wavepacket dynamics of electrons in solid

Wavepacket dynamics ofelectrons in solid

Dept. of PhysicsMing-Che Chang

• Wave packet dynamics in a single band

• Anomalous Hall effect

• Quantum Hall effect

• Nernst effect

• Wave packet dynamics in multiple bands

• Relativistic electron

• Spin Hall effect

Classical Hall effect (Hall, 1879)

,

; / 0 at steady state

e e

dv vm qE qv H m

dtH Hz dv dt

τ= + × −

= =

r rr rrrr r

//

x x

y y

v Em qHq

v EqH mτ

τ −

=

2

2

,x x xL H

y y yH L

m HE j jnq nqE j jH m

nq nq

ρ ρτρ ρ

τ

− − = ≡

rH

H12 2

1or L H

H LL H

ρ ρσ ρ

ρ ρρ ρ−

= = −+ % %

Hz

Jx

Ey

1L n

ρτ

∼

2H

HL

ρσ

ρ≈

( )j qnv=r r

Anomalous Hall effect in ferromagnet (Hall, 1880, 1881)

0 ' ( , )

' ( , ) ( ) ( ,

( ) ,

)

H

S

H

H

T H

T H R

H

M T H

R

T

T ρ

ρ

ρ

≡

= +

Berger and Bergmann, in The Hall effect andits applications, by Chien and Westgate (1980)

The usual Lorentzforce term

Anomalous term

( )' ( ) ,

1.5 for Ni 2.0 for Fe

nH H L T

nn

ρ ρ ρ≈

≈≈

∼

Temperature dependence(after saturation)

SO coupling required

2

2

H

then const (i

i

)

f

nHH

L

L

Tρ

σ

ρ

ρ

ρ

≈ ≈

∼

saturation

H

1( )

dkeE

dtdr Edt k

k k

= −

∂= −

∂× Ω

r rhr

rhr rr&

Anomalous velocity dueto Berry curvature ( )kΩ

rr

Modern view of the KL theory:

gives correct order of magnitude for Fe,also explains

Ex

Jy

2H Lρ ρ∼

• Smit’s skew scattering mechanism (1955)

• Berger’s side jump mechanism (1970)

“The difference of opinion between Luttinger and Smitseems never to have been entirely resolved.”CM Hurd, The Hall Effect in Metals and Alloys (1972)

“It is now accepted that two mechanisms are responsible forthe AHE: the skew scattering… and the side-jump… ”(Crepieux and Bruno, PRB 2001)

Actual situation is messy.

Relative importance differs between different materials.

Smit: KL’s (intrinsic) mechanism vanishes in a latticew/o disorder

Alternative mechanisms (extrinsic):

2H Lρ ρ∼H Lρ ρ∼

2H L La bρ ρ ρ+∼

1θ ≈ o

W k s T ks

T H HH

H

W n V k k

ks k s k k

ASs s i i s s k kks k s

r r r r

r r

hr r

Lr r r

r r

→= −

= +−

+

≈ ⋅ × −→

' ' '

( )' ' '

' ' ,

' ' '

'' '

2

1

2

0

3

πδ ε ε

ε

δ λ σ δ ε ε

c h

d i c h

Skew scattering(ª Mott scattering)

H V x S V x vmc

' ( ) ( ) ,= + ⋅∇ ×r r r r

λ λ = 1

2 2

r r h r r rr rk s H ks i

mk k Vs s s s k k

' ' ' '' ' '= +

FHG

IKJ ⋅ ×

LNM

OQP −

δ λ σ2

2

Transition rate:

(for d impurities, up to 2nd order Born approx.)

'

'H

H L

nσρ ρ

τ⇒ ≈∴ ∼

(Ref: Takahashi and Maekawa, PRL,2002, Landau and Lifshitz, QM)

(Ref: Crepieux and Bruno, PRB 2001)

0

2

1'

0

' ,4

ks

cs ss c

ks H ksH i

r r kmc

ψε

λσ λ

+= +− +

⇒ = + × ≡

rr r

r hr r r

Side jump

Side jump in materials

H V x S V x vmc

' ( ) ( ) ,= + × ∇ ⋅r r r r

λ λ = 1

2 2

1Aδ ≈

Microscopically it’s due tothe same anomalousvelocity in Luttinger’s theory

In real material, 2mc2 isreplaced by band gap(104 enhancement)

Mired in controversy from the start, it simmered for a long timeas an unsolved problem, but has now re-emerged as a topicwith modern appeal. -- Ong

Karplus-Luttinger mechanism:

Science 2001

Recent Theory

Science 2003

Science 2004

Recent experiment

In the 70's when experimentalists actively investigated the'Kondo problem', numerous Hall measurements wereperformed on nonmagnetic metals (e.g. Cu) with a diluteconcentration of magnetic impurities, e.g. Mn. The weakAHE observed was consistent with rxy being linear in r(skew scattering). These systems are paramagnetic ratherthan ferromagnetic (hence not really relevant).Nonetheless, the results fostered a collective, if unjustified,tilt towards the skew scattering mechanism and away fromLuttinger's theory. It is fair to say that most condensed-matter physicists today have some nodding acquaintancewith "skew scattering", but have a harder time recallingwhat the Luttinger anomalous velocity is. -- Ong

Semiclassical dynamics - outline of derivation

k Wr

( ) n nR k i u uk∂

≡∂

rr r Magnetization energyof 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

∂= −

∂

+ ⋅ − ⋅ −⋅r r&h

rr

r rr r && hr rr r& &h

1. Construct a wavepacket from oneBloch band that is localized in both the rand the k spaces.

2. Using the time-dependent variationalprinciple to get the effective Lagrangian

r Wr

( )( ) cm W r r vL k W= − ×r r rrr

Self-rotating angular momentum

Berry connection

Wavepacket energy

0 ( ) (, ( ))2

( )e

E k e r Bmc

E kk Lr φ= − + ⋅rrrrr r r

1( )

dk eeE r B

dt cdr E

kt

kd k

= − − ×

∂= −

∂× Ω

r r rr&hr r rr&rh

Anomalous velocity dueto Berry curvature

( ) ( )k R kΩ = ∇ ×r rr r

3. Using the Leff to get the equationsof motion

0( ) ( ) ( ) ( )2

eE k E k e r L k B

mcφ= − + ⋅

r r rr rr( )

23

filled

2

2

2 2

2

' ( )

= indep. of

' ' , ( 1/ )

' ,

'

/

and /

zH n

n BZ

z

H

H L

H L L

zH L

ed k k

en

An e

n

n

A

ρ ρ

σ

τ

ρ σ ρ ρ τ

ρ ρ

∈

= Ω

Ω

= ≈

∴ ≈ = Ω

∑ ∫r

h

h

∼

r

h

Anomalous Hall conductivity

KL’s intrinsic mechanism:

2

2

/

/

g F

c g F

eEa E E

E Eω

=h =

Negligible inter-band transition (one-band approximation)

“never close to being violated in a metal”

Limits of validity



• Quantum Hall effect (Hofstadter spectrum)

• Nernst effect




Hofstadter spectrum

The band structure of an electron subjects to botha lattice potential V(x,y) and a magnetic field B

¨ Can be studied using either the nearlyfree electron model or the tight-bindingmodel (TBM)

B

¨ The tricky part:

q=3 à q=29 upon a small change of B!Also, when B à 0, q can be very large!

1

31

10

1029

13

187−

= = +

¨ Surprisingly complex spectrum!Split of energy band depends onflux/plaquetteIf Fplaq/F0= p/q, where p, q are co-primeintegers, then a Bloch band splits to qsubbands (for TBM)

Hofstadter’s butterfly (Hofstadter, PRB 1976)

A fractal spectrum with self-similarity structure

Bà0 near band button, evenly-spaced LLs

The total band width for an irrational q is of measure zero (as in a Cantor set)

Self-similarity

1

−2

1

集異璧

著作者：Douglas R. Hofstadter翻譯者：郭維德

Reptiles, by M.C. Escher, 1943

• time-reversal symmetry

• lattice inversion symmetry

(assuming there is no SO coupling)

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

Three quantities required toknow your Bloch electron:

• Bloch energy

( )u u

k ik k

∂ ∂Ω = ×

∂ ∂

rr r r

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

( )0( )m u u

L k E Hi k k

∂ ∂= × −

∂ ∂

rr r rh

• Angular momentum(in Rammal-Wilkinson form)

0 ( )E kr

1( )

dk eeE r B

dt cdr E

kt

kd k

= − − ×

∂= −

∂× Ω

r r rr&hr r rr&rh

0 ( ) )) (( ( )2

Le

E k e r Bm

kc

E k φ= − + ⋅r rr rrr

Dispersion ofband energy

p/q=1/3

Chang and Niu, PRB 1996

Berry curvature and angular momentum21

( )2

Chern numberMBZ

C d k kπ

≡ Ω∫r r

C=1

C=−2

C=1

Chang and Niu, PRB 1996

Quantization of the hyper-orbits

Bohr-Sommerfeld quantizationcondition (Onsager, 1952)

( ) ( )

(

1 1ˆ 22 2 2

Berry phase )m

m

m

mC

C

C

C R d

e Bk dk dz m

k

δπ

π Γ

Γ =

× ⋅ =

⋅

+ −

∫

∫rr

r rhÑ

Ñ

0( ) ( ) ( ()2

)e

E k E k e Brmc

L kφ− + ⋅=r r r rr r

Crucial for accuracy

22/67=1/[3+(1/22)]




• Nernst effect (in FM material)




The “Tao” of materials

Electric(E,P)

Magnetic(B,M)

Thermal(T)

Mechanical(M)

Optical(O)

• E-B: Hall effect, magneto-electric material

• E-T: Thomson effect, Peltier/Seebeck effect

• E-B-T: Nernst/Ettingshausen effect, Leduc-Righieffect

• E-O, B-O: Kerr effect, Faraday effect,photovoltaic effect, photoelectric effect

• E-M, B-M: piezoelectric effect/electrostriction,piezomagnetic effect/magnetostriction

• M-O: photoelasticity

• ...

Landau and Lifshitz, Electrodynamics of continuous media

Scheibner, 4 review articles in IRE Transations on component parts, 1961, 1962

solid state sensor

solid state motor, artificial muscle

solid state refrigerator...

( )( )

( ) ( ) ,

( ) ( ) ,

e

q

j B E B T

j B E B T

σ α

β γ

= + −∇

= + −∇

r r rr% %r r rr% %

Thermo-galvano-magnetic phenomena

( ) ( )

( ) ( )

e

q

N T Bj E RE B T

j E NE B T L T B

σ α

β γ

= + × + −∇ +

=

−∇ ×

+ × + −∇ + −∇ ×

r r rr

r r r rr

r

Jx

-—yTBz

Ohm Hall Thomson Nernst 1826 1879 1851 1886

Thomson Fourier Leduc-RighiEttings-hausen

1851 1886 1807 1887(Thermal Hall effect)

Onsager relations

Berry curvature effect on DOS and magnetization

01 11 1e

k k

en B

V V

µ µ = → + ⋅Ω

∑ ∑r r

r rh

• Streda formula2 1

( )eH z

kz

n ee k

B V

µ

µ

σ∂

= − = − Ω∂ ∑

r

rh

• Total energy ( )0

11 ( )

k

eE B E k m B

V

µ = + ⋅Ω − ⋅

∑r

rr r rrh

( )0 0

1

kn

EM m

Ve

BE

µ

µ∂ = − = + ∂

Ω −

∑r

rh

r rr• Magnetization

T=0:

T>0:

Gat and Avron, PRL 2003

Thonhauser, Ceresoli, Vanderbilt, and Resta, PRL 2005

Xiao, Yao, and Niu, PRL 2005

0

00 0

( ) /

( ) /,

1 1( ),

1

( ) log 1 E kT

k

E kTkT

e kTk

GM f m k f

V eB

eV

µ

µµ

− −

−

∂= − = =

+∂

+ Ω +

∑

∑r

r

rh

rr rr

Berry curvature correction

0( ) /

=

log 1

e

k

E kT

k

j j

e f r

ekT e

M

µ− −

= −∇×

−

− ∇ + ×Ω

∑

∑

r

r

r rrr&

rh

3( )

ˆ( ,

)2

(

(

)

) ˆ

k

k

k

j

qf L k

m

f rr q rgd r

q f

r

c

r

r

k=

≈ + ∇ ×

−∑∫

∑∑

r

rr

r r r&

rr

r

r&

r rrr

Anomalous thermoelectric transport

Macroscopic chargecurrent density

Coarse-grain average

M0

Transport current density d

Transverse current

(Xiao et al, PRLs 2006)

At low T, one finds

0

2

1( )

' ( )3

xy xy

xy F

EfdE E

e E T

kkT E

e

µα σ

πσ

−∂ ∂ ∫;

;

Mott relation (valid for FM materials as well)

WL Lee et al, PRL 2004




• Nernst effect


• Relativistic electron (as a trial case)


Single band Multiple bands

0( , ) ( ) ( ) ( )2

r k E k e re

k Bmc

φ= − + ⋅H Lr rr r rr r

( )ij i jR k u i uk∂

=∂

rr r

( )1( ) ,

2k iγ αβγ α β β α α βε = ∂ − ∂ − F R R R Rr

Magnetization

( )

,

2

dk eeE r B

dt cdr

ek B

ikdt

di

dtk

k

mc

η η

η

η

η

η+ +

= − − ×

= −∂

−∂

− = ⋅ ⋅

×

L

R F

R

H

r

r r

r r&hrr

rr&hr

h

h

r

rhr

r

&

1

N

ηη

η

=

M

Covariantderivative

SO interaction

0( ) ( ) ( ) ( )2

E k E k e Bm

re

L kc

φ− + ⋅=r r r rr r

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

( )R k u i uk∂

=∂

rr r

( )

dk eeE r B

dt cd

kr E

dtk

k

=

× Ω

− − ×

∂= −

∂

r r rr&hr r rr&hrh

( )1( )

2kγ αβγ α β β αεΩ = ∂ − ∂R Rr

Basic quantities

Dynamics

Basics quantities

Dynamics

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

Construction of a Dirac wave packet

2mC2

2 4 2 2 20

2

( )

( )

E q m

q

c c q

mcε

= +

≡

h

( )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

ψ ψη η

η η

+=

= + =

∫∫

r r r rr r

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

If , then the negative-energy components ar

(Compton wave length)e not negligible.

/

p mc

x mc

δ

δ

≥

∴ ≥ h

This wave packet has a minimal size

10 12 150 : : 10 :10 :10c ea aλ − − −≈

Plane-wave solution

Center of mass

, i j iik

i i jre u u uψ δ⋅ ==

r r

2

2 ( 1)c k

ε ελ

σ= ×+

Rrr 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 kε ε

λσ

λσ

= − + ⋅ +

Fr rr r r

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

0

( )( ) ( )

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

cc c

c c c c c

ke gek k

mc mcr k E k e r k B

σε

φ

= − = −

= − − ⋅

M L

H M

rrr rr r hr r rr rr r

Energy of the wave packet

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

rr

( )2

2( ) ; 1

0or ,

0

c

j

c

i i j

k k k

L u u

σ σ

σ

ε ε

ε σ

λ = + ⋅ +

= Σ Σ =

Lr r rr h r r

rr r rh r

Ref: K. Huang, Am. J. Phys. 479 (1952).Angular momentum of the wave packet

2

2

1= 1+( / )

1 ( / )k mc

v cε =

−h

Semiclassical dynamics of Dirac electron

( )

2

2

2

1 B

c

c

dk eeE v B

dt cdr k e e

E F k B Fdt m

km

Emc

eB

λε ε

σµ

σλ

= − − ×

= + × + ⋅

⇒ − +

×

⋅

r r rrhrr rr r r

rrh

r

h rh hr

r

h

• Center-of-mass motion

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

Spin-dependenttransverse velocity⇔ side jump x 2 !

11

dS e kB E S

dt mc mcε ε

= + × × +

rr rr r h2

Sσ

η η+ =

rr

For v<<c

( / 2 )B e mcµ = h

Or,

“hidden momentum”

L

DL

62

for 1 GeV in 1 cm( )

10 !cELL mc

λ −∆∆≈∼

+ + + + + + + + + +

- - - - - - - - - -

To liner fields >

2B

2* , where * +* /m m

m Ec

k m r m c c Bm µ σ ×

+ = ⋅=rh r rr& rr

Trajectory is curved but transverse momentum is conserved!

( )

2g e

m Smc−

=rr

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

mGainenergy

Loseenergy

Larger m

Smaller m

E

A stationary current loop in an E field

Power flow andmomentum flow // m E×

rr

Force on a magnetic dipole

( )m B∇ ⋅rr

( )d m E

m Bdt c

×∇ ⋅ −

rrrr• 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 Bφ= − − ⋅H Mr r rr rr r

Energy of the wave packet

,i j ijr p δ=

†

†

( , )

= ( , )

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

dfi p r E r p

t

eA r

c

dt

k Rη

η

ηη

∂= + + ⋅ −⋅ ⋅ −

∂∂

+ ⋅ − +∂

r rr r&h hr r r r&h

r rr r& &h

new “canonical” variables,

( )

( )2

( ) ;

( ) ,

where 1/ 2( / ) ( )

c c

c c

c

c

r r R ke

p k A rc

G

G ke

B R

R k R B

kc

αα

= − −

= − −

≡ ∂ ∂ ⋅

×

×

rrrr

rrr rr rr rh

r r r

r

Conversely, one can write(correct to linear field)

( ) ;

( ) ,

wh

( )

( )

ere / ( )

c

c

r r R

ek p A r

G

eB R

ccp e cA r

π

π

π

π

= + +

= + ×+

≡ +

r rr

rr r rr rr r rh

rr r r

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

φφ

φ

λφ

+ ≅ + ≈∂

×⋅∂

⋅ ×−

R

S

R R Sr rr rr r h

r

rr

r rr This is the SO interaction with the correctThomas factor!

For Dirac electron, to linear order in fields

(Ref: Shankar and Mathur, PRL 1994)

(generalized Peierls substitution)

(Chuu, Chang, and Niu, to bepublished. Also see Duvar,Horvath, and Horvath, Int J ModPhys 2001)

Relativistic Pauli equation

2( ) ( )De

H c p A r mc e rc

α β ϕ = ⋅ + + −

rr r r r

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

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

B BPH mc E B e r

mcµ µ

ε π π σ σ ϕε π ε π ε π

= + × ⋅ + ⋅ −+

r rr r r r rr r r

†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 Bφ= − − ⋅H Mr r rr rr rSemiclassical energy

ˆ ˆ ˆ( ) ( );

ˆ ˆ( ).

c

c

r r R G

ek B R

c

π π

π π

→ + +

→ + ×

rrr r r rr r rr rh

/ ( )p e cA rπ ≡ +rr r r

C 2, 1

effC

Bd e e BE

dt mc mcβ ε

ω β ω βε ε ε

= × = = + × −

rr rr rrr

2

11a S C

eaB a E

mcω ω ω β

ε = − = + − × −

rr r

( )B

2( ) ( )

+ , / 2 1

De

H c p A r mc e rc

a ga B i E

α β ϕ

µ β α

= ⋅ + + −

= −Σ ⋅ − ⋅

rr r r r

r r rr

Anomalous magnetic moment

(Cf: eq. 5.64 of Brown and Gabrielse, RMP 1986)

( )

2 2

1 1

wh

, (1 ) 1

1

1en

S SdS e p B p

S a B a p a Edt mc m c mc

e B pE a B E B

mc mc

ω ωε ε ε ε ε

β β βε ε ε

⋅ = × = + − + + × + +

→ + × + + × ≡ ⊥ +

r rr rr r rr r r

r rr r rr r r r

• for muon, a=0.001165923. choose εmagic = 29.3 toeliminate the effect of confinement E field

• when E=0, ωa is velocity-independent

2

2 2

1( ) ( )

1

1

(1 )

BP

BB

H mc e r a

Ba a

m c

Emc

Bµ π σπ

ε

µε π ϕ π σ

ε ε

µ σε ε

= − + + × ⋅ +

+ + ⋅ −

⋅ ⋅+

r

rr r

r rr r r

r r

Pair production

n A heuristic model of the electron spinn Dynamics of electron spin precession (BMT)n Trajectory of relativistic electron (Newton-Wigner, FW )n Gauge structure of the Dirac theory, SO coupling (Mathur + Shankar)n Canonical structure, requantization (Bliokh)n 2-component representation of the Dirac equation (FW, Silenko)n Also possible: Dirac+gravity (Gossellin et al, hep-th/0604012), K-G eq,

Maxwell eq…

Relevant fields

n Relativistic beam dynamicsn Relativistic plasma dynamicsn Relativistic opticsn …

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

A coherent framework for




• Nernst effect




Hall effect (E.H. Hall, 1879)

From spin accumulation to charge accumulation

L< spin coherence length dsds ª130 mm at 36 K for Al

(Johnson and Silsbee, PRL 1985)

no magnetic field required

skew scattering byspinless impurities

Spin Hall effect

(J.E. Hirsch, PRL 1999,Dyakonov and Perel, JETP 1971.)

Anomalous Hall effect in ferromagnet

• spin-polarized incident current

• charge-polarized outgoing current

Spin Hall effect in semiconductor

• spin-unpolarized incident current

• charge-unpolarized outgoing current

• but spin-polarized outgoing current

Valence band of GaAs:

( )22

1 2 21 5

22 2

H k k Jm

γ γ γ = + − ⋅

r r

ˆ (helicity)is a good quantum number

k Jλ = ⋅r

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 in valence band,

r rΩλ λ λ( )

$k

kk

= − −FHG

IKJ2

74

22

(Non-Abelian) gauge potential

' '( )R k u i ukλλ λ λ∂

=∂

rr r

( )( )

dkeE

dtE dkkdx

dt kk

dtλ

λ

=

= −∂

×∂

Ω

r rhr r rr rr

h

Emergence of curvature by projection

• Free Dirac electron

• 4-band Luttingermodel (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

• spin-orbit coupling (current) tunable by gate voltage

• spin manipulation without using magnetic field

• not realized yet due to spin injection problem

(Initial spin eigenstate isnot energy eigenstate)

Datta-Das current modulator(aka spin FET, APL 1990)

QW with structure inversion asymmetry:Rashba coupling (Sov. Phys. Solid State, 1960)

Hpm

p z= + × ⋅2

2α

σhr r $

Berry curvature in conduction band

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

There is no curvature anywhereexcept at the degenerate point

Is there any curvature (henceRashba-like effect) as a result ofprojection?

( ) ( )k kπδ±Ω =r r

∓

Hpm

p z= + × ⋅2

2α

σhr r $

2

0

in Mathematica is zV V i S k Zm

= −hhEfros and Rosen, Ann. Rev. Mater. Sci. 2000

8-band Kane model

Gauge structure in conduction band, indeed!

( )2

0

22

1,

3

/

1

g

z

g

V

V S P Z m

kE E

σ = − × + ∆

=

Rrr

h

r

• Gauge potential,correct to k1

gE

∆

Chang et al, to be published

Gauge structures and angular momenta in other bands

• Angular momentum,correct to k0

202 1 1

,3 g g

VmE E

σ

= − − + ∆ Lr r

h

( )2

22

1

,

where

1

3

( = 0 if 0 )

g g

eE

eVE

E k

E

σα

α ≡ − +

⋅ = ⋅

∆

×

=

∆

Rrr r r r

Spin-orbit coupling

Ref: R. Winkler, SO couplingeffect in 2D electron and holesystems, Sec. 5.2

• Same form as Rashba

• In the absence of BIA/SIA

,i j ijr p δ=( ) ;

( ) ,

wh

( )

( )

ere / ( )

c

c

r r R

ek p A r

G

eB R

ccp e cA r

π

π

π

π

= + +

= + ×+

≡ +

r rr

rr r rr rr r rh

rr r r

r

Re-quantizing the semiclassical theory:

0 00

( ) ( )

( ) ( )

c

c

r r

E k E pEe

B Rc

E

p

φ φ∂

⋅ ×∂

⋅≅ −

≅ +

Rr r

r

r r

r

rr

r

generalized Peierls substitution:

00( )( , ) ( ) ( ) ( )

22

e EmeE

pr k E k e r B k

mckφ

= − + + ⋅

∂+ ×

∂

⋅ RH R Lr r rr

rr r rr r r

Effective Hamiltonian

Ref: Roth, J. Phys. Chem. Solids1962; Blount, PR 1962

• vanishes near band edge

• higher order in k

Agrees with those in Winkler’s (obtained using LÖ wdin partition)

Observation of non-Abelian Berry phase?• Energy splitting in nuclearquadruple resonance

• Conductance oscillationfor holes in valence bands

(Zee PRB, 1988;Zwanziger PRA 1990)

(Arovas and Lyanda-Geller,PRB 1998)

• Wave packet dynamics in single band



• (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 itsapplications, by Chien and Westgate (1980)

Not covered(among others):

Not related:

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

(Leduc-Righi effect, 1887)

wavepacket dynamics of electrons in solid

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