Theory challenges of semiconducting spintronics: spin-Hall effect and spin-dependent transport in spin-orbit coupled systems JAIRO SINOVA New Horizons in Condensed Matter Physics Aspen Center for Physics February 4th 2008 Research fueled by: NERC The challenges ahead in semiconductor spintronics Spin/charge transport in multi-band systems with interband coherence (Berrys phase dependent transport): SHE and AHE in strongly SO coupled systems, etc. What are the relevant length scales: spin-current connection to spinaccumulation. QSHE: transport in Z2 systems. Technological issues: how dissipative is it? Interplay between quasiparticle and collective degrees of freedom in a multiband system: Carrier mediated ferromagnetism: diluted magnetic semiconductors Magnetization dynamics: obtaining phenomenological LLG coefficients through microscopic calculations

Anomalous Hall effect: where things started, the long debate Spin-orbit coupling force deflects like-spin particles majority __ F _ SO FSO I H R0 B 4RRs M minority Simple electrical measurement of magnetization V controversial theoretically: semiclassical theory identifies three contributions (intrinsic deflection, skew scattering, side jump scattering) Spin Hall effect _ FSO

__ FSO non-magnetic I V=0 Spin-current generation in non-magnetic systems without applying external magnetic fields Spin accumulation without charge accumulation excludes simple electrical detection Carriers with opposite spin are deflected by the SOC to opposite sides. Intrinsic deflection Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure) E

Electrons have an anomalous velocity perpendicular to the electric field related to their Berrys phase curvature which is nonzero when they have spin-orbit coupling. Side jump scattering Related to the intrinsic effect: analogy to refraction from an imbedded medium Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step. Skew scattering Asymmetric scattering due to the spinorbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators. Spin Hall Effect (Dyaknov and Perel) Interband Coherent Response (EF) Intrinsic

`Berry Phase (e2/h) kF [Murakami et al, Sinova et al] 0 Occupation # Response `Skew Scattering (e2/h) kF (EF )1 X `Skewness Influence of Disorder `Side Jump [Hirsch, S.F. Zhang] [Inoue et al, Misckenko et al, Chalaev et al.] Paramagnets Quantum Spin Hall Effect (Kane et al and Zhang et al)

Future challenges in anomalous transport theory 1. Reaching agreement between different approaches (mostly AHE) 2. Connect spin current to spin accumulation for strongly SO system 3. Connect SHE to the inverse SHE in strongly SO coupled regime (charge based measurements of SHE) 4. Understanding weak localization corrections in SO coupled systems for Hall transport 5. Systematic treatment of microscopic calculations of AHE in strongly SO coupled ferromagnet (e.g. DMS) with complex band structure 6. Dissipation: answer the questions if a spin based device can really beat the kBTln2 limit of dissipation 1. Intrinsic + Extrinsic: Connecting Microscopic and Semiclassical approaches Sinitsyn et al PRL 06, PRB 07

Need to match Kubo to Boltzmann to Keldysh Kubo: systematic formalism Boltzmann: easy physical interpretation of different contributions Keldysh: microscopic version of Boltzmann AHE in Rashba systems with disorder: Dugaev et al PRB 05 + more Sinitsyn et al PRB 05 Inoue et al (PRL 06) Onoda et al (PRL 06) Borunda et al (PRL 07) All are done using same or equivalent linear response formulation different or not obviously equivalent answers!!! 2. From spin current to spin accumulation The new challenge: understanding spin accumulation

Spin is not conserved; analogy with e-h system Spin Accumulation Weak SO Quasi-equilibrium Parallel conduction Spin diffusion length Burkov et al. PRB 70 (2004 SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION STUDIES so>>/ Width>>mean free path Nomura, Wundrelich et al PRB 06 Key length: spin precession length!! Independent of !!

SHE experiment in GaAs/AlGaAs 2DHG p 1.5mm channel LED1 0 y -1 z n LED2 x 1 0 Polarization in % n

Polarization in % 1 -1 1.505 1.510 1.515 1.520 Energy in eV Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05 - shows the basic SHE symmetries 10mm channel - edge polarizations can be separated over large distances with no significant effect on the magnitude

- 1-2% polarization over detection length of ~100nm consistent with theory prediction (8% over 10nm accumulation length) Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '05 3. Charge based measurements of SHE Non-equilibrium Greens function formalism (Keldysh-LB) Advantages: No worries about spin-current definition. Defined in leads where SO=0 Well established formalism valid in linear and nonlinear regime Easy to see what is going on locally PRL 05

H-bar for detection of Spin-HallEffect (electrical detection through inverse SHE) E.M. Hankiewicz et al ., PRB 70, R241301 (2004) New (smaller) sample sample layout 200 nm 1 mm SHE-Measurement 7000 5 insulating n-conducting p-conducting Rnonlocal /

5000 4 Q2198H I (3,6) U (9,11) 4000 3 3000 2 I / nA 6000 2000 1000 1 0

0 -2 -1 strong increase of the signal in the p-conducting regime, with pronounced features 0 1 2 V Gate / V 3 4 5 no signal

in the n-conducting regime Mesoscopic electron SHE calculated voltage signal for electrons (Hankiewicz and Sinova) L/2 L/6 L Mesoscopic hole SHE calculated voltage signal (Hankiweicz, Sinova, & Molenkamp) L L/2 L/ 6

L more than 10 time larger! WHERE WE ARE GOING (THEORY) Theoretical achievements: Intrinsic SHE back to the beginning on a higher level 2003 Extrinsic SHE approx microscopic modeling Extrinsic + intrinsic AHE in graphene: two approaches with the same answer Theoretical challenges: GUT the bulk (beyond simple graphene) intrinsic + extrinsic SHE+AHE+AMR Obtain the same results for different equivalent approaches (Keldysh and Kubo must agree) Others

materials and defects coupling with the lattice effects of interactions (spin Coulomb drag) spin accumulation -> SHE conductivity 2006 EXTRAS WHERE WE ARE GOING (EXPERIMENTS) Experimental achievements Optical detection of current-induced polarization photoluminescence (bulk and edge 2DHG) Kerr/Faraday rotation (3D bulk and edge, 2DEG) Transport detection of the SHE Experimental (and experiment modeling) challenge General edge electric field (Edelstein) vs. SHE induced spin accumulation Photoluminescence cross section edge electric field vs. SHE induced spin accumulation free vs. defect bound recombination spin accumulation vs. repopulation angle-dependent luminescence (top vs. side emission)

hot electron theory of extrinsic experiments SHE detection at finite frequencies detection of the effect in the clean limit Scaling of H-samples with the system size L=90nm change of voltage [ mV] 5.0 4.5 4.0 Spin orbit coupling = 72meVnm n=1*1011cm-2 3.5 L/6 L=150nm 3.0 2.5

2.0 0.000 L=240nm L=120nm L=200nm 0.004 0.008 L 0.012 1/L [nm] Oscillatory character of voltage difference with the system size. Aharonov-Casher effect: corollary of Aharonov-Bohm effect with electric fields instead M. Koenig, et al, "Direct observation of the Aharonov-Casher phase", Phys. Rev. Lett. 96, 076804 (2006).

Alexey A. Kovalev, et al "Aharonov-Casher effect in a two dimensional hole ring with spin-orbit interaction", pre-print: cond-mat/0701534, submitted to Phys. Rev. B Control of conductance through a novel Berrys phase effect induced by gate voltages instead of magnetic fields HgTe Ring-Structures Three phase factors: Aharonov-Bohm Berry Aharonov-Casher s and , parallel and anti parallel to Btot b 1 for , Bext , Btot ; Btot Bext Beff THE THREE CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACH Skew scattering

n, q n, k m, p n, q m, p Skew HSkew (skew)-1 2~0 S where S = Q(k,p)/Q(p,k) 1~ V0 Side-jump scattering Intrinsic AHE Im[] Vertex Corrections Intrinsic Averaging procedures: = -1 / 0 n, q

nn, Intrinsic 0 /F = 0 Success of intrinsic AHE approach in strongly SO coupled systems DMS systems (Jungwirth et al PRL 2002) Fe (Yao et al PRL 04) Layered 2D ferromagnets such as SrRuO3 and pyrochlore ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19 (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003), Shindou and Nagaosa, Phys.

Rev. Lett. 87, 116801 (2001)] Colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999). Ferromagnetic Spinel CuCrSeBr: Wei-Lee et al, Science (2004) Berrys phase based AHE effect is quantitativesuccessful in many instances BUT still not a theory that treats systematically intrinsic and extrinsic contribution in an equal footing. Experiment sAH 1000 ( cm)-1 Theroy sAH 750 ( cm)-1 First experimental observations at the end of 2004 Kato, Myars, Gossard, Awschalom, Science Nov 04 Observation of the spin Hall effect bulk in semiconductors Local Kerr effect in n-type GaAs and InGaAs: (weaker SO-coupling, stronger disorder)

1.52 Wunderlich, Kstner, Sinova, Jungwirth, PRL 05 1 0 CP [%] Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system -1 1.505 Light frequency (eV) OTHER RECENT EXPERIMENTS Transport observation of the SHE by spin injection!! Saitoh et al APL 06 Valenzuela and

Tinkham condmat/0605423, Nature 06 Sih et al, Nature 05, PRL 05 demonstrate that the observed spin accumulation is due to a transverse bulk electron spin current SHE at room temperature in HgTe systems Stern et al PRL 06 !!! Kubo-Streda formula summary I xy xy = + II xy e 2 + df(f()Tr[vGv-) =- df()Tr[vGvTr[v x (G R -G A )v y G A 4d - df()Tr[vGv-v x G R v y (G R -G A )] Semiclassical Boltzmann equation

f l f l eE l 'l ( f l f l ' ) t k l' I xy e2 + dG R R = df()Tr[vGv-f(f()Tr[vGv-)Tr[v x G v y 4d df()Tr[vGvdG R dG A dG A R A -v x v y G -v x G v y

+v x v yG A ] df()Tr[vGvdf()Tr[vGvdf()Tr[vGvII xy Golden rule: 2 l 'l | Tl 'l |2 ( l ' l ) In metallic regime: Vl 'l ''Vl ''l Tl 'l Vl 'l ... l ' l '' i J. Smit (1956): l 'l ll ' Skew Scattering Semiclassical approach II Golden Rule: 2

l 'l | Vl 'l |2 ( l ' l ) l ( m , k ) Vl 'l Tl 'l Modified Boltzmann f l eEv f 0 ( l ) f 0 ( l ) eE r ( f f ) l ' l 'l l ' l 'l l l ' l l 'l l Equation: t l l l F eE l 'l rl 'l velocity: vl k l'

Sinitsyn et al PRL 06, PRB 06 u u ul ul l l l F Im Berry curvature: z k y k x k x k y Coordinate shift: current: J e f l vl l

rl 'l ul ' i ul ' ul i ul D k',karg Vl 'l k ' k Single K-band with spin up H K =v(k x x +k y y )+ so z Kubo-Streda formula: 2 I xy xy = + II xy e df(f()Tr[vGv-) R A

A df()Tr[vGvTr[v (G -G )v G x y - 4d df()Tr[vGv-v x G R v y (G R -G A )] Ixy =- + In metallic regime: I xy = -e2so 2 4d (vk F ) +so

2 e2 + dG R R = df()Tr[vGv-f(f()Tr[vGv-)Tr[v x G v y - 4d df()Tr[vGvdG R dG A dG A R A -v x v y G -v x G v y +v x v yG A ] df()Tr[vGvdf()Tr[vGvdf()Tr[vGvII xy II xy

=0 2 4 4(vk ) 3(vk ) 1+ F F + 2 (vk )2 +4 2 F so (vk F )2 +4so

e2 V3 2 2 2dn V so (vk F )4 2 2 (vk ) +4 2 F so Sinitsyn et al PRL 06, PRB 06 SAME RESULT OBTAINED USING BOLTMANN!!! For single occupied linear Rashba band; zero for both occupied !! Success in graphene EF

Armchair edge Zigzag edge Comparing Boltzmann to Kubo in the chiral basis f 0 ( l ) f l f 0 ( l ) eEvl l 'l eE rl 'l l 'l ( f l f l ' ) t l l l' l' The spintronics Hall effects: multiband transport with inter-band coherence SHE charge current gives spin current AHE

polarized charge current gives charge-spin current SHE-1 spin current gives charge current Anomalous Hall transport Commonalities: Spin-orbit coupling is the key Same basic (semiclassical) mechanisms Differences: Charge-current (AHE) well define, spin current (SHE) is not Exchange field present (AHE) vs. non-exchange field present (SHE-1) Difficulties: Difficult to deal systematically with off-diagonal transport in multiband system

Large SO coupling makes important length scales hard to pick Farraginous results of supposedly equivalent theories The Hall conductivities tend to be small Actual gated H-bar sample HgTe-QW R = 5-15 meV 5 mm GateContact ohmic Contacts First Data HgTe-QW R = 5-15 meV Signal due to depletion? Results... Symmetric HgTe-QW R = 0-5 meV I: 1->4 U:7-10

-1.00E-007 U_7-10 [V] Signal less than 10-4 -5.00E-008 -1.50E-007 -2.00E-007 -2.50E-007 -3.00E-007 -2 -1 0 1

2 3 -V_gate14 [V] Sample is diffusive: Vertex correction kills SHE (J. Inoue et al., Phys. Rev. B 70, 041303 (R) (2004)).