Wavepacket dynamics for Massive Dirac electron

Dept. of Physics Ming-Che Chang

C.P. Chuu Q. Niu

Semiclassical electron dynamics in solid (Ashcroft and Mermin, Chap 12)

1

dk eeE r B

dt cdr E

dt k

• 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)

• …2

2

/

/

g F

c g F

eEa E E

E E

Negligible inter-band transition (one-band approximation)

“never close to being violated in a metal”

• Lattice effect hidden in E(k)

• Derivation is non-trivial

Explains

Limits of validity

Semiclassical dynamics - wavepacket approach

k W

( ) n nR k i u uk

Magnetization energy

of the wavepacket

( , ; , )

( ) , =

eff c c

c c

c c

c c cc

L r k r k W i H Wt

ek r A r E rk

cR k

1. Construct a wavepacket that is localized in both the r and the k spaces.

2. Using the time-dependent variational principle to get the effective Lagrangian

r W

( ) cm W r r vL k W

Self-rotating angular momentum

Berry connection

Wavepacket energy

0 ( ) (, ( ))2

( )e

E k e r Bmc

E kk Lr

• time-reversal symmetry

• lattice inversion symmetry

(assuming there is no SO coupling)

Ω(k) and L(k) are zero when there are both

Three quantities required to know your Bloch electron:

• Bloch energy

( )u u

k ik k

• Berry curvature (1983), as an effective B field in k-space

0( )m u u

L k E Hi k k

• Angular momentum (in the Rammal-Wilkinson form)

0 ( )E k

1( )

dk eeE r B

dt cdr E

kt

kd k

Anomalous velocity due to the Berry curvature

( ) ( )k R k

3. Using the Leff to get the equations of motion

0( , ) ( )2

( ( ))e

E r k e r Bmc

LE kk

Single band Multiple bands

0( , ) ( ) ( ) ( )2

r k E k e re

k Bmc

H L

( )ij i jR k u i uk

1( ) ,

2k i F R R R R

Magnetization

,

( , )

dk eeE r B

dt cdr

dt

di r kd

ki

kt

k

R

FR H

H

1

N

Covariant derivative

SO interaction

0( ) ( ) ( ) ( )2

E k E k e Bm

re

L kc

Culcer, Yao, and Niu PRB 2005Shindou and Imura, Nucl. Phys. B 2005

( )R k u i uk

( )

dk eeE r B

dt cd

kr E

dtk

k

1( )

2k R R

Basic quantities

Dynamics

Basics quantities

Dynamics

Chang and Niu, PRL 1995, PRB 1996 Sundaram and Niu, PRB 1999

• Relativistic electron (as a trial case)

• Semiconductor carrier

Construction of a Dirac wave packet

2mC2

2 4 2 2 20

2

( )

( )

E q m

q

c c q

mc

31 2

3 2 2 21 2

1 2( , ) ( , ) ( , ) ,

| ( , ) | 1; | | | | 1

w d qa q t q t q t

d q a q t

23 and | ( , ) |c cw r w r d qq a q t q

c

If , then the negative-energy components

are

(Compton wave lengt

not negligible.

/ h )

p mc

x mc

This wave packet has a minimal size

10 12 150 : : 10 :10 :10c ea a

Plane-wave solution

Center of mass

, i j iik

i i jre u u u

Classical electron radius

2

2 ( 1)c k

R

• Gauge structure (gauge potential and gauge field, or Berry connection and Berry curvature)

SU(2) gauge potential SU(2) gauge field

2 2

32 1c c k k

F

Ref: Bliokh, Europhys. Lett. 72, 7 (2005)

0

( )( ) ( )

2 2 2

( , ) ( ) ( ) ( )

cc c

c c c c c

ke gek k

mc mc

r k E k e r k B

M L

H M

• Energy of the wave packet

The self-rotation gives the correct magnetic energy with g=2 !

r

2

2( ) ;

1

0or ,

0

c c c

ij i j

ck k k

L u u

L

Ref: K. Huang, Am. J. Phys. 479 (1952).

• Angular momentum of the wave packet

2

2

1= 1+( / )

1 ( / )k mc

v c

Semiclassical dynamics of Dirac electron

2

2

2

1 B

c

c

dk eeE v B

dt c

dr k e eE F k B F

dt m

k

mE

mc

eB

• Center-of-mass motion

• Precession of spin (Bargmann, Michel, and Telegdi, PRL 1959)

Spin-dependent transverse velocity

1

1

dS e kB E S

dt mc mc

2S

For v<<c

( / 2 )B e mc

Or, 2B

2* , where * +

( )

* /

2

m mmk m r m c mc

g em S

mc

Ec

B

“hidden momentum”

L

L

62

for 1 GeV in 1 cm

( )10 !cEL

L mc

+ + + + + + + + + +

- - - - - - - - - -

To liner fields >

Shockley-James paradox (Shockley and James, PRLs 1967)

A simpler version (Vaidman, Am. J. Phys. 1990)

A charge and a solenoid:

E

B

Sq

Resolution of the paradox• Penfield and Haus, Electrodynamics of Moving Media, 1967• S. Coleman and van Vleck, PR 1968

m

Gain energy

Lose energy

Larger m

Smaller m

E

A stationary current loop in an E field

Power flow and momentum flow // m E

Force on a magnetic dipole

( )m B

( )d m E

m Bdt c

• magnetic charge model

• current loop model

(Jackson, Classical Electrodynamics, the 3rd ed.)

Where is the spin-orbit coupling energy?

0( , ) ( ) ( ) ( )c c c c cr k E k e r k B H M

Energy of the wave packet

,i j ijr p

†

†

( , )

= ( , )

eff c c c cc cL i k r E r kt

dfi p r E r p

t

eA r

c

dt

k R

new “canonical” variables,

( )

( )2

( ) ;

( ) ,

where 1/ 2( / ) ( )

c c

c c

c

c

r r R k

ep k A r

c

G

G k

eB R

R k R B

kc

Conversely, one can write (correct to linear field)

( ) ;

( ) ,

wh

( )

( )

ere / ( )

c

c

r r R

ek p A r

G

eB R

cc

p e cA r

Re-quantizing the semiclassical theory:

(Non-canonical variables)

Standard form (canonical var.)

Effective Lagrangian (general)

2 2

1( ) ( ) ,

2

= )2

( c

r rr

km

E km

c

rc

R

S

R R S

This is the SO interaction with the correct Thomas factor!

For Dirac electron, to linear order in fields

(Ref: Shankar and Mathur, PRL 1994)

(generalized Peierls substitution)

(Chuu, Chang, and Niu, to be published. Also see Duvar, Horvath, and Horvath, Int J Mod Phys 2001)

Relativistic Pauli equation

2( ) ( )D

eH c p A r mc e r

c

Foldy-Wouthuysen transformation Silenko, J. Math. Phys. 44, 2952 (2003)

2( ) ( )( )[ ( ) 1] ( )

B BPH mc E B e r

mc

†P DH U H U

Pair production

Dirac Hamiltonian (4-component)

Pauli Hamiltonian (2-component)

Ref: Silenko, J. Math. Phys. 44, 1952 (2003)

correct to first order in fields, exact to all orders of v/c!

generalized Peierls substitution

0( , ) ( ) ( ) ( )c c c c cr k E k e r k B H M

Semiclassical energy

ˆ ˆ ˆ( ) ( );

ˆ ˆ( ).

c

c

r r R G

ek B R

c

/ ( )p e cA r

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…

Relevant fields Relativistic beam dynamics Relativistic plasma dynamics Relativistic optics …

Pair production

Why heating a cold pizza? advantages of the wave packet approach

A coherent framework for

• Relativistic electron (as a trial case)

• Semiconductor carrier

Hall effect (E.H. Hall, 1879)

• skew scattering by spinless impurities

• no magnetic field required

(Extrinsic) Spin Hall effect(J.E. Hirsch, PRL 1999, Dyakonov and Perel, JETP 1971.)

Valence band of GaAs:

22

1 2 2

1 52

2 2H k k J

m

ˆ (helicity)

is a good quantum number

k J

Intrinsic spin Hall effect in p-type semiconductor (Murakami, Nagaosa and Zhang, Science 2003; PRB 2004)

Luttinger Hamiltonian (1956) (for j=3/2 valence bands)

Berry curvature, due to monopole field in k-space

( )

k

k

k F

HGIKJ2

7

42

2

(Non-Abelian) gauge potential

' '( )R k u i uk

( )( )

dkeE

dt

E dkkdx

dt kk

dt

Emergence of curvature by projection

• Free Dirac electron

• 4-band Luttinger model (j=3/2)

0

( ) 0

F dR iR R

F d PRP iPRP PRP

Curvature for a subspace

Non-Abelian

x

y

z

vu

Analogy in geometry

Ref: J.E. Avron, Les Houches 1994

Curvature for the whole space

Berry curvature in conduction band?

8-band Kane model Rashba system (in asymm QW)

There is no curvature anywhere except at the degenerate point

Is there any curvature simply by projection?

( ) ( )k k

Hp

mp z

2

2

Efros and Rosen, Ann. Rev. Mater. Sci. 2000

8-band Kane model

Gauge structure in conduction band

2

022

1,

3/

1

gx

g

VV S P X mk

E E

R

• Gauge potential, correct to k1

gE

Chang et al, to be published

• Angular momentum, correct to k0

202 1 1

,3 g g

Vm

E E

L

Gauge structures and angular momenta in other subspaces

,i j ijr p

( ) ;

( ) ,

wh

( )

( )

ere / ( )

c

c

r r R

ek p A r

G

eB R

cc

p e cA r

Re-quantizing the semiclassical theory:

0 00

( ) ( )

( ) ( )

c

c

r r

E k E pEe

B Rc

E

p

R

generalized Peierls substitution:

0

0

( )

( , )

2

( ) (

)2

)

(

r

Em

p

k E k e r

eB k

mc

eE k

RH

L R

Effective Hamiltonian

Ref: Roth, J. Phys. Chem. Solids 1962; Blount, PR 1962

• vanishes near band edge

• higher order in k

2

22

1

,

where

1

3

( = 0 if 0 )

g g

eE

eV

E

E k

E

R

Spin-orbit coupling for conduction electron

Ref: R. Winkler, SO coupling effect in 2D electron and hole systems, Sec. 5.2

• Same form as Rashba

• In the absence of BIA/SIA

Effective Hamiltonian for semiconductor carrier

0 ((( , ) ( )2

) )q

r k E k qE Bmc

k k LRH

Effective H’s agree with Winkler’s obtained using LÖwdin partition

0

0

0

( , ) ( )2

( , ) ( , ) 2

( , ) ( ) 2

c B

H H H B

SO SO SO B

gr k E k E k B

r k E k J E J k B J

r k E k E k B

H

H

H

2 2

22 2

2 2

2 2

2 2

2 2

1 1 4 1 1,

3 3

1 1 4 1,

3 2 3

1 4 1

2

1,

3 2 3

g g gg

H Hg g

SO SOgg

eV mVg

E E EE

eV mV

E E

eV mV

EE

Yu and Cardona, Fundamentals of semiconductors, Prob. 9.16

Spin part orbital part

• Wave packet dynamics in multiple bands

• Relativistic electron

• Spin Hall effect

• Wave packet dynamics in single band

• Anomalous Hall effect

• Quantum Hall effect

• (Anomalous) Nernst effect

• optical Hall effect

(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 )

Covered in this talk:

• thermal Hall effect

• phonon Hall effect

Forward jump and “side jump”Berger and Bergmann, in The Hall effect and its applications, by Chien and Westgate (1980)

Not covered

Not related:

(Strohm, Rikken, and Wyder, PRL 2005, L. Sheng, D.N. Sheng, and Ting, PRL 2006)

(Leduc-Righi effect, 1887)

Thank you !