Variations on the Spin Hall Effect in a Two-dimensional Electron Gas

Transcription

1 Variations on the Spin Hall Effect in a Two-dimensional Electron Gas Roberto Raimondi Dipartimento di Matematica e Fisica Disorder and Correlations in Quantum Systems September 2013 Sapienza, University of Rome

2 Introduction Main motivation The spin Hall effect (SHE) has emerged over the last decade as one of the most promising transport paradigm in spintronics. Perhaps, the main reason is due to the potential for an all-electrical control of the electron spin via the spin-orbit interaction. Outline 1 Brief historical sketch of SHE 2 SHE in a 2DEG with Rashba SOC 3 First variation: Interplay of intrinsic and extrinsic mechanisms in a 2DEG 4 Second variation: Onsager relations in the presence of SOC 5 Third variation: spin Nernst effect 6 Conclusions and some personal recollections

3 The spin Hall effect (SHE) Dyakonov and Perel, JETP Lett. 13, 467 (1971); Phys. Lett. A 35, 459 (1971) E x z y j z y Symmetry argument Dyakonov PRL 99, (2007) Charge current: odd under T and P Spin current: odd under P, even under T For P-invariant systems Onsager relations imply one parameter γ proportional to Spin-orbit coupling (SOC) j yz = σ sh E x j i = σe i σ a Ei a 4 ε ija γ j ja j ia = σ 4e 2 E i a + σ a E i + ε iak γ j k, σ sh = γ σ SO Mechanisms: extrinsic (impurities) and intrinsic (band and device structure) (Cf. Nozières and Lewiner 1973 for AHE; π θreview: Nagaosa et al. 2010) θ p F = 2m θ π θ Mott skew scattering (Smit 1958) θ θ π θ L SO = 2m Side jump (Berger 1970) Rashba in 2DEG (Bychkov and Rashba 1984)

4 Biodiversity of the SHE (and ISHE) due to several mechanisms Theoretical proposals after Dyakonov and Perel Hirsch 1999 Zhang 2000 Murakami, Nagaosa, Zhang 2003 Sinova et al Review: Engel, Rashba, Halperin 2007 Review: Sinitsyn 2008 Review: Vignale 2010 Experiments interpretation (not exaustive list!) GaAs film, extrinsic, γ 10 4 Kato et al. Sci. 306, 1910 (2004) In x Ga 1 x As, extrinsic, γ 10 3 Garlid et al. PRL (2010) 2DHG in GaAs, intrinsic, γ 10 2 Wunderlich et al. PRL 94, (2005); 2DEG in ALGaAs extrinsic and intrinsic γ 10 3 Wunderlich et al. Nat. Phys. 5, 675 (2009) Al, extrinsic, γ 10 4 Valenzuela and Tinkham, Nat. 442, 176 (2006) Pt, Au, Mo, Pd, intrinsic, γ Mosendz et al. PRB 82, (2010) HgTe/(Hg,CdTe), intrinsic Brüne et al. Nat. Phys. 6, 448 (2010) Ir-doped Cu, extrinsic, γ 10 1 Niimi et al. PRL 106, (2011) β-ta, intrinsic, γ 10 1 Liu et al. Sci. 336, 555 (2012)

5 Rashba SOC in a 2DEG H = p2 2m + αê z σ p + V disorder (r) = 1 ( p + 1 ) 2 2m 2 A a σ a + const + V disorder (r) SU(2) language Mathur and Stone PRL 68, 2964 (1992); Fröhlich and Studer RMP 65, 733 (1993); Lyanda-Geller PRL 80, 4273 (1998); Hatano et al. PRA 75, (2007); Tokatly PRL 101, (2008) Tokatly and Sherman Ann. Phys. 325, 1104 (2010) Potentials Ψ = 1 2 Ψa σ a A = 1 2 A a σ a Fields Ei a = t Ai a xi Ψ a + i[ψ,a i ] a Bi a = 1 ( 2 ε ijk xj Ak a x k Aj a + i [ ] ) a A j,a k Rasba case Ψ = 0, Ay x = Ax y = 2mα Ei a = 0,Bi a = (2mα) 2

6 SHE in the presence of Rashba SOC (Sinova et al. 2004) Key arguments for the SHE Continuity-like equation for spin density Dimitrova 2005, Chalaev and Loss 2005 E x z y j z y t s y + j y = 2mαj z y No spin current in static and uniform conditions SHE still possible at edges and in transient regime Mishchenko et al. PRL 93, (2004); Raimondi et al. PRB 74, (2006) SU(2) vector potential language for SOC diffusion contribution from non-abelian covariant derivative drift contribution from Lorentz-like force due to SU(2) magnetic field j z y = D( ε zxy 2mαs y ) + σ sh E x diagrammatically correspond to vertex and bubble terms Raimondi and Schwab PRB 71, (2005) By product: vanishing of spin current implies in-plane spin polarization Edelstein, Solid. State Commun. 73, 233 (1990); Aronov and Lyanda-Geller, JETP Lett. 50, 431 (1989)

7 How to obtain SHE in the presence of Rashba SOC Add extra source of spin relaxation t s y + j y = 2mαj z y 1 τ s s y j z y = D( ε zxy 2mαs y ) + σ sh E x SOC from impurities H extso = λ 0 2 σ V (r) p 4 Scattering from magnetic impurities H magimp = S(r) σ Competition between τ s and Dyakonov-Perel mechanism l = v F p F = 2m mean free path L SO = m Dyakonov-Perel spin relaxation L 2 SO = D DP S = A + k k B x 2 0s k Side-jump ( s B x t s y 1 = + 1 ) s y 2mασ sh E x τ DP τ s skew-scattering R(AB ) 2 0 Elliott-Yafet spin relaxation E z y SHE

8 SU(2) formulation: Technical point Combine SU(2) with Keldysh GF: Gorini et al. PRB 82, (2010) Introduce gauge invariant Green function In the standard U(1) case G(R,t;p,ε) G(R,t;p + ea,ε) For the SU(2) case define a locally covariant Green function G(R,t;p,ε) G(R,t;p,ε) 1 { A p,g(r,t;p,ε) } 2 Define distribution function in terms of the covariant Green function f (R,p,t) = 1 [ ] dε πi G(R,t;p,ε)

9 Interplay of intrinsic and extrinsic SHE in the 2DEG Effective Boltzmann equation Raimondi et al. Ann. Phys. (Berlin) 524, 153 (2012) ( [ t + p x m f p + 1 λ 2 ] 0 2τ 4 {p σ,f p} + 1 ) 2 {F p,.} f p = I 0 [f p ] + I l [f p ] l=0,4 side-jump I λ0 2 W pp p 4 skew-scattering I 2 { ee σ (p p), f } p ε p W pp (πn 0 u) λ 2 0 { p p σ,f p 4 p } Elliott-Yafet I 3 I 3 [f p ] = λ0 4 W pp p 16 ( p p ) 2 ( fp σ z f z p σ z ) Extra term I 4 λ0 2 8 { } W pp ε abc ( x ) a σ b,p c f p p cf p p Key observation Separation of terms: intrinsic SU(2)-fields in the hydrodynamic derivative, while extrinsic in the collision integral Spin-continuity equation (almost) unchanged and extra extrinsic drift terms t s y + 1 τ s s y = 2mαj z y ( j z y = 2mαDs y + σ sh R + σ sh sj ) + σss sh E x

10 Spin Hall conductivity due to both intrinsic and extrinsic mechanisms The uniform steady-state σ sh = σ R sh + σ sj sh + σss sh 1 + τ s /τ DP when α 0, σ sh σ sh ss+sj since τ DP τ s /τ DP controlling parameter Raimondi and Schwab, EPL 87, (2009); Raimondi et al. Ann. Phys. 524, 153 (2012) Modified Edelstein effect s y = 2mατ ( DP 1 + τ DP /τ s λ 0 0 σr sh + σ sh sj s y = 2mατ DP σr sh E x Standard Edelstein effect ) + σss sh E x Some non trivial remarks Notice that σsj sh,σss sh λ0 2 as for standard AHE, while τ s 1 λ0 4 at α 0, it is enough to confine at order λ0 2 when α 0, one must consider terms at order λ0 4, otherwise one would get a paradoxical result that the limit α 0 is different from α 0

11 Spin current definition and Onsager relations Let me ask two questions: What is the correct definition of the spin current? Proposal to replace the conventional (and most natural) definition (which is not generally conserved because of e.g. spin-orbit interaction) J a i = 1 2 {v i,s a } with a conserved one Shi et al. PRL 96, (2006) J a i = J a i + P a i, i P a i = T a Are Onsager relations satisfied for SHE and ISHE? Observation that a spin current (which may induce a ISHE) may be driven by different external spin-dependent (SU(2)) potentials Wang et al. PRB 85, (2012)... and answer that the two issues are intimately related Gorini et al. PRL 109, (2012) Both definitions are legitimate (provided being careful) Connection between spin current definition and external driving field

12 How to see this Consider both the spin-vector gauge and spin-scalar gauge H V = H 0 + J z i A z i, A z i = te z i H S = H 0 + τz 2 Ψz, Ψ z = r i E z i, Perform a gauge transformation from the scalar to the vector one ( U = exp i 1 ) 2 τz χ In general: H S = H 0 + Ji z Ai a,ji z = ī [ τ z ] h 2 r i,h S Note: only if H S commutes with τ z the current reduces to the conventional one

13 Results in practice Rashba 2DEG Using the conventional current σ sh (ω) = γσ e Using the conserved current σ sh (ω) = γσ e iω + τs 1 iω + τdp 1 + τ 1 s iω iω τdp 1 + τ 1 s iω + 2τDP 1 iω + τdp 1 +. τ 1 s σ Drude conductivity γ = γ intr + γ ss + γ sj charge-spin coupling parameter γ intr = mα 2 τ, γ sj = λ 2 0 m 4τ γ ss = (λ 0p F ) 2 (2πN 0 v) 16 τdp 1 Dyakonov-Perel relaxation τs 1 Elliott-Yafet relaxation Rashba 2DHG Hughes et al. PRB 74, (2006) σ sh = γσ e, σ(ω) = γσ iω e iω + τdp 1, γ = 6 α 3 kf 4 mτ Experiment should tell which current is coupled to an external probe

14 Spin Caloritronics: Spin Nernst Effect Seebeck effect ΔT j Peltier effect j q2 6= j q1 j q1 1 2 j q1 j q2 Spin Seebeck effect ΔT Spin Peltier effect j q1 j q2 =0 FM N j s j q1 K. Uchida et al., Nat. Lett. 455, 778 (2008); Nat. Mater. 9, 894 (2010). C. M. Jaworski et al., Nat. Mat. 9, 898 (2010). Review: Adachi et al. Rep. Prog. Phys. 76, (2013). A. Slachter et al., Phys. Rev. B 84, (2011); J. Flipse et al., Nature Nanotechnology 7, 166 (2012). B. Scharf et al., Phys. Rev. B 85, (2012). giovedì 16 maggio 13 Nernst effect ΔT Spin Nernst effect ΔT Ballistic limit for Rashba model: Z. Ma, Solid State Communications, 150, 510 (2010 Present work: inclusion of disorder and interplay of several SO mechanisms

15 Phenomenological considerations and main result Charge and heat currents j x = L 11 E x + L 12 ( x T ) Electrical conductivity L 11 = σ and electric thermopower S L 12 /L 11 L 12 = el T σ with Lorenz number L = π 2 k 2 B /3e2 and σ = µ σ, µ Spin and heat currents in a simple situation Assume inversion symmetry for a homogeneous and non-ferromagnetic material in the absence of magnetic fields M. I. Dyakonov, Phys. Rev. Lett. 99, (2007): j z y = γj x j z y = L s 11 E x + L s 12 ( x T ), S s L s 12 /Ls 11 S s = S Spin and charge thermopowers coincide but what about when Rashba SOC is present? S s = SR s with R s depending on the various competing SO mechanisms R so 3 can be achieved, in principle, in GaAs samples with Rashba SO and extrinsic mechanisms heat-to-spin conversion could be more efficient than heat-to-charge

16 Detail: spin equivalent of the Mott formula Same range of validity of the Wiedemann-Franz law based on single-particle description of transport, Fermi statistics of carriers, elastic scattering (G. V. Chester and Thellung, Proc. Phys. Soc. London 77, 1005 (1961; M. Jonson and G. D. Mahan, Phys. Rev. B 21, 4223 (1980); J. S. Langer, Phys. Rev. B 128, 110 (1962).) The spin thermopower S s = el T σ sh /σ sh = el T σ σ [ 1 + γ ss γ ζ 1 + ζ ( 1 2τ )] s. τ EY γ = γ intr + γ sj + γ ss Combine the various spin-charge current couplings 1 τ DP = (2m) 2 (α 2 + β 2 )D Combine D-P relaxation from Rashba and Dresselhaus SO interaction 1 τ s = τ 1 EY + 3τ 4 sf Combine spin relaxation from SO from impurities and from magnetic impurities ζ = τ s τ DP 4 τ2 s /(τ R DP τd DP ) τ s /τ DP +1

17 The spin thermopower S s of a disordered 2D-electron gas Borge et al. PRB 87, (2013) Each panel shows the ratio S s /S as a function of the ratio α/β (Rashba / Dresselhaus ) for a given Elliott-Yafet scattering strength, strong to weak from top left to bottom right Ss/S Ss/S τ EY τ D DP τ EY τ D DP = 0.1 = α/β τ EY τ D DP τ EY τ D DP = 1 = α/β Magnetic scattering is strongest for the dotted curve, τ sf /τdp D = 1, and strong (weak) for the dashed (solid) curves, τ sf /τdp D = 2,3(10,20,30). Panel 3 corresponds to standard GaAs mobility µ = 10 4 cm 2 /Vs density n = cm 2 effective extrinsic wavelength λ 0 = cm Dresselhaus coupling constant hβ = evm Then γ ss γ intr,γ sj, τ EY τ D DP.

18 Conclusions and acknowledgements Non trivial interplay in the SHE between Rashba and impurity driven SOCs Spin current definition associated to the gauge choice for spin potentials Onsager relations hold irrespective of the choice Spin thermopower enhanced with respect to charge thermopower Collaborators Juan Borge Cosimo Gorini Andrei Shelankov Peter Schwab Giovanni Vignale Papers EPL 87, (2009) PRB 82, (2010) Ann. Phys. 524, 153 (2012) PRL 109, (2012) PRB 87, (2013)

19 When all this began PHYSICAL REVIEW B, VOLUME 64, Spin-orbit induced anisotropy in the magnetoconductance of two-dimensional metals R. Raimondi and M. Leadbeater INFM e Dipartimento di Fisica E. Amaldi, Università di Roma Tre, Via della Vasca Navale 84, Roma, Italy P. Schwab Institut für Physik, Universität Augsburg, Augsburg Germany E. Caroti and C. Castellani INFM e Dipartimento di Fisica, Università La Sapienza, piazzale A. Moro 2, Roma Italy Received 7 February 2001; revised manuscript received 18 May 2001; published 27 November 2001 Spin-orbit coupling due to structure inversion asymmetry leads to a characteristic anisotropy in the magnetoconductance of two-dimensional metals. By calculating the Drude conductivity in the case of a realistic spin-orbit coupling, for a system near the metal-insulator transition we find an anisotropy which is of the order of several percent. This agrees with recent experiments. p i /m i. The equation for the momentum independent part of the current vertex reads i i ss ss 1 2 N 0 p ab G sa R i ab G A bs. 14 One observes that for a weak magnetic field the quantities become very small, so that J RA p/m, i.e., the anomalous velocity is canceled by the vertex corrections. Only for fields which are at least of the order of the Fermi energy does an Happy Birthday Claudio!