nonequilibrium green's function (negf) method in thermal...

2371-1

Advanced Workshop on Energy Transport in Low-Dimensional Systems: Achievements and Mysteries

Jian-Sheng WANG

15 - 24 October 2012

National University of Singapore Dept. of Physics, 11754

Singapore

Nonequilibrium Green's function (NEGF) method in thermal transport and some applications

Nonequilibrium Green’s function (NEGF) method in thermal transport and some

applications

Jian-Sheng Wang National University of Singapore

ICTP workshop 2012 2

Outline of the talk

ವ Introduction ವ Method of nonequilibrium Green’s functions ವ Applications

ವ Thermal currents in1D chain and nanotubes

ವ Transient problems

ವ Full counting statistics

3

Fourier’s law for heat conduction

TJ

Fourier, Jean Baptiste Joseph, Baron (1768-1830)

[ ] ( ) i tf f t e dt

4

Thermal conductance

area section cross :tyconductivi :econductanc :

leadright andleft of re temperatu:,densitycurrent : current, thermal: where

,

)(

S

TTJI

SJISL

TTI

RL

RL


Thermal transport of a junction

Left Lead, TL

Right Lead, TR

Junction

semi-infinite leads


Models

1 1 , , , ,2 2

,

1 13 4

LC CRL C R n

T T

aC T CC j

C C C C C C Cn ijk i j k ijkl i j k l

ijk ijkl

H H H H H H H

H p p u K u u m x L C R

H u V u u u

H T u u u T u u u u

Left Lead, TL

Right Lead, TR

Junction


Force constant matrix

RR

RRR

RR

RC

CRCCL

LC

LL

LL

L

kkkkk

kkV

VKV

Vkkkk

k

1110

011110

0100

1110

0100

01

0

00

00

0

KR


Definitions of Green’s functions

ವ Greater/lesser Green’s function

ವ Time-ordered/anti-time ordered Green’s function

ವ Retarded/advanced Green’s function

( , ') ( ) ( ') ( ', )jk j k kjiG t t u t u t G t t

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

ttGttttGttttGttGttttGttttG

t

t

GGttttGGGttttG

a

r

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

See text books, e.g., H. Haug and A.-P. Jauho, or J. Rammer, or S. Datta, or Di Ventra


Contour-ordered Green’s function

( )

0 , , '

( , ') ( ) ( ')

Tr ( ) C

jk C j k

i H d

C j k

iG T u u

t T u u e

t0

’

Contour order: the operators earlier on the contour are to the right.


Relation to the real-time Green’s functions

'

( , ), or ,

( , ') ( , ') or

,,

t

t

t t

t t

G GG G t t

G G

G G G GG G G G

G G G G

t0

t+

t


Equations for Green’s functions (in ballistic systems)

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 GG G G

c and d can be fixed by initial/boundary condition.


Transform to interaction picture

( '') ''

0 , , '

( '') ''

( , ') ( ) ( ')

Tr ( )

Tr ( ) ( ')

C

InC

jk j k

i H d

C j k

i H dI II C j k

iG u u

t T u u e

T u u e

t0

’


Perturbative expansion of contour ordered Green’s function

( '') ''

2

1 1 1 1 2 2

3

1 2 3 1 2 3

( , ') ( ) ( ')

1( ) ( ') 1 ( ) ( ) ( )2!

1 1( ) ( ') ( ) ( ') ( , , ) ( ) ( ) ( )2 3

ni H d

jk C j k

C j k n n n

C j k C j k lmn l m nl

iG T u u e

i i iT u u H d H d H d

i iT u u T u u T u u u 1 2 3

4 5 6 4 5 6 4 5 6

01 2 3 4 5 6

1 2 5 3 6 4

1 ( , , ) ( ) ( ) ( )3

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

(Wick's theorem)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( '

mn

opq o p qopq

jk C j k l m n o p q

C j l C m p C n q C o k

d d d

T u u u d d d

G T u u u u u u u u

T u u T u u T u u T u u )

See, e.g., A. L. Fetter & J. D. Walecka.


Diagrammatic representation of the expansion

0 0 1 1 2 2 1 2( , ') ( , ') ( , ) ( , ) ( , ')conn n connG G G G d d

= + 2i + 2i

+ 2i

= +


Diagrammatic representation of the nonlinear self-energy


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

ICTP workshop 2012

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

GT

iG T u u u

Single line

3-line vertex

n-double line vertex


Junction systems ವ Three types of Green’s functions: ವ g for isolated systems when leads and centre are decoupled ವ G 0 for ballistic system ವ 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 T

Nonequilibrium steady state established


Three regions

RCLuuTiG

uuu

uuuu

u

TC

CL

L

L

R

C

L

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

,, 2

1


Dyson equations and solutions 0 0

0 0

120

0 0 0

110

0

,

( ) , 0

(Keldysh equation)

( ) ,

( ) ( )

( )

CL LC CR RCC C L R

n

r C r

r a

r r rn

r a r r a an n n

r an

G g g G V g V V g VG G G G

G i I K

G G G

G G

G G G I G G I GG G


Energy current (Meir-Wingreen)

1 Tr [ ]2

1 Tr [ ] [ ] [ ] [ ]2

T LCLL L C

LCCL

r aL L

dHI u V udt

V G d

G G d


Landauer/Caroli formula

0

Tr2

,2

,

( )

r aLL CC L CC R L R

r a

L RL

r aL L R R

a r r aL R

dH dI G G f fdt

i

I II

G G G i f fG G iG G


Ballistic transport in a 1D chain ವ Force constant matrix

ವ Equation of motion

kkkk

kkkkkkkk

k

K

0020

202

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 ವ Surface Green’s function 2

0

0

0

0

1 20

( ) , 0

2 02 0

0 20 0

, 0,1,2, ,

(( ) 2 ) / 0, | | 1

RR

R

jRj

i K g I

k k kk k k k

Kk k k k

k

g jk

i k k k


Lead self energy and transmission

2 1

| |

1

20 0

0 00 0 0 00 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[ ]

1


Heat current and “universal” thermal conductance

max

min

0

2 2

0

[ ]2

1lim ,2 1

, 0, 03

L R

L L R

LT T

L R

B

dI T f f

I f d fT T T e

k T k Th

Rego and Kirczenow, 1998.


1D nonlinear model Three-atom

junction with cubic nonlinearity (FPU- ). From JSW, Wang, Zeng, PRB 74, 033408 (2006). Cross + from QMD: JSW, Wang, Lü, Eur. Phys. J. B, 62, 381 (2008).


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

Lt t t

H H HH t

H tH u V u t

G t tI t kt


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 coupling is suddenly switched on. (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).


Finite lead problem

Top ඍ��QR�RQVLWH��k0=0), (a) NL = NR = 100, (b) NL=NR=50, temperature T=10, 100, 300 K (black, red, blue). TL=1.1T, TR = 0.9T. Right ඎ��ZLWK�RQVLWH�k0=0.1k and similarly for other parameters. From E. C. Cuansing, H. Li, and J.-S. Wang, Phys. Rev. E 86, 031132 (2012).


Full counting statistics (FCS)

• What is the amount of energy (heat) Q transferred in a given time t ?

• This is not a fixed number but given by a probability distribution P(Q )

• Generating function

• All moments of Q can be computed from the derivatives of Z.

• The objective of full counting statistics is to compute Z(˫).

dQQPeZ Qi )()(


A brief history on full counting statistics ವ L. S. Levitov and G. B. Lesovik proposed the concept

for electrons in 1993; rederived for noninteracting electron problems by I. Klich, K. Schönhammer, and others

ವ K. Saito and A. Dhar obtained the first result for phonon transport in 2007

ವ J.-S. Wang, B. K. Agarwalla, and H. Li, PRB 2011; B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 2012.


Definition of generating function based on two-time measurement

/

2

)(

)0,(e.g.,)0,(),0()(

||

||

' 'Tr

iHtLL

L

aa

aaa

tHiHi

etUtUHtUtH

aaaH

aaPP

PPeeZ LL Consistent

history quantum mechanics (R. Griffiths).


Approaches to compute Z ವ Express Z as expectation value of some effective

evolution operator over a contour ವ Evaluate the expression using

ವ Feynman path integral/influence functional

ವ Feynman diagrammatic expansion


Product initial state

, ,

( )

/ 2 / 2

/ 2 / 2

, [ , ] 0, '

Tr

Tr (0, ) ( ,0)

Tr (0, ) ( ,0)

Tr (0, ) ( ,0)

( , ') ( , ')

L L

L L

L L L

L L

HL L

L C R

i H i H t

i H i H

i H i H i H

ixH ixHx

e P H

Z e e

e U t e U t

e U t e U t e

U t U t

U t t e U t t e


Ux

)(

)(

)(

)()(

',

)',(

00

)(

)(

'

'

xueueu

HHuVuHHeHeHHHH

eHHHHHeetHetH

ttTe

eTeettU

LixHLixHLx

xI

CLCTLx

RC

ixHLCixHRCRCL

ixHRCLCRCL

ixH

ixHixHx

dttH

ixHdttHixHx

LL

LL

LL

LL

t

t xi

L

t

ti

L


Schrödinger, Heisenberg, and interaction pictures

tHitHi

I

dttHi

I

I

tHi

I

I

HHH

HH

i

AeetA

ttTettSttStHtttSi

tttStet

HHH

tHtAt

tAi

tAUtUtA

ttUtHt

ttUi

iiwtttUt

t

t I

00

'

0

)(

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

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

)](),([)()0(||),0,(),0()(

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

'')''(

0

Schrödinger picture Heisenberg picture Interaction picture


Compute Z in interaction picture

LxL

ALRLCC

AL

ALCR

xLC

LCxL

CLC

C C

xI

xIC

xIC

dHi

C

GggG

G

GggZ

VgVg

ddHHidHiT

eT

tUtUZ

CxI

,,

)]1ln[(Tr21

)]1)(1det[(ln21)](1[lnTr

21ln

][Tr211

')'()(21)(1Tr

]Tr[

)]0,(),0([Tr

00

0

0

2

)(

2/2/- /2

+ /2


Important result

otherwise00if2/)(0if2/)()(

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

,,

)1ln(Tr21ln

00

0

M

M

LLAL

LxL

ALRLCC

AL

tttxtttxx

xx

GggG

GZ


Long-time result, Levitov-Lesovik formula

)/(1),1/(1

,),()1]([),1]([

where

)2()1()1(detln4

01lnTr

21

)1ln(Tr21ln

00

0

0

0

22

22

TkefRLi

ebea

ffeefefeGGIdt

GGG

GZ

Ba

ar

iL

iL

RLii

Ri

Li

Ra

Lr

M

a

Kr

AL

baba

baba


Arbitrary time, transient result

time)long in(

)(ln

)(Tr

21

)(ln

)(ln

)1ln(Tr21ln

2

2222

00

0

0

ItQ

iZQQQ

iG

iZQQ

iZQ

GZ

M

AL

n

nn

AL


Numerical results, 1D chain

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. B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 85, 051142, 2012.


FCS for coupled leads

,

LC LRL

CL CRC

RL RCR

K V VK V K V G g gVG

V V K

1 1

ln ln det ( 1) (1 ) ( 1) (1 ) [ ]4

[ ] , , ,

If center is absent 0, then

[ ] [ ]

i iM L R R L g

r a a rg LR R RL L

LC RC

r ag I RR R RR L

dZ t I e f f e f f T

T G G i g g L R

V V

T T G G

From H. Li, B. K. Agarwalla, and J.-S. Wang, Phys. Rev. E 86, 011141 (2012); and arXiv:1208.4915


FCS in nonlinear systems

ವ Can we do nonlinear? ವ Yes we can. Formal expression generalizes Meir-

Wingreen ವ Interaction picture defined on contour

ವ Some example calculations


Vacuum diagrams, Dyson equation, interaction picture transformation on contour

,

[

Lj,

0 0

( )I1 2 1 2

0 0

( )

ln 1 Tr( ) 2 ( )

1( , ) Tr ( ) ( )

, (0 , ) ( ,0 ) /

( ) (0 , ) ( ) ( ,0 )

( ,0 )

InC

x T LC T CRL C C R

C

AL n

i H dI I TC

n

In M M

I

i u V u u V u d

C

Z Gi i

G G G G

iG T u u eZ

Z Z Z V t V t ZO V O V

V T e ,0 ]


Nonlinear system self-consistent results

Single-site (1/4)˨u4 model cumulants. k=1 eV/(uÅ2), k0=0.1k, Kc=1.1k, V LC-1,0=V CR0,1=-0.25k, TL=660 K, TR=410K. From H. Li, B. K. Agarwalla, B. Li, and J.-S. Wang, arxiv::1210.2798

( , ') 3 ( , ') ( , ')n i G


Other results

ವ We can also compute the cumulants for the projected steady state ˮ’

ವ Entropy production ವ Fluctuation theorem, Z(˫) = Z(-˫ +i(˟R-˟L))

ವ The theory is applied equally well to electron number of

electron energy transport


Summary remarks

ವ NEGF is a powerful tool to handle thermal transport problems in nanostructures

ವ Steady state current is obtained from Landauer and

Caroli formula ವ New results for transient and full counting statistics. A

key quantity is the self-energy ːLA


Group members/acknowledgements

Front (left to right) Mr. Thingna Juzar Yahya, Mr. Zhou Hangbo, Mr. Bijay Kumar Agarwalla, Dr. Leek Meng Lee, Dr/Ms Ni Xiaoxi, Mr. Tan Kuo Liang; Back: Dr. Jose Luis Garcia Palacios, Prof. Wang Jian, Dr. Jiang Jinwu, Prof. Wang Jian-Sheng, Dr. Zhang Lifa, Mr. Li Huanan, Dr. Eduardo C. Cuansing.

Thank you

see http://staff.science.nus.edu.sg/̃phywjs/ for more information

http://staff.science.nus.edu.sg/~phywjs/

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