In this paper, we present the numerical analysis and simulations of a multi-dimensional memristive device model. Memristive devices and memtransistors based on two-dimensional (2D) materials have demonstrated promising potential as components for next-generation artificial intelligence (AI) hardware and information technology. Our charge transport model describes the drift-diffusion of electrons, holes, and ionic defects self-consistently in an electric field. We incorporate two types of boundary models: ohmic and Schottky contacts. The coupled drift-diffusion partial differential equations are discretized using a physics-preserving Voronoi finite volume method. It relies on an implicit time-stepping scheme and the excess chemical potential flux approximation. We demonstrate that the fully discrete nonlinear scheme is unconditionally stable, preserving the free-energy structure of the continuous system and ensuring the non-negativity of carrier densities. Novel discrete entropy-dissipation inequalities for both boundary condition types in multiple dimensions allow us to prove the existence of discrete solutions. We perform multi-dimensional simulations to understand the impact of electrode configurations and device geometries, focusing on the hysteresis behavior in lateral 2D memristive devices. Three electrode configurations -- side, top, and mixed contacts -- are compared numerically for different geometries and boundary conditions. These simulations reveal the conditions under which a simplified one-dimensional electrode geometry can well represent the three electrode configurations. This work lays the foundations for developing accurate, efficient simulation tools for 2D memristive devices and memtransistors, offering tools and guidelines for their design and optimization in future applications.

翻译：本文提出了一种多维忆阻器件模型的数值分析与模拟方法。基于二维材料的忆阻器件与忆晶体管已展现出作为下一代人工智能硬件与信息技术组件的巨大潜力。我们的电荷传输模型自洽地描述了电子、空穴及离子缺陷在电场作用下的漂移-扩散过程。模型包含两种边界模型：欧姆接触与肖特基接触。耦合的漂移-扩散偏微分方程采用保持物理特性的Voronoi有限体积法进行离散化，该方法基于隐式时间步进格式与过剩化学势通量近似。我们证明了全离散非线性格式具有无条件稳定性，能够保持连续系统的自由能结构并确保载流子密度的非负性。针对多维情形下两种边界条件类型提出的新型离散熵耗散不等式，使我们能够证明离散解的存在性。通过执行多维模拟，我们研究了电极构型与器件几何结构的影响，重点关注横向二维忆阻器件的迟滞行为。针对不同几何结构与边界条件，我们对三种电极构型——侧面接触、顶部接触及混合接触——进行了数值比较。这些模拟揭示了简化的一维电极几何结构能够有效表征三种电极构型的适用条件。本工作为开发精确高效的二维忆阻器件与忆晶体管模拟工具奠定了基础，为其在未来应用中的设计与优化提供了方法与指导。