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„ ⎪
⎪ √ √

© 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

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

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