1. ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 6: Introduction to the Phonon Boltzmann Transport Equation J. Murthy Purdue University

2. ME 595M J.Murthy2 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

3. ME 595M J.Murthy3 Equilibrium Distribution At equilibrium, distribution function is Planck: Note that equilibrium distribution function is independent of direction, and requires a definition of “temperature”

4. ME 595M J.Murthy4 BTE Derivation Consider f(r,t, k) = number of particles in drd 3 k Recall d 3 k = d  k 2 dk = sin  d  d  k 2 dk Recall that dr =dx dy dz

5. ME 595M J.Murthy5 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 v g in the direction v g. Group velocity v g 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?

6. ME 595M J.Murthy6 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

7. ME 595M J.Murthy7 N and U Processes Determine thermal conductivity in bulk solids These processes are 3-phonon collisions Must satisfy energy and momentum conservation rules N processes U processes Reciprocal wave vector Energy conservation

8. ME 595M J.Murthy8 N and U Processes 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 k1k1 k2k2 k3k3 G k’ 3 k1k1 k2k2 k3k3 N processes change f and affect U processes indirectly

9. ME 595M J.Murthy9 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 f 0 ) It is in general, a non-linear function

10. ME 595M J.Murthy10 Relaxation Time Approximation Assume small departure from equilibrium for f; interacting phonons assumed at equilibrium Invoke Possible to show that Delta function Kronecker Delta Single mode relaxation time

11. ME 595M J.Murthy11 Relaxation Time Approximation (Cont’d)  Define single mode relaxation time  Thus, U and N scattering terms in relaxation time approximation have the form

12. ME 595M J.Murthy12 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 f 0 is independent of direction, but depends on  (same as f) So this form is incapable of directly transferring energy across frequencies

13. ME 595M J.Murthy13 Non-Dimensionalized BTE Say we’re solving the BTE in a rectangular domain Non-dimensionalize using f1f1 f2f2 Symmetry L Acoustic thickness:

14. ME 595M J.Murthy14 Energy Form Energy form of BTE

15. ME 595M J.Murthy15 Diffuse (Thick) Limit

16. ME 595M J.Murthy16 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

17. ME 595M J.Murthy17 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.