Downloaded by guest on September 30, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1906513116 superlattice 1:1 the variants polar in symmetry generate example, to with For ordering ordering cation (14–16). charge layered by by breaking breaking symmetry bining lowered- the in distortion lattice polar state. polar- a symmetry switching by the generated systems, is these ization In variants. phase polar Ruddlesden–Popper generating perovskite a a in in either or (10) layering, by symme- breaking with try combines an pattern) (typically rotation-tilt symmetry ferro- oxygen-octahedron inversion perovskite preserves a via of that (10), ferroelectricity distortion types systems improper lattice new hybrid oxide in several particular, of complex In electrics. been discovery of thus of has variety basis orderings the of wide combination The a (6–9). superlattices in Fur- achieved can (3–5). order be phases compositional symmetry-breaking layer-by-layer bulk atomic-scale techniques, in in growth progress mag- ordering recent exhibit the as orbital with thermore, They well and as principle. charge oxides instabilities netic, broad lattice Perovskite gov- abovementioned symmetry-lowering mechanisms variants. of various the polar realizations by for more high- search symme- erned to or a inversion systems with 2 ideal lifts are start generate separately to symmetry-breaking to which is 2 try, of combine 2, neither and and orderings, phase 1 reference refs. symmetry in proposed principle, T superlattices ferroelectricity charge-order–induced to lead order- can phase charge solution robust ferroelectricity. of solid with the oxides structuring in valence artificial ing mixed that FeO other and between reveal LSFO transfer calculations 39 electron Our is from is octahedra. arising polarization phase value switching significant The computed centrosymmetric a layering. the ground-state and superlattice a phase, charge with the the degenerate to of registry energetically respect the changing with by order broken inver- is which symmetry in (111)-layered phase sion charge-ordered cation ferroelectric A-site a exhibits that LSFO show calculations Our imentally. Park Young Se La charge-ordered in ferroelectricity Superlattice-induced otelyr,gnrtn oa ains(6.Tedistinctive The (16). variants ferroelectric- polar charge-order–driven normal 2 differentiating symmetry generating up–down characteristic break layers, to the Sr and to La of ordering ered of ordering charge layered mixed-valence the- on structuring density-functional artificial use of La effect we solid-solution the the Here, study motivating materials. to far, ory thus new realized for behav- this experimentally need exhibiting been systems have few a ior only of promise, ferroelectrics, frequency. their high at class conventional Despite occur can from emerging switching Picozzi) electron-dominated Silvia distinct where and an May Steven materials, by are reviewed functional 2019; 17, ferroelectrics May review for Charge-order–driven (sent 2019 14, October Rabe, M. Karin by Contributed 94720 CA Berkeley, California, of University Institute, 08854; NJ Piscataway, University, Korea; Rutgers of Republic 08826, Seoul University, a etrfrCreae lcrnSses nttt o ai cec,Sol086 eulco Korea; of Republic 08826, Seoul Science, Basic for Institute Systems, Electron Correlated for Center hreodrdie ereetiiycnb bandb com- by obtained be can ferroelectricity Charge-order–driven aeil a eetyatatdgetitrs.Abroad A interest. ferroelectric great for attracted principles recently design has new materials of formulation he 1/3 | Sr hreordering charge | est-ucinltheory density-functional a,b,c 1/ 2/3 3 Sr ai .Rabe M. Karin , 2/ ABO 3 FeO FeO V 3 3 3+ LF) ytmwl tde exper- studied well system a (LSFO), /A 0 and BO 3 e oeua onr,Lwec eklyNtoa aoaoy ekly A970 and 94720; CA Berkeley, Laboratory, National Berkeley Lawrence Foundry, Molecular 3 d,1 | V c eateto hsc,Uiest fClfri,Bree,C 94720; CA Berkeley, California, of University Physics, of Department 01 ueltie(11–13), superlattice (001) perovskite-oxide 4+ n efe .Neaton B. Jeffrey and , obnswt h lay- the with combines LaVO 3 /SrVO µC/cm 3 2 6 , , c,e,f 1073/pnas.1906513116/-/DCSupplemental hsatcecnan uprigifrainoln at online information supporting contains article This - 1 Ricerche the delle under Published Nazionale interest.y competing Consiglio no declare (CNR-SPIN). y S.P., authors institute The devices and and materials INnovative University; other and performed Drexel SuPerconducting S.Y.P. S.M., research; paper. designed y Reviewers: the wrote J.B.N. J.B.N. and and K.M.R., S.Y.P., and K.M.R., research; S.Y.P., contributions: Author +3, +2, assuming +3.67 is Fe of low-temperature valence and the average as The phase. state Mott-insulating charge-ordered ferrite hexagonal (20), (23). solution magnetite solid as such observed oxides LuFe are iron orderings down many charge solution robust in solid (19), the temperatures in La low observed is to Unlike ordering conditions. charge mate- no these of which class satisfies a is in that family already iron-oxide rials ordering The ordering, phase. charge solution charge robust solid to as the tendency manifested be strong disproportionation. could a which charge be to should tendency there system Further, strong the a ferroelectricity, exhibit charge-order–driven should to due phonon ization by limited not is 18). that (17, timescale given frequency switching devices switching ferroelectrics polarization electronic high-frequency the Such for phase. useful sig- the be to a of proxy might a nature by as polar used accompanied the be is can nal which the This distortion, polarization when lattice switched. switching polar electrons is small the of pattern transfer that ordering interionic is charge from type primarily displacive arises the from ity owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To ra lse fmxdvlnemtrasmyb promising be ferroelectrics. may electronic materials of discovery valence for other charge- mixed candidates that to of suggest lead classes results can broad and Our ordering, maintained ferroelectricity. and cation order–driven is layered ordering ordering by charge charge formed perovskite bulk superlattices robust bulk in with that realized find solution experimentally solid an iron-oxide explore the- to density-functional artificially use ory of to we Here, exploration extended superlattices. the be structured control, can composition ferroelectrics of charge-ordered level With high techniques far. a growth thus with layer-by-layer this realized of development exhibiting experimentally continued electron- systems been few have high-frequency a behavior only for However, conventional class from ferroelectrics. promise distinct switching, emerging with polarization dominated an materials are of ferroelectrics Charge-order–driven Significance eosiesldslto La solution solid Perovskite polar- switchable of observation experimental promote To hresae o r a n ,rsetvl.For respectively. O, and La, Sr, for states charge −2 2 O 4 2) n perovskite and (21), NSlicense.y PNAS b eateto hsc n srnm,SolNational Seoul Astronomy, and Physics of Department d eateto hsc Astronomy, & Physics of Department . CaFeO y 1/3 Sr 3 2/3 www.pnas.org/lookup/suppl/doi:10. 2)adLa and (22) NSLts Articles Latest PNAS f al nryNanoScience Energy Kavli FeO 3 1/2 LF)hsa has (LSFO) 1/3 Sr 1/2 Sr 2/3 VO | FeO 3 f5 of 1 in , 3

PHYSICS A B as a proof of concept for this design principle, which can be more broadly applied in the search for electronic ferroelectric- ity in chemically complex perovskite-oxide solid solutions with multiple valence cations. Our analysis of the symmetry breaking by charge ordering and cation ordering in this system starts with the atomic arrangement in the LSFO solid solution shown in Fig. 1. The corner-connected octahedra of the perovskite structure are rotated in the a−a−a− C pattern with La/Sr ions in between the Fe-centered octahedral cages, forming a triangular lattice as viewed along the c (pseu- docubic [111]) direction (Fig. 1B). This high-symmetry phase has space group R3¯c, with all Fe sites equivalent and all A cation sites equivalent. We analyze the symmetry lowering due to the charge and cation orderings in Fig. 2. First, we introduce the experimentally observed (111)-oriented charge ordering, which Fig. 1. (A) Atomic arrangement of La1/3Sr2/3FeO3 solid solution with lowers the space group symmetry to P3¯c1 in which there are 2 charge and magnetic ordering. Light blue and magenta colors are for Fe ions and surrounding oxygen octahedron of Fe3+ and Fe5+, respectively. Wyckoff positions for A-site ions (2a, 4d), represented as silver The silver and black spheres represent the La or Sr atoms occupying 2 distinct and black spheres in Fig. 1A. Next, we introduce layered A-site Wyckoff positions (2a and 4d) of the P¯3c1 space group with multiplicity of 2 cation orderings. Following the (111)-oriented charge ordering, and 4, respectively. (B) Top view showing the a−a−a− oxygen-octahedron relevant A-site orderings are expected to be (111) oriented and rotation pattern. (C) Side view showing the antiphase antiferromagnetic thus can be expressed within the 30-atom unit cell. To maintain ordering. the same average valence for Fe-d states, the superlattice needs to satisfy the constraint of 1:2 ratio of La and Sr. This yields 2 types of atomic arrangements: one maintaining the same symme- T > 210 K, LSFO is metallic with all Fe sites equivalent. Below try by assigning La to 2a (with multiplicity 2) and Sr to 4d (with 210 K, a metal–insulator transition is observed with the onset of multiplicity 4) Wyckoff positions and the other lowering symme- both antiferromagnetic (AF) and charge ordering (CO) where try by assigning more than one atomic species to the 2a and/or 4d 2 distinct charge states are stabilized by the breathing distor- Wyckoff position. The first type, illustrated in Fig. 2, Bottom Left, tion of the oxygen octahedra (23–25). Nominally this charge is clearly unique and does not break any additional symmetry. ordering corresponds to 3Fe3.67+ = Fe5+ + 2Fe3+, but due The latter type allows for many possible arrangements, but there to the strong hybridization of Fe-d eg states and the surround- is only one arrangement, shown in Fig. 2, Bottom Right, that has ing oxygen ligands, the configurations include some ligand hole uniformly separated La-(111) planes which, as observed for the character (26) and indeed the measured magnetic moments of (001)-oriented superlattice (29), maintains the charge-ordering Fe3+/Fe5+ states are lower than the nominal values (23, 25, pattern. For this arrangement, the space group symmetry is low- 27). However, for convenience we continue to refer to these ered to noncentrosymmetric P3c1, demonstrating the possibility charge states as Fe3+ and Fe5+. Fig. 1A describes the crystal of superlattice-driven ferroelectricity in the presence of charge structure with the observed charge-ordering pattern, in which ordering. each charge state forms a (111) plane stacked in the repeated Fig. 3 shows the charge-ordering patterns for the centrosym- pattern of Fe5+-Fe3+-Fe3+. The magnetic moments order fer- metric P3¯c1 phase (Fig. 3A) and the 2 polar variants of the romagnetically within each (111) plane. In the out-of-plane direction, the moments order in an antiphase antiferromagnetic (APAF) pattern in which the moments in adjacent planes with the same charge state are antialigned and the moments in adja- cent planes with different charge states are aligned, as illustrated in Fig. 1C. Experimentally, (001)- and (111)-oriented thin films and (001)-oriented superlattices are reported, with resistivity measurements consistent with a charge-ordered Mott-insulating state as the low-temperature phase (28, 29). Charge ordering in LSFO has been the subject of inves- tigation through first-principles calculations (26, 29–31), with disproportionation to Fe3+/Fe5+ observed when on-site electron correlation is included via a Hubbard U parameter (30). These studies have considered various ordering patterns for La/Sr and their effects on charge disproportionation, charge ordering, and magnetic ordering of the electronic ground state (29–31). The energetics of the system are found to be driven by magnetic exchange, with both lattice distortion and correlation needed to produce an insulating ground state. In this paper, using symmetry analysis and first-principles density-functional theory (DFT) results, we find that a layered superlattice ordering of La/Sr cations in charge-ordered La1/3Sr2/3FeO3 generates electronic ferroelectricity via the combined symmetry breaking of charge ordering and cation layering. Depending on the registry between the superlattice lay- Fig. 2. Space group analysis of symmetry lowering in the LSFO solid solu- ering and the charge-ordering patterns, both centrosymmetric tion. Starting from the high-temperature R¯3c structure, the charge ordering and ferroelectric phases can be stabilized. Electronic and struc- lowers the symmetry to the P¯3c1 space group. From there, different super- tural properties of the low-energy phases and details of switching lattice layering arrangements described in the text produce centrosymmetric between 2 polar branches are presented. This example serves and ferroelectric phases.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1906513116 Park et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 antcodrn PFAA AF APAF APAF m ordering Magnetic Phase al .Lweeg hsso SO(1)superlattice (111) states LSFO Charge of phases Low-energy 1. Table 3 (Fig. erntwr h apae h ereeti (FE) ferroelectric The plane. La the toward hedron The for reig epciey o h antcodrn atrs h rosrpeetsi ietosadmgiuei hc h obearw denote arrows double the which al. in et Park magnitude and directions spin antiphase–antiferromagnetic and represent ferrimagnetic, arrows antiferromagnetic, sites. the Fe ferromagnetic, 3 denote patterns, APAF magnetic-ordering and Fe the Fi, the AF, For F, respectively. symbols The ordering, respectively. metal, and lator ∆E patterns CO 9.5 is phase CS-APAF the of structure relaxed vol- octahedral the oxygen in The would ordering. ume ordering A-site magnetic the and to charge insensitive the be confirm- that solution, hypothesis solid the the in ing observed that magnetic-ordering to and identical charge-ordering is pattern This (32). eV) band (0.13 measured DFT+U gap experimentally the a with with agreement reasonable insulator in an is it ordering; (CS) centrosymmetric 3 the (Fig. is phase lowest-energy first-principles the of in and discussed analysis are this approach of symme- details systematic magnetic- The a analysis. through and try considered charge- be ordered to alternative A-site patterns the the ordering in generate charge-ordering maintained We is the solution structures. that solid assumption the the of pattern check phases. we superlattice these addition, of In parameters structural and energies occur can phase sites. Fe ferroelectric between the phase transfer in charge ferroelectric variants through the polar 2 to A- between centrosymmetric to or the respect electric-field–controlled with from that ordering switching suggests charge This the ordering. layered of site registry ferroelectric the and is centrosymmetric phases the between difference and main 5, 4, layers ferroelectric in those the to both opposite are For 3 6. and to 2, 1 respectively. 1, 6, numbered the layers phases, are in ferroelectric which layers octahedra the in Fe and variants The centrosymmetric polar inversion. 2 by with related B ( patterns pattern. charge-ordering charge-ordering tric Centrosymmetric (A) tices. 3. Fig. ABC [Fe1/Fe2/Fe3], o ahpae h hreodrn,ttleeg e e lcrnccaatr antcodrn,admgei oet r ie.IadMdnt insu- denote M and I given. are moments magnetic and ordering, magnetic character, electronic Fe, per energy total ordering, charge the phase, each For The identified. phases low-energy the summarizes 1 Table total the investigate now we calculations, first-principles With e meV /Fe, Fe Fe 3+ 5+ with A) B 3+ he hreodrn atrsfrActo ree superlat- ordered A-cation for patterns charge-ordering Three hresaeadsnl rosdnt h te hresae (Fe states charge other the denote arrows single and state charge n 10.5 and and ossitaot0.01 about shift ions P ihtesm PFmgei reigi only is ordering magnetic APAF same the with C) 3c Fe µ B 1 3+ A ˚ hs Fg 3 (Fig. phase 3 /Fe ././. ././. ././. ././. ././. ././. 4.0/4.0/4.0 3.9/4.1/3.3 2.7/4.2/4.2 4.1/2.9/4.1 2.8/4.1/4.1 4.0/3.5/4.0 3.5/4.0/4.0 for ⇑↓⇑⇑⇑↓ ⇓↓⇓⇑⇑↑ ⇓↑⇓⇑↑⇓ ↑↑↑↑↑↑ ⇑⇓↑⇑⇓↑ ↑⇓⇓↑⇓⇓ ⇑↓⇑⇓↑⇓ ↑⇓⇑↓⇑⇓ ⇑↑⇑⇓↓⇓ ↑⇑⇓↓⇓⇑ 5+ CS Fe 0 I IAppendix. SI hreodrn n PFmagnetic APAF and ordering charge 3+ hwn h rahn distortion. breathing the showing , rmtecne fteocta- the of center the from A ˚ B and .41419243 119 124 0.54 FE I .W a e htthe that see can We C). adgpo . eV, 0.3 of gap band c ai oain fthe of rotations -axis 2Fe and CS I 5+ P P C (d 3c ¯ 3c Ferroelec- ) P + 3 1 )/4Fe 1 /P phase phase − 3+ A are ˚ AF FE (d 3 I 5 ) 5+ sams dnia ota fteCS-APAF the of FE-APAF that the to difference, structural of identical and energy small almost PDOS the with is the consistent phase, 4B) that (Fig. find phase We phase, phase. FE-Fi2 CS-APAF the to leads order gap. charge band superlat- a the the opening of and registry layering the changing tice by breaking symmetry and inversion symmetry, the inversion in by detail has protected more phase are in degeneracies this discussed that the As is crossing note energy. lines which Fermi We high-symmetry Fe, along 1). per bands Table meV degenerate doubly 51 of by column energy (last total metallic in to higher phase, relative CS-Fi Fe per charge- disproportionation meV different 67 stable signif- (see about locally a phase of CS-APAF has a the energy that total find ordering higher magnetic we icantly APAF plane, with (111) phase and ordered the patterns in ordering with ordering other states with Considering states disproportionation. consider we ordering, 37). (36, state ground the as found pure is to state ferromagnetic contrast a in which meV phase, 24 CS-APAF in of the energy above an Fe at per metallic distortions breathing a or find ordering ferro- we charge For ordering, consistent (25). (F) measurements magnetic interactions scattering neutron magnetic inelastic with strong showing respectively, for constants coupling Fe exchange CS-Fi nearest-neighbor and the CS-AF, calculate CS-APAF, By assuming of ordering. and energies phases magnetic total APAF the the favoring comparing (33–35) than rule Fe) Goodenough–Kanamori– Anderson the per with magnetic- consistent the is meV patterns changing ordering on phases (>100 increase energy insulating The energy phase. charge-ordered CS-APAF in find higher we substantially ordering, (Fi) netic ferroelectric variant a polar struc- producing polar a removed, the is to loop. of hysteresis field system retention the with this when field switch ture external to applied an possible result, with be a As should (16). other it in materials implies seen also ferroelectric difference ligands, charge-order–induced oxygen energy the computed from screening tiny substantial The energy. trostatic a between layered an A-site phase, CS adjacent the 2 the to in that respect is of with ordering registries electrostatic different pattern the 2 charge-ordering the to the between respect difference With main the phase. distortions energy, CS-APAF octahedral the with in energy, those for in moments higher magnetic Fe and per meV 0.54 rFe or i.4 A Fig. the of robustness the investigate To ferrimag- and AF for patterns, magnetic-ordering Considering 3+ -Fe 4+ CS Fi 5+ I .Tecluae antcmmnspeetdaefrtefirst the for are presented moments magnetic calculated The ). hw h rjce est fsae PO)o the of (PDOS) states of density projected the shows Sr Sr and 2+ 2+ Fe aes hl nteF hs,the phase, FE the in while layers, n a and 3+ Fi2 42 FE -Fe I S La 5 = 3+ 3+ IAppendix SI Fe /2 ar obe to pairs Fe ae,wt oial ihrelec- higher nominally a with layer, 3+ Fe 6Fe for 3+ 3+ /Fe 3.67+ Fe and /Fe 5+ 24 CS M F 3+ Fe (d 4+ o eal) yasmn a assuming By details). for nuaigcharge-ordered insulating Fe 5+ 4.33 and . e n . meV, 9.5 and meV −8.3 efidalclystable locally a find we , Fe NSLts Articles Latest PNAS 5+ ln isbtenthe between lies plane 2Fe ) 3+ S P ital dnia to identical virtually /Fe 3c ¯ 3 = these Appendix, SI Fe 1 5+ 3+ /2 3+ Fe hs ihno with phase 4.1/3.7/3.7 /Fe 11 charge (111) (d ⇑↓↓⇑↓↓ 5+ for 5 5+ CS 51 )/4Fe M Fi ln lies plane CaFeO Fe charge | 5+ 4+ P f5 of 3 we , (d –E 3 4 ) ,

PHYSICS tribute significantly to the net polarization. We note that the A C shifts in Wannier centers associated with hybridized Fe-d O-p states do contribute to the calculated Born effective charges, as discussed in more detail in SI Appendix (43). The magnitude of 2 the switching polarization is 39 µC/cm , comparable to 2Ps = 2 D 54 µC/cm of BaTiO3. For extending this mechanism beyond LSFO, we note that robust checkerboard ((111)-layered) charge ordering is reported B in a wide range of perovskite oxides. In this case, (111)-layered A-cation ordering can remove the inversion center of the charge- E ordered structure. Alternating single A and A0 layers results in the noncentrosymmetric F 43¯ m (216) space group, the same as 0 0 that of the cubic double-perovskite AA BB O6. However, this space group is in fact nonpolar due to the presence of 2-fold rotation symmetries and the resulting phase is not ferroelectric. The combination of checkerboard charge ordering with 2 A lay- 0 ¯ Fig. 4. Total density of states (black line) and orbital PDOS for Fe-d (blue) ers alternating with A layers results in a centrosymmetric P3m1 and O-p (red) orbitals of CS-APAF (A) and FE-APAF (B) phases. (C) Close-up (164) space group, with an inversion center on one of the B sites. of PDOS for the unoccupied states. (D) PDOS of unoccupied t2g and eg states Other cation layering sequences and/or inclusion of additional 5+ for the spin-up Fe site. (E) PDOS of unoccupied t2g and eg states for the symmetry-breaking distortions such as oxygen octahedron rota- spin-up Fe3+ site. tions could result in a polar space group; this is the subject of future work. In conclusion, we propose a design principle for charge- noting that the band gap is slightly smaller (0.24 eV). The valence order–induced ferroelectricity by artificial structuring of mixed bands are mainly derived from O-p bands located above the valence oxides with robust charge ordering in the solid solu- occupied Fe-d bands around −7 eV and the conduction bands tion phase. We have demonstrated this principle for charge- consist mostly of Fe-d derived bands with a band gap about ordered Mott-insulating La1/3Sr2/3FeO3 solid solution in which 0.3 eV, showing that this is a charge-transfer insulator (38). a substantial switching polarization is predicted due to charge We find that the low-lying unoccupied bands around 1 eV have transfer between FeO6 octahedra. Our approach is not lim- strong hybridization between Fe-d and O-p states (Fig. 4C), ited to perovskite oxides and can be applied to broad classes supporting the strong screening by the oxygen ligand holes, of mixed valence materials to identify electronic ferroelectrics, previously discussed for the solid solution (26, 39). From the potentially useful for applications requiring high-frequency PDOS of Fe5+ (Fig. 4D), we find that the relevant Fe-d states switching properties. 3+ are eg . The PDOS of Fe (Fig. 4E) shows negligible occu- pation of spin-up states, consistent with a fully spin-polarized 5 d state. A Next, we consider the switching between the 2 inversion- related polar variants shown in Fig. 5A by computing energy, band gap, magnetic moments, and polarization for fixed atomic arrangements obtained by linearly interpolating between those of the 2 variants, parameterized by 0 ≤ λ ≤ 1. As shown in Fig. 5B, the calculated energy barrier along the path is 25 meV per Fe; this is smaller than the barriers calculated for small B C polaron hopping for Lix FePO3 (88 to 108 meV) and hematite (85 to 120 meV) (40, 41) and only slightly larger than the computed double-well potential depth in the conventional fer- roelectric BaTiO3 (20 meV) (42). For 0 ≤ λ ≤ 0.5, the charge- ordering pattern is that of the λ = 0 polar variant and the magnetic moments change by relatively small amounts. Similarly, for 0.5 ≤ λ ≤ 1, the charge-ordering pattern of the λ ≤ 1 variant DE is maintained. Exactly at λ = 0.5, with centrosymmetric atomic arrangements, we find 2 degenerate states with broken inversion symmetry, one with the change ordering of the λ = 0 polar vari- ant and the other with the charge ordering of the λ = 1 polar variant. The cusp in the Born–Oppenheimer energy surface indi- cates a first-order transition between the 2 polar branches, with a band gap > 70 meV maintained along the transition path (Fig. 5C). We note that due to the small band gap, leakage cur- rents may be generated during the switching process, which could Fig. 5. (A) Atomic arrangements of 2 inversion-related polar variants, P− hinder the polarization switching. The first-order nature of the and P+, showing the electron transfer relating the two. (B) Total energy transition can be also seen from the magnetic moments of the Fe per Fe for atomic arrangements linearly interpolated between those of the ions, with discontinuous jumps at λ = 0.5 from the nominal 2e− 2 polar variants shown in A. The zero of energy is taken as the energy of charge transfer (Fig. 5D). This coincides with the discontinuous the 2 polar variants. λ is the interpolation parameter ranging from λ = 0 for P to λ = 1 for P .(C) λ dependence of the DFT+U band gap. (D) λ jump in the polarization, shown in Fig. 5E corresponding to a − + − dependence of the magnetic moments of the 2 Fe sites adjacent to the charge transfer of 2e across the La layer. This is supported by La layer (A, Left and Right). (E) λ dependence of the polarization paral- the Wannier center interpretation of the switching polarization lel to the c axis assuming a branch choice corresponding to the electron in that 2 Wannier centers shift from one Fe site to the other at transfer in A. The dashed line is drawn at half the quantum of polarization, λ = 0.5, while the shifts in other Wannier centers do not con- for reference.

4 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1906513116 Park et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 6 .Matsuno J. 26. 4 .Q i .Mtu,S .Pr,Y oua hreodrdsae nLa McQueeney in J. states R. ordered Charge Tokura, 25. Y. Park, K. S. Matsui, Y. Li, Q. J. 24. disproportion- charge of consequences structural The Lightfoot, P. Gibb, T. in Battle, disproportionation P. Charge Takada, 23. T. Naka, S. Takeda, Y. Nakanishi, N. Takano, M. 22. orbital and charge by induced ferroelectricity Electronic Barone, P. Yamauchi, K. frustration. 18. and ferroelectricity Electronic Ishihara, S. 17. new to path A rotations: octahedral Polar Fennie, J. C. Mulder, T. A. for Benedek, A. mechanism N. A 12. ferroelectricity: improper Hybrid Fennie, J. C. Benedek, A. N. 10. 1 .Ikeda N. (Fe magnetite 21. of conduction Electronic Verwey, W. J. E. 20. properties electronic of Change Tokura, Y. Katsufuji, T. Ishikawa, T. Arima, T. Inaba, F. 19. LaVO in ferroelectricity Charge-order-induced Rabe, M. K. Kumar, A. Park, Y. S. 16. Kobayashi K. 15. Fe magnetite multiferroic in Ferroelectricity Picozzi, S. of Fukushima, control T. Yamauchi, electrical K. and Coupling 14. Ghosez, P. Bousquet, E. Bristowe, C. N. Varignon, J. 13. Bousquet E. 11. ihbetigdsotoscrepnigt h hreodrn pattern charge-ordering the structure to crystal corresponding (see starting distortions a construct breathing we with pattern, charge-ordering each For ake al. et Park relevant the accommodate to chosen is atoms Fe 6 with cell unit nal (2s O for electrons (3p valence Fe 6 for containing Fe pseudopotentials the of with to insensitive stability Appendix is (SI the range ordering Wannier90 that magnetic the APAF find using with We analysis center (49). Wannier package through determined is tern La and [U 31)] work (30, previous in the used used ues are and (47) interaction functional Coulomb on-site with exchange-correlation the of the form Perdew–Becke–Erzenhof invariant rotationally for The 45). (46) (44, package parameterization simulation initio ab plus Vienna the approximation with gradient calculations theory generalized density-functional first-principles perform We Methods and Materials .J hkain .W read .J ils .Pngpuo,J .Rondinelli, M. from J. perovskites noncentrosymmetric of Panagopoulos, Design Rondinelli, C. M. I. Lalkiya, P. Millis, Young, J. J. 9. A. Freeland, W. J. Chakhalian, J. 8. complex in Hwang Y. physics H. Interface Triscone, M. 7. J. Ghosez, P. Gabay, M. Gariglio, S. Zubko, P. 6. Khomskii, I. D. 5. oxides. transition-metal in physics Orbital Nagaosa, transitions. N. Tokura, Y. Metal-insulator Tokura, 4. Y. Fujimori, A. 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O Lwec eklyNtoa aoaoyLwecu)adteRutgers the the cluster. and Computer by Lawrencium) Parallel provided Laboratory University National Insti- were Berkeley the Research (Lawrence resources Naval by DOE of Computational supported Office IBS-R009-D1), also N00014-17-1-2770. (Grant is Korea Grant in work Science This by Basic Sciences, number. for Energy supported tute Basic contract of also same Office DOE, is the US under work the Divi- through This Engineering Foundry Molecular and DE-AC02-05CH11231. the Office Sciences Contract (DOE), Materials under Triscone, Energy Sciences, of sup- J.-M. sion Energy Department was Noh, Basic US W. Science, work the National T. by of This Berkeley funded Lawrence Millis, discussion. the is at J. which valuable Proposal Laboratory, Work A. for Field May, Theory Yang the S. by B.-J. ported Lee, and K. Vanderbilt, Lee, D. J.-H. Kim, H.-S. ACKNOWLEDGMENTS. and/or text main the in present are Availability. Data an with (51) method phase Berry the 8 using calculated is polarization neous eV/ 0.01 cut- of energy threshold An eV, patterns. 600 of charge-order off and distortions rotation octahedral × lvn Li olivine Matter Condens. Phys. lcrncsrcueo La of structure electronic insulators. Mott-Hubbard (1995). R5470 in ordering Orbital interactions: simple. method. wave set. basis plane-wave a using calculations structure. bilayer and pseudocubic (1992). hematite. in mobility small-polaron of calculation (2006). 104301 73, compounds. transition-metal ferrites. alkaline-earth (2018). in transition metal-insulator ordered oxides. perovskite ordering” Solids Chem. Phys. J. [La, (1950). 350–356 79, La in transition ordering La charge-ordered (2005). (2006). B (2014). functions. Wannier maximally-localised CaFeO in states ground (2016). 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