Pseudodielectric functions of uniaxial materials in certain symmetry directions.

The pseudodielectric function is often used to represent ellipsometric data and corresponds to the actual dielectric functions of materials when there is no surface overlayer and the material is isotropic. If a uniaxial material is oriented such that the optic axis is in the plane of incidence or is perpendicular to the plane of incidence, then the cross-polarization terms are zero and appropriate pseudodielectric functions can be determined from the ellipsometry data. We calculate the pseudodielectric functions for uniaxial crystals in three primary symmetry directions: (1) the optic axis is perpendicular to the plane of incidence, (2) the optic axis is in the plane of the sample surface and parallel to the plane of incidence, and (3) the optic axis is in the plane of the sample surface and perpendicular to the plane of incidence. These results are expanded in terms of the difference in the ordinary and extraordinary dielectric functions and compared with the approximation of Aspnes [J. Opt. Soc. Am. 70, 1275 (1980)]. Comparisons are made with experimental results on oriented crystals of rutile (TiO2), and a simple procedure is presented to determine the complex dielectric function from standard ellipsometry techniques.

Windows in ellipsometry measurements.

The effect of windows or lenses placed between the polarization-state generator and the polarization-state detector in a general ellipsometry measurement is examined. It is found that three parameters are required for describing the effects of the window retardation on the ellipsometry measurements. Two of these window parameters can be measured at the same time as the sample parameters if the sample is isotropic, but the third window parameter cannot be determined independently and must be measured separately. If the sample is anisotropic, then none of the windows parameters can be measured independently at the same time as the sample parameters. An example is given in which the strain-induced retardation in fused-silica focusing lenses is measured with a two-modulator generalized ellipsometer and the results are used to correct the sample data.

Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO(2)).

Two-modulator generalized ellipsometry is applied to determination of the optical functions of uniaxial rutile. For a nondepolarizing sample the two-modulator generalized ellipsometer determines all the elements of the normalized Jones matrix with one measurement and thereby totally characterizes light reflecting from the sample. If a uniaxial crystal is appropriately aligned, then determining its optical functions requires only a single measurement. We have used this new instrument to obtain optical functions of rutile that are the most accurate available for optical energies above the band edge.

Characterization and optimization of absorbing plasma-enhanced chemical vapor deposited antireflection coatings for silicon photovoltaics.

We have optimized plasma-enhanced chemical vapor deposition (PECVD) of SiN-based antireflection (AR) coatings with special consideration for the short-wavelength (<600 nm) parasitic absorption in SiN. Spectroscopic ellipsometry was used to measure the dispersion relation for both the refractive index n and the extinction coefficient k, allowing a precise analysis of the trade-off between reflection and absorption in SiN-based AR coatings. Although we focus on photovoltaic applications, this study may be useful for photodetectors, IR optics, and any device for which it is essential to maximize the transmission of light into silicon. We designed and optimized various AR coatings for minimal average (spectrally) weighted reflectance (? R(w) ?) and average weighted absorptance (? A (w) ?), using the air mass 1.5 global solar spectrum. In most situations ? R (w) ? decreased with higher n, but ? A (w) ? increased because k increased with n. For the practical case of a single-layer AR coating for silicon under glass, an optimum refractive index of ~2.23 (at 632.8 nm) was determined. Further simulations revealed that a double-layer SiN stack with an n = 2.42 film underneath an n = 2.03 film gives the minimum total photocurrent loss. Similar optimization of double-layer SiN/SiO(2) coatings for silicon in air revealed an optimum of n = 2.28 for SiN. To determine the allowable tolerance in index and film thickness, we generated isotransmittance plots, which revealed more leeway for n values below the optimum than above because absorption begins to reduce photocurrent for high n values.

Two-modulator generalized ellipsometry: experiment and calibration.

A two-modulator generalized ellipsometer is described that is capable of measuring all 16 elements of a sample Mueller matrix with four measurements made at different azimuthal orientations of the polarization state generator and polarization state detector. If the sample can be described with a Mueller-Jones matrix, only a single measurement is needed. Only two calibration steps are needed to determine the fundamental operating parameters of the instrument. A reflection measurement from silicon is presented as an example, which illustrates that the elements of the Mueller-Jones matrix can be measured to an accuracy of ~0.1-0.2%.

Two-modulator generalized ellipsometry: theory.

A new ellipsometer is described that uses two photoelastic modulator-polarizer pairs, where the photoelastic modulators are operating at differing resonant frequencies. The time-dependent intensity of the light beam is extremely complicated but can be analyzed so that all elements of the sample Mueller matrix are obtained. For a given configuration, nine of the Mueller matrix elements can be measured at any one time; the other seven elements are accessible when the azimuthal angles of the photoelastic modulators are changed. The single-configuration measurement is often sufficient to characterize a number of real situations completely, such as film growth in a vacuum environment, anisotropic samples, and simple depolarization.

Optical functions of transparent thin films of SrTiO(3), BaTiO(3), and SiO(x) determined by spectroscopic ellipsometry.

A procedure is presented for accurately determining the thickness, optical functions, and surfaceroughness characteristics of thin-film insulators from two-channel spectroscopic polarization-modulation ellipsometry data. For films with minimal surface roughness, the optical functions can be determined over the entire measured spectrum; for rougher films, the analysis of the spectroscopic ellipsometry data yields meaningful values of the optical functions only in the transparent region. In general, the films must be transparent in a given range of wavelengths sampled by the ellipsometer so that at least two interference oscillations can be observed. The use of the procedure is illustrated with the determination of the optical functions of SrTiO(3) and BaTiO(3) thin films grown on MgO, and of SiO(x) films grown on Si. For SrTiO(3) and BaTiO(3), the thin-film results are compared with the measured optical functions of the respective bulk materials.

Use of the biased estimator in the interpretation of spectroscopic ellipsometry data.

The use of the biased estimator in the fitting of spectroscopic ellipsometry data is examined and applied to data from two-channel polarization modulation ellipsometry experiments. It is pointed out that the use of the biased estimator, as opposed to the unbiased estimator that is usually found in the literature, allows the experimentalist to weight properly the more accurate parts of the spectrum, to switch among different representations of the data, and to calculate a goodness of fit. The fit to data taken on a 59-nm SiO(2) film on Si is examined with both the biased and the unbiased estimators.

Determination of the optical functions of transparent glasses by using spectroscopic ellipsometry.

Two-channel spectroscopic polarization-modulation ellipsometry measurements have been made on four different glasses (fused SiO(2), finely annealed BK-7, a lead-indium-phosphate glass, and a germanium-arsenic-selenium glass). We show that this technique is sensitive to thin surface layers and that these surface layers can be modeled by using the Bruggeman effective-medium theory with 50% glass and 50% voids. By correcting the experimental spectra for this surface layer, we determined the refractive index of the sample within an error of +/-0.002 in the transparent region. For wavelength regions where the material is normally opaque, the ellipsometric data can be corrected for this overlayer, thereby increasing the accuracy of the determination of both the refractive index and the extinction coefficient.

Two-channel polarization modulation ellipsometer.

A new wavelength-scanning two-channel polarization modulation ellipsometer is described, where a photo-elastic modulator is used and the analyzed light is separated into orthogonally polarized beams using a Wollaston prism. Both beams are detected using phototubes whose bias voltage is dynamically controlled for constant dc. The dc from each phototube is measured with a digital voltmeter, and the fundamental and second harmonic of the phototube current are measured using individual lock-in amplifiers. All three of the associated ellipsometric parameters (N = cos2psi, S = sin2psi sindelta, and C = sin2psi, cosDelta) can be determined simultaneously in a single scan. The versatility of the instrument is demonstrated by the determination of the optical functions of Si from 238 to 652 nm (5.3-1.9 eV).

Four-channel polarimeter for time-resolved ellipsometry.

A new four-channel partial polarimeter was designed for use in time-resolved ellipsometry measurements. The device outputs four parallel, equispaced beams from a single input beam. By detecting and digitizing the intensities of these four beams, the intensity, as well as two of the three elements of the reduced Stokes vector can be determined. Since there are no moving parts to the polarimeter, time resolution is limited only by the response time of the photodetectors and the digitizers. The linear array of output beams is ideal for streak-camera detection, yielding a potential time resolution of ~1 psec. A beam splitter is described that separates the input beam into two beams, which are parallel to the input beam and whose polarizations are nearly the same as the input beam. In addition, the polarimeter is achromatic, operating in the wavelength region from 220 to 1100 nm.