The prediction of the scaling theory of the absence of a true metallic state in

4 In this day and age, low-dimensional electron systems are routinely manufactured. For example,

a two-dimensional electron gas can be created in the inversion layer of an MBE grown GaAs“AlGaAs

heterostructure. Two-dimensional localization e¬ects provide a useful tool for probing material

characteristics, as we discuss in Section 11.2.

11.1. Localization 377

two dimensions was at variance with the previously conjectured theory of minimal

metallic conductivity. The classical conductivity obtained from the Boltzmann theory

has the form, in two and three dimensions (d = 2, 3),5

e2 kF l d’2

σ0 = k . (11.10)

dπ d’1 F

According to Mott [79], the conductivity in three (and two) spatial dimensions should

decrease as the disorder increases, until the mean free path becomes of the order of the

Fermi wavelength of the electron, l ∼ »F . The minimum metallic conductivity should

thus occur for the amount of disorder for which kF l ∼ 2π, and in two dimensions

should have the universal value e2 / . Upon further increasing the disorder, the

conductivity should discontinuously drop to zero.6 This is in contrast to the scaling

theory, which predicts the conductivity to be a continuous function of disorder. The

metal“insulator transition thus resembles a second-order phase transition, a quantum

phase transition at zero temperature, in contrast to Mott™s ¬rst-order conjecture

(corresponding to a scaling function represented by the dashed line in Figure 11.1).7

The phenomenological scaling theory o¬ers a comprehensive picture of the con-

ductance of disordered systems, and predicts that all states in two dimensions are

localized irrespective of the amount of disorder. To gain con¬dence in this surprising

result, one should check the ¬rst correction to the metallic limit. We therefore calcu-

late the ¬rst quantum correction to the scaling function and verify that it is indeed

negative.

11.1.2 Coherent backscattering

In this section we apply the standard diagrammatic impurity Green™s function tech-

nique to calculate the in¬‚uence of quenched disorder on the conductivity.8 In dia-

grammatic terms, the quantum corrections to the classical conductivity are described

by conductivity diagrams, as discussed in Section 6.1.3, where impurity lines connect-

ing the retarded and advanced propagator lines cross. Such diagrams are nominally

smaller, determined by the quantum parameter /pF l, than the classical contribu-

tion. The subclass of diagrams, where the impurity lines cross a maximal number

of times, is of special importance since their sum exhibits singular behavior. Such a

type of diagram is illustrated in Eq. (11.11).

5 In one dimension, the Boltzmann conductivity is σ0 = 2e2 l/π . However, the conclusion to

be drawn from the scaling theory is that even the slightest amount of disorder invalidates the

Boltzmann theory in one and two dimensions.

6 In three dimensions in the in¬nite volume limit, the conductance drops to zero at the critical

value according to the scaling theory.

7 The impressive experimental support for the existence of a minimal metallic conductivity in

two dimensions is now believed either to re¬‚ect the cautiousness one must exercise when attempting

to extrapolate measurements at ¬nite temperature to zero temperature, or to invoke a crucial

importance of electron“electron interaction in dirty metals even at very low temperatures.

8 For a detailed description of the standard impurity average Green™s function technique we refer

the reader to reference [1].

378 11. Disordered conductors

R

(11.11)

A

The maximally crossed diagrams describe the ¬rst quantum correction to the classical

conductivity, the weak-localization or coherent backscattering e¬ect, a subject we

discuss in detail in Section 11.2.

In the frequency and wave vector region of interest, each insertion in a maximally

crossed diagram is of order one.9 Diagrams with maximally crossing impurity lines

are therefore all of the same order of magnitude and must accordingly all be summed

( Q ≡ p + p ):

p+ p+ p+

p+ p+ p+ p+

+ + ... .

qω qω qω qω

p’ p’

p’

p’

Q’p+ Q’p+ Q’p+ (11.12)

From the maximally crossed diagrams, we obtain analytically, by applying the Feyn-

man rules for conductivity diagrams, the correction to the conductivity of a degen-

10

erate Fermi gas, ω, kT F,

e dp dp

2

˜

p± pβ Cp,p ( F , q, ω) GR (p+ ,

δσ±,β (q, ω) = + ω)

F

(2π )d (2π )d

m π

— GR (p + , + ω)GA (p ’ , A

F )G (p’ , F ) . (11.13)

F

To describe the sum of the maximally crossed diagrams, we have introduced the

9 This is quite analogous to the case of the ladder diagrams important for the classical conduc-

tivity, recall Exercise 6.1 on page 163, and for details see chapter 8 of reference [1].

10 In fact we shall in this section assume zero temperature as we shall neglect any in¬‚uence on the

maximally crossed diagrams from inelastic scattering. Interaction e¬ects will be the main topic of

Section 11.3.

11.1. Localization 379

˜ ≡

+

so-called Cooperon C,11 corresponding to the diagrams ( + ω):

F

F

p+ p +

˜

C

˜

Cp,p ( F , q, ω) ≡

p’ p ’

R R R

p+ p+ p+ p+

+ + +

F p+ F p+ F p+

≡ + + ...

A A A

p’ p’ p’ p’

Q’p+ Q’p+ Q’p+

F F F

R

p+ p+

+

F p+

=

A

p’ p’

Q’p+

F

R R

p+ p+

+ +

F p+ F p+

+ + ... . (11.14)

A A

p’ p’

Q’p+ Q’p+

F F

In the last equality we have twisted the A-line around in each of the diagrams, and

by doing so, we of course do not change the numbers being multiplied together.

Let us consider the case where the random potential is delta correlated12

= u2 δ(x ’ x ) .

V (x)V (x ) (11.15)

11 The nickname refers to the singularity in its momentum dependence being for zero total momen-

tum, as is the case for the Cooper pairing correlations resulting in the superconductivity instability

as discussed in Chapter 8.

12 For the case of a short-range potential, the only change being the appearance of the transport

time instead of the momentum relaxation time. For details we refer the reader to reference [1].

380 11. Disordered conductors

Since the impurity correlator in the momentum representation then is a constant,

u2 , all internal momentum integrations become independent. As a consequence, the

dependence of the Cooperon on the external momenta will only be in the combination

˜

p+p , for which we have introduced the notation Q ≡ p+p , as well as Cω (p+p ) ≡

˜ ˜

Cp,p ( F , 0, ω) ≡ Cω (Q), and we have

R R R

p+ p+ p+ p+

+ + +

F p+ F p+ F p+

˜

Cω (Q) = + + ...

A A A

p’ p’ p’ p’

Q’p+ Q’p+ Q’p+

F F F

⎛

R R

⎜

⎜

+ +

F p+ F p+

⎜

⎜

⎜

= 1 +

⎜

⎜

⎝

A A

Q’p+ Q’p+

F F

⎞

R R

⎟

⎟

+ +

F p+ F p+

⎟

⎟

...⎟

+ +

⎟

⎟

⎠

A A

Q’p+ Q’p+

F F

R

p+ p+

+

F p+

C

≡ . (11.16)

A

p’ p’

Q’p+

F

For convenience we have extracted a factor from the maximally crossed diagrams

which we shortly demonstrate, Eq. (11.24), is simply the constant u2 in the relevant

11.1. Localization 381

parameter regime. We shall therefore also refer to the quantity C as the Cooperon.

Diagrammatically we obtain according to Eq. (11.16)

R

p+

C = 1 + C . (11.17)

A

Q’p+

Analytically the Cooperon satis¬es the equation

dp

’ Q,

Cω (Q) = 1 + u2 GR (p + ω)GA (p

+, F F) Cω (Q) . (11.18)

+

d

(2π )

It is obvious that a change in the wave vector of the external ¬eld can be compensated

by a shift in the momentum integration variable, leaving the Cooperon independent

of any spatial inhomogeneity in the electric ¬eld, which is smooth on the atomic

scale.

The Cooperon equation is a simple geometric series that we can immediately sum

(1 + ζ(Q, ω) + ζ 2 (Q, ω) + ζ 3 (Q, ω) + ... )

Cω (Q) =

= 1 + ζ(Q, ω) Cω (Q)

1

= , (11.19)

1 ’ ζ(Q, ω)

where we have for the insertion

dp

+ ω)GA (p ’ Q,

ζ(Q, ω) = u2 GR (p , F) . (11.20)

F

d

(2π )

Diagrammatically we can express the result

1

Cω (Q) = . (11.21)

R

+

F p+

1’

A

Q’p+

F

The insertion ζ(Q, ω), Eq. (11.20), is immediately calculated for the region of

1, and we have13

interest, ω„, Ql

ζ(Q, ω) = 1 + iω„ ’ D0 „ Q2 (11.22)

13 Fordetails we refer the reader to [1], where the relation between the Di¬uson and its twisted

diagrams, the Cooperon, in the case of time-reversal invariance, is also established.

382 11. Disordered conductors

and for the Cooperon

1

„

Cω (Q) = . (11.23)

’iω + D0 Q2

The Cooperon exhibits singular infrared behavior.14

In the singular region the prefactor in Eq. (11.16) equals the constant u2 as

R

+

F p+

u2 ζ(Q, ω) u2

= (11.24)

A

Q’p+

F

˜

i.e. in the region of interest we thus have C = u2 C. As far as regards the singular

behavior we could equally well have de¬ned the Cooperon by the set of diagrams

R

p+ p+

+

F p+

˜

Cω (Q) = +

A

p’ p’

Q’p+

F

R R

p+ p+

+ +

F p+ F p+

+ + ... (11.25)

A A