Theoretical Consideration of the Potential Perturbation Induced by Localized Surface Charge
In Section 3.5 we have found that the shift on the voltage axis produced by a localized interface charge is smaller than that corresponding to a uniform interface charge (demonstrated in Figure 3.24). A rigorous analytical formulation of this effect is introduced in this appendix. The theoretical expressions are compared with the two-dimensional numerical calculations. Agreement and deviations between analytical and numerical models are argued in detail.
It is not possible to develope a rigorous analytical model of this effect in the
general case. For instance, the effect is also influenced by the current flow.
When adopting vanishing minority and majority-carrier currents, the problem
reduces to the Poisson equation solely, which is a nonlinear equation upon
the potential. For the charge pumping conditions considered in
Section 3.5, the carrier concentrations at the interface
typically satisfy the inequalities: 
, 
, where 
is
the bulk doping. Motivated by this fact, in order to be able to derive a
theoretical expression, a total depletion approximation is assumed to hold in
the active part of the bulk. A possible influence of the moving charge will
be noted when comparing theory with numerical results.
Two problems can be formulated:
