Topological Currents in Solids Multiband Effect and Band

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 Spin metals, spin superconductors etc. Look at “conventional” materials from the new eyes of strong correlation physics Hopefully predict new functions/phenomena

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

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

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

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

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

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

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

rxy = R 0 H ordinary term + 4 p. RSM Anomalous Hall Effect anomalous term M magnetization y v -e -e N. P. Ong Electric field E -e -e x spin-orbit interaction

Anomalous Hall Effect in Sr. Ru. O 3 - Magnetic Monopole in k-Space Z. Fang et al. Small energy scale 0. 02 e. V Behavior like quantum chaos

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

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

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

Resonant AHE S. Onoda-N. Sugimoto-NN, PRL 06 E-EF p Band crossing lifted by spin-orbit interaction Spin-orbit Coupling Intrinsic (without vertex Correction) is robust against scattering

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

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

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

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

Spin Hall Effect in p-Ga. As 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 Ga. As S. Murakami-N. N. -S. C. Zhang J. Sinova-Q. Niu-A. Mac. Donald

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

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

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

Quantum Spin Hall System S. Murakami, N. N. , S. C. Zhang (2004) Bernevig-S. C. Zhang Zero/narrow gap semiconductors Hg. Te, Hg. S, alpha-Sn Rocksalt structure: Pb. Te, Pb. S Finite spin Hall conductance but not quantized Kane-Mele Z 2 = # of helical edge mode pair Pfaffian time-reversal operation

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

Spin Current produces polarization - Multiferroic phenomena p-orbitals ｊs ：spin current Δ：ｄ－ｐ energy difference d-orbitals M 1 O M 2 V： transfer integral I：constant　 （Bohr radius） Katsura-Nagaosa-Balatsky PRL 05 Katsura-Balatsky-Nagaosa PRL 07

Tokura-Kimura group

Photon also has “spin” Optical Hall Effect Onoda-Murakami-Nagaosa PRL 04 Gigantic shift of X-ray beam in deformed crystals Sawada-Murakami-Nagaosa PRL 06

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

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

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