⎪ √ √

⎪

© 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