Predictions of Radiative Properties of Patterned Silicon Wafers by Solving Maxwell’s Equations in the Time Domain

A rigorous electromagnetic model is developed to predict the radiative properties of patterned silicon wafers. For nonplanar structures with 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 finite difference time-domain (FDTD) method. In the die area, only one period of the structure is modeled due to its periodicity in geometry. To truncate a computational domain, both the Mur condition and perfectly matched layer (PML) technique are available to absorb outgoing waves. With the steady state time-harmonic electromagnetic field known, the Poynting vector is used to calculate the radiative properties. Due to its importance, the reflection error is checked at first for two absorbing boundary conditions. As expected, the PML technique yields much lower errors than the Mur condition and it is thus used in this study. To validate the present model, radiative interactions with a planar structure and a nonplanar structure are investigated, and predicted reflectivities are found to match available other solutions very well. To demonstrate the importance of the present study, a patterned wafer consisting of periphery and die area is also investigated. While the thin film theory is accurate for the wafer periphery, the rigorously electromagnetic model described in this study is found to be necessary to accurately predict the radiative properties in the die area.