Multiwavelength ellipsometry fathoms materials
Ron Synowicki and James N. Hilfiker
Optical measurements used to determine the properties of thin-film and bulk materials in research and production environments can be done with high precision, sensitivity, and speed. The noncontact and nondestructive nature of optical testing is also suitable for either measurements ex situ on the tabletop, or in on-line measurements for film process control. Utilizing these advantages, a flexible acquisition-and-analysis-software package, WVASE32 from J. A. Woollam (Lincoln, NE), is capable of powerful analysis, including working with complex multilayer samples or anisotropic materials.
Need for software
Optical measurements on thin films are most commonly performed to determine film thickness and refractive index. However, the most widely used optical techniques of ellipsometry and spectrophotometry measure these properties indirectly. The instruments measure changes in reflected or transmitted light such as a reflected or transmitted intensity (spectrophotometry) or a change in polarization of the light beam (ellipsometry). These experimentally measured changes are then fit to a model that describes the sample under study. These models contain such information as the refractive indices of the substrate and films in the sample and nominal layer thicknesses. A model also specifies the Fresnel equations needed to describe the expected changes in reflectance or polarization of light.
These model-generated data are used for comparison with the experiment. Regression analysis is used to vary parameters in the model, for example, film thickness, to obtain a best-fit between the model generated and experimental data. The best-fit values in the model may represent film thickness, refractive index, alloy fractions, surface roughness, and other information useful to the coating engineer.
Program features
Written in C++ and assembly language, WVASE32 takes full advantage 32-bit processing power and runs on 486 or higher Windows-based PCs. The Levenburg-Marquardt Regression algorithm has been optimized for maximum speed. A true Windows program, WVASE32 allows the user to copy data, graphs, models, and statistics to the Windows Clipboard for easy transfer to other Windows programs. The package is also included with all Woollam ellipsometers. The code not only operates the company`s ellipsometer hardware but also can analyze optical data from other maker`s instruments.
The WVASE32 package can acquire and analyze many different optical data types, including reverse-side, transmission, and anisotropic ellipsometry; intensity reflectance and transmission, both polarized or unpolarized; and magneto-optic Kerr and Faraday spectroscopy. The program can simultaneously analyze any combination of the these different data-measurement types.
Acquisition can be tailored for maximum information content in the data. This flexibility permits complex models to extract the most information about the sample while still ensuring a unique best-fit solution. For example, ellipsometric data can be analyzed simultaneously with transmission or reflection data to take advantage of the information content in both data sets.
Correlation problem
In extracting useful information from optical data, the user is required to build a model of the sample and optimize the values using a regression algorithm (see Fig. 1). The WVASE32 software is supplied with more than 240 different reference materials to aid in building suitable models. However, because each sample is different and each experimental data set contains different information about the sample, the acceptable number of fit parameters in a model may vary widely for different types of materials. It is important to include enough parameters for a good data fit to give the most accurate results, while keeping the model as simple as possible.
A model becomes overcomplicated when there are more fit parameters in the model than there is information in the experimental data. When this occurs, two or more parameters become sensitive to the same information in the experimental data, and these parameters become coupled or correlated to each other--they become less sensitive to the experimental data and more sensitive to each other. Changes in one parameter are simply accounted for by changes in the other correlated parameters, with little or no improvement in the fit quality. This results in multiple sets of parameter values that will give the same quality fit to the experimental data, and it is not possible to determine a unique set of best-fit values.
Parameter correlation must be monitored carefully when fitting data. Eventually, all models will correlate when the number of fit parameters exceed the information content of the data. The user slowly adds parameters to improve the data fits, but must watch for the point where unacceptable correlation is reached. After the completion of each fit, WVASE32 makes available statistics to hel¥the user determine the degree of correlation present in the fit. In addition to the mean-squared error (a quality measure of a fit parameter), 90% confidence limits are given for each parameter, as well as the covariance matrix showing the degree of independence of each relative to the other fit parameters.
The best way to minimize parameter correlation is to minimize the number of fit parameters in the model. One way to minimize parameters is to use mathematical equations, known as dispersion models, to describe the refractive index (optical constants) of a material as a function of wavelength. Using dispersion models reduces the number of fit parameters while allowing the refractive index to be described over a wide wavelength range.
Dispersion relations and realities
The WVASE32 package supports a wide variety of commonly used dispersion relations including the Cauchy, Lorentz, effective medium approximation, Herzinger-Johs parametric semiconductor model, and even user-defined relations. This allows input of other dispersion models not currently supplied as dedicated layers in WVASE32, such as Sellmeier, Conrady, and Urbach, and those user inputs using standard algebraic notation.
Optical data can be very sensitive to the microstructure of the sample, and even small departures from the assumption of perfectly smooth and homogeneous layers can lead to errors in the model fit parameters. The WVASE32 code allows the model to account for the presence of realities such as back-surface reflections from transparent substrates, angular spread from curved samples, film-thickness nonuniformity, refractive-index gradients in a film, and optical anisotropy (uniaxial, biaxial, and arbitrary orientation of the optic axis) within a film or substrate. Thus, the sample can be analyzed in a quantitative manner rather than just a qualitative analysis from a marginal-quality data fit, expanding the types of samples that can be analyzed.
Often a grou¥of samples are processed together and contain the same layers, but these may have different layer thicknesses. It is possible to analyze all these data sets simultaneously to extract all the different layer thicknesses and fit the average optical constants over all samples. This is quite useful for analyzing thin overlayers on metal or other samples involving a coated substrate with unknown optical constants. Such multisample analysis also helps in reducing parameter correlation.
In situ, too
There has been a recent increase in using ellipsometry in situ during processing steps. Such measurements provide new insight to thin-film properties. The WVASE32 software can analyze data in real time to measure sample properties, including thickness, growth or etch rates, optical constants, alloy ratio, and surface temperature, as materials are being processed. The program has been optimized to compute complicated algorithms at high speeds allowing all types of deposition and etch processes to be monitored and controlled.
The WVASE32 software is accessible to other programs through an external-programming interface. This permits users to write their own programs that control WVASE32 functions, but more important, allows the Woollam VASEManager and GrowthManager software packages to use WVASE32. These manager programs have been written to automate the routine procedures of measuring and analyzing ellipsometry data for ex situ and in situ measurements, respectively, making spectroscopic ellipsometry a push-button tool (see Fig. 2). o
FIGURE 1. A good material sample model features a small enough number of fit parameters to avoid having some of these become sensitive to each other, rather than the experimental data, while still allowing a quality data fit and wavelength range.
FIGURE 2. Oxide thickness and other thin-film properties are capable of being assessed by VASEManger software, which automatically builds models and fits the raw data for answers, such as film thickness and refractive index. Such real-time capability allows monitoring processes and check for sample uniformity.
RON SYNOWICKI and JAMES N. HILFIKER are engineers with J. A. Woollam, 645 M St., Ste. 102, Lincoln, NE 68508; e-mail: [email protected].