The Monte Carlo simulation method [96][88][84], is a numerical technique to estimate solutions of problems with multi-dimensional integrals. As a stochastic technique the method can be used in various applications. The discussion in this section is limited however to its use to calculate simulated output statistics of a TCAD model when the parameters are sampled from appropriate distributions. In TCAD applications, a model is often an abstraction of all or a subset of the actual manufacturing process. Typical usage of the Monte Carlo method involves the prediction of manufacturing variations and tolerance analysis.
In essence, the basic procedure of Monte Carlo integration estimates the integral of a function over a multidimensional volume by substituting a summation over the discrete set of functional values at sampled inputs. A major feature of the Monte Carlo method is its dimensional independence [96]. Stated differently, the probability that the calculated integral using the Monte Carlo method is within plus or minus some distance of the true value can be readily determined from the sample size N, irrespective of the number of parameters.
Sampling consists of a statistical exploration of the parameter space using
computer generated pseudo-random numbers
. Random numbers are essentially uniform
deviates: Independent random variables distributed uniformly over the interval
. By evaluating prescribed functions of random numbers, random
variates from different distribution can be generated. In this section,
procedures for generating normal (i.e. Gaussian) and correlated multinormal
random deviates are described. These are the typical distributions used in
simulating random fluctuations of a semiconductor manufacturing process through
the equivalent TCAD model representation.
A brief review of some statistical background is given first.