Analysis of quasi-planar defects using the Thomas--Fermi--von Weizsäcker model

We analyze the convergence of the electron density and relative energy with respect to a perfect crystal of a class of volume defects that are compactly contained along one direction while being of infinite extent along the other two using the Thomas--Fermi--von Weizsäcker (TFW) model. We take advantage of prior work on the thermodynamic limit and stability estimates in the TFW setting, and specialize it to the case of quasi-planar defects. In particular, we prove that the relative energy of the defective crystal with respect to a perfect crystal is finite, and in fact conforms to a well-posed minimization problem. In order to show the existence of the minimization problem, we modify the TFW theory for thin films and establish convergence of the electronic fields due to the perturbation caused by the quasi-planar defect. We also show that perturbations to both the density and electrostatic potential due to the presence of the quasi-planar defect decay exponentially away from the defect, in agreement with the known locality property of the TFW model. We use these results to infer bounds on the generalized stacking fault energy, in particular the finiteness of this energy, and discuss its implications for numerical calculations. We conclude with a brief presentation of numerical results on the (non-convex) Thomas-Fermi-von Weizsäcker-Dirac (TFWD) model that includes the Dirac exchange in the universal functional, and discuss its implications for future work.