Spin-injection Hall effect Spin-injection Hall effect: A new member of the spintronic Hall family University of Maryland March 12 th, 2009 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research fueled by: 1 Hitachi Cambridge Jorg Wunderlich, A. Irvine, et al Institute of Physics ASCR Tomas Jungwirth,, Vít Novák, et al Texas A&M L. Zarbo, M. Borunda, et al Sanford University Shoucheng Zhang, et al

2 Anomalous Hall transport: lots to think about Wunderlich et al SHE Kato et al Fang et al Intrinsic AHE (magnetic monopoles?) AHE Taguchi et al AHE in complex spin textures Valenzuela et al Inverse SHE Brune et al

The family of spintronic Hall effects 3 AHE B=0 polarized charge current gives charge-spin current Electrical detection SHE B=0 charge current gives spin current Optical detection SHE -1 B=0 spin current gives charge current Electrical detection j s ––––––––––– iSHE

4 Towards a spin-based non-magnetic FET device: can we electrically measure the spin-polarization? Can we achieve direct spin polarization detection through an electrical measurement in an all paramagnetic semiconductor system? Long standing paradigm: Datta-Das FET Unfortunately it has not worked: no reliable detection of spin-polarization in a diagonal transport configuration No long spin-coherence in a Rashba SO coupled system

Spin-detection in semiconductors Ohno et al. Nature’99, others Crooker et al. JAP’07, others  Magneto-optical imaging  non-destructive  lacks nano-scale resolution and only an optical lab tool  MR Ferromagnet  electrical  destructive and requires semiconductor/magnet hybrid design & B-field to orient the FM  spin-LED  all-semiconductor  destructive and requires further conversion of emitted light to electrical signal

 non-destructive  electrical  nm resolution with current lithography  in situ directly along the SmC channel (all-SmC requiring no magnetic elements in the structure or B-field) Wunderlich et al. arXives: Spin-injection Hall effect

Utilize technology developed to detect SHE in 2DHG and measure polarization via Hall probes J. Wunderlich, B. Kaestner, J. Sinova and T. Jungwirth, Phys. Rev. Lett (2005) Spin-Hall Effect 7 B. Kaestner, et al, JPL 02; B. Kaestner, et al Microelec. J. 03; Xiulai Xu, et al APL 04, Wunderlich et al PRL 05 Proposed experiment/device: Coplanar photocell in reverse bias with Hall probes along the 2DEG channel Borunda, Wunderlich, Jungwirth, Sinova et al PRL 07

material Device schematic - material 8 i p n 2DHG

- i p n 9 trench Device schematic - trench

i p n 2DHG 2DEG 10 n-etch Device schematic – n-etch

VdVd VHVH 2DHG 2DEG VsVs 11 Hall measurement Device schematic – Hall measurement

2DHG 2DEG e h e e ee e h h h h h VsVs VdVd VHVH 12 SIHE measurement Device schematic – SIHE measurement

Photovoltaic Cell Reverse- or zero-biased: Photovoltaic Cell trans. signal Red-shift of confined 2D hole  free electron trans. due to built in field and reverse bias light excitation with = 850nm ( well below bulk band-gap energy) σoσoσoσo σ+σ+σ+σ+ σ-σ-σ-σ- σoσoσoσo VLVL 13 -1/2 +1/2 +3/2 -3/2 bulk Transitions allowed for ħω>E g Transitions allowed for ħω<E g -1/2 +1/2 +3/2-3/2 Band bending: stark effect Transitions allowed for ħω<E g

n2 + + - - Spin injection Hall effect: Spin injection Hall effect: experimental observation n1 (  4) n2 n1 (  4) n2 n3 (  4) Local Hall voltage changes sign and magnitude along the stripe 14

Spin injection Hall effect  Anomalous Hall effect 15

and high temperature operation Zero bias- + + -- + + -- Persistent Spin injection Hall effect 16

THEORY CONSIDERATIONS Spin transport in a 2DEG with Rashba+Dresselhaus SO 17 For our 2DEG system: The 2DEG is well described by the effective Hamiltonian: Hence

18 spin along the [110] direction is conserved long lived precessing spin wave for spin perpendicular to [110] What is special about ? Ignoring the term for now The nesting property of the Fermi surface:

The long lived spin-excitation: “spin-helix” An exact SU(2) symmetry Only Sz, zero wavevector U(1) symmetry previously known: J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, (2003). K. C. Hall et. al., Appl. Phys. Lett 83, 2937 (2003). Finite wave-vector spin components Shifting property essential 19

Spin configurations do not depend on the particle initial momenta. For the same x + distance traveled, the spin precesses by exactly the same angle. After a length x P =h/4mα all the spins return exactly to the original configuration. Physical Picture: Persistent Spin Helix Thanks to SC Zhang, Stanford University 20

21 Persistent state spin helix verified by pump-probe experiments Similar wafer parameters to ours

The Spin-Charge Drift-Diffusion Transport Equations For arbitrary α,β spin-charge transport equation is obtained for diffusive regime For propagation on [1-10], the equations decouple in two blocks. Focus on the one coupling S x+ and S z : For Dresselhauss = 0, the equations reduce to Burkov, Nunez and MacDonald, PRB 70, (2004); Mishchenko, Shytov, Halperin, PRL 93, (2004) 22

Steady state solution for the spin-polarization component if propagating along the [1-10] orientation 23 Steady state spin transport in diffusive regime Spatial variation scale consistent with the one observed in SIHE

Understanding the Hall signal of the SIHE: Anomalous Hall effect Simple electrical measurement of out of plane magnetization Spin dependent “force” deflects like-spin particles I _ F SO _ _ _ majority minority V InMnAs 24

25 Anomalous Hall effect (scaling with ρ) Dyck et al PRB 2005 Kotzler and Gil PRB 2005 Co films Edmonds et al APL 2003 GaMnAs Strong SO coupled regime Weak SO coupled regime

Intrinsic deflection Side jump scattering 26 Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling. ~τ 0 or independent of impurity density 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. They however come out in a different band so this gives rise to an anomalous velocity through scattering rates times side jump. independent of impurity density Skew scattering Asymmetric scattering due to the spin- orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators. ~1/n i STRONG SPIN-ORBIT COUPLED REGIME (Δ so >ħ/τ) V imp (r) 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 SO coupled quasiparticles

27 WEAK SPIN-ORBIT COUPLED REGIME (Δ so <ħ/τ) Side jump scattering from SO disorder 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. They however come out in a different band so this gives rise to an anomalous velocity through scattering rates times side jump. independent of impurity density  λ*  V imp (r) Skew scattering from SO disorder Asymmetric scattering due to the spin- orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators. ~1/n i  λ*  V imp (r) The terms/contributions dominant in the strong SO couple regime are strongly reduced (quasiparticles not well defined due to strong disorder broadening). Other terms, originating from the interaction of the quasiparticles with the SO- coupled part of the disorder potential dominate. Better understood than the strongly SO couple regime

28 AHE contribution Type (i) contribution much smaller in the weak SO coupled regime where the SO-coupled bands are not resolved, dominant contribution from type (ii) Crepieux et al PRB 01 Nozier et al J. Phys. 79 Two types of contributions: i)S.O. from band structure interacting with the field (external and internal) ii)Bloch electrons interacting with S.O. part of the disorder Lower bound estimate of skew scatt. contribution

Spin injection Hall effect: Theoretical consideration Local spin polarization  calculation of the Hall signal Weak SO coupling regime  extrinsic skew-scattering term is dominant 29 Lower bound estimate

30 Semiclassical Monte Carlo of SIHE Numerical solution of Boltzmann equation Spin-independent scattering: Spin-dependent scattering: phonons, remote impurities, interface roughness, etc. side-jump, skew scattering. AHE Realistic system sizes (  m). Less computationally intensive than other methods (e.g. NEGF).

31 Example: MC Transport Simulation in 2DEG Inject N particles with random momenta. Allow each particle to propagate from t to t+  t. Compute particle distribution function. Compute observables Repeat for each subhistory  t until T (simulation time). Time average results.

32 Single Particle Monte Carlo Particle with random momentum injected from drain. Randomly generate “free flight” times. Semiclassical particle propagates freely during t s and spin processes due to SO interaction. Randomly choose scattering mechanism at the end of “free flight”. Randomly choose new momentum and spin after scattering. Stop at time T and collect the observable values.

33 Ensemble Monte Carlo Obtain particle distribution at the end of each subhistory

34 Finding Distribution in Phase Space

Effects of B field 35

The family of spintronics Hall effects SHE -1 B=0 spin current gives charge current Electrical detection AHE B=0 polarized charge current gives charge-spin current Electrical detection SHE B=0 charge current gives spin current Optical detection 36 SIHE B=0 Optical injected polarized current gives charge current Electrical detection

37 SIHE: a new tool to explore spintronics nondestructive electric probing tool of spin propagation without magnetic elements all electrical spin-polarimeter in the optical range Gating (tunes α/β ratio) allows for FET type devices (high T operation) New tool to explore the AHE in the strong SO coupled regime

AHE in the strong SO regime 38

Why is AHE difficult theoretically in the strong SO couple regime? AHE conductivity much smaller than σ xx : many usual approximations fail Microscopic approaches: systematic but cumbersome; what do they mean; use non-gauge invariant quantities (final result gauge invariant) Multiband nature of band-structure (SO coupling) is VERY important; hard to see these effects in semi-classical description (where other bands are usually ignored). Simple semi-classical derivations give anomalous terms that are gauge dependent but are given physical meaning (dangerous and wrong) Usual “believes” on semi-classically defined terms do not match the full semi-classical theory (in agreement with microscopic theory) What happens near the scattering center does not stay near the scattering centers (not like Las Vegas) T-matrix approximation (Kinetic energy conserved); no longer the case, adjustments have to be made to the collision integral term Be VERY careful counting orders of contributions, easy mistakes can be made. 39

40 What do we mean by gauge dependent? Electrons in a solid (periodic potential) have a wave-function of the form Gauge dependent car BUT is also a solution for any a(k) Any physical object/observable must be independent of any a(k) we choose to put Gauge wand (puts an exp(ia(k)) on the Bloch electrons) Gauge invariant car

Boltzmann semiclassical approach: easy physical interpretation of different contributions (used to define them) but very easy to miss terms and make mistakes. MUST BE CONFIRMED MICROSCOPICALLY! How one understands but not necessarily computes the effect. Kubo approach: systematic formalism but not very transparent. Keldysh approach: also a systematic kinetic equation approach (equivalent to Kubo in the linear regime). In the quasi-particle limit it must yield Boltzmann semiclassical treatment. Microscopic vs. Semiclassical AHE in the strongly SO couple regime 41

Kubo microscopic approach to transport: diagrammatic perturbation theory Averaging procedures: = 1/  0 = 0  = + Bloch Electron Real Eigenstates Need to perform disorder average (effects of scattering) n, q Drude Conductivity σ = ne 2  /m * ~1/n i Vertex Corrections  1-cos(θ) Perturbation Theory: conductivity n, q 42

43 intrinsic AHE approach in comparing to experiment: phenomenological “proof” Berry’s phase based AHE effect is reasonably successful in many instances BUT still not a theory that treats systematically intrinsic and ext rinsic contribution in an equal footing n, q n’  n, q DMS systems (Jungwirth et al PRL 2002, APL 03) Fe (Yao et al PRL 04) layered 2D ferromagnets such as SrRuO3 and pyrochlore ferromagnets [Onoda et al (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003) colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999). CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004) Experiment  AH  1000 (  cm) -1 Theroy  AH  750 (  cm) -1 AHE in Fe AHE in GaMnAs

44 “Skew scattering” “Side-jump scattering” Intrinsic AHE: accelerating between scatterings n, q m, p n’, k n, q n’  n, q Early identifications of the contributions Vertex Corrections  σ Intrinsic ~  0 or n 0 i Intrinsic  σ 0 /ε F  ~  0 or n 0 i Kubo microscopic approach to AHE n, q m, p n’, k matrix in band index m’, k’

Armchair edge Zigzag edge EFEF “AHE” in graphene: linking microscopic and semiclassical theories 45

Single K-band with spin up In metallic regime: Sinitsyn, JS, et al PRB Kubo-Streda calculation of AHE in graphene Don’t be afraid of the equations, formalism can be tedious but is systematic (slowly but steady does it) Kubo-Streda formula: A. Crépieux and P. Bruno (2001)

47 Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current As before we do this in two steps: first calculate steady state non- equilibrium distribution function and then use it to compute the current. Set to 0 for steady state solution Only the normal velocity term, since we are looking for linear in E equation order of the disorder potential strength and symmetric and anti-symmetric components 1 st Born approximation 2 nd Born approximation (usual skew scattering contribution) To solve this equation we write the non-equilibrium component in various components that correspond to solving parts of the equation the corresponding order of disorder

48 Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current ~V 0 ~V ~V 2 2 nd step: (after solving them) we put them into the equation for the current and identify from there the different contributions to the AHE using the full expression for the velocity

Comparing Boltzmann to Kubo (chiral basis) 49 Kubo identifies, without a lot of effort, the order in n i of the diagrams BUT not so much their physical interpretation according to semiclassical theory Sinitsyn et al 2007

Intrinsic deflection 50 Popular believe: ~τ 1 or ~1/n i WRONG E ~n i 0 or independent of impurity density Skew scattering (2 contributions) term missed by many people using semiclassical approach Side jump scattering (2 contributions) Popular believe: ~n i 0 or independent of impurity density Origin is on its effect on the distribution function

51 Next simplest example: AHE in Rashba 2D system Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984) Only when ONE both sub-band there is a significant contribution When both subbands are occupied there is additional vertex corrections that contribute Nuner et al PRB08, Borunda et al PRL 07

52 Recent progress: full understanding of simple models in each approach Semi-classical approach: Gauge invariant formulation; shown to match microscopic approach in 2DEG+Rashba, Graphene Sinitsyn et al PRB 05, PRL 06, PRB 07 Borunda et al PRL 07, Nunner et al PRB 08 Sinitsyn JP:C-M 08 Kubo microscopic approach: Results in agreement with semiclassical calculations 2DEG+Rashba, Graphene Sinitsyn et al PRL 06, PRB 07, Nunner PRB 08, Inoue PRL 06, Dugaev PRB 05 NEGF/Keldysh microscopic approach: Numerical/analytical results in agreement in the metallic regime with semiclassical calculations 2DEG+Rashba, Graphene Kovalev et al PRB 08, Onoda PRL 06, PRB 08 – – – – – – – – – – – jqsjqs nonmagnetic Spin-polarizer current injected optically Spin injection Hall effect (SIHE) Up to now no 2DEG+R ferromagnetis: SIHE offers this possibility

53 CONCLUSIONS Spin-injection Hall effect observed in a conventional 2DEG - nondestructive electrical probing tool of spin propagation - indication of precession of spin-polarization - observations in qualitative agreement with theoretical expectations - optical spin-injection in a reverse biased coplanar pn- junction: large and persistent Hall signal (applications !!!)

EXTRA SLIDES 54

A2070 Kelvin Nanotechnology, University of Glasgow thicknesscompositiondopingfunction 5nmGaAsp=1E19 (Be)cap 2MLGaAsun 50nmAl x Ga 1-x As, x=0.5p=8E18 (Be) 3nmAl x Ga 1-x As, x=0.3un 90nmGaAsunchannel 5nmAl x Ga 1-x As, x=0.3unspacer 2MLGaAsun n=5E12 delta (Si)delta-doping 2MLGaAsun 300nmAl x Ga 1-x As, x=0.3un 50 period(9ML GaAs: 9ML AlGaAs, x=0.3)superlattice 1000nmGaAsun GaAsSIsubstrate 55

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: Hall discovers the Hall and the anomalous Hall effect The tumultuous history of AHE 1970: Berger reintroduces (and renames) the side-jump: claims that it does not vanish and that it is the dominant contribution, ignores intrinsic contribution. (problem: his side-jump is gauge dependent) Berger 58 Luttinger 1954: Karplus and Luttinger attempt first microscopic theory: they develop (and later Kohn and Luttinger) a microscopic theory of linear response transport based on the equation of motion of the density matrix for non-interacting electrons, ; run into problems interpreting results since some terms are gauge dependent. Lack of easy physical connection. Hall 1970’s: Berger, Smit, and others argue about the existence of side-jump: the field is left in a confused state. Who is right? How can we tell? Three contributions to AHE are floating in the literature of the AHE: anomalous velocity (intrinsic), side-jump, and skew contributions : Smit attempts to create a semi-classical theory using wave- packets formed from Bloch band states: identifies the skew scattering and notices a side-step of the wave-packet upon scattering and accelerating..Speculates, wrongly, that the side-step cancels to zero. The physical interpretation of the cancellation is based on a gauge dependent object!!

The tumultuous history of AHE: last three decades ’s: Spin-Hall effect is revived by the proposal of intrinsic SHE (from two works working on intrinsic AHE): AHE comes to the masses, many debates are inherited in the discussions of SHE. 1980’s: Ideas of geometric phases introduced by Berry; QHE discoveries 2000’s: Materials with strong spin-orbit coupling show agreement with the anomalous velocity contribution: intrinsic contribution linked to Berry’s curvature of Bloch states. Ignores disorder contributions ’s: Linear theories in simple models treating SO coupling and disorder finally merge: full semi-classical theory developed and microscopic approaches are in agreement among each other in simple models.

How does side-jump affect transport? 60 Side jump scattering The side-jump comes into play through an additional current and influencing the Boltzmann equation and through it the non-equilibrium distribution function VERY STRANGE THING: for spin-independent scatterers side-jump is independent of scatterers!! 1 st -It creates a side-jump current: 2 nd -An extra term has to be added to the collision term of the Boltzmann eq. to account because upon elastic scattering some kinetic energy is transferred to potential energy. full ω ll’ does not assume KE conserved, T-matrix approximation of ω ll’ (ω T ll’ ) does.

AHE in Rashba 2D system When both subbands are occupied the skew scattering is only obtained at higher Born approximation order AND the extrinsic contribution is unique (a hybrid between skew and side-jump) Kovalev et al PRB 08 Keldysh and Kubo match analytically in the metallic limit Numerical Keldysh approach (Onoda et al PRL 07, PRB 08) 61