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ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 7: The Gray Phonon BTE

ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 7: The Gray Phonon BTE. J. Murthy Purdue University. Relaxation Time Approximation . The BTE in the relaxation time approximation is

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ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 7: The Gray Phonon BTE

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  1. ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 7: The Gray Phonon BTE J. Murthy Purdue University ME 595M J.Murthy

  2. Relaxation Time Approximation • The BTE in the relaxation time approximation is • Recall units on f: number of phonons per unit volume per unit solid angle per unit wave number interval for each polarization • s is the direction of wave propagation (unit vector in k direction under isotropic assumption) ME 595M J.Murthy

  3. Definition of s z s   y x ME 595M J.Murthy

  4. Energy Form • Energy form of BTE ME 595M J.Murthy

  5. Phonon Dispersion Curves Frequency vs. reduced wave number in (100) direction for silicon ME 595M J.Murthy

  6. Energy Definitions ME 595M J.Murthy

  7. Temperature and Energy ME 595M J.Murthy

  8. Gray BTE in Energy Form ME 595M J.Murthy

  9. Diffuse (Thick) Limit of Gray BTE Gray BTE also gives the Fourier limit ME 595M J.Murthy

  10. Discussion • Gray BTE describes “average” phonon behavior • Does not distinguish between the different velocities, specific heats and scattering rates of phonons with different polarizations and wave vectors • Does capture ballistic effects unlike Fourier conduction. • Model has two parameters: vg and eff • We know that for bulk (i.e. in the thick limit): • We know k at temperature of interest. Choose vg.from knowledge of which phonon groups are most active at that temperature. Then find eff from above. • Which phonon group velocity to pick? Choice is a bit arbitrary. ME 595M J.Murthy

  11. Discussion (cont’d) • One way to estimate group velocity: • Choose longitudinal phonon velocity at the dominant frequency (why longitudinal?) • The model can capture broad bulk scattering and boundary scattering behaviors and is good for thermal conductivity modeling • Cannot accurately model situations where there are large departures from equilibrium, eg. high energy FETs. Here the different phonon groups behave very differently from each other and cannot be averaged in this way. ME 595M J.Murthy

  12. Boundary Conditions • Thermalizing boundaries • BTE region bounded by regions in which equilibrium obtains. • Temperature known and meaningful in these regions • Symmetry or specular boundaries • Phonons undergo mirror reflections • Energy loss at specular bc? • Resistance at specular bc? • Diffusely reflecting boundaries • Phonons reflected diffusely • Energy loss at diffuse bc? • Resistance at diffuse bc? • Partially specular • Interface between BTE/Fourier regions ME 595M J.Murthy

  13. Thermalizing Boundaries s Tb n • Computational domain is bounded by reservoirs in thermal equilibrium with known temperature Tb. • For directions incoming to the domain • For directions outgoing to the domain s.n>0 no boundary condition is required (why?) ME 595M J.Murthy

  14. Specular Boundaries s sr • Consider incoming direction s to domain • Need boundary value of e” for all incoming directions • Do not need boundary condition for outgoing directions to domain ME 595M J.Murthy

  15. Diffuse Boundaries n s • Consider incoming direction s to domain • Need boundary value of e” for all incoming directions • Average value of e” incoming to boundary is: • Set e” in all directions incoming to domain equal to the average incoming value • For all directions outgoing to domain, no boundary condition is needed ME 595M J.Murthy

  16. Partially Specular Boundaries • Frequently used to model interfaces whose properties are not known • Define specular fraction p. • A fraction p of energy incoming to boundary is reflected specularly, and (1-p) is reflected diffusely. • For directions s incoming to domain, set boundary value of e” as: • Again, no boundary condition is required for directions outgoing to the domain ME 595M J.Murthy

  17. BTE/Fourier Interfaces incoming reflected BTE emitted n Fourier conducted • An energy balance is required to determine the interface temperature. • For diffuse reflection: • Need e”interface. Assume interface is in equilibrium : ME 595M J.Murthy

  18. Conclusions • In this lecture, we developed the gray form of the BTE in the relaxation time approximation by summing the energies of all phonon modes • We described different boundary conditions that might apply • In the next lecture, we will develop a numerical technique that could be used to solve this type of equation. ME 595M J.Murthy

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