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Spin Hall effect and related issuesDept of PhysicsTaiwan Normal Univ.Ming-Che Chang8/22/2005

past/nowgoal magnetic memory, GMR, TMR generation, manipulation, and detection of spins inmetals, semiconductors on-going effort FM/semiconductor spin injection not easy magnetic semiconductor not easywish for integration with existing semiconductor technology control via electric field, instead of magnetic field more researches on the spin-orbit coupling insemiconductors

Spin-orbit interaction in semiconductor(Kittel, Quantum Theory of Solids)H SOr r1 r SV(x) v2mc 2(V(x) is the lattice potential energy) splitting of valence bands (GaAs, D 0.34 eV) change of g-factor (GaAs, g* -0.44) for materials without inversion symmetry,lift the spin degeneracy of energy bands(Dresselhaus, Rashba) skew scattering from impuritiesr rr λ SOσ s' s k ' ktransition rate,Wksr kr ' s 'For strong SO couplings, choose low-symm,narrow-gap materials formed from heavyelements (g*»-50 in InSb) (Rashba, condmat/0309441)

Generation of spin in semiconductorusing SO coupling (Rashba PRB 2004)[1] Hirsch, PRL 1999 spin Hall effect (SHE), skew scattering Voskoboynikov et al, PRB 1999and many others resonant tunneling related ideas Kiselev and Kim, APL 2001 T-shaped filter Ioniciociu and D’Amico, PRB 2003 Stern-Gerlach devicedevicedesign Ramaglia et al, Euro Phys J B 2003 quantum point contact Watson et al, PRL 2003 adiabatic pumping (need B field) Rokhinson et al, PRL 2004 electron focusing (need B field) Bhat and Sipe, PRL 2000 all-optical technique Mal’shukov et al, PRB 2003 AC gate[2] Murakami et al, Science 2003 SHE, in bulk p-type semiconductor[3] Sinova et al, PRL 2004 SHE, in n-type heterojunction (2DEG)

Hall effect (E.H. Hall, 1879)[1] Spin Hall effect(J.E. Hirsch, PRL 1999, S Zhang,PRL 2000, Dyakonov and Perel,JETP 1971.)skew scatteringby spinless impurities:no magnetic field requiredFrom spin accumulation to charge accumulationL spin coherence length dsds »130 mm at 36 K for Al(Johnson and Silsbee, PRL 1985)

[2] Intrinsic spin Hall effect in p-type semiconductor (I)(Murakami, Nagaosa and Zhang, Science 2003)Valence band of GaAs:Luttinger Hamiltonian (1956)(for j 3/2 valence bands)r r 2 1 5 2H γ 1 γ 2 k 2γ 2 k S 2m 2 r λ k S (helicity)( )is a good quantum number(Non-Abelian) gauge potentialrr rAλλ ' ( k ) i k , λ r k , λ ' kBerry curvature,due to monopole field in k-spaceFGHIJKr r7 k 2Ω λ ( k ) 2 λ λ 4 k2

Intrinsic spin Hall effect in p-type semiconductor (II)JySemiclassicalEOMrrdkh eEdtrrrdx E λ ( k ) dk r rr Ωλ (k )dtdth kExAnomalous velocitydue to Berry curvature(Chang and Niu, PRL 1995Sundaram and Niu, PRB 1999)Spin currentr1k FHz& nλ ( k ) 2 eE x ,HH J yS3 λ 3/ 2 ,kr4πr1k FLzz& nλ ( k ) LH J y eE x , yS3 λ 1/ 2 ,kr12π 2zySpin Hall conductivitycchhe3k FH k FL212πeHL k kFF12π 2e 2 k FH k FL6π σ zyx cJ yz σ zyx E x(semiclassical)(Q correction)hNo magnetic field requiredApplies to Si as well

[3] Intrinsic spin Hall effect in 2 dimensional electron gas (2DEG)(Sinova, Culcer, Niu, Sinitsyn, Jungwirth, and MacDonald, PRL 2004)Semiconductor heterojunctionzz» triangularquantum well

QW with structure inversion asymmetry (SIA):Rashba coupling (Sov. Phys. Solid State, 1960)p2 α r rH σ p z 2m h 1974 Ohkawa and Uemura, due to gradient of the confinement potential V / z 1976 Darr, V / z for a bound state is actually zero 1985 Lassnig, interface/valence band are crucialNo easy way tocalculate aZawadzki’s, Semi Sci Tech 2004)VG-dependence of the Bychkov-Rashba parameter Can be determined from thebeating of dHvA oscillation tunable by gate voltageEngels et al 1997 PRB,InP/In 0.77 Ga 0.23 As/InP

Intrinsic spin Hall effect in 2DEGRashba Hamiltonian (1960)p2 α r rH σ p z 2m hrλ (σ p ) z (helicity)r rr rασ k z µ Bσ Beffrr rBeff ( k ) λz kis a good quantum numberEigen-energiesrh2 k 2Eλ ( k ) λαk , λ 12mEl -1El(k) El(-k)Kramer degeneracy no space inversion symmetryk invariant under time reversal

Dynamics of spin under electric perturbation(l -1)dk -eEt // -xdBeff » lz dk // -lyLandau-Lifshitz eq.rr r rdS r r r S Beff ( k ) γS S BeffdtdidampingWhen both bands are filled,spin Hall conductivity:e σ zyx independent of α8π not so for non-parabolic bandsJyEx only for clean system not related to Berry curvatureNo magnetic field required

Effect of disorder on the intrinsic spin Hall effect (I) Rashba system with short-range impurities Inoue et al (2003) Sheng et al, cond-mat/0504218 Dimitrova (2004) Nomura et al cond-mat/0506189 Khaetskii (2004) Raimonde and Schwab (2004) σ SH σcleanSH σvertexSHFGHIJKee 0! 8π8π Perturbative calculations for other systems If H(k) H(-k), eg. Luttinger modelthen vertex correction is zero (Murakami, PRB 2004)rrrr For systems with H ( k ) E 0 ( k ) σ x d y ( k ) σ y d x ( k )r rIf E0 / k d , then perfect cancelation (eg. Rashba)otherwise σ s remains finite. (quoted from Murakami' s talk)Spin Hall effect is finite in general

Effect of disorder on the spin Hall effect in Rashba system (II) sSH robust against weak disorder in finite systems Nikolic et al, cond-mat/0408693 Hankiewicz et al, PRB 2004 Sheng et al, PRL 2005Stronger SO coupling

Spin Hall effect observed (I) (Kato et al, Science 2004) Local Kerr effect in strained n-typebulk InGaAs, 0.03% polarizationMostly likely extrinsic.

Spin Hall effect observed (II) (Wunderlich et al, PRL 2005) spin LED in GaAs 2D hole gas,1% polarizationpnmight be intrinsic?(Bernevig and Zhang, PRL July 2005)Spin Hall effect observed (III) (Sih et al, cond-mat/0506704) n-type GaAs [110] QW

Dresselhaus coupling (PRB 1955):III-V semiconductorwith bulk inversion asymm (BIA)(BIA)[001]r r QW,2 linearΩ( k ) k n ( k x , k y )rr r rH ( k ) S Ω( k )r rΩ( k ) k x ( k y2 k z2 ), k y ( k z2 k x2 ), k z ( k x2 k y2 )d[110] QWr rΩ( k ) k n2 ( k x / 2, k x / 2)RashbaRashba and Dresselhaus,p2 α[001] quantum well:H σ x p y σ y px*2mhβ σ x px σ y p yhddDresselhaus[111] QWr rΩ( k ) ( 2 / 3 ) k n2 ( k y , k x )(SIA)iii

Effective magnetic field:BIASIABIA SIABIA¹SIAGanichev and Prettl, cond-mat/0304266σ zxy eN8πFor 2D electron systems, with Rashba and Dresselhaus coupling,N 1 if Rashba DresselhuasN -1 if Dresselhaus Rashba (Shen, PRB 2004)For 2D hole system with (cubic) Rashba, N 9(Schliemann and Loss, PRB 2005)

Rashba-Dresselhaus system in an in-plane magnetic fieldp2 αγH σp σp σ x px σ y p y β xσ x β yσ yxyyx*2mhhdi dEigen-energies:rrEλ ( k ) E0 ( k ) λdγk x αk y β xii d2i2 αk x γk y β y ,λ Distorted Fermi surfaces (generic cases):(c)(b)(a)FGHrγβ x αβ y αβ x γβ yPoint of degeneracy k 0 , 2α2 γ 2α γ 2Parameters: α 1 eV A (tunable by gate voltage)γ of the same orderbgβ g * /2 µ B B, µ B 0.06 meV / Tk F 2πn 102 / A for n 1011 / cm2IJK

Effect of in-plane magnetic field on spin Hall conductivityσ ηµνBKubo formularrrf kr ,λ f kr ,λ ' r1ηk , λ jµ k , λ ' k , λ ' jν k , λ , r 2ih k ,λ ,λ ' ω λλ ' kJyb λ λ 'gExjµη dihvµσ η σ η v µ ;4dFor g 0 (pure Rashba)iE , minjν evνE(k0)E(k0)E-, minE , minE-, minrσ ( B ) could be changed by 100% simplyzxyby rotating the magnetic field

Spin Hall conductivity (electron density fixed)yσ xxy σ xy 0Boundary of plateau E(k0) mβ 4αγβ x β y β2x2yαc ζ2 γ(1) (3)h2 2(2)α2 γ 2(2)M.C. Chang, PRB 2005Acknowledgement: M.F. Yang

Existence of charge Hall effect?FGHThouless formula (PRL 1982)σλxye2 hIJKr11,k , iθ2 ierΩ λ ( k ),r tan θ k filledFGHIJKiθr1 ie,k , 2 1γk x αk y β xαk x γk y β yBerry curvaturerΩλ (k ) i λ ' λrrrrrrrrk , λ vx k , λ ' k , λ ' v y k , λ k , λ v y k , λ ' k , λ ' vx k , λr 02kω λλ()'R λπ for αr rr for αΓ z dk k , λ i r k , λ S0 k T λπ for αrr r Ω ( k ) sgncα γ hλπδ d k k iBerry phaseλλ22 γ 22 γ22 γ 2at every k, except atdegenerate point k0(c)(a)20Hall conductivity is zerowherever the chemicalpotential is0 0 0(-p) p 0

Issues on the spin current in SO coupled systems(Rashba, cond-mat/0408119) spin current is not well defined (total spin not conserved)rα αS J Re ψ s&αψ , Spin torque trαr rα αwhere S ψ s ψ , J (1 / 2) Re ψ sα v vsα ψ Spin fluxch existence of background spin current Rashba, PRB 2003(which produces no spin accumulation)rx rry rJ ( k , λ ) α / 2 y ; J ( k , λ ) α / 2 x no experimental procedure to measure it directly(accumulation? Induced electric field?) Meier and Loss, PRL 2003 connection with Maxwell eqs?(Bernevig and Zhang, PRL Aug 2005)

(Johnson and Silsbee, PRL 1985) no magnetic field required skew scattering by spinless impurities: [1] Spin Hall effect (J.E. Hirsch, PRL 1999, S Zhang, PRL 2000, Dyakonov and Perel, JETP 1971.)