Modeling and Simulation of Sub-Micron Thermal Transport Jayathi Y. - - PowerPoint PPT Presentation

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Modeling and Simulation of Sub-Micron Thermal Transport

Jayathi Y. Murthy School of Mechanical Engineering Purdue University

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• Review of phonon transport
• Phonon Boltzmann Transport Equation (BTE)
• Finite volume method
• Improved BTE models
• Future directions and closure

nanoHUB.org online simulations and more Sub-micron Heat Conduction

L Λ<<L L Λ>>L

Two regimes, (a) Fourier’s law is valid, (b) Fourier’s law is invalid, Λ is the mean free path of the heat carrier (phonons or electrons)

(a) (b)

Phonons are quanta of lattice vibrations. They are the main carriers of energy in semiconductors and dielectrics.

nanoHUB.org online simulations and more Heat Transport in Solids

2 1 1 2 2 2 1 2

( 2 ) ( 2 )

n n n n n n n n

d x m g y y x dt d y m g x x y dt

+ +

= +

• =

+

• m1

m2 xn yn xn+1

One-dimensional spring-mass system Dispersion relation for crystal vibrations, Majumdar (1998)

nanoHUB.org online simulations and more Frequency vs. reduced wave number in (100) direction for silicon Boltzmann Transport Equation

.

( )

scat

f f v f t t

• +

=

• s

Boltzmann transport equation for phonons:

Ballistic term

Phonons are characterized by (r, t, K) and polarization

nanoHUB.org online simulations and more Scattering Processes

• Phonons scatter on impurities, grain boundaries,

isotopes, physical boundaries, other phonons and carriers

• Three-phonon interactions are major contributors to

thermal resistance at room temperature in silicon

• Three-phonon interactions must satisfy momentum and

energy conservation rules:

nanoHUB.org online simulations and more Three-Phonon Interactions

General scattering term for 3-phonon processes very complex:

Valid only for phonons satisfying conservation rules

nanoHUB.org online simulations and more Relaxation Time Approximation

.

( ) ( ) ( )

scat

f f f t

• =
• K

r,K K

.

( )

scat

f f v f t t

• +

=

• s

Relaxation time Planck distribution function

nanoHUB.org online simulations and more Gray Phonon BTE

• Energy form:
• e” is energy per unit volume per unit solid angle
• There are as many pde’s as there are s directions. In

each direction, e” varies in space and time

• Different directions are related to each other because of

e0 in the scattering term:

• Notice that

( )

g eff

e e e v e t

• +

=

• s

1 1 ( , ) sin 4 4 e t e d e d d

• =

=

• r

4 eff

e e d

• =
• No net

energy source

nanoHUB.org online simulations and more When is a Gray Model Good?

Courtesy IBM

• BTE in relaxation time approximation incapable of

transferring energy across frequencies and polarizations

– Not useful in MOSFET modeling

• Too expensive at present to solve full BTE, though work

is in progress

• Gray approximations to BTE are good for problems with

small departures from equilibrium provided modeling is done well

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Overview of Finite Volume Method

• Divide spatial domain into control volumes of extent ΔxΔy
• Divide angular domain into control angles of extent ΔΩ
• Divide time into steps of Δt - but will only do steady here for simplicity
• Integrate gray BTE in each direction over control volume and control
• angle. Get discrete equation set.
• Solve each direction sequentially and iteratively
• Back out “temperature” from e0 upon convergence using

s

4

ref

e T T C

• =

+

nanoHUB.org online simulations and more Discretization

• Divide domain into rectangular control volumes of extent Δx and Δy.

Assume 2D, so that depth into page (z) is one unit.

• Divide angular domain of 4π in NθxNφ control angles per octant.

Centroid of each control angle is (θi , φi), extents are (Δθ, Δφ). For each control angle i:

• Important: The directions s are 3D even though we are considering 2D

x y y x z s

i

sin sin sin cos cos

i i i i i

• =

s i + j+ k

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Angular Discretization Nomenclature

• Control angle extent is
• In 2D, only directions in the “front” hemisphere are necessary. Thus

θ ranges from 0-π/2 and φ=0-2π

• Thus, increase control angle extent to:
• Define for future use:

( )

/2 /2 /2 /2 sin

2sin sin 0.5

i i i i

i

d d

• +
• =

=

• (

)

/2 /2 /2 /2

2* sin _ *2sin sin 0.5 _ 2 for 2D

i i i i

i

d d Weight Factor Weight Factor

• +
• =

=

• =
• /2

/2 /2 /2

i i i i

d d

• +
• =
• s

S

nanoHUB.org online simulations and more Formula for S

( ) ( ) ( )

( )

sin sin 0.5 cos 2 sin _ cos sin 0.5 cos 2 sin 0.5 sin 2 sin

i i i j i

Weight Factor

• =
• i

+ j + k S

nanoHUB.org online simulations and more Spatial Discretization

P E N W S

Δy Δx

e w n s e” stored at cell centroids

nanoHUB.org online simulations and more Control Volume Balance

• Integrate governing equation over control volume and control angle:

( )

" i , , " " " "

Look at LHS. Apply Divergence Theorem: =

i i g eff V V g i i g i i A A g i A g i

e e v e dVd dVd v e dA d v e dA d v e dA v e

• =
• =
• s

s n n s n

i

S

" " " " "

=

g if f faces A g ie xi g iw xi g in yi g is yi

dA v e A v e y v e y v e x v e x

• +
• f

n n

ii ii

S=S S=S SSSS SSSS

faces f n s w e

P s

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Control Volume Balance (cont’d)

• Now look at RHS
• Collecting terms:
• Control volume balance says that net rate of energy

entering the CV in direction si must be balanced by net in-scattering to the direction i in the CV

( )

" " 2 , i i iP iP eff eff V

e e e e dVd V O x

• =

+

• "

" " " " , , g ie xi g iw xi g in yi g is yi i P i P eff

v e y v e y v e x v e x e e V

• +
• =

SSSS SSSS

nanoHUB.org online simulations and more Upwinding

• e” is stored at cell centroids, but we need it on the CV faces
• Need to interpolate from cell centroid to face
• Can use a variety of schemes to perform interpolation

– Central difference scheme

• Second-order accurate, but wiggles in spatial solution

– Upwind difference scheme

• Upwind scheme is only first order accurate
• Higher order schemes are available

P E W e

( )

0.5

• n uniform mesh

e P E

• =

+

x x

if >= 0 if < 0

e P E

• =

= S S

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Discrete Equation

• Using upwinding and collecting terms, we obtain

an algebraic equation:

• We obtain one such equation for each grid point

P for each direction i

• The b term contains e0iP
• Once we have boundary conditions discretized,

we can solve the set

" " , , , , ,

Here, nb are the spatial neighbors E,W,N,S.

i P i P i nb i nb i P nb

a e a e b = +

nanoHUB.org online simulations and more Discussion

• Prefer to solve iteratively and if possible,

sequentially to keep memory requirements low

• For upwind scheme, diagonal dominance is

guaranteed, making it possible to use iterative schemes

• Conservation of energy is guaranteed

regardless of spatial and angular discretization

– Confirm that sum of all scattering source terms at a point is zero regardless of discretization

• Any linear solver can be used

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Overall Solution Algorithm

• 1. Initialize all e”

i values for all cell centroids and directions

• 2. Find e0

P for each point P from current e” values.

• 3. Start with direction i=1
• 4. For direction i:

– Find discretized equations for direction i, assuming e0 temporarily known – Solve for e”

i at all grid points using a linear solver

• 5. If (i.le. total number of directions) go to 4
• 6. If (all directions are done) check for convergence. If

converged, stop. Else, go to 2.

nanoHUB.org online simulations and more Improved BTE Models – Semi Gray Model

Semi-gray BTE: Divide phonons into two groups – Propagating mode which transports energy – Reservoir mode which accounts for capacitative effects

( )

R R L R R vol

T C T T C q t

• =

+

• P

1 ( ) 4 ( )

L ref P P p P

C T T e e v e t

• +

=

• s

Propagation mode Reservoir mode

nanoHUB.org online simulations and more Full-Dispersion Model

• Divide polarization

branches into bands

• BTE solved for each band
• All optical bands grouped

into single band with zero velocity

• Bulk dispersion curve for

silicon (100) direction is assumed

• Dispersion curve

assumed to be isotropic

nanoHUB.org online simulations and more Properties of Full-Dispersion Model 1-D transient diffusion, with 3X3X1 spectral bands

In acoustically- thick limit, full dispersion model

• Recovers

Fourier conduction in steady state

• Parabolic heat

conduction in unsteady state

nanoHUB.org online simulations and more Silicon Bulk Thermal Conductivity

Full-Dispersion Model

nanoHUB.org online simulations and more Thermal Conductivity of Undoped Silicon Thin Films

• Experimental

data is from Asheghi et. al (1998, 2002)

• For the numerical

results, a specularity parameter p=0.4 is used Full-Dispersion Model

nanoHUB.org online simulations and more Thermal Conductivity of Doped Silicon Thin Films

• 3.0 micron boron-doped

silicon thin films. Experimental data is from Asheghi et. al (2002)

• p=0.4 is used for

numerical predictions

• Boron dopings of

1.0e+24 and 1.0e+25 atoms/m3 considered

Full-Dispersion Model

nanoHUB.org online simulations and more Multi-Finger PD/SOI nMOSFET

STI BOX M1 Silicon substrate Metallization Layers

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Channel Max temperature: 332K

nanoHUB.org online simulations and more Comparison of Models

Fourier; max temp=313.25 K Gray; max temp= 320.85K Semi-gray; max temp= 456.31 K

nanoHUB.org online simulations and more What Phonons Responsible for Transport?

Silicon Silicon dioxide Heat generation region 1633 nm 100 nm 315 nm 900 nm 72 nm 10 nm

26.25 73.71 Branch totals 13.61 29.34 6 8.65 23.17 5 3.69 13.14 4 0.28 6.26 3 0.05 1.78 2 0.01 1 TA(%) LA (%) Band

nanoHUB.org online simulations and more Conclusions

• In this lecture, we developed the basics of a

finite volume method to solve the gray phonon BTE

• We briefly examined extensions and

improvements to this model which consider phonon dispersion and polarization

• New work considering the effects of phonon

confinement and electron-phonon scattering is underway

• Many opportunities for new and interesting

interdisciplinary research