Over the last decade, interest in the simulation of micro-and nano-scale heat transfer has lead to the development of a variety of models and numerical methods for phonon transport in semiconductors and dielectrics. These models span direct simulation using molecular dynamics, a range of models of varying fidelity based on the Boltzmann transport equation, as well as simpler hyperbolic extensions to the classical Fourier heat conduction equation. The paper reviews the basics of phonon transport in crystals, available models for phonon transport, as well as numerical methods for solving the equations resulting from these models. Recommendations are made for future work.
Thermal transport in metal-oxide-semiconductor field effect transistors (MOSFETs) due to electron-phonon scattering is simulated using phonon generation rates obtained from an electron Monte Carlo device simulation. The device simulation accounts for a full band description of both electrons and phonons considering 22 types of electron-phonon scattering events. Detailed profiles of phonon emission/absorption rates in the physical and momentum spaces are generated and are used in a MOS-FET thermal transport simulation with a recently-developed anisotropic relaxation time model based on the Boltzmann transport equation (BTE). Comparisons with a Fourier conduction model reveal that the anisotropic heat conduction model predicts higher maximum temperatures because it accounts for the bottlenecks in phonon scattering pathways. Heat fluxes leaving the boundaries associated with different phonon polarizations and frequencies are also examined to reveal the main modes responsible for transport. It is found that though the majority of the heat generation is in the optical modes, the heat generated in the acoustic modes is not negligible. The modes primarily responsible for the transport of heat are found to be medium-to-high frequency acoustic phonon modes.
A sub-micron thermal transport model based on the phonon Boltzmann transport equation (BTE) is developed using anisotropic relaxation times. A previously-published model, the full-scattering model, developed by Wang, directly computes three-phonon scattering interactions by enforcing energy and momentum conservation. However, it is computationally very expensive because it requires the evaluation of millions of scattering interactions during the iterative numerical solution procedure. The anisotropic relaxation time model employs a single-mode relaxation time, but the relaxation time is derived from detailed consideration of three-phonon interactions satisfying conservation rules, and is a function of wave vector. The resulting model is significantly less expensive than the full-scattering model, but incorporates directional and dispersion behavior. A critical issue in the model development is the role of three-phonon normal (N) scattering processes. Following Callaway, the overall relaxation rate is modified to include the shift in the phonon distribution function due to N processes. The relaxation times so obtained are compared with the data extracted from equilibrium molecular dynamics simulations by Henry and Chen. The anisotropic relaxation time phonon BTE model is validated by comparing the predicted thermal conductivities of bulk silicon and silicon thin films with experimental measurements. The model is then used for simulating thermal transport in a silicon metal-oxide-semiconductor field effect transistor (MOSFET) and leads to results close to the full-scattering model, but uses much less computation time.
Thermal transport in metal-oxide-semiconductor field effect transistors (MOSFETs) due to electron-phonon scattering is simulated using phonon generation rates obtained from an electron Monte Carlo device simulation. The device simulation accounts for a full band description of both electrons and phonons considering 22 types of electron-phonon scattering events. Detailed profiles of phonon emission/absorption rates in the physical and momentum spaces are generated and are used in a MOSFET thermal transport simulation with a recently-developed anisotropic relaxation time model based on the Boltzmann transport equation (BTE). Comparisons with a Fourier conduction model reveal that the anisotropic heat conduction model predicts higher maximum temperatures because it accounts for the bottlenecks in phonon scattering pathways. Heat fluxes leaving the boundaries associated with different phonon polarizations and frequencies are also examined to reveal the main modes responsible for transport. It is found that though the majority of the heat generation is in the optical modes, the heat generated in the acoustic modes is not negligible. The modes primarily responsible for the transport of heat are found to be medium-to-high frequency acoustic phonon modes.
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