nonequilibrium green’s function method: application to thermal transport and thermal expansion...
TRANSCRIPT
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Nonequilibrium Green’s Function Method: application to thermal transport and thermal
expansion
Wang Jian-Sheng
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Outline
• An introduction to nonequilibrium Green’s function (NEGF) method
• Heat transport, counting statistics
• Problem of thermal expansion
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NEGF
Our review: Wang, Wang, and Lü, Eur. Phys. J. B 62, 381 (2008); Wang, Agarwalla, Li, and Thingna, Front. Phys. (2013), DOI:10.1007/s11467-013-0340-x
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Evolution Operator on Contour
2
1
2 1 2 1
3 2 2 1 3 1 3 2 1
11 2 2 1 1 2
0 0
( , ) exp ,
( , ) ( , ) ( , ),
( , ) ( , ) ,
( ) ( , ) ( , )
c
iU T H d
U U U
U U
O U t OU t
Contour-ordered Green’s function
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( )
0 '
( , ') ( ) ( ')
Tr ( ) C
TC
iH dT
C
iG T u u
t T u u e
t0
τ’
τ
Contour order: the operators earlier on the contour are to the right. See, e.g., H. Haug & A.-P. Jauho.
Relation to other Green’s functions
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'
( , ), or ,
( , ') ( , ') or
,
,
t
t
t t
G GG G t t G
G G
G G G G
G G G G
t0
τ’
τ
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Heisenberg Equation on Contour
2
1
2 1 2 1
0 0
( , ) exp ,
( ) ( , ) ( , )
( )[ ( ), ]
c
iU T H d
O U t OU t
dOi O H
d
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Thermal conduction at a junction
Left Lead, TL Right Lead, TR
Junction Partsemi-infinite
Three regions
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RCLuuTi
G
uu
u
u
u
u
u
u
TC
CL
L
L
R
C
L
,,,,)'()()',(
,, 2
1
Dyson equations and solution
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an
r
aan
rn
ran
r
rn
rr
ar
rCr
n
CC
GG
GIGGIGGG
GG
GGG
KIiG
GGGG
GggG
)(
)()(
,)(
0,)(
,
0
110
000
120
00
00
Energy current
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0
( ', ) ( ', )( , ') ( , ') '
1Tr [ ]
2
1Tr [ ] [ ] [ ] [ ]
2
T LCLL L C
t ar L LCC CC
t
LCCL
r aCC L CC L
dHI u V u
dt
t t t ti G t t G t t dt
t t
V G d
G G d
Landauer/Caroli formula
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0
1Tr
2
,2
,
( )
r aLL CC L CC R L R
r a
L RL
r aL L R R
a r r aL R
dHI G G f f d
dt
i
I II
G G G i f f
G G iG G
Ballistic transport in a 1D chain
• Force constants
• Equation of motion13
k
kkk
kkkk
kkkk
k
K
00
20
2
02
0
0
0
0
1 0 1(2 ) , , 1,0,1,2,j j j ju ku k k u ku j
Lead self energy and transmission
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2 1
| |
1
20 0
0 0
0 0 0 0
0 0 0
0 0 0
( ) ,
1, 4[ ] Tr
0, otherwise
L
r CL R
j krjk
r aL R
k
G K
Gk
k k kT G G
T[ω]
ω
1
Heat current and conductance
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max
min
0
2 2
0
[ ]2
1lim ,
2 1
, 0, 03
L R
L L R
L
T TL R
B
dI T f f
I f df
T T T e
k TT k
h
Arbitrary time, transient result
16 time)longin (
)(
ln
)(Tr
2
1
)(
ln
2)(
ln
)',())'('),(()',(
)1ln(Tr2
1ln
2
2222
0
0
0
0
ItQ
i
ZQQQ
iG
i
ZQQ
xi
ZQ
xx
GZ
M
AL
n
nn
LLAL
AL
Numerical results, 1D chain
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1D chain with a single site as the center. k= 1eV/(uÅ2), k0=0.1k,TL=310K, TC=300K, TR=290K. Red line right lead; black, left lead.
From Agarwalla, Li, and Wang, PRE 85, 051142, 2012.
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Thermal Expansion
• Grüneisen theory
• NEGF – compute the displacement of each atom <uj>. It is obtained by the standard Feynman-diagrammatic expansion with respect to nonlinear interactions.
ln 1,
ln 3c
V BV
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One-Point Green’s Function
Average displacement, thermal expansion
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One-point Green’s function
0 0
( ) ( )
' '' ''' ( ', '', ''') ( ', '') ( ''', )
( 0) [ 0]
1
j C j
lmn lm njlmn
rj lmn lm nj
lmn
jj
j
iG T u
d d d T G G
G T G t G
dGi
M x dT
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Connection, see Jiang, Wang, Wang, Park, arXiv:1408.1450.
NEGF * 12
*2
1 NEGF
1
ln
ln 3
/
,
N lmn l m nNlmn
lmn l m nlmn
n n
n nN n N
cT S S K
L
LT S S
V
R L
FLu K F
B
Thermal expansion
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(a) Displacement <u> as a function of position x.
(b) as a function of temperature T. Brenner potential is used. From Jiang, Wang, and Li, Phys. Rev. B 80, 205429 (2009).
Left edge is fixed.
Graphene Thermal expansion coefficient
The coefficient of thermal expansion v.s. temperature for graphene sheet with periodic boundary condition in y direction and fixed boundary condition at the x=0 edge. is onsite strength. From Jiang, Wang, and Li, Phys. Rev. B 80, 205429 (2009).
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Phonon Life-Time
( )2
2 2,
,
,
2
1, ( )
[ ]
Re [ ] 2 ,
Im [ ]
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q qq
ti t
r rq qr
q n q
rn q q q q
qrn q q
q
q q qq
G G t ei
c v
For calculations based on this, see, Xu, Wang, Duan, Gu, and Li, Phys. Rev. B 78, 224303 (2008).
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Summary
• NEGF: powerful tool for steady state and transient, best for ballistic system, difficult for interaction systems
• Thermal expansion problem: NEGF does not need to assume uniform expansion, suited for any nanostructure or bulk
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Acknowledgements
• NEGF, transport: Wang Jian, Lü Jingtao, Eduardo C Cuansing, Zhang Lifa, Bijay Kumar Agarwalla, Li Huanan
• Thermal expansion: Jiang Jinwu (now at Shanghai Univ)