Structure and stability of copper clusters: A tight-binding molecular dynamics study

In this paper we propose a tight-binding molecular dynamics with parameters fitted to first-principles calculations on the smaller clusters and with an environment correction, to be a powerful technique for studying large transition-metal/noble-metal clusters. In particular, the structure and stability of ${\text{Cu}}_n$ clusters for $n=3–55$ are studied by using this technique. The results for small ${\text{Cu}}_n$ clusters $\left(n=3–9\right)$ show good agreement with ab initio calculations and available experimental results. In the size range $10⩽n⩽55$ most of the clusters adopt icosahedral structure which can be derived from the 13-atom icosahedron, the polyicosahedral 19-, 23-, and 26-atom clusters, and the 55-atom icosahedron, by adding or removing atoms. However, a local geometrical change from icosahedral to decahedral structure is observed for $n=40–44$ and return to the icosahedral growth pattern is found at $n=45$ which continues. Electronic “magic numbers” ($n=2$,$8$,$20$,$34$,$40$) in this regime are correctly reproduced. Due to electron pairing in highest occupied molecular orbitals (HOMOs), even-odd alternation is found. A sudden loss of even-odd alternation in second difference of cluster binding energy, HOMO-LUMO (LUMO, lowest unoccupied molecular orbital) gap energy and ionization potential is observed in the region $n\sim 40$ due to structural change there. Interplay between electronic and geometrical structure is found.

Figure 1

Most stable structures for copper clusters with $n=10–55$ atoms. Most of the clusters adopt icosahedral structures except for $n=40–44$, where the structures are decahedral.

Figure 2

(Upper panel) Binding energy per atom as a function of cluster size $n^{1∕3}$. Inset of the upper panel represents a comparison of binding energy per atom as a function of cluster size $n$, among the present TBMD $(◻)$, FP-LMTO $(엯)$, DF-LDA $(▵)$ calculations and experimental $(◇)$ values. (Lower panel) Variation of relative stability ${\Delta }_{2}E$ with cluster size $n$. Shell closing effect at $n=8,18,20,34,40$ and even-odd alternation up to $n\sim 40$ are found. However, due to geometrical effect this even-odd alternation is disturbed at $n=11,13$, and $15$.

Figure 3

Comparison of binding energies per atom as a function of cluster size $n$ among cuboctahedral, decahedral, and icosahedral structures. For the whole region most of the clusters prefer icosahedral structure. However, a local geometrical change from icosahedral to decahedral structure is found for $n=40–44$.

Figure 4

Highest occupied–lowest unoccupied molecular orbital (HOMO-LUMO) gap energy vs cluster size $n$. Electronic shell closure at $n=2$, $8$, $18$, $20$, $34$, $40$ and even-odd alternation are observed. However, sudden loss in even-odd alternation is found around $n\sim 40$ due to the structural change there.

Figure 5

Ionization potential vs cluster size $n$. Electronic shell closing effect and prominent even-odd alternation up to $n\sim 40$ are observed.