with parabolic confinement potential f.m....
TRANSCRIPT
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: [email protected]
(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
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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