Nonequilibrium Green’s Function Method for Thermal

Transport

Jian-Sheng Wang

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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

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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

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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

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Steady state, Fourier transform

][][

,)(][

),'()',(

ar

ti

GG

dtetGG

ttGttG

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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

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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

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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.

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Relation to other Green’s function

t

t

GGGG

GGGG

GG

GGttGG

itt

,

,

or)',()',(

0,,or),,(

'

t0

τ’

τ

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• 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 ])(),([

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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

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Equations for Green’s functions

0)',()',(

)'()',()',(

)'()',()',(

)'()',()',(

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

,,2

2

2

2

,,,,2

2

'''

2

2

2

2

ttGKttGt

IttttGKttGt

IttttGKttGt

IttttGKttGt

IGKG

tt

tartar

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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.

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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

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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

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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

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Diagrammatic representation of the expansion

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

= + 2i + 2i

+ 2i

= +

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Self -energy expansion

Σn

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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

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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

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Three regions

RCLuuTi

G

uu

u

u

u

u

u

u

TC

CL

L

L

R

C

L

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

,, 2

1

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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

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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

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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

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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

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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

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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

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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

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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

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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).

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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

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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

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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.

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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

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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