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

The Dependence of a Quantum Acoustoelectric Current on Some Qualitiesin a Cylindrical Quantum Wire with an In finite Potential GaAs/GaAsAl Nguyen Vu Nhan1,+, Nguyen Van Nghia2,4and Nguyen

The Dependence of a Quantum Acoustoelectric Current on Some Qualities

in a Cylindrical Quantum Wire with an In finite Potential GaAs/GaAsAl

Nguyen Vu Nhan1,+, Nguyen Van Nghia2,4and Nguyen Van Hieu2,3

1Faculty of Physics, Academy of Defence Force­ Air Force, Son Tay, Hanoi, Vietnam

2Faculty of Physics, Hanoi University of Science, Vietnam National University, 334-Nguyen Trai, Hanoi, Vietnam

3Faculty of Physics, Danang University, 459 Ton Duc Thang, Danang, Vietnam

4Faculty of 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

condition ½ ~q ¼ ½~kþ h
2 ðB 2

n 0 ;N 0 B 2 n;N Þ

electron confinement in the CQWIP GaAs/GaAsAl and transitions between mini-bands All these results are compared with those for normal

(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

low-dimensional 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

semi-conductors 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

electro-magnetic 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

submicron-separated 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 QAMEfield in a QW In the

present work, we use the quantum kinetic equation method

for external acoustic wave interaction and

electron-acoustic phonon (internal electron-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 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 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) super-lattice.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

¼n;N;~pzð~rÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffi1

³R2L

p expðinºÞ exp ipz

h
z

¼n;Nð~rÞ

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

of the CQWIP; L is the length of the CQWIP; ~p ¼ ð0; 0; pzÞ

Materials Transactions, Vol 56, No 9 (2015) pp 1408 to 1411

Special Issue on Nanostructured Functional Materials and Their Applications

is the electronỖs momentum vector along z-direction;

Ửn;Nđ~rỡ ỬJ n đB n;N r=Rỡ

J nợ1 đB n;N ỡ is the radial wave function of the

electron in the plane Oxy, with Bn,Nare the N level root of

Bessel function of the order n

The electron energy spectrum takes the form

ớn;N;~pz Ử h
2p2z

h
2B2n;N

where m is the effective mass of the electron

We assume that an external acoustic wave of frequencyơ~q

is propagating along the CQWIP axis (Oz) and the acoustic

wave will be considered as a packet of coherent phonons with

theấ-function distribution in ~k-space Nđ~kỡ Ửđ2Ỡỡơ~qvs3Ứấđ~k  ~qỡ,

whereỨ is the flux density of the external acoustic wave with

frequency ơ~q, vsis 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

n;N;~ p z

ớn;N;~pzaợn;N;~p

zan;N;~pz

n;N;n0;N0;~k

In;N;n0;N0C~kaợ

n0;N0;~ p z ợ~kan0 ;N0;~ p0zđb~kợ bợ

~kỡ

~k

h
ơ~kbợ~kb~k

n;N;n0;N0;~q

C~qUn;N;n0;N0aợn0 ;N0;~ p z ợ~qan0 ;N0;~ p0zb~qexpđiơ~qtỡ;

đ3ỡ where C~kỬ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik=đ2μvsSLỡ

is the electron-internal phonon interaction factor,μ is the mass density of the medium, $ is

the deformation potential constant, C~qỬ iv2

l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h
ơ3~q=đ2μFSỡ

p

is the electron-external phonon interaction factor, with F Ử

qơđ1 ợ ở2lỡ=đ2ởtỡ ợ đởl=ởt 2ỡđ1 ợ ở2

tỡ=đ2ởtỡ, ởlỬ đ1 

v2s=v2lỡ1=2, ởtỬ đ1  v2

s=v2tỡ1=2, S= ỠR2 is the surface area,

vl (vt) is the velocity of the longitudinal (transverse) bulk

acoustic wave, aợn;N;~pz (an;N;~ p z) is the creation (annihilation)

operator of the electron; bợ~k (b~k) is the creation (annihilation)

operator of internal phonon and b~q is the annihilation

operator of the external phonon The notation jn; ~ki is the

electron states before interaction and jn0; ~k ợ ~qi is the

electron states after interaction Un,N,nA,NAis the matrix element

of the operator U= exp(iqy Ự klz):

Un;N;n0 ;N0 Ử 2 expđklLỡ

R2L

ZR

0 Ử

n0;N0;~ p0zđ~rỡ

 Ửn;N;~pzđ~rỡ expđiq?rỡdr; đ4ỡ here kl= (q2Ự (ơq/vl)2)1/2is the spatial attenuation factor of

the potential part the displacement field and In,N,n A,NA is the

electronic form factor:

In;N;n0 ;N0Ử 2

R2

ZR

0 Jjnn0 jđq?RỡỬn0 ;N0;~ p0zđ~rỡỬn;N;~pzđ~rỡrdr; đ5ỡ 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;~ p zđtỡit

@t

QAEỬ hơaợ

n;N;~ p zan;N;~pz; Hit; đ6ỡ where the notation hXit is mean the usual thermodynamic average of the operator X and fn;N;~ p zđtỡ Ử haợ

n;N;~ p zan;N;~pzitis the 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

fđtỡ Ử 2Ỡị

h
2

X

n0;N0;~k

jCkj2jIn;N;n0 ;N0j2Nkfđfn;N;~pz fn0 ;N0;~ p z ợ~kỡ

 ấđớn0 ;N0;~ p z ợ~k ớn;N;~pz h
ơ~kỡ

ợ đfn;N;~pz fn0 ;N0;~ p z ~kỡấđớn0 ;N0;~ p z ~k ớn;N;~pzợ h
ơ~kỡg

ợ Ỡị

h
2

X

n0;N0;~q

jCqj2jUn;N;n0;N0j2Nqfđfn;N;~pz fn0 ;N0;~ p z ợ~qỡ

 ấđớn0 ;N0;~ p z ợ~q ớn;N;~pzợ h
ơ~k h
ơ~qỡ

 đfn0 ;N0;~ p z ~q fn;N;~pzỡ

 ấđớn0 ;N0;~ p z ~q ớn;N;~pz h
ơ~kợ h
ơ~qỡg; đ7ỡ where ị 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 ấ 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

jQAE Ử 2e 2Ỡh

X

n;N

Z

vp zfđtỡdpz; đ8ỡ here vp z is the average drift velocity of the moving charges and it is given by vp z Ử @ớn;N;~pz=@pz

Substituting eq (7) into eq (8) and takingị to be constant,

we obtain for the density of the QAE current in the CQWIP GaAs/GaAsAl

jQAE Ử  2eị

h
3

X

n;N;n 0 ;N 0 ;~k

Z

vpzjCkj2jIn;N;n0 ;N0j2Nk

 fđfn;N;~pz fn0 ;N0;~ p z ợ~kỡấđớn0 ;N0;~ p z ợ~k ớn;N;~pz h
ơ~kỡ

ợ đfn;N;~pz fn0 ;N0;~ p z ~kỡ

 ấđớn0 ;N0;~ p z ~k ớn;N;~pzợ h
ơ~kỡgdpz

ợ eị

h
3

X

n;N;n0;N0;~q

Z

vp zjCqj2jUn;N;n0;N0j2Nq

 fđfn;N;~pz fn0 ;N0;~ p z ợ~qỡ

 ấđớn0 ;N0;~ p z ợ~q ớn;N;~pzợ h
ơ~k h
ơ~qỡ

 đfn0 ;N 0 ;~ p z ~q fn;N;~pzỡ

 ấđớn0 ;N0;~ p z ~q ớn;N;~pz h
ơ~kợ h
ơ~qỡgdpz: đ9ỡ

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

jQAE Ử  eịjj2f0

2Ỡh
5μvsmơq

2m

h
ằ

eằớF

n;N;n0;N0

jIn;N;n0 ;N0j2exp ằh
2

2mB2n;N





ỗ3

ợeỗợ

 2mỗợ

h
ằ

K3đỗợỡ ợ 3K2đỗợỡ

ợ 3K1đỗợỡ ợ K0đỗợỡ



ợ ỗ3

eỗ

 2mỗ

h
ằ

K3đỗỡ ợ 3K2đỗỡ

ợ 3K1đỗỡ ợ K0đỗỡ



ợeịjj2v4lơ2

qf0ỨỠ2

h
6μFSvs

4m

ằ

eằớF

n;N;n0;N0

jUn;N;n0;N0j2exp ằh
2

2mB2n;N

 feỪ ợỪ5=2ợ ơK5đỪợỡ ợ 3K3đỪợỡ

ợ 3K1đỪợỡ ợ K 1đỪợỡ

 eỪ Ừ5=2

 ơK5đỪỡ ợ 3K3đỪỡ

here ỗỬh
2 ằ

2m

h
đB 2

2R 2  mơq , ỪỬ ỗh
ằơ k

2 , with

ằ = 1/kBT, kBis the Boltzmann constant, T is the temperature

of the system andớFis 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,7ễ10)

super-lattice.14,15) The cause of the difference between the bulk

semiconductor,7ễ10)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 timeị 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 ớF and the radius R of CQWIP GaAs/GaAsAl

The parameters used in the numerical calculations6,13) are

as follow: ị = 10Ự12s, Ứ = 104W mỰ2, μ = 5320 kg mỰ3,

vl= 2 ẹ 103m sỰ1, vt= 18 ẹ 102m sỰ1, vs= 5370 m sỰ1,

$ = 13.5 eV, m = 0.067 me(meis 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 ơ~q, respectively In Fig 1, 2 there is one peak

when the condition ơ~q Ử ơ~kợh
2 đB 2

2mR 2 (n 6Ử n0 and

N 6Ử N0) is satisfied The existent peak in the CQWIP

GaAs/GaAsAl may be due to the transition between mini-bands (n ! n0 and N ! N0) 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 jQAEỬ 0, where mean that only the intersubband transition (n 6Ử n0 and N 6Ử N0) contribute to the jQAE These results are different from those

in the normal bulk semiconductors,7ễ10) 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.7ễ10) 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 waveơ~qincreases In contrast, Fig 1 shows that the positions of the maxima nearly are not move as the temperature is varied because the condition ơ~qỬ

ơ~kợh
2đB 2

2mR 2 (n 6Ử n0 and N 6Ử N0) 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 electronỖs energy

N V Nhan, N Van Nghia and N Van Hieu 1410

Figure 3 shows the dependence of the QAE current on the

temperature and the Fermi energy¾F 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 ½q¼ 3  1011s¹1 From the results of research on the

absorption coefficient of electromagnetic wave in

super-lattice, 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 ! N0)

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 condition½~q¼ ½~kþh
2ðB 2

2mR 2 (n 6¼ n0and

N 6¼ N0) 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

mini-bands

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

CQWIP GaAs/GaAsAl has a maximum peak at a certain value R= Rmalthough 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

Acknowledgments This work is completed with financial support from the National Foundation for Science and Technology Develop-ment of Vietnam (NAFOSTED) under Grant no 103.01-2015.22

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