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

63 The minimum value of the magneto-resistance occurs exactly for zero magnetic ¬eld value, and

the weak localization e¬ect is thus one of the few e¬ects that can be used as a reference for zero

magnetic ¬eld.

428 11. Disordered conductors

i.e. the typical size of a di¬usive loop of duration t is proportional to D0 t.

For the ¬‚uctuations we have

∞

≡

2

dS S 2 ft (S) = 8π 2 (D0 t)2

S (11.228)

t

’∞

and we can write

π 1

ft (S) = . (11.229)

cosh2 √ πS 2

2 S2 t

2S t

The probability distribution for di¬usive loops is thus a steadily decreasing function

of the area.

The weak localization e¬ect in cylinders and rings leads through the Aharonov“

Bohm e¬ect to an amazing manifestation of the quantum mechanical superposition

principle at the macroscopic level. Furthermore, the weak localization e¬ect can

be reversed to weak anti-localization if the impurities, such as is the case in heavy

compounds, give rise to spin-orbit scattering. Discussion of these e¬ects can be found

in chapter 11 of reference [1].

11.5 Coulomb interaction in a disordered conductor

The presence of impurities changes the e¬ective electron“electron interaction. We

shall study this e¬ect in the weak disorder limit, F „ , which is the common situ-

ation in conductors such as metals and semiconductors. The change from ballistic to

di¬usive motion leads to di¬usion enhancement of the electron“electron interaction.

This leads to interesting observable e¬ects such as the temperature dependence of

the conductivity of a three-dimensional sample being proportional to the square root

√

of the temperature [99], σ ∝ T , instead of the usually unnoticeable T 2 -term due to

Umklapp processes in a clean metal. For experimental evidence of the square root

temperature dependence see references [100, 101].

Let us assume that the inverse screening length is much smaller than the Fermi

wavelength; i.e. the range of the screened Coulomb potential, V , is much larger than

the spacing between the electrons. The exchange correction to the electron energy

» due to electron“electron interaction is then much larger than the direct or Hartree

term. We shall use the method of exact impurity eigenstates and, since diagonal

elements dominate, Σ» ≡ Σ»» , we have for the exchange self-energy

— —

Σex = ’ dx dx V (x ’ x ) ψ» (x) ψ» (x ) ψ» (x ) ψ» (x) , (11.230)

»

» occ.

where the summation is over all occupied states » , i.e. all the states below the Fermi

level since for the moment we assume zero temperature. We are interested in the

mean energy shift averaged over all states with energy ξ (measured from the Fermi

energy)

1

δ(ξ ’ ξ» ) Σex

Σex (ξ) = (11.231)

»

N0 V

»

11.5. Coulomb interaction in a disordered conductor 429

for which we obtain the expression, say ξ > 0,

0

1

’ dξ dx dx V (x ’ x )

Σex (ξ) =

N0 V

’∞

— —

— δ(ξ ’ ξ» ) δ(ξ ’ ξ» ) ψ» (x) ψ» (x ) ψ» (x ) ψ» (x) , (11.232)

»,»

where the prime on the summation sign indicates that the sum is only over states »

occupied and states » unoccupied. The impurity-averaged quantity is the product of

two spectral weight functions in the exact impurity eigenstate representation, except

for the restrictions on the summations. However, these are irrelevant as the main

contribution comes from ξ ξ. In the standard impurity averaging technique we

encounter in the weak-disorder limit, 1/kF l 1, the di¬usion ladder, and we obtain

∞

D0 q 2

1 dq

’

ex

Σ (ξ) = dω V (q) 2 . (11.233)

(2π)d ω + (D0 q 2 )2

2π

ξ/

In the above model of a static interaction the average change in energy is purely

real. The result obtained can be used to calculate the change in the density of states.

To lowest order in the electron“electron interaction we have for the change in density

of states due to the electron“electron interaction

‚Σex (ξ)

δN (ξ) ≡ N (ξ) ’ N0 (ξ) = ’N0 (ξ)

‚ξ

D0 q 2

N0 dq

= V (q) (11.234)

2

(2π)d

2π ξ

q 2 )2

+ (D0

as the change in the density of states due to disorder is negligible in the weak-disorder

limit.

Exercise 11.3. Verify that if V is a short-range potential, the change in the density

of states near the Fermi surface due to electron“electron interaction is in the weak-

disorder limit

|ξ|

δN3 (ξ) V (q = 0)

√

= (11.235)

4 2π 2 ( D0 )3/2

N3 (0)

in three dimensions and, in two dimensions,

|ξ|„

δN2 (ξ) V (q = 0)

= ln . (11.236)

(2π)2 D0

N2 (0)

430 11. Disordered conductors

The singularity in the density of states is due to the spatial correlation of the exact

impurity wave functions of almost equal energy, as described by the singular behavior

of the spectral correlation function. The singularity in the density of states gives rise

to the zero-bias anomaly, a dip in the conductivity of a tunnel junction at low voltages

[102].

Quite generally the propagator in the energy representation satis¬es, in the pres-

ence of disorder and electron“electron interaction, the equation

(0)R (0)R

GR (E) = G»» (E) + G»»1 (E) ΣR1 »1 (E) GR1 » (E) , (11.237)

»» » »

»1 »1

where the propagator in the absence of electron“electron interaction is diagonal,

(0)R (0)R

G»» (E) = G» (E) δ»» , and speci¬ed in terms of the exact impurity eigenstates

(here in the momentum representation)

—

ψ» (p) ψ» (p ) —

R(A) (0)R(A)

≡

G0 (p, p , E) = ψ» (p) ψ» (p ) G» (E) .

E ’ » (’) i0

+

» »

(11.238)

Since energy eigenstates are only spatially correlated if they have the same energy,

only the diagonal terms, ΣR (E) ≡ ΣR (E), contribute in Eq. (11.237), and we obtain

» »»

the result that the propagator is approximately diagonal and speci¬ed by

1

GR (E) = . (11.239)

E ’ » ’ ΣR (E)

»

»

The imaginary part of the self-energy describes the decay of an exact impurity eigen-

state due to electron“electron interaction. When calculating the inelastic decay rate,

we should only count processes starting with the same energy, and on the average in

the random potential we are therefore interested in the quantity

1

δ(E ’ ξ» ) ΣR (E) .

ΣR (E) = (11.240)

E »

N0 V

»

To lowest order in the electron“electron interaction we can set E equal to E in

Eq. (11.240) because their di¬erence is the real part of the self-energy, and we get

for the inelastic electron“electron collision rate

1

’2 m ΣR (E) = i ΣR (E) ’ ΣA (E)

= E E E

„e’e (E, T )

1 (0)R (0)A

’ ΣR (E) ’ ΣA (E) (E) ’ G»

= G» (E) ,

» »

2π N0 V

»

(11.241)

11.5. Coulomb interaction in a disordered conductor 431

where we have expressed the delta function in Eq. (11.240) in terms of the spec-

tral function. We thus have to impurity average a product of a self-energy and

a propagator, say the retarded self-energy and the advanced propagator, presently

both expressed in the exact impurity eigenstate representation. In the weak-disorder

limit, kF l 1, the contributions to the collision rate are therefore speci¬ed in terms

of the Di¬uson and the e¬ective electron“electron interaction as depicted in Figure

11.6. For the case of the product of the retarded self-energy and the advanced prop-

agator there are contributions from the two diagrams depicted in Figure 11.6. In the

case of the retarded interaction, the Di¬uson occurs only for the case where the ki-

netic Green™s function appears right at the emission vertex since impurity correlators

e¬ectively decouple momentum integrations (recall the similar analysis in connection

with Eq. (11.82)).

R K

q, ω q, ω

R K R R

p’ q p’ q p’ q p’ q

E’ ω E’ ω E’ ω E’ ω

D D

+

A A A A

p, E p ,E p, E p ,E

Figure 11.6 Lowest order interaction diagrams for the inelastic collision rate.

We then obtain for the inelastic collision rate or energy relaxation rate in terms

of the Di¬uson and the electron“electron interaction

1 1 dq dω R A

’ D(q, ω)(V (q, ω) ’ V (q, ω)) u4

= m

2 V2 (2π)3

„e’e (E, T ) 2π

— GR (E ’ ω, p ’ q) GA (E, p ) GR (E ’ ω, p ’ q) GA (E, p)

pp

E’ ω ω

— tanh + coth . (11.242)

2kT 2kT

Here we have used that the e¬ective Coulomb interaction has similar statistics prop-

erties as bosons, and in arriving at Eq. (11.242) we have in fact used the ¬‚uctuation“

432 11. Disordered conductors

dissipation relation that relates the kinetic component of the e¬ective Coulomb in-

teraction to the spectral component

ω

K R A

V (q, ω) ’ V (q, ω) coth

V (q, ω) = (11.243)

2kT

accounting for the second term arising from the second diagram in Figure 11.6.64

At this point, we bene¬t in interpretation from an important feature of the de-

veloped real-time non-equilibrium diagram technique, viz. that for the choice of

propagators we have made, the quantum statistics of fermions and bosons manifest

itself in a distinct way in diagrams as noted in Section 5.4. In the ¬rst diagram in

Figure 11.6, where the retarded interaction appears, it leads (according to the dia-

grammatic rules of Section 5.4) to the appearance of the quantum statistics of the

fermions, accounting for the ¬rst term in Eq. (11.242). It is important that this term

occurs in combination with the term containing the boson statistical properties of

the e¬ective Coulomb interaction, and that the boson kinetic component couples to

the electrons as a classical external ¬eld. This feature is generic, and leads in the

present case to the physical feature that zero-point ¬‚uctuations do not cause dissipa-

tive e¬ects. In the present context it corresponds to the fact that the imaginary part

of the self-energy, the inelastic collision rate, for an electron on the Fermi surface,

E = 0, vanishes at zero temperature. Or equivalently, that in accordance with the

exclusion principle the lifetime of an electron on the Fermi surface, E = 0, at zero

temperature is in¬nite.65

The momentum integrals over the impurity-averaged propagators are immediately

performed and we obtain

E’ ω

dq dω ω

R

= mV (q, ω) eD(q, ω) tanh + coth

(2π)3

„e’e (E, T ) 2π 2kT 2kT

(11.244)

from which we can calculate the collision rate.

The e¬ective electron“electron interaction itself, specifying the electron self-energy,

is also changed owing to the presence of impurities. It is thus the dynamically

screened electron“electron interaction in the presence of impurities, as expressed by

the dielectric function, (q, ω),

V (q)

R

V (q, ω) = , (11.245)

(q, ω)

64 Inthe calculation in Section 11.3.2 of the weak localization phase-breaking rate due to electron“

K

electron interaction with small energy transfers, only the kinetic component of the interaction, V ,

was included, but this is justi¬ed by the presence of its quantum statistics factor making it the

dominant component in the low frequency regime. This is the reason for the success of the single-

particle description used for the calculation, where the electron“electron interaction is represented

by a Gaussian distributed classical stochastic potential since it has identical properties with respect

to the dynamical indices as the kinetic component.

65 Such spurious zero-point ¬‚uctuation e¬ects are with frequency conjectured in the literature for

various physical quantities. For an early rebuttal in the context of weak localization see reference

[91].

11.5. Coulomb interaction in a disordered conductor 433