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by energy transfers of the order of the temperature as a consequence of the exclu-
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


v
(•=0)
= F (x, t; x , 0) ≡
C(t) F (x, t; v , x , t ) . (11.148)
imp


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.

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