3.2 Discretization at the Interface

At the interface of two regions with different material properties rapid variations of different quantities can occur within the immediate vicinity of the interface. In these regions of the simulation domain special measures have to be taken for an appropriate discretization.

The simplest approach is to increase the density of the grid which is used for the discretization in the vicinity of the interface where the quantities vary rapidly. The same discretization method is used as for segment volume. Fig. 3.2 shows the continuous band edge energy at an interface. Outside the region in which the strong variation occurs the difference in the band edge energies is $\Delta$W. In most applications the region where strong variations occur is much smaller than the rest of the simulation domain. This leads to a considerable number of additional grid points and strong variations in the grid spacing which degrades the condition of the equation system [21]. An additional problem at material interfaces is that many quantities which were defined in a macroscopic context are not valid on a sub-atomic scale.

Figure 3.2: Rapid variation of a continuous band edge at the interface x₀.

Figure 3.3: Idealized discontinuous band edge model.

Figure 3.4: Model of discontinuous band edge using three distinct values.

An other method is to idealize the vicinity of interfaces where the strong variations occur into an infinitely thin region where the quantities on each side of the interface are related by a special interface model. Fig. 3.3 shows the idealized situation in which the interface is characterized by the difference of the band edge energies $\Delta$W and the transmission probability D(k). This enables the treatment of abrupt changes without the need of increasing the grid density at the interface. Thereby the increase of the number of grid points and the degradation of the condition of the equation system caused by strong variations of the grid spacing is avoided. On the other hand the functions describing the abrupt variation at the interface might be quite complicated and also can cause poor convergence.

Both discretization methods can be used in MINIMOS-NT for modeling interfaces. When the so-called segment split method is used the adjoining segments are connected by appropriate interface models. At the interface there are three discretization points located at the same geometric coordinate but with three distinct values. These are the left and right side limits of the quantity at the interface and an additional value directly at the interface (see Fig. 3.4). The value at the center point could for example be used to account for the effects of an interface charge. All currently implemented models in MINIMOS-NT use only the left and right side limits.

When different sets of differential equations are used for two adjoining segments, special interface models have to be used to connect the two segments. For example when the hydrodynamic model is used in one segment and the drift-diffusion model in the other segment an assumption for the carrier temperatures at the interface has to be made. Values for the carrier temperature at the interface can be calculated by equating the field dependent drift-diffusion mobility and the carrier temperature dependent hydrodynamic mobility and solving for the carrier temperature.

$$\mu \nu \mu$$ (3.1)