Second Harmonic Generation in Lithiated Silicon Nanowires: Derivations and Computational Methods

Abstract

This research will examine the computational methods to calculate the nonlinear optical process of second harmonic generation (SHG) that will be hypothesized to be present during lithium ion insertion into silicon nanowires. First it will be determined whether the medium in which SHG is conveyed is non-centrosymmetric or whether the medium is inversion symmetric where SHG as a part of the second-order nonlinear optical phenomenon does not exist. It will be demonstrated that the main interaction that determines SHG is multiphoton absorption on lithium ions. The quantum harmonic oscillator (QHO) is used as the background that generates coherent states for electrons and photons that transverse the length of the silicon nanowire. The matrix elements of the Hamiltonian which represents the energy of the system will be used to calculate the probability density of second-order nonlinear optical interactions which includes collectively SHG, sum-frequency generation (SFG) and difference-frequency generation (DFG). As a result it will be seen that at varies concentrations of lithium ions (Li+) within the crystallized silicon (c-Si) matrix the second-order nonlinear optical process has probabilities substantial enough to create second harmonic generation that could possibly be used for such applications as second harmonic imaging microscopy.