Topological insulators in the quaternary chalcogenide compounds and ternary famatinite compounds

We present first-principles calculations to predict several three dimensional (3D) topological insulators in quaternary chalcogenide compounds which are made of I$_2$-II-IV-VI$_4$ compositions and in ternary compositions of I$_3$-V-VI$_4$ famatinite compounds. Among the large members of these two families, we give examples of naturally occurring compounds which are mainly Cu-based chalcogenides. We show that these materials are candidates of 3D topological insulators or can be tuned to obtain topological phase transition by manipulating the atomic number of the other cation and anion elements. A band inversion can occur at a single point $\Gamma$ with considerably large inversion strength, in addition to the opening of a bulk band gap throughout the Brillouin zone. We also demonstrate that both of these families are related to each other by cross-substitutions of cations in the underlying tetragonal structure and that one can suitably tune their topological properties in a desired manner.

We present first-principles calculations to predict several three dimensional (3D) topological insulators in quaternary chalcogenide compounds which are made of I2-II-IV-VI4 compositions and in ternary compositions of I3-V-VI4 famatinite compounds. Among the large members of these two families, we give examples of naturally occurring compounds which are mainly Cu-based chalcogenides. We show that these materials are candidates of 3D topological insulators or can be tuned to obtain topological phase transition by manipulating the atomic number of the other cation and anion elements. A band inversion can occur at a single point Γ with considerably large inversion strength, in addition to the opening of a bulk band gap throughout the Brillouin zone. We also demonstrate that both of these families are related to each other by cross-substitutions of cations in the underlying tetragonal structure and that one can suitably tune their topological properties in a desired manner.

I. INTRODUCTION
FIG. 1. Crystal structure and the property of the topological band-inversion. (a), The crystal structure of ternary famatinite compounds is given for a representative material Cu3SbSe4. (b) The crystal structure of quaternary chalcogenide compounds is illustrated for Cu2CdSbSe4 as an example. (c)-(e) Band structures illustrated schematically near the time-reversal invariant point Γ for a trivial band insulator in (c), a non-trivial topological semimetal in (d) and a non-trivial topological insulator in (e). Dark-blue dots represent the twofold degenerate s−like states. In the non-trivial case, band inversion occurs when s-like states drops below the four-fold degenerate p−like states (J = 3/2) (blue lines). Owing to the lattice distortion, the degeneracy of the J = 3/2 states vanishes in the present non-cubic crystal structures. that 3D non-trivial topological insulating phase exists in the compounds of famatinite and quaternary chalcogenides families.

III. RESULTS
The bulk band structures along high symmetry path M (π, π) − Γ(0, 0) − X(π, 0) are shown in Fig. 2 for two representative compunds Cu 3 SbS 4 and Cu 3 SbSe 4 which belong to the famatinite family. The effect of the zonefolding is clearly evident at the high-symmetry point Γ. In the ground state of these compounds, the structural compression along c-axis (i.e. c < 2a) compared to cubic zinc-blende lattice lifts the four-fold degeneracy of the cation p−states at the Γ−point owing to the crystal-field splitting. In Cu 3 SbS 4 , an insulating energy gap between the p−states is present at the Fermi level throughout the Brillouin zone as shown in Fig. 2(a). However, as the nuclear charge of the anion is increased from S atom to Se atom, such substitution causes the conduction band to drop below the Fermi level at the X-point, yielding an electron pocket, see Fig. 2(b). Simultaneously, the valence band maximum also gradually moves above the Fermi-level making Cu 3 SbSe 4 a metal. Nevertheless, a finite direct gap persists throughout the Brillouin zone and thus the Z 2 topological invariant can still be defined for the valence bands in these two materials and the inverted band order is retained at only Γ−points. Therefore, the Cu 3 SbS 4 and Cu 3 SbSe 4 are topologically non-trivial insulator and metal, respectively.
The mutation of the crystal structure from the ternary to a quaternary compound is also evident in the electronic structure of these two families of materials as the overall changes in the electronic structure are only subtle near the energy region of present interests. Due to the tetragonal symmetry, all the Cu-based quaternary materials listed in Fig. 3 host a gap in the p−states close to the Fermi-level, where the gap magnitude depends on the extra cation cross-substitutions. With increasing atomic mass from Zn to Cd to Hg atom, this gap decreases mainly due to the downshift of the s−like conduction bands at Γ−point. The Fermi-level goes through the energy gap of the p−states in most materials, harboring a topological insulating state in these compounds. However, only in two compounds Cu 2 ZnGeTe 4 and Cu 2 CdGeTe 4 , the low-energy electronic properties reveal that the conduction band minimum is shifted to the X point and drops below the Fermi level to form bulk electron pockets. Simultaneously, the concaveupward shaped valence band maximum pops up above the Fermi level at Γ−point, making these systems intrinsically bulk metallic. We find very similar nature of the band-inversion across these materials in which the split p−states lie in energy above the twofold-degenerate s− states at a single time-reversal momentum Γ−point, representing a bandinversion relative to the natural order of s−and p−type orbital derived band structure. Therefore, these materials are intrinsically bulk non-trivial topological insulators or semimetals, except Cu 2 ZnGeS 4 , Cu 2 ZnSnS 4 , and Cu 2 HgGeS 4 which are trivial topological insulators.
Finally, we demonstrate our tuning procedure to generate a topological phase transition from a trivial band insulator to the non-trivial topological insulator. In the present study, we take advantage of the structural freedom of the solids in ternary famatinite and quaternary chalcogenides to achieve non-trivial topological phases. In the Fig. 4, we discuss one representative example of this route here in going from the quaternary chalcogenide Cu 2 ZnSnS 4 as the starting point to the ternary famatinite Cu 3 SbS 4 compound. Note that the topological phase transition is a generic feature and there exists numerous such route. Cu 2 ZnSnS 4 is a trivial band insulator whereas Cu 3 SbS 4 is a topological insulator in their pristine conditions as discussed before in Figs. 3(i) and 2(b), respectively. We systematically manipulate the overall crystal structure of our starting compound by making the atomic number Z of the constituent atoms, the lattice constant and the internal displacement of the anion as variable parameters.
To illustrate this process, we denote the atomic number Z of various elements by Z M1 , Z M2 , Z M3 , and Z M4 . For our starting element Cu 2 ZnSnS 4 , we have Z M1 =29, Z M2 =30-x, Z M3 =50+x, Z M4 =16, where x varies between 0 to 1 in the phase transformation process. At each step of changing the parameter x, we also pay attention to the lattice constant and the internal displacements of the anion by keeping the same ratio of changes corresponding to the change in x of atomic number Zs from Cu 2 ZnSnS 4 to Cu 3 SbS 4 . Note that variation of the atomic number Zs can also be considered as a doping effect in the virtual crystal approximation while the total nuclear charge is neutral during the whole process.
We monitor the band structure along high symmetry path M (π, π) → Γ(0, 0) → X(π, 0) at every step of the phase transition process from Figs. 4(a)-(e). While the overall band-structure remains very much the same throughout the process, the band characters and the band gap vary dramatically. In Fig. 4(b), on increasing x by 0.25, the size of band gap at Γ starts to shrink compared to the parent compound in Fig. 4(a), although the s−character continues to dominate in the conduction band (blue dots). The critical point of the phase transition is reached when the value of x is increased to 0.5 in Fig. 4(c), at which point the bottom of the conduction band and the top of the valence band touch at a single momentum point at Γ leading to a critical topological point. A full band inversion is achieved in Fig. 4(d) which recreates an insulating band gap but topologically non-trivial in nature and the s−like orbital character shifts much below the valence band. Finally, in Fig. 4(e), when x equals 1, Cu 3 SbS 4 is a non-trivial topological insulator.

IV. CONCLUSION
We have shown via first-principle calculations that the ternary famatinite and quaternary chalogenide compounds are non-trivial topological insulators in their pristine phase. The involvement of partially filled d− and f − electrons in these compounds naturally opens up the possibility of incorporating magnetism and superconductivity within the topological order. The large-scale tunability available in the quaternary compounds adds versatility in using topolgical insulators in multifunctional spin-polarized quantum and optical information procesing applications. Notably, the copper-based quaternary chalcogenides are known to have non-linear optoelectronics and thermodynamics applications. 36 .
During the final stages of our work, we became aware of the work of Su-Huai Wei et al. 54 who have also proposed that some of the multinary chalcogenides are potential candidates to be non-trivial topological insulators.