J. Murthy Purdue University

1. ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 6: Introduction to the Phonon Boltzmann Transport Equation J. Murthy Purdue University ME 595M J.Murthy

2. Introduction to BTE • Consider phonons as particles with energy and momentum • This view is useful if the wave-like behavior of phonons can be ignored. • No phase coherence effects - no interference, diffraction… • Can still capture propagation, reflection, transmission indirectly • Phonon distribution function f(r,t,k) for each polarization p is the number of phonons at position r at time t with wave vector k and polarization p per unit solid angle per unit wavenumber interval per unit volume • Boltzmann transport equation tracks the change in f(r,t,k) in domain ME 595M J.Murthy

3. Equilibrium Distribution • At equilibrium, distribution function is Planck: • Note that equilibrium distribution function is independent of direction, and requires a definition of “temperature” ME 595M J.Murthy

4. BTE Derivation • Consider f(r,t, k) = number of particles in drd3k • Recall d3k = dk2dk = sinddk2dk • Recall that dr =dx dy dz ME 595M J.Murthy

5. General Behavior of BTE • BTE in the absence of collisions: • This is simply the linear wave equation • The phonon distribution function would propagate with velocity vg in the direction vg. Group velocity vg and k are parallel under isotropic crystal assumption • Collisions change the direction of propagation and may also change k if the collision is inelastic (by changing the frequency) How would this equation behave? ME 595M J.Murthy

6. Scattering • Scattering may occur through a variety of mechanisms • Inelastic processes • Cause changes of frequency (energy) • Called “anharmonic” interactions • Example: Normal and Umklapp processes – interactions with other phonons • Scattering on other carriers • Elastic processes • Scattering on grain boundaries, impurities and isotopes • Boundary scattering ME 595M J.Murthy

7. N and U Processes Reciprocal wave vector • Determine thermal conductivity in bulk solids • These processes are 3-phonon collisions • Must satisfy energy and momentum conservation rules Energy conservation N processes U processes ME 595M J.Murthy

8. N and U Processes k2 k1 k3 k3 k’3 k1 k2 G • 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 N processes change f and affect U processes indirectly ME 595M J.Murthy

9. N and U Scattering Expressions • For a process k + k’ = k” +G or k + k’ = k” the scattering term has the form (Klemens,1958): • Only non-zero for processes that satisfy energy and momentum conservation rules • Notice that scattering rate depends on the non-equilibrium distribution function f (not equilibrium distribution funciton f0) • It is in general, a non-linear function ME 595M J.Murthy

10. Relaxation Time Approximation Kronecker Delta Delta function • Assume small departure from equilibrium for f; interacting phonons assumed at equilibrium • Invoke • Possible to show that Single mode relaxation time ME 595M J.Murthy

11. Relaxation Time Approximation (Cont’d) Define single mode relaxation time Thus, U and N scattering terms in relaxation time approximation have the form ME 595M J.Murthy

12. Relaxation Time Approximation (Cont’d) • Other scattering mechanisms (impurity, isotope…) may also be written approximately in the relaxation time form • Thus, the BTE becomes • Why is it called the relaxation time approximation? • Note that f0 is independent of direction, but depends on  (same as f) • So this form is incapable of directly transferring energy across frequencies ME 595M J.Murthy

13. Non-Dimensionalized BTE Symmetry f1 f2 L Symmetry Acoustic thickness: • Say we’re solving the BTE in a rectangular domain • Non-dimensionalize using ME 595M J.Murthy

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

15. Diffuse (Thick) Limit ME 595M J.Murthy

16. Energy Conservation • Energy conservation dictates that For small departures from equilibrium, we are guaranteed that the BTE will yield the Fourier conduction equation for acoustically thick problems ME 595M J.Murthy

17. Conclusions • We derived the Boltzmann transport equation for the distribution function • We saw that f would propagate from the boundary into the interior along the direction k but for scattering • Scattering due to U processes, impurities and boundaries turns the phonon back, causing resistance to energy transfer from one boundary to the other • We saw that the scattering term in the small-perturbation limit yields the relaxation time approximation • In the thick limit, the relaxation time form yields the Fourier conduction equation. ME 595M J.Murthy