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navdeep-bte - BOLTZMANN SOLVER FOR PHONON TRANSPORT N.S....

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BOLTZMANN SOLVER FOR PHONON TRANSPORT N.S. Dhillon Department of Mechanical Engineering Purdue University West Lafayette, Indiana 47906 USA Email: ndhillon@purdue.edu ABSTRACT Boltzmann Transport Equation is solved numerically to model phonon transport in a sub- continuum domain in order to study heat transfer in thin film semiconductors. The phonon distribution function is modified to get an Energy equation from the Boltzmann Transport Equation. Gray form of the Energy equation is solved in the Relaxation time approximation to get the Phonon Energy Density distribution. The phonon group velocity and the relaxation times are obtained using other methods. Structured Finite Volume Method is used to discretize the Energy equation and a recursive solution procedure is used to solve it. Temperatures in the domain are obtained by assuming statistical equilibrium. The temperature profiles and heat fluxes for different acoustic thicknesses agree with theoretical radiation results by Heaslet and Warming. Silicon bulk thermal conductivity is reproduced under the acoustically thick limit. Boundary scattering and confinement effects are studied by working with specularity and confinement parameters. NOMENCLATURE f phonon distribution function g v phonon group velocity k r phonon wave vector eff τ effective relaxation time ω phonon frequency h modified Planck’s constant ' ' e phonon energy density () D phonon density of states k Boltzmann constant Thermal conductivity s r unit direction vector r r position vector t time o e angular averaged phonon energy density C specific heat ref T reference temperature ΔΩ control angle INTRODUCTION During the last 10 years heat conduction at the sub-micron level has received increasing attention. One of the reasons for this has been the continued and aggressive scaling of micro- electronic devices bringing into sharp focus thermal among other issues [1]. The problems related to self-heating in microelectronic devices have exacerbated for example in some of the new device designs like SOI(Silicon On Insulator) and others involving low conductivity semiconductors and dielectrics. The thermal management of these devices has thus become critical to device performance. It has been observed that at the sub-micron level the classical Fourier model for heat conduction is not only suspect but in fact gives grossly erroneous results. This is not particularly surprising given the fact that the Fourier relation is an empirical relation obtained from experimental observations at large scales rather than from any concrete physical principles. Thermal conductivity is not a material property and it can depend on the geometry and length scales of the sample [7]. Heat conduction at small scales is of critical importance in not one but many areas such as microelectronics, thin- films, superlattices, nanomaterials, short pulse laser heating, etc. [5]. Heat in solids is carried by electrons and phonons. Electrons are the predominant carriers in conductors while
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navdeep-bte - BOLTZMANN SOLVER FOR PHONON TRANSPORT N.S....

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