Part 4

Report
Spintronics: How spin can act on charge carriers and vice versa
Tomas Jungwirth
Institute of Physics Prague
University of Nottingham
Two paradigms for spintronics
“Mott“ non-relativistic two-spin-channel model of ferromagnets
I
I
Mott, 1936
“Dirac“ relativistic spin-orbit coupling
I
Dirac, 1928
I
SHE & STT switching
SOT switching
Ralph, Buhrman,et al., Science ‘12
Miron et al., Nature ‘11
-We see (anti)damping-like torque
-We also see (anti)damping-like torque
-SOT is field-like so we exclude it
-SOT is field-like but maybe there is some
(anti)damping-like SOT as well
- non-relativistic STT in metals is
dominated by the (anti)damping torque
Ohmic “Dirac“ device: AMR
Kelvin, 1857
Magnetization-orientation-dependent scattering
Spin-orbit coupling
Spin-orbit coupling
Extraordinary magnetoresistance: AMR, AHE, SHE, SOT.....
Ordinary magnetoresistance:
response to external magnetic field
Acting via classical Lorentz force
Extraordinary magnetoresistance:
response to internal quantum-relativistic
spin-orbit field
anisotropic
magnetoresistance
B
Lord Kelvin 1857
_ _ _ _ _ _ _ _ _ _
_
FL
+++++++++++++
I

AMR

s

1
2
__
M
V
( ij   ji )


 ij ( M )   ji (  M )
FSO
I
ordinary Hall effect 1879
V

AHE

A

1
2
anomalous Hall
effect 1881
( ij   ji )

k
t
df n , k
dt

 E n , k

k
Classical Boltzmann equation
 f n , k
 
 f 0 ( E n , k )
 e E  v 0 n , k
 

t
E n ,k
n

d k
d
 ( 2 )
d
W n , k , n  , k  ( f n , k  f n  , k  )
Non-equilibrium distribution function
g n , k  f n , k  f 0 ( E n , k )
Steady-state current in linear response to applied electric field
j i   ij E j  e  
n

d
d k
( 2 )
d
i
v 0 n , k g n , k ( E j )
Linear response: g linear in Ej
Steady-state solution for elastic (impurity) scattering
Steady-state solution for elastic (impurity) scattering
if
g(i,k)=
Constant quasi-particle relaxation time solution
Steady-state solution for elastic (impurity) scattering
if
is isotropic: depends on | - ’|
g(i,k)=
Transport relaxation time solution: back-scattering dominates
Steady-state solution for elastic (impurity) scattering
if
is anisotropic: depends on k, k’
No relaxation time solution
AMR in Rashba 2D system

AMR

s

1
2
Rashba Hamiltonian
( ij   ji )
Eigenspinors
AMR in Rashba 2D system
QM: 1st order Born approximation
  i k  r  i k  r
  dr e
 (r )e
 const .
isotropic

V  1  ( r )


V  M x   x  (r )
anisotropic

M
AMR in Rashba 2D system
Heuristic picture from back-scattering matrix elements

V  M y   y  (r )

V  M x   x  (r )
Rashba SOI
Rashba SOI
current

M

M
Back-scattering  high resistivity
No back-scattering  low resistivity
Anomalous Hall effect in FMs
1881
Polarimetry of electrons in FMs
Spin Hall effect in PMs
jc
Mott, N. F. Proc. R. Soc. Lond. A 1929
Dyakonov and Perel 1971
Kato, Awschalom, et al., Science‘04
Electron spin-dependent scattering
off Coulomb field of heavy atoms due
to spin-orbit coupling
Polarimetry of high-energy electron
beams in accelerators
Wunderlich, Kaestner, Sinova, TJ, PRL‘05
Electron spin-dependent scattering
off Coulomb field of dopands in a
semiconductor due to spin-orbit
coupling
Proposal for electrical spin injection by the spin Hall effect
and electrical detection by the inverse spin Hall effect
jc
Hirsch PRL‘99
js
Proposal for electrical spin injection by the spin Hall effect
and electrical detection by the inverse spin Hall effect
jc
js
- index
Hirsch
PNAS‘05
Intrinsic anoumalous Hall effect
in (Ga,Mn)As
FM (Ga,Mn)As
TJ, Niu, MacDonald, PRL’02
Theoretical proposal of
intrinsic spin Hall effect
Non-magnetic GaAs
Murakami, Nagaosa, & S.-C. Zhang, Science’03
Proposed detection by polarized
electroluminescence
Sinova, TJ, MacDonald, et al. PRL’04
Proposed detection by magneto-optical
Kerr effect
Magneto-optical Kerr microscopy
Extrinsic SHE Kato, Awschalom, et al.,
Science‘04
Edge polarized electro-luminescence
Intrinsic SHE Wunderlich, Kaestner, Sinova,
TJ, PRL‘05
Optically generated spin current
Optically detected charge accummulation due to iSHE
fs pump-and-probe: iSHE generated before first scattering in the intrinsic GaAs
 intrinsic iSHE
Zhao et al., PRL‘06
Werake et al., PRL‘11
AHE and SHE

AHE

A

1
2
( ij   ji )
AHE and SHE
Skew scattering SHE
Mott (skew) scattering SHE
 ll '
(3a )
  ij   
 ll '   ij  
(2)
ji
ji
SHE
AMR
Skew scattering AHE (SHE)
 ll ' : not constant, not isotropic, not even symmetric  no relaxation time solution
(3a )
Approximation:
Skew scattering AHE (SHE)
Spin orbit torque
M
Ie
Field-like SOT

d k
d
si   
n
( 2 )
 0 n , k g n , k ( E j )
i
d

s
E=Ex ^x
Compare with AMR or skew-scattering SHE

d k
d
ji  e  
n
( 2 )
d
i
v 0 n , k g n , k ( E j )
Field-like SOT

d k
d
si   
n
( 2 )
d
 0 n , k g n , k ( E j )
i

s
E=Ex ^x
 (r)
  i k  r  i k  r
  dr e
 (r )e
 const .
isotropic

Field-like SOT

d k
d
si   
n
( 2 )
d
 0 n , k g n , k ( E j )
i
 (r)
  i k  r  i k  r
  dr e
 (r )e
 const .
isotropic
g(i,k)=

Field-like SOT

s
E=Ex ^x

s  
1  me  tr E x
2

3
yˆ
H
ex
H ex  H R

dM
dt

J ex



M  s
 
 J ex M  
Intrinsic anoumalous Hall effect in FMs
FM (Ga,Mn)As
TJ, Niu, MacDonald, PRL’02
Intrinsic spin Hall effect in PMs
Non-magnetic GaAs
Murakami, Nagaosa, & S.-C. Zhang, Science’03
Sinova, TJ, MacDonald, et al. PRL’04
Wunderlich, Kaestner, Sinova, TJ, PRL‘05
Werake et al., PRL‘11
Linear response I.
Boltzmann theory : non-equilibrium distribution function and equilibrium states
Linear response II.
Perturbation theory: equilibrium distribution function and non-equilibrium states
H 
pˆ
  i t
  Ee

2

 V ( rˆ )
2m

1 A
c t

A

cE
i
2
e  2
pˆ
e 
( pˆ  A ) 

A  pˆ    
2m
c
2 m mc
1
vˆ 
1
i
[ rˆ , H ] 
1
i
[ rˆ ,
2
pˆ
]
2m
pˆ
m

e
 i t
Uˆ ( t ) 
E  vˆ e
 c .c
i
|  l (t ) | l  e
 i l t

e
i 
 | l'
l ' l

 i t
 l ' | E  vˆ | l  e
l  l'  
e
 i l t
 
e
 i t
Linear response II.
Perturbation theory: equilibrium distribution function and non-equilibrium states
H SO 
pˆ
2
 
   [ zˆ  pˆ ]  V ( rˆ )
  i t
  Ee



2m

1 A
c t

A

cE
i
e
 i t
2
e  2  
e 
pˆ
e  pˆ  
( pˆ  A )    [ zˆ  ( pˆ  A )] 
 A  [  (  zˆ )]    
2m
c

c
2m c
m 
1
vˆ 
1
i
[ rˆ , H ] 
1
i
[ rˆ , (
2
pˆ
 
pˆ  
ˆ
   [ zˆ  p ]] 
 (  zˆ )

2m
m


e
 i t
Uˆ ( t ) 
E  vˆ e
 c .c
i
|  l (t ) | l  e
 i l t

e
i 
 | l'
l ' l

 i t
 l ' | E  vˆ | l  e
l  l'  
e
 i l t
 
Linear response II.
Perturbation theory: equilibrium distribution function and non-equilibrium states
Intrinsic SHE (AHE)
ˆj z
y
Ex
z
z
J y    l ( t ) | ˆj y |  l ( t )  f 0 ( l )
l
|  l (t ) | l  e
 i l t
0

e
i 
0
 | l'
l ' l

 i t
 l ' | E  vˆ | l  e
l  l'  
e
 i l t
 
Heuristic picture: Bloch equations
ds y
1


dt
 eq
( s z  B eff )
2
d sy
dt
2
0
B eff , y  0
py
ds z
0
dt
pz
pz
B eff , y ~  p x ~ E x t
py
px
 sz ~ E x
px
E=Ex ^x
 sz ~ E x
Field-like SOT

d k
d
si   
n
( 2 )
 0 n , k g n , k ( E j )
i
d

s
E=Ex ^x
Compare with AMR or skew-scattering SHE

d k
d
ji  e  
n
( 2 )
d
i
v 0 n , k g n , k ( E j )
Intrinsic antidamping SOT from linear response II.
0
0
Compare with intrinsic SHE
0
0
pz
pz Hex=0
B eff , y ~  p x ~ E x t
py
Intrinsic SHE: transverse spin current
pz
py
Hex >> HR
px
py
px
B eff , y ~  p x ~ E x t
px
Intrinsic SOT: spin polarization
py
 sz ~ E x
pz
 sz ~ E x
 sz ~ E x
px
 sz ~ E x
pz
Intrinsic SHE: transverse spin current
dB eff , y / dt
s z , p 
s
B eff , y ~  p x ~ E x t
px
py
2
2 p
2
2
 eE x sin  p
pz
equil
( B eff
)
dB eff , y / dt
s
2
sd
2J M
2
 sz ~ E x
2
Intrinsic SOT: spin polarization
sz 
 sz ~ E x
B eff , y ~  p x ~ E x t
py
 sz ~ E x
px
 eE x cos  M
(B
equil
eff
)
2
 sz ~ E x
Intrinsic SOT is antidamping-like
pz

M || xˆ
B eff , y ~  p x ~ E x t
px
py
pz

M || yˆ
px

dM
dt

J ex

px
py
B eff , y ~  p x ~ E x t
py
s z  0
py





M   s ~ M  ([ E  zˆ ]  M )
s z  0
px
SHE & STT switching
SOT switching
Ralph, Buhrman,et al., Science ‘12
Miron et al., Nature ‘11
-We see (anti)damping-like torque
-We also see (anti)damping-like torque
-SOT is field-like so we exclude it
-SOT is field-like but maybe there is some
(anti)damping-like SOT as well
and maybe we found it
 intrinsic SOT analogous to intrinsic SHE
- non-relativistic STT in metals is
dominated by the (anti)damping torque

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