1.2 Device Optimization Methods

The setup of an optimization task includes the definition of the optimization parameters which can be varied and the definition of an optimization goal. The optimization goal, for example, can be minimizing a target function (also called ``cost function'') with a number of constraints.

In general, there are two main optimization methods applicable to device optimization purposes [42]:

• gradient based methods and
• genetic algorithms.

Gradient based optimization methods utilize the gradient vector and the Hessian matrix of the target function. The target function is minimized, step by step, using the information about the derivatives to determine the new step direction and step length in the optimization parameter space. Convergence is achieved when the target function cannot be improved anymore [29].

The derivatives of the target function are usually not available in a closed form since the target function itself is evaluated by numerical device simulation for a requested set of parameters and, therefore, only known for discrete values. Thus the derivatives have to be computed using a discrete method which requires a lot of additional evaluations of the model function.

The genetic algorithm is a model of machine learning which derives its behavior from some of the mechanisms of evolution in nature. This is done by the creation of a population of individuals represented by chromosomes. The individuals in the population then go through a process of simulated ``evolution''. In case of device optimizations the chromosomes can be used to encode the values for the optimization parameters [71].

A new population is created by performing operations such as crossover, mutation, and fitness-proportionate reproduction on the individuals whose fitness has just been measured. Therefore, the population is gradually ``growing'' into an optimum.

There are several issues to consider when looking for an optimization method which fits best to a specific optimization task. First of all, global optima can only be found using global search methods such as genetic algorithms. With gradient based methods one can never guarantee that the optimum found is a global one, but in many cases the target function is ``well-behaved'' and there are only a few local minima. Different initial solutions can help to determine the local minimum with the best target.

Second, the computational effort has to be considered. Global search methods usually need a very large amount of evaluations of the model function which reduces their feasibility for device optimization tasks as the evaluation of the model function uses time consuming numerical simulations. A gradient based method is more efficient in this case.

Third, the level of parallelization of the model function evaluations can lead to a reduction of the total optimization time. This is especially important for global search methods which are highly parallelizable. If there is a very large number of workstations available, these methods can become attractive even for device optimization tasks.

In general the target function can be approximated using Response Surface Models (RSM's) [9]. This means that polynomial functions, usually of second order, are used to approximate the target function in the multi-dimensional optimization parameter space. To find the coefficients for the polynoms a Design Of Experiment (DOE) is launched and the necessary numerical device simulations are performed [33]. Once the response surface is defined, a model function evaluation takes considerably less time since only polynomial functions have to be computed and no numerical simulations are necessary.

This approach has one major disadvantage: Quadratic functions have exactly one local optimum which can be far away from the real one if the target function has contributions of higher order. This becomes obvious when looking at a simple one-dimensional case, as shown in Fig. 1.1. The three known points P$_1$, P$_2$, and P$_3$ of the target $t$ are fitted using an RSM. The real target function is also drawn. It can be seen that there is a big gap between the two optima received by the RSM $p_1$, $t_1$ and the real target curve $p_2$, $t_2$.

Figure 1.1: Comparing an RSM to the real target function.