Topological Currents in Solids - Multi-band Effect and Band Crossings - Naoto Nagaosa 永長 直人...
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Transcript of Topological Currents in Solids - Multi-band Effect and Band Crossings - Naoto Nagaosa 永長 直人...
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