup>h</sup>*<sup>2</sup> = *<sup>U</sup>*(**x**) − *<sup>p</sup>* on the level set *<sup>h</sup>*<sup>2</sup> = *<sup>E</sup>* describes the dynamics of holes. The corresponding classical trajectories can be obtained

<sup>2</sup> −

*<sup>p</sup>*<sup>2</sup> +

*αx*<sup>1</sup> 2

<sup>2</sup> +

*αx*<sup>1</sup> 2

<sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup>

<sup>2</sup> +

2*mU*0, with the magnetic length, as a function of the applied magnetic field, given

<sup>2</sup>*mE*<sup>0</sup> with *<sup>α</sup>* = *eB*/*c*. We now write the dimensionless forms of the one and

<sup>2</sup> *<sup>U</sup>*(**x**)

<sup>2</sup> )<sup>2</sup> *<sup>U</sup>*(**x**)

*<sup>x</sup>*, **<sup>x</sup>** = (*x*1, *<sup>x</sup>*2), **<sup>p</sup>** = (*p*1, *<sup>p</sup>*2), with an equivalent Hamiltonian

*E* − *U*(**x**)

*E* − *U*(**x**)

2 

2 

<sup>2</sup> −

<sup>2</sup>*<sup>m</sup>* [*h*¯(−*i∂x*<sup>1</sup> <sup>−</sup> *<sup>∂</sup>x*<sup>2</sup> ) <sup>−</sup> *<sup>i</sup> <sup>α</sup>x*<sup>1</sup>

<sup>2</sup> *U*(**x**)

<sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> 2 ] 2

<sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> 2

> <sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> <sup>2</sup> )<sup>2</sup>

(52)

(53)

 .

$$H^\hbar = \left[ (p\_1 - \frac{a\mathbf{x}\_2}{2})^2 + (p\_2 + \frac{a\mathbf{x}\_1}{2})^2 \right]^2 + (E - \mathcal{U}(\mathbf{x}))^2$$

The Hamiliton-Jacobi equation is satisfied in the two-layers of graphene case by,

$$[(\mathcal{S}\_{\mathbf{x}\_1} - \frac{a\mathbf{x}\_2}{2})^2 + (\mathcal{S}\_{\mathbf{x}\_2} + \frac{a\mathbf{x}\_1}{2})^2]^2 - (E - \mathcal{U}(\mathbf{x}))^2 = 0. \tag{55}$$

The solutions of the Hamiltonian-Jacobi equations, for monolayer and bilayers, electrons and holes, are given by the following curvilinear integrals over classical trajectories connecting points *<sup>M</sup>*(0) and *<sup>M</sup>* = (*x*1, *<sup>x</sup>*2) (*M*(0) is fixed and *<sup>M</sup>* is variable)

$$S(M) = \int\_{M^{(0)}}^{M} [E - \mathcal{U}(\mathbf{x}(s))] ds - \frac{e}{c} \int\_{M^{(0)}}^{M} \mathbf{A} d\mathbf{x} = \tag{56}$$

$$\int\_{M^{(0)}}^{M} \left( [E - \mathcal{U}(\mathbf{x}(t))] \sqrt{\dot{\mathbf{x}}\_1^2 + \dot{\mathbf{x}}\_2^2} - \frac{a}{2} (-x\_2 \dot{\mathbf{x}}\_1 + x\_1 \dot{\mathbf{x}}\_2) \right) dt,$$

$$S(M) = \int\_{M^{(0)}}^{M} \sqrt{E - \mathcal{U}(\mathbf{x}(s))} ds - \frac{e}{c} \int\_{M^{(0)}}^{M} \mathbf{A} d\mathbf{x} = \tag{57}$$

$$\int\_{M^{(0)}}^{M} \left( \sqrt{E - \mathcal{U}(\mathbf{x}(t))} \sqrt{\dot{\mathbf{x}}\_1^2 + \dot{\mathbf{x}}\_2^2} - \frac{a}{2} (-x\_2 \dot{\mathbf{x}}\_1 + x\_1 \dot{\mathbf{x}}\_2) \right) dt,$$

for mono and bi-layer, respectfully, where *s* is the arc length. This representation is correct in the neighbourhood of a regular family of classical trajectories emanating from *<sup>M</sup>*(0). For electrons and holes one can seek solution to the Dirac system zero-order problem in the form

$$
\psi\_0 = \sigma^{(0)}(\mathbf{x})\mathbf{e}\_1 \tag{58}
$$

with unknown amplitude *<sup>σ</sup>*(0)(**x**). Solvability of the problem

$$(H - EI)\psi\_1 = -\hat{R}\psi\_{0\prime} \qquad E = h\_{1,2}$$

requires that the orthogonality condition with complex conjugation

$$<\mathbf{e}\_1, \hat{R}(\sigma^{(0)}(\mathbf{x})\mathbf{e}\_1) > = 0$$

must hold, where

$$\begin{aligned} \text{Monolayer}: \quad \widehat{R} = \begin{pmatrix} 0 & \partial\_x - i\partial\_y \\ \partial\_x + i\partial\_y & 0 \end{pmatrix} \end{aligned} $$

$$Bilayer: \widehat{\mathbb{R}} = \left( \begin{array}{cc} 0 & \mathbf{Y}\_1 \\ \mathbf{Y}\_1^\* & 0 \end{array} \right).$$

where,

$$\mathbf{Y}\_1 = 2\left(\mathbf{S}\_{\mathbf{x}\_1} - i\mathbf{S}\_{\mathbf{x}\_2} - \frac{i\alpha\mathbf{x}\_1}{2} - \frac{i\alpha\mathbf{x}\_2}{2}\right)\left(\partial\_{\mathbf{x}\_1} - i\partial\_{\mathbf{x}\_2}\right) + \mathbf{S}\_{\mathbf{x}\_1\mathbf{x}\_1} - 2i\mathbf{S}\_{\mathbf{x}\_1\mathbf{x}\_2} - \mathbf{S}\_{\mathbf{x}\_2\mathbf{x}\_2}$$

Using the basic elements of the techniques described in [48], from the orthogonality condition one obtains the transport equation for *<sup>σ</sup>*(0)(**x**). The geometrical spreading for an electron or hole with respect to the Hamiltonian system with *<sup>h</sup>*1,2 = *<sup>U</sup>* ± *vF <sup>p</sup>* has a solution

$$Monolayer: \quad \sigma^{(0)} = \frac{\mathcal{C}\_0}{\sqrt{\mathcal{I}}} e^{-i\frac{\theta}{2}},\tag{59}$$

10.5772/52267

57

http://dx.doi.org/10.5772/52267

√

*h* = *O*(1),

(*s*) + **en**(*s*)*n*, (63)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

√

*h*) as ¯*h* → 0. We

**11. Construction of eigenfunctions, periodic orbit stability analysis and**

Let **<sup>x</sup>**<sup>0</sup> = (*x*1(*s*), *<sup>x</sup>*2(*s*)) be a particle (electron or hole) classical trajectory, where *<sup>s</sup>* is the arc length measured along the trajectory. Consider the neighborhood of the trajectory in terms of local coordinates *s*, *n*, where *n* is the distance along the vector, normal to the trajectory,

we seek an asymptotic solution to the Dirac system related to (2) where *<sup>S</sup>*0(*s*) and *<sup>S</sup>*1(*s*) are chosen similar to [55], [66] as they give a linear approximation for solution to the

*<sup>a</sup>*(*s*) = *<sup>E</sup>* − *<sup>U</sup>*0(*s*)

*<sup>U</sup>*(**x**) = *<sup>U</sup>*0(*s*) + *<sup>U</sup>*1(*s*)*<sup>n</sup>* + *<sup>U</sup>*2(*s*)*n*<sup>2</sup> + ... .

In the following *<sup>γ</sup>i*(*s*), *<sup>i</sup>* = 1, 2 are the Cartesian components of **en**(*s*). Following [54], [55], [65], and [66], we apply the asymptotic boundary-layer method to the Dirac system (2). We

assume that we deal with a continuous family of POs symmetric with respect to both axes (see Fig. 4). Thus, the trajectory of the PO consists of two symmetric parts between two reflection points A and B. We seek the asymptotic solution of the eigenfunction for electrons and holes localized in the neighborhood of a PO as a combination of two Gaussian beams

*<sup>ψ</sup>*(**x**, *<sup>E</sup>*) = *<sup>ψ</sup>*1(**x**, *<sup>E</sup>*) + *<sup>R</sup><sup>ψ</sup>*2(**x**, *<sup>E</sup>*),

<sup>0</sup> (*s*) + *<sup>S</sup>*1(*s*)*<sup>n</sup>* <sup>+</sup>

(0) <sup>2</sup> *<sup>γ</sup>*2) *ds*′

**e1**  1 2

1 + *O*(*h*¯ 1/2)

*<sup>p</sup>*1,2(*s*) *<sup>z</sup>*1,2(*s*)

> ,

*n*2) 

, 0 < *s* < *s*0,

*<sup>E</sup>* − *<sup>U</sup>*0(*s*)

**<sup>x</sup>** <sup>=</sup> **<sup>x</sup>**(0)

where **en**(*s*) is the unit vector normal to the trajectory. Introducing *ν* = *n*/

Hamilton-Jacobi equation (55) (see [55], [66])). The parameter for monolayers

*<sup>a</sup>*(*s*) =

allow that the width of the boundary layer is determined by |*n*| = *O*(

*h* (*S*(1,2)

<sup>−</sup>*iθ*/2 *Qm*(*z*1,2(*s*), *<sup>ν</sup>*) *z*1,2(*s*)

*<sup>ψ</sup>*1,2(**x**, *<sup>E</sup>*) = exp *<sup>i</sup>*

*e*

0

 *a*(*s* ′ ) + *<sup>α</sup>* 2 (*x* (0) <sup>1</sup> *<sup>γ</sup>*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*

*S*(1) <sup>0</sup> (*s*) = *<sup>s</sup>*

**quantization conditions**

and for Bernal bilayers,

described by

where

is obtained from the approximation

such that

$$
abla 
varphi : \quad \sigma^{(0)} = \frac{\mathfrak{c}\_0}{\sqrt{\mathfrak{I}}} \varepsilon^{-i\theta}, \quad \mathfrak{c}\_0 = const. \tag{60}$$

where

$$J(t, \gamma) = \left| \frac{\partial(\mathbf{x}\_1, \mathbf{x}\_2)}{\partial(t, \gamma)} \right| \tag{61}$$

where we have introduced *θ*, which is the angle made by the velocity of the particle trajectory with the *x*<sup>1</sup> axis:

$$i\left(p\_1 - \frac{\alpha x\_2}{2}\right) + i\left(p\_2 - \frac{\alpha x\_1}{2}\right) = pe^{i\theta} \tag{62}$$

Here −*θ*/2 is the adiabatic phase for monolayer graphene, as introduced by Berry [64]. Chirality results in a Berry phase of *θ* in bilayer graphene and the confinement of electronic states. Conservation of chirality in monolayer graphene means that the particles cannot backscatter and this leads to normal incidence transmission equal to unity. This is not the case in bilayer graphene and backscattering can occur.

## **11. Construction of eigenfunctions, periodic orbit stability analysis and quantization conditions**

Let **<sup>x</sup>**<sup>0</sup> = (*x*1(*s*), *<sup>x</sup>*2(*s*)) be a particle (electron or hole) classical trajectory, where *<sup>s</sup>* is the arc length measured along the trajectory. Consider the neighborhood of the trajectory in terms of local coordinates *s*, *n*, where *n* is the distance along the vector, normal to the trajectory, such that

$$\mathbf{x} = \mathbf{x}^{(0)}(\mathbf{s}) + \mathbf{e}\_{\mathbf{n}}(\mathbf{s})\mathbf{n},\tag{63}$$

where **en**(*s*) is the unit vector normal to the trajectory. Introducing *ν* = *n*/ √*h* = *O*(1), we seek an asymptotic solution to the Dirac system related to (2) where *<sup>S</sup>*0(*s*) and *<sup>S</sup>*1(*s*) are chosen similar to [55], [66] as they give a linear approximation for solution to the Hamilton-Jacobi equation (55) (see [55], [66])). The parameter for monolayers

$$a(s) = E - \mathcal{U}\_0(s)$$

and for Bernal bilayers,

28 Graphene - Research and Applications

must hold, where

<sup>Υ</sup><sup>1</sup> = <sup>2</sup> 

*Sx*<sup>1</sup> <sup>−</sup> *iSx*<sup>2</sup> <sup>−</sup> *<sup>i</sup>αx*<sup>1</sup>

where,

where

with the *x*<sup>1</sup> axis:

with unknown amplitude *<sup>σ</sup>*(0)(**x**). Solvability of the problem

requires that the orthogonality condition with complex conjugation

<sup>&</sup>lt; **<sup>e</sup>**1, *<sup>R</sup>*(*σ*(0)

*Monolayer* : *<sup>R</sup>* <sup>=</sup>

<sup>2</sup> <sup>−</sup> *<sup>i</sup>αx*<sup>2</sup> 2

hole with respect to the Hamiltonian system with *<sup>h</sup>*1,2 = *<sup>U</sup>* ± *vF <sup>p</sup>* has a solution

*Bilayer* : *<sup>σ</sup>*(0) <sup>=</sup> *<sup>c</sup>*<sup>0</sup>

case in bilayer graphene and backscattering can occur.

*<sup>p</sup>*<sup>1</sup> <sup>−</sup> *<sup>α</sup>x*<sup>2</sup> 2 + *i* 

*J*(*t*, *γ*) =

*Monolayer* : *<sup>σ</sup>*(0) <sup>=</sup> *<sup>c</sup>*<sup>0</sup>

√*J e*

> 

where we have introduced *θ*, which is the angle made by the velocity of the particle trajectory

Here −*θ*/2 is the adiabatic phase for monolayer graphene, as introduced by Berry [64]. Chirality results in a Berry phase of *θ* in bilayer graphene and the confinement of electronic states. Conservation of chirality in monolayer graphene means that the particles cannot backscatter and this leads to normal incidence transmission equal to unity. This is not the

*<sup>∂</sup>*(*x*1, *<sup>x</sup>*2) *∂*(*t*, *γ*)

*<sup>p</sup>*<sup>2</sup> <sup>−</sup> *<sup>α</sup>x*<sup>1</sup> 2 

 

*Bilayer* : *<sup>R</sup>* <sup>=</sup>

Using the basic elements of the techniques described in [48], from the orthogonality condition one obtains the transport equation for *<sup>σ</sup>*(0)(**x**). The geometrical spreading for an electron or

(*<sup>H</sup>* <sup>−</sup> *EI*)*ψ*<sup>1</sup> <sup>=</sup> <sup>−</sup>*<sup>R</sup><sup>ψ</sup>*0, *<sup>E</sup>* <sup>=</sup> *<sup>h</sup>*1,2

(**x**)**e**1) >= <sup>0</sup>

 0 *∂<sup>x</sup>* − *i∂<sup>y</sup> ∂<sup>x</sup>* + *i∂<sup>y</sup>* 0

> ,

 0 Υ<sup>1</sup> Υ∗ <sup>1</sup> 0

*∂x*<sup>1</sup> − *i∂x*<sup>2</sup>

√*J e* −*i <sup>θ</sup>*  ,

+ *Sx*<sup>1</sup> *<sup>x</sup>*<sup>1</sup> − <sup>2</sup>*iSx*<sup>1</sup> *<sup>x</sup>*<sup>2</sup> − *Sx*<sup>2</sup> *<sup>x</sup>*<sup>2</sup>

<sup>−</sup>*i<sup>θ</sup>* , *<sup>c</sup>*<sup>0</sup> <sup>=</sup> *const*, (60)

<sup>2</sup> , (59)

= *pei<sup>θ</sup>* (62)

(61)

$$a(\mathbf{s}) = \sqrt{E - U\_0(\mathbf{s})}$$

is obtained from the approximation

$$\mathcal{U}(\mathbf{x}) = \mathcal{U}\_0(\mathbf{s}) + \mathcal{U}\_1(\mathbf{s})n + \mathcal{U}\_2(\mathbf{s})n^2 + \dots$$

In the following *<sup>γ</sup>i*(*s*), *<sup>i</sup>* = 1, 2 are the Cartesian components of **en**(*s*). Following [54], [55], [65], and [66], we apply the asymptotic boundary-layer method to the Dirac system (2). We allow that the width of the boundary layer is determined by |*n*| = *O*( √*h*) as ¯*h* → 0. We assume that we deal with a continuous family of POs symmetric with respect to both axes (see Fig. 4). Thus, the trajectory of the PO consists of two symmetric parts between two reflection points A and B. We seek the asymptotic solution of the eigenfunction for electrons and holes localized in the neighborhood of a PO as a combination of two Gaussian beams

$$
\psi(\mathbf{x}, E) = \psi\_1(\mathbf{x}, E) + \widehat{R}\psi\_2(\mathbf{x}, E)\_{\prime\prime}
$$

described by

$$\psi\_{1,2}(\mathbf{x},E) = \exp\left(\frac{i}{\hbar}(S\_0^{(1,2)}(s) + S\_1(s)\eta + \frac{1}{2}\frac{p\_{1,2}(s)}{z\_{1,2}(s)}\eta^2)\right),$$

$$e^{-i\theta/2}\frac{Q\_m(z\_{1,2}(s),\nu)}{\sqrt{z\_{1,2}(s)}}\mathbf{e\_1}\left(1 + O(\hbar^{1/2})\right),$$

where

$$S\_0^{(1)}(s) = \int\_0^s \left( a(s') + \frac{a}{2} (\mathbf{x}\_1^{(0)} \gamma\_1 + \mathbf{x}\_2^{(0)} \gamma\_2) \right) ds', \quad 0 < s < s\_{0,1}$$

$$
\mathcal{S}\_0^{(2)}(s) = \mathcal{S}\_0^{(1)}(s\_0) + \int\_{s\_0}^s \left( a(s') + \frac{a}{2} (\mathbf{x}\_1^{(0)} \gamma\_1 + \mathbf{x}\_2^{(0)} \gamma\_2) \right) ds', \quad s\_0 < s < 2 \mathbf{s}\_0.
$$

$$
\mathcal{S}\_1(s) = \frac{a}{2} \left( \mathbf{x}\_2^{(0)} \gamma\_1 - \mathbf{x}\_1^{(0)} \gamma\_2 \right), \tag{64}
$$

where we have defined the Berry phase to be, *<sup>e</sup>i<sup>θ</sup>* <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>γ*1, and **e1**,**<sup>2</sup>** = (1,*ei<sup>θ</sup>* )/ √2. Here, each beam is related to the corresponding part of the periodic orbit. Namely, *<sup>ψ</sup>*<sup>1</sup> is determined by *<sup>z</sup>* = *<sup>z</sup>*1(*s*), *<sup>p</sup>* = *<sup>p</sup>*1(*s*) for 0 < *<sup>s</sup>* < *<sup>s</sup>*0, describing the electrons propagation along the lower part of the orbit from A to B, whereas *ψ*<sup>2</sup> is determined by *<sup>z</sup>* = *<sup>z</sup>*2(*s*), *<sup>p</sup>* = *<sup>p</sup>*2(*s*) for *<sup>s</sup>*<sup>0</sup> < *<sup>s</sup>* < <sup>2</sup>*s*0, for the electrons propagation along the upper part of the orbit from B to A. The complete derivation of the electronic Gaussian beams for monolayer graphene can be found in the work of Zalipaev [66]. Following the methodology developed in the previous work [66], we state the problem in terms of the function *z*(*s*) and write the Hamiltonian in its terms,

$$\dot{z} = \frac{p}{a(s)}, \quad \dot{p} = -a(s)d(s)z \tag{65}$$

10.5772/52267

59

, (69)

http://dx.doi.org/10.5772/52267

. (70)

The localized solution can be constructed if *z*(*s*), *p*(*s*) is a complex (in the complex

coefficients (see [50], [54]). Namely, for the monodromy 2 × 2 matrix *M*, describing the

where *<sup>M</sup>*<sup>1</sup> = *<sup>M</sup>*1(*s*0) and *<sup>M</sup>*<sup>2</sup> = *<sup>M</sup>*2(2*s*0) are fundamental matrices of the system (65) describing the evolution (*z*(*s*), *<sup>p</sup>*(*s*)) for 0 < *<sup>s</sup>* < *<sup>s</sup>*<sup>0</sup> and *<sup>s</sup>*<sup>0</sup> < *<sup>s</sup>* < <sup>2</sup>*s*0, correspondingly.

where *γ* is the angle of incidence of the trajectory at the points *A*, *B*. To attain *R<sup>A</sup>* and *RB*, the classical action *S* of the phase function at the reflecting boundary requires continuity to

In a general case, the entries of *M*1,2 are to be determined numerically as the Hamiltonian system has variable coefficients. It is worth to remark that all the multipliers in (71) are

The classical theory of linear Hamiltonian systems with periodic coefficients states that, if |*TrM*| < 2, we have a stable PO (elliptic fixed point, for example, see [27]), and ||*Mn*|| < *const* for arbitrary *n* ∈ **N**. Then, there exist two bounded complex Floquet's solutions for <sup>−</sup><sup>∞</sup> <sup>&</sup>lt; *<sup>s</sup>* <sup>&</sup>lt; <sup>+</sup>∞, namely, (*z*(*s*), *<sup>p</sup>*(*s*)) and (*z*¯(*s*), *<sup>p</sup>*¯(*s*)) with Floquet's multipliers *<sup>λ</sup>*1,2 <sup>=</sup> *<sup>e</sup>*±*i<sup>ϕ</sup>*

*λ*<sup>2</sup> − *TrMλ* + 1 = 0.

, *R<sup>A</sup>* =

, *R<sup>B</sup>* =

<sup>21</sup> <sup>=</sup> <sup>2</sup>*<sup>α</sup>* tan(*γ*),

*M*<sup>11</sup> *M*<sup>12</sup>

 *<sup>z</sup>*(*<sup>s</sup>* + <sup>2</sup>*s*0) *<sup>p</sup>*(*<sup>s</sup>* + <sup>2</sup>*s*0)

*M*

a Floquet solution for arbitrary *s* is defined as

 *<sup>z</sup>*1(0) *<sup>p</sup>*1(0)

 *<sup>z</sup>*2(*s*0) *<sup>p</sup>*2(*s*0)

(0 < *ϕ* < *π*), which are solutions of

 = *R<sup>A</sup>*

> = *R<sup>B</sup>*

> > *R<sup>A</sup>* <sup>21</sup> <sup>=</sup> *<sup>R</sup><sup>B</sup>*

be set between the incident and the reflected beams (see [50], [54], [51]).

symplectic matrices. Thus, the monodromy matrix *M* is symplectic.

 =

 *z*(*s*) *p*(*s*) = *λ*

The structure of the monodromy matrix *M* is given by the following product

The reflection matrices at points A and B (see Fig. 6), *R<sup>A</sup>* and *R<sup>B</sup>* are given by

 *<sup>z</sup>*2(2*s*0) *<sup>p</sup>*2(2*s*0)

> *<sup>z</sup>*1(*s*0) *<sup>p</sup>*1(*s*0)

*<sup>z</sup>*,*p*) quasi-periodic Floquet solution of Hamiltonian system (65) with periodic

*<sup>M</sup>*<sup>21</sup> *<sup>M</sup>*<sup>22</sup> *<sup>z</sup>*(*s*)

 *z*(*s*) *p*(*s*)  *p*(*s*) 

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

*<sup>M</sup>* = *<sup>M</sup>*2*RBM*1*RA*, **det***<sup>M</sup>* = 1, (71)

 −1 0 *R<sup>A</sup>* <sup>21</sup> <sup>−</sup><sup>1</sup>

 −1 0 *RB* <sup>21</sup> <sup>−</sup><sup>1</sup>  ,

 ,

phase space *C*<sup>2</sup>

mapping for the period 2*s*0,

with the Hamiltonian,

$$H(z,p) = a(s)\frac{\dot{z}^2}{2} - \frac{z^2}{2} \tag{66}$$

The above are the same for both mono and bilayer graphene, but with different *d*(*s*) (and *a*(*s*), as mentioned above),

$$Monolayer: d(s) = \frac{2}{a(s)} \left( \mathcal{U}\_2 - \frac{\mathcal{U}\_1}{\rho} \right) + \frac{a}{\rho a(s)}$$

$$Bilayer: d(s) = \frac{\mathcal{U}\_2}{E - \mathcal{U}\_0} + \frac{\mathcal{U}\_1^2}{4(E - \mathcal{U}\_0)^2} - \frac{\mathcal{U}\_1}{\rho(E - \mathcal{U}\_0)^2} - \frac{\kappa}{\rho a(s)}$$

where *<sup>ρ</sup>*(*s*) is the radius of curvature of a trajectory. Thus, (*z*1(*s*), *<sup>p</sup>*1(*s*)) and (*z*2(*s*), *<sup>p</sup>*2(*s*)) define (*z*(*s*), *<sup>p</sup>*(*s*)) for 0 < *<sup>s</sup>* < <sup>2</sup>*s*0. The asymptotic localized solution of Gaussian beam *ψ*(*s*, *n*) is constructed in an asymptotically small neighbourhood of the PO. This solution is to be periodic with respect to *<sup>s</sup>* ∈ **<sup>R</sup>** with the period 2*s*0, and satisfies the zigzag boundary conditions. The reflection coefficient *R* is derived in the short-wave approximation, and given by

$$Monolayer: \mathbb{R} = i \exp\left[i(2\gamma + \frac{\Delta}{2})\right],\tag{67}$$

where *<sup>γ</sup>* is the angle of incidence, and *<sup>δ</sup>*<sup>1</sup> = *<sup>θ</sup>*(*s*<sup>0</sup> + <sup>0</sup>) − *<sup>θ</sup>*(*s*<sup>0</sup> − <sup>0</sup>). In the bilayer graphene system the reflection coefficient is,

$$Bilayer: R = i \exp\left[i(4\gamma + \Delta)\right].\tag{68}$$

The localized solution can be constructed if *z*(*s*), *p*(*s*) is a complex (in the complex phase space *C*<sup>2</sup> *<sup>z</sup>*,*p*) quasi-periodic Floquet solution of Hamiltonian system (65) with periodic coefficients (see [50], [54]). Namely, for the monodromy 2 × 2 matrix *M*, describing the mapping for the period 2*s*0,

$$
\begin{pmatrix} z(\mathbf{s} + \mathbf{2s}\_0) \\ p(\mathbf{s} + \mathbf{2s}\_0) \end{pmatrix} = \begin{pmatrix} M\_{11} \ M\_{12} \\ M\_{21} \ M\_{22} \end{pmatrix} \begin{pmatrix} z(\mathbf{s}) \\ p(\mathbf{s}) \end{pmatrix} \tag{69}
$$

a Floquet solution for arbitrary *s* is defined as

30 Graphene - Research and Applications

*S*(2)

<sup>0</sup> (*s*) = *<sup>S</sup>*(1)

write the Hamiltonian in its terms,

with the Hamiltonian,

*a*(*s*), as mentioned above),

system the reflection coefficient is,

by

<sup>0</sup> (*s*0) +

*s*

 *a*(*s* ′ ) + *<sup>α</sup>* 2 (*x* (0) <sup>1</sup> *<sup>γ</sup>*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*

*<sup>S</sup>*1(*s*) = *<sup>α</sup>*

*<sup>z</sup>*˙ <sup>=</sup> *<sup>p</sup> a*(*s*)

*Monolayer* : *<sup>d</sup>*(*s*) = <sup>2</sup>

*<sup>E</sup>* − *<sup>U</sup>*<sup>0</sup>

+

*Bilayer* : *<sup>d</sup>*(*s*) = *<sup>U</sup>*<sup>2</sup>

*H*(*z*, *p*) = *a*(*s*)

The above are the same for both mono and bilayer graphene, but with different *d*(*s*) (and

*a*(*s*) *<sup>U</sup>*<sup>2</sup> <sup>−</sup> *<sup>U</sup>*<sup>1</sup> *ρ* + *α ρa*(*s*)

*U*<sup>2</sup> 1 <sup>4</sup>(*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*0)<sup>2</sup> <sup>−</sup> *<sup>U</sup>*<sup>1</sup>

where *<sup>ρ</sup>*(*s*) is the radius of curvature of a trajectory. Thus, (*z*1(*s*), *<sup>p</sup>*1(*s*)) and (*z*2(*s*), *<sup>p</sup>*2(*s*)) define (*z*(*s*), *<sup>p</sup>*(*s*)) for 0 < *<sup>s</sup>* < <sup>2</sup>*s*0. The asymptotic localized solution of Gaussian beam *ψ*(*s*, *n*) is constructed in an asymptotically small neighbourhood of the PO. This solution is to be periodic with respect to *<sup>s</sup>* ∈ **<sup>R</sup>** with the period 2*s*0, and satisfies the zigzag boundary conditions. The reflection coefficient *R* is derived in the short-wave approximation, and given

*Monolayer* : *R* = *i* exp [*i*(2*γ* +

where *<sup>γ</sup>* is the angle of incidence, and *<sup>δ</sup>*<sup>1</sup> = *<sup>θ</sup>*(*s*<sup>0</sup> + <sup>0</sup>) − *<sup>θ</sup>*(*s*<sup>0</sup> − <sup>0</sup>). In the bilayer graphene

*z*˙ 2 <sup>2</sup> <sup>−</sup> *<sup>z</sup>*<sup>2</sup>

2 *x* (0) <sup>2</sup> *<sup>γ</sup>*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*

where we have defined the Berry phase to be, *<sup>e</sup>i<sup>θ</sup>* <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>γ*1, and **e1**,**<sup>2</sup>** = (1,*ei<sup>θ</sup>* )/

Here, each beam is related to the corresponding part of the periodic orbit. Namely, *<sup>ψ</sup>*<sup>1</sup> is determined by *<sup>z</sup>* = *<sup>z</sup>*1(*s*), *<sup>p</sup>* = *<sup>p</sup>*1(*s*) for 0 < *<sup>s</sup>* < *<sup>s</sup>*0, describing the electrons propagation along the lower part of the orbit from A to B, whereas *ψ*<sup>2</sup> is determined by *<sup>z</sup>* = *<sup>z</sup>*2(*s*), *<sup>p</sup>* = *<sup>p</sup>*2(*s*) for *<sup>s</sup>*<sup>0</sup> < *<sup>s</sup>* < <sup>2</sup>*s*0, for the electrons propagation along the upper part of the orbit from B to A. The complete derivation of the electronic Gaussian beams for monolayer graphene can be found in the work of Zalipaev [66]. Following the methodology developed in the previous work [66], we state the problem in terms of the function *z*(*s*) and

(0) <sup>2</sup> *<sup>γ</sup>*2) *ds*′

(0) <sup>1</sup> *<sup>γ</sup>*<sup>2</sup>  , *s*<sup>0</sup> < *s* < 2*s*0.

, *p*˙ = −*a*(*s*)*d*(*s*)*z* (65)

*<sup>ρ</sup>*(*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*0)<sup>2</sup> <sup>−</sup> *<sup>α</sup>*

∆

*Bilayer* : *R* = *i* exp [*i*(4*γ* + ∆)], (68)

<sup>2</sup> (66)

*ρa*(*s*)

<sup>2</sup> )], (67)

, (64)

√2.

*s*0

$$M\begin{pmatrix} z(s) \\ p(s) \end{pmatrix} = \lambda \begin{pmatrix} z(s) \\ p(s) \end{pmatrix}.\tag{70}$$

The structure of the monodromy matrix *M* is given by the following product

$$M = M\_2 R^B M\_1 R^A , \qquad \det M = 1 , \tag{71}$$

where *<sup>M</sup>*<sup>1</sup> = *<sup>M</sup>*1(*s*0) and *<sup>M</sup>*<sup>2</sup> = *<sup>M</sup>*2(2*s*0) are fundamental matrices of the system (65) describing the evolution (*z*(*s*), *<sup>p</sup>*(*s*)) for 0 < *<sup>s</sup>* < *<sup>s</sup>*<sup>0</sup> and *<sup>s</sup>*<sup>0</sup> < *<sup>s</sup>* < <sup>2</sup>*s*0, correspondingly.

The reflection matrices at points A and B (see Fig. 6), *R<sup>A</sup>* and *R<sup>B</sup>* are given by

$$\begin{aligned} \begin{pmatrix} z\_1(0) \\ p\_1(0) \end{pmatrix} &= R^A \begin{pmatrix} z\_2(2s\_0) \\ p\_2(2s\_0) \end{pmatrix}, \quad R^A = \begin{pmatrix} -1 & 0 \\ R\_{21}^A & -1 \end{pmatrix}, \\\\ \begin{pmatrix} z\_2(s\_0) \\ p\_2(s\_0) \end{pmatrix} &= R^B \begin{pmatrix} z\_1(s\_0) \\ p\_1(s\_0) \end{pmatrix}, \quad R^B = \begin{pmatrix} -1 & 0 \\ R\_{21}^B & -1 \end{pmatrix}, \\\\ R\_{21}^A &= R\_{21}^B = 2a \tan(\gamma), \end{aligned}$$

where *γ* is the angle of incidence of the trajectory at the points *A*, *B*. To attain *R<sup>A</sup>* and *RB*, the classical action *S* of the phase function at the reflecting boundary requires continuity to be set between the incident and the reflected beams (see [50], [54], [51]).

In a general case, the entries of *M*1,2 are to be determined numerically as the Hamiltonian system has variable coefficients. It is worth to remark that all the multipliers in (71) are symplectic matrices. Thus, the monodromy matrix *M* is symplectic.

The classical theory of linear Hamiltonian systems with periodic coefficients states that, if |*TrM*| < 2, we have a stable PO (elliptic fixed point, for example, see [27]), and ||*Mn*|| < *const* for arbitrary *n* ∈ **N**. Then, there exist two bounded complex Floquet's solutions for <sup>−</sup><sup>∞</sup> <sup>&</sup>lt; *<sup>s</sup>* <sup>&</sup>lt; <sup>+</sup>∞, namely, (*z*(*s*), *<sup>p</sup>*(*s*)) and (*z*¯(*s*), *<sup>p</sup>*¯(*s*)) with Floquet's multipliers *<sup>λ</sup>*1,2 <sup>=</sup> *<sup>e</sup>*±*i<sup>ϕ</sup>* (0 < *ϕ* < *π*), which are solutions of

$$
\lambda^2 - \text{Tr}M\lambda + 1 = 0.
$$

These solutions (*z*(*s*), *p*(*s*)) and (*z*¯(*s*), *p*¯(*s*)) may be obtained as follows. If |*TrM*| < 2, the monodromy matrix has complex eigenvectors *w* = (*wz*, *wp*) and *w*¯ = (*w*¯ *<sup>z</sup>*, *w*¯ *<sup>p</sup>*)

10.5772/52267

61

http://dx.doi.org/10.5772/52267

. (73)

, (74)

<sup>Ω</sup><sup>3</sup> sin <sup>Ω</sup>*t*,

<sup>Ω</sup> sin <sup>Ω</sup>*t*, (75)

*απ*<sup>1</sup> + *<sup>E</sup><sup>β</sup>*

**12. Numerical results**

in the following form

In this section we concentrate on the example for monolayer graphene with piece-wise linear potential *<sup>U</sup>*(*x*2) = *<sup>β</sup>*|*x*2|. The numerical techniques used in this section are described in [55]. We deal with the Dirac system (2) by incorporating the Landau gauge **<sup>A</sup>** = *<sup>B</sup>*(−*x*2, 0, 0). Thus, using dimensionless *U*, *E*, *α* and dimensionless coordinates, the Dirac system is written

The energy in eV is given by *<sup>U</sup>*0*E*, where *<sup>U</sup>*<sup>0</sup> = *vFh*¯ /*hD* = 6.59*meV*/*<sup>h</sup>* is the characteristic scale of the potential *<sup>U</sup>*. Here we assume that *<sup>D</sup>* = <sup>10</sup>−7*m*. A new small dimensionless parameter *h* (0 < *h* << 1) is supposed to be predetermined. The magnetic induction amplitude is given by *<sup>B</sup>* <sup>=</sup> *<sup>α</sup>cU*0/*vFeD* <sup>=</sup> *<sup>α</sup>*/*<sup>h</sup>* 6.5910−2*T*. Consider, as an example, a family of continuous POs which are symmetric with respect to both axes, with two reflection points A, B (see Fig. 1). The formulas describing electronic POs as solutions of the corresponding

(*p*<sup>1</sup> − *<sup>α</sup>x*2)<sup>2</sup> + *<sup>p</sup>*<sup>2</sup>

*απ*<sup>1</sup> + *<sup>E</sup><sup>β</sup>* <sup>Ω</sup><sup>2</sup> )*<sup>t</sup>* <sup>+</sup>

*<sup>x</sup>*<sup>2</sup> <sup>=</sup> *<sup>f</sup>*2(*t*, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*) = *απ*<sup>1</sup> <sup>+</sup> *<sup>E</sup><sup>β</sup>*

 *u v* = 0 0 

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

<sup>2</sup> <sup>−</sup> (*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*(*x*2))<sup>2</sup>

<sup>Ω</sup><sup>2</sup> (cos <sup>Ω</sup>*<sup>t</sup>* <sup>−</sup> <sup>1</sup>) + *<sup>α</sup>*

*<sup>x</sup>*<sup>2</sup> = *<sup>f</sup>*2(*t*, −*π*1, −*π*2, −*β*), (76)

*π*2*α*

*<sup>x</sup>*<sup>1</sup> = *<sup>D</sup>* + *<sup>f</sup>*1(*t*, −*π*1, −*π*2, −*β*),

for the upper part 0 < *t* < *t*0. Here *π*<sup>1</sup> and *π*<sup>2</sup> are the initial values of the components of momentum *<sup>p</sup>*<sup>1</sup> and *<sup>p</sup>*<sup>2</sup> at the point A, and <sup>Ω</sup> = *α*<sup>2</sup> − *<sup>β</sup>*2. It is important that *<sup>α</sup>* > *<sup>β</sup>*. In this case a drift motion of electrons and holes takes place in the positive direction of the *x*<sup>1</sup> axis, from the point *A* to the point *B* (see Fig. 6). This fact helps to construct POs. We assume that everywhere in a domain, in which we construct asymptotic solutions for electronic eigenfunctions, that the inequality *<sup>E</sup>* > *<sup>U</sup>*(*x*2) holds. In equations (75-76) *<sup>π</sup>*<sup>1</sup> is a fixed parameter, whereas *π*<sup>2</sup> and *t*<sup>0</sup> as functions of *π*<sup>1</sup> are determined uniquely by the

<sup>Ω</sup><sup>2</sup> (<sup>1</sup> <sup>−</sup> cos <sup>Ω</sup>*t*) + *<sup>π</sup>*<sup>2</sup>

 *<sup>U</sup>*(*x*2) − *E h*(−*i∂<sup>x</sup>* − *<sup>∂</sup>y*) − *<sup>α</sup>x*<sup>2</sup> *<sup>h</sup>*(−*i∂<sup>x</sup>* + *<sup>∂</sup>y*) − *<sup>α</sup>x*<sup>2</sup> *<sup>U</sup>*(*x*2) − *<sup>E</sup>*

integrable system with the Hamiltonian in the Landau gauge

*<sup>H</sup>* <sup>=</sup> <sup>1</sup> 2 

on the level set *H* = 0, are easily obtained and given by

*<sup>x</sup>*<sup>1</sup> = *<sup>f</sup>*1(*t*, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*)=(*π*<sup>1</sup> − *<sup>α</sup>*

for the lower part 0 < *t* < *t*0, and

$$M\begin{pmatrix} w\_z \\ w\_p \end{pmatrix} = e^{i\boldsymbol{\wp}} \begin{pmatrix} w\_z \\ w\_p \end{pmatrix} , \quad M \begin{pmatrix} \bar{w}\_z \\ \bar{w}\_p \end{pmatrix} = e^{-i\boldsymbol{\wp}} \begin{pmatrix} \bar{w}\_z \\ \bar{w}\_p \end{pmatrix} .$$

Then, the first solution is determined by

$$M\_1(s)R^A\begin{pmatrix}w\_z\\w\_p\end{pmatrix} = \begin{pmatrix}z(s)\\p(s)\end{pmatrix}, \quad 0 \le s \le s\_{0\times}$$

$$M\_2(s)R^B\begin{pmatrix}z(s\_0-0)\\p(s\_0-0)\end{pmatrix} = \begin{pmatrix}z(s)\\p(s)\end{pmatrix}, \qquad s\_0 \le s \le 2s\_0.$$

Satisfaction of the solution to the zigzag boundary condition at the interface *<sup>L</sup>*<sup>1</sup> (*u*|*L*<sup>1</sup> <sup>=</sup> 0), as well as the requirement that the solution should be periodic with respect to *s* ∈ **R** with the period 2*s*<sup>0</sup> lead to a generalized Bohr-Sommerfeld quantization condition determining semiclassical asymptotics of the high energy spectrum. Namely, after the integration around the closed loop of PO, the total variation of the classical action *S* and the phase of the amplitude of *ψ*<sup>0</sup> must be equal to 2*πm*1. Thus, we obtain the quantization condition for electrons and holes in the form

$$\int\_{0}^{2s\_0} a(s)ds - aA = h[\pm 2\pi m\_1 + (m\_2 + 1/2)\varrho + \Delta],\tag{72}$$

$$Monolayer: \Delta = \pi - \frac{\gamma}{2}\gamma$$

$$Bilayer: \Delta = -\gamma$$

where *<sup>m</sup>*1,2 ∈ **<sup>N</sup>** are the longitudinal and the transversal quantization indexes, and for electrons we have +, for holes −. The index *<sup>m</sup>*<sup>2</sup> and factor <sup>∆</sup> appear due to the variation of the phase of *<sup>σ</sup>m*<sup>2</sup> (*s*, *<sup>ν</sup>*) (see the formulas in [66]). Here in the left-hand side in (72)

$$A = \frac{1}{2} \int\_0^{2s\_0} ( (\varkappa\_1^{(0)} \gamma\_1 + \varkappa\_2^{(0)} \gamma\_2) ) ds$$

is the area encircled by PO.

Assuming the presence of a continuous family of POs that depend on *E*, the quantization condition is satisfied only for a discrete set of energy levels *<sup>E</sup>* = *Em*1,*m*<sup>2</sup> . It is clear that the quantization condition may be fulfilled only if the longitudinal index *m*<sup>1</sup> is positive and large as *<sup>h</sup>* → 0. At the same time, the transversal index *<sup>m</sup>*<sup>2</sup> = 0, 1, 2, ... should be of the order 1 as very large values of *<sup>m</sup>*<sup>2</sup> would lead to the asymptotic solution *<sup>ψ</sup>* = *<sup>ψ</sup>*<sup>1</sup> + *<sup>R</sup>ψ*<sup>2</sup> becoming not localized.

## **12. Numerical results**

32 Graphene - Research and Applications

*M wz wp* = *e iϕ wz wp* 

Then, the first solution is determined by

electrons and holes in the form

is the area encircled by PO.

localized.

*<sup>M</sup>*1(*s*)*R<sup>A</sup>*

*<sup>M</sup>*2(*s*)*R<sup>B</sup>*

 2*s*<sup>0</sup>

0

These solutions (*z*(*s*), *p*(*s*)) and (*z*¯(*s*), *p*¯(*s*)) may be obtained as follows. If |*TrM*| < 2, the

, *M*

 *z*(*s*) *p*(*s*) 

> *z*(*s*) *p*(*s*)

Satisfaction of the solution to the zigzag boundary condition at the interface *<sup>L</sup>*<sup>1</sup> (*u*|*L*<sup>1</sup> <sup>=</sup> 0), as well as the requirement that the solution should be periodic with respect to *s* ∈ **R** with the period 2*s*<sup>0</sup> lead to a generalized Bohr-Sommerfeld quantization condition determining semiclassical asymptotics of the high energy spectrum. Namely, after the integration around the closed loop of PO, the total variation of the classical action *S* and the phase of the amplitude of *ψ*<sup>0</sup> must be equal to 2*πm*1. Thus, we obtain the quantization condition for

*Monolayer* : <sup>∆</sup> <sup>=</sup> *<sup>π</sup>* <sup>−</sup> *<sup>γ</sup>*

*Bilayer* : ∆ = −*γ* where *<sup>m</sup>*1,2 ∈ **<sup>N</sup>** are the longitudinal and the transversal quantization indexes, and for electrons we have +, for holes −. The index *<sup>m</sup>*<sup>2</sup> and factor <sup>∆</sup> appear due to the variation

Assuming the presence of a continuous family of POs that depend on *E*, the quantization condition is satisfied only for a discrete set of energy levels *<sup>E</sup>* = *Em*1,*m*<sup>2</sup> . It is clear that the quantization condition may be fulfilled only if the longitudinal index *m*<sup>1</sup> is positive and large as *<sup>h</sup>* → 0. At the same time, the transversal index *<sup>m</sup>*<sup>2</sup> = 0, 1, 2, ... should be of the order 1 as very large values of *<sup>m</sup>*<sup>2</sup> would lead to the asymptotic solution *<sup>ψ</sup>* = *<sup>ψ</sup>*<sup>1</sup> + *<sup>R</sup>ψ*<sup>2</sup> becoming not

of the phase of *<sup>σ</sup>m*<sup>2</sup> (*s*, *<sup>ν</sup>*) (see the formulas in [66]). Here in the left-hand side in (72)

((*x* (0) <sup>1</sup> *<sup>γ</sup>*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*

*<sup>A</sup>* <sup>=</sup> <sup>1</sup> 2 2*s*<sup>0</sup>

0

 *w*¯ *<sup>z</sup> w*¯ *p* = *e* −*iϕ w*¯ *<sup>z</sup> w*¯ *p* .

, 0 ≤ *<sup>s</sup>* ≤ *<sup>s</sup>*0,

*<sup>a</sup>*(*s*)*ds* − *<sup>α</sup><sup>A</sup>* = *<sup>h</sup>*[±2*πm*<sup>1</sup> + (*m*<sup>2</sup> + 1/2)*<sup>ϕ</sup>* + <sup>∆</sup>], (72)

2 ,

(0) <sup>2</sup> *<sup>γ</sup>*2))*ds*

, *<sup>s</sup>*<sup>0</sup> ≤ *<sup>s</sup>* ≤ <sup>2</sup>*s*0.

monodromy matrix has complex eigenvectors *w* = (*wz*, *wp*) and *w*¯ = (*w*¯ *<sup>z</sup>*, *w*¯ *<sup>p</sup>*)

 *wz wp* =

 *<sup>z</sup>*(*s*<sup>0</sup> − <sup>0</sup>) *<sup>p</sup>*(*s*<sup>0</sup> − <sup>0</sup>)

 = In this section we concentrate on the example for monolayer graphene with piece-wise linear potential *<sup>U</sup>*(*x*2) = *<sup>β</sup>*|*x*2|. The numerical techniques used in this section are described in [55]. We deal with the Dirac system (2) by incorporating the Landau gauge **<sup>A</sup>** = *<sup>B</sup>*(−*x*2, 0, 0). Thus, using dimensionless *U*, *E*, *α* and dimensionless coordinates, the Dirac system is written in the following form

$$
\begin{pmatrix}
\mathcal{U}(\mathbf{x}\_2) - E & h(-i\partial\_x - \partial\_y) - a\mathbf{x}\_2 \\
h(-i\partial\_x + \partial\_y) - a\mathbf{x}\_2 & \mathcal{U}(\mathbf{x}\_2) - E
\end{pmatrix}
\begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.
\tag{73}
$$

The energy in eV is given by *<sup>U</sup>*0*E*, where *<sup>U</sup>*<sup>0</sup> = *vFh*¯ /*hD* = 6.59*meV*/*<sup>h</sup>* is the characteristic scale of the potential *<sup>U</sup>*. Here we assume that *<sup>D</sup>* = <sup>10</sup>−7*m*. A new small dimensionless parameter *h* (0 < *h* << 1) is supposed to be predetermined. The magnetic induction amplitude is given by *<sup>B</sup>* <sup>=</sup> *<sup>α</sup>cU*0/*vFeD* <sup>=</sup> *<sup>α</sup>*/*<sup>h</sup>* 6.5910−2*T*. Consider, as an example, a family of continuous POs which are symmetric with respect to both axes, with two reflection points A, B (see Fig. 1). The formulas describing electronic POs as solutions of the corresponding integrable system with the Hamiltonian in the Landau gauge

$$H = \frac{1}{2} \left( (p\_1 - a\mathbf{x}\_2)^2 + p\_2^2 - (E - \mathcal{U}(\mathbf{x}\_2))^2 \right),\tag{74}$$

on the level set *H* = 0, are easily obtained and given by

$$\mathbf{x}\_1 = f\_1(t, \pi\_1, \pi\_2, \boldsymbol{\beta}) = (\pi\_1 - \boldsymbol{a}\frac{\boldsymbol{a}\pi\_1 + \boldsymbol{E}\boldsymbol{\beta}}{\Omega^2})t + \frac{\pi\_2 \boldsymbol{a}}{\Omega^2}(\cos\Omega t - 1) + \boldsymbol{a}\frac{\boldsymbol{a}\pi\_1 + \boldsymbol{E}\boldsymbol{\beta}}{\Omega^3}\sin\Omega t, \boldsymbol{\beta}$$

$$\mathbf{x}\_{2} = f\_{2}(t, \pi\_{1}, \pi\_{2}, \boldsymbol{\beta}) = \frac{a\pi\_{1} + E\boldsymbol{\beta}}{\Omega^{2}}(1 - \cos\Omega t) + \frac{\pi\_{2}}{\Omega}\sin\Omega t,\tag{75}$$

for the lower part 0 < *t* < *t*0, and

$$\mathbf{x}\_1 = D + f\_1(t\_\prime - \pi\_{1\prime} - \pi\_{2\prime} - \beta),$$

$$\mathbf{x}\_{2} = f\_{2}(t, -\pi\_{1\prime} - \pi\_{2\prime} - \beta),\tag{76}$$

for the upper part 0 < *t* < *t*0. Here *π*<sup>1</sup> and *π*<sup>2</sup> are the initial values of the components of momentum *<sup>p</sup>*<sup>1</sup> and *<sup>p</sup>*<sup>2</sup> at the point A, and <sup>Ω</sup> = *α*<sup>2</sup> − *<sup>β</sup>*2. It is important that *<sup>α</sup>* > *<sup>β</sup>*. In this case a drift motion of electrons and holes takes place in the positive direction of the *x*<sup>1</sup> axis, from the point *A* to the point *B* (see Fig. 6). This fact helps to construct POs. We assume that everywhere in a domain, in which we construct asymptotic solutions for electronic eigenfunctions, that the inequality *<sup>E</sup>* > *<sup>U</sup>*(*x*2) holds. In equations (75-76) *<sup>π</sup>*<sup>1</sup> is a fixed parameter, whereas *π*<sup>2</sup> and *t*<sup>0</sup> as functions of *π*<sup>1</sup> are determined uniquely by the equations *<sup>f</sup>*1(*t*0, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*) = *<sup>D</sup>*, *<sup>f</sup>*2(*t*0, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*) = 0. The formulas, describing PO holes with Hamiltonian

$$H = \frac{1}{2} \left( -\left( p\_1 - \alpha x\_2 \right)^2 - p\_2^2 + \left( E - \mathcal{U}(x\_2) \right)^2 \right), \tag{77}$$

10.5772/52267

63

http://dx.doi.org/10.5772/52267

1

<sup>2</sup> computed by semiclassical analysis for the state *m*<sup>1</sup> = 27.2, *m*<sup>2</sup> = 2

TrM/2

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> −1

π1

<sup>2</sup> is shown that

2

3

4

5

6

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

x1

0

(a) (b) **Figure 11.** Dependence of *TrM*/2 on *π*<sup>1</sup> - (a) and dependence of *Im*(Γ) on *s* - (b), for the state *m*<sup>1</sup> = 27, *m*<sup>2</sup> = 0 with

For the lens-shaped class of POs the high-energy semiclassical localized eigenstates were tested successfully against the energy eigenvalues and the eigenfunctions computed by the

*<sup>u</sup>*|*x*1=*<sup>D</sup>* = 0, *<sup>v</sup>*|*x*1=<sup>0</sup> = 0.

was computed semiclassically for the the states *m*<sup>1</sup> = 27, *m*<sup>2</sup> = 0, 1, 2 with *E* = 2.2538, 2.2812, 2.3078 for *h* = 0.025, *α* = 1.0, *β* = 0.5. It is worth to remark that in this

finite element method using COMSOL (see [70]). The boundary conditions

In Fig's. 9 and 10, the electronic eigenfunction density component |*u*|

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

x2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

**Figure 10.** Electronic eigenfunction density component |*u*|

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 2.1

*E* = 2.2538 for *α* = 1, *β* = 0.5, *h* = 0.025.

s

were used in the following numerical experiments.

with *E* = 2.3078 for *α* = 1, *β* = 0.5, *h* = 0.025.

Im(Γ)

2.2 2.3 2.4 2.5 2.6 2.7 2.8

on the level set *H* = 0, are given by

$$x\_1 = g\_1(t, \pi\_1, \pi\_2, \beta) = -(\pi\_1 - a \frac{a\pi\_1 - E\beta}{\Omega^2})t + \frac{\pi\_2 a}{\Omega^2}(\cos \Omega t - 1) - a \frac{a\pi\_1 - E\beta}{\Omega^3} \sin \Omega t, \beta$$

$$\mathbf{x}\_{2} = \mathbf{g}\_{2}(t, \pi\_{1}, \pi\_{2}, \boldsymbol{\beta}) = \frac{\mathbf{a}\pi\_{1} - E\boldsymbol{\beta}}{\Omega^{2}}(1 - \cos\Omega t) - \frac{\pi\_{2}}{\Omega}\sin\Omega t,\tag{78}$$

for the upper part 0 < *t* < *t*0, and

$$\mathbf{x}\_1 = D + \mathbf{g}\_1(t\_\prime - \pi\_{1\prime} - \pi\_{2\prime} - \beta),$$

$$\mathbf{x}\_2 = \mathbf{g}\_2(\mathbf{t}\_\prime - \boldsymbol{\pi}\_1, -\boldsymbol{\pi}\_2, -\boldsymbol{\beta}),\tag{79}$$

for the lower part 0 < *t* < *t*0. It is worth to remark that holes move along a clockwise direction of PO whereas electrons run counter-clockwise around the PO contour. Thus, we have for electrons and holes the continuous family of POs with respect to parameter *π*<sup>1</sup> which look like lens-shaped contours. As soon as the parameter *π*<sup>1</sup> has been determined from the generalized Bohr-Sommerfeld quantization condition (72), the semiclassical energy levels for electrons and holes are computed by

$$E = \pm \sqrt{\pi\_1^2 + \pi\_2^2}. \tag{80}$$

**Figure 9.** Electronic eigenfunction density component |*u*| <sup>2</sup> computed by semiclassical analysis for the state *m*<sup>1</sup> = 27, *m*<sup>2</sup> = 0 with *E* = 2.2538 - (a) and *m*<sup>1</sup> = 27.1, *m*<sup>2</sup> = 1 with *E* = 2.2812 - (b), for *α* = 1, *β* = 0.5, *h* = 0.025.

**Figure 10.** Electronic eigenfunction density component |*u*| <sup>2</sup> computed by semiclassical analysis for the state *m*<sup>1</sup> = 27.2, *m*<sup>2</sup> = 2 with *E* = 2.3078 for *α* = 1, *β* = 0.5, *h* = 0.025.

**Figure 11.** Dependence of *TrM*/2 on *π*<sup>1</sup> - (a) and dependence of *Im*(Γ) on *s* - (b), for the state *m*<sup>1</sup> = 27, *m*<sup>2</sup> = 0 with *E* = 2.2538 for *α* = 1, *β* = 0.5, *h* = 0.025.

For the lens-shaped class of POs the high-energy semiclassical localized eigenstates were tested successfully against the energy eigenvalues and the eigenfunctions computed by the finite element method using COMSOL (see [70]). The boundary conditions

$$u|\_{\mathbf{x}\_1=D} = \mathbf{0}, \quad \left. v|\_{\mathbf{x}\_1=0} = \mathbf{0}. \right|$$

were used in the following numerical experiments.

34 Graphene - Research and Applications

on the level set *H* = 0, are given by

for the upper part 0 < *t* < *t*0, and

*<sup>x</sup>*<sup>1</sup> = *<sup>g</sup>*1(*t*, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*) = −(*π*<sup>1</sup> − *<sup>α</sup>*

levels for electrons and holes are computed by

x1

**Figure 9.** Electronic eigenfunction density component |*u*|

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −0.4

x2

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 *<sup>H</sup>* <sup>=</sup> <sup>1</sup> 2 

with Hamiltonian

equations *<sup>f</sup>*1(*t*0, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*) = *<sup>D</sup>*, *<sup>f</sup>*2(*t*0, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*) = 0. The formulas, describing PO holes

*<sup>x</sup>*<sup>1</sup> = *<sup>D</sup>* + *<sup>g</sup>*1(*t*, −*π*1, −*π*2, −*β*),

for the lower part 0 < *t* < *t*0. It is worth to remark that holes move along a clockwise direction of PO whereas electrons run counter-clockwise around the PO contour. Thus, we have for electrons and holes the continuous family of POs with respect to parameter *π*<sup>1</sup> which look like lens-shaped contours. As soon as the parameter *π*<sup>1</sup> has been determined from the generalized Bohr-Sommerfeld quantization condition (72), the semiclassical energy

> *E* = ± *π*2 <sup>1</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup>

with *E* = 2.2538 - (a) and *m*<sup>1</sup> = 27.1, *m*<sup>2</sup> = 1 with *E* = 2.2812 - (b), for *α* = 1, *β* = 0.5, *h* = 0.025.

(a) (b)

x2

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

<sup>2</sup> + (*<sup>E</sup>* <sup>−</sup> *<sup>U</sup>*(*x*2))<sup>2</sup>

<sup>Ω</sup><sup>2</sup> (cos <sup>Ω</sup>*<sup>t</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> *<sup>α</sup>*

*<sup>x</sup>*<sup>2</sup> = *<sup>g</sup>*2(*t*, −*π*1, −*π*2, −*β*), (79)

<sup>2</sup>. (80)

x1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −0.5

<sup>2</sup> computed by semiclassical analysis for the state *m*<sup>1</sup> = 27, *m*<sup>2</sup> = 0

*π*2*α*

<sup>Ω</sup><sup>2</sup> (<sup>1</sup> <sup>−</sup> cos <sup>Ω</sup>*t*) <sup>−</sup> *<sup>π</sup>*<sup>2</sup>

, (77)

<sup>Ω</sup><sup>3</sup> sin <sup>Ω</sup>*t*,

<sup>Ω</sup> sin <sup>Ω</sup>*t*, (78)

*απ*<sup>1</sup> − *<sup>E</sup><sup>β</sup>*

− (*p*<sup>1</sup> − *<sup>α</sup>x*2)<sup>2</sup> − *<sup>p</sup>*<sup>2</sup>

*απ*<sup>1</sup> − *<sup>E</sup><sup>β</sup>* <sup>Ω</sup><sup>2</sup> )*<sup>t</sup>* <sup>+</sup>

*<sup>x</sup>*<sup>2</sup> <sup>=</sup> *<sup>g</sup>*2(*t*, *<sup>π</sup>*1, *<sup>π</sup>*2, *<sup>β</sup>*) = *απ*<sup>1</sup> <sup>−</sup> *<sup>E</sup><sup>β</sup>*

In Fig's. 9 and 10, the electronic eigenfunction density component |*u*| <sup>2</sup> is shown that was computed semiclassically for the the states *m*<sup>1</sup> = 27, *m*<sup>2</sup> = 0, 1, 2 with *E* = 2.2538, 2.2812, 2.3078 for *h* = 0.025, *α* = 1.0, *β* = 0.5. It is worth to remark that in this case the localization of eigenfunction density components takes place in close neighborhood of PO. In all shown figures computed by semiclassical analysis one can easily see a white contour of PO. In Fig. 9 (a) dependence of *TrM*/2 on *<sup>π</sup>*<sup>1</sup> - (a) and dependence of *Im*(Γ) on *<sup>s</sup>* - (b), for the state *<sup>m</sup>*<sup>1</sup> = 27, *<sup>m</sup>*<sup>2</sup> = 0 with *<sup>E</sup>* = 2.2538 *<sup>α</sup>* = 1, *<sup>β</sup>* = 0.5, *<sup>h</sup>* = 0.025 are shown.

10.5772/52267

65

http://dx.doi.org/10.5772/52267

. (83)

<sup>2</sup> = −1, (86)

(88)

<sup>1</sup> , *<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>1, (81)

<sup>1</sup> , *<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>3, (82)

2. (85)

. (87)

, (89)

2. (90)

*<sup>S</sup>*+*<sup>S</sup>* <sup>=</sup> *SS*<sup>+</sup> <sup>=</sup> *<sup>I</sup>*, (91)

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup> i <sup>h</sup> Sp* (*x*,*x*1) *J* + *p*

*<sup>ψ</sup>*<sup>3</sup> <sup>=</sup> *<sup>e</sup> i <sup>h</sup> Sp* (*x*,*x*2) *J* + *p*

*d* = *Ta*, *T* =

Thus, for the slope transfer matrix *T* it holds that

(see equation (8) in [17] or (18) in [18])

and

we obtain that

or

we have

and

*T*¯

From (90) we obtain that

<sup>21</sup>*T*<sup>22</sup> − *T*¯

*a*1*e* + <sup>1</sup> <sup>+</sup>

*d*1*e* + <sup>1</sup> <sup>+</sup> *e* −*i <sup>h</sup> Sp* (*x*,*x*2) *J* − *p*

 *T*<sup>11</sup> *T*<sup>12</sup> *<sup>T</sup>*<sup>21</sup> *<sup>T</sup>*<sup>22</sup>


<sup>11</sup>*T*<sup>12</sup> = 0, |*T*21|

*T*+ −1 0 0 1

<sup>2</sup> − |*a*2|

*e*− *i*

*<sup>h</sup> Sp* (*x*,*x*1) *J* − *p*

, *d* =

Taking into account the conservation of the *x*-component of the probability density current

<sup>2</sup> = |*d*2|

<sup>2</sup> − |*T*11|

*T* = 1 0 0 −1 

As a result we have |*T*11| = |*T*22|, |*T*12| = |*T*21|, |*det*(*T*)| = 1. For the scattering matrix

*S* = *S*<sup>11</sup> *S*<sup>12</sup> *<sup>S</sup>*<sup>21</sup> *<sup>S</sup>*<sup>22</sup>

*S a*<sup>1</sup> *d*1 = *a*<sup>2</sup> *d*2 

<sup>2</sup> + |*d*1|

<sup>2</sup> = |*d*2|

<sup>2</sup> + |*d*2|


*a*2*e* −

*d*2*e* −

 *d*1 *d*2 

<sup>2</sup> − |*d*1|

<sup>2</sup> = 1, |*T*22|

, *a* =

 *a*1 *a*2 

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

*Jx* = *vu*¯ + *uv*¯ , (84)

<sup>2</sup> − |*T*12|

### **13. Conclusion**

In this review we have outlined our work on the semiclassical analysis of graphene structures and introduced some new results for monolayer and bilayer graphene. We have outlined a range of new asymptotic methods and a semiclassical analysis of Dirac electron-hole tunneling through a Gaussian shaped barrier that represents an electrostatic potential. We started by analyzing the rectangular barrier and have found some important differences between it and the smooth barrier. Namely, the smooth barrier exhibits a quasi-discrete spectrum and complex bound states that do not exist for the rectangular barrier (in zone "3" in Fig. 3). In both types of barrier Klein tunneling occurs. The WKB approximation deals with the asymptotic analysis of matched asymptotic techniques and boundary layers for the turning points in the barrier. The main results of this work are eloquent WKB formulas for the entries in a smooth barrier transfer matrix. This matrix explains the mechanism of total transmission through the barrier for some resonance values of energy of skew incident electrons or holes. Moreover, it has been shown that the existence of modes localized within the barrier, and exponentially decaying away from it, for two discrete complex and real sets of energy eigenlevels can be determined by the Bohr-Sommerfeld quantization condition. It was shown that the total transmission through the barrier takes place when the energy of an incident electron or hole coincides with a real part of the complex energy eigenlevel of one among the set of localized modes. These facts were confirmed by numerical simulations done by finite elements methods and have been found to also be in excellent agreement with the results found using finite difference methods as in [71].

We have also applied the Gaussian beam methods, originated by Popov [51] and expanded by Zalipaev [66] to quantum problems, to describe monolayer and bilayer graphene. We have constructed eigenfunctions and defined the stable periodic orbit conditions and the quantization conditions. The reflection and transmission coefficients of monolayer and bilayer graphene have been derived within the context of semiclassical physics in full. It is clear that these methods can offer good insights into the behavior of the graphene Fabry-Perot resonator.

Such systems will find applications in plasmonics, and nanoribbon heterostructures made from graphene are promising to emerge. The kind of bilayer structure analyzed here can be created by chemical vapor deposition [32] and this opens up the road to a flurry of geometrically optimized graphene resonator systems, whether acting in isolation or as part of a composite, or array.

## **14. Appendix A. Transfer and scattering matrix properties for a smooth step**

Let us formulate this scattering problem in terms of transfer matrix *T* for the left slope of the entire barrier (see [67])

$$\psi\_1 = \frac{e^{\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^+}} a\_1 e\_1^+ + \frac{e^{-\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^-}} a\_2 e\_1^-, \qquad \mathbf{x} \in \Omega\_1. \tag{81}$$

$$\psi\_3 = \frac{e^{\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^+}} d\_1 e\_1^+ + \frac{e^{\frac{-i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^-}} d\_2 e\_1^- , \qquad \mathbf{x} \in \Omega\_{3\prime} \tag{82}$$

and

36 Graphene - Research and Applications

**13. Conclusion**

Fabry-Perot resonator.

of a composite, or array.

entire barrier (see [67])

**step**

case the localization of eigenfunction density components takes place in close neighborhood of PO. In all shown figures computed by semiclassical analysis one can easily see a white contour of PO. In Fig. 9 (a) dependence of *TrM*/2 on *<sup>π</sup>*<sup>1</sup> - (a) and dependence of *Im*(Γ) on *<sup>s</sup>* - (b), for the state *<sup>m</sup>*<sup>1</sup> = 27, *<sup>m</sup>*<sup>2</sup> = 0 with *<sup>E</sup>* = 2.2538 *<sup>α</sup>* = 1, *<sup>β</sup>* = 0.5, *<sup>h</sup>* = 0.025 are shown.

In this review we have outlined our work on the semiclassical analysis of graphene structures and introduced some new results for monolayer and bilayer graphene. We have outlined a range of new asymptotic methods and a semiclassical analysis of Dirac electron-hole tunneling through a Gaussian shaped barrier that represents an electrostatic potential. We started by analyzing the rectangular barrier and have found some important differences between it and the smooth barrier. Namely, the smooth barrier exhibits a quasi-discrete spectrum and complex bound states that do not exist for the rectangular barrier (in zone "3" in Fig. 3). In both types of barrier Klein tunneling occurs. The WKB approximation deals with the asymptotic analysis of matched asymptotic techniques and boundary layers for the turning points in the barrier. The main results of this work are eloquent WKB formulas for the entries in a smooth barrier transfer matrix. This matrix explains the mechanism of total transmission through the barrier for some resonance values of energy of skew incident electrons or holes. Moreover, it has been shown that the existence of modes localized within the barrier, and exponentially decaying away from it, for two discrete complex and real sets of energy eigenlevels can be determined by the Bohr-Sommerfeld quantization condition. It was shown that the total transmission through the barrier takes place when the energy of an incident electron or hole coincides with a real part of the complex energy eigenlevel of one among the set of localized modes. These facts were confirmed by numerical simulations done by finite elements methods and have been found to also be in excellent agreement with

We have also applied the Gaussian beam methods, originated by Popov [51] and expanded by Zalipaev [66] to quantum problems, to describe monolayer and bilayer graphene. We have constructed eigenfunctions and defined the stable periodic orbit conditions and the quantization conditions. The reflection and transmission coefficients of monolayer and bilayer graphene have been derived within the context of semiclassical physics in full. It is clear that these methods can offer good insights into the behavior of the graphene

Such systems will find applications in plasmonics, and nanoribbon heterostructures made from graphene are promising to emerge. The kind of bilayer structure analyzed here can be created by chemical vapor deposition [32] and this opens up the road to a flurry of geometrically optimized graphene resonator systems, whether acting in isolation or as part

**14. Appendix A. Transfer and scattering matrix properties for a smooth**

Let us formulate this scattering problem in terms of transfer matrix *T* for the left slope of the

the results found using finite difference methods as in [71].

$$d = \text{Ta}, \quad T = \begin{pmatrix} T\_{11} \ T\_{12} \\ T\_{21} \ T\_{22} \end{pmatrix}, \quad d = \begin{pmatrix} d\_1 \\ d\_2 \end{pmatrix}, \quad a = \begin{pmatrix} a\_1 \\ a\_2 \end{pmatrix}. \tag{83}$$

Taking into account the conservation of the *x*-component of the probability density current (see equation (8) in [17] or (18) in [18])

$$J\_{\lambda} = \overline{v}u + \overline{u}v\_{\prime} \tag{84}$$

we obtain that

$$\left| \left| a\_1 \right|^2 - \left| a\_2 \right|^2 = \left| d\_2 \right|^2 - \left| d\_1 \right|^2. \tag{85}$$

Thus, for the slope transfer matrix *T* it holds that

$$|\bar{T}\_{21}T\_{22} - \bar{T}\_{11}T\_{12} = 0, \quad |T\_{21}|^2 - |T\_{11}|^2 = 1, \quad |T\_{22}|^2 - |T\_{12}|^2 = -1,\tag{86}$$

or

$$T^{+}\begin{pmatrix}-1 \ 0 \\ 0 \ 1\end{pmatrix}T = \begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}.\tag{87}$$

As a result we have |*T*11| = |*T*22|, |*T*12| = |*T*21|, |*det*(*T*)| = 1. For the scattering matrix

$$\mathbf{S} = \begin{pmatrix} \mathbf{S}\_{11} \ \mathbf{S}\_{12} \\ \mathbf{S}\_{21} \ \mathbf{S}\_{22} \end{pmatrix} \tag{88}$$

we have

$$S\begin{pmatrix} a\_1\\d\_1 \end{pmatrix} = \begin{pmatrix} a\_2\\d\_2 \end{pmatrix},\tag{89}$$

and

$$|a\_1|^2 + |d\_1|^2 = |d\_2|^2 + |d\_2|^2. \tag{90}$$

From (90) we obtain that

$$\mathbf{S}^+\mathbf{S} = \mathbf{S}\mathbf{S}^+ = I,\tag{91}$$

thus, the scattering matrix is unitary. If the entries of *S* are known, then,

$$T = \begin{pmatrix} -\mathbb{S}\_{11}/\mathbb{S}\_{12} & 1/\mathbb{S}\_{12} \\ \mathbb{S}\_{21} - \mathbb{S}\_{11}\mathbb{S}\_{22}/\mathbb{S}\_{12} \ \mathbb{S}\_{22}/\mathbb{S}\_{12} \end{pmatrix}, \quad \det(T) = -\frac{\mathbb{S}\_{21}}{\mathbb{S}\_{12}}.\tag{92}$$

10.5772/52267

67

<sup>2</sup> = 1. (100)

http://dx.doi.org/10.5772/52267

<sup>2</sup> = 1. (101)

2, (102)

. (103)

, (104)

2. (105)

. (107)

*<sup>S</sup>*+*<sup>S</sup>* <sup>=</sup> *SS*<sup>+</sup> <sup>=</sup> *<sup>I</sup>*. (106)

*S*12

, *detT* = 1. (108)

, *det*(*T*) = *<sup>S</sup>*<sup>21</sup>

*<sup>t</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> *TL* 12

> *<sup>t</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> *TL* 12

> > |*a*1|

*T*<sup>+</sup> 1 0 0 −1 *T* = 1 0 0 −1 


*T* = *T*<sup>11</sup> *T*<sup>12</sup> *T*∗ <sup>12</sup> *<sup>T</sup>*<sup>∗</sup> 11

<sup>2</sup> − |*a*2|

*S a*<sup>1</sup> *d*2 = *a*<sup>2</sup> *d*1 

<sup>2</sup> + |*d*2|

*<sup>S</sup>*<sup>21</sup> − *<sup>S</sup>*11*S*22/*S*<sup>12</sup> *<sup>S</sup>*22/*S*<sup>12</sup> <sup>−</sup>*S*11/*S*<sup>12</sup> 1/*S*<sup>12</sup>

<sup>2</sup> = |*a*2|

Taking into account the time-reversal symmetry in scattering through the graphene barrier,

<sup>2</sup> + |*d*1|

If *<sup>a</sup>*<sup>1</sup> = 0, *<sup>a</sup>*<sup>2</sup> = *<sup>t</sup>*2, *<sup>d</sup>*<sup>1</sup> = 1, *<sup>d</sup>*<sup>2</sup> = *<sup>r</sup>*2, then

For the scattering matrix *S* we have

From (105) we obtain that

we obtain *S* = *ST*, and

If the entries of *S* are known, then,

*T* = 

**barrier**

we have

and

and

, *<sup>r</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>T</sup><sup>L</sup>*

, *<sup>r</sup>*<sup>2</sup> <sup>=</sup> *<sup>T</sup><sup>L</sup>*

11 *TL* 12

22 *TL* 12

**15. Appendix B. Transfer and scattering matrix properties for a smooth**

Let us formulate this scattering problem in terms of transfer matrix *T* for the entire barrier. The definition of *T* is given by (81),(82), and looks the same *Ta* = *d*. However, for the barrier

<sup>2</sup> = |*d*1|

<sup>2</sup> − |*d*2|

, |*r*1|

, |*r*2|

<sup>2</sup> + |*t*1|

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

<sup>2</sup> + |*t*2|

Time-reversal symmetry in scattering through the graphene barrier would mean that

$$(\sigma\_3 \psi\_1)^\* = \frac{e^{-\frac{i}{\hbar}S\_\mathbb{P}(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^-}} a\_1^\* e\_1^- + \frac{e^{\frac{i}{\hbar}S\_\mathbb{P}(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^+}} a\_2^\* e\_1^+ , \quad \mathbf{x} \in \Omega\_{1\prime} \tag{93}$$

$$\left(\sigma\_3 \psi\_3\right)^\* = \frac{e^{-\frac{i}{\hbar}S\_\rho(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^-}} d\_1^\* e\_1^- + \frac{e^{\frac{i}{\hbar}S\_\rho(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^+}} d\_2^\* e\_1^+ , \quad \mathbf{x} \in \Omega\_{3\nu} \tag{94}$$

are both asymptotic solutions to the Dirac system, and

$$
\sigma\_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.
$$

Thus, we have

$$S\begin{pmatrix} a\_2^\* \\ d\_2^\* \end{pmatrix} = \begin{pmatrix} a\_1^\* \\ d\_1^\* \end{pmatrix} \tag{95}$$

$$T\begin{pmatrix} a\_2^\* \\ a\_1^\* \end{pmatrix} = \begin{pmatrix} d\_2^\* \\ d\_1^\* \end{pmatrix}. \tag{96}$$

In what follows that

$$S = S^T \, , \quad \begin{pmatrix} 0 \ 1 \\ 1 \ 0 \end{pmatrix} T \begin{pmatrix} 0 \ 1 \\ 1 \ 0 \end{pmatrix} = T^\* \, . \tag{97}$$

Thus, *<sup>S</sup>*<sup>12</sup> = *<sup>S</sup>*21,

$$
\det T = -1,\tag{98}
$$

and

$$T = \begin{pmatrix} T\_{11} \ T\_{12} \\ T\_{12}^\* \ T\_{11}^\* \end{pmatrix}. \tag{99}$$

If *<sup>a</sup>*<sup>1</sup> = 1, *<sup>a</sup>*<sup>2</sup> = *<sup>r</sup>*1, *<sup>d</sup>*<sup>1</sup> = 0, *<sup>d</sup>*<sup>2</sup> = *<sup>t</sup>*1, then

$$t\_1 = \frac{1}{T\_{12}^L}, \quad r\_1 = -\frac{T\_{11}^L}{T\_{12}^L}, \quad |r\_1|^2 + |t\_1|^2 = 1. \tag{100}$$

If *<sup>a</sup>*<sup>1</sup> = 0, *<sup>a</sup>*<sup>2</sup> = *<sup>t</sup>*2, *<sup>d</sup>*<sup>1</sup> = 1, *<sup>d</sup>*<sup>2</sup> = *<sup>r</sup>*2, then

$$t\_2 = \frac{1}{T\_{12}^L}, \quad r\_2 = \frac{T\_{22}^L}{T\_{12}^L}, \quad \left|r\_2\right|^2 + \left|t\_2\right|^2 = 1. \tag{101}$$

#### **15. Appendix B. Transfer and scattering matrix properties for a smooth barrier**

Let us formulate this scattering problem in terms of transfer matrix *T* for the entire barrier. The definition of *T* is given by (81),(82), and looks the same *Ta* = *d*. However, for the barrier we have

$$|a\_1|^2 - |a\_2|^2 = |d\_1|^2 - |d\_2|^2,\tag{102}$$

and

38 Graphene - Research and Applications

Thus, we have

In what follows that

Thus, *<sup>S</sup>*<sup>12</sup> = *<sup>S</sup>*21,

If *<sup>a</sup>*<sup>1</sup> = 1, *<sup>a</sup>*<sup>2</sup> = *<sup>r</sup>*1, *<sup>d</sup>*<sup>1</sup> = 0, *<sup>d</sup>*<sup>2</sup> = *<sup>t</sup>*1, then

and

*T* =

(*σ*3*ψ*1)<sup>∗</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> *<sup>i</sup>*

(*σ*3*ψ*3)<sup>∗</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> *<sup>i</sup>*

are both asymptotic solutions to the Dirac system, and

thus, the scattering matrix is unitary. If the entries of *S* are known, then,

 −*S*11/*S*<sup>12</sup> 1/*S*<sup>12</sup> *<sup>S</sup>*<sup>21</sup> − *<sup>S</sup>*11*S*22/*S*<sup>12</sup> *<sup>S</sup>*22/*S*<sup>12</sup>

> *<sup>h</sup> Sp* (*x*,*x*1) *J* − *p*

*<sup>h</sup> Sp* (*x*,*x*2) *J* − *p*

Time-reversal symmetry in scattering through the graphene barrier would mean that

*a*∗ 1 *e* − <sup>1</sup> <sup>+</sup> *e i <sup>h</sup> Sp* (*x*,*x*1) *J* + *p*

*d*∗ 1 *e* − <sup>1</sup> <sup>+</sup> *e i <sup>h</sup> Sp* (*x*,*x*2) *J* + *p*

*<sup>σ</sup>*<sup>3</sup> = 1 0 0 −1 .

*S <sup>a</sup>*<sup>∗</sup> 2 *d*∗ 2 = *<sup>a</sup>*<sup>∗</sup> 1 *d*∗ 1 

*T a*∗ 2 *a*∗ 1 = *d*∗ 2 *d*∗ 1 

> 0 1 1 0 *T* 0 1 1 0

*T* = *T*<sup>11</sup> *T*<sup>12</sup> *T*∗ <sup>12</sup> *<sup>T</sup>*<sup>∗</sup> 11 

*S* = *ST*,

, *det*(*T*) = <sup>−</sup>*S*<sup>21</sup>

*a*∗ 2 *e* +

*d*∗ 2 *e* + *S*<sup>12</sup>

<sup>1</sup> , *<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>1, (93)

<sup>1</sup> , *<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>3, (94)

, (95)

. (96)

<sup>=</sup> *<sup>T</sup>*∗. (97)

. (99)

*detT* = −1, (98)

. (92)

$$T^{+}\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} T = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}.\tag{103}$$

For the scattering matrix *S* we have

$$S\begin{pmatrix} a\_1\\d\_2 \end{pmatrix} = \begin{pmatrix} a\_2\\d\_1 \end{pmatrix} \tag{104}$$

and

$$|a\_1|^2 + |d\_2|^2 = |a\_2|^2 + |d\_1|^2. \tag{105}$$

From (105) we obtain that

$$\mathcal{S}^+ \mathcal{S} = \mathcal{S} \mathcal{S}^+ = I. \tag{106}$$

If the entries of *S* are known, then,

$$T = \begin{pmatrix} \mathbb{S}\_{21} - \mathbb{S}\_{11}\mathbb{S}\_{22}/\mathbb{S}\_{12} \ \mathbb{S}\_{22}/\mathbb{S}\_{12} \\ -\mathbb{S}\_{11}/\mathbb{S}\_{12} & 1/\mathbb{S}\_{12} \end{pmatrix}, \quad \det(T) = \frac{\mathbb{S}\_{21}}{\mathbb{S}\_{12}}.\tag{107}$$

Taking into account the time-reversal symmetry in scattering through the graphene barrier, we obtain *S* = *ST*, and

$$T = \begin{pmatrix} T\_{11} \ T\_{12} \\ T\_{12}^\* \ T\_{11}^\* \end{pmatrix}, \quad \det T = 1. \tag{108}$$

## **16. Appendix C. WKB asymptotic solution for tunneling through a smooth step.**

#### **16.1. Left slope tunneling**

Let us assume that *E* > *Ec*, where *Ec* = |*py*| is the cut-off energy. In the case |*E*| < *Ec* there is no wave transmission through the barrier. On the other side, we assume that *E* < *U*<sup>0</sup> − *δE*, and *δE* is chosen such as to avoid coalescence of all four turning points. Consider a scattering problem through a smooth step that is the left slope of the barrier. Assume that the right slope in Fig. 1 does not exist, that is *U*(*x*) = *U*<sup>0</sup> if *x* > *xmax*. In this case we have three domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 2, 3 with the only difference for <sup>Ω</sup><sup>3</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>2</sup> < *<sup>x</sup>* < +∞}. Thus, to the leading order, in the domain Ω<sup>1</sup> we have a superposition of waves traveling to the left and to the right.

$$\psi\_1 = \frac{e^{\frac{i}{\hbar}S\_\mathbb{P}(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^+}} a\_1 e\_1^+ + \frac{e^{-\frac{i}{\hbar}S\_\mathbb{P}(\mathbf{x}, \mathbf{x}\_1)}}{\sqrt{J\_p^-}} a\_2 e\_1^-. \tag{109}$$

10.5772/52267

69

http://dx.doi.org/10.5772/52267

, (115)

(116)

*<sup>W</sup>*′ <sup>=</sup> 0. (112)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

. (113)

<sup>=</sup> 0, *<sup>w</sup>* <sup>=</sup> *<sup>W</sup>*√*α*, (114)

Next, differentiating the first equation with respect to *ξ* and eliminating *V*, we obtain a

*<sup>V</sup>* <sup>=</sup> *ihαW<sup>ξ</sup>* <sup>−</sup> *<sup>ξ</sup><sup>W</sup> ipy*

Both boundary layers for two turning points *ξ* = −|*py*| and *ξ* = |*py*| are determined by following scale, well-known in WKB asymptotics for turning points in 1D Schrödinger

*ξ* + |*py*| = *O*(*h*2/3), *ξ* − |*py*| = *O*(*h*2/3).

On the other side, this scattering problem for the equation (112) written as effective

*ih α* 

*<sup>a</sup>*1sgn(*py*)

*e i <sup>h</sup>* <sup>Φ</sup>−(*ξ*) *D*− *ξ*

*t* <sup>2</sup> − *p*<sup>2</sup> *y <sup>α</sup>*2(*t*) , *<sup>D</sup>*<sup>±</sup>

− *ia*2*e* −*i <sup>h</sup>* <sup>Φ</sup>−(*ξ*) *D*− *ξ* 

*<sup>ξ</sup>* <sup>=</sup> *<sup>ξ</sup>* <sup>+</sup>

− *d*2sgn(*py*)*e*

−*i <sup>h</sup>* <sup>Φ</sup>+(*ξ*) *D*+ *ξ*

*ξ*<sup>2</sup> − *p*<sup>2</sup> *y* ±|*py*<sup>|</sup> ,

*ih <sup>α</sup>* ) + *<sup>h</sup>*<sup>2</sup> *<sup>α</sup>*′ *α*

*ξ*<sup>2</sup> − *p*<sup>2</sup> *y <sup>α</sup>*<sup>2</sup> <sup>+</sup>

*<sup>h</sup>*2*W*′′ <sup>+</sup> *<sup>W</sup>*(

second order ODE for *W*

Schrödinger equation

for *ξ* < −|*py*|, where

and

Then, after we have found *W*, we have

equation as *h* → 0 (see for example [16]),

*<sup>h</sup>*2*w*′′ <sup>+</sup> *<sup>w</sup>*

*<sup>y</sup>*)1/4

may be represented to leading order as follows

*ξ* 

*<sup>y</sup>*)1/4

±|*py* |

*<sup>w</sup>* <sup>=</sup> <sup>1</sup> 2(*ξ*<sup>2</sup> − *p*<sup>2</sup>

Φ±(*ξ*) =

*<sup>w</sup>* <sup>=</sup> <sup>1</sup> 2(*ξ*<sup>2</sup> − *p*<sup>2</sup>  *<sup>ξ</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *y <sup>α</sup>*<sup>2</sup> <sup>+</sup>


*q*0(*t*)*dt*, *q*0(*t*) =


In the domain Ω<sup>2</sup> we have exponentially decaying and growing contributions. In the domain Ω<sup>3</sup> we have

$$\psi\_3 = d\_1 \frac{e^{\frac{i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^+}} e^+ + d\_2 \frac{e^{\frac{-i}{\hbar}S\_p(\mathbf{x}, \mathbf{x}\_2)}}{\sqrt{J\_p^-}} e^-. \tag{110}$$

where *d* = *TLa*. It is worth remarking that the electron state in *x* < *a* transfers into a hole state for *x* > *a*.

To determine the unknown entries of the transfer matrix *T<sup>L</sup>* (see Appendix A), we have to match the principal terms of all asymptotic expansions by gluing them through asymptotically small boundary layers at *<sup>x</sup>* = *<sup>x</sup>*<sup>1</sup> and *<sup>x</sup>* = *<sup>x</sup>*2. To perform matching asymptotics techniques in this case we introduce a new variable *U*(*x*) − *E* = *ξ* and derive an effective Schrödinger equation. Then, we have

$$
\begin{pmatrix}
\mathfrak{F} & -iha\partial\_{\mathfrak{F}} - ip\_{y} \\
\end{pmatrix}
\begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix},
\tag{111}
$$

where *α* = *α*(*ξ*) = *dξ*/*dx* > 0. Changing *u*, *v* as follows

$$W = \frac{(u+v)}{2}, \qquad V = \frac{(u-v)}{2}.$$

we obtain the system of

$$-i\hbar \alpha \mathcal{W}\_{\xi} + ip\_{\mathcal{Y}}V + \mathfrak{F}W = 0\_r$$

and

$$i\hbar \alpha V\_{\tilde{\xi}} - i p\_y W + \tilde{\xi} V = 0.$$

Next, differentiating the first equation with respect to *ξ* and eliminating *V*, we obtain a second order ODE for *W*

$$h^2 W'' + W(\frac{\xi^2 - p\_y^2}{a^2} + \frac{i\hbar}{a}) + h^2 \frac{a'}{a} W' = 0. \tag{112}$$

Then, after we have found *W*, we have

40 Graphene - Research and Applications

**16.1. Left slope tunneling**

**smooth step.**

to the right.

Ω<sup>3</sup> we have

state for *x* > *a*.

we obtain the system of

and

**16. Appendix C. WKB asymptotic solution for tunneling through a**

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup> i <sup>h</sup> Sp* (*x*,*x*1) *J* + *p*

*<sup>ψ</sup>*<sup>3</sup> = *<sup>d</sup>*<sup>1</sup>

effective Schrödinger equation. Then, we have

where *α* = *α*(*ξ*) = *dξ*/*dx* > 0. Changing *u*, *v* as follows

*e i <sup>h</sup> Sp* (*x*,*x*2) *J* + *p*

 *ξ* −*ihα∂ξ* − *ipy* −*ihα∂ξ* + *ipy ξ*

*<sup>W</sup>* <sup>=</sup> (*<sup>u</sup>* <sup>+</sup> *<sup>v</sup>*)

Let us assume that *E* > *Ec*, where *Ec* = |*py*| is the cut-off energy. In the case |*E*| < *Ec* there is no wave transmission through the barrier. On the other side, we assume that *E* < *U*<sup>0</sup> − *δE*, and *δE* is chosen such as to avoid coalescence of all four turning points. Consider a scattering problem through a smooth step that is the left slope of the barrier. Assume that the right slope in Fig. 1 does not exist, that is *U*(*x*) = *U*<sup>0</sup> if *x* > *xmax*. In this case we have three domains <sup>Ω</sup>*i*, *<sup>i</sup>* = 1, 2, 3 with the only difference for <sup>Ω</sup><sup>3</sup> = {*<sup>x</sup>* : *<sup>x</sup>*<sup>2</sup> < *<sup>x</sup>* < +∞}. Thus, to the leading order, in the domain Ω<sup>1</sup> we have a superposition of waves traveling to the left and

> *a*1*e* + <sup>1</sup> <sup>+</sup>

In the domain Ω<sup>2</sup> we have exponentially decaying and growing contributions. In the domain

*e* <sup>+</sup> <sup>+</sup> *<sup>d</sup>*<sup>2</sup> *e* −*i <sup>h</sup> Sp* (*x*,*x*2) *J* − *p*

where *d* = *TLa*. It is worth remarking that the electron state in *x* < *a* transfers into a hole

To determine the unknown entries of the transfer matrix *T<sup>L</sup>* (see Appendix A), we have to match the principal terms of all asymptotic expansions by gluing them through asymptotically small boundary layers at *<sup>x</sup>* = *<sup>x</sup>*<sup>1</sup> and *<sup>x</sup>* = *<sup>x</sup>*2. To perform matching asymptotics techniques in this case we introduce a new variable *U*(*x*) − *E* = *ξ* and derive an

*e*− *i*

*<sup>h</sup> Sp* (*x*,*x*1) *J* − *p*

 *u v* = 0 0 

<sup>2</sup> ,

<sup>2</sup> , *<sup>V</sup>* <sup>=</sup> (*<sup>u</sup>* <sup>−</sup> *<sup>v</sup>*)

−*ihαW<sup>ξ</sup>* + *ipyV* + *ξW* = 0,

*ihαV<sup>ξ</sup>* − *ipyW* + *ξV* = 0.

*a*2*e* −

*e*

<sup>1</sup> . (109)

<sup>−</sup>. (110)

, (111)

$$V = \frac{i\hbar\omega W\_{\tilde{\xi}} - \tilde{\xi}W}{ip\_y}.\tag{113}$$

Both boundary layers for two turning points *ξ* = −|*py*| and *ξ* = |*py*| are determined by following scale, well-known in WKB asymptotics for turning points in 1D Schrödinger equation as *h* → 0 (see for example [16]),

$$|\mathfrak{F} + |p\_y| = O(h^{2/3}), \quad \mathfrak{F} - |p\_y| = O(h^{2/3}).$$

On the other side, this scattering problem for the equation (112) written as effective Schrödinger equation

$$i\hbar^2 w'' + w(\frac{\vec{\xi}^2 - p\_y^2}{a^2} + \frac{i\hbar}{a}) = 0, \quad w = W\sqrt{a},\tag{114}$$

may be represented to leading order as follows

$$w = \frac{1}{2(\tilde{\xi}^2 - p\_y^2)^{1/4}} \sqrt{\frac{|p\_y|a}{2}} (a\_1 \text{sgn}(p\_y) \frac{e^{\frac{i}{\hbar}\Phi^-(\tilde{\xi})}}{\sqrt{D\_{\tilde{\xi}}^-}} - ia\_2 e^{\frac{i}{\hbar}\Phi^-(\tilde{\xi})} \sqrt{D\_{\tilde{\xi}}^-}) ,\tag{115}$$

for *ξ* < −|*py*|, where

$$\Phi^{\pm}(\xi) = \int\_{\pm |p\_y|}^{\xi} \sqrt{q\_0(t)} dt, \quad q\_0(t) = \frac{t^2 - p\_y^2}{a^2(t)}, \quad D\_{\xi}^{\pm} = \frac{\xi + \sqrt{\xi^2 - p\_y^2}}{\pm |p\_y|}.$$

and

$$w = \frac{1}{2(\xi^2 - p\_y^2)^{1/4}} \sqrt{\frac{|p\_y|a}{2}} \left( -i d\_1 \frac{e^{\frac{i}{\hbar} \Phi^+(\xi)}}{\sqrt{D\_\xi^+}} - d\_2 \text{sgn}(p\_y) e^{\frac{i}{\hbar} \Phi^+(\xi)} \sqrt{D\_\xi^+} \right) \tag{116}$$

for *ξ* > |*py*|. According to the method of comparison equations described in [61] and [62], we seek asymptotic solutions, uniform with respect to |*py*|, as follows

$$w = \frac{\sqrt{|p\_{\overline{\xi}}|}}{2} h^{\nu/2} \left( \frac{\frac{z^2}{4} - a^2}{q(\overline{\xi})} \right)^{1/4} \left( b\_1 D\_{\nu}(h^{-1/2} z) + b\_2 D\_{-\nu-1}(ih^{-1/2} z) \right), \tag{117}$$

and the function *z*(*ξ*) is determined by

$$z'^2(a^2 - z^2/4) = q(\xi). \tag{118}$$

10.5772/52267

71

http://dx.doi.org/10.5772/52267

−*q*0(*ξ*)*dξ*. (121)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

*<sup>ν</sup>* = *iQ*<sup>1</sup>

*i ξ ξ*2

*i ξ*1

*ξ*

*<sup>h</sup> log*(−*z*) <sup>≈</sup> *<sup>i</sup>*

*<sup>h</sup> log*(*z*) <sup>≈</sup> *<sup>i</sup>*

In case of the linear potential with constant *α* we obtain the substitute

*z* = 2 *α e*

functions in (123) are applied in a way similar to the case of the linear potential.

*h*

*h*

For *ξ* > |*py*| we have

for *ξ* << −|*py*|

and for *ξ* >> |*py*|

where

*z*2 <sup>4</sup>*<sup>h</sup>* <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

> *z*2 <sup>4</sup>*<sup>h</sup>* <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

whereas for *ξ* < −|*py*| we have

*<sup>π</sup><sup>h</sup>* , *<sup>Q</sup>*<sup>1</sup> <sup>=</sup>

*q*(*ξ*)*dξ* =

*q*(*ξ*)*dξ* =


−|*py* |

*z*

*z*2/4 − *a*2,

*z*2/4 − *a*2.

*<sup>i</sup>π*/4*ξ*, *<sup>ξ</sup>* ∈ **<sup>R</sup>** (122)

2*a*

− 2*a*

*z*

from (124) (see [18]). For a general case, we assume that our *z* belongs to a sector of the complex plane based on this central line (122), where *<sup>α</sup>* = *<sup>α</sup>*<sup>0</sup> is evaluated at the point *<sup>x</sup>* = *a*, (*E* = *U*(*a*)). Thus, we assume that the asymptotic expansions for the parabolic cylinder

Then, the following important techniques for matching asymptotic estimates may be obtained

*<sup>q</sup>*0(*ξ*)*d<sup>ξ</sup>* <sup>−</sup> <sup>1</sup>

*<sup>q</sup>*0(*ξ*)*d<sup>ξ</sup>* <sup>−</sup> <sup>1</sup>

2

2

*log* <sup>2</sup>*<sup>ξ</sup> py* 

*log* <sup>2</sup>*<sup>ξ</sup>* − *py* 

+ *γ* (123)

+ *γ* (124)

−|*py* | *ξ*

> *ξ*

*γ* = *a*2/2*h*(1 − log *a*2) + 1/2(*ν* + 1/2)log *ν* + 1/2/*ν* − 1/4

= 1/2(*ν* + 1/2)(1 − log (*hν*)) − 1/4.


where

$$q = \frac{\xi^2 - p\_y^2}{\alpha^2} + \frac{i\hbar}{\alpha'} \qquad a^2 = \hbar(\nu + \frac{1}{2});$$

The asymptotics include the parabolic cylinder function *Dν*(*z*) that is a solution to

$$h^2 y\_{zz} + (h(\nu + 1/2) - z^2/4)y = 0.$$

From (124) we obtain

$$\operatorname{i}\int\_{\xi\_1}^{\xi\_2} \sqrt{q(\xi)} d\xi = \int\_{-2a}^{2a} \sqrt{a^2 - \frac{z^2}{4}} dz = \pi a = \pi h(\frac{1}{2} + \nu),\tag{119}$$

where *<sup>ξ</sup>*1,2 are the complex roots of *<sup>q</sup>*(*ξ*) = 0. Using the estimate

$$\frac{1}{h} \int\_{\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{-q(t)} dt = \frac{1}{h} \int\_{-p\_y}^{p\_y} \sqrt{-q\_0(t)} dt - \frac{i\pi}{2} + \mathcal{O}(h)\_y$$

we obtain

$$i\int\_{-p\_y}^{p\_y} \sqrt{-q\_0(\xi)}d\xi + \frac{\pi\hbar}{2} = \pi a = \pi\hbar(\frac{1}{2} + \nu),\tag{120}$$

where,

$$q\_0 = \frac{\xi^2 - p\_y^2}{a^2}$$

Thus,

#### <sup>70</sup> New Progress on Graphene Research Localised States of Fabry-Perot Type in Graphene Nano-Ribbons 43 10.5772/52267 Localised States of Fabry-Perot Type in Graphene Nano-Ribbons http://dx.doi.org/10.5772/52267 71

$$\nu = \frac{iQ\_1}{\pi \hbar}, \quad \mathcal{Q}\_1 = \int\_{-|p\_\mathcal{Y}|}^{|p\_\mathcal{Y}|} \sqrt{-q\_0(\xi)} d\xi. \tag{121}$$

For *ξ* > |*py*| we have

42 Graphene - Research and Applications

*w* =

where

From (124) we obtain

we obtain

where,

Thus,

 |*py*| <sup>2</sup> *<sup>h</sup>ν*/2

and the function *z*(*ξ*) is determined by

*i ξ*2

*ξ*1

1 *h ξ*2

*ξ*1

*i py* −*py*

*q*(*ξ*)*dξ* =

where *<sup>ξ</sup>*1,2 are the complex roots of *<sup>q</sup>*(*ξ*) = 0. Using the estimate

<sup>−</sup>*q*(*t*)*dt* <sup>=</sup> <sup>1</sup>

 2*a*

*h*

−*q*0(*ξ*)*dξ* +

*py* −*py*

*πh*

*<sup>q</sup>*<sup>0</sup> <sup>=</sup> *<sup>ξ</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

−2*a*

for *ξ* > |*py*|. According to the method of comparison equations described in [61] and [62],

*<sup>b</sup>*1*Dν*(*h*−1/2*z*) + *<sup>b</sup>*2*D*−*ν*−1(*ih*−1/2*z*)

*<sup>z</sup>*′2(*a*<sup>2</sup> <sup>−</sup> *<sup>z</sup>*2/4) = *<sup>q</sup>*(*ξ*). (118)

1 2 );

1

<sup>2</sup> <sup>+</sup> *<sup>O</sup>*(*h*),

1

<sup>2</sup> <sup>+</sup> *<sup>ν</sup>*), (119)

<sup>2</sup> <sup>+</sup> *<sup>ν</sup>*), (120)

, (117)

1/4

we seek asymptotic solutions, uniform with respect to |*py*|, as follows

*<sup>q</sup>* <sup>=</sup> *<sup>ξ</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup>

*y <sup>α</sup>*<sup>2</sup> <sup>+</sup>

The asymptotics include the parabolic cylinder function *Dν*(*z*) that is a solution to

*ih*

*h*2*yzz* + (*h*(*ν* + 1/2) − *z*2/4)*y* = 0.

*<sup>a</sup>*<sup>2</sup> <sup>−</sup> *<sup>z</sup>*<sup>2</sup>

*<sup>α</sup>* , *<sup>a</sup>*<sup>2</sup> <sup>=</sup> *<sup>h</sup>*(*<sup>ν</sup>* <sup>+</sup>

<sup>4</sup> *dz* <sup>=</sup> *<sup>π</sup><sup>a</sup>* <sup>=</sup> *<sup>π</sup>h*(

<sup>−</sup>*q*0(*t*)*dt* <sup>−</sup> *<sup>i</sup><sup>π</sup>*

<sup>2</sup> <sup>=</sup> *<sup>π</sup><sup>a</sup>* <sup>=</sup> *<sup>π</sup>h*(

*y α*2

 *<sup>z</sup>*<sup>2</sup> <sup>4</sup> <sup>−</sup> *<sup>a</sup>*2) *q*(*ξ*)

$$i\int\_{\tilde{\xi}\_2}^{\tilde{\xi}} \sqrt{q(\tilde{\xi})}d\tilde{\xi} = \int\_{2a}^{z} \sqrt{z^2/4 - a^2} \, dz$$

whereas for *ξ* < −|*py*| we have

$$i\int\_{\tilde{\xi}}^{\tilde{\xi}\_1} \sqrt{q(\tilde{\xi})}d\tilde{\xi} = \int\_z^{-2a} \sqrt{z^2/4 - a^2}z$$

In case of the linear potential with constant *α* we obtain the substitute

$$z = \sqrt{\frac{2}{\alpha}} e^{i\pi/4} \tilde{\xi}\_{\prime} \ \tilde{\xi} \in \mathbf{R} \tag{122}$$

from (124) (see [18]). For a general case, we assume that our *z* belongs to a sector of the complex plane based on this central line (122), where *<sup>α</sup>* = *<sup>α</sup>*<sup>0</sup> is evaluated at the point *<sup>x</sup>* = *a*, (*E* = *U*(*a*)). Thus, we assume that the asymptotic expansions for the parabolic cylinder functions in (123) are applied in a way similar to the case of the linear potential. Then, the following important techniques for matching asymptotic estimates may be obtained for *ξ* << −|*py*|

$$\frac{z^2}{4h} - \frac{a^2}{h} \log(-z) \approx \frac{i}{h} \int\_{\mathbb{R}}^{-|p\_y|} \sqrt{q\_0(\xi)} d\xi - \frac{1}{2} \log \frac{2\xi}{-|p\_y|} + \gamma \tag{123}$$

and for *ξ* >> |*py*|

$$\frac{z^2}{4h} - \frac{a^2}{h} \log(z) \approx \frac{i}{h} \int\_0^{\frac{\pi}{2}} \sqrt{q\_0(\xi)} d\xi - \frac{1}{2} \log \frac{2\xi}{|p\_y|} + \gamma \tag{124}$$

where

$$\begin{aligned} \gamma &= a^2 / 2h(1 - \log a^2) + 1/2(\nu + 1/2)\log \nu + 1/2/\nu - 1/4 \\\\ &= 1/2(\nu + 1/2)(1 - \log(h\nu)) - 1/4. \end{aligned}$$

Using the asymptotic expansions of the parabolic cylinder functions for large argument (see the appendices in [9]), we obtain for *ξ* << −|*py*|

$$\begin{split} w \sim \frac{1}{2(\xi^2 - p\_y^2)^{1/4}} \sqrt{\frac{|p\_y|}{2}} \left( b\_1 [e^{-z^2/4h} z^\nu h^{-\nu/2} - e^{z^2/4h - i\pi\nu} z^{-\nu - 1} h^{\nu/2 + 1/2} \frac{\sqrt{2\pi}}{\Gamma(-\nu)}] + \right. \tag{125} \\\\ b\_2 e^{z^2/4h - i\frac{\pi}{\Gamma}(\nu + 1)} z^{-\nu - 1} h^{\nu/2 + 1/2} \Big), \end{split}$$

where Γ(*z*) is the Gamma function. For *ξ* >> |*py*| we have

$$w \sim \frac{1}{2(\xi^2 - p\_y^2)^{1/4}} \sqrt{\frac{|py|}{2}} \left( b\_1 e^{-z^2/4h} z^\nu h^{-\nu/2} + \right. \tag{126}$$

10.5772/52267

73

http://dx.doi.org/10.5772/52267

*<sup>t</sup>* , (131)

*<sup>t</sup>* . (132)

*<sup>h</sup>ν*+1/2

.

). (134)

Eliminating *b*<sup>1</sup> and *b*<sup>2</sup> from the system (130), we obtain the relations determining the transfer

*r*1 *<sup>t</sup>* <sup>+</sup> *<sup>a</sup>*<sup>2</sup> 1 *t* ,

1 2

*Q*<sup>1</sup> *πh*

(1 − log (

*iθ*1 

> *Q*<sup>1</sup> *πh*

−*iθ*<sup>1</sup> 

> −*r*<sup>1</sup> *t* 1 *t* 1 *<sup>t</sup> <sup>r</sup>*<sup>2</sup> *t*

1 − *e*−2*Q*1/*<sup>h</sup>* −*e*

*<sup>h</sup>* sgn(*py*)*e*−*iθ*1+*Q*1/*<sup>h</sup>*

)) <sup>−</sup> *<sup>π</sup>*

*<sup>t</sup>* ) + *<sup>a</sup>*<sup>2</sup>

<sup>21</sup> <sup>=</sup> *<sup>t</sup>* <sup>−</sup> *<sup>r</sup>*1*r*<sup>2</sup>

)(<sup>1</sup> <sup>−</sup> log (*hν*)) <sup>−</sup> <sup>1</sup>

*Q*<sup>1</sup> *πh* )) − <sup>1</sup>

2*Q*<sup>1</sup> *h*(*eQ*1/*<sup>h</sup>* − *e*−*Q*1/*h*)

<sup>4</sup> <sup>−</sup> arg <sup>Γ</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*

*Q*1 *h*

√

*r*2

*<sup>t</sup>* , *<sup>T</sup><sup>L</sup>*

2

<sup>2</sup> log *<sup>ν</sup>*

,

1 − *e*−2*Q*1/*h*, (133)

*Q*<sup>1</sup> *πh*

1 − *e*−2*Q*1/*h*. (135)

= (136)

(137)

1 − *e*−2*Q*1/*<sup>h</sup>*

<sup>22</sup> <sup>=</sup> *<sup>r</sup>*<sup>2</sup>

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

 <sup>√</sup>2*<sup>π</sup>* Γ(1 − *ν*)

> <sup>√</sup>2*<sup>π</sup>* Γ(1 − *ν*)

*<sup>d</sup>*<sup>2</sup> = *<sup>a</sup>*1(*<sup>t</sup>* − *<sup>r</sup>*1*r*<sup>2</sup>

*<sup>d</sup>*<sup>1</sup> = −*a*<sup>1</sup>

<sup>12</sup> <sup>=</sup> <sup>1</sup> *t* , *T<sup>L</sup>*

*iπν* + (*ν* +

 *πν* sin (*πν*) <sup>=</sup>

*<sup>r</sup>*<sup>1</sup> = sgn(*py*)*<sup>e</sup>*

Similarly, taking into account that arg Γ(1 + *ν*) = − arg Γ(1 − *ν*), we obtain

*r*<sup>2</sup> = −sgn(*py*)*e*

(1 − log (

*<sup>T</sup>L*(*Q*1) =

*πh*

√

−*e Q*1 − *<sup>Q</sup>*<sup>1</sup> <sup>2</sup>*<sup>h</sup>* <sup>+</sup> *<sup>i</sup>*

*<sup>t</sup>* , *<sup>T</sup><sup>L</sup>*

The expressions for *r*<sup>1</sup> and *r*<sup>2</sup> can be simplified as follows

matrix *T<sup>L</sup>*

that is

we derive

where

*TL* <sup>11</sup> <sup>=</sup> <sup>−</sup>*r*<sup>1</sup>

*<sup>r</sup>*<sup>1</sup> = −*<sup>i</sup>* sgn(*py*)*<sup>ν</sup>* exp

= −*i* sgn(*py*)*ν* exp

Using the properties of the Gamma function (see [63])


*<sup>θ</sup>*<sup>1</sup> <sup>=</sup> *<sup>θ</sup>*(*Q*1) = *<sup>Q</sup>*<sup>1</sup>

Hence, for the left slope transfer marix we obtain

sgn(*py*)*eiθ*1+*Q*1/*<sup>h</sup>*

$$\mathfrak{b}\_2[e^{z^2/4\hbar - i\frac{\pi}{2}(\nu+1)}z^{-\nu-1}\hbar^{\nu/2+1/2}+e^{-z^2/4\hbar - i\pi\nu/2}z^{\nu}\hbar^{-\nu/2}\frac{\sqrt{2\pi}}{\Gamma(-\nu)}] . $$

Matching these two asymptotic expansions with the asymptotics, correspondingly, leads to the following system

$$\begin{cases} \begin{aligned} a\_1 \text{sgn}(p\_\mathcal{Y}) &= b\_1 (-1)^\nu e^{-\gamma} \\ -i a\_2 &= (-b\_1 e^{-i\pi\nu} \frac{\sqrt{2\pi}}{\Gamma(-\nu)} + b\_2 e^{-i\frac{\pi}{2}(\nu+1)}) h^{\nu+1/2} (-1)^{-\nu-1} e^{\gamma} \\ -i d\_1 &= b\_2 e^{\gamma - i\frac{\pi}{2}(\nu+1)} h^{\nu+1/2} \\ -d\_2 \text{sgn}(p\_\mathcal{Y}) &= (b\_1 + b\_2 e^{-i\frac{\pi}{2}\nu} \frac{\sqrt{2\pi}}{\Gamma(\nu+1)}) e^{-\gamma} .\end{aligned} \end{cases} \tag{127}$$

Let us introduce new notations

$$t = -(-1)^{-\nu} = -e^{i\pi\nu} = -e^{-\frac{Q\_1}{\hbar}},\tag{128}$$

$$r\_1 = \text{sign}(p\_y)e^{i\pi\nu + 2\gamma} \frac{\sqrt{2\pi}}{\Gamma(-\nu)}h^{\nu + 1/2}, \quad r\_2 = -\text{sgn}(p\_y)e^{-2\gamma} \frac{\sqrt{2\pi}}{\Gamma(1+\nu)}h^{-\nu - 1/2}.\tag{129}$$

Then, the system (127) reads

$$\begin{cases} \begin{aligned} a\_1 \text{sgn}(p\_y) &= -b\_1 \frac{e^{-\gamma}}{t}, \\ -ia\_2 &= b\_1 \frac{i r\_1 \text{sgn}(p\_y)}{t} e^{-\gamma} - ib\_2 e^{\gamma - i \frac{\pi}{2} \nu} h^{\nu + 1/2} t, \\ -id\_1 &= -ib\_2 e^{\gamma - i \frac{\pi}{2} \nu} h^{\nu + 1/2}, \\ -d\_2 \text{sgn}(p\_y) &= b\_1 e^{-\gamma} + b\_2 e^{\gamma + i \frac{\pi}{2} \nu} h^{\nu + 1/2} \frac{r\_2 \text{sgn}(p\_y)}{t}. \end{aligned} \end{cases} \tag{130}$$

Eliminating *b*<sup>1</sup> and *b*<sup>2</sup> from the system (130), we obtain the relations determining the transfer matrix *T<sup>L</sup>*

$$\begin{cases} d\_1 = -a\_1 \frac{r\_1}{l} + a\_2 \frac{1}{l}, \\ d\_2 = a\_1 (t - \frac{r\_1 r\_2}{l}) + a\_2 \frac{r\_2}{l}, \end{cases} \tag{131}$$

that is

44 Graphene - Research and Applications

*<sup>w</sup>* <sup>∼</sup> <sup>1</sup> 2(*ξ*<sup>2</sup> − *p*<sup>2</sup>

the appendices in [9]), we obtain for *ξ* << −|*py*|

� |*py*| 2 � *<sup>b</sup>*1[*<sup>e</sup>*

*b*2*e*

where Γ(*z*) is the Gamma function. For *ξ* >> |*py*| we have

*<sup>w</sup>* <sup>∼</sup> <sup>1</sup> 2(*ξ*<sup>2</sup> − *p*<sup>2</sup>

<sup>2</sup> (*ν*+1)

*<sup>a</sup>*1*sgn*(*py*) = *<sup>b</sup>*1(−1)*νe*−*γ*, <sup>−</sup>*ia*<sup>2</sup> = (−*b*1*e*−*iπν* <sup>√</sup>2*<sup>π</sup>*

<sup>−</sup>*id*<sup>1</sup> <sup>=</sup> *<sup>b</sup>*2*eγ*−*<sup>i</sup> <sup>π</sup>*

<sup>−</sup>*d*2*sgn*(*py*)=(*b*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*2*e*−*<sup>i</sup> <sup>π</sup>*

√2*π* Γ(−*ν*)

*<sup>a</sup>*1sgn(*py*) = −*b*<sup>1</sup>

−*ia*<sup>2</sup> = *<sup>b</sup>*<sup>1</sup>

<sup>−</sup>*id*<sup>1</sup> <sup>=</sup> <sup>−</sup>*ib*2*eγ*−*<sup>i</sup> <sup>π</sup>*

<sup>−</sup>*d*2sgn(*py*) = *<sup>b</sup>*1*e*−*<sup>γ</sup>* <sup>+</sup> *<sup>b</sup>*2*eγ*+*<sup>i</sup> <sup>π</sup>*

*iπν*+2*γ*

 

*z*2/4*h*−*i <sup>π</sup>*

*<sup>y</sup>*)1/4

*<sup>z</sup>*−*ν*−1*hν*/2+1/2 <sup>+</sup> *<sup>e</sup>*

*<sup>y</sup>*)1/4

*b*2[*e*

the following system

 

*<sup>r</sup>*<sup>1</sup> = *isgn*(*py*)*<sup>e</sup>*

Then, the system (127) reads

Let us introduce new notations

*z*2/4*h*−*i <sup>π</sup>*

Using the asymptotic expansions of the parabolic cylinder functions for large argument (see

*<sup>z</sup>*2/4*h*−*iπνz*−*ν*−1*hν*/2+1/2

� ,

<sup>−</sup>*z*2/4*h*−*iπν*/2*zνh*−*ν*/2

√2*π* Γ(−*ν*)

<sup>−</sup>*z*2/4*hz<sup>ν</sup>h*−*ν*/2<sup>+</sup> (126)

√2*π* Γ(−*ν*) ] � .

<sup>2</sup> (*ν*+1))*hν*+1/2(−1)−*ν*−1*eγ*,

*<sup>h</sup>* , (128)

*<sup>h</sup>*−*ν*<sup>−</sup>1/2. (129)

]+ (125)

(127)

(130)

<sup>−</sup>*z*2/4*hz<sup>ν</sup>h*−*ν*/2 <sup>−</sup> *<sup>e</sup>*

<sup>2</sup> (*ν*+1)

� |*py*| 2 � *b*1*e*

Matching these two asymptotic expansions with the asymptotics, correspondingly, leads to

<sup>Γ</sup>(−*ν*) <sup>+</sup> *<sup>b</sup>*2*e*−*<sup>i</sup> <sup>π</sup>*

*<sup>h</sup>ν*<sup>+</sup>1/2, *<sup>r</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*sgn*(*py*)*<sup>e</sup>*

*e*−*<sup>γ</sup> t* ,

*ir*1*sgn*(*py* )

*<sup>i</sup>πν* = −*<sup>e</sup>*

*<sup>t</sup> <sup>e</sup>*−*<sup>γ</sup>* <sup>−</sup> *ib*2*eγ*−*<sup>i</sup> <sup>π</sup>*

<sup>2</sup> *<sup>ν</sup>hν*<sup>+</sup>1/2,

− *<sup>Q</sup>*<sup>1</sup>

−2*γ*

<sup>2</sup> *<sup>ν</sup>hν*+1/2 *<sup>r</sup>*2*sgn*(*py* )

√2*π* Γ(1 + *ν*)

<sup>2</sup> *<sup>ν</sup>hν*+1/2*t*,

*<sup>t</sup>* .

<sup>2</sup> (*ν*+1)*hν*<sup>+</sup>1/2,

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(−1)−*<sup>ν</sup>* <sup>=</sup> <sup>−</sup>*<sup>e</sup>*

<sup>2</sup> *<sup>ν</sup>* <sup>√</sup>2*<sup>π</sup>* <sup>Γ</sup>(*ν*+1))*e*−*γ*.

*<sup>z</sup>*−*ν*−1*hν*/2+1/2

$$T\_{11}^L = -\frac{r\_1}{t}, \quad T\_{12}^L = \frac{1}{t}, \quad T\_{21}^L = t - \frac{r\_1 r\_2}{t}, \quad T\_{22}^L = \frac{r\_2}{t}. \tag{132}$$

The expressions for *r*<sup>1</sup> and *r*<sup>2</sup> can be simplified as follows

$$r\_1 = -i \operatorname{sgn}(p\_\mathcal{Y}) \nu \exp\left(i\pi\nu + (\nu + \frac{1}{2})(1 - \log\left(h\nu\right)) - \frac{1}{2}\right) \frac{\sqrt{2\pi}}{\Gamma(1-\nu)} h^{\nu+1/2}$$

$$=-i\operatorname{sgn}(p\_{\mathcal{Y}})\nu\exp\left(-\frac{Q\_1}{2h} + i\frac{Q\_1}{\pi h}(1-\log\left(\frac{Q\_1}{\pi h}\right)) - \frac{1}{2}\log\nu\right)\frac{\sqrt{2\pi}}{\Gamma(1-\nu)}.$$

Using the properties of the Gamma function (see [63])

$$|\Gamma(1 \mp \nu)| = \sqrt{\frac{\pi \nu}{\sin \left( \pi \nu \right)}} = \sqrt{\frac{2Q\_1}{h(e^{Q\_1/h} - e^{-Q\_1/h})}} \nu$$

we derive

$$r\_1 = \text{sgn}(p\_y)e^{i\theta\_1}\sqrt{1 - e^{-2Q\_1/\hbar}},\tag{133}$$

where

$$\theta\_1 = \theta(Q\_1) = \frac{Q\_1}{\pi h} (1 - \log\left(\frac{Q\_1}{\pi h}\right)) - \frac{\pi}{4} - \arg\Gamma(1 - i\frac{Q\_1}{\pi h}).\tag{134}$$

Similarly, taking into account that arg Γ(1 + *ν*) = − arg Γ(1 − *ν*), we obtain

$$r\_2 = -\text{sgn}(p\_y)e^{-i\theta\_1}\sqrt{1 - e^{-2Q\_1/\hbar}}.\tag{135}$$

Hence, for the left slope transfer marix we obtain

$$T^L(Q\_1) = \begin{pmatrix} -\frac{r\_1}{l} & \frac{1}{l} \\ \frac{1}{l} & \frac{r\_2}{l} \end{pmatrix} = \tag{136}$$

$$
\begin{pmatrix}
\text{sgn}(p\_y)e^{i\theta\_1+Q\_1/\hbar}\sqrt{1-e^{-2Q\_1/\hbar}} & -e^{\frac{Q\_1}{\hbar}} \\
\end{pmatrix}
\tag{137}
$$

This is the main result of this section and this formula was originally stated in [18]. It is clear that the transfer matrix for the left slope satisfies all the properties in the Appendix A, namely *T<sup>L</sup>* <sup>22</sup> = (*T<sup>L</sup>* <sup>11</sup>)∗, *<sup>T</sup><sup>L</sup>* <sup>12</sup> = (*T<sup>L</sup>* <sup>21</sup>)∗, det *TL* <sup>=</sup> <sup>−</sup>1. Now it clear that the quantities *<sup>r</sup>*1,2 mean the corresponding reflection cefficients, *t* is the transmission coefficient. It is worth to remark that due to the asymptotics as *x* → +∞,

$$Im\log\left(\Gamma(-i\mathbf{x})\right) = \pi/4 + \pi(1 - \log\pi) + O(1/\pi)\_{\pi}$$

if *<sup>Q</sup>*1/*<sup>h</sup>* → +<sup>∞</sup> (the turning points *<sup>ξ</sup>* = ±|*py*| do not coalesce), we observe that

$$\arg \Gamma(1 - iQ\_1/\hbar \pi) = \arg \left( -iQ\_1/\hbar \pi \right) + \arg \Gamma(-iQ\_1/\hbar \pi)$$

$$=-\pi/4 + Q\_1/h\pi(1 - \log Q\_1/h\pi)\_{\prime\prime}$$

and, consequently from (134), we obtain that *θ*<sup>1</sup> → 0.

#### **16.2. Right slope tunneling**

Now let us formulate the scattering problem with transfer matrix *T<sup>R</sup>* for the right slope. Taking into account that *α* = | *dξ dx* <sup>|</sup>, the problem for transfer matrix written in terms of solution to the effective Schrödinger equation

$$h^2 W'' + w(\frac{\mathfrak{F}^2 - p\_y^2}{a^2} - \frac{i\hbar}{a}) = 0,\tag{138}$$

10.5772/52267

75

. (142)

http://dx.doi.org/10.5772/52267

= *<sup>T</sup>L*(*Q*2). (143)

*qx*(*x*)*dx*, *<sup>θ</sup>*<sup>2</sup> = *<sup>θ</sup>*(*Q*2). (144)

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

Hence, we have

where

Since *d* = *TRa*, we obtain

**Acknowledgments**

**Author details**

**17. References**

V.V. Zalipaev, D.M. Forrester, C.M. Linton and F.V. Kusmartsev

98, 236803 (2007).

 *d*1 *d*2 = 1 0 0 −1 

*<sup>Q</sup>*<sup>2</sup> =

commercial strategy for nanoblade technology".

 −*r*<sup>∗</sup> 2 *t* 1 *t* 1 *t r*∗ 1 *t*

 *p*2 *<sup>y</sup>* − *<sup>ξ</sup>*<sup>2</sup> *<sup>α</sup>*(*ξ*) *<sup>d</sup><sup>ξ</sup>* <sup>=</sup>

School of Science, Loughborough University, Loughborough, UK

[2] Cheianov, V.V, Fal'ko, V., Altshuler, B. L., Science. 315, 1252 (2007).

Solid State Communications. 143, 72 (2007).

Solid State Communications 115 1094 (2011).

[5] Allain, P.E. and Fuchs, J.N., Eur.Phys.J. B 83, 301 (2011).

*py* −*py*

*T<sup>R</sup>* = 1 0 0 −1 (*TL*(*Q*2))−<sup>1</sup><sup>∗</sup>

*x*4

*x*3

It is worth to remark that *<sup>Q</sup>*<sup>1</sup> and *<sup>Q</sup>*<sup>2</sup> differ as the function *<sup>α</sup>*(*ξ*) behave differently for the

The authors would like to thank Prof J.Ferapontov, Dr A.Vagov and Dr D. Maksimov for constructive discussions and valuable remarks. DMF would like to acknowledge support from the EPSRC (KTA) for funding under the Fellowship: "Developing prototypes and a

[1] Heersche, H.B., Jarillo-Herrero, P., Oostinga, J.B.,Vandersypen, L.M.K., Morpurgo,A.F.,

[3] Huard, B., Sulpizio, J.A., Stander, Todd, K., Yang, B. and Goldhaber-Gordon, D., PRL

[4] Tzalenchuk, A., Lara-Avila, S., Cedergren, K., Syvajarvi, K., Yakimova, R., Kazakova, O., Janssen, T.J.B.M, Moth-Poulsen, K. Bjornholm, T., Kopylov, S. Fal'ko, V. Kubatkin, S.,

same segment *ξ* → (−|*py*|, |*py*|) for left and right slopes of non-symmetric barrier.

 −1 0 0 1 = −*r*<sup>1</sup> *t* 1 *t* 1 *<sup>t</sup> <sup>r</sup>*<sup>2</sup> *t* 

 −1 0 0 1 *<sup>a</sup>*<sup>1</sup> *a*2 

may be represented as follows

$$w = \frac{1}{2(\xi^2 - p\_y^2)^{1/4}} \sqrt{\frac{|p\_y|}{2}} \left( -ia\_1 \frac{e^{-\frac{i}{\hbar} \Phi^+(\xi)} }{\sqrt{D\_{\xi}^+}} - a\_2 \text{sgn}(p\_y) e^{\frac{i}{\hbar} \Phi^+(\xi)} \sqrt{D\_{\xi}^+} \right) \tag{139}$$

for *ξ* > |*py*|,

$$w = \frac{1}{2(\mathfrak{F}^2 - p\_y^2)^{1/4}} \sqrt{\frac{|p\_y|}{2}} (d\_1 \text{sgn}(p\_y) \frac{e^{-\frac{i}{\hbar} \Phi^-(\xi)}}{\sqrt{D\_{\xi}^-}} - i d\_2 e^{\frac{i}{\hbar} \Phi^-(\xi)} \sqrt{D\_{\xi}^-}) \tag{140}$$

for *<sup>ξ</sup>* <sup>&</sup>lt; −|*py*|. If *<sup>w</sup>* is a solution to (138), then *<sup>w</sup>*<sup>∗</sup> is the solution to (114). Thus, the coefficients from (139), (140) are connected by

$$
\begin{pmatrix} -a\_1^\* \\ a\_2^\* \end{pmatrix} = T^L(Q\_2) \begin{pmatrix} d\_1^\* \\ -d\_2^\* \end{pmatrix} . \tag{141}
$$

Hence, we have

46 Graphene - Research and Applications

<sup>22</sup> = (*T<sup>L</sup>*

**16.2. Right slope tunneling**

Taking into account that *α* = |

may be represented as follows

for *ξ* > |*py*|,

*<sup>w</sup>* <sup>=</sup> <sup>1</sup> 2(*ξ*<sup>2</sup> − *p*<sup>2</sup>

> *<sup>w</sup>* <sup>=</sup> <sup>1</sup> 2(*ξ*<sup>2</sup> − *p*<sup>2</sup>

from (139), (140) are connected by

to the effective Schrödinger equation

<sup>11</sup>)∗, *<sup>T</sup><sup>L</sup>*

that due to the asymptotics as *x* → +∞,

<sup>12</sup> = (*T<sup>L</sup>*

and, consequently from (134), we obtain that *θ*<sup>1</sup> → 0.

*dξ*

*<sup>y</sup>*)1/4

*<sup>y</sup>*)1/4

 |*py*| 2 

> −*a*<sup>∗</sup> 1 *a*∗ 2

 |*py*| 2 − *ia*<sup>1</sup>

*<sup>h</sup>*2*W*′′ <sup>+</sup> *<sup>w</sup>*

namely *T<sup>L</sup>*

This is the main result of this section and this formula was originally stated in [18]. It is clear that the transfer matrix for the left slope satisfies all the properties in the Appendix A,

the corresponding reflection cefficients, *t* is the transmission coefficient. It is worth to remark

*Im* log (Γ(−*ix*)) = *π*/4 + *x*(1 − log *x*) + *O*(1/*x*),

arg <sup>Γ</sup>(<sup>1</sup> − *iQ*1/*hπ*) = arg (−*iQ*1/*hπ*) + arg <sup>Γ</sup>(−*iQ*1/*hπ*)

= −*π*/4 + *<sup>Q</sup>*1/*hπ*(<sup>1</sup> − log *<sup>Q</sup>*1/*hπ*),

Now let us formulate the scattering problem with transfer matrix *T<sup>R</sup>* for the right slope.

 *<sup>ξ</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *y <sup>α</sup>*<sup>2</sup> <sup>−</sup> *ih α* 

> *e*− *i <sup>h</sup>* <sup>Φ</sup>+(*ξ*)

*<sup>d</sup>*1sgn(*py*)

for *<sup>ξ</sup>* <sup>&</sup>lt; −|*py*|. If *<sup>w</sup>* is a solution to (138), then *<sup>w</sup>*<sup>∗</sup> is the solution to (114). Thus, the coefficients

= *TL*(*Q*2)

*e*− *i <sup>h</sup>* <sup>Φ</sup>−(*ξ*)

> *D*− *ξ*

 *<sup>d</sup>*<sup>∗</sup> 1 −*d*<sup>∗</sup> 2 

 *D*+ *ξ*

if *<sup>Q</sup>*1/*<sup>h</sup>* → +<sup>∞</sup> (the turning points *<sup>ξ</sup>* = ±|*py*| do not coalesce), we observe that

<sup>21</sup>)∗, det *TL* <sup>=</sup> <sup>−</sup>1. Now it clear that the quantities *<sup>r</sup>*1,2 mean

*dx* <sup>|</sup>, the problem for transfer matrix written in terms of solution

− *a*2sgn(*py*)*e*

− *id*2*e i <sup>h</sup>* <sup>Φ</sup>−(*ξ*) *D*− *ξ*

*i <sup>h</sup>* <sup>Φ</sup>+(*ξ*) *D*+ *ξ*

= 0, (138)

. (141)

(139)

(140)

$$
\begin{pmatrix} d\_1 \\ d\_2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \left( (T^L(Q\_2))^{-1} \right)^\* \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a\_1 \\ a\_2 \end{pmatrix}.\tag{142}
$$

Since *d* = *TRa*, we obtain

$$T^{R} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} -\frac{r\_2^\*}{l} & \frac{1}{l} \\ \frac{1}{l} & \frac{r\_1^\*}{l} \end{pmatrix} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -\frac{r\_1}{l} & \frac{1}{l} \\ \frac{1}{l} & \frac{r\_2}{l} \end{pmatrix} = T^L(Q\_2). \tag{143}$$

where

$$Q\_2 = \int\_{-p\_\mathbf{y}}^{p\_\mathbf{y}} \frac{\sqrt{p\_\mathbf{y}^2 - \xi^2}}{a(\xi)} d\xi = \int\_{\chi\_\mathbf{3}}^{\chi\_4} q\_\mathbf{x}(\mathbf{x}) d\mathbf{x}, \qquad \theta\_\mathbf{2} = \theta(Q\_\mathbf{2}). \tag{144}$$

It is worth to remark that *<sup>Q</sup>*<sup>1</sup> and *<sup>Q</sup>*<sup>2</sup> differ as the function *<sup>α</sup>*(*ξ*) behave differently for the same segment *ξ* → (−|*py*|, |*py*|) for left and right slopes of non-symmetric barrier.

#### **Acknowledgments**

The authors would like to thank Prof J.Ferapontov, Dr A.Vagov and Dr D. Maksimov for constructive discussions and valuable remarks. DMF would like to acknowledge support from the EPSRC (KTA) for funding under the Fellowship: "Developing prototypes and a commercial strategy for nanoblade technology".

## **Author details**

V.V. Zalipaev, D.M. Forrester, C.M. Linton and F.V. Kusmartsev

School of Science, Loughborough University, Loughborough, UK

#### **17. References**


10.5772/52267

77

http://dx.doi.org/10.5772/52267

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

[26] Mello, P. A., Kumar, N., 2004 *Quantum Transport in Mesoscopic systems*, Oxford University

[27] Stockmann, H. J., *Quantum Chaos. An Introduction*, Cambridge University Press,

[31] Castro Neto, A.H., Guinea, F., Peres, N.M.R., Geim, A.K., and Novoselov, K.S. Rev. Mod.

[37] Zhu, Y., Murali, S., Stoller, M.D., Ganesh, K.J., Cai, W., Ferreira, P.J., Pirkle, A., Wallace, R.M., Cychosz, K.A., Thommes, M., Su, D., Stach, E.A., Ruoff, R.S. Science 332(6037)

[38] Echtermeyer, T.J., Britnell, L., Jasnos, P.K., Lombardo, A., Gorbachev,R.V., Grigorenko,A.N., Geim, A.K., Ferrari, A.C., Novoselov, K.S., Nature Communications

[39] Brack, M. and Bhaduri R.K., 1997, *Semiclassical Physics. Frontiers in Physics*, Vol. 96,

[40] Belov, V.V, Dobrokhotov, S.Yu., and Tudorovskii, T. Ya., Theoretical and Mathematical

[41] Bruning, J., Dobrokhotov, S., Sekerzh-Zenkovich, S., and Tudorovskii, T. , Russian

[42] Bruning, J., Dobrokhotov, S., Nekrasov, R., and Tudorovskii, T., Russian Journal of

[44] Kormanyos, A., Rakyta, P., Oroszlany, L, and Cserti, J. Phys.Rev. B. 78, 045430 (2008).

[28] Schwieters, C.D., Alford, J.A. and Delos, J.B. Phys.Rev.B. 54, N15, 10652 (1996).

[30] Blomquist, T. and Zozoulenko, I. V. Phys.Rev.B. 61, N3, 1724 (2000).

[32] Yan, K., Peng, H., Zhou, Y. Li, H., Liu, Z. Nano Letters 11, 1106 (2011)

[33] Cheianov, V.V., Fal'ko, V.I. Phys.Rev. B. 74, 041403(R)(2006).

[36] Rafiq, R., Cai, D., Jin, J. and Song, M. Carbon, 48, 4309 (2010)

Journal of Mathematical Physics 13(4), 380-396 (2006).

[43] Carmier, P., Ullmo, D. Phys.Rev. B. 77, N24, 245413 (2008).

[45] Cserti, J., Hagymasi, I., Kormanyos, A., Phys.Rev. B. 80, 073404 (2009).

Press, New York.

Cambridge, (2000).

1537 (2011)

[29] Beenaker, C.W.J. Rev. Mod. Phys. 69, 731 (1997).

Phys. 81, Jan-March, 109-162, (2009).

[34] Fistul, M.V., Efetov, K.B. PRL 98, 256803 (2007).

2, 458 doi:10.1038/ncomms1464 (2011).

Addison-Wesley, Reading, MA.

Physics 141(2), 1562-1592 (2004).

Mathematical Physics 15, 1, 1-16 (2008).

[35] Silvestrov, P.G., Efetov, K.B. PRL 98, 016802 (2007).


48 Graphene - Research and Applications

665-9 (2001).

222 (2011).

[6] Young A. F. and Kim, P. , Nat. Phys. 5, 222-6 (2009)

[11] J. Velasco Jr et al, New J. Phys. 11, 095008 (2009).

[10] P. Barbara et al, (2012) unpublished.

*Engineers*, McGraw-Hill, Inc.

[17] Sonin, E.B. Phys.Rev. B. 79, 195438 (2009).

[7] Shytov, A. V., Rudner, M.S, and Levitov, L.S., Phys. Rev. Lett. 101, 156804 (2008).

Avouris, Nano Lett., 12 (3), pp 1417-1423, (2012). DOI: 10.1021/nl204088b

[8] Ramezani Masir, M., Vasilopoulos, P.,Peeters, F. M., Phys. Rev. B. 82, 115417 (2010).

[9] Yanqing Wu, Vasili Perebeinos, Yu-ming Lin, Tony Low, Fengnian Xia, and Phaedon

[12] Liang W J, Bockrath M, Bozovic D, Hafner J H, Tinkham M and Park H, Nature, 411,

[13] Miao F, Wijeratne S, Zhang Y, Coskun U, Bao W and Lau C N, Science 317 1530-3 (2007).

[15] Williams, J.R., Low, T., Lundstrom, M.S and Marcus, C.M., Nature Nanotechnology 6,

[16] Bender, C.M., Orszag, S.A. 1978, *Advanced Mathematical Methods For Scientists and*

[18] Tudorovskiy, T., Reijnders, K.J.A., Katsnelson, M.I. Phys. Scr. T146, 014010, 17pp, (2012).

[19] Rodriquez-Sensale, B., Yan, R., Kelly, M.M, Fang, T., Tahy, K., Hwang, W.S., Jena, D., Liu, L. and Xing, H.G Nature Communications. 3 780 doi:10.1038/ncomms1787 (2012).

[20] Crassee, I., Orlita, M. Potemski, M., Walter, A.L., Ostler, M., Seyller, Th. Gaponenko, I.,

[21] L. V. Berlyand and S. Yu. Dobrokhotov. Operator separation of variables in problems of short-wave asymptotics for differential equations with rapidly oscillating coefficients.

[22] V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskiy. Operator separation of variables for adiabatic problems in quantum and wave mechanics. Journal of Engineering

[24] V. P. Maslov and M. V. Fedoryuk. Semi-Classical Approximation in Quantum

[25] Datta, S., *Electronic transport in mesoscopic systems*, Cambridge University Press,

[23] V. P. Maslov. Perturbation Theory and Asymptotic Methods. Dunod, Paris, (1972).

[14] O'Hare, A.O. Kusmartsev, F.V, Kugel, K.I. Nano Letters 12, 1045 (2012).

Chen, J. and Kuzmenko, A.B. Nano Letters 12 2470 (2012).

Doklady Akad. Nauk SSSR, V32, p.714, (1987).

Mathematics, 55(1-4):183-237, (2006).

Mechanics. Reidel, Dordrecht, (1981).

Cambridge, (1995) .


[46] Keller, J.B. Corrected Borr-Sommerfeld quantum conditions for non-separable systems. Ann. Phys., 4, N12, 180-188 (1958).

10.5772/52267

79

http://dx.doi.org/10.5772/52267

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

[63] M. Abramowitz and I. Stegun, editors. *Handbook on Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Dover, New York, 1965. Chapter 19 on parabolic cylinder

[65] Popov M M 2002 *Ray Theory and Gaussian Beam Method for Geophysicists*, Edufba,

[67] P. Markos and C Soukoulis. Wave Propagation. From electrons to photonic crystals and

[69] Pereira Jr,J.M., Peeters,F.M., Chaves,A., Farias, G.A., Semicond. Sci. Technol 25, 033002,

[70] Zalipaev, V.V., High-energy localized eigenstates in Fabry-Perot graphene resonator in

[71] Zalipaev, V.V., Maksimov, D.N., Linton. C.M., Kusmartsev, F.V., submitted to Phys. Let.

[68] Pelligrino, F. M. D., Angilella, G. G. N., Pucci, R., Phys. Rev. B. 84, 195404 (2011).

a magnetic field, J.Phys.A.: Math. Theor 45, 215306, 20pp (2012).

functions.

(2010).

A (2012)

Salvador-Bahia, Brazil.

[64] Berry, M.V. Proc. R. Soc. Lond. A. 392, 45 (1984).

[66] V. V. Zalipaev, J.Phys. A: Math. Theor. 42, 205302, 14pp, (2009).

left-handed materials. Priceton University Press, 2008.


50 Graphene - Research and Applications

24-75 (1960).

Russian).

(1995).

(2008).

1993.

Reidel, Dordrecht.

Springer-Verlag, New York.

*problems*, Springer-Verlag, Berlin.

Ann. Phys., 4, N12, 180-188 (1958).

[46] Keller, J.B. Corrected Borr-Sommerfeld quantum conditions for non-separable systems.

[47] Keller, J.B. Rubinov, S. Asymptotic solution of eigenvalue problems. Ann. Phys., 9, N1,

[48] Maslov, V. P. and M.V.Fedoriuk, 1981, *Semiclassical approximation in quantum mechanics*,

[49] Maslov, V. P., 1977, *Complex WKB method in nonlinear equations*, Nauka, Moscow (in

[50] Babich, V. M., Buldyrev, V.S. 1991, *Asymptotic methods in shortwave diffraction problems*,

[51] Popov, M.M., The asymptotic behaviour of certain subsequences of eigenvalues of boundary problems for the Helmholtz equation in higher dimensions, English

[52] Dubnov, V.L., Maslov, V.P., Nazaikinskii, V.E. The complex lagrangian germ and the canonical operator. Russsian Journal of Mathematical Physics. 3, 2, pp.141-190, (1995).

[53] Dobrokhotov S Yu, Martinez-Olive V, Shafarevich A I, Closed trajectories and two-dimensional tori in the quantum problem for a three-dimensional resonant anharmonic oscillator. Russsian Journal of Mathematical Physics. 3, 1, pp.133-138,

[54] Babich, V. M., Kirpichnikova, N. Ya., 1979, *The boundary-layer method in diffraction*

[55] Zalipaev,V.V., Kusmartsev,F. V. and Popov,M. M., J. Phys. A: Math. Theor. 41, 065101

[60] Castro Neto, A.H., Guinea, F., Peres, N.M.R., Geim, A.K., and Novoselov, K.S. Rev. Mod.

[62] M. V. Fedoryuk. Asymptotic Analysis: Linear Ordinary Differential Equations. Springer,

Translation of Soviet Physics, Doklady, vol.14, pp.108-110,(1969).

[56] McCann, E., Fal'ko,V.I., Phys.Rev. Letters. 96, 086805 (2006).

[57] Yabana, K., Horiuchi, H., Prog. Theor. Phys. 75, 592 (1986).

[58] Duncan, K.P., Gyorffy, B.L., Ann. Phys.(N.Y.) 298, 273 (2002).

[61] F. W. J. Olver. *Asymptotics and Special Functions*, Academic Press, 1974.

[59] Keppeler, S., Ann. Phys.(N.Y.) 304, 40 (2003).

Phys. 81, Jan-March, (2009).


**Chapter 3**

**Electronic Properties of Deformed Graphene**

As early as 1947, the tight-binding electronic energy spectrum of a graphene sheet had been in‐ vestigated by Wallace (Wallace, 1947). The work of Wallace showed that the electronic proper‐ ties of a graphene sheet were metallic. A better tight-binding description of graphene was given by Saito et al. (Saito et al., 1998). To understand the different levels of approximation, Reich et al. started from the most general form of the secular equation, the tight binding Hamiltonian, and the overlap matrix to calculate the band structure (Reich et al., 2002). In 2009, a work including

It is common knowledge that a perfect grphene sheet is a zero-gap semiconductor (semime‐ tal) that exhibits extraordinarily high electron mobility and shows considerable promise for applications in electronic and optical devices, high sensitivity gas detection, ultracapacitors and biodevices. How to open the gap of graphene has become a focus of the study. Early in 1996, Fujita et al. started to study the electronic structure of graphene ribbons (Fujita et al., 1996; Nakada et al., 1996) by the numerical method. The armchair shaped edge ribbons can be either semiconducting (*n*=3*m* and *n*=3*m*+1, where *m* is an integer) or metallic (*n*=3*m*+2) de‐ pending on their widths, i. e., on their topological properties. First-principles calculations showed that the origin of the gaps for the armchair edge nanoribbons arises from both quan‐ tum confinement and the deformation caused by edge dangling bonds (Son et al., 2006; Rozhkov et al., 2009). This result implies that the energy gap can be changed by deforma‐ tion. In 1997, Heyd et al. studied the effects of compressive and tensile, unaxial stress on the density of states and the band gap of carbon nanotubes (Heyd et al., 1997). Applying me‐ chanical force (e.g., nanoindentation) on the graphene can lead to a strain of about 10%(Lee et al., 2008). Xiong et al. found that engineering the strain on the graphene planes forming a channel can drastically change the interfacial friction of water transport through it (Xiong et

> © 2013 Tong; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Tong; licensee InTech. This is a paper 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.

the non-nearest-neighbor hopping integrals was given by Jin et al. (Jin et al., 2009).

Additional information is available at the end of the chapter

**Nanoribbons**

Guo-Ping Tong

**1. Introduction**

http://dx.doi.org/10.5772/51348
