Numerical ellipsometry: A method for selecting a near-minimal infrared measurement set for β-gallium oxide

The objective of this work is to develop a method for defining, a priori, a set of minimum ellipsometry measurements that provide for a near-minimal data set (measurement set) sufficient for an analysis of optically anisotropic crystals with monoclinic symmetry with minimal prior knowledge. Example measurements are obtained by reflection from two differently oriented smooth, flat, anisotropic, monoclinic β-Ga₂O₃ crystals. A measurement may consist of any set of common type ellipsometry data such as selected sets of normalized Jones matrix elements or selected sets of normalized Mueller matrix elements measured at selected angles of incidence and/or rotation of the sample, also know as table rotation. The only prior knowledge used here is the crystallographic surface orientation. The four complex-valued permittivity parameters of the monoclinic symmetry crystal are then the object of interest and the only remaining unknown parameters. The motif for our investigation is the desire to better understand how to reduce measurement time for the otherwise extensive data acquisition considered necessary thus far to fully characterize low-symmetry anisotropic materials in substrates and semiconductor heterostructures. The near-minimal measurement set introduced here is then obtained by selection from a prior, larger data set. The larger data set thus far largely overdetermines the amount of necessary information. The first criterion is that there be sufficient intensity of the reflected light considering the p and s polarized reflections produced by p and s polarized incident light. The second criterion for inclusion is that the permittivity tensor is sensitive to measurement, that is, a significant change in a measurement set results in a significant change in the solution for permittivity. Finally, the near-minimal measurement set must result in solvable sets of equations. This is examined by computing the Jacobian of the system of equations for various sets of measurements in order to only keep measurements for which the condition of the Jacobian falls below the threshold for usability. We find that sets containing four measurements of on-diagonal Jones matrix elements alone, obtained across a wide spread of table rotations, provide most sensitive and intrinsically sufficient information to solve for the permittivity values. In summary, the scheme consists of determining the following measurement conditions: (1) reflections of high intensity to enable accurate measurements. (2) sensitivity to the unknown parameters, and (3) a solvable set of equations.