Effect of bundling on the stability, equilibrium geometry, and electronic structure of ${\text{Mo}}_6{\text{S}}_{9-x}{\text{I}}_x$ nanowires

We use ab initio density-functional theory calculations to determine the effect of bundling on the equilibrium structure, electronic, and magnetic properties of ${\text{Mo}}_6{\text{S}}_{9-x}{\text{I}}_x$ nanowires with $x=0,3,4.5,6$. Each unit cell of these systems contains two S- and I-decorated ${\text{Mo}}_6$ clusters that are connected by ${\text{S}}_3$ linkages to form an ordered linear array. Due to the bistability of the sulfur linkages, the total energy of the nanowires exhibits typically many minima as a function of the wire length. We find the optimum interwire distance depends on composition and to a smaller degree on the orientation of the wires. Structural order is expected in bundles with $x=0$ and $x=6$, since there is no disorder in the decoration of the Mo clusters. In bundles with other stoichiometries we expect structural disorder to occur. We find that the electronic structure of some nanowires can be switched from metallic to semiconducting by changing the interwire separation. Also, we find that selected stable or metastable nanowire geometries exhibit ferromagnetic behavior.

Figure 1

(Color online) (a) Atomic arrangement within an isolated ${\text{Mo}}_6{\text{S}}_{9-x}{\text{I}}_x$ nanowire. The unit cell of length $a$ contains 2 formula units, including 12 Mo atoms arranged in two octahedra. The ${\text{S}}_3$ linkages connecting the ${\text{Mo}}_6$ octahedra are shown at both ends of the unit cell. Sites decorating the Mo octahedra are denoted by ${\text{A}}_i$ and ${\text{B}}_i$ and may be occupied by either S or I atoms. (b) Binding energy per unit cell as a function of the lattice constant $a$ for different iodine concentrations $x$. For better visualization of energy differences within each stoichiometry, the energy axis has been split into four segments.

Figure 2

(Color online) Equilibrium packing of ${\text{Mo}}_6{\text{S}}_{9-x}{\text{I}}_x$ nanowires. (a) End-on view of the structural arrangement of ${\text{Mo}}_6{\text{S}}_{9-x}{\text{I}}_x$ nanowires on a triangular lattice with $b=\left|{\mathbf{b}}_1\right|=\left|{\mathbf{b}}_2\right|$. (b) Equilibrium lattice constant $b_{eq}$ as function of the wire orientation $\phi$ in four different stoichiometries with the lattice constant $a$ corresponding to the shortest optimum in the $E_b\left(a\right)$ curves in Fig. 1b. Contour plots of the bundling energy $E_{\text{bundle}}$ as a function of $b$ and $\phi$ for (c) ${\text{Mo}}_6{\text{S}}_9$ $\left(a=10.2\text{Å}\right)$, (d) ${\text{Mo}}_6{\text{S}}_6{\text{I}}_3$ $\left(a=11.0\text{Å}\right)$, (e) ${\text{Mo}}_6{\text{S}}_{4.5}{\text{I}}_{4.5}$ $\left(a=11.0\text{Å}\right)$, and (f) ${\text{Mo}}_6{\text{S}}_3{\text{I}}_6$ $\left(a=11.0\text{Å}\right)$. The bundling energy is the difference between the binding energies $E_b$ per formula unit of isolated and bundled nanowires. The equipotential lines are separated by 0.2 eV.

Figure 3

Electronic band structure $E\left(k\right)$ along the axial direction $\Gamma \text{-A}$ and the corresponding DOS of isolated ${\text{Mo}}_6{\text{S}}_{9-x}{\text{I}}_x$ nanowires with the lattice constant $a$ corresponding to the shortest optimum in the $E_b\left(a\right)$ curves in Fig. 1b. Results are presented for (a) ${\text{Mo}}_6{\text{S}}_9$ $\left(a=10.2\text{Å}\right)$, (b) ${\text{Mo}}_6{\text{S}}_6{\text{I}}_3$ $\left(a=11.0\text{Å}\right)$, (c) ${\text{Mo}}_6{\text{S}}_{4.5}{\text{I}}_{4.5}$ $\left(a=11.0\text{Å}\right)$, and (d) ${\text{Mo}}_6{\text{S}}_3{\text{I}}_6$ $\left(a=11.0\text{Å}\right)$. The densities of states have been convoluted with 0.02 eV wide Gaussians.

Figure 4

(Color online) [(a)–(d)] Electronic band structure $E\left(k\right)$, [(e)–(h)] the corresponding DOS, and [(i)–(l)] the Fermi surface of bundled ${\text{Mo}}_6{\text{S}}_{9-x}{\text{I}}_x$ nanowires with the lattice constant $a$ corresponding to the shortest optimum in the $E_b\left(a\right)$ curves in Fig. 1b. Results for $x=0$ with $a=10.2\text{Å}$ and $x=3,4.5,6$ with $a=11.0\text{Å}$ are displayed in columns. The densities of states have been convoluted with 0.02 eV wide Gaussians. Since ${\text{Mo}}_6{\text{S}}_6{\text{I}}_3$ is a semiconductor and does not have a Fermi surface, we depict the empty Brillouin zone and the high-symmetry points in (j).

Figure 5

Electronic band structure $E_{\uparrow }\left(k\right),E_{\downarrow }\left(k\right)$ of ${\text{Mo}}_6{\text{S}}_{4.5}{\text{I}}_{4.5}$ nanowire bundles with $a=11.0\text{Å}$ at interwire separations $9\le b\le 20\text{Å}$. The spin-resolved bands are represented by the solid and dashed lines.

Figure 6

(Color online) Nature of frontier states in ${\text{Mo}}_6{\text{S}}_{4.5}{\text{I}}_{4.5}$ nanowire bundles with $a=11.0\text{Å}$ at the equilibrium interwire separations $b=8.85\text{Å}$ and $\phi \approx 0°$. (a) Spatial distribution of states in the range $E_F-0.1\text{eV}\le E\le E_F+0.1\text{eV}$ in the optimized bundle. The associated charge density is presented as the value $\rho =5\times {10}^{-4}e/{\text{Å}}^3$ along with the structural model. (b) Orbital-decomposed density of states in the same energy range around $E_F$.