**3. Quantum dots for photovoltaics**

Quantum dots are one of the most interesting objects in modern fundamental and applied solid state physics, including applications for photovoltaics systems [10]. Typical sizes of the quantum dots of the order of 10 nanometers determine the majority of their physical properties, including the spectra of light

**345**

tum dots.

tion with the Hamiltonian

*IIIrd Generation Solar Cell*

*DOI: http://dx.doi.org/10.5772/intechopen.95289*

cell elements applications [10].

absorption and properties of light-injected carries. These spectra determine the application of quantum dots in photogalvanics and photovoltaics systems. In contrast to bulk materials, where free electron–hole pairs can be produced optically, a strong confinement of carriers in quantum dots and resulting interaction between them leads to formation of exciton-like states. This effect qualitatively modifies optical properties of quantum dots and can make them useful for solar

Various 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. There are also a number of possibilities for the architecture of photovoltaic cells. Their common feature is that the phenomenon of multiple exciton excitation in dots a is used the energized charges (electrons and holes) are conducted to the electrodes in various ways, ensuring, however, their spatial separation. One possibility is to use scatter dots. in a 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 be accomplished by coupling between the quantum dots. For regular networks of dots (single, double, or three-dimensional), discrete states of dots are transformed into mini-electron bands, ensuring charge transport. Photovoltaic cells with the use of regular quantum dot networks and their electronic mini-band structure (also called intermediate bands) have become one of the important directions in the development of photovoltaics. 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 containing dots quantum between which the distance is so small that an intermediate band is created in this area during the energy gap. 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 optical transitions from the valence band to the intermediate band and from the intermediate band to the conduction band. An important element is also that the recombination processes are in the case of the intermediate band much less likely 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 quantum suppositories instead of regular networks.

Semiconductor quantum dots are usually produced of two types of binary materials. First type is usually referred to as III-V semiconductors, where one of the elements belongs to the III group of the periodic table of elements and the other belongs to the V group such as GaAs, InAs, InSb, and similar ones. The other group, named in the same way, is the II-VI semiconductors such as ZnS, ZnSe, CdS, and similar ones. In addition, coated quantum dots, where the core and the coating layer are made of different materials of the same (usually II-VI) type can be produced and used for different applications. Conventionally produced quantum dots show a variety of sizes and shapes. On one hand, this variety of quantum dot geometries extends the ability to use them for various applications, including photovoltaics. On the other hand, this variety hampers controllability of their applications. This circumstance should be taken into account in the analysis of all applications quan-

Direct calculations of properties of quantum dots are very difficult, simple, but still highly efficient, approach relies on employing of the effective mass approxima-

( ) 2 2

<sup>0</sup> <sup>2</sup> <sup>∗</sup> = + *<sup>k</sup> H Ur m*

#### *IIIrd Generation Solar Cell DOI: http://dx.doi.org/10.5772/intechopen.95289*

*Solar Cells - Theory, Materials and Recent Advances*

ties of the order of 103

**Figure 4.**

cm2

*density of the incident light. Emin corresponds to the bottom of the conduction band.*

contributing to the electron energy as well.

mental studies and remains to be investigated.

**3. Quantum dots for photovoltaics**

phonons (with the frequency linear in the momentum) would lead to high mobili-

*Interband transitions caused by different photons, and electron distribution over the energy states, as injected. The behavior of the distribution at energies close to the minimum of the conduction band Ec is due to small density of states* ∼ *E E* − *<sup>c</sup> while the high-energy behavior is mainly determined by decrease in the spectral* 

factor for the observed low mobilities. Therefore, we concentrate on the relevant coupling to optical phonons. The coupling is due to the asymmetry of the field and change of the hopping integrals due to change in the interatomic distances. The value of the deformation potential constant is attributed to two main effects. First effect is the change in the site energy, corresponding to atomic displacement in the crystal field formed by its interaction with surrounding ions. Second effect is the changing in the overlap transfer integrals between the iodine and the lead orbitals,

The energy relaxation of the photoexcited electrons due to electron–phonon coupling with optical phonons, is relatively fast and occurs on the time scale of the order of 10 ps. This fast relaxation demonstrates that a thermalized room-temperature energy distribution is quickly produced. As a result, the performance of the photovoltaic elements with typical involved time scales of the order of 1–10 ns, is determined by the thermalized distributions, where the static local defects, either charged or not, structural disorder, and low-frequency optical phonons can play a role for the kinetics of the carriers distributions. The relation of these energy relaxation processes to the photovoltaic performance of real solar cells needs experi-

The light absorption is efficient due to the band structure of perovskite materials having a direct bandgap close to 1.5 eV in the vertex point of the Brillouin zone. As a result, almost the entire sunlight spectrum can be absorbed. The efficiency of the absorption, in addition, is enhanced by relatively large momentum matrix elements between the group-IV heavy metal and halogen atoms resulting from their spatial overlap what makes perovskite material promising material for III generation of

Quantum dots are one of the most interesting objects in modern fundamental and applied solid state physics, including applications for photovoltaics systems [10]. Typical sizes of the quantum dots of the order of 10 nanometers determine the majority of their physical properties, including the spectra of light

V−1 s−1 and, therefore, this coupling is not the limiting

**344**

photovoltaic.

absorption and properties of light-injected carries. These spectra determine the application of quantum dots in photogalvanics and photovoltaics systems. In contrast to bulk materials, where free electron–hole pairs can be produced optically, a strong confinement of carriers in quantum dots and resulting interaction between them leads to formation of exciton-like states. This effect qualitatively modifies optical properties of quantum dots and can make them useful for solar cell elements applications [10].

Various 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. There are also a number of possibilities for the architecture of photovoltaic cells. Their common feature is that the phenomenon of multiple exciton excitation in dots a is used the energized charges (electrons and holes) are conducted to the electrodes in various ways, ensuring, however, their spatial separation. One possibility is to use scatter dots.

in a 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 be accomplished by coupling between the quantum dots. For regular networks of dots (single, double, or three-dimensional), discrete states of dots are transformed into mini-electron bands, ensuring charge transport. Photovoltaic cells with the use of regular quantum dot networks and their electronic mini-band structure (also called intermediate bands) have become one of the important directions in the development of photovoltaics. 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 containing dots quantum between which the distance is so small that an intermediate band is created in this area during the energy gap. 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 optical transitions from the valence band to the intermediate band and from the intermediate band to the conduction band. An important element is also that the recombination processes are in the case of the intermediate band much less likely 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 quantum suppositories instead of regular networks.

Semiconductor quantum dots are usually produced of two types of binary materials. First type is usually referred to as III-V semiconductors, where one of the elements belongs to the III group of the periodic table of elements and the other belongs to the V group such as GaAs, InAs, InSb, and similar ones. The other group, named in the same way, is the II-VI semiconductors such as ZnS, ZnSe, CdS, and similar ones. In addition, coated quantum dots, where the core and the coating layer are made of different materials of the same (usually II-VI) type can be produced and used for different applications. Conventionally produced quantum dots show a variety of sizes and shapes. On one hand, this variety of quantum dot geometries extends the ability to use them for various applications, including photovoltaics. On the other hand, this variety hampers controllability of their applications. This circumstance should be taken into account in the analysis of all applications quantum dots.

Direct calculations of properties of quantum dots are very difficult, simple, but still highly efficient, approach relies on employing of the effective mass approximation with the Hamiltonian

$$H\_0 = \frac{\hbar^2 k^2}{2m^\*} + U(r)$$

where 2 2 *k* /2 m\* is the kinetic energy, k is the electron momentum, m\_ is the electron effective mass, and U (r) is the effective confining potential of the quantum dot. For model calculations the anisotropic oscillator form of the confinement

$$U(R) = \frac{m^\*}{2} \left( \Omega\_x^2 \mathcal{X}^2 + \Omega\_y^2 \mathcal{Y}^2 + \Omega\_z^2 \mathcal{Z}^2 \right)$$

where <sup>Ω</sup> i are the corresponding frequencies, is assumed. One-dimensional representation of this potential and corresponding wave functions are shown in **Figure 5**. This form, being useful for basic understanding, especially of the ground state of the system, has strong limitations for the analysis of applications of quantum dots in photovoltaics and photogalvanics, where highly excited states are involved.

Another form of the potential is given by:

U r 0 for r inside the quantum dot ( ) =

U r for r outside the quantum dot ( ) = ∞

and determines the quantum dot shape. Usual model shapes of quantum dot are ellipsoidal (in simple realizations, spherical) or coin-like cylindrical with the radius much larger than the width, as shown in **Figure 6**.

It is well-known that in semiconductors, although the band electron velocities, being of the order of v ∼ 108 cm/s, are much smaller than the speed of light c = 3 × 1010 cm/s, the relativistic effects, dependent on the v/c ratio, should be taken into account. In both types of semiconductors these relativistic effects lead to a coupling between electron momentum and electron spin, which appears due to the effect of the electric field in the unit cell of a binary semiconductor without inversion symmetry.

The Hamiltonian describing the spin-orbit coupling in bulk III-V materials has the form

$$H\_{\rm so} = \alpha\_{\rm D} \left( \sigma \kappa \right),$$

presenting the Dresselhaus realization of the spin-orbit coupling [11]. In this Hamiltonian σ is the vector of Pauli matrices, and κis defined by.

$$
\kappa\_x = k\_x \left( k\_y^2 - k\_x^2 \right).
$$

with the cyclic permutation of the indices defining the other two components. The coupling constant α *<sup>D</sup>* being of the order of 10 eVA3 leads to spin splitting of the band electron states of the order of 1 meV at electron concentrations of the order of 1018 cm−3.

For II-VI compounds the bulk Hamiltonian has the form

$$H\_{SO} = \alpha \left( \sigma\_{\gamma} k\_{\times} - \sigma\_{\times} k\_{\gamma} \right),$$

usually referred to as the" Rashba Hamiltonian".

The same form of the Hamiltonian describes the spin-orbit coupling in twodimensional systems with a structural inversion asymmetry. Nonzero values of acan be achieved, in addition, by applying an electric field across the two-dimensional structure or a quantum dot. Usually both (Rashba and Dresselhaus) terms are present in two-dimensional electron systems and quantum dots.

**347**

**Figure 6.**

**Figure 5.**

*modification of the charge densities.*

*IIIrd Generation Solar Cell*

as shown in **Figure 7**.

g-(Lande) factor.

*DOI: http://dx.doi.org/10.5772/intechopen.95289*

and external magnetic field in the form:

The corresponding spectrum of the Rashba Hamiltonian is given by:

( ) 2 2 <sup>2</sup> <sup>∗</sup> = ± *<sup>k</sup> Ek k m*

In the absence of an external magnetic field the presented states are doubledegenerate as dictated by the time-reversal symmetry of the spin-orbit coupled Hamiltonian. In the presence of such a field, the spin-orbit coupling terms should be augmented by the Zeeman coupling for the interaction between electron spin

> ( ) <sup>2</sup> <sup>=</sup> *<sup>g</sup> Hz B* σ

*(a) Typical spherical quantum dot with a coating layer. Typical value of the radius R is of the order of 10 nm. (b) Typical coin-like cylindrical model of a quantum dot. The width d is of the order or less than 10 nm.*

Here B is the magnetic field applied to the quantum dot, and g is the

*One-dimensional parabolic potential modeling a simple quantum dot. Schematic plots of the ground and first excited states with opposite spins (marked by up- and down-arrows) are presented. Spin-orbit coupling couples these two spatial states with opposite spins and, as result, leads to the spin-position entanglement and* 

α

The corresponding spectrum of the Rashba Hamiltonian is given by:

$$E(k) = \frac{\hbar^2 k^2}{2m^\*} \pm ak$$

as shown in **Figure 7**.

*Solar Cells - Theory, Materials and Recent Advances*

Another form of the potential is given by:

much larger than the width, as shown in **Figure 6**.

α

ties, being of the order of v ∼ 108

sion symmetry.

Hamiltonian

1018 cm−3.

σ

The coupling constant

the form

is the kinetic energy, k is the electron momentum, m\_ is the

electron effective mass, and U (r) is the effective confining potential of the quantum dot. For model calculations the anisotropic oscillator form of the confinement

∗

2

( ) ( *x yz* ) *<sup>m</sup> UR x y z*

U r 0 for r inside the quantum dot ( ) =

U r for r outside the quantum dot ( ) = ∞

It is well-known that in semiconductors, although the band electron veloci-

c = 3 × 1010 cm/s, the relativistic effects, dependent on the v/c ratio, should be taken into account. In both types of semiconductors these relativistic effects lead to a coupling between electron momentum and electron spin, which appears due to the effect of the electric field in the unit cell of a binary semiconductor without inver-

The Hamiltonian describing the spin-orbit coupling in bulk III-V materials has

*HSO D* =α σκ( )

is the vector of Pauli matrices, and

For II-VI compounds the bulk Hamiltonian has the form

usually referred to as the" Rashba Hamiltonian".

ent in two-dimensional electron systems and quantum dots.

κ

presenting the Dresselhaus realization of the spin-orbit coupling [11]. In this

2 2 = − (

*x xy z kk k* ).

with the cyclic permutation of the indices defining the other two components.

band electron states of the order of 1 meV at electron concentrations of the order of

= − ( ), *H kk SO* ασ

The same form of the Hamiltonian describes the spin-orbit coupling in twodimensional systems with a structural inversion asymmetry. Nonzero values of acan be achieved, in addition, by applying an electric field across the two-dimensional structure or a quantum dot. Usually both (Rashba and Dresselhaus) terms are pres-

 σ*yx xy*

*<sup>D</sup>* being of the order of 10 eVA3

and determines the quantum dot shape. Usual model shapes of quantum dot are ellipsoidal (in simple realizations, spherical) or coin-like cylindrical with the radius

cm/s, are much smaller than the speed of light

κ

is defined by.

leads to spin splitting of the

where <sup>Ω</sup> i are the corresponding frequencies, is assumed. One-dimensional representation of this potential and corresponding wave functions are shown in **Figure 5**. This form, being useful for basic understanding, especially of the ground state of the system, has strong limitations for the analysis of applications of quantum dots in photovoltaics and photogalvanics, where highly excited states are involved.

= Ω +Ω +Ω 2 2 22 22

where 2 2 *k* /2 m\*

**346**

In the absence of an external magnetic field the presented states are doubledegenerate as dictated by the time-reversal symmetry of the spin-orbit coupled Hamiltonian. In the presence of such a field, the spin-orbit coupling terms should be augmented by the Zeeman coupling for the interaction between electron spin and external magnetic field in the form:

$$H\mathbf{z} = \frac{\mathbf{g}}{2}(\sigma B)$$

Here B is the magnetic field applied to the quantum dot, and g is the g-(Lande) factor.

*One-dimensional parabolic potential modeling a simple quantum dot. Schematic plots of the ground and first excited states with opposite spins (marked by up- and down-arrows) are presented. Spin-orbit coupling couples these two spatial states with opposite spins and, as result, leads to the spin-position entanglement and modification of the charge densities.*

#### **Figure 6.**

*(a) Typical spherical quantum dot with a coating layer. Typical value of the radius R is of the order of 10 nm. (b) Typical coin-like cylindrical model of a quantum dot. The width d is of the order or less than 10 nm.*

#### **Figure 7.**

*Two branches of spectrum of two-dimensional carriers in the presence of the Rashba spin-orbit coupling with*  ( ) 2 2 <sup>2</sup> <sup>∗</sup> = ± *<sup>k</sup> E k k. <sup>m</sup>* α *This shape of the spectrum corresponds to the anomalous velocity in the presence of the spin-orbit coupling: The zeromomentum states have nonzero velocities while finite-momentum states at* <sup>∗</sup> <sup>=</sup> *<sup>m</sup> <sup>k</sup>* α *have zero velocities.*

Being a relativistic effect, the g − factor is strongly related to the spin-orbit coupling and the Zeeman term plays an important role in the physics of the quantum dot. The values of g- factor can vary by two orders of magnitude, e.g. between −0.45 for GaAs and approximately −50 in InSb. The corresponding spin splitting can reach the values of the order of 10 meV at realistic fields of the order of 10 T. We mention here that while the spin-orbit coupling for a given quantum dot is fixed by its material and shape [12], the Zeeman coupling can easily be modified by applying a magnetic field. Thus, the absorption spectra and optics-related properties of the quantum dot can be modified as well.

The geometry of a quantum dot plays the crucial role for its spectrum and spin-orbit coupling, and, therefore in the light absorption and photovoltaics effects. A qualitative effect of the spin-orbit coupling in quantum dots is the quantum entanglement of particle spin and its position, where the particle wavefunction cannot be presented as a product of the spin and coordinate states. This entanglement determines several processes in quantum dots.

Since in the optical absorption electrons and holes are produced, similar spinorbit coupling and Zeeman Hamiltonians should be defined for the holes as well. The Coulomb interaction between electrons and holes plays the crucial role in the light absorption in quantum dots. The spectrum of holes is described by 4 × 4 matrices due to degeneracy of the spectrum consisting of "light" and "heavy" hole branches. Dependent on the material and shape of the quantum dot, spin-orbit coupling and the Zeeman coupling for holes can be much stronger than that for electrons. In the presence of spin-orbit coupling, spin-defined states are prone to relaxation and loosing well-defined spin values. In bulk systems, where electron momentum is a well-defined quantum number, spins relax mainly due to the celebrated Dyakonov-Perel mechanism [13] with the spin relaxation rate of the order of 2 *<sup>k</sup>* α*k* τ , where *<sup>k</sup>* τ is the momentum relaxation time. In quantum dots the situation is qualitatively different since the localization suppresses the free motion of carriers. Here the relaxation is mainly due to the so-called" admixture mechanism" caused by the spin-orbit coupling between different orbital states with opposite spins and interaction of these orbital states with lattice vibrations. This mechanism strongly depends on the applied magnetic field. In relatively week magnetic fields, the spin relaxation rate is small, and spin states are conserved for

**349**

**Author details**

Paweł Kwaśnicki1,2

190G, 36-062 Zaczernie, Poland

provided the original work is properly cited.

1 Department of Physical Chemistry and Physicochemical Basis of Environmental Engineering, Institute of Environmental Engineering in Stalowa Wola, John Paul II Catholic University of Lublin, Kwiatkowskiego 3A, 37-450 Stalowa Wola, Poland

long times of the order of 1 ms or more. If the spin-orbit coupling is sufficient, the resulting spin-position entanglement influences the spin states, and, therefore, the orbital wave functions and the exchange interaction between the injected and original carriers in the quantum dots. In addition, spin states of the injected carriers can be controlled by polarization of the incident light [14]. These interactions should be clearly seen in the absorption spectra and, as a result, influence the

We mention here that a new very large class of very promising for the photovoltaics materials such as perovskites or quantum dots show the spin-orbit coupling in the general form of the Rashba and Dresselhaus terms. Therefore, the spin-orbit coupling-related aspects of their applications can be treated in general terms similarly to semiconductors. This coupling can increase the carrier's lifetime - the quantity important for solar cell applications. However, these rather complex compounds, demonstrating a great variety of different properties, are not yet reliably functionalized in the form of quantum dots. At the same time, whether the combined effects of the spin-orbit coupling lead to an increase or to a decrease in the efficiency of the light-to-voltage conversion in solar cells, is not yet clear even

This work was supported by the National Centre for Research and Development

2 Research and Development Centre for Photovoltaics, ML System S.A. Zaczernie

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: pawel.kwasnicki@mlsystem.pl

*IIIrd Generation Solar Cell*

on the qualitative scale.

**Acknowledgements**

*DOI: http://dx.doi.org/10.5772/intechopen.95289*

photovoltaics and photogalvanic application.

under the project No. POIR.01.02.00-00-0265/17-00.
