J. Murthy Purdue University ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 8: The Gray Phonon BTE J. Murthy Purdue University ME 595M J.Murthy

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) ME 595M J.Murthy

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

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

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

Energy Definitions ME 595M J.Murthy

Temperature and Energy ME 595M J.Murthy

Gray BTE in Energy Form ME 595M J.Murthy

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

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

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

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

Thermalizing Boundaries 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 ( ) no boundary condition is required (why?) s Tb n ME 595M J.Murthy

Specular Boundaries 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 s sr ME 595M J.Murthy

Diffuse Boundaries 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 n s ME 595M J.Murthy

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

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

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