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Nonequilibrium Green’s Function and Quantum Master Equation Approach to Transport

Wang Jian-Sheng

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Outline

• A quick introduction to nonequilibrium Green’s function (NEGF), applied to molecular dynamics with quantum baths

• Formulation of quantum master equation to transport (energy, particle, or spin)

• Dyson expansion and 4-th order results

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NEGF

Our review: 1. Wang, Wang, and Lü, Eur. Phys. J. B 62, 381 (2008); 2. Wang, Agarwalla, Li, and Thingna, Front. Phys. (2013), DOI:10.1007/s11467-013-0340-x

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Evolution Operator on Contour

2

1

2 1 2 1

3 2 2 1 3 1 3 2 1

11 2 2 1 1 2

0 0

( , ) exp ,

( , ) ( , ) ( , ),

( , ) ( , ) ,

( ) ( , ) ( , )

c

iU T H d

U U U

U U

O U t OU t

Contour-ordered Green’s function

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( )

0 '

( , ') ( ) ( ')

Tr ( ) C

TC

iH dT

C

iG T u u

t T u u e

t0

τ’

τ

Contour order: the operators earlier on the contour are to the right. See, e.g., H. Haug & A.-P. Jauho or J. Rammer.

Relation to other Green’s functions

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'

( , ), or ,

( , ') ( , ') or

,

,

t

t

t t

G GG G t t G

G G

G G G G

G G G G

t0

τ’

τ

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Heisenberg Equation on Contour

2

1

2 1 2 1

0 0

( , ) exp ,

( ) ( , ) ( , )

( )[ ( ), ]

c

iU T H d

O U t OU t

dOi O H

d

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Thermal conduction at a junction

Left Lead, TL Right Lead, TR

Junction Partsemi-infinite

Three regions

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RCLuuTi

G

uu

u

u

u

u

u

u

TC

CL

L

L

R

C

L

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

,, 2

1

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IMPORTANT RESULT

otherwise0

0if2/)(

0if2/)()(

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

,,

)1ln(Tr2

1ln

00

0

M

M

LLAL

LxL

ALRLCC

AL

tttx

tttxx

xx

GggG

GZ

Arbitrary time, transient result

11 time)long in(

)(

ln

)(Tr

2

1

)(

ln

)(

ln

)1ln(Tr2

1ln

2

2222

0

0

0

0

ItQ

i

ZQQQ

iG

i

ZQQ

i

ZQ

GZ

M

AL

n

nn

AL

Numerical results, 1D chain

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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.

From Agarwalla, Li, and Wang, PRE 85, 051142, 2012.

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Quantum heat-bath & MD• Consider a junction system with left and right harmonic leads at equilibrium

temperatures TL & TR, the Heisenberg equations of motion are

• The equations for leads can be solved, given

,

,

L LCL L C

C CL CRC L R

R RCR R C

u K u V u

u F V u V u

u K u V u

0

2 20

2 2

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

where

( ) 0, ( ) ( )

tLC

L L L C

L LL L

u t u t g t t V u t dt

d dK u t K g t t I

dt dt

1

2

j

u

u

u

u

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

See J.-S. Wang, et al, Phys. Rev. Lett. 99, 160601 (2007);Phys. Rev. B 80, 224302 (2009).

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Equilibrium simulation1D linear chain (red lines exact, open circles QMD) and nonlinear quartic onsite (crosses, QMD) of 128 atoms. From Eur. Phys. J. B, 62, 381 (2008).

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From ballistic to diffusive transport

1D chain with quartic onsite nonlinearity (Φ4 model). The numbers indicate the length of the chains. From JSW, PRL 99, 160601 (2007).

NEGF, N=4 & 32

4

16

64

256

1024

4096

Classical, ħ 0

,S

J TL

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Conductance of graphene strips

Sites 0 to 7 are fixed left lead and sites 28 to 35 are fixed right lead. Heat bath is applied to sites 8 to 15 at temperature TL and site 20 to 27 at TR. JSW, Ni, & Jiang, PRB 2009.

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Quantum Master Equation

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Quantum Master Equation

• Advantage of NEGF: any strength of system-bath coupling V; disadvantage: difficult to deal with nonlinear systems.

• QME: advantage - center can be any form of Hamiltonian, in particular, nonlinear systems; disadvantage: weak system-bath coupling, small system.

• Can we improve?

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Dyson Expansions

0 0 0

0

0

( )

0

( )

0

2 42 4 6

( ) Tr ( , ) ( ) ( , )

Tr ( ) ,

( ) Tr ( ) [ ( ), ( )] ,

( )2! 4!

| |,

C

C

H B

V d

B c

V d

H B c

T T T

nm

O t S t O t S t

iT O t e

dO t T O t O t V t e

dt

X X V X V O

X n m

0 1Tr ..B cT d

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Divergence

11 1,

is diagonal,

if | |

d d

T n T n

d

m nmn

X V X V Vn i n

E EX m n

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Unique one-to-one map, ρ0↔ρ; ordered cumulants

2

2

2

2 4 42 4 2

0 2

42 3

4

4 42 3 6

2! 4! 2!

[ , ] [ , ]3!

[ , ]2!

( )3! 2!

T

T

T

T T T

X V

T T

T m nmnX V

H X V

T LCL C

X V X V X V

di X V V X V V

dt

E EX V V

I VV VV VV

V p V u

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Order-by-Order Solution to ρ

(0)

(0)

( 2) ( 2) (0)

2(0)

(0) 2 (2) 4

(0)

(2)

3

( )

0

1[ , ]

...

[ , ] 0

1[ , ] [ , ] [ , ]

3!1

[ , ]2!

d

d f

T

T T Td f

f

Tf f

Td

T T Td d d

Td X V

O

X X X

X V Vi

X V V

X V V X V V X V V

X V V

|1 1 | 0 0 ...

0 | 2 2 | 0

0 0 | 3 3 |

... ...

0 |1 2 | |1 3 | ...

| 2 1| 0 | 2 3 |

| 3 1| | 3 2 | 0

... ... ...

d

f

X

X

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DiagrammaticsDiagrams representing the terms for current `V or [X T,V]. Open circle has time t=0, solid dots have dummy times. Arrows indicate ordering and pointing from time -∞ to 0. Note that (4) is cancelled by (c); (7) by (d).

From Wang, Agarwalla, Li, and Thingna, Front. Phys. (2013), DOI: 10.1007/s11467-013-0340-x.

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QD model to 4th order

†0

†, ,

, ,

†, , ,

h.c.

S

j jj L R

j j jj

H E d d

V g c d

H c c

The coefficients for the current IL = a2 η+a4η2, for the Lorentz-Drude bath spectrum J(ω) = ηħ/(1 + ω2/D2). 50% chemical potential bias, equal temperature. Curves from NEGF, dots from 4-th order master equation.

From Thingna, Zhou, and Wang, arxiv:1408.6996.

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Spin-Boson Model

, ,, ,

2 2 2, , ,

2 2

2

1

2

S z x

zj j

j L R

j j jj

EH

V g Q

H P Q

The coefficients for the current IL = a2 η+a4η2

For the spin-boson model with Rubin baths,J(ω) = (1/2)ħηω (4-ω2)1/2.TL = 1.5, TR=0.5, E=0.5. We see co-tunneling featuers.

From Thingna, Zhou, and Wang, arxiv:1408.6996.

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Summary

• NEGF: powerful tool to study transport in nanostructures for steady state and transient

• Application to molecular dynamics – quantum heat bath

• Quantum master equation approach – arbitrary strong interaction in the system