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)

2π

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

dˆ

p

dE μ(pF (ˆ ’ p ), E ’ E ) RE pˆ ,

Eˆ

Ie’ph = ’2π pˆ (7.138)

p

4π

where

RE pˆ

Eˆ

(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

dˆ

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)

˜

2π

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 •)

∞

dˆ

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