Quantum Spin Hall Effect - A New State of Matter ? -
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Quantum Spin Hall Effect
- A New State of Matter ? -
Naoto Nagaosa
Dept. Applied Phys. Univ. Tokyo
Collaborators:
M. Onoda (AIST), Y. Avishai (Ben-Grion)
Aug. 1, 2006 @Banff
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Bmagnetic field
Voltage
Hall effect
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(Integer) Quantum Hall Effect
Quantized Hall conductance in the unit of h
e2
Plateau as a function of magnetic field
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(Integer) Quantum Hall Effect
Quantized Hall conductance in the unit of h
e2
Plateau as a function of magnetic field
pure case
Disorder effect and localization
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pure case
Localized states do not contribute to xy
Extended states survive only at discrete energies
(Integer) Quantum Hall Effect
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Anderson Localization of electronic wavefunctions
xx
ximpurity
Extended Bloch waveLocalized state
EELeGLg d /)/()( 22 Thouless number= Dimensionless conductance
E
L
E
Periodic boundary condition
Anti-periodic boundary condition
quantum interference betweenscattered waves.
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Scaling Theory of Anderson Localization
)/),(()( LdLLgfdLLg The change of the Thouless numberIs determined only by the Thouless number Itself.
In 3D there is a metal-insulator transition
In 1D and 2D all the states are localized for any finite disorder !!
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Symplectic class with Spin-orbit interaction
Universality classes of Anderson Localization
Orthogonal: Time-reversal symmetric system without the spin-orbit interaction
Symplectic: Time-reversal symmetric system with the spin-orbit interaction
Unitary: Time-reversal symmetry broken Under magnetic field or ferromagnets Chern number extended states
Universality of critical phenomena Spatial dimension, Symmetry, etc. determine the critical exponents.
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xx k/
yy k/
0 2
2wave function
Chern number
cckk
kdi
Cheyx
xy .|4
)//( 22
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Chern number is carried only by extended states.
Topology “protects” extended states.
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Chiral edge modes
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M
vy
x
-e
-e
-e
-eE
Anomalous Hall Effect
magnetization
Electricfield
Hall, Karplus-Luttinger, Smit,Berger, etc.
Berry phase
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Electrons with ”constraint”
Projection onto positive energy stateSpin-orbit interaction
as SU(2) gauge connection
Dirac electrons
doublydegenerate
positive energy states.
E
k
Bloch electrons
Projection onto each bandBerry phase
of Bloch wavefunction
k
E
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Berry Phase Curvature in k-space
Bloch wavefucntion )()( ruer nkikr
nk
nkknkn uuikA ||)( Berry phase connection in k-space
)()( kAikArx nknii i covariant derivative
)())()((],[ kiBkAkAiyx nznxknyk yx Curvature in k-space
y
VkB
m
k
y
Vyxi
m
kHxi
dt
tdxnz
xx
)(],[],[)(
xk yk
zk
Anomalous Velocity andAnomalous Hall Effect
New Quantum Mechanics !!Non-commutative Q.M.
knku| nku|
k
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dt
tkdkB
k
k
dt
trdn
n )()(
)()(
dt
trdrB
r
rV
dt
tkd )()(
)()(
Duality between Real and Momentum Spaces
k- space curvature
r- space curvature
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Gauge flux density
M.Onoda, N.N.J.P.S.P. 2002
Chern #'s : (-1, -2, 3, -4, 5 -1)
Chern number = Integral of the gauge fluxover the 1st BZ.
Distribution of momentum space “magnetic field” in momentum spaceof metallic ferromagnet with spin-orbit interaction.
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M.Onoda-N.N. 2003
Localization in Haldane model -- Quantized anomalous Hall effect
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vy
x
-e
-e
-e
E
Spin Hall Effect
Electric field
v-e
-e
-e
spin currenttime-reversal even
D’yakonov-Perel (1971)
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Spin current induced by an electric field
x: current direction y: spin directionz: electric field
SU(2) analog of the QHE• topological origin• dissipationless • All occupied states in the valence ba
nd contribute.• 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
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Let us extend the wave-packet formalism to the case with time-reversal symmetry.
Adiabatic transport = The wave-packet stays in the same band, but can transform inside the Kramers degeneracy.
Wave-packet formalism in systems with Kramers degeneracy
),(),,(),(),,(),()( 22113 LHntxqtqatxqtqaqdt cncnn
),(
),(1
),(
),(
2
1
22
212
1
tqa
tqa
aatqz
tqz
zAkiz
LHnzFzkk
Ex
Eek
n
nljj
l
n
l
,
Eq. of motion
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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
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Recent focus of theories
Quantum spin Hall effect - A New State of Matter ?
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Spin Hall Insulator with real Dissipationless spin current
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.ZhangKane-Mele
rryxrryxr
rryrrxr
rrr
cccc
cccc
cMcH
H.c.3232
H.c.33
)(
52
33
52
33
52
4252
42
521
Quantum spin Hall GenericSpin Hall InsulatorM.Onoda-NN (PRL05)
0
Finite spin Hall conductance but not quantized
No edge modesfor generic spinHall insulator
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Two sources of “conservation law”
Rotational symmetry Angular momentumGauge symmetry Conserved current
Topology winding number
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Quantum Hall Problem
Quantized Hall Conductance
Localization problem
Topological Numbers
ChernEdge modes
TKNN
2-param. scalin
g
Gauge invariance
TKNN
Conserved charge current and U(1) gauge invariance
Landauer
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Issues to be addressed
Spin Hall Conductance
Localization problem
Topological Numbers
Spin Chern, Z2Edge modes
No conserved spin current !!
Kane-MeleXu-MooreWu-Bernevig-ZhangQi-Wu-Zhang
Sheng-Weng-Haldane
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Kane-Mele 2005
Kane-Mele Model of quantum spin Hall system
Stability of edge modes Z2 topological number = # of helical edge mode pairs
kk HH Lattice structureand/or inversion symmetry breakingGraphene, HgTe at interface, Bi surface (Bernevig-S.C.Zhang) (Murakami)
Pfaffian
time-reversal operation
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1st BZ
K
K
K
K’
K’
K’
Two Dirac Fermions at K and K’ 8 components
helical edge modes
SU(2) anomaly (Witten) ?
Stability against the T-invariant disorder due to Kramer’s theorem
Kane-Mele, Xu-Moore, Wu-Bernevig-Zhang
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Sheng et al. 2006Qi et al. 2006
Chern Number Matrix
CC : spin Chern number
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Generalized twisted boundary condition Qi-Wu-
Zhang(2006)
nn 4or 24 Spin Chern number
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Issues to be addressed
Spin Hall Conductance
Localization problem
Topological Numbers
Spin Chern, Z2Edge modes
?
No conserved spin current !!
Kane-MeleXu-MooreWu-Bernevig-ZhangQi-Wu-Zhang
Sheng-Weng-Haldane
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Two decoupled Haldane model(unitary)
Chern number =0
Chern number =1,-1
Z2 trivialZ2 non-trivial
xh
Generalized Kane-Mele Model
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Numerical study of localization MacKinnon’s transfer matrix method and finite size scaling
M
L
Localization length ),( WM
/),1( LeLG
MWMWM /),(),(
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(a-1)
(b-1)
(a-2) (a-3)
(b-2) (b-3)
(c-1) (c-2) (c-3)
2 copies of Haldane model
increasing disorder strength W
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Two decoupled unitary modelwith Chern number +1,-1
Symplectic model
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xh
Disappearance of the extended states in unitary model
hybridizes positive andnegative Chern number statesxh
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xh
Disappearance of the extended states in trivial symplectic model
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Scaling Analysis of the localization/delocalization transition
73.2symplectic 33.2unitary
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Conjectures
Spin Hall Conductance
Localization problem
Topological Numbers
Spin Chern, Z2Helical Edge modes
No conserved spin current !!
No quantized spin Hall conductancenor plateau
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Conclusions
Rich variety of Bloch wave functions in solids Symmetry classification Topological classification Anomalous velocity makes the insulator an active player.
Quantum spin Hall systems: No conserved spin current but Analogous to quantum Hall systems characterized by spin Chern number/Z2 number
Novel localization properties influenced by topology New universality class !? Graphene, HgTe, Bi (Murakami) Stability of the edge modes
Spin Current physics Spin pumping and ME effect
EE