Shuichi Murakami (Department of Applied Physics, University of Tokyo) Collaborators: Naoto Nagaosa (U. Tokyo, CREST, CERC) Shoucheng Zhang (Stanford)

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1 Spin Hall Effect: Bey Phase and Topology in Solids Intenational Meeting on Pespectives of Soliton Physics (U.Tokyo, Feb.17,007) Shuichi Muakami (Depatment of Applied Physics, Univesity of Tokyo) Collaboatos: Naoto Nagaosa (U. Tokyo, CREST, CERC) Shoucheng Zhang (Stanfod)

2 battey Magnetic field B Loentz foce Hall effect

3 battey Magnetic field -e -e -e -e Feomagnet Anomalous Hall effect

4 Spin cuent battey -e -e -e -e -e -e paamagnetic Spin Hall effect = tansvese spin cuent induced by electic field

5 Spin Hall effect (SHE) Electic field induces a tansvese spin cuent. Intinsic spin Hall effect (SHE) y x Bey phase in momentum space (Does not ely on impuity scatteing) E p-gaas z p-type semiconductos (SM, Nagaosa, Zhang, Science 301,1348(003)) j i j = σ s ε σ s E ijk k i: spin diection j: cuent diection k: electic field : even unde time evesal j S i j i v j Nonzeo in nonmagnetic mateials.

6 Extinsic spin Hall effect D yakonov and Peel (1971) Hisch (1999), Zhang (000) impuity scatteing = spin dependent (skew-scatteing) Spin-obit couping up-spin down-spin impuity = elativistic effect E (impuity potential) B if seen fom moving electons, Small Uncontollable Difficult to estimate couple with electon spin

7 Intinsic spin Hall effect Does not ely on impuity scatteing Bey phase in momentum space --- multiband effect Semiclassical eq. of motion (Sundaam,Niu,1999) E k x& 1 n( ) & = k Bn ( k ) h k Anomalous velocity & k = ee (detemined fom the Bloch wf.) Motion of a wavepacket acquies a tansvese shift A ni ( k ) = i nk nk = i u k i u * nk unit cell u k nk i d d x : Gauge field ( nk : peiodic pat of the Bloch wf.) ik x ψ ( x) = u ( x) e nk nk antimonopole B ( k ) = A ( k ) n k n : Connection n ( : band index) monopole

8 Wavepacket motion and Bey phase E k n x& 1 ( ) & = k Bn ( k ) h k & k = ee f (k) obtains a phase duing popagation F ( x) = dk f ( k) e ikx f ( k k a ( k) e c )

9 Valence band of GaAs p-obit (x,y,z) (, ) + spin-obit coupling split-off band (SO) heavy-hole band (HH) doubly degeneate light-hole band (LH) (Kames) Luttinge Hamiltonian (Luttinge(1956)) Helicity λ = k ˆ S h 5 H = γ 1 + γ k m γ ( k S ) ( S : spin-3/ matix) is a good quantum numbe. λ = ± 3 E γ 1 γ = m k : heavy hole (HH) λ = ± 1 E γ1 + γ = m k : light hole (LH)

10 Semiclassical eq. of motion & hk = ee, x& = hk m λ e E B h ( λ ) ( k ) B ( λ ) ( k ) λ = k ˆ S = λ λ 3 λ = ± : HH, λ = ± 1 : LH 7 k k 3 Anomalous velocity (pependicula to S and E ) j j H yx L yx h = 3 h = 3 λ=± λ =± 3 1, k, k λ < 0 λ > 0 Spin cuent (spin//x, velocity//y) ys & ys & x x n n λ λ ( k ) = EzkF 4π H L EzkF ( k ) = 1π,, σ s Hole spin // k e = (3k 1π E // H F z k L F Quantum coection impuity scatteing etc. ) Missing!

11 Spin Hall effect Can contol spins without magnetism o magnetic field Good fo high integation semiconducto spintonics Dissiplationless mechanism Diffeent fom Ohmic tanspot jchage = σe ohmic j spin = σ E s dissipationless Diven by spin-obit coupling, which can be lage than oom tempeatue Suvive even at oom tempeatue cf. Exp. : Sten et al.,physical Review Lettes( 06) n-znse Kimua et al., cond-mat ( 06) Pt

12 Disode effect : pominent in n-type Rashba model: (D n-type) H Sinova et al.(003) σ S = ( σ k σ k ) k = + λ x y m e 8π y x + spinless impuities (shot-ange pot.) + Vetex coection (Inoue, Baue, Molenkamp, PRB (004)) σ vetex S = e 8π + + σ S = 0 fagile against shot-anged impuities Luttinge model (3D p-type): vetex coection=0 fo -fn. impuities. (Muakami, PRB (004)) Robust against shot-anged impuities δ + + = 0

13 Quantum spin Hall phases bulk = gapped (insulato) gapless edge states caies spin cuent topologically potected Kane and Mele,PRL(005) Quantum spin Hall phase simple insulato : 1 pai of edge states : no edge states I = odd I = even I : Z topological numbe : even o odd = (Even o odd numbe of Kames pais of edge states) Even simple insulato Odd quantum spin Hall I dk k agpf ) half BZ π [ u ( k ) Θ u ( k ] i j No expeiments!

14 Z topological numbe N bands below E F L. Fu and C. L. Kane, Phys. Rev. Lett. 95, (005) Phys. Rev. B 74, (006) Insulato with time-evesal symmety Rank vecto bundle ove Billouin zone tous twisted eal K theoy Z topological numbe N N unitay (& antisymmetic at Γ i ) π k y Δ : gauge invaiant (mod ) π k x

15 D bismuth as a candidate fo QSH phase Band stuctue fo D stip SM, Phys.Rev. Lett. 97, ( 06) Z topological numbe zeos of the Pfaffian in half Billouin zone 1 pai of edge states at each edge I=odd Conclusion D bismuth = good candidate fo QSH phase! Suface states of Bi -- lage spin splitting -- Kooteev et al., PRL ( 04). (Calc.) (Exp.)

16 Expeiments on spin Hall effect E 3D n-type, Ke otation Y.K.Kato, R.C.Myes, A.C.Gossad, D.D. Awschalom, Science (004) Sih et al., Natue Phys. (005) Sih et al., PRL (006) Sten et al., PRL(006) D p-type, spin LED J. Wundelich et al., PRL(005) Metal (Pt, Al) -- Invese SHE E. Saitoh, M. Ueda, H. Miyajima, G. Tataa, APL (006) Valenzuela and Tinkham, Natue(006) Metal (Pt) -- SHE & Invese SHE T. Kimua, Y. Otani, T. Sato, S. Takahashi, S. Maekawa, cond-mat(006)

17 Why spintonics? Electonics Electon=Chage Spintonics Spin degee of feedom etc. Enhanced functionality Low powe consumption Useful fo memoy stoage, logic, quantum computing

18 Towad semiconducto spintonics devices (e.g.) Datta-Das spin tansisto(1990) souce dain Spin injection Spin manipulation Spin detection Poblem: effective spin injection into semiconductos cuent Feo. metal semiconducto cuent Feo. semicond. ( Ga 1-x Mn x As etc.) semiconducto Conductivity mismatch spin polaization lost at inteface No feomagnetism at oom tempeatue (Cuie tempeatue <140K)

19 semiconducto Spin Hall effect

20 Hall effect of light Onoda, SM, Nagaosa, Phys. Rev. Lett.93, (004) -- Analog of the spin Hall effect -- Semiclassical eq. of motion & v kˆ & = ( ) + k & k = k v( ) & z& = ik Λ( k ) z Ω( k ) z z Geometical optics Femat s pinciple Shift of a tajectoy of light beam Hall effect of light Polaization change Chiao,Wu( 86) : theoy Tomita,Chiao( 86) : expeiment c v( ) = n( ) : slowly vaying In the vacuum z : polaization Ω( k ) = 3 Λ (k k 1 ) : gauge field Ω (k in the helicity basis ) : Bey cuvatue k 1 Left cicula pol. ight

21 Tansvese shift linea polaization Left cicula polaization

22 Tajectoy of light beam in photonic cystals Simulation To see the anomalous velocity k should change in time. & k ( z Ω z) k ( In addition to peiodic modulation of ε( ) ), slow 1D modulation needed slow 1D modulation nea x=0 lage ε fo lage x Lage shift!! Onoda, SM, Nagaosa, Phys. Rev. Lett.93, (004)

23 Summay Spin Hall effect in semiconductos Can induce spin cuent without magnetic field o magnet Dissipationless mechanism Due to spin-obit coupling suvive even at oom tempeatue Quantum spin Hall effect: Edge states --- topologically potected Z topological numbe Candidate mateials