4.3.1 Open and Closed Boundary conditions

First, the closed-boundary matrix equation (3.103) is set up, as indicated by the one-dimensional energy barrier $W(x)$ in Fig. 4.6. There are closed boundary conditions at the points 0 and 9, respectively. If the system is coupled to a reservoir at so called connection points, injection points must be given which determine the values of $W$, $\ensuremath{{\mathcal{E}}_\mathrm{f}}$, and $m$ at the reservoir. As described in Section 3.5.4, the coupling entries are calculated by expressions such as (3.87) and (3.88), where the values of the wave vector are

$$k_j = \frac{1}{\hbar} \sqrt{2 m_j ({\mathcal{E}}- W_j)} \ .$$ (4.6)

Note that these values may be complex. Injection points are stored in a table which holds the information about the electrostatic potential, the electron mass, and the FERMI level at the injection point. If the transmission coefficient has to be calculated, the points which are considered for tunneling -- the boundary nodes and their partner nodes -- are used to set up these injection and connection points. The transmission coefficient is then calculated from the wave functions entering and leaving the simulation domain.

Figure 4.6: One-dimensional energy barrier: Injection points are coupled to connection points.

A. Gehring: Simulation of Tunneling in Semiconductor Devices