GATEPOT

The code GATEPOT computes the electrostatic potential produced inside a semiconductor structure by a generic arrangement of polygonal gates placed at the surface of the semiconductor and biased with appropriate voltages. Our algorithm is an implementation of the procedure proposed by Davies et al. [1], based on a semi-analytical solution of the Laplace equation, assuming an electrostatic potential equal to zero at the exposed surface of the semiconductor (this corresponds to assuming Fermi level pinning at the surface). The computed potential does not take into account screening effects and the numerical procedure in our code does not allow inclusion of variations of permittivity within the semiconductor: screening effects can be separately included, as a correction to the bare confinement potential obtained from our code, while permittivity variations can be, in many cases, ignored because they are relatively small. For example, in the case of a GaAs/AlGaAs heterostructure, the relative permittivity has a variation between the two materials that can be neglected in most calculations: 11.76 for AlGaAs and 12.9 for GaAs. The 2DEG is assumed to be much closer to the surface than to the conducting substrate, so that it is possible to neglect image effects from the substrate.

The input data required by the GATEPOT code are read from two files: poly.in and indatab.in. The file poly.in contains the description of the polygons defining the gates or the semiconductor surface area without gates: the former is actually the default assumption for our code, which considers the polygons provided in poly.in as the perimeters of the gates. The structure described in the example file poly_ex.in is represented in Fig. 1: two square gates 100 nm wide have been obtained on top of the otherwise exposed surface of the semiconductor. The poly.in file contains, in the first line, the total number of polygons, 2 in this case; then each of the polygons is described, providing the number of vertices, the applied voltage, and the coordinates of each vertex, listed according to clockwise ordering (if polygon vertices were listed counterclockwise, the current version of the program would not produce meaningful results). For example, the first polygon has 4 vertices, at (0,0), (0,100), (100,100) (100,0).

The other input file, indatab.in contains the remaining geometric parameters:

• the first line contains the distance between the surface of the heterostructure and the 2DEG, i.e. the depth of the 2DEG, in nanometers;
• the lines from the second to the fifth contain, in the order, the minimum and maximum coordinates and the minimum and maximum coordinates, in nanometers, for the grid on which the potential values are to be computed;
• the sixth line contains the number of discretization points for the potential along ;
• the seventh line contains the number of discretization points for the potential along .

The potential is computed as potential energy, in eV, for a positive unit test charge (i.e. it is the opposite of the potential for electrons) and is written in the output file pot.out. The content of pot.out consists in a vector obtained scanning the values of the potential over the grid on which it is defined, with the index for the coordinate running faster. A blank line is inserted at the end of the subvector corresponding to each horizontal row of potential values, in order to allow 3D plotting with gnuplot.

The result of the calculation performed with the data in the template file is shown in Fig. 2: we obtain two potential minima, corresponding to the two square gates (the voltage applied to the gates is negative and therefore they produce negative minima).

Figure 2: Potential due to the gate arrangement of Fig. 1 with an applied voltage of V

So far we have discussed usage of the program to compute the effect of polygonal gates on an otherwise exposed surface. It is possible to use the same code to compute the effect of polygonal ``holes'' in an otherwise full gate, i.e. swapping ``empty'' and ``full'' regions in Fig. 1. To do this, it is sufficient to consider the same voltage applied to all the polygonal gates (corresponding to the holes), equal in module to that of the full gate but opposite in sign. The we must subtract the same quantity from the the computed potential. For example, if we want to determine the potential produced by polygonal holes in a gate kept at a voltage of V, we can run the code defining gates in the regions were the holes are, with an applied voltage of V, and then subtract 1.5 eV from the computed potential.

Currently, the maximum number of polygons that can be treated by the code is 100 and the maximum size of the discretization mesh for the computation of the potential is . Each polygon can have at most 500 vertices. It is possible that error conditions occur if the vertex of a polygon is exactly on the vertical of one of the mesh points: to prevent such occurrences, it is sufficient to introduce a very small shift for the polygon coordinates.