There are different methods of how directed quantities can be represented in box discretized systems. In the initial divergence equation (7.1) the vector quantities is transformed into one-dimensional edge discretized values which are subsequently used to assemble the box and flux equations. The single fluxes are estimated using quantities defined at the end nodes of the edges, like in relation (7.6) for the dielectric flux and (7.7) for the current density. No vector quantities are required to assemble the basic equations. However, vector attributes like the electric field and the current density are needed for the calculation of physical models. The electric field and the current density are required, for example, to model the high-field mobility or the impact-ionization rate in the drift-diffusion framework. For the representation of attributes as vector quantity, additional methods are required, some are represented in this section.
In the work of Laux et al.  a discretization scheme is introduced for the accurate evaluation of the impact-ionization rate. The model assumes an unstructured triangular mesh. Since the potential and the electric field are based on linear functions, an electric field vector can be evaluated in a straightforward manner, consistent with the Poisson discretization. In a triangle each pairwise linear combination of and gives the same constant value for the electric field
Due to the non-linear dependence of the current density on the carrier concentrations in the Scharfetter-Gummel discretization, each linear combination of and gives a different current density vector The system in the triangle is overdetermined. Laux suggested to partition each triangle into three avalanche regions associated with each edge as shown in Fig. 7.8.
In this work, a current vector is defined for each avalanche region, i.e. for each edge. Note, that in the notations used here, the indices for the carrier type specification have been omitted. The current associated to the edge between the vertices and for example, is defined as
This approach for vector discretization gives good results as discussed in Section 7.3.3. The limitation to triangular grids (although an extension to tetrahedra is possible) and the higher complexity during equation assembly seem to be the main disadvantages of this method compared the box based vector discretization schemes presented in the next section.
Considering the calculation of the summand in the box approach (7.10), it seems to be most convenient to estimate the vectorial attribute for each box volume, i.e. for each vertex. The model evaluation within the box can then be performed straightforwardly, since all quantities, scalar and vectorial, are then available for the whole box. The results of model evaluation can be directly applied to the box integration equation. In the following, two schemes of vector discretization within boxes are presented. In addition to the simple coupling to the box discretization method, both approaches give accurate approximations for homogenous fields and are numerically stable.
The first scheme follows the derivation of the box discretization scheme itself . Similar to the discussion regarding the right hand side of the box integral, the discretized vector quantities are assumed constant over the whole Voronoi box volume. The derivation of an according discretization scheme is shown for the electric field and can be generalized to auxiliary gradient fields. The electric field is defined as
The discretization scheme is also analyzed in a one-dimensional formulation applying a linearly changing electric field and an corresponding quadratic electrostatic potential Using only the x-axis and the naming convention from Fig. 7.9, (7.21) can be reduced for the linearly changing field to
The second discretization scheme is an extension of the finite difference method and is based on a scheme suggested by Fischer . The evaluation concentrates on the vector at the vertex itself. Considering the box 0 in a non-equidistant orthogonal mesh depicted in Fig. 7.9 and its neighboring box 1 (not shown explicitly), the electric field along the edge can be expressed as
At the boundary between the two boxes, i.e. the midpoint between 0 and 1, the finite difference method gives
Note that (7.25) and (7.26) are still retained and can be extracted by using and respectively. at the left side of (7.34) can be taken out of the sum and the remaining part of the sum results in a pure geometry dependent matrix, which is calculated once in the beginning of the simulation. This allows the convenient formulation of the final discretization rule for a vector in point as
Similar to Scheme A, the validation for homogenous fields using and the electrostatic potential is shown. Again, the relations and are inserted into (7.35) which leads to
The geometry matrix introduced in this discretization scheme needs to be inverted for evaluation. The matrix results from a sum of symmetric matrices whose determinants equal to 0 and whose main diagonals are positive. The sum of matrices with these constraints and with non-negative determinants also result in a symmetric matrix with positive main diagonal and a non-negative determinant. If at least two of the participating matrices are linearly independent, the determinant of the geometry matrix is positive which is the case as long as the Delaunay criterion is fulfilled. The inverse of the geometry matrix can therefore always be calculated.
In this comparison, the vector discretization scheme proposed by Laux and the two box discretized schemes where implemented in MINIMOS-NT and used to calculate the impact-ionization rate. The generation integral on the right hand side of the continuity equation is assembled using (7.10) and (7.12), respectively. The implementation of the scheme by Laux is only limited to two-dimensional domains using triangular meshes, whereas the two other schemes are dimension and mesh independent and the same program code can be used for two- and three-dimensional simulations.
Simulation results from two devices are presented. The first device is a diode which was selected to investigate effects in a simple one-dimensional device. The second device used for the comparison is a parasitic n -p-n-n structure of a smart power device which is significantly influenced by the two-dimensional extension. The diode uses an equidistant mesh and is investigated in reverse-biased operating condition for different mesh spacings. The parasitic bipolar smart power structure is simulated in snap-back, a state the device can be driven in during voltage peaks on the power line.
Simulations on the diode structure clearly show that the mesh dependency is higher for the two box based discretization schemes. Using a high mesh density results, as expected, in a comparable output for all three discretization methods. Using a coarser mesh spacing, the results using Laux's scheme change very little, whereas the two box based schemes show larger deviations. In Fig. 7.10 an example using the diode clearly shows that an increased mesh spacing leads to a shift of the breakdown voltage using the box based schemes, whereas only a very small shift is observed using the scheme by Laux.
The results for the snap-back simulation in the smart power device are depicted in Fig. 7.11 and show only small deviations between the three schemes. The two box based schemes again show a voltage shift in comparison to the scheme by Laux. The latter one fits well to the reference solution which was generated using a high mesh density (not shown in the figure). The reason for the stronger mesh dependence of the box based schemes can be found in the implicitly finer discretization used in Laux's scheme.
The influence of the vector discretization scheme on the convergence behavior is also investigated on the reduced smart power structure as shown in Fig. 7.11(b). Only very little differences were noted and no trend favoring one or another scheme was observed. The convergence process is tracked using the norm of the right hand side. The investigated simulation step is a numerically critical current level step at the triggering phase of the snap-back. It can be clearly seen that the choice of discretization method has only very little influence, despite the critical simulation step.
Although the Laux scheme gives results with a higher accuracy, the influence on the convergence behavior seems to be negligible. On the other hand, the box based schemes can be coupled straightforwardly to the box discretization scheme. Additionally, it is possible to reuse the same implemented code for arbitrary meshes in all dimensions. Throughout this work, the box based Scheme B has been used for all drift-diffusion simulations.