Superlattice-Induced Ferroelectricity in Charge-Ordered La1/3Sr2/3Feo3

Superlattice-induced ferroelectricity in charge-ordered La1/3Sr2/3FeO3

Se Young Parka,b,c, Karin M. Rabed,1, and Jeffrey B. Neatonc,e,f

aCenter for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Republic of Korea; bDepartment of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea; cDepartment of Physics, University of California, Berkeley, CA 94720; dDepartment of Physics & Astronomy, Rutgers University, Piscataway, NJ 08854; eMolecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720; and fKavli Energy NanoScience Institute, University of California, Berkeley, CA 94720

Contributed by Karin M. Rabe, October 14, 2019 (sent for review May 17, 2019; reviewed by Steven May and Silvia Picozzi)

Charge-order–driven ferroelectrics are an emerging class of ity from the displacive type is that the switching polarization functional materials, distinct from conventional ferroelectrics, arises primarily from interionic transfer of electrons when the where electron-dominated switching can occur at high frequency. charge ordering pattern is switched. This is accompanied by a Despite their promise, only a few systems exhibiting this behav- small polar lattice distortion, which can be used as a proxy to sig- ior have been experimentally realized thus far, motivating the nal the polar nature of the phase. Such electronic ferroelectrics need for new materials. Here, we use density-functional the- might be useful for high-frequency switching devices given that ory to study the effect of artificial structuring on mixed-valence the polarization switching timescale is not limited by phonon solid-solution La1/3Sr2/3FeO3 (LSFO), a system well studied exper- frequency (17, 18). imentally. Our calculations show that A-site cation (111)-layered To promote experimental observation of switchable polar- LSFO exhibits a ferroelectric charge-ordered phase in which inver- ization due to charge-order–driven ferroelectricity, the system sion symmetry is broken by changing the registry of the charge should exhibit a strong tendency to charge disproportionation. order with respect to the superlattice layering. The phase is Further, there should be a strong tendency to charge ordering, energetically degenerate with a ground-state centrosymmetric which could be manifested as robust charge ordering already in phase, and the computed switching polarization is 39 µC/cm2, the solid solution phase. The iron-oxide family is a class of mate- a significant value arising from electron transfer between FeO6 rials that satisfies these conditions. Unlike La1/2Sr1/2VO3, in octahedra. Our calculations reveal that artificial structuring of which no charge ordering is observed in the solid solution down LSFO and other mixed valence oxides with robust charge order- to low temperatures (19), robust charge orderings are observed ing in the solid solution phase can lead to charge-order–induced in many iron oxides such as magnetite (20), hexagonal ferrite ferroelectricity. LuFe2O4 (21), and perovskite CaFeO3 (22) and La1/3Sr2/3FeO3 solid solution (23). ferroelectricity | density-functional theory | perovskite-oxide Perovskite solid solution La1/3Sr2/3FeO3 (LSFO) has a superlattices | charge ordering charge-ordered Mott-insulating state as the low-temperature phase. The average valence of Fe is +3.67 assuming +2, +3, and −2 charge states for Sr, La, and O, respectively. For he formulation of new design principles for ferroelectric Tmaterials has recently attracted great interest. A broad principle, proposed in refs. 1 and 2, is to start with a high- Significance symmetry reference phase and combine 2 symmetry-breaking orderings, neither of which separately lifts inversion symme- Charge-order–driven ferroelectrics are an emerging class try, to generate 2 or more polar variants. Perovskite oxides of materials with promise for high-frequency electron- are ideal systems to search for realizations of mechanisms gov- dominated polarization switching, distinct from conventional erned by the abovementioned broad principle. They exhibit ferroelectrics. However, only a few systems exhibiting this various symmetry-lowering lattice instabilities as well as mag- behavior have been experimentally realized thus far. With netic, charge and orbital ordering in bulk phases (3–5). Fur- continued development of layer-by-layer growth techniques thermore, with the recent progress in atomic-scale layer-by-layer with a high level of composition control, the exploration of growth techniques, symmetry-breaking compositional order can charge-ordered ferroelectrics can be extended to artificially be achieved in a wide variety of complex oxide systems via structured superlattices. Here, we use density-functional the- superlattices (6–9). The combination of orderings has thus been ory to explore an experimentally realized bulk perovskite the basis of discovery of several new types of perovskite ferro- iron-oxide solid solution with robust charge ordering and electrics. In particular, in hybrid improper ferroelectricity (10), a find that in superlattices formed by layered cation ordering, lattice distortion that preserves inversion symmetry (typically an bulk charge ordering is maintained and can lead to charge- oxygen-octahedron rotation-tilt pattern) combines with symme- order–driven ferroelectricity. Our results suggest that other try breaking by layering, either in a Ruddlesden–Popper phase broad classes of mixed valence materials may be promising 0 (10) or in a perovskite ABO3/A BO3 (001) superlattice (11–13), candidates for discovery of electronic ferroelectrics. generating polar variants. In these systems, the switching polarization is generated by a polar lattice distortion in the lowered- Author contributions: S.Y.P., K.M.R., and J.B.N. designed research; S.Y.P. performed symmetry state. research; and S.Y.P., K.M.R., and J.B.N. wrote the paper.y Charge-order–driven ferroelectricity can be obtained by com- Reviewers: S.M., Drexel University; and S.P., Consiglio Nazionale delle Ricerche - bining symmetry breaking by charge ordering with symmetry SuPerconducting and other INnovative materials and devices institute (CNR-SPIN).y breaking by layered cation ordering to generate polar variants The authors declare no competing interest.y (14–16). For example, in the 1:1 superlattice LaVO3/SrVO3, Published under the PNAS license.y 3+ 4+ layered charge ordering of V and V combines with the lay- 1 To whom correspondence may be addressed. Email: [email protected] ered ordering of La and Sr to break up–down symmetry normal This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. to the layers, generating 2 polar variants (16). The distinctive 1073/pnas.1906513116/-/DCSupplemental.y characteristic differentiating charge-order–driven ferroelectric- First published November 11, 2019.

23972–23976 | PNAS | November 26, 2019 | vol. 116 | no. 48 www.pnas.org/cgi/doi/10.1073/pnas.1906513116 Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 i.1. Fig. ewe oa rnhsaepeetd hseapeserves al. et example Park This presented. are branches switching polar of 2 details struc- and between phases and low-energy Electronic the of stabilized. centrosymmetric properties be tural both can phases patterns, cation ferroelectric charge-ordering and and the lay- charge-ordered superlattice and ordering the between ering charge registry in the of on Depending cations layering. breaking symmetry La/Sr combined layered of a La that ordering find we superlattice results, (DFT) theory magnetic density-functional by driven to state. be needed ground correlation to insulating and an found distortion produce lattice The are both (29–31). with system state exchange, the ground of electronic the energetics and of ordering, ordering charge and disproportionation, magnetic La/Sr charge for on patterns effects ordering their various considered Hubbard have a studies via with included is 29–31), correlation (26, calculations to disproportionation first-principles through tigation 29). (28, phase resistivity low-temperature the with Mott-insulating as state charge-ordered reported, a with are films consistent thin measurements superlattices (111)-oriented (001)-oriented and (001)- and illustrated Experimentally, as adja- 1C. aligned, in Fig. are moments states in the charge with different and planes with antialigned planes adjacent are cent in state out-of-plane moments charge same the the the which antiferromagnetic in In antiphase pattern an plane. in (APAF) order (111) moments repeated the the each direction, in within stacked plane romagnetically (111) which a of in forms pattern pattern, these state charge-ordering charge to observed each refer the to with continue structure as we states convenience charge for of However, moments 27). magnetic measured the hole Fe ligand indeed some and include (26) configurations character e the ligands, Fe-d charge oxygen of this ing hybridization strong Nominally distor- the 3Fe (23–25). breathing to to octahedra the corresponds by oxygen ordering stabilized the are of where states (CO) tion charge ordering of distinct charge onset and 2 the with (AF) observed antiferromagnetic is both transition metal–insulator a K, 210 T ordering. (C the showing pattern. view rotation Top (B) respectively. 4, distinct 2 and occupying atoms Sr Fe (2a or La positions for the Wyckoff Fe represent are spheres of colors black and octahedron magenta silver The oxygen and surrounding blue and Light ions ordering. magnetic and charge A > nti ae,uigsmer nlssadfirst-principles and analysis symmetry using paper, this In inves- of subject the been has LSFO in ordering Charge 3+ 1/3 1 ,LF smtli ihalF ie qiaet Below equivalent. sites Fe all with metallic is LSFO K, 210 /Fe Sr tmcarneeto La of arrangement Atomic (A) 2/3 5+ FeO Fe ttsaelwrta h oia aus(3 25, (23, values nominal the than lower are states 5+ 3 -Fe ieve hwn h nihs antiferromagnetic antiphase the showing view Side ) Fe eeae lcrncfrolcrct i the via ferroelectricity electronic generates n 4d and 3+ 3+ -Fe Fe and 3+ fthe of ) 3+ /Fe h antcmmnsodrfer- order moments magnetic The . Fe 3.67+ 5+ 5+ C B P ¯ 3c i.1A Fig. . bevdwe nst electron on-site when observed pc ru ihmlilct f2 of multiplicity with group space 1 1/3 = g Sr U Fe a ttsadtesurround- the and states 2/3 − 3+ aaee 3) These (30). parameter 5+ a FeO − ecie h crystal the describes a n Fe and 2Fe + 3 − oi ouinwith solution solid oxygen-octahedron 5+ 3+ respectively. , u due but , atr.Frti ragmn,tesaegopsmer slow- is symmetry noncentrosymmetric group to space ered the arrangement, charge-ordering this the For the pattern. maintains for observed (29), as superlattice which, (001)-oriented planes 2, La-(111) Fig. separated in symmetry. uniformly shown there arrangement, but additional arrangements, one any possible only many break is for not allows type does latter and The unique 2, Fig. clearly in is illustrated type, the first to The species position. atomic Wyckoff one symme- than lowering more assigning other by the 2 try and yields positions Wyckoff This 4) Sr. multiplicity to and La symme- assigning La same by of the try maintaining ratio one arrangements: 1:2 atomic of of types constraint maintain the To satisfy Fe-d cell. to for unit valence 30-atom average and the same oriented within the (111) expressed be be to can expected thus are A-site ordering, orderings layered charge A-site introduce (111)-oriented relevant we the Next, Following 1A. orderings. Fig. cation in spheres (2a black ions and A-site the for to introduce positions symmetry Wyckoff we group First, space which the ordering, 2. lowers charge Fig. the (111)-oriented to in observed due orderings experimentally lowering cation symmetry and the charge analyze We equivalent. sites n ereeti phases. centrosymmetric ferroelectric produce and text the in described arrangements layering lattice the to symmetry the lowers high-temperature the from Starting tion. 2. Fig. metric charge of presence the ordering. in ferroelectricity superlattice-driven of has phase the group high-symmetry along This space 1B). viewed (Fig. octahedral as direction Fe-centered [111]) lattice docubic the triangular a between forming in cages, ions La/Sr the in with rotated pattern are structure corner-connected perovskite The 1. the Fig. of in octahedra shown arrangement solution atomic solid the LSFO with the in starts system this in ordering cation with be solutions can solid which cations. ferroelectric- perovskite-oxide valence principle, electronic complex multiple for design chemically search this in the ity for in applied concept broadly of more proof a as i.3sostecag-reigpten o h centrosym- the for patterns charge-ordering the shows 3 Fig. and ordering charge by breaking symmetry the of analysis Our pc ru nlsso ymtylwrn nteLF oi solu- solid LSFO the in lowering symmetry of analysis group Space P 3c ¯ 1 R n h oa ainso the of variants polar 2 the and 3A) (Fig. phase PNAS 3c ¯ ihalF ie qiaetadalAcation A all and equivalent sites Fe all with , | 2a oebr2,2019 26, November P ¯ 3c wt utpiiy2 n rto Sr and 2) multiplicity (with pc ru.Fo hr,dfeetsuper- different there, From group. space 1 P 3c eosrtn h possibility the demonstrating 1, R tts h ueltieneeds superlattice the states, ¯ 3c P , tutr,tecag ordering charge the structure, 4d | 3c ¯ o.116 vol. ,rpeetda silver as represented ), 1 hthas that Right, Bottom nwihteeae2 are there which in | o 48 no. 2a otmLeft, Bottom and/or a 4d c − | (pseu- a (with 23973 − a 4d −

PHYSICS ABC0.54 meV per Fe higher in energy, with octahedral distortions and magnetic moments for Fe3+ and Fe5+ virtually identical to those in the CS-APAF phase. With respect to the electrostatic energy, the main difference between the 2 different registries of the charge-ordering pattern with respect to the A-site layered ordering is that in the CS phase, an Fe5+ plane lies between the 2 adjacent Sr2+ layers, while in the FE phase, the Fe5+ plane lies between a Sr2+ and a La3+ layer, with a nominally higher electrostatic energy. The tiny computed energy difference implies substantial screening from the oxygen ligands, also seen in other charge-order–induced ferroelectric materials (16). As a result, it should be possible to switch this system to a polar variant with an applied external field with retention of the polar structure when the field is removed, producing a ferroelectric P–E hysteresis loop. Fig. 3. Three charge-ordering patterns for A-cation ordered superlat- Considering magnetic-ordering patterns, for AF and ferrimag- tices. (A) Centrosymmetric charge-ordering pattern. (B and C) Ferroelec- netic (Fi) ordering, we find charge-ordered insulating phases tric charge-ordering patterns with 2 polar variants in which P+/P− are > related by inversion. The Fe layers are numbered 1 to 6. For both the substantially higher in energy ( 100 meV per Fe) than the centrosymmetric and the ferroelectric phases, the c-axis rotations of the CS-APAF phase. The energy increase on changing the magnetic- octahedra in layers 1, 2, and 3 are opposite to those in layers 4, 5, and ordering patterns is consistent with the Goodenough–Kanamori– 6, respectively. Anderson rule (33–35) favoring APAF magnetic ordering. By comparing the total energies of CS-APAF, CS-AF, and CS-Fi phases and assuming S = 5/2 for Fe3+ and S = 3/2 for Fe5+, we ferroelectric P3c1 phase (Fig. 3 B and C). We can see that the calculate the nearest-neighbor exchange coupling constants for main difference between the centrosymmetric and ferroelectric Fe3+-Fe5+ and Fe3+-Fe3+ pairs to be −8.3 meV and 9.5 meV, phases is the registry of the charge ordering with respect to A- respectively, showing strong magnetic interactions consistent site layered ordering. This suggests that electric-field–controlled with inelastic neutron scattering measurements (25). For ferro- switching from the centrosymmetric to the ferroelectric phase magnetic (F) ordering, we find a metallic P3¯c1 phase with no or between 2 polar variants in the ferroelectric phase can occur charge ordering or breathing distortions at an energy of 24 meV through charge transfer between Fe sites. per Fe above the CS-APAF phase, in contrast to pure CaFeO3, With first-principles calculations, we now investigate the total in which a ferromagnetic Fe3+/Fe5+ insulating charge-ordered energies and structural parameters of these superlattice phases. state is found as the ground state (36, 37). In addition, we check the assumption that the charge-ordering To investigate the robustness of the Fe3+/Fe5+ (111) charge pattern of the solid solution is maintained in the A-site ordered ordering, we consider states with other ordering patterns and structures. We generate the alternative charge- and magnetic- disproportionation. Considering states with Fe3+/Fe5+ charge ordering patterns to be considered through a systematic symme- ordering in the (111) plane, we find a locally stable charge- try analysis. The details of this analysis and of the first-principles ordered phase with APAF magnetic ordering that has a signif- approach are discussed in SI Appendix. icantly higher total energy of about 67 meV per Fe relative to Table 1 summarizes the low-energy phases identified. The the CS-APAF phase (see SI Appendix for details). By assuming a lowest-energy phase is the centrosymmetric (CS) P3¯c1 phase 3+ 4+ 3+ 5+ different disproportionation Fe /Fe , we find a locally stable (Fig. 3A) with Fe /Fe charge ordering and APAF magnetic CS-Fi phase, higher in total energy by 51 meV per Fe, which is ordering; it is an insulator with a DFT+U band gap of 0.3 eV, metallic (last column of Table 1). We note that this phase has in reasonable agreement with the experimentally measured band doubly degenerate bands along high-symmetry lines crossing the gap (0.13 eV) (32). This charge-ordering and magnetic-ordering Fermi energy. As discussed in more detail in SI Appendix, these pattern is identical to that observed in the solid solution, confirm- degeneracies are protected by inversion symmetry, and breaking ing the hypothesis that the charge and magnetic ordering would the inversion symmetry by changing the registry of the superlat- be insensitive to the A-site ordering. The oxygen octahedral vol- 3 tice layering and the charge order leads to the FE-Fi2 phase, ume in the relaxed structure of the CS-APAF phase is 9.5 A˚ opening a band gap. 5+ 3 3+ for Fe and 10.5 A˚ for Fe , showing the breathing distortion. Fig. 4A shows the projected density of states (PDOS) of the The Fe3+ ions shift about 0.01 A˚ from the center of the octa- CS-APAF phase. We find that the PDOS of the FE-APAF hedron toward the La plane. The ferroelectric (FE) P3c1 phase phase (Fig. 4B) is almost identical to that of the CS-APAF (Fig. 3 B and C) with the same APAF magnetic ordering is only phase, consistent with the small energy and structural difference,

Table 1. Low-energy phases of LSFO (111) superlattice Charge states 2Fe5+(d3)/4Fe3+(d5) 6Fe3.67+(d4.33) 2Fe3+(d5)/4Fe4+(d4) CO patterns CS FE CS FE CS FE CS CS ∆E/Fe, meV 0 0.54 124 119 243 42 24 51 Phase I I I I I I M M Magnetic ordering APAF APAF AF AF Fi Fi2 F Fi ↑⇑⇓↓⇓⇑ ⇑↑⇑⇓↓⇓ ↑⇓⇑↓⇑⇓ ⇑↓⇑⇓↑⇓ ↑⇓⇓↑⇓⇓ ⇑⇓↑⇑⇓↑ ↑↑↑↑↑↑ ⇑↓↓⇑↓↓

m [Fe1/Fe2/Fe3], µB 3.5/4.0/4.0 4.0/3.5/4.0 2.8/4.1/4.1 4.1/2.9/4.1 2.7/4.2/4.2 3.9/4.1/3.3 4.0/4.0/4.0 4.1/3.7/3.7

For each phase, the charge ordering, total energy per Fe, electronic character, magnetic ordering, and magnetic moments are given. I and M denote insulator and metal, respectively. The symbols F, AF, Fi, and APAF denote ferromagnetic, antiferromagnetic, ferrimagnetic, and antiphase–antiferromagnetic ordering, respectively. For the magnetic-ordering patterns, the arrows represent spin directions and magnitude in which the double arrows denote the Fe3+ charge state and single arrows denote the other charge states (Fe5+ or Fe4+). The calculated magnetic moments presented are for the ﬁrst 3 Fe sites.

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(Fig. transfer Fe the at charge of jumps moments discontinuous the magnetic with the of from ions, nature seen also first-order be The can cur- transition switching. leakage polarization gap, could which band the process, small switching hinder the the during to generated be due may that rents note We with 5C). branches, (Fig. polar gap 2 band the the between a transition of indi- first-order ordering surface a energy charge Born–Oppenheimer cates the the in cusp with The other variant. the the of and ordering change ant inversion the broken with with one states symmetry, degenerate 2 find we arrangements, at Exactly maintained. is the the Similarly, amounts. for of small than relatively by that change larger moments magnetic is slightly pattern only fer- ordering conventional and the 41) in small roelectric depth (40, for potential calculated meV) double-well barriers computed 120 the to than meV (85 25 smaller for is hopping is path polaron the this along Fe; barrier energy per calculated the 5B, Fig. fte2vrat,prmtrzdby parameterized variants, 2 those between the interpolating atomic of linearly fixed by for polarization obtained 5A and arrangements Fig. moments, magnetic in gap, shown band variants polar spin-polarized related fully a with consistent states, d spin-up the of From pation holes, 39). (26, ligand are solution oxygen solid the of the PDOS by for screening discussed previously (38). strong have insulator the eV Fe-d 1 charge-transfer supporting around between a bands hybridization unoccupied is low-lying strong the this that that find We showing eV, Fe-d 0.3 of mostly O-p consist from Fe-d derived occupied valence mainly The eV). (0.24 are smaller slightly bands is gap band the that noting Fe spin-up ake al. et Park Fe unoccupied spin-up of PDOS the (D) for states. unoccupied the for PDOS of O-p and 4. Fig. 5 0 = et ecnie h wthn ewe h inversion- 2 the between switching the consider we Next, state. 0.5 e . g hl h hfsi te ane etr ontcon- not do centers Wannier other in shifts the while 5, h DSof PDOS The . oa est fsae bakln)adobtlPO o Fe-d for PDOS orbital and line) (black states of density Total ≤ hss ( phases. (B) FE-APAF and (A) CS-APAF of orbitals (red) 3+ λ Fe BaTiO ≤ site. 5+ > h hreodrn atr fthe of pattern charge-ordering the 1, 5+ ad around bands ,w n htterlvn Fe-d relevant the that find we 4D), (Fig. 0mVmitie ln h rniinpath transition the along maintained meV 70 3 ie (E site. 2 e)(2.For (42). meV) (20 − Li costeL ae.Ti sspotdby supported is This layer. La the across x DSo unoccupied of PDOS ) Fe eie ad ihabn a about gap band a with bands derived FePO λ 3+ 0 = Vadtecnuto bands conduction the and eV −7 hw elgbeoccu- negligible shows 4E) (Fig. B A 3 ihcnrsmercatomic centrosymmetric with .5, 8 o18mV n hematite and meV) 108 to (88 λ λ 0 = 0 = n O-p and ad oae bv the above located bands .5 0 0 oa ain n the and variant polar ≤ ≤ ycmuigenergy, computing by rmtenmnl2 e nominal the from t λ 2g λ orsodn oa to corresponding ≤ ≤ and tts(i.4C), (Fig. states ssonin shown As 1. λ h charge- the 0.5, 0 = e t 2g g λ ttsfrthe for states λ ≤ and oa vari- polar C 1 = 1 Close-up ) e variant g states polar (blue) states E D C − oetal sflfrapiain eurn high-frequency requiring applications classes broad ferroelectrics, for properties. to switching electronic applied identify useful to be potentially materials can valence and mixed oxides of charge perovskite to to due ited predicted between is polarization transfer switching substantial a 39 54 is polarization as switching charges, in the effective detail Born more calculated in the discussed to contribute do states inpae ehv eosrtdti rnil o charge- for solu- principle solid this the demonstrated La in have Mott-insulating ordering ordered We charge phase. robust tion mixed with of oxides structuring valence artificial of by subject ferroelectricity the order–induced is this group; space polar a work. future in rota- result octahedron additional could oxygen of tions as inclusion such sites. and/or B distortions the sequences symmetry-breaking of one layering on cation center inversion Other an with group, lay- space 2-fold (164) A 2 of with with presence ordering alternating ferroelectric. ers charge the not checkerboard is to of phase combination due resulting The nonpolar the and fact symmetries in rotation is double-perovskite group cubic space the of that and noncentrosymmetric A the single (111)-layered charge- Alternating case, the of structure. this center ordered In inversion the oxides. remove perovskite can ordering of A-cation range wide reported a is in ordering charge ((111)-layered) checkerboard robust o reference. for in transfer the to lel eedneo h antcmmnso h estsajcn othe to adjacent sites Fe 2 the of (A, moments layer magnetic La the of dependence for variants. polar the 2 in of the shown those between variants interpolated polar linearly 2 arrangements atomic for Fe per and 5. Fig. rbt infiatyt h e oaiain ent htthe Fe-d that hybridized note with We associated centers polarization. Wannier net in the shifts to significantly tribute DE B A ncnlso,w rps einpicpefrcharge- for principle design a propose we conclusion, In that note we LSFO, beyond mechanism this extending For µC/cm P P − + oa energy Total ( B) two. the relating transfer electron the showing , to tmcarneet f2ivrinrltdplrvariants, polar inversion-related 2 of arrangements Atomic (A) c λ h ahdln sdana afteqatmo polarization, of quantum the half at drawn is line dashed The A. 2 xsasmn rnhcoc orsodn oteelectron the to corresponding choice branch a assuming axis = Left of for 1 BaTiO PNAS and P λ FeO + Right A (C . | steitroainprmtrrnigfrom ranging parameter interpolation the is 3 0 oebr2,2019 26, November . h eoo nryi ae steeeg of energy the as taken is energy of zero The A. aesrslsi centrosymmetric a in results layers 6 ) .(E ). λ caer.Orapoc sntlim- not is approach Our octahedra. F eedneo h DFT+U the of dependence 43m ¯ ) 1/3 IAppendix SI λ Sr eedneo h oaiainparal- polarization the of dependence C 26 pc ru,tesm as same the group, space (216) 2/3 µC/cm FeO AA | 2 3 4) h antd of magnitude The (43). oprbet 2P to comparable , 0 o.116 vol. BB oi ouini which in solution solid 0 A O 0 6 aesrslsin results layers oee,this However, . | adgp ( gap. band o 48 no. | P 23975 3m ¯ λ O-p D) s = P = − λ 1 0

PHYSICS Materials and Methods octahedral rotation distortions and charge-order patterns. An energy cut- We perform first-principles density-functional theory calculations with the off of 600 eV, k-point sampling on a Γ-centered 6 × 6 × 3 grid, and a force generalized gradient approximation plus U (GGA+U) method using the threshold of 0.01 eV/A˚ are used for full structural relaxation. The sponta- Vienna ab initio simulation package (44, 45). The Perdew–Becke–Erzenhof neous polarization is calculated using the Berry phase method (51) with an parameterization (46) for the exchange-correlation functional and the 8 × 8 × 4 k-point grid. rotationally invariant form of the on-site Coulomb interaction (47) are used with U = 5.4 eV and J = 1.0 eV for the iron d consistent with the val- Data Availability. All data needed to evaluate the conclusions in this paper are present in the main text and/or SI Appendix. ues used in previous work [Ueff (=U − J) = 4.4 eV (29), U = 5 eV, J = 1 eV (30, 31)] and Uf = 11 eV and Jf = 0.68 eV for the La f states to shift the ACKNOWLEDGMENTS. We thank J. Ahn, C. Dreyer, A. Georges, D. Khomskii, La f bands away from the Fermi energy (48). The charge-ordering pat- H.-S. Kim, J.-H. Lee, K. Lee, S. May, A. J. Millis, T. W. Noh, J.-M. Triscone, tern is determined through Wannier center analysis using the Wannier90 D. Vanderbilt, and B.-J. Yang for valuable discussion. This work was sup- package (49). We find that the stability of Fe3+/Fe5+ charge ordering ported by the Theory Field Work Proposal at the Lawrence Berkeley National with APAF magnetic ordering is insensitive to the U values over a broad Laboratory, which is funded by the US Department of Energy (DOE), Office range (SI Appendix). We used the projector-augmented wave method (50) of Science, Basic Energy Sciences, Materials Sciences and Engineering Divi- with pseudopotentials containing 6 valence electrons for O (2s22p4), 14 sion under Contract DE-AC02-05CH11231. This work is also supported by the Molecular Foundry through the US DOE, Office of Basic Energy Sciences, for Fe (3p63d74s1), 11 for La (5s25p65d16s2), and 10 for Sr (4s24d65s2). under the same contract number. This work is also supported by the Insti- For each charge-ordering pattern, we construct a starting crystal structure tute for Basic Science in Korea (Grant IBS-R009-D1), Office of Naval Research with breathing distortions corresponding√ to the√ charge-ordering√ pattern Grant N00014-17-1-2770. Computational resources were provided by the (see SI Appendix for further details). A 2 × 2 × 2 3 simple hexago- DOE (Lawrence Berkeley National Laboratory Lawrencium) and the Rutgers nal unit cell with 6 Fe atoms is chosen to accommodate the relevant University Parallel Computer cluster.

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