**4. Relaxation processes for energy and spin in quantum dots important for photovoltaics applications**

Possible applications of quantum dots in photovoltaics [6] call for the studies and understanding of their optical and transport properties. These properties are either controlled or strongly related to the energy and spin relaxation processes in these systems. The energy relaxation is important since it determines transport properties of the photovoltaic devices such as the probability of electron escape from the quantum dot to the glass layer and the transparent conductive oxide and, as a result, the integral photovoltaic efficiency. The spin relaxation is important since it is directly related to the coupling mechanisms responsible for the energy relaxation and allows one to find the efficient mechanisms of both the energy and the spin relaxation. In addition, the spin states are critically important for the formation of the spectra of quantum dots with several carriers (electrons and/or holes) and, thus, spin relaxation can determine the photovoltaic efficiency of the charged quantum dots in the regime of strong light irradiation, where the light intensity is sufficiently strong to produce more than one carrier per quantum dot [8].

There are different types of quantum dots applicable in photovoltaics. Let us consider ellipsoidal quantum dots serving as optical absorbers and, subsequently, source of the photo excited electrons upon their escape from the dot. The main source of the energy relaxation in single-electron quantum dots is due to various types of electronphonon coupling, which will be analysed below. In addition, we will analyse the mechanisms of the energy relaxation in multi-electron quantum dots taking into account the electron-electron interaction. We will demonstrate that a novel mechanism of spin relaxation, different from the conventional admixing mechanism, can be relevant for

the quantum dots based on III-V group materials. Let us begin with a model of a semiconductor quantum dot applicable in photovoltaics. These dots have a variety of different properties related to their sizes and shapes. Although this variety extends the ability to use the quantum dots for nonmonochromatic light sources such as sunlight, on the other hand, it decreases the controllability of their applications. Direct systemdependent analysis of properties of quantum dots is difficult and can basically be done with the effective mass approximation as included in the Hamiltonian

$$H\_0 = \frac{p^2}{2m^\*} + U\_\epsilon(r). \tag{9}$$

Where *<sup>p</sup>*<sup>2</sup> <sup>2</sup>*m*<sup>∗</sup> , is the electron kinetic energy, p is the electron momentum, m∗ is the electron effective mass, and U (r) is the effective confining potential of the quantum dot. Frequently, for model calculations the confinement is taken in the form of an anisotropic oscillator:

$$U\_{\varepsilon}(r) = 2m^\* \left[ \Omega\_{\parallel}^2 (\varkappa^2 + \jmath^2) + \Omega\_x^2 x^2 \right] / 2,\tag{10}$$

where Ω<sup>k</sup> and Ω*z*, are the corresponding frequencies of the anisotropic oscillator. However, this potential is well-suitable only for the description of the low-energy states of the photoexcited electrons, where wavefunction is well-localized near the potential minimum. Another form of the potential is given by *Ue*ð Þ*r* = 0 for r inside the quantum dot, *Ue*ð Þ*r* = U for r outside the quantum dot, where the inside/outside boundary determines the quantum dot shape. Usual model shapes of quantum dot are ellipsoidal with an example presented in **Figure 6**. The typical scale of the quantum kinetic energy is determined by the z � axis extension of the quantum dot *ω* as *ℏ*2 *<sup>m</sup>*∗*ω*<sup>2</sup> with the corresponding quantity of the order of 10 meV for *<sup>ω</sup>* � 10 nm and *m*<sup>∗</sup> of the order of 0.1 of the free electron mass. For highly excited states, with the energies above the excitation threshold of the order of 100 meV, but less than the confinement potential U, a classical description in terms of the electron trajectories becomes possible. Here emission of phonons leads to the relaxation to the lowerenergy quantum states. To study the relaxation, we present electron-phonon coupling Hamiltonian (assuming the crystal volume � 1) in the form:

#### **Figure 6.**

*(a) Two model ellipsoidal quantum dot on the surface of a glass. The vertical size of the quantum dot is denoted as w, typically of the order of 10 nm. (b) Classical trajectory of highly photoexcited electron in the quantum dot.*

*Quantum Dots - Recent Advances, New Perspectives and Contemporary Applications*

$$\mathcal{V}\_{e-ph} = \frac{D\sqrt{\hbar}}{\sqrt{2\zeta\_r}} \sum\_{q} \sqrt{q} (de) \left( e^{-iqr} a\_q^\dagger + H.c \right) \tag{11}$$

Where *a*† *<sup>q</sup>* is the creation operator for the phonon with momentum q, D is the deformation potential (D = �5.5 eV in GaAs), ζ is the crystal density, d = q/q is the phonon propagation direction, e is the phonon polarization, c is the speed of the longitudinal sound mode, and summation is taken over the momenta. We have chosen a single longitudinal phonon branch with (de) = 1, since for the transverse branches (de) = 0, taking into account only the strongest interaction with the deformational potential of the acoustic phonons.

There are different regimes of energy relaxation. The first regime is relevant for the highly-excited semiclassical states, leading to their energy loss and subsequent quantization with localization in the quantum states. The second regime is relevant for the localized states, where phonon-induced transitions occur between the low-energy quantum states in the dots. We begin with the most important for the photovoltaics semiclassical regime, where the electron states can be presented as plane waves. For the relaxation of these states shown in **Figure 6(b)** one can use the classical Boltzmann equation based on Fermi's golden rule. The energy-dependent phonon emission rate 1/τ (E) per single phonon mode with wavevector q leading to the energy relaxation is given by this rule as [9]:

$$\frac{1}{\pi(E)} = \frac{2\pi D^2 \hbar}{\hbar 2\zeta c} \left[ q \delta \left[ E\_f - E\_i + \hbar \Omega(q) \right] \frac{d^3 k\_f}{(2\pi)^3} \right] \tag{12}$$

where the initial and final energy is *Ei* <sup>¼</sup> *<sup>ℏ</sup>*2*k*<sup>2</sup> *i* <sup>2</sup>*m*<sup>∗</sup> and *Ef* <sup>¼</sup> *<sup>ℏ</sup>*2*k*<sup>2</sup> *f* <sup>2</sup>*m*<sup>∗</sup> respectively, final electron wavevector kf = ki � q, and δ [..] � function corresponds to the energy conservation with (q) = cq. Taking into account that the characteristic wave vector ki of a photoexcited electron is of the order of 106 cm�<sup>1</sup> or higher, at the corresponding phonon momentum its energy *ℏck* <sup>i</sup> is of the order of less than one meV, that is 10 K, much less than the electron energy. As a result, the electron scattering is quasi-elastic and the occupation number of the phonons *nB* ¼ exp *ℏ*Ωð Þ*q <sup>T</sup>* � 1 h i � �<sup>1</sup> , equal to T/*ℏ*Ω at the room temperature T, is large. Therefore, the emission probability is greatly enhanced by the factor nB + 1. However, this occupation factor increases the phonon absorption probability as well and these processes partially compensate each other. This process leads to the energy dependence of <sup>1</sup> *<sup>τ</sup>*ð Þ *<sup>E</sup>* determined by the electron density of states, proportional to ffiffiffi *E* <sup>p</sup> and by the absolute value of the phonon momentum, also behaving as ffiffiffi *E* <sup>p</sup> and resulting in:

$$\frac{1}{\pi(E)} \approx \frac{1}{\pi\_D} \frac{E}{\hbar \Omega\_D} \left(\frac{m^\* c^2}{\hbar \Omega\_D}\right)^{1/2} \tag{13}$$

where *τ<sup>D</sup>* is the nominal momentum relaxation time Ref. [9] (*τ<sup>D</sup>* = 2.5 ps in GaAs), and D is the Debye phonon frequency. The time-dependence of the energy due to nearly balanced quasi-elastic emission and absorption of phonon is given by: *dE t*ð Þ *dt* ¼ *ℏ*Ωð Þ *E <sup>τ</sup>*ð Þ *<sup>E</sup>* , where the energy goes to phonon subsystem. The numerical value of e<sup>τ</sup> is rather long, being of the order of 10�<sup>7</sup> s as a result. The resulting time dependence of energy has the form:

*Quantum Dots as Material for Efficient Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.106579*

$$E(t) = \frac{E(\mathbf{0})}{\left(\sqrt{\frac{E(\mathbf{0})}{2\sqrt{\hbar\tau}}}t + \mathbf{1}\right)^2} \tag{14}$$

For the initial energy E(0) = 100 meV this estimate yields the value τ � 10 ps, demonstrating that energy relaxation of photoexcited electrons is fast and can set up the initial condition for the diffusion through the glass layer. Taking into account that the electron velocity is of the order of 10<sup>8</sup> cm/s, electron during this time hits the quantum dot boundary at least 100 times. As a result, the electron can escape from the quantum dot if the escape probability per single boundary collision is higher than 10�<sup>2</sup> , thus, requiring for a relatively transparent boundary between the quantum dot and the glass (see **Figure 6(a)**).

Quantum dots are often single-electron charged since they can absorb electrons from the bulk of the semiconductors. Therefore, complex states built by a photoexcited electron-hole pair inside the dot and the resident electron, usually called 'trions' Ref. [9], can be formed, as shown in **Figure 7(a)**. The formation of trions extends the absorption range of quantum dots. The spectrum and the structure of a trion strongly depends on electron-electron and electron-hole coulomb interaction between electrons. The Hamiltonian of the system becomes:

$$H\_0 = \frac{p\_1^2}{2m^\*} + \frac{p\_2^2}{2m^\*} + \frac{P^2}{2m\_h^\*} + U\_\epsilon(r\_1) + U\_\epsilon(r\_2) + U\_h(R) + \frac{e^2}{\epsilon|r\_1 - r\_2|} + \left[\frac{e^2}{\epsilon|r\_1 - R|} + \frac{e^2}{\epsilon|r\_2 - R|}\right] \tag{15}$$

where r1 and r2 are positions of electrons (with momenta p1 and p2, respectively), R is the position of the hole (with mass *m*<sup>∗</sup> *<sup>h</sup>* and the momentum P with the confinement potential Uh (R)), and o is the quantum dot material dielectric constant. The process ˛ leading to the overall decrease in the escape probability is shown in **Figure 7(b)**. Note that this process reduces the energy of the upper electron and increases the energy of the lower one.

#### **Figure 7.**

*(a) Trion in a photoexcited quantum dot. Electron positions are r1 and r2, and the hole position is R. (b) Interaction-induced transition between initial (left) and final (right) states in a quantum dot. Note that, due to the Pauli principle, this transition is possible only for the singlet electron spin state.*

Despite this partial increase, the escape probability, strongly dependent on the electron energy, decreases. The corresponding energy and time scales are given by the following matrix element, where we neglected the exchange effects take only the direct interactions as: *VC* <sup>¼</sup> <sup>Ð</sup> *ψ*1 *<sup>f</sup>*ð Þ *<sup>r</sup>*<sup>1</sup> *<sup>ψ</sup>*<sup>2</sup> *<sup>f</sup>*ð Þ *<sup>r</sup>*<sup>2</sup> *<sup>ψ</sup><sup>h</sup> <sup>f</sup>* ð Þ *<sup>R</sup> HCψ*<sup>1</sup> *<sup>i</sup>*ð Þ *<sup>r</sup>*<sup>1</sup> *<sup>r</sup>*2Þ*ψ<sup>h</sup> <sup>i</sup>* ð Þ *<sup>R</sup> <sup>d</sup>*<sup>3</sup> *r*1*d*<sup>3</sup> *r*2*d*<sup>3</sup> *R* �

where the coulomb term

$$H\_C = \frac{e^2}{\varepsilon |r\_1 - r\_2|} + \left[ \frac{e^2}{\varepsilon |r\_1 - R|} + \frac{e^2}{\varepsilon |r\_2 - R|} \right] \tag{16}$$

The probability of this process is given by the time scale corresponding to the energy scale of the Hamiltonian (16), typically of the order of *<sup>e</sup>*<sup>2</sup> *ϵω*, that is higher than 1 meV. Therefore, these processes are very fast, being, in general, energy-conserving and dependent on the total spin of participating electrons. This energy value corresponds to short time scales of these processes, of the order of a picosecond. Then, on the top of these fast transitions, a relatively slow energy relaxation due to phonons occurs, complicated by the transitions between the localized states in the quantum dots. Yet, the energy relaxation weakly depends on these fast electron transitions since these states are localized and, therefore, the estimate of long τ[loc] given in the previous subsection is valid here as well.

Thus, understanding of energy and spin relaxation processes in quantum dots suitable for applications in photovoltaics is crucial. The energy relaxation is due to the phonon emission. It was found the relaxation times of semiclassical states of the order of 1 ps, sufficient for electron escape from the quantum dot and its contribution to the photovoltaic effects. In addition, the redistribution of electrons among the energy levels and subsequent energy evolution can be caused by the electron-electron interactions. Spin relaxation in quantum dots is caused by direct spin-phonon coupling and leads to the spin relaxation times of the order of 1�10<sup>2</sup> microsecond, being the longest relaxation process in quantum dots.

### **Acknowledgements**

This work was supported by the National Centre for Research and Development under the project No. POIR.01.02.00-00-0265/17-00.

*Quantum Dots as Material for Efficient Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.106579*
