Multistability of graphene nanobubbles

TL;DR

Graphene nanobubbles on flat substrates are shown to be multistable, with stable states distinguished by the number of inner layers $l$ and arranged as circular, flat layers forming an $l$-step pyramid. A two-component atomistic model combining graphene, inert gas, and substrate interactions is used to compute stationary states via energy minimization, yielding observables such as the height $H$, base radius $R$, density $d$, and internal pressure $P$. Thermal stability is explored with Langevin dynamics, revealing a ground state for each $N_g$ and gas type and temperature-driven transitions from excited $l$-layer states to the ground state or to a liquid, with notable examples such as Xe where the ground state is $l=4$. Importantly, there is no universal bubble shape across $l$-states; $H/R$ can range from 0 to 0.24, with a universal value of $H/R \approx 0.204$ arising only for the ground state, highlighting tunable high-pressure nanostructures in van der Waals heterostructures.

Abstract

Using the example of Ar, Kr, and Xe atoms, it is shown that graphene nanobubbles on flat substrates are multistable systems. A nanobubble can have many stable stationary states, each characterized by the number of layers, $l$, within the cluster of internal atoms. The layers are circular in shape, concentrically stacked on top of each other, forming an $l$-stepped pyramid with a flat top. The covering of this pyramid with a graphene sheet is achieved through its local stretching. The valence bonds of the sheet stretch only over the group of internal atoms; outside the coverage zone, the sheet remains undeformed and lies flush against the substrate. The maximum possible number of layers, $l_m$, increases monotonically with the number of atoms $N$ ($l_m=6$ for $N=4000$). The graphene sheet, interacting with the substrate, compresses the internal atom cluster against it, generating an internal pressure of $P\sim 1$ GPa. Numerical simulations of thermal vibrations reveal that among all $l$-layer configurations of a nanobubble, there is always one "ground"\ state. Upon heating, this ground state smoothly transitions into a layerless liquid state. All other stationary states transform into this ground state once a certain temperature is reached (for $N=4000$, the ground state corresponds to state with $l=4$). The coexistence of several stable states with different numbers of layers at low temperatures leads to the absence of a universal shape for the nanobubbles. In this scenario, the height-to-radius ratio, $H/R$, can vary from 0 to 0.24, depending on the number of layers.

Figures (10)

• Figure 1: A rectangular graphene sheet with $N_x= 10$, $N_y=12$. Black disks represent carbon atoms C, black solid lines represent valence bonds C--C. Small light disks indicate the positions of the centers of mass of the valence bond hexagons. Dashed lines illustrate the triangulation of the graphene sheet using these centers -- the inner surface of the sheet is divided into triangles, each containing one carbon atom (the triangle corresponding to the $n$-th atom is highlighted; $\mathbf{u}_n$ is the coordinate vector of the atom).
• Figure 2: Stationary $l$-layer states of encapsulated $N_g=2909$ argon atoms with $l = 1$ (a), 2 (b), 3 (c), 4 (d), and 5 (e). For clarity of presentation, only one half of the graphene sheet is shown. The graphene sheet is depicted in gray. For argon atoms, their vertical displacement is indicated by color (blue corresponds to the minimum displacement with coordinate $z = h_0$, red corresponds to the maximum displacement with $z=h_0+10$ Å).
• Figure 3: Distribution of vertical displacements $p(z)$ for the stationary states of encapsulated $N_g=3936$ krypton atoms with number of layers $l = 1$, ..., 5.
• Figure 4: Stationary $l$-layer states of a nanobubble with $N_g = 3936$ krypton atoms at $l = 1$ (a), 2 (b), 3 (c), 4 (d), and 5 (e). A top view is shown. Color indicates the distribution of deformation energy across the graphene sheet (blue corresponds to zero deformation, red corresponds to deformation with an energy of 0.008 eV per atom).
• Figure 5: Stationary $l$-layer states of a nanobubble with $N_g = 2909$ xenon atoms at $l = 1$ (a), 2 (b), 3 (c), 4 (d), and 5 (e). A top view is shown. Color indicates the distribution of pressure across the graphene sheet (blue corresponds to zero pressure, red corresponds to a pressure of $p_n = 5$ GPa). The average pressure values are $P = 1.09$, 0.90, 0.86, 0.84, and 0.85 GPa.