Julien Varignon

Manuel C Bibes

Alex Zunger

Mott gapping in 3 d A

come

I. Introduction

Transition metal oxide perovskites ABO3 with a 3d transition metal atom substituting the B site exhibit intriguing metal vs insulator characteristics as well as different forms of magnetism across the series both in their high-temperature (HT) spin-disordered paramagnetic (PM) phases and/or in the low-temperature (LT) spin-ordered phases [START_REF] Khomskii | Transition Metal Compounds[END_REF] . Some of the compounds are metallic (CaVO3, SrVO3 or LaNiO3 for instance), while others are insulating, such as titanates RTiO3 (d 1 ), vanadates RVO3 (d 2 ), manganites CaMnO3 (d 3 ) and RMnO3 (d 4 ), ferrites CaFeO3 (d 4 ) and RFeO3 (d 5 ), cobaltites RCoO3 (d 6 ), nickelates RNiO3 (d 7 ) or possibly cuprates RCuO3 (d [START_REF] Hubbard | [END_REF] ), where R is a rare-earth element or yttrium. Concomitantly with the opening of a band gap, one observes a variety of systematic symmetry-breaking modes, such as the Jahn-Teller (JT) distortions in RVO3 2 or RMnO3 3 compounds, propagating either in-phase (Q2 + mode) or in anti-phase (Q2 -mode), or the B-O bond disproportionation Boc observed in the insulating phase of RNiO3 4 and CaFeO3 5 (see sketches in Figure 1). Understanding and controlling such band gaps and the associated lattice distortions and forms of magnetism is central to the ability to dope these oxides, just as is designing specific band offsets in oxide heterojunctions, to the benefit of future oxide electronics. The crucial question here is what minimum theoretical framework is needed to explain and therefore design such gappingrelated phenomena. The standard explanation of gapping in these compounds despite the presence of partially filled d shells and the ensuing expected orbital degeneracy is generally formulated in terms of strong inter-electronic repulsions appearing in the celebrated Mott-Hubbard model [6][START_REF] Mott | Metal-Insulator Transitions[END_REF][START_REF] Hubbard | [END_REF][9] . This leads to a uniform explanation of gapping for all d-electron ABO3 compounds and degrees of spin order or disorder, based on the symmetry-conserving interelectronic repulsion, and ensuing localization. Within this framework, the experimental observations of a variety of different symmetry-breaking modes, such as those presented in Fig. 1, or magnetic moments is not related to the gapping mechanism but can appear afterwards as an additional effect.

Whereas Density Functional Theory (DFT) has been shown to be able to address numerous physical effects in such oxides, including ferroelectricity 10 , catalysis 11 and electrical battery voltage 12 , its use of a single Slater determinant and its mean-field treatment of electronelectron interactions (static correlations) has, according to numerous literature statements 13- 15 , disqualified it for the study such "strongly correlated" oxides, requiring far more computationally costly dynamically correlated methodologies, such as Dynamical Mean Field Theory (DMFT).

However, the DFT calculations in such demonstrations of failure [13][14][15] often used a nonspin polarized description and at times exchange correlation functionals that do not distinguish occupied from unoccupied orbitals (LDA or GGA functionals without U), and generally neglect sublattice displacements. For example, Ref. 14 16 or ABO3 materials [17][18][19][20][21][22][23][24] . In these calculations 16,24 , the PM phase, which is a collection of magnetic moments m """⃗ with random magnitude and direction on each site i but whose sum is zero (M """⃗ = ∑ m """⃗ ' ' = 0 "⃗ ), was represented by specially constructed supercells that has a total zero spin, but unlike the N-DFT implementations, there was no requirement that each transition metal ion have a zero spin. Such polymorphous representation lowered substantially the total energy relative to the N-DFT representations, while producing finite band gaps and local moments in all studied binary and ternary ABO3 compounds, in general accord with experiment. The need for an exchange-correlation functional that distinguishes occupied from unoccupied orbitals was satisfied by using a simplified self-interaction corrected functional in the form of "DFT+U" 25 . The use of "U" created sometimes the false impression that this approach owes its success in explaining gaps to the interelectronic repulsion and localization, just as in the Mott-Hubbard view.

To examine if the explanation of gapping requires an explicit Hubbard U, we have performed DFT calculations on several popular ABO3 perovskite oxides with d fillings from 1 to 8 electrons, using (unlike previous DFT+U calculations of e.g. Ref. 24 ) the recently developed SCAN meta-GGA functional 26 , without Mott-Hubbard interelectronic U. The use of the SCAN functional for transition metal and transition metal oxides is reported in many papers [27][28][29][30][31][32] .

Unlike these publications we do not just study how SCAN functional behaves with ternary oxides on gaps, structural features or electron localization, but we tackle a different problem:

is the Mott-Hubbard model required to capture trends in gapping, structural motions or magnetic moments in ABO3 materials? Here we show that (i) DFT, without any U parameter but with an exchange-correlation functional better representing self-interaction errors, is sufficient to explain trends in properties of ternary ABO3 oxides in both LT spin-ordered and HT PM phases; (ii) since DFT-no-U and its static mean-field treatment of electron interactions are largely sufficient to produce insulation, ABO3 oxide perovskites may certainly be complicated but they are not necessarily strongly dynamically correlated and (iii) thus, gapping in such specific cases may not sweepingly obey the celebrated Mott-Hubbard explanation of formation of two electron sites which depends on the existence of d like band edges and on interelectronic repulsion exceeding band width, neither of which are needed in the current explanation . However, our results do not imply that if DFT works for a material, then all forms of correlations do not play a role.

II. Method

We have performed DFT 33,34 total energy minimization with respect to lattice parameters and cell-internal atomic positions of different perovskite oxides with d fillings from 1 to 8 electrons. Structure types compared for their total energy included orthorhombic (Pbnm), monoclinic (P21/b and P21/n) and rhombohedral (R-3c), whereas spin configurations examined included ferromagnetic (FM), as well as classical A, C and G-type antiferromagnetic (AFM) orders, and more complex S-type AFM order based on ↑↑↓↓ spins chains in the (ab)-plane with different stackings along the c axis. We emphasize here that we did not explore the relative stability of the different magnetic orders and we just focused on the spin order experimentally observed at low temperature for each compound. The paramagnetic (PM) spin-disordered state has been modelled using the Special Quasirandom Structures (SQS) method 35 , following references 16,24 , which represents a random "alloy" of up and down (collinear) spins with total spin zero, within the 2x2x2 orthorhombic or monoclinic cells (32 ABO3 formula units containing 160 atoms). Convergence with SQS supercell size was tested and found adequate at 160 atoms 24 . All atomic displacements as well as breaking of degeneracies of partially occupied eg and t2g levels are allowed as long as they reduce the total energy. Amplitudes of the energy-minimizing distortions were determined by performing a symmetry adapted mode analysis 36,37 with a reference structure set to the ideal high symmetry cubic Pm-3m structure of perovskites. The lattice parameter of this hypothetical cubic structure is fixed to the ground state pseudo-cubic lattice parameter. and LaVO3 in the PM phase and CaFeO3 in the AFM phase. In the two vanadates, the strongly entangled spin and orbital degrees of freedom induce small lattice distortions on each octahedra in the PM phase (where each transition metal element experiences a unique potential), thus reducing the symmetry from P21/c to P-1 38 . However, the lattice parameters, B-O-B angles and B-O bond lengths are very similar to the respective quantities observed in experimental structures. For CaFeO3, the AFM-S magnetic order that we have used to approximate the experimentally observed AFM spiral 5 breaks the inversion center and induce some small lattice distortions such as polar displacements 39 , thus producing a polar P21 space group instead of a centrosymmetric P21/n symmetry. Jahn-Teller and bond disproportionation motions [34]. Ins. stands for insulating phases. Experimental values are taken from a: Ref. 40 , b: Ref. 41 , c: Ref. 42 , d: Ref. 43 , e: Ref. 44 , f: Ref. 45 ; g: Ref. 46 , h:

III. Results

Calculated structural and electronic properties:

Electronic properties

Ref. 47 , i: Ref. 48 , j: Ref. 49 , k: Ref. 50 , l: Ref. 51 , m: Ref. 52 , n: Ref. 53 , o: Ref. 54 , p: Ref. 5 , q: Ref. 55 , r: Ref. 56 , s: Ref. 57 , t: Ref. 58 , u: Ref. 4 , v: Ref. 59 , w: Ref. 60 Electronic structure as gleaned from DOS: Figure 3 Band gaps without U: All ABO3 compounds tested here are insulators in both their spin-ordered and spin-disordered PM solutions (Figure 2). These results agree with the insulating character observed experimentally (some experimental values available in literature are reported in Figure 2) and also reproduce trends observed with DFT+U and DMFT simulations on some of these materials (e.g. RTiO3, RVO3, RMnO3 and RNiO3, see references therein). For instance, we observe insulation in the yttrium nickelate YNiO3 compound in both AFM and PM phases, as DMFT does in LuNiO3 PM phase 14,65 . Likewise, we also observe an

LaTi 3+ O 3 (d 1 ) LaV 3+ O 3 (d 2 ) LaMn 3+ O 3 (d 4 ) LaFe 3+ O 3 (d 5 ) YNi 3+ O 3 (d 7 ) CaMn 4+ O 3 (d 3 ) d-d d-d p-d p-d p-d d-d E-E VBM (eV) a) d) b) e) c) f) B-d O-p
increase in the band gap when going from rare-earth titanates (d 1 ) to rare-earth vanadates (d 2 ) in agreement with experimental observations 41,45 . We emphasize here that experimental data of structural and electronic properties on bulk and stoichiometric LaCuO3 crystals are scarce and diverging hindering confirmation of the SCAN-DFT calculation. Finally, just as standard exchange and correlation functionals underestimate the band gap of the highly uncorrelated semi-conductors such as Si and GaAs, the SCAN functional behaves similarly for ABO3 materials and one may improve the band gap description and related quantities by using GW corrections 66 .

Magnetic moments: Our calculations provide magnetic moment values that are comparable to experimental quantities available in literature (see Figure 2). For instance, we capture the decrease of the magnetic moment of Ti cations when going from YTiO3 to LaTiO3 40,43 . Due to the presence of a DLE for Ni and Fe cations in YNiO3 and CaFeO3, respectively, two very different magnetic moments are extracted from our simulations. These quantities, compatible with experimental values, point towards a disproportionation of the unstable 3+ and 4+ formal oxidation states (FOS) of Ni and Fe cations, respectively, towards their more stable 2+/4+ and 3+/5+ FOS.

Trends in energy differences between different phases:

The energy gain in forming local moments is given by the total energy difference ENM -EAFM. As the number of unpaired 3d electrons increases, we see that this energy strongly increases, signaling the large energy gain obtained by forming local magnetic moments, and thus the irrelevance of the NM ansatz (Fig. 2). The energy cost for forming a random configuration from an ordered one is (EPM-EAFM)-TS where the first term is the contribution of the internal T=0 energy. For all compounds studied (EPM-EAFM)>0, i.e. the spin-ordered states are just slightly more stable than the PM solution. The entropy contribution will cause an order -disorder transition at finite TNéel. LaTiO3 is an exception in which (EPM-EAFM)<0, i.e. the experimentally observed AFM-G state is higher in energy than the PM state. This delicate balance could be because other magnetic configurations occur at low T. We checked the FM spin order (in the spirit of YTiO3) for this compound and found that the FM order now represents an energy gain of 10 meV/f.u over the PM phase. Interestingly LaTiO3 in the FM spin order is still an insulator (Eg=0.02 eV) with similar distortions to the PM and AFM-G solutions. It is possible that an exhaustive search for other spin configurations will change the result somewhat. Indeed, we did not perform such an exhaustive search.

Electronic Localization without U:

It is interesting to analyze the electronic localization of the d electrons in those perovskite oxides that are believed to be strongly dynamically correlated systems (d 1 , d 2 , d 4 and d 7 materials for instance). We report in Figure 4 the electronic charge density at the top of the valence band in the low temperature spin-ordered phase of LaTiO3 (d 1 ), YVO3 (d 2 ), LaMnO3 (d 4 ) and YNiO3 (d 7 ). In agreement with the pDOS reported in Fig. 3, one notices increasing density on the O atoms when going from light to heavier transition metal atoms, underlining the progressive shift from "Mott insulators" (LaTiO3, YVO3, Fig4. a and b) to "charge transfer insulator" (YNiO3, Fig4.d) behaviors as mentioned earlier. In LaTiO3 (Fig4.a), the single Ti 3+ d electron is localized in a linear combination of the three t2g levels, whose relative coefficients are alternating on neighboring Ti sites. In YVO3 (Fig4.b), the additional d electron sits in a combination of the dxy and dxz or dyz orbital, with alternating coefficients on neighboring sites due to the presence of an in-phase Q2 + JT motions in the low-temperature phase (see Figure 2). We observe a similar situation in LaMnO3 (Fig4.c): the Mn 3+ "eg" electron is localized either in dx [START_REF] Miyasaka | [END_REF] or dy 2 orbitals between neighboring Mn sites in the (ab)-plane but with similar stackings along the c axis. Finally, a "charge ordered" picture is observed in YNiO3 (Fig4.d) with Ni cations sitting in an extended (compressed) octahedra bearing approximately 2 (0) electrons on the eg levels. This is clearly proving a charge disproportionated insulating state with Ni ions adopting their more stable 2+ and 4+ formal oxidation state (FOS) instead of the unstable 3+ FOS. Similar observations were obtained for other members such as YTiO3, LaVO3 or CaFeO3, independently of the spinorder/disorder.

How displacements affect or create gapping:

To understand the role of JT and bond disproportionation motions on the gap opening, we use as a starting configuration a high symmetry Pm-3m phase and then apply to it successively all displacement modes appearing in the ground state AFM structure (i.e. O6 rotations, antipolar displacements of ions). We then freeze the JT or Boc modes and compute the potential energy surface vs amplitude of the modes for NM, AFM and PM solutions in LaMnO3 (Q2 + mode), LaVO3 (Q2 -mode) and YNiO3 (Boc mode). This protocol has been used in Ref. 15 to demonstrate the inability of naïve N-DFT to stabilize JT motions in LaMnO3 and the crucial role of strong dynamic correlation to obtain such displacements. Our results for the same NM ansatz are reported in Figure 5 where we confirm that this naïve model yields a single well potential whose energy minimum is located in zero amplitude, i.e. these structural distortions do not appear. Comparing with the full DFT calculation in Fig. 5 clearly shows, however that N-DFT is not what DFT can actually do, the latter producing distortions where they appear experimentally. Moving to AFM and PM spinpolarized solutions, we observe that the energy minimum of the different potentials is located

Ti O !" #$ ± &" #' ± (" $' z x V O !" #$ ± &" $' z x !" #$ ± &" #' Mn O " # ) y x y x * + , * + ) Ni L O Ni S Electron density 0 0.5 x LaTiO 3 YVO 3 YNiO 3 LaMnO 3 a) b) c) d) " $ )
to non-zero amplitude of JT and Boc modes. We conclude that DFT without an interelectronic U can stabilize these previously believed to be "correlation-induced lattice distortions" 14,15 if minimal ingredients are provided to the simulations. It is evident that dynamic correlation effects are not forcing these distortions whose role was previously hampered by a false initial hypothesis. The role of atomic displacements on the gap opening. Opened and closed symbols in Fig. 5 denote metallic or insulating solutions, respectively. Upon increasing the amplitude of the Q2 + and Boc modes in LaMnO3 and YNiO3, respectively, a band gap opens in the PM and AFM solutions. LaVO3 is different where rotations plus antipolar displacements of ions are sufficient to produce an insulating state since the Q2 -JT mode is not important for the gap opening. The present SCAN-no-U results are consistent with DFT+U and DMFT simulations in LaMnO3 13,23,67 and RVO3 18,62 , as well as the experimental observation of insulating states in RVO3 irrespective of the presence of the JT Q2 -mode 68 . Surprisingly, even without any the fact that the PM phase of YNiO3 without bond disproportionation is found metallic in experiments 4 . The observation of an electronic instability agrees with our recent DFT+U study 24 and with numerous DMFT calculations identifying a spontaneous tendency of Ni 3+ cations to undergo disproportionation effects 65,69 .

IV. Discussion

So far, we have shown the ability of the SCAN functional to capture the formation of basic physical quantities (e.g. band gap, magnetic moments, structural motions) of ABO3 materials upon various d fillings. We may now question the ability of this functional to reproduce physical trends within series showing isoelectronic configurations.

The dependence of the Jahn-Teller distortions on the existence of the local magnetic moments:

The case of SrVO3: For instance, SrVO3 (d 1 ) display a paramagnetic metallic state at all temperature while the isolectronic rare-earth titanates RTiO3 exhibit insulating states.

We have explored the paramagnetic state of SrVO3 using the constrained magnetic approach proposed by Franchini et al 70,71 (we restricted V 4+ spins to be aligned along z, since spins have the tendency to flip during the self-consistent field). Unlike the isoelectronic RTiO3 compounds, SrVO3 relaxes to a cubic metallic phase -thus showing no JT distortion. This can be explained using the 1926 Goldschmidt tolerance factor close to 1, which implies cubic stability in agreement with experiments. In contrast, we have shown that isovalent YTiO3 and LaTiO3 are insulating for both paramagnetic and spin-ordered phases. The case of PrNiO3: The RNiO3 compounds form another important family of ABO3 perovskites in which materials with an A site cation presenting a small ionic radius are insulating in both AFM and PM phases (R=Lu-Sm, Y) while remaining compounds (R=Nd, Pr) are only insulators in the AFM phase 72 . We correctly find that the geometry relaxation for PrNiO3 with the AFM order results in an insulating monoclinic cell showing a band gap of 0.78 eV and disproportionation effects (QBoc=0.15 Å, µNiL=1.38 µB and µNiS=0 µB), whose amplitudes are reduced with respect to YNiO3. Just as in the case of YNiO3, the stabilization of the breathing mode is not essential for the gap opening and the AFM order is already sufficient to produce a sizable band gap of 0.56 eV and an asymmetry of magnetic moments (µNiL=1.21 µB and µNiS=0 µB) (see the energy potential surface as a function of Boc amplitude when starting from a cell with only rotations and anti-polar displacement presented in Fig6). Using the AFM structure but with a PM order, PrNiO3 is also willing to adopt a disproportionated cell whose amplitude is a priori not enough to produce insulation (Fig6). In contrast to the AFM order and ins.

PrNiO 3 -bond disproportionation

This discrepancy could result from an exaggerated electron localization in SCAN that contributes to (i) bond disproportionation and (ii) Ni magnetic moments substantially larger in SCAN-no-U (QBoc=0.17 and 0.15 Å, µNiL=1.41 and 1.35 µB for R=Y and Pr, respectively) than in GGA+U calculations (QBoc=0.14 and 0.11 Å, µNiL=1.26 and 1.17 µB for R=Y and Pr, respectively) of Refs. 20,24 . Overestimated magnetic moments are not specific to ABO3 materials and instead seem inherent to the SCAN functional 27,73 that will have to be improved in this respect.

V. Conclusion

We have shown that a DFT using a functional without on-site correlation energy U but better amending self-interaction errors captures the basic properties of 3d electron transition metal oxide perovskites namely band gap, magnetic moments, relative B-d to O-p levels positions and all structural features, by using a polymorphous representation allowing energy lowering formation of (i) magnetic moments, (ii) atomic displacements and octahedral rotations, and (iii) breaking of crystal field symmetry of partially occupied orbitals. We further show that lattice distortions including, Jahn-Teller and bond disproportionation modes, are captured by single-determinant, mean field DFT without U parameter suggesting that dynamic correlations are not the universal controlling factor here. Success of DFT for a given case does not exclude the existence of other forms of correlation. Furthermore, the success of DFT does not imply Coulomb interactions are not important, because DFT certainly includes (mean field like Hartree) Coulomb interaction. However, this is very different than the highly complex treatment of non-mean field Hubbard like U interaction term in the Hubbard Hamiltonian. The latter effect is excluded in the much simpler present calculation which still provides good and material-specific results.

Figure 1 :

 1 Figure 1: Sketches of Jahn-Teller motions (a and b) and bond disproportionation (c) distortions appearing in some ABO3 materials.

Figure 2

 2 summarizes the calculated band gap ∆E (in eV), magnetic moment M3d (in µB) associated with the B cations and amplitudes of distortions (in Å) associated with JT and bond disproportionation motions, all done without Hubbard U, compared with experimental values available in literature. The calculated energy-minimizing lattice type agrees with experiments with the exception of YVO3

  depicts the B-atom (green) and Oxygen (red) projected density of states (in eV/states/f.u) averaged on all sites in the low temperature phase. Several observations are apparent: (i) only the perovskites with light B atoms (LaTiO3 and YVO3, Fig3.a and b) have band edge states with dominant d characters (as imagined in the original Mott-Hubbard model), while the O p levels lie below the B d levels in accord with Pavarini et al 61,62 ; (ii) for heavier 3d atoms the d bands move deeper in energy, towards the oxygen bands and the electronic structure becomes strongly p-d hybridized, yielding covalent charge transfer insulator behaviors (CaMnO3 to YNiO3 in accord with Bisogni et al 63 ). Thus, our calculation reproduces the correct position of transition metal d levels with respect to O p levels as deduced from experiments 46,64 and DMFT simulations on some perovskite compounds 1,61-63 .

Figure 3 :

 3 Figure 3: Projected density of states (in states/eV/f. u) on B-d levels (filled green) and O-p levels (red line) averaged on all sites in the low temperature phase. Band edges are shown with the vertical lines. The band gap is represented by the light grey area.

Figure 4 :

 4 Figure 4: Partial charge density maps of levels at the top of the valence band in some selected "correlated oxide perovskites".

Figure 5 :

 5 Figure 5: Energy gains ∆E (in meV/f.u with respect to the 0-mode amplitude) associated with Jahn-Teller distortions (a and b) and the bond disproportionation (c) modes as obtained by the naïve nonmagnetic N-DFT (black circles, upper panel), the PM described with a SQS supercell (blue triangles, middle panel) and AFM (red squares, lower panel). DMFT potential (orange diamonds) for the PM phase of LaMnO3 and LuNiO3 extrapolated from Refs.15,65 are also reported. Filled and unfilled symbols correspond to insulating (ins.) and metallic (met.) solutions. The reference structure at 0 amplitude of JT or Boc modes is set to a material displaying octahedra rotations and anti-polar motions of ions.

  disproportionation Boc mode, YNiO3 already exhibits a clear-cut of the Ni electronic structures with one Ni site bearing a magnetic moment of 1.06 (1.26) µB and the other one a value around 0.72 (0.00) µB in the PM (AFM) solution (Fig.5.c). It follows that YNiO3 has a spontaneous tendency to undergo disproportionation effects through an electronic instability transforming the unstable 3+ formal oxidation state (FOS) of Ni cations to the more stable 2+ and 4+ FOS in the insulating phase. Nevertheless, only the AFM order can render an insulating state without bond disproportionation Boc mode (Fig.5.c) -this is consistent with

Figure 6 :

 6 Figure 6: Properties of isoelectronic compounds. Energy difference as a function of the bond disproportionation in PrNiO3 in the AFM (squares) and PM (triangles) magnetic orders. Filled and unfilled symbols correspond to insulating and metallic solutions, respectively. The reference structure at 0 amplitude of JT or Boc modes is set to a material displaying octahedra rotations and anti-polar motions of ions.

  YNiO3, the PM solution without bond disproportionation does not show any asymmetry of magnetic moments, all Ni cations bear a spin of 0.93 µB. This observation signals the absence of a spontaneous electronic instability toward a disproportionation in the PM cell for PrNiO3, in agreement with experiments. Nevertheless, a full structural relaxation with the PM order yields an insulating disproportionated cell at 0 K for PrNiO3 with a narrow band gap of 0.30 eV in contrast with experimentsi.e. it becomes an insulator at the AFM transition (TN=135 K72 ).

Key properties of oxide perovskites with the SCAN meta-GGA functional without U using SQS for the PM phase (Experimental values are provided in parentheses

								Structural distortions
		d filling	Sym.	Mag.	∆ENM (mev/f.u)	Eg (eV)	M3d (µB)	Q2 + (Å)	Q2 -(Å)	Boc (Å)
	YTiO3	t2g 1 eg 0	Pbnm Pbnm	FM PM	-242 -218	0.08 0.33 (1.20 b )	0.92 (0.84 a ) 0.84	0.03 0.02 (0.00 c )	--	--
	LaTiO3	t2g 1 eg 0	Pbnm Pbnm	AFMG PM	-75 -84	0.05 0.14 (0.20 b )	0.67 (0.46 d ) 0.78	0.06 0.04 (0.04 c )	--	--
	YVO3	t2g 2 eg 0	Pbnm P-1	AFMG PM	-1052 -978	0.89 0.55 (1.60 f )	1.77 (1.72 e ) 1.82	0.19 (0.14 e ) NA	-NA	--
	LaVO3	t2g 2 eg 0	Pbnm P-1	AFMC PM	-864 -833	0.78 0.36 (1.10 i )	1.78 (1.30 g ) 1.80	0.00 (0.01 h ) NA	0.09 (0.08 h ) NA	--
	CaMnO3	t2g 3 eg 0	Pbnm	AFMG	-2049	0.79	2.62 (2.64 j )	0.01	-	-
			Pbnm	PM	-2036	1.46 (ins.)	2.62	0.01 (0.04 k )	-	-
	LaMnO3	t2g 3 eg 1	Pbnm Pbnm	AFMA PM	-1783 -1758	0.52 0.30 (0.24 m -1.70 n )	3.65 (3.70 l ) 3.60	0.28 0.32 (0.30 o )	--	--
	CaFeO3 t2g 3 eg 0 + t2g 3 eg 2 P21 P21/n	AFMS PM	-1474 -1449	0.20 0.07 (0.25 q )	2.72-3.67 (2.48-3.48 p ) 2.66-3.72	0.00 0.00 (0.04 p )	--	0.13 0.14 (0.18 p )
	LaFeO3	t2g 3 eg 2	Pbnm Pbnm	AFMG PM	-1073 -936	1.67 (2.10 i ) 0.52	3.94 (4.60 r ) 4.04	0.01 0.01 (0.00 s )	--	--
	YCoO3	t2g 6 eg 0	Pbnm	NM	-	1.48 (ins. t )	-	0.06 (0.05 t )	-	-
	YNiO3 t2g 6 eg 2 + t2g 6 eg 0 P21/n P21/n	AFMS PM	-353 -350	0.92 0.59 (0.20 i -1.00 v )	1.41-0.00 (1.70-0.40 u ) 1.38-0.28	0.05 0.05 (0.05 w )	--	0.18 0.17 (0.13 w )
	LaCuO3	t2g 6 eg 2	R-3c	AFMG	-143	0.48 (?)	0.70 (?)	-	-	-
	Figure 2:

) Energy difference (in meV/f.u) between spin polarized and N-DFT (NM) solution, band gap Eg (in eV), magnetic moment M3d (in µB) associated with the B cation and amplitudes of distortions (in Å) associated with

Acknowledgments: This work received support from the European Research Council (ERC) Consolidator grant MINT under the Contract No. 615759. Calculations took advantages of the Occigen machines through the DARI project EPOC A0020910084 and of the DECI resource FIONN in Ireland at ICHEC through the PRACE project FiPSCO. The work of AZ was supported by Department of Energy, Office of Science, Basic Energy Science, MSE division under Grant No. DE-FG02-13ER46959 to CU Boulder. JV acknowledges technical support from


Julien Varignon

Jacobo Santamaria

Manuel Bibes

Switchable and tunable Rashba-type spin splitting in covalent perovskite oxides

In transition metal perovskites (ABO3) most physical properties are tunable by structural parameters such as the rotation of the BO6 octahedra. Examples include the Néel temperature of orthoferrites, the conductivity of mixed-valence manganites, or the band gap of rare-earth scandates. Since oxides often host large internal electric dipoles and can accommodate heavy elements, they also emerge as prime candidates to display Rashba spin-orbit coupling, through which charge and spin currents may be efficiently interconverted. However, despite a few experimental reports in SrTiO3-based interface systems, the Rashba interaction has been little studied in these materials, and its interplay with structural distortions remain unknown. In this Letter, we identify a bismuthbased perovskite with a giant, electrically-switchable Rashba interaction whose amplitude can be controlled by both the ferroelectric polarization and the breathing mode of oxygen octahedra. This particular structural parameter arises from the strongly covalent nature of the Bi-O bonds, reminiscent of the situation in perovskite nickelates. Our results not only provide novel strategies to craft agile spin-charge converters but also highlight the relevance of covalence as a powerful handle to design emerging properties in complex oxides.

Spintronics exploits the spin degree of freedom of carriers in addition to their charge, giving rise to a very broad range of electronic applications including magnetic memories or magnetic sensors [1,2]. It relies on the exchange interaction of electrons with local magnetic moments of a ferromagnet to generate spin currents. Usually, the ferromagnets are simple transition metal elements such as Co, Ni or their alloys. Nevertheless, the control of their magnetization typically requires large energies and they generate undesired magnetic fields hindering high densification of devices. In the search for lower power spintronic devices, the spin-orbit interaction (SOI) has been identified as a promising pathway to achieve very efficient spin-to-charge current conversion and vice versa. The two phenomena at the core of this branch of spintronics, also called spin-orbitronics, are the spin Hall effect (SHE) [3,4] and inverse spin Hall effect (ISHE) [5,6]. For instance, the former effect allows efficient charge-tospin current conversion in non-magnetic materials based on heavy elements [7], therefore alleviating the need for ferromagnets to generate spin currents.

More recently, renewed interest for SOI based applications arise thanks to the Rashba effect [8]. In materials lacking inversion symmetry, the interplay between the electric field and the SOI lifts the degeneracy of electronic bands according to their spin. For a material exhibiting a polar axis oriented along the z direction, Rashba proposed the following spin-orbit interaction:

H R = α R (σ x k y -σ y k x ) (1)
where α R is the Rashba coefficient -proportional to the spin-orbit strength of the material λ SO and to the magni-tude of the electric field -, σ i (i=x,y) are Pauli matrices and k j (j=x,y) is the momentum of the electron. The Rashba interaction leads to a spin-locking of electrons according to their momentum k enabling efficient spinto-charge current interconversions through Edelstein and inverse Edelstein effects [9,10]. Although originally formulated at interfaces and surfaces [11][12][13][14], the Rashba interaction is now generalized to non-centrosymmetric bulk materials [15][16][17], although it may coexist with the Dresselhaus interaction [18].

In the context of SOI related devices, ABO 3 oxide perovskites remain largely unexplored. They encompass a wealth of properties -including ferroelectricity and magnetism -and functionalities originating from the coupling of their structural, electronic and magnetic degrees of freedom [19]. Most notably, the lattice distortions usually control the properties of the perovskite, such as the metal-insulator phase transition temperature in rareearth nickelates [20,21], the Néel temperature of orthoferrites [22,23] or the band gap value of rare-earth scandates [24,25]. Being largely ionic in most cases, many oxide perovskites host large internal electric dipoles. Moreover, they can also accommodate heavy elements, an important aspect for SOI based applications. So far, BiAlO 3 has been proposed theoretically to be the first oxide exhibiting ferroelectricity and Rashba physics [17] -although this coexistence was already observed in the semiconductor GeTe [16]. While BiAlO 3 harbors a heavy element and a large ferroelectric polarization -estimated around 90 µC.cm -2 -, the computed Rashba coefficient α R remains relatively moderate -around 0.39 and 0.74 eV. Åalthough comparable to conventional systems such as (111) oriented Bi [26]. In fact, other ingredients such as a narrow band gap and/or similar atomic and orbital characters for the electronic states located around the Fermi level are proposed to be additional key ingredients for reaching large α R [27]. Covalent oxides that are characterized by a strong hybridization of B cations and O anions electronic states around the Fermi level, are therefore an ideal platform to engineer materials undergoing large Rashba effects. Furthermore, oxides are usually characterized by numerous lattice distortions but the role of the latter on the Rashba physics remains unexplored.

In this letter, we use first-principles simulations to identify the existence of large Rashba effects with α R as large as 8 eV. Å in a ferroelectric phase of the covalent oxide strontium bismuthate SrBiO 3 reachable by epitaxial strain. We further reveal that the amplitude of the Rashba coefficient is tunable by a lattice distortion strongly connected to bands dispersion and to the level of hybridization between B and O electronic states. Our results therefore highlight a new pathway to control and optimize the Rashba interaction in oxides, further revealing the potential of these materials for spin-orbitronics.

Strontium bismuthate Sr 2+ Bi 4+ O 3 is an insulator adopting a monoclinic P 2 1 /n symmetry at low temperature [28,29]. The ground state structure is characterized by the usual oxygen cage rotations appearing in perovskites, a -a -c + antiferrodistortive motions in Glazer's notation, as well as by a breathing of the oxygen cage octahedra distortion B oc (see Figure 1.a) [28,29]. The latter distortion produces a rocksalt-like pattern of large and small O 6 groups resulting in Bi sites splitting. In the following, Bi cations sitting in compressed (extended) O 6 groups are labeled Bi S (Bi L ) (see Figure 1.a). Since SrBiO 3 shares several features with rare-earth nickelates RNiO 3 , the appearance of B oc in bismuthates is sometimes interpreted in terms of ionic charge disproportionation of unstable Bi 4+ valence state to the more stable Bi 3+ L + Bi 3+ S electronic configurations in the insulating phase [30][31][32]. Alternatively, it is also proposed to be a consequence of charge transfer effects from O anions to Bi cations [33]. Nevertheless, Tonnhauser and Rabe demonstrated that the appearance of B oc is responsible of the band gap opening in bismuthates [34].

Although very similar to rare-earth nickelates, bismuthates also exhibit many other striking properties such as high T c superconductivity upon hole doping [29,[35][START_REF] Sleight | Ten Years of Superconductivity: 1980-1990[END_REF][START_REF] Kim | [END_REF][38] or possible topological surface states accessible upon electron doping [39]. The latter point highlights the strength of the SOI in these materials, being de facto an ideal playground to study theoretically the interplay between lattice distortions, covalence and spin-orbit related phenomena.

First-principles calculations were carried out with Density Functional Theory (DFT) using the Vienna Ab initio Simulation Package (VASP) [40,41]. We used the PBE functional revised for solids [42] with a cutoff en- ergy of 500 eV and a 6 × 6 × 4 (4 × 4 × 4) Monkhorst-Pack kpoint mesh for 20 (40) atoms unit cells. We used Projector Augmented Waves (PAW) pseudopotentials [43] for core electrons with the following valence electron configurations: 4s 2 4p 6 5s 2 (Sr), 6s 2 6p 3 (Bi) and 2s 2 2p 4 (O). Geometry optimizations were performed until forces are lower than 0.01 meV. Å-1 without spin-orbit interaction included unless stated. Born effective charges and phonons frequencies were computed with the density functional perturbation theory [44,45]. Band structures and spin textures were plotted with the PyProcar script [46]. Maxi-Localized Wannier Functions (MLWFs) were extracted with the Wannier90 package [47][48][49].

We first inspect the properties of bulk strontium bismuthate. Geometry relaxation yields an insulating mon-oclinic P 2 1 /n ground state with a band gap of 0.3 eV, other polymorphs observed experimentally at higher temperature -F m 3m, R 3c or I 2 /m symmetries -are found higher in energies. Our optimized lattice parameters (a=5.9400 Å, b=6.1365 Å, c=8.4859 Å and β=90.0597 • ) are found in very good agreement with experiments from reference 29, yielding 0.55 % of error on the total volume. Regarding the electronic structure of the material, we extract for both Bi S and Bi L cations similar site occupancies N, although a weak charge order does exist but with a sign opposite to that expected ( N Bi S = 3.97e while N Bi L =3.55e). The quasi-absence of charge ordering can be further appreciated on the projected band structure on Bi s and O p levels plotted in Figure 1.d. Between -12 to -6 eV, we clearly observe four bands that have a dominant Bi s character rather indicating that both Bi S and Bi L cations have filled 6s levels. This is confirmed by our Wannier Functions (WFs) analysis based on these 4 Kohn-Sham states that yields one s-like WF per Bi cation (Figure 1.c). In constrast, we observe two bands above the Fermi level that have a dominant O p character. Building again the WFs for these two bands, we obtain two s-like WFs centered on Bi S cations with large contributions coming from neighboring O anions (Figure 1.b). These two WFs are in fact the two oxygen holes that are formed during the charge transfer from O to Bi S cations. This result is in close agreement with the work of Foyevtsova et al and the electronic structure of SrBiO 3 can be interpreted in terms of Bi L 6s 2 and Bi S 6s 2 L 2 [33] where the notation L labels a ligand hole.

Having now established that the structural and electronic properties of SrBiO 3 are in good agreement with experiments and recent theoretical works, we now turn our attention to spin-orbit related phenomena. When including the spin-orbit interaction in the calculation, we do not observe any sizable effect on the ground state properties, rather suggesting a marginal SOI in this material. However, the fact that SrBiO 3 is paraelectric in its bulk hinders possible Rashba phenomena. Surprinsingly, we observe the existence of a ferroelectric phonon mode at relatively low frequency (ω FE 50 cm -1 ) in the bulk. Therefore, through the coupling of this phonon mode with epitaxial strain [50], one may expect to unlock ferroelectricity in SrBiO 3 . This is reinforced by the ability of Bi-based oxide perovskites to exhibit ferroelectricity in bulk or under epitaxial strain. Since the pseudocubic lattice parameter of SrBiO 3 is rather large (a pc 4.26 Å), we only explored the effect of a compressive strain in our simulations. Additionally, the SOI having marginal effect on the structural aspect of the bismuthate, we ignored it during geometry relaxation under strain.

Figure 2 reports the energy of the most stable structures found in the calculations as a function of the substrate lattice parameter (lower scale)/compressive strain value (upper scale) [START_REF]different oxygen cage rotations pattern and/or breathing distortion Boc and/or polar axes. Two growth orientation were used[END_REF]. Between 0 and 5 % of compression, SrBiO 3 remains a paraelectric insulator adopting a P 1 structure described by a a -a -c + oxygen cage rotation patterns and a breathing of O 6 groups. This structure is equivalent to the bulk except that the in-phase oxygen cage rotation axis is aligned along the substrate -such a growth is sometimes referred to c -P 2 1 /n, the notation c-indicating that the c axis is aligned along the substrate [52]. Beyond 5% of compression, the ground state becomes ferroelectric and is associated with a P 1 structure exhibiting the aforementioned structural distortions plus a spontaneous polarization in all three directions. Focusing on the structure obtained at 6.1 % of compressive strain -i.e. a substrate =4 Å-, SrBiO 3 develops a polarization of 12.65 µC.cm -2 (P x =9.64, P y =2.73, P z =-7.73 in a pseudo P 2 1 /n 20-atom unit cell setting, c being the long axis) approaching the value of the popular ferroelectric BaTiO 3 . When including the SOI in the ferroelectric phase, we now observe several splittings of the "oxygen hole bands" located above the Fermi level E F (see Figure 3 Albeit smaller than values reported at the Γ point, these Rashba coefficients remain comparable to values observed in BiAlO 3 [17] or in heavy metals such as (111) oriented Bi [26]. We now turn our attention to possible functionalization of the Rashba interaction in this ferroelectric material. Firstly, as inferred by Figures 3.b and 3.c, switching the polarization reverses the spin textures therefore allowing their non-volatile electrical control. By freezing different amplitudes of the ferroelectric distortion while keeping other distortions to their ground state amplitude, we observe a linear dependence of α R with the ferroelectric mode amplitude (see Figure 4.a), in agreement with its dependence with the electric field amplitude. Eventually, such functionalities were already proposed theoreti-cally in GeTe [16] or in BiAlO 3 [17]. A key aspect of the bismuthate is the strong relationship between the breathing mode B oc and electronic properties. Although B oc is found to open the band gap in the ferroelectric phase upon its condensation in our simulations -all other lattice distortions are fixed to their ground state amplitude -, it also controls the amplitude of the Rashba coefficient as inferred by Figures 4.b, c, d, e andf. Consequently, at odds with usual assumptions, the Rashba coefficient is not only sensitive to the spin-orbit strength λ SO and the electric field amplitude, but it can also be strongly related to structural distortions altering the band dispersions and the level of hybridization between the B site cation and O states [53]. The latter observation is in close agreement with the work of Bahramy et al highlighting that hybridized electronic states around E F were a key aspect for large α R [27].

Although the ferroelectric phase is reached at relatively large compressive strain value, it might be achieved experimentally using scandate substrates for instance. Additionally, the control of the breathing mode may be unlocked by several external stimuli such as strain engineering, octahedra connectivity [54] or with hybrid improper ferroelectricity [50].

In conclusion, we have predicted in this Letter unprecedented Rashba spin splittings of oxygen hole bands in the covalent oxide perovskite SrBiO 3 . We have also demonstrated that the Rashba coefficient α R could be strongly related to lattice distortions irrespective of the amplitude of the electric field or of the spin-orbit interaction. The Rashba physics being at the core of the most recent spin-orbitronics based applications, our work unveils new pathways to optimize and control α R through the subtle interplay between lattice and electronic degrees of freedom appearing in oxide perovskites.

This work has been supported by the European Research Council (ERC) Consolidator grant MINT under the contract #615759. Calculations took advantages of the Occigen machines through the DARI project EPOC #A0020910084 and of the DECI resource FIONN in Ireland at ICHEC through the PRACE project FiPSCO. J. Varignon acknowledges technical help from Adam Ralph from ICHEC. with either the a and b lattice parameters aligned at 45 • of the a sub and b sub substrate lattice parameters or the c lattice parameter along the substrate. The eight most stable structures identified in the calculations are the following : P 21/n (a -a -c + + Boc), P c (a -a -c + + Boc + PxPyPz), P bnm (a -a -c + ), P 21/m (a + a -c -), P m (a + a -c -+Pz), P 1 (a + a -c -+ Boc) and P 1 (a + a -c - + Boc + PxPyPz).

[52] The P 1 and P 1 structures grow with the in-phase rotation axis along the substrate. We therefore have to use a 40 atoms unit cell to model the constraint in our calculations. Nevertheless, results regarding band structures and polarizations are reported with respect to a 20 atoms unit cell where the c axis is the long axis of the perovskite.

[53] Through the condensation of the breathing distortion in the ferroelectric ground state achieved under compressive strain, the asymmetry of the sphere integrated charges between BiS and BiL increases from 0e to 0.44e, rather suggesting a strong modification of the electronic structure.

[54] Z. Liao et al, Nat. Mater. 15, 425 (2016).

FIG. 1 .

 1 FIG. 1. Structural and electronic properties of SrBiO3 ground state. a) Ground state P 21/n monoclinic structure. b) s-like Wannier functions associated with the 2 bands lying above the Fermi level. c) s-like Wannier functions associated with the 4 low lying bands between -12 eV and -6 eV. d) Angular momentum resolved band structure of the monoclinic P 21/n phase. The Fermi level EF is set to 0. The high symmetry coordinates in the Brillouin zone of the P 21/n symmetry are the following: Γ (0,0,0), B (1/2,0,0), Y (0, 1/2,0), A (1/2,1/2,0), Z (0,0,1/2), C (0,1/2,1/2), E(1/2,1/2,1/2) and D(1/2,0,1/2).

FIG. 2 .

 2 FIG.2. Energy differences (in meV/f.u) between the different metastable phases as a function of the substrate lattice parameter (in Å, lower scale) or strain value (in %, upper scale). The reference energy is set to the P -1 structure energy (dashed horizontal black line). The vertical line represents the boundary between ferroelectric (FE) and paraelectric (PE) phases.

FIG. 3 .

 3 FIG. 3. Electronic properties of the ferroelectric ground state of SrBiO3 at 6.1 % of compressive strain. a) Orbital projected -on O p (blue character) and Bi s (red character) states -band structure in a monoclinic P 21/n setting. The Fermi level is set to 0 eV. b and c) Bands projected on the ±σy component along the (-1/2, 0, 0) to (+1/2, 0, 0) direction for +P (c) or -P (d) spontaneous polarization. d and e) Spin textures in the (kx, ky) plan obtained at E=+2 eV when projecting on the ±σy (d) or ±σx (e) spin component for a positive spontaneous polarization. f) and g) Spin textures in the (kz, kx) plan obtained at E=+0.75 eV when projecting on the ±σx (f) or ±σz (g) spin component for a positive spontaneous polarization.

FIG. 4 .

 4 FIG. 4. Evolution of the Rashba coefficient when freezing different amplitudes of the polar distortion (a) and the breathing of the oxygen cage octahedra (b).c, d, e and f) Bands projected on the ±σy component along the (-1/2,0,0) to (+1/2,0,0) direction for different amplitudes of the breathing distortion. We emphasize that all other lattice distortions are fixed to their amplitude appearing in the ferroelectric ground state.