F.2 Gate-Bias Shift

The dashed curve and the data and are connected with the second problem. In order to obtain the exact solution to this problem, a procedure similar to that performed for solving the first problem may be employed. This is not only cumbersome, but also not necessary. Hence, an approximate, but very simple solution to the second problem is developed, which relies on the solution to the first problem. Let us assume that the width of the depletion region in the bulk is for the gate bias . No interface charge is present in device at the moment. Suppose that the gate bias changes to a value . For small bias changes the boundary of the depletion region modulates only slightly ; the capacitance of the depletion region remains nearly constant. Under these conditions the surface potential variation may be found by

Let us insert a localized interface charge into the device. The charge perturb the surface potential by which is the solution to the first problem when and are assumed to be constant. The solution to the second problem follows from the principle of superposition: . This equality means that is the same at the gate bias with the presence of the interface charge as those at the gate bias and absence of any charge. A connection between the solutions to both problems can be easily established

Remark that holds. In typical conditions holds and both solutions come close to each other. In order to confirm relationship F.21, we have to find an efficient way to calculate when . This case is of interest because becomes comparable to . In the absence of a model which is more convenient than F.15, we will confirm relationship F.21 for a large width of the depletion region, where for relationship F.16 is used. As can be seen in Figure F.2, the theory agrees with the numerical results (dashed curve versus points).

When considering an infinitely long interface charge-sheet, expression F.21 together with F.17 reduces properly to the well known result: . This result is correct for arbitrary large charge densities, in spite of formula F.21 being derived for small bias changes. For a spatially uniform charge, the same conditions at the interface automatically result in the same and expression F.21 still remains valid.

As derived, the extreme value of the band-bending is , whereas for the gate-bias shift . From F.17 it follows

To confirm this equality we plotted the numerical data from Figure F.2 normalized with the extreme values (Figure F.3). The extremes are calculated by numerical simulation, assuming uniform charge along the whole channel. Additional calculations are performed for heavily doped bulk , depleted shallowly so that . These results are depicted by the points and . Numerical simulations nicely reflect relationship F.22. Small differences between the normalized and data for the same bulk dopant concentration originate due to different (just under the traps) in the calculations. In the main text, Section 3.5, we have adduced that the field lines from the interface surrounding the charge end in the space-charge region under the localized charge. Is has been heuristically concluded that this two-dimensional effect reduces the perturbation of the surface potential. This conclusion is not in coherence with Figure F.2, where for a finite falls below the characteristic for an infinite , because we expect that the two-dimensional effect attenuates with decreasing . However, the lowering of with stems from lowering the extreme , while the ratio increases as a consequence of the reduced two-dimensional effect. With decreasing , the factor increases. It turns out that increases with decreasing , approaching the value which corresponds to the uniform charge.

Figure F.4 shows the spatial distribution of the flat-band potential in our long channel MOSFET in the presence of interface charge, computed numerically. The bulk is uniform, heavily doped , of -type. The local flat-band potential is defined as the gate-bulk voltage which induces the flat-band condition at the point : . Here we deal with the typical second problem. At a specific gate bias , where is the relative coordinate , the part of the interface in the interval is accumulated due to negative charge, while it is depleted out of this region. Specially at , the interface is depleted almost in complete. Because of a high doping the depletion region is shallow; only at . Since is very small, increases significantly, easily approaching the extreme value. Therefore, the simple engineering estimate for the local shift is more accurate for the shallowly depleted bulk than for the largely depleted bulk. Moreover, may be well used when dealing with very shallow depletion, even for a quite localized interface charge (Figure F.4).

So far we have contemplated on the bulk depletion. In conditions comprised in the example shown in Figure 3.24 a considerable part of the interface is inverted. The inversion layer screens the bulk under the localized charge from possible influences of the gate-bias variations; the screening reduces the two-dimensional effect under the interface charge. Presently, we did not develop an appropriate theory to account for the screening (see [331][40]). Reasoning heuristically, one expects that the screening attenuates the induced , but increases with respect to its extreme value. The necessary to induce increases in comparison with the case when the inversion layer vanishes. The bulk capacitance, consisting of a large inversion-layer capacitance and the depletion capacitance , becomes much larger than . Rigorously considering, the problem is nonlinear and the moving charge cannot be neglected, as done by deriving the present theory for . We demonstrate the effect on two examples. In Figure 3.24 the interface is inverted in all points whose is below a specified gate bias, whereas it is depleted around the central point , due to negative interface charge.

In Figure F.5 the bulk is inverted in the area surrounding the point , else it is completely depleted for all points whose is larger than a specified gate bias, because of positive interface charge. As arising from numerical modeling, the attenuation of the local threshold-voltage shift is larger when the interface around the charge is depleted than when it is inverted, in spite of a larger depletion-layer width assumed in the calculation in the latter case. As already mentioned, larger leads to lower . We ascribe this finding to the screening in the former example.

Although the exact results can be obtained by means of numerical simulation only, because of the screening, the possible lateral current flow and the inhomogeneity in the two-dimensional doping profile, the given analytical models are still a good basis for evolution of the importance of the effect. Further work should cover finding an efficient representation of expression F.21 and simplifying the developed models, in order to make them more appropriate for engineering purposes.