Orbital Delocalization and Enhancement of Magnetic Interactions in Perovskite Oxyhydrides

Recent experiments showed that some perovskite oxyhydrides have surprisingly high magnetic-transition temperature. In order to unveil the origin of this interesting phenomenon, we investigate the magnetism in SrCrO2H and SrVO2H on the basis of first-principles calculations and Monte Carlo simulations. Our work indicates that the Cr-O-Cr superexchange interaction in SrCrO2H is unexpectedly strong. Different from the previous explanation in terms of the H− ion substitution induced increase of the Cr-O-Cr bond angle, we reveal instead that this is mainly because the 3d orbitals in perovskite oxyhydrides becomes more delocalized since H− ions have weaker electronegativity and less electrons than O2− ions. The delocalized 3d orbitals result in stronger Cr-O interactions and enhance the magnetic-transition temperature. This novel mechanism is also applicable to the case of SrVO2H. Furthermore, we predict that SrFeO2H will have unprecedented high Neel temperature because of the extraordinarily strong Fe-H-Fe σ-type interactions. Our work suggests the anion substitution can be used to effectively manipulate the magnetic properties of perovskite compounds.

is, the electron configuration is (d xy ) 1 (d yz ) 1 (d xz ) 1 . In average, each Cr 3+ ion in SrCrO 2 H has two Cr-H-Cr superexchange paths and four Cr-O-Cr superexchange paths. Since the t 2g orbitals of the Cr 3+ ion could not interact with the 1s orbitals of H − ions by symmetry, the Cr-H-Cr superexchange interaction is negligible. In LaCrO 3 , there are six Cr-O-Cr superexchange paths for each Cr 3+ ion. So the fact that the Neel temperature T N of SrCrO 2 H is higher than that of LaCrO 3 is rather unexpected.
To probe the origin of the high T N in SrCrO 2 H and SrVO 2 H, we studied the magnetic properties of SrCrO 2 H, SrVO 2 H, and LaCrO 3 based on the density functional theory (DFT). We show that due to weaker electronegativity and less electrons of H − ions than those of O 2− ions, the substitution of H − ions with O 2− ions lead to more delocalized 3d orbitals of Cr 3+ ions which make Cr-O-Cr superexchange in SrCrO 2 H much stronger and leads to a high T N . This new mechanism is also applicable to the case of SrVO 2 H.
Although the H − ion was experimentally reported to be randomly distributed in SrCrO 2 H, we will adopt an ordered structure (see Fig. 1a) to investigate the magnetism in SrCrO 2 H for the following reasons. First, by using the cluster expansion approach, we find the ground state structure of SrCrO 2 H is similar to the experimentally observed structure of SrVO 2 H. Second, we find that the magnetic properties of another SrCrO 2 H structure with a more random H − ion distribution are similar to those of the ordered structure (see Supplementary Material). The ground state structure (tetragonal P4/mmm symmetry) of SrCrO 2 H has one H − ion and two O 2− ions in the primitive cell. The Cr 3+ cations are located within square planes of oxide ions and form CrO 2 sheets. The CrO 2 sheets are connected by hydride ions, which occupy the remaining two coordination sites around each Cr 3+ center. Thus the ground state structure of SrCrO 2 H can be described by a CrO 2 -SrH-CrO 2 -SrH stacking sequence. The computed phonon frequencies 26 (see Supplementary Figure S3) indicate that the ground state structure of SrCrO 2 H is dynamically stable.
To examine the magnetic properties of SrCrO 2 H, we considered four ordered spin states, namely, the ferromagnetic (FM), A-type AFM, C-type AFM and G-type AFM states. The experimentally observed AFM structure is the G-type. Our GGA + U calculations show that all the AFM states have lower energy than the FM state, and the G-type AFM is indeed the ground state, in consistent with experimental observations 22 . To extract the values of the spin exchange parameters, we adopt the four-state mapping approach 27,28 . There are four relevant Cr-Cr spin exchange interactions (see Fig. 1a): (1) J O is the nearest neighbor (NN) superexchange interaction for the Cr-O-Cr path; (2) J H is the NN spin exchange interaction for the Cr-H-Cr path; (3) J 3 is next nearest neighbor (NNN) super-superexchange interaction in the plane; (4) J 4 is the NNN out of plane super-superexchange interaction. The calculated exchange parameters in the tetragonal structure of SrCrO 2 H are summarized in Table 1. As expected, the NN Cr-O-Cr path has the strongest spin exchange interaction (J O ) since it is mediated by the strong π -π hybridization between 3d and 2p orbitals. The NN spin exchange interaction J H is weakly AFM due to the direct through-space overlap between the t 2g orbitals of Cr 3+ ions. The NNN exchange interactions (J 3 and J 4 ) are negligible. The AFM nature of the NN interactions can result in the G-type AFM order. Based on the calculated spin exchange parameters, our Monte Carlo (MC) simulations indicate that the T N is around 285 K. If a smaller Hubbard U is adopted, we can get higher T N according to the theory of superexchange, in better agreement with the experimentally observed T N
For comparison, we also studied the magnetism in LaCrO 3 . Experiments show the ground state structure of LaCrO 3 is the GdFeO 3 -type distorted perovskite with Pbnm space group and its lattice constants are a = 5.478, b = 7.759, and c = 5.516 29 . The lattice constants of our optimized Pbnm structure are a = 5.468 Å, b = 7.758 Å,  Table 1. The NN interaction J O is AFM and the NNN interactions are negligible. This is in accord with the experimentally observed G-type AFM ground state in LaCrO 3 . Compared with SrCrO 2 H, it is clear that J O in LaCrO 3 is much weaker. Based on the calculated spin exchange parameters, our MC simulations indicate that the T N for LaCrO 3 is around 133 K. Therefore, our theoretical calculations confirm the experimental observation that SrCrO 2 H has a higher T N than LaCrO 3 , as shown in Fig. 3. Now we begin to understand the difference in the magnetic properties between SrCrO 2 H and LaCrO 3 . There are two significant differences between SrCrO 2 H and LaCrO 3 . First, there is a structural difference. The CrO 6 octahedron in LaCrO 3 is tilted due to a small tolerance factor, while there is no octahedron tilt in SrCrO 2 H. As a result, the Cr-O-Cr angle in SrCrO 2 H is 180° while the average Cr-O-Cr angle in LaCrO 3 is 167°. Second, a chemical difference exists since one third of the O 2− ions are replaced by H − ions.
To make clear whether the structural difference or the chemical difference is responsible for the stronger Cr-O-Cr exchange in SrCrO 2 H, we investigate the magnetic properties of SrCrO 2 H and LaCrO 3 with the same cubic perovskite crystal structure. The lattice constant is set to be the average lattice constant (3.85 Å) of experimental SrCrO 2 H structure 22 . Note that the hypothetic cubic LaCrO 3 phase can be regarded as a result of substituting Sr 2+ and H − in cubic SrCrO 2 H with La 3+ and O 2− , respectively. The computed spin exchange parameters of these two cubic structures are summarized in Table 1. Surprisingly and interestingly, the Cr-O-Cr superexchange (29.69 meV) in SrCrO 2 H is almost as twice as that (14.25 meV) in LaCrO 3 despite of the fact that the Cr-O bond length and the Cr-O-Cr bond angle are identical in cubic SrCrO 2 H and LaCrO 3 . As expected, the Cr-O-Cr exchange in cubic LaCrO 3 is stronger than that (8.88 meV) in Pbnm LaCrO 3 according to the Goodenough-Kanamori rule [30][31][32] . Previously, it was suggested that the structural difference is solely responsible for the high T N in SrCrO 2 H 22 . However, our calculations show that the structure difference and chemical difference enhance the Cr-O-Cr exchange interaction by 5.37 meV and 15.44 meV, respectively. Therefore, the effect  Fig. 1a. The spin-up and spin-down levels are denoted by black and blue colors, respectively. The effective orbitals hoppings responsible for the Cr1-Cr2 superexchange interaction are illustrated. of chemical difference on the Cr-Cr exchange interaction is much more important than that of the structural difference.
To account for why J O in the cubic SrCrO 2 H is as twice as that in the cubic LaCrO 3 , we examine their electronic structures in details. Figure 2a shows the partial density of states (PDOS) of the spin-up Cr 3+ ion in SrCrO 2 H with the G-type AFM order. We can see that, for SrCrO 2 H, the majority-spin t 2g (d xy , d xz , and d yz ) orbitals are occupied by three electrons but the majority-spin e g orbitals are unoccupied, which is similar to the orbital occupation in LaCrO 3 . The energy level and occupancy of the 3d orbitals of the tetragonal SrCrO 2 H are shown in Fig. 2b. Note that the t 2g orbitals split into the low-lying two-fold degenerate d xz /d yz level and a higher d xy level as a result of the tetragonal symmetry.
Without loss of generality, we consider the spin exchange J O for the Cr1-O-Cr2 path along the x axis, as shown in Fig. 1a. According to Anderson's superexchange theory 33 , the magnitude of the spin exchange can be estimated by /∆ t 2 , where t is the effective hopping between the d orbitals and Δ is the energy difference between majority-spin and minority-spin orbitals. In the cubic SrCrO 2 H or LaCrO 3  , where t m (m= xz, yz, xy) is effective hopping between the m orbitals of Cr1 and Cr2, and Δ m is the energy difference between the majority-spin m orbital and minority-spin m orbital of the Cr 3+ ion (see Fig. 2b). The Δ m parameters are estimated by constructing the maximally localized Wannier functions (MLWFs) based on the ferromagnetic electronic structure. It turns out that Δ m (about 5.3 eV) in SrCrO 2 H is rather close to that (about 5.5 eV) in LaCrO 3 . Therefore, we can regard Δ m as a constant. The effective hopping paramters t m between the Cr 3d orbitals is obtained by constructing the MLWFs using the spin-unpolarized Bloch wavefunctions. These hopping parameters are listed in Table 2. We can see that the hopping parameter t yz between the d yz orbitals of Cr1 and Cr2 is negligible since these two orbitals are almost parallel to each other. The π -π hopping parameter t xy in the cubic SrCrO 2 H are the same as that in the cubic LaCrO 3 . The striking result is that the π -π hopping parameter t xz between the d xz orbitals of Cr1 and Cr2 in the cubic SrCrO 2 H is almost 50% stronger than that in LaCrO 3 . Using these hopping parameters, we can estimate the ratio between J O in cubic SrCrO 2 H and that in LaCrO 3 as: Thus, our first-principles result can be well accounted for by this simple model. This analysis clearly shows that the stronger t xz hopping in SrCrO 2 H is responsible for its high T N . Figure 4a,b shows the real-space distribution of the d xz -like MLWFs in the cubic SrCrO 2 H and LaCrO 3 . It is clear that the effective d xz -like MLWF not only distributes around the Cr ion, but also has tails on the neighboring O 2− ions due to the anti-bonding π * hybridization between Cr-d xz and O-2p orbitals. It is the tails on the O 2− ions that mediate the effective hopping between the Cr-d xz orbitals. An interesting observation is that the lobes on the O 2− ion in the MLWF of SrCrO 2 H are bigger than those in LaCrO 3 . This suggests that the interaction between Cr-d xz orbital and O-p z orbital in SrCrO 2 H is stronger than that in LaCrO 3 , in agreement with our previous result that the effective hopping between the d xz orbitals of Cr1 and Cr2 in SrCrO 2 H is larger than that in LaCrO 3 .   We propose that the stronger interaction between Cr-d xz orbital and O-p z orbital in SrCrO 2 H results from the more delocalized Cr-d xz orbital in SrCrO 2 H (see Supplementary Figure S5). This is supported by a separate MLWF analysis which indicates that the spread of the atomic Cr-d xz orbital in the cubic SrCrO 2 H is larger than that in the cubic LaCrO 3 . The more delocalized Cr-d xz orbital in SrCrO 2 H can be reasoned by considering the electrostatic potential exerted on the Cr 3d electrons. The contour plots of the electrostatic potential on the x-z plane are displayed in the Fig. 4c,d. We can see that the electrostatic potential along the Cr-H direction is much weaker than that along the Cr-O direction. Thus, the d xz orbital in SrCrO 2 H is more delocalized along the z-axis, as can also be seen from Fig. 4a. It is the weaker repulsion between the H − ion and the 3d electrons that makes the d xz orbital in SrCrO 2 H more delocalized. The weaker electrostatic potential along the Cr-H direction results from the fact that H − ions have the weaker electronegativity and less charge than O 2− ions. Therefore, the replacement of O 2− ions by H − ions will not only change the hybridization type between the transition metal and the anions, but also affect the wavefunction distribution of d orbitals. Such novel mechanism revealed here for SrCrO 2 H can be also applied to SrVO 2 H. The only difference is that there is one d electron less than that in SrCrO 2 H, which makes the T N slightly lower (see Supplementary Material). We note that first-principles calculations 34 were recently carried out to study the electronic and magnetic properties of SrVO 2 H. However, the mechanism for the high Neel temperature in SrVO 2 H was not discovered.
The mechanism that the H − ion induced delocalization of the d orbitals is general and may have profound effect on the electronic and magnetic properties of other perovskite oxyhydrides. Below we will predict that SrFeO 2 H has an extremely high T N . With the cluster expansion approach, we predict that SrFeO 2 H takes the same ground state structure as that of SrCrO 2 H. This is reasonable since the ionic radius of the Fe 3+ ion is close to that of the Cr 3+ ion. The optimized lattice constants a and c for SrFeO 2 H are 3.997 Å and 3.645 Å, respectively. The computed phonon dispersion 26 (see Supplementary Figure S4) indicates that SrFeO 2 H is dynamically stable. The spin exchange parameters calculated for the optimized SrFeO 2 H structures are listed in Table 1  structure. In fact, the much stronger Fe-H-Fe interaction is mainly because the out-of-plane Fe-− d z r 3 2 2 orbital is more delocalized than the in-plane Fe-− d x r 3 2 2 and − d y r 3 2 2 orbitals. Therefore, the σ bond between the H 1s orbital and Fe-− d z r 3 2 2 orbital is much stronger than the σ bond between the O-2p orbitals and the in-plane Fe − d x r 3 2 2 / − d y r 3 2 2 orbitals (see Supplementary Figure S6). Similar to the cases of SrCrO 2 H and SrVO 2 H, the weaker electrostatic potential of H − ions exerting on the d electrons of Fe 3+ ions leads to more delocalized Fe-− d z r 3 2 2 orbitals, which results in an anomalously strong spin exchange J H . Our MC simulations indicate that the T N of SrFeO 2 H is around 826 K, which is even higher than that of BiFeO 3 (T N = 643 K) 35 and SrFeO 2 with a quasi-two-dimensional structure (T N = 473 K) 36,37 . Our result suggests that the replacement of O 2− ions by H − ions can enhance the magnetic interactions not only in t 2g d 2 and d 3 systems, but also in d 5 systems. Our work suggests that the high magnetic transition temperature in LaSrCoO 3 H 0. 7 18-20 and Sr 3 Co 2 O 4.33 H 0.84 21 should be also due to the H − ion induced delocalization of the 3d e g orbitals.
In summary, we perform a systematic theoretical study on the magnetic properties of perovskite oxyhydrides. The high magnetic transition temperature in SrCrO 2 H is revealed to be due to the delocalization of 3d orbitals in perovskite oxyhydrides. This is because H − ions have weaker electronegativity and fewer electrons than O 2− ions. The more delocalized 3d orbitals in SrCrO 2 H make Cr-O-Cr superexchange strong and T N high. This novel mechanism also applies to the case of SrVO 2 H. We predict that the σ -type Fe-H-Fe interactions in SrFeO 2 H are extraordinarily strong which also result from the delocalization of the 3d orbitals. The delocalization of d orbitals in oxyhydrides discovered in this work is universal and may also have profound effects on properties other than the magnetic properties.

Methods
Our DFT calculations are performed on the basis of the projector augmented wave method 38,39 encoded in the Vienna ab initio simulation package 40,41 (VASP) using the generalized-gradient approximation (GGA) of Perdew, Burke, and Ernzerhof 42 . The plane-wave cutoff energy is set to be 450 eV. To properly describe the strong electron correlation in the 3d transition-metal oxide, the GGA plus on-site repulsion U method (GGA + U) is employed 43 . U = 4 eV and J = 1eV are applied to the 3d electron of Cr 3+ ions. The maximally localized Wannier functions (MLWFs) are constructed with the Wannier90 program 44,45 . The spread functional is considered to be converged if the corresponding fractional change because two successive iterations is smaller than 10 −10 . To find the ground state structures of SrCrO 2 H, SrVO 2 H, and SrFeO 2 H, we adopt the cluster expansion approach 46 by using the alloy theoretic automation toolkit (ATAT) 47 .
We perform parallel tempering Monte Carlo (PTMC) simulations 48,49 to estimate the magnetic transition temperature. In PTMC simulations, many replicas with different temperature are simultaneously simulated and a virtual process exchanging configuration of these replicas is introduced. PTMC simulations can avoid a local minimum at low temperatures and can reduce relaxation time. We adopt a 10 × 10 × 10 supercell to perform PTMC simulations. Our test shows that the results obtained with a 12 × 12 × 12 supercell are almost the same as those with the 10 × 10 × 10 supercell. The number of replicas is set to 120.