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5.2 Balance bulk and slab surface energies

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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5.2.1 Scientific question

How much energy is required to create two equivalent surfaces from a bulk crystal? The calculation is a subtraction between large energies; a mismatch in bulk and slab settings can dominate the small excess being sought.

5.2.2 Model and prerequisites

Continue the symmetric, stoichiometric silicon slab in vacuum and thickness. For each slab containing N atoms, prepare a bulk reference with the same in-plane strain and functional. Obtain the per-atom bulk energy from a converged three-dimensional mesh. The surface calculation uses a two-dimensional mesh with similar in-plane reciprocal spacing, not the same literal mesh tuple. Exclude nonstoichiometric, polar and chemically decorated surfaces from this first derivation; those require explicit reservoirs and, often, electrostatic compensation.

5.2.3 Worked input delta

# Final single-point evaluation after slab relaxation.
calculation    scf
suffix         surface_final
cal_force      1
cal_stress     1
# Retain the converged basis, functional and smearing policy.

5.2.4 Physics and units

For two identical faces, the factor of two counts the surfaces created by the cut. The area is the cross product of the in-plane vectors and must correspond to the simulated surface repeat, including any reconstruction or lateral supercell. If the two faces differ, the same subtraction gives their sum, not either individual surface energy. A relaxed slab and an unrelaxed slab answer different questions. The unrelaxed cleavage excess describes the chosen initial termination; allowing ions to move includes its relaxation energy. Neither result is automatically the minimum among all possible reconstructions.

\[ \gamma=\frac{E_{\mathrm{slab}}-N e_{\mathrm{bulk}}}{2A} \]

5.2.5 Workflow and outputs

  1. Archive the bulk reference INPUT, STRU and KPT beside the slab directories; record the pseudopotential and numerical-orbital hashes once, then verify the same hashes in every branch.
  2. Relax the selected slab with a fixed cell and an explicitly documented set of mobile atoms. Confirm convergence on those atoms and inspect the fixed-center forces separately.
  3. Re-evaluate the final geometry using calculation scf; this creates a clean, consistent final electronic energy rather than accidentally choosing an intermediate ionic step. Save OUT.surface_final/running_scf.log.
  4. Read the final energy unit from the log. ABACUS can display energy information in more than one unit, so convert both references to eV before subtraction. Obtain A in Ų from the stored cell vectors.
  5. Fill a thickness series with N, area, slab energy, bulk energy per atom and the resulting excess. If reporting J/m², use 1 eV/Ų = 16.02176634 J/m². Retain the original eV/Ų column for auditability.
  6. Repeat the subtraction with tighter electronic convergence, denser in-plane k sampling and an enlarged compatible orbital basis. Judge changes in gamma, not only changes in the two raw energies.

5.2.6 Convergence and acceptance

A small error in bulk energy per atom is multiplied by N. Consequently, apparent thickness drift may reflect an inaccurate bulk reference rather than a physical interaction between the faces. Plot the slab excess against thickness only after bulk convergence. As a diagnostic, fit E_slab against N for a family with identical area and termination; compare its slope with the independently converged bulk energy. A linear fit is not a cure for inadequate slab thickness or a changing surface geometry. Require convergence in gamma and in a structural descriptor such as the first interlayer spacing. For metallic extensions, keep occupations and electronic-temperature definitions consistent and examine the smearing limit.

5.2.7 Pitfalls

Do not subtract a molecular chemical potential from a stoichiometric elemental slab. Do not divide by the projected cell area twice or omit the second face. Large negative gamma is a warning to inspect reference bookkeeping and competing phases, not evidence of a miraculous stable surface. A single frozen-layer convention cannot be mixed with a fully mobile convention in one convergence curve.

5.2.8 Exercises with guidance

  1. Derive how a bulk per-atom energy bias δe changes gamma for N atoms. Guidance: the shift is proportional to −Nδe/(2A), so thicker slabs amplify the bias.
  2. Explain what the same subtraction measures for different top and bottom terminations. Guidance: it gives gamma_top + gamma_bottom when divided by A; extra independent information is needed to separate them.

5.2.9 Primary sources