Photostimulated Nernst effect in two-dimensional compositional semiconductor superlattices under the influence of confined phonons

The photostimulated Nernst effect in compositional semiconductor superlattices under the influence of a confined acoustic phonon is studied by using the quantum kinetic equation. The case of the confined electron-confined acoustic phonon scattering process is examined in detail. The obtained analytical results reveal that the Nernst coefficient (NC) depends on the external field such as magnetic field B, the frequency Ω, and the amplitude E 0 of the electromagnetic wave in a complicated way but also like a function of the temperature of the system and the period of the superlattice. It also depends on the quantum number m-describing the confined acoustic phonon. The theoretical results are depicted and discussed for GaAs/AlxGa 1 − xAs compositional semiconductors superlattices. The results indicate that the Shubnikov-De hass oscillations have appeared. The confined phonons make the magnitude of the Nernst coefficient higher and more obvious than the case of a bulk phonon. In addition, the magneto-photon resonance condition is also proved. Besides, the Nernst coefficient decreases significantly as the temperature increases. The oscillations Nernst coefficient in a magnetic field are consistent with the previous experimental.


Introduction
The compositional semiconductor superlattices is a two-dimensional material that is made of two semiconductor materials that have different band gaps [1].In this structure, the movement of electrons is confined in one direction and free in the two directions in the lattice space.As a result, a variety of physical phenomena can appear, making it different compared to bulk semiconductors, specifically, when examining the kinetic effects under the influence of Electromagnetic waves and confined phonons.In some recent studies, the problems of this material have been considered in detail [2,3].The Shubnikov-de Haas (SdH) can be broken down in the region of a large magnetic field under the influence of the electromagnetic wave (EMW) in compositional semiconductor superlattice of GaN/AlGaN [2].The confined optical phonon plays an important role in making more resonance peaks in comparison with that for bulk phonons when examining the Hall effect in compositional semiconductor superlattice of Ga/Al x Ga 1−x As [3], and becomes important when the compositional semiconductor superlattice IOP Publishing doi:10.1088/1742-6596/2744/1/012004 2 period is smaller than 30 nm [3].In addition, the photo-stimulated thermo-magnetoelectric effects including the Ettingshausen effect and the Peltier effect in two-dimensional compositional semiconductors superlattices have been investigated in our recent work [4].It is demonstrated that the confined phonons not only change the magnitude of both the quantum Ettingshausen coefficient and the quantum Peltier coefficient compared to the case of the unconfined phonons but also make the appearance of the Shubnikov-de Hass oscillations due to the influence of confined acoustic phonon [4].Nevertheless, the effect of confined phonons and the presence of electromagnetic waves on the Nernst effect in this material has been ignored.
Nernst effect also known as Nernst-Ettingshausen is a thermoelectric phenomenon in which a transverse electric field is produced by a longitudinal thermal gradient in the presence of a magnetic field, which was originally discovered in 1886 by Ettingshausen and Walther Nernst while they were studying the Hall effect in Bismuth [5].Also, the influence of the external magnetic field on this effect plays a vital role in giving new properties.Since then, more and more research about the Nernst effect in two-dimensional systems has been conducted [6][7][8][9][10][11][12].The Nernst signal increased considerably in graphene in a quantizing magnetic field [6][7][8].The sign of the Nernst coefficient is changed in strong magnetic fields in twodimensional systems [7].A sharp peak was seen in the Nernst signal when a Landau level meets the Fermi level [8].Oscillation of the Nernst-Ettingshausen coefficient under the effect of semiclassically strong magnetic fields for a quasi-two-dimensional system with a parabolic or linear dispersion of carriers was observed [10].Besides, in the superlattice structure, the transverse Nernst-Ettingshausen effect under the influence of an external magnetic field with a temperature gradient in the Two-dimensional electron gas of a doubly periodic semiconductor superlattice has been studied [13].A considerable rise in the Nernst-Ettingshausen coefficient of a degenerate quasi-three-dimensional electron gas in superlattices in a weak magnetic field is determined [14].However, in the above-mentioned studies, no work has considered the Nernst effect under the influence of strong electromagnetic waves.In addition, the problem of the Nernst effect in compositional semiconductors superlattices superlattices (CSS) has not been noticed.
Therefore, in this study, we apply the quantum kinetic equation method [15] to investigate the influence of a confined acoustic phonon (confined AP) on the photostimulated Nernst effect in compositional semiconductor superlattices, especially the presence of electromagnetic waves.The analytical expression for the Nernst coefficient is determined in detail.In addition, the contribution of acoustic phonon as well as the effect of external fields, and the temperatures of the system on the Nernst coefficient in the case with and without a confined acoustic phonon have been considered and compared.The organization of this paper is as follows.The photostimulated Nernst effect in compositional semiconductors superlattices within a confined acoustic phonon is presented in section 2. In section 3, we present numerical results for GaAs/Al 0.25 Ga 0.75 As CSS to clarify the above dependence.Finally, section 4 is conclusions.
2. Photostimulated Nernst effect in two-dimensional compositional semiconductors superlattices under the influence of a confined acoustic phonon 2.1.The wave function and energy spectrum of an electron in compositional semiconductors superlattices A simple model of compositional semiconductors superlattice of GaAs/Al x Ga 1−x As is considered.In this structure, the type I semiconductor layer with thickness d I is established alternately with the type II semiconductor layers with thickness d II and the period of superlattice is determined by d = d I + d II .Also in this structure, we set up a temperature gradient ∇T along the y-direction, a strong EMW − → E = (0, E 0 sin Ωt, 0), and magnetic field − → B = (0, 0, B).Under these conditions, a constant electric field − → E 1 along the x-axis will be produced normally  [16,17] ψ N,n, in which N = 0, 1, 2, . . . is the Landau level index; N d is the number of superlattice period; − → k y and − → k z is the wave vector of the electron; ϕ N (x − x 0 ) is the harmonic wave functions, centerred is the radius of the Landau orbit in the x-y plane); is cyclotron frequency with m * being the effective mass of an electron, c is the velocity of light in vacuum.; ; is the eigenfunction of the sub-bands; and ∆ n is the half-width of the n-th mini-band determined by [18,19] ∆

The quantum kinetic equation for confined electrons in CSS
Hamiltonian of confined electron -confined phonon in compositional semiconductors superlattices in the second quantization representation can be written by [4] where and b m, − → q ⊥ (a + N,n,ky,kz and a N,n,ky,kz ) represent the creation and annihilation operators of phonon (electron); is the associated Laguerre polynomial [16,20] and the energy of a confined phonon and the confined electron-confined phonon interaction constant, respectively.In the confined electron-confined acoustic phonon scattering, the frequency of a confined phonon and the confined electron-confined AP is defined by From the Hamiltonian of confined electron -confined phonon in CSS, given by Eq. ( 4), we set up the kinetic equation for electron distribution function f N,n, Then, using transformations of operator algebra and after some calculations, we obtain where f N,n, kz is the nonequilibrium distribution function of the electron due to an external field; N m, − → q ⊥ is the phonon distribution function; J s (x) is the s th order the Bessel function of the argument x; δ(x) is the Dirac delta function.

Photostimulated Nernst effect in compositional semiconductors superlattices under the influence of a confined phonon
To determine the Nernst coefficient in compositional semiconductors superlattices under the influence of a confined phonon.Firstly, we multiply both sides of Eq. ( 8) with and take the sum in N, n, − → k y .In addition, skipping the process with more than one photon, we only focus on the absorption process of one photon, the problem is limited by the case of l = 0, ±1.Then, solving this equation, the expression for the partial current density is found by and is the Fermi level of the electron).The total current density is given by where σ ik (m) and β ik (m) is conductivity tensors, ▽T k is the temperature gradient with unit vector k.The quantum Nernst coefficient is determined by in which where (−1) s e −2πsΓ/(ℏωc) cos (2πsx i ) ; ; Equation (13) shows the NC not only is affected by the confined AP but also dependent on the external fields including the EMW, and magnetic field B is in a complicated way.The obtained results are different from the results obtained in bulk semiconductors [23].In the next section, we will give a deeper insight into this above dependence.

Numerical results
To illustrate the theoretical results of the Nernst coefficient, we examine a CSS of GaAs/Al x Ga 1−x As where x is the Al content in AlGaAs layers (the barriers).The conduction band offset between GaAs and AlGaAs is determined by as U = 0, 564x−0, 032x 2 ; The parameter used are [24]: L x = L y = 1nm; m * = 0, 067 m 0 (m 0 = 9, 1094.10−31 kg is the mass of free electron); ζ = 9.2 eV; ν s = 6560 m/s; d II = 5 nm.
The appearance of the typical Shubnikov de-Haas oscillations whose period is proportional to 1/B is observed in Fig. 1. Figure 1 gives information about the effect of the inverse magnetic field on the Nernst coefficient for the confined AP (the red line) and the bulk phonon (the blue-dashed line) with d=15 nm and T=5 K.It is obvious that the NC as a function of the inverse magnetic field.From Fig. 1(a), it can be seen that at low temperatures condition T=5 K, the magnitude of NC's oscillations increases with the rising magnetic field.In addition, the oscillation peaks of the NC become more obvious when the magnetic field is large (B > 2 T), corresponding to B −1 is lower 0.55 1/T.Besides, in the previous experiment when investigating the dependence of the Nernst coefficient on the inverse magnetic field in Bismuth across the quantum limit [11], Kamran Behnia and co-workers also observed the oscillations of the Nernst coefficient, which also become stronger when B −1 is smaller 0.6 T −1 [11].A similar trend was found when considering the NC on the inverse magnetic field in the graphene [25].However, when the magnetic field is small (B < 2 T), respectively B −1 > 0.55 1/T, the oscillations of the NC instead of reducing like the obtained results in Bismuth [11] and Graphene [25], the results in this study tend to increase with higher magnetic fields.Hence, the obtained results in two composition semiconductor superlattice in this study are nearly consistent with the previous experimental in two-dimensional system [11,25] when taking into account the B −1 is smaller 0.7  1/T.This can be explained by the difference in the structure, the influence of the electromagnetic wave, and the effect of confined AP.In addition, the confined AP also plays an important role in changing the amplitude of the NC, it not only makes the magnitude of the NC higher but also more obvious compared to the bulk phonon case.This importance of a confined AP in composition semiconductor superlattice can be demonstrated in Fig. 1(b).It can be seen that when the CSS period is small and under the effect of a confined AP, it makes the magnitude of the NC higher compared to the bulk phonon case.Figure 2 shows the impact of the EMW frequency on the Nernst coefficient for the confined AP (the red line) and bulk phonon (the dashed-black line) with T=5 K and B=3 T. It can be seen that the resonance peak has appeared and the given graph has one resonance peak in two phonon cases with and without confined AP.While in the confined AP case, the resonance peak stands at Ω = 4.5.10 11 Hz, the case of bulk phonon stands at Ω = 3.7.10 11Hz.Due to the influence of confined AP, the position of the resonance peak of the NC not only shifts to a higher frequency but also decreases the amplitude of the NC.The influence of temperature on the Nernst coefficient in two cases: with and without confined AP is compared in Fig. 3.It can be seen that when the temperature is less than 20 K, the NC decreases significantly and tends to approach a constant when the temperature is higher.In addition, the confined AP makes the magnitude of the NC higher.

Conclusion
In this paper, we focus on the Nernst effect in compositional superlattice by using a quantum kinetic equation and numerical computation for GaAs/Al 0.25 Ga 0.75 As compositional superlattice.We obtain the theoretical expression of kinetic tensor and Nernst effect.The numerical computation illustrated the dependence of the Nernst coefficient on external parameters (magnetic field, EMW frequency) and internal parameters (temperature).The result indicates that the form of the Shubnikov-De hass oscillation has appeared in electron-acoustic phonon interaction when examining the dependence of the Nernst coefficient on the magnetic field.The obtained results are nearly consistent with the previous experimental in Bismuth and Graphene.Besides, when we increase temperature, the Nernst coefficient decreases significantly.In addition, the magneto-phonon resonance peak can be seen by the dependence of the Nernst coefficient on EMW frequency.Finally, the influence of confined acoustic phonon on the Nernst effect in CSS has been examined in detail.It is an indispensable part that should not be ignored when investigating the Nernst effect in low dimensional.

Figure 1 .
Figure 1.Dependence of the Nernst on the inverse magnetic field for the confined AP (the red line) and the bulk phonon (the blue-dashed line) with T=5 K.

Figure 2 .
Figure 2. Dependence of the Nernst coefficient on the EMW frequency for the confined AP (the red line) and bulk phonon (the dashed-blue line) with T=5 K, B=3 T.

Figure 3 .
Figure 3. Dependence of the Nernst coefficient on the temperature for the confined AP (the red line) and bulk phonon (the dashed-blue line) with B=3 T.
Hence, the wave function and energy spectrum are found by