Assessing exchange-correlation functional performance in the chalcogenide lacunar spinels $\mathrm{Ga}{M}_4{Q}_8$ $(M=\mathrm{Mo}, \mathrm{V}, \mathrm{Nb}, \mathrm{Ta};$ $Q=\mathrm{S},\mathrm{Se})$

We perform systematic density functional theory (DFT) calculations to assess the performance of various exchange-correlation potentials $V_{xc}$ in describing the chalcogenide $\mathrm{Ga}{M}_4{Q}_8$ lacunar spinels ($M=\text{Mo}$, V, Nb, Ta; $Q=\text{S}$, Se). We examine the dependency of crystal structure (in cubic and rhombohedral symmetries), electronic structure, magnetism, optical conductivity, and lattice dynamics in lacunar spinels at four different levels of $V_{xc}$: The local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid with fractional Fock exchange. We find that LDA underperforms the Perdew-Burke-Ernzerhof (PBE) and PBE revised for solids (PBEsol) GGA functionals in predicting lattice constants as well as reasonable electronic structures. The performance of LDA and GGAs can be improved both quantitatively and qualitatively by including an on-site Coulomb interaction (LDA/$\mathrm{GGA}+U$) with a Hubbard $U$ value ranging from 2 to 3 eV. We find that the PBE functional is able to produce a semiconducting state in the distorted polar $R3m$ phase without on-site Coulomb interactions. The meta-GGA functional SCAN (strongly constrained and appropriately normed) predicts reasonable lattice constants and electronic structures; it exhibits behavior similar to the $\mathrm{GGA}+U$ functionals for small $U$ values of 1 to 2 eV. The hybrid functional HSE06 is accurate in predicting the lattice constants but leads to a band gap greater than the experimental estimation of 0.2 eV in this family. All of the lacunar spinels in the cubic phase are metallic at these levels of band theory; however, the predicted valence bandwidths are extremely narrow ($\approx 0.5\mathrm{eV}$). The DFT ground states of cubic vanadium chalcogenides are found to be highly spin polarized, which contrast with previous experimental results. With spin-orbit coupling (SOC) interactions and a Hubbard $U$ value of 2 to 3 eV, we predict a semiconducting cubic phase in all compounds studied. SOC does not strongly impact the electronic structures of the symmetry-broken $R3m$ phase. We also find that these $V_{xc}$ potentials do not quantitatively agree with the available experimental optical conductivity on ${\mathrm{GaV}}_4{\mathrm{S}}_8$; nonetheless, the LDA and GGA functionals correctly reproduce its lattice dynamical modes. Our findings suggest that accurate qualitative and quantitative simulations of the lacunar spinel family with DFT requires careful attention to the nuances of the exchange-correlation functional and considered spin structures.

Figure 1

(a) Derivation of the cubic phase lacunar spinel $A{M}_{4}Q{}_8$ from an ideal spinel structure; some anions are hidden in the figure to facilitate visualization of the $M_4$ cluster formation. (b) The primitive cell of $A{M}_4{Q}_8$ in both cubic and rhombohedral phases, with the interaxial rhombohedral angle ${\alpha }_{rh}$, intracluster metal-metal-bond angle ${\theta }_m$. (c) $M_4$ cluster connectivity in the cubic phase; they occupy the four octahedral holes created by the $A$ cation vacancies. (d) Schematic phase diagram of lacunar spinels exhibiting multiple phase transitions. (Key: FM, ferromagnetic; PM, paramagnetic).

Figure 2

Valence molecular orbital diagrams of ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaMo}}_4{\mathrm{S}}_8$. The valence $t_2$-symmetry orbitals are triply degenerate in the cubic phase. The orbital degeneracy is lifted by the accompanied Jahn-Teller distortion with a distortion sense that stabilizes and leads to filling of either the $a_1$ (${\mathrm{GaV}}_4{\mathrm{S}}_8$) or $e$ (${\mathrm{GaMo}}_4{\mathrm{S}}_8$) orbitals based on orbital occupancy of the metals forming the cluster. The dotted lines indicate the Fermi level in the distorted phases, whereas in the cubic phase the Fermi level intersects the triply degenerate valence bands.

Figure 3

Relative error in the unit cell volume of the cubic phase at different levels of DFT. PS is an abbreviation for the PBEsol functional and the number in parentheses is the value of the on-site Coulomb interaction used in the $\mathrm{GGA}+U$ method.

Figure 4

The volume of the tetrahedral ${\mathrm{V}}_4$ cluster with different DFT functionals and their corresponding ground-state magnetic moment per formula unit. The white area shows nonmagnetic results. The light-shaded and dark-shaded areas correspond to states with 5 and 7 ${\mu }_B$ magnetic moments, respectively. PS is an abbreviation for the PBEsol functional and the number in parentheses is the value of the on-site Coulomb interaction used in the $\mathrm{GGA}+U$ method.

Figure 5

DFT-PBE ground-state band structures and projected density of states (DOS) of the cubic lacunar spinels within a primitive cell. The Fermi level ($E_F$) is indicated by a broken line; the vertical energy axis is in units of eV. The gray shaded areas in the DOS panels correspond to the total electronic density of states. The second and third rows show results with SOC included, as indicated in the rightmost column. The orange curve in the DOSs represent the contribution from the transition metal cluster.

Figure 6

DFT-PBE band structure and DOS of metastable cubic ${\mathrm{GaV}}_4{\mathrm{S}}_8$ with 1 ${\mu }_B$ per formula unit. We define $\gamma$ as the exchange splitting between different spin channels and $\mathrm{\Delta }$ as the splitting of the three valence bands at the X point. Here, those bands are located approximately at $E_F$ and $E_F+0.3\mathrm{eV}$.

Figure 7

Relative error of the rhombohedral unit cell volume at different levels of DFT. PS is an abbreviation for the PBEsol functional and the number in parentheses is the value of the on-site Coulomb interaction used in the LDA/$\mathrm{GGA}+U$ method.

Figure 8

The $3a$ ($z_1$) Wyckoff position in rhombohedral ${\mathrm{GaMo}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{S}}_8$ at different $V_{xc}$. The gray dashed lines correspond to the experimental values. PS is an abbreviation for the PBEsol functional and the number in parentheses is the value of the on-site Coulomb interaction used in the $\mathrm{GGA}+U$ method.

Figure 9

DFT-PBE ground-state band structures and density of states (DOS) of the rhombohedral $R3m$ lacunar spinels. The second and third rows show simulation results with SOC. Color representation and the vertical energy axis are the same as that in Fig. 5.

Figure 10

DFT-functional dependence of the electronic band gaps for lacunar spinels in the rhombohedral $R3m$ structure.

Figure 11

Effect of on-site Coulomb interactions ($U$ values of 0.0, 1.0, 2.0, and 3.0 eV) and the amount of exact exchange interactions in the hybrid HSE functional (EX values of 0.05, 0.1, 0.2, and 0.25) on the electronic structures of rhombohedral $R3m$ ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$. $\mathrm{EX}=0.25$ corresponds to the standard amount of exact exchange in HSE06. The band structure panel on the left is obtained using the PBE functional.

Figure 12

DFT calculated optical conductivity of ${\mathrm{GaV}}_4{\mathrm{S}}_8$ in the rhombohedral $R3m$ structure compared with the experimental values obtained from Ref. [20].

Figure 13

Phonon frequencies of ${\mathrm{GaV}}_4{\mathrm{S}}_8$ in the cubic (left panel) and rhombohedral (right panel) phase with different DFT functionals. The experimental data from Ref. [57] is reproduced in the first column labeled “Raman/IR.”