2 Analytical Ion Implantation

  In order to obtain the depth distribution of an implanted dopant species, the most accurate results are issued from Monte Carlo simulations. Monte Carlo simulations are best suited to understand the underlying physics, but they require too much CPU time for systematic use needed for optimization purposes. For this reason the analytical simulation of the ion implantation process is commonly used. It is one of the simplest and time efficient ways to simulate the ion implantation process, and is implemented in many process simulators [Pic85a] [Law88] [Bac88b] [Mul89] [Hün90]. The basic idea is the approximation of the impurity profiles by statistic functions, where characteristic parameters, the so-called moments, are extracted from Monte Carlo calculations or from measurements [Rys81] [Tas89]. The exact shape of an arbitrary ion distribution can, of course, not be reproduced by the knowledge of only a number of characteristic parameters, but the difference between the purely mathematical statistical functions and the measured profiles is, however, small enough for most of the practical applications. In the following we present the most common statistical moments by means of one-dimensional probability density functions. To obtain multi-dimensional profiles one-dimensional probability density functions are convoluted in the space domain.

• 2.1 Statistical Moments and Distributions
  • 2.1.1 Central Moments and their Distribution Functions
  • 2.1.2 Probability Weighted Moments and their Distribution Functions
  • 2.1.3 Evaluation of Statistical Moments from Data Sets
• 2.2 Two-Dimensional Profiles in Multilayer Structures
  • 2.2.1 Numerical Range Scaling
  • 2.2.2 Convolution Method
• 2.3 Comparison Analytical - Monte Carlo Simulation Method