Persistent half-metallic ferromagnetism in a (111)-oriented manganite superlattice

Heterostructures of mixed-valence manganites are still under intense scrutiny, due to the occurrence of exotic quantum phenomena linked to electronic correlation and interfacial composition. For instance, if two anti-ferromagnetic insulators as LaMnO$_3$ and SrMnO$_3$ are grown in a (001)-oriented superlattice, a half-metallic ferromagnet may form, provided that the thickness is sufficiently small to allow tunneling across interfaces. In this article, we employ electronic structure calculations to show that all the layers of a (111)-oriented LaMnO$_3$|SrMnO$_3$ superlattice retain a half-metallic ferromagnetic character for a much larger thickness than in its (001) counterpart. This behavior is shown to be linked to the charge transfer across the interface, favored by the octahedral connectivity between the layers. This also results in a symmetry-induced quenching of the Jahn-Teller distortions, which are replaced by breathing modes. The latter are coupled to charge and spin oscillations, whose components have a pure e g character. Most interestingly, the magnetization reaches its maximum value inside the LaMnO$_3$ region and not at the interface, which is fundamentally different from what observed for the (001) orientation. The analysis of the inter-atomic exchange coupling shows that the magnetic order arises from the double-exchange mechanism, despite competing interactions inside the SrMnO$_3$ region. Finally, the van Vleck distortions and the spin oscillations are found to be crucially affected by the variation of Hund's exchange and charge doping, which allows us to speculate that our system behaves as a Hund's metal, creating an interesting connection between manganites and nickelates.

Much research on interfaces and surfaces has been focused on mixed-valence manganites.Bulk materials are primarily known for their colossal magnetoresistance [41], which is favoured by a disordered solid mixture of Mn +3 and Mn +4 ions [42][43][44].In superlattices, however, non trivial phenomena may be observed and ordered hetero-valent ions do not prevent the emergence of a large magnetoresistance [45].Particularly studied are (LaMnO 3 ) n |(SrMnO 3 ) m superlattices, where various magnetic and electronic ground states across the metal-insulator transition can be tuned [10,[46][47][48][49][50].This is achieved by varying their period (n + m) or component ratio (n/m) [47,48,51], which is a way to modulate the tunnelling of e g electrons across the interface [48]; orbital [52] and charge [53] order were also reported.More exotic phenomena, including correlated topological states, may be expected when passing from the (001) orientation to the (111) orientation, because of the large polarity and a peculiar symmetry-driven epitaxial strain [54,55].This is the reason why (111)-oriented superlattices remain under intense scrutiny, despite of the scarcity of suitable substrates and unfavourable thermodynamics [55][56][57].
In this context, the current article presents an ab-initio study on the structural, electronic and magnetic properties of a (111)-oriented (LaMnO 3 ) 12 |(SrMnO 3 ) 6 superlattice as illustrated in Fig. 1(a), which is isostochiometric to the colossal magnetoresistive La 2/3 Sr 1/3 MnO 3 .Our calculations will show that this superlattice has a half-metallic ferromagnetic (FM) ground state, whose character persists inside the innermost layers of the component regions.This behaviour is profoundly different from what reported for other orientations and is traced back to the cooperation of charge transfer across the interface, strain, structural distortions and electronic correlations.Magnetism will be shown to originate from a double-exchange interaction between the Mn atoms and to be pinned inside the LaMnO 3 region and not at the interface.Finally, the Mott-Hund character of the electronic correlations will also be analysed.

II. RESULTS AND DISCUSSION
Ground-state structure and spectral properties.The (LaMnO 3 ) 12 |(SrMnO 3 ) 6 superlattice is investigated via density functional theory (DFT) plus Hubbard U approach, labelled as sDFT+U.While tilting systems and angles are known for many perovskites in the bulk, their determination at surfaces and interfaces is not trivial.A previous ab-initio study [54] predicted that (111)-oriented manganite superlattices should adopt the a − a − a − tilting system instead of the native a − a − c + tilting system, as expressed in Glazer notation [58], used throughout this paper.The energy difference between these two structures depends on the lattice parameters.With a lattice constant of 3.860 A (hereby denoted as equilibrium or 0 % strain), the a − a − c + tilting system is favoured by 7.6 meV per formula unit (f.u., i.e. a AMnO 3 unit, where A may be La or Sr) with respect to the a − a − a − tilting system.A reasonably small compressive strain may change this structural order, as recently reported for nickelates [59].As shown in Fig. 1(b), we predict a transition to the a − a − a − tilting system for a compressive strain of ∼ 1.5 %, corresponding to epitaxial growth on SrMnO 3 (on LaAlO 3 the strain is ∼ 2 %).The energy difference between the two tilting systems remains below 7 meV (∼80 K) at all simulated strain values, whereas that between the ground state FM and anti-ferromagnetic (AFM) orders are larger than 25 meV (∼290 K), see Table I.Therefore, phase transitions and coexistence are more likely to occur in the structure than in the magnetic order.
The two tilting systems do not lead to qualitatively different results in terms of distribution of charges and magnetic moments, but the a − a − c + requires a large supercell, which may hinder a thorough analysis of the layer-resolved properties.Therefore, we focus on the a − a − a − tilting system at the equilibrium lattice constant, for simplicity.
The superlattice analysed has a half-metallic character, as inferred by the total den-sity of electronic states (DOS) in Fig. 1(d), with a band-gap in the minority-spin channel between Mn-d and O-p of ∼2.0 eV.The site-projected DOS in Fig. 1(c) reveals that the half-metallicity persists across all the layers.The curves show that the deeper the Mn inside the LaMnO 3 region, the lower the onset of the e g bands, because of the larger electronegativity of La with respect to Sr.The smooth variation of these onsets across the superlattice is a further signature of metallicity.Moreover, the e g states are well separated from the t 2g states in the LaMnO 3 region (see layers 5 and 6), whereas they are much closer in the SrMnO 3 region.Their separation is small also at the interface, note the larger upshift of the t 2g states with respect to the e g states in Fig. 1(b).
The features just outlined are unusual for manganite superlattices with this large thickness.For the (001) orientation, the superlattice becomes insulating if LaMnO 3 is thicker than 2 layers [48,53,60], whereas it is 12-layers thick in the present case.To understand the difference between the two orientations we should first understand what drives the formation of a half-metallic FM state.We recall that bulk LaMnO 3 (characterised by the a − a − c + tilting system) becomes half-metallic FM under a small compressive strain [61,62].
As discussed above, SrMnO 3 causes a small compressive strain on LaMnO 3 , and this should be sufficient to induce the transition.Interestingly, this situation is even more favourable in the a − a − a − tilting system, which is predicted by our calculations to be a FM half-metal for all in-plane lattice parameters hereby considered.
Once established that strain is the driving factor in determining the half-metallic FM state in the LaMnO 3 region, we need to understand why this state is more likely to survive in the (111)-oriented superlattice than in its (001)-oriented counterpart.Clarifying this issue requires a deeper analysis of the structural features and the magnetic properties, presented in the next sections.
Breathing distortions and spin-charge oscillations.As for most perovskites, the structural features are linked to the electronic and magnetic properties.In agreement with leading literature [63,64], we introduce the Jahn-Teller distortions and breathing distortions in terms of the variations of the octahedral lengths x, y, z (with respect to their average values).Breathing distortions are defined as Q R 1 = (∆x+∆y+∆z)/ √ 3, whereas the Jahn-Teller Breathing distortions are seldom found in manganites, which host orbital order and Jahn-Teller distortions (also defined in the Methods) instead.A recent study highlighted that Jahn-Teller modes arise from a steric effect that affects the electron-lattice coupling and are therefore dependent on the tilting system [65,66].In the bulk, the constraint imposed by the R 3c phase should lead to a total quenching of these (pseudo) Jahn-Teller modes.The Jahn-Teller distortions are shown in Fig. 2(a) and are quenched in agreement with the aforementioned literature.The quenching is not full because the relaxation of the superlattice modifies the pristine a − a − a − tilting pattern.
The quenching of the Jahn-Teller modes is accompanied by the presence of the breathing modes.The latter are lessened by a factor 4 in the structure without tilts (data not shown).
A similar relation between octahedral tilts and breathing distortions was recently found in rare-earth nickelates, where it leads to a structurally triggered metal-insulator transition [67].In addition, LaMnO 3 is mentioned as a case where a close competition between charge and orbital order is driven by a similar mechanism (in line with Refs.63 and [68]).In Exchange couplings and ferromagnetic order.As we argued that the magnetism is a consequence of structure and strain, we do not expect it to be interfacially driven.This is clearly visible in Fig. 2(b), where the largest magnetization is found in the innermost layers of the LaMnO 3 region.For a better insight into the magnetic properties, we analyse the interatomic exchange coupling, computed for a lattice constant of 3.892 A (which corresponds to a tensile strain below 1%).The largest contributions in the Mn sublattice are those connecting a Mn atom to its first nearest neighbours or fourth-nearest neighbours.The latter correspond to the second-nearest neighbours along Mn-O-Mn-• • • lines, consistently with the double-exchange mechanism [48,60].The relevant exchange couplings across the superlattice are illustrated in Fig. 4. In the LaMnO 3 region, the magnetic order is driven by the FM nearest neighbour coupling, which in the innermost layer takes the value of 17.7 meV and sharply decreases at the interface, exhibiting oscillations in phase with the magnetic moments.Interestingly, the maximum value is not reached at the innermost layer, but at intermediate layers, and amounts to 18.4 meV, which is 30% smaller than in the isostochiometric La 2/3 Sr 1/3 MnO 3 [75].This behaviour reflects a competition between the trend of the magnetization -see Fig. surprising if compared to the behaviour of (001)-oriented supercells, whose nearest neighbour exchange becomes bulk-like AFM for LaMnO 3 regions thicker than 2 unit cells [48,60].A smaller contribution to the magnetic order is given by the fourth-nearest neighbour exchange, whose values are noticeable at the interface (1.36 meV), but are totally quenched in the innermost layers of the LaMnO 3 region.
The situation is more complicated in the SrMnO 3 region.In the innermost layers, the nearest neighbour exchange is AFM, as in the bulk [76].However, The strength of the coupling is much weaker than in the bulk, i.e. −1.6 meV versus −7.5 meV [77], due to the combined effect of charge transfer and epitaxial strain (about 1%).Strain alone was shown to induce an AFM-FM transition at about 3% in bulk cubic SrMnO 3 [77] -while here the strain is virtually null on SrMnO 3 .Interestingly, the FM order inside the SrMnO 3 region is stabilised by the fourth-nearest neighbour coupling, which becomes even larger (1.8 meV) than the nearest neighbour one.This frustration due to competing FM and AFM interactions is likely to lead to a more complex magnetic structure, probably accompanied by non-collinearity.Exploring the magnetic phase diagram may be an interesting project, but outside the scope of the present work.We prefer, instead, to focus on the origin of the oscillations of charge, magnetic moments, exchange couplings and breathing distortions.
We can summarise what leads to the half-metallic FM state with the help of Fig. 5.
In the bulk, LaMnO 3 and SrMnO 3 behave as an AFM Mott insulator with Mn 3+ ions and an AFM band insulator with Mn +4 ions, respectively.In the superlattice, the local strain in the LaMnO 3 region induces the delocalisation of the Mn-3d states, which in turn suppresses the AFM super-exchange and favours the FM double-exchange [60,61].This effect is further enhanced by the charge transfer across the interface, which penalises the ionic picture and promotes the hopping between Mn sites.In (001)-oriented superlattices, the local strain is imposed in-plane -hence along two crystallographic directions -and allows different relaxations in different regions: SrMnO 3 recovers its G-type AFM order, blocking the tunnelling of e g electrons from LaMnO 3 and imposing a strong penalty on the double-exchange mechanism.For the (111) orientation, the strain acts on the same footing for all octahedral axes, and therefore the aforementioned phase separation is forbidden, the e g tunnelling survives and the FM coupling prevails.In summary, geometrical degrees of freedom affect the electronic ones, governing the magnetic and metallic properties of the superlattice.Further information can be inferred by the analysis of the bond angles, shown in Fig. 3.In the LaMnO 3 region, the Mn-O-Mn angles vary from 160°to 165°.These values are higher than the bulk LaMnO 3 [78] value of 155°and close to the La 2/3 Sr 1/3 MnO 3 value [79] of 165°.The analysis of the Mn-O bond lengths, see Fig. 3, is in line with the presence of breathing modes in the LaMnO 3 , but also with a tendency to ferroelasticity in SrMnO 3 , avoided by the equivalence of the interfaces.Such behaviour is typical of SrMnO 3 [74].
Charge doping and metallicity would anyway prevent the transition to a ferroelectric phase.
Role of Hund's coupling.The suppression of Jahn-Teller order in favour of breathing distortions was predicted a decade ago in nickelates as a consequence of Hund's coupling and was pointed out to be persistent well into the metallic side of the Mott transition [80].Later studies [67] pointed to structural distortions as the driving mechanism.In the present case, we observe an intermediate situation.On one hand, the structure has a primary importance, forbidding certain distortions, such as Jahn-Teller, and preserving metallicity.On the other hand, the breathing distortions are not large enough to induce a metal-insulator transition, but we still observe signs of charge order.The dependence of our results on the strength of Hund's exchange J is investigated via charge density functional plus U scheme (denoted as cDFT+U|J hereafter) [81,82] for a lattice constant of 3.892 A. It suggests to what extent Hund's coupling affects the properties of our system.The oscillations of the magnetic moments across the superlattice depend crucially on the value of Hund's coupling, see Fig. emphasising that the former drives the latter, to a large extent.Furthermore, electron/hole doping may lead to the disappearance of the breathing distortions as well as the magnetic moments oscillations (data not shown).Such a drastic change is surprising, considering the shape of the density of states in the corresponding doping range (±0.1 eV), and suggests that strong electronic correlations play an important role.Overall, the metallic character with spin and charge oscillations, the presence of strong correlation effects, and the key role of the Hund's exchange J suggest that our superlattice behaves as a Hund's metal [83,84].
Using our parameters U and J, as well as the effective bandwidth extracted from the DOS, we can obtain some information from existing phase diagrams of the Hubbard model [85][86][87].
The most accurate comparison is offered by Ref. 87, where Merkel et al. investigated a 5 orbital system with a d 4 occupation, including the level splittings associated to the presence of breathing modes (representative of CaFeO3).Using their phase diagram, we can confirm the regime of (homogeneous) Hund's metallicity, with an estimated quasiparticle weight between 0.4 and 0.6.By increasing J, we expect to get closer to a valence skipping metal phase, which finds correspondence in an increasing amplitude of charge and spin oscillations (see inset of Fig. 6(b)).A further increase of J would lead to a charge disproportionated insulator, which in Ref. 87 is predicted to happen for values larger than 2.3 eV.Despite signs of "Hundness" have been found in a variety of systems during the last decade [83,84], they had never been reported for a manganite superlattice.For a more quantitative analysis allowing for more precise conclusions, one would need to perform calculations beyond DFT, e.g. in combination with the dynamical mean-field theory (DMFT), including the structural response.Considering the size of the system as well as the very high number of degrees of freedom, such an analysis would be beyond the scope of the present study.
Outlook.In summary, we have investigated the electronic and magnetic properties of the (111)-oriented (LaMnO 3 ) 12 |(SrMnO 3 ) 6 superlattice using DFT+U|J.A half-metallic FM state is supported by the cooperation of charge, spin, orbital and lattice degrees of freedom, and is favoured with respect to an AFM state.The half-metallic FM character is found to persist across the entire superlattice, while the innermost layers of its (001)-oriented counterpart become AFM insulators for a thickness larger than 3 layers.The atomic volumes, charges and magnetic moments are correlated across the superlattice, in particular in the LaMnO 3 region, where adjacent sites display charge, spin, and volume oscillations.Breathing distortions arise and may be accompanied by the quenching of Jahn-Teller distortions in the presence of a − a − a − tilting of MnO 6 octahedra while may coexist with the Jahn-Teller distortions in other tilting systems such as a − a − c + .Overall, the present results suggest that the [111] epitaxial strain associated to the superlattice formation is a viable pathway to engineer a system analogous to La 2/3 Sr 1/3 MnO 3 without introducing doping-induced disorder.This is not only an advantage for avoiding alloy-related problems, as e.g.cation disorder, but does also widen the potential of interfacial engineering in oxide heterostructures.Finally, we also speculate that the fascinating physics exhibited by this superlattice may arise from Hundness, alongside with structural aspects, similarly to what happens in nickelates.Therefore, the system addressed here can provide further theoretical ground for the development of heterostructures hosting exotic magnetic phases and topological states.

III. METHODS
Framework and parameters.The superlattice is fully relaxed with an optimised lattice constant of 3.860 A (denoted as equilibrium or 0 % strain).Calculations are performed in density functional theory (DFT) using the projector-augmented wave method as implemented in the Vienna Ab-initio Simulation Package (VASP) [88,89].The exchangecorrelation functional is treated in the generalised gradient approximation (GGA) by Perdew-Burke-Ernzerhof [90,91].To improve the description of the Mn-3d states [92], we make use of on-site corrections for static correlation effects in the rotational invariant DFT+U approach by Liechtenstein et al. [93], denoted as sDFT+U|J.Further calculations to analyse Hund's coupling are performed by following the approach discussed in Ref. 94, which we label as cDFT+U|J.The Coulomb interaction parameters are chosen as U = 3.8 eV and J = 1.0 eV, in accordance with works on similar systems [95,96].A deeper analysis of the magnetic properties is then performed via the full-potential linear muffin-tin orbital (FP-LMTO) method as implemented in the RSPt code [97][98][99].We used also the SCAN parameter-free functional [100]: results (shown in the Supplemental Material) confirm the core results obtained with DFT+U|J.
The inter-atomic exchange interactions J ij are calculated by mapping the magnetic excitations onto an effective Heisenberg Hamiltonian Ĥ = − i =j J ij • ( e i • e j ), where i, j are atomic sites and e i , e j are unit vectors along the local magnetization direction.This calculation is performed via the magnetic force theorem, using the implementation of Ref.
101, which was also successfully applied to CaMnO3 [21].Due to the better accuracy of all-electron methods [102], these calculations also serve to confirm the validity of VASP results.
General considerations on the modelled system.The supercells used for the calculations consist of two types: the R 3c cell was used for the results presented throughout this work, whereas an orthorhombic cell was used for calculations to compare total energies and results on the charge and spin distributions.The two interfaces have the same stacking sequence (i.e.LaO 3 |Mn|SrO 3 or SrO 3 |Mn|LaO 3 ), thus they are equivalent; furthermore, the superlattice possess inversion symmetry with respect to the Mn atom at the interface and therefore there is no built-in electric field generated.Further details on the cells and the sampling of the Brillouin zones are given in the Supplemental Material.
The analysis on the on-site charges and magnetic moments is carried on by in agreement with state-of-the-art methods [69][70][71][72][73].The analysis of the electronic properties is performed with the aid of the post-processing code VASPKIT [103].Finally, images of structures and charge/spin distributions are produced with VESTA JP-Minerals [104].Further details on the calculations are given in the Supplemental Material.

IV. CODE AVAILABILITY
The calculations for this work have been performed with VASP and RSPt.The former is a licence product from the University of Vienna; the licence can be obtained upon submitting an application through the vasp portal (http://www.vasp.at).The latter is a free software distributed under GPL license after registration to a mailing list.More information about RSPt can be found at http://fplmto-rspt.org/ [98].
the superlattice under investigation, this mechanism has to compete with the high stability of the FM half-metallic phase, associated to the strained structure, and with the uniform shift of the band-edge, induced by the charge transfer -see again Fig.1(b).Therefore, it becomes unfavourable to induce a transition to an AFM insulator with orbital order.The site-projected charge and magnetic moment distributions, as computed à la Bader[69][70][71][72][73], is presented in Fig.2(b) and shows a hint of charge order, leading to oscillations in the LaMnO 3 region and a smooth behaviour in the SrMnO 3 region.The smooth variation of the Mn-O-Mn angles across the superlattice (which take approximately the same values for all layers in the central LaMnO 3 region) mirrors the uniform shift of the band-edge, compare Figs.3(b) and 1(b), whereas the Mn-O distances appear with oscillations in the inner LaMnO 3 region, see Fig.3(a).Where the Mn-O-Mn bond are closer to a flat angle (the SrMnO 3 region) the structure presents larger splitting in the Mn-O distances between increasing ẑ and decreasing ẑ, and where the Mn-O-Mn bonds are more bent (the LaMnO 3 region) the structure presents oscillating Mn-O distances and breathing modes, see Figs.2 and 3.In fact, the connection between breathing distortions and angles is even more apparent in the inset of Fig.3(b), where flat angles correspond to single values for the MnO 6 volumes (breathing distortion) and bent angles -in the LaMnO 3 regioncorrespond to large variance of the MnO 6 volumes distribution, in line with the above mentioned literature[67].The large Mn-O distance splitting, occurring in the SrMnO 3 region, reveals a typical tendency for SrMnO 3 to ferroelasticity[74], avoided by the symmetry with respect to the interface.
2(b) -and the potential induced by the charge transfer across the interface -see Fig. 1(b).Such relatively strong ferromagnetism is even more

6
(b).When J becomes as small as 0.6 eV, the magnetism is no longer pinned at the innermost layers of the LaMnO 3 region.Instead, it becomes pinned at the interface, similarly to what happens in (001)-oriented superlattices.The change in the trend of the magnetic moments across the superlattice is accompanied by an analogous change of the breathing distortions,

FIG. 1 .
FIG. 1.(a) Crystal structure of the superlattice.The interfacial layers are #0 and #12.(b) Curve total energy vs strain for FM solutions with a − a − c + (black) and a − a − a − (red); 'zero' strain means that the pseudo-cubic lattice constant is 3.860 A. (c) Projected DOS for Mn-t 2g and Mn-e g states for selected layers; additional information, covering all Mn and O layers, is provided in the Supplemental Material.(d) Total, Mn-d projected and O-p projected DOS for the FM solution.

FIG. 2 .FIG. 5 .
FIG. 2. (a) Layer-resolved van Vleck distortions.(b) Mn-projected Bader charges and magnetic moments.The dashed vertical line indicates the interfacial layer.The charge transfer across the interface is visible in having Bader charges larger (lower) than for the nominal oxidation state, i.e. 13 (14) in the SrMnO3 (LaMnO3) region.A more direct visualization of the structure with respect to the layers numbering is provided in Fig. 1.

FIG. 6 .
FIG.6.(a) Layer-resolved van Vleck distortions and (b) Mn-projected Bader magnetic moments for different values of Hund's exchange J (in eV).See the Methods section for details on cDFT+U|J and sDFT+U|J.Inset: difference ∆µ between the magnetic moments at layers #5 and #6, which provides a measure of the amplitude of the moment oscillations.