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The phonon Hall effect – NEGF and Green-Kubo treatments. Jian-Sheng Wang, National University of Singapore. Overview. The phonon Hall effect NEGF formulism Green-Kubo formula Conclusion. Phonon Hall effect. B. - PowerPoint PPT Presentation
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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
4
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