Weak Localizations:
This is a physical phenomena that gives rise to an early
recognition of dephasing time
 .
The system of concern is a
disordered system due to the presence of impurities positioned
randomly throughout the entire sample.
Magnetoresistance of a Mg film (d = 8.4 nm) as a function of the
magnetic field H. [Physics Reports, 107, 1 (1984), G. Bergmann]
According to the classical theory, the resistance would be
completely field independent because the product  c  is much
smaller than 1. Furthermore, the magnetoresistance would
increase with the magnetic field H , and the increment of order
( c  ) 2 . The classical theory of the resistance could not explain
the behavior in the curves.
Weak Localization
19-1
Conduction electrons propagate diffusively in the system:
bumping against the random impurities on their way.
In
contrast to the classical diffusion, the electrons still maintain
their capability to undergo interference.
That is, they maintain
their phase coherence: they have kept track of their phases, even
though their phases have changed a lot during their many
collisions with the impurities.
The electrons are said to
undergo quantum diffusion.
Diffusion path of a
conduction electron.
The starting point is
O. The solid and the
dashed lines denote
a pair of timereversed paths for
the
electron
to
return to its starting
point.
Key result:
Since the amplitudes add coherently, the quantum diffusion
probability to return to the origin is twice as in classical diffusion.
Weak Localization
19-2
Possible random paths between positions A and B due
to elastic scatterings:
Probability PAB of going from A to B is
2
PAB 
 Ai   Ai   Ai A*j
2
i
i j
i
where Ai is the probability amplitude for the i-th
path. The term
AA
i j
i
*
j
is the interference term.
Since Ai contains the random phases due to the
random positions of the impurities involved in the i-th
path, one might think that
Weak Localization
AA
i j
i
*
j
averages to zero.
19-3
Dropping the term
AA
i j
i
*
j
reproduces the classical
transport results from Boltzmann equation, and
Drude model.
But for the path that involve position O, there are
actually two time-reversal looping paths starting and
ending at O.
(In the momentum space, we can say


k

k
that the electron starts out with
and ends at
)
Since the impurities involved in these two paths are
the same, the interference term between them do not
average to zero.
It turns out that the probability of finding the
electron at O is twice that of the classical result
because of the quantum interference.
Therefore, the
probability of finding the electron at B is decreased,
indicating a drop in the conductance.
We will discuss in more detail in the following using
the momentum space point of view.
Weak Localization
19-4
Coherent backscatterings in disordered conductors:
Interference between two time-reversal scattering processes
An electron k
is scattered to  k
via time-reversal
scattering sequences A or A .
Sequence A ：
k  k1  k2  k3   k
Sequence A ：
k  k1   k2   k3     k
A’
A”
Weak Localization
19-5
Sequence A and sequence A have the momentum
transfer occurs in opposite order.
The scattering amplitudes A  A  A if the system has
time reversal invariance.
Total intensity for backscattering is ：
A  A  B  B    
2
2
2
 4 A  4 B  
In contrast, the classical (Boltzmann) result is
A  A  B  B    
2
2
2
2
2
2
 2 A  2 B  
Weak Localization
19-6
Weak Localization
19-7
Z ：assumed number of intermediate states available for
elastic scattering （within a thin shell around the Fermi
surface）
An ( k   k ) ：probability amplitude for n
scattering processes sequence A
An ( k   k )  An ( k   k ) 
1
Z n/ 2
i 
e 
Interference intensity is A A  c.c.  2/ Z
*
n
Total interference intensity for n event sequence：
 2
 n
Z
 n 1  1  1
Z
 2  Z

 
For a n event sequence, the electron are scattered into n-1
n 1
intermediate states, therefore there will be Z / 2 distinct
n event sequence.
The factor 1 / 2 is to correct for double
counting.
Weak Localization
19-8
Estimation for Z the number of intermediate states:
   / 
shell thickness
where
k 
 

v F lel
el is the elastic mean free path.
d
Z （for one spin state only）
1
   L
Z  2

 el  2
2

  A
Z   2 kF

2
el  (2 )

3

  
Z   4 kF2

3
el  (2 )

Weak Localization
19-9

 
Scattering from k   k  q :
Phase difference
 i 
where

( Ei  En  i ) 

(
 
 i   q  v Fi .
1  2
2
) pi    pi  q  

2m 
Total phase difference
n


 q   v Fi 
 i 1

 2   2


2
2
n
2
q

v



Fi
i 1

 v F2  / d q 2 ( n )
 D q2t
A A*  A A e i 
Total interference intensity for
n
scattering events
sequence：
1 i  ( q )
e
Z
Weak Localization
19-10
e
i
e
Effective number of states near
N coh 
d
(2 )
d
 Dq2t
 k that are still coherent：
d
d
qe
 Dq 2 t
Coherent intensity due to backscatterings
I coh
For
N coh

Z
d  2：
I c o h

 kF
e tl
for
  t  


0

 (  0)   p e  t /  dt   (  p )

1
 p 1 
  kF
ne 2
 (  0) 
m
el

 kF
el t
dt
 
ln 
 

1
1


  kF
el
 
ln 
 

ne 2
e2

 2 ln
m

2
Weak Localization
19-11

k
Contribution of the electron state
to the momentum as a
function of time: the original state and its momentum decay
exponentially within the time constant  0 .
But an echo

with the momentum  k is formed with depends on time as
1/ t .
This echo reduces the contribution of the electron to
the current and yields a correction to the resistance which is
proportional to ln(   /  ) .
Weak Localization
19-12
Dephasing time due to magnetic field:

: Phase difference between the two complementary
waves due to the presence of a magnetic field
H.
The probability amplitudes become
A'  A'0 e
i(e / )

 
Adr
A"  A"0 e
c
;
i(e / )

 
Adr
cc
such that
A' A"*  A'0 A'*0 e
where
  BA is
i ( 2e / )

 
Adr
c
 A'0 A'*0 e i ( 2e /  )
the magnetic flux (out of the page)
enclosed by the path.
  ( 2e / ) ;
  ( 2 Dt ) H
  1
H
Weak Localization


4eDH
19-13
Magnetosresistance of a Mg film (d=8.4 nm) as a function of
H. The points represent the experiment results and the solid
curves are fitted with the theoretical weak localization
expression.
The values for   are obtained from the
fitting.
Weak Localization
19-14
p-GaAs well on
(311)A
substrate:
2.5  k F l  5
(a) The magnetoconductivity
 xx just on the metallic side of the
transition for temperatures of 147, 200, 303, 510, and 705 mK. (b) A
plot of 1 /  
versus temperature for densities close to the
metal-insulator transition. Solid symbols are data obtained from this
study; open symbols are data from Si MOSFETs, Ref. [15].
[Ref.: Simmons et al Phys. Rev. Lett. 84, 2489 (2000)]
The solid curves in (a) are theoretical fit:
 ( B ) 
 e2
h
  1 Bo 
 1 B 
,
    
 
2
B
2
B





digamma function and Bn n 
Weak Localization
where  ( x ) is the

.
4eD
19-15
Effects of spin-orbit scattering:
Magnetoresistance of the same Mg film (in the previous
second graph) after superposition with 0.25 atomic layers of
Au. The points represent the experimental results. The
solid curves are calculated with the same set of   as in the
previous second graph but with a new time constant
Weak Localization
 so .
19-16

 00
 1 H 
 1 H  1  1 H  1  1 H 
      1      2      3      4 
 2 H  2  2 H  2  2 H 
 2 H 
where  00 
e2
2 2 
,
 ( x ) is digamma function, and
H1  H 0  H so  H s
H 2  ( 4 / 3) H so  ( 2 / 3) H s  H 
H 3  2H s  H
H 4  ( 2 / 3) H s  ( 4 / 3) H so  H 
Weak Localization
19-17
A physical picture for the spin-orbit effects:
[Ref: G. Bergmann, Solid State Commun. 42, 815 (1982)]
The form of the spin-orbit interaction

 
  
 
Vkk ' 1  i k  k '  s  Vkk ' 1  iK  s

yields a rotation of the spin s by angles K i around the

i -th axis. When the particle is scattered from state k s

to  k
s' , the accumulative effects on the spin add up to a
finite rotation R such that
s'  R s .
For the complementary (time-reversed) scattering process,


k
s" where
the final state becomes
s'  R 1 s .
Therefore the interference term will contain an additional
2
factor s" s'  s R s .
 cos(/2)e i(  )/2
R
i (  ) / 2
isin ( / 2)e
isin ( / 2)e i (  ) / 2 

cos( / 2)e i (  ) / 2 
where  ,  , and  are the angles rotating about the
original axes in the order z ,
x and then z axis.
[Ref.: Feynman Lecture III, p.6-12]
Weak Localization
19-18
2
2
 i (  )
The factor s" s'  sin ( / 2)  cos ( / 2) e
for spin up (+ sign) s
state or for spin down (- sign) s .
Case 1: If there is no spin flip, then all angles are zero.
s" s'  1 , we have the regular weak localization.
Case 2: If the spin-orbit coupling is strong, the spin state
s diffuses on the surface of the sphere in the
following diagram.
The orientation of the final
spin states are statistical.
s" s'
ave

Therefore the average
1
2
and we have the anti-localization.
Weak Localization
19-19