Lattice-distortion Induced Magnetic Transition from Low-temperature Antiferromagnetism to High-temperature Ferrimagnetism in Double Perovskites A2FeOsO6 (A = Ca, Sr)

High-temperature insulating ferrimagnetism is investigated in order to further reveal its physical mechanisms, as well as identify potentially important scientific and practical applications relative to spintronics. For example, double perovskites such as Sr2FeOsO6 and Ca2FeOsO6 are shown to have puzzling magnetic properties. The former is a low-temperature antiferromagnet while the latter is a high-temperature insulating ferrimagnet. In order to understand the underlying mechanisms, we have investigated the frustrated magnetism of A2FeOsO6 by employing density functional theory and maximally-localized Wannier functions. We find lattice distortion enhances the antiferromagnetic nearest-neighboring Fe-O-Os interaction, however weakens the antiferromagnetic interactions via the Os-O-O-Os and Fe-O-Os-O-Fe paths, so is therefore responsible for the magnetic transition from the low-temperature antiferromagnetism to the high-temperature ferrimagnetism as the decrease of the A2+ ion radii. Also discussed is the 5d3-3d5 superexchange. We propose that such superexchange is intrinsically antiferromagnetic instead of ferromagnetic as previously thought. Our work clearly illustrates the magnetic frustration can be effectively relieved by lattice distortion, thus paving the way for tuning of complex magnetism in yet other 3d–5d (4d) double perovskites.

neighboring (NN) Fe 3+ and Os 5+ ions. Consequently, the NN Fe 3+ and Os 5+ ions are coupled antiparallel and ferrimagnetism is experimentally observed. Simultaneously, corresponding AFM interactions via the Os-O-O-Os and Fe-O-Os-O-Fe paths are weak, so the Os-Os and Fe-Fe induced magnetic frustration is effectively relieved, and one observes a very high T C . Because SrCaFeOsO 6 is less distorted compared to Ca 2 FeOsO 6 , its magnetic frustration becomes stronger despite the FIM ground state being preserved. Accordingly, its T C is lowered. In the tetragonal I4/m structure of Sr 2 FeOsO 6 , lattice distortion vanishes along the c axis but it is very similar to that of Ca 2 FeOsO 6 in the ab plane. This special lattice distortion pattern results in both the in-plane NN Fe 3+ and Os 5+ ions being aligned antiparallel and the FM chains along the c axis. The resulting magnetic structure is just the strongly frustrated antiferromagnetism AF1 with a very low Neel temperature T N A F1 . Lastly, strong spin-lattice coupling leads to a transformation from AF1 to AF2. Our work illustrates the magnetic frustration can be effectively relieved by lattice distortion, which may well be responsible for the complex magnetism observed in other 3d-5d (4d) double perovskites as well.

Results
Lattice-Distortion dependence of magnetic interactions in Ca 2 FeOsO 6 . In order to understand why Ca 2 FeOsO 6 is FIM, and how lattice distortion affects this ferrimagnetism, we have systematically explored the effect of lattice distortion on the magnetic interaction of Ca 2 FeOsO 6 . Since the positions of O 2− ions are known to be vital, we performed a series of calculations using a linear superposition of the Wyckoff positions of the O 2− ions of both the relaxed and pseudo-cubic structure. In the pseudo-cubic structure, O 2− ions are artificially positioned to make Fe-O-Os angles straight, but lattice constants and the positions of the Fe 3+ , Os 5+ and Ca 2+ ions are fixed at their corresponding positions in the relaxed structure. The O 2− ions positions is computed as follows: In Eq. (1), R relax and R cubic are the position vectors of O 2− ions in the relaxed and pseudo-cubic structures respectively, and α x varies between 0 and 1. For example, α x = 0 corresponds to the relaxed structure and α x = 1 corresponds to the pseudo-cubic structure. Thus α x characterizes the lattice distortion induced by O 2− ions. The dominant magnetic interactions are divided into three groups (see Fig. 2a). The first group is the superexchange between the NN Fe 3+ and Os 5+ ions. The second involves super-superexchange between the next near-neighboring (NNN) Os 5+ ions. The third involves long-range Fe-Fe interactions via the four-bond Fe-O-Os-O-Fe path. Technically, we adopt the four-state mapping method to evaluate these various magnetic interactions 24 . Note that a positive exchange constant J corresponds to the AFM interaction, but a negative exchange constant J corresponds to the FM interaction. We find the magnetic interaction between the NN Fe 3+ and Os 5+ ions is intrinsically AFM. The calculated magnetic exchange constants of Fe-O-Os paths in the pseudo-cubic structure are shown in the Fig. 2b. They are all positive and thus AFM. The intrinsically AFM interaction of the Fe-O-Os path can be qualitatively understood based upon the extended Kugel-Khomskii model [25][26][27] . According to this model, magnetic interactions can be evaluated based on the hopping integrals and on-site energies, namely, In Eq. (2), U, J H and Δ ij are the on-site Coulomb interaction, Hund's coupling and the energy difference between the i th and j th energy levels, respectively, and t t ij afm ij fm ( ) is the hopping integral. The first term in J ij describes the AFM contribution due to the hybridization between the 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 of the pseudo-cubic Ca 2 FeOsO 6 as a typical example. Its detailed hopping integrals and energy levels are given in the right panel of Fig. S1 of supplemental material (SM). Compared with the FM interaction between the NN Mn 3+ ions in the cubic LaMnO 3 (LMO) 28 , two pivotal factors are seen to drive the magnetic interaction between the NN Fe 3+ and Os 5+ ions in the pseudo-cubic Ca 2 FeOsO 6 to be intrinsically AFM. The first factor is the very large energy difference Δ (up to 3.0 eV) between the occupied e g orbitals of Fe 3+ ion and the unoccupied e g orbitals of Os 5+ ion. This will give a weak FM contribution according to the Eq. (2). The second factor is the rather large hopping integrals between the occupied t 2g orbitals of the Fe 3+ and the Os 5+ ions. For instance, the leading hopping integral is 0.27 eV. This will give strong AFM contribution according to the Eq. (2). Therefore the AFM contribution dominates over the FM one, giving rise to the intrinsically AFM interaction between the NN Fe 3+ and Os 5+ ions, regardless of the magnitude of the Fe-O-Os angle.
In addition, we find lattice distortion can effectively relieve the magnetic frustration in Ca 2 FeOsO 6 and thereby raise its FIM phase transition temperature T C . Since Os 5+ ions form a face-centered sublattice with geometrically frustrated edge-sharing tetrahedrons, antiferromagnetically interacting Os 5+ ions are strongly frustrated. Figure 2c shows lattice distortion can dramatically weaken the NNN AFM interactions between the NNN Os 5+ ions, which implies that the Os 5+ ions' induced magnetic frustration can be relieved by lattice distortion. Besides, Fe 3+ ions can also be magnetically frustrated because of the following factor. The dominant NN Fe-O-Os AFM interactions require the magnetic moments of Fe 3+ ions to be aligned parallel, but the AFM magnetic interaction through the four-bond Fe-O-Os-O-Fe paths requires the magnetic moments of Fe 3+ ions to be antiparallel. Because lattice distortion slightly enhance the NN AFM interactions between the NN Fe 3+ and Os 5+ ions (see Fig. 2b axes (see Fig. 2d), it can effectively relieve the Fe 3+ ions induced magnetic frustration. And, we should note, accompanied with the relief of such magnetic frustration is the raising of the FIM phase transition temperature T C . Figure 2e shows the evolution of T C obtained by Monte Carlo (MC) as lattice distortion weakens. It clearly shows the T C of the relaxed structure 0 x α ( = ) (about 266 K, close to the experimentally measured one 7 T C ≈ 320 K) is higher than that of the less distorted one 0 25 x α ( = . ). Note that T C slightly increases with the weakening of lattice distortion for large α x . This is because the magnetic  ground state of Ca 2 FeOsO 6 with small lattice distortion is no longer FIM but AFM with the AF1 order as appearing in the Sr 2 FeOsO 6 (see Fig. 2e). Figure 3a demonstrates the mechanism by which lattice distortion enhances the NN Fe-O-Os AFM interaction. For illustration purposes, we consider the Fe-O-Os path along the c axis as an example. Fig. S1 of the SM shows the detailed leading hopping integrals and energy levels in the relaxed and pseudo-cubic structures, respectively. These hopping integrals clearly indicate lattice distortion tremendously reduces the electron hopping between the occupied e g orbitals of Fe 3+ ions and the unoccupied one of Os 5+ ions. Consequently, one can conclude based on the formula of J ij (see Eq. (2)) that lattice

. Mechanism by which lattice distortion enhances the NN AFM interaction between Fe 3+ and
Os 5+ ions (a), and weakens the NNN AFM interaction between Os 5+ ions (b) in Ca 2 FeOsO 6 . Solid (dashed) lines with double arrowheads indicate the electron hopping causing AFM (FM) contribution to the NN superexchange or NNN super-superexchange. S and W represent "strong" and "weak" words, respectively. In (a), the FM contribution of a x = 0 is weaker than that of a x = 1. However, the AFM contribution of a x = 0 is stronger than that of a x = 1. In (b), the AFM contribution of a x = 0 is weaker than that of a x = 1. Insets in distortion extraordinarily reduces the FM contribution to the NN superexchange. In contrast, lattice distortion has a rather minor effect on the AFM contribution, because it increases the electrons hopping between the occupied e g orbitals of Fe 3+ ions and the occupied t 2g orbitals of Os 5+ ions, although it reduces the hopping between the occupied t 2g orbitals of Fe 3+ and Os 5+ ions. Therefore, lattice distortion enhances the NN AFM interaction by dramatically reducing the FM contribution, and by maintaining the AFM contribution almost unchanged.
We find the NNN AFM interaction between the NNN Os 5+ ions is weakened by lattice distortion. This is because such NNN super-superexchange has a sensitive dependence on the geometry of the Low-temperature antiferromagnetism of Sr 2 FeOsO 6 . Sr 2 FeOsO 6 adopts two different magnetic and lattice structures depending on temperature 21 . With decreasing temperature, 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 the c axis. In both AF1 and AF2, moments of Fe 3+ and Os 5+ ions are coupled antiparallel in the ab plane ( Fig. 4a and Fig. 4c). In AF1 spins order as + + + + along the c axis (Fig. 4b). In AF2, spins order as + + − − + + − − (Fig. 4d).
Our study on the I4/m-AF1 phase ( Fig. 4a and Fig. 4b) shows that the out-of-plane NN AFM interaction J Fe Os compared with the others, and thus is omitted from our model.
Here we discuss how the competing magnetic interactions establish AF1 in the tetragonal I4/m structure of Sr 2 FeOsO 6 . First, it should be noted that the magnetic easy axis is the c axis 21 , that is, magnetic moments can only point up and down along it. The calculated results (see Fig. 4a is taken into consideration, the FIM Fe 3+ -Os 5+ layers should be coupled antiparallel along the c axis. In this case, the resulting magnetic structure is FIM (see Fig. 5a). If only the out-of-plane NNN AFM interaction J Os Os 2 − is taken into consideration, the FIM Fe 3+ -Os 5+ layers should be coupled parallel along the c axis. In this case, the resulting magnetic structure is AF1 (Fig. 5b), which is just the experimentally observed. Finally, if only the long-range four-bond Fe-O-Os-O-Fe AFM interaction J Fe Fe 3 − is taken into consideration, it gives rise to AF2 (Fig. 5c). Obviously, the out-of-plane NN J Fe Os This deduction can be confirmed as follows. In the FIM, all the out-of-plane Fe-Fe and Os-Os pairs are frustrated (see Fig. 5a). In the AF1, all the out-of-plane Fe-Os and Fe-Fe pairs are frustrated (see Fig. 5b). In the AF2, half of the out-of-plane Fe-Os and Os-Os pairs are frustrated (see Fig. 5c). In terms of the out-of-plane NN J Fe Os   Figures (a,b) correspond to the I4/m-AF1 phase. Figures  (c,d) correspond to the I4-AF2 phase. Figures (a,c) are the spin arrangement in the tetragonal ab plane. Figures (b,d) are the spin ordering along the c axis. Blue arrows represent spins. The relevant bond distances and angles obtained from DFT calculations are shown in (a-d).  − can all lower the T N because they are all frustrated. Moreover, the out-of-plane Fe-O-Os AFM interactions make the largest contribution to the lowering of T N for AF1. If all the dominating magnetic interactions are taken into consideration, the MC simulated T N is 155 K, which is very close to the experimental value.
By comparing the magnetic exchange constants of I4 structure with those of I4/m structure (see Fig. 4), one finds that the magnetic interactions in the former are very similar to the latter's, with the exception that the rather slight dimerization along the c axis in the I4 structure prominently enhances the out-of-plane NN AFM interactions J Fe Os 3 − (see Fig. 4d Low-temperature ferrimagnetism in the SrCaFeOsO 6 . Comparing SrCaFeOsO 6 with Sr 2 FeOsO 6 and Ca 2 FeOsO 6 , one can conclude that its mediate lattice distortion causes its ferrimagnetism to have a lower T C . Experiments show that SrCaFeOsO 6 has a rather similar lattice structure to that of Ca 2 FeOsO 6 18 . However, its Fe-O-Os bond angles reveal a more linear geometry than that of Ca 2 FeOsO 6 , because half of Ca 2+ ions are replaced by larger Sr 2+ ions 18 . So it can be inferred that SrCaFeOsO 6 can be readily ferrimagnetic. To confirm this, we studied three different types of arrangements of Ca 2+ and Sr 2+ ions. The first is where all the Ca 2+ (Sr 2+ ) are arranged in the ab plane (Fig. 6a). The second is where all Ca 2+ (Sr 2+ ) are arranged along the c axis (Fig. 6b). The third is where Ca 2+ and Sr 2+ ions are arranged in a checkerboard manner (Fig. 6c). For each arrangement, the FIM, AF1 and AF2 are considered. In all three of these cases, FIM always has the lowest total energy (see Fig. 6d). So the magnetic ground of SrCaFeOsO 6 should be FIM, consistent with experimental observations 18 Table I of the SM. Consequently, its magnetic frustration gets stronger and its T C should accordingly be lowered. Our MC simulated T C for SrCaFeOsO 6 is approximately 100 K, lower than the corresponding T C of 266 K for Ca 2 FeOsO 6 , consistent with experimental observations.

Discussions
Based on the present work, an important and general rule on the 3d 5 -5d 3 superexchange in the double perovskites can be proposed as follows. It is generally accepted that 29 the d 5 -d 3 superexchange changes from FM for θ > θ c to AFM for θ < θ c with 135° < θ c < 150°. However, we demonstrate the magnetic interaction between 3d 5 and 5d 3 TMs will be intrinsically AFM (this conclusion is independent of the particular choice of (a reasonable) U, see Table II of the SM) and further that this AFM interaction will increase as its angle θ decreases, as evidenced by the Fe-O-Os interactions in Ca 2 FeOsO 6 , SrCaFeOsO 6 and Sr 2 FeOsO 6 . This intrinsically AFM interaction results from both the large hopping integrals between the occupied t 2g orbitals and the large energy difference between the occupied e g orbitals of 3d TM, and the unoccupied orbitals of 5d TM, because the former gives rise to a strong AFM contribution and the latter gives rise to a relatively weak FM contribution to the 3d 5 -5d 3 superexchange. As the angle θ decreases, the electron hoppings between the occupied e g orbitals of the 3d 5 TM and the unoccupied ones of the 5d 3 TM will be substantially reduced, but the electron hopping between the occupied orbitals of 3d 5 TM and 5d 3 TM remain largely unchanged. Thus decreasing the angle θ means reducing the FM contribution, while leaving the AFM contribution largely unchanged. Consequently, the AFM interaction of the 3d 5 -O-5d 3 path increases with decreasing θ.

Conclusions
In conclusion, we have investigated the effect of lattice distortion on the frustrated magnetism of certain double perovskites: Ca 2 FeOsO 6 , Sr 2 FeOsO 6 and Sr 2 CrOsO 6 . In these cases, we find lattice distortion enhances the NN AFM Fe-O-Os interactions but weakens the AFM interactions of the Os-O-O-Os and Fe-O-Os-O-Fe paths. Because lattice distortions become increasingly severe from Sr 2 FeOsO 6 to SrCaFeOsO 6 to Ca 2 FeOsO 6 , the NN AFM Fe-O-Os interactions also become increasingly strong, but the AFM interactions of Os-O-O-Os and Fe-O-Os-O-Fe paths become increasingly weak. Consequently, the magnetic ground state transforms from antiferromagnetism to ferrimagnetism, and the magnetic transition temperature increases. We propose the 5d 3 -3d 5 superexchange is intrinsically antiferromagnetic, instead of being, as previously thought, ferromagnetic. Our work illustrates the magnetic frustration can be effectively relieved by lattice distortion in certain 3d-5d (4d) double perovskites.

Methods
First-principles calculations. 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) 30 . The projector-augmented wave method 31 , with 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 ion set to be less than 0 01eV Å . / . We use the simplified (rotationally invariant) coulomb-corrected density functional (DFT + U) method according to Dudarev et al. 32

Maximal localized Wannier functions calculations.
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 33 . In order to obtain the 3d/5d-like Wannier functions, we construct MLWFs in a suitable energy window, using primarily 3d/5d antibonding states. All MLWFs are considered to be well converged if the total spread over 50 successive iterations is smaller than 10 −9 Å 2 .
Monte Carlo simulations. The magnetic phase transition temperature T C or T N is obtained using parallel tempering Monte Carlo simulations 34,35 . These calcuations are performed on the 7 × 7 × 5