interference term38

xt / 2 =x t/ 2

x2

™

’ ’ i•(xcl ))

t

dt ( 4D ˜t

0

Dxt e ’t / 2

C t , ’t (x, x) = , (11.123)

2 2

x’t / 2 =x

2

where D0 is the di¬usion constant for a particle with energy F , D0 = vF „ /d.

Let us now obtain the equation satis¬ed by the Cooperon in the presence of a

time-dependent electromagnetic ¬eld. In that case we have for the interaction the

Lagrangian

L1 (xt , xt , t) = ext · A(xt , t) ’ eφ(xt , t) .

™ ™ (11.124)

Since the coherence between time-reversed trajectories is partially upset, it is con-

venient to introduce arbitrary initial and ¬nal times, and we have for the phase

di¬erence between a pair of time-reversed paths

1

{S[xcl ] ’ S[xcl+tf ’t ]}

•[xcl ] =

t t ti

tf

dt L1 (xcl , xcl , t) ’ L1 (xcl+tf ’t , xcl+tf ’t , t)

t ™t ™ ti

= (11.125)

ti

ti

37 An exception to this is discussed in Section 11.3.1.

38 In case the classical motion in the random potential is adequately described as the di¬usion

process, we immediately recover the result Eq. (11.93) for the return probability.

11.2. Weak localization 407

as the contributions to the phase di¬erence from L0 cancels, and we are left with

tf

e

dt xcl (t)·A(xcl (t), t) + φ(xcl (ti + tf ’t), ti + tf ’t) ’ φ(xcl (t), t)

•[xcl ] = ™

t

ti

’ xcl (ti + tf ’ t) · A(xcl (ti + tf ’ t), ti + tf ’ t) .

™ (11.126)

Introducing the shift in the time variable

1

t ≡t’T , T≡ (tf + ti ) (11.127)

2

we get

t f ’t i

2

e

xcl (t + T ) · A(xcl (t + T ), t + T )

•[xcl ] = ™

dt

t

t i ’t f

2

’ xcl (T ’ t ) · A(xcl (T ’ t ), T ’ t )

™

’ φ(xcl (t + T ), t + t) + φ(xcl (T ’ t ), T ’ t ) . (11.128)

The electromagnetic ¬eld is assumed to have a negligible e¬ect on determining the

classical paths, and we can shift the time argument specifying the position on the

path to be symmetric about the moment in time T , and thereby rewrite the phase

di¬erence, t ≡ tf ’ ti ,

t

2

e

dt xcl · AT (xcl , t) ’ φ(xcl , t) ,

¯ ™t ¯¯ ¯¯

•[xcl ] = (11.129)

¯

t t t

’2

t

where

φT (x, t) = φ(x, T + t) ’ φ(x, T ’ t) (11.130)

and

AT (x, t) = A(x, T + t) + A(x, T ’ t) . (11.131)

An electric ¬eld can be represented solely by a scalar potential, and we imme-

diately conclude that only if the ¬eld is di¬erent on time-reversed trajectories can

it lead to destruction of phase coherence. In particular, an electric ¬eld constant in

time does not a¬ect the phase coherence, and thereby does not in¬‚uence the weak-

localization e¬ect.

The di¬erential equation corresponding to the path integral, Eq. (11.123), there-

fore gives for the Cooperon equation for the case of a time-dependent electromagnetic

408 11. Disordered conductors

¬eld

2

‚ e ie

+ φT (xt , t) ’D0 ∇x ’ AT (x, t) Ct,t (x, x ) = δ(x ’ x ) δ(t ’ t ) .

T

‚t

(11.132)

When the sample is exposed to a time-independent magnetic ¬eld, we recover the

static Cooperon equation, Eq. (11.107).

11.3 Phase breaking in weak localization

The phase coherence between the amplitudes for pairs of time-reversed trajectories

is interrupted when the environment of the electron, besides the dominating random

potential, is taken into account. At nonzero temperatures, energy exchange due

to the interaction with the environment will partially upset the coherence between

time-reversed paths involved in the weak-localization phenomenon. The constructive

interference is then partially destroyed.

Quantitatively the e¬ect on weak localization by inelastic interactions with energy

transfers ”E of the order of the temperature, ”E ∼ kT , strongly inelastic processes,

can be understood by the observation that the single-particle Green™s function will

be additionally damped owing to interactions. If in addition to disorder we have an

interaction, say with phonons, the self-energy will in lowest order in the interaction

be changed according to

R

pE pE

pE

R R

’ pE pE pE pE

+

pE pE

(11.133)

and we will get an additional contribution to the imaginary part of the self-energy

mΣR = ’ ’ . (11.134)

2„ 2„in

11.3. Phase breaking in weak localization 409

Upon redoing the calculation leading to Eq. (11.22) for the case in question, we obtain

in the limit „in „

„

ζ(Q, ω) = 1 ’ + iω„ + D0 „ Q2 . (11.135)

„in

This will in turn lead to the change in the Cooperon equation, ω ’ ω + i/„in , and

we get the real space Cooperon equation39

1 1

’ iω ’ D0 ∇2 + δ(x ’ x ) .

Cω (x, x ) = (11.136)

x

„in „

The e¬ect on weak localization of electron“electron interaction and electron“

phonon interaction have been studied in detail experimentally [88, 89], and can phe-

nomenologically be accounted for adequately by introducing a temperature-dependent

phase-breaking rate 1/„• in the Cooperon equation, describing the temporal expo-

nential decay C(t) ’ C(t) exp{’t/„• } of phase coherence. In many cases the in-

elastic scattering rate, 1/„in , is identical to the phase-breaking rate, 1/„• . This is

for example the case for electron“phonon interaction, as we shortly demonstrate.

However, one should keep in mind that the inelastic scattering rate is de¬ned as the

damping of an energy state for the case where all scattering processes are weighted

equally, irrespective of the amount of energy transfer. In a clean metal the energy

relaxation rate due to electron“phonon or electron“electron interaction is determined

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

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.

Second, the term ’k 2 v2 /3 is connected with ’mv2 ∇ · u/3, and originates from the

interaction of the electrons with the lattice vibrations. Without this compensation,

each of the mechanisms would appear to be enhanced in an impure metal, and would

lead to an enhanced phase-breaking rate proportional to (kT )2 /(nM c3 l).

l

±βγδ

For the case of transverse vibrations, we note that DT is of similar form as

ˆ± kγ has to be replaced by (δ±γ ’ k± kγ ) and an addi-

ˆ ˆˆ

Eq. (11.154) where, however, k

tional factor of 2, which accounts for the multiplicity of transverse modes. We then

obtain a phase-breaking rate due to interaction with transverse phonons, „•,t , which

is similar to the expression in Eq. (11.158) with cl and φL replaced by ct and

3 2k 3 l3 + 3kl ’ 3(k 2 l2 + 1) arctan kl

¦T (kl) = (11.160)

k4 l4

π

respectively. In particular, we obtain the limiting behaviors for the phase-breaking

rate due to transverse phonons

§ 2 (kT )2

⎪ π mMc3 l ct /l kT ct kD

⎨2

1 t

= (11.161)

⎪ π4 (kT )4

„•,t ©

20 l 2 mMc5 kT ct /l .

t

We note that in the high-temperature region, ct /l kT ct kD , the transverse

contribution is negligible in comparison with the longitudinal one if ct cl . But the

transverse rate dominates in the case where the transverse sound velocity is much

smaller than the longitudinal one. Such a situation may quite well be realized in

some amorphous metals; then, it is possible to observe a phase-breaking rate of the

’1

form „• ∝ T 2 /l at higher, but not too high, temperatures.49 The predictions of the

theory are in good agreement with magneto-resistance measurements and carefully

conducted experiments of the temperature dependence of the resistance [92].

49 A quadratic temperature dependence of the phase-breaking rate is often observed experimen-

tally.

416 11. Disordered conductors

The physical meaning of the second term in Eq. (11.149) is as follows. It is

appreciable only if the lattice deformation stays approximately constant during the

time the electron spends on its path and leads, in this case, to a cancellation of the

¬rst term. Equivalently, electron“phonon interactions with small energy transfers do

not lead to destruction of phase coherence. The e¬ect of this term is thus e¬ectively

to introduce a lower cut-o¬ in the integral of Eq. (11.158) at wave vector k0 = 1/cl „•,l .

However, there are no realistic models of phonon spectra where this e¬ect is of

importance. We therefore have the relationships ω kT / ω 1/„• . It is

therefore no surprise that the calculated phase-breaking rates are identical to the

inelastic electron“phonon collision rates in a dirty metal [93]. When considering

phase breaking due to electron“electron interaction, which we now turn to, the small

energy transfer interactions are of importance.

11.3.2 Electron“electron interaction

In this section we consider the temperature dependence of the phase-breaking rate

due to electron“electron interaction.50 As already discussed at the beginning of this

section, special attention to electron“electron interaction with small energy transfer

must be exercised due to the di¬usion enhancement. In diagrammatic terms we

therefore need to take into account diagrams where the electron“electron interaction

connects also the upper and lower particle lines in the Cooperon.

In Section 11.5 we shall show that the e¬ective electron“electron interaction at low

energies can be represented by a ¬‚uctuating ¬eld. Its correlation function in a dirty

metal will be given by the expression in Eq. (11.269), which we henceforth employ.

We can therefore obtain the e¬ect on the Cooperon of the quasi-elastic electron“

electron interaction by averaging the Cooperon with respect to a time-dependent

electromagnetic ¬eld using the proper correlator. We therefore consider the equation

for the Cooperon in the presence of an electromagnetic ¬eld, Eq. (11.81),

2

‚ ie 1 1

2 ’D0 ∇x ’ AT (x, t) δ(x ’ x ) δ(t ’ t ) ,

T

+ Ct,t (x, x ) =

„ e’e

‚t „

(11.162)

where we have chosen a gauge in which the scalar potential vanishes, and 1/„ e’e is

the energy relaxation rate due to high-energy electron“electron interaction processes,

i.e. processes with energy transfers ∼ kT .51

To account for the electron“electron interaction with small energy transfers, we

must perform the Gaussian average of the Cooperon with respect to the ¬‚uctuating

¬eld. This is facilitated by writing the solution of the Cooperon equation as the path

integral

xt =x

1

Dxt e’S[xt ] ,

T

Ct,t (R, R ) = (11.163)

2„

xt =x

50 We follow reference [94].

51 As will become clear in the following, the separation in high- and low-energy transfers takes

place at energies of the order of the temperature. However, in the following we shall not need to

specify the separation explicitly.

11.3. Phase breaking in weak localization 417

where the Euclidean action consists of two terms

S = S0 + SA , (11.164)

where

t

x21

™t 1

S0 [xt ] = dt1 + e’e (11.165)

4D0 „

t

and

t

ie

dt1 xt1 · AT (xt1 , t1 ) .

™

SA [xt ] = (11.166)

t

In terms of diagrams, the Gaussian average corresponds to connecting the external

¬eld lines pairwise in all possible ways by the correlator of the ¬eld ¬‚uctuations,

thereby producing the e¬ect of the low-energy electron“electron interaction. Since the

¬‚uctuating vector potential appears linearly in the exponential Cooperon expression,

the Gaussian average with respect to the ¬‚uctuating ¬eld is readily done

rt =R

1

Drt e’( S0 [xt ] +

T SA [xt ] )

Ct,t (R, R ) = (11.167)

2„

rt =R

where the averaged action SA is expressed in terms of the correlator of the vector

potential

t t

e2

dt1 dt2 xμ (t1 ) xν (t2 ) AT (xt2 , t1 )AT (xt2 , t2 ) .

SA [xt ] = ™ ™ (11.168)

μ ν

2

2

t t

If we recall the de¬nition of AT (xt , t), Eq. (11.77), we have

dq dω iq·(xt ’xt )

AT (xt2 , t1 )AT (xt2 , t2 ) = 2 e Aμ Aν

1 2

qω

μ ν

(2π)d 2π

t1 ’ t2

t 1 + t2

— cos ω + cos ω , (11.169)

2 2

where we have introduced the notation

≡ Aμ (q, ω)Aν (’q, ’ω) .

Aμ Aν (11.170)

qω

The electric ¬eld ¬‚uctuations could equally well have been represented by a scalar

potential

1

Aμ (q, ω)Aν (’q, ’ω) Eμ (q, ω)Eν (’q, ’ω)

=

ω2

qμ qν

φ(q, ω)φ(’q, ’ω) .

= (11.171)

ω2

418 11. Disordered conductors

In Section 11.5 we show that the electron“electron interaction with small energy

transfers, ω kT , is determined by the temperature, T , and the conductivity of

the sample, σ0 , according to52

2kT qμ qν

Aμ Aν = . (11.172)

qω

ω 2 σ0 q 2

Upon partial integration we notice the identity (the boundary terms are seen to

vanish as x’t = xt )

t t

ω(t1 ’ t2 )

ω(t1 + t2 )

dt1 dt2 qμ qν xμ (t1 ) xν (t2 )eiq·(xt 1 ’xt 2 ) cos

™ ™ + cos

2 2

t t

t t

ω(t1 ’ t2 )

2

iq·(xt 1 ’xt 2 ) ω ω(t1 + t2 )

= ’ dt1 dt2 e ’ cos

cos (11.173)

4 2 2

t t

and obtain

t t

eiq·(xt 1 ’xt 2 ) ω(t1 ’ t2 )

e2 kT dq dω ω(t1 + t2 )

[xt ] = ’ ’ cos

SA dt1 dt2 cos .

2σ0 (2π)d 2π q2 2 2

t t

(11.174)

Performing the integration over ω and t2 , the expression for the Cooperon becomes

xt =x t xt

™ 2 dq

q’2 (1’cos(q·(xt 1 ’x’t 1 )))

’ dt1 + „ e’e + 2eσ k T

1

1

1 (2π )2

4D 0

Dxt e

0

T ’t

Ct,’t (x, x ) = .

2„

x’t =x

(11.175)

The singular term is regularized by remembering that in Eq. (11.174) the ω-integra-

tion actually should have been terminated, in the present context, at the large fre-

quency kT / . The factor exp{iq · (xt1 ’ xt2 )} does therefore not reduce strictly to

1 for the ¬rst term in the parenthesis in Eq. (11.174) as |xt1 ’ xt2 | ≥ (D0 /kT )1/2 ,

and this oscillating phase factor provides the convergence of the integral. We should

therefore cut o¬ the q-integral at the wave vector satisfying q = (kT / D0 )1/2 ≡ L’1 ,

T

as indicated by the prime on the q-integration in the two previous equations.

Introducing new variables

xt ’ x’t

xt + x’t

√ √

Rt = , rt = (11.176)

2 2

52 Since the time label T now has disappeared, no confusion should arise in the following where T

denotes the temperature. We recall Section 6.5, and note that the relation Eq. (11.172) is equivalent

to the statement that the low-frequency electron“electron interaction in a disordered conductor is

identical to the Nyquist noise in the electromagnetic ¬eld ¬‚uctuations.

11.3. Phase breaking in weak localization 419

the path integral separates in two parts53

√

Rt = 2R

∞ R2

t ™

’ dt t

1

√

2D 0

DRt e

Ct,’t (R, R) = dR0 0

2 2„

’∞ Rt =0 =R0

rt =0 √

r2

t ™

2e 2 k T dq

q’2 (1’cos( 2q·rt ))

’ dt 2

t + + (2π )2

„ e’e

4D 0 σ0

— Drt e (11.177)

0

r0 =0

The path integral with respect to Rt gives the probability that a particle started at

√

position R0 at time t = 0 by di¬usion reaches the point 2 R (recall Eq. (7.103)).

Integrating this probability over all possible starting points is identical to integrating

over all ¬nal points and by normalization gives unity. We are thus left with the

expression for the Cooperon

ρt =0 t

r™¯2

’ dt

¯ t

1 + V (rt )

¯

=√

4D 0

Drt e

Ct,’t , (11.178)

0

¯

2 2„

ρ0 =0

where we have introduced the notation

√

2e2 kT

2 dq

q ’2 1 ’ cos( 2 q · r)

V (r) = + . (11.179)

„ e’e (2π)d

σ0

As expected from translational invariance, the Cooperon is independent of position.

We have thus reduced the problem of calculating the quantum correction to the

conductivity,

∞

2

4e D0 „

δσ(ω) = ’ dt eiωt Ct,’t (r, r) , (11.180)

π

’∞

in the presence of electron“electron interaction, to solving for the Green™s function

the imaginary time Schr¨dinger problem

o

1

√

{‚t ’ D0 δ(r ’ r ) δ(t ’ t ) .

+ V (r)} Ct,t (r, r ) = (11.181)

r

2 2„

In the three-dimensional case the ¬rst term in the integrand of Eq. (11.179) gives

rise to a temperature dependence of the form T 3/2 . This is the same form as the one

we shall ¬nd in Section 11.5 for the inelastic scattering rate due to electron“electron

interaction in a dirty metal. This term can thus be joined with the ¬rst term of

Eq. (11.179). We note that the description of the low-energy behavior thus joins up

smoothly with the description of the high-energy behavior, as it should.

53 This is immediately obtained by using the standard discretized representation of a path integral.

420 11. Disordered conductors

We thus have for the potential in the three-dimensional case

2 ˜

V3 (r) = + V3 (r) (11.182)

„ e’e

§

where

1

⎨ r LT

’e 2 r

˜3 (r) = √ kT

V (11.183)

√

2π 2 σ0 © L’1

22

r LT .

T

π

Fourier-transforming Eq. (11.181) with respect to time and taking the static limit we

obtain

1

{’D0 r + V3 (r)} Cω=0 (r, r ) = √ δ(r ’ r ) . (11.184)

2 2„

˜

Solving this equation to ¬rst order in the potential V3 gives

√L

√L

e2 kT L ’2 2 L

C1 (0, 0, ω = 0) = ’ ’1 + Ei ’2 2

e T

2

4π 2 „ D0 σ0 πLT LT

(11.185)

where Ei is the exponential integral54 and

D0 „ e’e .

L= (11.186)

In accordance with the calculation of the inelastic lifetime in section 11.5 we have

( kT „ )1/4

LT

∼ . (11.187)

L kF l

We can therefore expand the expression in Eq. (11.185), and obtain for the quantum

correction to the conductivity

e2 4πe2 kT L LT

δσ = 1+ ln , (11.188)

2π 2 L 2D σ L

00

where the second term is the correction due to collisions with small energy transfer,

proportional to T 1/4 ln T . In the two-dimensional case we obtain from Eq. (11.179)

for the potential

L’1 √

T

1 ’ J0 ( 2 qr)

2

2 e kT

V2 (r) = + dq , (11.189)

„ e’e π 2 σ0 q

0

where J0 denotes the Bessel function. We observe the limiting behavior of the po-

tential

§ 2

⎪

⎪ 1 r

r LT

⎪

⎪ 4 LT

⎨

2

2 e kT

V2 (r) = e’e +

π 2 σ0 ⎪

„ ⎪

⎪ √ √

⎪

© 1 ’ J0 ( L2 r ) ln L2 r ’ C + ln 2 r LT ,

T T

(11.190)

et

54 Ei(x) x

= dt for x < 0.

’∞ t

11.3. Phase breaking in weak localization 421

where C is the Euler constant.

We then get the following equation for the Cooperon in the region of large values

of r

√

e2 kT

2 2r 1

Cω=0 (r, r ) = √ δ(r ’ r ) . (11.191)

’D0 r + e’e + 2 ln

„ π σ0 LT 2 2„

√

The electron typically di¬uses coherently the distance D0 „ e’e . According to Sec-

tion 11.5, for the relaxation time in two dimensions for processes with large energy

transfers, we have

2 2

D0 N2 (0)

∼ ∼ (kF l)1/2 LT ,

„ e’e

D0 (11.192)

kT

where N2 (0) denotes the density of states at the Fermi energy in two dimensions. The

electron thus di¬uses coherently far into the region where the potential is logarithmic,

and the slow change of the potential allows the substitution

√ √

e2 kT e2 kT 2„ e’e D0

2 2r

’

+ ln ln . (11.193)

„ e’e π 2 σ0 π 2 σ0

LT LT

Inserting into Eq. (11.191), we can read o¬ the phase-breaking rate due to electron“

electron interaction in a dirty conductor in two dimensions55

1 kT

= ln 2π D0 N2 (0) . (11.194)

4π 2 D0 N2 (0)

„•

The phase-breaking rate due to di¬usion-enhanced electron“electron interaction thus

depends in two dimensions linearly on the temperature at low temperatures, kT <

/„ .

The above result for the phase-breaking rate can be understood as a consequence

of the phase-breaking rate setting the lower energy cut-o¬, /„• , for the e¬ciency of

inelastic scattering events in destroying phase coherence. To show this we note that

the path integral expression for the Cooperon, Eq. (11.167), is the weighted average

with respect to di¬usive paths. Since this weight is convex, we have

e’ (•[xc l ])2

(0)

Ct ≥ Ct , (11.195)

ee imp

t

where the second bracket signi¬es the average with respect to di¬usive paths of the

phase di¬erence between the two interfering alternatives, Eq. (11.129),

xt / 2 =x

Dxt Pt [xt ] (•[xcl ])2 ee

t

x’t / 2 =x

(•[xcl ])2 = (11.196)

xt / 2 =x

ee imp

t

Dxt Pt [xt ]

x’t / 2 =x

55 Many experiments are performed on thin metallic ¬lms. For such a quasi-two-dimensional case

we can express the result for the phase breaking due to electron“electron interaction in a ¬lm of

e 2 kT

thickness a as „1 = 2πaσ 2 ln πaσ 0 .

e2

• 0

422 11. Disordered conductors

(0)

and Ct is the return probability in the absence of the ¬‚uctuating ¬eld, i.e. the

denominator in the above equation. The ¬rst bracket signi¬es the Gaussian average

over the ¬‚uctuating ¬eld, i.e. the low-energy electron“electron interaction,

t/2 t/2

2

e

φ(xcl ’ xcl , t1 ’ t2 ) φ(0, 0)

(•[xcl ])2 = dt1 dt2

ee ee

t t1 t2

2

’t/2 ’t/2

’ φ(xcl ’ xcl , t1 + t2 ) φ(0, 0) , (11.197)

ee

t1 t2

where we now choose to let the scalar potential represent the ¬‚uctuating ¬eld.

Fourier-transforming we encounter

dq dω iq·(xcl ’xcl )

φ(xcl ’xcl , t1 ’t2 ) φ(0, 0) =2 e φφ

t1 t2

qω

ee imp imp

t1 t2

(2π)d 2π

— (cos ω(t1 + t2 ) ’ cos ω(t1 ’ t2 )) , (11.198)

where the correlator for the ¬‚uctuating potential is speci¬ed in Eq. (11.269). For a

di¬usion process we have, according to Eq. (7.104),56

eiq·(xt 1 ’xt 2 ) (xcl ’xcl )

cl cl

= e’D0 q |t1 ’t2 |

2

= eiq· imp

(11.199)

t1 t2

imp

and we get

t/2 t/2

2e2 kT dq dω ’ 1 D0 q2 |t1 ’t2 |’iω(t1 ’t2 )

(•[xcl ])2 = dt1 dt2 e2 , (11.200)

ee imp

t

(2π)d 2π

πσ0

’t/2 ’t/2

where the ω-integration is limited to the region 1/„• ¤ |ω| ¤ kT / . The averaged

phase di¬erence is seen to increase linearly in time:

1 t

(•[xcl ])2 = (11.201)

ee imp

t

2 „•

at a rate in accordance with the previous result for the phase-breaking rate, Eq.

(11.194).

The lack of e¬ectiveness in destroying phase coherence by interactions with small

energy transfers is re¬‚ected in the compensation at small frequencies between the two

cosine terms appearing in the expression for the phase di¬erence, Eq. (11.198). In the

case of di¬usion-enhanced electron“electron interaction this compensation is crucial

as there is an abundance of scattering events with small energy transfer, whereas

the compensation was immaterial for electron“phonon interaction where the typical

energy transfer is determined by the temperature.

56 The last equality is an approximation owing to the constraint, x’t/2 = xt/2 , however, for large

times a very good one.

11.4. Anomalous magneto-resistance 423

Whereas the phase-breaking rate for electron“phonon interaction is model depen-

dent, i.e. material dependent, we note the interesting feature that the phase-breaking

rate for di¬usion-enhanced electron“electron interaction is universal. In two dimen-

sions we can rewrite

e2 σ0 kT

1 kF l

= ln . (11.202)

2π 2

„• 2

Phase-breaking rates in accordance with Eq. (11.194) have been extracted from

numerous magneto-resistance measurements; see, for example, references [88] and

[89]. We note that at su¬ciently low temperatures the electron“electron interaction

dominates the phase-breaking rate in comparison with the electron“phonon interac-

tion.

11.4 Anomalous magneto-resistance

From an experimental point of view, the disruption of coherence between time-

reversed trajectories by an externally controlled magnetic ¬eld is the tool by which

to study the weak-localization e¬ect. Magneto-resistance measurements in the weak-

localization regime has considerably enhanced the available information regarding

inelastic scattering times (and spin-¬‚ip and spin-orbit scattering times). The weak-

localization e¬ect thus plays an important diagnostic role in materials science.

The in¬‚uence of a magnetic ¬eld on the Cooperon was established in Section

11.2.4, and we have the Cooperon equation

2ie 1

’iω ’ D0 {∇x ’ δ(x ’ x ) .

A(x)}2 + 1/„• Cω (x, x ) = (11.203)

„

We can now safely study the d.c. conductivity, i.e. assume that the external electric

¬eld is static, so that its frequency is equal to zero, ω = 0, as the Cooperon in an

external magnetic ¬eld is no longer infrared divergent. The Cooperon is formally

identical to the imaginary-time Schr¨dinger Green™s function for a ¬ctitious particle

o

with mass equal to /2D0 and charge 2e moving in a magnetic ¬eld (see Exercise C.1

on page 515). To solve the Cooperon equation for the magnetic ¬eld case, we can

thus refer to the equivalent quantum mechanical problem of a particle in an external

homogeneous magnetic ¬eld. Considering the case of a homogeneous magnetic ¬eld,57

and choosing the z-direction along the magnetic ¬eld and representing the vector

potential in the Landau gauge, A = B (’y, 0, 0), the corresponding Hamiltonian is

D0 D0

(ˆx + 2eB y)2 + p2 + p2 .

H= p ˆ ˆy ˆz (11.204)

The problem separates

i i

px x pz z

ψ(x, y) = e e χ(y) , (11.205)

where the function χ satis¬es the equation

D0 d2 χ(y) 1 px 2

˜

’ ωc y ’

˜2

+ χ(y) = E χ(y) (11.206)

2

2 dy 2 2D0 2eB

57 The case of an inhomogeneous magnetic ¬eld is treated in reference [95].

424 11. Disordered conductors

the shifted harmonic oscillator problem where ωc is the cyclotron frequency for the

˜

˜

¬ctitious particle, ωc ≡ 4D0 |e|B/ , so that the energy spectrum is E = E + D0 Q2 =

˜ z

ωc (n + 1/2) + D0 Qz , n = 0, 1, 2, ...; Qz = 2πnz /Lz , nz = 0, ±1, ±2, ... . In the

2

˜

particle in a magnetic ¬eld analogy, n is the orbital quantum number and px is the

quantum number describing the position of the cyclotron orbit, and describes here the

possible locations of closed loops. The Cooperon in the presence of a homogeneous

magnetic ¬eld of strength B thus has the spectral representation

—

nmax

ψn,px (x) ψn,px (x )

dpx

C0 (x, x ) = , (11.207)

2π 4D0 |e|B„ ’1 (n + 1/2) + D0 „ Q2 + „ /„•

z

n=0

Qz

where the ψn,px are the Landau wave functions

1 i

ψn,px (x) = √ e px x eiQz z χn (y ’ px /2eB) (11.208)

Lz

and χn (y) is the harmonic oscillator wave function. In accordance with the derivation

of the Cooperon equation, we can describe variations only on length scales larger than

the mean free path. The sum over the orbital quantum number n should therefore

terminate when D0 „ |e|Bnmax ∼ , i.e. at values of the order of nmax lB /l2 , where

2

lB ≡ ( /|e|B)1/2 is the magnetic length.

To calculate the Cooperon for equal spatial values, C0 (x, x), we actually do not

need all the information contained in Eq. (11.207), since by normalization of the wave

functions in the completeness relation we have

∞ ∞

dpx — px px 2eB 2eB

χn y ’ y’ =’ dy |χn (y)|2 = ’

χn

’∞ 2π 2eB 2eB 2π 2π

’∞

(11.209)

and thereby

nmax

2eB 1

C0 (x, x) = ’ . (11.210)

4D0 |e|B„ ’1 (n + 1/2) + D0 „ Q2 + „ /„•

2π z

n=0

Qz

11.4.1 Magneto-resistance in thin ¬lms

We now consider the magneto-resistance of a ¬lm of thickness a, choosing the di-

rection of the magnetic ¬eld perpendicular to the ¬lm.58 Provided the thickness

of the ¬lm is smaller than the phase coherence length, a L• (the thin ¬lm, or

quasi-two-dimensional criterion), or the usually much weaker restriction that it is

smaller than the magnetic length, a lB , only the smallest value of Qz = 2π n/Lz ,

n = 0, ±1, ±2, ... contributes to the sum. Since the smallest value is Qz = 0, we

obtain, according to Eq. (11.59), for the quantum correction to the conductivity

nmax

e3 BD0 „ 1

δσ(B) = . (11.211)

4D0 |e|B„ ’1 (n + 1/2) + „ /„•

π2 2 a n=0

58 Thestrictly two-dimensional case can also be realized experimentally, for example by using the

two-dimensional electron gas accumulating in the inversion layer in a MOSFET or heterostructure.

11.4. Anomalous magneto-resistance 425

Employing the property of the di-gamma function ψ (see, for example, reference [96])

n’1

1

ψ(x + n) = ψ(x) + (11.212)

x+n

n=0

we get for the magneto-conductance

e2 ˜ ’1

f2 (4D0 |e|B

δG±β (B) = „ • ) δ±β , (11.213)

4π 2

where

1 1 3 1

˜

f2 (x) = ψ + +ψ + nmax + . (11.214)

2 x 2 x

The magneto-conductance of a thin ¬lm is now obtained by subtracting the zero ¬eld

conductance. In the limit B ’ 0, the sum can be estimated to become

nmax

1 ’1

’ ln(nmax 4D0 |e|B „• ) . (11.215)

’1 (n

4D0 |e|B„ + 1/2) + „ /„•

n=0

Using the property of the di-gamma function

3 1

lim ψ +n+ ln n (11.216)

2 x

n’∞

we ¬nally arrive at the low-¬eld magneto-conductance of a thin ¬lm

e2

”G±β (B) ≡ δG±β (B) ’ δG±β (B ’ 0) = f2 (B/B• ) δ±β , (11.217)

2π 2

where

11

f2 (x) = ln x + ψ + (11.218)

2x

and B• = /4D0|e|„ • , the (temperature-dependent) characteristic scale of the mag-

netic ¬eld for the weak-localization e¬ect, is determined by the inelastic scatter-

ing. This scale is indeed small compared with the scale for classical magneto-

resistance e¬ects Bcl ∼ m/|e|„ , as B• ∼ Bcl / F „ • .59 The weak-localization

magneto-conductance is seen to be sensitive to very small magnetic ¬elds, namely

when the magnetic length becomes comparable to the phase coherence length, lB ∼

L• , or equivalently, ωc „ ∼ / F „• . Since the impurity mean free time, „ , can be

much smaller than the phase coherence time „• , the above description can be valid

over a wide magnetic ¬eld range where classical magneto-conductance e¬ects are ab-

sent. Classical magneto-conductance e¬ects are governed by the orbit bending scale,

59 In terms of the mass of the electron we have for the mass of the ¬ctitious particle /2D0 ∼

m / F „ , and the low magnetic ¬eld sensitivity can be viewed as the result of the smallness of the

¬ctitious mass in the problem.

426 11. Disordered conductors

ωc „ ∼ 1, whereas the weak-localization quantum e¬ect sets in when a loop of typical

area L2 encloses a ¬‚ux quantum.60 We note the limiting behavior of the function

•

§

x2

⎨ for x 1

24

f2 (x) = (11.219)

©

ln x for x 1.

The magneto-conductance is positive, and seen to have a quadratic upturn at low

¬elds, and saturates beyond the characteristic ¬eld in a universal fashion, i.e. in-

dependent of sample parameters.61 The magneto-resistance is therefore negative,

”R = ’”G/G2 , which is a distinct sign that the e¬ect is not classical, since we are

cl

considering a macroscopic system.62

Weak localization magneto-conductance is also relevant for a three-dimensional

sample, and cleared up a long-standing mystery in the ¬eld of magneto-transport in

doped semiconductors. For details on the three-dimensional case we refer the reader

to chapter 11 of reference [1].

The negative anomalous magneto-resistance can be understood qualitatively from

the simple interference picture of the weak-localization e¬ect. The presence of the

magnetic ¬eld breaks the time-reversal invariance, and upsets the otherwise identical

values of the phase factors in the amplitudes for traversing the time-reversed weak-

localization loops. The quantum interference term for a loop c is the result of the

presence of the magnetic ¬eld changed according to

2ie

Ac A— ’ |A(B=0) |2 exp

2i e

d¯ · A(¯) = |A(B=0) |2 e ¦c

, (11.220)

x x

c

¯ c c

c

where ¦c is the ¬‚ux enclosed by the loop c. The weak-localization interference term

acquires a random phase depending on the loop size, and the strength of the magnetic

¬eld, decreasing the probability of return, and thereby increasing the conductivity.

The negative contribution from each loop in the impurity ¬eld to the conductance

is modulated in accordance with the phase shift prescription for amplitudes by the

oscillatory factor, giving the expression

e2

|A(B=0) |2 {1’cos(2π¦c /¦0 )} e’tc /„•

G(B) ’ G(O) = . (11.221)

imp

c

2π 2 c

The summation is over all classical loops in the impurity ¬eld returning to within a

distance of the mean free path to a given point, and tc is the duration for traversing

60 Beyond the low-¬eld limit, ωc „ < / F „ , the expression for the magneto-conductance can not

be given in closed form, and its derivation is more involved, since we must account for the orbit

bending due to the magnetic ¬eld, the Lorentz force [97]. When the impurity mean free time „

becomes comparable to the phase coherence time „• , we are no longer in the di¬usive regime, and

a Boltzmannian description must be introduced [98].

61 Experimental observations of the low ¬eld magneto-resistance of thin metallic ¬lms are in re-

markable good agreement with the theory. The weak-localization e¬ect is thus of importance for

extracting information about inelastic scattering strengths, which is otherwise hard to come at. For

reviews of the experimental results, see references [88] and [89].

62 The classical magneto-resistance of a macroscopic sample calculated on the basis of the Boltz-

mann equation is positive.

11.4. Anomalous magneto-resistance 427

the loop c, and ¦0 is the ¬‚ux quantum ¦0 = 2π /2|e|. The sum should be performed

weighted with the probability for the realization of the loop in question, as expressed

by the brackets. The weight of loops that are longer than the phase coherence length

is suppressed, as their coherence are destroyed by inelastic scattering. In weak mag-

netic ¬elds, only the longest loops are in¬‚uenced by the phase shift due to the mag-

netic ¬eld. It is evident from Eq. (11.221) that the low ¬eld magneto-conductance

is positive and quadratic in the ¬eld.63 The continuing monotonic behavior as a

function of the magnetic ¬eld until saturation is simply a geometric property of dif-

fusion, viz. that small di¬usive loops are proli¬c. Instead of verifying this statement,

let us turn the argument around and use our physical understanding of the weak

localization e¬ect to learn about the distribution of the areas of di¬usive loops in

two dimensions. Rewriting Eq. (11.210) we have in two dimensions

∞ nmax

B 4π B D 0

’

’t/„ • (n+1/2)t

C0 (x, x) = dt e e . (11.222)

¦0

„ ¦0

0 n=0

For times t > „ we can let the summation run over all natural numbers and we can

sum the geometric series to obtain

∞

e’t/„ • 2πBD0 t 1

C0 (x, x) = dt . (11.223)

sinh 2πBD0 t

4π„ D0 t ¦0

0 ¦0

The factors independent of the magnetic ¬eld are the return probability and the

dephasing factor. Representing the factors depending on the ¬eld strength, which

describes the in¬‚uence of the magnetic ¬eld on the quantum interference process, by

its cosine transform

∞

2πBD0 t 1 SB

= dS cos ft (S) (11.224)

sinh 2πBD0 t

¦0 ¦0

’∞

¦0

and inverting gives

1 1

ft (S) = . (11.225)

4D0 t cosh2 S

4D0 t

For the weak localization contribution to the conductance we can therefore write

∞ ∞

e’t/„ •

2e2 D0 „ BS

δG(B) = ’ dt dS ft (S) cos (11.226)

π 4π„ D0 t ¦0

0 0

and we note that ft (S) is normalized, and has the interpretation of the probability

for a di¬usive loop of duration t to enclose the area S.

For the average size of a di¬usive loop of duration t we have

∞

≡

S dS Sft (S) = 4D0 t ln 2 , (11.227)

t

0