Melting and Rippling Phenomenan in Two Dimensional Crystals with localized bonding

TL;DR

This paper investigates the atomic fluctuations and rippling phenomena in two-dimensional crystals, revealing size-dependent divergence of vertical deviations and the impact of substrate attraction on ripple suppression.

Contribution

It provides a comprehensive analysis of in-plane and out-of-plane fluctuations in 2D crystals, including effects of interlayer coupling and substrate attraction, supported by analytical and Monte Carlo methods.

Findings

In-plane atomic fluctuations grow logarithmically with system size.

Vertical ripples diverge rapidly with increasing system size, regardless of interlayer coupling.

Weak substrate attraction effectively suppresses ripple amplitude and wavelength.

Abstract

We calculate Root Mean Square (RMS) deviations from equilibrium for atoms in a two dimensional crystal with local (e.g. covalent) bonding between close neighbors. Large scale Monte Carlo calculations are in good agreement with analytical results obtained in the harmonic approximation. When motion is restricted to the plane, we find a slow (logarithmic) increase in fluctuations of the atoms about their equilibrium positions as the crystals are made larger and larger. We take into account fluctuations perpendicular to the lattice plane, manifest as undulating ripples, by examining dual layer systems with coupling between the layers to impart local rigidly (i.e. as in sheets of graphene made stiff by their finite thickness). Surprisingly, we find a rapid divergence with increasing system size in the vertical mean square deviations, independent of the strength of the interplanar coupling. We consider an attractive coupling to a flat substrate, finding that even a weak attraction significantly limits the amplitude and average wavelength of the ripples. We verify our results are generic by examining a variety of distinct geometries, obtaining the same phenomena in each case.