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

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 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**• 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**• 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**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**• 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**ME 595M J.Murthy**Discussion**• 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**• 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**Optical band Acoustic bands Each band characterized by its group velocity, specific heat and “temperature” ME 595M J.Murthy**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**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**• 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**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**Full-Dispersion Model ME 595M J.Murthy**Full Scattering Model**Elastic Scattering Inelastic Scattering Klemens, (1958) Valid only for phonons satisfying conservation rules Complicated, non-linear ME 595M J.Murthy**N and U Processes**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**• 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**Specularity Parameter p=0.4 Experimental data from Ju and Goodson (1999), and Asheghi et al. (1998, 2002) ME 595M J.Murthy**Conclusions**• 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