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Processing of electronic materials often produces graded refractive index profiles near the surface, as for instance in ion-implanted and diffused semiconductors. Analysis of these materials by means of optical reflectance and transmittance measurements, simulations, and least-squares fits is prevalent, but cumbersome. In order to facilitate the nondestructive characterization of materials containing inhomogeneous layers, we developed new theoretical solutions in integral form for the reflectance and transmittance. Two special cases are emphasized in this paper: a buried Gaussian refractive index profile, and a transition region described by a half Gaussian. The analytical study of Gaussian profiles provides us with new methods for the direct estimation of the average depth xp and the standard deviation σ from optical reflectance data. Our estimates of these parameters, derived from experimental reflectance data of ion-implanted silicon, germanium, and GaAs, agree well with the values estimated by means of elaborate curve-fitting procedures. Simulations performed by means of the well-known matrix method and our theory, not only show excellent agreement but also prove that our method is very time efficient. Therefore, this technique could be used advantageously as a diagnostic tool as well as in process controllers in the semiconductor manufacturing industry.

A new method for the characterization of ion-implanted silicon is proposed. It is based on analysis of the Fourier spectrum of bilinearly transformed infrared reflectance versus wavenumber data of ion-implanted samples. This non-destructive technique has been applied to previously published infrared reflectance data of 〈111〉 and 〈100〉 oriented Si samples which had been implanted with 2.7 MeV phosphorus and 380 keV silicon ions, respectively, and annealed at 500° C for various lengths of time. The refractive index and thickness of the amorphous layer of the as-implanted samples can be measured directly by means of this technique. The position of boundaries between the amorphous, recrystallized and substrate zones, as well as the position of the carrier concentration peak can be determined for the various annealing times. Depending on the annealing time, the recrystallized layer in 〈111〉 silicon has a refractive index which is between 2% and 4% higher than the substrate refractive index, while the difference in refractive index between the amorphous and recrystallized layers is in the order of 5%. In contrast to these results, the presence of the substrate/recrystallized material interface could not be detected in partially recrystallized 〈100〉 silicon by this method, implying that the refractive index step at the substrate/recrystallized material interface is less than 1%. The step in refractive index at the crystalline/amorphous interface in 〈100〉 silicon implanted with a dose of 0.5 x 1016 cm−2 silicon ions, was measured to be 12%, and it is reduced to 8% after partial regrowth has occurred. These results confirm the data obtained by a model-based least-squares analysis.

Thickness determination in layered structures by means of interferometric techniques and spectral estimation is limited by dispersion. It is shown that compensation for the dispersion of the bilinearly transformed reflectance, enhances the resolution and also increases the accuracy of the thickness estimator. Results are presented for thin and ultra-thin silicon-on-insulator structures.

Closed-form mathematical expressions are derived in this work enabling one to obtain both thickness and refractive index steps from the measured optical reflectance of an arbitrary number of layers on a substrate. The theory is based on the assumption that the normalized index steps are small, making this new technique particularly appropriate to the non-destructive analysis of hetero-epitaxial layers. Despite the fact that the derivation is quite involved, experimental data can be analyzed with very modest computer requirements. If the reflectance data is available at equi-spaced wavenumber intervals, each data point is pre-processed by an algebraic transform requiring one addition, one subtraction, and a division, followed by a fast Fourier transform. It is shown theoretically that the Fourier spectrum of the transformed reflectance possesses spectral peaks only at positions corresponding to the interfaces, and in the same sequence as the actual positions of the interfaces in the multilayer structure. The structural determination is therefore totally unambiguous, requiring no time consuming interactive procedures. It is also shown that the relative refractive index steps between adjacent layers may be calculated directly from the amplitude of the discrete Fourier components by direct substitution. The thickness and refractive index estimation technique is demonstrated by both simulated and experimental data on AlGaAs-GaAs structures comprising up to five layers on a substrate. It is demonstrated by these examples that the thickness dynamic range of the method is very wide, ranging in these examples from 1 to 50. The minimum thickness which can be resolved is determined by the frequency resolution of the Fourier transform, which for AlGaAs-GaAs is approximately 150 to 200 nm. However, if this new method is used in conjunction with a curvefit procedure, for the thin layers, it is possible to improve the resolution. Analysis of the simulated data shows that the accuracy in the determination of thickness and refractive index is better than 3% in most instances.

Knowledge of film thickness is essential for device design in silicon-on-insulator technology. A new thickness estimation technique, based on the calculation of the spatial frequencies of bilinearly transformed infrared reflectance data in a spectral window, is introduced. The assignment of dominant spectral peaks in the power spectral density curve to the optical thickness of the silicon, silicon dioxide and the combined layer, is also presented. Examples for silicon-on-silicon dioxide with the silicon layer ranging in thickness between 1000 nm and 50 nm, with fixed oxide thickness, are presented. Thickness values of both layers to better than a few percent accuracy, were obtained for silicon layers down to 100 nm and with reduced accuracy for layers as thin as 50 nm.

The oxidation rate and the oxide thickness of the hydrogen ion implanted silicon wafers were examined. It was observed that the native oxide thickness is higher for the H+ implanted Si(100) compared to the Si(111). Also the native oxide thickness depended on the implanted hydrogen distribution. The thickness increased with the hydrogen con-centration. The oxide thickness after wet oxidization of the H+ implanted Si(111) was higher than that of the unimplanted wafers. The oxide thickness also depended on the resistivity of the H+ implanted Si wafers. The suggested explanation is that in high energy H+ implanted Si the oxidation rate is higher as a result of the higher diffusion and reaction kinetics. All these measurements were done with the assumption that the implanted and the unimplanted samples have the same indices of refraction for the oxide as well as for the substrate.