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1K
ˆ ˆ
h(E, p, R, T ) = g (E, p, R, T ) (7.133)
2
and
iN0 |gq |2
(DR (E, q) ’ DA (E, q))
μ(q, E) = (7.134)

is the Eliashberg spectral weight function. Here we have allowed for a more gen-
eral longitudinal electron“phonon coupling than the jellium model. The coupling is
denoted gq , corresponding to momentum transfer q. The connection to the jellium
model is |gq | = g ωq /2, where ωq = c q is the energy of a phonon with momentum
q, c being the sound velocity.
We have further assumed that the phonons are in thermal equilibrium at temper-
ature T ,24 and have therefore used the ¬‚uctuation“dissipation relation for bosons,
Eq. (5.103),

E
DR (E, p) ’ DA (E, p)
DK (E, p) = coth . (7.135)
2kT
We note that the variables in the distribution function are quite di¬erent from
that of the classical Boltzmann equation for electron“phonon interaction, which is
the Wigner coordinates (p, R, T ). Here an energy variable and a position on the
Fermi surface appear separately (besides space and time). This feature of the quasi-
classical equation re¬‚ects the fact that we do not rely on a de¬nite relation between
24 This is not necessary, but would otherwise lead to the requirement of considering the kinetic
equation for the phonons also. For typical transport situations in a metal, the approximation, viz.
considering the phonons a heat reservoir, is applicable.
7.5. Quasi-classical Green™s function technique 205


the energy and momentum variables as is the case in the quasi-particle approximation
of Section 7.3.2.
Introducing the Fermi and Bose type distribution functions
1
1 ’ h(E, p, R, T )
ˆ ˆ
f (E, p, R, T ) = (7.136)
2
and
1 E
n(E) = ’ 1 ’ coth (7.137)
2 2kT
the collision integral takes the more familiar form


p
dE μ(pF (ˆ ’ p ), E ’ E ) RE pˆ ,

Ie’ph = ’2π pˆ (7.138)
p


where

RE pˆ

(1 + n(E ’ E ))f (E, p)(1 ’ f (E , p ))
ˆ ˆ
=
p


’ n(E ’ E )(1 ’ f (E, p))f (E , p ) .
ˆ ˆ (7.139)

Finally, introducing a gauge invariant distribution function by the substitution

f (E) ’ f (E ’ •(R, T )) (7.140)

we obtain the quantum kinetic equation

((1 ’ ‚E eσ)‚T + ‚T eσ ‚E + vF · (∇R + eE(R, T )) ‚E ) f = Ie’ph [f ] ,
(7.141)

where E(R, T ) = ’∇R •(R, T ) is the perturbing electric ¬eld. We note that the
self-energy terms on the right-hand side in the kinetic equation describe collision
processes, and we now turn to show that the self-energies on the left describe renor-
malization e¬ects, in particular mass renormalization due to the electron“phonon
interaction.
From the kinetic equation, Eq. (7.141), Prange and Kadano¬ [23] drew the con-
clusion that many-body e¬ects can be seen only in time-dependent transport prop-
erties and that static transport coe¬cients, such as d.c. conductivity and thermal
conductivity are correctly given by the usual Boltzmann results. However, there is a
restriction to the generality of this statement, viz. that in deriving the quasi-classical
equation of motion particle“hole symmetry was assumed. Within the quasi-classical
scheme, all thermoelectric coe¬cients therefore vanish, and no conclusion can be
drawn about many-body e¬ects on the thermo-electric properties. In Section 7.6.1,
we shall by not employing the quasi-classical scheme consider how thermo-electric
properties do get renormalized by the electron“phonon interaction.
206 7. Quantum kinetic equations


7.5.2 Renormalization of the a.c. conductivity
As an example of electron“phonon renormalization of time-dependent transport co-
e¬cients we shall consider the a.c. conductivity in the frequency range ω„e’ph 1,
where 1/„e’ph is the clean-limit electron“phonon scattering rate for an electron on
the Fermi surface
§ 7πζ(3) (kT )3
⎪ » (pF c)2 kT 2pF c

⎨ 2
1
= (7.142)

„ ( F, T ) ⎪
©
2π» kT kT 2pF c
ζ being Riemann™s zeta function.25 For de¬niteness we also consider the temperature
to be low compared to the Debye temperature θD . In this high-frequency limit the
collision integral can be neglected and the linearized kinetic equation takes the simple
form
(1 ’ ‚E eσ)‚T h + ‚T eσ ‚E h0 + evF · E(T ) ‚E h0 = 0 (7.143)
for a spatially homogeneous electric ¬eld. Except for a real constant, just renormal-
izing the chemical potential, we have according to the Feynman rules
1 dˆ
p
dE |gpF ’pF |2 h(E , p , R, T ) eD(pF ’ pF , E ’ E ) , (7.144)
ˆ
eσ = N0
2 4π
where gpF ’pF denotes the electron“phonon coupling, and
1R
(D + DA ) .
eD = (7.145)
2
For an applied monochromatic ¬eld, E(t) = E0 exp{’iωt}, the solution can be
sought in the form
h1 = aeE · vF ‚E h0 , (7.146)
where the constant a remains to be determined. Inserting Eq. (7.146) into the kinetic
equation we obtain
1
a= , (7.147)
’iω(1 + »— )
where
|gpF ’pF |2

p
»— = 2N0 (1 ’ p · p ) .
ˆˆ
dE (7.148)
4π ωpF ’pF
The current can now be evaluated and we obtain for the frequency dependence
of the conductivity
ne2
σ(ω) = (7.149)
’iωmopt
25 For
a calculation of the collision rate see Exercise 8.8 on page 237 and Section 11.3.1, and, for
example, chapter 10 of reference [1].
7.5. Quasi-classical Green™s function technique 207


where the optical mass is renormalized according to

mopt = m(1 + »— ) (7.150)

a result originally obtained by Holstein using a di¬erent approach, viz. linear response
theory [26]. We note that it is the non-equilibrium electron contribution to the real
part of the self-energy that makes the optical mass renormalization di¬erent from
the speci¬c heat mass renormalization, m ’ (1 + »)m (see also the result of Exercise
8.8 on page 237).
As a consequence of electron“phonon interaction, the physically observed mass of
the electron is not the mass or band structure e¬ective mass of the electron, but it has
been changed owing to the interaction.26 Furthermore, we note that the magnitude
of the mass renormalization depends on how the system is probed, the optical mass
being di¬erent from the speci¬c heat mass.

7.5.3 Excitation representation
The quasi-classical theory leads to equations which are more general than the Boltz-
mann equation, and the kinetic equation looks quite di¬erent. We have shown that
the basic variables, besides space and time, are the energy variable and the momen-
tum position on the Fermi surface. Although the electron“phonon interaction does
not permit the quasi-particle approximation a priori, we recapitulate the deriva-
tion of reference [23] showing that it is still possible to cast the electron“phonon
transport theory into the standard Landau“Boltzmann form. We start by de¬ning a
quasi-particle energy Ep , which is de¬ned implicitly by (we suppress the space-time
variables and use the short notation Ep = E(p, R, T ))

ˆ
Ep = ξp + eσ(Ep + e•(R, T ), p, R, T ) (7.151)

thereby satisfying the equations

∇p Ep = Zp ∇p ξp (7.152)

and
∇R Ep = Zp (e∇R • ‚E eσ + ∇R eσ) (7.153)
E=Ep +e•(R,T )

and
‚T Ep = Zp (e‚T •‚E eσ + ‚T eσ) (7.154)
E=Ep +e•(R,T )

where in Eq. (7.151), assuming for simplicity a spherical Fermi surface, any angular
dependence of the real part of the self-energy has been neglected, and the so-called
26 Thus interaction causes renormalization of observable quantities. This point of view is the
rationale for avoiding the ubiquitous in¬nities occurring in quantum ¬eld theories such as QED,
and being taken to an extreme since there the unobservable bare mass (and the bare coupling
constant, the bare electron charge) is taken, it turns out, to be in¬nite in order to provide the ¬nite
and accurate predictions of QED by phenomenologically introducing the observed mass and charge.
208 7. Quantum kinetic equations


wave-function renormalization constant

Zp = (1 ’ ‚E eσ)’1 (7.155)
E=Ep +e•(R,T )

has been introduced.
The energy variable E in the kinetic equation is now set equal to Ep + e• and
we introduce the distribution function (again suppressing the space-time variables)
ˆ
np = f (E, p, R, T ) . (7.156)
E=Ep +e•(R,T )

Using the relations
∇p n = ∇p Ep (‚E f ) (7.157)
E=Ep +e•(R,T )

and
∇R n = (∇R f + ∇R (Ep + e•(R, T ))‚E f ) (7.158)
E=Ep +e•(R,T )

and
‚T n = (‚T f + ‚T (Ep + e•(R, T ))‚E f ) (7.159)
E=Ep +e•(R,T )

and Eqs. (7.152“7.154), we obtain the kinetic equation of the form
’1 ˜
Zp (‚T + ∇p Ep · ∇R ’ ∇R (Ep + •(R, T )) · ∇p ) n(p, R, T ) = Ie’ph (7.160)
with the electron“phonon collision integral
2π dp p
ˆ
˜
Ie’ph = ’ Zp μ(p ’ p ) Rp ,
˜ (7.161)
ˆ
3
N0 (2π)
where
p
ˆ
Rp = (1 + N (Ep ’ Ep ))np (1 ’ np ) ’ N (Ep ’ Ep )(1 ’ np )np (7.162)
ˆ

and
iN0 |gp’p |2 R
μ(p ’ p ) = (D (p ’ p , Ep ’ Ep ) ’ DA (p ’ p , Ep ’ Ep )) . (7.163)
˜

In transforming the collision integral we have utilized the substitution
dˆ dˆ dEp dp
p p
dE ’ N0 ’
N0 dξp Zp . (7.164)
(2π)3
4π 4π dξp
Since the sound velocity is much smaller than the Fermi velocity, the phonon
damping is negligible, and the phonon spectral weight function has delta function
character

μ(p ’ p ) = N0 |gp’p |2 (δ(Ep ’ Ep ’ ωp’p ) ’ δ(Ep ’ Ep + ωp’p )) . (7.165)
˜
7.5. Quasi-classical Green™s function technique 209


The kinetic equation can then be written in the ¬nal form

(‚T + ∇p Ep · ∇R ’ ∇R (Ep + •(R, T )) · ∇p ) n(p, R, T ) = Ie’ph , (7.166)

where the electron“phonon collision integral is

dp p
ˆ
Ie’ph = ’2π Zp Zp |gp’p |2 Rp (δ(Ep ’ Ep ’ ωp’p ).
ˆ
3
(2π)


’ δ(Ep ’ Ep + ωp’p )) . (7.167)

This has the form of the familiar Landau“Boltzmann equation, except for the fact
that the transition matrix elements are renormalized.
We stress that only the quasi-classical approximation was used to derive the above
kinetic equation. In particular, we have not assumed any relation between the life-
time of a electron in a momentum state at the Fermi surface and the temperature.
This would have been necessary for invoking a quasi-particle description in order
to justify the existence of long-lived electronic momentum states. It has thus been
established from microscopic principles that the validity of the Landau“Boltzmann
description of the electron“phonon system is determined not by the Peierls criterion
(stating the upper bound is not the Fermi energy but the temperature), but by the
Landau criterion
F. (7.168)
„ ( F, T )
This is of importance for the validity of the Boltzmann description of transport in
semiconductors, for which the Peierls criterion would be detrimental.

7.5.4 Particle conservation
That an approximation for the quasi-classical Green™s function respects conservation
laws, say particle number conservation, is not in general as easily stated as for the
microscopic Green™s function. We therefore establish it here explicitly. The collision
integral, Eq. (7.167), has the invariant

dp
Ie’ph = 0 , (7.169)
(2π)3

which we shall see expresses the conservation of the number of particles, here the
electrons in question. Integrating the kinetic equation, Eq. (7.166), with respect to
momentum we obtain the continuity equation

‚T n + ∇R · j = 0 , (7.170)

where
dp
n(R, T ) = 2 n(p, R, T ) (7.171)
(2π)3
210 7. Quantum kinetic equations


and
dp
∇p Ep n(p, R, T )
j(R, T ) = 2 (7.172)
(2π)3

are the Landau“Boltzmann expressions for the density and current density and the
factor of two accounts for the spin of the electron.
In order to establish that these are indeed the correctly identi¬ed densities (in
the excitation representation), we should connect one of them with the microscopic
expression. Assuming that |e•| F , the microscopic expression for the density,
Eq. (7.108), is (suppressing space-time variables in quantities, here in •)


p
n(R, T ) = ’2N0 dE f (E + e•, p) . (7.173)
4π ’∞

In order to compare the density expression in the particle representation with the
excitation representation we transform Eq. (7.171) to the particle representation

dp dˆ
p
dE (1 ’ ‚E eσ) f (E, p).
ˆ
n(R, T ) = 2 n(p, R, T ) = 2N0 (7.174)
(2π)3 4π ’∞

Since Eq. (7.173) and Eq. (7.174) appear to be di¬erent, Eq. (7.172) is also trans-
formed to the particle representation

dp dˆ
p
∇p Ep n(p, R, T ) = 2N0 ˆ
2 dE vF f (E, p) . (7.175)
(2π)3 4π ’∞

Comparing the expression in Eq. (7.175) to that of Eq. (7.109), we observe that it
is identical to the quasi-classical current-density expression. The only possibility for
the above-mentioned apparent discrepancy not to lead to a violation of the continuity
equation is the existence of the identity

ˆ
‚T dˆ dE f (E, p) ‚E eσ = 0 (7.176)
p
’∞

which we now prove. Inserting the expression from Eq. (7.144) into the left side of
Eq. (7.176) we are led to consider

dˆ dE dˆ dE |gpF ’pF |2 ( eD(pF ’ pF , E ’ E )
p p


‚T f (E, p) ‚E f (E , p ) ’ ‚E f (E, p) ‚T f (E , p )) = 0
ˆ ˆ ˆ ˆ (7.177)

ˆ ˆ
which by interchanging the variables E, p and E , p is seen to vanish, and the identity
Eq. (7.176) is thus established. We have thus established that the approximations
made do not violate particle conservation.
7.6. Beyond the quasi-classical approximation 211


7.5.5 Impurity scattering
For electrons interacting with impurities in a conductor, the self-energy is given by
the diagram in Eq. (7.51), F „ , and we can immediately implement the quasi-
classical approximation. The equation for the kinetic component of the quasi-classical
Green™s function in the presence of an electric ¬eld becomes
1 dˆ K
p
(‚T + vF · ∇R + e‚T • ‚E ) g K = ’ g K (E, p, R, T ) +
ˆ ˆ
g (E, p, R, T ) ,
„ 4π
(7.178)

where for simplicity we have assumed that the momentum dependence of the impurity
potential can be neglected.
In the di¬usive limit the quasi-classical kinetic Green™s function will be almost
isotropic, and an expansion in spherical harmonics needs to keep only the s- and
p-wave parts

g K (E, p, R, T ) = gs (E, R, T ) + p · gp (E, R, T )
K
ˆK

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