Defects, orderings and interfaces

Tasks 9.5.1-9.5.4. References: pymatgen analysis.defects and transformations; Hart and Forcade (2009); Zunger, Wei, Ferreira and Bernard (1990); Zur and McGill (1984).

Point defects

A symmetry search of the P1 cell gives the orbits; one representative per orbit is enough because the other members differ by a symmetry operation. A vacancy removes the representative, a substitution replaces its element and an interstitial inserts a species at a candidate position.

Interstitial candidates are the Voronoi vertices of Task 9.3.1, relative to the central atom, filtered by a minimum clearance to every existing atom and deduplicated under the detected symmetry operations. The defect supercell repeats each lattice vector by ceil(min_image / d_i), where d_i is the interplanar spacing, so the shortest distance between defect images reaches the requested value.

Orderings of partial occupancies

A disordered site is a group of atoms at the same fractional position with different elements and partial occupancies (or an explicit disorder_group). The group is replicated into the requested supercell and every combination of species is enumerated up to max_orderings. The structure matcher of Task 9.1 removes decorations related by a lattice translation, rotation or origin shift. Each remaining ordered cell is ranked by the Ewald energy:

E = 1/2 sum_ij q_i q_j erfc(alpha r_ij) / r_ij
  + 2 pi / V sum_k exp(-k^2 / 4 alpha^2) / k^2 |S(k)|^2
  - alpha / sqrt(pi) sum_i q_i^2

with S(k) = sum_i q_i exp(i k . r_i) and the Coulomb constant 14.399645 eV Angstrom. The cell must be charge neutral; a non-neutral decoration reports energy: null instead of a meaningless number.

Special quasirandom structures

The binary disorder is decorated with a seeded SplitMix64 generator and fixed species counts. Pair correlations are xi_s = <sigma_i sigma_j> with sigma = +/-1 for each distance shell; the triplet correlation averages sigma_i sigma_j sigma_k over nearest-neighbor triangles. Simulated annealing swaps two opposite sites, accepts improvements and, with a decreasing temperature, some uphill moves. The objective is w_pair sum xi_s^2 + w_triplet xi_triplet^2, whose target is zero.

Interfaces

Zur-McGill matching enumerates integer 2x2 supercells of the two in-plane cells up to max_index, rejects determinants that leave the area bound, and compares the transformed lengths and angles. A match requires the maximum relative length mismatch and the angle mismatch to stay inside the tolerances; matches are ranked by strain and then area.

Stacking maps the film Cartesian coordinates onto the substrate in-plane cell (which applies the film strain), places the film gap Angstrom above the substrate and adds the requested vacuum. The in-plane shift is a translation in substrate fractional coordinates.