Two-Dimensional Numerical Approach



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Two-Dimensional Numerical Approach

With regard to the tunneling path, two approaches have been investigated and implemented in MINIMOS [160]:

To calculate the tunneling-generation rate which is associated with an individual tunneling path, the models for direct and indirect tunneling as function of the electric field at the starting point , an average field along the tunneling path and the field variable from to the field at the endpoint have been considered. We have assumed that the electric field strength linearly varies along the tunneling path [456][229][170]: . This assumption is correct for the totally-depleted region in one dimension. Our analysis show, however, that the linear field variation can only be considered as an approximation for the gate/drain overlap region. In Appendix H, an approximate model for direct tunneling in a linearly variable field is derived. The model reduces to Kane-Keldysh expression in a constant field [244][237][236], including the dependence of the transition probability on the transverse impulse component . The expressions , and are employed in both, the one-dimensional and the two-dimensional approach and compared with each other in the following section. Note that direct tunneling models are assumed in the comparison although the MOSFETs of interest are silicon devices. The reason for that is the present lack of expressions for phonon-assisted tunneling in a variable field.

 

The calculated generation rates of electrons and holes are separated in the position space, as can be seen in Figure 4.3. The proper spatial positions of the generated carriers could be of importance when the secondary effects are analyzed, like the acceleration of the generated carriers and thereby induced impact ionization and hot-carrier injection. Note that the local electric field strength can differ significantly at the places where the holes and electrons are generated ( versus ). The generation rates are coupled with the continuity equations via the generation terms in a selfconsistent manner after filtering. The total charge in devices is strictly conserved. The filtering is only applied for smoothing the distributions of the generated carriers, to allow an efficient and automatic grid adaption in the critical areas. Since the tunneling rate is strongly dependent on the electric field, tunneling analysis is done on the grid which is finer than the mesh for solving the basic semiconductor equations.

 

Terminal currents are calculated by an accurate technique which is derived assuming the local concentration-dependent weighting functions, as noted in Section 3.2.4.



next up previous contents
Next: The Potential and Up: 4.2 Numerical Analysis of Previous: 4.2 Numerical Analysis of



Martin Stiftinger
Sat Oct 15 22:05:10 MET 1994