Superconductivity in layered binary silicides: A density functional theory study

A class of metal disilicides (of the form $X$Si${}_2$, where $X$ is a divalent metal) crystallizes in the EuGe${}_2$ structure, formed by hexagonal corrugated silicon planes intercalated with metal atoms. These compounds are superconducting like other layered superconductors, such as MgB${}_2$. Moreover, their properties can be easily tuned either by external pressure or by negative chemical pressure (i.e., by changing the metal), which makes disilicides an ideal testbed to study superconductivity in layered systems. In view of this, we present an extensive density functional theory study of the electronic and phonon band structures as well as the electron-phonon interaction of metal disilicides. Our results explain the variation of the superconducting transition temperature with pressure and the species of the intercalating atom, and allow us to predict superconductivity for compounds not yet synthesized belonging to this family.

Figure 1

Crystal structure of the disilicides compounds in the trigonal phase ($P$-$3m1,164$). The metal atoms are the (green) dark gray larger balls, while the orange (light green) smaller balls represent Si atoms. (a) Projection of the crystal structure onto the $ab$ plane. (b) Layered structure where it is visible the buckling of the Si layer.

Figure 2

Total energy per $X$Si${}_2$ unit calculated at 0 GPa for the different disilicides at the lattice parameters $a$ and $c$ given in Table .

Figure 3

Range of $z$ values (left panel) and Si-Si distances (right panel) for which phonon-frequencies are real for the different systems studied. The black dots represent the fully relaxed $z_{\mathrm{opt}}$/Si-Si distance, while the empty square shows the “tuned $z$” positions $z_{\mathrm{tuned}}$ and the bar indicates the range of stable structures. In the case of CaSi${}_2$, we consider two different pressures.

Figure 4

Brillouin zone with high-symmetry points (a) and Fermi surfaces[39] of MgSi${}_2$ (b), CaSi${}_2$ at the external pressure of 0 GPa (c), CaSi${}_2$ at the external pressure of 16 GPa (d), SrSi${}_2$ (e), and RaSi${}_2$ (f). For RaSi${}_2$, note the third sheet appearing at $\Gamma$ composed purely of Si $3s$ states.

Figure 5

Electronic density of states at the Fermi level $N\left(E_{\mathrm{F}}\right)$ for the systems studied in this work. The (green) light gray bars represent calculations at the relaxed geometry $z=z_{\mathrm{opt}}$, while the dark gray bars are obtained for the tuned values $z=z_{\mathrm{tuned}}$.

Figure 6

The six optical phonon modes present in the trigonal phase (space group $P$-$3m1,164$) of disilicides. The $E_g$ modes involve the movement of Si atoms in the $xy$ plane. The $A_{1g}$ mode represents the out-of-plane motion of the Si atoms. The double degenerate $E_u$ modes are the displacements of individual metal and Si ions in the $xy$ plane and the $A_{2u}$ mode is the corresponding out-of-plane displacement.

Figure 7

Left panels: Phonon band structure of the disilicides studied in this work. Right panels: Eliashberg spectral function ${\alpha }^{2}F(\omega$) for each structure, calculated for the values of $z_{\mathrm{tuned}}$ reported in Table .