ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 11:Extensions and Modifications J. Murthy Purdue University

ME 595M J.Murthy2 Drawbacks 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.Murthy3 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 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.Murthy4 Propagating Mode Equations  Propagating model scatters to a bath at lattice temperature T L with relaxation time   “Temperature” of propagating mode, T P, is a measure of propagating mode energy in all directions together  C P is specific heat of propagating mode phonons

ME 595M J.Murthy5 Reservoir Mode Equation Note absence of velocity term No angular dependence – equation is for total energy of reservoir mode T R, the reservoir mode “temperature” is a measure of reservoir mode energy C R is the specific heat of reservoir mode phonons Reservoir mode also scatters to a bath at T L with relaxation time  The term q vol is an energy source per unit volume – can be used to model electron-phonon scattering

ME 595M J.Murthy6 Lattice Temperature

ME 595M J.Murthy7 Discussion Model contains two unknown constants: v g and  Can show that in the thick limit, the model satisfies: Choose v g 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.Murthy8 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, 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, : p

ME 595M J.Murthy9 Phonon Bands Optical band Acoustic bands Each band characterized by its group velocity, specific heat and “temperature ”

ME 595M J.Murthy10 Optical Mode BTE No ballistic term – no transport Energy exchange due to scattering with jth acoustic mode Electron- phonon energy source  oj is the inverse relaxation time for energy exchange between the optical band and the jth acoustic band T oj 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.Murthy11 Acoustic Mode BTE Ballistic term Scattering to same band Energy exchange with other bands  ij is the inverse relaxation time for energy exchange between bands i and j T ij 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.Murthy12 Lattice Temperature Lattice “temperature” is a measure of the energy in all acoustic and optical modes combined

ME 595M J.Murthy13 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.Murthy14 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.Murthy15 Silicon Bulk Thermal Conductivity Non-Gray Model

ME 595M J.Murthy16 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/m 3 considered Full-Dispersion Model

ME 595M J.Murthy17 Thin Layer Si Thermal Conductivity Specularity Factor p=0.6

ME 595M J.Murthy18 Thermal Modeling of SOI FET SiO nm 315 nm 72 nm Heat generation region (100nmx10nm) Si SiO 2 Heat source assumed known at 6x10 17 W/m 3 in heat generation region Lower boundaries at 300K Top boundary diffuse reflector BTE in Si layer Fourier in SiO 2 region Interface energy balance

ME 595M J.Murthy19 Temperature Prediction -- Full Dispersion Model T max =393.1 K  = 7.2 ps for optical to acoustic modes

ME 595M J.Murthy20 Low frequency LA mode Optical mode Mode Temperatures

ME 595M J.Murthy21 Maximum Temperature Comparison ModelHeat Gen. In O-Mode Heat Gen. In A-Modes Fourier320.7 K Gray326.4 K Semi-Gray504.9 K365.5 K Full- Dispersion K364.4 K

ME 595M J.Murthy22 Conclusions 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