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Topological Currents in Solids - Multi-band Effect and Band Crossings -

Topological Currents in Solids - Multi-band Effect and Band Crossings -. Dec. 20 ＠ HKU. Naoto Nagaosa 永長　直人 Department of Applied Physics The University of Tokyo. What I learned at MIT. Gauge field structure in strongly correlated electronic systems

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Topological Currents in Solids - Multi-band Effect and Band Crossings -

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  1. Topological Currents in Solids- Multi-band Effect and Band Crossings - Dec. 20 ＠HKU Naoto Nagaosa 永長　直人 Department of Applied Physics The University of Tokyo

  2. What I learned at MIT Gauge field structure in strongly correlated electronic systems Spin metals, spin superconductors etc. Look at “conventional” materials from the new eyes of strong correlation physics Hopefully predict new functions/phenomena

  3. wave packet Electron Wavepacket Dynamics in solids group velocity Boltzmann transport equation Totally-filled band does not contribute to current. Only energy dispersion matters ?

  4. Intra- and Inter-band matrix elements of current Wavefunction matters !! wave packet Even a filled band can support current e.g., polarization current quantum Hall current

  5. anomalous velocity k-space curvature r-space curvature Correct equation of motion taking into account inter-band matrix element Luttinger, Blount, Niu Origin of the k-space curvature = interband current matrix How the wavefunction is connected in k-space  Berry phase

  6. Geometry on sphere – Parallel transport of vector C Constrained onto sub-Hilbert-space

  7. 3 Kinds of Current in Solids 1. Ohmic (transport) Current Dissipation/Joul heating in nonequilibrium state 2. Topological Current Due to multi-band effect/Berry phase Dissipationless in equilibrium The occupied states contribute Berry phase 3. Superconducting Current / Diamagnetic Current Dissipationless in equilibrium Responding to A

  8. B(k) diverges at band crossing C Energy degeneracy point = Magnetic monopole Gauge Flux = Solid angle Breakdown of semi-classical Boltzmann approach

  9. When the band crossing occurs ?(with spin-orbit int.) • tune 5 parameters • Need for symmetry reason tune 3 parameters Kramer’s double degeneracy accidental degeneracy No degeneracy

  10. rxy = R0H + 4pRSM ordinary term anomalous term -e -e -e -e Anomalous Hall Effect M magnetization y v spin-orbit interaction N.P.Ong Electric field E x

  11. Anomalous Hall Effect in SrRuO3 - Magnetic Monopole in k-Space Z.Fang et al. Small energy scale 0.02eV Behavior like quantum chaos

  12. Kubo Formula Energy broadening Also A.H.MacDonald group for (Ga,Mn)As and Fe

  13. with (Skew scattering) Previous theories of AHE - 50 years of debates !! Karplus-Luttinger (1954) Interband effect Perturbation in s-o int. Intrinsic mechanism with dissipationless current Smit extrinsic mechanism with impurity scatt. and dissipation Engel et al. A rough estimation 3 energy scales in the problem Band width/gap Relaxation Skew scattering Spin-orbit interaction KL term

  14. Hardware: Gauge-covariant formalism of Keldysh Green’s function Wigner representation Operator commutation relation  Non-commutative geometry in Wigner space Dyson equation  separation into extrinsic and intrinsic contributions Diagram technique for self-energy -- including vertex correction

  15. Resonant AHE S.Onoda-N.Sugimoto-NN, PRL06 E-EF p Band crossing lifted by spin-orbit interaction Spin-orbit Coupling Intrinsic (without vertex Correction) is robust against scattering

  16. Global behavior of anomalous Hall effect hopping metallic Super clean Miyasato-Asamitsu c.f. N.P.Ong

  17. -e -e -e -e -e -e y v v E Electric field x Spin Hall Effect

  18. Classification of Order Parameters Time reversal odd even Inversion even charge density magnetization polarization odd spin current toroidal moment current

  19. Advantages of Spin Hall Effect Manipulation of spins by purely electric method without magnetic field/magnets Small scale spintronics devices with ordinary materials Spin current can be dissipationless in sharp contrast to charge current Functionality with low energy cost ohmic dissipationless Driven by the spin-orbit interaction with large energy scale Function at room temperature

  20. Spin Hall Effect in p-GaAs x: current direction y: spin direction z: electric field • SU(2) analog of the QHE • topological origin • dissipationless • Occupied HH and LH bands have • opposite contributions. • Spin current is time-reversal even GaAs S.Murakami-N.N.-S.C.Zhang J.Sinova-Q.Niu-A.MacDonald

  21. Wunderlich et al.2004 Experimental confirmation of spin Hall effect in GaAs D.D.Awschalom (n-type)UC Santa Barbara J.Wunderlich (p-type ) Hitachi Cambridge p-type n-type Y.K.Kato,et.al.,Science,306,1910(2004)

  22. spin density spin current Luttinger model Rashba model Mesoscopic Spin Hall Effect Impurity scattering, electrodes, leads, sample edgeKeldysh formalism voltage spin current Spin-orbit int. produces spin current but relaxes spin accumulation. Spin accumulation is due to the dissipation (charge current). Intrinsic one dominates in Luttinger (p-type) and is much larger than extrinsic one in n-type Spin accumulation Hitachi-Cambridge exp. Is consistent with the present calculation and intrinsic SHE. Relaxation rate M.Onoda and N.N. PRB(05 )

  23. Spin Hall effect in metals Otani-Maekawa E.Saitoh et al.

  24. Quantum Spin Hall System Bernevig-S.C.Zhang S.Murakami, N.N., S.C.Zhang (2004) HgTe, HgSe, HgS, alpha-Sn Zero/narrow gap semiconductors Rocksalt structure: PbTe, PbSe, PbS Finite spin Hall conductance but not quantized Kane-Mele Pfaffian Z2 = # of helical edge mode pair time-reversal operation

  25. Localization/delocalization is affected by topology M.Onoda-Avishai-Nagaosa

  26. Spin Current produces polarization - Multiferroic phenomena - p-orbitals ｊs：spin current Δ：ｄ－ｐ energy difference V： transfer integral I：constant　 （Bohr radius） d-orbitals d-orbitals Katsura-Nagaosa-Balatsky PRL05 M1 O M2 Katsura-Balatsky-Nagaosa PRL07

  27. Tokura-Kimura group

  28. Photon also has “spin” Optical Hall Effect Gigantic shift of X-ray beam in deformed crystals Onoda-Murakami-Nagaosa PRL04 Sawada-Murakami-Nagaosa PRL06

  29. To Summarize Transport in multi-band systems have different features from the single-band systems Topological current by occupied states Extension of quantum Hall physics to common materials  Room temperature quantum phenomena Band Crossing play essential roles Many phenomena related to the multi-band Anomalous Hall effect, Spin Hall effect, Dielectrics/Ferroelectrics, Magneto-electric effect/Multi-ferroics, Optical Hall effect……………… Application to Nano-Sciences -- Geometry drives electrons/light

  30. 多謝 Z.Fang G.Y. Guo H.Katsura S.Murakami M.Onoda S.Onoda K.Ohgushi K.Sawada R.Shindou N.Sugimoto G.Tatara K.Terakura S.C.Zhang Y.Oohara Y.Tokura Y.Taguchi H.Yoshizawa

  31. Lastly but not in the least………. Dec. 20/AP 20

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