Ability to increase a weak electromagnetic wave by confined electrons in doped superlattices in the presence of laser radiation modulated by amplitude

Abstract. The analytic expressions for the absorption coefficient (ACF) of a weak electromagnetic wave (EMW) caused by confined electrons in the presence of laser radiation modulated by amplitude in doped superlattices (DSL) are obtained by using the quantum kinetic equation for electrons in the case of electron–optical phonon scattering. The dependence of the ACF of a weak EMW on the temperature, frequency and superlattice parameters is analyzed. The results are numerically calculated, plotted and discussed for n-GaAs/p-GaAs DSL. The numerical results show that ACF of a weak EMW in a DSL can get negative values. So, by the presence of laser radiation modulated by amplitude, in some conditions, the weak EMW is increased. The results also show that in some conditions, the ability to increase a weak EMW can be enhanced in comparison with the use of non-modulated laser radiation. This is different from the case of the absence of laser radiation modulated by amplitude.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 113-123 This paper is available online at ABILITY TO INCREASE AWEAK ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN DOPED SUPERLATTICES IN THE PRESENCE OF LASER RADIATIONMODULATED BY AMPLITUDE Nguyen Thi Thanh Nhan1, Nguyen Vu Nhan2 and Nguyen Quang Bau1 1Faculty of Physics, College of Natural Sciences, Hanoi National University 2Faculty of Physics, Academy of Defence force - Air force Abstract. The analytic expressions for the absorption coefficient (ACF) of a weak electromagnetic wave (EMW) caused by confined electrons in the presence of laser radiation modulated by amplitude in doped superlattices (DSL) are obtained by using the quantum kinetic equation for electrons in the case of electron–optical phonon scattering. The dependence of the ACF of a weak EMW on the temperature, frequency and superlattice parameters is analyzed. The results are numerically calculated, plotted and discussed for n-GaAs/p-GaAs DSL. The numerical results show that ACF of a weak EMW in a DSL can get negative values. So, by the presence of laser radiation modulated by amplitude, in some conditions, the weak EMW is increased. The results also show that in some conditions, the ability to increase a weak EMW can be enhanced in comparison with the use of non-modulated laser radiation. This is different from the case of the absence of laser radiation modulated by amplitude. Keywords: Absorption coefficient, doped superlattices, weak electromagnetic wave, laser radiation. 1. Introduction In recent times, there has been a growing interest in studying and discovering the behavior of low-dimensional systems, in particular, DSL. The confinement of electrons in these systems considerably enhances electron mobility and leads to their unusual behaviors under external stimuli. As a result, the properties of low-dimensional systems, especially the optical properties, are very different in comparison with those of normal Received September 25, 2012. Accepted October 4, 2012. Physics Subject Classification: 62 44 01 03. Contact Nguyen Thi Thanh Nhan, e-mail address: nhan_khtn@yahoo.com.vn 113 Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau bulk semiconductors [1, 2]. The linear absorption of a weak EMW by confined electrons in low-dimensional systems has been investigated using the Kubo-Mori method [3, 4] and the nonlinear absorption of a strong EMW by confined electrons in low-dimensional systems has been studied by using the quantum kinetic equation method [5, 6]. The influence of laser radiation on the absorption of a weak EMW by free electrons in normal bulk semiconductors has been investigated using the quantum kinetic equation method [7, 8] and the presence of laser radiation and, in some conditions, the weak EMW is increased. The influence of laser radiation (non-modulated and modulated by amplitude) on the absorption of a weak EMW in compositional superlattices has been investigated using the Kubo-Mori method [9]. The influence of laser radiation on the absorption of a weak EMW in quantum wells has been investigated using the quantum kinetic equation method [10]. However, the influence of laser radiation modulated by amplitude on the absorption of a weak EMW in DSL is still open for study. Researching the influence of laser radiation on the absorption of a weak EMW plays an important role in experiments because it is difficult to directly measure the ACF of strong EMW (laser radiation) by experimental means. Therefore, in this paper, we study the ability to increase a weak EMW by confined electrons in DSL in the presence of laser radiation modulated by amplitude. The electron-optical phonon scattering mechanism is considered. The ACF of a weak EMW in the presence of a laser radiation field modulated by amplitude are obtained using quantum kinetic equation for electrons in a DSL. We then estimate numerical values for the specific n-GaAs/p-GaAs DSL to clarify our results. 2. Content 2.1. The absorption coefficient of a weak EMW in the presence of a laser radiation field modulated by amplitude in a DSL 2.1.1. The laser radiation field modulated by amplitude As in [9], here we also assume that the strong EMW (laser radiation) modulated by amplitude has the form: ~F (t) = ~F1(t) + ~F2(t) = ~F1 sin (Ω1t + α1) + ~F2 sin (Ω2t + α2) (2.1) where, ~F1 and ~F2 has same direction, Ω1 and Ω2 are a bit different from each other or Ω1 ≈ Ω2 ; |∆Ω| = |Ω1 − Ω2| ≪ Ω1,Ω2. After some transformations, we obtain: ~F (t) = ~E01 sin (Ωt + ϕ1) (2.2) with E01 = √ F 21 + F 2 2 + 2F1F2 cos(∆Ωt +∆α),∆Ω = Ω1 − Ω2,∆α = α1 − α2, Ω = Ω1 + Ω2 2 , ϕ1 = α + α ′ , α = α1 + α2 2 , tgα ′ = F1 − F2 F1 + F2 tg ( ∆Ω 2 t+ ∆α 2 ) . 114 Ability to increase a weak electromagnetic wave by confined electrons in doped superlattices... here, Ω is the reduced frequency (or the frequency of the laser radiation modulated by amplitude) and |∆Ω| is the modulated frequency. ~E01 is the intensity of the laser radiation modulated by amplitude. In the case that ~F1, ~F2, Ω1, Ω2, Ω satisfy the conditions: ~F1 Ω21 = ~F2 Ω22 = 1 2 ~F Ω2 , and ∆α = 0, the above formulas can be approximated as in [9]. When ∆Ω = 0, laser radiation modulated by amplitude becomes non-modulated laser radiation. 2.1.2. The electron distribution function in a DSL It is well known that the motion of an electron in a DSL is confined and that its energy spectrum is quantized into discrete levels. We assume that the quantization direction is the z direction. The Hamiltonian of the electron-optical phonon system in a DSL in an EMW field in the second quantization representation can be written as: H = ∑ n,~p⊥ εn ( ~p⊥ − e ~c ~A(t) ) a+n,~p⊥an,~p⊥ + ∑ ~q ~ω~qb + ~q b~q + ∑ n,n′,~p⊥,~q C~qIn,n′(qz)a + n′,~p⊥+~q⊥ an,~p⊥(b~q + b + −~q) (2.3) where n denotes the quantization of the energy spectrum in the z direction (n = 0, 1, 2, etc.); (n, ~p⊥) and (n′, ~p⊥ + ~q⊥) are electron states before and after scattering, respectively; ~p⊥(~q⊥) is the in plane xOy wave vector of the electron (phonon); a+n,~p⊥ and an,~p⊥ , (b + ~q and b~q) are the creation and the annihilation operators of the electron (phonon), respectively; ~q = (~q⊥, qz); ~A(t) = c Ω ~E01 cos(Ωt + ϕ1) + c ω ~E02 cos(ωt) is the vector potential of the EMW field (including two EMWs: a strong EMW with the intensity ~E01 and the frequency Ω; a weak EMW with the intensity ~E02 and the frequency ω); ω~q ≈ ω0 is the frequency of an optical phonon.C~q is the electron-optical phonon interaction constant [7]: |C~q|2 = 2πe 2~ω0 V ε0q2 ( 1 χ∞ − 1 χ0 ) (2.4) here V, e, ε0 are the normalization volume, the electron charge and the electronic constant, χ0 and χ∞ are the static and the high-frequency dielectric constants, respectively. The electron form factor In,n′(qz) is written as: In,n′(qz) = Nd∑ l=1 d∫ 0 eiqzzψn(z − ld)ψn′(z − ld)dz (2.5) In a DSL, the electron energy takes the simple form: 115 Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau εn(~p⊥) = ~2~p2⊥ 2m∗ + ~ωp ( n+ 1 2 ) (2.6) here, m∗ is the effective mass of electron, ψn(z) is the wave function of the n-th state for a single potential well which composes the DSL potential, d is the DSL period, Nd is the number of the DSL period, ωp = ( 4πe2nD χ0m∗ )1/2 is the frequency plasma caused by donor doping concentration and Nd is doped concentration. In order to establish the quantum kinetic equations for electrons in a DSL, we use the general quantum equation for statistical average value of the electron particle number operator (or electron distribution function) nn,~p⊥(t) = 〈 a+n,~p⊥an,~p⊥ 〉 t [7]: i~ ∂nn,~p⊥(t) ∂t = 〈[ a+n,~p⊥an,~p⊥, H ]〉 t (2.7) where 〈ψ〉t denotes a statistical average value at moment t and 〈ψ〉t = Tr(Wˆ ψˆ), with Wˆ being the density matrix operator. Starting from the Hamiltonian in Eq. (2.1) and using the commutative relations of the creation and the annihilation operators, we obtain the quantum kinetic equation for electrons in a DSL: ∂nn,~p⊥(t) ∂t = − 1 ~2 ∑ n′,~q |C~q|2 |In,n′(qz)|2 +∞∑ l,s,m,f=−∞ Jl(~a1~q⊥)Js(~a1~q⊥)Jm(~a2~q⊥)Jf(~a2~q⊥) × exp {i {[(s− l)Ω + (m− f)ω − iδ] t + (s− l)ϕ1}} × t∫ −∞ dt2 {[nn,~p⊥(t2)N~q − nn′,~p⊥+~q⊥(t2)(N~q + 1)] exp { i ~ [εn′(~p⊥ + ~q⊥)− εn(~p⊥)− ~ω~q − s~Ω−m~ω + i~δ] (t− t2) } + [nn,~p⊥(t2)(N~q + 1)− nn′,~p⊥+~q⊥(t2)N~q] exp { i ~ [εn′(~p⊥ + ~q⊥)− εn(~p⊥) + ~ω~q − s~Ω−m~ω + i~δ] (t− t2) } − [nn′,~p⊥−~q⊥(t2)N~q − nn,~p⊥(t2)(N~q + 1)] exp { i ~ [εn(~p⊥)− εn′(~p⊥ − ~q⊥)− ~ω~q − s~Ω−m~ω + i~δ] (t− t2) } − [nn′,~p⊥−~q⊥(t2)(N~q + 1)− nn,~p⊥(t2)N~q] 116 Ability to increase a weak electromagnetic wave by confined electrons in doped superlattices... exp { i ~ [εn(~p⊥)− εn′(~p⊥ − ~q⊥) + ~ω~q − s~Ω−m~ω + i~δ] (t− t2) } (2.8) If we consider a similar problem in normal bulk semiconductors that the authors V. L. Malevich and E. M. Epshtein published, we will see that Eq. (2.8) has a similarity to the quantum kinetic equation for electrons in the bulk semiconductor [8]. It is well known that to obtain explicit solutions from Eq. (2.8) is very difficult. In this paper, we use the first-order tautology approximation method to solve this equation [7, 8]. In detail, in Eq. (2.8), we choose the initial approximation of nn,~p⊥(t) as: n0n,~p⊥(t2) = n¯n,~p⊥, n 0 n,~p⊥+~q⊥ (t2) = n¯n,~p⊥+~q⊥, n 0 n,~p⊥−~q⊥(t2) = n¯n,~p⊥−~q⊥ where n¯n,~p⊥ is the balanced distribution function of electrons. We perform the integral with respect to t2, and then we perform the integral with respect to t of Eq. (2.8). The expression for the unbalanced electron distribution function can be written as: nn,~p⊥(t) = n¯n,~p⊥− 1 ~ ∑ n′,~q |C~q|2 |In,n′(qz)|2 +∞∑ k,s,r,m=−∞ Js(~a1~q⊥)Jk+s(~a1~q⊥)Jm(~a2~q⊥)Jr+m(~a2~q⊥) exp {−i {[kΩ + rω + iδ] t+ kϕ1}} kΩ + rω + iδ{ n¯n′,~p⊥−~q⊥N~q − n¯n,~p⊥(N~q + 1) εn(~p⊥)− εn′(~p⊥ − ~q⊥)− ~ω~q − s~Ω−m~ω + i~δ + n¯n′,~p⊥−~q⊥(N~q + 1)− n¯n,~p⊥N~q εn(~p⊥)− εn′(~p⊥ − ~q⊥) + ~ω~q − s~Ω−m~ω + i~δ − n¯n,~p⊥N~q − n¯n′,~p⊥+~q⊥(N~q + 1) εn′(~p⊥ + ~q⊥)− εn(~p⊥)− ~ω~q − s~Ω−m~ω + i~δ − n¯n,~p⊥(N~q + 1)− n¯n′,~p⊥+~q⊥N~q εn′(~p⊥ + ~q⊥)− εn(~p⊥) + ~ω~q − s~Ω−m~ω + i~δ } (2.9) where ~a1 = e ~E01 m∗Ω2 , ~a2 = e ~E02 m∗ω2 , ~E01 and Ω are the intensity and the frequency of a strong EMW (laser radiation), ~E02 and ω are the intensity and the frequency of a weak EMW, N~q is the balanced distribution function of phonons, ϕ1 is the phase difference between two electromagnetic waves and Jk(x) is the Bessel function. 117 Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau 2.1.3. Calculations of the absorption coefficient of a weak EMW in the presence of laser radiation modulated by amplitude in a DSL The carrier current density formula in a DSL takes the form: ~j⊥(t) = e~ m∗ ∑ n,~p⊥ ( ~p⊥ − e ~c ~A(t) ) nn,~p⊥(t). (2.10) Because the motion of electrons is confined along the z direction in a DSL, we only consider the in plane xOy current density vector of electrons ~j⊥(t). The ACF of a weak EMW by confined electrons in the presence of laser radiation modulated by amplitude in the DSL takes the simple form [7]: α = 8π c √ χ∞E202 〈 ~j⊥(t) ~E02 sinωt 〉 t . (2.11) Because the strong EMW (laser radiation) is modulated by amplitude, according to section (2.1.1), it is expressed by Eq. (2.2). According to the hypothesis, due to |∆Ω| ≪ Ω, then in a small amount of time there are about a few periods T = 2π Ω , we can presume that (∆Ωt +∆α) is changeless. Therefore, we let t get a certain specific value τ in such a small amount of time. Then, we have: E01 = √ F 21 + F 2 2 + 2F1F2 cos(∆Ωτ +∆α) = const;ϕ1 = α + α ′ = const. (2.12) From the Eqs. (2.9), (2.10), (2.11) and (2.12), we established the ACF of a weak EMW in the presence of laser radiation modulated by amplitude in DSL: α = n0ωpe 4 ~ω0 √ π√ 2χ∞(m∗kbT )3/2ε0cω3 ( 1 χ∞ − 1 χ0 ) +∞∑ n,n′=1 IIn,n′ {(D0,1 −D0,−1) −1 2 (H0,1 −H0,−1) + 3 32 (G0,1 −G0,−1)+1 4 (H−1,1 −H−1,−1 +H1,1 −H1,−1) − 1 16 (G−1,1 −G−1,−1 +G1,1 −G1,−1) + 1 64 (G−2,1 −G−2,−1 +G2,1 −G2,−1) } (2.13) 118 Ability to increase a weak electromagnetic wave by confined electrons in doped superlattices... where: Ds,m = e − ξs,m 2kBT ( 4m∗ 2 ξ2s,m ~4 )1/4 K1/2 ( |ξs,m| 2kBT )[ e − εn kBT (Nω0 + 1)− e− ε n′ −ξs,m kBT Nω0 ] Hs,m = a 2 1 ( 1 2 + 1 4 cos 2γ ) e − ξs,m 2kBT ( 4m∗ 2 ξ2s,m ~4 )3/4 K3/2 ( |ξs,m| 2kBT )[ e − εn kBT (Nω0 + 1)− e− ε n′ −ξs,m kBT Nω0 ] Gs,m = a 4 1 ( 3 8 + 1 4 cos 2γ ) e − ξs,m 2kBT ( 4m∗ 2 ξ2s,m ~4 )5/4 K5/2 ( |ξs,m| 2kBT )[ e − εn kBT (Nω0 + 1)− e− ε n′ −ξs,m kBT Nω0 ] εn = ~ωp ( n+ 1 2 ) , εn′ = ~ωp ( n′ + 1 2 ) , Nω0 = 1 e ~ω0 kBT − 1 , IIn,n′ = +∞∫ −∞ |In,n′(qz)|2 dqz a1 = eE01 m∗Ω2 , a2 = eE02 m∗ω2 , E01 = √ F 21 + F 2 2 + 2F1F2 cos(∆Ωτ +∆α) ξs,m = ~ωp (n ′ − n) + ~ω0 − s~Ω−m~ω,with s = −2,−1, 0, 1, 2;m = −1, 1. γ is the angle between the two vectors ~E01 and ~E02. ~F1 and ~F2 are the intensities of two laser radiations that creates laser radiation modulated by amplitude (with the intensity ~E01 and the frequency Ω). Equation (2.13) is the expression of the ACF of a weak EMW in the presence of external laser radiation modulated by amplitude in a DSL. From the expression of the ACF of a weak EMW, we see that ACF of a weak EMW is independent of E02 and dependent only on E01 , Ω, ω, T, d,Nd, nD. When ∆Ω = 0, the above results will come back the case of absorption of a weak EMW in the presence of non-modulated laser radiation. From epression (2.13), when we set E01 = 0, we will receive an expression of the ACF of a weak EMW in the absence of laser radiation in a DSL that has been investigated in [4] but by using the Kubo-Mori method. Expression (2.13) is similar to the expression of the ACF of a weak EMW in the presence of laser radiation in a quantum well that has been investigated in [10], but different from [10] in wave function, the energy spectrum and the electron form factor In,n′(qz), in addition to the laser radiation which in this case is modulated by amplitude. Here it is very difficult to calculate IIn,n′ = +∞∫ −∞ |In,n′(qz)|2 dqz by hand as in [10], so we have to program the calculation to be done on a computer. 119 Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau 2.2. Numerical results and discussion In order to clarify the mechanism for the absorption of a weak EMW in a DSL in the presence of laser radiation modulated by amplitude, in this section we will evaluate, plot and discuss the expression of the ACF for the case of a DSL with equal thickness dn = dp of the n-doped and p-doped layers, equal and constant doped concentration nD = nA: n-GaAs/p-GaAs. The parameters used in the calculations are as follows [4, 7]: χ∞ = 10.9, χ0 = 12.9, m = 0.067m0, m0 being the mass of free electron, d = 80 nm, n0 = 10 23m−3, ~ω0 = 36.25meV , γ = π3 . Figure 1. The dependence of α on T Figure 2. The dependence of α on ∆Ω Figure 1 describes the dependence of α on temperature T, with Nd = 15, nD = 1023m−3, Ω1 = 3.1013 Hz, Ω2 gets five different values: 2.6.1013 Hz, 3.1013 Hz, 3.4.1013 Hz, 3.8.1013 Hz, 4.1013 Hz; α1 = π 3 , α2 = π 6 , F1 = 10.10 6V/m, F2 = 15.10 6V/m, ω = 1013Hz. The five different values of Ω2 correspond to the five different values of ∆Ω: 0.4.1013 Hz, 0 Hz, -0.4.1013 Hz, -0.8.1013 Hz, -1013 Hz. Figure 1 shows that when the temperature T of the system rises from 30K to 400K, its ACF decreases, and then gradually increases to 0. From Figure 1 we also see that when T gets a value which is under 80K, the ACF of a weak EMW in the presence of non-modulated laser radiation is greater than one in the presence of laser radiation modulated by amplitude. This means that the absorption of a weak EMW is reduced when a strong EMW is modulated by amplitude and, when T gets a value which is over 100K, the ACF of a weak EMW gets values greater than one for the case of a non-modulated strong EMW. In addition, the ACF also gets negative values, i.e. the ACF of a weak EMW becomes increased coefficient of a weak EMW. So, when T gets a value which is over 100K, the ability to increase a weak EMW in the presence of laser radiation modulated by amplitude is decreased in comparison with that in the presence of non-modulated laser radiation; and when T gets the value about from 80K to 100K, the ACF of a weak EMW in the presence of laser radiation modulated by amplitude can be greater or smaller than one in the presence of non-modulated laser radiation. 120 Ability to increase a weak electromagnetic wave by confined electrons in doped superlattices... Figure 2 describes the dependence of α on∆Ω (|∆Ω| is the modulated frequency), and also with the above conditions and seven different values of T. From Figure 2 we see that the curves can have a maximum or a minimum in the investigative interval. Both Figures 1 and 2 show that in the high temperature region, the ACF is almost independent of ∆Ω, i.e. the amplitude modulation of laser radiation hardly affects the ability to increase a weak EMW in the presence of laser radiation. Figure 3. The dependence of α on Ω Figure 4. The dependence of α on ω Figure 3 describes the dependence of α on Ω (reduced frequency), with Nd = 15, nD = 10 23m−3, Ω1 = 2.5.1013Hz ÷ 8.1013Hz, ∆Ω = 0.5.1013Hz , α1 = π 3 , α2 = π 6 , F1 = 3.10 6V/m, F2 = 10.10 6V/m , ω = 1013Hz and five different values of T. The curves in this figure can have a minimum or no minimum in the investigative interval. Figure 4 describes the dependence of α on the frequency ω of the weak EMW, with Nd = 15, nD = 10 23m−3, Ω1 = 3.1013Hz, α1 = π 3 , α2 = π 6 , and five different values of Ω2 corresponding to the five different values of∆Ω, ω = 0.5.10 13Hz ÷ 20.1013Hz. This figure includes two subplots: the first subplot with F1 = 4.10 5V/m, F2 = 8.10 5V/m, T = 30K, the second subplot with F1 = 10.10 6V/m, F2 = 15.10 6V/m, T = 90K. From Figure 4 we see that the curves in the first subplot have a maximum where ω = ω0 while the curves in the second subplot have no maximum and can have a minimum or no minimum in the investigative interval. Figure 5 shows the ACF as a function of the number of DSL period Nd, with Ω1 = 3.10 13Hz, nD = 10 23m−3, α1 = π 3 , α2 = π 6 , F1 = 6.10 6V/m, F2 = 10.10 6V/m, T = 100K, ω = 1013Hz and five different values ofΩ2 corresponding to the five different values of ∆Ω. From this figure, we see that when ∆Ω = 0Hz, the ACF gets a negative value and smaller than one for the cases of ∆Ω = 0.4.1013Hz,−0.8.1013Hz,−1013Hz, and greater than one for the case of ∆Ω = −0.4.1013Hz. So, the ability to increase a weak EMW is enhanced when laser radiation is modulated by amplitude with ∆Ω = −0.4.1013Hz; and when laser radiation is modulated by amplitude with ∆Ω = 121 Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau 0.4.1013Hz,−0.8.1013Hz,−1013Hz, the ability to increase a weak EMW is not enhanced in comparison with the case of non-modulated laser radiation (∆Ω = 0Hz), but is decreased. Figure 5. The dependence of α on Nd Figure 6. The dependence of α on nD Figure 6 describes the dependence of α on the Nd, with Nd = 15, Ω1 = 3.10 13Hz , α1 = π 3 , α2 = π 6 , F1 = 6.10 6V/m, F2 = 10.10 6V/m , T = 100K, ω = 1013Hz and five different values of Ω2 corresponding to the five different values of ∆Ω. From this figure, we also see that the ability to increase a weak EMW is enhanced when laser radiation is modulated by amplitude with ∆Ω = −0.4.1013Hz, and when laser radiation is modulated by amplitude with ∆Ω = 0.4.1013Hz,−0.8.1013Hz,−1013Hz, the ability t