1. 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

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)
ME 595M J.Murthy

3. Definition of s x y z s  
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
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

14. 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

15. 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

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
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

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