Abstract
Using computational and theoretical approaches, we investigate the snapthrough 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 moleculardynamics (MD) simulations and elasticity theory calculations of the uptodown and downtoup snapthrough transitions for membranes of several sizes. We have obtained expressions for the threshold switching forces for both uptodown and downtoup transitions. Moreover, the uptodown 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 snapthrough transition.
Introduction
Memcapacitors^{1} 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 processing^{2,3} 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 microelectromechanical systems^{4}, ionic transport^{5}, electronic effects^{6}, superconducting qubits^{7}, etc^{8}.
Generally, voltagecontrolled memcapacitive systems (memcapacitors) are described by^{1}
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 ndimensional vector function. In some cases, it is more convenient to consider chargecontrolled memcapacitors^{1} such that the charge instead of voltage is considered as input.
Among several possible realizations of memcapacitors, the membranebased memcapacitors^{4} are of significant interest as their geometry makes them intrinsically suitable for nonvolatile 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 suggested^{4} 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 inmemory computing functionality^{3}.
In this work, we consider a possible realization of membranebased memcapacitor^{4} employing a single or multilayer graphene membrane^{9,10,11,12,13} as its bistable plate (see Fig. 1). Our aim is to understand the basic physical processes and parameters underlying the snapthrough 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 singlelayer graphene membrane with clamped boundary conditions. This choice of boundary conditions is justified by the typically strong adhesion of graphene to substrates. Our results extend our prior DFT investigation^{14} of the uptodown snapthrough 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 uptodown transition.
This paper is organized as follows. In Sec. “Molecular Dynamics Simulations” we investigate the snapthrough transition of graphene membranes using molecular dynamics simulations. In particular, MD simulations of the uptodown and downtoup transitions are reported in Subsec. “Uptodown transition” and “Downtoup 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 snappingthrough within the theory of elasticity”. A phenomenological analytical model of the snapthrough 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) and (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 = 16 eV  the bending rigidity of graphene
E_{2D} = 340 N/m  the 2D Young’s module
ζ  the deflection of membrane (see Eq. (3))
ζ_{c}  the maximum deflection of membrane (see Eq. (7))
θ_{ i }(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))
A_{ i } and c_{ i }  coefficients (see Eqs (16) and (17))
z_{ cm }  the center of mass position (see Eq. (19))
U_{ b }, U_{ str }, U_{ ext }  the bending, stretching and external potential contributions to the potential energy of membrane (see Eq. (18))
F^{↓}  the uptodown threshold switching force (see Eqs (12), (20), (21) and (22))
F^{↑}  the downtoup threshold switching force (see Eqs (14) and (28))
ε_{0}  the vacuum permittivity
Molecular Dynamics Simulations
MD simulations are a well established modeling tool frequently employed in studies of nanoscale carbonbased materials^{15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36}. We used classical MD simulations to investigate the dynamics of snapthrough 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 the distance between the fixed sides from L to d < L. The same external force was applied to each atom in the downward direction to simulate the attractive interaction between the plates.
See SI Sec. “MD Simulation Details” for the details of MD simulations.
Uptodown transition
In order to simulate the uptodown transition, the doubleclamped graphene nanoribbon buckled upwards was subjected to the force in the downward direction. The final state of membrane was found using MD simulations as described in SI Sec. “MD Simulation Details”. Figure 2 shows the final position of a central atom of membrane versus the applied force for several values of d/L and two membrane sizes. According to Fig. 2, at fixed L, the uptodown threshold switching force (the minimal force required for the uptodown transition) is larger for smaller values of d/L. Moreover, at fixed d/L, the threshold switching force is smaller for longer membranes. Additionally, some curves in Fig. 2 exhibit a noisy threshold (such as d/L = 0.8 and d/L = 0.98 in (a)), which can be related to thermal fluctuations of membranes. All these observations are intuitively reasonable.
Depending on the force magnitude, the membrane switching occurs either through the symmetric or nonsymmetric membrane profile (see Fig. 3). Figure 3 was obtained using overdamped simulations of membrane dynamics (additional results of these simulations can be found in SI Sec. “MD Simulation Details”). The nonsymmetric profile is associated with a smaller energy barrier^{14} and involved in switching by smaller forces. Larger applied forces result in the switching through the symmetric profile. According to our observations and previous work^{14}, in all cases, the membrane profile is symmetric at short times. If the force magnitude is sufficient to overcome the energy barrier for the symmetric profile, then, typically, the switching takes place through the symmetric path. Otherwise, a symmetry breaking occurs leading to the switching through the lower energy barrier associated with the nonsymmetric membrane profile. Thermal fluctuations help the symmetry breaking.
Downtoup transition
It was suggested in ref.^{4} that the memcapacitor membrane can be set into the buckled upwards state 0 (see Fig. 1) by also using the electrostatic attraction between the capacitor plates. When the pulleddown membrane is suddenly released, there are conditions such that the membrane overcomes the potential barrier and ends up in the buckled upwards state. As the kinetic energy plays an important role in the downtoup transition, this process can not be analyzed using the overdamped dynamics.
In order to simulate the downtoup transition, every atom of a doubleclamped buckled membrane was subjected to a constant force in z direction. After an equilibration period, the forces were removed, and the system was simulated for a time interval sufficient to reach the steady state. The final position of a central atom of membrane is presented in Fig. 4 as a function of the applied force for several values of d/L.
Figure 4 shows that the threshold forces for the downtoup transition are larger than those for the uptodown transition (see Fig. 2). Moreover, the threshold forces for the downtoup transition are larger for the shorter membrane A compared to the longer membrane B. It is interesting that the final state of membrane oscillates as a function of the applied force (see d/L = 0.98 curves in Fig. 4). Qualitatively, having a high kinetic energy, the membrane moves up and down while its energy dissipates. Finally, it gets trapped in one of two stable buckled states.
Overall, our MD results indicate the possibility of writing one bit of information into the state of membrane memcapacitor. In this interpretation, logic 1 corresponds to the membrane buckled downwards, and logic 0  to the membrane buckled upwards. In addition to 0 → 1 and 1 → 0 transitions presented in Figs 2 and 4, we have verified the occurrence of 1 → 1 and 0 → 0 transitions at the same values of forces required for 0 → 1 and 1 → 0 transitions, respectively. Therefore, logic 0 or 1 can be written into the memcapacitor just by selecting a suitable value of the applied force.
In our work, we considered relatively small nanoribbons, which are mechanically rigid. This is an important requirement for the memcapacitor application as the mechanical nonvolatile information storage is not possible with flexible graphene. From some previous studies^{37} it is known that local structural corrugations (ripples)^{38} disappear in the transition from flexible to rigid graphene. In agreement with this previous work^{37}, our molecular dynamics simulations have shown the absence of ripples (spontaneous height fluctuations) in nanoribbons of reported sizes and their existence in larger nanoribbons. The latter, however, are not suitable for the memcapacitor application because of their flexibility. To summarize, while our simulation tools provide a means for ripples detection, we did not observe these in the reported structures that are mechanically rigid.
Theory of Elasticity
Even though the graphene has a thickness of one atom, the graphene membranes are quantitatively good described by the theory of elasticity^{39,40,41,42}. This allows us, on one hand, to obtain analytical expressions for the buckled membranes^{43,44,45} and, on the other hand, perform a comparison with MD simulations.
There is a number of publications dealing with buckled beams and membranes under the transverse load (see, for example, refs^{46,47,48,49}). Such systems are frequently described in the framework of EulerBernoulli theory, which, however, leads to complex analytically unsolvable equations. BubnovGalerkin decomposition is a one of the best approaches to find the approximate analytical solution of these equations, although it still requires a complex phasediagram analysis. In particular, using such analysis, the authors of refs^{46,47,48,49} investigated an electrostatically loaded doubleclamped membrane above a rigid flat electrode and derived cumbersome conditions for the symmetric snapthrough transition, symmetry breaking, existence of bifurcation and pullin instability.
In this Section, we derive compact but sufficiently precise expressions for the snapthrough switching forces based on the theory of elasticity.
Buckling and snappingthrough within the theory of elasticity
Description of buckled membranes within the theory of elasticity
Consider a 2D membrane within the theory of elasticity^{40,41,50}. The total potential energy of membrane is defined by the deflection ζ (along the normal direction z) as
Here, Δ stands for the 2D Laplacian, T = T_{0} + δT is the total tensile force, T_{0} is the force applied by the support and δT is the bending tension resulted from the extension,
S_{0} = wL, and F is the external force density. The compression of membrane corresponds to T_{0} < 0.
Given the potential energy, one may also be interested in the dynamics of membrane, which is defined by the equation
where μ is the 2D mass density.
Eigenmodes and buckling
The spatial harmonics (eigenmodes) of clamped membranes can be written as
where the numbers b_{ n } are the solutions of the equation cosh b_{ n }cos b_{ n } = 1. One can find that b_{ n } ≈ 3π/2 + nπ (in particular, b_{0} = 4.73 is close to 3π/2 = 4.65). The functions ζ_{ n }(x) are orthonormal.
Consider a buckled membrane shown in Fig. 1. Its potential energy (Eq. (3)) calculated for the fundamental n = 0 mode (taking ζ(x) = ζ_{0}(x) with ζ(d/2) ≡ ζ_{c}) is
where α = π^{2}w(T_{0} − T_{c})/(4d), β = 3π^{4}Dw/(4ε^{2}d^{3}), and f = πFdw/6. Here, T_{c} = 4π^{2}D/d^{2} is the critical tension. At α > 0, that is at T_{0} > T_{c}, the potential is the doublewell potential with minima at symmetric deflections, ${\zeta}_{c}=\pm \phantom{\rule{.25em}{0ex}}\sqrt{\alpha \mathrm{/2}\beta}$, and the potential barrier δU = α^{2}/4β. (We note that a less realistic case of hinged boundary conditions would lead to simpler expressions for eigenmodes, $\sim \phantom{\rule{0.15em}{0ex}}\mathrm{sin}\left(\pi nx/d\right)$, compared to Eq. (6), making the hinged boundary conditions preferable for some calculations. However, this case results in quantitatively different values. For example, the critical tension differs four times.)
We note that the nanoribbon length is given by
Expanding this expression to the second order, one obtains the interrelation between L, d, and the maximal deflection ζ_{c}:
On the other hand, the maximal deflection of buckled membrane can be inferred from the minima of the potential energy, Eq. (7):
Equations (9) and (10) link the tensile force T_{0} to the parameter d/L, which can be considered as a measure of deformation of buckled membrane. These equations can be used to express the tensile force T_{0} through d/L.
Snapthrough transition
Here we present a convenient method to describe the dynamics of membrane subjected to an external force and find the minimal force causing its snapthrough transition. The main idea is to expand ζ(x, t) in harmonics ζ_{ n }(x) (given by Eq. (6)) with amplitudes q_{ n }(t) as
and limit the sum to few first terms. We found that in order to describe the symmetric and nonsymmetric transitions, it is sufficient to consider n = 0,2 and n = 0,1 terms in Eq. (11), respectively. The calculation consists in the following. First, we substitute the expansion (11) in Eq. (5). Second, the resulting equation is multiplied by ζ_{ m }(x) and integrated taking into account the orthogonality of harmonics. Here we emphasize that for the harmonics (6), the integrals are readily calculated, and instead of the integrodifferential equation one obtains a system of differential equations for the functions q_{ n }(t). Inserting the obtained functions q_{ n }(t) back into Eq. (11) gives us the description of membrane dynamics. In the following, the results of these calculations (performed using the time normalized by the membrane characteristic frequency ${\omega}_{\mathrm{c}}=\mathrm{(1/}{L}^{2})\sqrt{D/\mu}$ and t_{c} = 1/ω_{c})^{51} are analyzed for membrane B.
Consider the results presented in Fig. 5. Figure 5(a) demonstrates the case of fully symmetric switching, when the dynamics can be described by n = 0 and 2 harmonics. Next, we introduce a small asymmetry into the problem via a small nonsymmetric modification of the force (of the order of 0.1%). This results in the nonsymmetric dynamics of Fig. 5(b), which can be described in terms of n = 0 and 1 harmonics. We note that the force needed for the nonsymmetric snapthrough transition is smaller than that for the symmetric one, which will also be analyzed below in more detail. Finally, in Fig. 5(c) we illustrate a combination of the above regimes found using a symmetric force, when a tiny fluctuation in the numerical solution changes the symmetric dynamics to the nonsymmetric one. Note that this case is close to the one discussed in ref.^{14}.
Figure 6(a) depicts the dependence of the final position of membrane center, ζ_{c}(t_{ f }), on the applied force F. At a certain value of force, F = F^{↓}, the membrane switches from the buckled upwards state to the buckled downwards state. Our calculations show that the threshold force is proportional to the initial centralpoint deflection ζ_{c} and is different for the symmetric (s) and nonsymmetric (ns) dynamics. Namely, this force, being multiplied by the membrane area wL, reads
Since the nonsymmetric transition is energetically favorable, we expect that this value of force is the one to be used in the device design/experiment analysis. We note that estimations based on Eq. (12) are in a good agreement with results found by other methods as we discuss later and illustrate in Sec. “Discussion” below. Figure 6(b) presents the timedependence of harmonics’ amplitudes q_{ n }(t) in the uptodown transition. Importantly, the dynamics is well described by n = 0, 1, and 2 harmonics, while n = 3 and 4 harmonics amplitudes are negligibly small.
Our calculations (similar to the consideration of the uptodown force above) demonstrate that the minimal force needed for the downtoup transition is proportional to ${\zeta}_{\mathrm{c}}^{2}$ as
See also SI Subsec. “Downtoup transition”, where the downtoup transition is illustrated graphically.
Phenomenological elasticity theory
In this Section, in order to avoid all complications associated with the BubnovGalerkin decomposition, we elaborate a different approach describing the uptodown and downtoup transitions in a compact analytical form.
Ansatz
Following our observation of the existence of the symmetric and nonsymmetric membrane profiles in the snapthrough transition (see Fig. 3), we introduce two simplest polynomial functions to describe such symmetric (s) and nonsymmetric (ns) shapes of membrane:
Here θ_{ i }(s) is the angle that the membrane makes with the horizontal, s is the internal coordinate that changes between −1/2 and 1/2, A_{ i } and c_{ i } are coefficients, and i = {s, ns}. Clearly, Eqs (16) and (17) describe the doubleclamped membrane as θ_{ i }(±1/2) = 0.
Uptodown transition
Our goal here is to estimate the minimal force required for the snapthrough transition. In this subsection, we assume that the snapthrough transition is induced by a slowly increasing force such that the system always stays in the potential energy minimum. At zero applied force, there are two minima of the potential energy corresponding to the two stable states of membrane. The applied force modifies the potential energy landscape such that, starting at a certain force, the potential energy has a single minimum. Here, this value of force is found and considered as an estimation for the threshold switching force.
In the presence of a constant force F in the downward direction, the membrane potential energy is given by
where U_{ b } is the bending energy, U_{ str } is the stretching energy, U_{ ext } is the potential energy due to the applied force. Then, using U_{ ext } = Fz_{ cm }, where z_{ cm } is z coordinate of the center of mass, and neglecting U_{ str } term not important for the uptodown transition (see, e.g., Fig. S1 that indicates the insignificance of U_{ str } in buckled membrane), we get the following equation for the potential energy extrema:
where i = {0, 1} corresponds to {s, ns}, respectively. Equation (19) allows finding c_{0} and c_{1} for a given strength of the applied force.
The minimal force needed for the snapthrough transition corresponds to the maximum of F(z_{ cm }) and, for the transition through the symmetric shape, is given by
taking place at c_{0} = 0.3589 (the corresponding membrane profile is presented in Fig. S4). At this value of c_{0}, the bending energy is U_{b,s} = 108Dw(L − d)/L^{2}. Performing the same calculations for the nonsymmetric shape, one can find that the force needed to support the equilibrium nonsymmetric shape decreases with the progress of switching. A zero force is reached at ${c}_{1}=\phantom{\rule{.25em}{0ex}}{c}_{2}=\mathrm{1/}\left(2\sqrt{5}\right)$ that corresponds to the maximum of U_{b,ns} = 90Dw(L − d)/L^{2}. A possible (rough) estimation for the threshold switching force can be obtained taking the limit c_{1} → ∞ leading to
and U_{b,ns} = 42Dw(L − d)/L^{2}.
In fact, a realistic scenario of the switching through the nonsymmetric shape can be imaged as follows. We start with a symmetric membrane at F = 0 and slowly increase the force. The symmetry breaking occurs at a certain value of force, say, F^{↓}. The dynamics of switching is a complex process significantly relying on thermal fluctuations. As the switching dynamics can not be reached within the framework of our model, for estimation purposes, we assume that F^{↓} corresponds to the point where the bending energies and applied forces for the symmetric and nonsymmetric shapes are the same. One can find that both conditions are satisfied at c_{0} = 0.4683 and c_{1} = 0.6948, so that U_{ b } = 47.8Dw(L − d)/L^{2} and
Consequently, at F < F^{↓}, the membrane keeps the symmetric shape and switches to the nonsymmetric one as soon as F reaches F^{↓}. As there is no barrier involved in the nonsymmetric switching, no further increase in the applied force is required to complete the snapthrough transition through the nonsymmetric shape.
Downtoup transition
As the stretching energy U_{ str } (see Eq. (18)) plays an important role in the downtoup transition, it needs to be taken into account. For our purposes, it is sufficient to approximate U_{ str } as
where ΔL is the change in the membrane length. In the approximation of small elongation, $\mathrm{\Delta}L\ll L$, one can assume that both U_{ b } and the shape of the stretched membrane are not significantly modified compared to F = 0 case. In this situation, at the threshold of transition, the stretching energy (Eq. (23)) is equal to the difference between the maximal and minimal bending energies (the energy conservation condition). Using the energies given below Eq. (S8) we find
Moreover, at equilibrium
Under the condition of small $\mathrm{\Delta}L\ll L$, the center of mass position of membrane (buckled downwards) can be expressed as
with
Using Eqs (24) and (27), Eq. (25) can be rewritten as
We have verified that the threshold switching force for the downtoup transition given by Eq. (28) is in agreement with our typical MD simulations results. A very good agreement was obtained with MD simulations performed with very small energy dissipations.
Density Functional Theory
A DFT investigation of the uptodown transition was performed following the previously reported study of the uptodown switching of buckled graphene membrane^{14}. In the present calculations, we focused on the effects of boundary conditions and buckling strength on the uptodown threshold switching force. DFT computational details can be found in SI Sec. “DFT Calculations”.
We investigated a buckled zigzag graphene nanoribbon specified in SI. Two types of boundary conditions were examined: (1) hinged boundary conditions (configurations C1 with d/L = 0.96, Fig. 7(a), C2 with d/L = 0.78, and C3 with d/L = 0.63), and (2) clamped boundary conditions (configuration D1 with d/L = 0.96, Fig. 7(b,c)). The initial geometries of nanoribbons corresponded to the buckled upwards state. In order to simulate the uptodown transition, we used the approach developed in ref.^{14}. Specifically, we performed a series of DFT calculations with the central chain of membrane atoms (marked red in Fig. 7) gradually displaced in −z direction from its position in the buckled upwards state. The membrane geometry found at the preceding deformation step was used to build its subsequent modification. In contrast to ref.^{14}, in the present calculations only z coordinates of the central chain of atoms were constrained.
The deformation energies of buckled graphene are reported in SI. Our force estimation shows that the threshold switching force for the uptodown transition is about 3.8 pN/atom for C1, 11.1 pN/atom for C2, and 16.4 pN/atom for C3. Moreover, the nonsymmetric switching force for D1 is 3.7 pN/atom and the symmetric one is about 11.5 pN/atom. As the nonsymmetric switching forces for D1 and C1 are close to each other, we use the results for C1C3 for the orderofmagnitude comparison with MD and elasticity theory results.
Discussion
Four methods to describe the snapthrough transitions
As the key finding, Fig. 8 presents the dependence of the uptodown threshold switching force (calculated with different approaches) on the membrane deformation parameter d/L. In particular, this figure demonstrates that the threshold forces obtained by a number of different methods are in good agreement with each other.
We emphasize that:

(i)
MD is widely used approach to describe the dynamics of nanoscale systems. In our work, MD clearly shows the possibility of 0 → 1 and 1 → 0 transitions of graphene membrane. We have also verified the occurrence of 1 → 1 and 0 → 0 transitions at the same values of forces required for 0 → 1 and 1 → 0 transitions, respectively.

(ii)
In the framework of elasticity theory, the membrane is described by means of an integrodifferential equation for the deflection ζ(x, t). Expanding the function ζ(x, t) into membrane’s harmonics allows reducing the integrodifferential equation to a system of differential equations. This approach was utilized in Subsec. “Buckling and snappingthrough within the theory of elasticity”. There it was shown that the uptodown threshold switching force depends linearly on the initial centralpoint deflection, F^{↓} ∝ ζ_{c}, while the downtoup snapthrough force at low dissipation depends quadratically on the initial deflection, ${F}^{\uparrow}\propto {\zeta}_{\mathrm{c}}^{2}$. Besides, we have demonstrated that with a high accuracy, the relevant membrane’s dynamics can be described (and visualized) just by two harmonics.

(iii)
The phenomenological approach based on the theory of elasticity has allowed us to obtain analytical expressions for the uptodown and downtoup switching forces for buckled graphene membrane. The forces for 0 → 1 and 1 → 0 transitions are in a good agreement with MD simulations results.

(iv)
DFT is a nonstandard method for studying mechanical properties of membranes. Similarly to other methods, DFT has shown that the nonsymmetric switching pathway is the preferable one. This method also provides a consistent estimation for the uptodown threshold switching force.
Implications for memcapacitor design
From the application point of view, the membrane memcapacitor should have a strong ‘memory content’, namely, the device characteristics in its two logic states should be sufficiently different so as to provide a significant influence on other elements of electronic circuits^{52}. For this purpose, it is desirable that both the distance between the fixed edge of membrane and second electrode (h in Fig. 1) and the maximal deflection of membrane (z_{ s }(0)) are chosen (much) smaller than the membrane length. Moreover, the gap between electrodes should be larger than the maximal membrane deflection, h > z_{ s }(0) (see Eq. (S2) for the definition of z_{ i }(s)). For given capacitances C_{0} and C_{1} of the states 0 and 1 (logic 1 corresponds to the membrane buckled downwards, and logic 0  to the membrane buckled upwards) such that C_{1}/C_{0} < 3.01, one can find that
Next, we need an expression for the force applied to the membrane expressed through the input variable of memcapacitor such as the applied voltage V (see Eqs (1) and (2)). Since the energy of the plate can be written as
where ε_{0} is the vacuum permittivity and A = d · w is the plate area, the force is given by
The overall scheme of memcapacitor switching is schematically presented in Fig. 9. Let us consider first the uptodown transition (path (a–c,f) in Fig. 9). As mentioned in SI Sec. “Phenomenological Elasticity Theory”, the smallest threshold force for this transition is associated with the switching through the nonsymmetric г state and corresponds to ${c}_{1}^{\u204e}=0.6948$. In this way, the buckled upwards membrane (structure (a) in Fig. 9) evolves into the buckled downwards state according to (b) and (c) in Fig. 9. After the voltage removal, the membrane remains in the buckled downwards state ((e) in Fig. 9). In order to estimate the minimal (threshold) voltage required for this transition, Eq. (22) is rewritten accounting for Eq. (31):
Consequently,
where ${z}_{cm}({c}_{1}^{\u204e})=0.3203\sqrt{L(Ld)}$. Eq. (33) provides an easy way to find the threshold voltage necessary for the uptodown transition of membrane.
In the downtoup transition, the initial buckled downwards membrane ((e) in Fig. 9) is first stretched by the applied force ((c) in Fig. 9). When the force is removed, the membrane state becomes (d). If the potential energy of (d) is sufficient to overcome the potential barrier height symmetrically (f), then the membrane may end up in the buckled upwards state (a). These general processes are partially described by Eqs (24) and (25). Because of the membrane elongation in (c,d), a refined value of z_{ cm } should be used in relevant calculations.
The threshold voltage can be estimated based on the energy conservation (Eq. (24)) and force equilibrium (Eq. (25)) conditions accounting for ${z}_{cm}^{\u204e}$ from Eqs (26) and (27). One can find that
and
An additional aspect of membrane dynamics – an estimation of thermallyinduced switching time – is considered in SI Sec. “Stability of Buckled Membrane”.
Concluding Remarks
We have investigated the snapthrough transition of a buckled graphene membrane using a variety of computational and theoretical tools. The main results of this paper are the expressions for the threshold switching forces (Eqs (12 and 22) for the uptodown transition and Eqs (14 and 28) for the downtoup one) and corresponding voltages (Eqs (33) and (35)). Our analytical results are supported by the results of numerical simulations, MD and DFT calculations. We expect that our findings will find applications in the design, fabrication and analysis of membranebased memory devices.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Di Ventra, M., Pershin, Y. V. & Chua, L. O. Circuit elements with memory: Memristors, memcapacitors, and meminductors. Proc. IEEE 97, 1717–1724 (2009).
 2.
Traversa, F. L., Bonani, F., Pershin, Y. V. & Di Ventra, M. Dynamic computing random access memory. Nanotechnology 25, 285201 (2014).
 3.
Pershin, Y. V., Traversa, F. L. & Di Ventra, M. Memcomputing with membrane memcapacitive systems. Nanotechnology 26, 225201 (2015).
 4.
MartinezRincon, J. & Pershin, Y. V. Bistable nonvolatile elastic membrane memcapacitor exhibiting chaotic behavior. IEEE Trans. El. Dev. 58, 1809 (2011).
 5.
Lai, Q. et al. Analog memory capacitor based on fieldconfigurable iondoped polymers. Appl. Phys. Lett. 95, 213503 (2009).
 6.
MartinezRincon, J., Di Ventra, M. & Pershin, Y. V. Solidstate memcapacitive system with negative and diverging capacitance. Phys. Rev. B 81, 195430 (2010).
 7.
Shevchenko, S. N., Pershin, Y. V. & Nori, F. Qubitbased memcapacitors and meminductors. Phys. Rev. Applied 6, 014006 (2016).
 8.
Pershin, Y. V. & Di Ventra, M. Memory effects in complex materials and nanoscale systems. Advances in Physics 60, 145–227 (2011).
 9.
Lindahl, N. et al. Determination of the bending rigidity of graphene via electrostatic actuation of buckled membranes. Nano Lett. 12, 3526–3531 (2012).
 10.
Weber, P., Guttinger, J., Tsioutsios, I., Chang, D. E. & Bachtold, A. Coupling graphene mechanical resonators to superconducting microwave cavities. Nano Lett. 14, 2854–2860 (2014).
 11.
Lambin, P. Elastic properties and stability of physisorbed graphene. Appl. Sci. 4, 282 (2014).
 12.
Benameur, M. M. et al. Electromechanical oscillations in bilayer graphen. Nat. Comm. 6, 8582 (2015).
 13.
Davidovikj, D. et al. Visualizing the motion of graphene nanodrums. Nano Lett. 16, 2768–2773 (2016).
 14.
Sedelnikova, O. V., Bulusheva, L. G., Okotrub, A. V. & Pershin, Y. V. Spontaneous symmetry breaking during the switching of a buckled graphene membrane. JETP Letters 103, 244–247 (2016).
 15.
Yakobson, B. I., Brabec, C. J. & Bernholc, J. Nanomechanics of carbon tubes: Instabilities beyond linear response. Phys. Rev. Lett. 76, 2511–2514 (1996).
 16.
Berber, S., Kwon, Y.K. & Tománek, D. Unusually high thermal conductivity of carbon nanotubes. Phys. Rev. Lett. 84, 4613–4616 (2000).
 17.
Yao, Z., Zhu, C.C., Cheng, M. & Liu, J. Mechanical properties of carbon nanotube by molecular dynamics simulation. Comp. Mat. Sc. 22, 180–184 (2001).
 18.
Legoas, S. B. et al. Moleculardynamics simulations of carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 90, 055504 (2003).
 19.
Maruyama, S. A molecular dynamics simulation of heat conduction of a finite length singlewalled carbon nanotube. Micr. Thermophys. Eng. 7, 41–50 (2003).
 20.
Lee, G.D., Wang, C. Z., Yoon, E., Hwang, N.M. & Ho, K. M. Vacancy defects and the formation of local haeckelite structures in graphene from tightbinding molecular dynamics. Phys. Rev. B 74, 245411 (2006).
 21.
Shiomi, J. & Maruyama, S. Nonfourier heat conduction in a singlewalled carbon nanotube: Classical molecular dynamics simulations. Phys. Rev. B 73, 205420 (2006).
 22.
Wang, C. Y., Mylvaganam, K. & Zhang, L. C. Wrinkling of monolayer graphene: A study by molecular dynamics and continuum plate theory. Phys. Rev. B 80, 155445 (2009).
 23.
Hu, J., Ruan, X. & Chen, Y. P. Thermal conductivity and thermal rectification in graphene nanoribbons: A molecular dynamics study. Nano Letters 9, 2730–2735 (2009).
 24.
Jiang, J.W., Wang, J.S. & Li, B. Young’s modulus of graphene: A molecular dynamics study. Phys. Rev. B 80, 113405 (2009).
 25.
Martins, B. & Galvao, D. Curved graphene nanoribbons: structure and dynamics of carbon nanobelts. Nanotechnology 21, 075710 (2010).
 26.
NeekAmal, M. & Peeters, F. M. Graphene nanoribbons subjected to axial stress. Phys. Rev. B 82, 085432 (2010).
 27.
Ni, Z. et al. Anisotropic mechanical properties of graphene sheets from molecular dynamics. Phys. B: Cond. Matt. 405, 1301–1306 (2010).
 28.
Lebedeva, I. V., Knizhnik, A. A., Popov, A. M., Lozovik, Y. E. & Potapkin, B. V. Interlayer interaction and relative vibrations of bilayer graphene. Phys. Chem. Chem. Phys. 13, 5687–5695 (2011).
 29.
Bagri, A., Kim, S.P., Ruoff, R. S. & Shenoy, V. B. Thermal transport across twin grain boundaries in polycrystalline graphene from nonequilibrium molecular dynamics simulations. Nano Lett. 11, 3917–3921 (2011).
 30.
Min, K. & Aluru, N. R. Mechanical properties of graphene under shear deformation. Appl. Phys. Lett. 98, 013113 (2011).
 31.
Ng, T., Yeo, J. & Liu, Z. A molecular dynamics study of the thermal conductivity of graphene nanoribbons containing dispersed stone–thrower–wales defects. Carbon 50, 4887–4893 (2012).
 32.
Rajabpour, A. & Allaei, S. M. V. Tuning thermal conductivity of bilayer graphene by interlayer sp3 bonding: A molecular dynamics study. Appl. Phys. Lett. 101, 053115 (2012).
 33.
Kalosakas, G., Lathiotakis, N. N., Galiotis, C. & Papagelis, K. Inplane force fields and elastic properties of graphene. J. Appl. Phys. 113, 134307 (2013).
 34.
Berdiyorov, G., NeekAmal, M., Peeters, F. & van Duin, A. C. Stabilized silicene within bilayer graphene: A proposal based on molecular dynamics and densityfunctional tightbinding calculations. Phys. Rev. B 89, 024107 (2014).
 35.
Kang, J. W. & Hwang, Z. Positiondependent mechanical responses of nanoindented graphene nanoribbons: Molecular dynamics study. J. Korean Phys. Soc. 67, 625–633 (2015).
 36.
Yamaletdinov, R. D. & Pershin, Y. V. Finding stable graphene conformations from pull and release experiments with molecular dynamics. Scientific Reports 7, 42356 (2017).
 37.
Schoelz, J. K. et al. Graphene ripples as a realization of a twodimensional Ising model: A scanning tunneling microscope study. Phys. Rev. B 91, 045413 (2015).
 38.
Fasolino, A., Los, J. H. & Katsnelson, M. I. Intrinsic ripples in graphene. Nature Materials 6, 858 (2007).
 39.
Eriksson, A. M., Midtvedt, D., Croy, A. & Isacsson, A. Frequency tuning, nonlinearities and mode coupling in circular mechanical graphene resonators. Nanotechnology 24, 395702 (2013).
 40.
Tomi, M. et al. Buckled diamondlike carbon nanomechanical resonators. Nanoscale 7, 14747 (2015).
 41.
Abdi, M., DegenfeldSchonburg, P., Sameti, M., NavarreteBenlloch, C. & Hartmann, M. J. Dissipative optomechanical preparation of macroscopic quantum superposition states. Phys. Rev. Lett. 116, 233604 (2016).
 42.
Bonilla, L. L. & RuizGarcia, M. Critical radius and temperature for buckling in graphene. Phys. Rev. B 93, 115407 (2016).
 43.
Qiu, J., Lang, J. H. & Slocum, A. H. A curvedbeam bistable mechanism. J. Microelectromech. Sys. 13, 137 (2004).
 44.
Cazottes, P., Fernandes, A., Pouget, J. & Hafez, M. Bistable buckled beam: Modeling of actuating force and experimental validations. J. Mech. Des. 131, 101001 (2009).
 45.
Bitarafan, M. H. et al. Thermomechanical characterization of onchip buckled dome fabryperot microcavities. J. Opt. Soc. Am. B 32, 1214 (2015).
 46.
Medina, L., Gilat, R. & Krylov, S. Symmetry breaking in an initially curved prestressed micro beam loaded by a distributed electrostatic force. Int. J. Solid. Struct. 51, 2047–2061 (2014).
 47.
Chen, X. & Meguid, S. A. On the parameters which govern the symmetric snapthrough buckling behavior of an initially curved microbeam. Int. J. Solid. Struct. 66, 77–87 (2015).
 48.
Krylov, S., Ilic, B. R., Schreiber, D., Seretensky, S. & Craighead, H. The pullin behavior of electrostatically actuated bistable microstructures. J. Micromech. Microeng. 18, 055026 (2008).
 49.
Medina, L., Gilat, R. & Krylov, S. Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force. Int. J. Solid. Struct. 49, 1864–1876 (2012).
 50.
Landau, L. D. & Lifshitz, E. M. Theory of Elasticity (Pergamon Press 1975).
 51.
Gomez, M., Moulton, D. E. & Vella, D. Critical slowing down in purely elastic ‘snapthrough’ instabilities. Nat. Phys 13, 142 (2017).
 52.
Di Ventra, M. & Pershin, Y. V. The parallel approach. Nature Physics 9, 200 (2013).
Acknowledgements
This work has been supported by the Russian Science Foundation grant No. 151320021. OVI and SNS acknowledge a partial support by the State Fund for Fundamental Research of Ukraine (grant No. F66/952016).
Author information
Affiliations
Nikolaev Institute of Inorganic Chemistry SB RAS, Novosibirsk, 630090, Russia
 Ruslan D. Yamaletdinov
 , Olga V. Sedelnikova
 & Yuriy V. Pershin
Novosibirsk State University, Novosibirsk, 630090, Russia
 Ruslan D. Yamaletdinov
 & Olga V. Sedelnikova
B. I. Verkin Institute for Low Temperature Physics and Engineering, Kharkov, 61103, Ukraine
 Oleg V. Ivakhnenko
 & Sergey N. Shevchenko
V. N. Karazin Kharkov National University, Kharkov, 61022, Ukraine
 Oleg V. Ivakhnenko
 & Sergey N. Shevchenko
Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina, 29208, USA
 Yuriy V. Pershin
Authors
Search for Ruslan D. Yamaletdinov in:
Search for Oleg V. Ivakhnenko in:
Search for Olga V. Sedelnikova in:
Search for Sergey N. Shevchenko in:
Search for Yuriy V. Pershin in:
Contributions
R.D.Y. and Y.V.P. did the MD calculations and developed the Phenomenological elasticity theory. O.V.I. and S.N.S. developed the Buckling and snappingthrough within the theory of elasticity. O.V.S. did the DFT calculations. All authors discussed the results and cowrote the manuscript.
Competing Interests
The authors declare no competing interests.
Corresponding authors
Correspondence to Ruslan D. Yamaletdinov or Yuriy V. Pershin.
Electronic supplementary material
Rights and permissions
Springer Nature has partnered with Copyright Clearance Center's RightsLink service to offer a variety of options for reusing this content.
Visit RightsLink for more information about using this article.
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.