Charge density functional plus U calculation of lacunar spinel GaM{}_{4}Se{}_{8} (M = Nb, Mo, Ta, and W)

Charge density functional plus calculation of lacunar spinel GaMSe (M = Nb, Mo, Ta, and W)

Abstract

Charge density functional plus calculations are carried out to examine the validity of molecular =1/2 and 3/2 state in lacunar spinel GaMX (M = Nb, Mo, Ta, and W). With LDA (spin-unpolarized local density approximation), which has recently been suggested as the more desirable choice than LSDA (local spin density approximation), we examine the band structure in comparison with the previous prediction based on the spin-polarized version of functional and with the prototypical =1/2 material SrIrO. It is found that the previously suggested =1/2 and 3/2 band characters remain valid still in LDA calculations while the use of charge-only density causes some minor differences. Our result provides the further support for the novel molecular state in this series of materials, which can hopefully motivate the future exploration toward its verification and the further search for new functionalities.

I Introduction

A series of ‘lacunar spinel’ compounds, GaMX (M = V, Nb, Mo, and Ta; X = S, Se, and Te), have attracted great attention due to their interesting physical properties and promising material characteristics. For example, multiferroicity has been observed in GaVS and GaVSe carrying a great potential for memory device applications Ruff et al. (2015); Widmann et al. (2017); Reschke et al. (2017); Ruff et al. (2017). In GaTaSe, resistive switching phenomena which can be used for resistive random access memory (RRAM) have been reported Dubost et al. (2013). In the case of M=Nb and Ta, the insulator-to-metal transition followed by superconducting transition is known to occur by applying pressure Abd-Elmeguid et al. (2004); Pocha et al. (2005). Their intriguing low temperature behaviors in susceptibility and specific heat measurement Yaich et al. (1984); Pocha et al. (2005); Kawamoto et al. (2016) can possibly be related to the unconventional superconductivity. Further, novel ’molecular ’ ground states have been suggested recently. According to the first-principles band calculations, the molecular =1/2 and 3/2 state are realized in the case of M = Mo, W, and Nb, Ta, respectively, due to the crucial role of spin-orbit interaction while this effect has been ignored in earlier studies Kim et al. (2014a). For GaTaSe, the novel =3/2 ground state has been verified by resonant inelastic x-ray scattering (RIXS) experiment combined with theoretical calculations Jeong et al. (2017).

One important next step is therefore to study the other materials (i.e., GaNbSe, GaMoSe, and GaWSe) and to confirm their characteristic molecular states, which can provide a new exciting playground in search for the novel quantum phenomena Jackeli and Khaliullin (2009); Chaloupka et al. (2010); Watanabe et al. (2010); Wang and Senthil (2011); Kim et al. (2012); Dey et al. (2012); Kimchi and Vishwanath (2014); Kim et al. (2014b); Chun et al. (2015); Kim et al. (2016). While the similar type of experiments such as RIXS and RXMS (resonant x-ray magnetic scattering) can be utilized Kim et al. (2009); Jeong et al. (2017), only available at this moment is the band structure prediction Kim et al. (2014a). On the one hand, the successful verification of =3/2 for the case of M = Ta Jeong et al. (2017) supports the reliability of the previous theoretical prediction Kim et al. (2014a). On the other, a series of recent DFT (density functional theory)+ studies require the further investigation. According to recent careful studies, the use of charge-only density functional (such as LDA and GGA (spin-unpolarized generalized gradient approximation)) is highly desirable for DFT+ type of calculation rather than the use of spin density functional (such as LSDA and SGGA (spin-polarized GGA)) Chen et al. (2015); Park et al. (2015); Chen and Millis (2016); Ryee and Han (2018a, b); com (). In LSDA+ or SGGA+ scheme, the intrinsic Stoner type exchange interactions can likely cause the unphysical magnetic behaviors through the uncontrolled competition and double counting with the interaction term like Hund exchange. This feature has been noticed in some case studies Chen et al. (2015); Park et al. (2015); Chen and Millis (2016) and then further analyzed in a formal way Ryee and Han (2018a, b). Since the previous band structure prediction of molecular states has been based on SGGA+ calculation with the functional form suggested by Dudarev et al. Dudarev et al. (1998); Han et al. (2006); Kim et al. (2014a), it is necessary to confirm the validity of it.

In this paper, we performed LDA+ calculations and confirmed the robustness of band structure for the and lacunar spinels. It is found that and 3/2 Mott ground states are well maintained in the reasonably large range of Hubbard and Hund parameters. By introducing a new quantity , which is designed to measure the and 3/2 band separation, we present the quantitative argument in comparison to the previous SGGA+ results and the prototype material, SrIrO. By confirming the previous theoretical prediction, our result hopefully motivates the further research toward the verification of these exotic quantum states and the search for the new functionality.

Ii Computation details

Band structure calculations were carried out using ‘OpenMX’ software package which is based on the linear combination of pseudo-atomic orbital (LCPAO) formalism Ozaki (2003). LDA+ calculations were performed with the (so-called) fully localized limit (FLL) functional form Anisimov et al. (1993); Solovyev et al. (1994); Czyżyk and Sawatzky (1994); Liechtenstein et al. (1995). The spin-orbit coupling (SOC) was treated within the fully relativistic -dependent pseudopotential MacDonald (1983); Bachelet et al. (1982); Theurich and Hill (2001). We adopted the energy cutoff of 400 Rydberg for real space grid and -points for the primitive unit cell. For the reasonable values of Hubbard and Hund , we took the previous cRPA (constrained random-phase approximation) results for each transition-metal element as our reference Şaşıoğlu et al. (2011). Considering that our lacunar spinels are all insulating, we used the 25% larger values of than the cRPA results for elements since the Mott gap is opened at around this value for GaNbSe and GaTaSe: 3.4, 4.5, 3.0, and 4.4 eV   for GaNbSe, GaMoSe, GaTaSe and GaWSe, respectively. The crystal structures were optimized, and for the band structure, we present the 2-formula-unit cell results with the antiferromagnetic inter-cluster order.

In order to discuss the robustness of nature in a quantitative way, we introduce a new parameter:

(1)

Here

(2)
and

where and represent the band index and the momentum, respectively. Also, the atomic states are written as . The quantifies the ratio between and with a factor 2 which represents the statistical ratio, 2:1, for the ideal or molecular states. Thus, ideally, becomes 0 if the given state is a purely atomic or molecular state (i.e., well identified just by , , and ) while it becomes 1.0 when the eigenstate is identical with the pure states. Now the separation of and bands of a given material can be represented by taking the average for the entire space:

Figure 1: (a) Crystal structure of GaMSe (cubic F). The yellow, red, and grey spheres represent Ga, M (= Nb, Mo, Ta, and W) and Se atoms respectively. GaMSe is composed of MSe (red) and GaSe (grey) clusters. (b) Schematic energy level diagrams for M cluster with and without SOC. Without SOC, molecular orbital states are 6-fold degenerate. SOC splits them into 2-fold and 4-fold degenerate states.
(3)

This value therefore provides a single number with which the nature of band structure can be expressed. As an example, let us consider the prototypical =1/2 material SrIrO. The calculated based on the ’(so-called) Dudarev functional Dudarev et al. (1998)’ with yields . This is the case for the original calculation result by Kim et al. Kim et al. (2008). If we performed the calculation with the SOC turned off, becomes zero. Here we also performed LDA+ calculation for SrIrO with the 25% larger value of than the cRPA value for elemental Ir. The result remains same; . Here it should be noted that depends on the degree of orbitals hybridization with other orbitals (e.g., oxygen or chalcogen ), the local structure, and crystal field splitting, etc. Therefore the interpretation and the comparison of the absolute values of need to be careful.

Iii Results and discussion

Figure 2: The calculated -projected DOS and the band dispersions for (a) GaNbSe ( eV, eV), (b) GaMoSe ( eV, eV), (c) GaTaSe ( eV, eV) and (d) GaWSe ( eV, eV). The red, blue, green and violet colors in DOS represent the , , and character, respectively. In the band dispersion, the (red) and (blue) character are represented by the line thickness.

The crystal structure of lacunar spinel GaMSe (space group F3m) can be understood as deduced from the spinel, GaMSe, with half-deficient Ga atoms François et al. (1991); Pocha et al. (2005). A characteristic feature is that the well-defined molecular units of MSe and GaSe are arranged to be NaCl structure as shown in Fig. 1(a). The 12-fold M-M bonding complex is split into 6-fold , 4-fold and 2-fold degenerate states due to molecular symmetry Johrendt (1998); Pocha et al. (2000); Nakamura et al. (2005); Guiot et al. (2011); Ta Phuoc et al. (2013); Kim et al. (2014a); Camjayi et al. (2014) as shown in Fig. 1(b). The electronic structure near the Fermi level is governed by molecular states which are derived from the atomic orbitals of transition metals Johrendt (1998); Pocha et al. (2005); Ta Phuoc et al. (2013); Kim et al. (2014a); Camjayi et al. (2014). It is noted that the molecular orbitals have the same symmetry with atomic , and the SOC leads them to split into the ‘effective’ angular momentum doublet and quartet Kim et al. (2014a). Depending on the number of valence electrons, GaMoSe and GaWSe have while GaNbSe and GaTaSe carry moment; see Fig. 1(b). These ‘molecular’ ground states were first predicted by band structure calculation Kim et al. (2014a), and the case for has recently been confirmed for M = Ta Jeong et al. (2017).

Fig. 2 presents the projected density of states (PDOS; left panels) and the fat band dispersion (right panels) obtained by LDA+ calculations; (a) GaNbSe, (b) GaMoSe, (c) GaTaSe, and (d) GaWSe. First of all, we note that the characteristic molecular states are well maintained as in the previous calculation of SGGA+ functional Kim et al. (2014a). The upper/lower Hubbard bands are predominantly of and character for GaWSe and GaTaSe, respectively; see Fig. 2(c) and (d). For materials, the mixture between the two states is noticed in the upper and lower Hubbard part for GaNbSe and GaMoSe, respectively (see Fig.2(a) and (b)), which is a comparable feature with the case of SrIrO Arita et al. (2012).

In particular, for GaTaSe, the higher-lying peak is located at around +0.8 eV  and well identified (i.e., having a negligible mixture with other states). This feature together with the lower-lying ’’ bands at 0.3 eV   (see the green-colored PDOS in Fig.2(c)) is mainly responsible for the novel quantum interference observed in a recent RIXS experiment which is the first direct experimental evidence for moment in a real material Jeong et al. (2017). Thus, our current result provides the further confirmation of the characteristic electronic structure of GaTaSe and other lacunar spinels by using the charge-only LDA+ calculations which have recently been suggested as the more desirable functional choice than SGGA+ Chen and Millis (2016); Ryee and Han (2018a).

Figure 3: The calculated of four lacunar spinel compounds as a function of Hund . The violet, green, blue, and red color represents the results of GaWSe, GaTaSe, GaMoSe, and GaNbSe, respectively.

The differences in the two calculated band structure, namely by LDA+ and LSDA+, are relatively minor. For all four compounds, the calculated band gaps are smaller in LDA+ than LSDA+ by about 0.1–0.2 eV  at the same interaction parameters of and . It implies that the larger is required to open the Mott gap in LDA+. The band separation between and is slightly more pronounced in LDA+ results. This feature can also be seen by comparing the calculated which will be discussed in the below.

One obvious limitation of Dudarev formalism is its inability to calculate Hund--dependent electronic structure Dudarev et al. (1998); Kim et al. (2014a). In order to check the robustness of band character in lacunar spinels, we performed the calculations as a function of Hund ; see Fig. 3. The results show that the degree of separation of and bands as represented by is well maintained in the wide range of , especially for compounds. The calculated for GaTaSe and GaWSe are quite large () in the entire range of Hund considered in this study (see the violet and green line). Due to the smaller SOC, on the other hand, the calculated for compounds depends more severely on Hund . In the large limit, the calculated becomes as small as 0.49 and 0.25 for GaMoSe and GaNbSe, respectively (see the blue and red line). In the reasonable range of Hund eV, is about 0.5 for both Nb and Mo cases which is comparable with the value of SrIrO. While is just a simple measure of the degree of separation of and bands based on the calculated electronic structure, our calculation clearly supports the robustness of the molecular character of lacunar spinels.

In order to further check if the band character remains valid, we calculated as a function of with the fixed to the cRPA values: 0.45, 0.50, 0.40, and 0.45 eV   for GaNbSe, GaMoSe, GaTaSe, and GaWSe. In a wide range of value from 2.0 to 4.5 eV, we found that does not change much. For compounds of M = Ta and W, the calculated remains well above and mainly close to 0.7 for 4.0 eV. In the case of GaMoSe, the calculated remains not smaller than 0.5. For M = Nb, the calculated is gradually reduced from 0.43 at eV  to 0.33 at eV, which is noticeably smaller than the other three compounds. While the quantified is certainly smaller in materials, we think that GaNbSe can also be well identified as a molecular material especially considering that its material properties are quite similar with those of GaTaSe including the insulator-to-metal transition and superconductivity under pressure Abd-Elmeguid et al. (2004). It would be an interesting experimental challenge to confirm this exotic ground state for M = Nb just as in the recent report on GaTaSe Jeong et al. (2017).

Iv Conclusion

With LDA+ calculations, we confirm the previous theoretical prediction based on SGGA for the molecular band structures in / lacunar spinels. By introducing a new parameter, , we performed the quantitative examination of band separation as a function of which was not feasible in the previous study. It is found that compounds have the quite robust band character while both and materials have well-identified feature at around the realistic values. This nature is also quite well maintained in the reasonable range of . Our results provide the solid guidance for future study of this materials by strengthening the theoretical prediction of the novel material characteristic. In particular, the detailed magnetic property at low temperature and under pressure need to be further identified and understood, which can also elucidate the nature of superconductivity found in =3/2 materials.

Acknowledgements.
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1A2B2005204) and Creative Materials Discovery Program through the NRF funded by Ministry of Science and ICT (2018M3D1A1058754).

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