Statistical Phonon Transport Model of Thermal Transport in Silicon

2009 ◽  
Vol 1229 ◽  
Author(s):  
Thomas W Brown ◽  
Edward Hensel
Keyword(s):  
Thermal Transport ◽  
Lattice Dynamics ◽  
Transport Model ◽  
Silicon Nanowire ◽  
Boltzmann Transport Equation ◽  
Momentum Conservation ◽  
Phonon Transport ◽  
Boltzmann Transport ◽  
Crystalline Materials ◽  
Statistical Framework

AbstractThermal transport in crystalline materials at various length scales can be modeled by the Boltzmann transport equation (BTE). A statistical phonon transport (SPT) model is presented that solves the BTE in a statistical framework that incorporates a unique state-based phonon transport methodology. Anisotropy of the first Brillouin zone (BZ) is captured by utilizing directionally-dependent dispersion curves obtained from lattice dynamics calculations. A rigorous implementation of phonon energy and pseudo-momentum conservation is implemented in the ballistic thermal transport regime for a homogeneous silicon nanowire with adiabatic specular boundary conditions.

Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Chunjian Ni ◽  
Jayathi Y. Murthy
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Thermal Transport ◽  
Relaxation Times ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Oxide Semiconductor ◽  
Boltzmann Transport ◽  
Scattering Model ◽  
Anisotropic Relaxation

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.

Validation of a Unified Nondiffusive-Diffusive Phonon Transport Model for Nanoscale Heat Transfer Simulations

2014 ◽  
Author(s):  
Ashok T. Ramu ◽  
Yanbao Ma
Keyword(s):  
Heat Transfer ◽  
Spatial Dependence ◽  
Hot Spots ◽  
Transport Model ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
High Fidelity ◽  
Boltzmann Transport ◽  
Fourier Law ◽  
Limiting Case

Heat transfer in the vicinity of nanoscale hot-spots is qualitatively different from that in the macroscale, which effect stems from the breakdown of Fourier law due to phonon nondiffusive transport. In this work, we validate a recently proposed alternative, high-fidelity phonon transport model, the unified nondiffusive-diffusive (UND) model, which takes into account the mixed ballistic-diffusive nature of heat transport, as well as reduces to the Fourier law as a limiting case. In the UND model, the nondiffusive phonons are treated using the Boltzmann transport equation, while the diffusive phonon gas is treated by the Fourier law. The numerical results of Maznev et al. for the geometry and spatial dependence of variables corresponding to the transient gratings experiments of Johnson et al. are used for validation of the model.

A Fast Hybrid Fourier–Boltzmann Transport Equation Solver for Nongray Phonon Transport

2012 ◽  
Vol 135 (1) ◽  
Author(s):  
James M. Loy ◽  
Jayathi Y. Murthy ◽  
Dhruv Singh
Keyword(s):  
Transport Equation ◽  
Thermal Transport ◽  
Boltzmann Transport Equation ◽  
Iterative Solution ◽  
Phonon Transport ◽  
Computational Effort ◽  
Boltzmann Transport ◽  
Knudsen Numbers ◽  
Sequential Solution ◽  
Hybrid Solver

Nongray phonon transport solvers based on the Boltzmann transport equation (BTE) are being increasingly employed to simulate submicron thermal transport in semiconductors and dielectrics. Typical sequential solution schemes encounter numerical difficulties because of the large spread in scattering rates. For frequency bands with very low Knudsen numbers, strong coupling between other BTE bands result in slow convergence of sequential solution procedures. This is due to the explicit treatment of the scattering kernel. In this paper, we present a hybrid BTE-Fourier model which addresses this issue. By establishing a phonon group cutoff Knc, phonon bands with low Knudsen numbers are solved using a modified Fourier equation which includes a scattering term as well as corrections to account for boundary temperature slip. Phonon bands with high Knudsen numbers are solved using the BTE. A low-memory iterative solution procedure employing a block-coupled solution of the modified Fourier equations and a sequential solution of BTEs is developed. The hybrid solver is shown to produce solutions well within 1% of an all-BTE solver (using Knc = 0.1), but with far less computational effort. Speedup factors between 2 and 200 are obtained for a range of steady-state heat transfer problems. The hybrid solver enables efficient and accurate simulation of thermal transport in semiconductors and dielectrics across the range of length scales from submicron to the macroscale.

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

2017 ◽  
Vol 139 (10) ◽  
Author(s):  
Ajit K. Vallabhaneni ◽  
Liang Chen ◽  
Man P. Gupta ◽  
Satish Kumar
Keyword(s):  
Gallium Nitride ◽  
Transport Equation ◽  
Thermal Transport ◽  
Hot Spot ◽  
Boltzmann Transport Equation ◽  
Significant Loss ◽  
Computational Time ◽  
Boltzmann Transport ◽  
Phonon Modes ◽  
Fourier Model

Several studies have validated that diffusive Fourier model is inadequate to model thermal transport at submicron length scales. Hence, Boltzmann transport equation (BTE) is being utilized to improve thermal predictions in electronic devices, where ballistic effects dominate. In this work, we investigated the steady-state thermal transport in a gallium nitride (GaN) film using the BTE. The phonon properties of GaN for BTE simulations are calculated from first principles—density functional theory (DFT). Despite parallelization, solving the BTE is quite expensive and requires significant computational resources. Here, we propose two methods to accelerate the process of solving the BTE without significant loss of accuracy in temperature prediction. The first one is to use the Fourier model away from the hot-spot in the device where ballistic effects can be neglected and then couple it with a BTE model for the region close to hot-spot. The second method is to accelerate the BTE model itself by using an adaptive model which is faster to solve as BTE for phonon modes with low Knudsen number is replaced with a Fourier like equation. Both these methods involve choosing a cutoff parameter based on the phonon mean free path (mfp). For a GaN-based device considered in the present work, the first method decreases the computational time by about 70%, whereas the adaptive method reduces it by 60% compared to the case where full BTE is solved across the entire domain. Using both the methods together reduces the overall computational time by more than 85%. The methods proposed here are general and can be used for any material. These approaches are quite valuable for multiscale thermal modeling in solving device level problems at a faster pace without a significant loss of accuracy.

FREQUENCY DEPENDENT PHONON TRANSPORT IN TWO-DIMENSIONAL SILICON AND DIAMOND THIN FILMS

2012 ◽  
Vol 26 (17) ◽  
pp. 1250104 ◽  
Author(s):  
B. S. YILBAS ◽  
S. BIN MANSOOR
Keyword(s):  
Thin Films ◽  
Boltzmann Transport Equation ◽  
Diamond Films ◽  
Equilibrium Temperature ◽  
Phonon Transport ◽  
Two Dimensional ◽  
Boltzmann Transport ◽  
Frequency Dependent ◽  
Aluminum Films ◽  
The Difference

Phonon transport in two-dimensional silicon and aluminum films is investigated. The frequency dependent solution of Boltzmann transport equation is obtained numerically to account for the acoustic and optical phonon branches. The influence of film size on equivalent equilibrium temperature distribution in silicon and aluminum films is presented. It is found that increasing film width influences phonon transport in the film; in which case, the difference between the equivalent equilibrium temperature due to silicon and diamond films becomes smaller for wider films than that of the thinner films.

Modeling of Localized Heating Effect in Sub-Micron Silicon Transistors

2003 ◽  
Author(s):  
Keivan Etessam-Yazdani ◽  
Sadegh M. Sadeghipour ◽  
Mehdi Asheghi
Keyword(s):  
Transport Equation ◽  
Hot Spot ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Heat Diffusion ◽  
Normal Operation ◽  
Heating Effect ◽  
Boltzmann Transport ◽  
Heat Diffusion Equation ◽  
Localized Heating

The performance and reliability of sub-micron semiconductor transistors demands accurate modeling of electron and phonon transport at nanoscales. The continued downscaling of the critical dimensions, introduces hotspots, inside transistors, with dimensions much smaller than phonon mean free path. This phenomenon, known as localized heating effect, results in a relatively high temperature at the hotspot that cannot be predicted using heat diffusion equation. While the contribution of the localized heating effect to the total device thermal resistance is significant during the normal operation of transistors, it has even greater implications for the thermoelectrical behavior of the device during an electrostatic discharge (ESD) event. The Boltzmann transport equation (BTE) can be used to capture the ballistic phonon transport in the vicinity of a hot spot but many of the existing solutions are limited to the one-dimensional and simple geometry configurations. We report our initial progress in solving the two dimensional Boltzmann transport equation for a hot spot in an infinite media (silicon) with constant temperature boundary condition and uniform heat generation configuration.

Thermal Contact Resistance of a Silicon Nanowire on a Substrate

2007 ◽  
Author(s):  
Dhruv Singh ◽  
Timothy S. Fisher ◽  
Jayathi Y. Murthy
Keyword(s):  
Contact Resistance ◽  
Thermal Transport ◽  
Thermal Contact Resistance ◽  
Silicon Nanowire ◽  
Thermal Contact ◽  
Thermal Conductance ◽  
Boltzmann Transport ◽  
Constriction Resistance ◽  
The Wire ◽  
Volume Method

Increased interest in determining the thermal contact resistance encountered in various nanoscale devices derives from its role as a critical bottleneck to thermal transport at submicron scales. In the present work, the contact resistance of one such configuration — a silicon nanowire on a Si substrate — has been studied. A model based on the phonon Boltzmann transport equation (BTE) in the solid and Fourier conduction in the surrounding gas is used, in conjunction with a previously published finite volume method. This approach enables the accurate computation and examination of the relative magnitudes of constriction resistance, air thermal resistances, and the bulk resistance of the wire on transverse heat transport. The results confirm that the effective resistance is dominated by constriction resistance even for low and moderate wire acoustic thicknesses and that the constriction resistance is well represented by ballistic-limit models. In the mesoscale regime when the constriction is partially diffusive, a simple addition of diffusive and ballistic resistances is shown to work reasonably well, in agreement with previously published estimates. For higher wire acoustic thicknesses, the effects of bulk scattering in the wire cannot be ignored and a simple additive formula is inadequate. The fluid surrounding the wire can act as a complementary parallel pathway for thermal transport; however, in the high gas-phase Knudsen regime, the constriction resistance remains the controlling factor for overall thermal conductance.

Heat Conduction in Nanostructured Materials Predicted by Phonon Bulk Mean Free Path Distribution

2015 ◽  
Vol 137 (7) ◽  
Author(s):  
Giuseppe Romano ◽  
Jeffrey C. Grossman
Keyword(s):  
Free Path ◽  
Nanostructured Materials ◽  
Thermal Transport ◽  
High Efficiency ◽  
Theoretical Models ◽  
Mean Free Path ◽  
Phonon Transport ◽  
Computational Effort ◽  
Boltzmann Transport ◽  
Computational Framework

We develop a computational framework, based on the Boltzmann transport equation (BTE), with the ability to compute thermal transport in nanostructured materials of any geometry using, as the only input, the bulk cumulative thermal conductivity. The main advantage of our method is twofold. First, while the scattering times and dispersion curves are unknown for most materials, the phonon mean free path (MFP) distribution can be directly obtained by experiments. As a consequence, a wider range of materials can be simulated than with the frequency-dependent (FD) approach. Second, when the MFP distribution is available from theoretical models, our approach allows one to include easily the material dispersion in the calculations without discretizing the phonon frequencies for all polarizations thereby reducing considerably computational effort. Furthermore, after deriving the ballistic and diffusive limits of our model, we develop a multiscale method that couples phonon transport across different scales, enabling efficient simulations of materials with wide phonon MFP distributions length. After validating our model against the FD approach, we apply the method to porous silicon membranes and find good agreement with experiments on mesoscale pores. By enabling the investigation of thermal transport in unexplored nanostructured materials, our method has the potential to advance high-efficiency thermoelectric devices.

Validation of an Enhanced Dispersion Algorithm for Use With the Statistical Phonon Transport Model

2020 ◽  
Author(s):  
Michael P. Medlar ◽  
Edward C. Hensel
Keyword(s):  
Computer Simulations ◽  
Wave Vector ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Transport Model ◽  
Phonon Transport ◽  
Dispersion Relations ◽  
High Fidelity ◽  
High Symmetry ◽  
Energy And Momentum

Abstract Computer simulations of quasi-particle based phonon transport in semiconductor materials rely upon numerical dispersion relations to identify and quantify the discrete energy and momentum states allowable subject to quantum constraints. The accuracy of such computer simulations is ultimately dependent upon the fidelity of the underlying dispersion relations. Dispersion relations have previously been computed using empirical fits of experimental data in high symmetry directions, lattice dynamics, and Density Function Theory (DFT) or Density Functional Perturbation Theory (DFPT) approaches. The current work presents high fidelity dispersion relations describing full anisotropy for all six phonon polarizations with an adjustable computational grid. The current approach builds upon the previously published Statistical Phonon Transport Model (SPTM), which employed a first nearest neighbor lattice dynamics approach for the dispersion calculation. This paper extends the lattice dynamics approach with the use of both first and second nearest neighbors interactions that are quantified using published interatomic force constants calculated from DFT. The First Brillouin Zone (FBZ) is segmented into eight octants of high symmetry, and discretized in wave vector space with a 14 by 14 by 14 grid. This results in 65,586 states of unique wave vector and frequency combinations. Dispersion calculations are performed at each of the six faces of the wave vector space volume elements in addition to the centroid, resulting in 460,992 solutions of the characteristic equations. For the given grid, on the order of 108 computations are required to compute the dispersion relations. The dispersion relations thus obtained are compared to experimental reports available for high symmetry axes. Full anisotropic results are presented for all six phonon polarizations across the range of allowable wave vector magnitude and frequency as a comprehensive model of allowable momentum and energy states. Results indicate excellent agreement to experiment in high symmetry directions for all six polarizations and illustrate an improvement as compared to the previous SPTM implementation. Dispersion relations based on the lattice dynamic model with first and second nearest neighbor atomic interactions relying upon DFT calculated inter-atomic force constants provides an accurate high fidelity energy and momentum model for use in phonon transport simulations.

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