3.1 The Computation of the Vector Product

Proper simulation of semiconductor devices requires solving a set of coupled differential equations. Solving the Poisson and continuity equations implies partitioning the simulation domain into a finite number of subdomains where a solution can be approximated. The way in which a solution is approximated to a device geometry is called discretization. Any discretization needs a mesh, a group of points that defines lines, surfaces, and volumes which covers geometric information about the device geometry under consideration [11].

The discretization of the set of differential equations that describes the behavior of a semiconductor device without magnetic field can be found in several papers [11,39]. Via the Gauß theorem the divergence of a gradient on volumes is expressed in terms of the fluxes through the surfaces of the volumes [13]. Those fluxes are basically current projections along the meshing lines which are computed using the local information found at the end points of a mesh line. Additionally, the points are ordered in such a way that the current projections are diagonal entries in the Jacobian matrix. This eases the solution of the equation system.

Discretization of (2.18,2.19), the equations comprising the magnetic field for the current densities, introduces a serious complication in the discretization procedure: namely the computation of a vector product. With any discretization procedure the only available information is the projection of quantities along the mesh lines. There is no information about either the vector components or the direction of the components. Even worse, if an approximation of such vector components exists, the computation of the vector product will introduce non-diagonal entries in the Jacobian matrix, compromising the existence of a solution [13].

From the programming point of view, the discretization of (2.18,2.19) must be performed without modifying the discretization procedure, that is, the way how the projections are computed along the mesh lines in order to avoid non-diagonal entries in the Jacobian matrix. Any discretization procedure applied to (2.18,2.19) must prevail the details of the remaining code in any project, for example, the source code of MINIMOS-NT. It is obvious that the new discretization procedure must reduce to the original one when the magnetic field is zero. In the following section a discretization procedure taking into account the magnetic field will be developed based on the proposal of Gajewski and Gärtner [13]. The vectorial product is computed using only the information of the neighboring points, and actually it is not related to the discretization scheme used for computing the current projections along the mesh lines. That is, a drift-diffusion or a hydrodynamic approximation for computing the current projections along the mesh lines can be used together with the new discretization procedure to simulate magnetic effects in semiconductor devices.

The Hall mobilities are computed by means of the Hall scattering factors $r_n$ and $r_p$. Their values are 1.15 for electrons and 0.8 for holes [4] for bulk silicon at room temperature. Jungemann et. al [18] give values for the Hall scattering factor for electrons in inversion layers at different temperatures and electric fields (see Figure 3.1).

Figure 3.1: Hall scattering factor for electrons versus temperature (after [18]).