Switching of a closed mobile vacancy based memristor, whose specific resistance linearly depends on local vacancy concentration

The linear (proportional to local vacancy concentration) term in specific resistance of the material does not directly contribute to the change of memristor's total resistance when the vacancies are redistributed inside while keeping their total number constant. But it still changes kinetics of the vacancy drift under the influence of a passing electric current. These changes are especially significant in the presence of metal-insulator phase transition in the memristor's material. In this paper, kinetic equation for local vacancy concentration is obtained, and exact solutions for its steady states are analyzed. It is shown that not only in the weakly nonlinear case (when the dependence of the specific resistance on the vacancy concentration can be neglected), but also in a strongly non-linear memristor with phase transition, its kinetics can be reduced to the classical exactly solvable Burgers equation.