First-principles-based calculation of branching ratio for 5d, 4d, and 3dtransition metal systems

To calculate BR, we first calculate ⟨L ⋅ S⟩. With our basis set of localized pseudo-atomic orbital (PAO) ϕ_α,i (α: orbital index, i: site index) [36], Kohn–Sham eigenstate is decomposed $\vert {{\Psi}}_{n\boldsymbol{k}}\rangle ={{\Sigma}}_{i,\alpha }{c}_{\alpha ,i}^{n,\boldsymbol{k}}\vert {\phi }_{\alpha ,i}\rangle$ where n and k refer to band index and momentum, respectively. The SOC part of Hamiltonian is then estimated as

$$\begin{align}\hfill \langle \boldsymbol{L}\cdot \boldsymbol{S}\rangle & =\sum _{n\boldsymbol{k}}^{\mathrm{o}\mathrm{c}\mathrm{c}}\langle {{\Psi}}_{n\boldsymbol{k}}\vert \boldsymbol{L}\cdot \boldsymbol{S}\vert {{\Psi}}_{n\boldsymbol{k}}\rangle \hfill \\ \hfill & \hfill =\sum _{{{\epsilon}}_{n\boldsymbol{k}}}\left(1.0{\times}{P}_{J=5/2}\left({{\epsilon}}_{n\boldsymbol{k}}\right)-1.5{\times}{P}_{J=3/2}\left({{\epsilon}}_{n\boldsymbol{k}}\right)\right),\end{align} \tag{ 1 }$$

where P_J=5/2 and P_J=3/2 are densities of states for J = 5/2 and J = 3/2 state, respectively [23].

In reference [20], the core–valence interaction is assumed to be small enough in comparison to the core SOC, and then the core states are well described by total angular momentum quantum number J. The intensity ratio is

$$\begin{equation}\frac{{I}_{{L}_{3}}}{{I}_{{L}_{2}}}=\frac{2{n}_{h}-\langle \boldsymbol{L}\cdot \boldsymbol{S}\rangle }{{n}_{h}+\langle \boldsymbol{L}\cdot \boldsymbol{S}\rangle }=\frac{2-r}{1+r},\end{equation} \tag{ 2 }$$

where r = ⟨L ⋅ S⟩/n_h.

Note that BR can be calculated just from n_h and ⟨L ⋅ S⟩. It should also be noted that the correct estimation of n_h can be non-trivial typically due to the hybridization with neighboring atomic orbitals [23], and the value of n_h is dependent on the charge counting method. In the current study, we simply take the numerical integration of the partial density of states. The deviation ranges caused by this choice are represented by error bars in figures 1–3.

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For the electronic structure calculations, we used 'OpenMX' density functional theory software package [36], which takes the linear combination of numerical PAO as a basis set and the norm-conserving pseudo-potential [37, 38]. The cutoff radii for O, Cl, Ti, Cr, Fe, Ni, Cu, Sr, Ru, Rh, Pd, Hf, Ta, W, Re, Ir, Pt, Mn, Co, Sc, Mg, and Ca are 5.0, 7.0, 7.0, 6.0, 6.0, 6.0, 6.0, 10.0, 7.0, 7.0, 7.0, 7.0, 7.0, 7.0, 7.0, 7.0, 7.0, 6.0, 6.0, 7.0, 7.0 and 9.0 a.u., respectively. The Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional has been adopted [39]. For Mott insulators, we used spin-polarized PBE plus U scheme to treat on-site electronic correlations as our main data set. U_eff = U − J_H [40, 41] values for CuO, RuCl₃, IrO₂, and Sr₂CaIrO₆ are 4.0, 1.5, 2.0, and 2.0  eV, respectively [23, 30, 42–46]. We also checked with spin-unpolarized PBE +U considering the recent discussion on this issue [47–51]. It is found that the calculated values are in a reasonable agreement with each other, and any of our conclusion is not affected by this choice. The SOC was taken into account within a fully relativistic J-dependent pseudo-potential scheme in the non-collinear methodology [36]. While the experimental crystal structures are mainly used [52–65], we also checked the results with varying lattice parameters up to ±3% (see section 3.1).

First, we apply our method to 5d systems. A total of 18 different materials have been investigated; Hf, HfO₂, Ta, Ta₂O₅, W, WO₂, Re, ReO₂, ReO₃, Ir, IrO₂, Sr₂CaIrO₆, Sr₂MgIrO₆, Sr₂TiIrO₆, Sr₂ScIrO₆, Sr₂IrO₄, Pt, and PtO₂. Figures 1(a) and (b) present the calculation results of BR and ⟨L ⋅ S⟩, respectively, in comparison with experimental data [66–69]. The red and blue colors represent the calculation and experimental data, respectively. The error bars for the experimental values are taken directly from references. The overall good agreement between calculations and experiments is clearly noticed.

For elemental Hf, Ta, W, Re, and their oxides, BR is quite close to the statistical value, ${I}_{{L}_{3}}/{I}_{{L}_{2}}=2$ [66, 67, 70], and this feature is well reproduced by our calculation. For elemental Ir, the calculation result is located in between the two experimental values. For Pt and PtO₂, the BRs are notably greater than the statistical value in both experiment and calculation.

The BRs of iridium oxides are significantly larger than 2 which is attributed to the combined effect of charge transfer and crystal fields [66, 67]. For IrO₂, two experiments report BR ≃ 3.1 [67] and ≃ 6.9 [66], respectively. Our calculation shows that BR is close to 3.1 in good agreement with the data by Cho et al [67]. Also for Sr₂IrO₄, there are two experimental data available; BR ≃ 7.0 [66] and ≃ 5.3 [68]. Our calculation supports the latter. It is noted that the data of reference [68] is taken from 5% Rh-doped sample while reference [66] and our calculation measure the stoichiometric Sr₂IrO₄. According to reference [68], BR does not change much as a function of Rh concentration in the small doping regime [68]. For Ir double perovskites, Sr₂AIrO₆, Laguna-Marco et al recently performed experiments [69], and three compounds (A = Mg, Ti, Sc) were calculated in the previous study (also presented in figure 1). Our calculation not just reproduces the overall feature of BR for iridium oxides (see the trend from Sr₂CaIrO₆ to IrO₂), but it also gives the quantitative agreement with experiments. It is also noted that, as reported in reference [23], the difference between calculation and experiment can further be reduced (especially for the case of Sr₂MgIrO₆) by considering the possible oxygen vacancy in the experimental situation.

Figure 1(b) shows the calculated ⟨L ⋅ S⟩ in comparison with experiments. We take the experimental data as in the original papers if ⟨L ⋅ S⟩ values are presented. Otherwise, we estimated the experimental values from the nominal charges through ⟨L ⋅ S⟩ = n_h(2 − BR)/(1 + BR). Note that the measured quantity in the experiment is BR from which ⟨L ⋅ S⟩ is estimated, whereas, in our calculation, the more direct quantity is ⟨L ⋅ S⟩ as clearly seen in equation (2). Therefore the good agreement between theory and experiment can become a strong indication of the reliability of our method.

As expected, for the materials whose BR is close to the statistical value, ⟨L ⋅ S⟩ is close to 0. Overall, the calculation results are in reasonable agreement with experiments. In iridium oxides, |⟨L ⋅ S⟩| is significantly larger being consistent with figure 1(a). It is interesting to note that Sr₂IrO₄ has a larger BR than Sr₂TiIrO₆ while both have d⁵ configuration. It is attributed to the smaller crystal field in Sr₂IrO₄ (Δ = 3.61  eV) than that of Sr₂TiIrO₆ (Δ = 4.19  eV). It is known that |⟨L ⋅ S⟩| decreases as the crystal field increases [23].

Here we emphasize that our calculations consider a wide energy range for d-orbital states. In order to represent the ambiguity in taking the d-orbital contributions (or in other words, the ambiguity in the projection onto d orbitals) due to the hybridization with O-2p ligands (in the case of oxides) or with other orbitals (in the case of elements), the energy range is set to cover the entire non-negligible PDOS (projected density of states) contributions. It is the reason why the current results are different from the values in reference [23] for the case of double perovskites. In the current study, our energy range is typically tens of electron volts while it was set to cover only the main peaks in reference [23]. Considering that this type of ambiguity, the good agreement between calculation and experiment supports the reliability of the method. We also check the error range caused by varying lattice parameters. In figure 2, the calculation results of IrO₂ are presented in which its lattice parameter is changed by ±3%. It is found that the dependence is not significant. We also the same feature for other materials.

4d TM systems are of particular interest since our method has only been applied to 5d materials and its applicability is expected to be limited as the atomic number decreases. Here we select six different materials for which the experimental data are available; RuO₂, Sr₂RuO₄, Sr₄Ru₂O₉, RuCl₃, Rh, and Pd. Figures 2(a) and (b) show the calculated BR and ⟨L ⋅ S⟩ (red colors), respectively, along with experimental data (blue colors) [71–73] whose error bars are taken from the original reference papers. The experimental data are fairly well reproduced by our calculations.

For Sr₄Ru₂O₉, Sr₂RuO₄, RuO₂, Rh, and Pd, the experimental BRs are close to the statistical value which indicates that the effect of SOC is not significant as also reflected in figure 2(b). The calculation results are in good agreement with experimental data while they tend to overestimate. RuCl₃ is of great recent research interest for which the novel Kitaev physics can be realized due to the sizable SOC [30, 71, 74–76]. Indeed, |⟨L ⋅ S⟩| and BR are much larger in this material, see figures 2(a) and (b). The experimental values are in reasonable agreement with our calculation within the error bar ranges. Note that J = 5/2 state (not J = 3/2) is solely responsible for so-called J_eff = 1/2 in this material, which leads to the larger intensity at L₃ edge and therefore also to the larger BR. It is also noted that, within the simple ionic picture, Ru ion in this material has d⁵ configuration whose |⟨L ⋅ S⟩| value should be smaller than that of d⁴ according to the naive charge counting. This example clearly shows that the elaborate electronic structure information is important to predict the effect of SOC and the related experimental quantity (figure 4).

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Finally, we investigate 3d TM elements and compounds. A total of twelve systems are taken into account; Ti, TiO₂, SrTiO₃, Cr, MnO, Fe, FeO, CoO, Ni, NiO, Cu, and CuO. Our calculation results of BR (red colors) are represented in figure 3 in comparison with experiments (blue colors) [77–79]. Different from the case of 4d and 5d materials, the calculated results significantly differ from the experimental data. The calculated ⟨L ⋅ S⟩ is small and the BRs are all quite close to the statistical value. It is attributed to the assumption that the core–valence interaction is negligible which can hardly be relevant to light elements. In the case of 3d TMs, the SOC splitting of 2p_1/2 and 2p_3/2 core states is typically 5–20  eV which is comparable with the core–valence interaction of roughly a few eV [80–82]. For example, the core–valence interactions are shown to be 5.29–5.6  eV and 6.67–6.8  eV in Mn²⁺ and Ni²⁺, respectively [80–82]. These values are just slightly increased by considering the core hole states [82]. It is in sharp contrast to the case of 4d and 5d TMs where the SOC of core 2p electrons is several hundred eV or even more [67, 69–71] whereas the core–valence interaction is typically ⩽ 5  eV [3, 72]. In order to describe 3d TM systems within the first-principles framework, the more sophisticated techniques are needed to deal with the core holes directly [16, 83, 84].