An algorithm for smoothing three-dimensional Monte Carlo ion implantation simulation results

Clemens Heitzinger, Andreas Hössinger, Siegfried Selberherr

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We present an algorithm for smoothing results of three-dimensional Monte Carlo ion implantation simulations and translating them from the grid used for the Monte Carlo simulation to an arbitrary unstructured three-dimensional grid. This algorithm is important for joining various simulations of semiconductor manufacturing process steps, where data have to be smoothed or transferred from one grid to another. Furthermore different grids must be used since using ortho-grids is mandatory because of performance reasons for certain Monte Carlo simulation methods. The algorithm is based on approximations by generalized Bernstein polynomials. This approach was put on a mathematically sound basis by proving several properties of these polynomials. It does not suffer from the ill effects of least squares fits of polynomials of fixed degree as known from the popular response surface method. The smoothing algorithm which works very fast is described and in order to show its applicability, the results of smoothing a three-dimensional real world implantation example are given and compared with those of a least squares fit of a multivariate polynomial of degree 2, which yielded unusable results.

Original languageEnglish (US)
Pages (from-to)219-230
Number of pages12
JournalMathematics and Computers in Simulation
Issue number2-3
StatePublished - Jun 29 2004
Externally publishedYes


  • Bernstein polynomials
  • Ion implantation
  • Monte Carlo simulation
  • Smoothing operator

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science
  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics


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