Moldavian Journal of the Physical Sciences, Vol.9, N3-4, 2010

PHONON DRAG THERMOPOWER OF 2D ELECTRON GAS IN QUANTUM WELL

WITH PARABOLIC CONFINEMENT POTENTIAL

F.M. Hashimzade, M.M. Babayev, B.H. Mehdiyev, and Kh.A. Hasanov

Institute of Physics, National Academy of Sciences of Azerbaijan, 33, H. Javid ave., AZ 1143,

Baku, Azerbaijan; e-mail: firudin_hashimzade@yahoo.com

(Received 13 September 2010)

Abstract

We developed a quantitative theory of the phonon-drag thermopower for the 2D electron

gas in a quantum well with parabolic confinement potential. The temperature gradient is directed

across the confinement direction, i.e., along the plane of a two-dimensional electron gas. Our

theoretical results are in a good agreement with experimental data [1] in a range of 1-10 K. In

this temperature interval, the main contribution to the thermopower comes from phonon-drag.

1. Introduction

The thermoelectric and thermomagnetic effects in quantum wells were studied theoreti-

cally and experimentally in [1-8]. Here we study the phonon-drag thermopower in a quantum

well with parabolic confinement potential. In the case where the temperature gradient (and

therefore the current) is directed along the plane of the two-dimensional electron gas, it is suf-

ficient to consider the relaxation time approximation and to use the Boltzmann equation.

2. Theory

We consider a simple model for the quantum well, in which a two-dimensional electron

gas is confined in the x-direction and the temperature gradient is parallel to the y-axis. The

parabolic confining potential can be written as 2 2

0/2

xU m xω= , where m is the effective mass

of a conduction electron, and 0

ω is the parameter of the parabolic potential. For this model,

the one-particle Hamiltonian, its eigenvalues and eigenfunctions are given in [9].

As the temperature gradient is directed along the y-direction, the thermopower is ob-

tained from the equation 0yj = as α β σ= [10]. The coefficient β (and therefore the ther-

mopower α ) consists of two components: the diffusion and the phonon-drag: e ph

β β β= + .

Using the concept of generalized force given in [10],

ph Be E T A k TT

ε ς−Φ = − − ∇ − ∇

rr

, (1)

where the last term is related to the phonon-drag effect, Er

is the electric field, ε and ζ are

the energy and chemical potential of electrons, we can obtain the electrical conductivity (σ ),

diffusion (e

β ) and the phonon-drag ( phβ ) components of coefficient β :

( )2 20

e y

fe

γ

σ τ ε υ

ε

∂⎛ ⎞= −⎜ ⎟∂⎝ ⎠

∑ , (2)

Moldavian Journal of the Physical Sciences, Vol.9, N3-4, 2010

300

( ) 20

e e y

fe

Tγ

ε ζβ τ ε υ

ε

∂ −⎛ ⎞= − −⎜ ⎟∂⎝ ⎠

∑ , (3)

( ) 20

ph B e y ph

fe k A

γ

β τ ε υε

∂⎛ ⎞= − −⎜ ⎟∂⎝ ⎠

∑ . (4)

Here e is the elementary charge, υr

and ( )e

τ ε are the velocity and relaxation time of elec-

trons, respectively, 0f is the equilibrium electron distribution function, ( )y z

N k kγ = , , is a set

of quantum numbers that determine electronic states in a magnetic field, N is the oscillation

quantum number, ( )y zk k k,r

is the wave vector of electrons,

( ) ( )( ) ( )( )

( )( )

02 22

2 exp2

qxph da pa ph k q k

qB

dNq TR qsA w q w q q

k T dTτ δ ε ε

υ+

∇⎛ ⎞= − + − −⎜ ⎟

∇⎝ ⎠∑ r rr

Pr

r

h r , (5)

( ), ,x y z

q q q qr

and ( )ph qτ are the phonon wave vector and the relaxation time of phonons, re-

spectively, 0

qN is the occupation number (the Planck function) for phonons with frequency

qs qω = , s is the sound velocity, ρ is the density of the material, and ( ),

y zq q qP

r is the com-

ponent of wave vector of phonon in the plane of the free movement of electrons. In (5) both

the piezoelectric (pa) and the deformation (da) interaction of electrons with acoustic phonons

are taken into account:

( )( )

2 2

1

2da

q

E qw q

q

π

ε ρω

=

P

, ( )( )

2 2

2pa

q

ew q

q

π β

ε ρω=

P

, (6)

Here 1E is the deformation potential coefficient,

140 89eβ χ= . / ,

14e is the piezoelectric con-

stant [11], χ is the static dielectric constant.

In most experimental studies [1-3] the average energy of electrons satisfies the condi-

tion 0

ε ω< h , i.e., the electrons are mostly at the zeroth oscillator level ( 0N = ) and the den-

sity of states of the 2D electron gas becomes a constant: ( ) 2g mε π= h . In this case, the

expression for the chemical potential is [9]: 2

0

2

n

m

ω πζ = +

h h, (7)

where n is the areal density of electrons.

We obtained an analytical expression for the dielectric function that describes screening

of potential of electron-phonon interaction, as well as that of the elastic interaction of elec-

trons with ionized impurities, surface and alloy scatterings. The electron-electron interaction

in the quantum well is characterized by a Coulomb potential and a form-factor due to con-

finement. Therefore, for the dielectric function we have:

( )

2 21

2 2

2

22

1

R q Rqme e erfc

qq

ε

π χ

⎛ ⎞⎜ ⎟⎝ ⎠= +

P P

P

Ph

, (8)

where ( )1 2

/R mω= h is the “oscillator length”.

The scattering mechanisms of electrons explicitly considered in the present paper are

the acoustic phonon scattering via deformation and piezoelectric couplings, the impurity scat-

F.M. Hashimzade, M.M. Babayev, et al.

301

tering arising from ionized impurities in the quantum well, the interface roughness scattering,

and the alloy scattering.

The scattering rate accociated with da scattering is

( )2 22

2 212 2

1

3 2 2 20

22 211

1

y x

B

DA

me Re erfc y xmk T E xdx

s R y x xτ π π ρ π χ

−

⎛ ⎞⎜ ⎟= +⎜ ⎟ −⎝ ⎠∫

h h. (9)

The scattering rate accociated with pa scattering is

( ) ( )2 2

2 2

22 212 2

2

3 2 2 2 20

211 2

1

y x

y xB

PA

me Re erfc y xmk T e R xe erfc y x dx

s y y x x

β

τ π χ ρ π χ

−

⎛ ⎞⎜ ⎟= +⎜ ⎟ −⎝ ⎠∫

h h. (10)

where 2x q k= , y Rk= .

The scattering rate accociated with impurity scattering is

( )( )2 2

2 2

22 212 4 2

22

3 2 2 2 20

241 11

1

y x

y xi

I

me Re erfc y xmn Z e Re erfc y x dx

y y x x

π

τ χ π χ

−

⎛ ⎞⎜ ⎟= +⎜ ⎟ −⎝ ⎠

∫h h

, (11)

where Ze and in are the charge and areal density of impurity ions, respectively. The scatter-

ing rate accociated with interface roughness scattering is

( )2 2 2 2 2

2

22 2

12 2 2 2

2 2 20

21 41

1

y x x

R

IR

me Re erfc y xm xe dx

R y x x

ω

τ π π χ

−

Λ Δ−

⎛ ⎞Λ Δ ⎜ ⎟= +

⎜ ⎟ −⎝ ⎠∫

h h, (12)

where Δ is the average displacement of the interface and Λ is of the order of the range of its

spatial variation in the direction parallel to the surface [12].

Since 1 x x

Ga Al As−

is a disodered alloy, electron will be scattered by fluctuating poten-

tial. The scattering rate accociated with alloy scattering is

( ) ( )2 22

2 212 2

0

3/2 3 2 20

24 111

2 1

y x

all

all

me Re erfc y xm U x x xdx

R y x xτ π π χ

−

⎛ ⎞Ω − ⎜ ⎟= +

⎜ ⎟ −⎝ ⎠∫

h h, (13)

where 0

Ω is the volume of elementary cell, and all

U is the difference of screened Coulomb

potential of atoms Ga and Al.

The scattering mechanisms of phonons considered in the present paper are the Her-

ring’s, Simons’s, and boundary scattering mechanisms. For the different mechanisms of re-

laxation of phonons, we have

( )( )

( )( )

( )2 22

2 22 2 2 1 2

,2 22 20

2

1

1

y x

ph ph

B

me Re erfc y xe s m xA y F y x dx

y xk T R x

βτ

π χπ ρ

−

⎛ ⎞⎜ ⎟= +⎜ ⎟ −⎝ ⎠

∫l lhh

, (14)

where

( )22 2

1 212

t

B

E qsqF y x q Sinh e dt

k T e Rβ

−∞

−

−∞

⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ + ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠∫ l

l

h, ( )

1

2 2 2 2 24q R k x t= + . (15)

For the Herring’s ( 2=l ), Simons’s ( 1=l ), and boundary scattering ( 0=l ) mechanisms,

respectively,

Moldavian Journal of the Physical Sciences, Vol.9, N3-4, 2010

302

( )

2 3 2

,2 3ph

B

s R

k T

ρτ =

h;

( )

3 4

,1 4ph

B

s R

k T

ρτ =

h; ,0ph

L

sτ = , (16)

where L is the phonon mean free path due to boundary scattering.

For the degenerated electron gas (η = ζ/kBT >>1), we obtain 2 2

2

F Fe k

m

τ

σ

π

= , (17)

B

e

k

e

σβ

η

∂= −

∂, (18)

( )Bph ph F

kA y

eβ σ= − . (19)

where Fk and 2

/2F

n k π= are the Fermi’s wave number and the areal density of electrons.

3. Numerical results and discussion

In this section, we apply our theoretical results to the 2D degenerate electron gas in

GaAs/AlxGa1-xAs quantum wells. Numerical calculations for the thermopower were performed

for sample 1 in [1], with the electron density n=1.78·1015

m-2

, the mobility μ=22.7 m2/Vs, the

mean free path of phonons L=3·10–4

m, the width of the quantum well 810

xL m

−

= [1], the

mass density ρ=5.3·103 kg/m

3, the longitudinal sound velocity s=5·10

3 m/s, the effective mass

00 067m m= . (

0m is the free electron mass), the acoustic deformation potential

17 4E eV= . ,

and the piezoelectric constant 2

140 16e C m= . / [11].

Our analysis shows that the situations considered in [1-3] satisfied the conditions of the

quantum limit and the strong degeneracy of electron gas. Estimation shows that the dominant

mechanism of scattering of the electrons is the scattering by ionized impurities, and for the

phonons the dominant mechanism is the surface scattering.

Fig. 1. Variation of thermopower as a func-

tion of temperature T: the diffusion component

(curve 1), the phonon-drag thermopower (curve 2,

due to pa-coupling; curve 3, due to da-coupling;

curve 4, the total phonon-drag thermopower), and

the total thermopower (curve 5). Open and solid

circles are the experimental data [1].

Fig. 2. Plots of thermopower against T:

curve 1, unscreened e

α ; curve 2, screened e

α ;

curve 3, unscreened phα ; curve 4, screened

phα ; curve 5, unscreened total thermopower

α ; curve 6, screened total thermopower α .

F.M. Hashimzade, M.M. Babayev, et al.

303

We estimated parameter ω0=7·1013

s-1

of the parabolic potential using condition R~Lx/2.

Our theoretical results for the variation of thermopower as a function of temperature T are shown

in Fig. 1. The results are in a good agreement with experimental data in a range of 1-10 K. In this

temperature interval the diffusion thermopower is by 3 to 8 times smaller than the phonon-drag

thermopower. Thus, for example, at T=5 K, 215ph V Kα μ= / and 36e

V Kα μ= / . At tempera-

tures below 1.5 K, the main contribution to the thermopower comes from the piezoelectric inter-

action; above 1.5 K, from the deformation interaction with acoustic phonons.

The calculations indicate the importance of screening. In Fig. 2, we give screened and

unscreened results for diffusion thermopower (curves 1 and 2), phonon-drag thermopower

(curves 3 and 4) and total thermopower (curves 5 and 6). Values of the unscreened ther-

mopower are higher than the screened one. We find a 12% reduction in e

α . The effect of

screening on phonon-drag thermopower is significant, so the value of phα in a range of

1-10 K without taking into account the screening is 1.5-2 times higher.

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