The 21st Century COE International Symposium 2007.11.05~07
Linear Response Theory in Commemoration of its 50th Anniversary
Intrinsic Hall Effects of Electrons and Photons
– Geometrical Phenomena in Linear Response Theory
Masaru Onoda (CERC-AIST)
• 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 CERC
New 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
B
M
V
H  R0 B  Rs M
Current
M
Quantization
Our contribution not presented today
Disorder induced quantization of AHE : M. Onoda and N. Nagaosa, PRL 90, 206601 (2003)
Spin Hall effect
Current
Quantization
Our contributions not presented today
Real 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 CERC
New 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
Conventional mechanisms of AHE and SHE
sxy in Kubo formalism + diagrammatic technique
Spin dependent scattering
Skew scattering
Side jump
J.Smith, Physica 24, 39 (1958)
L.Berger, PRB 2, 4559 (1970)
Extrinsic spin Hall effect
•
•
•
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)
VH 
H
d
I
R0 B  Rs M
I
d
H
H
B
R0 B  Rs M
Conventional HE Hall element
Semiconductor with
high resistivity and high mobility
B
R0 B  Rs M
AHE high sensitive Hall element
(patent No. 2005-19894)
Material search and design are needed for optimization.
 Research on intrinsic mechanism
Intrinsic Mechanism
Multi-band effect
R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954)
J. M. Luttinger, Phys. Rev. 112, 739 (1958)
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)
s xy
e2

V

f ( nk ) nz k
n,k
Berry curvat ure: Ωnk  i  u nk u nk
Intrinsic anomalous Hall effect due to chiral spin order
K. Ohgushi, S. Murakami, N. Ngaosa, PRB 62, R6065 (2000)
Y. Taguchi et al., Science 291, 2573 (2001)
Intrinsic spin Hall effect
S. Murakami, N. Nagaosa., S.-C. Zhang, Science 301, 1348 (2003)
J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)
1
3
F
2
2
e
s xy 
V
 f (
nk
)
z
nk
n,k
Berry curvature ~ magnetic field in k-space
Geometrical aspect
only in
special situations?
EF
ky
kx
t2g model
px
py
l  s
Umz s z
dxy
dyz
dzx
M.Onoda and N. Nagaosa, JPSJ 71, 19 (2002)
Topological/Geometrical and Resonant aspects
n  1
n
n  1
n
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)
Small change of a parameter, e.g., Mz
 Drastic change of 
 Drastic change of sxy
Nearly degenerate (resonant) point Large 
SrRuO3, Sr0.8Ca0.2RuO3
Berry curvature of a t2g band
Sr1-xCaxRuO3
R. Mathieu et al., PRL 93, 16602 (2004)
Z. Fang et al., Science 302, 92 (2003)
Intrinsic spin Hall effect in p-type GaAs
 
  2
1 
5  2
H
  1   2 k  2 2 k  S 

2m 
2 

x: electric field
y: spin current
z: spin direction
y
x
z

E
GaAs
j yS z 


eE x
1
H
L
3
k

k

s s Ex
F
F
2
12 
2e
S. Murakami, N. Nagaosa., S.-C. Zhang, Science 301, 1348 (2003)
Intrinsic spin Hall effect in n-type GaAs
momentum
spin
J. Sinova et al., PRL 92, 126603 (2004)
Spin Hall Effect in n-type GaAs
Intrinsic
Spin Hall Effect in p-type GaAs
Y. K. Kato et al., Science 306,1910 (2004)
Extrinsic (Conventional)
J. Wunderlich et al., PRL 94, 047204 (2005)
Candidates of QSHE
Graphene
CdTe/HgTe/CdTe QW
C. L. Kane and E. J. Mele, PRL 95, 146802 (2005)
Bi bilayer
S. Murakami, PRL 97, 236805 (2006)
B. A. Bernevig, Science 314, 1757 (2006)
HgTe/Hg0.3Cd0.7Te QW
M. König, et al., Science 318, 766 (2007)
Semiclassical Interpretation
h p
Equations of motion of a wave-packet
F unit cell 
e q
in magnetic flux commensurate with lattice
M.-C. Chang and Q. Niu, PRL 75, 1348 (1995).
r   nk  k  Ωnk
k  E  er  B
r
r
Magnetic Bloch bands
Anomalous velocity  QHE
Effective Lorentz force
e2
 f

J  e f  i  ri    0  nk  nk ( nk  E )  f 0  nk E  Ωnk 
V n,k  

i
Spin-dependent  Intrinsic spin Hall effect
Ωk   Ωk
Berry curvature
Internal rotation
1st level
2nd level
3rd level
M.-C. Chang, Q. Niu, PRB 53, 7010 (1996)
E
Multi-band effect  Projection due to nk
Geometrical/T opological
Resonant enhancement
Ωnk  i  unk unk
Wave-particle duality
Noncommuta
tivity
[ X ,Y ]  0
Internal Rotation
Spin-orbit coupling in a broad sense
Wave optics  Eikonal  Fermat’s principle
 Geometrical optics
Semiclassical interpretation
Quantum mechanics  Path integral  least-action principle 
Classical mechanics
Equations of motion of optical packet
Anomalous velocity
Speed of light : vr
P olarizaton
i st at e:| z )
Λk    iek e k
P olarizaton
i vect or: ek
Ωk    Λk  iΛk  Λk 
k
s3
3
k
k 
r  vr  k  ( z | Ωk | z )
k
k  (vr )k
| z)  ik  Λ | z )
Neglecting the spin, i.e., polarization
→Conventional equation
of geometrical optics
M. Onoda, S. Murakami, N. Nagaosa, PRL 93, 083901 (2004)
k
Solid (transmission) and broken (reflection) lines:
conservation of angular momentum per photon
● ■ ：Maxwell’s equations
Imbert-Fedorov shift
Elliptical polarization
Linear polarization
M. Onoda, S. Murakami, N. Nagaosa, PRE 74, 066610 (2006)
Internal Angular momenta of light
Spin angular momentum
Linear S=0
Right circular S=1
Left circular S=-1
Orbital angular momentum
L=0
L=1
L=2
http://www.physics.gla.ac.uk/Optics/projects/singlePhotonOAM/
L=3
Transverse shift of Laguerre-Gaussian beam
Theory, V. G. Fedoseyev, Opt. Comm. 193, 9 (2001)
Experiment, R. Dasgupta, P. K. Gupta, Opt. Comm. 257, 91 (2006)
Experiment, H. Okuda, H. Sasada, Opt. Exp. 14, 8393 (2006)
Optical Hall effect in photonic crystals
Multi-band  Resonant enhancement
Berry curvature in PX
Trajectory of optical wave-packet in PX
Gradation  modulation  2 ( x)
1
 2 ( x)

ε (r )
ε (r )
r   ( x) k Enk  k  Ωk
k  [ ( x)] E
nk
Overhead view of the 2D PX.
The crystal structure is not shown.
Berry curvature and internal rotation (TE mode)
z
S
E
z
Applications of optical torque
(Extensions of the optical tweezer technology)
3m x 1.5m
calcite
1m dielectric
H. Ukita, 精密工学会誌 72, 977 (2006)
(in Japanese)
A. T. O’Neil et al., PRL 88, 053601 (2002)
S   dr (r  rc )  E (r )  B(r )
S E   dr (r  rc )  E (r )  H (r )

E
2
S  neff S , neff  1
E

S ~ 1  S  1
Beam with large angular momentum by photonic crystal
 optical mixer without fin
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

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