  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.
