Interplay between polarization, strain and defect-pairs in Fe-doped SrMnO$_{3-\delta}$

Defect chemistry, strain, and structural, magnetic and electronic degrees of freedom constitute a rich space for the design of functional properties in transition metal oxides. Here, we show that it is possible to engineer polarity and ferroelectricity in non-polar perovskite oxides via polar defect pairs formed by anion vacancies coupled to substitutional cations. We use a self-consistent site-dependent DFT+$U$ approach that accounts for local structural and chemical changes upon defect creation and which is crucial to reconcile predictions with the available experimental data. Our results for Fe-doped oxygen-deficient SrMnO$_3$ show that substitutional Fe and oxygen vacancies can promote polarity due to an off-center displacement of the defect charge resulting in a net electric dipole moment, which polarizes the lattice in the defect neighborhood. The formation of these defects and the resulting polarization can be tuned by epitaxial strain, resulting in enhanced polarization also for strain values lower than the ones necessary to induce a polar phase transition in undoped SrMnO$_3$. For high enough defect concentrations, these defect dipoles couple in a parallel fashion, thus enabling defect- and strain-based engineering of ferroelectricity in SrMnO$_3$.


I. INTRODUCTION
The interplay between electric polarization, magnetism, strain and the defect chemistry constitutes a rich phase diagram for the design and control of novel functional properties in transition-metal perovskites [1][2][3][4][5][6][7][8][9][10][11] . In particular, strain imposed, for example, by lattice matching with the substrate during coherent epitaxial growth of thin films is an established route to engineer polarity and ferroelectricity in non-polar complex oxides 5,7-10,12 . Defect engineering can tailor the ferroelectric response by introducing polar defect pairs. In particular, substitutional defects coupled to oxygen vacancies (V O ), such as Fe Ti -V O defect pairs, were shown to align with the direction of the lattice polarization in ferroelectric PbTiO 3 5 or to promote ferroelectricity in paraelectric SrTiO 3 13 . Polar distortions, strain and stoichiometry can couple or compete in determining the material properties as shown for oxygen-deficient SrMnO 3 (SMO) thin films 9 , the material we also use a model system in the present study.
Bulk SMO occurs in a hexagonal structure 14 , but the perovskite phase of SMO (space group Pnma, see Fig.  1) with G-type antiferromagnetic (AFM) order 15 can be stabilized at low temperature in thin films 16 . It was predicted from theory that biaxial epitaxial strain induces a polar distortion in SMO, mainly associated with Mn ions displacing from their high-symmetry positions, the magnitude of the distortion and hence the ferroelectric polarization increasing with increasing biaxial strain 8,9,12 . This is caused by softening of the in-plane polar modes that become unstable for tensile strains larger than about 2%. Compressive strains larger than 4% are, instead, necessary to induce ferroelectricity in the direction perpendicular to the strain plane. The strain response is different in the competing FM phase, requiring smaller compressive but larger tensile strains to trigger polar instabilities 17 . At the same time, tensile strain favors the formation of oxygen-vacancies, the pres- ence of which was, however, found to be detrimental to ferroelectricity 9 .
Fe-doping of oxygen deficient (SrMnO 3 -δ ) thin-films could be a strategy to reverse this behavior, since ferroelectricity is expected to be enhanced both by the decrease in crystal symmetry due to the aliovalent Fe ion and especially by the formation of a defect dipole due to the possible association of the substitutional Fe with an oxygen vacancy 13 . SrMn 1x Fe x O 3 -δ was synthesized in the past in an attempt to obtain manganites with a mixture of d 3 (Mn +4 ) and d 4 (Fe +4 ) cations 18 , as found in other manganites showing colossal magnetoresistance, e.g. the family of doped Ln 1x (Ca/Sr) x MnO 3 19,20 . At room temperature SrMn 1x Fe x O 3 -δ phases were found to adopt a cubic perovskite structure with a disordered arrangement of Mn and Fe transition-metal cations in the octahedral sites due to their similar ionic radii. Oxygen vacancies observed during synthesis in air are generally created to maintain charge balance after the aliovalent substitution and introduce Fe +3 cations in the structure as suggested by iodometric and Mössbauer measurements 18,21,22 . Mössbauer data suggest as well that oxygen vacancies are predominantly found in the vicinity arXiv:2106.13854v1 [cond-mat.mtrl-sci] 25 Jun 2021 of the Fe +3 ions 18,21,22 . SrMn 1x Fe x O 3 -δ samples show AFM behavior for both low (x ≤ 0.3) and high (x ≥ 0.9) Fe-doping, while a spin glass state caused by the interaction between Fe +3/+4 and Mn +3/+4 was observed for intermediate dopant concentrations 18,21 .
With the aim of understanding the mechanism underlying emerging polarization and the interaction between polar defect pairs, epitaxial strain and the electronic, structural and magnetic degrees of freedom, we used DFT+U calculations to investigate the properties of Fe-doped oxygen deficient SMO thin films. Our results suggest that defect engineering, through controlling the concentration and distribution of polar defect pairs formed by anion vacancies coupled to substitutional cations, constitutes a parameter to design multiferroic materials. Defect couples can, indeed, promote polarity and ferroelectricity in non-polar perovskites due to an off-center displacement of the defect: the spatially separated substitutional Fe Mn (negatively charged) and V O (positively charged) offset the charge center from the geometry center of the lattice, resulting in an electric dipole moment along the direction from Fe Mn to V O already in unstrained and hence non-polar SMO. Other effects related to the appearance of reduced Mn +3 , negatively charged with respect to the Mn lattice sites, should also be taken into account, since Mn +3 -V O pairs can result in additional dipoles that affect the overall ferroelectric response. Finally, the defect-pair dipole can couple with applied epitaxial strain favoring the transition to a polar phase, even for strains lower than the ones necessary to stabilize the polar structure in stoichiometric SMO.

II. METHODS
All DFT calculations were performed with the Quantum ESPRESSO distribution 23,24 . PBEsol 25 was used as exchange-correlation functional together with ultrasoft pseudopotentials 26 with Sr(4s, 4p, 5s), Mn(3p, 4s, 3d), and O(2s, 2p) valence states 27 . A kinetic-energy cutoff of 70 Ry for wave functions and 840 Ry for spin-charge density and potentials were applied. A Gaussian smearing with a broadening parameter of 0.01 Ry was used in all cases.
SMO was simulated as a 40-atom 2 × 2 × 2 supercell of the 5-atom primitive cubic cell. A shifted 6 × 6 × 6 Monkhorst-Pack 28 k-point grid was used to sample the Brillouin zone. Both bulk and thin film geometries of the G-type AFM and ferromagnetic (FM) phases of SMO were considered. For stoichiometric bulk calculations, both lattice parameters and atomic positions were relaxed, while thin-film geometries with biaxial epitaxial strain in the ac-plane imposed by a cubic substrate were computed following the procedure described in Ref. 29. Prior to defect creation, the atoms were displaced along the eigenvectors of the polar phonon modes computed for unstrained stoichiometric SMO. Defects were then created by removing one oxygen atom (V O , concentra-tion 4.2%) and at the same time substituting one Mn with a Fe ion (Fe Mn , concentration 12.5%). Different possible relative arrangements of the substitutional Fe with respect to the V O were taken into account (see Fig. 2). Mössbauer experiments have shown that iron is present as Fe +3 when associated with the doubly positively charged V O and the substitutional defect is thus negatively charged. The calculations were hence performed considering the positive charge state of this defect pair (Fe ' Mn -V O •• in Kröger-Vink notation 30 , where the prime and dot symbols indicate, respectively, a charge of -1 and +1 relative to the respective lattice site). This was obtained by adjusting the number of electrons and by applying a background charge to ensure neutrality of the unit cell, as required by calculations under periodicboundary conditions to avoid divergences in the electrostatic potential. For simplicity, we will refer to the defect pairs in this charge state simply as Fe Mn -V O . Finally, for defective cells, atomic positions were optimized while keeping the lattice vectors fixed at optimized values of the non-defective system. In all calculations, atomic forces were converged to within 5 × 10 −2 eV/Å, while energies were converged to within 1.4 × 10 −5 eV. Auxiliary calculations using 2 √ 2 × 2 √ 2 × 2 , 3 × 3 × 3, and 4 × 4 × 4 supercells with 80, 135, and 320 atoms (and 3 × 3 × 4, 3 × 3 × 3 k-meshes and Γ point sampling of the Brillouin zone, respectively) were performed to investigate the interaction of two defect pairs and the influence of the defect concentration on the predicted polarization.
A Hubbard U correction [31][32][33] was applied in all calculations. For stoichiometric bulk systems, where all Mn sites are crystallographically and chemically equivalent, we used global self-consistent U parameters (U SC ) computed for the G-AFM and FM phases of SMO in Ref. 34 using density-functional perturbation theory (DFPT) 35 , as implemented in hp.x of Quantum ESPRESSO 23,24 . Selfconsistent site-dependent U parameters (U SC−SD ) were instead computed for defective systems by perturbing the inequivalent sites resulting from defect formation (atoms were selected to be perturbed if their unperturbed atomic occupations differed by more than 10 −3 ) 34 . DFPT calculations were performed with a Γ-point sampling of the q-space 35 in the 40-atom cell. A convergence threshold of 0.01 eV was applied for the self-consistence of U values. In all cases, atomic orbitals were used to construct occupation matrices and projectors in the DFT+U scheme. For simplicity, U SC−SD values (see SI Sec. S1) have only been computed for the unstrained stoichiometric and defective geometries, since even 4% tensile strain changes U SC by only 0.01 eV compared to zero strain 34 .
The strain-dependent formation energy (E f ) of a Fe Mn -V O defect-pair in the q = +1 charge state was calculated as described in Ref. 36: Here E tot,def and E tot,stoic are the DFT total energies of the defective system with a charge q and of the stoichiometric cell, respectively. is the applied strain, E Fermi is the Fermi energy relative to the valence band maximum of the defect-free system, which can take values between zero and the band-gap of the material, while n i indicates the number of atoms of a certain specie i that is added (n i > 0) or removed (n i < 0) from the supercell to form the defect, while µ i is its chemical potential. Finally, E corr is a corrective term necessary to align the electrostatic potential of the defective cell with the one of the neutral stoichiometric system obtained by calculating the difference in electrostatic potential between the neutral defect-free cell and the charged defective one averaged in spheres around atomic sites located far from the defect 37 . No further finite-size corrections were applied since the defect concentrations we simulate are realistic for this material. Different synthesis conditions can be accommodated by adjusting the set of chemical potentials µ i = µ 0 i + ∆µ i for each element by assuming equilibrium with a physical reservoir such as a gas or a bulk phase. We expressed µ Fe and µ Mn as a function of µ O . For this latter, we used O 2 as a reference, µ O = 1 2 E(O 2 ) + ∆µ O , while bounds on ∆µ O were derived imposing the stability of SMO (∆µ Sr + ∆µ Mn + 3∆µ O ≤ ∆H f (SMO)) against decomposition into elemental Sr/Mn species (∆µ Sr/Mn ≤ 0) and against the formation of competing phases like SrO (∆µ Sr + ∆µ O ≤ ∆H f (SrO)), and MnO (∆µ Mn + ∆µ O ≤ ∆H f (MnO)). For the Fe impurity, stability against solubility-limiting phases, such as FeO, were instead considered to relate µ Fe to µ O 38 . The computed heat of formation (∆H f ) of the transition-metal oxides were corrected according to Ref. 39 to account for the mixing of DFT and DFT+U total energies in the derivation of the formation enthalpy. We will show results in the oxygenpoor limit with ∆µ Mn = -1.77 eV and ∆µ Fe = -1.43 eV and for a Fermi energy equal to the band gap of unstrained SMO.
The polarization P was computed using a point-charge model: where r i is the position of atom i and q i is its nominal charge: +2 for Sr, -2 for O, and +4 or +3 for stoichiometric or reduced Mn and Fe sites. The charge applied on each Mn and Fe ion was defined on the base of its oxidation state computed through the method introduced by Sit et al. 40 . The polarization, being a multivalued quantity, has been corrected by an integer number of polarization quanta Q, computed as: with a, b, and c being the lattice parameters, V the volume of the unit cell, and e the elementary charge. Results obtained with this method include polarization contributions of the lattice and the defect dipole, but neglect electronic redistribution effects compared to other approaches such as the Berry Phase formalism 41,42 . However, the metallic nature of some defective SMO cells did not allow the application of the Berry phase approach. pair two dipoles exist in the structure, one pointing from the Fe ' Mn to the V O and one pointing from the Mn '

III. RESULTS AND DISCUSSION
Mn to the V O •• as discussed in more detail in SI Sec. S1. This will have implications on the magnetism and polarization as discussed in Secs. III B and III C. In the FM phase a partial reduction of one or two Mn sites is observed, which we associated with its metallic nature.

A. Relative Stability and Formation Energy
We begin by investigating the relative stability of the Fe Mn -V O configurations in bulk AFM and FM SMO. We note that we are mainly interested in relative formationenergy differences for the different configurations and in strain-induced changes, rather than absolute defect pair formation energies, which have been derived in O-poor conditions and thus correspond to a lower limit for E f . These results indicate that the V O are preferentially in the neighborhood of the substitutional iron, as suggested by spectroscopic results 22 , even though not necessarily in its first coordination shell. More importantly, our data suggest that some disorder is expected, as also indicated by experiments 18 . This is particularly interesting because different configurations correspond to different orientations of the electric dipole associate to the Fe Mn -V O pair, which -as we will discuss in Sec. III C -is responsible for the polarization in unstrained SrMn 1x Fe x O 3 -δ . Hence, the presence of different energetically similar configurations could potentially lead to a switchable polarization and defect-induced ferroelectricity if the defect dipoles couple in a parallel fashion, which we will explore in Sec. III D.
A different behavior is observed, instead, in the FM phase (see Fig. 3b), where not only are Fe Mn -V O IP found to be generally more stable regardless of the distance between the substitutional iron and the vacancy, but where the most stable configuration is a NNNN Fe Mn -V O IP defect pair, likely because these configurations lead to Mn 3+ /Mn 4+ interactions that stabilize the FM phase, as we will discuss in more detail in Sec. III B. Furthermore, the average difference in E f between the defect configurations in FM SMO is only 0.2 eV and E f in the FM phase are on average 0.8 eV lower than in the AFM phase. This can be explained by the smaller energetic cost to accommodate the two excess electrons associated with the V O on delocalized Mn/Fe states in the metallic FM phase.
In SMO thin films, the defect formation energy and consequently the defect concentration depend on volume changes induced by biaxial strain 9,43,44 . Biaxial strain also breaks the symmetry 29 and could thus allow straincontrolled ordering of defects on inequivalent sites 9,43,44 . Hence, we now consider the interplay between Fe Mn -V O defects, strain, and magnetism in SMO. Fig. 4 shows the changes in E f for defect pairs as a function of the applied strain. In the AFM phase, tensile strain results in a reduction of the formation energy of Fe Mn -V O , consistent with the chemical expansion 45 SI Fig. S8, the FM phase exhibits a reduced sensitivity of the formation energy to strain, which is rationalized by its metallicity. We also note that for the FM phase, under compressive strain, the formation energy of Fe Mn -V O IP defects increases as expected from volume arguments, which can be explained in terms of a reduced sensitivity of the metallic FM phase to crystal field effects, allowing volume effects to dominate 43 .

B. Magnetic Order
Bulk stoichiometric SMO has a G-type AFM ground state. DFT+U calculations have shown a 4.2% oxygen vacancy concentration to induce a magnetic phase transition from AFM to FM 9,34 , which is explained by the vacancy-induced Mn 4+ -Mn 3+ double exchange coupling. The properties of Fe-doped oxygen-deficient SMO are more complex due to the presence of Fe transitionmetal atoms. Indeed, for the Mn 4+ , Mn 3+ , and Fe 3+ ions present in the simulated cells, the interactions between neighboring Mn 4+ -Mn 4+ and Mn 3+ -Mn 3+ are AFM, between Mn 4+ -Mn 3+ are FM, while those between Mn 4+ -Fe 3+ are AFM through the π orbitals and FM through the σ orbitals 18 . Fig. 5a shows the totalenergy difference between the unstrained AFM and FM phases with different Fe Mn -V O defect pairs as a function of the distance between the two defects. The most stable configurations (see Fig. 3) favour the AFM order. Indeed, all the configurations with a V O OP , and the NN and the majority of the NNN Fe Mn -V O IP configurations prefer this magnetic phase. Only the NNNN Fe Mn -V O IP defects strongly favor the FM phase since the Mn 4+ -Mn 3+ interactions are promoted due to the larger distance between the Fe 3+ ion and the reduced Mn 3+ site. This result is in line with the experimental data reporting SrMn 1x Fe x O 3 -δ with Fe concentration close to the one of our study (x = 0.125) to show AFM behavior. Interestingly, the DFT+U SC-SD approach including local chemical changes on the transition-metal atoms around the defect is fundamental to predict defect-induced magnetic properties, since DFT+U SC with global U SC of sto- ichiometric SMO predicts a preferential FM order for all configurations (see SI Fig. S9). This result can be explained by increasing U favoring the FM order, which conversely implies that the decreased U SC-SD values on the reduced Mn 3+ will locally destabilize the FM order 34 .
Biaxial strain beyond a critical value of 2% is known to induce a AFM to FM transition in stoichiometric SMO 9,34 . We now consider the interplay between this magnetic phase transition and the Fe Mn -V O defect pair (see Fig. 5b). Unsurprisingly, tensile strain stabilizes the FM phase even in presence of the defect pair, affecting all configurations in almost the same way and in an approximately linear fashion up to 4% strain. Larger tensile strain result, instead, in a stabilization of the AFM phase for the most stable configurations and in a reduction of the FM stabilization for the others. This observation can be explained considering the stronger sensitivity of the AFM phase to volume changes, which results, as previously discussed, in a larger reduction of the formation energy and consequently in a stabilization of the AFM phase for large tensile strains. Compressive strain up to -2% favors the AFM order: at -2% the majority of the configurations show an AFM ground state and for the remaining cases the two magnetic orders are very close in energy. The preference for the AFM, instead, decreases for larger compressive strain due to the increased stability of the FM phase in compressively strained stoichiometric SMO films 34 .
We now consider the polarization in Fe-doped oxygen deficient SMO by starting from the unstrained geometries. Polarization can arise due to symmetry reduction by the defect pair but also due to the formation of a defect dipole ( D): the spatially separated substitutional Fe ' Mn (negatively charged) and V O •• (positively charged) result in the charge center being offset from the geometry center of the lattice and hence an electric dipole moment along the direction from Fe Mn to V O 13 . As detailed in SI Sec. S1, the situation is further complicated in the AFM phase by the reduction of one Mn ion (Mn ' Mn ) adjacent to V O , which induces an additional defect dipole ( D ) from the negatively charged reduced Mn ' Mn to the positively charged V O •• . For simplicity, we will concentrate the following discussion mainly on results obtained for the AFM order. Indeed, the metallic nature of the FM phase and the consequent partial reduction of more than one Mn (cf. SI Sec. S1) result in a more complex behavior, which may not be properly described within the simple approach we use to estimate the polarization based on nominal charges (see SI Sec. S4). Results for the FM phase are reported in SI Sec. S4 B The computed polarization P is roughly aligned with the vector sum D+ D = D tot , forming with D tot an angle ranging from about 6 • to 50 • for the different configurations as shown in Fig. 6a. The alignment between D tot and P depends on the relative geometric arrangement of Fe ' Mn , Mn ' Mn , and V O . In particular, due to geometric constraints, the angle between D and D can either be around 60 • (see Fig. 6b), 120 • (see Fig. 6c), or 180 • (see Fig. 6d). The smaller this angle, the stronger the combination of the two dipoles and consequently the larger the alignment between P and D tot (see Fig. 6a and also SI Sec. S4). For example, for NN Fe Mn -V O configurations, the Fe ' Mn and Mn ' Mn ions are located at the two sites adjacent to V O , resulting in antiparallel D and D dipoles (see Fig. 6d). As a result of this peculiar arrangement of the defect dipoles and of the smaller distance between the substitutional iron and the vacancy, NN Fe Mn -V O configurations are associated with the smallest total polarization (P tot ) of about 2 µC/cm 2 (see Fig. 7  a). Unsurprisingly, P tot slightly increases with increasing Fe Mn -V O distance up to 15-20 µC/cm 2 (see Fig. 7a). These polarizations are of similar magnitude than the ones in conventional ferroelectrics such as BaTiO 3 46 . The in-plane component of the polarization (P ac , see Fig. 7b) These results suggest that the mechanism underlying the observed polarization in unstrained SMO is the off-centering of the charge due to the separation of the Fe ' Mn /Mn ' Mn and the V O •• defects.

D. Polarization and defect concentration
In order to further investigate the mechanism underlying the polarization induced by the defect pair, we now examine how the defect concentration impacts the polarization of Fe-doped oxygen-deficient SMO. SI Fig. S12 shows that the polarization decreases with increasing cell size (i.e. with decreasing the defect concentration). This suggests the polarization to originate from a local change in symmetry around the defect pair due to the defect dipoles. Indeed, Fe Mn -V O defects induce small displacements of the atoms in the vicinity of the defect pair from their high-symmetry positions. For example, Fig. 8 shows the Mn displacements along the b-axis (with the largest polarization component) for the defect configuration considered in SI Fig. S12 and computed in supercells of different size. These off-centerings have been computed excluding the two Mn adjacent to the oxygen vacancy to avoid artifacts due to the relaxation of these undercoordinated sites. In general, the larger the polarization in SI Fig. S12, the larger the Mn off-centerings. More importantly, the Mn displacements are generally larger and constant for sites lying within 6Å from the defect pair and decrease afterwards, pointing to a spatially limited effect of the defect pair.
High defect concentrations may thus promote macroscopic polarization but the possibility of different orientations of neighboring defect dipoles should be taken into account. For this reason, we performed additional calculations in a 4 × 2 × 2 supercell containing two Fe Mn -V O defect pairs and investigated the cases in which the Fe Mn -V O defect dipoles lie parallel or anti-parallel to one another. Due to the importance of elastic effects for defects in close proximity 5,47 , both atomic positions and lattice parameters were allowed to relax in these calculations. Fig. 9 schematically illustrates the structures and orientation of the defect dipoles for the most stable NNN Fe Mn -V O OP configuration in which the defect dipoles lie mainly in the ac-plane (Figs. 9a and b) or along the b-axis (Figs. 9c and d)

E. Interplay between polarization and strain
It is established that epitaxial strain imposed for example by lattice matching with a substrate during coherent epitaxial growth of thin films breaks the symmetry and affects competing energy contributions. Hence, it constitutes a viable strategy to induce ferroelectric properties in non-polar oxides 8,12 . In this section, we discuss how the Fe Mn -V O defect chemistry interacts with strain and with the magnetic properties in determining the polar properties of SMO thin films.
For tensile strained AFM Fe-doped oxygen-deficient SMO, we observe a general increase of the in-plane components of the polarization (P a and P c , see at +4% strain along the a-and c-axis, respectively (see Fig. 11a, b and e, f). Conversely, compressive strain is associated with a increase (up to 3 µC/cm 2 , see Fig. 10c and d) of the out-of-plane component of the polarization associated with Mn off-centerings of about 0.03-0.1Å along the b-axis (see Fig. 11c and d) already for -4% strain. Interestingly, this strain is smaller than the -6% predicted necessary to destabilize the polar out-of-plane phonon mode in AFM SMO 17 . Indeed, the larger Mn displacements computed in presence of defect pairs compared to the stoichiometric case (see Fig. 11) in a strain range between -4% and 2% strain further highlight the ability of defect pairs to favor the polar phase transition.
Interestingly, in the FM phase, polarization was found to be roughly constant as a function of the applied strain (cf. SI Fig. S13), between -2 and 4 % strain with small Mn off-centerings of the same magnitude as in the unstrained structure (cf. SI Fig. S14). A smaller increase of P b /P ac and of the Mn off-centerings along b-axis/acplane starting for strain of about -4%/6% are observed also in the FM phase. The different behavior of the FM phase can be explained by the larger electronic screening of the defect dipole in the metallic FM phase and the strain dependence of the polar modes in stoichiometric SMO, where the IP modes soften only for large tensile strains beyond 6% and the OP mode becomes unstable at 2% compressive strain 17 .
In summary, not only do these results suggest that doping SMO thin films with Fe can reverse the suppression of the ferroelectricity by oxygen vacancies generally present in the samples 9 , but also that Fe Mn -V O defect pairs can, depending on the magnetic order, couple with strain to favor the polar phase transition.

IV. CONCLUSIONS
In the present work we used DFT+U SC-SD calculations to investigate the potential of inducing ferroelectricity in SrMnO 3 (SMO) through polar defect pairs, formed by a substitutional Fe atom and an oxygen vacancy. We further studied the interplay of these defect pairs with epitaxial strain and the magnetic phase. DFT+U SC-SD is fundamental to describe electronic-structure changes upon defect formation and to reconcile predicted magnetic properties with the available experimental data.
Our results suggest that defect engineering via polar defect pairs constitutes a parameter to design multiferroic materials. Ferroelectricity in nominally non-polar SMO can arise due to an off-center displacement of the defect charge resulting in a net electric dipole moment along the direction from the negatively charged substitutional Fe ' Mn and Mn ' Mn sites to the positively charged V O . Furthermore, the defect pairs lead to a small offcentering of the Mn atoms in the defect neighborhood from their high-symmetry positions.
Epitaxial strain strain can couple with Fe Mn -V O not only reducing the defect formation energy (and hence increasing the defect concentration) under tensile strain, but more importantly inducing polarity either in-plane or out-of-plane for tensile and compressive strain respectively, already for strains smaller than those required to induce a polar phase transition in the defect-free material.
These results and the fact that local defect-induced dipoles couple in a parallel fashion, establishes polar defect pairs as a promising route to engineer ferroelectricity in nominally non-polar transition metal oxides. Defect formation in transition-metal materials can result in local perturbations of the chemical environment of Hubbard sites around the defect, upon which the Hubbard parameters physically depend. For this reason, we recently proposed a self-consistent and site-dependent DFT+U SC-SD approach in which the U values are computed for all inequivalent Hubbard sites 1 . DFT+U SC-SD was found to be promising to predict properties of defective systems, in particular when excess-charge localization is restricted to atoms around the defect site and can be properly captured by site-dependent U values. For example, in the case of V O in SMO, U values were found to depend on the distance of the Hubbard site from the defect, its coordination number, its oxidation state, and on the magnetic order of the host material 1 .

AFM SMO
Results for AFM SMO suggest that the distance from the V O has the strongest impact on the computed U values (cf. Fig. S1a). In fact, we observe deviations from the self-consistent U computed for the stoichiometric cell (U SC ), mainly for Mn atoms adjacent to V O at a distance of about 1.90Å, while Mn sites at larger distances recover U SC (the deviations are as small as 0.02 eV). For NNN and NNNN configurations, the two Mn sites adjacent to the vacancy show different behaviors. The Mn atom farthest from the Fe Mn defect show U values slightly higher than U SC (+0.08 eV) with a behavior similar to the one of the Mn sites adjacent to a doubly charged V O x in SMO, which can be explained in terms of a change in coordination number 1  We determine the oxidation state according to the method of Ref. 2 which is based on the occupation matrix of the d orbitals of each Hubbard atom that is available when performing DFT+U calculations. Each d orbital is considered to be fully occupied if the corresponding matrix element is closer to unity than a given threshold. In the AFM phase, we use a threshold of 0.9 for the occupation. These oxidation states confirm the above picture (see Fig. S4) where only one of the Mn sites in nearest-neighbor position to the oxygen vacancy is reduced to Mn 3+ , in line with results of Mössbauer spectroscopy suggesting a partial reduction of the Mn atoms in SrMn 1x Fe x O 3 -δ samples, presumably in the vicinity of the vacant anion 3 . The Fe ion which is always in the Fe 3+ charge state, independently of is position relatively to the V O .

FM SMO
For the FM phase (cf. Figs. S1b and S5), the U SC-SD values deviate less from U SC than for AFM SMO. Interestingly, all the Mn adjacent to the V O show the same behavior with a small reduction of their U SC-SD value with respect to U SC , suggesting a partial reduction of these sites (Mn (3+δ)+ ). These observations can be explained considering the metallic nature of FM SMO, in which one expects the defect state and the corresponding changes in the local chemical environment to be more delocalized over the whole structure, compared to the semiconducting AFM phase, where a more confined impact of the defect on the local chemical environment was already observed in the case of oxygen-deficient SMO 1 .
These observations of a partial reduction of multiple Mn sites accompanying the Fe 3+ formation is again supported by the determined oxidation states according to the method of Ref. 2 . As opposed to the AFM phase, we use a lower threshold of about 0.7 for the FM order, to be able to discern one (for configurations in which the V O is adjacent to one Mn and to the Fe ion) or two (in all the other cases) reduced Mn atoms as shown in Fig. S6. We also determined the U value on the Fe using the SC-SD approach as shown in Fig. S7 for each configuration as a function of the Fe Mn -V O distance. The horizontal dashed lines in Fig. S7 indicate the U SC value computed for iron in LaFeO 3 , a perovskite material where Fe has the same octahedral coordination environment and the same oxidation state (Fe 3+ ) as in Fe-doped SMO. Unsurprisingly, larger deviations from these U SC values are generally observed for configurations (in violet in Fig. S7) in which the Fe Mn is nearest-neighbor position to the V O and in the insulating AFM phase. The larger deviation of the U value for the Fe in the NNNN Fe Mn -V O IP observed only in the AFM case can be explained considering the interaction of the Fe atoms in these configurations with the V O in the neighboring cell, which is weaker in the more screened FM phase.  Figure S9 shows that for DFT+U SC , in which the global U SC computed for stoichiometric bulk SMO in the corresponding magnetic phase is applied on all the Mn sites, all the considered Fe Mn -V O configurations are more stable in the FM phase and that the stability of the FM order increases with increasing distance of Fe Mn from V O . Indeed, the increasing distance between the Fe 3+ and the reduced Mn 3+ ions promotes the ferromagnetic Mn 3+ /Mn 4+ interactions.

S3. MAGNETIC ORDER IN DFT+U SC
FIG. S9. Total energy differences (∆E(FM-AFM)) per formula unit between the defective cells with FM and AFM order computed with the DFT+USC approach. AFM is more stable for positive and FM for negative differences. ∆E(FM-AFM) is reported with respect to the the FeMn -VO distance in each defective configuration in the unstrained SMO structure. Circle and square symbols refer to data obtained for VOO OP and VOO IP , respectively. See color code in Fig. 2 of the main text.

S4. DEFECT-INDUCED POLARIZATION
A. Polarization in the unstrained AFM phase Figure S10(a) shows that the angle between the polarization P and the Fe ' Mn , and V O sites, the stronger the coupling between D and D and consequently the larger the deviation of P from D. Indeed, the polarization is mainly aligned along the direction of the vector sum of the D and D dipoles ( D tot ) as shown by the smaller angles between D tot and P (see Figure S10(b)).

FIG. S10. Angle between the (a) Fe '
Mn -VO •• defect dipole D or (b) the total defect dipole Dtot and the polarization P as a function of the distance between FeMn and VO for the different configurations in the AFM phase. Circle and square symbols refer to data obtained for VO OP and VO IP , respectively. See Fig. 2 in the main text for the color code.

B. Polarization in the unstrained FM phase
The polarization for the FM phase was computed using the nominal charges of +2 for Sr, -2 for O, +3 for Fe, and +4 or +3 for stoichiometric-like or reduced Mn sites, even though results of Sec. S1 A 2 indicated that, due to the metallic nature of this phase, the reduction of the Mn sites is only partial. While NN defect configurations, with only one reduced Mn site adjacent to the V O , exhibit similar or slightly higher total polarization ( P tot ) compared to the AFM phase, the NNN and NNNN Fe Mn -V O defect pairs show almost the same P tot , regardless of the Fe Mn -V O distance, contrarily to the increase of P tot with increasing Fe Mn -V O distance reported for the AFM phase (cf. Fig. 7 in the main text). This behavior can be explained considering that for these configurations the polarization is computed assuming the presence of two Mn As shown in Fig. S12, we generally observe a reduction of the polarization with increasing cell size, deviations like in the 80-atom cell likely being associated with cell-anisotropy effects. This suggests a local polarizing effect that becomes less important as the cell size increases and hence the defect-pair concentration decreases. The strain dependent polarization of Fe-doped oxygen deficient SMO is the result of a complex interplay between defect chemistry and electronic and magnetic degrees of freedom. While for the insulator AFM phase, polarization can be enhanced by strain, in the FM phase the polarization is fairly constant with respect to the applied epitaxial strain (see Fig. S13 and the average Mn off-centerings remain very small (about 0.02-0.04Å in the strain range between -2 and 4%, see Fig. S14) similar in magnitude to the unstrained structure. Interestingly, compressive strain values of 4% can result in an enhancement of the P b component, as confirmed also by the larger Mn off-centerings observed at this strain along the b-axis. Also at 6% tensile strain an increase of the components of the polarization and of the Mn off-centerings in the ac-plane are observed. This behavior can be understood considering the larger electronic screening of the defect dipole in the metallic FM phase, resulting in the lower sensitivity of the polarization of the defective cells to the applied strain and evolution with strain of the frequency of the polar modes in the stoichiometric FM order, where the IP modes soften only for large tensile strain of 6% and the OP modes becomes slightly unstable at 2% compressive strain 5 .