Relationship Between Surface Morphology and Effective Medium Roughness

Abstract

The modeling of surface and interface roughness is a key issue in the interpretation of ellipsometric measurements. Materials properties are often extracted from ellipsometry measurements in an indirect way by modeling the optical response of the material. Since roughness is known to affect the scattering of light on an interface, how roughness is incorporated into these models can affect the outcome of the fitting procedure.

A. Terms of the Rayleigh-Rice Theory

Following the notation used in Ref. [14], the kernel functions in the Rayleigh-Rice theory are given by:

$$\begin{aligned} f_p(q_x, q_y)&= A_p +(b-c) \frac{B_{31p}q^2+B_{62p}(q_y^2 + bc)}{q^2+bc} \nonumber \\&\quad + q_x\frac{B_{61p}\left[q^2(b-c)^2+(q^2+bc)^2\right]-B_{32p}}{q^2+bc} \end{aligned}$$

(4.45)

$$\begin{aligned} f_s(q_x, q_y)&= A_s + B_{5s}\frac{(b-c)(q_x^2-bc)}{q^2+bc} \end{aligned}$$

(4.46)

where:

$$\begin{aligned} b&= \sqrt{n_0^2k_0^2-q^2} \end{aligned}$$

(4.47)

$$\begin{aligned} c&= \sqrt{n_1^2k_0^2-q^2} \end{aligned}$$

(4.48)

and

$$\begin{aligned} A_p&= k_0^2n_0n_1^2X/D \end{aligned}$$

(4.49)

$$\begin{aligned} B_{31p}&= k_0n_0^2n_1^2\sin \theta _0 W/D \end{aligned}$$

(4.50)

$$\begin{aligned} B_{62p}&= k_0n_0n_1\cos \theta _1 X/D \end{aligned}$$

(4.51)

$$\begin{aligned} B_{61p}&= n_0n_1\cos \theta _1 W/(Dk_0) \end{aligned}$$

(4.52)

$$\begin{aligned} B_{32p}&= k_0^3n_0^2n_1^2\sin \theta _0 X/D \end{aligned}$$

(4.53)

$$\begin{aligned} A_s&= -2k_0^2n_0n_1\cos \theta _0\cos \theta _1r_s \end{aligned}$$

(4.54)

$$\begin{aligned} B_{5s}&= -2k_0n_0\cos \theta _0 r_s \end{aligned}$$

(4.55)

with:

$$\begin{aligned} D&= (n_0\cos \theta _0+n_1\cos \theta _1)(n_0^2\sin ^2\theta _0+n_0n_1\cos \theta _0\cos \theta _1)\end{aligned}$$

(4.56)

$$\begin{aligned} X&= (n_0^2-n_1^2)\cos \theta _1t_p \end{aligned}$$

(4.57)

$$\begin{aligned} W&= (n_1/n_0^2-1/n_1)n_0 \sin \theta _0t_p \end{aligned}$$

(4.58)

Here, $\theta _0$ is the incidence angle, $k_0$ is the light wavenumber, $n_0$ and $n_1$ are the air and the material complex refractive index, and $r_s$, $r_p$, $t_p$, $t_s$ are the Fresnel reflection and transmission coefficients for the flat interface.