Charge density functional plus $U$ calculation of lacunar spinel GaM$_4$Se$_8$ (M = Nb, Mo, Ta, and W)

TL;DR

This study uses charge density functional plus U calculations to validate the molecular J_eff=1/2 and 3/2 states in lacunar spinel GaM4Se8 compounds, supporting the existence of these states with implications for future research.

Contribution

It demonstrates that LDA+U calculations confirm the molecular J_eff states in GaM4Se8, providing a theoretical basis for their validity beyond previous spin-polarized approaches.

Findings

J_eff=1/2 and 3/2 states remain valid in LDA+U calculations

Charge-only density causes minor differences in band structure

Supports the existence of molecular J_eff states in these materials

Abstract

Charge density functional plus $U$ calculations are carried out to examine the validity of molecular $J_\text{eff}$=1/2 and 3/2 state in lacunar spinel GaM$_4$X$_8$ (M = Nb, Mo, Ta, and W). With LDA (spin-unpolarized local density approximation)$+U$, which has recently been suggested as the more desirable choice than LSDA (local spin density approximation)$+U$, we examine the band structure in comparison with the previous prediction based on the spin-polarized version of functional and with the prototypical $J_\text{eff}$=1/2 material Sr$_2$IrO$_4$. It is found that the previously suggested $J_\text{eff}$=1/2 and 3/2 band characters remain valid still in LDA$+U$ calculations while the use of charge-only density causes some minor differences. Our result provides the further support for the novel molecular $J_\text{eff}$ state in this series of materials, which can hopefully motivate the future exploration toward its verification and the further search for new functionalities.

Equations

$$b_{n,\mathbf{k}}$$

$$P_{1/2}^{n,\mathbf{k}}=$$

$$P_{3/2}^{n,\mathbf{k}}=$$

$$+{\left|⟨\frac{3}{2}-\frac{1}{2}⟩{\psi }_{n,\mathbf{k}}\right|}^2+{\left|⟨\frac{3}{2}-\frac{3}{2}⟩{\psi }_{n,\mathbf{k}}\right|}^2,$$

$$b_{\text{avg}}=\frac{{\sum }_{\mathbf{k}}{\sum }_n\left(P_{1/2}^{n,\mathbf{k}}+P_{3/2}^{n,\mathbf{k}}\right)b_{n,\mathbf{k}}}{{\sum }_{\mathbf{k}}{\sum }_n\left(P_{1/2}^{n,\mathbf{k}}+P_{3/2}^{n,\mathbf{k}}\right)}.$$

Full text

I Introduction

A series of ‘lacunar spinel’ compounds, GaM4X8 (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 GaV4S8 and GaV4Se8 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 GaTa4Se8, 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 $J_{\text{eff}}$’ ground states have been suggested recently. According to the first-principles band calculations, the molecular $J_{\text{eff}}$=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 GaTa4Se8, the novel $J_{\text{eff}}$=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., GaNb4Se8, GaMo4Se8, and GaW4Se8) and to confirm their characteristic molecular $J_{\text{eff}}$ 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 $J_{\text{eff}}$=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)+$U$ 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+$U$ 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+$U$ or SGGA+$U$ 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 $J_{\text{eff}}$ states has been based on SGGA+$U$ 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+$U$ calculations and confirmed the robustness of $J_{\text{eff}}$ band structure for the $4d$ and $5d$ lacunar spinels. It is found that $J_{\text{eff}}=1/2$ and 3/2 Mott ground states are well maintained in the reasonably large range of Hubbard $U$ and Hund $J$ parameters. By introducing a new quantity $b_{\text{avg}}$, which is designed to measure the $J_{\text{eff}}=1/2$ and 3/2 band separation, we present the quantitative argument in comparison to the previous SGGA+$U$ results and the prototype $J_{\text{eff}}=1/2$ material, Sr2IrO4. 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+$U$ 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 $j$-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 $12\times 12\times 12$ $k$-points for the primitive unit cell. For the reasonable values of Hubbard $U$ and Hund $J$, 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 $U$ than the cRPA results for elements since the Mott gap is opened at around this value for GaNb4Se8 and GaTa4Se8: $U=$ 3.4, 4.5, 3.0, and 4.4 eV   for GaNb4Se8, GaMo4Se8, GaTa4Se8 and GaW4Se8, 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 $J_{\text{eff}}$ nature in a quantitative way, we introduce a new parameter:

[TABLE]

Here

[TABLE]

where $n$ and ${\bm{k}}$ represent the band index and the momentum, respectively. Also, the atomic $J_{\text{eff}}$ states are written as $\bra{J_{\text{eff}}\ m_{J_{\text{eff}}}}$. The $b_{n,{\bm{k}}}$ quantifies the ratio between $J_{\text{eff}}=1/2$ and $J_{\text{eff}}=3/2$ with a factor 2 which represents the statistical ratio, 2:1, for the ideal $t_{\text{2g}}$ or molecular $t_{2}$ states. Thus, ideally, $b_{n,{\bm{k}}}$ becomes 0 if the given state is a purely atomic $t_{\text{2g}}$ or molecular $t_{2}$ state (i.e., well identified just by $\ket{xy\ \pm\sigma}$, $\ket{xz\ \pm\sigma}$, and $\ket{yz\ \pm\sigma}$) while it becomes 1.0 when the eigenstate is identical with the pure $J_{\text{eff}}$ states. Now the separation of $J_{\text{eff}}=1/2$ and $J_{\text{eff}}=3/2$ bands of a given material can be represented by taking the average for the entire space:

[TABLE]

This value therefore provides a single number with which the $J_{\text{eff}}$ nature of band structure can be expressed. As an example, let us consider the prototypical $J_{\text{eff}}$=1/2 material Sr2IrO4. The calculated $b_{\text{avg}}$ based on the ‘(so-called) Dudarev functional Dudarev et al. (1998)’ with $U=2.0$ yields $b_{\text{avg}}=0.486$. 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, $b_{\text{avg}}$ becomes zero. Here we also performed LDA+$U$ calculation for Sr2IrO4 with the 25% larger value of $U$ than the cRPA value for elemental Ir. The result remains same; $b_{\text{avg}}=0.487$. Here it should be noted that $b_{\text{avg}}$ depends on the degree of $d$ orbitals hybridization with other orbitals (e.g., oxygen or chalcogen $p$), the local structure, and crystal field splitting, etc. Therefore the interpretation and the comparison of the absolute values of $b_{\text{avg}}$ need to be careful.

III Results and discussion

The crystal structure of lacunar spinel GaM4Se8 (space group F$\bar{4}$3m) can be understood as deduced from the spinel, GaM2Se4, with half-deficient Ga atoms François et al. (1991); Pocha et al. (2005). A characteristic feature is that the well-defined molecular units of M4Se4 and GaSe4 are arranged to be NaCl structure as shown in Fig. 1(a). The 12-fold M-M bonding complex is split into 6-fold $t_{2}$, 4-fold $e$ and 2-fold degenerate $a_{1}$ states due to $T_{d}$ 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 $t_{2}$ states which are derived from the atomic $t_{\text{2g}}$ 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 $t_{2}$ orbitals have the same symmetry with atomic $t_{\text{2g}}$, and the SOC leads them to split into the ‘effective’ angular momentum $J_{\text{eff}}=1/2$ doublet and $J_{\text{eff}}=3/2$ quartet Kim et al. (2014a). Depending on the number of valence electrons, GaMo4Se8 and GaW4Se8 have $J_{\text{eff}}=1/2$ while GaNb4Se8 and GaTa4Se8 carry $J_{\text{eff}}=3/2$ moment; see Fig. 1(b). These ‘molecular’ $J_{\text{eff}}$ ground states were first predicted by band structure calculation Kim et al. (2014a), and the case for $J_{\text{eff}}=3/2$ 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+$U$ calculations; (a) GaNb4Se8, (b) GaMo4Se8, (c) GaTa4Se8, and (d) GaW4Se8. First of all, we note that the characteristic molecular $J_{\text{eff}}$ states are well maintained as in the previous calculation of SGGA+$U$ functional Kim et al. (2014a). The upper/lower Hubbard bands are predominantly of $J_{\text{eff}}=1/2$ and $J_{\text{eff}}=3/2$ character for GaW4Se8 and GaTa4Se8, respectively; see Fig. 2(c) and (d). For $4d$ materials, the mixture between the two $J_{\text{eff}}$ states is noticed in the upper and lower Hubbard part for GaNb4Se8 and GaMo4Se8, respectively (see Fig.2(a) and (b)), which is a comparable feature with the case of Sr2IrO4 Arita et al. (2012).

In particular, for GaTa4Se8, the higher-lying $J_{\text{eff}}=1/2$ 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 ‘$e$’ 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 $J_{\text{eff}}=3/2$ moment in a real material Jeong et al. (2017). Thus, our current result provides the further confirmation of the characteristic electronic structure of GaTa4Se8 and other lacunar spinels by using the charge-only LDA+$U$ calculations which have recently been suggested as the more desirable functional choice than SGGA+$U$ Chen and Millis (2016); Ryee and Han (2018a).

The differences in the two calculated band structure, namely by LDA+$U$ and LSDA+$U$, are relatively minor. For all four compounds, the calculated band gaps are smaller in LDA+$U$ than LSDA+$U$ by about 0.1–0.2 eV  at the same interaction parameters of $U$ and $J$. It implies that the larger $U$ is required to open the Mott gap in LDA+$U$. The band separation between $J_{\text{eff}}=1/2$ and $J_{\text{eff}}=3/2$ is slightly more pronounced in LDA+$U$ results. This feature can also be seen by comparing the calculated $b_{\text{avg}}$ which will be discussed in the below.

One obvious limitation of Dudarev formalism is its inability to calculate Hund-$J$-dependent electronic structure Dudarev et al. (1998); Kim et al. (2014a). In order to check the robustness of $J_{\text{eff}}$ band character in lacunar spinels, we performed the calculations as a function of Hund $J$; see Fig. 3. The results show that the degree of separation of $J_{\text{eff}}=1/2$ and $J_{\text{eff}}=3/2$ bands as represented by $b_{\text{avg}}$ is well maintained in the wide range of $J$, especially for $5d$ compounds. The calculated $b_{\text{avg}}$ for GaTa4Se8 and GaW4Se8 are quite large ($b_{\text{avg}}>0.68$) in the entire range of Hund $J$ considered in this study (see the violet and green line). Due to the smaller SOC, on the other hand, the calculated $b_{\text{avg}}$ for $4d$ compounds depends more severely on Hund $J$. In the large $J$ limit, the calculated $b_{\text{avg}}$ becomes as small as $\sim$0.49 and $\sim$0.25 for GaMo4Se8 and GaNb4Se8, respectively (see the blue and red line). In the reasonable range of Hund $J\approx 0.5$ eV, $b_{\text{avg}}$ is about 0.5 for both Nb and Mo cases which is comparable with the value of Sr2IrO4. While $b_{\text{avg}}$ is just a simple measure of the degree of separation of $J_{\text{eff}}=1/2$ and $J_{\text{eff}}=3/2$ bands based on the calculated electronic structure, our calculation clearly supports the robustness of the molecular $J_{\text{eff}}$ character of lacunar spinels.

In order to further check if the $J_{\text{eff}}$ band character remains valid, we calculated $b_{\text{avg}}$ as a function of $U$ with the fixed $J$ to the cRPA values: $J=$0.45, 0.50, 0.40, and 0.45 eV   for GaNb4Se8, GaMo4Se8, GaTa4Se8, and GaW4Se8. In a wide range of $U$ value from 2.0 to 4.5 eV, we found that $b_{\text{avg}}$ does not change much. For $5d$ compounds of M = Ta and W, the calculated $b_{\text{avg}}$ remains well above $b_{\text{avg}}=0.6$ and mainly close to 0.7 for $U\leq$ 4.0 eV. In the case of GaMo4Se8, the calculated $b_{\text{avg}}$ remains not smaller than 0.5. For M = Nb, the calculated $b_{\text{avg}}$ is gradually reduced from 0.43 at $U=3.0$ eV  to 0.33 at $U=4.5$ eV, which is noticeably smaller than the other three compounds. While the quantified $b_{\text{avg}}$ is certainly smaller in $4d$ materials, we think that GaNb4Se8 can also be well identified as a molecular $J_{\text{eff}}=3/2$ material especially considering that its material properties are quite similar with those of GaTa4Se8 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 GaTa4Se8 Jeong et al. (2017).

IV Conclusion

With LDA+$U$ calculations, we confirm the previous theoretical prediction based on SGGA$+U$ for the molecular $J_{\text{eff}}$ band structures in $4d$/$5d$ lacunar spinels. By introducing a new parameter, $b_{\text{avg}}$, we performed the quantitative examination of $J_{\text{eff}}$ band separation as a function of $J$ which was not feasible in the previous study. It is found that $5d$ compounds have the quite robust $J_{\text{eff}}$ band character while both $4d$ and $5d$ materials have well-identified $J_{\text{eff}}$ feature at around the realistic $J$ values. This $J_{\text{eff}}$ nature is also quite well maintained in the reasonable range of $U$. 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 $J_{\text{eff}}$=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).