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

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

k

E

wave packet

nkn vk

k

dt

trd

)()(

dt

trdrB

r

rV

dt

tkd )()(

)()(

nknkvfdk

eJ2

02

nn

nk

k

dkeJ

Totally-filled band does not contribute to current.

Boltzmann transport equation

group velocity

Only energy dispersion matters ?)(kn

Intra- and Inter-band matrix elements of current

k

E

wave packetnkevnkJnk ||

mkJnk ||Even a filled band can support currente.g., polarization current quantum Hall current

mkk

nkemkk

kHnkemkJnk

k

kenk

k

kHnkenkJnk

mknk

n

||)(|)(

|||

)(|

)(|||

Wavefunction matters !!

Correct equation of motion taking into account inter-band matrix element

dt

tkdkB

k

k

dt

trdn

n )()(

)()(

dt

trdrB

r

rV

dt

tkd )()(

)()(

k-space curvature

r-spacecurvature

anomalous velocity

)()( kAkB nn nknkikAn ||)(

Origin of the k-space curvature = interband current matrix

Luttinger,Blount,Niu

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

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

+-

-eE

Js

B

Berry phase

EkB )(

)(f

)(f

s

Energy degeneracy point　　　 = Magnetic

monopole

Gauge Flux = Solid angle

C

3,2,1a

aakH matrices Pauli :a

B(k) diverges at band crossing

Breakdown of semi-classical Boltzmann approach

3||2)(

k

kkBn

When the band crossing occurs ?　 (with spin-orbit int.)

)()(

symmetry reversal-Time

kk

T

)()(

symmetry inversion -Space

kk

I

)()(

case symmetric ,Both

kk

TI

k

E

k

E

Kramer’sdouble degeneracy accidental degeneracy

case symmetric ,TI broken and/or TI

tune 3 parameters

tune 5 parametersNeed for symmetry reason

No degeneracy

M

vy

x

-e

-e

-e

-eE

Anomalous Hall Effect

magnetization

Electricfield

spin-orbit interaction

xy = R0H + 4RSMordinary term anomalous term

N.P.Ong

Anomalous Hall Effect in SrRuO3 - Magnetic Monopole in k-Space

Small energy scale 0.02eV Behavior like quantumchaos

)(kb zn

Z.Fang et al.

Kubo FormulaEnergy broadening

meVi 5020/

Also A.H.MacDonald groupfor (Ga,Mn)As and Fe

Previous theories of AHE - 50 years of debates !!

Karplus-Luttinger (1954) Interband effect Perturbation in s-o int.

'

'''

'' )1()1(k

nknknknkk

nknknknknknk ffWffWk

feE

t

f

kkk fff )0(

anomalous ,anomalous,group,

)(k

nkkk v

k

kvvv

kkkk WW ''group,kkvf with (Skew scattering)

anomalous ,)0(kk vf Intrinsic mechanism with dissipationless current

extrinsic mechanism with impurity scatt. and dissipation

Smit

Skew scattering KL term

) 1

1(2

F

xy ha

eA rough estimation

3 energy scales in the problemWF or /h

Fk

Band width/gapRelaxation

Spin-orbit interaction

Engel et al.

Hardware: Gauge-covariant formalism of Keldysh Green’s function

mass ofcenter :2/)( 21 xxX relative :21 pxxr

tionrepresenta Wigner :),( pX

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

S.Onoda-N.Sugimoto-NN, PRL06

Resonant AHE

Spin-orbitCoupling

Intrinsic (without vertex Correction) is robust against scattering

imp2v)0(/1 nN

p

E-EF

2

Band crossing lifted by spin-orbit interaction

-1132 cm10/ hae

xy

hopping metallic

Super clean

Miyasato-Asamitsuc.f. N.P.Ong

Global behavior of anomalous Hall effect

)/)(/(

/2

Fxx he

h

1-13

-1132

cm)(10

cm10/

Fxx

hae310/ F

imp2v)0(/1 nN

vy

x

-e

-e

-e

E

Spin Hall Effect

Electric field

v-e

-e

-e

even odd

even

odd

M

P

T

,jmagnetization

polarizationtoroidal moment

Timereversal

Inversion

Classification of Order Parameters

,sj

currentspin current

charge density

E

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

Ej chargeEj sspin

ohmic dissipationless

Driven by the spin-orbit interaction with large energy scale

Function at room temperature

Spin Hall Effect in p-GaAs

x: current direction y: spin directionz: 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

zsLF

HF

zxy E

ekk

eEj

2

1

4 2

GaAs

E

x

y

z

S.Murakami-N.N.-S.C.ZhangJ.Sinova-Q.Niu-A.MacDonald

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

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

210

32122

103 --

55.1 through 100

,102,106.1

cmS

AmAI

cmcmn

B

p

D

Hitachi-Cambridge exp. Is consistent withthe present calculation and intrinsic SHE.

M.Onoda and N.N. PRB(05 )Relaxation rate

Sp

in a

ccu

mu

lati

on

voltage spincurrent

Spin-orbit int. produces spin current but relaxes spin accumulation.Spin accumulation is due to the dissipation (charge current).

Spin Hall effect in metals

E.Saitoh et al. Otani-Maekawa

Quantum Spin Hall System

Zero/narrow gap semiconductors

S.Murakami, N.N., S.C.Zhang (2004)

Rocksalt structure: PbTe, PbSe, PbSHgTe, HgSe, HgS, alpha-Sn

s

Bernevig-S.C.Zhang

Finite spin Hall conductance but not quantized

Pfaffian time-reversal operation

Kane-Mele

Z2 = # of helical edge mode pair

Localization/delocalization is affected by topology

73.2symplectic 33.2unitary

M.Onoda-Avishai-Nagaosa

V

d-orbitals d-orbitals

p-orbitals

P

sj

O M2M1

1e 2e

12e Δ ：ｄ－ｐ energy difference

V ： transfer integral

I ： constant 　 （ Bohr radius ）

0a

ｊ s ： spin current

sjeP

12

Katsura-Nagaosa-Balatsky PRL05

Spin Current produces polarization - Multiferroic phenomena -

)( SEAspin

spinspin AjLint

cjB

spinjE

e

mq

mq

mjE

mq

S

Katsura-Balatsky-Nagaosa PRL07

Tokura-Kimura group

Gigantic shift of X-ray beam in deformed crystals

Optical Hall Effect

)|)|

)]([

)||()(

ckcc

ccc

ckccc

ccc

zkiz

krvk

zzkk

krvr

c

c

Photon also has “spin”

Onoda-Murakami-Nagaosa PRL04

Sawada-Murakami-Nagaosa PRL06

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.OoharaY.TokuraY.Taguchi H.Yoshizawa

Lastly but not in the least……….

Dec. 20/AP 20