Van Roosbroeck System Research Articles

A uniqueness theorem for weak solutions of the stationary semiconductor equations

In this paper we prove a uniqueness theorem for weak solutions of a mixed boundary-value problem for the stationary semiconductor equations (van Roosbroeck's system) under the assumption that the deviation of the carrier potentials from an equilibrium solution is sufficiently small.

A new approach to the modeling of pnpn structures

An approach for the derivation of stationary voltage-current characteristics of pnpn-structures is presented. In contrast to the classical method of the two-transistor analog, the van Roosbroeck system is considered throughout the device. The problem is simplified by using the methods of asymptotic analysis: a suitable normalisation leads to the identification of small dimensionless parameters which are used in a formal perturbation procedure. Neglecting these small parameters is equivalent to making certain simplifying assumptions such as charge neutrality, depletion or low injection. The approximate problems are in general much easier to solve than the full van Roosbroeck system. The main results are explicit formulae for the stationary voltage-current characteristic for moderate currents, expressions for the holding current and the statement that, in the absence of impact ionisation, the breakover voltage can be explained by finite domain effects.

EXISTENCE AND REGULARITY FOR VAN ROOSBROECK SYSTEMS WITH GENERAL MIXED BOUNDARY CONDITIONS

Using the topological degree of Leray‐Shauder, and Grisvard's results for elliptic equations with mixed boundary conditions, we extend Mock's results for the steady‐state Van Roosbroeck system, with the change from Neuman to Dirichlet boundary conditions occuring at a flat angle. Similar results are obtained for continuity equations that include a general recombination rate.

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 /sup -1/2/, we obtain an asymptotic dependence of d /sup 5/2/ as d -> 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 /sup -2/3/ 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.

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.

Consistency of Semiconductor Modeling: An Existence/Stability Analysis for the Stationary Van Roosbroeck System

This paper examines the solution map of the stationary version of the Van Roosbroeck model for the flow of electrons and holes in a crystalline semiconductor. Thermodynamic equilibrium and the Einstein relations linking mobility and diffusion are assumed to hold for this model. Incorporated into the model is the quantum-statistical assumption that the carrier densities satisfy Fermi-Dirac statistical laws with localization in the “Boltzmann tail.” This latter approximation, which is seen to be inessential, characterizes the conduction and valence bands as nondegenerate. The boundary conditions are of two types, and correspond to applied potential differences and doping on the contact portions of the device, and insulation on the remainder. The complete system, then, involves a coupled set of three nonlinear partial differential equations, with their boundary conditions, for the electrostatic potential, and for the quasi-Fermi levels corresponding to the electron and hole concentrations. In the absence of the classical nondegeneracy approximation, the model remains valid, but without a natural physical interpretation for two of the three dependent variables. We demonstrate the existence of solutions for this system by associating a solution mapping T, such that the fixed points of T define solutions of the system. “A priori” estimates, or weak maximum principles, permit T to act invariantly on a naturally chosen subset of a Hilbert product space. The definition of T roughly follows the Gummel procedure for uncoupling systems; the Schauder fixed point theorem is used to prove the existence of a fixed point for T. Uniqueness is not demonstrated, and very possibly does not hold for these systems, without special restrictions on the boundary conditions. We also introduce a discrete version of the mapping T which is defined in terms of a family of finite difference approximations. This mapping enjoys the stability property that it acts invariantly on a rectangular region in Euclidean space; the dimensions are the same as those specified in the pointwise bounds which define the domain of T. A second existence theory framework, with quasi-constructive aspects is also studied. This is based upon pseudomonotone operators. The presentation is technology independent except for a section on MOS-FET devices. These typically induce a breakdown of uniform ellipticity, and may be handled by modifications of the major arguments. They also exemplify a class of devices for which the electrostatic field is defined on a strictly larger domain than the carriers themselves.