Optical functions of silicon from reflectance and ellipsometry on silicon-on-insulator and homoepitaxial samples

Optical measurements on silicon and Si-based structures are important in both basic and applied research, and in many branches of current technology. The optical properties of silicon are of considerable interest for characterization purposes for solar cells and photodetectors, in microelectronics, etc. Efforts to replace the international kilogram prototype using silicon spheres recently revealed that almost a quarter of the uncertainties of the determination of the Avogadro number has been related to the surface,¹ with surface mapping provided by spectroellipsometry.

A number of studies have been devoted to the analysis of reported optical functions of silicon, e.g., Refs. 2–4. The evolution of spectroellipsometry, one of the most useful techniques of obtaining the optical response from infrared to ultraviolet, has recently been reviewed in Ref. 5. The most used measurements of Si covering broad spectral ranges are probably those of Refs. 6–8. The pioneering work of Aspnes and Studna⁶ has been superseded by the rotating-analyzer data on hydrogen-terminated (111) samples.⁹ Here, we aim at significantly improving the knowledge of the optical functions of Si, using the unique optical behavior of specially designed SOI samples and broadband ellipsometric measurements of homoepitaxial layers. We have been able to utilize silicon layers as thin as 87 nm in the Si/SiO₂/Si (silicon on insulator, SOI) structure, in order to quantify the light attenuation, the most straightforward way to measure the extinction coefficient. Simultaneously, the spectral positions of the Fabry-Perot-type fringes in layers of varying thicknesses allow us to precisely measure the real part of refractive index. In order to avoid uncertainties related to the surface preparation of grinded/polished/etched samples for ellipsometric measurements, we have measured homoepitaxial material. The ellipsometric spectra were subsequently corrected for the presence of overlayers, using the SOI data.

Bond and grind back silicon-on-insulator (“BGSOI,” we use “SOI” throughout the paper) structure contains a buried thermal oxide (BOX) layer separating the substrate (handle) wafer from the thin (“active,” ACT) Si layer, called the device layer in the semiconductor industry. The latter is covered by an overlayer, as shown in Fig. 1, which is mostly a natural surface (oxide) film; in some cases we have also used a thin (∼30 nm) thermal oxide. Here, we always assume the response functions of the BOX and overlayer to be the same. In this simplified picture, the stack consists of alternating Si and SiO₂ materials. We have selected a thin BOX of 95 nm for reasons described below. Various ACT thicknesses were obtained by grinding, chemical–mechanical polishing, and wet chemical treatment of the sample surface, followed by thermal oxidation and removal of the oxide in HF. The SOI structure forms a Fabry-Perot type resonator, specifically suitable for the determination of the Si optical functions. Homoepitaxial layers of silicon are frequently used in the fabrication of devices, due to the possibility of controlling doping profiles and improved structural quality, absence of interstitial oxygen- and vacancy-related defects present in Czochralski grown material.¹⁰ We have chosen a 15 μm, (100) oriented, 9.3 Ω cm n-type layer for ellipsometric measurements.

Near-normal incidence reflectance spectra of SOI structures were measured in the MIR (NIR) range using a Bruker IFS66 vacuum FTIR spectrometer, with a globar source, and KBr (quartz) beamsplitters and DTGS (InGaAs) detectors, respectively. Two auxiliary mirrors in a reflection accessory were used to obtain the average angle of incidence of 10°. Rough backsides of measured samples prevented the detection of reflexes within the substrate, possibly occurring below the bandgap of Si. For reflectivity measurements in the NIR–VIS–UV range, Avantes 3648 and 2048 pixel spectrometers (with spectral ranges from 1.2 to 5.1 eV) were equipped with the fiber reflectance probe shown in Fig. 1. With a probe-sample distance of ∼15 mm, the angles of incidence cover the range of 0.2°–1.7° (0.1°–0.9°), and the nearly circular measured spot has a diameter of 0.8 (0.4) mm for the 400 (200) μm fibers, respectively. The small spot has been found advantageous as it suppresses the influence of lateral thickness inhomogeneities. Samples have been put horizontally on a circular aperture, facing the reflectance probe placed underneath. A fair (better than 1%) reproducibility of detected signals has been achieved in this configuration. The samples could be pushed horizontally by a micrometer screw, allowing us to measure at selected spots. The latter were usually chosen at local maxima or minima of the ACT thicknesses. The relative reflectance, R_SOI/R_Si, is the ratio of signals produced by the samples and bulk Si reference. Care has to be taken to avoid large signals, as the reflectivity maxima of SOI samples are fairly high. The NIR–VIS–UV ellipsometric data were acquired with a dispersive, rotating-compensator ellipsometer (Woolam VASE) at angles of incidence from 65° to 75°, at an equidistant (0.01 eV) photon energy mesh. Energy scales were calibrated using Hg and Ar discharge lamps. All measurements were performed at temperatures of 22–24 °C.

The interference pattern observed in SOI samples can easily be understood by considering the field amplitudes after passing through a homogeneous layer of the refractive index N = n + ik and thickness d. The corresponding diagonal transfer matrix has the form¹¹ 

where λ is the vacuum wavelength, E the photon energy, and φ the angle between the propagation vector and the surface normal. For negligible attenuation (N ≈ n), the matrix remains unchanged for the changes of the phase θ by multiples of 2π, leading to interference patterns in photon energies E and/or thicknesses d. In particular, the spectral distance of neighboring fringes is approximately

which is obtained for ΔE ≪ E and the dispersion n(E) of refractive index limited to the linear term with the derivative n′. The group refractive index, n + En′, entering Eq. (2) is much larger for Si than for SiO₂; this leads to a denser pattern due to the light interference in the ACT layer. Further, the transfer matrix of Eq. (1) is reduced to unity or its negative

Consequently, the ACT (BOX) layer is effectively absent from the stack (“optically invisible”) at a dense (sparse) mesh of photon energies, because of its large (small) refractive index and/or thickness. Since the reduced sensitivity to the presence of the ACT layer is unwanted here, we have selected a rather small BOX thickness of 95 nm. This places the first spectral point of Eq. (3) above the photon energy of 4 eV, deep inside the range of strong absorption in Si.

Reflectance spectra of one of our thickest SOI samples are shown in Fig. 2. The thickness of the oxide overlayer was determined ellipsometrically in UV, and those of the BOX and ACT layers by fitting the MIR–NIR reflectance for photon energies below 0.8 eV. We have used the refractive indices from Palik's handbook¹² (glassy SiO₂) and Frey¹³ (Si) in the fitting. As expected, the BOX and ACT thicknesses are correlated: a change of the former produces about three times smaller change of the latter in the opposite direction, due to the difference in refractive indices. Similarly, the thicknesses are correlated with refractive indices of the corresponding layers. Choosing slightly larger values of the refractive indices for the BOX and surface layers compared to the glassy SiO₂ (due to a possibly larger Si content) results in a slight lowering of their thicknesses; the influence on the sought optical functions of Si has been found negligible.

In Fig. 2, we see the dense interference pattern due to reflexes inside the ACT layer modulated by the presence of the thin BOX layer; it would disappear above 4 eV if there was no absorption in Si. The magnitudes of reflectivity in the ranges of overlap agree within ∼1%; a slight nonlinearity of the detection using the InGaAs detector influenced the FTIR data in NIR. The agreement of spectral positions, in particular, of the sharp reflectivity minima is excellent in the FTIR data. However, we have observed a small systematic shift due to the difference in the angles of incidence. In fact, the observed upwards shift of ∼1.4 meV seen in the range of 1.28 eV in Fig. 2 (the right inset) is consistent with the prediction of Eq. (2) for an angle of incidence of 10°. We have therefore used the actual angles of incidence in all of the data evaluation.

The gradual vanishing of the interference effect towards larger photon energies is typically due to the increasing attenuation in the ACT layer. For the thick layer of Fig. 2, the spectral averaging related to the limited spectral resolution is also important. Note also the slight suppression of the relative reflectivity at higher photon energies, due to the 33 nm thermal SiO₂ overlayer. The minima of reflectivity remain sharp, and their spectral position is very sensitive to the refractive index. We have used the polynomial representation

to fit the positions of reflectance minima in the range of NIR FTIR (0.7–1.32 eV) measurements. The n₀ coefficient was kept fixed, as it plays a decisive role in determining the ACT thickness. The mean square deviation of 0.012 eV (0.1 cm⁻¹) for 22 minima positions was obtained with the coefficients from Table I. This polynomial also represents the refractive indices of Frey¹³ well; the deviation from the data of Table II of Ref. 13 is smaller than 0.0001 in the rage of overlap. An alternative dataset¹⁴ discussed by Smith⁴ leads to ACT thicknesses larger by 2 nm (mainly due to the smaller n₀ value, 3.4161), and 3-times larger mean square deviation of minima positions. Although we cannot decide which of the refractive indices is closer to the actual one, the dispersion of Ref. 13 is preferred by our measurements. Finally, the sensitivity of the interference measurements on SOI structures using FTIR seems to be comparable with that of the prism method in the transparent range.

With thinner ACT layers, the interference pattern extends towards higher energies, displaying a reduced number of maxima and minima. The latter are sharper, and their positions at lower photon energies provide ACT thicknesses with precisions of the order of 0.1 nm owing to the known refractive index. With k above ∼0.1 (i.e., above the photon energy of ∼2.8 eV) the positions of extrema are influenced slightly by the attenuation. We have used the spectra around the (flatter) maxima to determine k, which was the single variable in fitting the single measured value of R_SOI/R_Si. With a suitable set of ACT thicknesses, reasonable coverage of the spectral range is obtained. Shown in Fig. 3 are the results for two samples with similar ACT thicknesses close to 780 nm. The uncertainty of the k values are mainly due to the relatively large (∼1%) fluctuations of the measured intensities. However, flat k(E) dependencies result from the smoothing spline approximation, as also shown in Fig. 3. We have used cubic splines¹⁵ with adjustable inner nodes and function values at the boundaries for the logarithm of k. The typical mean deviation between the scattered data points and the splines was in the 0.5–1% range with a proper number of spline nodes (e.g., 5 for the data of Fig. 3). After determining k(E) in a given spectral range, the positions of minima were used to refine the n(E) dependence and vice versa. These iterative refinements converged fairly quickly.

We have extended the use of spectral positions of minima and the heights of maxima up to a photon energy of 3.2 eV. Figure 4 shows the measured extinction coefficient obtained from our thinnest ACT layers. The four spectral segments overlap partially, and are in good agreement with our ellipsometry data discussed below.

The five interference minima above 3 eV allowed us to determine the real part of the refractive index. They are listed for 3.2 and 3.0 eV in Table II together with the value of Eq. (4) and the coefficients of Table I for 1.3 eV; the imaginary part from Fig. 4 is also listed. The indicated error intervals result from uncertainties of the spectral positions and magnitudes of reflectances, and of the film thicknesses.

Our measurements of extinction using SOI samples covered the 2.25–3.2 eV range; a comparison with selected results from literature is shown in Fig. 5 on the logarithmic scale.

In our ellipsometric studies, the homoepitaxial Si sample has been left in air after an HF etch for ∼12 h and measured at angles of incidence of 70° and 72.5°; the same sample was remeasured 10 months later (having a thicker, stabilized overlayer) at angles of incidence of 65°, 70°, and 75°. The measurements covered the spectral range from 1.1 to 6.5 eV. Figure 6 shows the results at 3 and 3.2 eV in the form of the (pseudo)refractive indices retrieved using the model of a single SiO₂ overlayer with varying thickness. Our data from columns 2 and 3 of Table II agree fairly well with these ellipsometric results, when assuming a thinner overlayer of 1.45 nm and a thicker (stabilized) one of 2.7 nm. Jellison's results⁷ are the closest from the three literature entries shown in Fig. 6.

With the overlayer fixed at 1.45 nm, we have fitted the pair of ellipsometric spectra (at the angles of incidence of 70° and 72.5°), on a point-by-point basis, to obtain the optical functions of Si. They are listed in columns 4 and 5 of Table II for three selected photon energies. Let us note the consistency of the results of the reflectance and ellipsometry technique. The former is rather insensitive to the surface overlayers, while the latter is very sensitive to them. The error estimates of ellipsometric results in Table II are based on the measurements at different angles of incidence on samples differing in overlayer thicknesses. Uncertainties are, in general, larger in the ranges of strong spectral variations of optical response as discussed in the next paragraph. Finally, the agreement of reflectance and ellipsometric data confirms negligible deviations of the optical response of our thinnest films (with thicknesses of the order of 100 nm) from the bulk Si.

We provide the refractive index spectra for the 0.2– 6.5 eV range, see the supplementary material,¹⁷ in the text file Si.txt. They were formed by merging the refractive index data as follows. Real part: polynomial of Eq. (4) below 1.3 eV; imaginary part: set to zero below 1.1 eV, smoothing spline of Weakliem's¹⁶ data in the 1.1–2.35 eV range, SOI data in the 2.35–3 eV; the rest from the ellipsometric measurements. We have included Weakliem's absorption data, as our measurements on thick ACT layers probably lead to inferior results below 2.35 eV. We also provide,¹⁷ besides the complex refractive index, several related spectral functions (complex permittivity, its negative inverse, conductivity, normal-incidence reflectance, penetration depth of light, and the wavelength inside Si) in the text file Si_OptSpe.txt; the listed quantities are defined in Ref. 11, p. 24.

Finally, we show the differences between data selected from the literature and our results above the bandgap energy. As expected, the differences are significant mainly in the ranges of strong variations of the optical functions, E₁ (∼3.4 eV) and E₂ (∼4.3 eV), critical points of the joint density of states, see, e.g., Ref. 18. The measured sharp spectral structures depend on the structural quality of the bulk and surface, on temperature and stress, and also on the surface overlayers. Even the quality of the monochromators used and their calibration might contribute to differences between various datasets. We show the deviations of previously published results from our refractive index in Fig. 7. Let us also recall the comparison of extinction coefficients shown in Fig. 5 with the underestimated values of Herzinger.⁸ Ellipsometric results also differ significantly from each other in the real part of the refractive index; for example, the (maximum) deviation of 0.18 between the data of Refs. 6 and 8 occurs at 3.3 eV. These deviations are well above the assumed precision and accuracy of both datasets; it cannot be attributed to the presence of the 0.18 nm overlayer in Aspnes' measurements,⁶ contrary to the suggestion in Ref. 8.

As the complex refractive index mixes the absorptive and dispersive response, assessing the differences in permittivity is more appropriate, see Fig. 8.

In the range above 4 eV shown in detail in Fig. 9, the differences from Yasuda's and Jellison's data are fairly simple in their spectral lineshapes. On the other hand, the results of Herzinger are very probably influenced adversely by the applied mixing of results on different Si and SiO₂/Si samples, and modeling of the spectral response of Si and the involved layers/interlayers; note, in particular, several sharp spectral structures, the most prominent one (absent in the remaining datasets) is located in the 4.5–4.8 eV range.

The differences between lineshapes in the E₁ range are shown in detail in Fig. 10. Again, there is apparently a spurious spectral structure in the spectra of Ref. 8; it is located in the low-energy part and seems to be related to the overestimated refractive indices shown in Fig. 6.

Thus, at least a part of the observed differences between these datasets can be related to the evaluation of measured data. Assessing possible genuine differences of measured samples is beyond the scope of the present study. Let us note a possible role of surface orientation via chemical termination of etched surfaces.⁵ In spite of the current level of structural perfection of Si single crystals, the preparation of surfaces is apparently the decisive factor for the discrepancies, typically exceeding the uncertainties of state-of-the-art ellipsometric measurements.

In conclusion, we have used SOI structures with properly chosen BOX and ACT thicknesses to obtain the spectra of refractive index from reflectance measurements, covering the 0.2–3.2 eV range. In infrared, we were able to extend the range of minimum—deviation data taken on prisms up to 1.3 eV (i.e., above the bandgap of Si) with remarkable precision. The attenuation of light was measured below the onset of the strong E₁ interband transitions at ∼3.2 eV, owing to the high-quality Fabry-Perot resonators with very thin ACT layers. Ellipsometric measurements on homoepitaxial Si surfaces with reduced and stabilized surface overlayers were performed, and the former were used to determine the optical functions of Si up to a photon energy of 6.5 eV. The substantial advantage of the homoepitaxial material is the absence of damage related to surface polishing. At the same time, ellipsometric data are in very good agreement with reflectance measurements on SOI structures, where the sensitivity to the surface quality is low. Thus, we believe the knowledge of optical functions of silicon has been improved substantially.

This work was supported by the project “CEITEC– Central European Institute of Technology” (CZ.1.05/1.1.00/02.0068) from the European Regional Development Fund, and Grant Nos. TA01010078 and TH01010419 from the Technology Agency of the Czech Republic. We wish to acknowledge M. Lorenc, J. Celý, M. Kučera, and S. Valenda for technical help.