An algorithm for smoothing three-dimensional Monte Carlo ion implantation simulation results
Introduction
After a Monte Carlo simulation of ion implantation on an ortho-grid, the question arises how to translate the resulting values, i.e., concentrations, to an unstructured grid. In the Monte Carlo simulation an ortho-grid is commonly used in order to achieve workable simulation times, since calculating point locations, i.e., tracing the position of ions, dominates performance. For other, subsequent simulations via, e.g., the finite element method, it is mandatory to use different, unstructured grids. Furthermore, the resulting values have to be smoothed in order to provide suitable input for the simulation of subsequent process steps like diffusion.
Thus an algorithm for smoothing Monte Carlo ion implantation results has to meet the following demands: it has to work with unstructured target grids, it must provide suitable smoothing, and since the number of grid points in the target grid is usually large, it must not be computationally expensive.
One simple approach is to perform a least squares fit of a multivariate polynomial of fixed degree, usually two, and to hope that this polynomial is a suitable approximation providing proper smoothing. This is known as the response surface methodology (RSM) [5] approach and has been used to a great extent in TCAD applications, but it does often not work satisfactorily (c.f. Fig. 4). In order to solve this problem, generalizations of Bernstein polynomials were devised and their properties proven. Hence a fast algorithm based on these polynomials was developed and applied to a real world example. The RSM approach will be compared to the proposed algorithm since least squares fits are a popular method: RSM has been used extensively in TCAD applications, e.g. in [2], [4], [6], [9], [12], [14], [16], [19], [20].
Although it can be argued that the RSM approximation is based on a truncated Taylor series expansion for a multivariate functionf, it is important to note that this is a local approximation and quite different from a least squares fit for several points. In the Taylor series expansion convergence occurs when the number of terms and thus the degree of the polynomial increases, whereas in the RSM approach the degree of the approximating polynomial is fixed to an arbitrary low value. Increasing the degree is possible of course, but the choice is still arbitrary and the number of coefficients and thus the number of points required for the least squares fit increases abundantly.
Furthermore, the RSM suffers from the fact that a polynomial of fixed degree cannot preserve the global properties of the original function: the set of all polynomials of a certain fixed maximal degree is not dense in C(X), compact.
Although the RSM approach can be improved by transforming the variables before fitting the polynomials, it has to be known a priori which transformations are useful and should be considered. If this knowledge is available, it can of course be applied to other approximation approaches as well.
Finally, an advantage of the RSM approach is the simple structure of the approximations: it is easy to deal with polynomials of degree 2. However, in the algorithm proposed in the following no polynomials have to be constructed explicitly and the computational effort for doing least squares fits is eliminated as well.
Section snippets
Properties of multivariate Bernstein polynomials
The Weierstraß Approximation Theorem states that continuous functions on compact intervals can be arbitrarily well approximated by polynomials. One constructive way to obtain such polynomials are Bernstein polynomials which were first introduced by Sergei N. Bernstein in the univariate case. A generalization to multidimensional intervals and its properties is presented in this section. Generalizations to multidimensional simplices using barycentric coordinates and other properties of Bernstein
The algorithm
The algorithm works by constructing approximating multivariate Bernstein polynomials in the neighborhood of the points of the unstructured, new grid. Let A be the initial isotropic homogeneous grid, where values are associated with the volume cells, as is usually the case in Monte Carlo simulations of ion implantations, and B an arbitrary grid where values are associated with the grid points. This grid is to be used in following simulations and hence it is determined by their demands. It is
A three-dimensional example
The example is a three-dimensional CMOS structure which consists of poly-silicon in the upper part, of silicon dioxide in the middle part, and silicon in the lower part. A boron dose of 1013 cm−2 with an energy of 15 keV was implanted in a Monte Carlo simulation [10], [11] using an isotropic homogeneous grid. The resulting concentration of boron interstitial atoms in (cm−3) is shown in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6. The new anisotropic inhomogeneous grid with 78,651 grid points was
Conclusion
In summary, the properties of polynomials of fixed degree arising from least square fits were compared to those of multivariate Bernstein polynomials. The Bernstein polynomials fulfill the requirements for approximations needed for smoothing Monte Carlo simulation results and translating them from ion implantation ortho-grids to arbitrary, unstructured grids.
The polynomials and the algorithm devised provide the following benefits. First, they converge uniformly when the number of base points
Acknowledgements
The authors acknowledge support from the “Christian Doppler Forschungsgesellschaft,” Vienna, Austria, and by Austrian Program for Advanced Research and Technology (APART) from the “Österreichische Akademie der Wissenschaften.”
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An order-statistics-based method for constructing multivariate distributions with fixed marginals
2008, Journal of Multivariate AnalysisCitation Excerpt :For example, Petrone [18] used them to approximate prior distributions, and Sancetta and Satchell [21] used them to approximate the Kimeldorf–Sampson copula and as histogram estimators. Heitzinger et al. [9] used them for smoothing Monte Carlo results. Approximation of multivariate distributions by Bernstein polynomials has been discussed by Babu and Chaubey [2].