Transcript 投影片 1 - 國立臺灣師範大學 物理學系

03/10/09 @ Juelich
Berry phase in solid state physics
－ a selected overview
Ming-Che Chang
Department of Physics
National Taiwan Normal University
Qian Niu
Department of Physics
The University of Texas at Austin
1
Taiwan
2
Paper/year with the title “Berry phase” or
“geometric phase”
80
70
60
50
40
30
20
10
0
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
3
 Introduction (30-40 mins)
 Quantum adiabatic evolution and Berry phase
 Electromagnetic analogy
 Geometric analogy
 Berry phase in solid state physics
4
Fast variable and slow variable


H (r , p; Ri , Pi )
H+2 molecule
e－
electron; {nuclei}
nuclei move thousands of times
slower than the electron
Instead of solving time-dependent Schroedinger eq., one uses
Born-Oppenheimer approximation
• “Slow variables Ri” are treated as parameters λ(t)
(Kinetic energies from Pi are neglected)
• solve time-independent Schroedinger eq.
H (r , p;  ) n , ( x )  En , n , ( x )
“snapshot” solution
5
Adiabatic evolution of a quantum system H (r , p;  )
• Energy spectrum:
E(λ(t))
• After a cyclic evolution
 (T )   (0)
n+1
x
n
x
 n ,  (T )
n-1
0
T
dt ' En (t ')

0
e
 n, ( 0 )
i
Dynamical phase
λ(t)
• Phases of the snapshot states at different λ’s are
independent and can be arbitrarily assigned
 n , (t )  ei
n ( )
 n , ( t )
• Do we need to worry about this phase?
6
No!
Pf :
• Fock, Z. Phys 1928
• Schiff, Quantum Mechanics (3rd ed.) p.290
Consider the n-th level,
t
dt ' En (t ')

0
e
 n ,
i n (  ) i
 (t )  e

H   (t )  i
  (t )
t
H n,  En n,

 n ,    0

 n  i  n ,
Redefine the phase,
Stationary,
snapshot state
≡An(λ)
 'n ,  ei
n ( )
An’(λ)  An(λ)
 n ,

n

Choose a  (λ) such that,
An’(λ)=0
Thus removing
the extra phase
7
One problem:
  A( )
Vector flow
does not always have a
well-defined (global) solution
A
A
Vector flow
Contour of 
 is not
defined here
Contour of 
C

C
A d  0
C

C
A d  0
8
M. Berry, 1984 :
Parameter-dependent phase NOT always removable!
  (T )
T
dt ' E ( t ')

0
e e
  (0)
i C
i
• Berry phase (path dependent)
C 

C
Index n neglected
Berry’s face

 i
   d  0

• Interference due to the Berry phase
if

C
 
1
2
 0, then

1

2
 
1

2
0
Phase difference
a
C
2
a
b
－2
interference
1
1
9
Some terminology
• Berry connection (or Berry potential)
A( )  i      
3
 (t )
• Stokes theorem (3-dim here, can be higher)
C 

C
S
A  d       A  da
S
• Berry curvature (or Berry field)
1
2
C
F ( )     A( )  i       
• Gauge transformation (Nonsingular gauge, of course)
   ei (  )  
A( )  A( )  
F ( )  F (  )
C  C
Redefine the phases of
the snapshot states
Berry curvature nd Berry phase
not changed
10
Analogy with magnetic monopole
Berry potential (in parameter space)
A( )  i      
Berry field (in 3D)
F ( )    A( )
Berry phase
C 

C
A( )  d 
S
Chern number

S
A(r )
Magnetic field
B(r )   A(r )
Magnetic flux
=  F  da
1
2
Vector potential (in real space)
F ( )  da  integer


C
A( r )  dr
=  B  da
S
Dirac monopole
1
4

S
B ( )  da  integer
11
Example: spin-1/2 particle in slowly changing B field
• Real space
• Parameter space
z
Bz
y
By
B(t )
x
B(t )
S
C
C
Bx
(a monopole at
the origin)
H   B  B B  
Level crossing at B=0
E(B)
Berry curvature

F ( B)  i  B  , B   B  , B 
B
Berry phase

  =  F  da 
S
1 Bˆ
2 B2
1
(C )
2
spin × solid angle
12
Experimental realizations :
Bitter and Dubbers , PRL 1987
Tomita and Chiao, PRL 1986
13
Geometry behind the Berry phase
Why Berry phase is often called geometric phase?
base space
fiber space
Fiber bundle
Examples:
• Trivial fiber bundle
(= a product space)
• Nontrivial fiber bundle
Simplest example: Möbius band
R1 x R1
fiber
R1 R1
base
15
Fiber bundle and quantum state evolution
(Wu and Yang, PRD 1975)
Fiber space:
inner DOF, eg., U(1) phase

Base space:
parameter space
• Berry phase = Vertical shift along fiber
(U(1) anholonomy)
• Chern number n
n
1
2
 da F
For fiber bundle
S
χ=2
χ=0
～ Euler characteristic χ

1
2

S
da G For 2-dim closed
surface
χ =－216
 Introduction
 Berry phase in solid state physics
19
20
Berry phase in condensed matter physics, a partial list:
 1982 Quantized Hall conductance (Thouless et al)
 1983 Quantized charge transport (Thouless)
 1984 Anyon in fractional quantum Hall effect (Arovas et al)
 1989 Berry phase in one-dimensional lattice (Zak)
 1990 Persistent spin current in one-dimensional ring (Loss et al)
 1992 Quantum tunneling in magnetic cluster (Loss et al)
 1993 Modern theory of electric polarization (King-Smith et al)
 1996 Semiclassical dynamics in Bloch band (Chang et al)
 1998 Spin wave dynamics (Niu et al)
 2001 Anomalous Hall effect (Taguchi et al)
 2003 Spin Hall effect (Murakami et al)
 2004 Optical Hall effect (Onoda et al)
 2006 Orbital magnetization in solid (Xiao et al)
…
21
Berry phase in condensed matter physics, a partial list:
 1982 Quantized Hall conductance (Thouless et al)
 1983 Quantized charge transport (Thouless)
 1984 Anyon in fractional quantum Hall effect (Arovas et al)
 1989 Berry phase in one-dimensional lattice (Zak)
 1990 Persistent spin current in one-dimensional ring (Loss et al)
 1992 Quantum tunneling in magnetic cluster (Loss et al)
 1993 Modern theory of electric polarization (King-Smith et al)
 1996 Semiclassical dynamics in Bloch band (Chang et al)
 1998 Spin wave dynamics (Niu et al)
 2001 Anomalous Hall effect (Taguchi et al)
 2003 Spin Hall effect (Murakami et al)
 2004 Optical Hall effect (Onoda et al)
 2006 Orbital magnetization in solid (Xiao et al)
…
22
Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect (QHE)
 Anomalous Hall effect (AHE)
 Spin Hall effect (SHE)
Spin
• Persistent
spin current
• Quantum
tunneling
Bloch state
• AHE
• Electric
polarization
• SHE
• QHE
23
Electric polarization of a periodic solid
P
1
3
d
r r  (r )

V
 well defined only for finite system
(sensitive to boundary)
 or, for crystal with well-localized dipoles
(Claussius-Mossotti theory)
• P is not well defined in, e.g., covalent crystal:
P
Unit cell
Choice 1
… +
－
…
P
Choice 2
－
…
+
• However, the change of P is well-defined
ΔP
…
…
Experimentally, it’s ΔP
that’s measured
…
33
Modern theory of polarization
One-dimensional lattice (λ=atomic displacement in a unit cell)
 nk (r )  eikr unk (r )
q
P    nk r  nk
L nk
Iℓℓ-defined
Resta, Ferroelectrics 1992
However, dP/dλ is well-defined, even for an infinite system !
P  
dP
d   P (2 )  P (1 )
d
King-Smith and
Vanderbilt, PRB 1993
dk   
where P     q  
unk i
unk
BZ 2
k
n

 q n
n 2
Berry potential
• For a one-dimensional lattice with inversion symmetry
(if the origin is a symmetric point)
 n  0 or 
(Zak, PRL 1989)
• Other values are possible without inversion symmetry
34
Berry phase and electric polarization
g1=5 g2=4
Dirac comb model
…
…
0
b
Rave and Kerr,
EPJ B 2005
a
P1  q
Lowest
energy
band: γ1
 1
2
← g2=0
γ1=π
similar formulation in 3-dim
using Kohn-Sham orbitals
r =b/a
Review: Resta, J. Phys.: Condens. Matter 12, R107 (2000)
35
Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect
 Anomalous Hall effect
 Spin Hall effect
36
Semiclassical dynamics in solid
Limits of validity: one band approximation
E(k)
n+1
x
Negligible inter-band transition.
n
x
“never close to being violated in a metal”
n-1
dk
 eE  er  B
dt
dr 1 En

dt
k
0
2π
• Lattice effect hidden in En (k)
• Derivation is harder than expected
Explains (Ashcroft and Mermin, Chap 12)
• Bloch oscillation in a DC electric field,
quantization → Wannier-Stark ladders
• cyclotron motion in a magnetic field,
quantization → LLs, de Haas - van Alphen effect
…
37
Semiclassical dynamics - wavepacket approach
rW
1. Construct a wavepacket that is
localized in both r-space and k-space
(parameterized by its c.m.)
rc (t )
2. Using the time-dependent variational
principle to get the effective Lagrangian
for the c.m. variables
k W
Leff (rc , kc ; rc , kc )  W i

H W
t
3. Minimize the action Seff[rc(t),kc(t)] and
determine the trajectory (rc(t), kc(t))
kc (t )
→ Euler-Lagrange equations
Wavepacket in Bloch band:
Leff 


kc  eA  rc  kc  Rn  En (rc , kc )
Berry potential
(Chang and Niu, PRL 1995, PRB 1996)
38
Semiclassical dynamics with Berry curvature
dk
 eE  er  B
dt
dr 1 En

 k  n (k )
dt
k
“Anomalous”
velocity
Cell-periodic
Bloch state
Berry curvature
 n (k )  i  k unk   k unk
Wavepacket energy
En (rc , kc )  En0 (kc ) 
e
Ln (kc )  B
2m
Bloch energy
Zeeman energy due to
spinning wavepacket
Ln (kc )  m W  r  rc   v W
If B=0, then dk/dt // electric field
→ Anomalous velocity ⊥ electric field
Simple
and
Unified
• (integer) Quantum Hall effect
• (intrinsic) Anomalous Hall effect
• (intrinsic) Spin Hall effect
39
 Why the anomalous velocity is not found earlier?
In fact, it had been found by
• Adams, Blount, in the 50’s
 Why it seems OK not to be aware of it?
For scalar Bloch state (non-degenerate band):
• Space inversion
symmetry
n (k )  n (k )
• Time reversal
symmetry
n (k )  n (k )
both symmetries
 When do we expect to see it?
n (k )  0,  k
n (k )  0
• SI symmetry is broken
← electric polarization
• TR symmetry is broken
← QHE
• spinor Bloch state (degenerate band)
← SHE
Also, • band crossing
← monopole
40
Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect
 Anomalous Hall effect
 Spin Hall effect
41
Quantum Hall effect (von Klitzing, PRL 1980)
classical
σH (in e2/h)
2 DEG
3
quantum
2
1
Increasing B
1/B
Each LL contributes one e2/h
z
AlGaAs
GaAs
EF
LLs
EF
energy
CB
Increasing B
B=0
2 DEG
Density of states
VB
42
Semiclassical formulation
Equations of motion
(In one Landau subband)
dk
 eE
dt
dr 1 E

 k  ( k )
dt
k
Magnetic field effect
is hidden here
d 2k
J  e 
r
2
(2 )
filled
=0 
e2
d 2k
E 
( k )
2
(2

)
filled
 e2 1
 Jx =  
 h 2


d k  z (k )  E y


filled

2
Hall conductance
H
Quantization of Hall conductance (Thouless et al 1982)
1
2

d 2 k  z (k )  integer n
BZ
e2
 H  n
h
Remains quantized even with
disorder, e-e interaction
(Niu, Thouless, Wu, PRB, 1985)
43
Quantization of Hall conductance (II)
For a filled Landau subband
Brillouin zone

d 2 k  z (k ) 

dk  A
BZ
BZ
Counts the amount of
vorticity in the BZ
due to zeros of Bloch state
(Kohmoto, Ann. Phys, 1985)
In the language of differential geometry,
this n is the (first) Chern number that
characterizes the topology of a fiber bundle
(base space: BZ; fiber space: U(1) phase)
44
Berry curvature and Hofstadter spectrum
2DEG in a square lattice + a perpendicular B field
(Hofstadter, PRB 1976)
Landau
subband
energy
Width of a Bloch band when B=0
tight-binding model:
LLs
 1

0 3
Magnetic flux (in Φ0) / plaquette
45
Bloch energy E(k)
Berry curvature Ω(k)
C1 = 1
C2 = 2
C3 = 1
46
Re-quantization of semiclassical theory
Leff 


k  eA  r  k  R  E (r , k )
Bohr-Sommerfeld quantization
1
1  (Cm )  eB

ˆ
k

dk

dz

2

m




2 Cm
2
2 



Berry phase  (Cm ) 

R  dk
Cm
• Bloch oscillation in a DC electric field,
re-quantization → Wannier-Stark ladders
• cyclotron motion in a magnetic field,
re-quantization → LLs, dHvA effect
•…
Would shift
quantized cyclotron
energies (LLs)
Now
with
Berry
phase
effect!
48
cyclotron orbits (LLs) in graphene ↔ QHE in graphene
E
Dirac cone
B
…
ρL
σH
Cyclotron
orbits
k
(k )   (k )   C  
E (k )   F k


1
2
 k 2  2  m  
 C  eB
2 
Novoselov et al, Nature 2005
1  

 En  vF 2eB  n   C 
2 2 

49
Berry phase in solid state physics
 Persistent spin current
 Quantum tunneling in a magnetic cluster
 Modern theory of electric polarization
 Semiclassical electron dynamics
 Quantum Hall effect
 Anomalous Hall effect
 Spin Hall effect
Mokrousov’s talks
this Friday
Buhmann’s next Thu
QSHE)
(on
Poor men’s, and women’s,
version of QHE, AHE, and SHE
50
Anomalous Hall effect (Edwin Hall, 1881):
Hall effect in ferromagnetic (FM) materials
FM material
ρH
saturation
slope=RN
RAHMS
H
The usual Lorentz
force term
Ingredients required for a
successful theory:
 H  RN H   AH ( H ),
• magnetization (majority spin)
Anomalous term
• spin-orbit coupling
(to couple the majority-spin
direction to transverse orbital
direction)
 AH ( H )  RAH M ( H )
51
Intrinsic mechanism (ideal lattice without impurity)
• Linear response
• Spin-orbit coupling
• magnetization
gives correct order of magnitude
of ρH for Fe, also explains that’s
AH
observed in some data
L2
52
Smit, 1955: KL mechanism should be annihilated by
(an extra effect from) impurities
Alternative
scenario:
• Skew scattering (Smit, Physica 1955)
～ Mott scattering
Spinless
impurity
Extrinsic
mechanisms
(with
impurities)
e－
 AH
L
 AH
L2
• Side jump (Berger, PRB 1970)
anomalous velocity due to
electric field of impurity ～
anomalous velocity in KL
(Crépieux and Bruno,
PRB 2001)
e
－
  1A
2 (or 3) mechanisms:  AH  a(M ) L  b(M ) L2
In reality, it’s not so clear-cut !
Review: Sinitsyn, J. Phys: Condens. Matter 20, 023201 (2008)
53
CM Hurd, The Hall Effect in Metals and Alloys (1972)
“The difference of opinion between Luttinger and
Smit seems never to have been entirely resolved.”
30 years later:
Crepieux and Bruno, PRB 2001
“It is now accepted that two mechanisms are
responsible for the AHE: the skew scattering…
and the side-jump…”
54
However,
Science 2001
Science 2003
And
many
more …
Karplus-Luttinger mechanism:
Mired in controversy from the start, it simmered for a long time
as an unsolved problem, but has now re-emerged as a topic
with modern appeal. – Ong @ Princeton
55
Old wine in new bottle
Berry curvature of fcc Fe
(Yao et al, PRL 2004)
Karplus-Luttinger theory (1954)
= Berry curvature theory (2001)
 AH 
e2
d 3k
(k )  0
3

(2 )
filled
→ intrinsic AHE
• same as Kubo-formula result
• ab initio calculation
Ideal lattice without impurity
56
• classical Hall effect
charge
B
EF
↑↓
 Lorentz force
+++++++
y
0
L
• anomalous Hall effect
↑↑↑↑
 Berry curvature
EF
↑
↑↑↑↑↑↑↑
↓
B
charge
spin
 Skew scattering
y
↑↑
↑↑↑↑
0
L
• spin Hall effect
spin
↑↑↑↑
↑↑↑↑↑↑↑
No magnetic field required !
EF
↑
↑↑↑↑
 Berry curvature
↓
↑↑↑↑↑↑↑
 Skew scattering
y
0
L
57
Murakami, Nagaosa, and Zhang, Science 2003:
Intrinsic spin Hall effect in semiconductor
Band structure
• Spin-degenerate Bloch state
due to Kramer’s degeneracy
→ Berry curvature becomes a
2x2 matrix (non-Abelian)
• (from Luttinger model)
Berry curvature for HH/LH
3 kˆ
Ω HH (k )  
σz
2
2k
4-band
Luttinger
model
The crystal has both
space inversion symmetry
and time reversal symmetry !
dx En (k ) e

 E  Ωn
dt
k
Spin-dependent
transverse velocity
→ SHE for holes
58
Only the HH/LH can have SHE?
: Not really
• Berry curvature for
conduction electron:
Ω   σ  O(k1 )
8-band
Kane
model
spin-orbit
coupling strength
• Berry curvature for
free electron (!):
Dirac’s
theory
electron
mc2
positron
Ω
C2
2
σ  O(k 1 )
C  / mc
 1012 m
Chang and Niu, J Phys, Cond Mat 2008
59
Observations of SHE (extrinsic)
Science 2004
Nature 2006
Nature Material 2008
Observation of Intrinsic SHE?
60
• Summary
Spin
• Persistent
spin current
• Quantum
tunneling
Bloch state
• AHE
• Electric
polarization
• SHE
• QHE
• Three fundamental quantities in any crystalline solid
E(k)
Berry curvature
Ω(k)
Bloch energy
L(k)
Orbital moment
(Not in this talk)
61
Thank you!
Slides :
http://phy.ntnu.edu.tw/~changmc/Paper
Reviews:
• Chang and Niu, J Phys Cond Matt 20, 193202 (2008)
• Xiao, Chang, and Niu, to be published (RMP?)
63