DSpace at VNU: The quantum acoustomagnetoelectric field in a quantum well with a parabolic potential

DSpace at VNU: The quantum acoustomagnetoelectric field in a quantum well with a parabolic potential tài liệu, giáo án,...

Superlattices and Microstructures 52 (2012) 921–930 Contents lists available at SciVerse ScienceDirect Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices The quantum acoustomagnetoelectric field in a quantum well with a parabolic potential Nguyen Quang Bau, Nguyen Van Hieu ⇑, Nguyen Vu Nhan Faculty of Physics, Hanoi University of Science, Vietnam National University, 334-Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received 17 March 2012 Received in revised form July 2012 Accepted 30 July 2012 Available online August 2012 Keywords: Parabolic quantum well Quantum acoustomagnetoelectric field Electron-external phonon interaction Quantum kinetic equation a b s t r a c t The acoustomagnetoelectric (AME) field in a quantum well with a parabolic potential (QWPP) has been studied in the presence of an external magnetic field The analytic expression for the AME field in the QWPP is obtained by using the quantum kinetic equation for the distribution function of electrons interacting with external phonons The dependence of the AME field on the temperature T of the system, the wavenumber q of the acoustic wave and external magnetic field B for the specific AlAs/GaAs/AlAs is achieved by using a numerical method The problem is considered for both cases: The weak magnetic field region and the quantized magnetic field region The results are compared with those for normal bulk semiconductor and superlattices to show the differences, and we use the quantum theory to calculate the AME field in the QWPP Ó 2012 Elsevier Ltd All rights reserved Introduction The acoustic waves propagate along the stress-free surface of an elastic medium has attracted much attention in the past two decades because of their utilization in acoustoelectronics Considerable interest in such waves has also been stimulated by the possibility of their use as a powerful tool for studying the electronic properties of the surfaces and thin layers of solids It is well known that the propagation of the acoustic wave in conductors is accompanied by the transfer of the energy and momentum to conduction electrons which may give rise to a current usually called the acoustoelectric current, in the case of an open circuit called acoustoelectri field Presently this effect has been studied in detail both theoretically and experimentally and has been found in wide application in radioelectrionic systems [1–8] The presence of an external magnetic field ⇑ Corresponding author E-mail address: nguyenvanhieu@gmail.com (N Van Hieu) 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.spmi.2012.07.023 922 N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930 applied perpendicularly to the direction of the sound wave propagation in a conductor can induce another field, the so-called AME field It was predicted by Galperin and Kagan [9] and observed in bismuth by Yamada [10] Calculations of the AME field in bulk semiconductor [11–14] and the Kane semiconductor [15] in both cases The weak and the quantized magnetic field regions have been investigated In recent years, the AME field in low-dimensional structures have been extensively studied [16,17] So far, however, almost all these works obtained by using the Boltzmann kinetic equation method, and are, thus, limited to the case of the weak magnetic field region and the high temperature, in the case of the quantized magnetic field (strong magnetic field) region and the low temperature using the Boltzmann kinetic equation is invalid Therefore, we use quantum theory to investigate both the weak magnetic field and the quantized magnetic field region The AME field is similar to the Hall field in the bulk semiconductor where the sound flux U plays the role of electric current ~ j The essence of the AME effect is due to the existence of partial current generated by the different energy groups of electrons, when the total acoustoelectric (longitudinal) current in specimen is equal to zero When this happens, the energy dependence of the electron momentum relaxation time causes average mobilities of the electrons in the partial current, in general, to differ, if an external magnetic field is perpendicular to the direction of the sound flux, the Hall currents generated by these groups will not compensate one another, and a non-zero AME effect will result Here it must be emphasized that the direction of the AME field depends on the carrier scattering mechanism The direction of the AME field is opposite depending upon whether the deformation potential scattering or the ionized impurity scattering is dominant In the present paper, we study the AME field in a QWPP by using the quantum kinetic equation for the distribution function of electrons interacting with external phonons We restricted our consideration to the case of specular reflection of electron at the surface of bulk crystal We assumed the deformation mechanism of electron-acoustic phonon interaction We also supposed that the mechanism that limits the electron mean free path is scattered on randomly distributed point defects (impurities) in the bulk of the crystal Within the framework of this model we analyzed the magnetic field dependence of the AME field in the weak magnetic field region (Xc ( kB T; Xc ( g), and for the quantized magnetic field region (Xc ) kB T; Xc ) g), (Xc is the cyclotron frequency; g is the frequency of the electron collisions and in this paper, we select h  ¼ 1) Numerical calculations are carried out for a specific quantum well AlAs/GaAs/AlAs to clarify our results This paper is organized as follows In Section 2, we calculate the AME field in a QWPP, in Section we find analytic expression for the AME field in the QWPP, in Section we discuss the results, and in Section we come to a conclusion The AME Field in a QWPP 2.1 Electronic Structure in a QWPP When the magnetic field is applied in the x-direction, in that case the vector potential is chosen as: A ¼ Ay ¼ ÀzB If the confinement potetial is assumed to take the form Vzị ẳ mx2 z2 =2 the eigenfunction of an unperturbed electron is expressed as  wN;~p ðrÞ ¼ Lx Ly 1=2 /N ðz À z0 Þ Â expðipx xÞ expðipy yÞ; ð1Þ where Lx and Ly are the normalization length in the x and y direction, respectively, /N ðz À z0 Þ is the oscillator wavefunction centred at z0 ẳ py Xc =ẵmX2c ỵ x2 ị; m is the effective mass of a conduction electron, x and Xc are the characteristic frequency of the potential and the cyclotron frequency, respectively, N ¼ 0; 1; is the azimuthal quantum number; ~ p ¼ ðpx ; py ; 0Þ is the electron momentum vector The electron energy spectrum takes the form eN ð~ pÞ ẳ XN ỵ 1=2ị ỵ p2y x2 p2x ; ỵ 2m 2m X with X ẳ X2c ỵ x2 ị1=2 and Xc ẳ eB=m 2ị N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930 923 2.2 General expression for the AME Field in a QWPP Let us suppose that the acoustic wave of frequency xq is propagated along the z QWPP axis (along the z direction, the energy spectrum of electron is quantizied or the motive direction of electron is limited) and the magnetic field is oriented along the x axis We consider the most realistic case from the point of view of a low-temperature experiment, when xq =g ẳ cs jqj=g ( 1; ql ) 1; 3ị where cs is the velocity of the acoustic wave and q is the modulus of the acoustic wave vector and l is the electron mean free path The compatibility of these conditions is provided by the smallness of the sound velocity in comparison with the characteristic velocity of the Fermi electrons We also suppose that inequalities (3) hold, i.e the quantization of the electron motion in the magnetic field is essential If the conditions (3) are satisfied, a macroscopic approach to the description of the acoustoelectric effect is inapplicable and the problem should be treated by using quantum mechanical methods The acoustic wave will be considered as a packet of coherent phonons with the delta-like distribution function N~k ẳ 2pị3 Udð~ k À~ qÞ=xq cs in the wavevector ~ k space, U is the sound flux density The Hamiltonian describing the interaction of the electron-phonon system in the QWPP, which can be written in the secondary quantization representation as H ẳ H0 ỵ Hep ; Hep ẳ X H0 ẳ X eN ~kịaỵN;~k aN;~k ; 4ị N;~ k C~q U N;N0 ~ qịaỵN0 ;~k aN0 ;~kỵ~q b~q expix~q tị; 5ị N;~ k;N ;~ q with C~q is the electron-phonon interaction factor and takes the form [11] ỵ r2l þ 2r À Á1=2 ¼ À c2s =c2t ; C~q ẳ iKc2l hx~3q =2q0 NSị1=2 ; rl ẳ À c2s =c2l Á1=2 rt N¼q   ! rl ỵ r2t ; rt 2rt 6ị 7ị K is the deformation potential constant; aỵN;~k and aN;~k are the creation and the annihilation operators of ki and jN ; ~ k ỵ~ qi the electron, respectively; b~q is the annihilation operator of the external phonon jN; ~ ~ are electron states before and after interaction, U N;N0 ðqÞ is the matrix element of the operator  1=2 U ẳ expiqy kl zị; kl ẳ q2 x2q =c2l is the spatial attenuation factor of the potential part the displacement field; cl and ct are the velocities of the longitudinal and the transverse bulk acoustic wave; q0 is the mass density of the medium and S ¼ Lx Ly is the surface area Using expression it is straightforward to evaluate the matrix elements of the operator U We obtain U N;N0 ~ qị ẳ Àk2l ð2pÞ2 LNÀN N Lx Ly mX !  exp z0 kl ỵ ! k2l d0 d d 0; 4mX ky ;ky ỵq kx ;kx N;N 8ị d is the Kronecker delta symbol and LNÀN ðxÞ is the associated Laguerre polynomials The quantum kiN netic equation for electrons in the single (constant) scattering time approximation takes the form:  @f @fN;~p  ~ fN;~p À f0 N;~ p h; ~ p ; eE ỵ Xc ẵ~ ẳ @t @p s 9ị where ~ h ẳ~ B=B is the unit vector along the direction of the external magnetic field, f0 is the equilibrium electron distribution function, fN;~p is an unknown distribution function perturbed due to the external fields, and s is the electron momentum relaxation time In order to find fN;~p , we use the quantum equation for the particle number operator or the electron distribution function fN;~p ẳ haỵ a p it : N;~ p N;~ i @fN;~p ẳ hẵaỵN;~p aN;~p ; Hit ; @t 10ị 924 N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930 cW c is the density mab Þ (W where hWit denotes a statistical average value at the moment t; hWit ¼ Trð W trix operator) From Eq (10), using the Hamiltonian in Eqs (4) and (5) and realizing operator algebraic calculations, we find @fN;~p ¼À @t Z t dt À1 h  h i X jC~k j2 jU N;N0 j2 ẵfN0 ;~pỵ~k N~k ỵ 1ị fN;~p N~k exp ieN0 ;~pỵ~k eN;~p x~k ịt À t Þ N ;~ k h i þ fN0 ;~pÀ~k N~k À fN;~p ðN~k þ 1ފ  exp ieN0 ;~p~k eN;~p ỵ x~k ịt t1 Þ ; ð11Þ substituting Eq (11) into Eq (9) and realizing calculations, we obtained the basic equation of the problem which is that equation for the distribution function of electrons interacting with external phonons in the presence of an external magnetic fields in QWPP:   @f  fN;~p À f0 X N;~ p 2 ¼À À e~ E þ Xc ½~ h;~ pŠ þ jC~k j jU N;N0 j ẵfN0 ;~pỵ~k N~k ỵ 1ị fN;~p N~k deN0 ;~pỵ~k eN;~p x~k ịỵ @p s N0 ;~ k  ỵẵfN0 ;~p~k N~k fN;~p N~k ỵ 1ịdeN0 ;~p~k eN;~p ỵ x~k ị : 12ị Eq (12) is fairly general and can be applied for any mechanism of interaction In the limit of x ! 0, i.e., the electron confinement vanishes, it gives the same results as these obtained in bulk semiconductor [13,14] Multiply both sides of Eq (12) by ðe=mÞ~ pdðe À eN;~p Þ and carry out the summation over N and ~ p, we have the equation for the partial current density ~ RN;N0 ðeÞ (the current caused by electrons which have energy of e): ~ RN;N0 eị ~ N eị ỵ ~ h; ~ RN;N0 eị ẳ Q SN;N0 eị; ỵ Xc ẵ~ seị 13ị where ~ N eị ẳ Q   X ~ p ~ @fN0 ;~p e dðe À eN;~p Þ; E; m @~ p N;~ p ~ ð2pÞ3 jC~q j2 U X p ~ jU N;N0 j2 dðe eN;~p ịd~ k ~ qị SN;N0 eị ẳ m x~q cs ~~ N;N ;p;k    ðfN0 ;~pỵ~k fN;~p ịdeN0 ;~pỵ~k eN;~p x~k ị þ ðfN0 ;~pÀ~k À fN;~p ÞdðeN0 ;~pÀ~k À eN;~p þ x~k Þ : Solving the Eq (13), we obtained the partial current ~ RN;N0 eị ~ RN;N0 eị ẳ n    seị ~ N eị ỵ ẵ~ ~ N eị ỵ ~ SN;N0 eị seị Xc seị ½~ h; Q h; ~ SN;N0 ðeފ Q 2 ỵ Xc s eị   o ~ N eị ỵ ~ SN;N0 eị; ~ h ~ h ; þ X2c s2 ðeÞ Q ð14Þ the total current density is generally expressed as ~j ¼ Z ~ RN;N0 eịde; 15ị we nd the current density ji ẳ aij Ej ỵ bij Uj ; 16ị where aij and bij are the electrical conductivity and the acoustic conductivity tensors, respectively aij ¼ o e2 n0 n a1 dij À Xc a2 ijk hk ỵ X2c a3 hi hj ; m n o c b3 hi hj bij ¼ A b1 dij Xc b2 ijk hk ỵ X ; ð17Þ ð18Þ N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930 925 here ijk is the unit antisymmetric tensor of third order, n0 is the carrier concentration, and al ; bl (l ¼ 1; 2; 3) are given as al ¼ bl ¼ m2 pn0 Z Z sl eị @f0 e XN ỵ 1=2ịị de; @e ỵ X2c s2 eị sl eị @f0 de; ỵ X2c s2 eị @ e " !#2 ! 8ep3 jC q j2 X q k2l 2kl X2c kl Aẳ 2pị LN  exp À À q xq cs N;N0 ðLy Lx Þ2 mX 4mX mX3 mð2pÞ2 È À Á À ÁÉ d ðN À NÞX À xq À d ðN NịX ỵ xq : We considered a situation whereby the sound is propagating along the x axis and the magnetic field B is parallel to the z axis and we assume that the sample is opened in all directions, so that ji ¼ Therefore, from Eq (16) we obtained the expression of the AME field EAME , which appeared along the y axis of the sample Ey ¼ EAME ¼ bzz ayz À byz ayy U: a2yy þ a2yz ð19Þ Eq (19) is the general expression to calculate the AME field in a QWPP in the case of the relaxation time of carrier s dependent on carrier energy Analytic expression for the AME field in the QWPP We can see that Eq (19) in the general case is very complicated, so that we only examined the relaxation time of carrier s depending on carrier energy as follow: s ẳ s0  e kB T m 20ị ; by using the Eqs (17) and (18) and carrying out manipulations, we derived the expression for the AME field as follows: EAME ¼ pXc As0 U È É : F 2m;2m F mỵ1;2m F m;2m F 2mỵ1;2m e2 mkB T ( 2  2 )1 XN ỵ 1=2ị XN ỵ 1=2ị ; F m;2m ỵ X2c s20 F 2mỵ1;2m F 2m;2m F mỵ1;2m kB T kB T 21ị where F m;m0 ẳ Z xm @f0 dx: ỵ X2c s20 xm0 @x The Eq (21) is the AME field in the QWPP in the case of the external magnetic field We can see that the dependence of the AME field on the external magnetic field and the frequency x~q is nonlinear We will carry out further analysis of the Eq (21) separately for the two limiting cases: the weak magnetic field region and the case of quantized one 3.1 The case of weak magnetic field region In the case of the weak magnetic field Xc ( kB T; Xc ( g; in this case, the expression of EAME in the Eq (20) takes the form ð22Þ 926 N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930 EAME ¼ pXc As0 U ẩ e2 mkB T F 2m;2m F mỵ1;2m F m;2m F 2mỵ1;2m o1 ẫn ; F mỵ1;2m ỵ X2c s20 F 22mỵ1;2m 23ị we calculated for f0 ẳ ð1 À expðx À zÞÞÀ1 which is the Fermi-Dirac distribution function, x ¼ e=kB T; z ¼ eF =kB T, and by carrying out a few manipulation we obtained an analysis expression for the AME field as follows: EAME ¼ ð2pÞ5 UqX4 jC q j2 exp À ! k2l 2kl X2c q À 4mX mX3 exq kB Tcs k3l Xc " !#2 X Àk2 È À Á À ÁÉ l  LN d ðN À NÞX À xq d N0 NịX ỵ xq m X N;N0      91À1       2 1 2 1 < = 2si ci cos ỵ sin si À ci Xc s0 Xc s0 Xc s0 Xc s0 Xc so Xc so X N ỵ 1=2ịs0 A ;      @1 À 2 : ; 1 kB T ỵ si ci Xc so Xc so ð24Þ with h L0N ðyÞ i2 ẳ N CN ỵ 1ị X 2kị!2N 2kị!L02k yị 2N N! kẳ0 N kị!Ck ỵ 1ị and siyị ẳ p X ỵ 1ịk y2k1 ; 2k 1ị2k 1ị! kẳ1 ciyị ẳ lnxị ỵ X 1ịk y2k kẳ1 2k2kị! ; where L02k ðyÞ is the associated Laguerre polynomials 3.2 The case of quantized magnetic field region In the case of quantized magnetic field region Xc ) kB T; Xc ) g; ð25Þ in this case, the expression of EAME in Eq (20) takes the form EAME ¼ pXc As0 kB T U ẩ 2 e2 mX N ỵ 1=2ị o1 ẫ n F 2m;2m F mỵ1;2m F m;2m F 2mỵ1;2m F 2m;2m ỵ X2c s20 F 22m;2m ; ð26Þ by carrying out a few manipulation we obtained an analysis expression for the AME field as follow: EAME ¼ ð2pÞ5 UqX2 jC q j2 exp À ! k2l 2kl X2c q À 4mX mX3 exq cs k3l Xc " !#2 X Àk2 È À Á À ÁÉ l  LN d ðN À NÞX À xq d N0 NịX ỵ xq N ỵ 1=2ị m X N;N      91À1       2 1:8T and below 4K, carriers in the samples satisfy the quantum limit conditions: Xc ) kB T and Xc s ) 1, and in the QWPP the energy spectrum of electron is quantized Also, the result is different from those in superlattice [16,17] In [16,17] by using the Boltzmann kinetic equation, AME field is proportional to B with all regions of temperature By using the quantum kinetic equation method, our result indicate that it is only linear to B in case of the weak magnetic field and higher temperature, N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930 929 while in case of the strong magnetic field and low temperature AME field is not proportional to B, but there are many peaks in Fig This is our new development Conclusions In this paper, we have obtained analytical expressions for the AME field in a QWPP for both the case of quantized magnetic field and the weak magnetic field region There is a strong dependence of AME field on the cyclotron frequency Xc of the magnetic field, x~q of the acoustic wave, and the temperature T of system The result showed that the cause of the AME effect is the existence of partial current generated by the different energy groups of electrons, and the dependence of the electron energy due to the momentum relaxation time In addition, the absorption of acoustic quanta by electron is accompanied by the electrons confinement and quantized magnetic field which led to the increase of the AME effect The numerical result obtained for AlAs/GaAs/AlAs QWPP shows that in the quantized magnetic field region, the dependence of AME field on the magnetic B is nonlinear, and there are many distinct maxima This dependence has differences in comparison with that in normal bulk semiconductors [12–14] and the Kane semiconductor [15] The AME field in the QWPP is bigger The results show a geometrical dependence of AME field due to the electrons confinement in the QWPP In the limit of x ! 0, i.e., the electron confinement vanishes, EAME increases linearly with the magnetic field, it gives the same results as these obtained in bulk semiconductor [12–14] In addition, this results are quite interesting as a similar result in the superlattice [16,17] for the case of the weak magnetic field and the higher temperature, but in the case of the strong magnetic field and the low temperature the result is different from that in superlattice [16,17] From the numerical result, we have EAME ¼ 2:5  10À6 V=m at T=170K, B=0.08(T) (in the case of the weak magnetic field) and EAME ¼ 3:2  10À3 V=m at T=4K, B=1.9(T) (in the case of quantized magnetic field region), Which are small but should be possible to measure experimentally Acknowledgement This work is completed with financial support from the Vietnam NAFOSTED (No 103.01-2011.18) Appendix A Appendix Here we add some brief explanations about deriving AME field in Section The electrical conductivity tensors aij and the acoustic conductivity tensors bij have the form e2 n0 Àe2 n0 Xc a2 ; azy ¼ Àayz ; a1 ; ayz ¼ m m bxz ¼ 0; bzz ¼ Ab1 ; bzy ¼ Àbyz ¼ AXc b2 : ayy ẳ 28ị 29ị Substituting Eqs (28, 29) into Eq (19) we obtain EAME AmXc ¼ e n0 b2 a1 b1 a2 ! a21 ỵ X2c a22 U; ð30Þ here Z    sðeÞ @f0 eX Nỵ de; 2 @e ỵ Xc s eị     Z m2 s2 eị @f0 eX Nỵ de; a2 ẳ pn0 ỵ X2c s2 ðeÞ @e Z Z sðeÞ @f0 s2 ðeÞ @f0 d e ; b ¼ de; Á Á b1 ẳ 2 2 @ e ỵ Xc s eị ỵ Xc s eị @ e 0 a1 ẳ m2 pn0 31ị 32ị 33ị 930 N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930 substituting a1 ¼ a2 ¼ m2 pn0 m2 pn0 b1 ¼ s ¼ s0 Z 1 b2 ¼ Z m e kB T  e m kB T  Z s0 Z  1 ỵ X2c s20 kBeT  2m s0 into Eqs (31)–(33), with x ¼ kBeT 2m e kB T ỵ X2c s20  e 2m   !   Z Z 1 @f0 m2 s0 kB Txmỵ1 @f0 s xm @f0 de ẳ dx X N ỵ dx ; 2 2 ỵ Xc s20 x2m @x @e pn0 ỵ Xc s20 x2m @x ð34Þ   !   Z Z @f0 m2 s0 kB Tx2mỵ1 @f0 s20 x2m @f0 dx X N ỵ dx ; de ¼ Á Á 2 2 þ Xc s20 x2m @x @e pn0 þ Xc s20 x2m @x 35ị eX N ỵ eX N þ kB T Z s0 ðkBeT Þm @f0 s Á xm @f0 Á de ¼ Á dx; 2 e 2m @ e ỵ X2c s20 x2m @x ỵ Xc s0 kB T ị Z s0 ðkBeT Þ2m @f0 s20 Á x2m @f0 Á d e ¼ Á dxÁ 2 e 2m @ e ỵ X2c s20 x2m @x ỵ Xc s0 ðkB T Þ ð36Þ ð37Þ We used following notations F m;m0 ¼ Z xm @f0 dx: ỵ X2c s20 xm0 @x We obtain a1 ẳ m2 pn0 s0 kB T F mỵ1;2m XN þ 1=2Þs0 Á F m;2m Þ; m À ð38Þ s2 k T F 2mỵ1;2m XN ỵ 1=2ịs20 F 2m;2m ; pn0 B b1 ẳ s0 Á F m;2m ; b2 ¼ s20 F 2m;2m : a2 ẳ 39ị 40ị Substituting Eqs (38)(40) into Eq (30) and realizing calculations, we obtain Eq (21) & mXc AU m2 Á s20 F 2m;2m Á ðs k TF XN ỵ 1=2ịs0 F m;2m ị s0 F m;2m e n0 pn0 B mỵ1;2m (  2 ' Á m2 s0 kB T À2 m2 À XN ỵ 1=2ị s0 kB TF 2mỵ1;2m XN ỵ 1=2ịs0 F 2m;2m F mỵ1;2m F m;2m kB T pn0 pn0  2 )À1 É XN ỵ 1=2ị pXc As0 U ẩ ẳ F 2m;2m F 2m;2m F mỵ1;2m F m;2m F 2mỵ1;2m ỵ X2c s20 F 2mỵ1;2m kB T e mkB T ( 2  2 )1 XN ỵ 1=2ị XN ỵ 1=2ị 2 F mỵ1;2m : 41ị F m;2m ỵ Xc s0 F 2mỵ1;2m F 2m;2m kB T kB T EAME ¼ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] S.V Biryukov, Yu V Gulyaev, V.V Krylov, V.P Plesskij, Surface Acoust Waves Inhomogenerous Media (1991) J.M Shilton, D.R Mace, V.I Talyanskii, J Phys.: Condens Matter (1996) 337 O.E Wohlman, Y Levinson, Yu M Galperin, Phys Rev B62 (2000) 7283 N.A Zimbovskaya, G Gumbs, J Phys.: Condens Matter 13 (2001) 409 Yu M Galperin, O.E Wohlman, Y Levinson, Phys Rev B63 (2001) 153309 J Cunningham, M Pepper, V.I Talyanskii, Appl Phys Lett 86 (2005) 152105 S.Y Mensah, F.K.A Allotey, N.G Mensah, H Akrobotu, G Nkrumah, J Phys.: Superlattices microstruct 37 (2005) 87 M.R Astley, K Kataoka, C.J.B Ford, J Appl Phys 103 (2008) 096102 Yu M Galperin, B.D Kagan, Phys Stat Sol 10 (1968) 2037 Yamada Toshiyuki, J Phys Soc Japan 20 (1965) 1424 A.D Margulis, V.A Margulis, J Phys.: Condens Matter (1994) 6139 M Kogami, S Tanaka, J Phys Soc Japan 30 (1970) 775 E.M Epshtein, JETP Lett 19 (1974) 332 G.M Shmelev, G.I Tsurkan, N.Q Anh, Phys Stat Sol 121 (1984) 97 N.Q Anh, N.Q Bau, N.V Huong, J Phys VN (1990) 12 S.Y Mensah, F.K.A Allotey J Phys.: Condens Matter (1996) 1235 N.Q Bau, N.V Hieu, in: PIERS Proceeding, vol 51, 2010, p 342  ... the deformation potential constant; a N;~k and aN;~k are the creation and the annihilation operators of ki and jN ; ~ k ỵ~ qi the electron, respectively; b~q is the annihilation operator of the. .. Numerical calculations are carried out for a specific quantum well AlAs/GaAs/AlAs to clarify our results This paper is organized as follows In Section 2, we calculate the AME field in a QWPP, in Section... frequency xq of ultrasound, the temperature T of system, and the parameters of the AlAs/GaAs/AlAs quantum well The parameters used in the numerical calculations are as follow: s0 ¼ 10À12 s; U