Wavepacket dynamics for Massive
Dirac electron
C.P. Chuu
Q. Niu
Dept. of Physics
Ming-Che Chang
Semiclassical electron dynamics in solid
(Ashcroft and Mermin, Chap 12)
dk
e
 eE  r  B
dt
c
dr 1 E

dt
k
• Lattice effect hidden in E(k)
• Derivation is non-trivial
Explains • oscillatory motion of an electron in a DC field
(Bloch oscillation, quantized energy levels are known as
Wannier-Stark ladders)
• cyclotron motion in magnetic field
(quantized orbits relate to de Haas - van Alphen
effect)
•…
Limits of validity
eEa
Eg2 / EF
c
Eg2 / EF
Negligible inter-band transition (one-band approximation)
“never close to being violated in a metal”
Semiclassical dynamics - wavepacket approach
1. Construct a wavepacket that is
localized in both the r and the k spaces.
rW
k W
2. Using the time-dependent variational
principle to get the effective Lagrangian

Leff (rc , kc ; rc , kc )  W i
H W
t
e
= kc  R  kc  rc  A  rc  E (rc , kc )
c
Berry connection
R(k )  i un

un
k
Magnetization energy
of the wavepacket
Wavepacket energy
E (r , k )  E0 (k )  e (r ) 
e
L (k )  B
2mc
Self-rotating angular momentum
L(k )  m W  r  rc   v W
3. Using the Leff to get the equations
of motion
• Bloch energy E0 (k )
dk
e
 eE  r  B
dt
c
dr 1 E

 k  ( k )
dt
k
• Berry curvature (1983), as an
effective B field in k-space
( k )  i
Anomalous velocity due
to the Berry curvature
( k )    R ( k )
E (r , k )  E0 (k )  e (r ) 
Three quantities required to
know your Bloch electron:
u
u

k
k
• Angular momentum
(in the Rammal-Wilkinson form)
e
L (k )  B
2mc
L (k ) 
m u
u
  E0  H 
i k
k
Ω(k) and L(k) are zero when there are both
• time-reversal symmetry
• lattice inversion symmetry
(assuming there is no SO coupling)
 1 
   
 
 N
Single band
Multiple bands
Basic quantities
Basics quantities
E (k )  E0 (k )  e (r ) 

R( k )  u i
u
k
e
L (k )  B
2mc
1
 (k )      R     R 
2
Dynamics
dk
e
 eE  r  B
dt
c
dr E

 k  ( k )
dt k
H (r , k )  E0 (k )  e (r ) 

Rij (k )  ui i
uj
k
e
L(k )  B
2mc
Magnetization

1
F (k )     R     R  i  R , R  
2

Dynamics
Covariant
dk
e
 eE  r  B derivative
dt
c
dr


     iR , H    k    F 
dt
 k

d 
i
 H (r , k )  k  R  

dt 
SO interaction
Chang and Niu, PRL 1995, PRB 1996
Sundaram and Niu, PRB 1999
Culcer, Yao, and Niu PRB 2005
Shindou and Imura, Nucl. Phys. B 2005
• Relativistic electron (as a trial case)
• Semiconductor carrier
Construction of a Dirac wave packet
E0 ( q)  m c  c
2 4
  ( q)mc 2
Plane-wave solution
 i  eik r ui ,
2 2 2
q
ui u j   ij
w   d 3 qa (q , t ) 1 (q , t )  1  2 ( q , t )  2  ,
3
2
2
2
d
q
|
a
(
q
,
t
)
|

1;
|

|

|

|
1
1
2

Center of mass
w r w  rc and
2mC2
2
3
d
 qq | a(q, t ) |  qc
If  p  mc, then the negative-energy components
are not negligible.
 x  / mc (Compton wave length c )
This wave packet has a minimal size
a0 : c : ae  1010 :1012 :10 15
Classical
electron radius
• Angular momentum of the wave packet


c2
L ( kc )  2   
kc   kc  ;
 
 1


or
Lij 

 0 
ui  u j ,   


0 
 = 1+( k/mc) 2 
1
1  (v / c ) 2
Ref: K. Huang, Am. J. Phys. 479 (1952).
• Energy of the wave packet
r
r
e
ge  (kc )
L( kc )  
2mc
2 mc 2
H(rc , kc )  E0 (kc )  e (rc )  M(kc )  B
M ( kc )  
The self-rotation gives the correct
magnetic energy with g=2 !
• Gauge structure (gauge potential and gauge field,
or Berry connection and Berry curvature)
SU(2) gauge potential
c2
R
k 
2 (  1)
SU(2) gauge field

c2 
c2
F   3  
k  k 
2 
 1



Ref: Bliokh, Europhys. Lett. 72, 7 (2005)
Semiclassical dynamics of Dirac electron
• Precession of spin (Bargmann, Michel, and Telegdi, PRL 1959)
dS
e 
1
k

B

E


S
dt  mc 
 1
mc 


 

S 
2 

L
• Center-of-mass motion
To liner fields >
For v<<c
dk
e
 eE  v  B
dt
c
dr
k e
e

 E F  c k BF
dt  m

2
k
B
 ec

  B 
 E
1 
m  mc 2
2


( B  e / 2mc)
Or,
++++++++++

Spin-dependent
transverse velocity
L
---------for 1 GeV in 1 cm
L E (c )
 106 !
2
L
mc
 m*/ m 
2
2
k  m*r  
 m  E , where m * c  mc +B   B
 c 
g ( e)
“hidden momentum”
m
S
2mc
Shockley-James paradox
(Shockley and James, PRLs 1967)
A simpler version (Vaidman, Am. J. Phys. 1990)
A charge and a solenoid:
q
S
B
E
Resolution of the paradox
• Penfield and Haus, Electrodynamics of Moving Media, 1967
• S. Coleman and van Vleck, PR 1968
A stationary current loop in an E field
Smaller m
m
Gain
energy
Lose
energy
E
Power flow and
momentum flow
// m  E
Larger m
Force on a magnetic dipole
(Jackson, Classical Electrodynamics,
the 3rd ed.)
• magnetic charge model
( m  B )
• current loop model
( m  B ) 
d  m E 


dt  c 
Energy of the wave packet
H(rc , kc )  E0 (kc )  e (rc )  M(kc )  B
Where is the spin-orbit coupling energy?
Re-quantizing the semiclassical theory:
Effective Lagrangian (general)
(Chuu, Chang, and Niu, to be
published. Also see Duvar,
Horvath, and Horvath, Int J Mod
Phys 2001)

e
(Non-canonical variables)
 kc  R  kc  rc  A  rc  E (rc , kc )
t
c

df
Standard form (canonical var.) ri , p j   ij
=i  †
 p  r  E (r , p ) 
t
dt
Leff  i  †

Conversely, one can write
(correct to linear field)
new “canonical” variables,
r  rc  R(kc )  G (kc );
rc  r  R ( )  G ( );
e
e
A( rc )  B  R( kc ),
c
2c
where G  1/ 2(R / k  )  ( R  B)
e
e
A(r )  B  R ( ),
c
c
where   p  e / cA(r )
p  kc 
kc  p 
(generalized Peierls substitution)
For Dirac electron, to linear order in fields

 R,
r
1


R

k S

2 2
2m c


This is the SO interaction with the correct

= ( r )  c E  k  S
Thomas factor!
2mc
 (r  R)   (r ) 
(Ref: Shankar and Mathur, PRL 1994)

Relativistic Pauli equation
Pair production
Dirac Hamiltonian (4-component)
e


H D  c   p  A( r )    mc 2  e ( r )
c


Foldy-Wouthuysen
transformation Silenko, J.
Semiclassical energy
H(rc , kc )  E0 (kc )  e (rc )  M(kc )  B
Math. Phys. 44, 2952 (2003)
generalized Peierls substitution
rc  rˆ  R (ˆ )  G (ˆ );
  p  e / cA(r )
e
kc  ˆ  B  R (ˆ ).
c
H P  U † H DU
Pauli Hamiltonian (2-component)
H P   ( )mc 2 
B

    E  B   B  e (r )
 ( )[ ( )  1]mc
 ( )
correct to first order in fields,
exact to all orders of v/c!
Ref: Silenko, J. Math. Phys. 44, 1952 (2003)
Why heating a cold pizza?
advantages of the wave packet approach
A coherent framework for







A heuristic model of the electron spin
Dynamics of electron spin precession (BMT)
Trajectory of relativistic electron (Newton-Wigner, FW )
Gauge structure of the Dirac theory, SO coupling (Mathur + Shankar)
Canonical structure, requantization (Bliokh)
2-component representation of the Dirac equation (FW, Silenko)
Also possible: Dirac+gravity, K-G eq, Maxwell eq…
Pair production
Relevant fields




Relativistic beam dynamics
Relativistic plasma dynamics
Relativistic optics
…
• Relativistic electron (as a trial case)
• Semiconductor carrier
Hall effect (E.H. Hall, 1879)
(Extrinsic) Spin Hall effect
(J.E. Hirsch, PRL 1999,
Dyakonov and Perel, JETP 1971.)
• skew scattering by spinless impurities
• no magnetic field required
Intrinsic spin Hall effect in p-type semiconductor
(Murakami, Nagaosa and Zhang, Science 2003; PRB 2004)
Valence band of GaAs:
Luttinger Hamiltonian (1956)
(for j=3/2 valence bands)
2
1 
5  2
H
  1   2  k  2 2 k  J 
2m 
2 



  kˆ  J (helicity)
is a good quantum number
(Non-Abelian) gauge potential
R ' (k )  u i

u '
k
Berry curvature,
due to monopole field in kspace r r
7 k
  ( k )  2 2 
4 k2
F
IJ
G
H K
dk
 eE
dt
dx E (k ) dk


  (k )
dt
dt
k
Emergence of curvature by projection
Non-Abelian
• Free Dirac electron
Curvature for the whole space
F  dR  iR  R  0
Curvature for a subspace
F  d ( PRP )  iPRP  PRP  0
• 4-band Luttinger
z
model (j=3/2)
Analogy in geometry
u
Ref: J.E. Avron, Les Houches 1994
x
v
y
Berry curvature in conduction band?
8-band Kane model
Rashba system (in asymm QW)
p2  r r
H
   p  z
2m 
Is there any curvature simply
by projection?
There is no curvature anywhere
except at the degenerate point
 (k )   (k )
8-band Kane model
Efros and Rosen, Ann. Rev. Mater. Sci. 2000
Gauge structure in conduction band
• Gauge potential, correct to k1
Eg


V2  1
1
  k , V 
R

2
2
3  Eg  E    
g



• Angular momentum, correct to k0
S Px X / m0
2m0V 2  1
1 
L


  ,
3
E
E


g
 g

Gauge structures and angular momenta in other subspaces
Chang et al, to be published
Re-quantizing the semiclassical theory:
generalized Peierls substitution:
Effective Hamiltonian
H (r , k )  E0 (k )  e (r )  eE  R (k )
rc  r  R ( )  G ( );

e
e
kc  p  A(r )  B  R ( ),
c
c
where   p  e / cA(r )
ri , p j   ij


Ref: Roth, J. Phys. Chem. Solids
1962; Blount, PR 1962
 (rc )   (r )  E  R
E0 (kc )  E0 ( p) 

E 
e
B   L( k )  2R  m 0 
2mc 
p 
• vanishes near band edge
e E0
 B R
c p
• higher order in k
Spin-orbit coupling for conduction electron
eE  R   E    k ,
• Same form as Rashba


eV 2  1
1

where  

2
2

3 Eg  E    
g


( = 0 if   0 )
• In the absence of BIA/SIA
Ref: R. Winkler, SO coupling
effect in 2D electron and hole
systems, Sec. 5.2
Effective Hamiltonian for semiconductor carrier
q
B  L( k )
2mc
g
H c (r , k )  E0 (k )   E    k   B B  
2
H H (r , k )  E0 (k , J )   H E  J  k  2 H  B B  J
H (r , k )  E0 (k )  qE  R (k ) 
H SO (r , k )  E0 (k )   SO E    k  2 SO  B B  
Spin part
orbital part


2 
eV 2  1
1
4
mV
1
1 



, g  2



3  Eg2  E   2 
3 2  Eg Eg   
g


eV 2 1
1 4 mV 2 1
H  
, H  
3 Eg2
2 3 2 Eg
 SO
Yu and Cardona,
Fundamentals of
semiconductors,
Prob. 9.16
eV 2
1
1 4 mV 2 1

,  SO  
3  E   2
2 3 2 Eg  
g
Effective H’s agree with Winkler’s obtained using LÖwdin partition
Covered in this talk:
• Wave packet dynamics in multiple bands
• Relativistic electron
• Spin Hall effect
Not covered
• Wave packet dynamics in single band
• Anomalous Hall effect
• Quantum Hall effect
• (Anomalous) Nernst effect
• optical Hall effect
Forward jump and “side jump”
Berger and Bergmann, in The Hall effect and
its applications, by Chien and Westgate (1980)
(Picht 1929+Goos and Hanchen1947, Fedorov 1955+Imbert 1968,
Onoda, Murakami, and Nagaosa, PRL 2004; Bliokh PRL 2006)
• wave packet in BEC
(Niu’s group: Demircan, Diener, Dudarev, Zhang… etc )
Not related:
• thermal Hall effect
(Leduc-Righi effect, 1887)
• phonon Hall effect
(Strohm, Rikken, and Wyder, PRL 2005,
L. Sheng, D.N. Sheng, and Ting, PRL 2006)
Thank you !