Snap-through transition of graphene membranes for memcapacitor applications: A combined study using MD, DFT and elasticity theory

Using computational and theoretical approaches, we investigate the snap-through transition of buckled graphene membranes. Our main interest is related to the possibility of using the buckled membrane as a plate of capacitor with memory (memcapacitor). For this purpose, we performed molecular-dynamics (MD) simulations and elasticity theory calculations of the up-to-down and down-to-up snap-through transitions for membranes of several sizes. We have obtained expressions for the threshold switching forces for both up-to-down and down-to-up transitions. Moreover, the up-to-down threshold switching force was calculated using the density functional theory (DFT). Our DFT results are in general agreement with MD and analytical theory findings. Our systematic approach can be used for the description of other structures, including nanomechanical and biological ones, experiencing the snap-through transition. Introduction Memcapacitors1 are an emerging type of circuit elements with memory whose instantaneous response depends on the internal state and input signal. Such devices are prospective candidates for applications in information storage and processing2, 3 technologies as their states can be manipulated by the applied voltages or charges and can store information for long intervals of time. Several possible realizations of memcapacitors were suggested by using micro-electro-mechanical systems4, ionic transport5, electronic effects6, superconducting qubits7, etc.8 Generally, voltage-controlled memcapacitive systems (memcapacitors) are described by1 q(t) = C (x,V, t)V (t), (1) ẋ = f (x,V, t) , (2) where q(t) is the charge on the capacitor at time t, V (t) is the applied voltage, C is the memcapacitance (memory capacitance), x is a set of n internal state variables, and f is a continuous n-dimensional vector function. In some cases, it is more convenient to consider charge-controlled memcapacitors1 such that the charge instead of voltage is considered as input. Among several possible realizations of memcapacitors, the membrane-based memcapacitors4 are of significant interest as their geometry makes them intrinsically suitable for non-volatile storage of binary information. Indeed, the buckled membrane used as the top capacitor plate (see Fig. 1 for schematics) has two stable buckled states corresponding to two distinct values of capacitance. It was suggested4 that the switching between these states can be performed using the attractive interaction of oppositely charged capacitor plates. Moreover, it was demonstrated theoretically that simple circuits of membrane memcapacitors offer an in-memory computing functionality3. In this work, we consider a possible realization of membrane-based memcapacitor4 employing a singleor multi-layer graphene membrane9–13 as its bistable plate (see Fig. 1). Our aim is to understand the basic physical processes and parameters underlying the snap-through transition of such membrane including details of the membrane dynamics, threshold forces, etc. For this purpose, we perform a combined study using MD, DFT and elasticity theory focusing on a single-layer graphene membrane with clamped boundary conditions. This choice of boundary conditions is justified by the typically strong adhesion ar X iv :1 70 7. 07 63 9v 2 [ co nd -m at .m es -h al l] 2 5 D ec 2 01 7 of graphene to substrates. Our results extend our prior DFT investigation14 of the up-to-down snap-through transition of graphene membrane with hinged boundary conditions. The combination of computational/theoretical methods adds breadth and depth to our analysis. Using MD simulations, we were able to understand main features of the membrane dynamics in the presence of an external force and after the force removal. This understanding has helped us to develop analytical models that resulted in compact algebraic expressions for the threshold switching forces. DFT calculations were used to validate MD results for the up-to-down transition. This paper is organized as follows. In Sec. ”Molecular Dynamics Simulations” we investigate the snap-through transition of graphene membranes using molecular dynamics simulations. In particular, MD simulations of the up-to-down and down-to-up transitions are reported in Subsec. ”Up-to-down transition” and ”Down-to-up transition”, respectively, while MD simulation details can be found in Supplementary Information (SI) Sec. ”MD Simulation details”. The standard elasticity theory is applied to the membrane switching in Subsec. ”Buckling and snapping-through within the theory of elasticity”. A phenomenological analytical model of the snap-through transition is presented in Subsec. ”Phenomenological elasticity theory” and in SI Sec. ”Phenomenological Elasticity Theory”. Our DFT calculations are summarized in Sec. ”Density Functional Theory”. The results obtained within different approaches as well as their implications are discussed in Sec. ”Discussion”. In this paper, the following notations are used: q the charge on capacitor (see Eq. (1)) V the applied voltage (see Eqs. (1), (2)) C the (memory) capacitance (see Eq. (1)) d the distance between fixed sides of membrane (see Fig. 1) h the distance between the bottom plate and the level of fixed sides (see Fig. 1) L the membrane length w the membrane width D = 1.6 eV the bending rigidity of graphene E2D = 340 N/m the 2D Young’s module ζ the deflection of membrane (see Eq. (3)) ζc the maximum deflection of membrane (see Eq. (7)) θi(s) the angle that the membrane makes with the horizontal (see Eqs. (16) and (17)), s the internal coordinate that changes between −1/2 and 1/2 (see Eqs. (16) and (17)) Ai and ci coefficients (see Eqs. (16) and (17)) zcm the center of mass position (see Eq. (19)) Ub, Ustr, Uext the bending, stretching and external potential contributions to the potential energy of membrane (see Eq. (18)) F↓ the up-to-down threshold switching force (see Eqs. (12), (20), (21), and (22)) F↑ the down-to-up threshold switching force (see Eqs. (14) and (28)) ε0 the vacuum permittivity Molecular Dynamics Simulations MD simulations are a well established modeling tool frequently employed in studies of nanoscale carbon-based materials15–36. We used classical MD simulations to investigate the dynamics of snap-through transition of buckled graphene membranes. Zigzag graphene nanoribbons (membranes) of two lengths were considered: the nanoribbon A, L = 54 Å (22 rings), and nanoribbon B, L = 103 Å (42 rings). Both nanoribbons were of the same width (w = 41 Å). In order to implement the clamped boundary conditions, the first two lines of carbon atoms at shorter sides were kept fixed. The buckling was realized by changing Figure 1. Schematics of the membrane memcapacitor employing a buckled graphene membrane as its top plate4.