6. Aluminium (111): a work function converged with slab thickness

This tutorial shows how LatticeMind builds a surface slab from a bulk crystal, reads a work function from the electrostatic potential, and runs several slab thicknesses in one request so you can see the value converge.

6.1. The prompt

Calculate the work function of the aluminium (111) surface and check how it
converges with slab thickness. Build symmetric Al(111) slabs from fcc
aluminium with lattice constant 4.05 A, using a 2x2 surface cell, 18 A of
vacuum, and 7, 8 and 9 atomic layers. For each thickness run a PBE SCF with
Gaussian smearing and a 6x6x1 k-point grid, then a potential calculation on
the saved density, and compare the three work functions.

6.2. What LatticeMind does

  • Builds a symmetric Al(111) slab for each requested thickness, holding the surface cell and the vacuum fixed so thickness is the only thing that changes.

  • Centres each slab in its cell, giving the same vacuum on both faces — the potential must reach a flat plateau on either side for a vacuum level to be well defined.

  • Samples the surface plane densely and the vacuum direction at \(\Gamma\) only, and applies Gaussian smearing because aluminium is a metal.

  • Runs each SCF, then a potential calculation that reads that SCF’s density.

  • Reads the vacuum level from the plateau of the plane-averaged electrostatic potential, subtracts the Fermi energy, and reports the work function for each thickness with the measured value alongside.

The 9-layer cell comes out as 36 atoms in a 5.73 × 5.73 × 36.71 Å cell, with 9 Å of vacuum above and below the slab:

domain.latvec = [...
    5.7276 0.0 0.0;
    2.8638 4.9602 0.0;
    0.0 0.0 36.7061;
];
kpoint.gridn = [6 6 1];
kpoint.sampling = 'gauss';

The nearest-neighbour distance in every slab is 2.864 Å and the interlayer spacing is 2.338 Å — the values fcc aluminium at \(a\) = 4.05 Å requires, so the surface is built from the bulk crystal rather than approximated.

6.3. Results

Layers

Atoms

Work function

Change

7

28

3.96 eV

—

8

32

4.11 eV

+0.15 eV

9

36

4.07 eV

−0.04 eV

The converged value of 4.07 eV compares with a measured 4.24 eV for Al(111). A few tenths of an electronvolt is the expected agreement for a semi-local functional on a simple metal.

The alternation between odd and even layer counts is physical: it is the quantum-size effect in thin metal films, and it damps as the slab thickens — from +0.15 eV between 7 and 8 layers to −0.04 eV between 8 and 9. The energy per atom tells the same story: it moves 425 meV between 7 and 8 layers and only 11 meV between 8 and 9, so the interior of the slab has become bulk-like. This is why the thickness study is worth running rather than trusting a single slab.

Note

The work function is read from the plateau of the plane-averaged potential far from the surface, not from an average over the cell — inside the slab the potential is dominated by the ionic cores. LatticeMind reports the plateau it used, so you can confirm the vacuum region was flat enough for the value to mean something.

6.4. Try next

  • Add a fourth thickness to the same request to tighten the convergence.

  • Ask for a wider surface cell (3x3) at fixed thickness to check the in-plane sampling.

  • Run the same study for another close-packed metal surface and compare the trend.