### Abstract

An algorithm for smoothing results of three-dimensional (3-D) Monte Carlo ion implantation simulations and translating them from the grid used for the Monte Carlo simulation to an arbitrary unstructured 3-D grid is presented. This algorithm is important for joining various process simulation steps, where data have to be smoothed or transferred from one grid to another. Furthermore, it is important for integrating the ion implantation simulator into a process flow. One reason for using different grids is that for certain Monte Carlo simulation methods, using orthogrids is mandatory because of performance reasons. The algorithm presented sweeps a small rectangular grid over the points of the new tetrahedral grid and uses approximation 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 adverse effects of least squares fits of polynomials of fixed degree as known from the response surface method. The most important properties of Bernstein polynomials generalized to cuboid domains are presented, including uniform convergence, an asymptotic formula, and the variation diminishing property. The smoothing algorithm which works very fast is described and, in order to show its applicability, the resulting values of a 3-D real world implantation example are given and compared with those of a least squares fit of a multivariate polynomial of degree two, which yielded unusable results.

Original language | English (US) |
---|---|

Pages (from-to) | 879-883 |

Number of pages | 5 |

Journal | IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems |

Volume | 22 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

- Approximation methods
- Integrated circuit ion implantation
- Monte Carlo methods
- Polynomials

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Hardware and Architecture
- Computer Science Applications
- Computational Theory and Mathematics

### Cite this

*IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems*,

*22*(7), 879-883. https://doi.org/10.1109/TCAD.2003.814259

**On smoothing three-dimensional Monte Carlo ion implantation simulation results.** / Heitzinger, Clemens; Hössinger, Andreas; Selberherr, Siegfried.

Research output: Contribution to journal › Article

*IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems*, vol. 22, no. 7, pp. 879-883. https://doi.org/10.1109/TCAD.2003.814259

}

TY - JOUR

T1 - On smoothing three-dimensional Monte Carlo ion implantation simulation results

AU - Heitzinger, Clemens

AU - Hössinger, Andreas

AU - Selberherr, Siegfried

PY - 2003/7

Y1 - 2003/7

N2 - An algorithm for smoothing results of three-dimensional (3-D) Monte Carlo ion implantation simulations and translating them from the grid used for the Monte Carlo simulation to an arbitrary unstructured 3-D grid is presented. This algorithm is important for joining various process simulation steps, where data have to be smoothed or transferred from one grid to another. Furthermore, it is important for integrating the ion implantation simulator into a process flow. One reason for using different grids is that for certain Monte Carlo simulation methods, using orthogrids is mandatory because of performance reasons. The algorithm presented sweeps a small rectangular grid over the points of the new tetrahedral grid and uses approximation 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 adverse effects of least squares fits of polynomials of fixed degree as known from the response surface method. The most important properties of Bernstein polynomials generalized to cuboid domains are presented, including uniform convergence, an asymptotic formula, and the variation diminishing property. The smoothing algorithm which works very fast is described and, in order to show its applicability, the resulting values of a 3-D real world implantation example are given and compared with those of a least squares fit of a multivariate polynomial of degree two, which yielded unusable results.

AB - An algorithm for smoothing results of three-dimensional (3-D) Monte Carlo ion implantation simulations and translating them from the grid used for the Monte Carlo simulation to an arbitrary unstructured 3-D grid is presented. This algorithm is important for joining various process simulation steps, where data have to be smoothed or transferred from one grid to another. Furthermore, it is important for integrating the ion implantation simulator into a process flow. One reason for using different grids is that for certain Monte Carlo simulation methods, using orthogrids is mandatory because of performance reasons. The algorithm presented sweeps a small rectangular grid over the points of the new tetrahedral grid and uses approximation 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 adverse effects of least squares fits of polynomials of fixed degree as known from the response surface method. The most important properties of Bernstein polynomials generalized to cuboid domains are presented, including uniform convergence, an asymptotic formula, and the variation diminishing property. The smoothing algorithm which works very fast is described and, in order to show its applicability, the resulting values of a 3-D real world implantation example are given and compared with those of a least squares fit of a multivariate polynomial of degree two, which yielded unusable results.

KW - Approximation methods

KW - Integrated circuit ion implantation

KW - Monte Carlo methods

KW - Polynomials

UR - http://www.scopus.com/inward/record.url?scp=0037698165&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037698165&partnerID=8YFLogxK

U2 - 10.1109/TCAD.2003.814259

DO - 10.1109/TCAD.2003.814259

M3 - Article

AN - SCOPUS:0037698165

VL - 22

SP - 879

EP - 883

JO - IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

JF - IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

SN - 0278-0070

IS - 7

ER -