4.2 Transient Simulation

At the beginning of a transient simulation in MINIMOS-NT a stationary simulation is performed, i.e. all time derivatives are set to zero. The result of this simulation is used as starting point of the subsequent transient simulation steps.

MINIMOS-NT uses the backward Euler method for time integration. In this method for example the derivative of the electrostatic potential $\varphi$ for the time step n is discretized as

$${\frac{\partial \ensuremath{\varphi}_{n}}{\partial t}} {\frac{\ensuremath{\varphi}_{n} - \ensuremath{\varphi}_{n - 1}}{t_{n} - t_{n - 1}}}$$ (4.1)

Here the subscript n denotes the values of the present time step and the subscript n - 1 the values of the previous time step. To calculate the values of the unknowns for time step n only the values of the previous time step n - 1 are necessary.

For each time step n the step size $\Delta$t_n has to be chosen based on the result of the last time step. To ensure a certain simulation accuracy the step size must not be too big. On the other hand the step size also should not be too small to minimize the CPU time required for the simulation and to avoid numerical problems. The algorithms used in MINIMOS-NT to determine the step size for the transient simulation of CCDs are described in the next section.

Figure 4.4: Control flow of a transient simulation.

After choosing the time step size the contact quantities are interpolated for this step from the specified values as described in Section 4.3.3. Using these values the transient simulation step is performed. Then the norm of the potential update from the last time step to the present is calculated and compared to a predefined threshold. In case the norm of the potential update exceeds this threshold the time step has to be repeated using a reduced step size. Otherwise the next time step is calculated. The control flow of a transient simulation is shown in Fig. 4.4.