A new algorithm for layer thickness and index step estimation in multi-layer hetero-epitaxial structures

Abstract

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.