Use of the Richardson extrapolation in optics of inhomogeneous layers: Application to optical characterization

Abstract

A new approach to calculation of optical quantities of inhomogeneous layers is presented. In this approach, the Richardson extrapolation is used to improve the accuracy of the method, in which the inhomogeneous layer is approximated by a stack of thin homogeneous layers. The results presented in this paper are based on the assumption that the media are isotropic and the inhomogeneity is along the axis normal to the boundary. The results obtained by the new method are compared with those obtained without the Richardson extrapolation. The Richardson extrapolation brings significant improvement in accuracy, especially if the number of approximating layers is large. Moreover, the method using the Richardson extrapolation proceeds in steps with an error estimate available in each step; thus, the calculation can be stopped when the desired accuracy is reached. The use of the method is illustrated by means of the optical characterization of strongly inhomogeneous film of non-stoichiometric silicon nitride. KEYWORDS ellipsometry, inhomogeneous layers, optical characterization, silicon nitride 1 INTRODUCTION Within the optical characterization and optical synthesis of thin films, inhomogeneous layers play an important role. While there are many works devoted to methods usable in optical characterization and optical synthesis of homogeneous thin films (see, eg, previous studies 1-18 ), less attention has been devoted to the methods enabling us to characterize or perform optical synthesis of inhomogeneous thin films exhibiting refractive index profiles. Therefore, it is necessary to develop new procedures usable in the optics of inhomogeneous layers.In the literature, many procedures for calculating of the optical quantities of inhomogeneous layers have been published, eg, previous studies. 11,[19][20][21][22][23][24][25][26][27][28][29][30] The theoretical procedures can be divided into those providing the exact solution and those providing approximate solution.The exact solution is known only for few special refractive index profiles (eg, linear profile and exponential profile 24 ); thus, the approximate methods are usually used in practice. As more and more sophisticated methods of the optical characterization and optical synthesis are developed, the approximate methods with increased precision and reduced calculation time are required.The general approximate method enabling us to calculate the optical quantities of inhomogeneous layers with arbitrary refractive index profile is based on dividing these layers into a stack of thin homogeneous layers. 19,20,22,24 We will refer to this method as to homogeneous layers approximation (HLA). Unfortunately, to calculate the optical quantities with high accuracy, a very large number of approximating layers is required in this method. In this paper, we will show that the Richardson extrapolation (RE) can be used to increase the accuracy of this method. The RE 31,32 is a mathematical method that can be used to increase the rate of convergence of a series, provided certain criteria ...

Approximate methods for the optical characterization of inhomogeneous thin films: Applications to silicon nitride films

“…In paper [24] an improvement of the method presented in this section is performed. This improvement consists in improving the convergence of the method using the Richardson extrapolation.…”

“…If the Richardson extrapolation is used then the error of calculation falls approximately as N − log 2 N , where N is the number of the approximating sub-layers. However, it should be noted that the faster convergence can be achieved using the Richardson extrapolation only if the number of the sub-layers is considerably large (for details see [24]).…”

“…The non-stoichiometric silicon nitride film, whose results of the optical characterization are introduced in this sub-section, exhibited a relatively large gradient of the refractive index profile and therefore the approximation by the multilayer system was utilized (WKBJ approximation completely failed). Within its optical characterization the simultaneous processing of reflectance and ellipsometric data was utilized [24]. The formulae corresponding to the matrix formalism were used to calculate the values of reflectance and associated ellipsometric parameters.…”

“…In other words, functions of single variable describe courses of the refractive index and extinction coefficient from the lower to upper boundaries of these inhomogeneous thin films (single variable corresponds to axis perpendicular to parallel boundaries of these films). As for optics of inhomogeneous thin films one can recommend, for example, these papers [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Within the last three decades an interest concerning optics of inhomogeneous thin films was intensified.…”

“…This is especially true if inhomogeneous thin films exhibit unusual and complicated refractive index profiles. Representatives of the complex inhomogeneous thin films are silicon nitride films that are rich in silicon (see eg [23,24]). In general it is possible to state that chemical and plasma chemical technologies frequently produce the thin films exhibiting more or less degree of optical inhomogeneity.…”