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 />
m 2<br />
ql im c1 m,0,<br />
m n1<br />
<br />
4<br />
m,n1V xiVy m,n1V xiVy n1<br />
<br />
4<br />
m,n1V xiVy m,n1V xiVy, n2,3..., m...2,1,0,1,2... . 16<br />
A vector is <strong>in</strong>troduced which is def<strong>in</strong>ed as V n 0 “T/n .<br />
Mirl<strong>in</strong> and Wölfle observed 20,21 that equations <strong>of</strong> the type <strong>of</strong><br />
Eq. 15, albeit with no driv<strong>in</strong>g term ( m0), are satisfied<br />
by Bessel functions <strong>of</strong> complex order. The solutions <strong>of</strong> Eq.<br />
15 with m0 are: m(i) m J mi(1m,0 )/ c (qR c).<br />
These may be computed <strong>in</strong> the form <strong>of</strong> cont<strong>in</strong>ued fractions. 23<br />
1 0 <br />
1<br />
1 2 / 1<br />
1 1<br />
2 3 1 4 / 3<br />
4¯ Simplifications occur by observ<strong>in</strong>g that, for m0, Eq. 15<br />
gives 1 1 . Second, coefficients with negative values<br />
<strong>of</strong> m are elim<strong>in</strong>ated us<strong>in</strong>g: m m * and m m * .<br />
, 1<br />
0<br />
Third 3 ( 3) relates to 0 and 1 via 3( 1 2<br />
1)1 2 0 2 3( 1 * 2 *1)1 2 * 0<br />
2 *. These manipulations reduce Eqs. 18 to a system <strong>of</strong><br />
two equations with two complex unknowns 0 and 1 .<br />
Once 1 is known, the thermoelectric current immediately<br />
follows from<br />
j nS 00<br />
a dy<br />
a 0<br />
2 d<br />
2 F ny,yu 1 . 19<br />
The Fermi velocity (y) F1 cos(qy)F F(/2)cos(qy)O(2 ) is well approximated by its two<br />
lead<strong>in</strong>g terms s<strong>in</strong>ce 1. Equation 19 is then easily <strong>in</strong>tegrated,<br />
and I f<strong>in</strong>d<br />
<br />
j xnS 0<br />
2<br />
4 Re 1,<br />
<br />
j ynS 0<br />
2<br />
4 Im 1.<br />
20<br />
It should be remarked that Eq. 6 cannot be used here due to<br />
the coupl<strong>in</strong>g to higher (n1) harmonics. This is the reason<br />
why the current must be calculated from 1 rather than from<br />
0 . 2,21 Consider<strong>in</strong>g Eqs. 17 and 18, one notes that the<br />
125302-4<br />
i J1 <br />
J 1<br />
1<br />
1<br />
2 1<br />
3¯<br />
, i J1 <br />
J 1<br />
1<br />
2<br />
1<br />
1<br />
3¯<br />
17<br />
where the shorthand notation J 1J 1i/(c )(qR c) and J <br />
J i/(c )(qR c) is be<strong>in</strong>g used. 21 If electron-<strong>phonon</strong> <strong>in</strong>teraction<br />
has a tw<strong>of</strong>old symmetry, the Fourier coefficients <strong>in</strong> a 2<br />
and b 2 dom<strong>in</strong>ate <strong>in</strong> the Fourier expansion 9. In the first<br />
<strong>in</strong>stance, one may neglect the contribution <strong>of</strong> higher angular<br />
harmonics Eqs. 4, 8, etc. and solve Eq. 15 driven by<br />
the second harmonic only. This approach limits the number<br />
<strong>of</strong> fitt<strong>in</strong>g parameters to two one if b 20 In these conditions,<br />
V will have non zero components if n3 n1 is<br />
already accounted for. Equation 16 shows that m will be<br />
f<strong>in</strong>ite for m2 and 4. The solutions <strong>of</strong> Eq. 15 therefore<br />
take the form<br />
<br />
1<br />
2<br />
1<br />
1 2 / 1<br />
1<br />
3 1 4 / 3<br />
4¯<br />
. 18<br />
sequence <strong>of</strong> ratios cont<strong>in</strong>u<strong>in</strong>g beyond 4 ( 4) is common<br />
to both equations. The sequence <strong>of</strong> equations 17 is then<br />
substituted <strong>in</strong>to Eq. 18 which, after some algebra, reduces<br />
to a nonl<strong>in</strong>ear system <strong>in</strong> 0 and 1 :<br />
2 2<br />
r 1r 0ii 0 1 r 1i i 0 r0,<br />
21<br />
2 2<br />
i 1i 0ir 0 1 i 1i r 0 i0,<br />
where r , i , r , i ...Zr , Zi are the real and imag<strong>in</strong>ary<br />
parts <strong>of</strong><br />
1 21,<br />
i 2J 1 /J ,<br />
2/ 2 ,<br />
21Z, 22<br />
2/ 2 2Z,<br />
2 2 Z,<br />
Z 1<br />
2 3 2 33 1 3/ 2.<br />
4 has been replaced us<strong>in</strong>g the fact that, for n3, Eq. 16<br />
gives 40.5 2 . The real and imag<strong>in</strong>ary parts r , i , r ,