Intrinsic Hall Effects of Electrons and Photons – Geometrical Phenomena in Linear Response Theory

Masaru Onoda (CERC-AIST)

The 21st Century COE International Symposium 2007.11.05~07

Linear Response Theory in Commemoration of its 50th Anniversary

• Collaborators

– S. Murakami (TIT) N. Nagaosa (Univ. of Tokyo)

• Special thanks to

– Y. Tokura (Univ. of Tokyo) H. Aoki (Univ. of Tokyo)

Outline

• Motivation

• Intrinsic mechanism of Hall effects

• Semiclassical Interpretation

• Optical Hall Effect

• Summary

Mission of the theory team of CERCNew functionalities based on geometrical phase of electron systems

• Anomalous Hall effect (AHE) Quantum AHE

• Spin Hall effect (SHE) Quantum SHE

• Optical Hall effect (OHE)

• AHE high-sensitive Hall element

• QAHE resistance standard without external magnetic field

• Edge states of QSHE spin filter, control of nuclear spin

• OHE + tunable photonic crystal (PX) optical switch

• Beam with internal rotation in PX optical mixer without fin

Anomalous Hall Effect

Current

M MRBR s 0H

M

B

Quantization

Our contribution not presented today

Disorder induced quantization of AHE : M. Onoda and N. Nagaosa, PRL 90, 206601 (2003)

V

Spin Hall effect

Current

Quantization

Our contributions not presented todayReal space simulation: 　 M. Onoda and N. Nagaosa, PRB 72, 081301(R) (2005);PRL 95, 106601 (2005)Disorder effect: M. Onoda, Y.Avishai, N. Nagaosa, PRL 98, 076802 (2007)

Optical Hall effect

Mission of the theory team of CERCNew functionalities based on geometrical phase of electron systems

• Anomalous Hall effect (AHE) Quantum AHE

• Spin Hall effect (SHE) Quantum SHE

• Optical Hall effect (OHE)

• AHE high-sensitive Hall element

• QAHE resistance standard without external magnetic field

• Edge states of QSHE spin filter, control of nuclear spin

• OHE + tunable photonic crystal (PX) optical switch

• Beam with internal rotation in PX optical mixer without fin

Skew scattering Side jump

L.Berger, PRB 2, 4559 (1970)J.Smith, Physica 24, 39 (1958)

Conventional mechanisms of AHE and SHE

xy in Kubo formalism + diagrammatic technique

Spin dependent scattering

• M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971)

• J. E. Hirsch, PRL 83, 1834 (1999)

• S. Zhang, PRL 85, 393 (2000)

Extrinsic spin Hall effect

Id

MRBRI

dV sH

H

0

B

H

B

H

MRBR s0 MRBR s0

Conventional HE Hall elementSemiconductor withhigh resistivity and high mobility

AHE high sensitive Hall element(patent No. 2005-19894)

Material search and design are needed for optimization. Research on intrinsic mechanism

Intrinsic Mechanism

kkk

kkk

Ω nnn

n

znnxy

uui

fV

e

:curvatureBerry

)(,

2

Quantum Hall effect

K. v. Klitzing, G. Dorda, M. Pepper, PRL 45, 494 (1980)

TKNN, PRL 49, 405 (1982)

H. Aoki and T. Ando, PRL 57, 3039 (1987)

Intrinsic anomalous Hall effect due to chiral spin orderK. Ohgushi, S. Murakami, N. Ngaosa, PRB 62, R6065 (2000)Y. Taguchi et al., Science 291, 2573 (2001)

Intrinsic spin Hall effectS. Murakami, N. Nagaosa., S.-C. Zhang, Science 301, 1348 (2003)J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)

Multi-band effectR. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954)J. M. Luttinger, Phys. Rev. 112, 739 (1958)

1

2

3

Berry curvature ~ magnetic field in k-space

k

kk,

2

)(n

znnxy f

V

e

EF

kx

ky

Geometrical aspect only in

special situations?

t2g model

dxy dyz dzx

px

py

M.Onoda and N. Nagaosa, JPSJ 71, 19 (2002)

zzsUm

sl

Nearly degenerate (resonant) point Large

Topological/Geometrical and Resonant aspects

n

n

1n

1n

Small change of a parameter, e.g., Mz

Drastic change of Drastic change of xy

Topological transition in QHE• J. E. Avron, R. Seiler, B. Simon, PRL 51, 51 (1983)• B. Simon, PRL 51, 2167 (1983)• Y. Hatsugai and M. Kohmoto, PRB 42, 8282 (1990)• M. Oshikawa, PRB 50, 17357 (1994)

Berry curvature of a t2g band

Z. Fang et al., Science 302, 92 (2003)

SrRuO3, Sr0.8Ca0.2RuO3

Sr1-xCaxRuO3

R. Mathieu et al., PRL 93, 16602 (2004)

x: electric fieldy: spin currentz: spin direction

xsLF

HF

xSy E

ekk

eEj z

2

13

12 2

S. Murakami, N. Nagaosa., S.-C. Zhang, Science 301, 1348 (2003)

Intrinsic spin Hall effect in p-type GaAs

GaAs

E

xy

z

2

22

21 22

5

2

1Skk

mH

momentum

spin J. Sinova et al., PRL 92, 126603 (2004)

Intrinsic spin Hall effect in n-type GaAs

Y. K. Kato et al., Science 306,1910 (2004)

Spin Hall Effect in n-type GaAs

J. Wunderlich et al., PRL 94, 047204 (2005)

Spin Hall Effect in p-type GaAs

Extrinsic (Conventional)

Intrinsic

S. Murakami, PRL 97, 236805 (2006)

Bi bilayer

B. A. Bernevig, Science 314, 1757 (2006)

Graphene

C. L. Kane and E. J. Mele, PRL 95, 146802 (2005)

Candidates of QSHECdTe/HgTe/CdTe QW

M. König, et al., Science 318, 766 (2007)

HgTe/Hg0.3Cd0.7Te QW

rr

kk

BrEk

Ωkr

e

nnAnomalous velocity QHE

Effective Lorentz force

Spin-dependent Intrinsic spin Hall effect

M.-C. Chang and Q. Niu, PRL 75, 1348 (1995).

Semiclassical Interpretation

Equations of motion of a wave-packet in magnetic flux commensurate with lattice

k

kkkkk ΩEErJ,

00

2

)(n

nnnnni

ii ff

V

efe

kk ΩΩ

q

p

e

h cellunit

Magnetic Bloch bands

M.-C. Chang, Q. Niu, PRB 53, 7010 (1996)

1st level

2nd level

3rd level

Berry curvature Internal rotation

EE

Multi-band effect Projection due to nk

kkkΩ nnn uui

call/TopologiGeometrica Resonant enhancement

0],[

tivityNoncommuta

YX

Internal RotationSpin-orbit coupling in a broad sense

Wave-particle duality

Wave optics Eikonal Fermat’s principle Geometrical optics

Quantum mechanics Path integral least-action principle Classical mechanics

Semiclassical interpretation

)|)|

)(

)||(

ziz

kv

zzk

v

k

r

kr

Λk

k

Ωkk

r

Equations of motion of optical packet

Anomalous velocity

33

:on vectorPolarizati

)|:stateon Polarizati

:light of Speed

ki

i

z

v

kΛΛΛΩ

e

eeΛ

kkkk

k

kkk

r

Neglecting the spin, i.e., polarization →Conventional equation of geometrical optics

M. Onoda, S. Murakami, N. Nagaosa, PRL 93, 083901 (2004)

Solid (transmission) and broken (reflection) lines: conservation of angular momentum per photon● ■ ：Maxwell’s equations

Linear polarizationImbert-Fedorov shift

Elliptical polarization

M. Onoda, S. Murakami, N. Nagaosa, PRE 74, 066610 (2006)

Internal Angular momenta of light

Linear S=0 Right circular S=1 Left circular S=-1

http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/

Spin angular momentum

Orbital angular momentum

L=0 L=1 L=2 L=3

Transverse shift of Laguerre-Gaussian beam

Experiment, H. Okuda, H. Sasada, Opt. Exp. 14, 8393 (2006)

Experiment, R. Dasgupta, P. K. Gupta, Opt. Comm. 257, 91 (2006)

Theory, V. G. Fedoseyev, Opt. Comm. 193, 9 (2001)

Optical Hall effect in photonic crystals

Multi-band Resonant enhancement

Berry curvature in PX

Trajectory of optical wave-packet in PX

k

kkk

k

Ωkr

n

n

Ex

Ex

)]([

)(

Overhead view of the 2D PX. The crystal structure is not shown.

)(

)(

)(

1

)( modulationGradation2

2

rr ε

x

ε

x

Berry curvature and internal rotation (TE mode)

z EzS

A. T. O’Neil et al., PRL 88, 053601 (2002)

Applications of optical torque

(Extensions of the optical tweezer technology)

H. Ukita, 精密工学会誌 72, 977 (2006)(in Japanese)

3　 x　1.5

calcite 　dielectric

Beam with large angular momentum by photonic crystal

optical mixer without fin

)()()(

)()()(

rHrErrrS

rBrErrrS

cE

c

d

d

11~

1, eff2eff

SS

nSnSE

E

Summary

• Intrinsic mechanism of Hall effects (IHE) in electron systems– Suitable for material search and design– Multi-band effect– Geometrical/topological– Resonant enhancement– Internal rotation + spin-orbit interaction in a broad sense

• Optical Hall effect (OHE)– Counterpart of electronic IHE in optical systems– Imbert-Fedorov shift as a simple example of OHE– Resonant enhancement of OHE in photonic crystals– Optical state with large angular momentum in PX