Development of Parallel Codes for a Nonlinear Heat Equation in Problems of Phase-Change Memory Simulation
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lopment of Parallel Codes for a Nonlinear Heat Equation in Problems of Phase-Change Memory Simulation G. N. Shumkin* Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991 Russia Received March 12, 2020; in final form, May 20, 2020; accepted May 20, 2020
Abstract—Results are presented from developing a parallel code for a joint solution to the nonlinear three-dimensional heat equation and the Poisson equation in creating a multiscale model for phasechange memory based on amorphous carbon nanofilms (a-C). Their numerical convergence is investigated. The parallel effectiveness of the new code is examined on the IBM BlueGene/P supercomputer. Keywords: nonlinear three-dimensional heat equation, phase-change memory, BlueGene/P, multiscale model, amorphous carbon nanofilms. DOI: 10.3103/S027864192003005X
1. INTRODUCTION Materials for phase transitions are now being actively studied. Of these, carbon materials are of particular interest [1]. An S-shaped current-voltage characteristic is observed in a nanodot with a diameter of 20 nm inside thin nanofilms of amorphous carbon, as was shown by the experiment in [1]. This corresponds to resistance switching inside the nanodot. The thickness of the nanofilms in the experiment [1] varied from 20 to 100 nm. The possibility of switching the resistance in a nanodot of amorphous carbon allows us to consider this material as a promising candidate for phase-change memory. In this work, we consider a multiscale model based on a heat equation with a nonlinear heat source and the Poisson equation for calculating electric potential. The heat source in the heat equation is the Joule heating of an amorphous carbon nanofilm, and the nonlinear source is related to temperature dependence σ(T ) of electrical conductivity. For conductivity factor σ(T ) of a-C, we applied the following temperature dependence in the model: −Egap (T ) , (1) σ(T ) = σ0 exp kB T where kB is the Boltzmann constant, T is the temperature (Kelvin) of the sample, and Egap (T ) is the bandgap of the amorphous carbon nanofilm, depending on temperature. This type of conductivity corresponds to the conductivity in an amorphous semiconductor. It should be noted that in this case, we consider the transition from a low-conductivity amorphous state to a highly conductive graphitelike state resulting from heating to a critical temperature of 2200 K, at which there is a transition to a highly conductive state occurs, as was confirmed by quantum-mechanical calculations in [2]. Thermal breakdown in a highly conductive graphite-like state without restoration of the current-voltage characteristic was considered in [3], and a two-dimensional heat equation was examined. To solve the problem, a set of parallel programs was developed to solve the heat equation jointly with the Poisson equation. The Douglas–Reckford scheme [6, 7] was applied to approximate the nonlinear heat equation, which requires successive three tridiagonal matrix algorithms along three directions. Such a scheme is simple to paralle
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