Metallic clusters on a model surface: Quantum versus geometric effects

We determine the structure and melting behavior of supported metallic clusters using an ab initio density-functional-based treatment of intracluster interactions and an approximate treatment of the surface as an idealized smooth plane yielding an effective Lennard-Jones interaction with the ions of the cluster. We apply this model to determine the structure of sodium clusters containing from 4 to 22 atoms, treating the cluster-surface interaction strength as a variable parameter. For a strong cluster-surface interaction, the clusters form two-dimensional (2D) monolayer structures; comparisons with calculations of structure and dissociation energy performed with a classical Gupta interatomic potential show clearly the role of quantum shell effects in the metallic binding in this case, and evidence is presented that these shell effects correspond to those for a confined 2D electron gas. The thermodynamics and melting behavior of a supported Na${}_{20}$ cluster is considered in detail using the model for several cluster-surface interaction strengths. We find quantitative differences in the melting temperatures and caloric curve from density-functional and Gupta treatments of the valence electrons. A clear dimensional effect on the melting behavior is also demonstrated, with 2D structures showing melting temperatures above those of the bulk or (at very strong cluster-surface interactions) no clear meltinglike transition.

Figure 1

Monolayer structures found for a cluster-surface interaction strength $\varepsilon =0.5$ eV using intracluster interactions from the second-moment approximation (SMA) [Eq. (1)] and the Kohn-Sham (KS) approach.

Figure 2

Dissociation energies $\Delta E_{\text{diss}}$ of the supported monolayer structures of Fig. 1, corrected for the cluster-surface interaction strength $\varepsilon =0.5$ eV.

Figure 3

Total energy of a Na${}_{20}$ cluster on a surface (upper panel) as a function of cluster-surface interaction strength $\varepsilon$ [see Eq. (2)] for the structures (a)–(e) shown in the lower panel. Structure (a) (short-dashed line) is the lowest-energy structure for $0⩽\varepsilon ⩽0.02$ eV; structure (b) (full line) for $0.02⩽\varepsilon ⩽0.18\text{eV}$; structure (c) (dotted line) for $0.18⩽\varepsilon ⩽0.21\text{eV}$; structure (d) (full line) for $0.21⩽\varepsilon ⩽0.36\text{eV}$; and structure (e) (dashed line) for $\varepsilon ⩽0.36\text{eV}$.

Figure 4

In Fig. 4, $\Delta$ E value for structures (b) and (c) are 0.8 meV and 1.3 meV respectively. In the figure 0.8 becomes 0 $▵$ 8 and same for 1.3 . Our original figure does not have this problem. Please correct it. Ground-state (left column) and first three excited (right columns) monolayer isomers of Na${}_{20}$ found by the basin-hopping procedure for a cluster-surface interaction strength $\varepsilon =0.5$ eV. Notation: SMA, second-moment approximation; KS, Kohn-Sham; PAW, projected-augmented plane wave; GGA, generalized gradient approximation; LDA, local density approximation; KS-soft, the simplified Kohn-Sham model in the LDA with a soft, phenomenological pseudopotential (see text). Excitation energies $\Delta E$ are shown relative to the ground-state structure.

Figure 5

Canonical ionic specific-heat capacities of Na${}_{20}$ for surface-cluster interaction strength [see Eq. (2)] $\varepsilon =0.1$ eV and for the free cluster ($\varepsilon =0$, taken from Ref. 30) for KS descriptions of the intracluster interactions. The specific heat is expressed as a multiple of its zero-temperature (classical) limit $C_0$.

Figure 6

Canonical ionic specific heat capacity of Na${}_{20}$ with surface-cluster interaction strength [see Eq. (2)] $\varepsilon =0.38$ eV for SMA and KS descriptions of the intracluster interactions. The specific heat is expressed as a multiple of its zero-temperature (classical) limit $C_0$.

Figure 7

Canonical ionic specific heat capacity of Na${}_{20}$ with surface-cluster interaction strength $\varepsilon =0.5$ eV. See Fig. 6 for other notation.