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Mesoscopic Anisotropic
Magnetoconductance Fluctuations in
Ferromagnets
Shaffique Adam
Cornell University
PiTP/Les Houches Summer School on Quantum
Magnetism, June 2006
For details: S. Adam, M. Kindermann, S. Rahav and P.W. Brouwer,
Phys. Rev. B 73 212408 (2006)
Mesoscopic Anisotropic
Magnetoconductance Fluctuations in
Ferromagnets
PiTP/Les Houches Summer School on Quantum
Magnetism, June 2006
Mesoscopic Anisotropic
Magnetoconductance Fluctuations in
Ferromagnets
PiTP/Les Houches Summer School on Quantum
Magnetism, June 2006
Quantum
Magnetism
Mesoscopic Anisotropic
Magnetoconductance Fluctuations in
Ferromagnets
PiTP/Les Houches Summer School on Quantum
Magnetism, June 2006
Quantum
Magnetism
Electron Phase Coherence
Mesoscopic Anisotropic
Magnetoconductance Fluctuations in
Ferromagnets
PiTP/Les Houches Summer School on Quantum
Magnetism, June 2006
Quantum
Magnetism
Electron Phase Coherence
Ferromagnets
Mesoscopic Anisotropic
Magnetoconductance Fluctuations in
Ferromagnets
PiTP/Les Houches Summer School on Quantum
Magnetism, June 2006
Quantum
Magnetism
Electron Phase Coherence
Ferromagnets
Phase Coherent Transport in Ferromagnets
Mesoscopic Anisotropic
Magnetoconductance Fluctuations in
Ferromagnets
PiTP/Les Houches Summer School on Quantum
Magnetism, June 2006
Quantum
Magnetism
Electron Phase Coherence
Ferromagnets
Phase Coherent Transport in Ferromagnets
• Motivation (recent experiments)
• Introduction to theory of disordered metals
• Analog of Universal Conductance Fluctuations in nanomagnets
Motivation:
Picture taken from Davidovic group
Cu-Co interface are good contacts
 ~ 5 nm
L ~ 30 nm


C ( )   G(m)G(
 
cos( )  m  m
  L , disordered regime


 2
C ( )   G(m)G(m)    G(m) 
 
Physical System we are Studying [2]



C ( )   G(m)G(m)    G(m)  2
 
cos( )  m  m
10
  [0,  / 2]
B(T)
•Aharanov-Bohm
contribution
• Spin-Orbit Effect
- 10
V ( mV)
Data/Pictures: Y. Wei, X. Liu, L. Zhang and D. Davidovic, PRL (2006)
Physical System we are Studying [3]


 2
 c is correlatio n angle
C ( )   G(m)G(m)    G(m) 
 
cos( )  m  m
Aharanov-Bohm
Spin-Orbit Effect
contribution
c ~  2
 
 ~ m  a
 c ~  10
VSO
  
~ (k  k )  S
  
~ (k  k )  m
Introduction to Phase Coherent Transport
Smaller and colder!
V
I
A
Image Courtesy (L. Glazman)
Sample dependent fluctuations are
reproducible (not noise)
Ensemble Averages
Need a theory for the mean <G> and
fluctuations <GG>
[Mailly and Sanquer, 1992]
Introduction to Phase Coherent Transport [2]
Electron diffusing in a dirty metal
Impurities
  path
 ~ path, flux
Electron
A  C e
i

PA B |
PA B
A |    A A
  | A |    A A
*
2
,
2
*

Classical Contribution
Quantum Interference
Diffuson
Cooperon
Introduction to Phase Coherent Transport [3]
Weak Localization in Pictures
Introduction to Phase Coherent Transport [3]
Weak Localization in Pictures
Introduction to Phase Coherent Transport [3]
Weak Localization in Pictures
For no magnetic field, the phase depends
only on the path.
Every possible path has a twin that is
exactly the same, but which goes
around in the opposite direction.
Because these paths have the same flux
and picks up the same phase, they can
interfere constructively.
Therefore the probability to return to the
starting point in enhanced (also called
enhanced back scattering).
In fact the quantum probability to return
is exactly twice the classical probability
Introduction to Phase Coherent Transport [4]
Weak Localization in Equations
A  C e
i

PA B |
2
A |   ,
PA B   |
2
A |
*
A A
   A A
*
  path

Classical Contribution Quantum Interference
Cooperon
Diffuson
A  C  e
i
 ~ path, flux

PA B |
2
|
A   ,
PA B   |
PA B
  reversepath
A |
  | A |
2
*
A A
  A A     A A
  | A |    A A
*

2
2
,
*

,
*
Universal Conductance Fluctuations
[Mailly and Sanquer (1992)]
[C. Marcus]
Theory: Lee and Stone (1985), Altshuler (1985)
Review of Diagrammatic Perturbation
Theory (Kubo Formula)
Conductance G ~
Diffuson
1
 ( Dq 2  i )
Cooperon
1
 ( Dq~ 2  i )
Cooperon defined similar to Diffuson upto normalization
Calculating Weak Localization and UCF
G
G
Weak Localization
 GG  c
Universal Conductance Fluctuations
Calculation of UCF Diagrams
2
e 
 GG    
h
2 2
e 
  
 h
2
3
Tr[ JDJD  JCJC ]
2
2 Lz
3
1
 4 
 4 n 2

C ,D
2
n
 
Sum is over the Diffusion Equation Eigenvalues scaled by Thouless Energy
L 
n  nz2  nx2  z 
 Lx 
2

2  Lz
 ny
L
 y
2
 iE
 
,

ET

nz  1,2,
nx , n y  0,1,2,

ET  D
 Lz



2
Quasi 1D can be done analytically,
and 3D can be done numerically: Var G = 0.272
1 4
n n 4  90
e
 GG   
h
2



2
1
2
C ,D 15 ~ 15
x 4 for spin
Effect of Spin-Orbit
(Half-Metal example) [1]
Energy
Ferromagnet
Fermi Energy
Spin Up
DOS(E)
Spin Down
DOS(E)
Half Metal
Effect of Spin-Orbit
(Half-Metal example) [2]
Without S-O
H
Vk k '
With S-O
V
 Vq Vq '  
 (q  q' )
2 V
1
 ( Dq 2  i )
1
 ( Dq 2  i )
 
k k '
 iV
so  
k k '
 Vqso Vqso'  
  
m  (k '  k ) / k F2
 (q  q' )
2 so V
1
 

1  m  m' 

  Dq 2  i 
 so 

1
 
 2
1  m  m' 

  Dq  i 
 so 

NOTE: For m=m’, Spin-Orbit does not affect the Diffuson (classical motion)
but large S-O kills the Copperon (interference)
Calculation of C(m,m’) in Half-Metal
Without S-O
With S-O



C ( )   G(m)G(m)    G(m) 2
 GG 
1
m
m
m’
m
(   )
( ) 
 ( Dq 2  i )
m’
=
m’
=
=
1

  Dq 2  i 

1 
n n 4  90
4
1  cos  

 so 
1
4
n (n2  A)2  4 x 4 [2 x coth( x) x 2 sinh 2 ( x)]
A
1 cos( )
ET so
x 
1 cos( )
ET so
Results for Half-Metal
D=1, Analytic Result
3  e2 
C ( )   
2 h 
2
  1  cos 
 F  
ET s 0
 
  1  cos 
  F 
 
ET s 0
 
 2 x coth( x) x 2 sinh 2 ( x)
F ( x) 
x4
D=3, Done Numerically




     , changes definition of , 
We can estimate correlation angle
for parameters and find about five
UCF oscillations for 90 degree change
Full Ferromagnet
Half Metal
H 
V
 
k k '
 iV
so  
k k '
 Vqso Vqso'  
  
m  (k '  k ) / k F2
 (q  q' )
2 so V
Ferromagnet
z
Vk  k '  E z 
 

 iV so k k ' (m z  e1 x  e2 y )  (k '  k ) / k F2
1
 
 2
1  m  m' 

  Dq  i 
 so 

n
n 2  A( ) are Eigenvalue s
of Diffusion Equation
A
1 cos 
ET so
n

2
 a ( ) are Eigenvalue s
of 2  2 Diffusion Equation
Results for C(m,m’) in Ferromagnet
2
3  e2 
C ( )    F a ( )   F a (   ) 
2 h 
 2 x coth( x) x 2 sinh 2 ( x)
F ( x) 
x4
Limiting Cases for m = m’
m=m’
SO
C
D
spin
Total
-
1/15
1/15
4
8/15
Half Metal
No
1/15
1/15
1
2/15
Half Metal
Strong
0
1/15
1
1/15
Ferromagnet
Weak
1/15
1/15
2
4/15
Ferromagnet
Strong
0
1/15
1
1/15
Normal Metal
Conclusions:
Showed how spin-orbit scattering causes Mesoscopic Anisotropic
Magnetoconductance Fluctuations in half-metals (This is the analog of UCF
for ferromagnets)
This effect can be probed experimentally
10
VSO
  
~ (k  k )  S
  
~ (k  k )  m
5 c ~ 
2
B(T)
- 10
V ( mV)
Backup Slides
Magnetic Properties of Nanoscale
Conductors
Shaffique Adam
Cornell University
Backup Slide


 2
 c is correlatio n angle
C ( )   G(m)G(m)    G(m) 
 
cos( )  m  m
Aharanov-Bohm
Spin-Orbit Effect
contribution
0
c ~
~ L 2

c ~  2
 so 
c ~  so ET ~
~ L1
 L
 c ~  10
Backup Slide
Density of States quantifies how closely packed are energy levels.
DOS(E) dE = Number of allowed energy levels per volume in energy window
E to E +dE
DOS can be calculated theoretically or determined by tunneling experiments
DOS(E)
Energy
Fermi Energy
Fermi Energy is energy of adding one more electron to the system (Large energy because
electrons are Fermions, two of which can not be in the same quantum state).
Backup Slide
Energy
DOS(E)
Fermi Energy
Spin Up
Spin Down
Energy
Fermi Energy
DOS(E)
DOS(E)
Backup Slide
• Magnetic Field shifts the spin up and spin
down bands
Energy
Ferromagnet
Fermi Energy
Spin Up
DOS(E)
Spin DOS
Spin Down
DOS(E)
Half Metal
Backup Slide
Weak localization (pictures)
For no magnetic field, the phase depends
only on the path.
Every possible path has a twin that is
exactly the same, but which goes
around in the opposite direction.
Because these paths have the same flux
and picks up the same phase, they can
interfere constructively.
Therefore the probability to return to the
starting point in enhanced (also called
enhanced back scattering).
In fact the quantum probability to return
is exactly twice the classical probability
Backup Slide
Weak localization (equations)
A  C e
i

PA B |
PA B
  path
i
*

Quantum Interference
Diffuson
 ~ path, flux
A  C  e
,
2
Classical Contribution
  reversepath
Cooperon

PA B |
2
|
A   ,
PA B   |
PA B
A |    A A
  | A |    A A
*
2
A |
  | A |
2
*
A A
  A A     A A
  | A |    A A
*

2
2
,
*

,
*
Backup Slide
Weak Localization and UCF in Pictures
<G>
<G G>
Backup Slide
Backup Slide
Backup Slide
Backup Slide
Introduction to Quantum Mechanics
Energy is Quantized
Wave Nature of Electrons (Schrödinger Equation)
Wavefunctions of electrons in the
Hydrogen Atom (Wikipedia)
Scanning Probe Microscope Image of
Electron Gas (Courtesy A. Bleszynski)