ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 10:Higher-Order BTE ModelsBTE ModelsSemi-Gray BTEPropagating Mode EquationsReservoir Mode EquationDiscussionFull-Dispersion BTEPhonon BandsOptical Mode BTEAcoustic Mode BTEModel AttributesProperties of Full-Dispersion ModelSilicon Bulk Thermal ConductivityFull Scattering ModelN and U ProcessesGeneral Computation Procedure for Three-phonon Scattering RatesThermal Conductivity of Bulk SiliconThermal Conductivity of Undoped Silicon FilmsConclusionsME 595M J.Murthy 1ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 10:Higher-Order BTE ModelsJ. MurthyPurdue UniversityME 595M J.Murthy 2BTE Models•Gray BTE drawbacksCannot distinguish between different phonon polarizationsIsotropicRelaxation time approximation does not allow direct energy transfers between different frequencies even if “non-gray” approach were takenVery simple relaxation time model•Higher-order BTE modelsTry to resolve phonon dispersion and polarization using “bands”But finer granularity requires more information about scattering ratesVarious approximations in finding these rates•Will look atSemi-gray modelsFull dispersion modelFull scattering modelME 595M J.Murthy 3Semi-Gray BTE•This model is sometimes called the two-fluid model (Armstrong, 1981; Ju, 1999).•Idea is to divide phonons into two groups“Reservoir mode” phonons do not move; capture capacitative effects“Propagation mode” phonons have non-zero group velocity and capture transport effects. Are primarily responsible for thermal conductivity.•Model involves a single equation for reservoir mode “temperature” with no angular dependence•Propogation mode involves a set of BTEs for the different directions, like gray BTE•Reservoir and propagation modes coupled through energy exchange termsArmstrong, B.H., 1981, "Two-Fluid Theory of Thermal Conductivity of Dielectric Crystals", Physical Review B, 23(2), pp. 883-899.Ju, Y.S., 1999, "Microscale Heat Conduction in Integrated Circuits and Their Constituent Films", Ph.D. thesis, Department of Mechanical Engineering, Stanford University.ME 595M J.Murthy 4Propagating Mode Equations''''P''1( )4( )L ref ppp pC T T eev etpt- -�+�� =�s''P4( )P ref pC T T e dp- = W�Propagating model scatters to a bath at lattice temperature TL with relaxation time “Temperature” of propagating mode, TP, is a measure of propagating mode energy in all directions togetherCP is specific heat of propagating mode phononsME 595M J.Murthy 5Reservoir Mode Equation•Note absence of velocity term•No angular dependence – equation is for total energy of reservoir mode•TR, the reservoir mode “temperature” is a measure of reservoir mode energy•CR is the specific heat of reservoir mode phonons•Reservoir mode also scatters to a bath at TLwith relaxation time •The term qvol is an energy source per unit volume – can be used to model electron-phonon scattering ( ) ( ) ( )R ref R L ref R R refR volT T C T T C T TC qt t� - - - -= +�ME 595M J.Murthy 6Discussion•Model contains two unknown constants: vg and •Can show that in the thick limit, the model satisfies:•Choose vg as before; find  to satisfy bulk k.•Which modes constitute reservoir and propagating modes?Perhaps put longitudinal acoustic phonons in propagating mode ? Transverse acoustic and optical phonons put in reservoir mode ?Choice determines how big  comes out•Main flaw is that  comes out very large to satisfy bulk kCan be an order-of-magnitude larger than optical-to-acoustic relaxation times•In FET simulation, optical-to acoustic relaxation time determines hot spot temperature•Need more detailed description of scattering rates( )21 with 3LP R L vol P gTC C k T q k C vtt�+ =��� + =�ME 595M J.Murthy 7Full-Dispersion BTE•Details in Narumanchi et al (2004,2005).•Objective is to include more granularity in phonon representation.• Divide phonon spectrum and polarizations into “bands”. Each band has a set of BTE’s in all directions•Put all optical modes into a single “reservoir” mode with no velocity.•Model scattering terms to allow interactions between frequencies. Ensure Fourier limit is recovered by proper modeling•Model relaxation times for all these scattering interactions based on perturbation theory (Han and Klemens,1983)•Model assumes isotropy, using [100] direction dispersion curves in all directionsNarumanchi, S.V.J., Murthy, J.Y., and Amon, C.H.; Sub-Micron Heat Transport Model in Silicon Accounting for Phonon Dispersion and Polarization; ASME Journal of Heat Transfer, Vol. 126, pp. 946—955, 2004.Narumanchi, S.V.J., Murthy, J.Y., and Amon, C.H.; Comparison of Different Phonon Transport Models in Predicting Heat Conduction in Sub-Micron Silicon-On-Insulator Transistors; ASME Journal of Heat Transfer, 2005 (in press). Han, Y.-J. and P.G. Klemens, Anharmonic Thermal Resistivity of Dielectric Crystals at Low Temperatures. Physical Review B, 1983. 48: p. 6033-6042.ME 595M J.Murthy 8Phonon BandsEach band characterized by its group velocity, specific heat and “temperature”Acoustic bandsOptical bandME 595M J.Murthy 9Optical Mode BTE0et�=�10 01ojrefTNbandsojjTC dT eg-=� �� �� �� �-� �� �� �� ����No ballistic term – no transportEnergy exchange due to scattering with jth acoustic modeElectron-phonon energy sourceoj is the inverse relaxation time for energy exchange between the optical band and the jth acoustic bandToj is a “bath” temperature shared by the optical and j bands. In the absence of other terms, this is the common temperature achieved by both bands at equilibriumvolq+ME 595M J.Murthy 10Acoustic Mode BTE01( ) ( )14ijrefii i i i iiTNbandsi i ijjTj iev e e etC dT eggp=������ ��+�� = - +�� �� �� ���� �-� �� �� �� ����sBallistic termScattering to same bandEnergy exchange with other bandsij is the inverse relaxation time for energy exchange between bands i and j Tij is a “bath” temperature shared by the i and j bands. In the absence of other terms, this is the common temperature achieved by both bands at equilibriumME 595M J.Murthy 11Model Attributes•Satisfies energy conservation•In the acoustically thick limit, the model can be shown to satisfy( )Ltotal L volTC K T qt�=�� � +�2111(