Stability analysis of a bulk material built from silicon cage clusters: A first-principles approach

We predict a stable bulk material whose constituent units are the exceptionally stable $\mathrm{Ti}@{\mathrm{Si}}_{16}$ clusters. We use first-principles density functional theory. Our results provide compelling evidence of a stable, wide-band-gap material crystallizing in a hexagonal close packed structure in which cages bind weakly, similar to fullerite. We further characterize the structural and electronic properties of this material.

Figure 1

(Color online) The Frank-Kasper cage structure, corresponding to the equilibrium shape of the $\mathrm{Ti}@{\mathrm{Si}}_{16}$ nanoparticle. This highly symmetric structure, exhibiting several $C_3$ symmetry axes, will be used as the building block of a molecular solid.

Figure 2

(Color online) Cohesive energy per cluster as a function of distance between clusters in bulk $\mathrm{Ti}@{\mathrm{Si}}_{16}$. Upper panel: The lowest solid (black) line with circles depicts the change of cohesive energy as a function of the distance between titanium atoms of neighboring cages. For all crystal structures, nearest neighbor cages are all at the same distance from any focal cage (for the HCP structure, $c∕a=\sqrt{8∕3}$; see main text for details). The corresponding curves for the FCC, BCC, and SC are drawn with solid (blue), dashed (red), and dotted (green) lines, respectively. The only curve which exhibits no bound state corresponds to the DIA structure, drawn with a (black) dotted line. Lower panel, left scale: Solid circles (black) correspond to the values of the cohesive energy per cluster drawn with a solid (black) line in the upper panel (HCP structure). The solid (red) line passing through the points illustrates the quality of the cubic spline fit to the data, which was subsequently used in computing the pressure, drawn with respect to the scale on the right of the lower panel. Lower panel, right scale: The pressure curve as a function of cage-cage distance is shown with a dashed (blue) line. It was obtained by computing the numerical derivative of the spline fit to the solid (red) line. The open (orange) squares represent the pressure curve as a function of cage-cage distance obtained from a fit of the Birch-Murnaghan equation of state (Ref. 19) to the energy points (Ref. 20) (solid black circles). The fit leads to a bulk modulus of $1.25\mathrm{GPa}$.

Figure 3

(Color online) Upper panel: Equilibrium structure of bulk $\mathrm{Ti}@{\mathrm{Si}}_{16}$. Each cage occupies the site of a HCP lattice, with parameters at equilibrium $a=16.54\text{bohr}$ and $c=27.13\text{bohr}$, such that the ratio $c∕a$ deviates only by 0.5% from the ideal packing value of $\sqrt{8∕3}$. Lower panel: Band structure of bulk $\mathrm{Ti}@{\mathrm{Si}}_{16}$. This molecular material is predicted to be an indirect gap semiconductor. It is well known that DFT underestimates the true gap in semiconductors. Hence, we expect that our computed value of $1.3\mathrm{eV}$ correspondingly underestimates the experimental gap.

Figure 4

(Color online) Upper panel: Change of total energy per cage (with respect to equilibrium HCP configuration) as a function of time for variable cell-shape quantum Langevin molecular dynamics of bulk $\mathrm{Ti}@{\mathrm{Si}}_{16}$. Simulation started from the HCP structure at a temperature of $300\mathrm{K}$. Lower panel: Time dependence of the percentual deviation (with respect to the equilibrium value) of the average radius of each cage. The results show the small amplitude of the oscillations taking place at room temperature, and confirm the predicted stability of this material. The time step used in each iteration is $2\times {10}^{-15}\mathrm{s}$, and the simulation ran for a total of $1\times {10}^{-12}\mathrm{s}$.