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5.3 Neutral vacancy formation and finite-size checks

These are unexecuted teaching templates, with no computed ABACUS results. The baseline is ABACUS 3.9.0; verify your executable version and data files before use.

Original model and balanced energy workflow schematic

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5.3.1 Scientific question

What is the energetic cost of removing one atom from bulk silicon and returning it to a bulk reservoir? This case concerns a neutral supercell only. It teaches atom counting, symmetry breaking and image interactions without adding unsupported charged-defect corrections.

5.3.2 Model and prerequisites

Repeat the validated silicon primitive cell to create a pristine N-atom supercell. Remove one Si coordinate and change the species count to N−1 in STRU. Keep the same cell vectors and compare pristine and defective cells with the same supercell KPT mesh. Permit local relaxation near the vacancy and document any constraints. Try more than one slightly perturbed initial neighborhood: high-symmetry starting coordinates can trap a symmetry-constrained stationary structure. Use the converged bulk per-atom energy as the Si reservoir, with the same functional and data files.

5.3.3 Worked input delta

# Delta relative to the pristine-supercell calculation.
calculation    relax
suffix         neutral_vacancy
symmetry       0
cal_force      1
relax_nmax     100
force_thr_ev   0.01
# STRU: remove one Si position and decrement its atom count.

5.3.4 Physics and units

Removing a neutral atom removes the valence electrons associated with its pseudopotential as well; do not manually retain the pristine electron count. A neutral defect may still have localized spin or several electronic solutions, so compare justified spin initializations rather than interpreting neutrality as proof of a nonmagnetic state. The reservoir term restores the energy of the removed atom in bulk form. In an elemental host the bulk chemical potential equals the selected phase energy per atom; in a compound, allowable chemical potentials are constrained by competing phases and this simple formula must be generalized.

\[ E_f^{0}=E_{N-1}^{\mathrm{vac}}-E_N^{\mathrm{perfect}}+\mu_{\mathrm{Si}} \]

5.3.5 Workflow and outputs

  1. Generate pristine and vacancy STRU files from one coordinate list; archive an atom-index map so the removed site and its neighbors remain traceable. Check that the density corresponds to an N−1 atom cell, not a duplicated or overlapping coordinate.
  2. Run pristine and defect references at the same cell shape and reciprocal spacing. Transfer only settings, not a pristine charge density with incompatible species or atom bookkeeping.
  3. Relax the vacancy using the INPUT delta above. The 0.01 eV/Å force threshold is a proposed initial target; repeat with a tighter target if the formation-energy accuracy demands it.
  4. Perform clean final SCF evaluations on both relaxed references. Read total energies, forces and, if spin-polarized, final moments from their own OUT.<suffix>/running_scf.log files.
  5. Compute the neutral formation energy with the same bulk chemical potential for every supercell. Record nearest-neighbor distances around the vacant site and identify whether several starts reach the same distortion.
  6. Increase supercell size and adjust KPT to approximately preserve reciprocal spacing. Repeat at least one size with a denser mesh and larger compatible numerical orbitals to distinguish finite-size effects from basis or sampling errors.

5.3.6 Convergence and acceptance

The defect is periodically repeated. Local strain and defect-state overlap can persist even when the vacancy looks visually isolated. Compare formation energy, neighbor displacements and defect-state localization across sizes. A defect band that disperses strongly may indicate interactions between images. Use the bands case only after the defect SCF state has converged, and state that the supercell bands are folded. Near-degenerate magnetic or geometric branches must be compared using identical numerical settings. If a larger cell changes the preferred branch, report both branches rather than fitting a smooth finite-size trend through unlike states.

5.3.7 Pitfalls

Setting an arbitrary excess electron count turns this into a charged calculation with a different thermodynamic problem. Do not add Makov–Payne or other correction formulas without verifying their assumptions and the actual electrostatic implementation. A vacancy created by leaving a zero-occupancy species line is not equivalent to removing a coordinate. Reusing primitive-cell k integers in a large supercell wastes effort and changes sampling density.

5.3.8 Exercises with guidance

  1. Show that using the pristine supercell energy per atom as the reservoir produces E_vac − (N−1)E_perfect/N. Explain when it approaches the bulk-reference expression.
  2. Propose a test for symmetry trapping. Guidance: compare small, opposite distortions of vacancy neighbors with symmetry disabled and compare final forces and energies, not just SCF iteration counts.

5.3.9 Primary sources