Intrinsic Spin Hall Effect

SMR 1646 - 11

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Conference on Higher Dimensional Quantum Hall Effect, Chern-Simons Theory and Non-Commutative Geometry in Condensed Matter Physics and Field Theory 1 - 4 March 2005

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Intrinsic Spin Hall Effect

Shuichi MURAKAMI Department of Applied Physics, University of Tokyo Tokyo, Japan

These are preliminary lecture notes, intended only for distribution to participants. March 3,2005, ICTP, Trieste, Italy

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) Spin Hall effect (SHE) r 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

up-spin down-spin impurity

Cf. Mott scattering

• Intrinsic spin Hall effect Berry phase in momentum space Independent of impurities ! y Intrinsic spin Hall effect x z • p-type semiconductors (SM, Nagaosa, Zhang, Science (2003)) r E p-GaAs Luttinger model γ γ 2 ⎡ 5 r r 2 ⎤ r = h ⎛ + ⎞ 2 − γ ()⋅ ( S : spin-3/2 matrix) H ⎢⎜ 1 2 ⎟k 2 2 k S ⎥ 2m ⎣⎝ 2 ⎠ ⎦

• 2D n-type semiconductors in heterostructure r (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL (2003)) E z Rashba model k 2 r y H = + λ()σr × k 2m z x Berry phase in momentum space ( U(1) gauge field)

∂ ∂ r r r r * unk d A (k ) = −i nk nk = −i u r d x : Gauge field ni ∂ ∫ nk ∂ ki unit cell ki

r antimonopole ( u n k : periodic part of the Bloch wf.) r ik⋅xr ψ r r = r r nk (x) unk (x)e

r r r r = ∇ r × Bn (k ) k An (k ) : Field strength

( n : band index) monopole

Intrinsic Hall conductivity 2 σ = − e ()r r xy nF En (k ) Bnz (k ) ∑r h n,k Hall conductivity due to Berry phase (intrinsic contribution) 2 σ = − e ()r r xy nF En (k ) Bnz (k ) ∑r h n,k

Berry phase

Kubo formula (Thouless et al. (1982))

Semiclassical eq. of motion (Sundaram,Niu (1999)) r 1 ∂E (k ) r r r xr& = n + k& × B (k ) ∂ r n Anomalous velocity due to Berry phase h k • Quantum Hall effect r r r k& = −eE − exr& × B(xr) • Anomalous Hall effect • Spin Hall effect

r r = Bn (k ) “magnetic field in k-space” Valence band of semiconductors (diamond (Si, Ge) or zincblende (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 Hamiltonianγ (Luttinger(1956)) γ 2 = h ⎡⎛ + 5 ⎞ 2 − γ ()r ⋅ r 2 ⎤ H ⎢⎜ 1 2 ⎟k 2 2 k S ⎥ 2m ⎣⎝ 2 ⎠ ⎦ r ( S : spin-3/2 matrix) r Helicityλ = kˆ ⋅ S is a good quantum number. r 3 γ − 2γ λ = kˆ⋅ S = ± ⇒ E = 1 2 2k 2 : heavy hole (HH) 2 2m h Helicity r 1 γ + 2γ λ = kˆ ⋅ S = ± ⇒ E = 1 2 2k 2 : light hole (LH) 2 2m h Semiclassical Equation of motion

r r& r r hk λ e r r (λ) r hk = eE , x& = + E × B (k ) ( λ : helicity= band index) m h

∂ = E Anomalous velocity Drift velocity ∂ ki (due to Berry phase)

Two bands touch at k=0 r k = 0 Æ monopole at k=0 r r (λ ) r λ⎛ 2 7 ⎞ k λB (k ) = ⎜2λ − ⎟ ⎝ 2 ⎠ k 3 3 1 = ± : HH, λ = ± : LH 2 2

A hole obtains a velocity perpendicular to both k and E. r r λ r λ⎛ 7 ⎞ k Real-space trajectory B( ) (k ) = ⎜2λ2 − ⎟ λ ⎝ 2 ⎠ k 3 r = ± 3 λ = ± 1 r& = r r = hk λ+ e r × r (λ) r : HH, : LH hk eE , x& E B (k ) 2 2 m r h λ = kˆ ⋅ S

r r Anomalous velocity (perpendicular to andS )E

λ < 0

r // k

λ > 0 r E // z Hole spin

Spin current (spin//x, velocity//y)

H λ λ r E k H = h =π z F jyx y&Sxn (k ) 2 , ∑ r 3 =± 3,k 4 σ 2 = e H − L L s (3kF kF ) λ r E k π 2 j L = h λ yS n (k ) = − z F , 12 yx ∑ & x 2 3 =± 1 r 12π 2 ,k Intrinsic spin Hall effect in p-type semiconductors y x In p-type semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field. z (SM, Nagaosa, Zhang, Science (2003)) r E i: spin direction p-GaAs i = σ ε j: current direction j j s ijk Ek k: electric field σ s : even under time reversal = reactive response (dissipationless) • Nonzero in nonmagnetic materials.

Cf. Ohm’s law: j = σE σ : odd under time reversal = dissipative response • topological origin (Berry phase in momentum space) • dissipationless Spin analog of the quantum Hall effect • All occupied state contribute. Order estimateπ (at room temperature) : GaAs eE x = z ()H − L ≡ h σ −1 -1 j y 2 3k F k F s E z σ (Ω cm ) 12 2e s : Unit of conductivity

carrier mobility Charge Spin (Hall) density conductivity conductivity −3 µ 2 σ Ω−1 -1 σ Ω−1 -1 n (cm ) (cm /Vs) ( cm ) s ( cm ) 1019 50 80 73

1018 150 24 34

1017 350 5.6 16

1016 400 0.64 7.3

σ = enµ σ ∝ ∝ 1/3 S kF n

σ σ As the hole density decreases, both and decrease.S σ σ decreases faster than .S Intrinsic spin Hall effect for 2D n-type semiconductors in heterostructure (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003))

Rashba Hamiltonian λ 2 z σ ⎛ k + ⎞ 2 ⎜ (k y ik x )⎟ = k + ()r × r = ⎜ 2m ⎟ y H k z λ 2 2m ⎜λ − k ⎟ ⎜ (k y ik x ) ⎟ x ⎝ 2m ⎠

Effective electric field along z

e Kubo formula : J J Sz σ = x y s 8π 1 J S z = {}J , S independent of λ 2D heterostructure y 2 y z

σ Note: Sis not small even when the spin splitting is λsmall.

interband effect spin Hall effect in the Rashba model ≈ Spin precession by “k-dependent Zeeman field”

k 2 r H = + λ()σr × k 2m z

r = λ × r Bint (zˆ k )

• Semiclassical theory Culcer et al., PRL(2004) • Rashba + Dresselhaus

• Sinitsyn et al., PRB(2004) • Shen, PRB(2004) Disorder effect, edge effect

Green’s function method λ σ k 2 δ Rashba model: H = + ()k −σ k + spinless impurities ( -function pot.) 2m x y y x σ = e z Intrinsic spin Hall conductivity (Sinova et al.(2003)) J y J x S 8π

+ Vertex correction in the clean limit σ vertex = − e + S +⋅⋅⋅ (Inoue, Bauer, Molenkamp(2003)) 8π z J y J x

σ = S 0 • Inoue, Bauer, Molenkamp (2004) • Rashba (2004) • Raimondi, Schwab (2004) • Dimitrova (2004) • Calculation by Keldysh formalism (Mishchenko, Shytov, Halperin (2004)) σ = Spin Hall current does not flow at the bulk – consistent with S 0 Spin current only flows near the electrodes Spin accumulation

SHE No SHE SHE σ •(ω = 0) = 0 in Rashba model in the clean limit S σ ω = = • (several papers incorrectly claiming S ( 0 ) in0 general.)

Question Intrinsic spin Hall effect is so fragile to impurities that it vanishes at the bulk?

Answer NO! Not in general.

• Rashba model is an exception. σ ω = there are models with nonzero S ( . 0)

• 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. Intrinsic spin Hall effect is nonzero in general.

r r •If Hˆ (k ) = Hˆ (−k ) e.g. Luttinger model (p-type semicond.)

Æ in the clean limit the vertex correction is ZERO. (SM (2004)) SHE is finite.

σ r = r + r − σ r • For models with H (k ) E0 (k ) x d y (k ) y d x (k ) : n-type semicond. in heterostructure

r k 2 r r e.g. Rashba model: E ( k ) = , d = λk 0 2m ∂E r (a) If r 0 = A d for constant ,A SHE vanishes. ∂k

(b) Otherwise, SHE is finite in general.

λ σr ×βr Rashba model ( k ) σz Æ (a) −σ Dresselhaus model ( x k x y k yÆ) (b)

In real materials SHE is finite in general. Tight-binding model on a square lattice

− V3

−V + iV σ 1 2 x − − σ V1 iV2 y without vertex correction

with vertex correction

SHE -- nonzero in general

EF Definition of spin current is not unique in the presence of spin-orbit coupling

• Spin-orbit coupling Æ spin is not conserved Æ no unique def. of spin current

• Noether’s theorem cannot be applied. ∂S ∂ • i + ∇⋅ J (spin) = 0 ⇐ 0 = S d d r = i[]H, S d d r ∂t i ∂t ∫ i ∫ i Eq. of continuity requires conservation of spin, but the spin is not conserved in these models

• In some models, spin current is covariantly conserved. (Zhang)

= 1 ()r + r 2 r = k k : gauge field associated with the spin-orbit coupling H p A , A Aj t 2m (e.g.) Rashba model

= ∂ +ψ : covariant derivative D j j iAψj ψ i + + J i = []()D t i − t i D ψ (): spin current j 2m j j ∂ i + ()i = t S D j J j 0 Criterion for nonzero spin Hall conductivity r r r • Different filling for bands in the same multiplet of J = L . + S ≠ c HS.O. v 0

(Example) : GaAs

Valence band: J=3/2 Conduction band: J=1/2

Hole-doping gives a Electron-doping does not give different filling for HH and rise to different filling for two LH bands . conduction bands Æ Spin Hall effect Æ NO spin Hall effect

In 2D heterostructure, Rashba coupling lifts the degeneracy Æ Spin Hall effect (Intrinsic) spin Hall effect should occur in wide range of materials. Experiments on spin Hall effect

• 2D electron gas, spin accumulation at the edges Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science 306, 1910 (2004)

• 2D hole gas, spin LED J. Wunderlich, B. Kästner, J. Sinova, T. Jungwirth, cond-mat/0410295, to appear in PRL Experiment -- Spin Hall effect in a 2D electron gas --

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 : measured by Kerr rotation Sz Spatial profile

Spin density maximum ≈ µ -3 n0 10 m

µ ≈ −2 Spin current js 10nA m Very small

≈ µ µ −2 Charge current jc 50 A m Y.K.Kato et al., Science (2004)

• unstrained GaAs -- (Dresselhaus) spin splitting negligibly small ( )∝ k 3 • strained InGaAs -- no crystal orientation dependence

It should be extrinsic!

Bernevig, Zhang, cond-mat (2004) • Dresselhaus term is relevant. It should be intrinsic!

• 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 : ∆ ≈ 0.025meV, /τ ≈ 1.6meV 2 h ⎛ ∆ ⎞ − Æ SHE suppressed by some factor, which is larger than ⎜ ⎟ ≈10 4 ⎝ h /τ ⎠ Consistent with experiments 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 : τ h / ≈1.2meV much smaller than spin splitting

• vertex correction =0 (Bernevig, Zhang (2004))

It should be intrinsic! Spin Hall insulator

• Nonzero spin Hall effect in band insulators

- SM, Nagaosa, Zhang, Phys. Rev. Lett.93, 156804 (2004) Nonzero spin Hall effect in band insulators

1) Zero-gap semiconductors : α-Sn, HgSe, HgTe, β-HgS…

conventional zero-gap α-Sn under uniaxial stress = 3.2×109 dyn/cm2 Æ energy gap 44meV

( e/a) 0.2 CB “LH” σ Estrain =0 0.15 E EF s strain =0.002 EF 0.1 Estrain =0.02 Estrain =0.2 HH “HH” 0.05 0 LH “CB” -0.05 -0.1

-0.15 k = 0 k k = 0 k -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 γ γ 2 / 3

• Spin Hall effect is nonzero () ≈ 0 . 1 e / ain band insulators • Uniaxial strain Æ finite gap at k=0

Spin Hall insulator 2) Narrow-gap semiconductors : PbS, PbSe, PbTe

( e/a) Doubly degenerate 0.05 σ Mva=0.15 0.04 Mva=0.25 s 0.03 Mva=0.35 E 0.02 F 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -4 -3 -2 -1 0 1 2 3 4 π k = (1,1,1) λ a τ λ Mva λ Direct gap (0.15eV-0.3eV) at 4 σequivalentτ L-points r r PbS 0.26 1.2 H = vk ⋅ pˆ + vk ⋅ ( pˆ × r ) + Mv 2τ 1 2 3 PbSe 0.16 1.4 σr : spin τr :orbital PbTe 0.35 3.3 • Spin Hall effect is nonzero () ≈ 0 .04 e /ina band insulators Spin Hall insulator Conclusion

In semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field.

y eE x: spin direction j x = z ()3k H − k L x y 12π 2 F F y: current direction : semiclassical result z: electric field z r • Topological origin E Spin analog • Dissipationless p-GaAs of the QHE • All occupied states contribute. • Large even at room temperature

Spin Hall effect can be nonzero in band insulators. (examples) • Zero-gap semiconductors (α-Sn, HgTe, HgSe, β-HgS) • Narrow-gap semiconductors (PbS, PbTe, PbSe) Hall effect of light

• Anomalous velocity due to Berry phase Æ interference of waves Common for every wave phenomenon.

How about “light” ?

YES!

Hall effect of light Hall effect of light Onoda, SM, Nagaosa, Phys. Rev. Lett. (2004)

-- Analog of the spin Hall effect -- Isotropic medium, slowly varying refractive index n(rr) Æ pick up terms up to O()λ∇ln n

Semiclassical eq. of motion r r r Shift of a trajectory of light rr& = v(rr)kˆ + k& × z Ω(k ) z “Hall effect of light” r k& = −k∇v(rr) = − r& ⋅Λr r z& ik (k ) z Chiao,Wu(’86) : theory Polarization change Tomita,Chiao(’86) : experiment

r = c v(r ) r : slowly varying In the vacuum n(r) r r r k ⎛1 ⎞ right z : polarization Ω(k ) = ⎜ ⎟ 3 ⎜ ⎟ k ⎝ −1⎠ left r r r + r Λ(k ) = −ie ∇ re : gauge field ij i k j in the basis of circular polarization r r r r r Ω = ∇ r × Λ + Λ × Λ (k ) k i : curvature c c r r r r = ⎡r −σ r ×∇ ⎤ r = r ˆ + & × Ω r& ⎢s s ln n⎥ r& v(r )k k z (k ) z n ⎣ ωn ⎦ r& = − ∇ r c k k v(r ) sr& = []∇ln n − sr()sr ⋅∇ln n r r r n z = −ik& ⋅Λ(k ) z & equivalent c er& = − sr()er ⋅∇ ln n Onoda, SM, Nagaosa, PRL (2004) n r sr = k / k er : polarization Zel’dovich, Liberman (1990) Bliokh (2004)

“Optical magnus effect”

Experiment Dugin, Zel’dovich, Kundikova, Liberman (1991)

Optical fiber

or 92cm Beam profile Æ rotated by 3.2 degrees Hall effect of light in interface refraction/reflection z r r r rr& = v(rr)kˆ + k& × z Ω(k ) z n x r r 2 r changes k& = −k∇v(r ) k⊥ n1 r& r r z& = −ik ⋅Λ(k ) z

At the interface, the refractive index changes Æ the trajectory is transversely shifted due to Berry phase (to y-direction) Opposite for right & left circular polarization r r r k Ω(k ) = ± for left (right) circular polarization k 3 Imbert shift Theory: Fedorov (1955) Experiment: Imbert(1972) Cf. Conservation of total angular momentum = + J z Sz Lz Imbert shift Imbert shift for Left circular polarization

Magnitude of the shift ≈ λ Width of the beam is much larger Æ not easy to observe. Experimental measurement of the Imbert shift C. Imbert, PRD (1972)

(Small shift) * (28 successive reflection)

Shift is opposite for the two circular polarizations.

right circular pol. left circular pol. 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 in photonic crystals? Æ YES! (Example) 2D photonic crystals (PC)

Caution : Berry curvature is zero for 2D PC with inversion symmetry.

We use a 2D PC without inversion symmetry. 2D photonic crystals

(“Photonic Crystals”, Joannopoulous et al.) Simulation : Dielectric constant and its band structure

Dielectric constant Photonic band structure Berry phase in photonic crystals

Large Berry curvature when the band approach other bands in energy Trajectory of light beam in photonic crystals Simulation

r r To see the anomalous velocity k& ×(z | Ω r | z) r k k should change in time.

( In addition to periodic modulation of ε ),(rr) slow 1D modulation needed

slow 1D modulation near x=0

larger ε for larger x

-- Analog of the electric field in the SHE Large shift !!

× envelope function linear in x Enhancement of Berry curvature in photonic crystals

r 1 Ω r ≈ ≈ λ • vacuum : k 2 Shift (e.g.: Imbert shift) k very small

• photonic crystals: ω 2∆ Ω ≈ v z r r 3/ 2 ()∆2 + v2 (k − k )2 Slope (velocity) 0 v ∆ v2 maximum at the gap : Ω ≈ z ∆2 v Maximum shift of the beam ∆

r r k Bigger for smaller gap k0 Shift of the light beam ≈ Distance from a monopole in k space

• Vacuum: ω

(shift) ≈ λ << (width of the beam) Small shift

k • Photonic crystal:

(shift) ≈ v ∆ ω

>> 1 (width of the beam) r k ∆ Shift can be large (for small gap) k