DSpace at VNU: The Dependence of a Quantum Acoustoelectric Current on Some Qualities in a Cylindrical Quantum Wire with an Infinite Potential GaAs GaAsAl

Materials Transactions, Vol. 56, No. 9 (2015) pp. 1408 to 1411
Special Issue on Nanostructured Functional Materials and Their Applications
© 2015 The Japan Institute of Metals and Materials

The Dependence of a Quantum Acoustoelectric Current on Some Qualities
in a Cylindrical Quantum Wire with an Infinite Potential GaAs/GaAsAl
Nguyen Vu Nhan1,+, Nguyen Van Nghia2,4 and Nguyen Van Hieu2,3
1

Faculty
Faculty
3
Faculty
4
Faculty
2

of
of
of
of

Physics, Academy of Defence Force ­ Air Force, Son Tay, Hanoi, Vietnam
Physics, Hanoi University of Science, Vietnam National University, 334-Nguyen Trai, Hanoi, Vietnam
Physics, Danang University, 459 Ton Duc Thang, Danang, Vietnam
Energy, Water Resources University, 175 Tay Son, Hanoi, Vietnam

The quantum acoustoelectric (QAE) current is studied by a quantum kinetic equation method and we obtain analytic expression for QAE in
a cylindrical quantum wire with an infinite potential (CQWIP) GaAs/GaAsAl. The computational results show that the dependence of the QAE
current on the radius of CQWIP GaAs/GaAsAl, the Fermi energy ¾F and temperature T is non-monotonic, and the appearance of peak when the
h" 2 ðB20

ÀB2 Þ

n ;N
n;N
(n 6¼ n0 and N 6¼ N 0 ) is satisfied. Our result indicates that the dominant mechanism for such a behavior is the
condition ½q~ ¼ ½k~ þ
2mR2
electron confinement in the CQWIP GaAs/GaAsAl and transitions between mini-bands. All these results are compared with those for normal
bulk semiconductors and superlattice to show the differences. The dependence of a QAE current on some qualities in a CQWIP GaAs/GaAsAl
is newly developed. [doi:10.2320/matertrans.MA201514]
0

(Received January 22, 2015; Accepted July 1, 2015; Published August 25, 2015)
Keywords: cylindrical quantum wire, quantum acoustoelectric current, electron-external acoustic wave interaction, electron-acoustic phonon
scattering, quantum kinetic equation

1.

Introduction

When an acoustic wave propagating in a conductor creates
a net drag of electrons and hence an acoustoelectric (AE)
current or, if the circuit is disconnected, a acoustoelectric
potential difference. The study of this effect is crucial because
of the complementary role it may play in the understanding
of the properties of low-dimensional systems (quantum wells,
superlattices, quantum wires+).
As we know, low-dimensional structure is the structure in
which the charge carriers are not free to move in all three

dimensions. The motion of electrons is restricted in one
dimension (quantum wells, superlattices), or two dimensions
(quantum wires), or three dimensions (quantum dots). In lowdimensional systems, the energy levels of electrons become
discrete and the physical properties of the electron will be
changed dramatically and in which the quantum rules began
to take effect. Thus, the electron-phonon interaction and
scattering rates1) are different from those in bulk semiconductors. The linear absorption of a weak electromagnetic
wave have been studied in the low-dimensional structure.2­4)
The quantum kinetic equation was used to calculate the
nonlinear absorption coefficients of an intense electromagnetic wave in quantum wells5) and in quantum wires.6)
Also, study on the effect of AE in the normal bulk
semiconductor has received a lot of attention.7­10) Further,
the AE effect was measured experimentally in a submicronseparated quantum wire11) and in a carbon nano-tube.12)
However, the calculation of the QAE current in a CQWIP
by using the quantum kinetic equation method is unknown.
Throughout,5,6) the quantum kinetic equation method have
been seen as a powerful tool. So, in a recent work13) we have
used this method to calculate the QAME field in a QW. In the
present work, we use the quantum kinetic equation method
for electron-external acoustic wave interaction and electron+

Corresponding author, E-mail:

acoustic phonon (internal acoustic wave) scattering in the
CQWIP GaAs/GaAsAl to study the QAE current. The
present work is different from previous works7­10) because:
1) the QAE current is a result of not only the electronexternal acoustic wave interaction but also the electronacoustic phonon scattering in the sample; 2) we use the
quantum kinetic equation method; 3) we show that the
dependence of QAE current on the Fermi energy ¾F, the
temperature T of system and the characteristic parameters

of CQWIP GaAs/GaAsAl is nonlinear; 4) we discussed for
the CQWIP GaAs/GaAsAl, which is a one-dimensional
system (the CQWIP GaAs/GaAsAl) and these results are
compared with those for the bulk semiconductor,7­10) superlattice.14,15)
This paper is organized as follows: In Section 2, the QAE
current is calculated through the use of the quantum kinetic
equation method. In section 3, the QAE current is discussed
for specific CQWIP GaAs/GaAsAl. Finally, we present a
discussion of our results in section 4.
2.

The Analytical Expression for QAE Current in a
CQWIP GaAs/GaAsAl

We consider a CQWIP structure of the radius R and length
L with an infinite confinement potential. Due to the
confinement potential, the motion of electrons in the Oz
direction is free while the motion in (x-y) plane is quantized
into discrete energy levels called subbands. Then the
eigenfunction of an unperturbed electron in the CQWIP is
expressed as


1
pz
¼n;N;~pz ð~rÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi expðinºÞ exp i z ¼n;N ð~rÞ
h"
³R2 L
ðr < RÞ;
ð1Þ

here N = 1, 2, 3, + is the radial quantum number; n =
0, «1, «2, + is the azimuth quantum number; R is the radius
~ ¼ ð0; 0; pz Þ
of the CQWIP; L is the length of the CQWIP; p


The Dependence of a Quantum Acoustoelectric Current on Some Qualities in a Cylindrical Quantum Wire

is the electron’s momentum vector along z-direction;
ðBn;N r=RÞ
¼n;N ð~rÞ ¼ JJnnþ1
ðBn;N Þ is the radial wave function of the
electron in the plane Oxy, with Bn,N are the N level root of
Bessel function of the order n.
The electron energy spectrum takes the form
¾n;N;~pz ¼

h" 2 p2z h" 2 B2n;N
þ
;
2m
2mR2

ð2Þ

where m is the effective mass of the electron.
We assume that an external acoustic wave of frequency ½q~
is propagating along the CQWIP axis (Oz) and the acoustic
wave will be considered as a packet of coherent phonons with
3

~
~ ¼ ð2³Þ
~ ~ Þ,
the ¤-function distribution in k-space
NðkÞ
½q~ vs º¤ðk À q
where º is the flux density of the external acoustic wave with
frequency ½q~, vs is the speed of the acoustic wave, q is the
external acoustic wave number. We also consider the external
acoustic wave as a packet of coherent phonons. Therefore,
we have the Hamiltonian describing the interaction of the
electron-internal and external phonons system in the CQWIP
in the secondary quantization representation can be written
as
X
H¼
¾n;N;~pz aþ
a pz
n;N;~
pz n;N;~
n;N;~
pz

X

þ

In;N;n0 ;N 0 Ck~ aþ0

a 0 0 p0z ðbk~

pz þk~ n ;N ;~
n ;N 0 ;~

n;N;n0 ;N 0 ;k~

þ

X
k~

þ

þ bþ ~ Þ
Àk

h" ½k~ bþ
b
k~ k~

X

n;N;n0 ;N 0 ;~q

Cq~ Un;N;n0 ;N 0 aþ
a
~ tÞ;
p0z bq~ expðÀi½ q
pz þ~q n0 ;N 0 ;~
n0 ;N 0 ;~

ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where Ck~ ¼ Ã k=ð2μvs SLÞ is the electron-internal phonon
interaction factor, μ is the mass density of thepmedium,
$ isffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
the deformation potential constant, Cq~ ¼ iÃv2l h" ½3q~ =ð2μFSÞ
is the electron-external phonon interaction factor, with F ¼
q½ð1 þ · 2l Þ=ð2· t Þ þ ð· l =· t À 2Þð1 þ · 2t Þ=ð2· t ފ,
· l ¼ ð1 À
v2s =v2l Þ1=2 , · t ¼ ð1 À v2s =v2t Þ1=2 , S = ³R2 is the surface area,
vl (vt) is the velocity of the longitudinal (transverse) bulk
acoustic wave, aþ
(an;N;~pz ) is the creation (annihilation)
n;N;~
pz
operator of the electron; bþ
(bk~ ) is the creation (annihilation)
k~
operator of internal phonon and bq~ is the annihilation
~ is the
operator of the external phonon. The notation jn; ki
0 ~
electron states before interaction and jn ; k þ q~ i is the
electron states after interaction. Un,N,nA,NA is the matrix element
of the operator U = exp(iqy ¹ klz):
Z
2 expðÀkl LÞ R Ã
Un;N;n0 ;N 0 ¼
¼n0 ;N 0 ;~p0 ð~rÞ

z
R2 L
0
 ¼ n;N;~pz ð~rÞ expðiq? rÞdr;
ð4Þ
here kl = (q2 ¹ (½q/vl)2)1/2 is the spatial attenuation factor of
the potential part the displacement field and In,N,nA,NA is the
electronic form factor:
Z
2 R
In;N;n0 ;N 0 ¼ 2
JjnÀn0 j ðq? RÞ¼ Ãn0 ;N 0 ;~p0 ð~rÞ¼n;N;~pz ð~rÞrdr; ð5Þ
z
R 0
with q? is the wave vector in the plane Oxy.
To set up the quantum kinetic equation for electrons in the
presence of an ultrasound, we use equation of motion of
statistical average value for electrons

ih"

@hfn;N;~pz ðtÞit
@t

!
QAE

¼ h½aþ
a pz ; HŠit ;
n;N;~

pz n;N;~

1409

ð6Þ

where the notation hXit is mean the usual thermodynamic
average of the operator X and fn;N;~pz ðtÞ ¼ haþ
a pz it is the
n;N;~
pz n;N;~
particle number operator or the electron distribution function.
Use the Hamiltonian in the eq. (3) replaced into the eq. (6)
and realizing operator algebraic calculations like in Ref. 13),
we obtain the solution of the quantum kinetic equation for
electrons in CQWIP GaAs/GaAsAl in the form of the
function f (t) as follows
2³¸ X
fðtÞ ¼ À 2
jCk j2 jIn;N;n0 ;N 0 j2 Nk fðfn;N;~pz À fn0 ;N 0 ;~pz þk~ Þ
h" 0 0 ~
n ;N ;k
 ¤ð¾n0 ;N 0 ;~pz þk~ À ¾n;N;~pz À h" ½k~ Þ
þ ðfn;N;~pz À fn0 ;N 0 ;~pz Àk~ Þ¤ð¾n0 ;N 0 ;~pz Àk~ À ¾n;N;~pz þ h" ½k~ Þg
³¸ X
jCq j2 jUn;N;n0 ;N 0 j2 Nq fðfn;N;~pz À fn0 ;N 0 ;~pz þ~q Þ
þ 2
h" n0 ;N 0 ;~q
 ¤ð¾n0 ;N 0 ;~pz þ~q À ¾n;N;~pz þ h" ½k~ À h" ½q~ Þ
À ðfn0 ;N 0 ;~pz À~q À fn;N;~pz Þ

 ¤ð¾n0 ;N 0 ;~pz À~q À ¾n;N;~pz À h" ½k~ þ h" ½q~ Þg;

ð7Þ

where ¸ is relaxation time of momentum, fn;N;~pz is the
electron distribution function, Nq is the particle number
external phonon, Nk is the particle number internal phonon
and ¤ is the Kronecker delta symbol. We found that the
expression (7) has the same form as the expression obtained
in Ref. 13), but the quantities of expressions additional
indicators specific to quantum wires and they also have the
completely different values.
The density of the QAE current is generally expressed as
Z
2e X
QAE
j
vpz fðtÞdpz ;
¼
ð8Þ
2³h" n;N
here vpz is the average drift velocity of the moving charges
and it is given by vpz ¼ @¾n;N;~pz =@pz.
Substituting eq. (7) into eq. (8) and taking ¸ to be constant,
we obtain for the density of the QAE current in the CQWIP
GaAs/GaAsAl
Z
2e¸ X
QAE
j

vpz jCk j2 jIn;N;n0 ;N 0 j2 Nk
¼À 3
h"
0
0
n;N;n ;N ;k~
 fðfn;N;~pz À fn0 ;N 0 ;~pz þk~ Þ¤ð¾n0 ;N 0 ;~pz þk~ À ¾n;N;~pz À h" ½k~ Þ
þ ðfn;N;~pz À fn0 ;N 0 ;~pz Àk~ Þ
 ¤ð¾n0 ;N 0 ;~pz Àk~ À ¾n;N;~pz þ h" ½k~ Þgdpz
Z
e¸ X
þ 3
vpz jCq j2 jUn;N;n0 ;N 0 j2 Nq
h" n;N;n0 ;N 0 ;~q
 fðfn;N;~pz À fn0 ;N 0 ;~pz þ~q Þ
 ¤ð¾n0 ;N 0 ;~pz þ~q À ¾n;N;~pz þ h" ½k~ À h" ½q~ Þ
À ðfn0 ;N 0 ;~pz À~q À fn;N;~pz Þ
 ¤ð¾n0 ;N 0 ;~pz À~q À ¾n;N;~pz À h" ½k~ þ h" ½q~ Þgdpz :

ð9Þ

By carrying out manipulations, we have received analytic
expressions for the density of the QAE current in the CQWIP
GaAs/GaAsAl as follows:


1410

N. V. Nhan, N. Van Nghia and N. Van Hieu

 3
e¸jÃj2 f0
2m
e¢¾F
2³h" 5 μvs m½q h" ¢


X
¢h" 2 2
2
B
Â
jIn;N;n0 ;N 0 j exp À
2m n;N
n;N;n0 ;N 0
&


2m² þ 3
 ²3þ eÀ²þ
K3 ð² þ Þ þ 3K2 ð² þ Þ
h" ¢
!
þ 3K1 ð² þ Þ þ K0 ð² þ Þ

jQAE ¼ À

þ ² 3À eÀ²À



2m² À
h" ¢

3

K3 ð² À Þ þ 3K2 ð² À Þ
!'

þ 3K1 ð² À Þ þ K0 ð² À Þ

Fig. 1 The dependence of the QAE current on the radius R of the CQWIP
GaAs/GaAsAl at different values of the temperature T = 290 K (dot line),
T = 295 K (dashed line), T = 300 K (solid line). Here ½q ¼ 1 Â 1011 s¹1.

3
4m 2 ¢¾F
e
¢
h" 6 μFSvs


X
¢h" 2 2
Bn;N
Â
jUn;N;n0 ;N 0 j2 exp À
2m
n;N;n0 ;N 0

þ

e¸jÃj2 v4l ½2q f0 º³ 2



 feÀ»þ »5=2
þ ½K 52 ð»þ Þ þ 3K 32 ð»þ Þ
þ 3K 12 ð»þ Þ þ KÀ 12 ð»þ ފ
À eÀ»À »5=2
À ½K 52 ð»À Þ þ 3K 32 ð»À Þ
þ 3K 12 ð»À Þ þ KÀ 12 ð»À ފg;
ð10Þ
2
À 2
Á
h" 2 ¢ h" ðBn0 ;N 0 ÀBn;N Þ
h" ¢½k
here ²Æ ¼ 2m
Æ m½q , »Æ ¼ ² Æ Æ 2 , with
2R2
¢ = 1/kBT, kB is the Boltzmann constant, T is the temperature
of the system and ¾F is the Fermi energy.
The eq. (10) is the expression of the QAE current in the
CQWIP GaAs/GaAsAl. The results show the dependence of
the QAE current on the temperature of system, the Fermi
energy and the radius of the CQWIP GaAs/GaAsAl are
nonlinear. These results are different from the results of other
authors have obtained in the bulk semiconductor,7­10) superlattice.14,15) The cause of the difference between the bulk
semiconductor,7­10) superlattice14,15) and the CQWIP GaAs/

GaAsAl is characteristics of a one-dimensional system, in
one-dimensional systems, the energy spectrum of electron
is quantized in two dimensions and exists even if the
relaxation time ¸ of the carrier does not depend on the carrier
energy.
3.

Numerical Results and Discussions

To clarify the results obtained, in this section, we consider
the QAE current in the CQWIP GaAs/GaAsAl. This quantity
is considered to be a function of the temperature T, the Fermi
energy ¾F and the radius R of CQWIP GaAs/GaAsAl.
The parameters used in the numerical calculations6,13) are
as follow: ¸ = 10¹12 s, º = 104 W m¹2, μ = 5320 kg m¹3,
vl = 2 © 103 m s¹1, vt = 18 © 102 m s¹1, vs = 5370 m s¹1,
$ = 13.5 eV, m = 0.067 me (me is the mass of free electron).
Figures 1, 2 present the dependence of the QAE current
on the radius R of the CQWIP GaAs/GaAsAl at different
values for the temperature T and the external acoustic wave
frequency ½q~, respectively. In Fig. 1, 2 there is one peak
when the condition ½q~ ¼ ½k~ þ

h" 2 ðB2n0 ;N 0 ÀB2n;N Þ
2mR2

(n 6¼ n0 and

N 6¼ N 0 ) is satisfied. The existent peak in the CQWIP

Fig. 2 The dependence of the QAE current on the radius R of the CQWIP
GaAs/GaAsAl at different values of the acoustic wave frequency ½q ¼
1 Â 1011 s¹1 (dot line), ½q ¼ 2 Â 1011 s¹1 (dashed line), ½q ¼ 3 Â
1011 s¹1 (solid line). Here T = 295 K, ¾F ¼ 0:048 eV.

GaAs/GaAsAl may be due to the transition between minibands (n ! n0 and N ! N 0 ). When we consider the case
n = nA and N = NA. Physically, we merely consider transitions
within sub-bands (intrasubband transitions), and from the
numerical calculations we obtain jQAE ¼ 0, where mean that
only the intersubband transition (n 6¼ n0 and N 6¼ N 0 )
contribute to the jQAE . These results are different from those
in the normal bulk semiconductors,7­10) in the limit of R
approximates micrometer-sized, the electron confinement
ignore, there does not appear peaks, this result is similar to
the results obtained in the normal bulk semiconductors.7­10)
These results are also different from those in superlattice.14,15)
Here, the difference is about shape graph and number of
peaks. In addition, Fig. 2 shows that the peaks move to
the larger frequency of the radius when the frequency of
external acoustic wave ½q~ increases. In contrast, Fig. 1 shows
that the positions of the maxima nearly are not move as
the temperature is varied because the condition ½q~ ¼
½k~ þ

h" 2 ðB2n0 ;N 0 ÀB2n;N Þ
2mR2

(n 6¼ n0 and N 6¼ N 0 ) do not depend on

the temperature. Therefore, We can use these conditions to

determine the peak position at the different value of the
acoustic wave frequency or the parameters of the CQWIP
GaAs/GaAsAl. This means that the condition is determined
mainly by the electron’s energy.


The Dependence of a Quantum Acoustoelectric Current on Some Qualities in a Cylindrical Quantum Wire

1411

CQWIP GaAs/GaAsAl has a maximum peak at a certain
value R = Rm although we change the temperature of system.
However, if the frequency of acoustic wave varies, the peaks
position have a shift. The QAE exists even if the relaxation
time ¸ of the carrier does not depend on the carrier energy,
and the results are similar to those for two-dimensional
systems.14,15) This differs from bulk semiconductors, because
in bulk semiconductors,7­10) the QAE current vanishes for a
constant relaxation time. These results are also different from
the results of other authors have superlattice.14,15) So, the
dependence of a QAE current on some qualities in a CQWIP
GaAs/GaAsAl is newly developed.
Fig. 3 The dependence of the jQAE current on the temperature T and the
Fermi energy ¾F. Here ½q ¼ 3 Â 1011 s¹1.

Figure 3 shows the dependence of the QAE current on the
temperature and the Fermi energy ¾F. The dependence of the
QAE current on the temperatures and the Fermi energy are
not monotonic have a maximum at T = 295 K, ¾F = 0.044 eV
for ½q ¼ 3 Â 1011 s¹1. From the results of research on the

absorption coefficient of electromagnetic wave in superlattice, quantum well, quantum wire3­6) was explained by
transition between the mini-bands and electron confinement
in the low-dimensional structures. This is basic to conclude
the existent peak in the CQWIP GaAs/GaAsAl may be due
to the electron confinement in one-dimensional structures and
transition between mini-bands (n ! n0 and N ! N 0 ).
4.

Conclusion

In this paper, we have theoretically investigated the QAE
in the CQWIP GaAs/GaAsAl. We found the strong nonlinear
dependence of the QAE current on the temperature T, the
Fermi energy and the radius of the CQWIP GaAs/GaAsAl.
The importance of the present work is the appearance of
peak when the condition ½q~ ¼ ½k~ þ

h" 2 ðB2n0 ;N 0 ÀB2n;N Þ
2mR2

(n 6¼ n0 and

N 6¼ N 0 ) is satisfied. Our result indicates that the dominant
mechanism for such a behavior is the electron confinement in
the CQWIP GaAs/GaAsAl and transitions between minibands.
The result of the numerical calculation was done for the
CQWIP GaAs/GaAsAl. This result have shown that the
dependence of the QAE current on the radius R of the

Acknowledgments

This work is completed with financial support from the
National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under Grant no. 103.012015.22.
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