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

Superlattices and Microstructures 52 (2012) 921–930

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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 7 July 2012
Accepted 30 July 2012
Available online 5 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.

1. 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: (N. Van Hieu).
0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>

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 3 we find analytic expression for the AME field in
the QWPP, in Section 4 we discuss the results, and in Section 5 we come to a conclusion.
2. 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 VðzÞ ¼ mx2 z2 =2 the eigenfunction of an unperturbed electron is expressed as


wN;~p ðrÞ ¼

2
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 ¼ py Xc =½mðX2c þ x2 ފ; 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; 2 . . . 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 ¼ XðN þ 1=2Þ þ

p2y x2
p2x
;
þ
2m 2m X

with X ¼ ðX2c þ x2 Þ1=2 and Xc ¼ eB=m.

ð2Þ


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 ¼ cs jqj=g ( 1; ql ) 1;

ð3Þ

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 ¼ ð2pÞ3 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 ¼ H0 þ HeÀp ;
HeÀp ¼

X

H0 ¼

X

eN ð~kÞaþN;~k aN;~k ;

ð4Þ

N;~
k

C~q U N;N0 ð~
qÞaþN0 ;~k aN0 ;~kþ~q b~q expðÀix~q tÞ;

ð5Þ

N;~
k;N 0 ;~
q

with C~q is the electron-phonon interaction factor and takes the form [11]

1 þ r2l
þ
2r
À
Á1=2

¼ 1 À c2s =c2t
;

C~q ¼ iKc2l ðhx~3q =2q0 NSÞ1=2 ;
À

rl ¼ 1 À c2s =c2l

Á1=2

rt

N¼q





!

rl
1 þ r2t
;
À2
rt
2rt

ð6Þ
ð7Þ

K is the deformation potential constant; aþN;~k and aN;~k are the creation and the annihilation operators of
ki and jN 0 ; ~
k þ~
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 ¼ expðiqy À kl zÞ; kl ¼ q2 À 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 2 it is straightforward to evaluate the matrix elements of the operator U. We obtain

U N;N0 ð~
qÞ ¼

0
2
Àk2l
ð2pÞ2 LNÀN
N
Lx Ly
mX

!
 exp Àz0 kl þ

!
k2l

d0
d 0 d 0;
4mX ky ;ky þq kx ;kx N;N

ð8Þ

0

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; ~
pŠ
;
À eE þ Xc ½~
¼À
@t
@p
s

ð9Þ

where ~
h ¼~

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 ¼ haþ
a p it :
N;~
p N;~

i

@fN;~p
¼ h½aþN;~p aN;~p ; HŠit ;
@t

ð10Þ


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 1

À1

h


h
i
X
jC~k j2 jU N;N0 j2 ½fN0 ;~pþ~k ðN~k þ 1Þ À fN;~p N~k Š  exp iðeN0 ;~pþ~k À eN;~p À x~k Þðt À t 1 Þ
N 0 ;~
k

h
i
þ fN0 ;~pÀ~k N~k À fN;~p ðN~k þ 1ފ  exp iðeN0 ;~pÀ~k À eN;~p þ x~k Þðt À 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
2
¼À
À e~
E þ Xc ½~
h;~
pŠ
þ
jC~k j jU N;N0 j  ½fN0 ;~pþ~k ðN~k þ 1Þ À fN;~p N~k ŠdðeN0 ;~pþ~k À eN;~p À x~k Þþ
@p
s
N0 ;~
k

þ½fN0 ;~pÀ~k N~k À fN;~p ðN~k þ 1ފdðeN0 ;~pÀ~k À eN;~p þ x~k Þ :
ð12Þ
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 ðeÞ
~ N ðeÞ þ ~

h; ~
RN;N0 ðeފ ¼ Q
SN;N0 ðeÞ;
þ Xc ½~
sðeÞ

ð13Þ

where

~ N ðeÞ ¼ À
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 À eN;~p Þdð~
k À~
qÞ
SN;N0 ðeÞ ¼
m
x~q cs
0 ~~
N;N ;p;k


 ðfN0 ;~pþ~k À fN;~p ÞdðeN0 ;~pþ~k À eN;~p À x~k Þ 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 ðeÞ

~
RN;N0 ðeÞ ¼

n



sðeÞ
~ N ðeފ þ ½~
~ N ðeÞ þ ~
SN;N0 ðeÞ sðeÞ À Xc sðeÞ ½~
h; Q
h; ~
SN;N0 ðeފ
Q
2 2

1 þ Xc s ðeÞ

 o
~ N ðeÞ þ ~
SN;N0 ðeÞ; ~
h ~
h ;
þ X2c s2 ðeÞ Q

ð14Þ

the total current density is generally expressed as

~j ¼

Z

1

~
RN;N0 ðeÞde;

ð15Þ

0

we find the current density

ji ¼ aij Ej þ bij Uj ;

ð16Þ

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 þ X2c a3 hi hj ;
m
n
o
2
c b3 hi hj

bij ¼ A b1 dij À Xc b2 ijk hk þ 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 1
0

Z

1

sl ðeÞ
@f0
ðe À XðN þ 1=2ÞÞ
de;
@e
1 þ X2c s2 ðeÞ
0
sl ðeÞ
@f0
de;
1 þ X2c s2 ðeÞ @ e

"
!#2
!
2
8ep3 jC q j2 X 4
q

k2l
2kl X2c
4
0 Àkl
A¼
ð2pÞ LN
 exp À
q
xq cs N;N0 ðLy Lx Þ2
mX
4mX mX3
mð2pÞ2
È À 0
Á
À
ÁÉ
d ðN À NÞX À xq À d ðN 0 À NÞX þ 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 0.
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.
3. 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 ¼ s0



e

kB T

m
ð20Þ

;

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 mþ1;2m À F m;2m F 2mþ1;2m
e2 mkB T

(
2

2 )À1
XðN þ 1=2Þ
XðN þ 1=2Þ
;
F m;2m þ X2c s20 F 2mþ1;2m À
F 2m;2m
Â
F mþ1;2m À
kB T
kB T

ð21Þ

where

F m;m0 ¼

Z

1

0

xm

@f0
dx:

1 þ 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


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N.Q. Bau et al. / Superlattices and Microstructures 52 (2012) 921–930

EAME ¼

pXc As0 U È
e2 mkB T

F 2m;2m F mþ1;2m À F m;2m F 2mþ1;2m

oÀ1

Én 2
;
F mþ1;2m þ X2c s20 F 22mþ1;2m

ð23Þ

we calculated for f0 ¼ 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 0 Àk2
È À 0
Á À
ÁÉ
l

Â
LN
d ðN À NÞX À xq À d ðN0 À NÞX þ xq
m
X
N;N0
   
 91À1
8      
0
2
2
1
1
2
2
1
1
2
<
=
2si
ci
cos
þ
sin
si
À
ci
Xc s0

Xc s0
Xc s0
Xc s0
Xc so
Xc so
X ðN þ 1=2Þs0
A ;
 
 
 @1 À 2
2
2
:
;
1
1
kB T
þ si
ci
Xc so

Xc so

ð24Þ
with

h

L0N ðyÞ

i2

¼

N
CðN þ 1Þ X
ð2kÞ!ð2N À 2kÞ!L02k ðyÞ
2N
2 N! k¼0 ðN À kÞ!Cðk þ 1Þ

and

siðyÞ ¼ À

1
p X

2

þ

ðÀ1Þk y2kÀ1
;
ð2k À 1Þð2k À 1Þ!
k¼1

ciðyÞ ¼ À lnðxÞ þ

1
X

ðÀ1Þk y2k
k¼1

2kð2kÞ!

;

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 È
2

2

e2 mX ðN þ 1=2Þ

oÀ1
É n

F 2m;2m F mþ1;2m À F m;2m F 2mþ1;2m  F 2m;2m þ 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 0 Àk2
È À 0
Á À
ÁÉ
1
l
Â
LN

d ðN À NÞX À xq À d ðN0 À NÞX þ xq
ðN
þ
1=2Þ
m
X
0
N;N
   
 91À1
8      
0
2
2
2
<2si X 1s ci X 1s cos X 2s þ sin X 2s
si Xc1so À ci Xc1so =
X
s
ðN
þ
1=2Þ
c 0
c 0
c 0
c 0
0
A :
 
 

Â@
À2
2
2
;
:
1
1
kB T
þ si
ci
Xc so

Xc so

ð27Þ
From Eq. (24) and Eq. (27), we see that in both cases, the weak magnetic field and the quantized magnetic field, the dependence of AME field on external magnetic field B is nonlinear. These results are
different from those obtained in bulk semiconductor [11–14] and the Kane semiconductor [15].


927

N.Q. Bau et al. / Superlattices and Microstructures 52 (2012) 921–930

4. Numerical results and discussion
To clarify the results that have been obtained, in this section, we considered the AME field in two
limited cases weak magnetic field region and the case of quantized magnetic field region in QWPP.
This quantity is considered as a function of an external magnetic field B, the 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 ¼
4
À2
10 W m ; m ¼ 0:067m0 ; m0 being the mass of free electron, q0 ¼ 5320 kg mÀ3 ; cl ¼ 2Â
103 msÀ1 ; ct ¼ 18 Â 102 msÀ1 ; cs ¼ 8 Â 102 msÀ1 ; K ¼ 13:5 eV; xq ¼ 109 sÀ1 .

4.1. The case of weak magnetic field region
Fig. 1 shows the dependence of AME field on the magnetic field for the case of the weak magnetic
field at different values of the lattice temperature. The dependence of the AME field on the magnetic
field at different values of the temperature shows that when magnetic field rises up, the AME field increases monotonically. However, it reached a maximum value at B is about 0.08 T, and decreased again
above 0.08 T. The AME field is very small, approximates 2:5 Â 10À6 V=m. On the other hand, EAME increases nonlinearly with the magnetic field. This result is different from those for bulk semiconductor
[11–14] and the Kane semiconductor [15] under condition ql ) 1, and the weak magnetic field region.
Because the AME field expression EAME in bulk semiconductor [11–14] and the Kane semiconductor
[15] is proportional to B. In other words, EAME increases linearly with the magnetic field. Our result
indicates that the dominant mechanism for such a behaviour is attributed to the electron confinement
in the QWPP. From Fig. 1, we see that the EAME depends significantly on the lattice temperature. See
Fig. 2, 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 [11–14].
Fig. 3 investigated the dependence of AME field on the magnetic field in the quantized magnetic
field region which have many distinct maxima. The result showed the different behaviour from results
in bulk semiconductor [11–14] and the Kane semiconductor [15]. Different from the bulk semiconductor, these peaks in this case are much sharper. According to the result in the bulk semiconductor
[11–14] and the Kane semiconductor [15] in the case of strong magnetic field EAME which is proportional to 1B. There are two reasons for the difference between our result and other results: one is that
−6

3

x 10

AME field (V/m)

2.5

2

1.5

1

0.5

0

T=250K
T=270K
0

0.02

0.04

0.06
0.08
0.1

Magnetic field B(T)

0.12

0.14

Fig. 1. The dependence of AME field on the magnetic field B for the case of the weak magnetic field and high temperature.


928

N.Q. Bau et al. / Superlattices and Microstructures 52 (2012) 921–930
−7

1.8

x 10

1.6

AME field (V/m)

1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.02

0.04

0.06
0.08
0.1
Magnetic field B(T)

0.12

0.14

Fig. 2. The dependence of AME field on the magnetic field B for the case of the weak magnetic field and high temperature
T=270K, in the limit of x ! 0 .

−3

3.5

x 10

3

AME field (V/m)

2.5
2
1.5
1

0.5
0
1.8

2

2.2

2.4
2.6
2.8
Magnetic field B(T)

3

3.2

Fig. 3. The dependence of AME field on the magnetic field B for the case of the strong magnetic field and low temperature;
T ¼ 4K.

in the presence of the quantum magnetic field, the electron energy spectrum was affected by quantized magnetic field and the other is the effect of the electrons confinement in the QWPP, that means
above B > 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. 3. This is our new development.
5. 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 Â 10À6 V=m at T=170K, B=0.08(T) (in the case of the weak magnetic field) and
EAME ¼ 3:2 Â 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 3. 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 ¼

ð28Þ
ð29Þ

Substituting Eqs. (28, 29) into Eq. (19) we obtain

EAME

AmXc
¼ 2
e n0

b2 a1 À b1 a2

!

a21 þ X2c a22

U;

ð30Þ

here

Z

1







sðeÞ
1
@f0
Á
eÀX Nþ
de;
2 2
2
@e
1 þ Xc s ðeÞ
0




Z
1
m2
s2 ðeÞ

1
@f0
Á
eÀX Nþ
de;
a2 ¼
2
pn0 0 1 þ X2c s2 ðeÞ
@e
Z 1
Z 1
sðeÞ
@f0
s2 ðeÞ
@f0
d
e
;
b
¼
de;
Á
Á
b1 ¼
2
2 2
2 2
@
e
1 þ Xc s ðeÞ

1 þ Xc s ðeÞ @ e
0
0

a1 ¼

m2
pn0

ð31Þ
ð32Þ
ð33Þ


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
0

1
0

b2 ¼

Z

m

e

kB T



e

m

kB T



0

Z

s0

1

Z



1
0

1 þ X2c s20 kBeT
 2m

s0

into Eqs. (31)–(33), with x ¼ kBeT
2m

e

kB T

1 þ X2c s20



e

2m





!


Z 1
Z 1
1
@f0
m2
s0 kB Txmþ1 @f0
1
s 0 xm
@f0
Á
de ¼
Á
Á
dx À X N þ
dx ;
2
2
2
2 0 1 þ Xc s20 x2m @x

@e
pn0 0 1 þ Xc s20 x2m @x

ð34Þ





!


Z 1
Z 1 2
1
@f0
m2
s0 kB Tx2mþ1 @f0
1
s20 x2m
@f0
Á
dx À X N þ
dx ;
de ¼
Á
Á
2
2
2

2 0 1 þ Xc s20 x2m @x
@e
pn0 0 1 þ Xc s20 x2m @x

ð35Þ

eÀX N þ

eÀX N þ

kB T

Z 1
s0 ðkBeT Þm
@f0
s 0 Á xm
@f0
Á
de ¼
Á
dx;
2 2 e 2m @ e
1 þ X2c s20 x2m @x
1 þ Xc s0 ðkB T Þ
0
Z 1
s0 ðkBeT Þ2m
@f0
s20 Á x2m
@f0

Á
d
e
¼
Á
dxÁ
2 2 e 2m @ e
1 þ X2c s20 x2m @x
1 þ Xc s0 ðkB T Þ
0

ð36Þ

ð37Þ

We used following notations

F m;m0 ¼

Z

1

0

xm

@f0
dx:
1 þ X2c s20 xm0 @x

We obtain

a1 ¼

m2

pn0

ðs0 kB T Á F mþ1;2m À XðN þ 1=2Þs0 Á F m;2m Þ;

m À
2

ð38Þ
Á

s2 k T Á F 2mþ1;2m À XðN þ 1=2Þs20 Á F 2m;2m ;
pn0 0 B
b1 ¼ s0 Á F m;2m ; b2 ¼ s20 F 2m;2m :
a2 ¼

ð39Þ
ð40Þ

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
À XðN þ 1=2Þs0 F m;2m Þ À s0 F m;2m
2
e n0
pn0 0 B mþ1;2m
(

2
'
Á m2 s0 kB T À2
m2 À 2
XðN þ 1=2Þ
2
Á
s0 kB TF 2mþ1;2m À XðN þ 1=2Þs0 F 2m;2m
Â
F mþ1;2m À
F m;2m
kB T
pn0
pn0

2 )À1
É
XðN þ 1=2Þ
pXc As0 U È
¼ 2
F 2m;2m
Á F 2m;2m F mþ1;2m À F m;2m F 2mþ1;2m

þ X2c s20 F 2mþ1;2m À
kB T
e mkB T
(
2

2 )À1
XðN þ 1=2Þ
XðN þ 1=2Þ
2 2
Â
F mþ1;2m À
:
ð41Þ
F m;2m þ Xc s0 F 2mþ1;2m À
F 2m;2m
kB T
kB T

EAME ¼

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