4.3.2 System HAMILTONian

The HAMILTONian for a system consisting of $n$ grid points coupled to a reservoir at $m$ of these points is described by the following matrix equation, where a simple one-dimensional finite-difference discretization was performed. The lines indicate non-zero elements.

Figure 4.7: Matrix equation for the system HAMILTONian.

The applicability of this solver is therefore not limited to the calculation of transmission coefficients or the calculation of eigenvalues and life times, but can perform both operations based on the same data. If the right hand side of one of the points is set to a value $\ne 0$, the transmission coefficient can be calculated according to the value of the wave function at the corresponding node. Furthermore, the module allows to calculate the wave function and the carrier concentration for both open- and closed-boundary cases.

Fig. 4.8 shows a flowchart of a possible application of the Schrödinger solver module where optional modules are indicated by dotted boxes. At the beginning the constructor is invoked to initialize the variables and the memory for the barrier is allocated. In the next step the closed-boundary HAMILTONian is set up. This step has an interface to read the energy barrier from MINIMOS-NT, but it is also possible to specify the barrier manually. Optionally the values in the barrier can be checked and printed to a file. By means of the open flag the open- and closed-boundary solver is distinguished.

If injection points are added, the equation system is solved by means of a complex solver. Otherwise the eigenvalues of the closed system are found using an eigenvalue solver (see Section 4.3.3). Both solvers are part of the numerical library of MINIMOS-NT. In both cases the carrier concentration and the wave function can be calculated, while the transmission coefficient can only be calculated for the open system and is directly returned to the tunneling model in MINIMOS-NT.

The output of the program consists of eigenvalues, wave functions, the transmission coefficient, and the carrier concentration. It can either be written in CRV-format (one-dimensional, for use in the program XCRV [224]), in PIF-format (two-dimensional, for use in the program XPIF2D [224]), in DX-format (three-dimensional, for use in the program DATA EXPLORER [236]), or in WSS-format (three-dimensional, for use in the program SMARTVIEW [237]).

Figure 4.8: Flow chart of the SCHRÖDINGER solver. Optional modules are indicated by dotted boxes.

A. Gehring: Simulation of Tunneling in Semiconductor Devices