Theoretical study of isoelectronic ${\mathrm{Si}}_{n}M$ clusters ($M={\mathrm{Sc}}^{-},\mathrm{Ti},{\mathrm{V}}^{+}$; $n=14–18$)

We study, from first-principles quantum mechanical calculations, the structural and electronic properties of several low-lying energy equilibrium structures of isoelectronic ${\mathrm{Si}}_{n}M$ clusters $(M={\mathrm{Sc}}^{-},\mathrm{Ti},{\mathrm{V}}^{+})$ for $n=14–18$. The main result is that those clusters with $n=16$ are more stable than its neighbors, in agreement with recent experimental mass spectra. By analyzing the orbital charge distribution and the partial orbital density of states, that special stability is rationalized as a combination of geometrical (near spherical cagelike structure for $n=16$) and electronic effects ($l$-selection rule of the spherical potential model). The structures of the two lowest energy isomers of ${\mathrm{Si}}_{16}M$ are nearly degenerate, and consist of the Frank-Kasper polyhedron and a distortion of that polyhedron. The first structure is the ground state for $M={\mathrm{V}}^{+}$, and the second is the ground state for Ti and ${\mathrm{Sc}}^{-}$. For the lowest energy isomers of clusters ${\mathrm{Si}}_{n}M$ with $n=14–18$, we analyze the changes with size $n$, and impurity $M$ of several quantities: binding energy, second difference of total energy, HOMO-LUMO gap, adiabatic electron affinity, addition energy of a Si atom, and addition energy of an $M$ impurity to a pure ${\mathrm{Si}}_n$ cluster. We obtain good agreement with available measured adiabatic electron affinities for ${\mathrm{Si}}_n\mathrm{Ti}$.

Figure 1

(Color online) Geometry of the few lowest energy isomers of ${\mathrm{Si}}_{14}M$ clusters. The index I in the label 14-I indicates the different geometries. Below each structure is given, for the three different types of dopant, the total energy difference with respect to the lowest energy configuration (eV), the HOMO-LUMO gap (eV), and the ordinal number of the isomer. In all structures the spin state is a singlet except for $14\text{−}{\mathrm{VIII}}^{*}$ which is a triplet (indicated by an asterisk).

Figure 2

(Color online) Same as Fig. 1 for $n=15$.

Figure 3

(Color online) Same as Fig. 1 for $n=16$. The structure 16-III is not stable for ${\mathrm{Si}}_{16}{\mathrm{V}}^{+}$, and evolves towards structure 16-II (see text).

Figure 4

(Color online) Same as Fig. 1 for $n=17$.

Figure 5

(Color online) Same as Fig. 1 for $n=18$.

Figure 6

(Color online) Different properties of ${\mathrm{Si}}_{n}M$ and ${\mathrm{Si}}_n$ clusters are represented versus the number of Si atoms $n$: (a) the binding energy per atom; (b) the second difference of the total energy; (c) the HOMO-LUMO gap; (d) the addition energy of $M$ to pure ${\mathrm{Si}}_n$ clusters (filled symbols), and the addition energy of Si to doped ${\mathrm{Si}}_{n-1}M$ clusters (empty symbols). Circles, squares, and triangles represent ${\mathrm{Sc}}^{-}$, Ti, and ${\mathrm{V}}^{+}$ doped clusters, respectively, and crosses represent pure ${\mathrm{Si}}_n$ clusters. The lines stand only to guide the eye.

Figure 7

(Color online) The total charge accumulation per Si atom (circles), and the $s$-(squares), $p$-(triangles), and $d$-orbital (rhombus) charge accumulation per Si atom, in units of one electron, are represented for ${\mathrm{Si}}_{16}M$ versus the radial distance to the center of cluster. The different panels correspond to the four isomeric geometries 16-I, 16-II, 16-III, and 16-IV of Fig. 3, and the different points correspond to well defined radial atomic shells. The lines connecting points stand only to guide the eye. Filled, empty, and crossed symbols stand for ${\mathrm{Sc}}^{-}$, Ti, and ${\mathrm{V}}^{+}$ doped clusters, respectively.

Figure 8

(Color online) For the isomers 16-I, 16-II, 16-V of ${\mathrm{Si}}_{16}{\mathrm{Sc}}^{-}$, and 17-I of ${\mathrm{Si}}_{17}{\mathrm{Sc}}^{-}$, is shown the density of states (upper panel), the projected density of states of ${\mathrm{Sc}}^{-}$ (middle panel), and orbital projected density of states of ${\mathrm{Sc}}^{-}$ (lower panel): (a) structure 16-I; (b) 16-II; (c) 16-V; (d) 17-I. The unit in the $x$ axis is eV, and the Fermi energy was set to zero. The splitting of the levels, which are grouped according to the labels of the spherical potential model, is caused by the different molecular symmetry of the isomers (see text).

Figure 9

(Color online) The same as in Fig. 8 for the isomers 16-I (left panels), 16-II (middle panels), and 16-IV (right panels) of ${\mathrm{Si}}_{16}{\mathrm{V}}^{-}$.