Nonequilibrium Green’s Function

(NEGF) Modeling of Phonon Transport

ME 595MWei Zhang and T.S. Fisher

Recall Lattice Dynamics• Equation of motion for a 1D atomic chain

• Plane wave assumption

• Combine

• Re-arrange and write in matrix form

{ }2

1 12 2nn n n

d xm g x x xdt − += − − −

( ){ }( ) ~ expnx t i Kna t−ω

{ }21 12n n n n

gx x x xm − +−ω = − − −

2 0⎡ ⎤−ω =⎣ ⎦k I xI is the identity matrix

Green’s Function• The matrix k~ is a form of the dynamical matrix,

and contains the spring constants• Earlier, the matrix equation gave us the

dispersion relation • In general, such equations can be written in

operator form• Green’s functions are often used in such

situations to determine general solutions of (usually) linear operators

[ ] 0=L x

Green’s Function, cont’d• The Green’s function g is the solution that

results from the addition of a perturbation to the problem

• In the present (matrix) problem, the unperturbed Green’s function becomes

– Where δ is called the broadening constant, and i is the unitary imaginary number

– See Eq. (11) of Zhang and Fisher

[ ]L g = δ

( ) 12 i−

⎡ ⎤= ω + δ −⎣ ⎦g I k

Why Green’s Functions?• So far, we have not made much progress in solving real

problems (we have only found a way to arrive at a general solution)

• To solve practical problems, we need to incorporate real materials and material interfaces

• We can do this through the use of a different Green’s function G

– This matrix function includes self-energy matrices (Σ1, Σ2) thatinvolve unperturbed Green’s functions (g’s) associated with contacts (i.e., boundaries) in a transport problem

– The full derivation is beyond the scope of this course, so we will simply use the results for computational purposes

NEGF Background• Initially developed to simulate electron

ballistic transport• Very efficient in the ballistic regime but

requires significant effort to implement scattering

• Recently being applied to phonon transport

Applications• Electron/phonon transport in advanced

semiconductor technology– sub-10 nm silicon integrated circuits– strained gate/source/drain

• Carrier transport in solid-state energy conversion devices– Superlattice thermoelectric power generator

or refrigerator

Phonon Transport through a “Device” between Two Contacts

Reservior 1 Reservoir 2

“Device”

Transmission function, Ξ

Cold T2Hot T1

1D Atomic Chain

• Can be visualized as an atomic chain between two bulk contacts

T1 T2

Transmission and Heat Flux

0

( ) ( )2

J N dω ω ω ωπ

∞

= ∆ Ξ∫Phonon occupationdifference

TransmissionPhonon energy

(units = W)

( )/

2 2/ 1

B

B

k T

k TB

eN N N Tk T e

+ −∆ = − = ∆−

ω

ω

ω

We need to evaluate transmission in order to calculate heat flux

Transmission and Green’s Function

• After a lengthy mathematical derivation

Dynamical matrix element

Unperturbed Green’s functions Green’s functions

“Tr” means trace operator on a matrix

Dynamical Matrix• Recall dynamic oscillator result

• Finally, define the modified dynamical matrix as

{ }1 1

2

, 1 , 1

2

now, define the matrix as:

2

harm

n n nn

harm

iji j

nn

n n n n

Uforce g x x xx

Ukx x

k gk k g

− +

+ −

∂= − = − − −

∂

∂= −

∂ ∂

→ = −

= =

k

Here, g is the spring constant

Dynamical Matrix

22

2

aa am

ma mm mb

bm bb

k k f fk k k f f f

f fk k

⎡ ⎤ −⎡ ⎤⎢ ⎥ ⎢ ⎥= = − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦⎢ ⎥⎣ ⎦

k

f is spring constant divided by atomic mass

1. k is not the same dynamical matrix used to determine dispersion curve (that matrix is the Fourier transform of k).

2. k is symmetric.3. Sum of all elements in any row or sum of all elements in any

column is zero, except in the first and last row/column.

Unperturbed Green’s Function

k1,00=k1,01=fb/(atomic mass of bulk material), spring constant in the bulk contact

Represents the response of the bulk contacts to incident phonon waves; its imaginary part is proportional to the density of states in the contacts

Self Energy Matrices

Represent the changes that need to be made to atomic chain’s dynamical properties due to presence of bulk contacts

/b b c bk k f m mβ β β= =

/a a c ak k f m mα α α= =

Green’s Function

Represent the response of atom chain to “point-source excitation”. In this case, “point-source excitation” refers to an infinitely small perturbation (phonon wave) from the bulk contacts

Homework Parameters

142 10 Hzω = ×

40 N/mc bf f f= = =

Use 4.65e-19 kg for all atomic mass

Assignment: Write a code to calculate the transmission of five-atom linear chain if all atoms are the same