Universal Conductance Fluctuations
Phase coherent specimens show sample-specific,
aperiodic structure due to their non-self-averaging
nature.
In contrast, macroscopically similar classical
specimens – having the same dimensions, electron
density, and average density of scatterers – should
show negligible specimen-to-specimen conductance
fluctuations in the diffusive (or quasi-ballistic)
transport regime.
Illustration 1:
Dimensionless
conductance vs B at
three gate voltages
for the inversion
layer deivce below.
The aperiodic g
variation is of order
e2 /  .
[PRL 56,2865 (1986)]
Typical geometry of
a MOSFET inversion
layer with six voltage
probes.
UCF
1
Illustration 2:
[PRL 58,2347 (1987)]
Resistance measured between various pairs of probes for the
short device with 0.15 m probe spacing.
UCF
2
Illustration 3:
[Physical Review Letters, 55, 2987 (1985)]
Aperiodic Magnetoresistance Oscillations in
Narrow Inversion Layers in Si
Change of resistance with magnetic field at four gate
voltages VG (threshold at VG  3.4 V ).
Note that the
resistance scales are different at each VG .
The devices used are metal-oxide-Si field-effect transistors
(MOSFET’s) in which the metal gate electrode is a narrow
( ~ 70  50 nm ) tungsten wire. A positive voltage VG applied
to the gate electrode creates an n -type inversion layer at the
surface of the p -type Si and simultaneously confines the
inversion layer to a narrow strip under the wire.
UCF
3
Illustration 4:
Observation of h / e Aharonov-Bohm oscillations in
Normal-Metal rings
[Physical Review Letters, 54, 2696 (1985)]
(a)
Magnetoresistance of the ring measured at T  0.01 K .
(b)
Fourier power spectrum in arbitrary units containing
peaks at h / e and h / 2e . The inset is a photograph
of the larger ring. The inside diameter of the loop is
784 nm, and the width of the wires is 41 nm.
UCF
4
Illustration 5:
Fluctuation independent of the background conductance
[Physical Review B, 35, 1039 (1987)]
Comparison of aperiodic magnetoconductance fluctuations
in two different systems.
(a) g (B ) in 0.8 - m - diam gold
ring in the previous illustration (the rapid Aharonov-Bohm
oscillations have been filtered out).
(b) g (B ) for a
quasi-1D silicon MOSFET (illustration 2). Conductance is
measured in units of e 2 / h , magnetic field in Tesla.
Note
the 3 order-of-magnitude variation in the background
conductance while the fluctuations remain order unity.
UCF
5
Illustration 6:
Spin degeneracy and Conductance Fluctuations in
Open Quantum Dots
[Physical Review Letters, 86, 2102 (2001)]
A sample of conductance fluctuations in one of the 8 m 2
dots, as a function of Bperp and at (a) zero parallel field
Bpar  0 ,
and (b)
Bpar  4 T .
The variance of fluctuations in (b)
is reduced from that of (a) by a factor of 4.
A factor of 2 is
expected for Zeeman splitting of spin-degenerate channels.
The larger suppression might arise from a field-dependent
spin-orbit scattering.
UCF
6
A summary about the major features of UCF
Provided that the incoherent length l is larger than any of
the sample dimensions (or the temperature is sufficiently
low), then
e2
1. G 
independent of the degree of disorder and the
h
size of the sample
2. G fluctuations are not time-dependent noise
3. G fluctuations is a deterministic, albeit fluctuating,
functions of its arguments for a given realization of the
impurity configuration.
UCF
7
In the regime where Universal Conductance Fluctuations
are important, the conductance is most naturally given by
the Landauer-B ü ttiker formula, which relates the
conductance to the transmission coefficients of modes at the
Fermi energy. We present, in the following, a simple
derivation of the Landauer-Büttiker formula in a two-probe
single-channel case.
A conductor (represented by the barrier in the middle) is
connected via leads to two electrodes. The electrodes are
particle and heat reservoirs within which incoherent and
inelastic processes occur.
They are characterized by
chemical potentials  L and  R . Here we assume L  R .
I  e
k
dn
(L  R )  F  T ,
dE
m
dn
m

dE 2 2 k F ,

Conductance
UCF
G
e
I   ( L  R ) T
h
I
[(  L   R ) /(  e)]

e2
G T .
h
8
Note that G   even if the conductor is a perfect
conductor, with T  1 .
The resistance comes from the
contact resistance because the incoherent processes occur
within the electrodes.
When the conductor has a finite width, the electrons can
traverse from left to right via more than one channel. The
corresponding Landauer-Büttiker formula is
e2 N
G
t

h  ,  1
2
where  ,  are, respectively, the incoming (say on the left)
and the outgoing (on the right) channels.
N
is the number of channels given approximately by
N  ( Lk F ) d 1 where L is the transverse dimension.
Here we have not included the spin factor 2 just for the sake
of simpler presentation.
[R. Landauer, IBM J. Res. Dev. 1, 223 (1957);
M. Büttiker, Physical Review Letters, 57, 1761 (1986)]
Equipped with this expression for the conductance G , we
turn to the discussion about the phenomena of UCF.
UCF
9
The following discussion focuses upon the diffusive regime:
when L, l  le .
A theoretical understanding to UCF has to invoke the
technique of Green’s function [Ref.: P.A. Lee, A.D. Stone,
Physical Review Letters, 55, 1622 (1985)].
Instead, we
present a simplified explanation due to the insightful
heuristic argument of Lee [Ref.: P.A. Lee, 140A, 169 (1986)].
Suppose that we start out with considering the fluctuation of
2
each transmission probability t .
M
t   A (i )
i 1
where A (i ) is the transmission amplitude from channel
 to
 , and there are
M such Feynman paths.
The
presence of disorder causes the important Feynman paths to
be that of diffusive motion and covering much of the sample.
There should be many such important Feynman paths.
Assuming that A (i ) are independent complex random
variables (We can always group the Feynman paths into sets
such that the paths are correlated only within a set. Then
we use the set labels as our new labels for uncorrelated
2
paths.), we can calculate the fluctuation in t :
 t
UCF
2

t
4
 t
2
2
10
t
4
*
*
  A
(i ) A ( j ) A
(k ) A (l )
ijkl
*
*
  A
(i ) A ( j ) A
(k ) A (l )
ijkl
*
*
  A
(i ) A (l ) A
(k ) A ( j )
ijkl
2
 2 t
  t
2
2
 t
2
To estimate the lower bound G' of G , we further assume
that different channels are uncorrelated. Thus we have
e2
G' 
h
Now we need to estimate
t
N 2 t
2
2
:
Ohm' s Law gives G  Ld 2 ;   ( e 2 / h )k F l ; N  ( Lk F ) d 1
2
2

l 1
LN

t

e2 l
G' 
h L
which is much smaller than the observed results.
2
Thus the correlation in the transmission probability t for
different channels  and  may not be neglected.
UCF
11
Since the contributions to G are from transmitting channels,
the reflection coefficients may then have much smaller
correlations (i.e. more easily averaged).
Another reason
supporting this is that the reflection would seem to be
dominated by only a few scattering events. Whereas t
must involve multiple scatterings in order to traverse the
sample, and sequence of scattering events might be shared
by different channels, therefore it is not a surprise to see
more correlations among channels in the transmission.
e2
G
h
t


e2
 G
h
2
e2

h
N

2
1

r

   
 1 
 1

N

2
N

r


 



var( G)  G 2  G
 e2 
 var( G )   
h
2
2

2
N

r


 



2

 e2 
   var   r
h
 
 e2 
  
h
2
 e2 
  
h
2
2
N   r
2





 
 var r

2
2
2
we have assumed that r and r '  ' are uncorrelated.
UCF
12
 .
2
 e2  2
 var( G )    N var r
h
 
var r
r
4
2
4
r
 r
2
2
2
*
  B
(i ) B ( j ) B* (k ) B (l )
ijkl
 2 r
2
 
 var r
2
2
r
2
2
 2
 N r

e
 G  
h

 N r

2
UCF
2
e
 var( G )  
h
2
2
2
2
13
Estimation of the reflection probability coefficient
r
2
:
From the conservation of current, we have
1   t   r
2


 1  N t

r
2
2
1
N

2
 N r
2
1  N t


2


According to Landauer-Büttiker formula:
e2
G
h
 t

2
e2 2

N t
h
2
and the order of magnitude of G can also be estimated from
the Drude conductance: (for the 2D case)
 ne 2 V  1
G  W
m L  V

W ne 2 W e 2 k F2
 G

L m
L m 4 2
W e2 kF l e2   l 
 G
  N
L h 2
h  2L 
Therefore
t
UCF
2

l
2 NL
14
 
var r
2
r
2 2
1

N
 e2 
 var( G )   
h
2
  l 
1  2 L  

2
  l
1  L 
e2
l
 G   O 
h
 L
2
2
The zero temperature conductance has a variance (e / h )
independent of l (i.e. disorder of the sample) or L (the
size of the sample) as long as we are in the diffusive regime
and the mesoscopic regime ( l  L ).
The correction is of
order l / L .
The numerical prefactors have to be determined by
diagrammatic analysis, the result is
g s g v 1 / 2 e 2
G 

C
2
h
where
C is a constant of order one and weakly dependent
on the shape of the sample.
UCF
15
Typically:
C  0.73
C
gS
W
L
in a narrow channel with L  W
if W  L .
is the spin degeneracy. And if the spin degeneracy is
lifted, such as by a magnetic field, then g s will be
replaced by
gs .
The applied magnetic field must
be large enough so that spin up and spin down
electrons at the Fermi energy will have sufficient
energy difference to render their reflection processes
become uncorrelated (or statistically independent).
gv
is the valley degeneracy.
  1 in a zero magnetic field when time-reversal
symmetry holds.
  2 when time-reversal symmetry is broken by a magnetic
field.
UCF
16
Nonzero temperatures (T  0) :
Two length scales l and lT are of importance here.
First, the phase coherence length
l  D 
are of
importance because it varies with temperature.
Second, the thermal length lT  D / k BT characterizes the
effect of thermal averaging.
Together, these two effects bring in partial restoration of
self-averaging.
In the following, we limit our discussion to the 1D
(W  l  L) regime.
Case 1:
l  lT
In this case we can neglect the thermal averaging.
The system can be thought of as subdivided into
uncorrelated segments of length l .
The conductance fluctuation of each segment will be of order
e 2 / h , according to the previously discussed UCF.
Furthermore, the segments are in series and their resistances
add according to Ohm’s law.
UCF
17
For an individual segment (of length l ), the resistance is
R1
denoted by
G1  1/ R1 .
and the corresponding conductance
2
Assuming that R1  h / e  25.8 k , we have
R1 
1
1  G1 

1 

G1  G1
G1 
G1 
var R1  
var R1   R1
1
G1
4
4
var  G1  G1 
 1
var    R1
 R1 
4
 e2 
 
h
2
R of the system is
The average of the total resistance
L
R    R1
l 
 
 L
 L
var R     var R1     R1
l 
l 
 
 
R
L
 
scales as  l 
 
4
 e2 
 
h
2
1/ 2
for L  l
How about G ?
UCF
18
From
G  1 / R , and assuming that
R  h / e 2 , we have
1
1  G 
R

1 

G  G
G 
G 
 G 
  R 4 var( G )
 var( R)  var 
 G 2


 var( G )  R
4
 l 
 var( G )    R1
L
4
4
var( R )
L
   R1
l 
 
 l 
 G  constant   
L
G scales as
UCF
 l 
 
L
3/ 2
3/ 2
4
 e2 
 
h
2
e2
h
in the regime when l  lT .
19
Case 2: l  lT
Two interfering Feynman paths, traversed with an energy
difference E , have to be considered as uncorrelated after a
time t1 , if the acquired phase difference t1E /  is of order
unity. The distance diffused by the electrons in time t1 is
L1  Dt1 ~ D / E .
In this case the total energy interval k BT around the Fermi
level that is available for transport is divided into
sub-intervals of width E c (l )   /   . Phase coherence is
maintained in each such sub-intervals.
The reason for
doing this is that at finite temperatures we are actually doing
the energy averaging.
Suppose there are N such sub-intervals:
 D 

N  k BT / Ec (l )  
 lT 
2
2
 D 
l 

   
l 
 l 
 T
  
2
then the var( G1 ) in the previous case will be affected in the
following way:
var( G1 )  N 2 var( G1i ) if all N sub - intervals were coherent
var( G1 )  N var( G1i ) if all N sub - intervals were incoherent
where G1i is the conductance due to the i -th sub-interval.
UCF
20
var( G )  R
4
var( R )  R
4
L
  var( R1 )  R
l 
 
4
l 
 var( G )     R1
 L
4
4
L
  R1
l 
 
4
var( G1 )
 L
4
    R1 var( G1 )
l 
 
4
 l 
 var( G )    var( G1 )
L
Therefore G would be reduced by a factor
e 2 lT  l 
G  constant     
h l  L 
l
1
 T
N l
3/ 2
1/ 2
e 2 lT l
G  constant   3 / 2
h
L
UCF
21
Experiment vs Theory
[Physical Review Letters, 56, 2865 (1986)]
(W.J. Skocpol, P.M. Mankiewich, R.E. Howard, L.D. Jackel, D.M. Tennant, and A.S. Stone)
l  0.25 m
Normalized correlation
function vs. displacements of
magnetic field and gate voltage.
Measured vs. predicted rms
fluctuation amplitude in units of
e 2 / h for many data sets with a wide
range of experimental parameter
values. (Open symbols, 4.2 K; solid
2 K.)
UCF
22
Measured vs. predicted magnetic field correlation half-width.
The 1D and 2D theoretical predictions (for the case l  lT )
1
2
 max (l , W )   l 
 

L

 L
g  
2.4(h / e)
Bc 
l  min (W , l )
UCF
23
Experiment vs. Theory
[Physical Review Letters, 58, 2347 (1987)]
(W.J. Skocpol et al, and A.D. Stone)
Amplitude of resistance fluctuations as a function of probe spacing
for the long and short devices: showing distinctly different
 l 
dependence (i.e. G varies as  
L
UCF
2
 e2 
  in the L  l regime).
h
24