Nguyen Thi Thanh Nhan * and Dinh Quoc Vuong

Transcription

1 Progress In Electromagnetics Research M, Vol. 38, 73 8, 014 Negative Absorption Coefficient of a Weak Electromagnetic Wave Caused by Electrons Confined in Rectangular Quantum Wires in the Presence of Laser Radiation Modulated by Amplitude Nguyen Thi Thanh Nhan * and Dinh Quoc Vuong Abstract The analytic expressions for the absorption coefficient ACF of a weak electromagnetic wave EMW by confined electrons in rectangular quantum wires RQWs in the presence of laser radiation modulated by amplitude are calculated by using the quantum kinetic equation for electrons with the electron-optical phonon scattering mechanism. Then, the analytic results are numerically calculated and discussed for GaAs/GaAsAl RQWs. The numerical results show that the ACF of a weak EMW in a RQW can have negative values, which means that in the presence of laser radiation non-modulated or modulated by amplitude, under proper conditions, the weak EMW is increased. This is different from the similar problem in bulk semiconductors and from the case of the absence of laser radiation. The results also show that in some conditions, when laser radiation is modulated by amplitude, ability to increase a weak EMW can be enhanced in comparison with the use of nonmodulated laser radiation. 1. INTRODUCTION Quantum wires are one-dimensional semiconductor structures. In quantum wires, many the physical properties of the material changes significantly from the properties of normal bulk semiconductors, including optical properties [1, ]. The linear absorption of a weak EMW and the nonlinear absorption of a strong EMW in low-dimensional systems have been studied [3 6]. The influence of laser radiation on the absorption of a weak EMW in normal bulk semiconductors, quantum wells and cylindrical quantum wires have been investigated using the quantum kinetic equation method [7 10]. The influence of laser radiation non-modulated and modulated by amplitude on the absorption of a weak EMW in component superlattices has been investigated by using the Kubo-Mori method [11]. However, the influence of laser radiation modulated by amplitude on the absorption of a weak EMW in RQW are still unsolved. Therefore, in this paper, we theoretically calculate the ACF of a weak EMW caused by electrons confined in a RQW in the presence of laser radiation modulated by amplitude by using the quantum kinetic equation for electrons. The electron-optical phonon scattering mechanism is considered. The results are numerically calculated for the specific case of GaAs/GaAsAl RQW. We show that the ACF of a weak EMW in a RQW can have negative values. That means that in the presence of laser radiation non-modulated or modulated by amplitude, under proper conditions, the weak EMW is increased; and in some conditions, when laser radiation is modulated by amplitude, ability to increase a weak EMW can be enhanced in comparison with the use of non-modulated laser radiation. Received 3 June 014, Accepted 14 August 014, Scheduled August 014 * Corresponding author: Nguyen Thi Thanh Nhan nhan khtn@yahoo.com.vn. The authors are with the Department of Physics, College of Natural Sciences, Hanoi National University, 334-Nguyen Trai, Thanh Xuan, Hanoi, Vietnam.

2 74 Nhan and Vuong. THE ABSORPTION COEFFICIENT OF A WEAK EMW IN RQWS IN THE PRESENCE OF LASER RADIATION FIELD MODULATED BY AMPLITUDE.1. The Laser Radiation Field Modulated by Amplitude As in [11], here we also assume that the strong EMW laser radiation modulated by amplitude has the form: F t = F 1 t+ F t = F 1 sin Ω 1 t + α 1 + F sin Ω t + α 1 where, F1 and F has same direction, Ω 1 and Ω are a bit different from each other or Ω 1 Ω ; ΔΩ = Ω 1 Ω Ω 1, Ω. After some transformations, we obtain: F t = E 01 sin Ωt + ϕ 1 with E 01 = F1 + F +F 1F cosδωt +Δα, ΔΩ = Ω 1 Ω,Δα = α 1 α,ω= Ω 1+Ω, ϕ 1 = α+α, α = α 1+α,tgα = F 1F F 1 +F tg ΔΩ t + Δα. Here, Ω is the reduced frequency or the frequency of the laser radiation modulated by amplitude, ΔΩ the modulated frequency, and E 01 the intensity of the laser radiation modulated by amplitude. In the case that F 1, F F,Ω 1,Ω, Ω satisfy the conditions: 1 F = = 1 F Ω 1 Ω,andΔα = 0, the above formulas Ω can be approximated as in [11]. When ΔΩ = 0, laser radiation modulated by amplitude becomes non-modulated laser radiation... Calculations of the Absorption Coefficient of a Weak EMW in RQWs in the Presence of Laser Radiation Modulated by Amplitude We consider a wire of GaAs with a rectangular cross section L x L y andalengthl z, embedded in GaAsAl. The carriers electron gas are assumed to be confined by infinite potential barriers in the xoy plane and to be free along the wire s axis the Oz-axis, where O is the origin. The EMW is assumed to be planar and monochromatic, to have a high frequency, and to propagate along the x direction. In a RQW, the state and the electron energy spectrum have the forms [1] e ipzz sin nπx sin lπy { 0 x Lx ψ n,l, pz = Lx L y L z L x L y 0 y L y 3 0 otherwise ε n,l p z = p z m + π n m L + l x L, 4 y where n and l n, l = 1,, 3,... denote the quantization of the energy spectrum in the x and the y directions, respectively. p z =0, 0,p z is the wave vector of an electron along the wire s z axis, and m is the effective mass of an electron. We consider a field of three EMWs: two laser radiations as two strong EMWs that creates laser radiation modulated by amplitude with the intensities F 1, F and the frequencies Ω 1,Ω Ω 1 Ω ; andaweakemwwithanintensitye 0 and a frequency ω: Et = F t+ E t = F 1 sin Ω 1 t + α 1 + F sin Ω t + α + E 0 sin ωt 5 The Hamiltonian of the electron-optical phonon system in the RQW in that field of three EMWs in the second quantization representation can be written as H = ε n,l p z e c A z t a + n,l, p z a n,l, pz + ω q b + q b q n,l, p z q + C q I n,l,n,l q a + n,l, p z+ q z a n,l, pz b q + b + q, 6 n,l,n,l, p z, q

3 Progress In Electromagnetics Research M, Vol. 38, where e is the elemental charge, c is the velocity of light, ω q ω 0 is the frequency of an optical phonon, n, l, p z andn,l, p z + q z are the electron states before and after scattering, respectively, a + n,l, p z a n,l, pz is the creation annihilation operator of an electron, b + q b q is the creation annihilation operator of an phonon for a state having wave vector q =q x,q y,q z, and q z =0, 0,q z. C q is the electron optical phonon interaction constant [3, 4], C q = e ω 0 1 ε 0 Vq χ 1 χ 0,whereV and ε 0 are the normalization volume and the electronic constant, χ 0 and χ are the static and the high-frequency dielectric constants, respectively. At isthevectorpotentialoffieldofthethreeemws.in,l,n,l q is the electron form factor which characterizes the confinement of electrons in a RQW. This form factor canbewrittenas[1] I n,l,n,l q = 3π 4 q x L x nn 1 1 n+n cosq x L x [q x L x 4 π q x L x n + n +π 4 n n ] 3π 4 q y L y ll 1 1 l+l cosq y L y [q y L y 4 π q y L y l + l +π 4 l l ]. 7 The current density vector of electrons along the z direction in the RQW has the form: j z t = e m p z e A c z t n n,l, pz t. 8 n,l, p z The ACF of a weak EMW caused by the confined electrons in the presence of laser radiation modulated by amplitude in the RQW takes the form [7] 8π α = c χ E0 j z t E 0 sin ωt. 9 t The general quantum equation for the statistical average value of the electron particle number operator or electron distribution function n n,l, pz t = a + n,l, p z a n,l, pz t [7]: i n n,l, p z t [ ] = a + t n,l, p z a n,l, pz,h. 10 t Because the strong EMW laser radiation is modulated by amplitude, according to Section.1, it is expressed by formula. According to the hypothesis, due to ΔΩ Ω, then in a small amount of time there are about a few periods T = π Ω, we can presume that ΔΩt +Δα is changeless. Therefore, we let t get a certain specific value τ in such a small amount of time. Then, we have: E 01 = F1 + F +F 1F cos ΔΩτ +Δα =const; ϕ 1 = α + α =const. 11 Using Equation 10 and the Hamiltonian in Equation 6 and Equation 11, we obtain the quantum kinetic equation for electrons in the RQW see Appendix A [5, 6, 8]: n n,l, pz t t = 1 exp t n,l, q C q In,l,n,l q + u,s,m,f= { {[ i s u Ω 1 +Ω +m fω iδ dt {[ nn,l, pz t N q n n,l, p z+ q z t N q +1 ] J u a 1z q z J s a 1z q z J m a z q z J f a z q z ] }} α1 + α t +su + α { [ i exp ε n,l p z + q z ε n,l p z ω q s Ω ] } 1 +Ω m ω + i δ t t + [ ] n n,l, pz t N q +1n n,l, p z+ q z t N q

4 76 Nhan and Vuong { [ i exp ε n,l p z + q z ε n,l p z + ω q s Ω ] } 1 +Ω m ω + i δ t t [ n n,l, p z q z t N q n n,l, pz t N q +1 ] [ ε n,l p z ε n,l p z q z ω q s Ω 1 +Ω m ω + i δ { i exp [ ] n n,l, p z q z t N q +1n n,l, pz t N q exp { i [ ε n,l p z ε n,l p z q z + ω q s Ω 1 +Ω m ω + i δ ] ] } t t }} t t, 1 4eE where a 1z and a z are the z-components of a 1 = 01 and a m Ω 1 +Ω = e E 0, respectively. N m Ω q is the balanced distribution function of phonons, α =arctan F 1F F 1 +F tan ΔΩ τ + Δα = const and J kx isthe Bessel function. In Equation 1, the quantum numbers n and l characterize the quantum wire. Similar equations can be found in bulk semiconductors, quantum wells and cylindrical quantum wires [5,8 10]. The firstorder tautology approximation method is used to solve this equation [7, 8]. The initial approximation of n n,l, pz t ischosenas n 0 n,l, p z t = n n,l, pz,n 0 n,l, p z+ q z t = n n,l, pz+ q z,n 0 n,l, p z q z t = n n,l, pz q z. The first-order tautology approximation method is implemented as follows: instead of the initial approximation of n n,l, pz t to the right hand side of the Equation 1, then integrating the Equation 1, we will receive first-order approximation of n n,l, pz t and we stop here. Then, the expression for the unbalanced electron distribution function n n,l, pz t is received as follows: n n,l, pz t = n n,l, pz 1 { exp i n,l, q C q I n,l,n,l q {[ k Ω 1 +Ω + rω + iδ k Ω 1+Ω + rω + iδ + k,s,r,m= J s a 1z q z J k+s a 1z q z J m a z q z J r+m a z q z ] }} α1 + α t + k + α n n,l, p z q z N q n n,l, pz N q +1 ε n,l p z ε n,l p z q z ω q s Ω 1 +Ω m ω + i δ n n +,l, p z q z N q +1 n n,l, pz N q ε n,l p z ε n,l p z q z + ω q s Ω 1 +Ω m ω + i δ n n,l, pz N q n n,l, p z+ q z N q +1 ε n,l p z + q z ε n,l p z ω q s Ω 1 +Ω m ω + i δ n n,l, pz N q +1 n n,l, p z+ q z N q ε n,l p z + q z ε n,l p z + ω q s Ω 1 +Ω m ω + i δ, 13 where n n,l, pz is the balanced distribution function of electrons, and the quantity δ is an infinitesimal and appears due to the assumption of an adiabatic interaction of the EMW. Substituting n n,l, pz t into the expression for j z t, we calculate the ACF of the weak EMW by using Equation 9. The resulting ACF of a weak EMW in the presence of laser radiation modulated by amplitude in a RQW can be written as: e 4 n 0 ω 0 1 α = πε 0 c πχ m k B Tm ω 3 1 Z 1 Z χ χ 0 cos β + n,l,n,l =1 A n,l,n,l B 1 B 1 14

5 Progress In Electromagnetics Research M, Vol. 38, With: B 1 = Q 0,1 1 F 0, M 0, F 1,1 + F 1, M 1,1 + M 1, M,1 + M,1 Quantity B 1 includes the contributions of the absorption processes of a photon of weak EMW; the absorption processes and emission processes of none photon, a photon, two photons of strong EMW modulated by amplitude. B 1 = Q 0,1 1 F 0, M 0, F 1,1 + F 1, M 1,1 + M 1, M,1 + M,1 Quantity B 1 includes the contributions of the emission processes of a photon of weak EMW; the absorption processes and emission processes of none photon, a photon, two photons of strong EMW modulated by amplitude. In the above formulas: Q s,m = e Rs,m Rs,m k B T K 0 e π m n π k B T L + l x L y N ω0 +1e m n L + l x L Rs,m y k B T N ω0 k B T F s,m = a 1 cos β 1 e Rs,m k B T 4m R s,m e π m k B T M s,m = a 4 1cos 4 β 1 e Rs,m k B T e π m k B T n L + l x L y 4 1 K 1 Rs,m k B T N ω0 +1 e 4m Rs,m Rs,m 4 K k B T n L + l x L y N ω0 +1 e π m n L + l x L Rs,m y k B T N ω0 π m n L + l x L Rs,m y k B T N ω0 In formulas for Q s,m, F s,m, M s,m, we obtain contribution of the Bose-Einstein distribution function for 1 optical phonons N ω0 = ω 0. e k B T 1 R s,m = π n n m L + l l x L + ω 0 s Ω 1+Ω y m ω, withs =, 1, 0, 1, ; m = 1, 1. Quantity R s,m includes contributions of the quantized energy in the restricted directions before and after scattering, phonon energy, photon energy of three EMWs. a 1 = 4e F1 +F +F 1F cosδωτ+δα ; m Ω 1 +Ω A n,l,n,l =[C 11 δ n,n +L 1 δ n,n ][C 1 δ l,l +L δ l,l ], where C 1 = 1 L x [ π 3 + n +n + 5n +n ], πn n πn n C = 1 L y [ π 3 + l +l + 5l +l ], L πl l πl l 1 = 1 L x 3π πn, L = 1 L y 3π πl, Z 1 = + e π n m k B TL x ; Z = + e π l m k B TL y. l=1 Quantities Z 1, Z appear while we standardize balance distribution function of the electron. k B is the Boltzmann constant, n 0 is the electron density in RQW. β 1 is the angle between the vector E 01 and the positive direction of the Oz axis, β is the angle between the vector E 0 and the positive direction of the Oz axis. F1 and F are the intensities of two laser radiations that create laser radiation modulated by amplitude with the intensity E 01 and the frequency Ω. Equation 14 is the expression of ACF of a weak EMW in the presence of external laser radiation modulated by amplitude in a RQW. From this expression, we see that ACF of a weak EMW is n=1

6 78 Nhan and Vuong independent of E 0 ; only dependent on F 1, F,Ω 1,Ω, ω, ΔΩ,T, L x and L y. This expression is different from expressions in the published works for the normal bulk semiconductors, quantum wells and cylindrical quantum wires [8 10]. When ΔΩ = 0, the above results will come back the case of absorption of a weak EMW in the presence of non-modulated laser radiation. From expression 14, when we set F 1 =0andF = 0, we will receive expression of ACF of a weak EMW in the absence of laser radiation in RQW. In Section 3, we will show clearly that under the influence of laser radiation non-modulated or modulated by amplitude, under proper conditions, ACF of a weak EMW can gets negative values, i.e., the weak EMW can be increased; and in some conditions, when laser radiation is modulated by amplitude, ability to increase a weak EMW can be enhanced in comparison with the use of non-modulated laser radiation. 3. NUMERICAL RESULTS AND DISCUSSIONS In this section, in order to show clearly that ACF of a weak EMW can gets negative values and in some conditions, when laser radiation is modulated by amplitude, ability to increase a weak EMW can be enhanced in comparison with the use of non-modulated laser radiation, we numerically calculated the ACF for the specific case of a GaAs/GaAsAl RQW. The parameters used in the calculations are as follows [13]: χ =10.9, χ 0 =13.1, m =0.066m 0, m 0 being the mass of free electron, n 0 =10 3 m 3, ω 0 =36.5 mev, β 1 = π 3, β = π 6, α 1 = π 3, α = π 6. Figure 1 describes the dependence of α on the temperature T,withL x =4nm,L y =6nm, ω = Hz, F 1 = V/m, F = V/m, Ω 1 = Hz, Ω get five different values: Hz, Hz, Hz, Hz, Hz. The five different values of Ω corresponding to the five different values of ΔΩ: Hz, 0 Hz, Hz, Hz, Hz. From the Figure 1, we see that there are the temperature regions at which the ACF of a weak EMW in the presence of laser radiation modulated by amplitude is smaller than one in the presence of non-modulated laser radiation and that there are the temperature regions at which the ACF of a weak EMW in the presence of laser radiation modulated by amplitude gets values greater than one for the case of non-modulated laser radiation. In addition, ACF can gets negative values, i.e., ACF of a weak EMW becomes increased coefficient of a weak EMW. For example, when T > 108 K, the ACF of a weak EMW gets negative values, and the ACF of a weak EMW in the presence of laser radiation modulated by amplitude with four cases of ΔΩ 0 Hz gets values greater than one for the case of non-modulated laser radiation ΔΩ = 0 Hz. It means that ability to increase a weak EMW in the presence of laser radiation modulated by amplitude is decreased in comparison with one for the case of non-modulated laser radiation. When 94 K <T <108 K, the ACF of a weak EMW in the presence of laser radiation modulated by amplitude with ΔΩ = Hz is more negative than one for the case of non-modulated laser radiation ΔΩ = 0 Hz, i.e., ability to increase a weak EMW is enhanced when strong EMW is modulated by amplitude with ΔΩ = Hz. Figure describes the dependence of α on ΔΩ ΔΩ is the modulated frequency, also with the above conditions and seven different values of T. From the Figure, we see that the curves can have a maximum or a minimum in the investigative interval. Both Figures 1 and show that at high temperature region, ACF is almost independent of ΔΩ, i.e., the amplitude modulation of laser radiation hardly affects ability to increase a weak EMW in the presence of laser radiation. Figure 3 describes the dependence of α on the frequency Ω 1 of either laser radiation with L x = 4nm, L y = 6nm, T = 95K, F 1 = V/m, F = V/m, ω = Hz and two different values of ΔΩ. The curves in this figure have a maximum and a minimum in the investigative interval. This figure show that the ACF of a weak EMW can have negative values. When Ω 1 [ , ] Hz, ACF of a weak EMW in the presence of strong EMW modulated by amplitude with ΔΩ = Hz is more negative than one in the presence of non-modulated strong EMW ΔΩ = 0 Hz, i.e., ability to increase a weak EMW is enhanced when strong EMW is modulated by amplitude with ΔΩ = Hz. Figure 4 describes the dependence of α on the frequency ω of the weak EMW, with L x =4nm, L y =6nm,T =30K,F 1 = V/m, F = V/m, Ω 1 = Hz, and five different values of Ω corresponding to the five different values of ΔΩ. From Figure 4, we see that the curves have a

7 Progress In Electromagnetics Research M, Vol. 38, maximum peak at ω = ω 0 and smaller maxima peaks at ω ω 0. In this figure, we also see that the ACF of a weak EMW can have negative values and that there are the frequency ω regions at which the ACF of a weak EMW in the presence of laser radiation modulated by amplitude is smaller or greater than one in the presence of non-modulated laser radiation. Figure 5 describes the dependence of α on the intensity F 1 of either laser radiation with L x =4nm, L y =6nm,T =95K,F = V/m, ω =10 13 Hz, Ω 1 = Hz, and five different values of Ω corresponding to the five different values of ΔΩ. From Figure 5, we also see that when strong EMW is modulated by amplitude, there are the F 1 regions at which ability to increase a weak EMW is enhanced Figure 1. The dependence of α on T. Figure. The dependence of α on ΔΩ. Figure 3. The dependence of α on Ω 1. Figure 4. The dependence of α on ω. Figure 5. The dependence of α on F 1. Figure 6. The dependence of α on L x.

8 80 Nhan and Vuong and that there are the F 1 regions at which ability to increase a weak EMW is decreased in comparison with the case of non-modulated laser radiation. Figure 6 describes the dependence of α on the L x with L y =6nm,T =90K,F 1 = V/m, F = V/m, ω = Hz, Ω 1 = Hz, and five different values of Ω corresponding to the five different values of ΔΩ. From this figure, we also see that the curves have many maxima peaks. Figure 6 also shows that the ACF of a weak EMW can have negative values and that in some conditions, ability to increase a weak EMW can be decreased when strong EMW is modulated by amplitude. When L x [1, 16] nm or L x [4, 100] nm, the ACF of a weak EMW in the presence of non-modulated strong EMW gets negative values, i.e., the weak EMW is increased; but when strong EMW is modulated by amplitude with ΔΩ = Hz, Hz, Hz, the ACF of a weak EMW gets positive values, i.e., the weak EMW is absorbed. Summary, under influence of laser radiation non-modulated or modulated by amplitude, ACF of a weak EMW in a RQW can have negative values. Negative ACF of a weak EMW speaks ability to increase a weak EMW. This is different from the similar problem in bulk semiconductors and from the case of the absence of laser radiation. This effect has also appeared in quantum wells and cylindrical quantum wires [9, 10]. In addition, if laser radiation is modulated by amplitude, in some conditions, ability to increase a weak EMW can be enhanced. 4. CONCLUSIONS In this research, we investigate negative absorption coefficient of a weak EMW caused by electrons confined in RQWs in the presence of laser radiation modulated by amplitude. We obtain an analytical expression of the ACF of a weak EMW in the presence of laser radiation modulated by amplitude in a RQW for the case of electron-optical phonon scattering. The expression shows that the ACF of a weak EMW is independent of E 0 and is only dependent on F 1, F,Ω 1,Ω, ω, ΔΩ,T, L x and L y.fromthis expression, we can receive the expression of the ACF of a weak EMW in the absence of laser radiation in RQW by setting F 1 =0andF = 0. The ACF is numerically calculated for the specific case of GaAs/GaAsAl RQW. These results show that under the influence of laser radiation non-modulated or modulated by amplitude, the ACF of a weak EMW in a RQW can have negative values. Negative ACF of a weak EMW speaks ability to increase a weak EMW in the presence of laser radiation. This is different from a similar problem in bulk semiconductors and from the case without laser radiation. If laser radiation is modulated by amplitude, in some conditions, the ACF can get more negative values than one for the case of non-modulated laser radiation, i.e., ability to increase a weak EMW can be enhanced in comparison with the case of non-modulated laser radiation. So, when we want to enhance ability to increase a weak EMW, we need only to modulate amplitude of laser radiation and chose proper conditions of system. ACKNOWLEDGMENT This research is done with financial support from Vietnam NAFOSTED number APPENDIX A. In this appendix, we write out some steps to obtain Equation 1 as follows. Replacing Hamiltonian H into Equation 10, we will obtain the following equation: where: n n,l, pz t t = 1 n,l, q C q I n,l,n,l q t H 1 + H + H 3 + H 4 dt A1 H 1 = C q1 I n,l,n 4,l 4 q 1 a + a n4,l4, pz+ qz+ q1z n,l, pz b q1 + b + b q1 q n 4,l 4, q 1 t

9 Progress In Electromagnetics Research M, Vol. 38, H H 3 H 4 = C q1 I n,l,n3,l 3 q 1 a + n,l a, pz+ qz n3,l 3, p z q 1z b q b q1 + b + q1 n 3,l 3, q 1 i [ε n,l p z+ q zε n,l p z ω q]tt ie t m e q c zazt 1 dt 1 t = C q 1 In,l,n 4,l 4 q 1 n 4,l 4, q 1 a + n,l, p z+ q z a n4,l 4, p z+ q 1z b + q b + + b q1 q1 C q 1 In,l,n 3,l 3 q 1 a + a n3,l3, pz+ qz q1z n,l, pz b + + b q1 q1 n 3,l 3, q 1 i [ε n,l p zε n,l p z+ q z ω q]tt ie t m e q c zazt 1 dt 1 t = C q1 I n,l,n4,l 4 q 1 a + a n4,l4, pz+ q1z n,l, p z q z b q1 + b + b q1 q n 4,l 4, q 1 C q1 I n,l,n 3,l 3 q 1 a + a n,l, pz n3,l 3, p z q z q 1z b q b q1 + b + q1 n 3,l 3, q 1 i [ε n,l p zε n,l p z q z ω q]tt ie t m e q c z A zt 1 dt 1 t n 4,l 4, q 1 C q 1 I n,l,n 4,l 4 q 1 b + q a + n,l, p z a n4,l 4, p z q z+ q 1z b + q b + + b q1 q1 C q 1 In,l,n 3,l 3 q 1 a + a n3,l3, pz q1z n,l, p z q z b + + b q1 q1 n 3,l 3, q 1 i [ε n,l p z q zε n,l p z ω q]tt ie t m e q c zazt 1 dt 1 t Equation A1 is the quantum kinetic equation for electrons in the RQW. We will approximate this equation to the second order of C q. So, in the expression H 1 we keep only term with q 1 = q, n 4 = n, l 4 = l, n 3 = n, l 3 = l ; in the expression H we keep only term with q 1 = q, n 4 = n, l 4 = l, n 3 = n, l 3 = l; in the expression H 3 we keep only term with q 1 = q, n 4 = n, l 4 = l, n 3 = n, l 3 = l; in the expression H 4 we keep only term with q 1 = q, n 4 = n, l 4 = l, n 3 = n, l 3 = l ;andwemakethe following approximation: b + q b q t N q, b + q b q t N q = N q, b q b + q t = 1+b + q b q t 1+N q ;and ignore b q b q t, b + q b+ q t. Then, we will get Equation A1 in the approximation to second order of C q. The vector potential At of field of the three EMWs is found from the following equation: 1 c d At dt = Et. We replace A z t = 0, 0,A z into the approximate equation of n n,l, pz t that has been just found above, where A z is the z-component of At. In addition, we use expansion: e ±iz sin ϕ = + m= J m ze ±imϕ. Then, we will obtain Equation 1, and it is the quantum kinetic equation for electrons in the RQW in the approximation to second order of C q. b + q t t t t t t t

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