Thermal Transport in Nanostrucutures

Jian-Sheng WangCenter for Computational Science and Engineering

and Department of Physics, NUS; IHPC & SMA

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Outline

• Heat transport using classical molecular dynamics

• Nonequilibrium Green’s function (NEGF) approach to thermal transport, ballistic and nonlinear

• QMD – classical molecular dynamics with quantum baths

• Outlook and conclusion

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Approaches to Heat Transport

Molecular dynamics Strong nonlinearity Classical, break down at low temperatures

Green-Kubo formula Both quantum and classical

Linear response regime, apply to junction?

Boltzmann-Peierls equation

Diffusive transport Concept of distribution f(t,x,k) valid at nanoscale?

Landauer formula Ballistic transport T→0, no nonlinear effect

Nonequilibrium Green’s function

A first-principle method Perturbative. A theory valid for all T?

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Fourier’s Law of Heat Conduction

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

Fourier proposed the law of heat conduction in materials as

J = - κ T

where J is heat current density, κ is thermal conductivity, and T is temperature.

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Normal & Anomalous Heat Transport

TL THJ

3D bulk systems obey Fourier’s law (insulating crystal: Peierls’ theory of Umklapp scattering process of phonons; gas: kinetic theory, κ = 1/3 cvl )

In 1D systems, variety of results are obtained and still controversial. See S Lepri et al, Phys Rep 377, 1 (2003); A Dhar, arXiv:0808.3256, for reviews.

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Carbon Nanotubes

Heat conductivity of carbon nanotubes at T = 300K by nonequilibrium molecular dynamics.

From S Maruyama, “Microscale Thermophysics Engineering”, 7, 41 (2003). See also Z Yao at al cond-mat/0402616, G Zhang and B Li, cond-mat/0403393.

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A Chain Model for Heat Conduction

m

ri = (xi,yi)

Φi

22

1

1( , )

2 2

cos( )

ir i i

i

ii

H K am

K

pp r r r

TL

TR

Transverse degrees of freedom introduced

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Nonequilibrium Molecular Dynamics

• Nosé-Hoover thermostats at the ends at temperature TL and TR

• Compute steady-state heat current: j =(1/N)i d (i ri)/dt,

where i is local energy associated with particle i

• Define thermal conductivity by <j> = (TR-TL)/(Na)

N is number of particles, a is lattice spacing.

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Conductivity vs size N

Model parameters (KΦ, TL, TR):

Set F (1, 5, 7), B (1, 0.2, 0.4), E (0.3, 0.3, 0.5), H (0, 0.3, 0.5), J (0.05, 0.1, 0.2) ,

m=1, a=2, Kr=1.

From J-S Wang & B Li, Phys Rev Lett 92, 074302 (2004). ln N

slope=1/3

slope=2/5

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Nonequilibrium Green’s Function 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 Current

( 0)

1Tr [ ]

2

1Tr [ ] [ ] [ ] [ ]

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 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

See, e.g., Mingo & Yang, PRB 68, 245406 (2003); JSW, Wang, & Lü, Eur. Phys. J. B, 62, 381 (2008).

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Contour-Ordered Green’s Functions

( '') ''

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 Interactions

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 Equations

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 an nI G G I G

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Feynman 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 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

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

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Energy 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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Thermal Conductance of Nanotube Junction

Con

d-m

at/0

6050

28

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Molecular Dynamics

• Molecular dynamics (MD) for thermal transport– Equilibrium ensemble, using Green-Kubo formula

– Non-equilibrium simulation• Nosé-Hoover heat-bath• Langevin heat-bath

• Disadvantage of classical MD– Purely classical statistics

• Heat capacity is quantum below Debye temperature of 1000 K

• Ballistic transport for small systems is quantum

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Quantum Corrections

• Methods due to Wang, Chan, & Ho, PRB 42, 11276 (1990); Lee, Biswas, Soukoulis, et al, PRB 43, 6573 (1991).

• Compute an equivalent “quantum” temperature by

• Scale the thermal conductivity by

1 1

exp( / ) 2BB Q

Nk Tk T

MDQ

dT

dT

(1)

(2)

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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

(3)

(4)

1

2

j

u

u

u

u

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Quantum Langevin Equation for 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

(5)

(6)

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Properties of Quantum Noise

† 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).

(7)

(8)

(9)

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Quasi-Classical Approximation, Schmid (1982)

• Replace operators uC & by ordinary numbers

• Using the quantum correlation, iħ∑> or iħ∑< or their linear combination, for the correlation matrix of .

• Since the approximation ignores the non-commutative nature of , quasi-classical approximation is to assume, ∑> = ∑<.

• For linear systems, quasi-classical approximation turns out exact! See, e.g., Dhar & Roy, J. Stat. Phys. 125, 805 (2006).

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Implementation

• Generate noise using fast Fourier transform

• Solve the differential equation using velocity Verlet• Perform the integration using a simple rectangular

rule• Compute energy current by

2 /1( ) i lk M

kk

t lh ehM

( ') ( ') 't

TLL C L c L

dHI u t t u t dt

dt

(10)

(11)

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Comparison of QMD with NEGF

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 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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Electron Transport & Phonons

• For electrons in the tight-binding form interacting with phonons, the quantum Langevin equations are

†

( ') ( ') ' ,

( ') ( ') '

tk

k

t

ic Hc t t c t dt M u c

u Ku t t u t dt c Mc

† †

†

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

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

T

TT

t t i t t t t i t t

t t i t t t t i t t

(12)

(13)

(14)

(15)

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Ballistic to Diffusive

Electronic conductance vs center junction size L. Electron-phonon interaction strength is m=0.1 eV. From Lü & JSW, arXiv:0803.0368.

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Conclusion & Outlook

• MD is useful for high temperature thermal transport but breaks down below Debye temperatures

• NEGF is elegant and efficient for ballistic transport. More work need to be done for nonlinear interactions

• QMD for phonons is correct in the ballistic limit and high-temperature classical limit. Much large systems can be simulated (comparing to NEGF)

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Collaborators

• Baowen Li• Pawel Keblinski• Jian Wang• Jingtao Lü• Nan Zeng

• Lifa Zhang• Xiaoxi Ni• Eduardo Cuansing• Jinwu Jiang

• Saikong Chin• Chee Kwan Gan• Jinghua Lan• Yong Xu