Understanding the Magnetic Puzzles of Double Perovskites A2FeOsO6 (A=Ca, Sr)

Double perovskites Sr2FeOsO6 and Ca2FeOsO6 show puzzling magnetic properties, the former a low-temperature antiferromagnet while the later a high-temperature insulating ferrimagnet. Here, in order to understand the underlying mechanism, we have investigated the frustrated magnetism of A2FeOsO6 by employing density functional theory and maximally-localized Wannier functions. We find that lattice distortion enhances the antiferromagnetic nearest-neighboring Fe-O-Os interaction but weakens the antiferromagnetic interactions through the Os-O-O-Os and Fe-O-Os-O-Fe paths, which is responsible for the magnetic transition from the low-temperature antiferromagnetism to the high-temperature ferrimagnetism with the decrease of the radius of the A2+ ions. We also discuss the 5d3-3d5 superexchange and propose such superexchange is intrinsically antiferromagnetic instead of the expected ferromagnetic. Our work illustrate that the magnetic frustration can be effectively relieved by lattice distortion, which provides another dimension to tune the complex magnetism in other 3d-5d (4d) double perovskites.

140K T  and AF2 N 67K T  [6], respectively. Theoretically, the mechanisms of the occurrence of the AF1 and AF2 are under debate. For the AF1, it was widely accepted that the ferrimagnetic (FIM) ab planes are coupled to the neighboring planes by a ferromagnetic (FM) Fe-O-Os superexchange [3,8,9]. However, it was recently suggested that these FIM ab planes may be coupled by the antiferromagnetic (AFM) Os-O-O-Os interactions [10]. For the AF2, Morrow et al. proposed that the long-range showed that the long-range Os-Os AFM interaction through the four-bond Os-O-Fe-O-Os path is responsible [9]. For the magnetic ordering temperature, there is no quantitative understanding why the T N of AF1 is low up to now.
Surprisingly, SCFOO is a ferrimagnet in comparison with the antiferromagnetic SFOO and has a higher magnetic ordering temperature 210K C T  [10] than SFOO, although the main lattice structure difference between SCFOO and SFOO is that the Fe-O-Os bond angle along the c axis in SCFOO is smaller than that in SFOO. More interestingly, CFOO is an insulating ferrimagnet with an even higher magnetic ordering temperature 320K C T  [3,10], which could not be accounted for by the generalized double exchange mechanism [12]. To the best of our knowledge, there is still a lack of comprehensive and unified understandings on how the low-temperature (LT) antiferromagnetism of A 2 FeOsO 6 transforms to the HT ferrimagnetism with the decrease of the radius of the A 2+ ions.
In this paper, to obtain a unified insight into all these puzzles, we have systematically investigated the frustrated magnetism of the DPs CFOO, SCFOO and SFOO by employing density functional theory (DFT) and maximally-localized Wannier functions (MLWFs). We find that lattice distortion enhances the AFM

II. COMOPUTATIONAL DETAILS
First-principles calculations based on DFT are performed within the generalized gradient approximation (GGA) according to the Perdew-Burke-Ernzerhof (PBE) parameterization as implemented in Vienna Ab initio Simulation Package (VASP) [13].
The projector-augmented wave method [14], an energy cutoff of 500 eV and a gamma-centered k-point mesh grid are used. Ion positions are relaxed towards equilibrium with the Hellmann-Feynman forces on each ions less than 0.01 eV/Å.
We use the simplified (rotationally invariant) coulomb-corrected density functional (DFT+U) method according to Dudarev et al. [15].  [3]. It has been demonstrated [7] that spin-orbit coupling (SOC) in CFOO is insignificant, so SOC is not taken into account in the present work. Hopping integrals between 3d/5d orbitals are extracted from the real-space Hamiltonian matrix elements in the non-spin-polarized MLWFs basis. MLWFs are obtained by employing the vasp2wannier90 interface in combination with the wannier90 tool [16]. In order to obtain the 3d/5d-like Wannier functions, we construct MLWFs in a suitable energy window mainly containing the 3d/5d antibonding states. All MLWFs are considered to be well converged if the total spread change within 50 successive iterations is smaller than 10 −9 Å 2 .

III. RESULTS AND DISCUSSIONS
We first demonstrate the magnetic interaction between the NN Fe 3+ and Os 5+ ions through the Fe-O-Os path in CFOO is intrinsically AFM and that lattice distortion can effectively relieve the magnetic frustration and increase c T . Then we show that the competing magnetic interactions in the tetragonal I4/m structure of SFOO give rise to the frustrated AF1 antiferromagnetism with a low Neel temperature N T . We give explanations to the ferrimagnetism of SCFOO with a lower T C . Last, we propose an important and general rule on the 3d 5 -5d 3 superexchange, which will help one to understand the complex magnetism in other DPs.

A. Intrinsically AFM interaction between the NN Fe 3+ and Os 5+ ions and the effect of lattice distortion on the ferrimagnetism in CFOO
The lattice of CFOO is severely distorted. It has the monoclinic structure with the space group of P2 1 /n [3]. Lattice distortion of DPs Lattice distortion has been suggested to cause the ferrimagnetism in CFOO [3].
According to Goodenough-Kanamori rules [17], the superexchange between the NN (2) the former has a weaker dependence on the Fe-O-Os angle than the latter [18]. However, it is highly possible that the 5d 2g t electrons of the Os 3+ ions are spatially extended so that the former is stronger than the latter, which leads to an AFM interaction between the NN Fe 3+ and Os 5+ ions.
In order to unveil why CFOO is FIM and how lattice distortion affects this  Technically, we adopt the four-states mapping method [19] to evaluate these magnetic interactions. Note that a positive exchange constant J corresponds to the AFM interaction whereas a negative one to the FM interaction.
We find the magnetic interaction between the NN Fe 3+ and Os 5+ ions is intrinsically The intrinsically AFM interaction of the Fe-O-Os path can be qualitatively understood based on the extended Kugel-Khomskii model [20][21][22]. According to this model, magnetic interactions can be evaluated based on the hopping integrals and on-site is the hopping integral. The first term in ij J describes the AFM contribution due to the hybridization between two occupied orbitals. The second term describes the FM contribution due to the hybridization between the occupied and empty orbitals. In order to elucidate why the magnetic interaction between the NN Fe 3+ and Os 5+ ions is intrinsically AFM, we take the Fe-O-Os path along the c axis in the pseudo-cubic CFOO as a typical example. Its detailed hopping integrals and energy levels are given in the right panel of FIG. S1 of supplemental material (S M) [23]. Compared with the FM interaction between the NN Mn 3+ ions in the cubic LaMnO 3 (LMO) [24], two pivotal factors drive the magnetic interaction between the NN Fe 3+ and Os 5+ ions in the pseudo-cubic CFOO to be intrinsically AFM. The first one is the very large energy difference  (up to 3.0 eV) between the occupied g e orbitals of Fe 3+ ion and the unoccupied one of Os 5+ ion. This gives weak FM contribution. The second one is the rather large hopping integrals between the occupied 2g t orbitals of Fe 3+ and Os 5+ ions.
For instance, the leading hopping integral is 0.27 eV. This gives strong AFM contribution. Therefore AFM contribution dominates over the FM one, giving rise to the intrinsically AFM interaction between the NN Fe 3+ and Os 5+ ions.
In addition, we find that lattice distortion can effectively relieve the magnetic frustration in CFOO, thereby raising the FIM phase transition temperature C T . Shown in the FIG. 1a and 1b

B. Competing magnetic interactions lead to the AF1 antiferromagnetis m with a low T N in SFOO
SFOO adopts two different magnetic and lattice structures depending on temperature [6]. With temperature decreasing, its magnetic structure transforms from AF1 into AF2 antiferromagnetism and its lattice structure transforms from I4/m into I4 with a dimerization between the NN Fe 3+ and Os 5+ ions along c axis. In both AF1 and AF2, moments of Fe 3+ and Os 5+ ions are antiparallelly coupled in the ab plane 19.8meV / f.u.
This indicates that FIM should have the highest energy, AF2 the media one and AF1 the lowers one. Such estimation is in accord with our DFT calculations: 34.139meV / f.u.  from both the large hopping integrals between the occupied 2 g t orbitals and the large energy difference between the occupied g e orbitals of 3d TM and the unoccupied ones of 5d TM, for the former gives rise to strong AFM contribution and the latter gives rise to weak FM contribution to the 3d 5 -5d 3 superexchange. As the angle  decreases, the electron hoppings between the occupied g e orbitals of the 3d 5 TM and the unoccupied ones of the 5d 3 TM will substantially reduce but the electron hoppings between the occupied orbitals of 3d 5 TM and 5d 3 TM are maintained almost unchanged. Thus decreasing the angle  means reducing the FM contribution but maintaining the AFM contribution almost unchanged. Consequently, the AFM interaction of the 3d 5 -O-5d 3 path enhances with  decreasing.

IV. SUMMARY
In