PHYSICAL REVIEW B, VOLUME 64, 235325
Random Berry phase magnetoresistance as a probe of interface roughness in Si MOSFET’s
H. Mathur1 and Harold U. Baranger2
1
Department of Physics, Case Western Reserve University, Cleveland Ohio 44106-7079
Department of Physics, Duke University, Box 90305, Durham, North Carolina 27708-0305
共Received 25 August 2000; revised manuscript received 17 July 2001; published 28 November 2001兲
2
The effect of silicon-oxide interface roughness on the weak-localization magnetoconductance of a silicon
metal-oxide-semiconductor field-effect transistor in a magnetic field, tilted with respect to the interface, is
studied. It is shown that an electron picks up a random Berry’s phase as it traverses a closed orbit. Effectively,
due to roughness, the electron sees a uniform field parallel to the interface as a random perpendicular field. At
zero parallel field the dependence of the conductance on the perpendicular field has a well-known form, the
weak-localization line shape. Here the effect of applying a fixed parallel field on the line shape is analyzed.
Many types of behavior are found including homogeneous broadening, inhomogeneous broadening, and a
remarkable regime in which the change in line shape depends only on the magnetic field, the two length scales
that characterize the interface roughness, and fundamental constants. Good agreement is obtained with experiments that are in the homogeneous broadening limit. The implications for using weak-localization magnetoconductance as a probe of interface roughness, as proposed by Wheeler and co-workers, are discussed.
DOI: 10.1103/PhysRevB.64.235325
PACS number共s兲: 73.20.Fz, 73.40.Qv, 05.40.Fb, 72.10.Fk
I. INTRODUCTION
Disorder has a profound effect on electron transport at
low temperature. The scaling theory of localization applies
over an enormous domain and is a keystone in our understanding of disorder effects.1,2 For two-dimensional samples
that are weakly disordered and at low temperature 共‘‘weaklocalization regime’’兲 the theory predicts the precise dependence of the conductance on an applied magnetic field
共‘‘weak-localization line shape’’兲.3 The exquisite agreement
of the predicted line shape with experiment constitutes an
important confirmation of scaling theory.4
Silicon metal-oxide-semiconductor field-effect transistors
共MOSFET’s兲 are an important experimental realization of a
two-dimensional electronic system. In a MOSFET electrons
are confined to the interface between layers of oxide and
semiconductor. In this paper we analyze the effect of interface roughness on the weak-localization line shape in MOSFET’s. Although the effects are small they are of interest
from various points of view:
共i兲 The deviations from the known line shape of an ideal
interface are small but measurable: our results agree well
with the experiments that stimulated our work.5–7
共ii兲 The relevant effect of roughness on the electrons can
be traced to a subtle quantum interference effect: Berry’s
phase.8,9 There has been considerable theoretical interest in
the influence of Berry’s phase on quantum electron
transport10 and recent experiments have reported a detection
of it.11 Here we show that the earlier experiments of Refs.
5–7 constitute an observation of Berry’s phase in the quantum transport context.
共iii兲 A remarkable feature of many quantum transport phenomena is their universality, in the sense that the observed
effects are independent of microscopic sample properties
such as the mean mobility. The effects of roughness have this
feature. We find a particularly striking regime in which the
effect is determined entirely by the geometric parameters of
the interface and the applied magnetic field while being in0163-1829/2001/64共23兲/235325共20兲/$20.00
dependent of both sample mobility and temperature 共which
controls the dephasing length兲.
共iv兲 There has been much experimental and theoretical
work on the motion of electrons in a random magnetic field,
particularly in the strong-field limit12–18 共motivated in part
by possible relevance to cuprates and the quantum Hall system at filling factor 1/2兲. We find that due to roughness electrons see a uniform in-plane magnetic field as a random perpendicular field; hence it may be possible to use this system
to study electron motion in a random magnetic field. Here
our analysis is restricted to the weak-field limit appropriate
for calculating the line shape in MOSFET’s. Experiments on
other realizations of the random-field problem are briefly discussed in Sec. VI.
共v兲 Finally, this work may have practical implications.
MOSFET’s are the building blocks of modern electronics.
Roughness of the interface between the oxide and semiconductor influences the mobility of the device and has been
correlated with device failure due to dielectric breakdown of
the oxide.19,20 It is therefore of technological interest to characterize the roughness. R. G. Wheeler and co-workers have
proposed that magnetoresistance measurements on a MOSFET in the weak-localization regime can be used as a nondestructive probe of its interface roughness.5,6 They and others have carried out experiments that demonstrate the
feasibility of making the needed measurements.5–7 The
present work contributes to this program by providing the
precise relationship between the interface roughness parameters and the measured magnetoresistance.
Figures 1 and 2 summarize our findings. Following the
experiments of Refs. 5 and 6 we imagine that the device is
placed in a magnetic field tilted with respect to the plane of
the interface. The in-plane component of the field, B 储 , is
kept fixed and the conductance is plotted as a function of the
perpendicular component B⬜ . To understand these curves it
is useful to recall that there are two important length scales
that determine the transport properties of the sample: l e , the
elastic mean free path of electrons 共which determines the
64 235325-1
©2001 The American Physical Society
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
FIG. 1. MOSFET with short-ranged correlated roughness (L
Ⰶl e ). The conductance 共in units of 10⫺5 mhos) is plotted as a
function of the applied perpendicular field, B⬜ /B ␾ . The dotted
curve is the classic weak-localization line shape. It corresponds to
zero in-plane field, B 储 ⫽0. The solid curve shows the effect of applying a fixed in-plane magnetic field. It corresponds to B e /B ␾
⫽400 and B 储 /B ␾ ⫽( 冑8/ ␲ 1/4) 冑l e l ␾2 /L⌬ 2 .
mobility兲, and l ␾ , the distance over which electrons maintain phase coherence and are thus able to interfere. In the
weak-localization regime, l ␾ Ⰷl e . Also, atomic force microscope images reveal that statistically the rough interface can
be characterized by two parameters: ⌬⫽ the root-meansquare height fluctuations and L⫽ the distance over which
the fluctuations are correlated.
For B 储 ⫽0 we obtain the classic weak-localization line
shape appropriate for an ideal interface 共dotted curves in
Figs. 1 and 2兲. The height 共measured relative to the Drude
conductance given by the large-field asymptotic value of the
conductance corrected for classical magnetoresistance兲 and
the width of the peak are controlled by the dephasing length
l ␾ . The solid curves show possible line shapes when the
parallel field is turned on. Figure 1 shows that when the
roughness is correlated over a very short length scale (L
Ⰶl e ) the effect of the parallel field is to decrease the dephasing length; the line shape is otherwise unaltered. Borrowing
the terminology of magnetic resonance and atomic physics
the effect of the in-plane field in this limit may be described
as homogeneous line broadening. In the opposite limit when
the interface fluctuates slowly (LⰇl e ), the effects are more
dramatic. The deviation from the ideal line shape changes
sign as a function of perpendicular field 共Fig. 2, inset兲.
Again, by analogy to magnetic resonance and atomic physics, the extreme limit, LⰇl ␾ Ⰷl e , can be interpreted as inhomogeneous broadening 共see Sec. V兲. The sign change is
then traceable to a necessary inflection point in the ideal
weak-localization line shape. The intermediate regime l ␾
ⰇLⰇl e is particularly interesting although it cannot be so
simply interpreted. In this limit the deviation from the ideal
line shape has a universal form independent of l e and l ␾ . For
example, the sign change occurs at a value of the perpendicular field determined by the purely geometric condition
B⬜ L 2 ⫽(1.79•••)h/e. Note that in all cases the deviation
from the ideal line shape grows with B 储 and ⌬ 共in proportion
to B 2储 and ⌬ 2 in the experimentally relevant regime兲.
Further discussion and a detailed summary of our results
are given in Sec. VII.
II. BORN-OPPENHEIMER ANALYSIS
As a model of the silicon-oxide interface, for many purposes it is sufficient to regard the oxide as an impenetrable
hard wall and to assume that the electrons are bound to the
interface by a uniform perpendicular electric field.21 It will
become apparent in the sequel that for the present purpose it
is only necessary to assume that the electrons are firmly
bound to the interface: it is not necessary to commit to a
specific form of the confinement potential U conf . If the axes
are chosen so that the interface lies in the plane z⫽0, for an
ideal interface the confinement potential would depend only
on z; but for a rough interface it would vary from point to
point and hence also depend on x and y. The strength of the
confining potential controls the extent of the wave function
in the z direction, denoted l. We also allow for the possibility
of electron scattering by impurities in the semiconductor by
introducing a potential U imp . The magnetic field is assumed
to lie in the x-z plane: thus B x ⫽B 储 ,B y ⫽0, and B z ⫽B⬜ . It is
convenient to work in a Landau gauge and to choose A x
⫽0, A y ⫽B⬜ x, and A z ⫽B 储 y. The Schrödinger equation is
⫺
冉
ប2 ⳵
Az
⫺ie
2m ⳵ z
ប
冊
2
⌿⫺
冉
冊
ប2
A 2
ⵜ⫺ie
⌿⫹U conf共 z,x,y 兲 ⌿
2m
ប
⫹U imp共 z,x,y 兲 ⌿⫽E⌿ 共 x,y,z 兲 .
共1兲
FIG. 2. Same as Fig. 1 for a MOSFET with
long-range correlated roughness (LⰇl e ). The
dotted curve corresponds to B 储 ⫽0; the solid
curve, to B 储 ⌬L⫽ ␾ sc/10. Also B e ⫽400B ␾ and L
⫽l ␾ / 冑5. The crossing of the two curves is highlighted by a plot of their difference 共inset兲.
235325-2
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
For later convenience ⵜ is used to denote the twodimensional gradient in the x-y plane. Similarly A denotes
the x-y component of the vector potential.
For an ideal interface the electronic motion in the plane of
the interface and in the perpendicular direction separate. The
motion in the perpendicular z direction is quantized by the
confining potential. Provided the temperature is sufficiently
low and the density of electrons not too high, only the lowest
subband mode in the z direction is populated and the electronic motion is essentially two dimensional. For a rough
interface the motion no longer separates but we find that it is
an excellent approximation to integrate out the motion perpendicular to the interface and to obtain an effective Hamiltonian for motion in the plane of the interface using the
Born-Oppenheimer method.22
For simplicity first suppose there is no magnetic field.
Assume that the motion in the x-y plane is slow; the motion
perpendicular to it, fast. The first step in the BornOppenheimer method is to analyze the motion of the fast
coordinates treating the slow coordinates as parameters. In
this case it is necessary to solve
ប d
␾ 共 z;x,y 兲 ⫹U conf共 z,x,y 兲 ␾ n 共 z;x,y 兲
⫺
2m dz 2 n
2
PHYSICAL REVIEW B 64 235325
E g (x,y) appears as a potential in the Born-Oppenheimer effective Hamiltonian. The effective potential W g (x,y) is also
determined by the solution to the fast problem. It is given by
a more complicated expression,
W g 共 x,y 兲 ⫽
共2兲
Here ␾ n (z;x,y) denotes the ‘‘local subband’’ wave function
and E n (x,y) is the energy of the nth subband wave function.
Sometimes it will be convenient to write the subband states
using Dirac notation: ␾ n (z;x,y)→ 兩 n;x,y 典 . Both the subband
wave function and the subband energy vary with location in
the x-y plane because the confinement potential varies from
point to point. It is assumed that the subband wave function
is normalized everywhere so that 具 n;x,y 兩 n;x,y 典
⬁
⫽ 兰 ⫺⬁
dz 兩 ␾ n (z;x,y) 兩 2 ⫽1. For a given (x,y) Eq. 共2兲 only
fixes the subband wave function up to a phase. It is convenient to choose the wave functions to be real and to vary
smoothly with x and y.
According to the Born-Oppenheimer method an approximate solution to Eq. 共1兲 is
⌿ 共 x,y,z 兲 ⬇ ␺ 共 x,y 兲 ␾ g 共 z;x,y 兲 .
共3兲
Next suppose that the magnetic field is turned on. The fast
coordinate is now governed by
⫺
冉
e
ប2 d
⫺i B 储 y
2m dz
ប
ប2 2
ⵜ ␺ 共 x,y 兲 ⫹Ũ imp共 x,y 兲 ␺ 共 x,y 兲 ⫹E g 共 x,y 兲 ␺ 共 x,y 兲
2m
⫹W g 共 x,y 兲 ␺ 共 x,y 兲 ⫽E ␺ 共 x,y 兲 .
共4兲
Equation 共4兲 describes the motion of the electrons in the
plane of the interface after the transverse motion has been
integrated out. Ũ imp , the effect of impurities, now depends
only on x and y. For impurities that are sufficiently far from
the interface so that their potential does not vary significantly
over the confinement scale l, Ũ imp⫽U imp , while for shortrange impurities Ũ imp incorporates their effect on electrons in
the lowest subband. Note that the local subband energy
冊
2
␾ n 共 z;x,y 兲 ⫹U conf共 z,x,y 兲 ␾ n 共 z;x,y 兲
⫽E n 共 x,y 兲 ␾ n 共 z;x,y 兲 .
共6兲
Let ␰ n (z;x,y) denote a normalized solution to Eq. 共6兲 when
the magnetic field is turned off. This solution is chosen to be
real and to vary smoothly with x and y. It is easy to verify
that a solution to Eq. 共6兲 with the magnetic field is
冉
冊
e
␾ n 共 z;x,y 兲 ⫽exp i B 储 yz ␰ n 共 z;x,y 兲 .
ប
共7兲
Turning on the magnetic field leaves the subband energy
E n (x,y) unchanged; but it introduces a nontrivial twist in the
subband wave functions due to the phase factor in Eq. 共7兲.
Following Berry,8,9 this can be made explicit by defining the
geometric
vector
potential
Ag (x,y)⬅i(ប/e)
⫻具 g;x,y 兩 ⵜ 兩 g;x,y 典 and its curl, the geometric magnetic
field. Straightforward calculation reveals that the Cartesian
components of A are
A gx 共 x,y 兲 ⫽0,
A gy 共 x,y 兲 ⫽⫺B 储 Zg 共 x,y 兲 ,
共8兲
and the geometric magnetic field is
B g 共 x,y 兲 ⬅
⳵
⳵
⳵
A gy ⫺ A gx ⫽⫺B 储 Zg 共 x,y 兲 .
⳵x
⳵y
⳵x
共9兲
Here Zg (x,y) is the mean z coordinate for the lowest subband wave function,
Here ␾ g denotes the lowest energy subband wave function
and ␺ (x,y) is governed by
⫺
具 g;x,y 兩 共 ⵜ 兩 n;x,y 典 兲 • 具 n;x,y 兩 ⵜ 兩 g;x,y 典 .
共5兲
2
⫽E n 共 x,y 兲 ␾ n 共 z;x,y 兲 .
兺
n⫽g
Zg 共 x,y 兲 ⬅
冕
⬁
⫺⬁
dz z 兩 ␰ g 共 z;x,y 兲 兩 2 .
共10兲
Now, according to the Born-Oppenheimer method, an approximate solution to Eq. 共1兲 is given by Eq. 共3兲. Here ␾ g
denotes the lowest energy subband wave function and
␺ (x,y) is governed by
⫺
冉
e
ប2
ⵜ⫺i 关 A共 x,y 兲 ⫹Ag 共 x,y 兲兴
2m
ប
冊
2
␺ 共 x,y 兲
⫹E g 共 x,y 兲 ␺ 共 x,y 兲 ⫹W g 共 x,y 兲 ␺ 共 x,y 兲
⫹Ũ imp共 x,y 兲 ␺ 共 x,y 兲 ⫽E ␺ 共 x,y 兲 .
共11兲
Equation 共11兲 is the central result of this section. It reveals
that after the fast motion perpendicular to the interface is
235325-3
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
integrated out, the electrons essentially move in two dimensions under the influence of an effective potential, Ũ imp
⫹E g ⫹W g , and an effective perpendicular magnetic field,
B⬜ ⫹B g .
The effective magnetic field is seen to be the component
of the applied field that is perpendicular to the surface defined by the mean z coordinate of the subband wave functions Zg (x,y) provided that the gradients in this surface are
small. Note that the geometric magnetic field B g given by
Eq. 共9兲 is proportional to the in-plane component of the applied field and to the gradient of the surface Zg . It vanishes
if B 储 ⫽0 or for an ideal interface for which the surface Zg
would be flat. The effective potentials E g and W g also owe
their existence to the roughness of the interface but are independent of B 储 .
Note that Eq. 共6兲 only defines the subband wave functions
␾ n up to a phase. A specific choice given by Eq. 共7兲 is made
in the calculation above. This amounts to choosing a gauge
for the geometric vector potential: a different choice of phase
would transform the geometric vector potential Ag , but
would leave the geometric magnetic field B g unchanged.
Thus the calculation above uses a specific gauge for both the
applied and geometric magnetic field—the Landau gauge defined before Eq. 共1兲 and the gauge given by Eq. 共8兲, respectively. The gauges are chosen for their convenience but the
results are, of course, independent of the choice of gauge. A
fuller discussion of the gauge invariance of the BornOppenheimer method is given in Chap. 3.7 of Ref. 9.
Finally we briefly discuss the applicability of the BornOppenheimer approximation. In Appendix A it is shown that
the approximation should work provided lⰆ␭ f , ⵜZg Ⰶ1,
and lⵜ 2 Zg Ⰶ1. Here l is the typical extent of the subband
wave function in the z direction and ␭ f is the Fermi wavelength of the electrons. The first condition is to ensure that
the electron density is not so high that more than one subband is occupied. The other two inequalities are adiabaticity
requirements on the surface roughness needed to ensure that
the motion in the plane of the interface can be approximately
decoupled from the transverse motion. They stipulate that the
roughness must vary slowly on a scale determined by the
strength of the confining potential.
To check the validity of the adiabaticity conditions we can
use the atomic force microscope images of the silicon-oxide
interface presented in Ref. 6. These images reveal that the
height fluctuations of the interface follow a Gaussian distribution parameterized by ⌬ AFM , the root-mean-square height
fluctuation, and L, the correlation distance. The discussion of
Appendix A suggests that the surface Zg should be similarly
distributed with essentially the same parameters. Hence the
adiabaticity conditions can be expressed in terms of the observable parameters as ⌬ AFM /LⰆ1 and (l⌬ AFM)/L 2 Ⰶ1 and
are seen to be very well satisfied for both samples of Ref. 6.
lating the tilted field magnetoresistance of a MOSFET with a
rough interface.
In the previous section it was shown that effectively the
electrons move in two dimensions under the influence of a
random potential V(x,y) and a perpendicular magnetic field.
The magnetic field is the sum of the applied perpendicular
field and the geometric field, which is due to interface roughness and the applied parallel field. It is assumed that the
surface Zg (x,y) is a Gaussian random surface with zero
mean and variance given by
冉
具 Zg 共 r兲 Zg 共 r⬘ 兲 典 rough⫽⌬ 2 exp ⫺
兩 r⫺r⬘ 兩 2
L2
冊
.
共12兲
As noted above, atomic force microscope images of the
silicon-oxide interface presented in Ref. 6 reveal a Gaussian
random surface, which makes it extremely plausible that the
surface Zg (x,y) must also be Gaussian. An argument to this
effect is given in Appendix A. Equations 共9兲 and 共12兲 thus
determine the statistics of the random magnetic field B g .
The random potential V is caused by impurities and by the
roughness of the interface 共in the notation of the previous
section V⫽U imp⫹E g ⫹W g 兲. Here we shall assume that the
random potential is Gaussian white noise with zero mean and
variance 具 V(r)V(r⬘ ) 典 imp⫽V 20 ␦ (2) (r⫺r⬘ ). Since we assume
that the interface roughness is correlated over a length scale
L, strictly the assumption that the random potential is Gaussian white noise is justified only under special circumstances
共if the impurity potential is white noise and it dominates
interface roughness scattering, or if L is much smaller than
all relevant length scales and the impurity scattering is either
also white noise or is dominated by interface roughness scattering兲, but in practice it is reasonable to believe that our
results will be more broadly applicable since localization effects are believed to be insensitive to microscopic details of
the random potential. We estimate that for both samples studied in Ref. 6 impurity scattering dominates interface roughness scattering, and for one sample L is shorter than all relevant length scales in addition; but it is expected that the
results should apply under less favorable circumstances also.
It is useful to consider the electron Green function which
obeys the Schrödinger equation
冋 冉
⫺
ប2
e
“⫺i 关 A⫹Ag 兴
2m
ប
冊
册
2
⫹V 共 x,y 兲 ⫺E G共 r,r⬘ ;E 兲
⫽⫺ ␦ (2) 共 r⫺r⬘ 兲 .
共13兲
For the retarded Green function G R , the energy E has a positive infinitesimal imaginary part; for the advanced, G A , a
negative part. Linear-response theory allows us to express
the conductance in terms of the Green functions23
g⫽⫺
III. WEAK-LOCALIZATION LINE SHAPE
The purpose of this section is to review the calculation of
the weak-localization line shape. The calculation is reformulated in a way that is suitable for the eventual goal of calcu235325-4
⫻
e 2ប 3
8 ␲ m 2L 2
冕 冕
dr
dr⬘ ⌬G共 r,r⬘ ,E f 兲
J
⳵ J⳵
⌬G共 r⬘ ,r,E f 兲 .
⳵y ⳵y⬘
共14兲
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
PHYSICAL REVIEW B 64 235325
C 共 r,r⬘ 兲 ⫽C (0) 共 r,r⬘ 兲 ⫹
FIG. 3. Cooperon diagrams. Solid lines denote impurity averaged Green functions. The upper lines are retarded Green functions;
the lower lines, advanced. Dotted lines represent scattering from
impurities 共shown as crosses兲.
Here ⌬G⬅G R ⫺G A , E F ⫽ Fermi energy, and m⫽ effective
mass of electrons. We have assumed that the sample is rectangular 共dimensions L⫻W) and current is driven along the
length of the sample which is oriented along the y axis. The
two-sided derivative in Eq. 共14兲 is defined as
J
⳵
⳵
⳵
g⫺g
f.
f g⬅ f
⳵y
⳵y
⳵y
共15兲
We are interested in the disorder averaged conductance. To
this end it is necessary to average products of Green functions over the random potential and the interface roughness
under the statistical assumptions made above.
In the two subsections below we review the calculation of
the conductance at zero magnetic field and in a uniform perpendicular field, circumstances under which there is no random magnetic field and it is only necessary to perform an
average over the random potential. The effect of the random
magnetic field is analyzed in the following sections.
A. Zero field
The average over the random potential can be calculated
perturbatively2,24 in an expansion in the small parameter
(E f ␶ e ) ⫺1 . Here ␶ e , the elastic scattering time for electrons,
is given by ប/2␲␳ (E f )V 20 , where ␳ (E f )⫽ density of states
for spinless electrons. In this approximation the average
Green function is given by24
G R 共 r,r⬘ ,E f 兲 ⫽
冕
dk
共 2␲ 兲2
冉
G R 共 k,E f 兲 exp ik• 共 r⫺r⬘ 兲 ,
G R 共 k,E f 兲 ⫽ E f ⫺
ប
ប 2k 2
⫹i
2m
2␶e
冊
⫺1
.
共16兲
Here G(r,r⬘ ;E f )⫽ 具 G(r,r⬘ ;E f ) 典 imp is the Green function averaged over the random potential.
The average of a product of Green functions does not
factorize into a product of the averages; it is correlated by the
underlying random potential. The correlation between G R
and G A responsible for weak localization is called the cooperon, C(r,r⬘ ). Semiclassically this correlation can be understood to arise from the constructive interference of classical
paths related by time-reversal symmetry.1– 4 Figure 3 shows
the Feynman diagrams that contribute to the cooperon.1– 4
From these diagrams we see that it obeys the integral equation
冕
dr⬙ C (0) 共 r,r⬙ 兲 C 共 r⬙ ,r⬘ 兲 . 共17兲
Here C (0) (r,r⬘ )⫽ 关 ប/2␲␳ (E f ) ␶ e 兴 兩 具 G R (r,r⬘ ,E f ) 典 imp兩 2 . Using
Eqs. 共14兲 and 共16兲, the weak-localization contribution to the
conductance, g WL , can be expressed in terms of the full
cooperon:
2 e2
D␶e
g WL⫽⫺
␲ L2
冕
共18兲
dr C 共 r,r兲 .
Here D⬅ v 2f ␶ e /2 is the electron diffusion constant. Calculation of g WL therefore reduces to solution of Eq. 共17兲.
The customary procedure is to obtain eigenfunctions of
C (0) which obey
冕
dr⬘ C (0) 共 r,r⬘ 兲 Q ␭ 共 r⬘ 兲 ⫽␭Q ␭ 共 r兲 .
共19兲
The solution to Eq. 共17兲 is then
C 共 r,r⬘ 兲 ⫽
兺␭
冉 冊
␭
Q 共 r兲 Q ␭ 共 r⬘ 兲 .
1⫺␭ ␭
共20兲
It turns out that eigenfunctions that vary slowly on the scale
of l e ⫽ v f ␶ e dominate the conductance; hence it is only necessary to accurately calculate the slowly varying solutions of
Eq. 共19兲. Asymptotic evaluation of the Green function, Eq.
共16兲, shows that
C (0) 共 r,r⬘ 兲 ⬇
1
2 ␲ l 2e
冉
exp ⫺
兩 r⫺r⬘ 兩
le
冊
共21兲
provided 兩 r⫺r⬘ 兩 Ⰷ␭ f and k f l e Ⰷ1. C (0) is therefore sharply
peaked about r⬇r⬘ and for the slowly varying eigenfunctions the integral in Eq. 共19兲 can be performed approximately
by expanding Q ␭ (r⬘ ) in a Taylor’s series about r⬘ ⫽r. Keeping terms to second order transforms the integral Eq. 共19兲
into a diffusion equation
D ␶ e ⵜ 2 Q ␭ 共 r兲 ⫹Q ␭ 共 r兲 ⫽␭Q ␭ 共 r兲 .
共22兲
It is sufficient to use the solutions of the diffusion equation
共22兲 to construct the cooperon, Eq. 共20兲, and to calculate the
conductance using Eq. 共18兲. This completes the usual calculation at zero field.
We now introduce an alternative formulation more suitable for analyzing the effects of roughness. Consider the kernel
K (0) 共 r,r⬘ 兲 ⬅
1
冉
exp ⫺
2
2␲le
兩 r⫺r⬘ 兩 2
2l 2e
冊
.
共23兲
By analogy with C (0) and C introduce K given by
K 共 r,r⬘ 兲 ⫽K (0) 共 r,r⬘ 兲 ⫹
冕
dr⬙ K 共 r,r⬙ 兲 K (0) 共 r⬙ ,r⬘ 兲 . 共24兲
Note that K (0) is not equal to C (0) 关compare Eqs. 共21兲 and
共23兲兴; but the slowly varying eigenfunctions of K (0) also
obey the diffusion Eq. 共22兲. For this reason K and C have the
235325-5
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
same long-distance behavior. Thus we may use K instead of
C in Eq. 共18兲 to calculate the weak-localization conductance
since it is dominated by the long-distance behavior.
The chief virtue of K compared to C is that it can be
expressed in terms of tractable Gaussian integrals. A useful
expression for K is obtained from Eq. 共24兲 by iteration
⬁
K 共 r,r⬘ 兲 ⫽
K (n) 共 r,r⬘ 兲 ⫽
冕
兺
n⫽0
K (n) 共 r,r⬘ 兲 ,
dr1 •••drn K (0) 共 r,r1 兲 K (0)
⫻ 共 r1 ,r2 兲 •••K (0) 共 rn ,r⬘ 兲 .
共25兲
Evidently K (n) (r,r⬘ ) may be interpreted as the probability
for a Gaussian random walker to go from r⬘ to r in n⫹1
steps.
An important physical ingredient must now be added. As
noted above, weak localization is a quantum interference effect. It is therefore damped by electron-electron and electronphonon scattering.25 In the conventional analysis the damping effect of these processes is included phenomenologically
by adding ␶ e / ␶ ␾ to the eigenvalue ␭ in Eqs. 共19兲, 共20兲, and
共22兲. Here ␶ ␾ is the dephasing time and it is assumed that
␶ ␾ Ⰷ ␶ e in the weak-localization regime. The equivalent procedure in our formulation is to replace
冉
K (n) →K (n) exp ⫺
共 n⫹1 兲 ␶ e
␶␾
冊
共26兲
.
e2 W
g WL⫽⫺
ប L
⬇⫺
冕
dr
兺n K
(n)
1
冉 冊
n␶e
. 共27兲
共 r,r兲 exp ⫺
␶␾
冉
exp ⫺
2
共 n⫹1 兲 l e
冉 冊
⫺
n␶e
␶␾
e2 W ␶␾
e2 W l␾
ln ⫽⫺
ln ,
ប L ␶e
ប L le
兩 r⫺r⬘ 兩 2
共 n⫹1 兲 l 2e
冊
,
共28兲
共29兲
B. Uniform perpendicular field
We consider magnetic fields that are weak compared to
impurity scattering. The precise condition we assume is B
ⰆB e , where B e ⫽ ␾ sc/2␲ l 2e and ␾ sc⫽h/2e is the superconducting flux quantum. In this limit the classical paths are not
modified by the magnetic field. Its main effect is to change
the phase of the paths 共the Aharonov-Bohm effect兲. This has
no effect on the classical Drude conductance, but the weaklocalization correction, which is an interference effect, is
suppressed in a manner analyzed in Ref. 3.
The starting point for this analysis is to assume that the
effect of turning on a perpendicular field on the Green function is to multiply it by a phase factor:
冉 冕 冊
R
G BR 共 r,r⬘ ;E f 兲 ⬇G B⫽0
共 r,r⬘ ;E f 兲 exp i
⬜
In summary, Eqs. 共23兲, 共25兲, and 共27兲 provide the tools to
calculate the weak-localization conductance.
Equation 共27兲 has an appealing form. g WL is expressed as
a sum over closed random walks, consistent with the physical idea that it is due to the interference of closed paths and
their time reversed counterparts. The integral over K (n) gives
the weight of n-bounce paths; the exponential factor shows
that the contribution of very long paths is cut off by dephasing. For a different formulation of weak localization in terms
of random walks, see Ref. 26.
Let us explicitly calculate g WL . Successive integration
over intermediate points shows
K (n) 共 r,r⬘ 兲 ⫽
1
兺 exp
n⫽1 n
a celebrated result.1–3 Here we have introduced l ␾ ⫽ 冑D ␶ ␾ ,
the dephasing length, and made use of the inequality ␶ ␾
Ⰷ ␶ e . The logarithmic divergence of g WL as l ␾ →⬁ can be
traced to the slow 1/n decay of the weight of long paths. In a
field theory formulation it indicates that the theory flows to
strong coupling on long length scales but on shorter scales it
is ‘‘asymptotically free.’’27 Provided that the cutoff l ␾ is
shorter than the localization length, perturbation theory is
accurate.
Eqs. 共18兲, 共25兲, and 共26兲 then yield
e2 D␶e
g WL⫽⫺
ប L2
⬁
e
ប
r
r⬘
dl•A . 共30兲
Here G B⫽0 is the disorder averaged Green function at zero
field, Eq. 共16兲; G B⬜ , the Green function with the perpendicular magnetic field turned on; and the integral in the phase
factor is evaluated along a straight line from r⬘ to r. The
zeroth order cooperon becomes
冉 冕 冊
(0)
C B(0) 共 r,r⬘ 兲 ⫽C B⫽0
共 r,r⬘ 兲 exp i
⬜
2e
ប
r
r⬘
dl•A .
共31兲
Once again we are interested in the eigenfunctions of C B(0)
⬜
which obey
冕
冉 冕 冊
(0)
dr⬘ C B⫽0
共 r,r⬘ 兲 exp i
2e
ប
r
r⬘
dl•A Q ␭ 共 r⬘ 兲 ⫽␭Q ␭ 共 r兲 .
共32兲
For the slowly varying eigenfunctions Eq. 共32兲 may be transformed into a differential equation by expanding both the
phase factor and Q ␭ (r⬘ ) about r⬘ ⫽r to second order. The
result is
a well-known result. K (n) has the same form as K (0) but with
l e → 冑n⫹1 l e in agreement with the general proposition that
the displacement of a random walker grows as the square
root of the number of steps. Now
235325-6
冋 冉
D ␶ e ⵜ⫺i
2e
A共 r兲
ប
冊 册
2
冉
⫹1 Q ␭ 共 r兲 ⫽ ␭⫹
冊
␶e
Q 共 r兲 .
␶␾ ␭
共33兲
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
In Eq. 共33兲 we have explicitly inserted a damping term
␶ e / ␶ ␾ . The customary solution3 is to construct the cooperon
from the Landau-level solutions to Eq. 共33兲.
As in the previous subsection we now describe an alternative formulation. Consider the kernel
冉
(0)
K B(0) 共 r,r⬘ 兲 ⫽K B⫽0
共 r,r⬘ 兲 exp i
⬜
2e
ប
冕
r
r⬘
dl•A
冊
共34兲
(0)
given by Eq. 共23兲. The slowly varying eigenfuncwith K B⫽0
tions of K B(0) also obey Eq. 共33兲; hence if we define K B⬜ via
⬜
K B⬜ 共 r,r⬘ 兲 ⫽K B(0) 共 r,r⬘ 兲 ⫹
⬜
冕
dr⬙ K B(0) 共 r,r⬙ 兲 K B⬜ 共 r⬙ ,r⬘ 兲 ,
⬜
共35兲
then K B⬜ has the same long-distance behavior as C B⬜ and
may be used in its place to calculate the conductance.
A useful expression for K B⬜ is obtained from Eq. 共35兲 by
iteration,
PHYSICAL REVIEW B 64 235325
Imagine that n-sided polygons are drawn on a plane
starting from a fixed point r. Let the probability density
of drawing a polygon with vertices at r,r1 ,r2 , . . . ,rn ,r
be given by the Gaussian expression 关 2 ␲ l 2e (n
(0)
(0)
(0)
(r,r1 )K B⫽0
(r1 ,r2 )•••K B⫽0
(rn ,r). The factor
⫹1) 兴 K B⫽0
2
2 ␲ l e (n⫹1) is included to normalize the distribution. Let a
denote the directed area of the polygon 共defined in Appendix
C兲. Then Eq. 共37兲 may be rewritten as
K Bn 共 r,r兲 ⫽
⬜
1
2 ␲ l 2e 共 n⫹1 兲
P n共 a 兲 ⫽
K B(n) 共 r,r⬘ 兲 ⫽
⬜
冕
兺 K B(n)共 r,r⬘ 兲 ,
n⫽0
⬜
冉 冕 冊
2e
ប
r
r⬘
K B(n) 共 r,r兲 ⫽
⬜
共36兲
dl•A .
The integral in the phase factor is evaluated along the path
from r⬘ to r obtained by joining the points r⬘ ,rn , . . . ,r1 ,r
with line segments in the given order. For r⫽r⬘
K B(n) 共 r,r兲 ⫽
⬜
冕
冉
g WL⫽⫺
冊
␾n
.
␾ sc
共37兲
Here ␾ n ⫽ flux through the polygon defined by the closed
path from r to r and ␾ sc⫽h/2e is the superconducting flux
quantum.
Using Eqs. 共18兲 and 共36兲
e D␶e
ប L2
2
g WL⫽⫺
冕
⬁
兺 K B(n)共 r,r兲 exp
n⫽0
dr
⬜
冉
⫺ 共 n⫹1 兲
冊
B⬜ a
.
␾ sc
共39兲
␲
2nl 2e
sech2
冉 冊
␲a
nl 2e
共40兲
,
冊
冊册
⫺1
共41兲
.
e2 W
ប L
⬁
兺
n⫽1
冋 冉 冊册 冉 冊
B⬜
nB⬜
sinh
2B e
2B e
⫺1
exp ⫺
n␶e
.
␶␾
共42兲
Since B⬜ ⰆB e , the sum may be evaluated asymptotically.28
Using an integral representation of the digamma function,29
␺共 x 兲⫽
冕
⬁
0
dq
冉
冊
e ⫺q
e ⫺xq
⫺
,
q
1⫺e ⫺q
共43兲
we obtain
␶e
.
␶␾
共38兲
In summary, Eqs. 共23兲, 共37兲, and 共38兲 provide the tools
needed to calculate the perpendicular field magnetoconductance.
Let us now explicitly calculate the magnetoconductance.
It is necessary to first evaluate K B(n) (r,r). The integrals are
⬜
performed in Appendix B and will be used in later sections.
For the present, more insight is gained by noticing that
K B(n) (r,r) has a natural interpretation in terms of random
⬜
walks.
冋 冉
n ␲ B⬜ l 2e
B⬜
sinh
2 ␾ sc
␾ sc
Equation 共41兲 is derived in Appendix B and is shown to be
sufficiently accurate for our purposes for all n provided B⬜
ⰆB e . For further discussion of the directed area distribution,
see Appendix C.
Substitution of Eq. 共41兲 in Eq. 共38兲 leads to
(0)
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••K B⫽0
共 rn ,r兲
⫻exp i2 ␲
⫺⬁
冉
da P n⫹1 共 a 兲 exp 2 ␲ i
and its Fourier transform is
(0)
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••K B⫽0
共 rn ,r⬘ 兲
⫻exp i
⫹⬁
Here P n (a) is the probability distribution of the directed area
of an n-sided polygon drawn from the Gaussian polygon
ensemble defined above. Equation 共39兲 shows that K B(n) (r,r)
⬜
is the Fourier transform of the distribution of directed areas.
The directed area distribution is discussed in Appendix C.
For nⰇ1
⬁
K B⬜ 共 r,r⬘ 兲 ⫽
冕
g WL⫽⫺
冋冉 冊 冉
Be
1 B␾
1 e2 W
ln
⫺
␺
⫹
B⬜
2 B⬜
2␲2 ប L
冊册
.
共44兲
Here B ␾ is the field corresponding to a flux quantum through
the phase-breaking length, B ␾ ⬅ ␾ sc/4␲ l ␾2 . This is the ideal
weak-localization line shape derived by Hikami, Larkin, and
Nagaoka3 and shown as the dotted curves in Figs. 1 and 2.
It is instructive to summarize the main points of the derivation. First the conductance is expressed as a sum of closed
random walks, Eq. 共38兲. Equation 共37兲 shows that the contribution of a fixed random walk oscillates with the enclosed
flux with a period ␾ sc , revealing the Aharonov-Bohm origin
of the weak-localization magnetoresistance. The contribution
235325-7
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
冓 冉
of all n-bounce paths is the Fourier transform of their directed area distribution; it decays exponentially for B⬜
Ⰷ ␾ sc /nl 2e 关see Eq. 共42兲兴. Summing over n leads to the gentler decay described by Eq. 共44兲.
exp i2 ␲
␾g
␾ sc
冊冔
冉
⬇exp ⫺2 冑␲
rough
⫹•••⫹
e2
ប
B 2⌬ 2L
2 储
共 y n ⫺y 兲 2
兩 rn ⫺r兩
IV. SHORT-RANGE ROUGHNESS
In this section we assume that the distance over which the
roughness is correlated is short compared to the elastic mean
free path (LⰆl e ). Under this circumstance the effect of turning on an in-plane magnetic field is comparatively straightforward: it enhances the dephasing rate and hence produces
‘‘homogeneous broadening’’ of the weak-localization line
shape.
For simplicity we first assume that the applied field lies in
the plane of the interface. According to the analysis of Sec.
II, the interface electrons see a uniform in-plane field as a
random perpendicular field. The typical strength of the random field is B g ⬃B 储 ⌬/L 关see Eqs. 共5兲 and 共12兲兴. We assume
that the random magnetic field is weak compared to impurity
scattering (B g ⰆB e ), a condition that is well satisfied in practice. As in the previous section, the only effect of the magnetic field in this limit is to multiply the Green function by a
phase factor, so the zeroth-order cooperon becomes
(0)
C B(0)储 共 r,r⬘ 兲 ⫽C B⫽0
共 r,r⬘ 兲 exp
冉 冕
2e
i
ប
r
r⬘
冊
dl•Ag .
⬁
C B 储 共 r,r兲 ⫽
C B(n)储 共 r,r兲 ⫽
冕
兺
n⫽0
C B(n)储 共 r,r兲 ;
冉
CB 储 共 r,r兲 ⫽
CB(n)储 共 r,r兲 ⫽
r
r
dl•Ag .
CB(n)储 共 r,r兲 ;
共48兲
dr1 •••drn CB(0)储 共 r,r1 兲 •••CB(0)储 共 rn ,r兲 ;
冉
e2
共 y⫺y ⬘ 兲 2
ប
兩 r⫺r⬘ 兩
B 2⌬ 2L
2 储
共46兲
The integral in Berry’s phase factor ␾ g is to be evaluated
around the perimeter of the polygon obtained by joining
r,r1 , . . . ,rn ,r in that order. We average the Berry phase
factor using the gauge given by Eq. 共8兲 and assuming that
Zg (x,y) is a Gaussian random surface with statistics given
by Eq. 共12兲. To avoid digression we shall return to the evaluation of this average below. For the moment we give the
result for LⰆl e :
冊
.
For brevity 具 C B 储 (r,r) 典 rough is written as CB 储 (r,r).
Evidently Eq. 共48兲 is the formal iterative solution to the
integral equation
CB 储 共 r,r⬘ 兲 ⫽CB(0)储 共 r,r⬘ 兲 ⫹
冕
dr⬙ CB(0)储 共 r,r⬙ 兲 CB 储 共 r⬙ ,r⬘ 兲 .
共49兲
In principle the roughness averaged cooperon is determined
by Eq. 共49兲 with CB(0)储 (r,r⬘ ) given by Eqs. 共21兲 and 共48兲. The
expression for C B(0)储 (r,r⬘ ) shows that after averaging over
roughness, the effect of the in-plane field is to multiply the
zeroth-order cooperon by an exponential factor.
To solve Eq. 共49兲 it is most convenient to look for eigenfunctions of C B(0)储 which obey
冉
(0)
dr⬘ C B⫽0
共 r,r⬘ 兲 exp ⫺2 冑␲
e2
共 y⫺y ⬘ 兲 2
ប
兩 r⫺r⬘ 兩
B 2⌬ 2L
2 储
冊
⫽␭Q ␭ 共 r兲 .
冊
冕
冕
兺
n⫽0
(0)
CB(0)储 共 r,r⬘ 兲 ⫽C B⫽0
共 r,r⬘ 兲 exp ⫺2 冑␲
冕
␾g
;
␾ sc
␾ g⫽
共47兲
.
⬁
(0)
(0)
dr1 •••drn C B⫽0
共 r,r1 兲 •••C B⫽0
共 rn ,r兲
⫻exp i2 ␲
共 y⫺y 1 兲 2
兩 r⫺r1 兩
From Eqs. 共46兲 and 共47兲 we obtain the average cooperon,
共45兲
The cooperon is determined in principle by Eqs. 共17兲 and
共45兲; our objective is to average it over the interface roughness.
To this end it is convenient to formally solve Eq. 共17兲 by
iteration:
册冊
冋
Q ␭ 共 r⬘ 兲
共50兲
CB 储 may be expanded in terms of these eigenfunctions as in
Eq. 共20兲. We are interested in eigenfunctions that vary slowly
compared to l e . For these eigenfunctions Eq. 共50兲 may be
transformed into a diffusion equation by expanding Q ␭ and
the exponential factor in powers of r⫺r⬘ . Keeping the leading terms 共second order in Q ␭ and first order in the exponential factor兲 we obtain
冉
共 1⫹D ␶ e ⵜ 2 兲 Q ␭ 共 r兲 ⫽ ␭⫹
冊
␶e ␶e
⫹
Q 共 r兲 .
␶␾ ␶储 ␭
共51兲
In Eq. 共51兲 we have put a dephasing term ␶ e / ␶ ␾ by hand.
The in-plane field is responsible for the ␶ e / ␶ 储 term. Here ␶ 储
is defined by
235325-8
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
␶e
e2
⫽2 冑␲ 2 B 2储 ⌬ 2 L
␶储
ប
⬇ 冑␲
e2
ប
2
冕
(0)
dr⬘ C B⫽0
共 r,r⬘ 兲
共 y⫺y ⬘ 兲 2
兩 r⫺r⬘ 兩
B 2储 ⌬ 2 Ll e .
共52兲
Equation 共51兲 shows that, upon averaging, the only effect of
an in-plane magnetic field on the long-distance behavior of
the cooperon, and hence on the conductance, is to enhance
the dephasing rate by
1 1
1
→ ⫹ ,
␶␾ ␶␾ ␶储
共53兲
where ␶ 储 is given by Eq. 共52兲.
in Eq. 共52兲 can be obtained from the
The form of ␶ ⫺1
储
following simple argument. We wish to find the typical phase
associated with an n-sided random polygon due to the random vector potential. Consider a single side of the polygon:
it has typical length l e , and the vector potential is approximately constant over segments of length L but of either
sign—A⬃⫾B 储 ⌬. Then the typical phase accumulated on
this side is rms关 (2e/ប) 兰 A•dl兴 ⬃(2e/ប) 冑l e /L)B 储 ⌬L. Since
the phase along each side is uncorrelated, the typical total
phase is rms关 ␾ n⫺gon兴 ⬃ 冑n(2e/ប) 冑l e /L)B 储 ⌬L. The contribution of n-gons will be cut off when this phase is of order 1,
giving the condition for ␶ ⫺1
储 . This gives the condition for
⫺1
␶ 储 if we set n→ ␶ 储 / ␶ e . We find ␶ e / ␶ 储 ⬃4(e/ប) 2 B 2储 ⌬ 2 Ll e ,
the same form as Eq. 共52兲.
Now we indicate how to generalize this analysis when the
applied magnetic field has a nonzero perpendicular component. Assuming this component is weak compared to impurity scattering (B⬜ ⰆB e ), an Aharonov-Bohm phase factor
must be included in Eqs. 共45兲 and 共46兲. After averaging over
interface roughness CB 储 ,B⬜ obeys Eq. 共49兲 with
冉 冕 冊
(0)
CB(0)储 ,B 共 r,r⬘ 兲 ⫽C B⫽0
共 r,r⬘ 兲 exp i
⬜
冉
⫻exp ⫺2 冑␲
2e
ប
PHYSICAL REVIEW B 64 235325
Since the form of the line shape is not changed, this effect is
analogous to homogeneous broadening in magnetic resonance and atomic physics.
The idea that an in-plane field produces homogeneous
broadening of the line was anticipated by the authors of Refs.
5 and 6. Our result differs in two important respects: First,
we find that homogeneous broadening occurs only if the
roughness is correlated over short distances (LⰆl e ), and second, the dependence of ␶ 储 on the interface roughness parameters that we derive 关Eq. 共52兲兴 is quite different from that
conjectured in Ref. 6.
Figure 6 shows that Eqs. 共44兲 and 共55兲 provide a good fit
to data with essentially no adjustable parameters. More details are given in Sec. VI.
To complete the analysis we must now average Berry’s
phase factor exp(i2␲␾g /␾sc) over interface roughness to obtain Eq. 共47兲. Recall that the distribution of a linear combination of correlated Gaussian random variables is also
Gaussian. The distribution of ␾ g is therefore Gaussian. Evidently it has zero mean since we have assumed
具 Zg (x,y) 典 rough⫽0. For a Gaussian random variable s with
zero mean it is easy to show
具 exp共 i ␰ s 兲 典 ⫽exp共 ⫺ 21 ␰ 2 具 s 2 典 兲 .
Hence averaging Berry’s phase factor reduces to the calculation of the variance of the phase. Explicitly we must calculate
n⫹1 n⫹1
具 ␾ 2g 典 rough⫽
冓冉 冕
dl•A
e2
共 y⫺y ⬘ 兲 2
ប
兩 r⫺r⬘ 兩
B 2⌬ 2L
2 储
冊
Keeping track of the phase factor, we find that the slowly
varying eigenfunctions of CB(0)储 ,B obey Eq. 共33兲 with the sub⬜
stitution given in Eq. 共53兲.
Hence we arrive at the central result of this section. For
short-range correlated roughness, application of an in-plane
field increases the dephasing rate according to Eq. 共53兲 but
does not otherwise alter the weak-localization line shape.
The dependence of the conductance on the perpendicular
field is still given by Eq. 共44兲 but with
eB 2储 ⌬ 2 L
.
h le
冓冕
共55兲
dli •Ag
冊冔
2
dli •Ag
⫽B 2储 ⌬ 2 sin2 ␪
rough
⫻
.
共54兲
B ␾ →B ␾ ⫹ ␲ 3/2
兺 兺
i⫽1 j⫽1
冕
dl j •Ag
冔
.
共57兲
rough
Here 兰 dli •Ag denotes the line integral along the edge joining
ri⫺1 to ri 共with the understanding that r0 ⫽rn⫹1 ⫽r).
Let us first evaluate the diagonal terms in Eq. 共57兲. Let the
ith edge make an angle ␪ with the x axis and let s be the
length parameter along that edge. Using Eq. 共8兲 for Ag and
Eq. 共12兲 for the correlation of Zg (x,y) we obtain
r
r⬘
共56兲
冕
冕
兩 ri ⫺ri⫺1 兩
0
⬇ 冑␲ B 2储 ⌬ 2 L
兩 ri ⫺ri⫺1 兩
0
ds
冋 冉 冊册
s⫺s ⬘
ds ⬘ exp ⫺
L
共 y i ⫺y i⫺1 兲 2
.
兩 ri ⫺ri⫺1 兩
2
共58兲
We have assumed that the edge is much longer than the
correlation length L. Since the length of a typical edge is l e ,
this is justified in the short-range limit LⰆl e . We have also
used 兩 ri ⫺ri⫺1 兩 sin ␪⫽yi⫺yi⫺1.
Evidently the only important off diagonal terms in Eq.
共57兲 are those for which the two edges pass within a correlation length of each other. The contribution of such edge
pairs to the sum will be smaller than the diagonal terms,
typically by a factor of L/l e 共except in the rare circumstance
that the two edges meet at a glancing angle and overlap for a
considerable part of their length兲. Furthermore, the number
of contributing off-diagonal terms is comparable to n, the
235325-9
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
number of diagonal terms; hence the contribution of the offdiagonal terms may be safely neglected in the short-range
limit LⰆl e .
That the number of important off-diagonal terms is comparable to n can be established by an argument commonly
used in polymer physics.30 Suppose that line segments of
typical length l e are drawn at random on a plane with a
density ␳ segments per unit area. If a fresh segment of length
l e is now drawn on this plane, typically it will suffer ⬃ ␳ l 2e
intersections. Working in units where l e ⫽1, the area of a
typical polygon in Eq. 共46兲 is n and hence the density of
edges ␳ ⬃n/n⫽1. Each edge therefore typically encounters
one intersection and the total number of intersections is comparable to n.
Adding the diagonal contributions 关Eq. 共58兲兴 and neglecting the off-diagonal ones we obtain
具 ␾ 2g 典 rough⬇ 冑␲ B 2储 ⌬ 2 L
冋
册
共 y 1 ⫺y 兲 2
共 y⫺y n 兲 2
⫹•••⫹
.
兩 r1 ⫺r兩
兩 r⫺rn 兩
␾ g⫽
共59兲
K B(n)储 共 r,r兲 ⫽
冕
⫻exp关 ⫺V 共 r,r1 , . . . ,rn 兲兴
V 共 r,r1 , . . . ,rn 兲 ⫽
dl•Ag
冊
in place of Eq. 共45兲 and
K B(n)储 共 r,r兲 ⫽
冕
共60兲
n
兺兺
j⫽1 k⫽1
1
4L 2
共 y j ⫺y j⫺1 兲共 y k ⫺y k⫺1 兲
册
共 r j ⫹r j⫺1 ⫺rk ⫺rk⫺1 兲 2 .
(n)
␦ K B(n)储 共 r,r兲 ⬅K B(n)储 共 r,r兲 ⫺K B⫽0
共 r,r兲
冕
(0)
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••K B⫽0
共 rn ,r兲
⫻V 共 r,r1 , . . . ,rn 兲 .
共64兲
V(r,r1 , . . . ,rn ) is a sum over pairs of links according to Eq.
共63兲. To illustrate the evaluation of Eq. 共64兲 focus on the
particular term for which j⫽1 and k⫽3,4, . . . ,n. It is convenient to perform the integrals over the end points of the jth
and kth links last. Integrating over the remaining intermediate points yields
兺 K B(n)共 r,r兲 ,
n⫽0
␦ K B(n)储 共 r,r兲 ⫽
储
1
␲2
⫻
(0)
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••K B⫽0
共 rn ,r兲
冉
ប2
n
Here K (n) ⫽ 具 K (n) 典 rough and r0 ⫽rn⫹1 ⫽r. K (n) is thus an integral over (n⫹1)-bounce closed random walks just as it is
at zero field but the weight of each polygon is no longer
Gaussian. The links of the polygon now ‘‘interact,’’ and the
interaction, given by Eq. 共63兲, is anisotropic and long ranged.
K (n) is difficult to calculate. For weak in-plane magnetic
fields it is sufficient to analyze the interaction perturbatively.
To first order
⬁
K B 储 共 r,r兲 ⫽
共62兲
共63兲
⫽⫺
Since B g ⰆB e is automatically ensured by the weak-field
condition on B 储 that we will impose below, the cooperon is
given by Eq. 共45兲. Since the Berry phase factor varies on the
scale of LⰇl e , we may work with K instead of C and write
r⬘
4e 2 B 2储 ⌬ 2
冋
A. Zero perpendicular field applied
r
共61兲
dl•Ag
(0)
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••K B⫽0
共 rn ,r兲
⫻exp ⫺
We now consider the roughness correlation length to be
long compared to the mean free path (LⰇl e ). This circumstance is more difficult to analyze; hence the discussion is
limited to the circumstance that the in-plane magnetic field is
weak in a sense made precise below. Roughly it is assumed
that the random Berry phase is small for all contributing
paths. This is more stringent than the condition B g ⬃B 储 ⌬/L
ⰆB e imposed in the previous section 关see discussion following Eq. 共45兲兴. The experiments of Refs. 5–7 meet this condition for most part. For simplicity, we first assume the applied field lies in the plane of the interface, B⬜ ⫽0 共Sec.
V A兲. In the next subsection no restriction is placed on B⬜
共other than B⬜ ⰆB e ).
2e
ប
r
in place of Eq. 共46兲. Note that we could not make this replacement in the previous section since the Berry phase fluctuated rapidly on the scale of l e . Our objective now is to
average K B 储 over interface roughness and then use Eq. 共27兲
to evaluate the conductance.
The Berry phase factor is a Gaussian random variable so
it is easy to perform the average. Using Eq. 共8兲 for the vector
potential, Eq. 共12兲 for its statistics and Eq. 共56兲, we obtain
V. LONG-RANGE ROUGHNESS
冉 冕
r
with
Use of Eqs. 共56兲 and 共59兲 finally leads to Eq. 共47兲.
(0)
K B(0)储 共 r,r⬘ 兲 ⫽K B⫽0
共 r,r⬘ 兲 exp i
冕
冕
drdr1 drk⫺1 drk
1
共 y ⫺y 兲共 y k ⫺y k⫺1 兲
共 n⫹1⫺k 兲共 k⫺2 兲 1
冋
冊
␾g
⫻exp i2 ␲
,
␾ sc
⫻exp ⫺
235325-10
共 r⫺rk 兲 2
2l 2e 共 n⫹1⫺k 兲
⫺
共 rk⫺1 ⫺r1 兲 2
2l 2e 共 k⫺2 兲
册
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
冋
冋
共 rk ⫺rk⫺1 兲 2
⫻exp ⫺
⫻exp
2l 2e
1
4L 2
⫺
共 r1 ⫺r兲 2
2l 2e
册
PHYSICAL REVIEW B 64 235325
册
共 r⫹r1 ⫺rk⫺1 ⫺rk 兲 2 .
共65兲
This is a low-dimensional Gaussian integral and can be explicitly evaluated.
Proceeding in this manner we obtain
␦ K B(n)储 共 r,r兲 ⫽
冉 冊
f 1 共 n,d 兲 ⬇
eB 储 ⌬
ប
2
关 f 1 共 n,d 兲 ⫹ f 2 共 n,d 兲 ⫹ f 3 共 n,d 兲兴 ,
n 共 1⫹nd 兲
16d
2
n
FIG. 4. Plot of f ( ␣ )/ ␣ as a function of ␣ ⫽l ␾2 /L 2 . f ( ␣ ) describes the dependence of the in-plane magnetoresistance, at zero
perpendicular field, on the roughness correlation length L 关see Eq.
共67兲兴.
1
,
兺
k⫽3 共 k 2 ⫺nk⫺ 关 n/4d 兴 兲 2
1⫹nd
,
f 2 共 n,d 兲 ⬇
n
␦ g⬇⫺
f 3 共 n,d 兲 ⬇1
共66兲
with d⫽l 2e /(2L 2 ). The three contributions to K (n) arise from
the interaction of disconnected links, adjacent links, and
from the self-interaction of links, respectively. We have used
nⰇ1 and dⰆ1 to simplify Eq. 共66兲. This is justified because
weak localization is dominated by long paths and we are
concerned with long-range correlated roughness in this section.
Using Eq. 共27兲 for the conductance and Eq. 共66兲 for
␦ K (n) , after considerable simplification we obtain the parallel field magnetoconductance
␦ g⫽⫺
e2 1
共 B 储 ⌬L 兲 2 f 共 ␣ 兲
2
ប ␾ sc
with ␣ ⫽l ␾2 /L 2 and
f 共 ␣ 兲⫽␣
共67兲
⬇⫺
e 2 B 2储 ⌬ 2 l ␾2
2
ប ␾ sc
l ␾ ⰇL
for
2 e 2 B 2储 ⌬ 2 l ␾4
2 2
3 ប ␾ sc
L
for
共70兲
LⰇl ␾ .
These formulas have a simple interpretation. The random
magnetic field has a typical magnitude B 储 ⌬/L and is correlated over a distance L. For LⰇl ␾ , the typical flux through
an area l ␾2 is therefore (B 储 ⌬/L)l ␾2 ; for LⰆl ␾ , it is B 储 ⌬l ␾
关The area l ␾2 can be broken up into l ␾2 /L 2 correlated squares
of area L 2 . The typical flux through each square is
(B 储 ⌬/L)L 2 . Since it can be of either sign, the flux through
the total area l ␾2 grows as the square root of the number of
squares.兴. Eq. 共70兲 then shows that ␦ g 共in units of e 2 /h) is
the square of the flux through a phase-coherent region in
units of the flux quantum.
B. Perpendicular field applied
冕
⬁
dxe
⫺x
0
冉
1⫹
The calculation with a perpendicular field applied closely
parallels the calculation at B⬜ ⫽0. We write
1
2 关 x ␣ 共 1⫹x ␣ 兲兴 1/2
冑1⫹x ␣ ⫺ 冑x ␣
⫻ln
冑1⫹x ␣ ⫹ 冑x ␣
冊
It is striking that ␦ g is independent of the mean free path l e .
In appropriate units it is a product of two factors: the flux
through an area ⌬L, determined entirely by the geometry of
the rough interface, and f ( ␣ ). f ( ␣ ) describes the crossover
as l ␾ is varied relative to L. It is plotted in Fig. 4 and has the
asymptotic behavior
f 共 ␣ 兲⬇␣
for
2
⬇ ␣2
3
␣ Ⰶ1.
2e
ប
r
r⬘
dl• 共 Ag ⫹A兲
册
共71兲
in place of Eq. 共60兲 and
⬁
K B⬜ ,B 储 共 r,r兲 ⫽
K B(n) ,B 储 共 r,r兲 ⫽
␣ Ⰷ1,
for
⬜
共68兲
.
冋 冕
(0)
K B(0) ,B 储 共 r,r⬘ 兲 ⫽K B⫽0
共 r,r⬘ 兲 exp i
⬜
共69兲
Substituting Eq. 共69兲 in Eq. 共67兲 gives the asymptotic behavior of the parallel field magnetoconductance,
235325-11
冕
兺
n⫽0
K B(n) ,B 储 共 r,r兲 ,
⬜
(0)
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••K B⫽0
共 rn ,r兲 •••
冉
⫻exp i2 ␲
冊 冉
冊
␾g
␾⬜
exp i2 ␲
,
␾ sc
␾ sc
␾ g⫽
冕
r
r
dl•Ag ,
H. MATHUR AND HAROLD U. BARANGER
␾⬜ ⫽
冕
PHYSICAL REVIEW B 64 235325
r
冉
⫻ coth关 k 冑␥ 兴 ⫹coth关共 n⫺k 兲 冑␥ 兴 ⫹4
共72兲
dl•A
r
in place of Eq. 共61兲. Here ␾⬜ is the Aharonov-Bohm flux
through the polygon that goes from r to r via r1 •••rn due to
the applied perpendicular field; ␾ g is the Berry phase around
the same polygon.
The Berry phase factor is a Gaussian random variable.
Performing the average as in the previous subsection yields
KB(n) ,B 储 共 r,r兲 ⫽
⬜
冕
(0)
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••K B⫽0
共 rn ,r兲
⫻exp关 ⫺V 共 r,r1 , . . . ,rn 兲兴
冉
⫻exp i2 ␲
冊
␾⬜
.
␾ sc
共73兲
Here K (n) ⫽ 具 K (n) 典 rough and V is given by Eq. 共63兲. As in Sec.
III B, K (n) is the Fourier transform of the directed area distribution of n-sided polygons; but the weight of the polygons
is no longer Gaussian. The links of the polygons now ‘‘interact’’ and the interaction given by Eq. 共63兲 is anisotropic
and long ranged.
For weak in-plane fields a full analysis of K (n) is not
needed. It is sufficient to analyze the interaction perturbatively. To first order
冕
(0)
⫻K B⫽0
共 rn ,r兲 V 共 r,
冉
冊
2 ␲ ␾⬜
. . . ,rn 兲 exp i
.
␾ sc
共74兲
V(r,r1 , . . . ,rn ) is a sum over pairs of links according to Eq.
共63兲. To evaluate Eq. 共74兲 it is convenient to focus on the
particular term corresponding to the jth and kth links. As in
the corresponding evaluation of Eq. 共64兲, it is convenient to
perform the integrals over the end points of the jth and kth
links last. The integral over the other intermediate points is
facilitated by Eqs. 共B10兲 and 共B15兲 of Appendix B. The expression that results is a low-dimensional Gaussian integral
over the end points of links j and k that can be explicitly
computed.
Proceeding in this manner we obtain
␦ K B(n)
共 r,r兲 ⫽
⬜ ,B 储
n␥
f 1 共 n, ␥ ,d 兲 ⬇
␲
⫹
n
兺
k⫽3
d
冑␥
冉
冉 冊
eB 储 ⌬
ប
2
关 f 1⫹ f 2⫺ f 3兴 ;
coth关 k 冑␥ 兴 coth关共 n⫺k 兲 冑␥ 兴 ⫺1
coth关 k 冑␥ 兴 ⫹
d
冑␥
f 3 共 n, ␥ ,d 兲 ⬇
coth关共 n⫺k 兲 冑␥ 兴
冊
冊
1
n
冑␥
,
␲ sinh关 n 冑␥ 兴
共75兲
where ␥ ⬅(B⬜ /2B e ) 2 is a measure of the dynamical strength
of the applied perpendicular field and we have taken nⰇ1,
␥ Ⰶ1, and dⰆ1. This is justified because weak localization
is dominated by long paths, the applied field is dynamically
weak, and we are concerned with long-ranged roughness in
this section. We have also taken n ␥ Ⰶ1, justifiable a posteriori, because we find ␦ K (n) decays exponentially for n
Ⰷ1/冑␥ .
Using Eq. 共27兲 for the conductance and Eq. 共75兲 for
␦ K B(n)
, after considerable simplification, we obtain a
⬜ ,B 储
lengthy expression for the parallel field magnetoconductance,
冉 冊
e 2 B 储 ⌬L
␦ g⫽⫺2
ប ␾ sc
u 共 ␩ ,B ␾ /B⬜ 兲 ⫽ ␩
(0)
dr1 •••drn K B⫽0
共 r,r1 兲 •••
冑␥
⫺2
冑␥
2n
兵 冑␥ coth关 n 冑␥ 兴 ⫹d 其
␲ sinh关 n 冑␥ 兴
2
u 共 ␩ ,B ␾ /B⬜ 兲 .
Here ␩ ⬅ ␾ sc/2␲ B⬜ L 2 and
␦ K B(n)
共 r,r兲 ⬅K B(n)
共 r,r兲 ⫺K B(n)
共 r,r兲
⬜ ,B 储
⬜ ,B 储
⬜
⫽⫺
f 2 共 n, ␥ ,d 兲 ⬇
d
冕
⬁
dy
1
ln y
y
2B ␾ /B⬜
冋
v共 y, ␩ 兲 ⫺
共76兲
2
y ⫺1
2
册
,
共77兲
with
v共 y, ␩ 兲 ⫽
2
关共 1⫹2 ␩ ⫹2 ␩ 2 兲共 y 2 ⫺1 兲 ⫹4 ␩ 兴
共 y 2 ⫺1 兲
关共 1⫹2 ␩ 兲 2 共 y 2 ⫺1 兲 ⫹8 ␩ 兴
⫺
冉
␩
f 3/2
⫻ln
再
关共 1⫹2 ␩ 兲共 y 2 ⫺1 兲 2 ⫹ 共 8⫹4 ␩ 兲共 y 2 ⫺1 兲 ⫹8 兴
关 冑 f ⫺2 ␩ 共 y 2 ⫺1 兲兴 2 ⫺ 共 y 2 ⫺1 兲 2
关 冑 f ⫹2 ␩ 共 y 2 ⫺1 兲兴 2 ⫺ 共 y 2 ⫺1 兲 2
冎冊
,
f 共 y, ␩ 兲 ⫽ 共 y 2 ⫺1 兲关共 1⫹2 ␩ 兲 2 共 y 2 ⫺1 兲 ⫹8 ␩ 兴 .
共78兲
Equations 共76兲–共78兲 are the main results of this section.
They give the shift in conductance from the standard weaklocalization line shape due to the applied in-plane field B 储 .
Remarkably ␦ g is independent of l e .
Since the expression for ␦ g is complicated, it is instructive to examine several special cases. First, by taking the
limit B⬜ →0 it is possible to recover the result of the previous subsection, Eq. 共67兲. Next consider the strong perpendicular field limit B⬜ l ␾2 Ⰷ ␾ sc . In this limit
␦ g⫽⫺2
⫻ 兵 sinh关 k 冑␥ 兴 sinh关共 n⫺k 兲 冑␥ 兴 其 ⫺1
235325-12
冉 冊
e 2 B 储 ⌬L
h共 ␩ 兲
ប ␾ sc
共79兲
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
PHYSICAL REVIEW B 64 235325
fied formula, Eq. 共79兲, applies corresponds to very small ␩
and does not include the geometric crossing point, ␩
⫽0.558 . . . . We must therefore use the general expression
Eq. 共76兲 to determine the crossing point. Nonetheless in the
inhomogeneous broadening regime the deviation from the
conventional weak-localization line shape and the existence
of a crossing point has a simple interpretation. Since the
random magnetic field is correlated over a length scale L, the
sample breaks up into blocks of size l ␾ each of which sees a
slightly different 共but uniform兲 perpendicular magnetic field,
B⬜ ⫹ ␦ B. For small ␦ B, the shift in conductance of a particular block may be computed from the conventional weaklocalization line shape g WL(B⬜ ) by expanding in a Taylor
series:
␦ g⬇
FIG. 5. Plot of h( ␩ )/ ␩ as a function of ␩ ⫽ ␾ sc /(2 ␲ L 2 B⬜ ). The
inset shows the small ␩ behavior and the zero of h( ␩ ) in detail.
h( ␩ ) describes the perpendicular field dependence of the in-plane
magnetoresistance in the geometric limit, l e ⰆLⰆl ␾ ; see Eq. 共79兲.
with
h共 ␩ 兲⫽␩
冕
⬁
1
dy ln y v共 y, ␩ 兲 ⫺
␲2
␩.
4
共80兲
A plot of h( ␩ ) is shown in Fig. 5. It has the asymptotic
behavior
h共 ␩ 兲⬇␩2
for
␩ Ⰶ1⬇⫺
␲2
8
for
␩ Ⰷ1.
共81兲
Note that h( ␩ ) changes sign; it vanishes for ␩ ⬇0.558.
Eq. 共79兲 reveals that for B⬜ l ␾2 Ⰷ ␾ sc , ␦ g depends only on
the magnetic fields B 储 and B⬜ and geometric parameters that
characterize the rough interface, ⌬ and L. It is independent
not only of l e it but also of l ␾ . The intermediate regime in
which l e ⰆL, but LⰆl ␾ , is particularly interesting. In this
limit B ␾ Ⰶ ␾ sc /L 2 and hence Eq. 共79兲 applies in a range of
B⬜ over which ␩ varies from large to small. Thus the perpendicular field magnetoconductance curves at different values of B 储 cross at a point determined by the purely geometric
condition
h
B⬜ L 2 ⬇1.79 .
e
共82兲
For this reason we call this intermediate regime, l e ⰆL
Ⰶl ␾ , the geometric regime.
The opposite limit of extremely long-ranged roughness,
LⰇl ␾ Ⰷl e , we call the inhomogeneous broadening regime.
In this limit, too, perpendicular magnetoconductance curves
corresponding to different small B 储 cross, but the crossing
point is not determined by the purely geometric condition,
Eq. 共82兲. In this case the range of B⬜ over which the simpli-
⳵
1 ⳵2
g WL共 B⬜ 兲 ␦ B⫹
g 共 B 兲␦ B 2⫹ . . . .
⳵ B⬜
2 ⳵ B⬜2 WL ⬜
共83兲
Bearing in mind that the average value of the random field is
zero and the typical value, B 储 ⌬/L, the average deviation is
␦ g⬃B 2储
⌬2 ⳵2
L 2 ⳵ B⬜2
g WL共 B⬜ 兲 .
共84兲
The curvature of g WL(B⬜ ) is positive at B⬜ ⫽0. At large
B⬜ Ⰷ ␾ sc /l ␾2 , g WL(B⬜ )⬃(e 2 /h)ln(B⬜ /B␾); hence the curvature is negative. Thus there must be an inflection point at
which the curvature of g WL(B⬜ ) vanishes. According to Eq.
共84兲 this is the crossing point. Substitution of the known
asymptotic forms of g WL(B⬜ ) in Eq. 共84兲 allows a check on
the consistency of this interpretation. For small B⬜ we recover the estimate given in the second line of Eq. 共70兲. For
B⬜ Ⰷ ␾ sc /l ␾2 we obtain
␦ g⬃
e 2 B 2储 ⌬ 2
h B⬜2 L 2
共85兲
in agreement with the small ␩ limit of Eq. 共79兲.
VI. COMPARISON TO EXPERIMENT
AND RELATED WORK
In their experiments Wheeler and collaborators apply a
fixed B 储 and measure the conductance as a function of B⬜ .
They have reported such line-shape measurements for many
devices and have also explored the effect of varying electron
density on the line shape. The data of Anderson et al.6 are
particularly interesting because they have independently
measured the interface roughness by etching away the oxide
and imaged the exposed silicon surface using atomic force
microscopy. Hence we focus on the measurements reported
in that paper.
Figure 6 shows the measured magnetoconductance of the
control device at B 储 ⫽0 共gray兲 and B 储 ⫽1.5 T 共black兲. By
fitting the B 储 ⫽0 data to a weak-localization line shape
共dashed curve in Fig. 6兲, Eq. 共44兲, Anderson et al. infer l e
⫽0.085 ␮ m, l ␾ ⫽0.76 ␮ m. Atomic force microscope measurements yield ⌬⫽1.35 Å and L⫽0.055 ␮ m. Thus the
235325-13
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
device is in the short-range correlated regime. The black
curve shows the calculated line shape using the independently measured values of ⌬, L, l e , and l ␾ . No parameters
are adjusted except for a constant offset. The fit is good,
comparable, for example, to the fit to weak-localization line
shape at B 储 ⫽0.
Anderson et al. also report measurements on a second device 共called the textured device兲. This device is fabricated on
a silicon wafer that has been roughened by argon sputtering
prior to gate oxidation. As a result the textured device has
atypical values of ⌬ and L. L for the textured sample is
comparable to l e . Consistent with this we do not obtain a
good zero-parameter fit to either the short- or long-range
roughness expressions derived here. Following Anderson
et al. we can force fit the data to the homogeneously broadened line shape; but the value of ⌬ 2 L we then obtain is
inconsistent with the atomic force microscope images.
Finally it is interesting to compare the experiments of
Wheeler et al. to other work on random or periodic magnetic
fields. Such fields have been realized by gating a GaAs/
AlGaAs heterostructure with type-II superconducting
film;12,13 by randomly depositing type-I superconducting
grains14 or ferromagnetic dysprosium dots15 on the heterostructure surface; and by covering the device with a rough
macroscopic magnet.16 In work most closely related to
Wheeler and co-workers, Gusev et al. have prepatterned the
substrate by drilling a lattice of holes so that the interface at
which the two-dimensional electron gas forms is periodically
modulated.17 An inhomogeneous magnetic field leads to
many interesting features in the magnetoresistance even in
the classical and ballistic regimes; many experiments have
focused on these effects. Except for the work of Rammer and
Shelankov18 noted below, much of the theoretical literature
too explores these regimes or focuses on the fundamental
question of whether all eigenstates are localized in a random
magnetic field.
Two experiments study quantum corrections in the weaklocalization limit: In the work of Bending, von Klitzing, and
Ploog12 the inhomogeneous magnetic field is produced by an
Abrikosov lattice of flux lines in the superconducting gate;
hence it is periodic. However the magnetoresistance is not
sensitive to the arrangement of the vortices since the dephasing length is shorter than the distances between the vortices.
This circumstance was analyzed by Rammer and
Shelankov.18 In the experiment of Gusev et al. too the magnetic field is essentially periodic rather than random allowing
them to analyze it via a chessboard model.17 Thus in both
these works the inhomogeneous magnetic field is rather different from the random Gaussian correlated field studied in
the experiments of Wheeler and co-workers and analyzed
here. It is also worth noting that the parallel field weaklocalization magnetoresistance for disordered films 共width
greater than l e but less than l ␾ ) has previously been studied
by a number of authors.31 In contrast to these works, the
system we consider here is a truly two-dimensional electron
gas 共width smaller than the Fermi wavelength兲 with correspondingly different physics.
FIG. 6. Plot of the conductance 共in units of 10⫺5 mhos兲 against
perpendicular field, B⬜ 共in G兲, for the control device of Anderson
et al.6 Data points for B 储 ⫽0 are grey; for B 储 ⫽1.5 T, black. The
dashed line is a fit to the weak-localization line shape, from which
Anderson et al. infer l e ⫽0.085 ␮ m, l ␾ ⫽0.76 ␮ m. The solid
curve shows the line shape at B 储 ⫽1.5 T calculated from the expression for homogeneous broadening derived in this paper, Eqs.
共44兲 and 共55兲. All the parameters for this curve are independently
determined. The values of ⌬⫽1.35 Å and L⫽0.055 ␮ m are obtained from atomic force microscope images; for l e and l ␾ , we use
the values cited above.
VII. SUMMARY AND CONCLUSION
In this paper we have been concerned with the influence
of interface roughness on the magnetoconductance of MOSFET’s. Using a Born-Oppenheimer approximation we have
shown that effectively electrons respond to an applied inplane magnetic field as though it were a random perpendicular field. Technically, the random magnetic field appears as a
Berry phase term in the Born-Oppenheimer effective Hamiltonian. Equation 共9兲 shows that the random magnetic field is
the component of B 储 normal to the rough surface Zg (x,y).
The Born-Oppenheimer analysis also gives precise meaning
to the surface Zg (x,y): it is the location of the center of the
local subband wave function.
The next step is to analyze the magnetoconductance. This
is controlled by the familiar physics of weak localization: at
zero field there is a contribution to the conductance due to
constructive interference of closed electron paths and their
time-reversed counterparts. This contribution is suppressed
by application of a perpendicular magnetic field because the
electron acquires an Aharonov-Bohm phase as it traverses a
closed path. The analysis of this paper shows that an in-plane
field too has an effect due to interface roughness. The electron acquires a Berry’s phase equal to the flux of the random
field through the closed path. The effect of the random Berry’s phase on the conductance is the main focus of this paper.
To calculate the effect it is necessary to sum the standard
divergent series of cooperon diagrams 共Fig. 3兲. We find it
useful to reformulate this sum in a manner reminiscent of the
central limit theorem. Feynman graphs of high order control
the sum. In these graphs we are able, without significant
235325-14
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
error, to replace pairs of Green-function lines with suitable
Gaussian factors. At zero field, the nth-order diagram is then
essentially the weight of a closed n-step Gaussian random
walk 关Eq. 共27兲兴. For the known case of an applied perpendicular field, this formulation brings out clearly the relationship between the classic weak-localization line shape 关Eq.
共44兲兴 and the distribution of directed area for closed random
walks 关see Eqs. 共38兲 and 共39兲 and Appendix C兴. The method
also proves useful when an in-plane field is applied for longrange correlated roughness (LⰇl e ). In this case the magnetoconductance is related to the directed area distribution of
an interacting random walk 关see Eqs. 共63兲 and 共73兲兴. To
make contact with experiment it is sufficient to analyze this
problem perturbatively. We do not attempt a complete solution of the interacting random walk or calculation of the full
magnetoconductance for long-range correlated roughness.
These problems are left open for future work.
For short-range correlated roughness, however, we are
able to obtain the full magnetoconductance. The dependence
of the conductance on perpendicular field B⬜ still has the
classic weak-localization line shape 关Eq. 共44兲兴 when an inplane field is applied, just as it does at B 储 ⫽0. The only effect
of the in-plane field is to enhance the dephasing rate 关Eq.
共52兲兴. In terms of field scales, B ␾ becomes larger; the effective B ␾ is given by Eq. 共55兲. Hence we say that short-range
correlated roughness produces homogeneous broadening of
the weak-localization line shape.
The departure from the conventional line shape is more
striking in case of long-range correlated roughness 共although
quantitatively smaller if all parameters except L are held
fixed兲. The change in conductance ␦ g when a parallel field is
applied 共‘‘parallel field magnetoconductance’’兲 is given by
Eq. 共76兲. The most remarkable feature of this expression is
that it is independent of l e . ␦ g is determined entirely by the
magnetic field, geometric parameters of the interface, ⌬ and
L, and the dephasing length l ␾ . It may be of either sign
共positive near B⬜ ⫽0 and negative for large B⬜ ). The expression for ␦ g is lengthy and cannot be more compactly expressed in terms of known special functions. It can be numerically computed and plotted with ease and precision;
however, it is worthwhile to examine several special cases.
In the limit of extremely long-ranged roughness, LⰇl ␾
Ⰷl e , the parallel field magnetoconductance has a simple interpretation: the sample breaks up into independent blocks of
size l ␾2 each of which sees a slightly different magnetic field.
Hence in this limit we say that the effect of an in-plane field
and roughness is to inhomogeneously broaden the weaklocalization line. Pursuing this interpretation we can understand the sign change in ␦ g and recover its form in various
circumstances 关see the discussion following Eqs. 共70兲 and
共82兲兴.
The limit of intermediate-range correlated roughness, l ␾
ⰇLⰇl e , is particularly striking. For sufficiently large perpendicular field, B⬜ l ␾2 Ⰷ ␾ sc , ␦ g becomes independent not
only of l e but also l ␾ 关see Eq. 共79兲兴. Essentially the parallel
magnetoconductance is controlled by the function h( ␩ )
where ␩ ⫽ ␾ sc /(2 ␲ L 2 B⬜ ) is a measure of the perpendicular
flux 共see Fig. 5兲.
PHYSICAL REVIEW B 64 235325
Another special case, useful for making contact with
experiment7 is the parallel field magnetoconductance for
B⬜ ⫽0 关Eq. 共67兲兴. In this case, ␦ g, in units of e 2 /h, is a
product of the squared flux (B 储 ⌬L) 2 , determined entirely by
the field and the geometry of the interface, and the function
f ( ␣ ). f ( ␣ ) describes the crossover as l ␾ is varied relative to
L ( ␣ ⫽l ␾2 /L 2 ; see Fig. 4兲.
Wheeler and co-workers have shown experimentally in a
number of cases that the effect of an in-plane field is to
produce homogeneous broadening. Our analysis of the shortrange correlated regime provides a theoretical justification of
such fits to data and allows quantitative information about
the interface roughness to be extracted. It is encouraging that
for the control sample studied by Anderson et al.,6 for which
the interface parameters are independently measured via
atomic force microscopy, our formulas give a satisfactory
zero parameter fit 共Fig. 6兲. Unfortunately, our analysis shows
that in the short-range regime, magnetoresistance measurements will not separately yield ⌬ and L: instead they provide
the combination ⌬ 2 L. Thus it will be necessary to combine
magnetoconductance measurements with other techniques to
separately obtain ⌬ and L.
The roughness of the interface is controlled by its processing. For example, Anderson et al.6 are able to increase
both ⌬ and L by argon sputtering the silicon surface prior to
gate oxidation. It would be desirable to create devices with
long-range correlated roughness in order to experimentally
observe, for example, the geometric regime described above.
At the same time, for long-range correlated roughness, the
prospects are much better for extracting both ⌬ and L from
magnetoconductance measurements alone, provided the
measurements can be made with sufficient precision. Equation 共76兲 provides the means to study the feasibility of such
experiments.
Another potential application of our analysis is related to
efforts to engineer the MOSFET interface to have specific
structure, for example, a structure that is periodic in one
direction like a corrugated sheet.32 Weak-localization magnetoconductance measurements would provide a nondestructive way to check that structures have been successfully fabricated. It should be a straightforward and interesting
extension of our analysis to calculate the magnetoresistance
signature of various simple periodic structures.
ACKNOWLEDGMENTS
It is a pleasure to acknowledge many stimulating conversations with Albert Chang, Rachel Lombardi, Satya Majumdar, Don Monroe, and Bob Wheeler. H. Mathur was supported by NSF Grant No. DMR 98-04983 and the Alfred P.
Sloan Foundation and acknowledges the hospitality of the
Aspen Center for Physics.
APPENDIX A:
ADIABATIC APPROXIMATION
Here we briefly discuss the conditions for the validity of
the Born-Oppenheimer approximation used in Sec. II.
Roughly, we want only the low-lying states of the lowest
subband to be occupied and we want the gap between sub-
235325-15
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
bands to be large. For this purpose, and to ensure the two
dimensionality of the electron gas, we need the Fermi energy
to be small compared to the subband spacing. The subband
spacing is of order ប 2 /ml 2 where l is the length scale over
which the electron is confined. Hence this condition may be
phrased as
␭ f Ⰷl.
l⌬ AFM
共A1兲
To obtain additional conditions we return to the Schrödinger
Eq. 共1兲 and seek a solution of the form
⌿ 共 x,y,z 兲 ⫽
the same conditions can in fact be derived under less restrictive assumptions than Eq. 共A5兲.
If we assume that Zg is statistically the same as the surface imaged by atomic force microscopy, Eq. 共A7兲 may be
rewritten
兺n ␺ n共 x,y 兲 ␾ n共 z;x,y 兲 .
共A2兲
L
冋
⫺
册
ប
ⵜ 2 ⫹U imp共 x,y 兲 ⫹E n 共 x,y 兲 ⫹W n 共 x,y 兲 ␺ n 共 x,y 兲
2m
⫹
兺 H nm ␺ m共 x,y 兲 ⫽E ␺ n共 x,y 兲 .
m⫽n
Ⰶ1,
APPENDIX B:
The purpose of this appendix is to calculate the integral,
Eq. 共36兲. It is useful to first calculate
共A3兲
ប2
ប2
H nm ⫽ 具 n;x,y 兩 ⵜ 兩 m;x,y 典 •ⵜ⫹
具 n;x,y 兩 ⵜ 2 兩 n;x,y 典 .
m
2m
共A4兲
Equation 共A4兲 constitutes an exact reformulation of the
Schrödinger Eq. 共1兲. The Born-Oppenheimer approximation
consists of keeping only one term in the sum 共A2兲 and hence
omitting the off-diagonal terms in Eq. 共A3兲.
In a commonly used model of the interface21 it is assumed
that the confining potential has the form
D (n) 共 y,y ⬘ 兲 ⫽
冕
⫹⬁
⫺⬁
dy 1 •••dy n D (0) 共 y,y 1 兲 •••D (0) 共 y n ,y ⬘ 兲 ,
共A6兲
Using Eq. 共A6兲 we can estimate the off-diagonal elements
given by Eq. 共A4兲. Requiring these to be small compared to
subband spacing ប 2 /ml 2 , leads to
lⵜ 2 Zg Ⰶ1,
ⵜZg Ⰶ1.
D (0) 共 y,y ⬘ 兲 ⫽
共A7兲
In summary, Eqs. 共A1兲 and 共A7兲 are the conditions for the
validity of the Born-Oppenheimer approximation. Note that
1
冑␲
e ⫺(y⫺y ⬘ )
2 ⫺g(y⫹y ) 2
⬘
.
共B1兲
It is easy to show by induction that D (n) is of the form
D (n) 共 y,y ⬘ 兲 ⫽
1 ⫺a (y⫺y ) 2 ⫺b (y⫹y ) 2
⬘
⬘
n
e n
冑␲ A n
1
共B2兲
with a n b n ⫽g. By completing squares we find
D (n⫹1) 共 y,y ⬘ 兲 ⫽
共A5兲
Here ␨ (x,y) is the elevation of the silicon-oxide boundary. In
this model the mean elevation of the local subband wave
function exactly tracks the interface; hence the surface
Zg (x,y), probed by weak-localization magnetoresistance,
has the same statistics as the silicon-oxide interface ␨ (x,y),
which is presumably imaged by atomic force microscopy.
For a more general form of the confinement potential this
precise correspondence between Zg (x,y) and ␨ (x,y) is lost
but it seems reasonable that the two surfaces would have
similar statistics, comparable correlation lengths, and meansquare fluctuations.
Equation 共A5兲 implies that the local subband wave function is of the form
␾ n 共 z;x,y 兲 ⫽ ␰ n 关 z⫺ ␨ 共 x,y 兲兴 .
共A8兲
COOPERON PATH INTEGRAL
Here
U conf共 z;x,y 兲 ⫽U关 z⫺ ␨ 共 x,y 兲兴 .
⌬ AFM
Ⰶ1.
L
For the control sample studied by Anderson et al.6 ⌬ AFM
⫽1.35 Å, L⫽0.055 ␮ m, ␭ f ⫽100 Å; for the textured
sample, ⌬ AFM⫽7.5 Å, L⫽0.09 ␮ m, ␭ f ⫽70 Å. Taking l
⬇30 Å, it is easily seen that the Born-Oppenheimer approximation is applicable.
For simplicity we limit discussion to the case of zero magnetic field. ␺ n (x,y) then obeys9,22
2
2
⫽
冕
⫹⬁
⫺⬁
dy 1 D (0) 共 y,y 1 兲 D (n) 共 y 1 ,y ⬘ 兲
1
2
2
e ⫺a n⫹1 (y⫺y ⬘ ) ⫺b n⫹1 (y⫹y ⬘ )
A
冑␲ n⫹1
1
共B3兲
provided a n b n ⫽g. Here
1
1
1
⫹
,
⫽
a n⫹1 a n ⫹g b n ⫹1
1
b n⫹1
⫽
1
1
⫹
,
a n ⫹1 b n ⫹g
A n⫹1 ⫽A n 冑a n ⫹b n ⫹1⫹g.
共B4兲
Using Eq. 共B4兲 we can verify that a n⫹1 b n⫹1 ⫽g ensuring
D (n⫹2) will also be of the form given in Eq. 共B2兲. In principle the recurrence relations, Eq. 共B4兲, determine a n , b n
and A n ; but in practice it is easier to follow a different
method.
Experience with paths integrals suggests that the eigen2
functions of the kernel D (0) are H m ( 冑2 ␣ y)e ⫺ ␣ y with ␣
suitably adjusted—of the same form as the eigenfunctions of
235325-16
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
a harmonic oscillator. To verify this conjecture and to fix ␣
we use a generating function for the Hermite polynomials
exp共 ⫺x 2 ⫹2x 冑2 ␣ y⫺ ␣ y 2 兲 ⫽
⬁
兺
m⫽0
PHYSICAL REVIEW B 64 235325
amounts to n⫹1 repeated applications of D (0) ). The result is
given in the fourth line. Comparison of the third and fourth
lines reveals
xm
2
H 共 冑2 ␣ y 兲 e ⫺ ␣ y .
m! m
共B5兲
a n ⫽ 冑g
Upon completing squares we find
冕
b n ⫽ 冑g
dy ⬘ D (0) 共 y,y ⬘ 兲 exp共 ⫺x 2 ⫹2x 冑2 ␣ y ⬘ ⫺ ␣ y ⬘ 2 兲
⫽
1⫹ 冑g
⫹2
exp ⫺
1⫺ 冑g
1⫹ 冑g
1⫺ 冑g
2
x
1⫹ 冑g
x2g
1/4
A n⫽
2
y⫺2 冑gy 2
H m 共 2g 1/4y 兲 exp共 ⫺2 冑gy 2 兲
兺
m⫽0 共 1⫹ 冑g 兲 m⫹1 m!
共B6兲
provided ␣ ⫽2 冑g. This condition is imposed by requiring
that the coefficient of the y 2 term in the second line of Eq.
共B6兲 should be ␣ . Comparing Eqs. 共B5兲 and 共B6兲 we con2
clude that H m (2g 1/4y)e ⫺2 冑gy are eigenfunctions of D (0)
with eigenvalue (1⫺ 冑g) m /(1⫹ 冑g) m⫹1 for m⫽0,1,2, . . . .
Now let us evaluate
冕
⫹⬁
⫺⬁
dy ⬘ D
(n)
再
2
冋
⫹4g
⬁
⫽
兺
m⫽0
⬁
⫽
兺
m⫽0
冋
1
m!
共 a n ⫺ 冑g 兲
共 a n ⫹ 冑g 兲
冑␲ a n
An
册冎
,
,
关共 1⫹ 冑g 兲 n⫹1 ⫹ 共 1⫺ 冑g 兲 n⫹1 兴 1/2
册
1
2l 2e
⬜
共B8兲
E (n) 共 r,r⬘ 兲
共B9兲
with
E (n) 共 r,r⬘ 兲 ⫽
1/4
E (0) 共 r,r⬘ 兲 ⫽
2
冕
dr1 •••drn E (0) 共 r,r1 兲 •••E (0) 共 rn ,r⬘ 兲 ,
1
exp关 ⫺ 兩 r⫺r⬘ 兩 2 ⫺i ␤ 共 x⫺x ⬘ 兲共 y⫹y ⬘ 兲兴 ,
␲
共B10兲
and ␤ ⫽B⬜ /B e . In the Landau gauge E (0) depends only on
(x⫺x ⬘ ) and has the Fourier transform
xy
共 a n ⫺ 冑g 兲 m
共 a n ⫹ 冑g 兲
共 1⫹ 冑g 兲 n⫹1 ⫹ 共 1⫺ 冑g 兲 n⫹1
K B(n) 共 r,r⬘ 兲 ⫽
1 冑␲ a n
共 a n ⫺ 冑g 兲
⫽
x2
exp ⫺2 冑gy 2 ⫺
A n a n ⫹ 冑g
冑
共 a n⫹ g 兲
1/4
2g 1/4
共 1⫹ 冑g 兲 n⫹1 ⫺ 共 1⫺ 冑g 兲 n⫹1
册
册
These expressions are the solution to the recurrence relation
in Eq. 共B4兲 with the initial condition a 0 ⫽1,b 0 ⫽g,A 0 ⫽1.
In summary the integral D (n) , defined in Eq. 共B1兲 is given
by Eqs. 共B2兲 and 共B8兲.
Now we evaluate Eq. 共36兲 for K B(n) (r,r⬘ ). Rescaling the
⬜
co-ordinates we obtain
共 y,y ⬘ 兲 exp共 ⫺2 冑gy ⬘ ⫺x ⫹4g xy 兲
2
冑␲
共 1⫹ 冑g 兲 n⫹1 ⫺ 共 1⫺ 冑g 兲 n⫹1
⫻ 关共 1⫹ 冑g 兲 n⫹1 ⫺ 共 1⫺ 冑g 兲 n⫹1 兴 1/2.
共 1⫺ 冑g 兲 m x m
⬁
⫽
冋冉 冊
冉 冊
册
1
冋
冋
共 1⫹ 冑g 兲 n⫹1 ⫹ 共 1⫺ 冑g 兲 n⫹1
m⫹1
x m H m 共 2g 1/4y 兲 e ⫺2 冑gy
1
2
冑␲
1
共 1⫺ 冑g 兲 m(n⫹1)
2
x m H m 共 2g 1/4y 兲 e ⫺2 冑gy .
m! 共 1⫹ 冑g 兲 (m⫹1)(n⫹1)
exp关 ⫺ 共 x⫺x ⬘ 兲 2 ⫺i ␤ 共 y⫹y ⬘ 兲共 x⫺x ⬘ 兲兴
⫽
冕
⫹⬁ dk
⫺⬁
2␲
冉
exp ⫺
冊
关 k⫹ ␤ 共 y⫹y ⬘ 兲兴 2 ik(x⫺x )
⬘.
e
4
共B11兲
共B7兲
The integral in the first line of Eq. 共B7兲 has been analyzed in
two ways. First we use the ansatz for D (n) , Eq. 共B2兲, perform
the integral by completing squares, and expand the result
using the generating formula for the Hermite polynomials,
Eq. 共B5兲. The results are given in the second and third lines.
Alternatively, we expand the exponential in the first line in
terms of the eigenfunctions of D (0) using the generating formula, Eq. 共B5兲, and note that the eigenfunctions of D (0) are
also eigenfunctions of D (n) but with the eigenvalue raised to
the (n⫹1)st power 共considered as an integral operator, D (n)
By virtue of translational invariance
E (n) 共 r,r⬘ 兲 ⫽
冕
冕
␲
⫹⬁ dk
⫺⬁
2
⫹⬁
⫺⬁
dy 1 •••dy n
1
冑␲ n⫹1
e ik(x⫺x ⬘ )
⫻e ⫺[k⫹ ␤ (y⫹y 1 )] /4•••e ⫺[k⫹ ␤ (y n ⫹y ⬘ )]
2
⫻e ⫺(y⫺y 1 ) •••e ⫺(y n ⫺y ⬘ ) .
2
If we shift y i →y i ⫺k/2␤
235325-17
2
2 /4
共B12兲
H. MATHUR AND HAROLD U. BARANGER
E (n) 共 r,r⬘ 兲 ⫽
冕
冕
␲
⫹⬁ dk
⫺⬁
2
冋
⫹⬁
⫺⬁
⫻exp ⫺
dy 1 •••dy n
PHYSICAL REVIEW B 64 235325
1
冑␲
e ik(x⫺x ⬘ )
n⫹1
␤2
␤2
共 ȳ⫹y 1 兲 2 ⫹•••⫺ 共 y n ⫹ȳ ⬘ 兲 2
4
4
⫻e ⫺(ȳ⫺y 1 )
2 ⫹•••⫺(y ⫺ȳ ) 2
⬘
n
册
共B13兲
.
Here ȳ⫽y⫺k/2␤ , ȳ ⬘ ⫽y ⬘ ⫺k/2␤ , and the k integral must be
performed after the y integrals. Evidently
E (n) 共 r,r⬘ 兲 ⫽
冕
⫹⬁ dk
⫺⬁
2␲
冉
e ik(x⫺x ⬘ ) D (n) y⫺
冊
k
k
,y ⬘ ⫺
.
2␤
2␤
共B14兲
D (n) is given by Eqs. 共B2兲 and 共B8兲 with g→ ␤ 2 /4. Performing the remaining Gaussian integral over k yields
1 ⫺a 兩 r⫺r 兩 2 ⫺i ␤ (y⫹y )(x⫺x )
⬘
⬘
⬘;
e n
␲qn
E (n) 共 r,r⬘ 兲 ⫽
a n⫽
关 1⫹ 共 ␤ /2兲兴 n⫹1 ⫹ 关 1⫺ 共 ␤ /2兲兴 n⫹1
;
关 1⫹ 共 ␤ /2兲兴 n⫹1 ⫺ 关 1⫺ 共 ␤ /2兲兴 n⫹1
1
q n⫽
␤
冋冉 冊 冉 冊 册
␤
1⫹
2
n⫹1
␤
⫺ 1⫺
2
⬜
⬇
冋冉 冊 冉
冋 冉 冊册
B⬜
2 B
2␲le e
1
1⫹
B⬜
2B e
nB⬜
B⬜
sinh
2 ␾ sc
2B e
1
1
a⫽⫺ 共 x 1 ⫺x 兲共 y 1 ⫹y 兲 ⫺ 共 x 2 ⫺x 1 兲共 y 2 ⫹y 1 兲
2
2
n⫹1
.
共B15兲
Equations 共B9兲, 共B10兲, and 共B15兲 constitute the final expression for K B(n) (r,r⬘ ).
⬜
The calculation of K B(n) is similar to the solution of Schrö⬜
dinger’s equation for an electron in a magnetic field because
the slowly varying eigenfunctions of K B(0) obey the differen⬜
tial Eq. 共33兲. However, the derivation of Eq. 共B15兲 goes beyond solution of the differential equation because we also
obtain the short-distance behavior of K B(n) .
⬜
Setting r⫽r⬘ in Eq. 共B15兲 we obtain
K B(n⫺1) 共 r,r⬘ 兲 ⫽
gon correspond to the two senses in which its perimeter may
be circumnavigated.
Now suppose a magnetic field is applied perpendicular to
the plane. Roughly, the directed area is the quantity by which
the magnetic field must be multiplied in order to obtain the
flux through the oriented polygon 共in magnitude and sign兲.
For a simple polygon with no self-intersection the directed
area is equal in magnitude to the area; the sign is positive or
negative depending on whether the orientation of the polygon is anticlockwise or clockwise viewed from the direction
in which the magnetic-field points. For a polygon with selfintersection, the directed area is obtained by subdividing the
polygon into infinitesimal area elements and weighting the
area of each element with its winding number before adding
them. Here the winding number of an area element is computed by drawing a vector from the area element to a point
on the boundary of the polygon and counting the number of
turns made by the vector as its tip then circumnavigates the
perimeter in the sense dictated by the orientation of the polygon. Clockwise turns count as negative turns 共see Fig. 7兲. For
a polygon with successive vertices at r,r1 , . . . ,rn⫺1 ,r immersed in a magnetic field pointing in the z direction, the
directed area is given by
n
⫺ 1⫺
B⬜
2B e
冊册
n ⫺1
1
⫹•••⫺ 共 x⫺x n 兲共 y⫹y n 兲 .
2
共C1兲
Let the probability density of drawing an n-sided oriented
polygon with successive vertices at r,r1 , . . . ,rn⫺1 ,r be
(n/ ␲ n⫺1 )exp关⫺兩r⫺r1 兩 2 ⫹•••⫹ 兩 rn⫺1 ⫺r兩 2 兴 . Equations 共C1兲
and 共B10兲 show that
E (n⫺1) 共 r,r兲 ⫽
1
n␲
冕
⫹⬁
⫺⬁
da P 共 a 兲 exp共 i2 ␤ a 兲 .
共C2兲
Here P(a)⫽ probability distribution of the directed area for
the Gaussian polygon ensemble defined above. By inverting
⫺1
共B16兲
provided that B⬜ /B e is sufficiently small that not only is
B⬜ /B e Ⰶ1 but also (B⬜ /B e ) 2 nⰆ1. The approximate expression in Eq. 共B16兲 is the same as Eq. 共41兲. For a given large
B e /B⬜ Eq. 共41兲 is accurate only for nⰆ(B e /B⬜ ) 2 ; but since
K B(n) (r,r⬘ ) is exponentially small for nⰇB e /B⬜ , no signifi⬜
cant error is made by taking Eq. 共41兲 to apply for all n.
APPENDIX C:
DIRECTED AREA DISTRIBUTION
First let us make precise the meaning of directed area.
Imagine an n-sided polygon drawn on a plane. The polygon
is endowed with an orientation by assigning each edge a
direction such that each vertex has one incoming edge and
one outgoing. Roughly, the two orientations of a given poly-
FIG. 7. An oriented polygon. Area elements in the white region
have winding number ⫹1; in the light grey region, ⫹2; and in the
dark region, ⫺1.
235325-18
RANDOM BERRY PHASE MAGNETORESISTANCE AS A . . .
the Fourier transform in Eq. 共C2兲 and making use of Eq.
共B15兲 for E (n⫺1) (r,r⬘ ) we obtain
P共 a 兲⫽
⬇
冕
冕
冉 冊冋冉 冊 冉 冊 册
冉 冊冋 冉 冊册
冉 冊
⫹⬁ d ␣
⫺⬁
n␣
2␲ 2
⫹⬁ d ␣
⫺⬁
n␣
2␲ 4
␲
2␲a
⫽ sech2
.
n
n
1⫹
sinh
␣
4
n␣
4
n
⫺ 1⫺
⫺1
␣
4
n ⫺1
e ⫺i ␣ a
e ⫺i ␣ a
共C3兲
The approximate form of the integrand in the second line
above is valid only for ␣ Ⰶ1/冑n; but it may be extended to
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J. Rammer, Quantum Transport Theory 共Perseus Books, Reading,
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Similar experiments have been performed by A.M. Chang 共private communication兲.
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M.V. Berry, Proc. R. Soc. London, Ser. A 292, 45 共1984兲.
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A.F. Morpargo, J.P. Heida, T.M. Klapwijk, and B.J. van Wees,
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A. Smith, R. Taboryski, L.T. Hansen, C.B. So” rensen, P. Hedegård,
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P.D. Ye, D. Weiss, K. von Klitzing, K. Eberl, and H. Nickel, Appl.
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F.B. Mancoff, R.M. Clarke, C.M. Marcus, S.C. Zhang, K. Campman, and A.C. Gossard, Phys. Rev. B 51, 13 269 共1995兲.
PHYSICAL REVIEW B 64 235325
all ␣ since the integrand is already exponentially small for
␣ Ⰷ1/n.
Equation 共C3兲 is the formula for the directed area distribution. It coincides with Eq. 共41兲 if we take the step size of
the random walker to be 冑2l e instead of 1 as we have done
in this appendix.
The distribution of directed area for Brownian motion was
analyzed by Lévy33 and restudied more explicitly in connection with electron transport by Argaman et al.34 Here, however, we require the area distribution P n (a) for a random
walk with a finite number of steps n. The large n behavior of
P n (a) was recently obtained by Bellissard et al. for random
polygons on a square lattice by an interesting application of
noncommutative geometry.35 Their result agrees with the
large n limit of Eq. 共C3兲.
17
G.M. Gusev, U. Gennser, X. Kleber, D.K. Maude, J.C. Portal, D.I.
Lubyshev, P. Basmaji, M.de P.A. Silva, J.C. Rossi, and Yu.V.
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18
J. Rammer and A.L. Shelankov, Phys. Rev. B 36, 3135 共1987兲;
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G. Timp, R.E. Howard, and P.M. Mankiewich, in Nanotechnology, edited by G. Timp 共Springer-Verlag, New York, 1999兲 pp.
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20
D. Monroe 共private communication兲.
21
For a thorough discussion see T. Ando, A.B. Fowler, and F. Stern,
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22
For a modern textbook discussion, see R. Shankar, Principles of
Quantum Mechanics, 2nd edition 共Plenum, New York, 1994兲,
Chap. 21.
23
The general principles of linear-response theory are discussed in
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Quantum Theory of Many Particle Systems 共McGraw-Hill, New
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Phys. Rev. B 40, 8169 共1989兲.
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Impurity averaged perturbation theory is discussed by A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics 共Dover, New York,
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For a microscopic derivation, see I.L. Aleiner, B.L. Altshuler and
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26
S. Chakravarty and A. Schmid, Phys. Rep. 140, 193 共1986兲.
27
A.J. McKane and M. Stone, Ann. Phys. 共Leipzig兲 131, 36 共1981兲.
28
The lower limit of the sum requires delicate handling. Introduce
a cutoff NⰇ1 but with NB⬜ /B e Ⰶ1 and N ␶ e / ␶ ␾ Ⰶ1. The sum
over n from 1 to N is approximately ln N since the summand is
approximately 1/n over this range. The sum from N to infinity
may be turned into an integral in the limit B⬜ /2B e →0. The
integral diverges logarithmically at its lower limit. The divergence may be extracted by integration by parts. This cancels the
235325-19
H. MATHUR AND HAROLD U. BARANGER
PHYSICAL REVIEW B 64 235325
log dependence of the first sum on the arbitrary cutoff N and
leaves a finite expression that can be expressed in terms of the
digamma function using Eq. 共43兲.
29
P.M. Morse and H. Feshbach, Methods of Theoretical Physics
共McGraw-Hill, New York, 1953兲 Vol. 1, pp. 422-423.
30
S.F. Edwards and M. Doi, Introduction to Polymer Dynamics
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B.L. Altshuler and A.G. Aronov, JETP Lett. 33, 499 共1981兲; V.I.
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E. Kapon in Epitaxial Microstructures, Semiconductors and Semimetals, edited by A.C. Gossard 共Academic, New York, 1994兲;
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33
P. Lévy, Processus Stochastiques et Mouvement Brownien
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N. Argaman, Y. Imry, and U. Smilansky, Phys. Rev. B 47, 4440
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J. Bellissard, C.J. Camacho, A. Barelli, and F. Claro, J. Phys. A
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235325-20