The role of semiconductor device diameter and energy-band bending in convergence of Picard iteration for Gummel's map

Steady-state semiconductor-device simulation is studied. A mathematical analysis of the so-called Lipschitz constant of the Gummel map, in the Slotboom variables, is given in terms of the device diameter
d
and the band-bending. This constant is directly related to the convergence of Picard iteration, within a convex, compact solution set, determined by the steady state, Van Roosbroeck system maximum principles. Allowing for specified gradient singularities, of order
cr^{-1/2}
we obtain an asymptotic dependence of
d^{5/2}
as
d \rightarrow 0
. Functions of sup-inf boundary values of the data also appear, particularly as related to doping concentrations, in the representation of the band-bending. For simplicity, constant mobilities and zero recombination are assumed, at thermodynamic equilibrium. The setting is general, but motivated by FET devices. For micron devices at room temperature, convergence and unique solutions are expected up to, perhaps, band-bending of 0.42 - ⅔ \log c(kT) electronvolts in width across the intrinsic energy level. The definition of the Gummel map used herein employs the potential equation as a fractional step, based upon given values of the Slotboom variables. The continuity equations are then solved for the values associated with the Gummel map. The condition given for convergence of Picard iteration also guarantees uniqueness of solutions of the device model for prescribed doping and contact conditions. Finally, the constant
c
is estimated on the basis of heuristic considerations.