4.2 The Tunneling Model

The main problem for the integration of tunneling current models in a device simulator such as MINIMOS-NT is that tunneling is a non-local effect. In contrast to the current density described by the drift-diffusion (2.4) or energy-transport model (2.8), the current density at a certain point does not only depend on quantities at the same point, but on geometrical properties such as the thickness of the segment considered for tunneling. Thus, the tunneling current contribution cannot be simply derived from local quantities alone. In MINIMOS-NT the tunneling current is calculated between two boundaries of insulator or semiconductor segments. The boundaries are either specified by the user (see Appendix D) or found automatically. In the latter case the tunneling boundaries are identified as the first two boundaries of the specified segment to neighboring non-insulating materials which have the smallest distance^4.1. For each grid node at the specified boundary, the node on the other boundary with minimum distance is selected as partner node. It may happen that some nodes share their partner nodes, such as the nodes $i$, and $i+1$ in Fig. 4.1. Thus, this implementation is valid for non-orthogonal grids, too.

Figure 4.1: Boundary node - partner node pairs. The considered boundaries are indicated by bold lines.

The physical quantities at the neighboring segments, such as the carrier concentration, the electrostatic potential, and the carrier temperature, are passed to the tunneling model which is evaluated for each boundary grid point. Then, the tunneling current density is calculated by one of the models described in Section 3 and the total tunneling current is found by summation of the current density along the boundary and multiplication with the area of the grid element. A projection factor $\alpha_i$ is calculated for every node $i$ to account for pair nodes which do not lie directly opposite to each other:

$$\alpha_i = \left\vert \frac{{\mathbf{x_1}} \cdot {\mathbf{x_2}}}{x_1 x_2} \right\vert = \left\vert\cos(\gamma_i)\right\vert \ ,$$ (4.1)

where ${\mathbf{x_1}}$ points from the boundary node to the partner node and ${\mathbf{x_2}}$ to the next node on the boundary. In Fig. 4.1, for example, the tunneling current is calculated for the boundary nodes $i$ and $i+1$ with respect to the partner node $j$.

The total tunneling current is calculated by a summation along the boundary with length $L$ which consists of $N$ segments

$$I = w \int_0^{L} J(x) \, \ensuremath {\mathrm{d}}x \approx w \sum_{i=1..N} J_{i} \Delta x_i \cos(\gamma_i) \ ,$$ (4.2)

where $w$ is the gate width, $J_i$ the local tunneling current density, and $\Delta x_i$ the interface length associated with the node $i$. The local tunneling current density $J_i$ is added self-consistently to the continuity equation of the neighboring segments by means of an additional recombination term $\ensuremath{R_\mathrm{tun}}= \ensuremath{J_\mathrm{tun}}/ \ensuremath {\mathrm{q}}w$

$$\renewedcommand{arraystretch}{2.2}\begin{array}{l} \displaystyle ... ...rtial t} - \ensuremath {\mathrm{q}}\ensuremath{R_\mathrm{tun,p}}\ . \end{array}$$ (4.3)

In MINIMOS-NT the NEWTON^4.2 method is used to calculate the solution vector consisting of $n$, $p$, and $\phi$ at step $k+1$ from the matrix equation

$$\left( \begin{array}{c} \phi_{k+1} \\ n_{k+1} \\ p_{k+1} \\ \end{... ..._k, n_k, p_k\right) \\ f_p\left(\phi_k, n_k, p_k\right) \\ \end{array} \right),$$ (4.4)

where $f_\phi$, $f_n$, and $f_p$ denote the control equations determining the electrostatic potential, the electron concentration, and the hole concentration. Since $\ensuremath{R_\mathrm{tun}}$ modifies all solution variables, the JACOBI an^4.3 $\ensuremath{{\underline{J}}}$ must be modified to achieve better convergence of the NEWTON solver. Therefore, the derivatives of the additional recombination term with respect to the potential, electron concentration, and hole concentration

$$\begin{array}{c@{\quad\quad}c@{\quad\quad}c} \displaystyle\fr... ...c{\partial \ensuremath{R_\mathrm{tun}}}{\partial p} \end{array}$$

have to be calculated. For the FOWLER-NORDHEIM, SCHUEGRAF, and FRENKEL-POOLE model, the derivatives are calculated analytically while for all other models they are calculated numerically.

Subsections

• 4.2.1 Single Segment Tunneling
• 4.2.2 Stacked Segment Tunneling
• 4.2.3 Trap-Assisted Tunneling

A. Gehring: Simulation of Tunneling in Semiconductor Devices