Determining Film Properties from Spectral Reflectance
The amplitude and periodicity of the reflectance of a thin film is determined by the film’s thickness, optical constants, and other properties such as interface roughness. In cases where there is more than one interface, it is not possible to solve for film properties in closed form, nor is it possible to solve for n and k at each wavelength individually. In practice, mathematical models are used that describe n and k over a range of wavelengths using only a few adjustable parameters. A film’s properties are determined by calculating reflectance spectra based on trial values of thickness and the n and k model parameters, and then adjusting these values until the calculated reflectance matches the measured reflectance.
Models for n and k
There are many models for describing n and k as a function of wavelength. When choosing a model for a particular film, it is important that the model be able to accurately describe n and k over the wavelength range of interest using as few parameters as possible. In general, the optical constants of different classes of materials (e.g., dielectrics, semiconductors, metals, and amorphous materials) vary quite differently with wavelength, and require different models to describe them (see below.) Models for dielectrics (k=0) generally have three parameters, while nondielectrics generally have five or more parameters. Therefore, as an example, to model the two-layer structure shown below, a total of 18 adjustable parameters must be considered in the solution.
Number of Variables, Limitations of Spectroscopic Reflectance
Spectral reflectance can measure the thickness, roughness, and optical constants of a broad range of thin films. However, if there is less than one reflectance oscillation (i.e. the film is very thin), there is less information available to determine the adjustable model parameters. Therefore, the number of film properties that may be determined decreases for very thin films. If one attempts to solve for too many parameters, a unique solution cannot be found; more than one possible combination of parameter values may result in a calculated reflectance that matches the measured reflectance.
An example of the reflectance from a very thin film, 50Å of SiO2 on silicon is shown below, where it is compared to the reflectance from a bare silicon substrate. In this case, measuring the thickness, roughness, and n of the SiO2 requires five parameters to be determined. Clearly, the change in the spectra caused by adding 50Å of SiO2 does not require five parameters to describe, and a unique solution cannot be found unless some additional assumptions are made.
Depending upon the film and the wavelength range of the measurement, the minimum single-film thickness that can be measured using spectral reflectance is in the 10Å to 300Å range. If one is trying to measure optical constants as well, the minimum thickness increases to between 100Å and 2000Å, unless minimal parameterization models can used. When solving for the optical properties of more than one film, the minimum thicknesses are increased even further.
Spectroscopic Reflectance versus Ellipsometry
Given the restrictions listed above, spectral reflectance can be used to measure a large percentage of technologically important films. However, when films are too thin, too numerous, or too complicated to be measured with spectral reflectance, oftentimes they can be measured with the generally more powerful technique of spectroscopic ellipsometry. By measuring reflectance at non-normal incidence (typically around 75° from normal) ellipsometry is more sensitive to very thin layers, and the two different polarization measurements provide twice as much information for analysis. To carry the idea even further, variable-angle ellipsometry can be used to take reflectance measurements at many different incidence angles, thereby increasing the amount of information available for analysis.
For more information:
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