Intrinsic Spin Hall Effect - International Centre for...
Transcript of Intrinsic Spin Hall Effect - International Centre for...
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SMR 1646 - 11
___________________________________________________________
Conference onHigher Dimensional Quantum Hall Effect, Chern-Simons Theory and
Non-Commutative Geometry in Condensed Matter Physics and Field Theory1 - 4 March 2005
___________________________________________________________
Intrinsic Spin Hall Effect
Shuichi MURAKAMIDepartment of Applied Physics, University of Tokyo
Tokyo, Japan
These are preliminary lecture notes, intended only for distribution to participants.
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Intrinsic Spin Hall Effect
Shuichi Murakami(Department of Applied Physics, University of Tokyo)
Collaborators:Naoto Nagaosa (U.Tokyo)Shoucheng Zhang (Stanford)
Masaru Onoda (AIST, Japan)
March 3,2005, ICTP, Trieste, Italy
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Spin Hall effect (SHE)Er
Electric field induces a transverse spin current.
• Extrinsic spin Hall effect
Spin-orbit couping
D’yakonov and Perel’ (1971)Hirsch (1999), Zhang (2000)
up-spin down-spinimpurity
• Intrinsic spin Hall effect Berry phase in momentum space
impurity scattering = spin dependent (skew-scattering)
Independent of impurities !
Cf. Mott scattering
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Er
p-GaAsEr
x
y
zIntrinsic spin Hall effect• p-type semiconductors (SM, Nagaosa, Zhang, Science (2003))
• 2D n-type semiconductors in heterostructure(Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL (2003))
y
z
( ) ⎥⎦
⎤⎢⎣
⎡ ⋅−⎟⎠⎞
⎜⎝⎛ +=
2
22
21
2
225
2Skk
mH
rrh γγγ
( )zkm
kHrr×+= σλ
2
2
Sr
( : spin-3/2 matrix)
Luttinger model
Rashba model
x
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xdku
uiknk
knikA d
i
knkn
ini ∂
∂−=
∂∂−= ∫
r
rrrr
cellunit
*)(
)()( kAkB nkn
rrrrr ×∇=
: Gauge field
: Field strength
n( : band index)
antimonopole
monopole
( : periodic part of the Bloch wf.)knu r
Berry phase in momentum space( U(1) gauge field)
xkiknkn exux
rr
rrrr ⋅= )()(ψ
Intrinsic Hall conductivity
( )∑−=kn
nznFxy kBkEnhe
r
rr
,
2
)()(σ
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)(
)()(1
xBxeEek
kBkk
kEx nn
rr&r
r&r
rr&rr
r
h&r
×−−=
×+∂
∂=
“magnetic field in k-space”
Semiclassical eq. of motion (Sundaram,Niu (1999))
=)(kBn
rr
Anomalous velocity due to Berry phase• Quantum Hall effect• Anomalous Hall effect• Spin Hall effect
( )∑−=kn
nznFxy kBkEnhe
r
rr
,
2
)()(σ
Hall conductivity due to Berry phase (intrinsic contribution)
Kubo formula (Thouless et al. (1982))
Berry phase
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p-orbit (x,y,z)×(↑,↓)
split-off band (SO)heavy-hole band (HH) doubly degeneratelight-hole band (LH) (Kramers)
Valence band of semiconductors (diamond (Si, Ge) or zincblende (GaAs))
Luttinger Hamiltonian (Luttinger(1956))
( ) ⎥⎦
⎤⎢⎣
⎡ ⋅−⎟⎠⎞
⎜⎝⎛ +=
2
22
21
2
225
2Skk
mH
rrh γγγ
( : spin-3/2 matrix)Sr
Helicity is a good quantum number.
2221
22
23ˆ k
mESk h
r γγλ −=⇒±=⋅=
2221
22
21ˆ k
mESk h
r γγλ +=⇒±=⋅=
: heavy hole (HH)
: light hole (LH)
+ spin-orbit coupling
Helicity
Skr
⋅= ˆλ
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32)(
272)(
kkkBr
rr⎟⎠⎞
⎜⎝⎛ −= λλλ
LH:21HH,:
23 ±=±= λλ
)(, )( kBEemkxEek
rrr
h
rh&r
r&rh λ
λ
×+==
Semiclassical Equation of motion
ikE
∂∂=Drift velocity
Anomalous velocity(due to Berry phase)
Two bands touch at k=0monopole at k=0
( : helicity= band index)λ
A hole obtains a velocity perpendicular to both k and E.
0=kr
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Anomalous velocity (perpendicular to and )
Real-space trajectory
kr
//
zE //r
Hole spin
Sr
0>λ
0<λ
,12
)(3
,4
)(3
2,
2,
21
23
π
π
λ
λ
λ
λ
LFz
xk
Lyx
HFz
xk
Hyx
kEknSyj
kEknSyj
−==
==
∑
∑
±=
±=
r&
h
r&
h
r
r
Spin current (spin//x, velocity//y)
Skr
⋅= ˆλ)(, )( kBEe
mkxEek
rrr
h
rh&r
r&rh λ
λ
×+==
32)(
272)(
kkkBr
rr⎟⎠⎞
⎜⎝⎛ −= λλλ
LH:21HH,:
23 ±=±= λλ
Er
)3(12 2
LF
HFs kke −=
πσ
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: even under time reversal = reactive response(dissipationless)
i: spin directionj: current directionk: electric field
p-GaAsEr
x
y
zIn p-type semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field.
kijksij Ej εσ=
sσ
• Nonzero in nonmagnetic materials.
Cf. Ohm’s law: Ej σ=σ : odd under time reversal
= dissipative response
Intrinsic spin Hall effect in p-type semiconductors
• topological origin(Berry phase in momentum space)
• dissipationless• All occupied state contribute.
Spin analog of the quantum Hall effect
(SM, Nagaosa, Zhang, Science (2003))
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7.30.644001016
165.63501017
34241501018
7380501019
Spin (Hall) conductivity
Charge conductivity
mobilitycarrier density
)cm( 3−n )cm( -11−Ωσ/Vs)cm( 2µ )cm( -11−Ωsσ
3/1nken
FS ∝∝
=
σµσ
As the hole density decreases, both and decrease.decreases faster than .
σσ Sσ
Sσ
Order estimate (at room temperature) : GaAs
( ) zsLF
HF
zxy E
ekkeEj σ
π 23
12 2
h≡−= : Unit of conductivity)cm( -11−Ωsσ
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(Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003))
πσ
8e
s =
Rashba Hamiltonian
( )⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−
+=×+=
mkikk
ikkm
k
km
kHxy
xy
z
2)(
)(2
2 2
2
2
λ
λσλ
rr
Intrinsic spin Hall effect for 2D n-type semiconductors in heterostructure
Kubo formula : zSyx JJ
independent of λ zySy SJJ z ,
21= 2D heterostructure
x
y
z
Effective electric field along z
Note: is not small even when the spin splitting is small.
interband effect
Sσ λ
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• Semiclassical theory • Rashba + Dresselhaus
Culcer et al., PRL(2004)
spin Hall effect in the Rashba model ≈ Spin precession by “k-dependent Zeeman field”
( )zkm
kHrr ×+= σλ
2
2
)ˆ(int kzBrr
×= λ
• Sinitsyn et al., PRB(2004)• Shen, PRB(2004)
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Rashba model:
Intrinsic spin Hall conductivity (Sinova et al.(2003))
+ Vertex correction in the clean limit (Inoue, Bauer, Molenkamp(2003))
Disorder effect, edge effect
0=Sσ
πσ
8e
S =
+ spinless impurities ( -function pot.)
πσ
8vertex e
S −=
( )xyyx kkm
kH σσλ −+=2
2
• Inoue, Bauer, Molenkamp (2004)• Rashba (2004)• Raimondi, Schwab (2004)• Dimitrova (2004)
Green’s function method
xJzyJ
+ ⋅⋅⋅+xJ
zyJ
δ
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• Calculation by Keldysh formalism (Mishchenko, Shytov, Halperin (2004))
Spin current only flows near the electrodes
0=SσSpin Hall current does not flow at the bulk – consistent with
No SHE SHESHE
Spin accumulation
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Intrinsic spin Hall effect is so fragile to impurities that it vanishes at the bulk?
Question
NO! Not in general.
• in Rashba model in the clean limit• (several papers incorrectly claiming in general.)
0)0( ==ωσ S0)0( ==ωσ S
• Rashba model is an exception.there are models with nonzero . )0( =ωσ S
• two experimental reports• 2D electron gas, spin accumulation at the edges
Y.K.Kato et al., Science (2004)• 2D hole gas, spin LED
J. Wunderlich et al., cond-mat (2004), to appear in PRL.
Answer
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• If
Intrinsic spin Hall effect is nonzero in general.
)(ˆ)(ˆ kHkHrr
−=
in the clean limit the vertex correction is ZERO. (SM (2004))SHE is finite.
e.g. Luttinger model (p-type semicond.)
• )()()()( 0 kdkdkEkH xyyx
rrrrσσ −+=For models with
e.g. Rashba model: kdm
kkErrr
λ== ,2
)(2
0
dAkE rr =
∂∂ 0(a) If for constant , SHE vanishes.
(b) Otherwise, SHE is finite in general.
A
: n-type semicond. in heterostructure
Rashba model (a)Dresselhaus model (b)
zk )(rr ×σλ
)( yyxx kk σσβ −
In real materials SHE is finite in general.
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Tight-binding model on a square lattice
without vertex correction
withvertex correction
yiVV σ21 −−xiVV σ21 +−
3V−
FE
SHE -- nonzero in general
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Definition of spin current is not uniquein the presence of spin-orbit coupling
• Noether’s theorem cannot be applied.
• [ ]∫∫ =∂∂=⇐=⋅∇+
∂∂ rdSHirdS
tJ
tS d
id
iii ,00(spin)
Eq. of continuity requires conservation of spin, but thespin is not conserved in these models
• Spin-orbit coupling spin is not conserved no unique def. of spin current
• In some models, spin current is covariantly conserved. (Zhang)
( ) kkj tAAAp
mH =+=
rrr ,21 2 : gauge field associated with the spin-orbit coupling
(e.g.) Rashba model
jjj iAD +∂= : covariant derivative
( ) ( )[ ]ψψψψ jii
jij DttD
miJ ++ −=
2: spin current
( ) 0=+∂ ijj
it JDS
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Criterion for nonzero spin Hall conductivity
• Different filling for bands in the same multiplet of . SLJrrr
+=
Valence band: J=3/2 Conduction band: J=1/2
(Example) : GaAs
Hole-doping gives a different filling for HH and LH bands.
Spin Hall effect
Electron-doping does not give rise to different filling for two conduction bands
NO spin Hall effect
In 2D heterostructure,Rashba coupling lifts the degeneracy
Spin Hall effect
0vc S.O. ≠H
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(Intrinsic) spin Hall effect should occurin wide range of materials.
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Experiments on spin Hall effect
• 2D electron gas, spin accumulation at the edgesY.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science 306, 1910 (2004)
• 2D hole gas, spin LEDJ. Wunderlich, B. Kästner, J. Sinova, T. Jungwirth, cond-mat/0410295, to appear in PRL
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Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science 306, 1910 (2004)
Experiment -- Spin Hall effect in a 2D electron gas --
(i) Unstrained n-GaAs(ii) Strained n-In0.07Ga0.93As
-316 cm103×T=30K, Hole density:: measured by Kerr rotation
zS
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Spatial profile
Spin density maximum -3
0 m10µ≈n
2
2
m50
m10
−
−
≈
≈
µµ
µ
Aj
nAj
c
sSpin current
Charge current
Very small
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Y.K.Kato et al., Science (2004)
• unstrained GaAs -- (Dresselhaus) spin splitting negligibly small ( )• strained InGaAs -- no crystal orientation dependence
3k∝
It should be extrinsic!
• Dresselhaus term is relevant.
• Dresselhaus term is small, but induced SHE is not small.Rather, experimental value is 10-3 times smaller than theory.
• For Dresselhaus term the vertex correction is zero.
• Dirty limit : SHE suppressed by some factor, which is larger than
It should be intrinsic!
Bernevig, Zhang, cond-mat (2004)
∆
meVmeV 6.1/,025.0 ≈≈∆ τh4
2
10/
−≈⎟⎠⎞
⎜⎝⎛ ∆
τh
Consistent with experiments
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• Circular polarization %1≈
meV2.1/ ≈τh
• Clean limit :
much smaller than spin splitting
• vertex correction =0(Bernevig, Zhang (2004))
It should be intrinsic!
Experiment -- Spin Hall effect in a 2D hole gas --
J. Wunderlich, B. Kästner, J. Sinova, T. Jungwirth, PRL (2005)
• LED geometry
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• Nonzero spin Hall effect in band insulators
Spin Hall insulator
- SM, Nagaosa, Zhang,Phys. Rev. Lett.93, 156804 (2004)
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1) Zero-gap semiconductors : α-Sn, HgSe, HgTe, β-HgS…
conventional
• Spin Hall effect is nonzero in band insulators• Uniaxial strain finite gap at k=0
Spin Hall insulator
zero-gap
( )ae /1.0≈
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
γ γ
σ
( e/a)
s
Estrain=0.02
/2 3
Estrain=0
Estrain=0.2
Estrain=0.002
CB
HH
LH “CB”
“HH”
“LH”
kk
FEFE
Nonzero spin Hall effect in band insulators
α-Sn underuniaxial stress =
energy gap
29 dyn/cm102.3 ×
meV44
0=k 0=k
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-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-4 -3 -2 -1 0 1 2 3 4
λ
σ
( e/a)Mva=0.15
sMva=0.25Mva=0.35
2) Narrow-gap semiconductors : PbS, PbSe, PbTe
FE
)1,1,1(a
k π=
Direct gap (0.15eV-0.3eV) at 4 equivalent L-points
Doubly degenerate
3.30.35PbTe
1.40.16PbSe
1.20.26PbS
Mva λ
32
21 )ˆ(ˆ ττσλτ MvpkvpkvH +×⋅+⋅= rrr
σr τr: spin :orbital
Spin Hall insulator
• Spin Hall effect is nonzero in band insulators( )ae /04.0≈
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x: spin directiony: current directionz: electric field
p-GaAs
Er
x
y
z
In semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field.
Conclusion
• Topological origin• Dissipationless• All occupied states contribute.• Large even at room temperature
Spin analog of the QHE
: semiclassical result
( )LF
HF
zxy kkeEj −= 3
12 2π
Spin Hall effect can be nonzero in band insulators.
• Zero-gap semiconductors (α-Sn, HgTe, HgSe, β-HgS)• Narrow-gap semiconductors (PbS, PbTe, PbSe)
(examples)
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Hall effect of light
• Anomalous velocity due to Berry phase interference of wavesCommon for every wave phenomenon.
How about “light” ?
YES!
Hall effect of light
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zkkiz
rvkk
zkzkkrvr
)(
)(
)(ˆ)(
rr&r&
r&r
rr&rr&r
Λ⋅−=
∇−=
Ω×+=
Hall effect of light -- Analog of the spin Hall effect --
Onoda, SM, Nagaosa, Phys. Rev. Lett. (2004)
)()(
rncrv r
r =
Semiclassical eq. of motion
: slowly varying
: curvature
: gauge field
Shift of a trajectory of light“Hall effect of light”
Polarization change
z : polarization
Isotropic medium, slowly varying refractive indexpick up terms up to
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=Ω1
1)( 3k
kkr
rr
in the basis of circular polarization
In the vacuum
right
left
Chiao,Wu(’86) : theoryTomita,Chiao(’86) : experiment
)(rn r
( )nO ln∇λ
Λ×Λ+Λ×∇=Ωrrrrr
r ik k)(
jkiij eeik rrrrr∇−=Λ +)(
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zkkiz
rvkk
zkzkkrvr
)(
)(
)(ˆ)(
rr&r&
r&r
rr&rr&r
Λ⋅−=
∇−=
Ω×+=
Onoda, SM, Nagaosa, PRL (2004)
( )[ ]
( )nesnce
nssnncs
nsn
csncr
ln
lnln
ln
∇⋅−=
∇⋅−∇=
⎥⎦⎤
⎢⎣⎡ ∇×−=
rr&r
rr&r
rr&r
ωσ
Zel’dovich, Liberman (1990)Bliokh (2004)
kks /rr = er : polarization
“Optical magnus effect”
equivalent
ExperimentDugin, Zel’dovich, Kundikova, Liberman (1991)
Optical fiber
or
Beam profile rotated by 3.2 degrees
92cm
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Imbert shiftTheory: Fedorov (1955)Experiment: Imbert(1972)
Hall effect of light in interface refraction/reflection
zkkiz
rvkk
zkzkkrvr
)(
)(
)(ˆ)(
rr&r&
r&r
rr&rr&r
Λ⋅−=
∇−=
Ω×+=
At the interface, the refractive index changesthe trajectory is transversely shifted due to Berry phase (to y-direction)
Cf. Conservation of total angular momentum
zzz LSJ +=
Opposite for right & left circular polarization
x
z
3)(kkkr
rr±=Ω for left (right) circular polarization
2n1n ⊥k
rchanges
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Imbert shift for Left circular polarization
Imbert shift
Magnitude of the shift Width of the beam is much larger not easy to observe.
λ≈
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C. Imbert, PRD (1972)Experimental measurement of the Imbert shift
(Small shift) * (28 successive reflection)
right circular pol. left circular pol.
Shift is opposite for the two circular polarizations.
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Photonic crystals and Berry phase
Electrons in condensed matter:
Periodic lattice enhances the Hall effect by some orders of magnitude
Will the “Hall effect of light” enhanced inphotonic crystals?
(Example) 2D photonic crystals (PC)
Caution :Berry curvature is zero for 2D PC
with inversion symmetry.
We use a 2D PC without inversion symmetry.
YES!
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2D photonic crystals
(“Photonic Crystals”, Joannopoulous et al.)
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Simulation : Dielectric constant and its band structure
Dielectric constant Photonic band structure
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Large Berry curvature when the band approach other bands in energy
Berry phase in photonic crystals
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( In addition to periodic modulation of ),slow 1D modulation needed
)(rrε
To see the anomalous velocityshould change in time.
Large shift !!
)||( zzk kr
r&r Ω×kr
SimulationTrajectory of light beam in photonic crystals
slow 1D modulation near x=0
larger for larger ε x
× envelope function linear in x
-- Analog of the electric field in the SHE
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maximum at the gap :
Maximum shift of the beam
Enhancement of Berry curvature in photonic crystals
• vacuum : 2
1kk ≈Ω r
r
• photonic crystals:
Shift (e.g.: Imbert shift)very small
∆
kr
ω
Slope (velocity)v( ) 2/32
022
2
)( kkvv
z rr−+∆
∆≈Ω
0kr
2
2
∆≈Ω v
z
∆v
Bigger for smaller gap
λ≈
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Shift of the light beam ≈ Distance from a monopole in k space
• Vacuum:
(shift) <<≈ λ (width of the beam) Small shift
• Photonic crystal:
(shift)
(width of the beam)
∆≈ v
kr1>>
Shift can be large(for small gap)
ω
ω
k
k
∆