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ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 11:Extensions and Modifications. J. Murthy Purdue University. Drawbacks. Gray BTE Cannot distinguish between different phonon polarizations Isotropic

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ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 11:Extensions and Modifications

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Me 595m computational methods for nanoscale thermal transport lecture 11 extensions and modifications

ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 11:Extensions and Modifications

J. Murthy

Purdue University

ME 595M J.Murthy



  • Gray BTE

    • Cannot distinguish between different phonon polarizations

    • Isotropic

    • Relaxation time approximation does not allow energy transfers between different frequencies even if “non-gray” approach were taken

    • Very simple relaxation time model

  • Numerical Method

    • “Ray” effect and “false scattering”

    • Sequential procedure fails at high acoustic thicknesses

  • We will consider remedies for each of these problems in the next two lectures

ME 595M J.Murthy

Semi gray bte

Semi-Gray BTE

  • This model is sometimes called the two-fluid model (Armstrong, 1981).

  • 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.

  • Ju (1999) used this idea to devise a model for nano-scale thermal transport

  • 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

Propagating Mode Equations

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

Reservoir Mode Equation

  • 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

Lattice Temperature

ME 595M J.Murthy



  • 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

Non gray bte

Non-Gray BTE

  • 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.

  • 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

Phonon Bands

Optical band

Acoustic bands

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

ME 595M J.Murthy

Optical mode bte

Optical Mode BTE

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

Acoustic Mode BTE

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

Lattice temperature1

Lattice Temperature

  • Lattice “temperature” is a measure of the energy in all acoustic and optical modes combined

ME 595M J.Murthy

Model attributes

Model Attributes

  • 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

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

ME 595M J.Murthy

Silicon bulk thermal conductivity

Silicon Bulk Thermal Conductivity

Non-Gray Model

ME 595M J.Murthy

Thermal conductivity of doped silicon thin films

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

ME 595M J.Murthy

Thin layer si thermal conductivity

Thin Layer Si Thermal Conductivity

Specularity Factor p=0.6

ME 595M J.Murthy

Thermal modeling of soi fet

Thermal Modeling of SOI FET


Heat generation region (100nmx10nm)

72 nm

315 nm


1633 nm


  • Heat source assumed known at 6x1017 W/m3 in heat generation region

  • Lower boundaries at 300K

  • Top boundary diffuse reflector

  • BTE in Si layer

  • Fourier in SiO2 region

  • Interface energy balance

ME 595M J.Murthy

Temperature prediction full dispersion model

Temperature Prediction -- Full Dispersion Model

Tmax =393.1 K

  • = 7.2 ps for optical to acoustic modes

ME 595M J.Murthy

Mode temperatures

Mode Temperatures

Optical mode

Low frequency LA mode

ME 595M J.Murthy

Maximum temperature comparison

Maximum Temperature Comparison

ME 595M J.Murthy



  • In this lecture, we considered two 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

  • Current models still employ temperature-like concepts not in keeping with non-equilibrium transport

  • Newer models are being developed which do not employ relaxation time approximations, and admit direct computation of the full scattering term

ME 595M J.Murthy

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