Download Topologoical Aspects of the Spin Hall Effect

Topological Aspects of
the Spin Hall Effect
Yong-Shi Wu
Dept. of Physics, University of Utah
Collaborators:
Xiao-Liang Qi and Shou-Cheng Zhang
(XXIII International Conference on
Differential Geometric Methods in Theoretical Physics
Nankai Institute of Mathematics; August 21, 2005)
Motivations
• Electrons carry both charge and spin
• Charge transport has been exploited in
Electric and Electronic Engineering:
Numerous applications in modern technology
• Spin Transport of Electrons
Theory: Spin-orbit coupling and spin transport
Experiment: Induce and manipulate spin currents
Spintronics and Quantum Information processing
• Intrinsic Spin Hall Effect:
Impurity-Independent Dissipation-less Current
The Spin Hall Effect
Electric field induces transverse spin
current due to spin-orbit coupling
y
x
z

E
J j   Hs  ijk Ek
p-GaAs
i
• Key advantages:
• Electric field manipulation, rather than magnetic field
• Dissipation-less response, since both spin current
and electric field are even under time reversal
• Intrinsic SHE of topological origin, due to Berry’s phase
in momentum space, similar to the QHE
• Very different from Ohmic current:
e2 2
J j   c E j where  c  k F l
h
Family of Hall Effects
• Classical Hall Effect
Lorentz force deflecting like-charge carriers
• Quantum Hall Effect
Lorentz force deflecting like-charge carriers
(Quantum regime: Landau levels)
• Anomalous (Charge) Hall Effect
Spin-orbit coupling deflecting like-spin carriers
(measuring magnetization in ferromagnetic materials)
• Spin Hall Effect
Spin-orbit coupling deflecting like-spin carriers
(inducing and manipulating dissipation-less spin currents
without magnetic fields or ferromagnetic elements)
Time Reversal Symmetry and Dissipative Transport
• Microscopic laws in solid state physics are T invariant
• Most known transport processes break T invariance
due to dissipative coupling to the environment
• Damped harmonic oscillator
mx  x  kx
• Ohmic conductivity is dissipative:
under T, electric field is even
e2 2
J j  E j where   k F l
charge current is odd
h
(only states close to the Fermi energy contribute!)
• Charge supercurrent and Hall current are non-dissipative:
 1 A j
J j   S Aj , E j 
c t
J    H   E ,  H  1 / B
under T vector potential is odd, while magnetic field is odd
Spin-Orbit Coupling
• Origin:
 ``Relativistic’’ effect in atomic, crystal, impurity
or gate electric field
 = Momentum-dependent magnetic field
 Strength tunable in certain situations
• Theoretical Issues:
 Consequences of SOC in various situations?
 Interplay between SOC and other interactions?
• Practical challenge:
 Exploit SOC to generate,manipulate and transport spins
The Extrinsic Spin Hall effect
(due to impurity scattering with spin-orbit coupling)
D’yakonov and Perel’ (1971)
Hirsch (1999), Zhang (2000)
impurity scattering = spin dependent (skew) Mott scattering
plus side-jump effect
Spin-orbit couping
up-spin
down-spin
impurity
Cf. Mott scattering
•
The Intrinsic Spin Hall Effect
Berry phase in momentum space
Independent of impurities
Berry Phase (Vector Potential) in Momentum
Space from Band Structure

 

unk d
*
Ani (k )  i nk
nk  i  unk
d x
ki
ki
unit cell
( u nk : periodic part of the Bloch
wf.


 ik  x


 nk ( x )  u nk ( x ) e
)
 
 
Bn (k )   k  An (k ): Magnetic field
in momentum space
n : Band index
Wave-Packet Trajectory in Real Space
Chang and Niu (1995); P. Horvarth et al. (2000)


k  eE ,

 k e   (  ) 
x
 E  B (k )
m 


Anomalous velocity (perpendicular to S and E )
 0

// k
 0

E // z
Hole spin
Spin current (spin//x,velocity//y)
Intrinsic Hall conductivity (Kubo Formula)
Thouless, Kohmoto, Nightingale, den Nijs (1982)
Kohmoto (1985)
 xy
e2

h





nF En (k ) Bnz (k )

n,k
 
 
Bn (k )   k  An (k ): field strength; n : band index
(Degeneracy point
Magnetic monopole)
Field Theory Approach
• Electron propagator in momentum space
with p  (p, p0 )
S F ( p)
• Ishikawa’s formula (1986):
i e2
σ xy 
24π 2 h

1
1
1
d p ε tr[  μ S F (p) ν S F (p) λ S F (p)]
3
μνλ
• Hall Conductance in terms of momentum space topology
Intrinsic spin Hall effect in p-type semiconductors
In p-type semiconductors (Si, Ge, GaAs,…),
spin current is induced by the external electric field.
y
x
(Murakami, Nagaosa, Zhang, Science (2003))
z

E
j   s ijk Ek
i
j
i: spin direction
j: current direction
k: electric field
p-GaAs
 s : even under time reversal = reactive response
(dissipationless)
• Nonzero in nonmagnetic materials.
Cf. Ohm’s law: j  E

: odd under time reversal
= dissipative response
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 
Luttinger model
Expressed in terms of the Dirac Gamma matrices:
Spin Hall Current (Generalizing TKNN)
ji   ijk Ek
j
4
 ijk   [nL (k )  nH (k )]Gij k (k )
V k
e
 2 k FH  k FL  ijk
6

J yz

Jx
• Of topological origin (Berry phase
in momentum space)
Spin Analog of the
• Dissipation-less
Quantum Hall Effect
• All occupied state contribute At Room Temperature
Intrinsic spin Hall effect for 2D n-type
semiconductors in heterostructure
(Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003))
Rashba Hamiltonian


k2

 (k y  ik x ) 
 
k2
2m

H
  k z  
2
k


2m

(
k

ik
)


y
x
2m




Effective magnetic field
Bint   ( zˆ  k )

Kubo formula :
J
Sz
y

J x J yS z
1
 J y , S z 
2
e
s 
8
independent of
z
y
x
2D heterostructure

SHE: Spin precession by “k-dependent Zeeman field”
Note:  S is not small even when the spin splitting  is small.
due to an interband effect
Spin Hall insulator
•
Motivation: Truly dissipationless transport
Gapful band insulator
(to get rid of Ohmic currents)
• Nonzero spin Hall effect in band insulators:
- Murakami, Nagaosa, Zhang, PRL (2004)
• Topological quantization of spin Hall conductance:
- Qi, YSW, Zhang, cond-mat/0505308 (PRL)
• Spin current and accumulation:
- Onoda, Nagaosa, cond-mat/0505436 (PRL)
Theoretical Approaches
• Kubo Formula (Berry phase in Brillouin Zone)
Thouless, Kohmoto, Nightingale, den Nijs (1982)
Kohmoto (1985)
• Kubo Formula (Twisted Phases at Boundaries)
Niu, Thouless, Wu (1985)
(No analog in SHE yet!)
• Cylindrical Geometry and Edge States
Laughlin (1981)
Hatsugai (1993) (convenient for numerical study)
Cylindrical Geometry and Edge States
Laughlin Gauge Argument (1981):
•Adiabatically changing flux
•Transport through edge states
Bulk-Edge Relation:
(Hatsugai,1993)
(Spectral Flow of Edge States)
Topological Quantization of the AHE (I)
Magnetic semiconductor with SO coupling in 2d
(no Landau levels)
Model Hamilatonian:
d x  sin k y , d y   sin k x
d z  c(2  cos k x  cos k y  es )
Topological Quantization of the AHE (II)



E (k )   (k ) Vd (k )
Two bands:
Band Insulator: a band gap, if V is large enough,
and only the lower band is filled
Charge Hall effect of a filled band:
Charge Hall conductance is quantized to be n/2
n  1,
if
0  es  4
  1,
if
2  es  4
 0,
if
es  0 or es  4
(c>0)
Topological Quantization of the AHE (III)
Open boundary condition in x-direction (c  1,V / t  3, es  0.5)
Two arrows: gapless edge states
The inset: density of (chiral) edge states at Fermi surface
Topological Quantization of Spin Hall Effect I
Paramagnetic semiconductors such as HgTe and a-Sn:
 a are Dirac 4x4 matrices (a=1,..,5)
With symmetry z->-z, d1=d2=0. Then, H
becomes block-diagonal:
SHE is topologically quantized to be n/2
Topological Quantization of Spin Hall Effect II
For t/V small: A gap develops
between LH and HH bands.
LH
d 3 (k )   3 sin k x sin k y
d 4 (k )  3 (cos k x  cos k y )
d 5 (k )  2  es  cos k x  cos k y
Conserved spin quantum number
12
1
2
is
  [ ,  ] / 2
HH
n  1,
if
0  es  4
  1,
if
2  es  4
 0,
if
es  0 or es  4
Topological Quantization of Spin Hall Effect III
• Physical Understanding: Edge states I
In a finite spin Hall insulator system, mid-gap edge states
emerge and the spin transport is carried by edge states
Laughlin’s Gauge
Argument:
When turning on a
flux threading a
cylinder system,
the edge states will
transfer from one
edge to another
Energy spectrum for
cylindrical geometry
Topological Quantization of Spin Hall Effect IV
• Physical Understanding: Edge states II
Apply an electric field
n edge states with 1211
transfer from left (right) to
right (left).
12 accumulation
Spin accumulation
Conserved
=
Non-conserved
+
Effect due to disorder (Green’s function method)
Rashba model:
k2
H
   x k y   y k x 
2m
+ spinless impurities ( -function pot.)
S 
Intrinsic spin Hall conductivity (Sinova et al.,2004)
+ Vertex correction in the clean limit
vertex
(Inoue et al (2003), Mishchenko et al,  S
Sheng et al (2005))

e
8
e
8
J
z
y
J yz

Jx
 
Jx
S  0
k2
2


H



k

S
 2 k y  S x  k x  S y 
Luttinger model:
1
2m
+ spinless impurities ( -function pot.
Intrinsic spin Hall conductivity (Murakami et al,2003)  S 
Vertex correction vanishes identically!
(Murakami (2004), Bernevig+Zhang (2004)
e
6
H
L
z
J
(
k

k
)
y
F
F
2
 S vertex  0
J
z
y

Jx
Jx
 
Topological Orders in Insulators
• Simple band insulators: trivial
• Superconductors: Helium 3 (vector order-parameter)
• Hall Insulators: Non-zero (charge) Hall conductance
2d electrons in magnetic field: TKNN (1982)
3d electrons in magnetic field:
Kohmoto, Halperin, Wu (1991)
• Spin Hall Insulators: Non-zero spin Hall conductance
2d semiconductors: Qi, Wu, Zhang (2005)
2d graphite film: Kane and Mele (2005)
• Discrete Topological Numbers: in 2d systems
Z_2: Kane and Mele (2005);
Z_n: Hatsigai, Kohmoto , Wu (1990)
• 2d Spin Systems and Mott Insulators:
Topological Dependent Degeneracy of the ground states
Fisher, Sachdev, Sethil, Wen etc (1991-2004)
Conclusion & Discussion
• Spin Hall Effect: A new type of dissipationless quantum
spin transport, realizable at room temperature
• Natural generalization of the quantum Hall effect
• Lorentz force vs spin-orbit forces: both velocity dependent
• U(1) to SU(2), 2D to 3D
• Instrinsic spin injection in spintronics devices
• Spin injection without magnetic field or ferromagnet
• Spins created inside the semiconductor,
no interface problem
• Room temperature injection
• Source of polarized LED
• Reversible quantum computation?
• Many Theoretical Issues:
Effects of Impurities
Effects of Contacts
Random Ensemble with SOC
Topological Order of Quantized Spin Hall Systems