sion principle (at temperatures below the Debye temperature).40 In Section 11.5

we shall soon learn that in a three-dimensional sample the energy relaxation rate

in a dirty metal is larger than in a clean metal owing to a strong enhancement of

the electron“electron interaction with small energy transfer. When calculating the

weak localization phase-breaking rate we must therefore pay special attention to the

low-energy electron“electron interaction. In a thin ¬lm or in the two-dimensional

case the energy relaxation rate even diverges in perturbation theory, owing to the

abundance of collisions with small energy transfer. However, the physically measur-

able phase-breaking rate does of course not su¬er such a divergence since the phase

change caused by an inelastic collision is given by the energy transfer times the re-

maining time to elapse on the trajectory. Collisions with energy transfer of the order

of (the phase-breaking rate) ω ∼ /„• or less are therefore ine¬cient for destroy-

ing the phase coherence between the amplitudes for traversing typical time-reversed

trajectories of duration the phase coherence time „• .41 In terms of diagrams this is

re¬‚ected by the fact that interaction lines can connect the upper and lower particle

lines in the Cooperon, whereas there are no such processes for the diagrammatic

representation of the inelastic scattering rate, as discussed in Section 11.5. This dis-

tinction is of importance in the case of a thin metallic ¬lm, the quasi two-dimensional

39 In the Cooperon, contributions from diagrams where besides impurity correlator lines interac-

tion lines connecting the retarded and advanced particle line also appear should be included for

consistency. However, for strongly inelastic processes these contributions are small.

40 For details see, for example, chapter 10 of reference [1].

41 A similar situation is the di¬erence between the transport and momentum relaxation time.

The transport relaxation time is the one appearing in the conductivity, re¬‚ecting that small angle

scattering is ine¬ective in degrading the current.

410 11. Disordered conductors

case, where there is an abundance of scatterings with small energy transfer due to

di¬usion-enhanced electron“electron interaction.

In the time-reversal invariant situation, the Cooperon is equal to the classical

probability that an electron at the Fermi level in time t returns to its starting point.

If coherence is disrupted by interactions, the constructive interference is partially de-

stroyed. This destruction of phase coherence results in the decay in time of coherence,

described by the factor exp{’t/„•} in the expression for the Cooperon, the probabil-

ity of not su¬ering a phase-breaking collision, described by the phase-breaking rate

1/„• . In view of the quantum interference picture of the weak localization e¬ect, we

shall also refer to „• as the wave function phase relaxation time.

A comprehensive understanding of the phase coherence length in weak local-

ization, the length scale L• ≡ D0 „• over which the electron di¬uses quantum

mechanically coherently, has been established, and this has given valuable informa-

tion about inelastic scattering processes. The phase coherence length L• is, at low

temperatures, much larger than the impurity mean free path l, explaining the slow

spatial variation of the Cooperon on the scale of the mean free path, which we have

repeatedly exploited.

11.3.1 Electron“phonon interaction

In this section we calculate the phase-breaking rate due to electron“phonon inter-

action using the simple interference picture described in the previous section.42 We

start from the one-electron Lagrangian, which is given by

1

mx2 ’ V (x) ’ eφ(x, t) ,

™ ™

L(x, x) = (11.137)

2

where V is the impurity potential, and the deformation potential is speci¬ed in terms

of the lattice displacement ¬eld, Eq. (2.72),

n

∇x · u(x, t) .

eφ(x, t) = (11.138)

2N0

It is important to note that the impurities move in phase with the distorted

lattice; hence the impurity potential has the form

Vimp (x ’ (Ri + u(x, t)) ,

V (x) = (11.139)

i

where Ri is the equilibrium position of the ith ion. The impurity scattering is thus

only elastic in the frame of reference that locally moves along with the lattice. We

therefore shift to this moving frame of reference by changing the electronic coordinate

according to x ’ x + u. The impurity scattering then becomes static on account of

generating additional terms of interaction. Expanding the Lagrangian Eq. (11.137)

in terms of the displacement, and neglecting terms of relative order m/M , such as

the term mu · v/2, the transformed Lagrangian can be written as L = L0 + L1 , where

™

42 We follow references [87] and [90].

11.3. Phase breaking in weak localization 411

L0 is given in Eq. (11.111), and43

12

L1 (xt , xt ) = mxt · (xt · ∇) u(xt , t) ’ x ∇ · u(xt , t) .

™ ™ ™ ™ (11.140)

3t

2

In the last line we have used the relation n/2N0 = mvF /3, and the fact that the

magnitude of the velocity is conserved in elastic scattering. We therefore obtain for

the phase di¬erence44

t/2

1 1

dt {∇β u± (xcl , t) ’ ∇β u± (xcl , ’t)} x± xβ ’

•[xcl ] = δ±β x2

™t

™t ™t , (11.141)

t t t

3

’t/2

where summation over repeated Cartesian indices is implied, and we have chosen the

classical paths to satisfy the boundary condition, xcl = 0 = xcl .

’t/2 t/2

We must now average the quantum interference term as given in Eq. (11.116)

with respect to the lattice vibrations, and with respect to the random positions of

the impurities. Since the Lagrangian for the lattice vibrations is a quadratic form

in the displacement u, and the phase di¬erence •[xcl ] is linear in the displacement,

t

the phonon average can be computed by Wick™s theorem according to (see Exercise

4.108 on page 103)45

ei•[xt ] ph = e’ 2 •[xt ] ph .

cl cl 2

1

(11.142)

For the argument of the exponential we obtain (vt ≡ xcl )

™t

t/2 t/2

m2

(±)D±βγδ (xcl ’ xcl , t1 “ t2 )

•[xcl ]2 = dt1 dt2

ph

t t1 t2

2

±

’t/2 ’t/2

1 1

— vt1 vt1 ’

±β

vt2 vt2 ’

γ

2 δ 2

δ±β vt1 δγδ vt2 , (11.143)

3 3

where the phonon correlator

∇β u± (x, t)∇δ uγ (0, 0)

D±βγδ (x, t) = (11.144)

is an even function of the time di¬erence t.

Concerning the average with respect to impurity positions, we will resort to an

approximation which, since the exponential function is a convex function, can be

expressed as the inequality

•=0 ’ 1 •[xcl ]2

≥

C(t) C(t) imp e , (11.145)

ph imp

t

2

imp

43 This result can also be obtained without introducing the moving frame of reference. By simply

Taylor-expanding Eq. (11.139) and using Newton™s equation we obtain a Lagrangian which di¬ers

from the one in Eq. (11.140) by only a total time derivative, and therefore generates the same

dynamics.

44 In neglecting the Jacobian of the nonlinear transformation to the moving frame, we neglect the

in¬‚uence of the lattice motion on the paths.

45 We have suppressed the hat on u indicating that the displacement is an operator with respect

to the lattice degrees of freedom (or we have envisaged treating the lattice vibrations in the path

integral formulation).

412 11. Disordered conductors

where we have introduced the notation for the impurity average

xt / 2 =x

Dxt Pt [xt ] (•[xcl ])2 ph

t

x’t / 2 =x

(•[xcl ])2 = . (11.146)

xt / 2 =x

ph imp

t

Dxt Pt [xt ]

x’t / 2 =x

The phase di¬erence Eq. (11.141) depends on the local velocity of the electron,

which is a meaningless quantity in Brownian motion.46 It is therefore necessary

when considering phase breaking due to electron“phonon interaction to consider the

time-reversed paths involved in the weak-localization quantum interference process

as realizations of Boltzmannian motion. At a given time, a Boltzmannian path is

completely speci¬ed by its position and by the direction of its velocity as discussed in

Section 7.4.1. We are dealing with the Markovian process described by the Boltzmann

propagator F (v, x, t; v , x , t ), where we now use the velocity as variable instead of

the momentum as used in Section 7.4.1. On account of the Markovian property, the

four-point correlation function required in Eq. (11.146) (the start and (identical) end

point and two intermediate points according to Eq. (11.143)) may be expressed as a

product of three conditional probabilities of the type Eq. (7.70), and we obtain

t/2 t/2

4m2 dˆ 1 dˆ 2

vv

•[xcl ]2 = dt1 dt2 dx1 dx2

ph imp

t 2 (4π)2

’t/2 ’t/2

t t

— F (0, ; x1 , v1 , t1 )F (x1 , v1 , t1 ; x2 , v2 , t2 ) F (x2 , v2 , t2 ; 0, ’ )

2 2

1 1

±β γδ

— (±)D±βγδ (xcl ’ xcl , t1 “ t2 ) vt1 vt1 ’ vt2 vt2 ’

2 2

δ±β vt1 δγδ vt2 .

t1 t2

3 3

±

(11.147)

We use the notation that an angular average of the Boltzmann propagator F with

respect to one of its velocities is indicated by a bar. For example, we have for the

return probability

dˆ

v

(•=0)

= F (x, t; x , 0) ≡

C(t) F (x, t; v , x , t ) . (11.148)

imp

4π

The space-dependent quantities may be expressed by Fourier integrals according to

Eq. (7.72). Since the Boltzmann propagator is retarded, F (v, x, t; v , x , t ) vanishes

for t earlier than t , we can expand the upper t1 -integration to in¬nity and the

46 The velocity entering in the Wiener measure, Eq. (7.103), is not the local velocity, but an

average of the velocity on a Boltzmannian path; recall Exercise 7.6 on page 197.

11.3. Phase breaking in weak localization 413

lower t2 -integration to minus in¬nity. Only thermally excited phonons contribute to

the destruction of phase coherence, and we conclude that D±βγδ (xcl ’ xcl , t1 “ t2 )

t1 t2

is essentially zero for |t1 ± t2 | ≥ /kT . We can therefore extend the domain of

integration to in¬nity with respect to |t1 ± t2 | provided that |t| /kT , and obtain

in the convex approximation

2m2 dkdk dωdω dˆ 1 dˆ 2

vv

(•=0)

’

C(t) = C(t) exp

imp imp

(2π)8 (4π)2

(•=0)

2 C(t) imp

— F (v1 ; k, ω)F (v1 , v2 , k + k , ω + ω )D±βγδ (’k , ’ω )

F (v2 , k, ω) e’iωt ’ F (v2 , k, ω + 2ω ) e’i(ω+ω )t

—

1 1

±β γδ

— vt1 vt1 ’ vt2 vt2 ’

2 2

δ±β vt1 δγδ vt2 . (11.149)

3 3

We expect that the argument of the exponential above increases linearly in t for

large times. Since the classical return probability in three dimensions has the time

∝ t’3/2 (recall the form of the di¬usion propagator), the

(•=0)

dependence C(t) imp

integral above should not decrease faster than t’1/2 . Such a slow decrease is obtained

from the (k, ω)-integration only from the combination F (v1 ; k, ω) F (v2 ; k, ω), which

according to Eq. (7.76) features an infrared singular behavior (’iω + D0 k 2 )’2 for

small k and ω. In fact, it is just this combination that leads to a time-dependence

proportional to t’1/2 and, compared with that, all other contributions may be ne-

glected. For the important region of integration we thus have ω ω , since ω

is determined by the phonon correlator, which gives the large contribution to the

integral for the typical value ω kT . We are therefore allowed to approximate

F (v1 , v2 ; k + k , ω + ω ) by F (v1 , v2 ; k , ω ). In addition, the same arguments show

that the second term in the square bracket may be omitted. We thus obtain

e’t/„• ,

(•=0)

C(t) = C(t) (11.150)

imp imp

where the phase-breaking rate due to electron“phonon interaction is given by

2m2

1 dk dω dˆ 1 dˆ 2 1

vv ±β

F (v1 , v2 ; k , ω ) D±βγδ (k , ω ) v1 v1 ’ δ±β v1

2

= 2 (2π)4 2

„• (4π) 3

1

±β

— v2 v2 ’ δ±β v2

2

. (11.151)

3

For simplicity we consider the Debye model where the lattice vibrations are spec-

i¬ed by the density ni and the mass M of the ions, and by the longitudinal cl and the

transverse ct sound velocities.47 We assume the phonons to have three-dimensional

47 The jellium model does not allow inclusion of Umklapp processes in the electron“phonon scat-

tering.

414 11. Disordered conductors

character. In case of longitudinal vibrations, we have the normal mode expansion of

the displacement ¬eld

i

u(r, t) = √ ˆ

k Qk (t) eik·r , (11.152)

N k

where N is the number of ions in the normalization volume. For the phonon average

we have

H(ωk ) cos ωk (t ’ t ) ,

Qk (t) Qk (t ) = δk,’k (11.153)

2M ωk

where ωk = cl k, provided that k is less than the cut-o¬ wave vector kD , and we

obtain for the Fourier transform of the longitudinal phonon correlator

1 ±βγδ

±βγδ

k k k k H(ωk ) [δ(ω ’ ωk ) + δ(ω + ωk )] .

DL (k, ω)] = (11.154)

2

Strictly speaking, we encounter in the above derivation H(ω) = 2n(ω) + 1, where

n is the Bose distribution function. However, the present single electron theory

does not take into account that the fermionic exclusion principle forbids scattering

of an electron into occupied states. Obedience of the Pauli exclusion principle is

incorporated by the replacement48

1 ω ω 2

H(ω) ’ coth ’ tanh = . (11.155)

ω

2 2kT 2kT sinh kT

Upon inserting Eq. (11.154) in the expression Eq. (11.151) for the phase-breaking

rate, we encounter the directional average of expressions of the type

2 2

v 2v

’2

ˆˆ

k± kβ v± vβ ’ δ±β (k · v) ’ k

2

=k . (11.156)

3 3

Altogether the angular averages appear in the combination

§ «

⎡ ¤2

⎪ ⎪

18 ⎨ dˆ [(k · v)2 ’ k 2 v ]2 ¬

dˆ (k · v)2 ’ k 2 v ¦

2 2

v v

I(k, ω)⎣ 3 3

¦L (kl) = +

πvF k 3 ⎪ 4π ’iω + iv · k + 1/„ ⎪

4π ’iω + iv · k + 1„

3

© ⎭

2 kl arctan kl 3

’

= , (11.157)

kl ’ arctan kl kl

π

48 The argument is identical to the similar feature for the inelastic scattering rate or imaginary part

of the self-energy. In terms of diagrams, we recall that, in the above discussion, we have included

only the e¬ect of the kinetic or Keldysh component of the phonon propagator. Including the

retarded and advanced components makes the electron experience its fermionic nature introducing

the electron kinetic component which carries the tangent hyperbolic factor. As a consequence, a

point also elaborated in reference [91], the zero-point ¬‚uctuations of the lattice can not disrupt the

weak-localization phase coherence. A detailed discussion of the Pauli principle and the inelastic

scattering rate is given in Section 11.5 in connection with the electron“electron interaction.

11.3. Phase breaking in weak localization 415

where the result in the last line is obtained since ω = cl k vF k. For the phase-

breaking rate due to longitudinal phonons we thus obtain

kD

π2

1 1

dk k 2 ¦L (kl)

= . (11.158)

„•,l 6mM cl sinh cl k/kT

0

We note the limiting behaviors

§ 7πζ(3) (kT )3

⎪ 12 cl /l kT cl kD

⎨ nMc4

1 l

= (11.159)

⎪ π4 (kT )4

„•,l ©

30 l nMc5 kT cl /l .

l

The expression Eq. (11.157) for the function ¦L demonstrates in a direct way

the important compensation that takes place in the case of longitudinal phonons

between the two mechanisms contained in L1 . First, the term (k · v)2 corresponds to

mv · (v · ∇)u and represents the coupling of the electrons to the vibrating impurities.