Rigorous Electromagnetic Modeling of Radiative Interactions with Microstructures Using the Finite Volume Time-Domain Method

Abstract

A rigorous electromagnetic model is developed to predict the radiative properties of patterned silicon wafers. For nonplanar structures with a characteristic length close to the wavelength of incident radiation, Maxwell's equations must be used to describe the associated radiative interaction and they are solved by the unstructured finite volume time-domain (FVTD) method. The basic idea of the FVTD method is to cast the two Maxwell curl equations in a conservative form, and then treat the six scalar components of the electromagnetic fields as conserved quantities via a finite volume approach. In the die area, only one period of the structure is modeled due to its periodicity in geometry. To truncate a computational domain in an open space, the Mur boundary condition is applied to absorb outgoing waves. With the steady state time-harmonic electromagnetic fields known, the Poynting vector is used to calculate the radiative properties. To validate the present model, a wave scattering problem from a cylinder is first considered and the predicted results are found to be essentially identical to the analytical solution. After that, radiative interactions with a nonplanar structure and a patterned wafer consisting of the periphery and die area are investigated, and predicted reflectivities and absorptivities are found to match other available solutions very well, indicating that the present finite volume approach in the time domain is accurate to predict radiative interaction with microstructures.