SMR 1646 - 11

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Conference onHigher Dimensional Quantum Hall Effect, Chern-Simons Theory and

Non-Commutative Geometry in Condensed Matter Physics and Field Theory1 - 4 March 2005

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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.

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

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

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

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

)(

)()(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

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

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

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

: 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))

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

(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

• 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)

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

• 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

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

• 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.

Tight-binding model on a square lattice

without vertex correction

withvertex correction

yiVV σ21 −−xiVV σ21 +−

3V−

FE

SHE -- nonzero in general

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

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

(Intrinsic) spin Hall effect should occurin wide range of materials.

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

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

Spatial profile

Spin density maximum -3

0 m10µ≈n

2

2

m50

m10

−

−

≈

≈

µµ

µ

Aj

nAj

c

sSpin current

Charge current

Very small

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

• 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

• Nonzero spin Hall effect in band insulators

Spin Hall insulator

- SM, Nagaosa, Zhang,Phys. Rev. Lett.93, 156804 (2004)

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

-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≈

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)

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

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 +)(

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

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

Imbert shift for Left circular polarization

Imbert shift

Magnitude of the shift Width of the beam is much larger not easy to observe.

λ≈

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.

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!

2D photonic crystals

(“Photonic Crystals”, Joannopoulous et al.)

Simulation : Dielectric constant and its band structure

Dielectric constant Photonic band structure

Large Berry curvature when the band approach other bands in energy

Berry phase in photonic crystals

( 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

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≈

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

∆