DSpace at VNU: Calculations of the acoustoelectric current in a quantum well by using a quantum kinetic equation

Calculations of the Acoustoelectric Current in a Quantum Well by Using aQuantum Kinetic Equation Nguyen Quang Bau, Nguyen Van Hieu and Nguyen Vu Nhan∗ Faculty of Physics, Hanoi Universit

Calculations of the Acoustoelectric Current in a Quantum Well by Using a

Quantum Kinetic Equation

Nguyen Quang Bau, Nguyen Van Hieu and Nguyen Vu Nhan∗ Faculty of Physics, Hanoi University of Science, Vietnam National University,

334-Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

(Received 18 June 2012, in final form 18 September 2012)

An analytic expression for the acoustoelectric current (AC) j ac induced by electron-external

acoustic wave interactions and electron-internal acoustic wave (internal phonons) scattering in a

quantum well (QW) is calculated by using the quantum kinetic equation for electrons The physical

problem is investigated in the region ql  1 (where q is the acoustic wave number and l is the

electrons mean free path) The dependence of the ACj acon the external acoustic wave frequency

ω
q, the width of the QW L and the temperature T for a specific QW of AlGaAs/GaAs/AlGaAs

is achieved by using a numerical method The computational results show that the dependence of

the ACj acon the temperatureT , the external acoustic wave frequency ω
q, the width of the QWL

is non-monotonic and that the peaks can be attributed to transitions between mini-bandsn → n
.

The dependence of the ACj acon the temperatureT and the Fermi energy F is obtained, and a

maximum of the ACj ac forF = 0.038 eV and ω
q= 3× 1011s−1seen atT = 50 K, which agrees

with the experimental results for AlGaAs/GaAs/AlGaAs QWs All these results are compared

with those for normal bulk semiconductors and superlattices to show the differences Finally, the

quantum theory of the acoustoelectric effect in a quantum well is newly developed

PACS numbers: 84.40.Ik, 84.40.Fe

Keywords: Quantum well, Quantum acoustoelectric current, Electron-external acoustic wave interaction,

Electron-internal phonons scattering, Quantum kinetic equation

DOI: 10.3938/jkps.61.2026

I INTRODUCTION

When an external acoustic wave is absorbed by a

con-ductor, the transfer of the momentum from the

acous-tic wave to the conduction electron may give rise to a

current that is usually called the acoustoelectric (AE)

current j ac 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 ), which, we believe,

should find an important place in the acoustoelectronic

devices

In low-dimensional systems, the energy levels of

elec-trons become discrete and to be different from other

di-mensionalities [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

opti-cal properties of semiconductor quantum wells (QWs),

superlattices (SLs), quantum wires, and quantum dots

(QDs) have attracted much attention in the past few

years For example, the linear absorption of a weak

elec-∗E-mail: nguyenquangbau54@gmail.com; Tel: +84-913-348-020

tromagnetic wave caused by confined electrons in low-dimensional systems has been investigated [3–5] Cal-culations of the nonlinear absorption coefficients of an intense electromagnetic wave by using the quantum ki-netic 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 in bulk semiconductors by using both the Boltz-mann classical kinetic equation and the quantum method [9–15]

In recent years, the AE effect in low-dimensional struc-tures has been extensively studied experimentally and theoretically So far, however, almost all those works [16-24] have been studied theoretically by using the Boltz-mann classical kinetic equation method, and are, thus, limited to the case of the electron-external acoustic wave interaction The AE effect in superlattices [16–18], the

AE current in one-dimensional channel [19], the AE ef-fect in a finite-length ballistic quantum channel [20], the

AC in a ballistic quantum point contact [21], the AC through a quantum wire containing a point impurity, the

AC in submicron-separated quantum wires [22,23], and the AE effect in a carbon nanotube [24] have also been studied In addition, the AE effect has been studied ex-perimentally in a SL [25] and in a QW [26] However,

-2026-the calculation of -2026-the AE current j ac in a QW by using

the quantum kinetic equation method is still open for

study In the present work, we use the quantum kinetic

equation method to study the AC current j acinduced by

the electron-external acoustic wave interactions and the

electron-internal acoustic wave (internal phonons)

scat-tering in a QW The present work is different from

previ-ous works [16-24] because 1) the AE current is a result of

not only the electron-external acoustic wave interaction

but also the electron-internal phonons scattering in the

sample, 2) we use the quantum kinetic equation method,

3) we show that the present results can explain the

ex-perimental results [26]

This paper is organized as follows: In Sec II, we

out-line the quantum kinetic equation for electrons confined

in a QW The analytical expression for the AE

cur-rent in the case of the electron-external acoustic wave

and electron-internal phonon scattering is obtained in

Sec III The numerical results and a brief discussions

are presented for a specific QW AlGaAs/GaAs/AlGaAs

in Sec IV Finally, we present conclusions in Sec V

II QUANTUM KINETIC EQUATION FOR

ELECTRONS IN A QUANTUM WELL

We use a simple model for a QW, in which an

elec-tron gas is confined by an infinite potential along the Oz

direction (along the Oz direction, the energy spectrum

of the electron is quantized, or the motive direction of

electron is limited); electrons are free on the (x-y) plane.

The motion of an electron is confined a QW and its en-ergy spectrum is quantized into discrete levels in the Oz direction Let us suppose that an external acoustic wave

of frequency ω q is propagating along the quantum well

axis (Oz) When

ω q /η = c s q/η
1 and ql  1, (1)

where η is the frequency of the electron collisions, q is

the modulus of the external acoustic wave-vector and

l is the electrons mean free path The acoustic wave

is considered the region ql  1 Under such

circum-stances, the external acoustic wave can be interpreted

as monochromatic phonons having the 3D phonon

distribution function N (k), and the acoustic flux can

be presented as a δ-function distribution in k-space

N (k) = (2π) ω
c s3Φδ(k −q), where Φ is the flux density of the

external acoustic wave (external phonon) with frequency

ω q In the presence of an external acoustic wave with frequency ω q, the Hamiltonian of the electron-external phonon and electron-internal phonon system in a QW

in second quantization representation can be written as (we select
= 1)

H = H0+ H e−ph; H0=

n,p ⊥

ε n (p ⊥ )a+n,p ⊥ a n,p ⊥+

k

ω k b+k b k , (2)

H e−ph =

n,p ⊥ ,n  ,q

C q U n,n  (q)a+n,p ⊥ +q ⊥ a n  ,p ⊥ c qexp(−iω q t) +

n,p ⊥ ,n  ,k

D k n,n  ( k z )a+n,p

⊥ +k ⊥ a n  ,p ⊥ (b k + b+−k ), (3)

where n denotes the quantization of the energy spectrum

in the Oz direction, n = 1, 2, ; a+n,p ⊥ and a n,p ⊥ (b+k

and b k) are the creation and the annihilation operators

of the electron (internal phonon), respectively, c q is the

annihilation operator of the external phonon | n, p ⊥ 

and | n  , p ⊥ +  k ⊥  are electron states before and after

scattering The electron energy takes the simple form

 n,p ⊥= p2⊥

2m+

n2π2

Here, m is the effective mass of the electron, L is the width of the QW, and p ⊥ is the transverse component of the quasi-momentum in the (x-y) plane U n,n  (q) is the matrix element of the operator U = exp(iqy − λ l z):

U n,n  (q) = (−1) n+n 

exp(−λ l L) − 1

λ l L + (n + n )2π2

λ l L

−(−1) n−n



exp(−λ l L) − 1

λ l L + (n − n )2π2

λ l L

where λ l = (q2−ω2

q /c2l)1/2is the spatial attenuation

fac-tor of the potential part of the displacement field, C qand

D k are the external phonon and the

electron-internal phonon interaction factors, respectively and take

the form

C q = iΛc2l (ω q3/2ρ0ΞS) 1/2 ,

Ξ = q



1 + σ l2 2σ l +



σ l

σ t − 2



1 + σ t2 2σ t



,

σ l = 

1− c2

s /c2l1/2

, σ t=

1− c2

s /c2t1/2

,

| D k |2 = Λ2k

Here, Λ is the deformation potential constant, c l and c t

are the velocities of the longitudinal and the transverse

bulk acoustic waves, respectively, c s is the velocity of

the acoustic wave, ρ0is the mass density of the medium,

S = L x L y is the surface area, and

I n  ,n (k z) = 2

L

 L

0 dz sin



n  π

L z

 sin

nπ

L z exp(ik z z).

(7)

In order to establish the quantum kinetic equations for electrons in a QW, we use the electron distribution

function f n,p ⊥ =a+

n,p ⊥ a n,p ⊥  t:

i



∂f n,p ⊥

∂t



ac

=[a+

n,p ⊥ a n,p ⊥ , H] t , (8)

whereΨ tdenotes a statistical average value at the

mo-ment t;Ψ t = T r( W Ψ) (W is the density matrix

oper-ator) Starting from the Hamiltonian, Eqs (2), (3), and (8), and realizing operator algebraic calculations, we ob-tain the quantum kinetic equation for electrons in a QW:



∂f n,p ⊥

∂t



ac

= −π

n  ,q

| C q |2| U n,n  (q) |2N (q) [f ( n,p ⊥)− f( n  ,p ⊥ +q ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ − ω q)

+[f ( n,p ⊥)− f( n  ,p ⊥ +q ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ + ω q ) + [f ( n,p ⊥)− f( n  ,p ⊥ −q ⊥)]

× δ( n  ,p ⊥ −q ⊥ −  n,p ⊥ + ω q ) + [f ( n,p ⊥)− f( n  ,p ⊥ −q ⊥ )]δ( n  ,p ⊥ −q ⊥ −  n,p ⊥ − ω q)



+π

n  ,k

| D k |2| I n,n  (k z)|2N (k) [f ( n,p ⊥)− f( n  ,p ⊥ +k ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ + ω q − ω k)

+[f ( n,p ⊥)− f( n  ,p ⊥ −k ⊥ )]δ( n  ,p ⊥ −k ⊥ −  n,p ⊥ − ω q + ω k)



Equation (9) is fairly general and can be applied for any

mechanism of the interaction In the case of vanishing

electron-internal phonon interaction, it gives the same

results as those obtained Refs 16 – 18

III ACOUSTOELECTRIC CURRENT

The external acoustic wave is assumed to propagate

perpendicular to the Oz axis of the QW After a new

equilibrium has been established, the distribution

func-tion f of the electrons will obey the condifunc-tion

∂f n,p ⊥ /∂t = (∂f n,p ⊥ /∂t) ac + (∂f n,p ⊥ /∂t) th = 0, (10) where (∂f n,p ⊥ /∂t) acis the rate of change caused by the

electron-external acoustic wave and inertial phonons

in-teraction and (∂f n,p ⊥ /∂t) th is the rate of change due

to the interaction of the electron with thermal phonons, impurities, and one another Substituting Eq (9) into

Eq (10) we obtain the basic equation of the problem:

(∂f n,p ⊥ /∂t) th = π

n,n  q

| C q |2| U n,n  (q) |2N (q) [f ( n,p ⊥)− f( n  ,p ⊥ +q ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ − ω q) +[f ( n,p ⊥)− f( n  ,p ⊥ +q ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ + ω q ) + [f ( n,p ⊥)− f( n  ,p ⊥ −q ⊥)]

× δ( n  ,p ⊥ −q ⊥ −  n,p ⊥ + ω q ) + [f ( n,p ⊥)− f( n  ,p ⊥ −q ⊥ )]δ( n  ,p ⊥ −q ⊥ −  n,p ⊥ − ω q)



−π
n,n  k

| D k |2| I n,n  (k z)|2N (k) [f ( n,p ⊥)− f( n  ,p ⊥ +k ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ + ω q − ω k)

+[f ( n,p ⊥)− f( n  ,p −k )]δ( n  ,p −k −  n,p ⊥ − ω q + ω k)



We linearize Eq (11) by replacing f ( n,p ⊥) with

f F ( n,p ⊥ ) + f1, where f F ( n,p ⊥) is the equilibrium

Fermi contribution function As indicated in Ref 27,

(∂f n,p ⊥ /∂t) th=−f1/τ p ; τ pis the momentum relaxation time Thus,

f1 = −πτ

n  ,q

| C q |2| U n,n  (q) |2N (q) [f F ( n,p ⊥)− f F ( n  ,p ⊥ +q ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ − ω q) +[f F ( n,p ⊥)− f F ( n  ,p ⊥ +q ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ + ω q ) + [f F ( n,p ⊥)− f F ( n  ,p ⊥ −q ⊥)]

× δ( n  ,p ⊥ −q ⊥ −  n,p ⊥ + ω q ) + [f F ( n,p ⊥)− f F ( n  ,p ⊥ −q ⊥ )]δ( n  ,p ⊥ −q ⊥ −  n,p ⊥ − ω q)



−πτ

n  ,k

| D k |2| I n,n  (k z)|2N (k) [f F ( n,p ⊥)− f( n  ,p ⊥ +k ⊥ )]δ( n  ,p ⊥ +q ⊥ −  n,p ⊥ + ω q − ω k)

+[f F ( n,p ⊥)− f F ( n  ,p ⊥ −k ⊥ )]δ( n  ,p ⊥ −k ⊥ −  n,p ⊥ − ω q + ω k)



The density of the AE current j ac in the direction of

the external acoustic wave vector q is expressed

j ac=

n

2e

(2π)2



v p f1dp ⊥ , (13)

where v p is the electron velocity given by v p = ∂ n,p /∂p.

Substituting Eq (12) into Eq (13) and solving for a

non-degenerate electron gas, and taking τ pto be constant, we

obtain for the AE current

j ac = A1

n,n 

| U n,n  |2exp(− π2n2

2mL2k B T )(B+− B − ) + A2

n,n 

| I n,n  |2exp(− π2n2

2mL2k B T )(C+− C − ), (14)

where

A1 = (2π)2eΦ ∧2τ c4l ω q2

ρ0c s exp



 F

k B T



,

A2 = (2π)2e ∧2τ (2mk B T π) 1/2

(2π)5ρ0c s mω q

exp



 F

k B T



,

B ± =

1 + D ±2

mk B T exp(−

D ±2 2mk B T ),

D ± = q

2+

m∆ n,n 

q ± m(ω k − ω q)

q ,

C ± = (m∆ n,n  ± ω k)2π 1/2exp(−2(b ± c) 1/2)

4c 3/2

× [2c + 2a ± (b ± c) 1/2 + a ±]

+b ± K 5/2 [2(b ± c) 1/2]

a ± = mk B T ± ∆ n,n  ± ω k

m∆ n,n  ± mω k exp



−∆n,n  ± ω k 2k B T



,

b ± = (m∆ n,n  ± mω k)2

2mK B T , c =

1

8mk B T ,

∆n,n  = π2

2mL2(n

2− n 2 ). (15)

Here,  F is the Fermi energy, k B is the Boltzmann

con-stant, and K n (x) is the Bessel function of 2nd order.

Equation (14) is the analytical expression for the AE current in a QW when the momentum relaxation time is

a constant In the case of the vanishing electron-internal phonon interaction, this result is the same as that ob-tained by using the Boltzmann kinetic equation in a QW

IV NUMERICAL RESULTS AND

DISCUSSION

To clarify the results obtained so for, in this section,

we consider the AE current This quantity is considered

to be a function of the temperature T , the acoustic wave number q, the acoustic intensity Φ, and the parameters of

the AlGaAs/GaAs/AlGaAs QW The parameters used

in the calculations are σ = 5300 kg m −3 , τ = 10 −12 s,

m = 0.067 m0, m0 being the mass of a free electron,

Φ = 104 W m−2 , and c l = 2× 103 ms−1 , c t= 18× 102

ms−1 , c s= 8× 102 ms−1.

Figure 1 shows the dependence of the AE current on the temperature and the Fermi energy The dependence

of the AE current on the temperatures and the Fermi

energy are not monotonic have a maximum at T = 50

K,  F = 0.038 eV for ω q = 3× 1011 s−1 This result

agrees with the experimental results [26] According to

[26] the peak appears at T = 50 K for ω q ≈ 3 × 1011

(s−1). However, Refs 26 contains no explanation for

Fig 1 Dependence of thej accurrent on the temperature

T and the Fermi energy (ωq = 3× 1011 (s−1), n = 1 → 2,

n
= 1→ 2).

Fig 2 Dependence of thej accurrent on the frequencyωq

of the external acoustic wave at different values of the QW’s

widths L = 30 nm (solid line), L = 31 nm (dot line), and

L = 32 nm (dashed line) Here T = 50 K, F = 0.038 eV,

n = 1 → 3, n
= 1→ 3.

this behavior From our calculation, we conclude that

the dominant mechanism for such a behavior is electron

confinement in the QW

Figure 2 presents the dependence of the AE current

on the frequency ω q of the external acoustic wave at

dif-ferent values for the QW’s widths Figure 2 shown some

maxima when the condition ω q = ω k ± ∆ n,n  (n = n )

is satisfied This result is different from the AE

cur-rent in a bulk semiconductor [9–15], because in a bulk

semiconductor, when the q increase, the AE current

in-creases linearly The cause of the difference between the

bulk semiconductor and the QW is characteristics of

a low-dimensional system, in low-dimensional systems,

the energy spectrum of electron is quantized and

ex-ists even if the relaxation time τ of the carrier does

Fig 3 Dependence of thej accurrent on the width of the

quantum well at different values of the external acoustic wave frequencyωq= 32× 1010s−1(solid line),ωq= 31× 1010 s−1

(dot line), andωq= 30× 1010s−1nm (dashed line) HereT

= 50 K,F = 0.038 eV,n = 1 → 3, and n
= 1→ 3.

Fig 4 Dependence of thej accurrent on the width of the

quantum well at different values of the temperatureT = 50

K (solid line), T = 52 K (dot line), and T = 54 K (dashed

line) Hereωq= 3× 1011(s−1),F = 0.038 eV,n = 1 → 3,

andn
= 1→ 3.

not depend on the carrier energy In Fig 2, there are two peaks This is attributed to the transitions between

mini-bands (n → n ), and the two peaks correspond to

(n = 1 → n  = 2) and (n = 2 → n  = 3) transitions or

intersubband transitions as the main contribution to j ac

When we consider the case n = n  Physically, we merely consider transitions within subbands (intrasubband tran-sitions), and from the numerical calculations we obtain

j ac = 0, where mean that only the intersubband

tran-sition (n = n  ) contribute to the j ac In superlattices [16–18], the AC appears even if the intrasubband transi-tions In addition, the positions of the peaks change

sig-nificantly when the quantum well’s width increases This

can be explained by assuming that only electrons whose

momenta comply with the condition ω q = ω k ± ∆ n,n 

(n = n ) contribute considerably to the effect

Figure 3 shows the dependence of the j acon the width

of the QW for different values of the external acoustic

wave frequency ω q and Fig 4 shows the dependence of

the j acon the width of the QW for differen values of the

T The figures show that the dependence of j ac on the

width of well is strong and nonlinear The dependence

of the j acon the width of the QW has maximum values

(peaks) The figures also show that when we consider

different transitions, we obtain peaks at different values

of L when the condition ω q = ω k ± ∆ n,n  (n = n ) is

sat-isfied Moreover, Fig 3 shows that at small QW width,

peaks are smaller and that the peaks move to the smaller

QW width when the frequency of the acoustic wave

in-creases In contrast, Fig 4 shows that the positions of

the maxima nearly are not move as the temperature is

varied because the condition ω q = ω k ± ∆ n,n  (n = n )

do not depend on the temperature This means that the

condition is determined mainly by the electron’s energy

V CONCLUSION

In this paper, we have obtained analytical expressions

for the j acin a QW by using the quantum kinetic

equa-tion for the distribuequa-tion funcequa-tion of electrons interacting

with an external acoustic wave and internal phonons We

have shown the strong nonlinear dependence of j acon the

temperature T, the frequency ω q of the external acoustic

wave and the width L of the QW The importance of the

present work is the appearance of peaks when the

con-dition ω q = ω k ± ∆ n,n  (n = n ) is satisfied, the results

are complex and different from those obtained in bulk

semiconductors [9–15] and the superlattices [16–18]

The numerical results obtained for the AlGaAs/GaAs/

AlGaAs QW show that a peak exists at T = 50 K,

ω q = 3×1011s−1 and  F = 0.038 eV, which fits with the

experimental results [26] Our result indicates that the

dominant mechanism for such a behavior is electron

con-finement in the QW and transitions between mini-bands

n → n  The j ac exists even if the relaxation time τ of

the carrier does not depend on the carrier energy, and the

results are similar to those for superlattices [16,18] This

differs from bulk semiconductors, because in bulk

semi-conductors [9–15], the AE current vanishes for a constant

relaxation time

ACKNOWLEDGMENTS

This work was completed with financial support

from the Vietnam National University, Ha Noi (No

QGTD.12.01) and Vietnam NAFOSTED (No

103.01-2011.18)

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oper-ator) Starting from the Hamiltonian, Eqs (2), (3), and (8), and realizing operator algebraic calculations, we ob-tain the quantum kinetic equation for... the analytical expression for the AE current in a QW when the momentum relaxation time is

a constant In the case of the vanishing electron-internal phonon interaction, this result is the. .. general and can be applied for any

mechanism of the interaction In the case of vanishing

electron-internal phonon interaction, it gives the same

results as those obtained