Transcript Document 7152469

Quantum Spin Hall Effect
- A New State of Matter ? Aug. 1, 2006 @Banff
Naoto Nagaosa
Dept. Applied Phys. Univ. Tokyo
Collaborators:
M. Onoda (AIST), Y. Avishai (Ben-Grion)
magnetic field
B
Voltage
Hall effect
(Integer) Quantum Hall Effect
Quantized Hall conductance in the unit of
Plateau as a function of magnetic field
e2
h
(Integer) Quantum Hall Effect
pure case
Quantized Hall conductance in the unit of
Plateau as a function of magnetic field
Disorder effect and localization
e2
h
(Integer) Quantum Hall Effect
pure case
Localized states do not contribute to  xy
Extended states survive
only at discrete energies
Anderson Localization of electronic wavefunctions
x
Extended Bloch wave
Localized statex
g ( L)  G /( e2 Ld 2 )  E / E
Periodic boundary
condition
impurity
quantum interference between
scattered waves.
L
E
x
Thouless number
= Dimensionless conductance
E
Anti-periodic boundary
condition
Scaling Theory of Anderson Localization
g ( L  dL)  f ( g ( L), dL / L)
The change of the Thouless number
Is 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 !!
Universality classes of Anderson Localization
Orthogonal:
Time-reversal symmetric system
without the spin-orbit interaction
Symplectic class with
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.
Chern number
wave function
2
y / ky
0
 x / kx
2
i
 xy /(e / h)  C 
4
2
 
 d k  k x | k y  c.c
2
Chern number is carried
only by extended states.
Topology “protects” extended states.
Chiral edge modes
M magnetization
y
v
-e
-e
-e
Electric
field E
-e
x
Anomalous Hall Effect
Hall, Karplus-Luttinger, Smit,
Berger, etc.
Berry phase
Electrons with ”constraint”
E
E
doubly
degenerate
positive
energy states.
k
k
Dirac electrons
Projection onto positive energy state
Spin-orbit interaction
as SU(2) gauge connection
Bloch electrons
Projection onto each band
Berry phase
of Bloch wavefunction
Berry Phase Curvature in k-space
 nk (r )  eikr unk (r )
Bloch wavefucntion
An (k )  i  unk |  k | unk 
Berry phase connection in k-space
xi  ri  An (k )  i ki  An (k )
covariant derivative
[ x, y]  i( k x Any (k )   k y Anx (k ))  iBnz (k )
Curvature in k-space
k
dx(t )
V k x
V
 i[ x, H ]  x  i[ x, y ]
  Bnz (k )
dt
m
y m
y
Anomalous Velocity and
Anomalous Hall Effect
kz
New Quantum Mechanics !!
Non-commutative Q.M.
 | unk  k 
k  | u 
nk
kx
ky
Duality between
Real and Momentum Spaces



d r (t )  n ( k )   d k (t )

 Bn (k ) 

dt
dt
kspace
curvature
k



d k (t ) V ( r )   d r (t )

 B( r ) 

dt
dt
rspace
curvature
r
Distribution of momentum space “magnetic field” in momentum space
of metallic ferromagnet with spin-orbit interaction.
Gauge flux density
Chern #'s : (-1, -2, 3, -4, 5 -1)
Chern number =
Integral of the gauge flux
over the 1st BZ.
M.Onoda, N.N.
J.P.S.P. 2002
Localization in Haldane model -- Quantized anomalous Hall effect
M.Onoda-N.N. 2003
spin current
time-reversal even
y
v
-e
v
-e
-e
-e
-e
-e
E
Electric field
x
Spin Hall Effect
D’yakonov-Perel (1971)
Spin current induced by an electric field
j xy 


eE z H
1
L
k

k

 s Ez
F
F
4 2 
2e
x: current direction
y: spin direction
z: electric field
SU(2) analog of the QHE
• topological origin
• dissipationless
• All occupied states in the valence
band contribute.
• Spin current is time-reversal even
z

E
y
x
GaAs
S.Murakami-N.N.-S.C.Zhang
J.Sinova-Q.Niu-A.MacDonald
Wave-packet formalism in systems with Kramers degeneracy
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.
 n (t )   d 3q a1 (q , t )  n1 (q , xc , t )  a2 (q , t )  n 2 (q , xc , t )


 z1 (q , t ) 
1
   
a12  a22
 z 2 (q , t ) 
 

 a1 (q , t ) 

 
a
(
q
 2 , t) 
Eq. of motion


k  eE
E n   n
xl 
 k j z Flj z
kl
 
z  i k  An  z




n  H, L

 

( n  H , L)
Experimental confirmation of spin Hall effect in GaAs
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
Recent focus of theories
Quantum spin Hall effect - A New State of Matter ?
Spin Hall Insulator with real Dissipationless spin current
S.Murakami, N.N., S.C.Zhang (2004)
Bernevig-S.C.Zhang
Kane-Mele
Zero/narrow gap semiconductors HgTe, HgSe, HgS, alpha-Sn
Rocksalt structure: PbTe, PbSe, PbS
Finite spin Hall
conductance
but not quantized
s
0
No edge modes
for generic spin
Hall insulator
H   cr (1  2 )  M5 cr

r








  cr x  2 4  2 5  cr  cr y   2 4  2 5  cr  H.c.

3 
3 


r 







 

  cr x  y  3 3  2 5  cr  cr x  y  3 3  2 5  cr  H.c.

3 
3 
2
 2
r 

M.Onoda-NN (PRL05)
Quantum spin Hall
Generic
Spin Hall Insulator
Two sources of “conservation law”
Rotational symmetry
 Angular momentum
Gauge symmetry
 Conserved current
Topology
 winding number
Quantum Hall Problem
Quantized Hall
Conductance
TKNN
Localization
problem
TKNN
Topological
Numbers
Chern
Edge modes
Gauge invariance
Conserved charge current and U(1) gauge invariance
Issues to be addressed
Spin Hall
Conductance
Localization
problem
Sheng-Weng-Haldane
Topological
Numbers
Spin Chern, Z2
Kane-Mele
Xu-Moore
Wu-Bernevig-Zhang
Qi-Wu-Zhang
Edge modes
No conserved spin current !!
Kane-Mele Model of quantum spin Hall system
H k  H k
Lattice structure
and/or inversion symmetry breaking
Graphene, HgTe at interface, Bi surface
(Bernevig-S.C.Zhang) (Murakami)
Pfaffian

time-reversal operation
Stability of edge modes
Z2 topological number = # of helical edge mode pairs
Kane-Mele 2005
Two Dirac Fermions at K and K’  8 components
1st BZ
K
K’
K’
K
K
K’
SU(2) anomaly (Witten) ?
helical edge modes
Stability against the T-invariant disorder due to Kramer’s theorem
Kane-Mele, Xu-Moore, Wu-Bernevig-Zhang
Sheng et al. 2006
Qi et al. 2006
Chern Number Matrix
C   C 
: spin Chern number
Generalized twisted boundary condition
Qi-Wu-Zhang(2006)
Spin Chern number
4n  2 or 4n
Issues to be addressed
Spin Hall
Conductance
Localization
problem
Sheng-Weng-Haldane
Topological
Numbers
Spin Chern, Z2
?
Kane-Mele
Xu-Moore
Wu-Bernevig-Zhang
Qi-Wu-Zhang
Edge modes
No conserved spin current !!
Generalized Kane-Mele Model
Z2 non-trivial
Z2 trivial
Chern number =1,-1
Chern number =0
Two decoupled Haldane model
(unitary)
hx
Numerical study of localization
MacKinnon’s transfer matrix method and finite size scaling
M
L
G(1, L)  e  L / 
Localization length  ( M , W )
( M ,W )   ( M ,W ) / M
(a-1)
(a-2)
(a-3)
(b-1)
(b-2)
(b-3)
(c-1)
(c-2)
(c-3)
2 copies of Haldane model
increasing disorder strength W
Two decoupled unitary model
with Chern number +1,-1
Symplectic model
Disappearance of the extended states in unitary model
hx hybridizes positive and
negative Chern number states
hx
Disappearance of the extended states in trivial symplectic model
hx
Scaling Analysis of the localization/delocalization transition
 symplectic  2.73
 unitary  2.33
Conjectures
No quantized
spin Hall conductance
nor plateau
Spin Hall
Conductance
Localization
problem
Topological
Numbers
Spin Chern, Z2
Helical Edge modes
No conserved spin current !!
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



E







E

