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ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 10:Higher-Order BTE ModelsPowerPoint Presentation

ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 10:Higher-Order BTE Models

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### ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 10:Higher-Order BTE Models

J. Murthy

Purdue University

ME 595M J.Murthy

BTE Models Transport

- Gray BTE drawbacks
- Cannot distinguish between different phonon polarizations
- Isotropic
- Relaxation time approximation does not allow direct energy transfers between different frequencies even if “non-gray” approach were taken
- Very simple relaxation time model

- Higher-order BTE models
- Try to resolve phonon dispersion and polarization using “bands”
- But finer granularity requires more information about scattering rates
- Various approximations in finding these rates

- Will look at
- Semi-gray models
- Full dispersion model
- Full scattering model

ME 595M J.Murthy

Semi-Gray BTE Transport

- This model is sometimes called the two-fluid model (Armstrong, 1981; Ju, 1999).
- Idea is to divide phonons into two groups
- “Reservoir mode” phonons do not move; capture capacitative effects
- “Propagation mode” phonons have non-zero group velocity and capture transport effects. Are primarily responsible for thermal conductivity.

- Model involves a single equation for reservoir mode “temperature” with no angular dependence
- Propogation mode involves a set of BTEs for the different directions, like gray BTE
- Reservoir and propagation modes coupled through energy exchange terms

Armstrong, B.H., 1981, "Two-Fluid Theory of Thermal Conductivity of Dielectric Crystals", Physical Review B, 23(2), pp. 883-899.

Ju, Y.S., 1999, "Microscale Heat Conduction in Integrated Circuits and Their Constituent Films", Ph.D. thesis, Department of Mechanical Engineering, Stanford University.

ME 595M J.Murthy

Propagating Mode Equations Transport

Propagating model scatters to a bath at lattice temperature TL with relaxation time

“Temperature” of propagating mode, TP, is a measure of propagating mode energy in all directions together

CP is specific heat of propagating mode phonons

ME 595M J.Murthy

Reservoir Mode Equation Transport

- Note absence of velocity term
- No angular dependence – equation is for total energy of reservoir mode
- TR, the reservoir mode “temperature” is a measure of reservoir mode energy
- CR is the specific heat of reservoir mode phonons
- Reservoir mode also scatters to a bath at TLwith relaxation time
- The term qvol is an energy source per unit volume – can be used to model electron-phonon scattering

ME 595M J.Murthy

Lattice Temperature Transport

ME 595M J.Murthy

Discussion Transport

- Model contains two unknown constants: vg and
- Can show that in the thick limit, the model satisfies:
- Choose vg as before; find to satisfy bulk k.
- Which modes constitute reservoir and propagating modes?
- Perhaps put longitudinal acoustic phonons in propagating mode ?
- Transverse acoustic and optical phonons put in reservoir mode ?
- Choice determines how big comes out

- Main flaw is that comes out very large to satisfy bulk k
- Can be an order-of-magnitude larger than optical-to-acoustic relaxation times

- In FET simulation, optical-to acoustic relaxation time determines hot spot temperature
- Need more detailed description of scattering rates

ME 595M J.Murthy

Full-Dispersion BTE Transport

- Details in Narumanchi et al (2004,2005).
- Objective is to include more granularity in phonon representation.
- Divide phonon spectrum and polarizations into “bands”. Each band has a set of BTE’s in all directions
- Put all optical modes into a single “reservoir” mode with no velocity.
- Model scattering terms to allow interactions between frequencies. Ensure Fourier limit is recovered by proper modeling
- Model relaxation times for all these scattering interactions based on perturbation theory (Han and Klemens,1983)
- Model assumes isotropy, using [100] direction dispersion curves in all directions

Narumanchi, S.V.J., Murthy, J.Y., and Amon, C.H.; Sub-Micron Heat Transport Model in Silicon Accounting for Phonon Dispersion and Polarization; ASME Journal of Heat Transfer, Vol. 126, pp. 946—955, 2004.

Narumanchi, S.V.J., Murthy, J.Y., and Amon, C.H.; Comparison of Different Phonon Transport Models in Predicting Heat Conduction in Sub-Micron Silicon-On-Insulator Transistors; ASME Journal of Heat Transfer, 2005 (in press).

Han, Y.-J. and P.G. Klemens, Anharmonic Thermal Resistivity of Dielectric Crystals at Low Temperatures. Physical Review B, 1983. 48: p. 6033-6042.

ME 595M J.Murthy

Phonon Bands Transport

Optical band

Acoustic bands

Each band characterized by its group velocity, specific heat and “temperature”

ME 595M J.Murthy

Optical Mode BTE Transport

Electron-phonon energy source

Energy exchange due to scattering with jth acoustic mode

No ballistic term – no transport

oj is the inverse relaxation time for energy exchange between the optical band and the jth acoustic band

Toj is a “bath” temperature shared by the optical and j bands. In the absence of other terms, this is the common temperature achieved by both bands at equilibrium

ME 595M J.Murthy

Acoustic Mode BTE Transport

Scattering to same band

Ballistic term

Energy exchange with other bands

ij is the inverse relaxation time for energy exchange between bands i and j

Tij is a “bath” temperature shared by the i and j bands. In the absence of other terms, this is the common temperature achieved by both bands at equilibrium

ME 595M J.Murthy

Model Attributes Transport

- Satisfies energy conservation
- In the acoustically thick limit, the model can be shown to satisfy

Fourier heat diffusion equation

Thermal conductivity

ME 595M J.Murthy

Properties of Full-Dispersion Model Transport

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

ME 595M J.Murthy

Full Scattering Model Transport

Elastic Scattering

Inelastic Scattering

Klemens, (1958)

Valid only for phonons satisfying conservation rules Complicated, non-linear

ME 595M J.Murthy

N and U Processes Transport

k2

k1

k3

k3

k’3

k1

k2

G

- N processes do not offer resistance because there is no change in direction or energy
- U processes offer resistance to phonons because they turn phonons around

N processes change f and affect U processes indirectly

ME 595M J.Murthy

General Computation Procedure for Three-phonon Scattering Rates

- 12 unknowns
- 7 equations
- Set 5, determine 7
- Specify K (Kx, Ky, Kz) and direction of K’ (K’x, K’y)

- Bisection algorithm developed to find all sets of 3-phonon interactions

One energy conservation equation

Three components of momentum conservation equation

Three dispersion relations for the three wave vectors

Wang, T. and Murthy, J.Y.; Solution of Phonon Boltzmann Transport Equation Employing Rigorous Implementation of Phonon Conservation Rules; ASME IMECE Chicago IL, November 10-15, 2006.

ME 595M J.Murthy

Thermal Conductivity of Bulk Silicon Rates

- 2-10K, boundary scattering dominant;
- 20-100K, impurity scattering important, as well as N and U processes;
- Above 300K, U processes dominant.

Experimental data from Holland (1963)

ME 595M J.Murthy

Thermal Conductivity of Undoped Silicon Films Rates

Specularity Parameter p=0.4

Experimental data from Ju and Goodson (1999), and Asheghi et al. (1998, 2002)

ME 595M J.Murthy

Conclusions Rates

- In this lecture, we considered three extensions to the gray BTE which account for more granularity in the representation of phonons
- More granularity means more scattering rates to be determined – need to invoke scattering theory
- Models like the semi-gray and full-dispersion models still employ temperature-like concepts which are not satisfactory.
- Newer models such as the full scattering model do not employ relaxation time approximations, and temperature-like concepts

ME 595M J.Murthy

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