ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 6: Introduction to the Phonon Boltzmann Transport EquationIntroduction to BTEEquilibrium DistributionBTE DerivationGeneral Behavior of BTEScatteringN and U ProcessesSlide 8N and U Scattering ExpressionsRelaxation Time ApproximationRelaxation Time Approximation (Cont’d)Slide 12Non-Dimensionalized BTEEnergy FormDiffuse (Thick) LimitEnergy ConservationConclusionsME 595M J.Murthy 1ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 6: Introduction to the Phonon Boltzmann Transport EquationJ. MurthyPurdue UniversityME 595M J.Murthy 2Introduction 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 pis 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 domainwhkhME 595M J.Murthy 3Equilibrium Distribution•At equilibrium, distribution function is Planck:•Note that equilibrium distribution function is independent of direction, and requires a definition of “temperature”01exp 1fkTw=� �-� �� �hME 595M J.Murthy 4BTE 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( , , ) ( , , ) (1)collisionsdfdf f d t dt d f t dtdt� �= + + + - =� �� �r r k k r kIf acceleration is zero (4)gcollisionsf dfft dt�� � � �+ �� =� � � ��� � � �v (2)r kcollisionsf dfdt d f d f dtt dt�� � � �+ �� + �� =� � � ��� � � �r k (3)g r kcollisionsf d dff ft dt dt�� � � �+ �� + �� =� � � ��� � � �kvME 595M J.Murthy 5General 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)0gfft�� �+ �� =� ��� �vHow would this equation behave?ME 595M J.Murthy 6Scattering•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 scatteringME 595M J.Murthy 7N and U Processes•Determine thermal conductivity in bulk solids •These processes are 3-phonon collisions•Must satisfy energy and momentum conservation rulesN processesU processesReciprocal wave vectorEnergy conservationME 595M J.Murthy 8N 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 aroundk1k2k3Gk’3k1k2k3N processes change f and affect U processes indirectlyME 595M J.Murthy 9N 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: Gruneisen constantM: atomic massG: number of atoms per unit cellv: sound velocitygME 595M J.Murthy 10Relaxation Time Approximation•Assume small departure from equilibrium for f; interacting phonons assumed at equilibrium•Invoke•Possible to show that 0'0 " "0 f ff f f fd= +�� �01 ; x= ; x +x =xexp( ) 1fx kTw� ��=-hDelta functionKronecker DeltaSingle mode relaxation timeME 595M J.Murthy 11Relaxation Time Approximation (Cont’d)Define single mode relaxation timeThus, U and N scattering terms in relaxation time approximation have the form0.scatUf f ft t� -� �=� ��� �ME 595M J.Murthy 12Relaxation 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 frequencies01 1 1 1effeff U I isotopef f fft tt t t t� -+ �� =�= + + +LgvME 595M J.Murthy 13Non-Dimensionalized BTE( )* * * *12 1** 0* ** ; x ; y ; twe obtaingg efftvf fx yff f L L Lf Lf f ft v t-= = = =-�+ �� = -�ggsvs = v•Say we’re solving the BTE in a rectangular domain•Non-dimensionalize using f1f2SymmetrySymmetryLg effLv tAcoustic thickness:ME 595M J.Murthy 14Energy Form•Energy form of BTE0 (1)gefff f fv ft t� -+ �� =�s( )" 0 ""Multiply by to obtain energy form: (2)geffDe e ev etw w www wt� -+ �� =�sh( )"Here is the phonon energy density=energy per unit volume per unit solid angle per unit frequency intervale D fww w=hME 595M J.Murthy 15Diffuse (Thick) Limit" 0 "" (2)geffe e ev etw w wwt� -+ �� =�s" 0" in the acoustically thick limit4e e Ce T T TT Tw w wwp� ��=ѻ�=�� �g2Multiply (2) by (v ) and integrate over 443 4geffCv Ttw w wppp t� -+ � =�sq qeff2Assume t>> . Hence drop :13g efftC v Tww wtt��=- �qq2Integrate over and all polarizations to get total heat flux:where13polarizationsg effpolarizationsd k Tk C v dwwwwt w=- �=����q = qME 595M J.Murthy 16Energy Conservation•Energy conservation dictates thatTC Str�=- �� +�qFor small departures from equilibrium, we are guaranteed that the BTE will yield the Fourier conduction equation for acoustically thick problemsBut - . Thusq k TTC k T Str= ��=��� +�2with13g effpolarizationsk C v dwt w=��ME 595M J.Murthy 17Conclusions•We derived the Boltzmann transport equation for the distribution function •We saw that f would propagate from