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

1. ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 11:Extensions and Modifications J. Murthy Purdue University ME 595M J.Murthy

2. 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.Murthy

3. 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

4. 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

5. 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

6. Lattice Temperature ME 595M J.Murthy

7. 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

8. 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

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

10. 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

11. 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

12. Lattice Temperature • Lattice “temperature” is a measure of the energy in all acoustic and optical modes combined ME 595M J.Murthy

13. 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

14. 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

15. Silicon Bulk Thermal Conductivity Non-Gray Model ME 595M J.Murthy

16. 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

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

18. Thermal Modeling of SOI FET Si Heat generation region (100nmx10nm) 72 nm 315 nm SiO2 1633 nm SiO2 • 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

19. Temperature Prediction -- Full Dispersion Model Tmax =393.1 K • = 7.2 ps for optical to acoustic modes ME 595M J.Murthy

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

21. Maximum Temperature Comparison ME 595M J.Murthy

22. 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 ME 595M J.Murthy