Multistability of graphene nanobubbles

Using He, Ne, Ar, Kr, and Xe atoms as a model system, it is demonstrated that graphene nanobubbles on flat substrates are multistable systems. A nanobubble can adopt multiple stable stationary states, each characterized by the number of layers $l$ within the cluster of encapsulated atoms. The layers are circular, concentrically stacked, and form an $l$-stepped pyramid with a flat top. Encapsulation of this pyramid by the graphene sheet is achieved through local stretching of the membrane: the valence bonds elongate only directly above the confined atoms. Outside this coverage zone, the sheet remains undeformed and lies flush against the substrate. The maximum number of possible layers, $l_m$, increases monotonically with the number of encapsulated atoms $N$, reaching $l_m=6$ for $N=4000$. The graphene membrane, through van der Waals interaction with the substrate, compresses the internal atomic cluster, generating pressures on the order of $P\sim 1$~GPa. Numerical simulations of thermal vibrations reveal that among all $l$-layer configurations, one ground state always exist. Upon heating, this state smoothly transitions into a layerless liquid configuration. All other stationary states transform into this ground state once a characteristic temperature $T_l$ is reached. For $N=4000$, the ground state corresponds to the four-layer packing ($l=4$). The coexistence of multiple stable states with distinct layer numbers at low temperatures leads to the absence of a universal shape for the nanobubbles. In this scenario, the height-to-radius ratio, $H/R$ is not constant and can vary from 0 to 0.28, depending on the number of layers.