Optical Spectroscopy with dispersive Spectrometers
Basics - Building Blocks - Systems - Applications

This page summarizes chapter 2 of the book
"Applications of Dispersive Optical Spectroscopy Systems",
ISBN 9781628413724, SPIE monographs, Bellingham, WA, USA

Application – E1
Spectroscopic Ellipsometry and Polarisation Measurement

This page presents the directory, the signs and symbols, conversions, and equations of the book, while the details are an exclusive part of  the book.

E1.0 Introduction
This page differs remarkably from others of the Spectra-Magic Basics.
The page on Ellipsometry is the only one, not specially written for Spectra-Magic. The German version is based on the paper 02 "Spektroskopische Ellipsometrie; Grundlagen, Stand der Technik und Anwendungen (D)". Another difference is the endorsement of a company name, which is not the case in all other Basics pages. But, in order to stay compatible with the other Basics and Application pages, the list of formulas and equations is maintained. In comparison to the German version, the English one appears summarized.

E1.0.1 Applications of spectroscopic Ellipsometry (SE)
If the wavelength of the instrument is correct, angle of incidence, polarization, and collimation of the light are perfect, an SE system needs no further calibration to measure the refractive index n, and the absorption coefficient k correctly. Both are basic parameters for the characterization of materials. In the case of measuring unknown samples, but based on knowlegde of the structure system, SE software is able to define the real structure. That in turn needs the precise n & k values of the materials involved in the reduction and fit library. Theoretically there are no calibration samples required. In practice, they are needed to check the above mentioned parameters, and for day-to-day comparison. But that does not diminish the application power of the method.

E1.0.2 SE System for Research, Material, and Product Definition
Block diagram of a reserach gerade SE system
Graph E1.1A
Principle of a research oriented SE system.
Graph E1.1B polarized and circular light
Graph E1.1B demonstrates the behaviour of a polarized light beam after the rotating polarizer.


Suggestion: an excellent tutorial on the topic of polarization is found in the homepage of András Szilágyi: http://www.enzim.hu/~szia/emanim/emanim.htm. It is available in Hungarian, English, and German, and can be loaded down.

E1.2 Basics of SE and relevant data:
SE data are based on the Fresnel equations:

F43:   also represented as

with
r
p and rs, the absolute parallel, and vertical, polarization values
d, the actual phase shift
r,  the complex result, which itself leads to tg Y and cos D.
The illumination angle at the sample is marked by
F.

The complex value of r, in a system with polarizer - sample - compensator - analyzer, is created by

F44

Whereas A, C, P are the angles of polarizer, compensator, and analyzer. For all standard SE measurements, it is satisfying to recover the real part only, which needs no algebraic sign. That in turn needs no compensator.

In an SE system with rotating polarizer, programmed analyzer, but no compensator, the measured data are created by:

F45
where the coefficients
a and b are recovered by Hadamard reduction, that leads to tg Y, and cos D.

F46:

Data, acquired with optimum working instruments, and well suited samples, can be reduced to absolute thicknesses of 0.1 nm, and values of 0.0005 in n and k.

E1.3 When is SE required in favour of single wavelength ellipsometry (SWE)?
E1.4 Components of an SE system
E1.5 SE with parallel detection
E1.6 Data origin and reduction
E1.7 Limits of the SE method
F47:  

with
S0 through S3, the 4 Stokes parameters
I0, the sum of all measured combinations S0 through S3
Ip und Is,  the two planes of polarization
p, s, +
p/4 and -p/4, the 4 parameters of linear polarization, and
Il and Ir, the values of circular polarization.

One more important parameter is the Degree of Polarization P of the light after the sample. Because it defines the ultimate detection limit, and is also important for the analysis of sample roughness.

F48:    

E1.8 SE examples:
Example Silicon wafer with native oxide
Graph E1.8.1 Ellipsometric data
of a bare silicon wafer (crystalline silicon) with a thin layer of native oxide (SiO2), which was reduced to be 1.38 nm thick. The solid curve is the data acquired, the dotted line is the fit curve of the reduced data.
Example Wafer with 151 nm Silicon Oxide
Graph E1.8.2 Ellipsometric data
of a calibration sample, consisting of a silicon wafer (crystalline silicon) with a known layer of silicon oxide (SiO2). Thickness of the layer is 151 nm. Again, the solid curve is the data acquired, the dotted line is the fit curve of the reduced data. As they both appear identical, there is a perfect fit.
Example Wafer with 2517 nm Silicon Oxide
Graph E1.8.3 Ellipsometric data
of a calibration sample, consisting of a silicon wafer (crystalline silicon) with a known layer of silicon oxide (SiO2). Thickness of the layer is 2.517 µm. Again, the solid curve is the data acquired, the dotted line is the fit curve of the reduced data. The peaks of the fit curve appear higher, because the rather thick layer has lower reflection as it should have theoretically. But that has no impact on the reduced result. As the interferences get very close, the measurement is near the upper thickness limit of the method in the UV-Vis range.

E1.11 A Selection of References on SE:
E1.12 Extensions to the instrumentation for spectroscopic ellipsometry (SE)
E1.13 SE system for the deep UV
SE system for the deep UV
Graph E1.13: UV SE system
.

E1.14 SE system for the IR range
Diagram of an IR-SE system
Graph E1.14: IR SE System
.

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Status April 2012