Ab initio prediction of nontrivial topological band and superconductivity in stable metallic Si allotropes at ambient pressure

Based on first-principles density functional calculations, we report the existence of both nontrivial band topology and superconductivity in three metallic Si allotropes, termed $Cmcm\text{−}{\mathrm{Si}}_4$, $Cmmm\text{−}{\mathrm{Si}}_4$, and $I4/mmm\text{−}{\mathrm{Si}}_4$. $Cmcm\text{−}{\mathrm{Si}}_4$ and $I4/mmm\text{−}{\mathrm{Si}}_4$ are predicted in this study, whereas $Cmmm\text{−}{\mathrm{Si}}_4$ was proposed in previous theoretical calculations. These metastable allotropes retain their crystalline structure at ambient pressure and exhibit superconductivity at the critical temperatures of 1.2–11.4 K. We investigate the topological characteristics of the electronic states and find the weak topological nature for all three allotropes. For $Cmcm\text{−}{\mathrm{Si}}_4$, in particular, the formation of the surface Dirac point is clearly identified. Our result provides a promising platform for realizing a topological superconducting state in all-Si systems at ambient pressure.

Figure 1

Top and side views of the atomic structures of (a) $Cmcm\text{−}{\mathrm{Si}}_4$, (b) $Cmmm\text{−}{\mathrm{Si}}_4$, and (c) $I4/mmm\text{−}{\mathrm{Si}}_4$. The $x$, $y$, and $z$ axes follow the [100], [010], and [001] directions of the conventional cell, respectively, so the $y$ and $z$ axes are converted to each other in $I4/mmm\text{−}{\mathrm{Si}}_4$. Dark and light circles in red and blue colors represent different planes made up of atoms. (d) The bulk BZ of the orthorhombic structure of $Cmcm\text{−}{\mathrm{Si}}_4$ and $Cmmm\text{−}{\mathrm{Si}}_4$ and the surface BZ projected onto the (001) plane drawn with high-symmetry points.

Figure 2

Total energies of various Si allotropes including $Cmcm\text{−}{\mathrm{Si}}_4$, $Cmmm\text{−}{\mathrm{Si}}_4$, and $I4/mmm\text{−}{\mathrm{Si}}_4$ plotted as a function of volume.

Figure 3

Phonon spectra, phonon densities of states (PDOS in arbitrary units), and Eliashberg spectral functions ${\alpha }^{2}F\left(\omega \right)$ for (a) $Cmcm\text{−}{\mathrm{Si}}_4$, (b) $Cmmm\text{−}{\mathrm{Si}}_4$, and (c) $I4/mmm\text{−}{\mathrm{Si}}_4$.

Figure 4

(a) The variation of enthalpy during the structural transition from $Cmcm\text{−}{\mathrm{Si}}_4$ to $sh\text{−}\mathrm{Si}$ is plotted as a function of configuration coordinate for different pressures using the generalized solid-state nudged elastic band method. The enthalpy barriers are calculated to be 28, 10, and 1 meV at 0, 3, and 6 GPa, respectively. (b) The atomic structures projected onto the (001) plane of $Cmcm\text{−}{\mathrm{Si}}_4$ are shown for the initial, intermediate, and final configurations.

Figure 5

Band structures without SOC of (a) $Cmmm\text{−}{\mathrm{Si}}_4$ and (b) $I4/mmm\text{−}{\mathrm{Si}}_4$ along the high-symmetry lines in the BZ of the primitive cell, with the Fermi level setting to zero.

Figure 6

(a) Band structure of $Cmcm\text{−}{\mathrm{Si}}_4$ in the presence of SOC and (b) dispersive band on the $\mathrm{\Gamma }-Z-T$ plane, with the Fermi level setting to zero. In (a), insets show the enlarged view of the energy splitting at the Dirac points near the Fermi level. In (b), the Dirac nodal line along the $Z-T$ symmetry line is highlighted in yellow.

Figure 7

(a) Surface electronic structure of $Cmcm\text{−}{\mathrm{Si}}_4$ along the $\overline{\mathrm{S}}\text{−}\overline{\mathrm{Y}}$ line in the (001) surface BZ. The Kramers pairs switch partners between the $\overline{\mathrm{S}}$ and $\overline{\mathrm{Y}}$ points. In (b), the ${\overline{\mathrm{s}}}_1$ and ${\overline{\mathrm{s}}}_2$ points represent $\pm 0.2\overline{\mathrm{S}}$ with respect to the $\overline{\mathrm{Y}}$ point, respectively.

Figure 8

Top and side views of the atomic structures of (a) $Amm2\text{−}{\mathrm{ASi}}_4$, (b) $Cmmm\text{−}{\mathrm{ASi}}_4$, and (c) $I4/mmm\text{−}{\mathrm{ASi}}_4$ with $A$ = Li, Na, K, Rb, and Cs. Dark and light circles in red and blue colors represent different planes made up of atoms.