Predicted Realization of Cubic Dirac Fermion in Quasi-One-Dimensional Transition-Metal Monochalcogenides

We show that the previously predicted “cubic Dirac fermion,” composed of six conventional Weyl fermions including three with left-handed and three with right-handed chirality, is realized in a specific, stable solid state system that has been made years ago, but was not appreciated as a “cubically dispersed Dirac semimetal” (CDSM). We identify the crystal symmetry constraints and find the space group $\mathrm{P}{6}_3/\mathrm{m}$ as one of the two that can support a CDSM, of which the characteristic band crossing has linear dispersion along the principle axis but cubic dispersion in the plane perpendicular to it. We then conduct a material search using density functional theory, identifying a group of quasi-one-dimensional molybdenum monochalcogenide compounds ${\mathrm{A}}^{\mathrm{I}}{({\mathrm{MoX}}^{\mathrm{VI}})}_3$ (${\mathrm{A}}^{\mathrm{I}}=\mathrm{Na}$, K, Rb, In, Tl; ${\mathrm{X}}^{\mathrm{VI}}=\mathrm{S}$, Se, Te) as ideal CDSM candidates. Studying the stability of the $\mathrm{A}{(\mathrm{MoX})}_3$ family reveals a few candidates such as $\mathrm{Rb}{(\mathrm{MoTe})}_3$ and $\mathrm{Tl}{(\mathrm{MoTe})}_3$ that are predicted to be resilient to Peierls distortion, thus retaining the metallic character. Furthermore, the combination of one dimensionality and metallic nature in this family provides a platform for unusual optical signature—polarization-dependent metallic vs insulating response.

Popular Summary

The study of topological properties—ones that stay the same even if the material is continually changed in some way—shows a lot of promise for exotic applications as well as insight into fundamental physics. However, identifying specific topological properties among the hundreds of thousands of known inorganic compounds is a monumental challenge if one were to guess or try all possibilities one at a time. One example of a recently proposed topological property is the elusive “cubically dispersed Dirac fermion,” a quasiparticle (a mathematical construct that simplifies emergent behavior in a complex system) that describes specific ways in which energy bands cross each other within a solid. This complex quasiparticle, which has not yet been identified in any recognizable compound, is unusual and has no analogue in the standard model of particle physics. We propose a set of theoretically derived design principles that allow us to quickly sift through candidate compounds and identify a promising group.

Our method searches specifically through the 230 “space groups,” descriptions of symmetries within a crystal. We quickly identify which of the space groups could support the properties of a cubically dispersed Dirac fermion and then add to this condition another set of theoretical filters (including thermodynamic stability) aimed at narrowing down the base. This hierarchical “inverse design” sets up a simple procedure for identification. The “best of class” candidates are then subject to a detailed examination of band-structure degeneracy and band-crossing dispersion. We identify a group of quasi-one-dimensional molybdenum monochalcogenide compounds as ideal candidates for cubically dispersed Dirac fermions.

We expect that our findings will trigger further interest in the material realization of various novel fermion types, including the prediction of unreported but chemically plausible compounds, followed by laboratory synthesis and characterization.

Figure 1

(a) Structure of the (${\mathrm{Mo}}_6{\mathrm{X}}_8$) cluster unit with a ${\mathrm{Mo}}_6$ octahedral surrounded by a cubic ${\mathrm{X}}_8$ cage. (b) Top and (c) side view of the quasi-1D chain ${({\mathrm{Mo}}_3{\mathrm{X}}_3)}^{1-}$. (d) Side view of the quasi-1D chain ${({\mathrm{Mo}}_3{\mathrm{X}}_3)}^{1-}$ with Peierls distortion. (e) Crystal structure of the cluster compound $\mathrm{A}{(\mathrm{MoX})}_3$. The blue, orange, and green balls denote Mo, X, and A atoms, respectively. (f) Hexagonal Brillouin zone and the high-symmetry $k$ path for band structure calculation. The dots mark the inequivalent Dirac points (one A point, three L points, and two $\mathbf{\Lambda }$ points representing ${\mathbf{\Lambda }}_1$ and ${\mathbf{\Lambda }}_2$) in undistorted structure.

Figure 2

(a) DFT calculated band structure of $\mathrm{Tl}{(\mathrm{MoTe})}_3$ as a representative of stable $\mathrm{A}{(\mathrm{MoX})}_3$ compounds. The different types of Dirac points are marked by red, blue, and purple circles. (b) In-plane band splitting in the vicinity of the four types of Dirac fermions at A, L, ${\mathbf{\Lambda }}_1$, and ${\mathbf{\Lambda }}_2$ points. The solid (dashed) lines denote DFT band splitting along the $k_x$ ($k_y$) direction, within the coordinate system defined in Fig. 1.

Figure 3

(a) Phase diagram of DSM vs Peierls distortion as a function of anisotropy parameter $\lambda$ for 15 $\mathrm{A}{(\mathrm{MoX})}_3$ compounds. The black lines connecting compounds with the same A cation are a guide to the eye.

Figure 4

Calculated phonon spectrum of undistorted $\mathrm{K}{(\mathrm{MoSe})}_3$. The red phonon dispersion indicates soft phonon modes. (b) Vibration modes of the soft phonon at the Γ point in (a). (c) Calculated phonon spectrum of $\mathrm{K}{(\mathrm{MoSe})}_3$ under Peierls distortion with the soft mode eliminated. (d) Band dispersion of distorted $\mathrm{K}{(\mathrm{MoSe})}_3$ showing a band gap throughout the $k_z=\pi$ plane (A-H-L).

Figure 5

Absorption coefficient (${\alpha }^{xx}={\alpha }^{yy}$; ${\alpha }^{zz}$) of (a) $\mathrm{Tl}{(\mathrm{MoTe})}_3$ and (b) $\mathrm{K}{(\mathrm{MoSe})}_3$ from DFT calculations.

Figure 6

Phonon spectra of 15 $\mathrm{A}{(\mathrm{MoX})}_3$ compounds with undistorted $\mathrm{P}{6}_3/\mathrm{m}$ structure. The red and blue dispersions indicate spectra with soft phonons and without soft phonons, respectively.

Figure 7

Band structure of 15 $\mathrm{A}{(\mathrm{MoX})}_3$ compounds with undistorted $\mathrm{P}{6}_3/\mathrm{m}$ structure.