Hierarchical Modeling of Heat Transfer in Silicon-Based Electronic Devices

Abstract

A hierarchical model of heat transfer for the thermal analysis of electronic devices is presented. The integration of participating scales (from nanoscale to macroscales) is achieved by (i) estimating the input parameters and thermal properties to solve the Boltzmann transport equation (BTE) for phonons using molecular dynamics (MD), including phonon relaxation times, dispersion relations, group velocities, and specific heat, (ii) applying quantum corrections to the MD results to make them suitable for the solution of BTE, and (iii) numerically solving the BTE in space and time subject to different boundary and initial conditions. We apply our hierarchical model to estimate the silicon out-of-plane thermal conductivity and the thermal response of an silicon on insulator (SOI) device subject to Joule heating. We have found that relative phonon contribution to the overall conductivity changes as the dimension of the domain is reduced as a result of phonon confinement. The observed reduction in the thermal conductivity is produced by the progressive transition of modes in the diffusive regime (as in the bulk) to transitional and ballistic regimes as the film thickness is decreased. In addition, we have found that relaxation time expressions for optical phonons are important to describe the transient response of SOI devices and that the characteristic transport regimes, determined with Holland and Klemens phonon models, differ significantly.

On the lattice Boltzmann method for phonon transport

“…Extending the Gray LBM to the Dispersion LBM is straightforward, provided the group velocity and relaxation time of each phonon mode are known. Details regarding the implementation of the Dispersion LBM can be found elsewhere [37,53].…”

“…For an ideal adiabatic boundary, every phonon which hits the boundary will be reflected back into the domain and the bins of the absorbed and transmitted phonons will be empty. An ideal reflecting boundary condition can be treated in two different ways, namely: diffuse and specular reflections [37,53]. The unknown outgoing distributions are based on the incoming distributions and are calculated differently in each of these two methods.…”

“…Their LBM for phonons was further developed in the works of Escobar et al [34][35][36] and Goicochea et al [37][38][39]. These latter works employed the D2Q9 lattice with multiple phonon modes.…”

Cross-plane phonon transport in thin films

“…Although previous works have used the LBM to model phonon transport in thin films, these studies assumed forms for the frequency dependence of phonon relaxation times that require fitting parameters [13][14][15][16][17][18] and/or were limited to isotropic material properties. 19 Our approach does not rely on the isotropic approximation or any fitting parameters. Methods other than the LBM, such as the equation of radiative phonon transport 20 and the discrete ordinate method, 21 have also been used to solve the BTE.…”

Quantifying Uncertainty in Multiscale Heat Conduction Calculations

“…In thermal analysis of microelectronics, for example, phonon relaxation times used in transistor heat trans fer calculations are computed using molecular dynamics simula tions [1][2][3][4][5][6][7][8]. In the analysis of multimaterial thermal transport problems, interface transmissivity and reflectivity may be com puted from atomistic simulations [9].…”