Abstract
A method for determining properties of a sample (12) by ellipsometry includes positioning the sample (12) in an ellipsometer (10) so that a surface normal (n) of a measurement region of the sample surface is tilted relative to a reference axis (z) of the ellipsometer (10) and measuring a Mueller matrix for the measurement region. The method then includes creating an equation system by equating the measured Mueller matrix and a matrix product formed of: a rotation matrix about an input rotation angle (γ); an isotropic Mueller matrix in normalized NCS form and a rotation matrix about an output rotation angle (−δ). The method then solves the equation system for the parameters representing the sample properties to be determined. The input rotation angle (γ) and the output rotation angle (—δ) are set as parameters independent of one another when setting up the equation system.
Claims
1. A method for determining properties of a sample (12) by ellipsometry, comprising the steps of: positioning the sample (12) in an ellipsometer (10) so that a surface normal (n) of a measurement region of the sample surface is tilted relative to a reference axis (z) of the ellipsometer (10), measuring a Mueller matrix for the measurement region, creating an equation system, which at least is not underdetermined, equating the measured Mueller matrix and a matrix product formed of a rotation matrix about an input rotation angle (γ), an isotropic Mueller matrix in normalized NCS form and a rotation matrix about an output rotation angle (−δ), solving the equation system for the parameters representing the sample properties to be determined, characterized in that when setting up the equation system, the input rotation angle (γ) and the output rotation angle (−δ) are set as parameters independent of one another.
2. The method of claim 1, wherein as part of solving the equation system, the Mueller parameters N, S, C and/or the ellipsometric transfer variables Δ and Ψ equivalent to these are explicitly calculated.
3. The method of claim 2, further comprising the step of: determining optical and/or mechanical sample properties from the calculated ellipsometric transfer variables Δ and Ψ by a numerical error minimization method by variation of parameters of a sample model.
4. The method of claim 3, wherein as part of the error minimization method the angle of incidence (φ) is used as an additional parameter to be varied.
5. The method of claim 1, wherein as part of solving the equation system, the input rotation angle (γ) and the output rotation angle (−δ) are calculated explicitly.
6. The method of claim 5, wherein the rotation angles (γ, −δ) determined by solving the equation system are used to calculate the alignment of the surface normal (n) of the sample surface in the measurement region relative to the reference axis (z) of the ellipsometer (10).
7. The method of claim 1, wherein the solving of the equation system is performed by a numerical error minimization method by variation of parameters of a sample model.
8. The method of claim 1, wherein the ellipsometer (10) is an imaging ellipsometer (10).
Description
BRIEF DESCRIPTION OF THE DRAWINGS
[0054] FIG. 1 is a diagram of the basic design of an ellipsometer,
[0055] FIG. 2 is a diagram illustrating a sample correctly aligned in the conventional manner,
[0056] FIG. 3 is a diagram illustrating a sample aligned in tilted manner,
[0057] FIG. 4 is a diagram illustrating the input rotation angle γ,
[0058] FIG. 5 is a diagram illustrating the output rotation angle δ according to approximation model 1.
[0059] FIG. 6 is a diagram illustrating the output rotation angle δ according to approximation model 2.
DETAILED DESCRIPTION
[0060] FIGS. 1 to 5 are intended to illustrate and check the geometric plausibility of variables relevant in the context of the present invention.
[0061] FIG. 1 shows the basic design of an ellipsometer 10 as it is known from prior art and can also be used within the scope of the present invention. A sample 12 with a surface normal n is illuminated by a light source 14 with an input beam k.sub.in (illumination beam). The incident input beam k.sub.in assumes an angle of incidence φ to the sample normal n. By means of a polarizer 18 and a downstream compensator 20, the incident input beam k.sub.in is given a defined, adjustable polarization state. After reflection from the surface of the sample 12, the light is fed to a detector 24 in the form of detection beam k.sub.out. The polarization state of the light has changed due to reflection from the sample surface in a way that is characteristic of the physical properties of the sample, such as a coating thickness and/or optical refractive index. Upstream of the detector 24, the output beam k.sub.out (detection beam) is passed through an analyzer 26, the variable setting of which affects the polarization-dependent intensity reaching the detector 24.
[0062] The diagram of FIG. 1 shows a non-imaging ellipsometer 10. An imaging ellipsometer differs from this only in having imaging optics between the sample 12 and the analyzer 26, or optionally between the analyzer 26 and the detector 24, and in having an imaging detector 24.
[0063] FIG. 2 shows in even more schematic form the geometric relationships within an ellipsometer 10 when the sample 12 is “correctly” aligned in the usual manner. The reference axis of the ellipsometer 10, denoted here by z, coincides with the sample normal n. At the same time, the input beam k.sub.in and the output beam k.sub.out run in a common plane, namely the yz-plane. With this alignment of the sample, the actual angle of incidence φ.sub.eff, which the input beam k.sub.in assumes in relation to the sample normal n, is identical with the nominal angle of incidence φ.sub.0, which the input beam k.sub.in assumes in relation to the reference axis z of the ellipsometer 10.
[0064] The situation is different for the tilted alignment of the sample 12 outlined in FIG. 3. The diagram of FIG. 3 can also be applied to curved sample surfaces in that each curved surface can be thought of as a grid of very small, tilted surfaces. In the case of a tilted sample surface, the actual angle of incidence φ.sub.eff and the nominal angle of incidence φ.sub.0 diverge, the sample normal n no longer coincides with the reference axis z of the ellipsometer 10. The alignment of the sample normal n can be described in the intrinsic system of the ellipsometer 10 with the polar angle Φ and the azimuthal angle Θ.
[0065] FIG. 4 is a sketch to illustrate and check the plausibility of the input rotation angle γ discussed in the preceding description of the invention. γ corresponds to the angle by which the actual plane of incidence, i.e. the plane created by the input beam k.sub.in and the actual output beam k.sub.out, is tilted in relation to the nominal plane of incidence, i.e. the yz-plane of the ellipsometer (calibration plane), as measured in the plane perpendicular to the input beam k.sub.in.
[0066] FIG. 5 serves to illustrate and check the plausibility of the output rotation angle δ according to approximation model 1 discussed in the description above. δ corresponds to the angle by which the “virtual output plane” defined below is tilted in relation to the nominal output plane, i.e. the plane in the case of the correctly aligned sample, as measured in the plane perpendicular to the nominal output beam k.sub.out,0. The “virtual output plane” corresponds to the plane created by the nominal output beam k.sub.out,0 and that vector which would be obtained from the sample normal n if the latter were mentally carried along as a rigid body during a three-dimensional rotation of the actual output beam k.sub.out onto the nominal output beam k.sub.out,0.
[0067] FIG. 6 serves to illustrate and check the plausibility of the output rotation angle δ according to the approximation model 2 discussed in the above description. δ corresponds to the angle which the auxiliary vector defined below forms with the nominal output plane, i.e. the plane in the case of the correctly aligned sample, as measured in the plane perpendicular to the nominal output beam k.sub.out,0. In this case, said auxiliary vector is obtained from the vector perpendicular to the output beam k.sub.out and lying in the actual output plane, by projection onto said plane, in which δ is measured, projected along the direction of the nominal output beam k.sub.out,0.
REFERENCE SYMBOL LIST
[0068] 10 Ellipsometer [0069] 12 Sample [0070] 14 Light source [0071] 18 Polarizer [0072] 20 Analyzer [0073] 24 Detector [0074] k.sub.in Input beam [0075] k.sub.out Output beam [0076] k.sub.out,0 Nominal output beam [0077] n Sample normal [0078] φ Angle of incidence [0079] φ.sub.0 Nominal angle of incidence [0080] φ.sub.eff Actual angle of incidence [0081] Φ Polar angle [0082] Θ Azimuthal angle [0083] γ Input rotation angle [0084] −δ Output rotation angle [0085] x, y, z Cartesian coordinates of the ellipsometer intrinsic system
