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

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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.

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

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 113-123 This paper is available online at http://stdb.hnue.edu.vn ABILITY TO INCREASE A WEAK ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN DOPED SUPERLATTICES IN THE PRESENCE OF LASER RADIATION MODULATED BY AMPLITUDE Nguyen Thi Thanh Nhan1, Nguyen Vu Nhan2 and Nguyen Quang Bau1 1 Faculty of Physics, College of Natural Sciences, Hanoi National University 2 Faculty 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) = F~1 (t) + F~2 (t) = F~1 sin (Ω1 t + α1 ) + F~2 sin (Ω2 t + α2 ) (2.1) where, F~1 and F~2 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) = E ~ 01 sin (Ωt + ϕ1 ) (2.2) p with E01 = F12 + F22 + 2F1 F2 cos(∆Ωt + ∆α), ∆Ω = Ω1 − Ω2 , ∆α = α1 − α2 , Ω1 + Ω2 ′ α1 + α2 ′ F1 − F2 ∆Ω ∆α Ω= , ϕ1 = α + α , α = , tgα = tg t+ 2 . 2 2 F1 + F2 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. E ~ 01 is the intensity of the laser radiation modulated by amplitude. F~1 F~2 1 F~ In the case that F~1 , F~2 , Ω1 , Ω2 , Ω satisfy the conditions: 2 = 2 = , and Ω1 Ω2 2 Ω2 ∆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: X e ~ + X H= εn p~⊥ − A(t) an,~p⊥ an,~p⊥ + ~ω~qb+ q b~ ~ q ~c n,~ p⊥ q ~ X + C~qIn,n′ (qz )a+ n′ ,~ p⊥ +~q⊥ an,~p⊥ (b~ + q + b−~q) (2.3) n,n′ ,~ p⊥ ,~ q 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+ p⊥ and an,~ n,~ + p⊥ , (b~ q and b~q ) are the creation and the annihilation operators of the electron (phonon), respectively; ~ c ~ c~ ~q = (~q⊥ , qz ); A(t) = E 01 cos(Ωt + ϕ1 ) + E02 cos(ωt) is the vector potential of Ω ω the EMW field (including two EMWs: a strong EMW with the intensity E ~ 01 and the frequency Ω; a weak EMW with the intensity E ~ 02 and the frequency ω); ω~q ≈ ω0 is the frequency of an optical phonon. C~q is the electron-optical phonon interaction constant [7]: 2 2πe2 ~ω0 1 1 |C~q| = − (2.4) V ε0 q 2 χ∞ χ0 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: Nd Z X d In,n′ (qz ) = eiqz z ψn (z − ld)ψn′ (z − ld)dz (2.5) l=1 0 In a DSL, the electron energy takes the simple form: 115
Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau ~2 p~2⊥ 1 εn (~p⊥ ) = + ~ωp n + (2.6) 2m∗ 2 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 1/2 4πe2 nD number of the DSL period, ωp = is the frequency plasma caused by donor χ0 m∗ 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 D of the electron E particle number + operator (or electron distribution function) nn,~p⊥ (t) = an,~p⊥ an,~p⊥ [7]: t ∂nn,~p⊥ (t) Dh + iE i~ = an,~p⊥ an,~p⊥ , H (2.7) ∂t t where hψit denotes a statistical average value at moment t and hψit = T r(W ˆ with W ˆ ψ), ˆ 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) 1 X 2 2 X =− 2 |C~q| |In,n′ (qz )| Jl (~a1 ~q⊥ )Js (~a1 q~⊥ )Jm (~a2 q~⊥ )Jf (~a2 ~q⊥ ) ∂t ~ ′ l,s,m,f =−∞ n ,~ q × exp {i {[(s − l)Ω + (m − f )ω − iδ] t + (s − l)ϕ1 }} Zt × dt2 {[nn,~p⊥ (t2 )N~q − nn′ ,~p⊥ +~q⊥ (t2 )(N~q + 1)] −∞ i exp [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] i exp [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)] i exp [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... i exp [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⊥ , n0n,~p⊥ +~q⊥ (t2 ) = n ¯ n,~p⊥ +~q⊥ , n0n,~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: +∞ 1X X nn,~p⊥ (t) = n ¯ n,~p⊥ − |C~q|2 |In,n′ (qz )|2 Js (~a1 ~q⊥ )Jk+s (~a1 ~q⊥ )Jm (~a2 q~⊥ )Jr+m (~a2 q~⊥ ) ~ ′ n ,~ q k,s,r,m=−∞ 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 − (2.9) εn′ (~p⊥ + ~q⊥ ) − εn (~p⊥ ) + ~ω~q − s~Ω − m~ω + i~δ ~ 01 eE eE~ 02 where ~a1 = , ~ a 2 = ~ 01 and Ω are the intensity and the frequency of a strong ,E m∗ Ω2 m∗ ω 2 EMW (laser radiation), E ~ 02 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: X e ~ ~j⊥ (t) = e~ p~⊥ − A(t) nn,~p⊥ (t). (2.10) m∗ ~c n,~ p⊥ 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]: 8π D E α= √ ~ 02 sin ωt . ~j⊥ (t)E (2.11) 2 c χ∞ E02 t 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, 2π then in a small amount of time there are about a few periods T = , we can presume Ω 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: q ′ E01 = F12 + F22 + 2F1 F2 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: √ X +∞ n0 ωp e4 ~ω0 π 1 1 α= √ − IIn,n′ {(D0,1 − D0,−1 ) 2χ∞ (m∗ kb T )3/2 ε0 cω 3 χ∞ χ0 n,n′=1 1 3 1 − (H0,1 − H0,−1 ) + (G0,1 − G0,−1 )+ (H−1,1 − H−1,−1 + H1,1 − H1,−1 ) 2 32 4 1 1 − (G−1,1 − G−1,−1 + G1,1 − G1,−1 ) + (G−2,1 − G−2,−1 + G2,1 − G2,−1 ) 16 64 (2.13) 118
Ability to increase a weak electromagnetic wave by confined electrons in doped superlattices... where: !1/ 22 4m∗ ξs,m 4 ξ − 2ks,mT |ξs,m | − kεnT − ε ′ −ξs,m n Ds,m = e B 4 K1 e B (Nω0 + 1) − e kB T N ω0 ~ /2 2kB T !3/ 22 4m∗ ξs,m 4 1 1 ξ − s,m |ξs,m | − kεnT ε ′ −ξs,m n a21 − Hs,m = + cos 2γ e 2kB T K3 e B (Nω0 + 1) − e kB T N ω0 2 4 ~4 /2 2kB T !5/ 22 4m∗ ξs,m 4 3 1 ξ − s,m |ξs,m | − kεnT ε ′ −ξs,m n a41 − Gs,m = + cos 2γ e 2kB T K5 e B (Nω0 + 1) − e kB T N ω0 8 4 ~4 /2 2kB T Z+∞ 1 1 1 εn = ~ωp , εn′ = ~ωp n′ + n+ , Nω0 = ~ω0 , IIn,n′ = |In,n′ (qz )|2 dqz 2 2 e kB T − 1 −∞ q eE01 eE02 a1 = ∗ 2 , a2 = ∗ 2 , E01 = F12 + F22 + 2F1 F2 cos(∆Ωτ + ∆α) mΩ mω ′ 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 E ~ 01 and E ~ 02 . F~1 and F~2 are the intensities of two laser radiations that creates laser radiation modulated by amplitude (with the intensity E ~ 01 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. R +∞ 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 = 1023 m−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 = 10 m−3 , Ω1 = 3.1013 Hz, Ω2 gets five different values: 2.6.1013 Hz, 3.1013 Hz, 3.4.1013 23 Hz, 3.8.1013 Hz, 4.1013 Hz; α1 = π3 , α2 = π6 , F1 = 10.106 V /m, F2 = 15.106 V /m, ω = 1013 Hz. 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, π π nD = 1023 m−3 , Ω1 = 2.5.1013 Hz ÷ 8.1013 Hz, ∆Ω = 0.5.1013 Hz , α1 = , α2 = , 3 6 F1 = 3.106 V /m, F2 = 10.106 V /m , ω = 1013 Hz 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 π π Nd = 15, nD = 1023 m−3 , Ω1 = 3.1013 Hz, α1 = , α2 = , and five different values of 3 6 Ω2 corresponding to the five different values of ∆Ω, ω = 0.5.1013 Hz ÷ 20.1013 Hz. This figure includes two subplots: the first subplot with F1 = 4.105 V /m, F2 = 8.105 V /m, T = 30K, the second subplot with F1 = 10.106 V /m, F2 = 15.106 V /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 π π Ω1 = 3.1013 Hz, nD = 1023 m−3 , α1 = , α2 = , F1 = 6.106V /m, F2 = 10.106 V /m, 3 6 T = 100K, ω = 1013 Hz and five different values of Ω2 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 = 0.4.1013 Hz, −0.8.1013 Hz, −1013 Hz, and greater than one for the case of ∆Ω = −0.4.1013 Hz. So, the ability to increase a weak EMW is enhanced when laser radiation is modulated by amplitude with ∆Ω = −0.4.1013 Hz; and when laser radiation is modulated by amplitude with ∆Ω = 121
Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau 0.4.1013 Hz, −0.8.1013 Hz, −1013 Hz, 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 = 3.1013 Hz π π , α1 = , α2 = , F1 = 6.106 V /m, F2 = 10.106V /m , T = 100K, ω = 1013 Hz 3 6 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 = −0.4.1013 Hz, and when laser radiation is modulated by amplitude with ∆Ω = 0.4.1013 Hz, −0.8.1013 Hz, −1013 Hz, 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. From dependence of the ACF of a weak EMW on the doped concentration nD of DSL, if we measure the value of the ACF of a weak EMW by experimental means, we can infer a value of doped concentration nD (of DSL) which corresponds to that value of the ACF. These figures show that under the influence of laser radiation, the ACF of a weak EMW in a DSL can get negative values, and it means that the ACF becomes an increased coefficient. So, due to the presence of a strong EMW, in some conditions the weak EMW is increased. This is different from a situation in which there is an absence of laser radiation. In addition, if a strong EMW is modulated by amplitude, in some conditions the ability to increase a weak EMW can be enhanced. 3. Conclusion In this paper, we investigated the ability to increase a weak EMW by confined electrons in a DSL in the presence of laser radiation modulated amplitude. We obtained an analytical expression of the ACF of a weak EMW in the DSL. This expression shows that the ACF of a weak EMW is independent of E02 and dependent only on E01 , Ω, ω, 122
Ability to increase a weak electromagnetic wave by confined electrons in doped superlattices... T, d, Nd , nD . We numerically calculated and graphed the ACF for n-GaAs/p-GaAs DSL to clarify. These results showed that due to the presence of a strong EMW modulated by amplitude, in some conditions, the weak EMW is increased and this ability to increase a weak EMW can also be enhanced in some conditions in comparison with the case of non-modulated laser radiation. Acknowledgements. This research which has been completed thanks to the financial support of the Program of Basic Research in Natural Science of VNU and NAFOSTED of Vietnam (Number 103.01-2011.18), TN-12-07. REFERENCES [1] Tsu, R. and L. Esaki, 1973. Tunneling in a finite superlattice. Appl. Phys. Lett., Vol. 22, No. 11, pp. 562-564. [2] Harris, Jr., J. S, 1990. From bloch functions to quantum wells. J. Mod. Phys. B, Vol. 4, pp. 1149-1179. [3] Bau, N. Q. and T. C. Phong, 1998. Calculations of the absorption coefficient of a weak electromagnetic wave by free carriers in quantum wells by the Kubo-Mori method. J. Phys. Soc. Jpn., Vol. 67, pp. 3875-3880. [4] Bau, N. Q., N. V. Nhan and T. C. Phong, 2002. Calculations of the absorption coefficient of a weak Electromagnetic wave by free carriers in doped superlattices by using the Kubo-Mori Method. J. Korean. Phys. Soc., Vol. 41, pp. 149-154. [5] Bau, N. Q. and H. D. Trien, 2011. The nonlinear absorption of a strong electromagnetic wave in low-dimensional systems. Wave propagation, Ch. 22, pp. 461-482, Intech. [6] Bau, N. Q., L. T. Hung and N. D. Nam, 2010. The nonlinear absorption coefficient of a strong electromagnetic wave by confined electrons in quantum wells under the influences of confined phonons. Journal of Electromagnetic Waves and Applications, Vol. 24, No. 13, pp. 1751-1761. [7] Pavlovich, V. V. and E. M. Epshtein, 1977. Quantum theory of absorption of electromagnetic wave by free carries in semiconductors. Sov. Phys. Solid State, Vol. 19, pp. 1760-1764. [8] Malevich, V. L. and E. M. Epshtein, 1974. Nonlinear optical properties of conduction electrons in semiconductors. Sov. Quantum Electronic, Vol. 1, pp. 1468-1470. [9] Bau, N. Q., Chhoumn Navy, 1997. Influence of laser radiation (non-modulated and modulated) on the absorption of a weak electromagnetic wave by free electrons in semiconductor superlattices. Journal of Science, Nat. Sci. (VNU) T. 13, No. 2, pp. 26-31. [10] Nhan, N. V., N. T. T. Nhan, N. V. Nghia, S. T. L. Anh and N. Q. Bau, 2012. Ability to increase a weak electromagnetic wave by confined electrons in quantum wells in the presence of laser radiation. PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27-30, pp. 1054-1059. 123