nonequilibrium green’s function method for thermal transport jian-sheng wang

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).


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


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


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

)'()',(

,)'()',(


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.


Relation to other Green’s function

t

t

GGGG

GGGG

GG

GGttGG

itt

,

,

or)',()',(

0,,or),,(

'

t0

τ’

τ


Equations for Green’s functions

0)',()',(

)'()',()',(

)'()',()',(

)'()',()',(

)',()',()',(

,,2

2

2

2

,,,,2

2

'''

2

2

2

2

ttGKttGt

IttttGKttGt

IttttGKttGt

IttttGKttGt

IGKG

tt

tartar


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.


Contour-ordered Green’s function

C

In

C

dHi

Ik

IjCI

dHi

kjC

kjjk

euuT

euuTt

uui

G

)(

)(

',,0

)'()(Tr

)(Tr

)'()()',(

t0

τ’

τ


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


Diagrammatic representation of the expansion

21221100 )',(),(),()',()',( ddGGGG n

= + 2i + 2i

+ 2i

= +


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


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


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


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


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


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


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


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


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).


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


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


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


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

nonequilibrium green’s function method for thermal transport jian-sheng wang

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