Signature of phonon drag thermopower in periodically modulated ...
Signature of phonon drag thermopower in periodically modulated ...
Signature of phonon drag thermopower in periodically modulated ...
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ALAIN NOGARET PHYSICAL REVIEW B 66, 125302 2002<br />
dence <strong>of</strong> the <strong>thermopower</strong> is calculated and results are discussed<br />
<strong>in</strong> the third section.<br />
II. ISOTROPIC ELECTRON PHONON SCATTERING<br />
The temperature gradient is applied along the potential<br />
modulation taken as V(y)V 0 cos(qy) where q2/a. This<br />
was experimentally realised by gat<strong>in</strong>g a 2DEG with 200-nmwide<br />
metal f<strong>in</strong>ger gates with a periodicity a500 nm. The<br />
electron density <strong>in</strong> the 2DEG was 2.410 15 m 2 and the<br />
mobility 930 000 cm 2 V 1 s 1 . The modulation had a piezoelectric<br />
orig<strong>in</strong>, 19 and its amplitude, calculated from the width<br />
<strong>of</strong> the positive magnetoresistance, was found to be V 0<br />
1 mV. The <strong>phonon</strong> <strong>drag</strong> <strong>thermopower</strong> S yy and the Nernst-<br />
Ett<strong>in</strong>gshausen coefficient S yx are obta<strong>in</strong>ed by measur<strong>in</strong>g the<br />
voltage drop parallel and perpendicular to the temperature<br />
gradient. The electron-<strong>phonon</strong> scatter<strong>in</strong>g rate responsible for<br />
<strong>phonon</strong> <strong>drag</strong> is given by 11<br />
f k<br />
<br />
t<br />
ph<br />
f k 0<br />
k <br />
m*s <br />
<br />
ep v k<br />
“T<br />
, 1<br />
T<br />
where f k 0 /k is the derivative <strong>of</strong> the Fermi-Dirac function,<br />
m*0.067m 0 is the electron effective mass, s is the <strong>phonon</strong><br />
velocity <strong>of</strong> the acoustic mode , is the <strong>phonon</strong> mean<br />
free path, and v k is the velocity <strong>of</strong> an electron <strong>in</strong> state k. At<br />
4 K, the 2DEG is highly degenerate hence f k 0 /k may be<br />
replaced by ( k F) <strong>in</strong> Eq. 1. An electron on the<br />
Fermi surface travell<strong>in</strong>g with velocity v(y) along the direction<br />
u 1(cos ,s<strong>in</strong> ) will be subject to a <strong>phonon</strong> <strong>drag</strong> W 0 ,<br />
W 0 ,y 0<br />
<br />
dkk 2e<br />
4 2<br />
f<br />
t ph<br />
y<br />
2<br />
0S 0<br />
u<br />
D<br />
1•“T,<br />
0<br />
2<br />
where 0n se 2 /m* is the Drude conductivity, D 0<br />
1/2 F 2 and<br />
S 0 <br />
m*s <br />
<br />
e epT<br />
is the isotropic <strong>phonon</strong> <strong>drag</strong> <strong>thermopower</strong>, F is the<br />
Fermi velocity <strong>of</strong> the un<strong>modulated</strong> 2DEG and e is the<br />
electron charge. The thermoelectric current is j<br />
a 2 0<br />
0(dy/a)0 d f (y,)u1 where f 0 is the charge distribution<br />
function solution <strong>of</strong> Boltzmann equation:<br />
Lf 0 yu 1•“ rycos c <br />
<br />
1 2 d<br />
1<br />
0<br />
2 f 0 y,W 0 y,. 3<br />
ceB/m* is the cyclotron frequency. Equation 3 is<br />
mathematically identical to that govern<strong>in</strong>g the resistivity<br />
problem previously solved by Beenakker; 2 therefore, only<br />
key steps <strong>of</strong> the calculation will need to be recalled here. A<br />
function F 0 (y,) is <strong>in</strong>troduced as<br />
2 f 0 y, y 0<br />
S<br />
D<br />
0u11“T 0<br />
F 0 y, 1S 0 cT xT y, 4<br />
which has to satisfy<br />
LF 0 eEy/ F<br />
where E(y)dV(y)/dy is the periodic electric field. n 0<br />
is the unperturbed conductivity tensor <strong>of</strong> order n (n1,2...)<br />
which has components xx yy n and xy yx<br />
n c n where n 0 /1(n c) 2 . It will also be useful<br />
to <strong>in</strong>troduce the vector u ncos(n),s<strong>in</strong>(n). The current<br />
<strong>in</strong>tegrals are simplified by observ<strong>in</strong>g that the only non<br />
vanish<strong>in</strong>g contributions arise from the terms <strong>in</strong> cos and<br />
s<strong>in</strong> <strong>in</strong> the Fourier expansion <strong>of</strong> F 0 . Insert<strong>in</strong>g this expansion<br />
<strong>in</strong>to Eq. 5 gives the Fourier coefficients <strong>of</strong> cos and s<strong>in</strong> ,<br />
and one obta<strong>in</strong>s<br />
5<br />
a dy 2 d 0 a 0 2 yF0 y,u 1K c,1. 6<br />
The ratio eV 0 / F is small (0.12) and, to first order<br />
<strong>in</strong> , Eq. 5 has an exact solution whose oscillatory dependence<br />
is given by Bessel functions <strong>of</strong> complex order, 20,21<br />
K<br />
125302-2<br />
D 0<br />
1 c 2 0<br />
2<br />
4<br />
ql 2<br />
1 c 2<br />
a dy<br />
a 0<br />
eEy<br />
F<br />
2<br />
0 y,<br />
2 d<br />
F<br />
J i/c qR cJ i/c qR c<br />
s<strong>in</strong>h/ c<br />
/ c J i/ c qR cJ i/c qR c<br />
7<br />
where Rcm* F /eB. Comb<strong>in</strong><strong>in</strong>g Eqs. 4 and 6 one f<strong>in</strong>ds<br />
the thermoelectric current to be: jS01“T, where<br />
1 1 1K c 2 1K c<br />
1K c 1K 8<br />
is the Drude conductivity tensor modified by the oscillatory<br />
contribution from the grat<strong>in</strong>g. The thermoelectric tensor is<br />
identified as L 1S 0 1 and, recall<strong>in</strong>g the condition 1E<br />
L 1“T0, the <strong>phonon</strong> <strong>drag</strong> <strong>thermopower</strong> follows as S 1<br />
1 1 L1S 01.<br />
This result demonstrates that the periodic modulation has<br />
no effect on the <strong>phonon</strong> <strong>drag</strong> <strong>thermopower</strong>. The lack <strong>of</strong> oscillatory<br />
dependence suggests that some <strong>in</strong>gredient may be<br />
miss<strong>in</strong>g <strong>in</strong> the <strong>in</strong>itial assumptions. For <strong>in</strong>stance, the formation<br />
<strong>of</strong> Brillou<strong>in</strong> m<strong>in</strong>izones due to the periodicity <strong>of</strong> the superlattice<br />
potential is assumed to have no effect on the scatter<strong>in</strong>g<br />
rate. The fold<strong>in</strong>g <strong>of</strong> energy dispersion curves fractures<br />
the Fermi surface <strong>in</strong>to a complex pattern <strong>of</strong> arcs <strong>of</strong> circle<br />
exhibit<strong>in</strong>g tw<strong>of</strong>old symmetry as shown <strong>in</strong> Fig. 1. Such an<br />
anisotropy will be passed onto the electron-<strong>phonon</strong> <strong>in</strong>teraction<br />
s<strong>in</strong>ce it depends strongly on the density <strong>of</strong> available<br />
states on the Fermi surface. In particular, the scatter<strong>in</strong>g rate<br />
<strong>in</strong>creases dramatically for transitions between opposite edges<br />
<strong>of</strong> the Fermi surface where the density <strong>of</strong> states is divergent<br />
,