Setting up an aluminium surface

In this exercise, we build an Al(100) surface. For this surface, we calculate the surface energy and other properties.

Fcc Surfaces

A real fcc surface has a large number of atomic layers, but most surface properties are well reproduced by a slab with just 2 - 20 layers, depending on the material and what properties you are looking for.

The most important cubic surfaces are (100), (110), and (111). For face centered cubic, (111) has the most compact atomic arrangement, whereas (110) is most open. Here we’ll focus on (100).

  • What is the coordination number Z (number of nearest neighbors) of an fcc(100) surface atom? What is it for a bulk atom? Start the Python interpreter and try this:

    from ase.visualize import view
    from ase.build import fcc100
    s = fcc100('Al', (1, 1, 5))
    view(s, repeat=(4, 4, 1))
    

    Read more here: ase.build.fcc100().

  • Answer the same questions for the (111) surface.

Aluminum fcc(100) surface energetics

One surface property is the surface tension \(\sigma\) defined implicitly via:

\[E_N = 2A\sigma + NE_B\]

where \(E_N\) is the total energy of a slab with \(N\) layers, \(A\) the area of the surface unit cell (the factor 2 because the slab has two surfaces), and finally \(E_B\) is the total energy per bulk atom. The limit \(N \rightarrow \infty\) corresponds to the macroscopic crystal termination.

Estimate the surface tension using an expression from the simplest Effective Medium Theory (EMT) description:

\[A\sigma \simeq [1 - (Z/Z_0)^{1/2}] E_{coh}\]

where \(Z\) and \(Z_0\) are the coordination numbers (number of nearest neighbors) of a surface and a bulk atom, respectively, and \(A\) is the surface area per surface atom, and \(E_{coh} = E_{atom}-E_B > 0\) is the cohesive energy per bulk atom. For Aluminium we have \(E_{coh}\) = 3.34 eV.

  • Derive the following equation:

    \[\sigma = \frac{NE_{N-1} - (N-1)E_N}{2A}\]
  • The script Al100.py defines an energy() function for calculating \(E_N\). Use it to calculate \(\sigma\) for \(N\) = 5. Use a two-dimensional Monkhorst-Pack k-point sampling (kpts=(k, k, 1)) that matches the size of your unit cell.

    Hint

    A rule of thumb for choosing the initial k-point sampling is, that the product, ka, between the number of k-points, k, in any direction, and the length of the basis vector in this direction, a, should be:

    • ka ~ 30 Å, for d band metals
    • ka ~ 25 Å, for simple metals
    • ka ~ 20 Å, for semiconductors
    • ka ~ 15 Å, for insulators

    Remember that convergence in this parameter should always be checked.

  • How well is the EMT estimate satisfied?

Note

The experimental value of \(\sigma\) is 54 meV/Å2. However, this was obtained from the curvature of an aluminium drop and is more likely to represent the value for a closepacked Al(111) surface.

Work function

Run the work_function.py script and estimate the work function for a Al(100) surface (this script does not run in parallel). A typical experimental value for the work function of the Al(100) surface is 4.20 eV.