Modeling of high energy ion implantation based on splitting of the Boltzmann transport equation

For the straight-ahead form of the Boltzmann transport equation, describing high energy heavy ion transport in matter, the splitting in the Boltzmann–Fokker–Planck equation is considered. The splitting consists of decomposition of differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. An approximation of the scattering integral with singular differential cross-section by the Fokker–Planck operator of second order, including the energy straggling term, is considered. For this form of the Fokker–Planck equation boundary conditions, preserving the number of particles in flux, and the energy dissipated in target are suggested. An effective algorithm for numerical integration of the Boltzmann–Fokker–Planck equation, which preserves the number of particles in flux, and the energy deposited in target is developed. The convergence of the implantation profile, nuclear and electronic energy deposition, calculated from the Boltzmann–Fokker–Planck equation, to the respective exact distributions, calculated from the Monte-Carlo method, was investigated for various values of the splitting parameter for the different ion-target mass ratios and for a large energy interval. For the universal potential, it is shown the existence of an optimal value of splitting parameter, at which solutions of the Boltzmann–Fokker–Planck equation accurately approximate solutions of the exact equation with minimal computational efforts.