Quantum Thermal Transport

1

Quantum Quantum Thermal Thermal

TransportTransportJian-Sheng Wang,Jian-Sheng Wang,

Dept of Physics, NUSDept of Physics, NUS

2

OverviewOverview• Diffusive and ballistic thermal

transport• Universal thermal conductance• NEGF formulism• Classical MD with quantum bath• Phonon Hall effect

3

Fourier’s LawFourier’s Law

T J

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

[ ] ( ) i tf f t e dt

4

Diffusive Transport vs Diffusive Transport vs Ballistic TransportBallistic Transport

2x t

t

t

5

Thermal ConductanceThermal Conductance

,

L RI T T

LI SJ

S

6

Experimental Report of Z Experimental Report of Z Wang et al (2007)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).

7

Landauer FormulaLandauer Formula

0

/( )

[ ]( ) ,2

1

1B

L R

L k T

dI T f f

fe

8

““Universal” Thermal Universal” Thermal ConductanceConductance

2 2

3Bk T

Mh

Rego & Kirczenow, PRL 81, 232 (1998).

M = 1

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Schwab et al ExperimentsSchwab et al Experiments

From K Schwab, E A Henriksen, J M Worlock and M L Roukes, Nature, 404, 974 (2000).

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Nonequilibrium Green’s Nonequilibrium Green’s Function ApproachFunction Approach

, ,

,

1 1,

2 21

3

T TL LC C C CR Rn

L C R

T T

C C Cn ijk i j k

ijk

H H u V u u V u H

H u u u K u

H T u u u

Left Lead, TL Right Lead, TR

Junction Part

T for matrix transpose

mass m = 1,

ħ = 1

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Heat CurrentHeat Current

( 0)

1ReTr [ ]

2

1ReTr [ ] [ ] [ ] [ ]

2

L L

LCCL

r aL L

CL LCL L

I H t

V G d

G G d

V g V

Where G is the Green’s function for the junction part, ΣL is self-energy due to the left lead, and gL is the (surface) Green’s function of the left lead.

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Landauer/Caroli FormulaLandauer/Caroli Formula• In systems without nonlinear interaction the heat

current formula reduces to that of Laudauer formula:

0

/( )

1[ ] ,

2

[ ] Tr ,

,

1

1B

L R L R

r aL R

r a

k T

I I d T f f

T G G

i

fe

JSW, Wang, & Lü, Eur. Phys. J. B, 62, 381 (2008).

(6,0) carbon nanotube

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Contour-Ordered Green’s Contour-Ordered Green’s FunctionsFunctions

( '') ''

0

'

0

( , ') ( ) ( ') ,

( , ') lim ( , ' '),

, , , ,

,

ni H dT

t t

r t a t

G i T u u e

G t t G t i t i

G G G G G G G G

G G G G G G

τ complex plane

See Keldysh, or Meir & Wingreen, or Haug & Jauho

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Adiabatic Switch-on of Adiabatic Switch-on of InteractionsInteractions

t = 0

t = −

HL+HC+HR

HL+HC+HR +V

HL+HC+HR +V +Hn

gG0

G

Governing Hamiltonians

Green’s functions

Equilibrium at Tα

Nonequilibrium steady state established

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Contour-Ordered Dyson Contour-Ordered Dyson EquationsEquations

0 1 2 1 1 2 0 2

0 1 2 0 1 1 2 2

†

0 0 2

0 0 0

1

0

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

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

Solution in frequency domain:

1, 0

( )

,

1,

C C

n

r aC r

r a

r

r rn

r an

G g d d g G

G G d d G G

G Gi I K

G G G

GG

G G G 0( ) ( ) ( )r r a a r an n nI G G I G G G

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Feynman DiagramsFeynman Diagrams

Each long line corresponds to a propagator G0; each vertex is associated with the interaction strength Tijk.

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Leading Order Nonlinear Leading Order Nonlinear Self-EnergySelf-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

σ = ±1, indices j, k, l, … run over particles

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Energy TransmissionsEnergy Transmissions

The transmissions in a one-unit-cell carbon nanotube junction of (8,0) at 300 Kelvin. From JSW, J Wang, N Zeng, Phys. Rev. B 74, 033408 (2006).

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Quantum Heat-Bath & MDQuantum 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

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Quantum Langevin Equation Quantum Langevin Equation for the Centerfor the Center

• Eliminating the lead variables, we get

where retarded self-energy and “random noise” terms are given as

( ') ( ') 't

CC C L Ru F t t u t dt

0

, ,

, ,

C CL R

CL R

V g V

V u

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Properties of Quantum Properties of Quantum NoiseNoise

† 0 0

†

†

( ) 0,

( ) ( ') ( ) ( ')

( ') ( '),

( ') ( ) ( '),

( ') ( ) [ ] 2 ( )Im [ ]

CL T LCL L L L

CL LCL L

T

L L L

T i tL L L L

t

t t V u t u t V

V i g t t V i t t

t t i t t

t t e dt i f

For NEGF notations, see JSW, Wang, & Lü, Eur. Phys. J. B, 62, 381 (2008).

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Comparison of QMD with Comparison of QMD with NEGFNEGF

QMD ballistic

QMD nonlinear

Three-atom junction with cubic nonlinearity (FPU-). From JSW, Wang, Zeng, PRB 74, 033408 (2006) & JSW, Wang, Lü, Eur. Phys. J. B, 62, 381 (2008).

kL=1.56 kC=1.38, t=1.8 kR=1.44

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From Ballistic to Diffusive From Ballistic to Diffusive TransportTransport

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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Electronic, Ballistic to Electronic, Ballistic to DiffusiveDiffusive

Electronic conductance vs center junction size L. Electron-phonon interaction strength is 0.1 eV. From Lü & JSW, J. Phys.: Condens. Matter, 21, 025503 (2009).

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Phonon Hall EffectPhonon Hall Effect

T

T3

T4

B

Tb3Ga5O12

Experiments by C Strohm et al, PRL (2005), also confirmed by AV Inyushkin et al, JETP Lett (2007). Effect is small |T4 –T3| ~ 10-4 Kelvin in a strong magnetic field of few Tesla, performed at low temperature of 5.45 K.

5 mm

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Thermal Hall conductivity, Thermal Hall conductivity, Green-Kubo formulaGreen-Kubo formula

J S Wang and L Zhang, arXiv:0902.1219

' ', , '

†'

/( )

1 '( ) ( ) ,

16 ( ' ) '

' ( )( ) ',

'

1

1B

a bab

aa

k T

f fF F

VT i

DF

k

fe

k

k k

kk

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Four-Terminal Junction Four-Terminal Junction Structure, NEGFStructure, NEGF

R=(T3 -T4)/(T1 –T2).

From L Zhang, J-S Wang, and B Li, arXiv:0902.4839

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Our GroupOur Group

From left to right, front: Dr. Lan Jinghua (IHPC), Prof. Wang Jian-Sheng, Ms Ni Xiaoxi, back: Dr. Jiang Jinwu, Mr. Teo Zhan Rui (Honours student), Mr. Zhang Lifa, Dr. Eduardo Chaves Cuansing Jr, Mr. Janakiraman Balachandran, Mr. Siu Zhuo Bin. Sep 2008.