KITP Program “Spintronics” May 2, 2006
Spin Hall Effect of Electrons and Photons Shuichi Murakami (Department of Applied Physics, University of Tokyo) Collaborators: Univ. of Tokyo: N. Nagaosa N. Sugimoto K. Sawada S. Onoda AIST, Tsukuba: M. Onoda Stanford Univ.: S.-C. Zhang
Spin Hall effect (SHE)
G E
Electric field induces a transverse spin current. • Extrinsic spin Hall effect
D’yakonov and Perel’ (1971) Hirsch (1999), Zhang (2000)
impurity scattering = spin dependent (skew-scattering) Spin-orbit couping down-spin
up-spin
= relativistic effect
G E
(impurity potential)
impurity
G B if seen from moving electrons, couple with electron spin
y
Intrinsic spin Hall effect • p-type semiconductors
(SM, Nagaosa, Zhang, Science (2003))
x
z
G E p-GaAs
Luttinger model
G G 2⎤ 5 ⎞ 2 = 2 ⎡⎛ H= ⎜ γ 1 + γ 2 ⎟k − 2γ 2 k ⋅ S ⎥ 2m ⎢⎣⎝ 2 ⎠ ⎦ G ( S : spin-3/2 matrix)
( )
• 2D n-type semiconductors in heterostructure (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL (2004)) G E
Rashba model
G G k2 H= +λ σ ×k 2m
(
• • •
)
z
z
y
x
Intrinsic spin Hall effect
Does not rely on impurity scattering
Berry phase in momentum space --- multiband effect Semiclassical eq. of motion (Sundaram,Niu,1999) G G 1 ∂En (k ) G G G G + k × Bn (k ) x= = ∂k G G Determined from the Bloch wf. k = −eE Æ Motion of a wavepacket acquires a transverse shift
G G ∂ G ∂u G Ani (k ) = −i nk nk = −i ∫ un*kG nk d d x ∂ki ∂ki unit cell
: Gauge field antimonopole
( u nkG : periodic part of the Bloch Gwf.) G G G ψ nkG ( x ) = unkG ( x )e ik ⋅ x
G G G G Bn (k ) = ∇ kG × An (k )
: Connection
( n : band index)
monopole
Valence band of GaAs p-orbit (x,y,z)×(↑,↓) + spin-orbit coupling
split-off band (SO) heavy-hole band (HH) doubly degenerate light-hole band (LH) (Kramers)
Luttinger Hamiltonian (Luttinger(1956)) G G 2⎤ 5 ⎞ 2 = 2 ⎡⎛ H= ⎜ γ 1 + γ 2 ⎟k − 2γ 2 k ⋅ S ⎥ 2m ⎢⎣⎝ 2 ⎠ ⎦ G ( S : spin-3/2 matrix) G ˆ Helicity λ = k ⋅ S is a good quantum number.
( )
G
γ − 2γ 2 2 2 3 ⇒ E= 1 = k : heavy hole (HH) 2 2m G γ + 2γ 2 2 2 1 ˆ λ = k ⋅S = ± ⇒ E= 1 = k : light hole (LH) 2 2m λ = kˆ ⋅ S = ±
Helicity
Semiclassical eq. of motion G G G G =k e G G ( λ ) G =k = eE , x = + E × B (k ) mλ =
G G (λ ) G 7⎞ k ⎛ B (k ) = λ ⎜ 2λ2 − ⎟ 3 2⎠ k ⎝ 3 2
λ = ± : HH,
G ˆ λ = k ⋅S
1 2
λ = ± : LH
(Cf. A.Zee, PRA38,1(’88))
G G Anomalous velocity (perpendicular to S and E ) λ0
G E // z Hole spin
Spin current (spin//x, velocity//y) G E kH = λ H e H L j yx = ∑ G y S x n (k ) = z F2 , = ( 3 k − k ) σ s F F 2 3 λ =± ,k 4π 12π G = E z k FL L λ j yx = ∑ G y S x n (k ) = − , • Quantum correction 2 3 λ =± ,k 12π • impurity scattering 3 2
1 2
etc.
Missing !
Intrinsic spin Hall effect in semiconductors
y
x
z
G E j ij = σ sε ijk Ek
p-GaAs
i: spin direction j: current direction k: electric field
σ s : even under time reversal = reactive response (dissipationless) σ s ≈ S iv j • Nonzero in nonmagnetic materials. Cf. Ohm’s law: j = σE σ : odd under time reversal = dissipative response
Intrinsic spin Hall effect for 2D n-type semiconductors in heterostructure (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2004)) Rashba Hamiltonian
G G k2 H= +λ σ ×k 2m
(
z
)
y
z
Structural inversion asymmetry (SIA) Effective electric field along z
Kubo formula : J x J yS J yS z =
1 {J y , S z } 2
z
e (clean limit) 8π independent of λ
x
σs =
2D heterostructure
Spin Hall effect in the Rashba model
≈ Spin precession by “k-dependent Zeeman field” G G k2 H= +λ σ ×k 2m
(
)
z
G G Beff = λ ( zˆ × k )
G k -dependent Zeeman field
G E = 0:
G G S // Beff
G G E ≠ 0 : Beff
varies due to electric field
G S
G is no longer parallel to Beff Spin precession Æ spin Hall effect
σs =
e 8π
Experiments on spin Hall effect • 3D n-type
G E
• Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science (2004) • Sih et al. , Nature Physics (2005)
Spin accumulation due to spin Hall current Æ observe by Kerr effect Δ SO
1
τ
: mostly intrinsic
(Q1): Disorder effect? Rashba model: (2D n-type)
k2 H= + λ (σ x k y − σ y k x ) 2m
Sinova et al.(2003)
σS =
e 8π
+ Vertex correction in the clean limit
(Inoue, Bauer, Molenkamp, PRB (2003))
σS = 0
+ spinless impurities (short-range pot.)
σ S vertex = −
e 8π
+
Inoue, Bauer, Molenkamp, PRB (2004)
• clean limit • δ -fn. impurity
Luttinger model (3D p-type): vertex correction=0 for δ -fn. impurities. (Murakami, PRB (2004))
+
+ ⋅⋅⋅ = 0
+ ⋅⋅⋅
• Calculation by Keldysh formalism (Mishchenko, Shytov, Halperin (2004)) Spin Hall current does not flow at the bulk – consistent withσ S = 0 Spin current only flows near the electrodes
SHE
Spin accumulation
No SHE
SHE
SHE in disordered Rashba model -- Green’s function-σ s = 0 in the following cases: E Fτ
Δτ
Potential
∞
∞
δ-fn.
any
any
δ-fn.
∞
∞
large
large
Inoue, Bauer, Molenkamp (2004) Mishchenko, Shytov,Halperin (2004) Raimondi,Schwab (2004)
Dimitrova (2004)
∞
Liu, Lei (2004,2005) Chalaev Loss (2004)
large
Dimitrova; Chalaev,Loss : J s ≡
∞ large
1 {v y , S z }∝ dS y 2 dt
any
δ-fn any
δ -fn
•diffusion equation •Might be nonzero for finiteEFτ •Arbitrary dispersion (?) •Arbitrary form of Rashba coupling (?) •Zero even with Dresselhaus (’05) •Might be nonzero for T>0 •Zero even with Dresselhaus. •Weak localization correction=0
J s ∝ S y = 0 for steady state
If J s ∝ S y, spin Hall current J s = 0 for any type of spinless impurities. (e.g. Rashba model)
Js
1 dω d 2 p K : Keldysh formalism = ∫ tr J G s 2 ∫ 2i 2π (2π )
(
)
Js ≡
Js
1 1 dω d 2 p K = tr σ H , G y 2i 4imλ ∫ 2π ∫ (2π )2
( [
[H , G ] K
[
1 {v y , S z } = 1 H , σ y 2 4imλ
]
]) No contribution for • 1st Born • higher Born • weak localization corr. for any type of spinless impurities
G i G K = −ieE ⋅ ∂ pG G − eE ⋅ ∂ pG H , ∂ ω G K 2 i G − eE ⋅ (∂ pG Σ∂ ω G − ∂ ω Σ∂ pG G )K 2 i G + eE ⋅ (∂ pG G∂ ω Σ − ∂ ω G∂ pG Σ )K − ([Σ, G ])K 2
{
No contribution to J s
}
Js = 0
Definition of the spin current “Spin current” is not directly measurable. Instead we can measure the spin accumulation at the edge Does spin accumulation reflect the “bulk spin current” due to the SHE?
“Conventional” definition of the spin current operator: Js ≡
1 {v y , S z } 2
not satisfy the eq. of continuity
∂Si + ∇ ⋅ J i(spin) = 0 ∂t
(Spin non-conservation due to spin-orbit coupling) Æ not directly related with spin accumulation
Conserved spin current G ∂ d Sz + ∇ ⋅ J s = Sz dt ∂t
Spin current
• Zhang, Shi, Xiao, Niu, cond-mat/0503505) • Entin-Wohlman et al. PRL95, 086603(2005)
Local spin precession due to SO coupling Let us write it as − ∇ ⋅ Pτ Eq. of continuity for spin
G ∂ S z + ∇ ⋅ (J s + Pτ ) = 0 ∂t J s : conserved spin current Æ measurable as spin accumulation Calculation of Pτ
in response to electric field ? G G 1) Spatial modulation of E : E : = ( Ee iqy −iωt ,0,0)
2) Calculate S z 3) Calculate Pτ
1 Sz ω →0 q →0 iq
Pτ = − lim lim
Spin Hall effect for the Rashba model -- conserved spin current •1st Born approx.
Js
1st
Js
=0
• 2nd Born approx
Js
2 nd
≠0
σ s = 0 in the clean limit σ s = 0 for theδ –fn. Impurity
τ impurities σ s ≠ 0 for longer-ranged with finite
β : range of impurity pot. (short but finite)
Spin Hall effect in the Rashba model
Conventional spin current
Conserved spin current
Js
Js G δ (r )
G G V ( p − p′) G G V ( p, p′)
1st
0
0
higher
0
0
0
0
0
finite
0
finite
0
finite
1st higher 1st higher
Nonzero for general spin-orbit-coupled system
Depends on impurity = extrinsic.
(Q2) SHE in insulators A) Spin Hall insulator
SM, Nagaosa, Zhang, PRL 93, 156804 (2004)
: no edge states
B) Quantum Spin Hall systems • • • • • •
: edge states
Kane, Mele, PRL95, 146802, 226801(‘05) Bernevig, Zhang, PRL96,106802 (’06) Qi, Wu, Zhang, cond-mat/0505308 Onoda, Nagaosa, PRL 95,106601 (’05) Xu, Moore, cond-mat/0508291 Wu, Bernevig, Zhang, PRL96,106401(’06)
• helical spin current at the edge Outline Calculate spin Hall conductivity σ s Å Streda formula ( cf. QHE ) • Relation with edge states • Candidate materials
Spin Hall effect and Streda formula Středa formula for Hall effect Středa (1982) σ
I xy
σ xy = σ xyI + σ xyII σ xyII
⎤ ie 2 ⎡ 1 v jδ (EF − H ) − h.c.⎥ : intraband (Fermi level) = Tr ⎢vi 2 ⎣ EF − H + i0 ⎦ zero for insulator ∂N : interband =e ∂B EF can be nonzero for insulator
N : Number of states below EF
G B∝t
G B∝t
G j
G E ∂N ∂B
electrons flow in. EF
j = Ee
∂N ∂B
: Hall current EF
σ xy
Středa formula for spin Hall effect Expected result
σ sI σ s = σ sI + σ sII
: intraband (Fermi level) zero for insulator
σ sII =
∂S z ∂B
G B∝t
EF
Spin : interband Spin-orbital susceptibility Å spin-orbit coupling
G B∝t
Orbital
G E ∂S z ∂B
spins flow in. EF
js = E
∂S z ∂B
G js
: spin Hall current EF
σs
“Středa-like” formula for spin Hall effect Yang, Chang, PRB73,073304 (2006)
• intraband (Fermi level) -- zero for insulator
• interband -- nonzero for insulator
Spin Å non-conservation unwanted terms
Definition of spin current
“Conventional” spin current Js ≡
not satisfy the eq. of continuity
1 {v y , S z } 2
∂Si + ∇ ⋅ Js = 0 ∂t
Æ not directly related with spin accumulation
Conserved spin current
• Shi, Zhang, Xiao, Niu, PRL96,076604 (2006) • Entin-Wohlman et al. PRL95, 086603(2005)
G ∂ d Sz + ∇ ⋅ J s = Sz ∂t dt Spin current
Local spin precession due to SO coupling write it as − ∇ ⋅ Pτ Eq. of continuity for spin
G ∂ S z + ∇ ⋅ (J s + Pτ ) = 0 ∂t J s : conserved spin current
• conserve • satisfy Onsager relation • have conjugate force
Středa formula for spin Hall effect (I ) ( II ) σ sH = σ sH + σ sH
• intraband -- zero for insulator (I ) = σ sH
ie df dε tr[(G+ − G− ) ⋅ ([ H , s z ]G+ {v x , y}− {v x , y}G− [ H , s z ] ∫ 8πV dε − 2[ H , s z y ]G+ v x + 2v x G− [ H , s z y ] + [ y, [ s z , v x ]])]
• interband -- nonzero for insulator (I ) σ sH =
ie 1 ds z dε f (ε )tr (2 s z G+ ( yv x − xv y )G+ − (+ ↔ −) ) = ∫ 8πV V dBorb.
Cf. N. Sugimoto For insulators …
σs =
1 dS z 1 dLz = V dBorb V dBZeeman
: Středa formula for spin Hall effect
How to calculate orbital magnetization? M=
et al., cond-mat/0503475 (to appear in PRB) P19.00009
“orbital-spin” susceptibility
G 1 Im d k ∑n ∫ ∂ kGunkG × (H + ε nkG − 2μ ) ∂ kGunkG 2c(2π ) 3 ε nk ≤ μ
• • • •
Resta et al., ChemPhysChem 6, 1815 (2005) Xiao et al., PRL 95, 137204 (2005) Thonhauser et al., PRL 95, 137205 (2005) Ceresoli et al., cond-mat/0512142
Honeycomb-lattice model for the QSHS C. L. Kane and E. J. Mele, PRL 95, 146802 ,226801 (2005) G G H = t ∑ ci+ c j + iλSO ∑ν ij ci+ s z c j + iλR ∑ ci+ s × d ij z c j + λv ∑ ξ i ci+ ci
(
ij
ij
kinetic
Spin-orbit
)
ij
i
Rashba
Staggered on-site energy
Z2 topological index Quantum spin Hall system (QSHS)
Spin Hall insulator (SHI)
Numerical calculation of σ s for Kane-Mele model .
Spin Hall insulator (SHI)
σs
σs ≈ 0 (e / 4π )
λR
λV
Quantum spin Hall system (QSHS)
σs ≈ 2⋅ • deviation from Æ
e - quantization 4π
enhanced near QSHS-SHI phase boundary = Band crossing
−e 4π
# edge states
Spin polarization for bulk & edge states Edge states :almost fully polarized ⎛⎜ ≥ 0.95 ⋅ = ⎞⎟ 2⎠ ⎝
σs =
( S 2π= e
z L
− Sz
)
R EF
Kane, Mele, PRL95 (2005) Sheng et al., cond-mat/0603054
⇒
e σs ≈ 2⋅ 4π
Helical spin current
Candidate materials for QSHE ? • Graphene Æ Kane-Mele model ? Estimate for spin-orbit coupling
Δ ≈ 0.2meV Δ ≈ 0.001meV
(Kane,Mele, PRL95,226801 (’05)) (Sinitsyn et al, cond-mat/0602598) : very small!
QSHE requires
• Bi – semimetal
Δ >> 1
τ
• sample quality • low temperature
hole pocket at T point 3 electron pockets at L points
Bi1-xSbx -- insulator (semiconductor) for x>0.07
T
T L
L Semimetal (x