nonequilibrium green’s function method for thermal transport jian-sheng wang
Post on 03-Jan-2016
33 Views
Preview:
DESCRIPTION
TRANSCRIPT
Nonequilibrium Green’s Function Method for Thermal
Transport
Jian-Sheng Wang
Xian Symposium 2010 2
Outline of the talk• Introduction
• Models
• Definition of Green’s functions
• Relation to transport (heat current)
• Applications
•1D chain and nanotubes
•Transient problem
•Disordered systems
3
Fourier’s law for heat conduction
T J
Fourier, Jean Baptiste Joseph, Baron (1768-1830)
[ ] ( ) i tf f t e dt
4
Thermal conductance
,
L RI T T
LI SJ
S
5
Experimental report of Z Wang et al (2007)
The experimentally measured thermal conductance is 50pW/K for alkane chains at 1000K. From Z Wang et al, Science 317, 787 (2007).
6
“Universal” thermal conductance in the low temperature limit
Rego & Kirczenow, PRL 81, 232 (1998).
M = 1
h
TkM B
3
22
7
Schwab et al experiments
From K Schwab, E A Henriksen, J M Worlock and M L Roukes, Nature, 404, 974 (2000).
Xian Symposium 2010 8
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
Xian Symposium 2010 9
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
Xian Symposium 2010 10
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
)'()',(
,)'()',(
Xian Symposium 2010 11
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.
Xian Symposium 2010 12
Relation to other Green’s function
t
t
GGGG
GGGG
GG
GGttGG
itt
,
,
or)',()',(
0,,or),,(
'
t0
τ’
τ
Xian Symposium 2010 13
Equations for Green’s functions
0)',()',(
)'()',()',(
)'()',()',(
)'()',()',(
)',()',()',(
,,2
2
2
2
,,,,2
2
'''
2
2
2
2
ttGKttGt
IttttGKttGt
IttttGKttGt
IttttGKttGt
IGKG
tt
tartar
Xian Symposium 2010 14
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.
Xian Symposium 2010 15
Contour-ordered Green’s function
C
In
C
dHi
Ik
IjCI
dHi
kjC
kjjk
euuT
euuTt
uui
G
)(
)(
',,0
)'()(Tr
)(Tr
)'()()',(
t0
τ’
τ
Xian Symposium 2010 16
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
Xian Symposium 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
Xian Symposium 2010 18
Diagrammatic representation of the expansion
21221100 )',(),(),()',()',( ddGGGG n
= + 2i + 2i
+ 2i
= +
Xian Symposium 2010 19
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
Xian Symposium 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
Xian Symposium 2010
Three regions
RCLuuTi
G
uu
u
u
u
u
u
u
TC
CL
L
L
R
C
L
,,,,)'()()',(
,, 2
1
Xian Symposium 2010 22
Dyson equations and solution
an
r
aan
rn
ran
r
rn
rr
ar
rCr
n
RCR
CRLCL
CLCC
GG
GIGGIGGG
GG
GGG
KIiG
GGGG
VgVVgVGggG
)(
)()(
,)(
0,)(
,
0
110
000
120
00
00
Xian Symposium 2010 23
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
Xian Symposium 2010 24
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
Xian Symposium 2010 25
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
Xian Symposium 2010 26
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
Xian Symposium 2010 27
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
Xian Symposium 2010 28
Heat current and conductance, Landauer formula
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
Xian Symposium 2010 29
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).
Xian Symposium 2010 30
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
Xian Symposium 2010 31
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
Xian Symposium 2010 32
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 PRE 82, 021116 (2010).
Treatment of mass disorder
Instead of massive efforts required in brute force calculations, configuration averaging of disordered systems can be efficiently handled in a self-consistent manner by setting up the phonon version of nonequilibrium vertex correction (NVC) theory.
Generate configuration randomly, and compute the transmission of each, and average the results.
Brute Force
Generate a configuration, but treat mass matrix as a variable
Constitute two self-consistent equations to solve effective mass
We have the statistically averaged thermal properties
Coherent Potential Approximation
2( ) ( )( )
[ ] ( ) ( )
h r ds s s s s s
r a r aL R L NVC R
M M M M G M M i
T Tr G G Tr G G
, , , ,, , '
, , '
Q Q r r a Q a Q Q r r a Q aNVC s s s L s s s NVC s s
Q h d Q h d s s
C t G G t C t G G t
N
Results for disordered systems
Results:1. The accuracy of this theory is then tested with Monte Carlo experiments on one-dimensional
disordered harmonic chains, the early proposed power law form of thermal conductivity has been recovered and we also indicate the possibility of varying the exponent for larger system size.
2. Anomalous thermal transport has been shown and also, we observe the transition between different transport regimes due to the scattering of phonons by impurities.
3. This method of considering mass disorder can also be extended to include force constant disorder.
X. Ni, M. L. Leek, J.-S. Wang, Y. P. Feng, and B. Li, ``Anomalous thermal transport in disordered harmonic chains and carbon nanotubes,'' submitting.
nanotubes1D chains
Xian Symposium 2010 35
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, transient time-dependent problems, and mass disordered systems
• Nonlinear problems are still hard to work with
Thank you
top related