Research fueled by: Universität zu Köln November 12 th, 2010 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Echoes of special relativity in condensed matter physics: anomalous Hall effect, spin-helix transistors, and topological thermoelectrics Hitachi Cambridge Joerg W ü nderlich, A. Irvine, et al Institute of Physics ASCR Tomas Jungwirth, Vít Novák, et al University of W ü rzburg Laurens Molenkamp, E. Hankiewiecz, et al University of Nottingham Bryan Gallagher, Richard Campion, et al.

2 Nanoelectronics, spintronics, and materials control by spin-orbit coupling I. Introduction: using the dual personality of the electron Internal coupling of charge and spin: origin and present use Control of material and transport properties through spin-orbit coupling Overview of program II. Anomalous Hall effect: from the metallic to the insulating regime Anomalous Hall effect basics, history, progress in the metallic regime Anomalous Hall effect in the hopping regime III.Spin injection Hall effect: a new paradigm in exploiting SO coupling Spin based FET: old and new paradigm in charge-spin transport Theory expectations and modeling Experimental results Topological thermoelectrics:Thermoelectric figure of merit Increase of ZT in topological insulators. Echoes of special relativity in condensed matter physics: anomalous Hall effect, spin-helix transistors, and topological thermoelectrics

3 Nanoelectronics, spintronics, and materials control by spin-orbit coupling The electron: the key character with dual personalitiesCHARGE Easy to manipulate: Coulomb interaction SPIN 1/2 Makes the electron antisocial: a fermion quantum mechanics E=p 2 /2m E→ iħ d/dt p→ -iħ d/dr “Classical” external manipulation of charge & spin special relativity E 2 /c 2 =p 2 +m 2 c 2 (E=mc 2 for p=0) + particles/antiparticles & spin Dirac equation =

4 e-e-e-e- Nanoelectronics, spintronics, and materials control by spin-orbit coupling Internal communication between spin and charge: spin-orbit coupling interaction (one of the few echoes of relativistic physics in the solid state) This gives an effective interaction with the electron’s magnetic moment Classical explanation (in reality it arises from a second order expansion of Dirac equation around the non-relativistic limit) “Impurity” potential V(r) Produces an electric field In the rest frame of an electron the electric field generates an effective magnetic field Motion of an electron

5 e-e-e-e- Nanoelectronics, spintronics, and materials control by spin-orbit coupling (one of the few echoes of relativistic physics in the solid state) This gives an effective interaction with the electron’s magnetic moment Classical explanation (in reality it arises from a second order expansion of Dirac equation around the non-relativistic limit) “Impurity” potential V(r) Produces an electric field ∇V∇V B eff p s In the rest frame of an electron the electric field generates an effective magnetic field Motion of an electron Consequence #1: Spin or the band- structure Bloch states are linked to the momentum. Internal communication between spin and charge:spin- orbit coupling interaction

6 e-e-e-e- Nanoelectronics, spintronics, and materials control by spin-orbit coupling (one of the few echoes of relativistic physics in the solid state) This gives an effective interaction with the electron’s magnetic moment Classical explanation (in reality it arises from a second order expansion of Dirac equation around the non-relativistic limit) “Impurity” potential V(r) Produces an electric field ∇V∇V B eff p s In the rest frame of an electron the electric field generates an effective magnetic field Motion of an electron Consequence #2 Mott scattering Internal communication between spin and charge:spin- orbit coupling interaction

7 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Control of materials and transport properties via spin-orbit couplingAs Ga Mn New magnetic materials Nano- transport Spintronic Hall effects Magneto- transport Topological transport effects Effects of spin-orbit coupling in multiband systems Caloritronics

8 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Effects of spin-orbit coupling in multiband systems As Ga Mn New magnetic materials Nano- transport Spintronic Hall effects Magneto- transport Topological transport effects Control of materials and transport properties via spin-orbit coupling Anomalous Hall effects I F SO majority minority V Nagaosa, Sinova, Onoda, MacDonald, Ong, RMP 10

9 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Simple electrical measurement of out of plane magnetization (or spin polarization ~ n ↑ -n ↓ ) InMnAs Spin dependent “force” deflects like-spin particles ρ H =R 0 B ┴ +4π R s M ┴ Anomalous Hall Effect: the basics I _ F SO _ _ _ majority minority V M⊥M⊥ AHE is does NOT originate from any internal magnetic field created by M ⊥ ; the field would have to be of the order of 100T!!!

Nanoelectronics, spintronics, and materials control by spin-orbit coupling Anomalous Hall effect (scaling with ρ for metals) Material with dominant skew scattering mechanism Material with dominant scattering-independent mechanism σ xx >10 6 (Ωcm) -1 σ xx ~ (Ωcm) -1

Nanoelectronics, spintronics, and materials control by spin-orbit coupling Anomalous Hall effect (scaling for insulators) Diagonal hopping conductivity for most systems showing approximate scaling σ xy ~σ xx over a few decades for σ xx <10 4 (Ωcm) -1

Nanoelectronics, spintronics, and materials control by spin-orbit coupling Nagaosa et al RMP 10 Tentative phase diagram of AHE 2

13 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Valenzuela et al Nature 06 Inverse SHE Anomalous Hall effect: more than meets the eye Wunderlich, Kaestner, Sinova, Jungwirth PRL 04 Kato et al Science 03 Intrinsic Extrinsic V Mesoscopic Spin Hall Effect Intrinsic Brune,Roth, Hankiewicz, Sinova, Molenkamp, et al Nature Physics 2010 Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09 Spin-injection Hall Effect Anomalous Hall Effect I _ FSOFSO FSOFSO _ _ majority minority V Spin Hall Effect I _ FSOFSO FSOFSO _ _ V Topological Insulators Kane and Mele PRL 05

14 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Cartoon of the mechanisms contributing to AHE independent of impurity density 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. 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 Intrinsic deflection B 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 Side jump scattering V imp (r) (Δso>ħ/τ)  ∝ λ* ∇ V imp (r) (Δso<ħ/τ) B Skew scattering Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. Known as Mott scattering. ~σ~1/n i V imp (r) (Δso>ħ/τ)  ∝ λ* ∇ V imp (r) (Δso<ħ/τ) A

15 Nanoelectronics, spintronics, and materials control by spin-orbit coupling : 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 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!!

16 Nanoelectronics, spintronics, and materials control by spin-orbit coupling The tumultuous history of AHE: last three decades 2004’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 treating SO coupling and disorder finally merge: full semi-classical theory developed and microscopic approaches are in agreement among each other in simple models.

17 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Contributions understood in simple metallic 2D models Semi-classical approach: Gauge invariant formulation Sinitsyn, Sinvoa, et al PRB 05, PRL 06, PRB 07 Kubo microscopic approach: in agreement with semiclassical Borunda, Sinova, et al PRL 07, Nunner, Sinova, et al PRB 08 Non-Equilibrium Green’s Function (NEGF) microscopic approach Kovalev, Sinova et al PRB 08, Onoda PRL 06, PRB 08 Restriction to homogeneous magnetization. Magnetic textures lead to the so called topological AHE.

18 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Generalization to 3D: scattering in dependent AHE Step 1. Use linearized version of Keldysh formalism to obtain where Kovalev, Sinova, Tserkovnyak PRL 2010

19 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Scattering-independent contribution to the AHE Well known intrinsic contribution Side jump contribution related to Berry curvature Remaining side jump contribution Kovalev, Sinova, Tserkovnyak PRL 2010 expressed through band structure only !!

20 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Results for Luttinger model: simplest model of GaMnAs Kovalev, Sinova, Tserkovnyak PRL 2010 It gives a formalism that allows for a systematic study of AHE through ab- initio calculations (ongoing collaboration with S. Blügel and Y. Mokrousov)

21 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Nagaosa et al RMP 10 Tentative phase diagram of AHE Terra incognita Previous Theories: S. H. Chun etal., PRL 2000; Lyanda-Geller et al PRB 2001 (Theory for Manganites; no scaling) A. A. Burkov and L. Balents, Phys. Rev. Lett. 91, (2003),

Nanoelectronics, spintronics, and materials control by spin-orbit coupling Phonon-assisted hopping between localized states The Hamiltonian describing localized states and e-ph interaction (Holstein 1961) with Localization: phonon-assisted hopping Electric current between two sites: RiRi RjRj RkRk : direct conductance due to two-site hopping. Responsible for longitudinal conductance. : off-diagonal conductance due to three-site hopping. i j k phonon

Nanoelectronics, spintronics, and materials control by spin-orbit coupling Transverse charge transport: Three-site hopping (Holstein, 1961) Interference term Hall transition rate Typical two real phonon process: Typical one real (two virtual) phonon process: Direct hopping Indirect hopping Geometric phase: break chirality m : the number of real phonons included in the whole transition.

Nanoelectronics, spintronics, and materials control by spin-orbit coupling Macroscopic anomalous Hall conductivity/resistivity via percolation theory In the thermodynamic limit, we get the AHC: Unfortunately, too complicated to get an analytical calculation!!! Critical path/cluster appears when:

Nanoelectronics, spintronics, and materials control by spin-orbit coupling The lower and upper limits are given by (note the differences in the constrain) For the Mott hopping, we find Physically, the upper limit (γ=1.38) corresponds to the situation that most triads in the system are equilateral triangles (this is actually a regular distribution), while for the lower limit (γ=1.76) the geometry of triads is randomly distributed. Since the impurity sites are randomly distributed, we expect the realistic result is usually close to the lower limit in most systems. Xiong-Jun Liu, Sinova, unpub Anomalous Hall conductivity via percolation theory

26 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Introduction: using the dual personality of the electron Electronics, ferromagnetism, and spintronics Internal coupling of charge and spin: origin and present use Control of material and transport properties through spin-orbit coupling Overview of program Anomalous Hall effect: from the metallic to the insulating regime Anomalous Hall effect basics, history, progress in the metallic regime Anomalous Hall effect in the hopping regime Spin injection Hall effect: a new paradigm in exploiting SO coupling Spin based FET: old and new paradigm in charge-spin transport Theory expectations and modeling Experimental results Topological thermoelectrics:Thermoelectric figure of merit Increase of ZT in topological insulators. Echoes of special relativity in condensed matter physics: anomalous Hall effect, spin-helix transistors, and topological thermoelectrics

27 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Towards a realistic spin-based non-magnetic FET device [001] [100] [010] Can we achieve direct spin polarization injection, detection, and manipulation by electrical means in an all paramagnetic semiconductor system? Long standing paradigm: Datta-Das FET (1990) Exploiting the large Rashba spin-orbit coupling in InAs Electrons are confined in the z-direction in the first quantum state of the asymmetric trap and free to move in the x-y plane. gate k y [010] k x [100] Rashba effective magnetic field ⊗⊗⊗

28 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Can we achieve direct spin polarization injection, detection, and manipulation by electrical means in an all paramagnetic semiconductor system? Long standing paradigm: Datta-Das FET (1990) Exploiting the large Rashba spin-orbit coupling in InAs Towards a realistic spin-based non-magnetic FET device High resistance “0” Low resistance “1” BUT l MF << L S-D at room temperature

29 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Dephasing of the spin through the Dyakonov-Perel mechanism L SD ~ μm l MF ~ 10 nm

30 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Problem: Rashba SO coupling in the Datta-Das SFET is used for manipulation of spin (precession) BUT it dephases the spin too quickly (DP mechanism). New paradigm using SO coupling: SO not so bad for dephasing 1) Can we use SO coupling to manipulate spin AND increase spin-coherence? Can we detect the spin in a non-destructive way electrically?

31 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Spin-dynamics in 2D electron gas with Rashba and Dresselhauss spin-orbit coupling a 2DEG is well described by the effective Hamiltonian:  > 0,  = 0 [110] _ k y [010] k x [100] Rashba: from the asymmetry of the confinement in the z-direction  = 0,  < 0 [110] _ k y [010] k x [100] Dresselhauss: from the broken inversion symmetry of the material, a bulk property 1) Can we use SO coupling to manipulate spin AND increase spin-coherence?

32 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Spin-dynamics in 2D electron gas with Rashba and Dresselhauss spin-orbit coupling Something interesting occurs when spin along the [110] direction is conserved long lived precessing spin wave for spin perpendicular to [110] The nesting property of the Fermi surface: Bernevig et al PRL 06, Weber et al. PRL 07 Schliemann et al PRL 04

33 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Effects of Rashba and Dresselhaus SO coupling  = -  [110] _ k y [010] k x [100]  > 0,  = 0 [110] _ k y [010] k x [100]  = 0,  < 0 [110] _ k y [010] k x [100]

34 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Spin-dynamics in 2D systems with Rashba and Dresselhauss SO coupling For the same distance traveled along [1-10], the spin precesses by exactly the same angle. [110] _ _

35 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Persistent state spin helix verified by pump-probe experiments Similar wafer parameters to ours

36 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Spin-helix state when α ≠ β Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09 For Rashba or Dresselhaus by themselves NO oscillations are present; only and over damped solution exists; i.e. the spin-orbit coupling destroys the phase coherence. There must be TWO competing spin-orbit interactions for the spin to survive!!!

37 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Problem: Rashba SO coupling in the Datta-Das SFET is used for manipulation of spin (precession) BUT it dephases the spin too quickly (DP mechanism). New paradigm using SO coupling: SO not so bad for dephasing 1) Can we use SO coupling to manipulate spin AND increase spin-coherence? Can we detect the spin in a non-destructive way electrically? Use the persistent spin-Helix state and control of SO coupling strength (Bernevig et al 06, Weber et al 07, Wünderlich et al 09) ✓

38 Nanoelectronics, spintronics, and materials control by spin-orbit coupling 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) Bloch electrons interacting with S.O. part of the disorder Lower bound estimate of skew scatt. contribution AHE contribution to Spin-injection Hall effect in a 2D gas Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09

39 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Local spin-polarization → calculation of AHE signal Weak SO coupling regime → extrinsic skew-scattering term is dominant Lower bound estimate Spin-injection Hall effect: theoretical expectations 1) Can we use SO coupling to manipulate spin AND increase spin-coherence? Can we detect the spin in a non-destructive way electrically? Use the persistent spin-Helix state and control of SO coupling strength Use AHE to measure injected current polarization electrically ✓ ✓

40 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Spin-injection Hall device measurements VLVL SIHE ↔ Anomalous Hall Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09

41 Nanoelectronics, spintronics, and materials control by spin-orbit coupling T = 250K Further experimental tests of the observed SIHE

42 Nanoelectronics, spintronics, and materials control by spin-orbit coupling V H2 I VbVb V H1 x V H2 VbVb V H1 x (a) (b) SiHE: new results Spin Hall effect transitor: Wunderlich, Irvine, Sinova, Jungwirth, et al, arXiv: SiHE inverse SHE

43 Nanoelectronics, spintronics, and materials control by spin-orbit coupling VHVH VgVg I VbVb VHVH VgVg VbVb x Δ x=1  m ++ SiHE transistor Spin Hall effect transitor: Wunderlich, Irvine, Sinova, Jungwirth, et al, arXiv:

44 Nanoelectronics, spintronics, and materials control by spin-orbit coupling -- -- -- -- H1 H Vg2 [V] R H1 [ ] 0 12 R H2 [ ] SHE transistor AND gate

45 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Introduction: using the dual personality of the electron Electronics, ferromagnetism, and spintronics Internal coupling of charge and spin: origin and present use Control of material and transport properties through spin-orbit coupling Overview of program Anomalous Hall effect: from the metallic to the insulating regime Anomalous Hall effect basics, history, progress in the metallic regime Anomalous Hall effect in the hopping regime Spin injection Hall effect: a new paradigm in exploiting SO coupling Spin based FET: old and new paradigm in charge-spin transport Theory expectations and modeling Experimental results Topological thermoelectrics:Thermoelectric figure of merit Increase of ZT in topological insulators. Echoes of special relativity in condensed matter physics: anomalous Hall effect, spin-helix transistors, and topological thermoelectrics

46 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Control of materials and transport properties via spin-orbit couplingAs Ga Mn New magnetic materials Nano- transport Spintronic Hall effects Magneto- transport Topological transport effects Effects of spin-orbit coupling in multiband systems Caloritronics Topological thermoelectrics

47 Nanoelectronics, spintronics, and materials control by spin-orbit coupling From AHE to topological insulators to thermoelectrics Vishwanath et al Nature Physics 09 Dislocations have 1D channels which also protected Topological Insulators: edge (2D) or surface states (3D) survive disorder effects when the bulk gap is produced by spin-orbit coupling Kane, Zhang, Molenkamp, Moore, et al Zhang, Physics 1, 6 (2008) X X QSHE in HgTe

48 Nanoelectronics, spintronics, and materials control by spin-orbit coupling From AHE to topological insulators to thermoelectrics Best thermoelectrics Can we obtain high ZT through the topological protected states; are they related to the high ZT of these materials? Vishwanath et al 09 Dislocations have 1D channels which also protected ? ? Seebeck coefficient electrical conductivity electric thermal conductivity phonon thermal conductivity

49 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Bi 1−x Sb x (0.07 < x < 0.22) where the L’s are the linear Onsager dynamic coefficients Localized bulk states Possible large ZT through dislocation engineering Tretiakov, Abanov, Murakami, Sinova APL 2010

50 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Possible large ZT through dislocation engineering Remains very speculative but simple theory gives large ZT for reasonable parameters Tretiakov, Abanov, Murakami, Sinova APL 2010

51 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Anomalous Hall effect: from the metallic to the insulating regime I. Established a consistent theory of Anomalous Hall effect for metallic regime with homogeneous magnetization II. Theory of observed scaling in Anomalous Hall effect in the hopping regime Spin injection Hall effect: a new paradigm in exploiting SO coupling I. Modeled and constructed a spin FET in the diffusive regime II. First spin AND gate with pure spin currents III. Topological thermoelectrics:Speculative theory of how to increase of ZT in topological insulators via line dislocations. SUMMARY: anomalous Hall effect, spin-helix transistors, and topological thermoelectrics Control of materials and transport properties via spin-orbit coupling A long list of challenges: DMSs, Antiferromagnetic semiconductors, current driven magnetization dynamics, pseudo-spintronics in double layer systems, spin-caloritronics (Spin Seebeck effect)

52 Nanoelectronics, spintronics, and materials control by spin-orbit coupling Allan MacDonald U of Texas Tomas Jungwirth Texas A&M U. Inst. of Phys. ASCR U. of Nottingham Joerg Wunderlich Cambridge-Hitachi Laurens Molenkamp Würzburg Xiong-Jun Liu Texas A&M U. Mario Borunda Texas A&M Univ. Harvard Univ. Nikolai Sinitsyn Texas A&M U. U. of Texas LANL Alexey Kovalev Texas A&M U. UCLA Liviu Zarbo Texas A&M Univ. Xin Liu Texas A&M U. Ewelina Hankiewicz (Texas A&M Univ.) Würzburg University Sinova’s group Principal Collaborators Gerrit Bauer TU Delft Bryan Gallagher U. of Nottingham and many others Oleg Tretiakov Texas A&M U.