Numerical analysis and simulation of lateral memristive devices: Schottky, ohmic, and multi-dimensional electrode models

TL;DR

The paper develops a vacancy-assisted drift-diffusion model for 2D TMDC memristive devices with Schottky and ohmic contacts and multiple electrode configurations. It contributes a physics-preserving, implicit finite-volume discretization and an entropy-dissipation framework that yields unconditional stability and existence of discrete solutions in multiple dimensions. Through 1D and 2D simulations, it analyzes boundary-model accuracy and the impact of electrode geometry on hysteresis, showing that 1D models can be adequate for small electrode regions while 2D modeling is essential for realistic contact layouts. The results provide rigorous numerical foundations and practical guidelines for simulating and optimizing TMDC-based memristors and memtransistors.

Abstract

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 for neuromorphic computing and next-generation memory technologies. 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.

Figures (10)

• Figure 1.1: (a) Simulated I-V curve of an MoS$_2$ memristive device showing a pinched hysteresis if mobile ionic defects are present (darkblue) and no hysteresis if the defects are immobile (red). (b) Typical representation of the curve in (a) on a semilogarithmic scale. (c) Atomic structure of 2D MoS$_\mathrm{2}$ (monolayer), as an example for a TMDC widely used for lateral memristive devices and memtransistors with indicated sulfur vacancy. (d) Cross sectional illustration of a three terminal memtransistor based on 2D MoS$_\mathrm{2}$, with two top electrodes and one bottom gate electrode. In contrast to the top electrodes, the gate electrode is electrically insulated from the 2D TMDC in the center. (e-g) Illustration of three different electrode configurations investigated in this work that are used in lateral memristive devices and memtransistors: side contacts, top contacts, and mixed contacts.
• Figure 2.1: Illustration of a three-dimensional geometry of a memristive device for a (a) side, (b) top, and (c) mixed contact with indicated substrate and TMDC layer $\mathbf{\Omega}$. Furthermore, the contact boundary $\mathbf{\Gamma}^C$, defined as the interfaces of the contacts (gold) and the TMDC material (blue), are highlighted in red for each configuration.
• Figure 3.1: Neighboring control volumes in (a) the interior of the device domain and (b) near outer boundaries $\mathbf{\Gamma}^C$ and $\mathbf{\Gamma}^N$ (right). For our numerical analysis, we assume that the cell centers (black points) of a boundary control volume are located in the interior of the computational domain. The boundary of the control volumes are highlighted in red.
• Figure 3.2: Summary of the proof schematic showing the different layers to prove \ref{['thm.exresult-memristor']}. Lemmas and theorems without red border can be easily adapted from Abdel2023Existence while the ones highlighted in red are the additional auxiliary results established in this work to prove the existence of discrete solutions.
• Figure 4.1: Comparison of charge carrier densities and I-V curves for the two contact boundary models \ref{['eq:BC-Schottky-time-dependent']} and \ref{['eq:BC-Ohmic-time-dependent']}. (a) Two consecutive voltage cycles applied at the right contact. All subsequent simulations correspond to the shaded second cycle. (b) Measured I-V curve from DaLi.2018 compared with second-cycle I-V simulations using Schottky boundary conditions \ref{['eq:BC-Schottky-time-dependent']} as well as ohmic boundary conditions \ref{['eq:BC-Ohmic-time-dependent']}. Arrows indicate the direction of hysteresis of all curves. (c) Electron, hole and defect densities $n_\text{n}$, $n_\text{p}$, $n_\text{a}$, respectively, during the second cycle at selected times. Colored lines correspond to the Schottky model \ref{['eq:BC-Schottky-time-dependent']}. In contrast, black dotted lines represent the densities computed with ohmic boundary conditions \ref{['eq:BC-Ohmic-time-dependent']}.

Theorems & Definitions (16)

• Theorem 2.1
• Lemma 3.1
• proof
• Theorem 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Remark 3.5
• Remark 3.6: Cell centers on boundary