Ultrafast Laser-Induced Elastodynamics in Single Crystalline Silicon Part I: Model Formulation
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The various responses of a silicon wafer excited by a femtosecond pulsed laser are investigated. A multi-time scale axisymmetric model that governs the transport dynamics in silicon is presented based on the relaxation-time approximation of the Boltzmann equation. Temperature-dependent multi-phonons, free-carrier absorptions, and the recombination and impact ionization processes are considered using a set of balance equations. The mechanical response of the lattice is described by momentum equations. To solve the model of 17 coupled time-dependent partial differential equations without having to be concerned with non-physical oscillations in the solution, an implicit finite difference scheme on a staggered grid is developed. The staggered finite difference scheme allows velocities and first-order spatial derivative terms to be calculated at locations midway between two consecutive grid points, and shear stresses to be evaluated at the center of each element. A multi-time-scale approach involving the use of varying time steps ranging from 5fs to 5ps is implemented to successfully obtain time integration results up to 10ns.
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