are smaller by the factor / F „ .

Displaying a maximally crossed kinetic propagator diagram on twisted form we

have (we use the notation 1 ≡ (x1 , t1 ) etc.); the diagram in depicted in Figure 11.4.

1 t10 t9 t8 t7

6

5 4 3 2 1

Figure 11.4 Twisted maximally crossed kinetic propagator diagram.

Because of the four di¬erent places where the kinetic propagator can occur we

explicitly keep the four outermost impurity correlators, and obtain for the quantum

11.2. Weak localization 397

correction to the kinetic propagator

eu8

K

d“ GR (x1 , t1 ; x5 , t10 ) GR (x5 , t10 ; x4 , t9 ) G0 (x3 , t8 ; x2 , t7 )

δG (x1 , t1 , x1 , t1 ) =

2im

”

— A(x6 , t6 ) · G0 (x2 , t7 ; x6 , t6 ) ∇x6 G0 (x6 , t6 ; x5 , t5 )

R

δg12 (x4 , t9 , t4 )

— 0

G (x5 , t5 ; x4 , t4 ) R

δV12 (x3 , t8 , t3 )

— GA (x3 , t3 ; x2 , t2 ) GA (x2 , t2 ; x1 , t1 ) , (11.85)

where the propagators labeled by a zero as the superscript index indicate where the

kinetic propagator can appear (i.e. we have a sum of four terms, and the kinetic

propagator is always sandwiched in between retarded propagators to the left and

advanced propagators to the right), and we have introduced the abbreviation

6

d“ = dt7 dt8 dt9 dt10 dxi dti . (11.86)

i=1

Since the propagators carry the large momentum pF , we can take for the explicitly

appearing linear response vector potential

A(t) = Aω1 e’iω1 t . (11.87)

The eight exhibited propagators in Eq. (11.85) can be taken to be the equilibrium

ones, and by Fourier transforming the propagators, and performing the integration

over the momenta, we obtain for the quantum correction to the current density at

±

frequency ω2 , E1 = E1 ± ω1 /2,

∞

4e2 D0 „ dE1 ’

f0 (E1 ) ’ f0 (E1 ) dt1 dt1 dt2 dt2 δ(t2 ’ t1 )

+

δj(x, ω2 ) = A(ω1 )

iπ 2π

’∞

’ +

(E1 t1 ’E1 t2 ’ ω2 t1 )

i

— e C(x, x; t1 , t1 , t2 , t2 ) (11.88)

or equivalently

∞

4e2 D0 „ dE1 ’

f0 (E1 ) ’ f0 (E1 )

+

δj(x, ω2 ) = A(ω1 )

iπ 2π

’∞

∞∞

dt dT Ct,’t (x, x) eiT (ω1 ’ω2 )+i 2 (ω1 +ω2 ) .

t

— T

(11.89)

’∞ ’∞

398 11. Disordered conductors

For the quantum correction to the conductivity in the presence of a time-dependent

electromagnetic ¬eld

δj(x, ω2 ) = δσ(x, ω2 , ω1 ) E(ω1 ) (11.90)

we therefore obtain29

∞

4e2 D0 „ dE1 ’

δσ(x, ω2 , ω1 ) = ’ f0 (E1 ) ’ f0 (E1 )

+

πω 2π

’∞

∞∞

dt dT eiT (ω2 ’ω1 )+ 2 t(ω1 +ω2 ) Ct,’t (x, x) .

i

— T

(11.91)

’∞ ’∞

In the degenerate case we have

∞∞

4e2 D0 „

dt dT Ct,’t (x, x) eiT (ω1 ’ω2 )+i 2 (ω1 +ω2 ) .

t

δσ(x, ω2 , ω1 ) = ’ T

(11.92)

π

’∞ ’∞

In the event that the included e¬ect of an electromagnetic ¬eld on the Cooperon is

caused by a time-independent magnetic ¬eld, we recover the expression Eq. (11.59)

for the quantum correction to the conductivity.

We shall exploit the derived formula when we consider the in¬‚uence of electron“

electron interaction on the quantum correction to the conductivity.

11.2.3 Quantum interference and the Cooperon

In this section, we shall elucidate in more detail than in Section 11.1.2 the physical

process in real space described by the maximally crossed diagrams, and in addition

consider the in¬‚uence of external ¬elds. The weak-localization e¬ect can be under-

stood in terms of a simple kind of quantum mechanical interference. By following the

scattering sequences appearing in the diagrammatic representation of the Cooperon

contribution to the conductivity, see Eq. (11.47), we realize that the quantum cor-

rection to the conductivity consists of products of the form “amplitude for scattering

sequence of an electron o¬ impurities in real space times the complex conjugate of the

amplitude for the opposite scattering sequence.” The quantum correction to the con-

ductivity is thus the result of quantum mechanical interference between amplitudes

for an electron traversing a loop in opposite directions. To lowest order in »F /l we

need to include only the stationary, i.e. classical, paths determined by the electron

bumping into impurities, as illustrated in Figure 11.2 where the trajectories involved

in the weak-localization quantum interference process are depicted. The solid line,

say, in Figure 11.2 corresponds to the propagation of the electron represented by

29 For an electron gas in thermal equilibrium f0 is the Fermi function, but in principle we could

at this stage have any distribution not violating Pauli™s exclusion principle. However, that would

then necessitate a discussion of energy relaxation processes tending to drive the system toward the

equilibrium distribution.

11.2. Weak localization 399

the retarded propagator in the conductivity diagram, and the dashed line to the

propagation represented by the advanced propagator, the complex conjugate of the

amplitude for scattering o¬ impurities in the opposite sequence. The starting and

end points refer to the points x and x in Eq. (11.52), respectively.30 According to

the formula, Eq. (11.59), for the quantum correction to the conductivity, we need to

consider only scattering sequences which start and end at the same point on the scale

of the mean free path, as demanded by the impurity-averaged propagators attached

to the maximally crossed diagrams in Eq. (11.52).

In the time-reversal invariant situation, the contribution to the return probabil-

ity from the maximally crossed diagrams equals the contribution from the ladder

diagrams, and the return probability including the weak-localization contribution is

thus twice the classical result31

d/2

1

Pcl+wl (x, t; x, t ) = 2 Pcl (x, t; x, t ) = 2 , (11.93)

4πD0 (t ’ t )

where the last expression is valid in the di¬usive limit. To see how this comes about

in the interference picture, let us consider the return probability in general. The

quantity of interest is therefore the amplitude K for an electron to arrive at a given

space point x at time t/2 when initially it started at the same space point at time

’t/2. According to Feynman, this amplitude is given by the path integral expression

xt / 2 =x

i

K(x, t/2; x, ’t/2) = Dxt e ≡

S[xt ]

Ac (11.94)

c

x’t / 2 =x

where the path integral includes all paths which start and end at the same point.

For the return probability we have

Ac A—

P = |K|2 = | Ac |2 = |Ac |2 + (11.95)

c

c c c=c

where Ac is the amplitude for the path c. In the sum over paths we only need to

include to order »F /l the stationary, i.e. classical, paths determined by the electron

bumping into impurities. The sum of the absolute squares is then the classical con-

tribution to the return probability, and the other terms are quantum interference

terms. In the event that the particle only experiences the impurity potential, we

have for the amplitude for the particle to traverse the path c,

t

2 dt { 1 mx2 (t) ’ V (xc (t))}

¯ ™c ¯ ¯

i

’t 2

Ac = e . (11.96)

2

30 The angle between initial and ¬nal velocities is exaggerated since we recall that in order for the

Cooperon to give a large contribution the angle must be less than 1/kF l.

31 The fact that impurity lines cross, does not per se make a diagram of order 1/k l relative to a

F

non-crossed diagram. In case of the conductivity diagrams this is indeed the case for the maximally

crossed diagrams because the circumstances needed for a large contribution set a constrain on the

correlation of the initial and ¬nal velocity, p ’p + Q (recall also when estimating self-energy

diagrams the importance of the incoming and outgoing momenta being equal, see reference [1]).

However, in the quantity of interest here the position is ¬xed.

400 11. Disordered conductors

Owing to the impurity potential, the amplitude has a random phase. A ¬rst con-

jecture would be to expect that, upon impurity averaging, the interference terms in

general average to an insigni¬cant small value, and we would be left with the clas-

sical contribution to the conductivity. However, there are certain interference terms

which are resilient to the impurity average. It is clear that impurity averaging can

not destroy the interference between time-reversed trajectories since we have for the

amplitude for traversing the time-reversed trajectory, xc (t) = xc (’t),

¯

t t

2 dt { 1 mx2 (t) ’ V (xc (t))} 2 dt { 1 m[’x (’t)]2 ’ V (xc (’t))}

¯ ™c ¯ ¯¯ ¯ ¯ ¯

i i

™c

¯

’t ’t

2 2

Ac = e =e = Ac .(11.97)

2 2

¯

In this time-reversal invariant situation the amplitudes for traversing a closed loop in

opposite directions are identical, Ac = Ac , and the corresponding interference term

¯

contribution to the return probability is independent of the disorder, Ac A— = 1!c

¯

The two amplitudes for the time-reversed electronic trajectories which return to the

starting point thus interfere constructively in case of time-reversal invariance. In

correspondence to this enhanced localization, there is a decrease in conductivity

which can be calculated according to Eq. (11.59).

The foregoing discussion based on the physical understanding of the weak lo-

calization e¬ect will now be substantiated by deriving the equation satis¬ed by the

Cooperon. The Cooperon Cω (x, x ) is generated by the iterative equation

R

x x x x

x x=

Cω + + + ...

x x

A

x x

x x

= + Cω (11.98)

x

where we have introduced the diagrammatic notation

x

≡ δ(x ’ x ) . (11.99)

x

The Cooperon equation, Eq. (11.98), is most easily obtained by adding the term

11.2. Weak localization 401

x

u2 δ(x ’ x )

= (11.100)

x

˜

to the in¬nite sum of terms represented by the function C, Eq. (11.51). Alterna-

tively, one can proceed as in Section 11.1.2, now exploiting the local character of the

˜

propagators. In any event, we have in the singular region C u2 C.

The Cooperon equation in the spatial representation is

˜C

Cω (x, x ) = δ(x ’ x ) + dx Jω (x, x ) Cω (x , x ) , (11.101)

where according to the Feynman rules the insertion is given by

˜C

Jω (x, x ) = u2 GR + ) GA (x, x ) .

ω (x, x (11.102)

F F

The Cooperon is slowly varying on the scale of the mean free path, the spatial range

˜C

of the function Jω (x, x ), and a low-order Taylor-expansion of the Cooperon on the

right-hand side of Eq. (11.101) is therefore su¬cient. Upon partial integration, the

integral equation then becomes, for a second-order Taylor expansion, a di¬erential

equation for the Cooperon

1

’ iω ’ D0 ∇2 Cω (x, x ) = δ(x ’ x ) . (11.103)

x

„

This equation is of course simply the position representation of the equation for the

Cooperon already derived in the momentum representation, Eq. (11.23). Indeed we

recover, that the Cooperon only varies on the large length scale Lω = (D0 /ω)1/2 .

The typical size of an interference loop is much larger than the mean free path, and

we only need the large-scale behavior of the Boltzmannian paths of Figure 11.2, the

smooth di¬usive loops of Figure 11.5.

The fact that in the time-reversal invariant case we have obtained that the

Cooperon satis¬es the same di¬usion-type equation as the Di¬uson is not surprising.

The Di¬uson is determined by a similar integral equation as the Cooperon, however,

with the important di¬erence that one of the particle lines, say the advanced one, is

reversed (recall Exercise 7.8 on page 197). The Di¬uson will therefore be determined

˜C

by the same integral equation as the Cooperon, except for Jω now being substituted

˜D

by the di¬usion insertion Jω , given by

˜D

Jω (x, x ) = u2 GR + ) GA (x , x) .

ω (x, x (11.104)

F F

˜C ˜D

In a time-reversal invariant situation the two insertions are equal, Jω = Jω , and we

recover that the Di¬uson and the Cooperon satisfy the same equation (and we have

hereby re-derived the result, Eq. (11.93)).

402 11. Disordered conductors

R—

R

Figure 11.5 Di¬usive loops.

In the time-reversal invariant situation the amplitudes for traversing a closed loop

in opposite directions are identical, and in such a coherent situation one must trace

the complete interference pattern of wave re¬‚ection in a random medium, and one

encounters the phenomenon of localization discussed earlier. However, we also realize

that the interference e¬ect is sensitive to the breaking of time-reversal invariance. By

breaking the coherence between the amplitudes for traversing time-reversed loops the

tendency to localization of an electron can be suppressed.32 In moderately disordered

conductors we can therefore arrange for conditions so that the tendency to localiza-

tion of the electronic wave function has only a weak though measurable in¬‚uence on

the conductivity. The ¬rst quantum correction then gives the dominating contribu-

tion in the parameter »F /l, and we speak of the so-called weak-localization regime.

The destruction of phase coherence is the result of the interaction of the electron with

its environment, such as electron“electron interaction, electron“phonon interaction,

interaction with magnetic impurities, or interaction with an external magnetic ¬eld.

From an experimental point of view the breaking of coherence between time-reversed

trajectories by an external magnetic ¬eld is of special importance, and we start by

discussing this case.

11.2.4 Quantum interference in a magnetic ¬eld

The in¬‚uence of a magnetic ¬eld on the quantum interference process described by

the Cooperon is readily established in view of the already presented formulas. In the

weak magnetic ¬eld limit, l2 < lB , where lB = ( /2eB)1/2 is the magnetic length, or

2

equivalently ωc „ < / F „ , the bending of a classical trajectory with energy F can be

32 By disturbance, the coherence can be disrupted, and the tendency to localization can be sup-

pressed, thereby decreasing the resistance. Normally, disturbances increase the resistance.

11.2. Weak localization 403

neglected on the scale of the mean free path. Classical magneto-resistance e¬ects are

then negligible, because they are of importance only when ωc „ ≥ 1. The amplitude

for propagation along a straight-line classical path determined by the impurities is

then changed only because of the presence of the magnetic ¬eld by the additional

phase picked up along the straight line, the line integral along the path of the vector

potential A describing the magnetic ¬eld. In the presence of such a weak static

magnetic ¬eld the propagator is thus changed according to

x

ie

GR (x, x ) ’ GR (x, x ) exp d¯ · A(¯) . (11.105)

x x

E E

x

The resulting change in the Cooperon insertion is then

x

2ie

˜C ˜C

Jω (x, x ) ’ Jω (x, x ) exp d¯ · A(¯)

x x

x

2ie

˜C (x ’ x ) · A(x)

= Jω (x, x ) exp . (11.106)