Download The spin Hall effect

The spin Hall effect
The quantum AHE and the SHE
The persistent spin helix
Shou-cheng Zhang, Stanford University
Les Houches, June 2006
Credits
Collaborators:
• Andrei Bernevig (Stanford)
• Taylor Hughes (Stanford)
• Shuichi Murakami (Tokyo)
• Naoto Nagaosa (Tokyo)
• Xiaoliang Qi (Tsinghua and Stanford)
• Congjun Wu (Stanford and KITP/Santa Barbara)
• Yongshi Wu (Utah)
The spin Hall effect
Can Moore’s law keep going?
Power density (W/cm2)
Power dissipation=greatest obstacle for Moore’s law!
Modern processor chips consume ~100W of power of which about 20%
is wasted in leakage through the transistor gates.
The traditional means of coping with increased power per generation
has been to scale down the operating voltage of the chip but
voltages are reaching limits due to thermal fluctuation effects.
500
500
400
400
Passive Power (Device Leakage)
300
300
200
200
Active Power
100
100
00
500
0.5
350
130 100
0.35 250
0.25 180
0.18 0.13
0.1
Technology node (nm)
70 0.05
50
0.07
Spintronics
• The electron has both charge and spin.
• Electronic logic devices today only used
the charge property of the electron.
• Energy scale for the charge interaction
is high, of the order of eV, while the
energy scale for the spin interaction is
low, of the order of 10-100 meV.
• Spin-based electronic promises a radical
alternative, namely the possibility of
logic operations with much lower power
consumption than equivalent charge
based logic operations.
• New physical principle but same
materials! In contrast to nanotubes and
molecular electronics.
Manipulating the spin using the Stern-Gerlach experiment
• Problem of using the magnetic
field:
• hard for miniaturization on
a chip.
• spin current is even while
the magnetic field is odd
under time reversal =>
dissipation just as in
Ohm’slaw.
Relativistic Spin-Orbit Coupling

E



E
• Relativistic effect: a particle in an
electric field experiences an
internal effective magnetic field in
its moving frame

 
Beff ~ v  E
• Spin-Orbit coupling is the
coupling of spin with the internal
effective magnetic field

v
 
H ~  S  Beff
Using SO: spin FET
V/2
V
-
v
v
-
Beff
Beff
•Das-Datta proposal.
•Animation by Bernevig and Sinova.
Generalization of the quantum Hall effect
• Quantum Hall effect exists in D=2, due to Lorentz force.
J i   H  ij E j
p e2
H 
q h
B
z
y
J
E
GaAs
x
• Natural generalization to D=3, due to spin-orbit force:
z
x: current direction
y: spin direction
z: electric field
J j   spin ijk Ek
i
y

E
 spin  ek F
GaAs
• 3D hole systems (Murakami, Nagaosa and Zhang, Science 2003)
• 2D electron systems (Sinova et al, PRL 2004)
x
Valence band of GaAs
S
S
P
P3/2
P1/2
Luttinger Hamiltonian
 
  2
1 
5  2
H
  1   2 k  2 2 k  S 
2m 
2 


( S : spin-3/2 matrix, describing the P3/2 band)



Sx  



0
3i / 2
0
0
3i / 2
0
i
0 
0
i
0
3i / 2
 0
0 


 3/2
0 
S

 y 
3i / 2 
 0
 0
0 

3/2
0
1
0
0
1
0
3/2
3/ 2
0 


0 
 0
 Sz   0
3 / 2

 0

0 

0 0
0 

1/ 2 0
0 
0 1/ 2 0 

0 0  3 / 2 
Unitary transformation
 
  2

1 
5  2
H



k

2

k

S

V
(
x
)


1
2
2

2m 
2 

Diagonalize the first term with a local unitary transformation
U (kˆ)
  

i S
U (k )k  SU (k )  kSz , U (k )  e y e i S z
Helicity basis
U (kˆ' )

  kˆ  S



  
1 
5  2
2
2
H   U (k ) HU  (k ) 



k

2

k
S

U
(
k
)
V
(
x
)U (k )


1
2
2
z 
2m 
2 

  1  2 2
   32 : HH


1
 1  2 2
k2 
   2 : LH
    1 : LH
 1  2 2
2m 
2




 1  2 2     32 : HH





U (k )V (i k )U (k )  V ( D)

 Ai
ki
   
Ai  iU (k )
U (k ) : gauge field in k!
ki
Di  i
Local gauge field in k space
Adiabatic transport = potential V does not cause inter-band transitions
 only retain the intra-band matrix elements



Ai dki  



 32 cosd
3
2 (sin d  id  )
3
2
(sin d  id )
 12 cos d
sin d  id
sin d  id 
1
2 cos d
3
2 (sin d  id  )
   32 : HH

   12 : LH
    1 : LH
3
2
2 (sin d  id  ) 
3
3

2 cos d
    2 : HH
Abelian approximation = retain only the intra-helicity matrix elements



Ai dki  



 32 cos d
3
2 (sin d  id  )
3
2
(sin d  id )
 12 cos d
sin d  id
sin d  id
1
2 cos d
3
2 (sin d  id  )
   32 : HH

   12 : LH
    1 : LH
3
(sin

d


id

)
2
2

3
    3 : HH
2
2 cos d

Effective Hamiltonian for adiabatic transport
H
eff

k2

 V (x)
2m
 ~ 
xi  Di  i
 Ai (k )
ki
Nontrivial spin dynamics comes from the
Dirac monopole at the center of k space, with
eg=:
(Dirac monopole)
Fij   ijk 
kk
k3
[ki , k j ]  0, [ xi , k j ]  i ij , [ xi , x j ]  iFij
Eq. of motion
ki   Ei ,
xi 
ki

 3  ijk E j k k
m k
Drift velocity 
E
ki
Topological term  
eE j

Fij
Dissipationless spin current induced by the electric field
The intrinsic spin Hall effect
e
6
2
L
• Key advantage:
• electric field manipulation, rather than
magnetic field.
• dissipationless response, since both spin
current and the electric field are even under
time reversal.
• Topological origin, due to Berry’s phase in
momentum space similar to the QHE.
• Contrast between the spin current and the
Ohm’s law:
I  V / R or
Bulk GaAs
(k F  k F )
H
Energy (eV)
J j   spin ijk Ek ,  spin 
i
e2 2
J j   E j where   k F l
h
Time reversal and the dissipationless spin current
T
-
v
v
-
v
v
T
-
v
v
-
-
Effect due to disorder
k2
2
Luttinger model: H 
 1 k  S   2 k y  S x  k x  S y  + spinless impurities ( -function pot.)
2m
Intrinsic spin Hall conductivity (Murakami et al.(2003))
Vertex correction vanishes identically!
2DHG Bernevig+Zhang (PRL 2004)
Rashba model:
e
H
L
z
J
(
k

k
)
y
F
F
6 2
S 
 S vertex  0
k2
H
   x k y   y k x 
2m
J
(Inoue, Bauer, Molenkamp(2003))
 S vertex  
 
Jx
+ spinless impurities ( -function pot.)
Intrinsic spin Hall conductivity (Sinova et al.(2004))  S 
+ Vertex correction in the clean limit

z
y
Jx
e
8
J
z
y
e
8

Jx
J yz
 
Jx
S  0
Mott scattering or the extrinsic Spin Hall effect
Electric field induces a transverse spin current.

E
• Extrinsic spin Hall effect Mott (1929), D’yakonov and Perel’ (1971)
Hirsch (1999), Zhang (2000)
impurity scattering = spin dependent (skew-scattering)
Spin-orbit couping
up-spin
down-spin
impurity
Cf. Mott scattering
• Intrinsic spin Hall effect
Independent of impurities !
Berry phase in momentum space
Experiment -- Spin Hall Effect in a 3D Electron Film
Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science 306, 1910 (2004)
(i) Unstrained n-GaAs
(ii) Strained n-In0.07Ga0.93As
T=30K, Hole density: 31016 cm -3
S z : measured by Kerr rotation
Experiment -- Spin Hall Effect in a 3D Electron Film
Y.K.Kato et al., Science (2004)
•
•
•
•
unstrained GaAs -- no strain spin-orbit coupling
strained InGaAs -- no crystal orientation dependence
extrinsic quantum spin hall calculation (Engel, Rashba, Halperin)
sign mismatch? but right ballpark value
It should be extrinsic!
Bernevig, Zhang, cond-mat (2004)
• Dresselhaus term is relevant, opposite sign.
It could be intrinsic!
• Dresselhaus term  is small, but induced SHE is not small.
• For Dresselhaus term the vertex correction does not cancel the intrinsic SHE.
• Dirty limit :
2
 SHE suppressed by some factor, which is roughly     10 4
  0.025meV ,  /   1.6meV
  / 
Experiment -- Spin Hall Effect in a 2D Hole Gas
J. Wunderlich, B. Kästner, J. Sinova, T. Jungwirth, PRL (2005)
• LED geometry
• Circular polarization  1%
• Clean limit :
 /   1.2meV
much smaller than spin splitting
• vertex correction =0
(Bernevig, Zhang (2004))
• should be intrinsic
Direct measurement of the spin current?
A modified version of the standard
drift-diffusion experiment in
semiconductor physics. Optically
inject up or down spin carriers, and
observe the longitudinal charge drift
and the spin-dependent transverse
drift.
z
E
y
x
Spin – Orbit Coupling in Two Dimensions
General Hamiltonian for spin ½ systems:
Rashba Hamiltonian
E
Strong out-of plane junction electric field
GaAs
Transport In Spin ½ Systems: Two Dimensions
• Upon momentum integration continuity equations:
• Rashba coupling (2D Asymmetric Quantum Wells):
Burkov Nunez and MacDonald;
Mishchenko, Shytov and Halperin
Rashba SO Coupling: 2D Photon
Bernevig, Yu and Zhang, PRL 95, 076602 (2005)
Spin-Orbit Coupling
Maxwell’s Equations
Spintronics without spin injection and spin detection
+
S
C
E
R
V
+
-
With strong spin-orbit coupling, injected charge packet spontaneously splits
Ina E
conventional
charge
dynamics,
injecteddirections
charge packets
Withtwo
field,packets,
the charge
packets
also
drifts.
into
spin
propagating
in opposite
at the simply
Rashbadiffuses.
speed,
Drift-diffusion
is theEfundamental
process underlying all conventional electronics.
without
any applied
field.
This effect can be used to construct a spin bus.
Conclusions
• Spin Hall effect is a profoundly deep effect in solid state physics,
Natural generalization of the Hall effect and quantum Hall effect.
• Natural extensions of the spin Hall effect: orbitronics, spintronics without
Spin injection and spin detection, quantum spin Hall effect.
• Need close interaction among theory, experiments and materials science.
• Frontier of science and technology.