Boundaries Determine the Formation Energies of Lattice Defects in Two-Dimensional Buckled Materials

TL;DR

This paper reveals that in buckled two-dimensional materials, the formation energy of lattice defects depends on boundary conditions and scales differently with system size compared to flat materials, affecting the interpretation of defect energies.

Contribution

It demonstrates that boundary conditions influence defect formation energies in buckled 2D materials and introduces a new scaling law for approaching the thermodynamic limit.

Findings

Boundary conditions cause significant variations in defect energies in buckled 2D materials.

The approach to the thermodynamic limit follows a logarithmic inverse scaling.

Universal features explained by simple bead-spring models.

Abstract

Lattice defects are inevitably present in two-dimensional materials, with direct implications on their physical and chemical properties. We show that the formation energy of a lattice defect in buckled two-dimensional crystals is not uniquely defined as it takes different values for different boundary conditions even in the thermodynamic limit, as opposed to their perfectly planar counterparts. Also, the approach to the thermodynamic limit follows a different scaling: inversely proportional to the logaritm of the system size for buckled materials, rather than the usual power-law approach. In graphene samples of $\sim 1000$ atoms, different boundary conditions can cause differences exceeding 10 eV. Besides presenting numerical evidence in simulations, we show that the universal features in this behavior can be understood with simple bead-spring models. Fundamentally, our findings imply that it is necessary to specify the boundary conditions for the energy of the lattice defects in the buckled two-dimensional crystals to be uniquely defined, and this may explain the lack of agreement in the reported values of formation energies in graphene. We argue that boundary conditions may also have impact on other physical observables such as the melting temperature.