1. ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 10:Higher-Order BTE Models J. Murthy Purdue University

2. ME 595M J.Murthy2 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

3. ME 595M J.Murthy3 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.

4. 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

5. 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

6. ME 595M J.Murthy6 Lattice Temperature

7. 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

8. ME 595M J.Murthy8 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.

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

10. 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

11. 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

12. ME 595M J.Murthy12 Model Attributes Satisfies energy conservation In the acoustically thick limit, the model can be shown to satisfy Fourier heat diffusion equation Thermal conductivity

13. ME 595M J.Murthy13 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

14. ME 595M J.Murthy14 Silicon Bulk Thermal Conductivity Full-Dispersion Model

15. ME 595M J.Murthy15 Full Scattering Model Elastic ScatteringInelastic Scattering Valid only for phonons satisfying conservation rules Complicated, non-linear Klemens, (1958)

16. ME 595M J.Murthy16 N and U Processes 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 k1k1 k2k2 k3k3 G k’ 3 k1k1 k2k2 k3k3 N processes change f and affect U processes indirectly

17. ME 595M J.Murthy17 General Computation Procedure for Three-phonon Scattering Rates 12 unknowns 7 equations Set 5, determine 7  Specify K (K x, K y, K z ) and direction of K ’ (K ’ x, K ’ y ) Bisection algorithm developed to find all sets of 3-phonon interactions Three dispersion relations for the three wave vectors One energy conservation equation Three components of momentum conservation equation 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.

18. ME 595M J.Murthy18 Thermal Conductivity of Bulk Silicon Experimental data from Holland (1963) 2-10K, boundary scattering dominant; 20-100K, impurity scattering important, as well as N and U processes; Above 300K, U processes dominant.

19. ME 595M J.Murthy19 Thermal Conductivity of Undoped Silicon Films Experimental data from Ju and Goodson (1999), and Asheghi et al. (1998, 2002) Specularity Parameter p=0.4

20. ME 595M J.Murthy20 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