Multiple Types of Topological Fermions in Transition Metal Silicides

Exotic massless fermionic excitations with nonzero Berry flux, other than the Dirac and Weyl fermions, could exist in condensed matter systems under the protection of crystalline symmetries, such as spin-1 excitations with threefold degeneracy and spin-$3/2$ Rarita-Schwinger-Weyl fermions. Herein, by using the ab initio density functional theory, we show that these unconventional quasiparticles coexist with type-I and type-II Weyl fermions in a family of transition metal silicides, including CoSi, RhSi, RhGe, and CoGe, when spin-orbit coupling is considered. Their nontrivial topology results in a series of extensive Fermi arcs connecting projections of these bulk excitations on the side surface, which is confirmed by (001) surface electronic spectra of CoSi. In addition, these stable arc states exist within a wide energy window around the Fermi level, which makes them readily accessible in angle-resolved photoemission spectroscopy measurements.

Figure 1

Energy dispersions for multiple types of topological fermions. (a) The Weyl fermion with $\mathbf{S}=1/2$. (b) The excitation with $\mathbf{S}=1$. (c) The Rarita-Schwinger-Weyl fermion with $\mathbf{S}=3/2$. (d) The double Weyl fermion. The red arrows indicate that two energy crossings should be at the same point for the double Weyl fermion. Chern numbers for upper and lower bands are marked in blue for topological fermions.

Figure 2

The crystalline lattice structure, Brillouin zone (BZ), and electronic properties for CoSi without spin-orbit coupling (SOC). (a) The lattice structure and (b) BZ for the CoSi family. The projected BZ of the (001) surface is marked in red lines. (c) The bulk band structure for CoSi along high symmetry lines. The inset is a 3D Fermi surface at the calculated Fermi level. (d) The corresponding electronic spectra for the (001) surface. Bulk bands along lines of $R\text{−}\mathrm{\Gamma }$ and $\mathrm{\Gamma }\text{−}X$ are plotted in black, in which the lines along $R\text{−}\mathrm{\Gamma }$ are rescaled to fit the distance between $\overline{M}$ and $\overline{\mathrm{\Gamma }}$ points. (e),(f) The Fermi surface contours on the (001) surface at different energies. The calculated Fermi level is set to be zero.

Figure 3

Electronic structures for CoSi with SOC. (a) The bulk band structure for CoSi along high symmetry lines. (b)–(d) Enlargement of bands in regions marked by red boxes. Weyl fermions are marked by red arrows (including type I and type II). The Rarita-Schwinger-Weyl fermion is marked by a blue arrow. The calculated Fermi level is set to be zero. (e) Schematics of the distribution of Weyl fermions along lines of $\mathrm{\Gamma }\text{−}X$ and $\mathrm{\Gamma }\text{−}R$ around the Fermi level. The red (blue) dots stand for the Weyl points with a positive (negative) topological charge.

Figure 4

The projected electronic spectra for CoSi with SOC. (a) The electronic spectra along high symmetry lines projected to the (001) surface. Bulk bands along lines of $R\text{−}\mathrm{\Gamma }$ and $\mathrm{\Gamma }\text{−}X$ are plotted as black lines, in which the lines along $R\text{−}\mathrm{\Gamma }$ are rescaled to fit the distance between $\overline{M}$ and $\overline{\mathrm{\Gamma }}$ points. Type-I and type-II Weyl fermions are marked by blue dots. (b)–(f) The corresponding Fermi surface contours on the (001) surface at different energies. The calculated Fermi level is set to be zero.