DSpace at VNU: The quantum acoustoelectric current in a doped superlattice GaAs:Si GaAs:Be

DSpace at VNU: The quantum acoustoelectric current in a doped superlattice GaAs:Si GaAs:Be tài liệu, giáo án, bài giảng...

Superlattices and Microstructures 63 (2013) 121–130 Contents lists available at ScienceDirect Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices The quantum acoustoelectric current in a doped superlattice GaAs:Si/GaAs:Be Nguyen Quang Bau a,⇑, Nguyen Van Hieu a,b a b Faculty of Physics, Hanoi University of Science, Vietnam National University, 334-Nguyen Trai, Thanh Xuan, Hanoi, Viet nam Faculty of Physics, Danang University of Education, 459-Ton Duc Thang, Danang, Viet nam a r t i c l e i n f o Article history: Received 13 February 2013 Received in revised form 24 August 2013 Accepted 26 August 2013 Available online September 2013 Keywords: Doped superlattice Quantum acoustoelectric current Electron-external acoustic wave interaction Electron-acoustic phonon scattering Quantum kinetic equation a b s t r a c t The quantum acoustoelectric (QAE) current is studied theoretically in a doped superlattice (DSL) The physical problem is investigated in the region ql ) (where q is the acoustic wave number and l is the electrons mean free path) We obtain analytical expression for the QAE current in the DSL by using the quantum kinetic equation for electron-external acoustic wave interaction and electronacoustic phonon (internal acoustic wave) scattering A nonlinear dependence of the QAE current on the frequency of external acoustic wave x~q , the temperature T of the system and the characteristic parameters of DSL is achieved The computational results for a specific GaAs:Si/GaAs:Be DSL indicates that the existent peaks in the DSL may be due to the transition between mini-bands n ? n0 All these results are compared with those for normal bulk semiconductors to show the differences Finally, the quantum theory of the QAE current in the DSL is newly developed Ó 2013 Elsevier Ltd All rights reserved Introduction When an external acoustic wave is absorbed by a conductor, the transfer of the momentum from the acoustic wave to the conduction electron may give rise to a current that is usually called the acoustoelectric (AE) current, in the case of an open circuit called AE field The study of this effect is crucial because of the complementary role it may play in the understanding of the properties of bulk semiconductors and low-dimensional systems (quantum wells, superlattices, quantum wires, etc.), which, we believe, should find an important place in the acoustoelectronic devices ⇑ Corresponding author E-mail address: nguyenquangbau54@gmail.com (N.Q Bau) 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.spmi.2013.08.026 122 N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121–130 In low-dimensional systems, the energy levels of electrons become discrete and to be different from other dimensionalities [1] Under certain conditions, the decrease in dimensionality of the system for semiconductors can lead to dramatically enhanced nonlinearities [2] Thus the nonlinear properties, especially electrical and optical properties of semiconductor quantum wells (QWs), compositional superlattices (CSLs), quantum wires, and quantum dots (QDs) have attracted much attention in the past few years For example, the linear absorption of a weak electromagnetic wave caused by confined electrons in low-dimensional systems has been investigated [3–5] Calculations of the nonlinear absorption coefficients of an intense electromagnetic wave by using the quantum kinetic equation for electrons in bulk semiconductors [6], in quantum wells [7] and in quantum wires [8] have also been reported Also, the AE effect has been studied in detail both theoretically and experimentally in bulk semiconductors [9–15] In recent years, the AE effect in low-dimensional structures has been extensively studied both theoretically and experimentally So far, however, almost all those works [16–23] have been studied theoretically by using the Boltzmann classical kinetic equation method, and are, thus, limited to the case of the electron-external acoustic wave interaction at high temperature So, a macroscopic approach to the description of the acoustoelectric effect is inapplicable and the problem should be treated by using quantum methods The AE effect in CSLs [16–18], the AE current in one-dimensional channel [19], the AE effect in a finite-length ballistic quantum channel [20], the AE current in a ballistic quantum point contact [21], the AE current through a quantum wire containing a point impurity, the AE current in submicron-separated quantum wires [22,23] In addition, the AE effect has been studied experimentally in a CSL [24] and in a QW, SL [25,26] However, the calculation of the QAE current in a DSL by using the quantum kinetic equation method is still open for study Throughout [6–8], the quantum kinetic equation method have been seen as a powerful tool So, in a recent work [27] we have used this method to calculate the quantum acoustomagnetoelectric field in a QW with a parabolic potential In the present work, we use the quantum kinetic equation method for electron-external acoustic wave interaction and electronacoustic phonon (internal acoustic wave) scattering in the DSL to study the QAE current The present work is different from previous works [16–24] because (1) the QAE current is a result of not only the electron-external acoustic wave interaction but also the electron-acoustic phonon scattering in the sample, (2) we use the quantum kinetic equation method, (3) we show that the dependence of QAE current on the frequency of external acoustic wave x~q , the temperature T of system and the characteristic parameters of DSL is nonlinear, (4) we show that the present results can explain the experimental results [26] This paper is organized as follows: In Section 2, we outline the quantum kinetic equation for electron-external acoustic wave interactions and electron-acoustic phonon scattering in the DSL The analytical expression for the QAE current in Section The numerical results and a brief discussions are presented for specific DSL of GaAs:Si/GaAs:Be in Section Finally, we present conclusions in Section Quantum kinetic equation for electrons in a doped superlattice 2.1 Electronic structure in a doped superlattice The superlattice potential in DSLs is created solely by the spatial distribution of the charge We use a simple model for a DSL, in which an electron gas is confined by the superlattice potential along the Oz direction and electrons are free on the (x–y) plane Consider a compensated n-p DSL with equal thicknesses dn = dp = d/2 of the n-doping and the p-doping layers (d is the period of the DSL) and equal constant doping concentrations nd = na = nD in the respective layers (nd and na are the donor and acceptor doping concentrations, respectively, nD is the doping concentration) The eigenfunction and eigenvalues of an unperturbed electron is expressed [28] We will restrict in the following to the case of homogeneous doping as shown in Fig The space-charge potential of the impurities Ui(z) is obtained by integrating Poisson’s equation N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121–130 123 Fig n-i-p-i Crystal with constant impurity concentration in the n- and p-type layers (a) Periodic doping profile and (b) modulation of conduction- and valence- band edge d U i zị dz ẳ 4pe2 v0 ẵnd zị na zị; and the boundary condition dU i zị ẳ U i 0ị ẳ 0; dz with v0 is the static dielectric constant (permittivity) of the semiconductor Ui(z) consists of parabolic parts in doping layers U i zị ẳ 2 < 2pe nd z ; jzj dn =2 v0 : 2U À 2pe2 na z2 ðd=2 À jzjÞ2 ; ðd=2 À jzjÞ dp =2; v and linear parts in the intrinsic regions U i zị ẳ 2pe2 nD z2 v0 ðjzj À dn =4Þ; dn =2 jzj d À dn ; here 2U0 is the maximum height of Ui(z), ! pe2 nd d2n na dp U0 ẳ ỵ ỵ nd dn di : v0 4 And together with the effective mass approximation this formalism lead to a Schrodinger-type equation of the form " #  h ỵ Uzị /n;~p ~ rị ẳ en ~ pị/n;~p ~ rị; 2m and    p  py /n;~p ~ rị ẳ pffiffiffi wn;pz ðzÞ Â exp i x x exp i y ; h   h A ð1Þ 124 N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121–130 where (px, py) is the electron momentum vector in (x, y) plane, n denotes quantization of the energy spectrum, m and e are the effective mass and charge of the electron, respectively, A is the normalization area and wn;pz ðzÞ is the superlattice Bloch function of subband n Here, Uzị ẳ U i zị ỵ U H zị ỵ U xc ðzÞ: UH(z) the Hartree contribution of the electron to the self-consistent potential is given by solution of Poisson’s equation d U H zị dz ẳ 4pe2 v0 nðzÞ; with n(z) is the electron density, Uxc is the local exchange and correction potential is obtained [28] U xc zị ẳ exc nzịị ỵ nzịdexc =dnị; where exc((n(z)) is the exchange and correlation energy per electron of a homogeneous electron gas of (local) density n Since the potential does not depend on x, y and is periodic in z with the periodicity d We can write for the eigenvalues en;~p?  en ~ pị ẳ en pz ị þ p2? ; 2m ~ py Þ ? Oz and parallel to the doping layer, the Bloch functions wn;pz ðzÞ and the eigenvalues p? ¼ ð~ px ; ~ en(pz) are to be determined self consistently, " À #  h 52 ỵ Uzị wn;pz zị ẳ en pz Þwn;pz ðzÞ: 2m ð2Þ In the most cases of interest only the lowest electronic sub-bands are partially occupied For these low subband and typical doping parameters an extreme tight-binding approach is correct The Bloch function can be written X wn;pz zị ẳ p expipz jd=hịwn z jdị; Nsl j here, Nsl is the number of superlattice periods in the crystal en(pz)  en is a eigenvalue for a singer potential well From Eq (2) when it ignores Hatree and exchange energies Uzị ẳ U i zị ẳ 2pe2 nD z2 v0 ¼ mx2p z2 ; with xp ¼  1=2 4pe2 nD : v0 m Because the eigenfunction wn(z) does not depend on x, y We can write for the 52 = d2/dz2 Substitution of U(z) back into Eq (2) gives d wn;pz ðzÞ d z ỵ 2m h   en mx2p z2 wn;pz zị ẳ 0: It is convenient to introduce the dimensionless variable n by the definition n ẳ the derivative to give d wn nị d n ỵ 2k ỵ n2 wn nị ẳ 0; here we dene k by the relation Àmxp Á1=2 h  z and then take ð2:aÞ 125 N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121130 2k ỵ ẳ 2en :  xp h We first investigate the asymptotic behavior of wn(n) For large value of n the constant 2k + may be neglected in comparison with n2 and Eq (2.a) become d wn ðnÞ d n À n2 wn nị ẳ 0: The approximate solutions are wn nị ẳ C expn2 =2ị ỵ C expn2 =2Þ; The function exp(n2/2) is not a satisfactory solution because it becomes infinite as n ? ± ,but the function exp(Àn2/2) is well-behaved This asymptotic behavior of wn(n) suggests that a satisfactory solution of Eq (2.a) has the form wn nị ẳ expn2 =2ịf nị; where f(n) is a function to be determined Substitution of wn(n) into Eq (2.a) gives f 00 nị 2nf nị ỵ 2kf nị ẳ 0: 2:bị We solve Eq (2.b) by the series solution method f nị ẳ X an nn ; f nị ẳ nẳ0 X an n nn1 ; nẳ0 f 00 nị ẳ X an nn 1ị nn2 ; nẳ0 Substitution into Eq (2.b) we obtain anỵ2 2n kị ẳ ; n ỵ 2ịn ỵ 1ị an as n ? ±1 to a finite polynomial f(n) Therefore, an – and an + = Since the parameter k is equal to a positive integer n en ¼ hxp n ỵ 1=2ị; The electron energy spectrum separate two part and it takes the form en;~p?  en ð~ pị ẳ p2? ỵ hxp n ỵ 1=2ị: 2m 3ị 2.2 Hamiltonian of the electron-external phonon and electron-acoustic phonon system in a DSL Let us suppose that an external acoustic wave of frequency x~q is propagating perpendicular to the Oz axis of the DSL We consider the most realistic case from the point of view of a low-temperature x~q =g ¼ cs q=g ( and ql ) 1; ð4Þ where g is the frequency of the electron collisions, cs is the velocity of the acoustic wave, q is the modulus of the external acoustic wave-vector and l is the electrons mean free path If the conditions (4) are satisfied, a macroscopic approach to the description of the acoustoelectric effect is inapplicable and the problem should be treated by using quantum method The acoustic wave will be considered as a packet of coherent phonons with the d-function distribution in ~ k-space N~ kị ẳ 2pị Ud~ k ~ qị, where x~q cs U is the flux density of the external acoustic wave with frequency x~q In the presence of an external acoustic wave with frequency x~q , the Hamiltonian of the electron-external phonon interaction and electron-acoustic phonon scattering system in a DSL in second quantization representation can be written as (we select ⁄ = 1) 126 N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121–130 H ẳ H0 ỵ Heph ; X Heph ẳ X X n;~ p? ~ k en ~ p? ịaỵn;~p? an;~p? ỵ H0 ẳ x~k b~ỵk b~k ; X qịaỵn;~p? ỵ~q? an0 ;~p? c~q expix~q tị ỵ C~q U n;n0 ~ n;~ p? ;n0 ;~ q 5ị kz ịaỵn;~p D~k In;n0 ~ ~ ? ỵk? n;~ p? ;n0 ;~ k þ Á an0 ;~p? b~k þ bÀ~k ; ð6Þ À þ Á where aþ and an;~p? b~k and b~k are the creation and the annihilation operators of the electron n;~ p? (acoustic phonon), respectively, c~q is the annihilation operator of the external phonon jn; ~ p? i and k? i are electron states before and after scattering U n;n0 ð~ jn0 ; ~ qÞ is the matrix element of the operp? ỵ ~ ator U = exp(iqy klz) We evaluate the matrix elements of the operator U We obtain qị ẳ U n;n0 ~ 1ịnỵn expkl dị kl d ỵ nỵn0 ị2 p ðÀ1ÞnÀn expðÀkl dÞ À 2 kl d þ ðnÀnk dÞ p kl d ð7Þ ; l  1=2 where kl ¼ q2 À x2q =c2l is the spatial attenuation factor of the potential part of the displacement field, C~q and D~k are the electron-external phonon and the electron-acoustic phonon interaction factors, respectively and take the form  1=2 C~q ¼ iKc2l x~3q =2q0 NS ; À rl ¼ À c2s =c2l Á1=2 ; N¼q À þ r2l þ 2rl rt ¼ À c2s =c2t 1=2 ;   ! rl ỵ r2t ; À2 rt 2r t jD~k j2 ¼ k K2~ : 2q0 cs ð8Þ Here, K is the deformation potential constant, cl and ct are the velocities of the longitudinal and the transverse bulk acoustic waves, respectively, cs is the velocity of the acoustic wave, q0 is the mass density of the medium, S = LxLy is the surface area, and In0 ;n kz ị ẳ N sl Z X jẳ1 d expðikz dÞwn ðz À jdÞwn0 ðz À jdÞdz: ð9Þ 2.3 Quantum kinetic equation for electrons in a DSL In order to establish the quantum kinetic equation for electron-external acoustic wave interaction and electron-acoustic phonon scattering in a DSL, we use the electron distribution function fn;~p? ẳ haỵ a p? i t n;~ p? n;~ i   Dh iE @fn;~p? ẳ aỵn;~p? an;~p? ; H ; t @t AE ð10Þ cW c is the density b Þð W where hWit denotes a statistical average value at the moment t; hWit ¼ Trð W matrix operator) Starting from, Eqs (5),(6) and (10), and realizing operator algebraic calculations, we obtain the quantum kinetic equation for electron-external acoustic wave interaction and electron-acoustic phonon scattering in a DSL   X È @fn;~p? qịj2 N~ qị ẵf en;~p? ị f en0 ;~p? ỵ~q? ịden0 ;~p? ỵ~q? en;~p? x~q Þ ¼ Àp jC~q j2 jU n;n0 ð~ @t AE n ;~ q ỵ f en;~p? ị f en0 ;~p? ỵ~q? ị den0 ;~p? ỵ~q? en;~p? ỵ x~q ị ỵ f en;~p? ị f en0 ;~p? ~q? ịden0 ;~p? ~q? en;~p? ỵ x~q ị ẫ ỵ f en;~p? ị f ðen0 ;~p? À~q? ފdðen0 ;~p? À~q? À en;~p? À x~q Þ n X kÞ ½f ðen;~p? Þ À f ðen0 ;~p? ỵ~k? ịden0 ;~p? ỵ~q? en;~p? ỵ x~q x~k ị ỵ p jD~k j2 jIn;n0 kz ịj2 N~ h n0 ;~ k ỵ f en;~p? ị f ðen0 ;~p? À~k? ފdðen0 ;~p? À~k? À en;~p? À x~q ỵ x~k ị o 11ị N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121–130 127 Eq (11) is fairly general and can be applied for any mechanism of the interaction In the case of vanishing electron-acoustic phonon (internal acoustic wave) scattering, Eq (11) gives the same results as those obtained [16–18] Quantum acoustoelectric current in a DSL The external acoustic wave of frequency x~q is assumed to propagate perpendicular to the Oz axis of the DSL After a new equilibrium has been established, the distribution function fn;~p? of the electrons will obey the condition @fn;~p? =@t ẳ @fn;~p? =@tịAE ỵ @fn;~p? =@tịth ẳ 0; 12ị where ð@fn;~p? =@tÞAE is the rate of change caused by the electron-external acoustic wave interaction and electron-acoustic phonon (internal acoustic wave) scattering, and ð@fn;~p? =@tÞth is the rate of change due to the interaction of the electron with thermal phonons, impurities, and one another Substituting Eq (11) into Eq (12) we obtain the basic equation of the problem ð@fn;~p? =@tÞth ẳ p X ẩ qịj2 N~ qị ẵf en;~p? ị f en0 ;~p? ỵ~q? ị den0 ;~p? ỵ~q? À en;~p? À x~q Þ jC~q j2 jU n;n0 ð~ q n;n0~ ỵ f en;~p? ị f en0 ;~p? ỵ~q? ịden0 ;~p? ỵ~q? n;~p? ỵ x~q ị ỵ f n;~p? ị f en0 ;~p? ~q? ị den0 ;~p? ~q? en;~p? ỵ x~q ị ẫ ỵ f en;~p? ị f en0 ;~p? À~q? ފdðen0 ;~p? À~q? À en;~p? À x~q Þ n X kị ẵf en;~p? ị f en0 ;~p? þ~k? ފdðen0 ;~p? þ~q? À en;~p? þ x~q À x~k Þ À p jD~k j2 jIn;n0 ðkz Þj2 Nð~ h n;n0~ k o ỵ f en;~p? ị f en0 ;~p? ~k? ịden0 ;~p? ~k? en;~p? x~q ỵ x~k ị : 13ị h e l i ỵ is We linearize Eq (13) by replacing f ðen;~p? Þ with fF en;~p? ị ỵ f1 , where fF en;~p? Þ ¼ 1= ðexp n;~kp?B T the equilibrium Fermi contribution function, l is the chemical potential, at absolute zero the chemical potential is equal to the Fermi energy As indicated in [29], @fn;~p? =@tịth ẳ f1 =s~p ; s~p is the momentum relaxation time Thus, X È qÞj2 Nð~ qÞ ẵfF en;~p? ị fF en0 ;~p? ỵ~q? ị den0 ;~p? ỵ~q? en;~p? x~q ị f1 ẳ Àps jC~q j2 jU n;n0 ð~ n0 ;~ q  þ fF ðen;~p? Þ À fF ðen0 ;~p? þ~q? ފdðen0 ;~p? ỵ~q? en;~p? ỵ x~q ị ỵ fF ðen;~p? Þ À fF ðen0 ;~p? À~q? ފdðen0 ;~p? À~q? en;~p? ỵ x~q ị ẫ ỵ fF en;~p? Þ À fF ðen0 ;~p? À~q? ފdðen0 ;~p? À~q? À en;~p? x~q ị n X kị ẵfF en;~p? ị f en0 ;~p? ỵ~k? ịden0 ;~p? ỵ~q? en;~p? ỵ x~q x~k ị jD~k j2 jIn;n0 kz Þj2 Nð~ À ps  n0 ;~ k h o þ fF ðen;~p? Þ À fF ðen0 ;~p? À~k? ފdðen0 ;~p? ~k? en;~p? x~q ỵ x~k ị : ð14Þ q is expressed The density of the QAE current jQAE in the direction of the external acoustic wave vector ~ j QAE ẳ X 2e n 2pị2 Z v~p f1 d~p? ; ð15Þ where v~p is the electron velocity given by v ~p ¼ @ en;~p =@~ p Substituting Eq (14) into Eq (15) and taking s~p to be constant, we obtain for the QAE current j QAE "  1=2 # X n ỵ 1=2ị 4pe2 nD Bỵ À BÀ Þ jU n;n0 j2 exp À kB T v0 m n;n0 "  1=2 # X n ỵ 1=2ị 4pe2 nD C ỵ C ị; ỵ A2 jIn;n0 j2 exp À kB T v0 m n;n0 ¼ A1 ð16Þ 128 N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121130 where A1 ẳ 2pị2 eU^2 sc4l x~2q q0 c s Bặ ẳ 1ỵ D2ặ mkB T  exp ! exp À l kB T  ; A2 ẳ ! D2ặ ; 2mkB T 2pị2 e^2 s2mkB T pị 2pị5 q0 cs mx~q Dặ ẳ 1=2 Cặ ẳ mDn;n0 ặ x~k ị2 p1=2 exp2bặ cị 4c3=2 aặ ẳ   mkB T ặ Dn;n0 ặ x~k Dn;n0 Ỉ x~k ; exp À mDn;n0 Ỉ mx~k 2kB T Dn;n0 ¼  1=2 4pe2 nD ðn À n0 ị; v0 m jIn;n0 j2 ẳ  1=2 exp l kB T  ; q mDn;n0 mðx~k À x~q ị ỵ ặ ; q q 1=2 ị ẵ2c ỵ 2aặ bặ cị Z bặ ẳ 1=2 ỵ aặ ỵ bặ K 5=2 ẵ2bặ cị 4c mDn;n0 ặ mx~k ị2 ; 2mK B T cẳ ; ; 8mkB T jIn;n0 ðkz Þj2 dkz : À1 Here, kB is the Boltzmann constant, and Kn(x) is the Bessel function of 2nd order Eq (16) is the analytical expression for the QAE current in a DSL when the momentum relaxation time is a constant Numerical results and discussion To clarify the results obtained, in this section, we consider the QAE current This quantity is considered to be a function of the temperature T, the frequency of external acoustic wave x~q , and the parameters of the GaAs:Si/GaAs:Be DSL The parameters used in the number calculations are as follow [26,30,31]: r = 5300 kg mÀ3,s = 10À12 s, m = 0.067 m0, m0 being the mass of a free electron, U = 104 W mÀ2, and cl =  103 m sÀ1, ct = 18  102 m sÀ1, cs =  102 m sÀ1, nD = 1023 mÀ3, d = 2dn = 2dp = 80 nm Figs and present the dependence of the QAE current on the frequency x~q of the external acoustic wave at different values for the temperature and the concentration nD, respectively Figs and shown some maxima when the condition x~q ẳ x~k ỵ Dn;n0 (n n0 ) is satisfied This result is different from the AE current in a bulk semiconductor [9–15], because in a bulk semiconductor, when the x~q increase, the AE current increases linearly The cause of the difference between the bulk Current Density [arb units] 2 ωq [s −1] 10 12 14 11 x 10 Fig Dependence of the jQAE current on the frequency xq of the external acoustic wave at different values of the temperature T = 45 K (solid line), T = 50 K (dot line), T = 55 K (dashed line) Here nD =  1023 mÀ3 N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121–130 129 Current Density [arb units] 0 10 15 11 ω q [s−1] x 10 Fig Dependence of the jQAE current on the frequency x~q of the external acoustic wave at different values of the doping concentration nD =  1023 mÀ3 (solid line), nD = 1.2  1023 mÀ3 (dot line), nD = 1.4  1023 mÀ3 (dashed line) Here T = 50 K, eF = 0.038 eV semiconductor and the DSL is characteristics of a low-dimensional system, in low-dimensional systems, the energy spectrum of electron is quantized and exists even if the relaxation time s of the carrier does not depend on the carrier energy In Figs and there are three peaks This is attributed to the transitions between mini-bands (n ? n0 ) When we consider the case n = n0 Physically, we merely consider transitions within sub-bands (intrasubband transitions), and from the numerical calculations we obtain jQAE = 0, where mean that only the intersubband transition (n – n0 ) contribute to the jQAE In CSL [16–18], the AE current appears even if the intrasubband transitions In addition, in Fig shows that the peaks move to the larger frequency of the external acoustic wave when the doping concentration nD increases In contrast, Fig shows that the positions of the maxima nearly are not move as the temperature is varied because the condition x~q ẳ x~k ỵ Dn;n0 (n n0 ) not depend on the temperature Therefore, We use can use these conditions to determine the peaks position at the different value of the acoustic wave frequency or the parameters of the GaAs:Si/GaAs:Be DSL This means that the condition is determined mainly by the electron’s energy Fig shows the dependence of the QAE current on the temperature and the Fermi energy The dependence of the QAE current on the temperatures and the Fermi energy are not monotonic have a maximum at T = 48 K, eF = 0.038 eV for xq =  1011 sÀ1, it gives the same results as the Current Density [arb units] 0.042 0.04 0.038 EF [eV] 0.036 20 30 40 50 60 T [K] Fig Dependence of the jQAE current on the temperature T and the Fermi energy Here x~q ẳ 1011 s1 ị 130 N.Q Bau, N Van Hieu / Superlattices and Microstructures 63 (2013) 121–130 experimental result obtained in two-dimensional structures [26] According to [26] the acoustoelectric current show a non-monotonic temperature dependence with a maximum 40–50 K However, Refs [26] contains no explanation for this behavior From our calculation, we conclude that the dominant mechanism for such a behavior is electron confinement in two-dimensional structures Conclusions In this paper, we have obtained analytical expressions for the jQAE in a DSL by using the quantum kinetic equation for the distribution function of electrons interacting with an external acoustic wave and electron-acoustic phonon scattering We have shown the strong nonlinear dependence of jQAE on the temperature T, the frequency x~q of the external acoustic wave The importance of the present work is the appearance of peaks when the condition x~q ẳ x~k ỵ Dn;n0 (n – n0 ) is satisfied, the results are complex and different from those obtained in bulk semiconductors [9–15] and the CSL [16–18] The numerical results obtained for the GaAs:Si/GaAs:Be DSL show that a peak exists at T = 48 K, eF = 0.038 eV for x~q ¼  1011 sÀ1, it gives the same results as the experimental result obtained in two-dimensional structures [26] Our result indicates that the dominant mechanism for such a behavior is electron confinement in the DSL and transitions between mini-bands n ? 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