1. Research fueled by: International Symposium High Performance Computing in Nano-Spintronics Hamburg, November 30 th, 2011 Transport theory and simulations in hybrid structures: the need for a bottom-up approach in new semiconductor spintronic systems JAIRO SINOVA Texas A&M University Institute of Physics ASCR Hitachi Cambridge Joerg W ü nderlich, A. Irvine, et al Institute of Physics ASCR Tomas Jungwirth, Vít Novák, Karel Vyborny, et al Hamburg University Jan Jacob, Bodo Krause-Kyora, et al

2. 2 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova I. Introduction: using the dual personality of the electron Internal coupling of charge and spin: origin and present use II.Spintronic devices and the need to understand them microscopically III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling Spin based FET: old and new paradigm in charge-spin transport II. Spin filters in InAs nano-wires III. A bottom to top approach: from mesoscopic to microscopic Why is it important to approach the problem from the mesoscopic regime V. Theoretical tools of the mesoscopic world: Landauer-Büttiker (coherent regime) Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq. Beyond coherent transport: the basic master equation of the non- equilibrium density matrix VI. Computational Challenges of NEGF approach Transport theory and simulations in hybrid structures: the need for a bottom-up approach in new spintronic systems

3. 3 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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. 4 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Using charge and spin in information technology HIGH tunablity of electronic transport properties the key to FET success in processing technology substrate semiconductor insulator SD gate Vg >0 Using charge to create a field effect transistor: work horse of information processing Using spin: Pauli exclusion principle and Coulomb repulsion →ferromagnetism work horse of information storage total wf antisymmetric = orbital wf antisymmetric × spin wf symmetric (aligned) Robust (can be as strong as bonding in solids) Robust (can be as strong as bonding in solids) Strong coupling to magnetic field Strong coupling to magnetic field (weak fields = anisotropy fields needed (weak fields = anisotropy fields needed only to reorient macroscopic moment) only to reorient macroscopic moment) e-e-e-e- What about the internal communication between charge & spin? (spintronics)

5. 5 e-e-e-e- International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova (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 Internal communication between spin and charge:spin- orbit coupling interaction

6. 6 e-e-e-e- International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova (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 Internal communication between spin and charge:spin- orbit coupling interaction

7. 7 e-e-e-e- International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova (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

8. 8 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova How spintronics has impacted your life: Metallic spintronics Appreciable sensitivity, simple design, cheap BUT only a 2-8 % effect Anisotropic magnetoresistance (AMR): In ferromagnets the current is sensitive to the relative direction of magnetization and current direction 1992 - dawn of (metallic) spintronics e-e-e-e- current magnetization Fert, Grünberg et al. 1998 Giant magnetoresistance (GMR) read head - 1997 High sensitivity, very large effect 30-100% ↑↑ and ↑↓ are almost on and off states: “1” and “0” & magnetic → memory bit Nobel Price 2007 Fert and Grünberg e-e-e-e- ×

9. 9 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Circuit heat generation is one key limiting factor for scaling device speed Industry has been successful in doubling of transistor numbers on a chip approximately every 18 months (Moore’s law). Although expected to continue for several decades several major challenges will need to be faced. What next? The need for basic research

10. 10 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova I. Introduction: using the dual personality of the electron Internal coupling of charge and spin: origin and present use II.Spintronic devices and the need to understand them microscopically III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling Spin based FET: old and new paradigm in charge-spin transport II. Spin filters in InAs nano-wires III. A bottom to top approach: from mesoscopic to microscopic Why is it important to approach the problem from the mesoscopic regime V. Theoretical tools of the mesoscopic world: Landauer-Büttiker (coherent regime) Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq. Beyond coherent transport: the basic master equation of the non- equilibrium density matrix VI. Computational Challenges of NEGF approach Transport theory and simulations in hybrid structures: the need for a bottom-up approach in new spintronic systems

11. 11 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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 Towards a realistic spin-based non-magnetic FET device Unfortunately it has not worked : no reliable detection of spin-polarization in a diagonal transport configuration No long spin-coherence in a Rashba spin- orbit coupled system (Dyakonov-Perel mechanism)

12. 12 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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, Wunderlich et al 09, 10) Use AHE to measure injected current polarization at the nano-scale electrically (Wunderlich, et al 09, 04)

13. 13 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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?

14. 14 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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]

15. 15 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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] _ _

16. 16 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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!!! In addition we performed MC Boltzmann transport calculations incorporating spin physics

17. 17 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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 ┴ “Local” magnetization measurement through the Anomalous Hall Effect 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!!!

18. 18 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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

19. 19 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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

20. 20 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Spin-injection Hall device measurements trans. signal VLVL SIHE ↔ Anomalous Hall Local Hall voltage changes sign and magnitude along a channel of 6 μm

21. 21 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova T = 250K Further experimental tests of the observed SIHE

22. 22 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova VHVH VgVg I VbVb VHVH VgVg VbVb x Δ x=1  m ++ SiHE transistor Spin Hall effect transitor: Wunderlich, Irvine, Sinova, Jungwirth, et al, Science 2010

23. 23 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova -- -- -- -- H1 H2 0 +0.1 -0.1 Vg2 [V] R H1 [ ] 0 12 R H2 [ ] 0 6 6 3 SiHE transistor AND gate

24. 24 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Next SiHE transistor: all electrical spin FET Key features: Spin injection through thin Fe contacts into GaAs Detection via SHE and Non-Local Hanle effect Spin current control through charge current bias K. Olejnik, Wunderlich, et al

25. 25 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Semiconductor spin filters in InAs nanowires Splits an unpolarized current into two oppositely spin-polarized currents A. A. Kiselev and K. W. Kim, Appl. Phys. Lett. 78, 775 (2001); J. Appl. Phys. 94, 4001 (2003). J. I. Ohe, M. Yamamoto, T. Ohtsuki, and J. Nitta, Phys. Rev. B 72, 041308 (2005). M. Yamamoto, T. Ohtsuki, and B. Kramer, Phys. Rev. B 72, 115321 (2005). A. W. Cummings, R. Akis, and D. K. Ferry, Appl. Phys. Lett. 89, 172115 (2006). M. Yamamoto, K. Dittmer, B. Kramer, and T. Ohtsuki, Physica E 32, 462 (2006). 12 Jan Jacob’s group

26. 26 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova In-plane magnetic fields Detect spin-Hall signal in filter outputs Top-gated spin-filter cascades Single quantum-point contacts Top/Backgate-controlled spin-orbit coupling Ferromagnet-Semiconductor Spin-Valves (together with OSU) 30 Semiconductor spin filters in InAs nanowires

27. 27 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova I. Introduction: using the dual personality of the electron Internal coupling of charge and spin: origin and present use II.Spintronic devices and the need to understand them microscopically III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling Spin based FET: old and new paradigm in charge-spin transport II. Spin filters in InAs nano-wires III. A bottom to top approach: from mesoscopic to microscopic Why is it important to approach the problem from the mesoscopic regime V. Theoretical tools of the mesoscopic world: Landauer-Büttiker (coherent regime) Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq. Beyond coherent transport: the basic master equation of the non- equilibrium density matrix VI. Computational Challenges of NEGF approach Transport theory and simulations in hybrid structures: the need for a bottom-up approach in new spintronic systems

28. 28 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova quantitative modeling of spin-current based devices from bottom up Microscopic: Ab-initio (tight-binding) Molecular level modeling Connection to effective Hamiltonians and boundary conditions of different phenomena Interface effects of spin-injection and bulk spin coherence scale ~ 1Å Semiclassical: Boltzmann-Monte-Carlo (2010) scale ~ 1μ Mesoscopic: Non-Equil. Green’s Function (2007) scale ~ 1nm ✓ ✓ COMPETING LENGTH SCALES OF THE SAME ORDER

29. 29 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Formalism Semiclassical bulk transport theory: Boltzmann “diagonal” transport Electrons are treated as Newtonian pinballs, but having an effective mass and g-factor determined by the band structure and undergo random quantum scattering Dynamics of the occupation number ASSUMES COLLISIONS ARE INSTANTANEOUS AND UNEVENTFUL Very successful for charge transport and micro-device modeling Key feature: l.h.s. classical – r.h.s. quantum collision term Main shortcoming: Does not treat systematically inter-band coherence or spin transport Applicable in the diffusive metallic regime 29

30. 30 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Golden Rule: Coordinate shift (Side jump): Modified Boltzmann Equation: Berry curvature: velocity:current: Semiclassical approach to bulk Hall transport: breakdown of collision assumption Sinitsyn et al PRB 06

31. 31 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Formalism a) Intrinsic deflection E b) Side jump scattering c) Skew scattering What is what in the modified Boltzmann treatment

32. 32 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Formalism Quantum mesoscopic transport Nikolic et al 04 1.When the mean free path is of the same order as the system size then quantum interference effects appear 2.In the presence of SO coupling bands are mixed: occupation number is not the only thing needed to describe transport 3.Need to explore the non-equilibrium dynamics of the whole density matrix NEGF formalism is in a sense the quantum Boltzmann Equation: both the collisions AND dynamics between collisions are treated quantum mechanically 32

33. 33 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Formalism WHAT CAN WE ADDRESS FROM THIS MICROSCOPIC Advantages of NEGF approach: No assumptions on spin-currents Deals with intrinsic and extrinsic on an equal footing Multi-band treatment comes out naturally No assumed length scales Inelastic processes can be incorporated Limitations: system sizes and extrapolation to the bulk regime (1) What is the effect of the nature of the scattering on the induced spin-currents and spin coherence in general? (2) What is the nature of the spin-current? (3) Can these induced currents lead to strong enough spin-accumulation? (4) Does coherent transport matter? (5) What is the loss of spin-coherence in relation to scattering? (6) Is the effect more readily controlled at the mesoscopic scales? (7) What is the dissipative ratio in spintronic devices analogous to current devices?

34. 34 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Mesoscopic transport via Landauer-Büttiker Formalism (Coherent limit of the NEGF formalism) Uncorrelated electrons injected from left lead. Single channel conductance: Sharvin conductance T transmission probability

35. 35 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 1.Calculate the leads self-energies (f-numberelt by the sample): 2.Compute the sample retarded GF: 3.Obtain transmission coefficients: Landauer-Büttiker Procedure in a Nutshell (real space) 4. Calculate currents with Büttiker formula: See, e.g., Electronic Transport in Mesos. Systems by Datta Usually known or easily calculable

36. 36 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova How to actually compute things: Tight-Binding Approximation Continuous 2DEG Effective mass Hamiltonian Tight-binding Hamiltonian Tight-binding lattice Artificial length scale

37. 37 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Example: Quantum Point Contact Conductance Van Wees et al, PRL 60, 848 (1988) Experimental evidence of conductance quantization: Conductance of a ballistic 2DEG QPC: 37

38. 38 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Limitations of Landauer-Büttiker Limited to relatively small systems. Must rely on finite size scaling to reach bulk values Limited to coherent transport Expresses physical quantities via correlation functions. Studies non-coherent transport. Can include the effect of interactions and dissipation gradually Can make clearer connection to the bulk transport theories However, it still has same size limitation as LB NEGF formalism advantages

39. 39 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Nonequilibrium Green Function Formalism Classical transport:Quantum transport: distribution function density matrix Key correlation function Generalizes the density matrix.

40. 40 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Boltzmann vs. NEGF Functions (but not literally; r.h.s. knows about interband coherence) Electron distribution function Hole distribution function Scattering functions Electron correlation function Hole correlation function Scattering functions Do not take into account phase correlations. Include phase correlations.

41. 41 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Formalism Connection to semiclassical Boltzmann when looking in real space Using the relation shown and G in real space

42. 42 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Algorithm - Interactions 1. Compute self-energies 2. Calculate the retarded GF 3. Find the correlation function 4. Calculate interaction self-energy 5. Check for convergence If not 6. Compute physical quantities only first iteration

43. 43 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Algorithm – Molecular Transport The sample Hamiltonian depends on charge density: 1. Calculate self-energies of the electrodes: 2. Calculate the retarded GF: 3. Compute the correlation function: 4. Compute the charge density for biased sample: 5. Check for convergence: no 6. Compute physical quantities:

44. 44 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova NEGF Formalism Nonequilibrium Spin Hall Accumulation in mesoscopic Rashba 2DEG: non-linear transport Spin density (NEGF): +eV/2 -eV/2 eV=0 PRL 95, 046601 (2005)

45. 45 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova I. Introduction: using the dual personality of the electron Internal coupling of charge and spin: origin and present use II.Spintronic devices and the need to understand them microscopically III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling Spin based FET: old and new paradigm in charge-spin transport II. Spin filters in InAs nano-wires III. A bottom to top approach: from mesoscopic to microscopic Why is it important to approach the problem from the mesoscopic regime V. Theoretical tools of the mesoscopic world: Landauer-Büttiker (coherent regime) Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq. Beyond coherent transport: the basic master equation of the non- equilibrium density matrix VI. Computational Challenges of NEGF approach Transport theory and simulations in hybrid structures: the need for a bottom-up approach in new spintronic systems

46. 46 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Usuki-transfer matrix method with GPUs (two terminal)

47. 47 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Key computational challenges The key issue with NEGF for realistic size microscopic calculations is how to best deal with large matrices (both multiplication and inversion) In order to obtain bulk realistic values one needs to do finite size scaling of more than three small sizes Can one dynamically deal with sparsity? Not all eginevectors are created equal - how to select? 2-terminal vs. multi-terminal? Conductance (two terminal) vs. G < (much more computationally expensive but contains everything)

48. 48 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova 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 Principal Outside Collaborators Gerrit Bauer TU Delft Bryan Gallagher U. of Nottingham and many others Jan Jacob U. Hamburg

49. 49 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Xiong-Jun Liu Texas A&M U. Now: PD- U. Maryland Mario Borunda Texas A&M Univ. Now: PD-Harvard Univ. Nikolai Sinitsyn Texas A&M U. Now: Staff LANL Alexey Kovalev PD-Texas A&M Now: PD-UC Riverside Liviu Zarbo PD-Texas A&M Univ. Now: PD- ASCR Xin Liu Texas A&M U. Ewelina Hankiewicz (PD-Texas A&M Univ.) Now: W2-Professor at Würzburg University Sinova’s group Oleg Tretiakov (main PI Abanov) Texas A&M U. H. Gao Texas A&M U. Vivek Amin Texas A&M U. Erin Vehstedt Texas A&M U. Jacob Gyles Texas A&M U. Previous members

50. 50 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova What next?

51. 51 International Symposium High Performance Computing in Nano-Spintronics, Hamburg Sinova Inversion of large sparse matrices