Calculating branching ratio and spin-orbit coupling from first-principles: A formalism and its application to iridates
We present a simple technique to calculate spin-orbit coupling, , and branching ratio measured in x-ray absorption spectroscopy. Our method is for first-principles electronic structure calculation and its implementation is straightforward for any of standard formulations and codes. We applied this technique to several different large spin-orbit coupling iridates. The calculated and branching ratio of a prototype =1/2 Mott insulator, SrIrO, are in good agreement with recent experimental data over the wide range of Rh-doping. Three different double perovskite iridates (namely, SrMgIrO, SrScIrO, and SrTiIrO) are also well described. This technique can serve as a promising tool for studying large spin-orbit coupling materials from first-principles and for understanding experiments.
pacs:75.70.Cn, 75.47.Lx, 71.15.Mb, 78.70.Dm
Recently the role of spin-orbit coupling (SOC) in solids has attracted tremendous attention. In many cases, SOC drastically changes the electronic band structure and results in a fundamentally different material property. A class of materials, called topological insulators, is an outstanding example Hasan and Kane (2010); Qi and Zhang (2011). SOC can also play together with on-site electronic correlation, , as often found in transition-metal oxides. In iridates, for example, the cooperation of SOC and drives materials to be a novel ’=1/2 Mott insulator’ Kim et al. (2008, 2009). Due to the characteristic hopping integrals caused by =1/2 nature (instead of =1/2), some interesting new possibilities have been proposed and still under active investigations Shitade et al. (2009); Pesin and Balents (2010); Jackeli and Khaliullin (2009); Wang and Senthil (2011). The basically same features can also be found in the non-oxide and transition-metal compounds Kim et al. (2014).
The spin-orbit Hamiltonian is represented by . While is known from the atomic nature of a given species, the direct estimation of is not always straightforward from experiment nor by theoretical calculation. For topological insulators, the observed band structure (e.g., by angle-resolved photoemission spectroscopy (ARPES)) is regarded as a strong evidence of the characteristic band dispersion caused by SOC Hasan and Kane (2010); Qi and Zhang (2011). For iridates, the data from resonant x-ray magnetic scattering (RXMS) and/or resonant inelastic x-ray scattering (RIXS) have been accepted as a confirmation of the novel SOC physics because the interpretation of the data seems consistent only with theoretical models that take strong SOC into account Kim et al. (2009, 2012). However it is noted that sometimes a different interpretation can be made and then the conclusion might be changed (for an example of iridates, see Ref. Moretti Sala et al. (2014)). Further, from the theoretical point of view, it is unsatisfactory that there is no simple and well-defined way to directly calculate SOC strength and to compare with experiments. In the standard first-principles calculations, can be calculated when the atomic wavefunctions are constructed by solving the relativistic Dirac equation. , however, is not just determined by atomic nature but depends on the electronic structure of solids.
In this paper, we point out that the calculation of can be performed in a simple and straightforward way within the standard first-principles framework and be directly compared with experiment. One possible reason that the calculation of has not been often made from first-principles may be partly because of no direct reference data available from the experimental side. We note that the branching ratio, typically measured in x-ray absorption spectroscopy (XAS), can be used to estimate the strength of SOC. Instead of calculating XAS spectrum itself, a simple technique can be used to directly calculate branching ratio through . Our formalism is implemented into our localized pseudo-atomic orbital (PAO) basis code and applied to several different iridium oxide compounds. For a =1/2 system, SrIrO (see Fig. 1(a)), we considered Rh doping (namely, SrRhIrO) and found that the calculated SOC and branching ratio are in good agreement with XAS data over the wide range of doping ratio . The iridate double perovskites (see Fig. 1(b)), SrIrO (: Mg, Sc, Ti), are also calculated and the results are in good agreement with recent experiments. We emphasize the formalism and implementation are simple enough to be adoptable for any type of first-principles code and method. This technique can be a useful tool for study large SOC materials by providing a direct estimation of and branching ratio.
Ii Computation Method
ii.1 Calculation details
For the electronic structure calculations, we used OpenMX software package _op (); Ozaki and Kino (2004); Ozaki (2003) which is based on the linear combination of numerical PAO and norm-conserving pseudopotential Ozaki (2003). The cutoff radii for Sr, Ir, Rh, Mg, Sc, Ti, and O are 10.0, 7.0, 7.0, 7.0, 7.0, 7.0, and 5.0 a.u. respectively. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional Perdew et al. (1996) and 300 Ry energy cutoff. and k-meshes were taken for Rh-doped SrIrO and double perovskites, respectively. have been used. The SOC was treated within a fully relativistic -dependent pseudopotential scheme in the non-collinear methodology _op (). The on-site electronic correlations were taken into account within DFT+ formalism Dudarev et al. (1998); Han et al. (2006). The reasonable value of may be about 2.0 – 3.0 as noticed by the previous studies on SrIrO and BaIrO Kim et al. (2008); Zhang et al. (2013); Martins et al. (2011); Arita et al. (2012). Throughout the manuscript, we present eV results as our main data both for Rh-doped SrIrO and double perovskites. After scanning the region of 2.0 – 3.0 eV, we found that any of our conclusion does not change by choosing different values. For Rh-doped iridates, the lattice constant and internal coordinates are optimized with the force criteria of 0.01 eV/Å. For double perovskites, we used the experimental lattice parameters Kayser et al. (2014); Jung and Demazeau (1995) of Å (SrMgIrO), Å (SrScIrO) and Å (SrTiIrO).
In this section, we present our formalism to calculate and branching ratio from first-principles. The localized atomic orbitals are assumed to be the basis set in the below. However, it is straightforward to extend our method to any other type. We used our pseudopotential-based DFT (density functional theory) package, OpenMX _op (), which takes the linear combination of numerical PAO basis Ozaki (2003); Ozaki and Kino (2004). The single particle energy eigenstate is decomposed into PAO; where is Khon-Sham eigenstate with momentum , the band index, and is PAO with orbital index at position . With , state as a basis set for a given Ir- orbitals,
Now the expectation value of is estimated within this basis as follow:
where and are the eigenvalue of the operator for and , respectively.
Note that it is crucial to use the state as a basis and the is not suitable although it is often adapted to describe the low energy electronic structure of iridates. Fig. 1(c) shows the calculated as a function of crystal field splitting with the basis set of total angular momentum eigenstates (blue) and states (red). For a reasonable value of eV for the iridates Haskel et al. (2012), the two lines differ significantly due to the coupling between and states. It is noted that, even in a large limit, and can noticeably differ from each other. Here is the same quantity with in Eq.(2), and is obtained from only space taken into account.
The ‘line strength’ can be expressed as the expectation value of an operator
where is the dipole operator with polarization , is the total angular momentum of the core hole, and denotes all quantum numbers other than . The sum is taken over the for the isotropic ‘line strength’ considered.
Branching ratio can also be estimated without calculating the full XAS spectra. We first note that the relative intensity of and edge ‘white lines’ can be related to the spin-orbit coupling via
where , , and refers to the orbital angular momentum of valence electron, the orbital angular momentum of core hole, and the total angular momentum of the core hole, respectively Van der Laan and Thole (1988); Thole and Van der Laan (1988). The number of holes in valence orbitals is denoted by . In our case,
This relation is expected to be exact for the dipole transition in which the core-hole interaction with the valence electrons is small enough in comparison to the spin-orbit interaction of the core hole. Therefore our case of iridates is suitable for this formalism to be applied. From Eq.(4), the branching ratio can be written as
with . Therefore the branching ratio can be estimated by calculating the number of holes (which is straightforward in the electronic structure calculation) and (which can also be estimated as described above).
Due to recent progress, calculating XAS spectra from first-principles becomes feasible. Some codes are already available for this capability Giannozzi et al. (2009); Pardini et al. (2012). However the calculation of the whole spectra is quite demanding in general. For example, a certain type of pseudopotential (such as projector augmented wave) should be prepared for describing the core holes. Also, the generalization to the non-collinear spin configuration space is sometimes not well prepared while the non-collinear spin order is actually stabilized in many of large SOC materials as in the case of SrIrO. Our technique has a clear advantage in these regards. First of all, it is much simpler in the implementation and calculation, and does not require any special type of pseudopotential. One can just use the original code as it is and the only required information is the final band structure that is properly transformed into -space. In spite of its simplicity, the quantitative comparison can still be made with experiment through the branching ratio, and the direct estimation of SOC is also provided although the full XAS spectra is not accessible.
Iii Result and Discussion: Application to Iridates
iii.1 Rhodium-doped SrIrO
The calculation results of Rh-doped iridates, SrRhIrO, are summarized in Fig. 2 (see the filled blue squares; corresponding to the upper -axis) where XAS data is also presented (open blue squares).The good agreement between calculation and experiment is clearly noticed over the wide range of Rh-doping ratio, . Note that the error can be caused in both theoretical and experimental estimation as marked by the error bars. In calculations, the one important source of error is the range of integration; namely, how to deal with the small portion of Ir- states hybridized with oxygen states while the major Ir peaks are clearly identified. According to our estimation, this intrinsic ambiguity can cause the deviation of branching ratio by up to 0.25 which is %. Counting the number of holes (or electrons) in Ir-5d orbitals is another source of errors. This is related to the long-standing issue of charge decomposition in the electronic structure calculation, and the number of holes depends on the electron counting method. In this study, we used the standard Mulliken charge analysis. For experimental data, we simply take the error limits presented in Ref. Laguna-Marco et al. (2015); Chikara et al. (2015).
Note that the and branching ratio are basically unchanged over the wide range of , and clearly larger than the Rh value of 0.8 and 3, respectively Chikara et al. (2015). This result is therefore in contrast to the previously suggested picture of ‘SOC tuning’ in which Rh-doping is assumed to reduce the SOC strength of Ir sites Lee et al. (2012); Qi et al. (2012). It is one example to show the importance of calculating SOC from the realistic electronic structure.
iii.2 Iridium oxide double perovskites
Another system we take to test our method is iridate double perovskites, SrIrO (: Mg, Sc, Ti). This series of materials are studied recently with XAS Chikara et al. (2015) while no theoretical investigation has been reported yet. Among many different double perovskite iridates, we chose = Mg, Sc, and Ti in which has configuration, and therefore we could avoid the additional ambiguity in determining values for sites. The nominal Ir valence in these compounds are , , and for SrMgIrO, SrScIrO, SrTiIrO, respectively, serving as a good test case to check the reliability of our method.
The calculation results of projected density of states (PDOS) is presented in Fig. 3. The so-called effective total angular momentum -character is well identified (=1/2 and 3/2 is in red and blue color, respectively). The gradual increase of Ir- band filling is clearly observed as we go from Mg (-filled =3/2) to Sc (fully-filled =3/2) and to Ti (half-filled =1/2). The Ti case has the same nominal valence with SrIrO. Among these three compounds, SrMgIrO is reported first by Jung and Demazeau Jung and Demazeau (1995) and its semiconducting behavior in the transport Jung and Demazeau (1995) and antiferromagnetic ordering below K Kayser et al. (2014) are consistent with our results.
The results of and branching ratio are summarized in Fig. 2(a) and (b), respectively (see the magenta diamond symbols corresponding to the lower axis). Again, the overall good agreement with XAS data is clearly noticed. Here we also note several possible sources of deviation, when compared to the experiments, such as the material dependent values, long PDOS tails due to the hybridization, and the electron number counting with the Mulliken analysis. Considering the uncertainties related to all of these factors, the agreement within the error bar is quite impressive. According to the previous studies Jung and Demazeau (1995); Kayser et al. (2014), it is noted that the amount of oxygen vacancy is likely non-negligible especially for =Mg sample (SrMgIrO: 0.35 at 1 bar). It is another factor that can cause the difference between experiment and calculation. In fact, the difference is distinguishable (considering the error bars) only in the =Mg case. In order to see the effect of oxygen vacancy, we performed the rigid-band shift (= 0, 0.1, 0.2, 0.3, 0.4) and the supercell calculation (=0.4). In the former, it is assumed that the role of vacancy is just electron doping. The results are summarized as the insets of Fig. 4. An excellent agreement with experiments for both and branching ratio is clearly noticed.
While our main results are obtained from 20-atom unitcells with antiferromagnetic order, we also performed the 10-atom cell calculations which correspond to ferromagnetic order. The calculated SOC and branching ratio are not noticeably different. It indicates that the effect of magnetic order and structural distortion is not significant. It is also found that the dependence is not significant in the range we considered.
We introduce a technique to calculate SOC and branching ratio. It is simple enough to be adoptable basically for any type of first-principles calculation methods and formalisms. Large SOC iridates were taken to be a test case for this method. The calculation results of Rh-doped SrIrO and double perovskites, SrIrO, are shown to be in good agreement with recent XAS experiments. This technique can serve as a new promising tool for studying large SOC materials from first-principles and for understanding related experiments.
MJH is greatly thankful to Michel van Veenendaal for helpful discussion. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2057202). The computing resource is supported by National Institute of Supercomputing and Networking / Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2014-C2-015).
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