Optimization of the algorithm of the mathematical model of the charge and electric field distribution stabilization in a multilayer semiconductor structure with metal contacts

Background. At the present time the capacities of computer systems allow using mathematical modeling as one of the major methodological approaches to solve various scientific and engineering problems. The increasing complexity of research objects inevitably leads to complication of mathematical models, so the search for new computation algorithms optimization methods is an important task. This work is dedicated to a topical issue of design, analysis and optimization of mathematical models of multilayer semiconductor structures. The aim of this work is to develop a method of sequential adjustment and adaptation of local-field mathematical model algorithm which describes stabilization dynamics of the charge and the electric field distribution in multilayer silicon structures with non-ohmic metal contacts. Materials and methods. The authors carried out simulation in a one-dimensional coordinate system. The system of the model equations includes the equation of continuity, Poisson's equation with the appropriate boundary and initial conditions, and the equation for the total current density through the structure. The metal-semiconductor contact with a potential barrier of 0.3 eV is considered as a non-ideal ohmic contact. The algorithm optimization method consists of several main items. The relevant initial and boundary conditions are selected on the basis of the known physical concepts. Adjustment of the appropriate conditions improves the solution accuracy and convergence. Tradeoff between the time steps and the coordinate steps provides stability and fast setting of the stationary solution. Optimization is performed in stages for several types of this structure with a sequential increasing of complexity. Results. The authors developed a method of sequential adjustment and adaptation of local-field mathematical model algorithm which describes the stabilization dynamics of the charge and the electric field distribution in multilayer n${^+}$-n-n${^+}$ silicon structures with non-ohmic metal contacts. This method improves the accuracy of the solution and reduces the computation time and required computing power. The validity of the calculation results (electron density distributions, electric field and potential distributions, current-voltage characteristics) is confirmed by its qualitative agreement with known physical concepts. Conclusions. The developed method has both methodological and practical values and can be used for other mathematical models of more complex structures, considering the different external physical factors effect.