nonequilibrium green’s function method for thermal transport jian-sheng wang
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Nonequilibrium Green’s Function Method for Thermal
Transport
Jian-Sheng Wang
TIENCS 2010 2
Outline of the lecture• Models
• Definition of Green’s functions
• Contour-ordered Green’s function
• Calculus on the contour
• Feynman diagrammatic expansion
• Relation to transport (heat current)
• Applications
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Models
Ck
Cj
ijk
Ciijkn
jCCTaC
TT
nRCLC
RCL
uuuTH
uuuVuH
RCLxmuuKuuuH
HHHHHHH
3
1
,
,,,,2
1
2
1
Left Lead, TL
Right Lead, TR
Junction
TIENCS 2010 4
Force constant matrix
RR
RRR
RR
RC
CRCCL
LC
LL
LL
L
kk
kkk
kk
V
VKV
V
kk
kk
k
1110
011110
0100
1110
0100
01
0
0
0
0
0
0
KR
TIENCS 2010 5
Definitions of Green’s functions
• Greater/lesser Green’s function
• Time-ordered/anti-time ordered Green’s function
• Retarded/advanced Green’s function
)()'()',(,)'()()',( tutui
ttGtutui
ttG jkjkkjjk
)',()'()',()'()',(
),',()'()',()'()',(
ttGttttGttttG
ttGttttGttttGt
t
GGttttG
GGttttGa
r
)'()',(
,)'()',(
TIENCS 2010 6
Relations among Green’s functions
),'()',(
),'()',(
,
,
ttGttG
ttGttG
GGGGGGG
GGGGGGG
GGGG
akj
rjk
kjjk
taartt
trtt
ar
TIENCS 2010 7
Steady state, Fourier transform
][][
,)(][
),'()',(
ar
ti
GG
dtetGG
ttGttG
TIENCS 2010 8
Equilibrium systems, Lehmann representation
• The average is with respect to the density operator exp(-β H)/Z
• Heisenberg operator
• Write the various Green’s functions in terms of energy eigenstate H |n> = En |n>
• Use the formula )(11
xix
Pix
//)( iHtiHt AeetA
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Fluctuation dissipation theorem
[ ] ( ) [ ] [ ] ,
[ ] 1 ( ) [ ] [ ] ,
[ ] [ ]
where
1 1( ) ,
1
r a
r a
B
G f G G
G f G G
G e G
fe k T
TIENCS 2010 10
Computing average (nonequilibrium)
• Average < ... > over an arbitrary density matrix ρ
• ρ = exp(-βH)/Z in equilibrium
• Schrödinger picture: A, (t)• Heisenberg picture: AH(t) = U(t0,t)AU(t,t0) , ρ0 ,
where operator U satisfies
'( )
( , ')( ) ( , '),
( , ') , 't
t
iH d
U t ti H t U t t
t
U t t Te t t
TIENCS 2010 11
Calculating correlations
'
0 0 0 0 0
0 0 0
( )
0 '
( )
( ) ( ') Tr ( ) ( ') '
Tr ( ) ( , ) ( , ) ( , ') ( ', )
Tr ( ) ( , ) ( , ') ( ', )
Tr ( ) ,
( , ') ,
( , ') ( ', '') ( , '')
C
t
t
iH d
C t t
iH d
A t B t A t B t t t
t U t t AU t t U t t BU t t
t U t t AU t t BU t t
t T e AB
U t t Te
U t t U t t U t t
t0 t’ t
B A
TIENCS 2010 12
Contour-ordered Green’s function
( )
0 , , '
( , ') ( ) ( ')
Tr ( ) C
jk C j k
iH d
C j k
iG T u u
t T u u e
t0
τ’
τ
Contour order: the operators earlier on the contour are to the right.
TIENCS 2010 13
Relation to other Green’s function
t
t
GGGG
GGGG
GG
GGttGG
itt
,
,
or)',()',(
0,,or),,(
'
t0
τ’
τ
TIENCS 2010
• Integration on (Keldysh) contour
• Differentiation on contour
14
Calculus on the contour
dttfdttfdttfdf )()()()(
dt
tdf
d
df )()(
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• Theta function
• Delta function on contour
where θ(t) and δ(t) are the ordinary theta and Dirac delta functions
15
Theta function and delta function
1)',(
0)',(
)'()',(
)'()',(
)',()',(
otherwise0
contour thealong ' later than is if1)',(
'
tt
tt
tttt
tttt
tt
)'()',()',(
)',( '' tttt
d
d
TIENCS 2010 16
Express contour order using theta function
),'()()'()',()'()(
)'()()',(
TTT
TC
uui
uui
uuTi
G
Operator A(τ) is the same as A(t) as far as commutation relation or effect on wavefunction is concerned
Iiuu T ])(),([
TIENCS 2010 17
Equation of motion for contour ordered Green’s function
IKG
IuKuTi
uui
uuTi
uui
uui
uui
G
uuTi
uui
uui
uui
uui
G
TC
TTC
TTT
T
TC
TTT
TTT
)',()',(
)',())'()((
)',(])'(),([)'()(
),'()()'()',()'()(
)',()'()()',(
)'()(
),'()()'()',()'()(
),'()()'()',()'()()',(
2
2
• Consider a harmonic system with force constant K
TIENCS 2010 18
Equations for Green’s functions
0)',()',(
)'()',()',(
)'()',()',(
)'()',()',(
)',()',()',(
,,2
2
2
2
,,,,2
2
'''
2
2
2
2
ttGKttGt
IttttGKttGt
IttttGKttGt
IttttGKttGt
IGKG
tt
tartar
TIENCS 2010 19
Solution for Green’s functions
2, , , ,
2
2 , , , ,
1, , 2
12
( , ') ( , ') ( ')
using Fourier transform:
[ ] [ ]
[ ] ( ) ( )
[ ] [ ] ( ) , 0
( ),
r a t r a t
r a t r a t
r a t
r a
r a
t r
G t t KG t t t t It
G KG I
G I K c K d K
G G i I K
G f G G G e G
G G G
c and d can be fixed by initial/boundary condition.
TIENCS 2010 20
Handling interactions
• Transform to interaction picture, H = H0 + Hn
0 00 0
0 00 0
0 00 0
0 0 00 0 0
( ) ( )
0 0 0
( ) ( ' )
( ) ( )
0 0
( ) ( ) ( )
0 0
( ) ( ) ( , ) ( , ) ( )
( , ') ( , ')
( ) ( , ) ( , )
( ) ( , ) ( ) (
H Hi t t i t t
I S H I
H Hi t t i t t
H Hi t t i t t
I H
H H Hi t t i t t i t t
I s H
t e t e U t t S t t t
S t t e U t t e
t e U t t U t t e
A t e A e e U t t A t U t
0
0( ), )
Hi t t
t e
TIENCS 2010 21
Scattering operator S
• Transform to interaction picture
• The scattering operator satisfies:
0 0 0 0
0 0 0 0( ) ( ) ( ' ) ( ' )
0 0 0 0
0 0
( ) ( ') Tr ( ) ( ') '
Tr[ ( , ) ( ) ( , ) ( , ') ( ') ( ', )]
Tr[ ( , ) ( ) ( , ') ( ') ( ', )]
H H H
H H H Hi t t i t t i t t i t t
I
I
A t B t A t B t t t
U t t e A t e U t t U t t e B t e U t t
S t t A t S t t B t S t t
0 00 0
'
( ) ( )
( '') ''
( , ') ( ) ( , '), ( )
( , ') , '
tIn
t
H Hi t t i t tI I S
n n n
iH t dt
i S t t H t S t t H t e H et
S t t Te t t
TIENCS 2010 22
Contour-ordered Green’s function
C
In
C
dHi
Ik
IjCI
dHi
kjC
kjjk
euuT
euuTt
uui
G
)(
)(
',,0
)'()(Tr
)(Tr
)'()()',(
t0
τ’
τ
TIENCS 2010 23
Perturbative expansion of contour ordered Green’s function
)'()()()()()()()(
em)Wick theor(
)()()()()()()'()(...)',(
)()()(),,(3
1
)()()(),,(3
1)'()()'()(
)()()(1)'()(
)'()()',(
463521
6543210
654654654
321321321
3
2211
2
11
'')''(
koCqnCpmCljC
qponmlkjCjk
opqqpoopq
lmnnmllmnkjCkjC
nnnkjC
dHi
kjCjk
uuTuuTuuTuuT
uuuuuuuuTG
ddduuuT
ddduuuTuuTi
uuTi
dHdHi
dHi
uuTi
euuTi
Gn
TIENCS 2010
General expansion rule
1 2 1 2
0
1 2 1 2
( , ')
( , , )
( , , , ) ( ) ( ) ( )n n
ijk i j k
j j j n C j j j n
G
T
iG T u u u
Single line
3-line vertex
n-double line vertex
TIENCS 2010 25
Diagrammatic representation of the expansion
21221100 )',(),(),()',()',( ddGGGG n
= + 2i + 2i
+ 2i
= +
TIENCS 2010 26
Self -energy expansion
Σn
TIENCS 2010 27
Explicit expression for self-energy
' ' ', 0, 0,
'' '' '', ' 0, 0,
, ''
4
'[ ] 2 [ '] [ ']
2
'2 '' [0] [ ']
2
( )
n jk jlm rsk lr mslmrs
jkl mrs lm rslmrs
ijk
di T T G G
di T T G G
O T
TIENCS 2010
Junction system• Three types of Green’s functions:
• g for isolated systems when leads and centre are decoupled• G0 for ballistic system• G for full nonlinear system
t = 0
t = −
HL+HC+HR
HL+HC+HR +V
HL+HC+HR +V +Hn
g
G0
G
Governing Hamiltonians
Green’s function
Equilibrium at Tα
Nonequilibrium steady state established
TIENCS 2010
Three regions
RCLuuTi
G
uu
u
u
u
u
u
u
TC
CL
L
L
R
C
L
,,,,)'()()',(
,, 2
1
TIENCS 2010
Heisenberg equations of motion in three regions
,
1 1,
2 2
1 1 1[ , ], [ , ],
, ,
T LC T RCL C R L C R C n
T T
C CL CRC C C L R C n
CC
H H H H u V u u V u H
H u u u K u
u u H H K u V u V u u Hi i i
u K u V u L R
TIENCS 2010 31
Relation between g and G0
Equation of motion for GLC
IgKg
dGVgG
GVGK
uuTi
G
uuTi
G
LL
L
CCLC
LLC
CCLC
LCL
TCLCLC
TCLCLC
)',()',()',(
,'')',''()'',()',(
),',()',(
)'()()',(
,)'()()',(
2
2
2
2
TIENCS 2010 32
Dyson equation for Gcc
RCR
CRLCL
CL
CCCCCC
CCRC
RCR
CCLC
LCL
CCC
RCCR
LCCL
CCC
TCCCCC
TCCCCC
VgVVgV
ddGggG
IdGVgV
dGVgVGK
IGVGVGK
IuuTi
G
uuTi
G
)',()',()',(
,)',(),(),()',()',(
),',('')',''()'',(
'')',''()'',()',(
)',()',()',()',(
)',()'()()',(
,)'()()',(
212211
2
2
TIENCS 2010 33
The Langreth theorem
aaarrr
rrrr
ar
ar
rrrrrr
CBACBACBAD
CBAD
ddCBAD
BABAC
dtttBttAdtttBttAttC
BACdtttBttAttC
dtttBttAdBAC
,
)',(),(),()',(
][][][][][
'')',''()'',('')',''()'',()',(
][][]['')',''()'',()',(
'''')',''()'',('')',''()'',()',(
212211
''
'''''
TIENCS 2010 34
Dyson equations and solution
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
TIENCS 2010 35
Energy current
1Tr [ ]
2
1Tr [ ] [ ] [ ] [ ]
2
T LCLL L C
LCCL
r aCC L CC L
dHI u V u
dt
V G d
G G d
TIENCS 2010 36
Caroli formula
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
TIENCS 2010 37
Ballistic transport in a 1D chain• Force constants
• Equation of motion
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
TIENCS 2010 38
Solution of g• Surface Green’s function
2
0
0
0
0
1 20
( ) , 0
2 0
2 0
0 2
0 0
, 0,1,2, ,
(( ) 2 ) / 0, | | 1
RR
R
jRj
i K g I
k k k
k k k kK
k k k k
k
g jk
i k k k
TIENCS 2010 39
Lead self energy and transmission
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
TIENCS 2010 40
Heat current and conductance
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
TIENCS 2010 41
General recursive algorithm for surface Green’s function
10
1110
011110
0100
0
0
0
0
k
kk
kkk
kk
K R
1200
12
01
11
00
)(
)|(| while}
)(
{ do
sIig
g
ggee
gss
eIig
k
ke
ks
T
14
5
10
10
TIENCS 2010 42
Carbon nanotube (6,0), force field from Gaussian
Dis
pers
ion
rela
tion
Tra
nsm
ission
TIENCS 2010 43
Carbon nanotube, nonlinear effect
The transmissions in a one-unit-cell carbon nanotube junction of (8,0) at 300K. From J-S Wang, J Wang, N Zeng, Phys. Rev. B 74, 033408 (2006).
TIENCS 2010 44
1D chain, nonlinear effect
Three-atom junction with cubic nonlinearity (FPU-). From J-S Wang, Wang, Zeng, PRB 74, 033408 (2006) & J-S Wang, Wang, Lü, Eur. Phys. J. B, 62, 381 (2008). Squares and pluses are from MD.
kL=1.56 kC=1.38, t=1.8 kR=1.44
TIENCS 2010 45
Molecular dynamics with quantum bath
2
2
, , , ,
( )( ) ( ') ( ') '
( ) ( ') ( '), , ,
2
,
tC rC
C L R
T
r C r C r r rL R
d u tF t t t u t dt
dt
t t i t t L R
V g V
TIENCS 2010 46
Average displacement, thermal expansion
• 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
TIENCS 2010 47
Thermal expansion
(a) Displacement <u> as a function of position x. (b) as a function of temperature T. Brenner potential is used. From J.-W. Jiang, J.-S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009).
Left edge is fixed.
TIENCS 2010 48
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 J.-W. Jiang, J.-S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009).
TIENCS 2010 49
Transient problems
1 2
0
0
0
,1 2
2
, 0( )
, 0
( , )( ) Im
L R
T LRL R
RL
L
t t t
H H H
H tH t
H u V u t
G t tI t k
t
TIENCS 2010 50
Dyson equation on contour from 0 to t
1 1 1
1 2 1 1 2 2,
( , ') ( ) ( ')
( ) ( ) ( ')
RL R RL L
C
R RL L LR RL
C C
G d g V g
d d g V g V G
Contour C
TIENCS 2010 51
Transient thermal current
The time-dependent current when the missing spring is suddenly connected. (a) current flow out of left lead, (b) out of right lead. Dots are what predicted from Landauer formula. T=300K, k =0.625 eV/(Å2u) with a small onsite k0=0.1k. From E. C. Cuansing and J.-S. Wang, Phys. Rev. B 81, 052302 (2010). See also arXiv:1005.5014.
TIENCS 2010 52
Summary
• The contour ordered Green’s function is the essential ingredient for NEGF
• NEGF is most easily applied to ballistic systems, for both steady states and transient time-dependent problems
• Nonlinear problems are still hard to work with
TIENCS 2010 53
References
• H. Haug & A.-P. Jauho, “Quantum Kinetics in Transport and …”
• J. Rammer, “Quantum Field Theory of Non-equilibrium States”
• S. Datta, “Electronic Transport in Mesoscopic Systems”
• M. Di Ventra, “Electrical Transport in Nanoscale Systems”
• J.-S. Wang, J. Wang, & J. Lü, Europhys B 62, 381 (2008).
TIENCS 2010 54
Problems for NGS students taken credits
• Work out the explicit forms of various Green’s functions (retarded, advanced, lesser, greater, time ordered, etc) for a simple harmonic oscillator, in time domain as well in frequency domain
• Consider a 1D chain with a uniform force constant k. The left lead has mass mL, center mC, and right lead mR. Work out the transmission coefficient T[ω] using the Caroli formula.
• Work out the detail steps leading to the Caroli formula.
TIENCS 2010 55
Website
• This webpage contains the review article, as well some relevant codes/thesis:
http://staff.science.nus.edu.sg/~phywjs/NEGF/negf.html
Thank you
nus.edu.sg