Variation of magnetic properties of Sr$_2$FeMoO$_6$ due to oxygen vacancies

Oxygen vacancies can be of utmost importance for improving or deteriorating physical properties of oxide materials. Here, we studied from first-principles the electronic and magnetic properties of oxygen vacancies in the double perovskite Sr$_2$FeMoO$_6$ (SFMO). We show that oxygen vacancies can increase the Curie temperature in SFMO, although the total magnetic moment is reduced at the same time. We found also that the experimentally observed valence change of the Fe ions from $3+$ to $2+$ in the x-ray magnetic circular dichroism (XMCD) measurements is better explained by oxygen vacancies than by the assumed mixed valence state. The agreement of the calculated x-ray absorption spectra and XMCD results with experimental data is considerably improved by inclusion of oxygen vacancies.


I. INTRODUCTION
Double perovskites A 2 BB O 6 (A = alkaline earth or rare earth and BB are heterovalent transition metals such as B = Fe, Cr, Mn, Co, Ni; B = Mo, Re, W) often demonstrate intrinsically complex magnetic structures and a wide variety of physical properties (see Ref. 1 for a review article on these materials).
Several experimental and theoretical studies have demonstrated that the double perovskite system Sr 2 FeMoO 6 (SFMO) and other related materials exhibit a ferrimagnetic half-metallic ground state with a high Curie temperature of 420 K. 2 The physical origin of the magnetoresistance in SFMO is half-metallicity, 3 i.e., the material is an insulator in one of the spin channels, but a metal in the other.This leads to a complete spin polarization at the Fermi level, which immediately suggests their application as a source of spin polarized charge carriers in spintronic devices.
5][6][7] The electronic structure of SFMO was investigated theoretically in Refs.3,8-16.Despite of many experimental and theoretical studies of the electronic and magnetic structures of SFMO, serious controversies remain, such as the value of the spin magnetic moment at the Mo site or the valency of iron.Ray et al. 4 did not observe a spin magnetic moment on Mo in their XMCD measurements.Later Besse et al. 5 using the same XMCD technique obtained the spin magnetic moments of 3.05 µ B on Fe and −0.32 µ B on Mo.Neutron diffraction measurements also produced contradictory observations, either no spin mo-ment at the Mo site in Ref. 17 or a spin moment of 1 µ B in Ref. 18. Concerning the valency of Fe, Ray et al. 4 concluded on the basis of x-ray absorption spectroscopy with linear polarized light that Fe is in the 3+ state.Mössbauer spectroscopy data from Sarma et al. 19 confirmed this result.On the other hand, Linden et al. 20 presented the evidence for the formation of a valence-fluctuation state of iron, formally denoted as Fe 2.5+ on the basis of Mössbauer measurements.Similar conclusions were made by Kapusta et al. 21from NMR measurements.Kuepper et al. 6 studied the electronic and magnetic properties of SFMO using x-ray diffraction (XRD), xray photoelectron spectroscopy (XPS), and XAS.The XAS measurements as well as XPS of the Mo 3d and Fe 3s core levels reveal a mixed valence state of Fe 2+ and Fe 3+ .In order to provide a quantitative description of the spectral features of the XAS results at the Fe L 2,3 edges Besse et al. 5 carried out calculations in the frame of the ligand-field atomic formalism.They found the best agreement between the calculated and the measured spectra for a linear combination of 34 % Fe 3+ (3d 5 ) and 66 % Fe 2+ (3d 6 ).Density functional calculations of the XAS and XMCD spectra of SFMO were performed by Kanchana et al. 16 They obtained a reasonably good agreement between the theory and the experiment for the Mo L 2,3 XAS and XMCD spectra.However, their calculations did neither reproduce the double peak structure of XAS at the Fe L 2,3 edges nor the high energy shoulder at the Fe L 3 edge or the three peak structure of the Fe L 2 edge in the XMCD spectrum.They also strongly overestimated the dichroism at the Fe L 2 edge.
Therefore, we reconsidered the XAS and XMCD spectra with respect to the different valency of Fe and the occurrence of lattice defects.Our results indicate that the valency change of Fe from 3+ to 2+ is strongly connected with the occurrence of defects.Such defects may alter also the magnetic coupling and the critical temperature, which is another important issue for experimental investigations and possible applications.
The ferrimagnetic ground state and the magnetic coupling were already extensively discussed by Solovyev by means of the LMTO method, 13 but simulations of the Curie temperature were done only via model Hamiltonians in a Monte Carlo method, fitted to the experimental T C values, 22 or using band structure parameters. 23In contrast, we calculated as well the critical temperature of SFMO from ab initio.For the description of such a complex oxide system, we have to deal in particular with strong correlation effects.For example, in the calculations of Szotek et al. 24 the self-interaction correction (SIC) method was applied and the half-metallic ground state with a valency of Fe 3+ was found.Other valencies of iron or localized Mo d states were energetically less favorable but the energy difference was small.The results of Ref. 24 are therefore not in contradiction to the picture of valency mixture.In this study we used the SIC approach, which is implemented within our Green's function method 25 , to compare ground state energies.However, the SIC overestimates usually the localization of the d states in the density of states (DOS) and, therefore, is not always adequate to describe excited state properties.A local density approximation (LDA) or a generalized gradient approximation (GGA) method supplemented by a Hubbard U term allows a continuous change of the correlation contribution.So, it is possible to study the transition from strong band magnetism in metallic SFMO to the ferrimagnetism in halfmetallic SFMO and the corresponding critical temperature.
In the following section, we describe the lattice structure of SFMO used in our calculations.Details about the calculation techniques are given in the next section.The influence of the electronic correlations and the lattice defects on the electronic structure is discussed in Sec.IV.Their effect to the magnetic properties, such as magnetic moment or Curie temperature, is shown in Sec.V. We present in the Sec.VI the XAS and XMCD calculations for the SFMO compound and compare them with previously measured spectra.We conclude in the last section.

II. LATTICE STRUCTURE
For our calculations of SFMO, we adopt the experimentally found double perovskite structure, where the oxygen atoms, which surround the Fe and Mo sites provide an octahedral environment.The FeO 6 and MoO 6 octahedra alternate along the three cubic axes, while Sr atoms occupy the hollow site formed by the corners of FeO 6 and MoO 6 octahedra at the body-centered positions (see Fig. 1).
SFMO was found to be cubic (F m 3m) in the paramagnetic phase, but changes into a tetragonal-type structure below a critical temperature. 9,18A representative choice of experimentally observed lattice constants is collected in Tab.I. Their variation between the smallest and largest value is quite small (0.25 % and 0.17 % for a and c, respectively) and the cubic symmetry would be represented by a c/a ratio of √ 2. So, we see from the data in Tab.I that the deviation from the TABLE I: Variation of the experimental results for the lattice constants and oxygen positions of SFMO.Fe occupies consistently (0, 0, 0).The Wyckoff positions of the oxygen atoms are (0, 0, z) and (x,y, 0).The positions of the other atoms are described in the text.cubic symmetry is small as well.
Since, the changes of the lattice constants are small, we used for simplicity the more symmetric body centered tetragonal structure type and the lattice constant from Ref. 28 as input for our study.As long it is not stated otherwise, we used the primitive unit cell with one functional unit of SFMO (see blue arrows in Fig. 1).

III. THEORETICAL BACKGROUND
For the microscopic understanding of the SFMO compound, we combined the theoretical results of three computational methods, namely the multiple scattering Green's function (GF) method, [32][33][34] the spin-polarized fully relativistic linear-muffin-tin-orbital (SPR-LMTO) method [35][36][37] and the Vienna ab initio simulation package (VASP). 38,39. Electronic structure The main calculations were done with the Green's function method within the full-charge density approximation to the crystal potential.For the Green's function, the angular momentum cutoff was l max = 3 and the Brillouin zone integration was done with a k-point mesh of 12 × 12 × 12.The complex energy contour was integrated over 24 Gaussian quadrature points.With the Green's function G(E) of the system, all quantities of interest like the local density of states (LDOS) follow in a straightforward way.
The exchange-correlation functional of a GGA-type was used very successful in the context of SFMO. 3,40Within the Green's function method, we used in particular the version of Perdew, Burke and Ernzerhof (PBE). 41,42In order to deal with the strong correlations in the oxide system, the selfinteraction correction (SIC) 25 and the GGA+U approach of Dudarev et al. 43 implemented in the Green's function method were used (U eff = U − J).For the SIC, only the LDA exchange-correlation functional can be used.
b. Magnetic properties We calculated the interatomic exchange coupling parameters J ij from ab initio using the magnetic force theorem implemented within the Green's function method. 44Since, we have also induced moments at the Mo atoms and the oxygen atoms, we applied also the disordered local moment (DLM) theory. 45,46The DLM simulates a paramagnetic state, where the induced moments become zero.These induced moments are expected to disappear above the magnetic phase transition, which should allow a better estimation of the critical temperature.
In any case, the J ij can be used to obtain the critical temperatures using a Monte Carlo (MC) simulation (details in Refs.47-49).Therefore, the unit cell in Fig. 1 was repeated 20 × 20 × 20 to form a large cluster.We considered also periodic boundary conditions and restricted the calculation only to the magnetic atoms of the unit cell.The starting point was a high-temperature disordered state above the critical temperature T C .In the course of the simulations, the temperature was stepwise reduced until magnetic ordering was reached.For each temperature T , the thermal equilibrium was assumed to be reached after 20 000 MC steps.The thermal averages were determined over 20 000 additional MC steps.T C was then obtained from the temperature dependency of the magnetic susceptibility and the heat capacity within an uncertainty range of ±5 K.
c. Simulation of x-ray spectra The x-ray absorption and dichroism spectra were calculated with the SPR-LMTO method taking into account the exchange splitting of the core levels.The approach for the XAS and XMCD simulations is described in our previous papers. 50The finite lifetime of a core hole was accounted for by folding the spectra with a Lorentzian.The widths of core level spectra Γ L2 = 1.14 eV and Γ L3 = 0.41 eV for Fe, Γ L2 = 1.83 eV and Γ L3 = 1.69 eV for Mo and Γ M2,3 = 2.1 eV for Mo were taken from Ref. 51.The finite apparative resolution of the spectrometer was accounted for by a Gaussian of width 0.6 eV.
In our SPR-LMTO simulations, the basis set consisted of the s, p, and d LMTO's for the Sr, Fe, Mo and O sites.The k-space integrations were performed with the improved tetrahedron method 52 and the self-consistent charge density was obtained with 1063 irreducible k-points.We used a relativistic generalization of the LDA+U method, which takes into account the spin-orbit coupling so that the occupation matrix of localized electrons become non-diagonal in spin indexes.This method is described in detail in our previous paper 53 including the procedure to calculate the screened Coulomb U and exchange J integrals, as well as the Slater integrals F 2 , F 4 , and F 6 .The constrained LSDA calculations produce J = 0.85 eV for the Fe site in SFMO.In order to be consistent with the other methods, the Hubbard U has been used as an external parameter.
d. Defects We investigated the influence of two types of crystal defects, namely, oxygen-deficiency δ and the antisite disorder (ASD).The latter means a swap of an Fe ion with a Mo ion and vice versa in the double perovskite lattice, which keeps the stoichiometry of SFMO to be the same.
Our Green's function method allows us also to deal with chemical disorder by using the coherent potential approximation (CPA).The ASD was modeled with the CPA by interchanging randomly Fe and Mo.The lattice structure was kept static because the basis position changes due to ASD are tiny. 54For the oxygen vacancies, a certain percentage of empty spheres were introduced at the lattice sites of the oxygen ions.The typical range of the total oxygen-deficiency δ ranges between 0.006 to 0.36. 15,55,56This represents 0.1 at.% to 6 at.% of the total oxygen amount in stoichiometric SFMO.
For the other methods, we used a supercell approximation with a cell of two functional units of SFMO (see red arrows in Fig. 1).This supercell was relaxed with the VASP including either ASD or a single oxygen vacancy close to Fe2.For the antisite defects, we found, similarly as in Ref. 54, no significant relaxation.The relaxation with VASP for a single oxygen vacancy led to a distance of 1.9722 Å to the vacancy.The correspondent distance between the Fe1 ion and the oxygen vacancy was 4.4054 Å.

IV. ELECTRONIC STRUCTURE
Due to the slight tetragonal distortion, the typical t 2g and e g parts of the transition metal d states are changed and the crystal field at the Fe (Mo) site (now D 4d point symmetry) splits the Fe (Mo) 3d (4d) orbitals into three singlets A 1g (d 3z 2 ), B 1g (d x 2 −y 2 ), and B 2g (d xy ) and a doublet e g (d yz and d xz ).However, due to the only slight deviation from the ideal c/a ratio the states d xy and d 3z 2 , as well as the states d x 2 −y 2 , d yz and d xz , form two groups of states, which showed each a quite similar DOS and we discuss exemplarily in the following only one state for each of them (d xy and d x 2 −y 2 , respectively).
Our calculations with the Green's function method pro- duce a metallic solution with a nonzero density of states in both spin channels at the Fermi energy (E F ) independent of the use of LSDA or GGA (GGA: see the upper panels of Figs. 2 and 3).We plotted only the states of the Fe and Mo ions, since, those states are the only ones lying at the Fermi energy.We also crosschecked the DOS calculation with the other methods and found similar results.
As already discussed in previous publications, the band gap in the majority spin channel opens when correlation corrections are applied (LDA(GGA)+U or SIC). 24,54The experimental values for this band gap range from 0.5 eV to 1.3 eV in Refs.57, and 58, respectively.On the other hand, when we calculated the DOS including SIC, the position of the Fe d states of the majority spin channel is far too deep below the Fermi energy in comparison with the experimental photoemission spectrum. 57To solve the problem, we investigated the influence of a continuously changed correlation parameter U Fe eff (applied only to Fe d states) on the DOS (see lower panels in Figs. 2 and 3).By increasing the U Fe eff parameter the main changes at the Fermi energy occur for the Fe d xy states and those of similar symmetry (d 3z 2 , DOS not shown).They are pushed towards to lower energies and all other states in the spin up channel are not crossing E F , therefore, at U Fe eff 1 eV a band gap in the majority spin channel opens.On the contrary, the minority spin channel at E F is always occupied by the d x 2 −y 2 states of Fe and Mo and those of similar symmetry (d yz and d xz , DOS not shown), which results in the halfmetallic behavior of the SFMO.We observed small traces of hybridization between the Mo and Fe d states in the local density of states.Otherwise, all d ↑ (d ↓ ) states of Fe are occupied (unoccupied), which is represented by a formal Fe 3+ valence state.For Mo, almost all states are unoccupied, except the delocalized d states.This can be considered as a Mo 5+ state.This situation was found in all self-consistent solutions with GGA+U .An Fe 2+ ion in the primitive cell could be simulated only with fixed occupation numbers in the SPR-LMTO, which can be used later for the discussion of the XAS and XMCD simualtions or with the SIC within the Green's function approach. 24The latter showed a small energy difference for the two different ground states of different Fe valency with the lowest energy for 3+ in agreement with Ref. 24.Therefore, we discuss here the DOS of the ground state and note that a mixed valency might still a possible solution but is not possible to obtain self-consistently.
In order to determine an appropriate U Fe eff value, we aim to match the band gap in the majority spin channel.The measurement of the photoelectron spectrum (PES) by Saitoh et al. 57 allows to compare with the complete DOS.The size of the band gap is determined from the position of the states involved in the first possible electron transition, which should be the d ↑ xy and d ↑ 3z 2 states, which lie roughly 1.3 eV below E F (see dashed line in Figs. 2 or 4).On the other hand, the measurement of Tomioka et al. 58 showed only a small optical gap of 0.5 eV, which was attributed to a transition from the Fe d ↑ xy and d ↑ 3z 2 to the Mo d x 2 −y 2 , d yz and d xz states by a comparison to their perfect bulk calculation.Such a small experimental gap would indicate in our calculation that U Fe eff should lie in a range of 1 eV to 2 eV (see Fig. 2).On top of that, those states of the measured spectra extent over more than 1 eV and might be altered as well by additional defects, which could be only scarcely considered in the experimental comparisons.For example, Saitoh et al. 57 noticed a possible occurance of up to 10 at.% antisite disorder in their measured samples, but they could not account for any kind of defects in their theoretical interpretation.When we took into account antisite disorder within the single site CPA approach, we obtained Mo d states at the Fermi energy for a Mo ion at the Fe site (not shown).With increasing the ASD, the number of those states increases as well and the full spin polarization of the halfmetallic SFMO is lost, as it is observed experimentally. 59n the other hand, oxygen vacancies are also very likely.The direct influence of oxygen vacancies to the DOS was al- ready discussed by Muñoz-García et al. 54 in a supercell approach.For a single oxygen vacancy, the Fe d states showed additional states in the spin down channel below the Fermi energy, which indicated a higher occupation of the Fe ion and valency change.However, the spin up band gap was very wide in all their calculations.We included oxygen vacancies in our unit cell with the CPA.We show exemplary the DOS for U Fe eff = 2 eV and 8 at.% of oxygen vacancies (green dashed line in Fig. 4).The contribution in the spin down channel right below the Fermi energy is increased similar to Ref. 54.This might also indicate the valency change as discussed above.The d ↑ states are shifted to lower energies and the vacancies show a similar effect as the correlation corrections with U Fe eff (see Fig. 2).The missing oxygen orbitals increase the localization of the transition metal ions.On the other hand, the averaging of the CPA leads to a general broadening of the states.
TABLE II: The calculated spin ms and orbital mo magnetic moments (in µB) of Fe and Mo in defect-free SFMO compared with the measured magnetic moments.The moments ms and mo were obtained with (1) the SPR-LMTO.In addition, we show ms determined with (2) the GF method.The sum rules are applied for the theoretically calculated XMCD spectra in the LDA+U approximation (SPR-LMTO).Experimental uncertainties are given in brackets behind the values.
Fe Mo method ms mo ms mo LSDA (1) 3.503 0.046 −0.456 0.034 LDA+U (1) 4.104 0.041 −0.504 0.047 Sum rules (1) 4.283 0.052 −0.464 0.041 GGA (2) 3.090 −0.555 GGA+U (2) 3 Both effects compete with each other in the GGA calculations for the band gap opening, while U Fe eff or the oxygen deficiency δ vary.If the band gap due to U Fe eff is not large enough, the broadening closes the gap again and the spin polarization of the system is reduced as well.
Taken as a whole, a direct comparison between the theoretical and the experimental band gap is complicated when considering all the various influences of electron correlations or possible defects.In order to have a point of reference for the later discussions, we believe that with respect to all available data and our discussion above U Fe eff = 2 eV is a suitable compromise (see Fig. 4) and is used in the following if it is not states otherwise.

V. MAGNETIC PROPERTIES
The observed transition from metallic to half-metallic state influences as well the magnetic properties.Coming back at first to the defect-free material, we investigated the magnetic moments, the magnetic exchange coupling, and the Curie temperature of SFMO in dependence of the electron correlation and the defects.

A. Magnetic moments
We calculated the local magnetic moments of Fe and Mo with the three methods and obtained always the ferrimagnetic ground state where the moments of Fe and Mo are antiparallel aligned (see Tab. II).The methods showed a good qualitative agreement with respect to the extent to which this comparison was reasonable with different exchange function-Ref.[22]  Ref. [60]    als.Still, we want to consider also the calculated orbital moments from the SPR-LMTO.The Fe spin and orbital moments are parallel, whereas the spin and orbital Mo moments are antiparallel to each other, in accordance with Hund's third rule.Our calculations for the metallic solution (weak electron correlation regime) provide for Fe a lower spin magnetic moment than in earlier calculations (3.72 µ B to 3.97 µ B cf. Refs.3,11,12,14,16).As soon as we observe half-metallic SFMO (U Fe eff > 1 eV), the Fe spin moment of ≈ 4 µ B is in a good agreement with the experimental neutron diffraction measurements of Ref. 17 and the theoretical ideal value for the Fe 3+ /Mo 5+ or Fe 2+ /Mo 6+ valency configuration.It remains quite stable and insensitive to the strength of the electron correlations (see also the total moment for U Fe eff > 1 eV and δ = 0 in Fig. 5 ).Additionally, the orbital Fe magnetic moment of 0.041 µ B is very close to the FPLMTO result of Jeng and Guo 14 (0.043 µ B ) and the LMTO results of Kanchana et al. 16 (0.042µ B ).The calculated spin moment of the Mo atom is always larger in comparison with earlier calculations −0.29 µ B to −0.39 µ B (Refs.3,11,12,14,16).We found that the orbital moment at the Mo site is also larger than in previous calculations.
The lower reported spin moments of the XMCD experiments compared to the theoretical value for Fe 3+ (4 µ B ) was always attributed to antisite disorder. 5,22,55We used again the CPA to investigate randomly distributed defects.The introduced Fe ions at the Mo sites became strongly antiferromagnetically coupled to the intrinsic Fe ions and oriented their moments antiparallel.Independent of the strength of the electron correlations, the total magnetic moment of SFMO is indeed reduced linearly for an ASD up to 15 at.% (see lhs in Fig. 5).The slope of this reduction is in good agreement with the estimations by previous Monte Carlo simulations 22 and experiments. 60n the other hand, oxygen vacancies reduce also the total magnetic moment as seen in the measurements by Kircheisen et al. 55 Our theoretical results with the CPA, depending on the strength of the electron correlations, show again a reduction of the total magnetic moments when the oxygen content is decreased.Above U Fe eff = 1 eV), the slope of the reduction is linear and remains independent of the strength of U Fe eff .We note the change of the slope for U Fe eff = 0.75 eV.As observed for the DOS, the randomly distributed oxygen vacancies act similar as the correlation parameter U Fe eff , since they both reduce the screening of the Coulomb interaction.After few percent of oxygen vacancies the slope of the magnetic moment changes is similar as for higher U Fe eff .With respect to the experimental results, we have a good qualitative agreement while the general reduction of the absolute magnetic moments might result from additional ASD.
It has to be mentioned here, that the spin and orbital magnetic moments were obtained from the XMCD experiments by using the sum rules, which relate the integrated signals over the spin-orbit split core edges of the circular dichroism to the ground state orbital and spin magnetic moments. 61It is well known that the application of the sum rules sometimes results in an error up to 50 %. 62To investigate the possible error of the sum rules in the case of SFMO we compare the spin and orbital moments obtained from the theoretically calculated XAS and XMCD spectra through sum rules with directly calculated LDA+U values in order to avoid additional experimental problems.The number of the transition metal 3d electrons is calculated by integrating the occupied d partial density of states inside the corresponding atomic spheres, which gives the values n Fe d = 6.737 and n Mo d = 4.552.The sum rules reproduce the spin magnetic moments within 4 % and 8 % and the orbital moments within 21 % and 14 % for Fe and Mo, respectively (Tab.II).

B. Magnetic exchange interactions and Curie temperatures
In contrast to the magnetic moment, the calculated magnetic exchange parameters between the magnetic atoms and the critical temperatures are more sensitive to changes in the DOS.The most prominent coupling constants of the order of several meV have, thereby, only a very restricted range up to 7.9 Å, which include only the interactions up to the next nearest neighbor Fe ions J Fe-Fe 02 (sketched in Fig. 6(a)).As it is shown below, only these magnetic couplings dominate the magnetic behavior with respect to the electron correlations.The more distance exchange constants up to 12.49 Å were one order of magnitude smaller and more reduced with increasing U Fe eff .All other could be considered to be zero.Due to the tetragonal structure, all magnetic exchange interactions show a slight asymmetry with respect to those with a component in z direction.For clarity, only the coupling constants in the x-y-plane are shown while the other varied only slightly.
We considered two different calculation schemes: (i) On the one hand side, the calculations of J ij are performed at the ground state (T = 0 K, red and blue in Fig. 6(b)).In this case, we observed the induced magnetic moment at the Mo sites as discussed before and tiny contributions at the oxygen sites.Thus, there appears a strong antiferromag- (meV) netic (AFM) coupling between Fe and Mo ions, which size becomes reduced only around U Fe eff ≈ 1 eV.An antiferromagnetic behavior was also expected from former studies. 54The closest Fe-Fe coupling constants J Fe-Fe 01 decrease linearly from 4 meV to 1 meV with increasing U Fe eff .It might be correlated to the linear changes in the d ↑ xy and d ↑ x 2 −y 2 states.This be-havior is typical for the application of U .The localization of the orbital leads to a decrease of orbital overlap and, thereby, to a decrease in the magnetic coupling strength.The behavior of the next nearest Fe-Fe coupling is more complicated (see J Fe-Fe 02 in Fig. 6

(b))
. There is a strong linear reduction until the coupling switches even to a negative value (AFM).At U Fe eff ≈ 1 eV the slope is reversed but the J Fe-Fe 02 coupling constants remain negative.Only the absolute values are decreased.
(ii) On the other hand, we estimated the exchange coupling within the DLM approach, which models the paramagnetic state high above the Curie temperature.In this case, the induced moment vanish.The obtained magnetic coupling constants (now all J Fe-Mo vanish) are also shown in Fig. 6(b).In principle, they have a similar tendency as the magnetic exchange interactions at 0 K, which are discussed at first.
We considered the J ij between Fe-Fe and Fe-Mo up to the distance of 12.49 Å to calculate the Curie temperature, using either the mean-field approximation (MFA) or the Monte Carlo (MC) method.Both methods show qualitatively a similar non-linear variation with increasing electron correlation (see Fig. 6(c)).Since, the MFA usually overestimates T C , we discuss in the following only the results from the Monte Carlo simulations.
A pure GGA calculation, including the induced moments at 0 K, returned a T C of ≈ 604 K, i.e. 200 K above the measured values.With an increasing U Fe eff and the metallic DOS, the Curie temperature decreases linearly until U Fe eff = 1 eV.This corresponds with the sign change of J Fe-Fe 02 and the decrease of J Fe-Mo

01
. For half-metallic SFMO, the theoretical critical temperatures stayed always 200 K below the experimental T C (compare shaded area in Fig. 6(c)) but increased slightly up to 250 K at U Fe eff ≈ 4 eV.This is observed despite the the decrease of J Fe-Fe 01 .So, the stronger antiferromagnetic coupling between Fe and Mo sites mediates an additional FM coupling and increased T C .For the sake of completeness, the T MC C for a SIC calculation is plotted in green and matches with those of the GGA+U method for the limit of large U Fe eff .The slight reduction follows the tendency of J Fe-Mo 01 .We found for those Monte Carlo simulations always an ordered ferrimagnetic (FiM) ground state with a nonzero saturation magnetization (Fig. 6(c)).
We note the importance of the induced magnetic moments, when they vanish in the DLM model.It led to an increase of the magnitude of the nearest neighbor Fe coupling constants with the similar tendency of the 0 K coupling constants.Above U Fe eff = 1 eV, the first and second neighbor interactions become equal in magnitude but with different sign.Such a behavior alters the critical temperature dramatically and we observed only low ordering temperature to a ground state with vanishing averaged saturation magnetization (non-magnetic (NM), open circles in Fig. 6(c)).Although, the DLM is expected to give a better description of the magnetic phase transition, its possible failure is not completely new.The induced moments of Mo might remain also above the transition temperature and the current single-site approximation of the local moments might not cover those correctly.An example of this issue was found during the investigation of the magnetic properties of Ni where only a non-local DLM approach could describe the moments sufficiently. 63,64n summary, the theoretical description of the Curie temperature of half-metallic SFMO is below the experimental results.But the simulation of the variation of T C with a changing defect concentration seems to be more interesting with respect to experimental studies rather than the exact absolute values.

C. The Curie temperature and defects
We found no literature about previous experimental studies, which considered also lattice defects in their measurements of T C .So, we simulated with the CPA the effect of ASD and oxygen vacancies up to 15 at.% and 10 at.%, respectively, as a prediction for future experiments.
The inclusion of ASD appeared again independent of the strength of the electron correlation.Although, the magnetic transition temperatures varied strongly with U Fe eff for defectfree SFMO, ASD always reduces T C at about 100 K (see lhs of Fig. 7).In every case, we observed large negative coupling constants between the intrinsic Fe ions and the ones sitting at the Mo positions.
In contrast, the variation of T C depends on the electron correlations for the oxygen vacancies (see rhs of Fig. 7).Already for the GGA, we noticed that the strong reduction of T C in dependence on the oxygen amount follows a similar nonlinear tendency as T C (U Fe eff ) (compare also Fig. 6).We observed a turning point for U Fe eff = 0.75 eV, close to half-metallic SFMO.Here, the oxygen vacancies open finally the gap in the majority spin channel as it was seen for the DOS (see Fig. 4).After that turning point, T C increases linearly with a similar slope as for the fully half-metallic SFMO (U Fe eff = 2 eV).This increase is in the order of 80 K per 5 at.% oxygen vacancies.So, it is possible that the appearance of oxygen vacancies solve the above described discrepancy between the theoretical results for defect-free SFMO and the experimental observations.

D. The Curie temperature and electron doping
More generally, the oxygen vacancies dope the material system with additional free electrons.This n doping might also result from other defects, e.g.Navarro et al. 65 observed in their experiments an increase of T C by substituting in SFMO Sr with La.This was related with an increase of the density of states at the Fermi energy D(E F ).
In theoretical calculations, such doping can be achieved by a shift of the Fermi energy E F .For the 0 K calculation and U Fe = 2 eV, we indicate such a variation ∆E F = 68 meV in Fig. 8(a).The dashed line represents an n doping with additionally 0.15 electrons, which is related with an increase of D(E F ) and T C at about 10 K.This results from a stronger magnetic coupling (see inset in Fig. 6(b)).Unfortunately, there were stronger variations in the magnetic coupling constants beyond the next nearest neighbors (not shown) and additional influences to the magnetic coupling, e.g. the positions of unoccupied Mo d ↑ x 2 −y 2 states (and of similar symmetry) from the Fermi energy, which affects also the hybridization with the occupied states.So, the T C could not be increased more strongly by more electron doping.
However, also the exchange splitting for the Mo d states might be a source of error.We found that the magnetic interaction and the Curie temperature are very sensitive to the positions of Mo d states.The Mo d electrons of the majority spin channel are sharp and above the Fermi level, while in the minority spin channel the d electrons are located at the Fermi level forming a relatively broad band (see Fig. 8(b)).A small shift of spin up electrons can substantially change the Curie temperature.Since the DFT approach can not describe adequately unoccupied states, the positions of the d electrons of the majority spin channel could be not correct.We did not observe any change of D(E F ) once we applied the U Mo to all Mo d states: main peak of the Mo d ↓ x 2 −y 2 states (and of similar symmetry) retains its position above E F but the unoccupied Mo d ↑ states are shifted to higher energies, which led, however, T C = 250 K with U Mo = 2 eV due to substantial changes in the DOS of the majority spin channel (see Fig. 8(b)).

VI. X-RAY ABSORPTION AND XMCD SPECTRA
Until now, the above discussion point towards the importance of the oxygen vacancies and the different valency states of Fe while we still lack a final experimental comparison.In a complex transition metal ionic compound such as the SFMO, the x-ray absorption and XMCD spectra at the L 2,3 absorption edges can be used as fingerprints of the ground state.The experimentally measured Fe L 2,3 absorption spectrum of the SFMO single crystal (Ref.5, average of left and right circularly polarized light) displays at both absorption edges a weak lower-energy shoulder together with a doublet structure at the white line position with almost the same magnitude (see Fig. 9).The complex fine structure of the Fe L 2,3 XAS is not compatible with a pure Fe 3+ valency state.In order to provide a quantitative description of the spectral features, we took into account both Fe valencies 3+ and 2+, separately, in a primitive unit cell.The theoretically calculated Fe L 2,3 XAS agrees most closely with the experimental data by using those results in a linear combination of 60 % Fe 3+ and 40 % Fe 2+ (see Fig. 9), which is opposite to the calculated proportion found in Ref. 5.
One of the possible reasons of mixed valency states are again lattice defects.We investigate the influence of the two types of defects in the tetragonal supercell.For the antisite defects, we observed only the Fe 3+ solution.This might be connected with the high defect concentration modeled within the supercell.However, a single vacancy among twelve oxygen atoms is a more realistic concentration.Self-consistent calculations in the tetragonal supercell produce the valency of the Fe ions being equal to 2.9+ and 2.4+ at the Fe1 and Fe2 sites, respectively.Therefore, the existence of the vacancy shifts the valency of the nearest Fe ion (Fe2) towards 2+.This valency changed could be also observed in the CPA calculation of the DOS including oxygen vacancies (see Fig. 4).
Thus, the full explanation of the experimental spectra is only possible by taking into account these crystal imperfections.The Fe L 3 x-ray absorption spectrum for left circularly polarized light (σ + ) possesses four major fine structures a, b, c and d (see Fig. 10).We found that the calculations for the ideal crystal structure with Fe 3+ ground state solution (dashed blue line) provide the x-ray absorption intensity σ + only at the major peaks b, c, and d and do not reproduce the low energy shoulder (peak a) as well as the low energy peak e at the L 2 edge.The calculations with the Fe 3+ solution produce only one high energy peak structure in the L 3 σ − spectrum (middle panel).However, the experimental measurements exhibit  a double-peak structure.The x-ray absorption from the Fe2 atoms with the oxygen vacancy (solid red line) contributes to the low energy peak of σ − absorption (see Fig. 10).The relative intensity of the peaks depends on the relative concentration of the Fe1 and Fe2 ions in SFMO, in other words, the concentration of defects such as oxygen vacancies.It is similar for the oxygen K edge.The calculations including an oxygen vacancy are in better agreement with the experimental measurements in the x-ray absorption as well as in the XMCD (see Fig. 11).
In contrast, the XAS and XMCD spectra at the Mo L 2,3 and M 2,3 edges are less sensitive for the crystal defects.For both edges the agreement with the experimental measurements is quite good (see Figs. 12 and 13).

VII. CONCLUSIONS
We performed an extensive study of the electronic and magnetic properties of Sr 2 FeMoO 6 with the main focus on the influence of oxygen vacancies and on the impact of the electronic correlations.To get an agreement with experiments and to describe the half metallic behavior of the SFMO, we had to apply an LDA+U approach on the d states of Fe.The band gap in the upper spin channel opens already at a small value of U Fe eff = 2 eV.Otherwise, the system remains metallic.The best agreement with available experimental data for XAS, XMCD and T C was achieved at U Fe eff = 2 eV.As it was already discussed in other works, the main contribution at the Fermi energy originates from the d states of Mo, particularly, d ↓ yz , d ↓ zx and d ↓ x 2 −y 2 .Those states have the same symmetry but only d ↓ yz and d ↓ zx are degenerated.
We observe that the total magnetic moment is linearly reduced by both kinds of lattice defects, which is in good agreement with the experimental studies.In contrast, the estimation of the Curie temperature for the half-metallic Sr 2 FeMoO 6 with the DLM model was below 100 K with a non-magnetic ground state.Only the inclusion of the induced magnetic moments at the Mo sites stabilized the ferrimagnetic ground state and yielded a higher T C than the DLM calculations.
We found several mechanisms, which affected the value of T C .If the measured sample is doped with additional electrons due to some unknown defects or we considered also electron correlation effects for the Mo ions, T C would be higher than the simple theoretical 0 K prediction.On the other hand, the inclusion of oxygen vacancies and a reduction of the Fe valency might also raise the Curie temperature.Antisite disorder might have the opposite effect and reduces T C .
The calculated magnetic dichroism spectra for the Fe L 2,3 edges had a good agreement with the measured ones only, if we take into account different Fe valencies, which seems to follow from oxygen deficiency.The spectra for the Mo edges was in a good agreement as well.
In conclusion, we showed that the consideration of oxygen vacancies and the change of the Fe valency in the theoretical description is crucial to improve the agreement with the experimental observations of the magnetic properties of Sr 2 FeMoO 6 , be it for the x-ray absorption spectroscopy or for the magnetic transition temperature.Oxygen vacancies might be one reason for the experimentally observed change in the Fe valency.

FIG. 1 :
FIG. 1: (Color online) The double perovskite structure of SFMO.The colored polyhedra visualize the octahedral surroundings of the Fe and Mo atoms (orange and blue).Following from the tetragonal symmetry, two different oxygen positions appear (marked with Oxy and Oz).The tetragonal supercell is shown by the black solid lines and the red arrows.It contains two functional units with two Fe sites (Fe1, Fe2).The black dashed lines and the blue arrows indicate the primitive unit cell.

FIG. 2 : 57 FIG. 3 :
FIG.2: Orbital resolved density of the Fe d states for a GGA calculation (upper panels) and in dependence on the correlation correction U Fe eff parameter as contour plots (lower panels).The variation of the peak height of the DOS is color coded in shades of orange and red for the similar groups of the symmetry splitted states dxy and d x 2 −y 2 , respectively (see text).The dashed line in the left hand side panel indicates the peak position in the experimental PES.57

2 8 % V O 1 FIG. 4 :
FIG. 4: (Color online) DOS for SFMO calculated with the Green function method for U Fe eff = 2 eV (lower panel) and scaled to the completely taken Fig. 5(a) from Ref. 57 (upper panel, for explanations of the labels see the reference) and connected at the Fermi energy (black vertical line at zero).The dashed line framed gray area shows the total DOS for the defect-free SFMO.The colored regions indicate the partial DOS for the d states of Fe (reddish) and Mo (bluish).The colors are chosen correspondingly to Figs. 2 and 3.The green dotted DOS includes 8 at.% of randomly distributed oxygen vacancies.The dashed vertical line indicates a possible position of the Fe d ↑ xy and d ↑ 3z 2 in the experiment.

1 FIG. 5 :
FIG.5: Total magnetic moment mtot in dependence on the ASD (lhs) and oxygen deficiency δ (rhs) obtained with the GF method for GGA and varying U Fe eff .They are compared to previous references.The oxygen amount is also given in absolute values (label above).The experimental results of Ref. 55 (green circles) are plotted together with a linear regression (green solid line).

1 FIG. 6 :
FIG. 6: (a) Sketch of the orientation of the magnetic exchange interactions Jij between the Fe sites with their nearest and next nearest neighbors and with the nearest neighbor Mo sites.(b) Jij in dependence on the correlation parameter U Fe eff 0 K in the ground state and within the DLM picture.The gray shaded inset shows the Jij at UFe = 2 eV (connected by the dotted line) for varied Fermi energies.(c) Curie temperature calculated from the corresponding Jij with the mean-field approximation or the Monte Carlo method and compared with the experimental range.The observed ground state change in the DLM model is marked with full and open circles (see text).

1 FIG. 7 :
FIG.7: Curie temperatures in dependence on the ASD (lhs) and oxygen deficiency δ (rhs) obtained with the GF method for GGA and varying U Fe eff .The oxygen amount is also given in absolute values (label above).

2 E 1 FIG. 8 :
FIG. 8: LDOS of Fe and Mo around the Fermi energy calculated with UFe = 2 eV for (a) simulated electron doping and (b) with an additional UMo = 2 eV.The legend is provided above the plots.

1 FIG. 9 :
FIG. 9: (Color online) The x-ray absorption spectra (Ref.5, open circles) at Fe L2,3 edges as average of left and right circularly polarized light in SFMO measured at 10 K with 5 T magnetic field compared with the theoretically calculated ones for Fe 3+ (dashed blue line) and Fe 2+ (solid red line).

1 FIG. 10 :
FIG. 10: (Color online) The x-ray absorption spectra (Ref.5, open circles) at Fe L2,3 edges in SFMO measured with left (σ + , top panel) and right circularly polarized light (σ − , middle panel) measured at 10 K with 5 T magnetic field and XMCD experimental spectrum (Ref.5, lower panel) compared with the theoretically calculated spectra with an Fe ion far away (dashed blue line) and near to an oxygen vacancy (solid red line).

1 FIG. 11 : 1 FIG. 12 :
FIG. 11: (Color online) The experimental x-ray absorption spectra (Ref.7, open circles) at O K edge (top panel) in SFMO and experimental XMCD spectra (lower panel) measured at T = 20 K with B = 1.1 T compared with the theoretical simulations carried out for the ideal crystal structure (solid red line) and with an oxygen vacancy (dashed blue line).