A study of processed electronic materials containing inhomogeneous refractive index profiles

Abstract

Processing of electronic materials often produces graded refractive index profiles near the surface, as for instance in ion-implanted and diffused semiconductors. Analysis of these materials by means of optical reflectance and transmittance measurements, simulations, and least-squares fits is prevalent, but cumbersome. In order to facilitate the nondestructive characterization of materials containing inhomogeneous layers, we developed new theoretical solutions in integral form for the reflectance and transmittance. Two special cases are emphasized in this paper: a buried Gaussian refractive index profile, and a transition region described by a half Gaussian. The analytical study of Gaussian profiles provides us with new methods for the direct estimation of the average depth x_p and the standard deviation σ from optical reflectance data. Our estimates of these parameters, derived from experimental reflectance data of ion-implanted silicon, germanium, and GaAs, agree well with the values estimated by means of elaborate curve-fitting procedures. Simulations performed by means of the well-known matrix method and our theory, not only show excellent agreement but also prove that our method is very time efficient. Therefore, this technique could be used advantageously as a diagnostic tool as well as in process controllers in the semiconductor manufacturing industry.