First-principles cluster expansion study of missing-row reconstructions of fcc (110) surfaces

Theoretical prediction of surface reconstructions is difficult and rare owing to the extremely large phase space of possible two-dimensional atomic surface configurations. Here, we demonstrate how a first-principles cluster expansion (CE) method can be used to identify a particular class of stable surface reconstructions involving the surface ordering of atoms and vacancies without any empirical input. We apply the method to late transition-metal (110) surfaces and correctly demonstrate the reconstruction tendency for $5d$ metals to reconstruct in the “missing row” ($1\times 2$) structure, but not $3d$ or $4d$ metals. In addition to providing physical insight into the origin of the reconstruction tendency, the CE also allows us to predict the finite-temperature stability of the reconstruction, the order-disorder $(1\times 2)\to (1\times 1)$ transition temperature, and the equilibrium shape of the surface islands.

Figure 1

(a) Side view and (b) top view of the (1 × 2) MR reconstruction on Au and Pt (110) surfaces. The arrows illustrate the relaxation of surface atoms. The dashed circles are the positions where the “missing” atoms from an unreconstructed (1 × 1) surface.

Figure 2

Ground-state diagrams for (a) Ag, (b) Cu, (c) Au, and (d) Pt(110) surfaces with predicted formation energies for ordered (blue dots) and random (black line) surface from first-principles CE. The green lines show the lowest-energy reconstruction configurations as a function of coverage. Au and Pt(110) surfaces show (1 × 2) MR reconstruction, but Ag and Cu(110) surfaces do not.

Figure 3

Clusters used in CE for Au(110) surface. For each cluster, the first number is the number of sites within the cluster and the second number differentiates clusters of the same class

Figure 4

Change of energy and LRO parameters with temperature for a Au(110) MR structure from MC simulations. Together with the snapshots of the surface configuration at different temperatures, the results show an order-disorder transition of the MR structure.

Figure 5

Change of cluster correlations with temperature for a Au(110) surface. The strong pair correlations are related to the short-range order of the surface above transition temperature.