Quantum Dots as Material for Efficient Energy Harvesting

*Paweł Kwaśnicki*

### **Abstract**

The essence of the photovoltaic effect is the generation of electric current with the help of light. Absorption of a quantum of the energy of light (photon) generates the appearance of an electron in the conduction band and holes in the valence band. The illumination of the material, in general, is not uniform, which leads to the appearance of spatially inhomogeneous charge in the band valence and conductivity. Besides, electrons and holes generally diffuse with different velocities, which leads to the creation of a separated space charge and generation of an electric field (sometimes called the Dember field). This field inhibits further separation of cargo. The reverse processes also take place in the system, i.e. electron recombination and holes. These processes are destructive from the point of view of photovoltaics and should be minimized, which is achieved; thanks to the spatial separation of electrons and holes. The point is that electrons and holes were carried away from the area where they formed as quickly as possible, yes to prevent their spontaneous recombination. The use of semiconductor quantum dots introduced into the photoelectric material is currently a very important and effective way to increase the efficiency of photoelectric devices and photovoltaic cells. This is due to the fact that in semiconductor photoelectric materials with no quantum dots, there is always some upper limit of the wavelength *λgr*½ � *gr* ≃1, 24*=Eg*½ � *eV* for absorbed light, above which the light is not absorbed.

**Keywords:** quantum dots, photovoltaic, QDSC, transparent PV

## **1. Introduction**

The maximum coefficient of light to electric conversion in semiconductor systems is determined by the so-called Shockley-Queisser limit, which is approximately 32% with the optimal width of the band gap of 1.2�1.3 eV. However, performance can be improved by using solutions based on semiconductor nanostructures [1, 2]. Such cells are called Third Generation Photovoltaic Cells (KFTG). One type of nanostructure that plays an important role is the so-called quantum dots, that is, small semiconductor grains with sizes in the order of nm. As bounded systems in all three dimensions, quantum dots are characterized by a discrete energy spectrum. The advantage of such systems is the fact that due to the lack of translational symmetry, the limitation on quantum transitions resulting from the behaviour of the wave vector (equality of the photon and electron wave vector) is removed. Exclusion of this limitation leads to the fact that absorption can arise from deep discrete levels to high energy states. As a result, the useful range of the light spectrum expands considerably towards violet light. However, the width of the energy gap of the output semiconductor limits the possibility of using the spectral part from the low-energy light side [3].

Another beneficial element resulting from the location of an electron and a hole (exciton) in the area of a quantum dot is the slowing down of thermalization processes and increasing the effective number of generated carriers through transitions in which the exciton while relaxing to lower energy, generates another exciton (the phenomenon of multiple exciton generation) [1]. Thermalisation occurs when carriers transfer their excess energies to the crystal lattice through interaction with phonons. In this way, large populations of non-equilibrium (hot) charges are created and the carriers lose their energy to the lattice vibrations. Thermalisation times depend on many factors such as carrier concentration and lattice temperature but are usually in the range below 100 ps. The multiple generations of an exciton can only occur for highenergy photons, due to the conservation of energy principle. This effectively means that one photon can generate two (or more) carriers. Of course, there are also recombination processes in which the exciton generated in the photon absorption process disappears. All these factors mean that the performance of such systems may exceed the Shockley-Queisser limit.

Numerous types of quantum dots can be used in photovoltaics: semiconductor polycrystalline and granular materials, quantum dots obtained by epitaxial methods or from colloidal solutions, nanoparticles of organic dyes, etc. There are also many possibilities for the architecture of photovoltaic cells. Their common feature is that the phenomenon of multiple exciton excitation in dots is used, and the generated charges (electrons and holes) are discharged in various ways to the electrodes while ensuring their spatial separation. One possibility is to use dots dispersed in the conductive material (e.g. in organic polymers). With the appropriate concentration of the dots, the discharge of the charge from the dots to the electrodes can take place due to the coupling between the quantum dots. In the case of regular networks of dots (one, two or three-dimensional), discrete dot states are formed into mini-electron bands, ensuring charge transport. This problem has been and is still widely studied in the literature [4, 5].

Photovoltaic cells using regular quantum dot networks and their electronic miniband structure (also called intermediate bands) have become one of the significant directions of photovoltaics development [6]. The essence of this type of solution is the fact that in the area between the electrodes in the p-n junction there is a layer with quantum dots between which the distance is so small that an intermediate band is created in this area. This allows the use of low-energy photons (with energy lower than the width of the output semiconductor gap) to generate electrons in the conduction band and holes in the valence band. This is due to the optical transitions from the valence band to the intermediate band and from the intermediate band to the conduction band. An important element is also the fact that recombination processes are much less likely in the case of the intermediate band than in the case of isolated quantum dots. In this case, it is enough for the wave functions of the dots to be quite delocalized. This can be achieved in systems with complexes of quantum dots instead of regular lattices [7].

#### **2. A single quantum dot**

First, consider a single semiconductor quantum dot. In general, the quantization associated with the limited size of the quantum dot leads to discrete energy values,

ε<sup>n</sup> (n = 1,2,3 … ). Each eigenvalues value corresponds to an orbital wave function ψ<sup>n</sup> (**r**). For the sake of simplicity, we ignore the magnetic and spin-orbital interactions in this consideration. Then, we deal with a two-fold spin degeneration of all discrete levels of a quantum dot. However, if there are spin-orbital or other magnetic forces in the system, then this degeneration is generally cancelled out. A diagram of the discrete levels of a semiconductor quantum dot in some other semiconductor matrix is shown in **Figure 1**. The energy gap, both in the matrix and in the material of which the dot is made of, is marked with a dashed line (the bottom of the conductivity band and the upper limit of the valence band).

Both the own energy values and the corresponding wave functions fulfil the Schrodinger equation:

$$\left[-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) + V(r)\right]\psi\_n(r) \equiv \hat{H}\psi\_n(r) = \varepsilon\_n \mathfrak{Q}\_n(r) \tag{1}$$

where V (**r**) is the limiting potential of the quantum dot, and *H*^ is the operator of the total energy, that is kinetic and potential energy.

#### **2.1 Two interacting quantum dots**

Let us now consider two quantum dots at some distance from each other. Let us assume that the separation of levels on the dots is large enough and consider only one discrete level in a selected dot and the closest discrete level in the second dot. Let us assume that the other levels are sufficiently distant on the energetic scale. Let us denote this level in the first dot as ε<sup>1</sup> and in the second dot as ε2. If the dots were identical, then ε<sup>1</sup> = ε2, but in general these energies may be different. We denote the corresponding normalized wave functions as ψ1(**r**) and ψ2(**r**). If these dots are close enough and their wave functions partially overlap, then the wave functions of a system composed of two dots can be taken in the form

$$\mathfrak{w}(\mathbf{r}) = \mathfrak{a}\mathfrak{w}\_1(\mathbf{r}) + \mathfrak{b}\mathfrak{w}\_2(\mathbf{r}) \tag{2}$$

#### **Figure 1.**

*A diagram of the energy structure of a single semiconductor quantum dot placed in a matrix made of another semiconductor material. The dashed line corresponds to the location of the bottom of the conduction band and the upper limit of the valence band in the semiconductor matrix and in the material of which the semiconductor quantum dot is made. The solid lines represent the discrete energy levels of the quantum dot.*

where *a* and *b* are constants. When the wave functions are not too large, one can put the normalization condition |a|<sup>2</sup> + |b|2 = 1. The energies of the system, E= *H*^� �, can then be written in the form

$$E = \mathfrak{e}\_1 + \mathfrak{e}\_2 + a \ast b \int dr \psi\_1^\*(r) V\_1(r) \psi\_2(r) + ab^\* \int dr \psi\_2^\*(r) V\_2(r) \psi\_1(r) \tag{3}$$

where V1(r) and V2(r) are the limiting potentials for both quantum dots.

If the dots are the same, so they have the same limiting potential, V1(**r**)=V2(**r-d**) � V(**r**), where d is the vector connecting the dots 1 and 2, and the same levels energy ε<sup>1</sup> = ε<sup>2</sup> � ε, and the same corresponding wave functions, ψ1(**r**) = ψ2(**r-d**) � ψð Þ**r** , then the energies of the system can be written in the form

$$E = 2\varepsilon + (a \ast b + ab \ast) \int dr \psi^\*(r) V(r) \psi(r - d) \tag{4}$$

The state corresponding to a = b (|a| = |b| 1*=* ffiffi 2 <sup>p</sup> ) is a symmetric state with energy ε2- t and the antisymmetric state, a = �b, is a state with energy ε<sup>2</sup> + t, where

$$t = -\int dr \psi^\*(r)V(r)\psi(r-d)\tag{5}$$

The hoping parameter t is positive because V (r) as the attracting potential is negative. The symmetric state is the binding (lower energy) state, while the antisymmetric state is the anti-binding (higher energy) state.

#### **2.2 Periodic networks of quantum dots**

Now, let us consider the periodic networks of quantum dots and first consider the one-dimensional periodic chain of equal quantum dots. The wave function then satisfies Bloch's theorem, ψ(**r+d**) = ψ(**r**) exp. iQ∙d where Q is a wave vector, Q = (Q, 0, 0) and d = (d, 0, 0). Consider a chain with N dots and assume periodic Born-Karman conditions. From the considerations, so far, it follows that the energies of such a chain can be written in the form

$$E = \sum\_{l} \mathbf{e}\_{l} + \frac{1}{2} \sum\_{l} (a\_{l} a\_{l+1} t\_{l,l+1} + a\_{l} a\_{l-1} t\_{l,l-1}) \tag{6}$$

where al is the amplitude of the wave function at node l, l = 1, .... N, and summing over l means summing over all N nodes. Factor 1/2 in the second term eliminates double counting of hoping terms. In turn tl, l � <sup>1</sup> means the element hoping to the nearest neighbours. It follows from Bloch's theorem that al � <sup>1</sup> = al exp. � iQd. Considering the constant value of the hoping term, tl, l � <sup>1</sup> = t, and the constant value of the discrete energies, ε<sup>l</sup> =ε, the total energy can be written as

$$E = \text{Ne} + \text{t}\text{cosQd} \tag{7}$$

The one-dimensional wave vector takes the following values: Q = (2π/d) l for l = 0, � 1, � � N. The electronic states thus form an energy band with a width of 2 t. This can be interpreted in such a way that, in the case of the interacting dots, the discrete levels *Quantum Dots as Material for Efficient Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.106579*

#### **Figure 2.**

*Diagram of the energy structure of a one-dimensional network of interconnected quantum dots. In the case of a large number of dots in the chain, a continuous energy band with a width of 2 t is created, corresponding to the shaded area.*

blur to form an energy band, as shown schematically in **Figure 2**. This figure shows a finite (and relatively small) number of quantum dots, so the energy levels of the entire system are a discrete structure. The original degeneracy is abolished. However, in the case of a long chain, the states resulting from the interaction between the dots create a continuous (at the limit of an infinitely long one-dimensional network) energy band with a width of 2 t. In **Figure 2**, this band is marked with a shaded area in which discrete levels are visible.

In the case of a two-dimensional rectangular quantum dot network, where the interaction between the dots is the same for all four closest neighbours, the bandwidth resulting from the interaction is greater due to the greater number of the nearest neighbours and amounts to 4 t. This is shown schematically in **Figure 3**. Similar to **Figure 2**, the energy structure diagram is shown for a finite number of quantum dots (49 to be exact). The energy band resulting from the discrete levels corresponds to the shaded area. The discrete structure resulting from a finite number of dots is now virtually indistinguishable in this figure. In the case of a network with N � N dots, the band becomes continuous for large values of N (more precisely for N ➔ ∞).

Similarly, the three-dimensional network of quantum dots can be described. As in the above-discussed cases of the one-dimensional and two-dimensional dot network, the interaction between the dots leads to the formation of an energy band from the discrete level of individual dots. However, the width of this band in the threedimensional case is greater and amounts to 6 t for the cubic quantum dot network.

In summary, discrete levels in periodic quantum dot networks blur to form an energy band. The width of this band depends on the value of the hoping parameter t, and it decreases to zero with t ! 0. Besides, the bandwidth also depends on the number of the nearest neighbours, which is 2 t for a one-dimensional chain, 4 t for a two-dimensional square lattice, and 6 t for a three-dimensional cubic lattice.

An important aspect of the resulting band structure is the fact that the electronic states are stretched as opposed to the localized states of a single dot. Electrons and holes are therefore mobile in these bands and can be led relatively easily to the appropriate electrode, even if their generation takes place at a considerable distance. Moreover, the value of the parameter t depends on the wave function of the given energy level of the quantum dot. Therefore, the widths of the mini-bands resulting from the levels of a quantum dot can vary significantly from level to level.

#### **2.3 Non-periodic systems**

Let us now consider non-periodic systems in which quantum dots of various shapes and sizes are arranged randomly. In such systems, the connection between the

#### **Figure 3.**

*Diagram of the electronic structure in the case of a two-dimensional network of interconnected quantum dots. In the case of a large (unlimited) two-dimensional network, a continuous 4 t wide energy band is created (discrete structure from a finite number of dots is now virtually indistinguishable).*

dots is also described by random hoping parameters. Moreover, if the dots are arranged quite rarely, then only some of them are connected together. In the simplest case, such a system includes complexes of two or three interconnected quantum dots. In the simplest model system, the complexes of double dots are arranged randomly, and it can be assumed that the coupling parameter between two dots and the eigen energies of individual dots in the complex are statistically different. If the energies of states in dot 1 and 2 are ε<sup>1</sup> and ε2, respectively, and the coupling parameter is t, then the eigen energies of the interacting complex, ε� for the bonding state and ε<sup>+</sup> for the anti-bonding state, are respectively

$$\mathbf{e}\_{\pm} = \frac{1}{2} (\mathbf{e}\_1 + \mathbf{e}\_2) \pm \frac{1}{2} \sqrt{\left(\mathbf{e}\_1 - \mathbf{e}\_2\right)^2 + 4t^2} \tag{8}$$

The appropriate wave functions are located on the dots and in the space between the dots. If the colon complexes are different, then a spectrum of discrete levels located in different areas of the matrix is effectively obtained.

In the case of triple complexes, in which three dots are connected together to form one complex, the coupling is described by two hoping parameters, and the interaction result produces three different discrete levels located on the complex and in the space between the dots.

In the case of networks with relatively small deviations from the translational symmetry, the interaction between the dots leads to the formation of a continuous band. However, the corresponding wave functions may be localized. The length of the location, that is, figuratively speaking of the number of quantum dots in the area of

which the wave function related to a given electron state extends, may be large enough, comparable to the size of the structure, so effectively such states meet the mobility conditions needed for pairing electrons and holes.
