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

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

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

5. Correct equation of motion taking into account inter-band matrix element k-space curvature r-space curvature anomalous velocity Origin of the k-space curvature = interband current matrix Luttinger, Blount, Niu 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 3. Superconducting Current / Diamagnetic Current Dissipationless in equilibrium Responding to A 2. Topological Current Due to multi-band effect/Berry phase Dissipationless in equilibrium The occupied states contribute Berry phase

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

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

10. M v y x -e E Anomalous Hall Effect magnetization Electric field spin-orbit interaction  xy = R 0 H + 4  R S M ordinary termanomalous term N.P.Ong

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

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

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

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

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

17. hoppingmetallic Super clean Miyasato-Asamitsu c.f. N.P.Ong Global behavior of anomalous Hall effect

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

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

20. 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 ohmicdissipationless Driven by the spin-orbit interaction with large energy scale Function at room temperature

21. 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

22. 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 Y.K.Kato,et.al.,Science,306,1910(2004) n-type p-type

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

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

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

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

27. d-orbitals p-orbitals O M2M1 Δ ：ｄ－ｐ energy difference V ： transfer integral I ： constant （ Bohr radius ） ｊ s ： spin current Katsura-Nagaosa-Balatsky PRL05 Spin Current produces polarization - Multiferroic phenomena - Katsura-Balatsky-Nagaosa PRL07

28. Tokura-Kimura group

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

30. 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

31. 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

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