**3. Physics of quantum dots**

ation (MEG), as opposed to only one for standard crystalline silicon solar cell.Theoretically,

Generally speaking, there are three important parameters that characterize the performance of a photovoltaic cell. These are the open-circuit voltage (*Voc*), the short circuit current (*Isc*), and the fill factor (*F F*). However, the fill factor is also a function of *Voc* and *Isc*. Therefore, these last two parameters are the key factors for determining the cell's power conversion efficiency. Under ideal conditions, each photon incident on the cell with energy greater than the band gap will produce an electron flowing in the external circuit. The fill factor is determined from the maximum area of the I-V characteristics under illumination and the short circuit current

> <sup>=</sup> mI *mp p oc sc*

where Vmp and Imp are the operating point that will maximize the power output. In this case,

*in V FF*

This chapter main objective is to give an introductory coverage of a more sophisticated subject. After we review the physics, designs, structures, and some growth/synthesis techniques of quantum dots. We will give a comprehensive description of some architectures of QD solar cells (e.g., Schottky cell, p-i-n configuration, depleted heterojunction, and quantum dots sensitized solar cell. Also, challenges and opportunities of quantum dots solar cells will be

Since the early days of 1960s colloidal semiconductor crystallites or quantum dots concept has been suggested as a new structure of semiconductor materials. In 1981 Ekimove [1] reported the existence of semiconductor crystallitesin a glass matrix. In 1985 Louis Brus developed a quantum model of spherical quantum dots based on effective mass model [2]. Quantum dot term was coined in 1988 by Reed's group [3]. Smith and his coworkers [4] successfully reported the growth of quantum film by depositing 3D silver Ag islands on gallium arsenide GaAs substrate in a two-step process. By the end of the 1990s, commercial productions of colloidal quantum dots becomefeasible. In 2004 a research group at Los Alamos [5] reported that a quantum dotis capable of emitting up to three electrons per photon, as opposed to onlyone for

*V I* (1)

*<sup>P</sup>* (2)

*V*

*FF*

h<sup>=</sup> <sup>I</sup> *oc sc*

this could boost solar power efficiencyfrom 20 % to as high as 65 %.

and open circuit voltage, or

304 Solar Cells - New Approaches and Reviews

where *Pin* is the input power.

**2. Brief history of quantum dots**

discussed.

the energy conversion efficiency is given by:

In a bulk semiconductor, electrons and holes are free to move and there is no confinement and hence they have continuous energy values, where energy levels are so close to each other and packed such that energy bands are formed. Occupied bands called valance band and empty ones called conduction bands. The highest occupied band (valance band) and the lowest unoccupied band (conduction band) are separated by what is called energy bandgap Eg. Exciton is formed when electron hole pair are generated. The bond electron-hole system (exciton) form a hydrogen like atom. The separation of between the electron-hole is called Bohr's radius. Table 1 presents examples of exciton Bohr radius for some semiconductors.


**Table 1.** Exciton Bohr radius and bandgap energy of some common semiconductors.

Dimensionality of a material specifies how many dimensions do the carriers of thematerial act as free carriers. In bulk semiconductor continuous density of states results in both conduction and valence bands. However, when the number of atoms in the lattice is very few, the density of states becomes discrete, and loses the continuous 'band' likefeature. Generally speaking, when a material has one or more dimensions small enough to affect itselectronic density of state as illustrated in Figure 1, then the material is said to be confined. Accordingly we can have quantum wells, quantum wires, and quantum dots. Bulk semiconductor materials are example of three dimensional systems where density of states is proportional to (E-Ec/v) 1/2. Quantum well system is a two dimensional system where electrons are confined in one dimension and therefore possess step like density of states. Quantum wire system is a one dimensional systemwhere electrons are confined in two dimensions and therefore possess density of states proportional to (E-Ec/v) -1/2. Quantum dot is a zero dimensional system where electron motion is confined in three dimensions. Therefore, a quantum dot possess atomic like density of states that is described mathematically by a delta function δ(E-Ec/v).

**Figure 1.** Schematic of density of states as system dimensionality is reduced. The density of states in different confine‐ ment configurations: (a) bulk; (b) quantum well; (c) quantum wire; (d) quantum dot. The conduction and valence bands split into overlapping subbands that become successively narrower as the electron motion is restricted in more dimensions. dimension. Adopted from [7].

In fact, quantum confinement is mainlybecause of relatively few atoms presentin a quantum dot (see Figure 2),where excitons get confined to a much smaller space, on the order of the material's exciton Bohr radius. This pronounced confinement leads to discrete, quantized energy levels more like those of an atom than the continuous bands of a bulk semiconductor. For this reason in some literatures, quantum dots have sometimes been referred to as "artificial atoms."

**Figure 2.** TEM of colloidal lead selenide PbSe quantum dot, from [8].

In order to specify energy eigenvalues of electrons and holes in a quantum dot, a good approximation is the infinite potential well illustrated in Figure 3. Therefore, a quantum dot can be imagined as quantum box. Assuming Ψn(r) is the wave function of the nth state. Schrödinger equation for an electron confined in one-dimensional infinite square potential well of size L is:

Figure 1

306 Solar Cells - New Approaches and Reviews

dimensions. dimension. Adopted from [7].

**Figure 2.** TEM of colloidal lead selenide PbSe quantum dot, from [8].

atoms."

**Figure 1.** Schematic of density of states as system dimensionality is reduced. The density of states in different confine‐ ment configurations: (a) bulk; (b) quantum well; (c) quantum wire; (d) quantum dot. The conduction and valence bands split into overlapping subbands that become successively narrower as the electron motion is restricted in more

In fact, quantum confinement is mainlybecause of relatively few atoms presentin a quantum dot (see Figure 2),where excitons get confined to a much smaller space, on the order of the material's exciton Bohr radius. This pronounced confinement leads to discrete, quantized energy levels more like those of an atom than the continuous bands of a bulk semiconductor. For this reason in some literatures, quantum dots have sometimes been referred to as "artificial

$$-\frac{\hbar^2}{2m}\frac{d^2\nu\_n}{dx^2} + V(\mathbf{x})\nu\_n = E\_n\nu\_n\tag{3}$$

The energy eigenvalues En and eigen-wavefunction Ψn(r) of the Schrödinger equation are given

$$\Psi\_n(\mathbf{x}) = \sqrt{\frac{2}{L}} \sin \left( \frac{n\pi}{L} \mathbf{x} \right) + \sqrt{\frac{2}{L}} \cos \left( \frac{n\pi}{L} \mathbf{x} \right) \tag{4}$$

$$E\_n = \frac{\pi^2 \hbar^2}{2mL^2} n^2 \text{ , } n = 1, 2, 3, \dots \text{ } \tag{5}$$

Now if we extend the confinements of electron in three-dimensional potential well (box with dimensions Lx, Ly, and Lz) its momentum and energy will be quantized in all dimensions and we have:

$$E\_{n\_{x,y,z}} = \frac{\pi^2 \hbar^2}{2m} \left( \frac{n\_x^2}{L\_x^2} + \frac{n\_y^2}{L\_y^2} + \frac{n\_z^2}{L\_z^2} \right) \tag{6}$$

With quantum dot (cubic box) of side dimension Lx=Ly=Lz=L,then En is written as:

$$E\_{n\_{x,y,z}} = \frac{\pi^2 \hbar^2}{2mL^2} \left( n\_x^2 + n\_y^2 + n\_z^2 \right) = \frac{\pi^2 \hbar^2}{2mL^2} n^2 \tag{7}$$

Similar energy eigenvalues can be written for holes. One must specify the me for electron and mh for hole. Considering spherical shape of quantum dot with radius R, based on the effective mass model developed by Louis Brus [2] for colloidal quantum dots. The band gap EQD can be approximated by:

$$E^{QD} = E\_g^{bulk} + \frac{\hbar^2 \pi^2}{2\mathcal{R}^2} \left(\frac{1}{m\_e} + \frac{1}{m\_h}\right) - \frac{1.8e^2}{4\pi\varepsilon x\_0 R} \tag{8}$$

where ε is the relative permittivity, and ε0 = 8.85410-14 F.cm-1 the permittivity of free space.

**Figure 3.** Infinite quantum well model.

In Equation (8), the first term on the right hand side describes the energy bandgap value of the bulk, the second term represents particle in a box quantum confinement model, and the third term details the Coulomb attraction between electron and hole (exciton). As the radius of the quantum dot decreases the Coulomb attraction term could be neglected compared to the second term in calculations. Therefore, Equation (8) indicates that bandgap energy eigenvalues increases as the quantum dot size decreases.

**Figure 4.** Variation of quantum dot energy bandgap vs. dot size for some common semiconductors. From [9].

Figure 4 presents few examples of well-known quantum dots, only the first and the second terms have been considered. Other detailed models such as the strong confinement model [10] have been adopted in determining the quantum dot sizes such as CdS.

Figure 3

308 Solar Cells - New Approaches and Reviews

**Figure 3.** Infinite quantum well model.

**0**

**PbS**

**1**

**2**

**3**

**Energy bandgap (eV)**

**4**

**5**

increases as the quantum dot size decreases.

In Equation (8), the first term on the right hand side describes the energy bandgap value of the bulk, the second term represents particle in a box quantum confinement model, and the third term details the Coulomb attraction between electron and hole (exciton). As the radius of the quantum dot decreases the Coulomb attraction term could be neglected compared to the second term in calculations. Therefore, Equation (8) indicates that bandgap energy eigenvalues

**0 5 10 15 20 25 30 35 40 45 50**

**InP PbTe**

**GaAs CTe**

**GaN**

**CdS**

**CdSe**

**InAs**

**Quantum dot radius (nm)**

**Figure 4.** Variation of quantum dot energy bandgap vs. dot size for some common semiconductors. From [9].

Figure 5 shows the measured absorbance of three different sizes of lead sulfide (PbS) quantum dots suspended in toluene using dual beam spectrophotometer. Since a quantum dot bandgap is tunable depending on its size, the smaller the quantum dot the higher energy is required to confine excitons into its volume. Also, energy levels increase in magnitude and spread out more. Therefore, exciton characteristic peak is blue shifted.

**Figure 5.** Measured absorbance of three different sizes PbS quantum dot suspended in toluene.

Quantum dot structured materials can be used to create the so called intermediate band. The large intrinsic dipole moments possessed by quantum dots, lead torapid charge separation.So‐ lar cells based on intermediate bandgap materials expected to achieve maximum theoretical efficiency as high as 65%.

Nozik and his coworkwers [11-13] invesitigated Multiple exciton generation in semiconductor quantum dots. As illustrated in Figure 6, unlike bulk semiconductors such as crystalline silicon, quantum dots can generate multiple exciton (electron-hole pairs) after collisionwith one photon of energy exceeding the bandgap.In bulk semiconductor absorption of photon with energy exceeding the bandgap promotes an electron from the valance band to higher level in the conduction band these electrons are called hot carrier. The excited electron (hot carrier) undergoes many nonradiative relaxation (thermalization: multi-phonon emission) before reaching the bottom of the conduction band. However, in a quantum dot the hot carrier undergoes impact ionization process (carrier multiplication). Therefore, absorption of a single photon generates multiple electron-hole pairs.This phenomena is called multiple exciton generation MEG. Therefore, absorption of UV photons in quantum dots produces more electrons than near infrared photons.

**Figure 6.** Thermalization of hot carriers in a bulk semiconductor (a) and a quantum dot (b), from [12].

After excitation of charge carriers the timing of the processes leading to generation of multiple charge carriers are detailed in [14] and schematically illustrated in Figure 7. After absorption of photon it takes hot carries 0.1 ps to go for impact ionization, then after 2 ps carries cool down. It takes around 20 ps for Auger recombination and finally 2ps are needed for carries to cool down and become ready for new excitation.

Experimentally carrier multiplication process-one photon generates more than one exciton via impact ionization or inverse of Auger recombination-in quantum dot has been investigated and quantified using the transient absorption spectroscopy technique. In short, [5] laser pulses are directed on the sample one for excitation and another for absorption. Measuring time of relaxation indicates whether there are single excitons recombining radiatively or biexcitons recombining via Auger recombination.

It has been discovered that there is linear proportionality between absorption change for number of generated electron-hole pairs less than 3 and number of generated electron-hole pairs. For example, investigations using PbSe and PbS QDs; confirmed 300% photo-generated carrier density (quantum yield QY)in PbSe QDs by a photon with energy of 4 times the energy spacing between the HOMO and LUMO (Eg) of the quantum dot [5, 15] as shown in Figure 8 a.Further investigations by Schaller and his coworkers [6] reported higher carrier multiplica‐ tion efficiencies of 700% of photo-generated carrier density at higher photon energy to bandgap

**Figure 7.** Illustration of the time of the processes leading to generation of multiple charge carriers in a quantum dot, from [14].

(Eg) ratios (photon energy 8×Eg).Multiplication onset starts at photon with energy equals to 3 times Eg as shown in Figure 8-b.

**Figure 8.** Experimental confirmation of carrier multiplication efficiencies. From(a) [5] and (b) [6].
