Unified analysis and mathematical representation of film-thickness behavior of film-substrate systems

Abstract

The ellipsometric function p of a film-substrate system is studied as the film thickness d is kept constant and the angle of incidence phi is changed. The generated constant-thickness contours (CTCs) are characterized by an introduced mathematical behavior indicator that represents a group of CTCs. The behavior of each group is developed and studied in the four planes phi-d, X, Z, and p, where X is the film-thickness exponential function and Z is a previously introduced intermediate plane. In the phi-d plane the film-thickness domain is identified and divided into a sequence of disconnected thickness subdomains (DTSs), depending on only N0 and N1, and their number depending on the range in which N0/N1 lies. The behavior of the CTCs in the successive planes X, Z, and p is then studied in each DTS, and the CTC's space is divided into disconnected subfamilies according to the behavior indicator. Equivalence classes that reduce the infinite number of subfamilies into a finite number are then introduced. The transformation from each plane to the next is studied with the origin of the Z plane mapped onto the point at infinity of the p plane, forming a singularity. A multiple-film-thickness inequality is derived to determine the unique solution of the film thickness. The type of reflection being internal or external at both ambient-film and film-substrate interfaces affects the analysis and is also considered. To conclude we introduce the design of polarization-preserving devices and a novel oscillating single-element ellipsometer to fully characterize zero film-substrate systems as examples of applying the knowledge developed here.

Single-instrument morphology: single-element polarizer (SEP) ellipsometers, reflecsometers, elliptometers, and reflectometers

“…Figure 1 shows the dependence on the two basic experimental parameters; the film thickness d and the angle of incidence 0 φ [1][2][3]. Note the uniqueness of the intersection point between any constant-angle-of-incidence contour (CAIC) and any constant-thickness contour (CTC) [3].…”

“…Fig. 2 The constant-angle-of-incidence contours (solid) and constant-thickness contours (dashed) for a Film-substrate system of N 0 = 1, N 1 = 1.46, and N 2 = 2.1316, at λ = 632.8 nm [2]. Note that the domain of the complex ρ -plane is the unit circle for the zero system, and an ellipsoid within the unit circle for the positive system, Figs.…”

“…The constant-angle-of-incidence contours (solid) and constant-thickness contours (dashed) for a Film-substrate system of N 0 = 1, N 1 = 1.46, and N 2 = 2, at λ = 632.8 nm[2].Proc. of SPIE Vol.…”

Single-angle-of-incidence single-element rotating-polarizer (Single SERP) ellipsometer for film-substrate systems

“…6 In addition, the negative and zero systems are both single-valued functions, where the positive system is a two-valued function. Therefore, for the negative .…”

“…Accordingly, inverting experimental data leads to a single correct value for the negative and zero systems, but leads to two possible mathematically-correct solutions where more information is needed to select the actual correct value, such as an added experimental set, for example. 6 For the zero system, the point of (0,1) in the complex ρ -plane represents multiple ( ϕ , d ) experimental sets, which would also present a challenge for the inversion of experimental results. The points on the unit circle of the ρ -plane all represent a special mathematical condition of ρ , where its magnitude is unity.…”