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The phonon Hall effect The phonon Hall effect – NEGF and Green-– NEGF and Green-Kubo treatmentsKubo treatments

Jian-Sheng Wang,Jian-Sheng Wang,National University of SingaporeNational University of Singapore

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OverviewOverview• The phonon Hall effect• NEGF formulism• Green-Kubo formula• Conclusion

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Phonon Hall effectPhonon Hall effect

T

T3

T4

B

Tb3Ga5O12

Experiments by C Strohm et al, PRL (2005), also confirmed by AV Inyushkin et al, JETP Lett (2007). Effect is small |T4 –T3| ~ 10-4 Kelvin in a strong magnetic field of few Tesla, performed at low temperature of 5.45 K.

5 mm

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Previous theoriesPrevious theories• L. Sheng, D. N. Sheng, & C. S. Ting,

PRL 2006, give a perturbative treatment

• Y. Kagan & L. A. Maksimov, PRL 2008, appears to say nonlinearity is required

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Ballistic model of phonon Ballistic model of phonon Hall effectHall effect

1 1

2 2

where , e.g.,

( )

T T T

T

n nn

H p p u Ku u Ap

A A

V

Λ U P

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Four-terminal junction Four-terminal junction structure, NEGFstructure, NEGF

R=(T3 -T4)/(T1 –T2).

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Hamiltonian for the four-Hamiltonian for the four-terminal junctionterminal junction

4 4

0 0 0 00 1

,

1 1,

2 2

0 0 0

0 0 0

0 0 0

0 0 0

T T

T T

H H u V u u Ap

H p p u K u

h

hA

h

h

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The energy currentThe energy current

2 20

1Re Tr( ) ,

2

1[ ] ,

( ) [ ] 2

r a

rr

r a

I d G G

Gi I K A i A

G G G

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Linear response regimeLinear response regime

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1

0

, small

( ),

Tr( ) ,2

1

exp[ /( )] 1

r a

B

T T

I

d fG G

T

fk T

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Ratios of transverse to Ratios of transverse to longitudinal temperature longitudinal temperature

differencedifference

R=(T3 -T4)/(T1 –T2).

From L Zhang, J-S Wang, and B Li, arXiv:0902.4839.

No Hall effect on square lattice with nearest neighbor couplings.

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RR vs vs BB or or TT

The relative Hall temperature difference R vs (a) magnetic field B, (b) vs temperature T at B = 1 Tesla.

Red line is σ13 – σ14

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Green-Kubo methodGreen-Kubo method• Work on periodic lattices• Find the phonon eigenmodes (turns

out not othonormal)• Derive the energy density current• Compute equilibrium correlation

function of the energy density current

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EigenmodesEigenmodes2

, ' ''

† †

( ) ( ) 0,

or

( ) exp[ ( ) ],

exp( ) . .2

1

l l l ll

l k l kk k

i A I D

A Di

I A

D K i

u i a H cN

iA

k

k R R k

R k

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Effect of Effect of AA to phonon to phonon dispersiondispersion

Phonon-dispersion relation of a triangular lattice. (a) longitudinal mode as a function of kya with kx = 0. black (h=0), red (h=5x1012 rad s-1.) (b) as a function of h at ka=(0,1).

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Current density vector Current density vector (Hardy 1963)(Hardy 1963)

' , ' ', '

† †'' ' , '

, ' '

1( )

2

( )

4

( , )

c c c Tl l l l l l

l l

k kk k k kc

k k k k

J R R u K uV

Da a

V k

k

k k

k

k

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Green-Kubo formulaGreen-Kubo formula

eq0 0

† †

eq

( ) ( ) ,

( 1) ,

1/[exp( ) 1]

a bab

i j k l i k ij kl i j il jk

i i

Vd dt J i J t

T

a a a a f f f f

f

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Thermal Hall conductivity, Thermal Hall conductivity, Green-Kubo formulaGreen-Kubo formula

J S Wang and L Zhang, PRB 80, 012301 (2009).

' ', , '

†'

/( )

1 '( ) ( ) ,

16 ( ' ) '

' ( )( ) ',

'

1

1

B

a bab

aa

k T

f fF F

VT i

DF

k

fe

k

k k

kk

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Hall conductivity vs hHall conductivity vs h

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A symmetry principleA symmetry principle• If there is a symmetry transformation

S, such that SDST =D, SAST=-A,

then the off-diagonal elements of the thermal conductivity tensor κab = 0

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Mirror reflection symmetryMirror reflection symmetry

x, -T

y

J=-κ T

J(D,A)=J(D,-A)

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ConclusionConclusion• Both NEGF and Green-Kubo

approaches give phonon Hall effect in the ballistic models, provided that a symmetry is not fulfilled.

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AcknowledgementsAcknowledgements• This work is in collaboration with Lifa

Zhang and Baowen Li

• Support by NUS faculty research grants