For a discussion of the classical Boltzmann transport coe¬cients for a degenerate

Fermi system, electrical and thermal conductivities, we refer the reader to chapter 5 of

reference [1]. Here we just note that, for the case of a time-independent electric ¬eld,

the solution to the Boltzmann equation, Eq. (7.56), to linear order is immediately

obtained giving for the conductivity, σ0 , the Boltzmann result

ne2 „tr

σ0 = (7.64)

m

where „tr ≡ „tr ( F ) is the transport relaxation time in the Born approximation

dˆ F

p

|Vimp (pF ’ pF )|2 (1 ’ pF · pF ) .

ˆˆ

= 2πni N0 (7.65)

„tr ( F ) 4π

The appearance of the transport time expresses the simple fact that small angle

scattering is not e¬ective in degrading the current. For isotropic scattering the mo-

mentum and transport relaxation times are identical, as each scattering direction is

14 This so-called Landau criterion is not su¬cient for the applicability of the Boltzmann equation

in low-dimensional systems, d ¤ 2. This is a subject we shall discuss in detail in Chapter 11.

192 7. Quantum kinetic equations

weighted equally. The transport relaxation time is the characteristic time a particle

can travel before the direction of its velocity is randomized.

Exercise 7.4. Show that the retarded impurity self-energy, Eq. (7.51), in equilibrium

and for |E ’ F | F and |p ’ pF | pF just becomes the constant

ΣR (E, p) = ’i (7.66)

2„

where

dˆ F

p

|Vimp (pF ’ pF )|2 .

= 2πni N0 (7.67)

„ 4π

For later use, we end this section on dynamics due to impurity scattering by

considering Boltzmannian motion and its large scale features, Brownian motion.

7.4.1 Boltzmannian motion in a random potential

In later chapters we shall discuss quantum corrections to classical transport. How-

ever, in many cases we often still need to know only the classical kinetics of the

particle motion. We therefore take this opportunity to discuss the Boltzmannian

motion of a particle scattered by impurities, although we shall not need these results

before we discuss destruction of phase coherence due to electron“phonon interaction

in Chapter 11. The Boltzmann theory is a stochastic description of the classical mo-

tion of a particle in a weakly disordered potential. At each instant the particle has

attributed a probability for a certain position and velocity (or momentum). In the

absence of external ¬elds the Boltzmann equation for a particle in a random potential

has the form

‚f (x, p, t) ‚f (x, p, t) dp

=’

+ v· W (p, p ) [f (x, p, t)’f (x, p , t)] , (7.68)

(2π )3

‚t ‚x

where we have introduced the notation v = vp = p/m for the particle velocity.

The Boltzmann equation is ¬rst order in time (the state of a particle is completely

determined in classical mechanics by specifying its position and momentum), and

the solution for such a Markov process can be expressed in terms of the conditional

probability F for the particle to have position x and momentum p at time t given it

had position x and momentum p at time t

dˆ

p

f (x, p, t) = dx F (x, p, t; x , p , t ) f (x , p , t ) . (7.69)

4π

For elastic scattering only the direction of momentum can change, and consequently

we need only integrate over the direction of the momentum. In the absence of ex-

ternal ¬elds the motion in between scattering events is along straight lines, and the

conditional probability describes how the particle by impurity scattering, is thrown

between di¬erent straight-line segments, i.e. a Boltzmannian path.

7.4. Impurity scattering 193

We de¬ne the Boltzmann propagator as the conditional probability for the initial

condition that it vanishes for times t < t , the retarded Green™s function for the

Boltzmann equation. The equation obeyed by the Boltzmann propagator is thus,

assuming isotropic scattering,

‚ ‚ 1 1 dˆ

p

+ vp · F (p, x, t; p , x , t ) ’

+ F (p, x, t; p , x , t )

‚t ‚x „ „ 4π

ˆp ˆ

δ(ˆ ’ p ) δ(x ’ x ) δ(t ’ t ) ,

= (7.70)

ˆ

where δ is the spherical delta function

dˆ ˆ

p

δ(ˆ ’ p ) f (p ) = f (p) .

pˆ (7.71)

4π

The equation for the Boltzmann propagator is solved by Fourier transformation, and

we obtain

dq dω iq·(x’x )’iω(t’t )

F (p, x, t; p , x , t ) = e F (p, p ; q, ω) , (7.72)

(2π)4

where

1 1/„ ˆp ˆ

I(q, ω) + δ(ˆ ’ p )

F (p, p ; q, ω) =

’iω + p · q/m + 1/„ ’iω + p · q/m + 1/„

(7.73)

and

ql

I(q, ω) = , (7.74)

ql ’ arctan ql/(1 ’ iω„ )

where l = v„ is the mean free path.

We note, by direct integration, the property

dˆ

p

F (x, p, t; x , p , t ) = dx F (x, p, t; x , p , t ) F (x , p , t ; x , p , t )

4π

(7.75)

15

the signature of a Markov process. This property will be utilized in Section 11.3.1

in the calculation of the dephasing rate in weak localization due to electron“phonon

interaction.

7.4.2 Brownian motion

If we are interested only in the long-time and large-distance behavior of the particle

motion, |x ’ x | l, t ’ t „ , the wave vectors and frequencies of importance in

15 For a Markov process, the future is independent of the past when the present is known, i.e.

the causality principle of classical physics in the context of a stochastic dynamic system, here the

process in question is Boltzmannian motion.

194 7. Quantum kinetic equations

the Boltzmann propagator, Eq. (7.73), satisfy ql, ω„ 1, and we obtain the di¬usion

approximation

1/„

I(q, ω) , (7.76)

’iω + D0 q 2

where D0 = vl/3 is the di¬usion constant in the considered case of three dimensions

(and isotropic scattering). By Fourier transforming we ¬nd that, in the di¬usion

approximation, the dependence on the magnitude of the momentum (velocity) in the

momentum directional averaged Boltzmann propagator appears only through the

di¬usion constant, t > t ,

dqdω eiq·(x’x )’iω(t’t )

dˆdˆ

pp

D(x, t; x, t ) ≡ F (p, x, t; p , x , t ) =

’iω + D0 q 2

(4π)2 (2π)4

e’(x’x ) /4D0 (t’t )

2

= . (7.77)

(4πD0 (t ’ t ))d/2

This di¬usion propagator describes the di¬usive or Brownian motion of the particle,

the conditional probability for the particle to di¬use from point x to x in time span

t ’ t , described by the one parameter, the di¬usion constant. The absence of the

explicit appearance of the magnitude of the velocity re¬‚ects the fact that the local

velocity is a meaningless quantity in Brownian motion.

Exercise 7.5. Show that

2

≡ dx x2 D(x, t; x , t ) = x + 2dD0 (t ’ t ) ,

x2 (7.78)

t,x ,t

where the d on the right-hand side is the spatial dimension.

If we are interested only in the long-time and large-distance behavior of the Boltz-

mannian motion we can, as noted above, get a simpli¬ed description of the classical

motion of a particle in a random potential. We are thus not interested in the zigzag

Boltzmannian trajectories, but only in the smooth large-scale behavior. It is instruc-

tive to relate the large-scale behavior to the velocity (or momentum) moments of the

distribution function, and the corresponding physical quantities, density and current

density. Expanding the distribution function on spherical harmonics

f (x, p, t) = f0 ( p , x, t) + p · f ( p , x, t) + · · · (7.79)

we have the particle current density given in terms of the ¬rst moment

1 dp 1 dp

p p · f ( p , x, t) = p2 f ( p , x, t)

j(x, t) = (7.80)

(2π )3 3

m 3m (2π )

and the density given in terms of the zeroth moment

dp

n(x, t) = f0 ( p , x, t) . (7.81)

(2π )3

7.4. Impurity scattering 195

Taking the spherical average

dˆ

p

≡

... ... (7.82)

4π

of the force-free Boltzmann equation, Eq. (7.68), we obtain the zeroth moment equa-

tion

p2

‚f0 ( p , x, t)

∇x · f ( p , x, t) = 0 .

+ (7.83)

‚t 3m

Integrating this equation with respect to momentum gives the continuity equation

‚n(x, t)

+ ∇x · j(x, t) = 0 . (7.84)

‚t

This result is of course independent of whether external ¬elds are present or not.

This is seen directly from the Boltzmann equation by integrating with respect to

momentum as we have the identity

dˆ

p

Ix,p,t [f ] = 0 (7.85)

4π

simply re¬‚ecting that the collision integral respects particle conservation.

Taking the ¬rst moment of the Boltzmann equation, p . . . ,

dˆ ‚f (x, p, t) ‚f (x, p, t)

p

+ vp · ’ Ix,p,t [f ] =0 (7.86)

p

4π ‚t ‚x

we obtain the ¬rst moment equation

p2 p2 ‚f0 (x, p, t)

‚ 1

+ f (x, p, t) + = 0, (7.87)

3 ‚t „ ( p ) 3m ‚x

where we have repeatedly used the angular average formulas

p2

dˆ dˆ

p p

p± pβ = δ±β , p± pβ pγ = 0 . (7.88)

4π 3 4π

We have thus reduced the kinetic equation to a closed set of equations relating the

two lowest moments of the distribution function, f0 and f , and we get the equation

satis¬ed by the zeroth moment f0 :

p2

‚ 1 ‚f0 (x, p, t)

’

+ x f0 (x, p, t) = 0. (7.89)

3m2

‚t „ ( p ) ‚t

In a metal the derivatives of the zeroth harmonic of the distribution function for

the conduction electrons, ‚t f0 ( p , x, t) and ”x f0 ( p , x, t), are peaked at the Fermi

energy, and we can use the approximations

dp dp

p2 ”x f0 ( p , x, t) p2 ”x f0 ( p , x, t) (7.90)

F

3 (2π )3

(2π )

196 7. Quantum kinetic equations

and

dp ‚ 1 ‚f0 ( p , x, t) ‚ 1 ‚n(x, t)

+ + , (7.91)

(2π )3 ‚t „ ( p ) ‚t ‚t „ ‚t

where as usual „ ≡ „ ( pF ). Assuming only low-frequency oscillations in the density,

ω„ 1,

‚2n 1 ‚n

(7.92)

‚t2 „ ‚t

and we obtain from Eq. (7.89) the continuity equation on di¬usive form

‚

’ D0 n(x, t) = 0 . (7.93)

x

‚t

Since ∇x f0 ( p , x, t) is peaked at the Fermi energy, we can use the approximation

dp dp

p2 ∇x f0 ( p , x, t) ∇x f0 ( p , x, t)

p2 (7.94)

F

3 (2π )3

(2π )

and assuming only low-frequency current oscillations

‚j(x, t) 1

|j(x, t)| (7.95)

‚t „

we obtain from the ¬rst moment equation, Eq. (7.87), the di¬usion expression for

the current density

‚n(x, t)

j(x, t) = ’D0 . (7.96)

‚x

If we assume that the particle is absent prior to time t , at which time the particle

is created at point x , the di¬usion equation, Eq. (7.93), gets a source term, and we

obtain for the conditional probability or di¬usion propagator D(x, t; x , t )

n(x, t) = dx D(x, t; x , t ) n(x , t ) (7.97)

the equation

‚

’ D0 D(x, t; x , t ) = δ(x ’ x ) δ(t ’ t ) (7.98)

x

‚t

with the initial condition

D(x, t; x , t ) = 0 , for t<t . (7.99)

We can solve the equation for the di¬usion propagator, the retarded Green™s function

for the di¬usion equation, by referring to the solution of the free particle Schr¨dinger

o

Green™s function equation, Eq. (C.24), and letting it ’ t, and /2m ’ D0 , and we

obtain 2 (x’x )

’

e 4D 0 (t ’t )

D(x, t; x , t ) = θ(t ’ t ) . (7.100)

(4πD0 (t ’ t ))d/2

7.4. Impurity scattering 197

Exercise 7.6. Show that the Di¬uson or di¬usion propagator has the path integral

representation

xt =x xt =x

t

Dxt e’SE [xt¯] = Dxt e’ d t L E (x t )

¯ ™¯

D(x, t; x , t ) = (7.101)

t

¯ ¯

xt =x xt =x

where the Euclidean action SE [xt ] is speci¬ed by the Euclidean Lagrangian

¯

x2

™t

™

LE (xt ) = . (7.102)

4D0

The probability density of di¬usive paths is therefore given by

x2

™¯

’ t ¯

’SE [xt ] t

dt

PD [xt ] ≡ ¯

e =e . (7.103)

4D 0

t

¯

Note that the velocity entering the above Wiener measure is not the local velocity

but the velocity averaged over Boltzmannian paths.16

Exercise 7.7. Show that, for a di¬using particle, we have the Gaussian property for

the characteristic function

Dxt PD [xt ] eiq·(x(t)’x(t ))

¯ ¯

= e’D0 q |t’t |

2

iq·(x(t)’x(t ))

<e >D = . (7.104)

Dxt PD [xt ]

¯ ¯

Exercise 7.8. Consider the Di¬uson or di¬usion propagator speci¬ed by the ladder

diagrams

R R R

p+ p+ p+ p+

E+ p+ E+ p+ E+ p+

+ ···

DE (q, ω) = + +

A A A

p’ p’ p’ p’

Ep’ Ep’ Ep’

⎛

R

p+

⎜

⎜ E+ p+

⎜

2⎜

u⎜ 1