Charge transfer and hybrid ferroelectricity in (YFeO 3 ) n /(YTiO 3 ) n magnetic superlattices
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Charge transfer and hybrid ferroelectricity in (YFeO ) n /(YTiO ) n magneticsuperlattices Huimin Zhang, Yakui Weng, Xiaoyan Yao, and Shuai Dong ∗ Department of Physics, Southeast University, Nanjing 211189, China (Dated: May 25, 2015)Interfaces in oxide heterostructures always provide a fertile ground for emergent properties.Charge transfer from a high energy band to a low energy opponent is naturally expected, as oc-curring in semiconductor p - n junctions. In this study, several exceptional physical phenomena havebeen predicted in (YFeO ) n /(YTiO ) n superlattices. First, the charge transfer between these Mottinsulators is in opposite to the intuitive band alignment scenario. Second, hybrid ferroelectricitywith a moderate polarization is generated in the n = 2 magnetic superlattice. Furthermore, theferroelectric-type distortion can persist even if the ( AB O ) /( AB ’O ) system turns to be metallic,rending possible metallic ferroelectricity. PACS numbers: 73.20.At, 75.70.Cn, 77.55.Nv
I. INTRODUCTION
Oxide heterostructures provide a unique and fertileground to study emergent physics of correlated electronsand a promising route to design new devices using quan-tum effects. At the interfaces between oxides, collec-tive behaviors of electrons can be contrastively differentfrom their original roles in parent materials . For ex-ample, previous studies have revealed plethoric phenom-ena relevant to electronic reconstructions, e.g. metal-insulator transition and enhanced N´eel temperaturesin LaMnO /SrMnO superlattices , two-dimensionalelectronic gas and superconductivity in LaAlO /SrTiO heterostructures , orientation-dependent magnetism inLaNiO /LaMnO and LaFeO /LaCrO superlat-tices.Charge transfer is one important driving force for elec-tronic reconstructions, which tunes the local electrondensity as well as the interfacial electrostatic field. Inthe light of the band alignment scenario, the heights ofFermi levels are the decisive factor for charge transfer.Naturally, to reach a uniform chemical potential acrossthe interface, electrons will leak from the high energyband to the low energy opponent, which is well known insemiconductor p - n junctions.Different from the nearly-free electrons, the bandsof correlated electrons are not strictly rigid but some-what “soft” , which may lead to emergent phenom-ena beyond the simple band theory. In this work, the(YFeO ) n /(YTiO ) n superlattices have been studied us-ing the density functional theory (DFT). Several un-expected physical phenomena, e.g. the charge transferagainst the band alignment scenario and hybrid mul-tiferroicity, have been predicted. In addition, due tothe origin of improper component of polarization, theferroelectric-type distortion in ( AB O ) /( AB ’O ) su-perlattices can persist even if the system turns to bemetallic. II. MODEL & METHOD
Both YFeO and YTiO are Mott insulators . Byneglecting the weak ferromagnetism due to tiny spincanting, the magnetic ground state of YFeO is antifer-romagnetic (AFM) . In contrast, YTiO is ferromag-netic (FM) . These two materials share the commonA-site cation Y , as well as the identical space groupof crystal structure (orthorhombic P bnm , see Fig. 1(a)).Their lattice constants ( a , b , c ) in unit ˚A are: (5 . . . and (5 . . . . The proximate structures allow a high possi-bility for epitaxial growth of multilayers. In the follow-ing, the (YFeO ) n /(YTiO ) n superlattices are assumedto be grown on the mostly-used SrTiO (001) substrate.To match the substrate, the in-plane lattice constants ofYFeO and YTiO are fixed as 3 . × √ . ab initio Simulation Package (VASP) basedon the generalized gradient approximation (GGA). TheHubbard U eff (= U − J ) is imposed on Fe’s and Ti’s d orbitals using the Dudarev implementation . In theGGA+ U calculation, the plane-wave cutoff is 550 eV. A7 × × k -point mesh centered at Γ pointis adopted for (YFeO ) /(YTiO ) and the parent mate-rials, while it is 6 × × ) /(YTiO ) . Boththe lattice constant along the c -axis and inner atomic po-sitions are fully relaxed till the Hellman-Feynman forcesare all below 0 .
01 eV/˚A.For comparison, the hybrid functional calcula-tions based on the Heyd-Scuseria-Ernzerhof (HSE)exchange are also performed. Due to its extremedemand of CPU-time, the plane-wave cutoff is reducedto 400 eV. And the k -point mesh is reduced to 3 × × ) /(YTiO ) and the parent materials, whileit is 3 × × n = 2 superlattice. III. RESULTS & DISCUSSIONA. Band alignment
Before the study on superlattices, the parent materialsare checked. According to the previous literature ,proper U eff values U Fe = 4 eV and U Ti = 3 . andYTiO are G-type AFM (G-AFM) and FM, respectively.Both Fe and Ti are in the high-spin states. Therelaxed lattice constants also agree with the experimentalvalues quite well . All these results guarantee thereliability of following calculations on superlattices.Then the substrate strain is imposed. The G-AFMorder persists in strained YFeO on SrTiO substrate.However, for YTiO , the strain can drive a magnetictransition to A-type antiferromagnet . The atomic pro-jected density of states (PDOS) of strained YFeO andYTiO are displayed in Fig. 1(b) and 1(c), respectively.Both materials retain insulating with energy gaps of ∼ . and ∼ . .According to Fig. 1(b-c), the topmost valence bandof YTiO is from Ti’s one t orbital (whose position isdenoted as µ Ti ), and the bottommost conducting bandof YFeO is formed by the spin-down t orbitals ofFe (whose position is denoted as µ Fe ). Both these twobands are very narrow, implying localized states. Byaligning the deep energy bands of Y’s 4 p and O’s 2 s orbitals which should be identical in these two materi-als, a band alignment can be obtained straightforwardly.Ti’s occupied t band just locates within the forbiddengap of YFeO , lower than the unoccupied conductingband of YFeO . Mathematically, it can be expressedas µ Ti < µ Fe . Thus an intuitive conclusion is that thecharge transfer should be forbidden between these twomaterials, keeping the original Fe -Ti configurationacross the interface. This band alignment is further con-firmed in the HSE calculations (Fig. 1(d-e)).Even if the above U Fe = 4 eV and U Ti = 3 . U eff in a wider parameter space. By varying U eff ’s, the alignedband positions are summarized in Fig. 1(f). In general, µ Ti decreases with increasing U eff , while µ Fe increaseswith increasing U eff . This tendency can be understoodbased on the Hubbard model, since the occupied Ti’s t band is the lower-Hubbard band while the unoccupiedFe’s band is the upper-Hubbard one. Then, the intensityof charge transfer can be determined by comparing µ Ti and µ Fe , as summarized in Fig. 1(g). When both U Ti and U Fe are large, there is no charge transfer, as illustrated inFig. 1(b-c). In contrast, while in the small U Ti and U Fe limit, a partial or complete charge transfer should occur. B. Charge transfer
Above analyses on charge transfer were based onthe band alignment scenario. To verify this scenario,(YFeO ) n /(YTiO ) n superlattices ( n = 1 and 2) arestudied. After relaxing the crystal structures, the mag-netic ground states of superlattices are determined. Inboth superlattices, the magnetic moment of Ti is (al-most) quenched, while the YFeO layers retain the G-AFM configuration with a suppressed moment ∼ . µ B /Fe. This magnetic reconstruction is due to the chargetransfer, since Ti is non-magnetic and the moment ofhigh-spin Fe is 4 µ B /Fe. This scenario is further con-firmed by the PDOS’s (Fig. 2(a-b)). One of Fe’s upperHubbard bands is occupied by one electron, while Ti’s 3 d bands are fully empty, implying a complete charge trans-fer. Such a complete charge transfer is further confirmedusing the HSE calculation (Fig. 2(c) and (d)).For physical comparison, the GGA+ U calculationswith lower U eff ’s have also been done. The charge trans-fer always occurs in superlattices’ calculation despite thevalue of U eff ’s. In other words, the prediction of chargetransfer is not U eff -sensitive. In addition, our GGA+ U and HSE calculations give consistent results regardingthe charge transfer, which are also in agreement with theprevious LDA/LDA+ U calculations and X-ray photoe-mission experiment on similar system LaTiO /LaFeO superlattices . Thus the prediction is not an artefact ofthe level of treatment of electronic correlations.Then how to understand such an unexpected chargetransfer, which violates the band alignment scenario?Previously, similar charge transfer in LaTiO /LaFeO su-perlattices was attributed to the transition of Fe fromthe high-spin state to low-spin state . However, in ourcase, Fe remains in the high-spin state, giving ∼ µ B per Fe. In this sense, such a theoretical argument basedon the low-spin state can not interpret the “anomalous”charge transfer predicted here.One may suspect that such an unexpected chargetransfer is due to the reduced dimension of YFeO andYTiO layers in superlattices. The band alignmentshown in Fig. 1(b-c) is obtained according to the three-dimensional bulks, which may be reshaped into ultra-thinlayers. Taking a tight-binding model for illustration, thereduced dimension can only tune the bandwidths but notthe (central) positions of bands. According to Fig. 1(f),the shrinking of bandwidths can not reverse the bandalignment when U Ti = 3 . U Fe = 4 eV. Thus,pure dimension reduction is not enough.Dimension reduction can also tune the Mottness bytuning the ratio of bandwidth and Hubbard U . Then thesplitting between the upper and lower Hubbard bandscan be shifted, which may alter the band alignment.However, this possibility can also be simply ruled out.The dimension reduction can only suppress the kineticenergy and thus prefers the Mottness, equivalent to in-crease Hubbard U . According to Fig. 1(f), the chargetransfer is suppressed to zero with enhanced U . In short, -4-2024-4-2024-4-2024 -24 -22 -20 -18 -8 -6 -4 -2 0 2 4 6-4-2024 U Fe =4 eV (e)(d) (b) Y Ti O U Ti = 3.2 eV (c) Y Ti OHSEHSE
Y Fe O Y Fe O P D O S Energy(eV) 0.0000.25000.50000.75001.000 U Fe (eV) U Fe (eV)U Ti (eV) (g) -2-1012 U T i ( e V ) E ne r g y ( e V ) YFO
YTO(f) Ti Fe FIG. 1. (Color online) Properties of YTiO and YFeO . (a) Sketch of the common structure. (b-e) Projected density of states(PDOS) for YTiO [(b) and (d)] and YFeO [(c) and (e)]. The Fermi level in (b) is set as zero and the deep energy bands ofY’s 4 p (around −
23 to −
22 eV) and O’s 2 s orbitals (around −
19 to − . U . (d-e)Obtained using HSE. (f) After the treatment of band alignment, the energy positions of near-Fermi-level bands of YTiO (greenupper-triangles, µ Ti ) and YFeO (pink lower-triangles, µ Fe ) as functions of U Ti (upper horizontal axis) or U Fe (lower horizontalaxis). In addition, the band edges of 1) topmost valence band of YFeO (pink rhombs) and 2) bottommost conducting band ofYTiO (green circles) are also presented. Occupied/empty bands are marked by solid/open symbols, respectively. (g) Contourof charge transfer as a function of U Fe and U Ti according to the band alignment. The numerical label is on the top of box. -4-2024-4-2024-4-2024-26 -24 -22 -20 -18 -8 -6 -4 -2 0 2-4-2024 P D O S Y Fe Ti O n=2
Energy(eV) (d) n=1n=1(a)(c) U Fe =4 eVU Ti =3.2 eV(b) HSE n=2
FIG. 2. (Color online) PDOS of superlattices. (a-b) GGA+ U .(c-d) HSE. The Fermi level is positioned at zero (blue brokenline). For n = 1, the deep Y’s 4 p and O’s 2 s bands are similarto their bulk’s correspondences, while for n = 2, clear split-tings are observed due to the electrostatic potential createdby the charge transfer. the dimension reduction can not explain the unexpectedcharge transfer.All above evidences suggest that the reconstruction ofHubbard bands is the intrinsic origin for such an “anoma- lous” charge transfer. As illustrated in Fig. 2, for bothTi and Fe, the energy splittings between upper and lowerHubbard bands become much smaller than those in par-ent materials. The Fe , with half-filling 3 d orbitals, isthe most prominent candidate for the Hubbard repulsionbetween the spin-up and spin-down bands. However, theHubbard repulsion will be suppressed when the electrondensity deviates from the half-filling, as in Fe . Similarmechanism works for Ti ions. In other words, the bandsof YFeO and YTiO are not rigid neither but ratherfragile against (virtual) hopping of electrons. Namely,although µ Fe > µ Ti , both ∂µ Fe /∂n and ∂µ Ti /∂n are neg-ative, where ∂n denotes charge transfer from Ti to Fe.Similar mechanism should also work inLaFeO /LaTiO , which induces the charge transferand low-spin state of Fe . The low-spin state of Fe does not occur in our calculation due to the structuraldifference between LaFeO and YFeO . FollowingRef. 29, we can reproduce the transition from thehigh-spin state to low-spin state in LaFeO by addingone more electron to Fe. However, the same treatmentshows that the high-spin state of YFeO is more stable.The highly distorted structure of YFeO reduces thebandwidth of 3 d orbitals, which prefers the Mottnessand Hubbard splitting between spin-up and spin-down.In fact, the high-pressure experiment also found thehigh-spin state of orthoferrites is more stable when theA-site ion is small .Finally, it should be noted that although the Fe -Ti configuration also occurs in other systems, e.g. FeTiO and Ti doped Fe O , the involved mechanism maybe not simply identical. Although the electronegativity ofdifferent ions usually plays a crucial role , other factors,especially the coordination environment and correlation,will also tune the charge transfer especially when thealignment between involved bands is subtle, as demon-strated in the present work. C. Hybrid ferroelectricity
It was predicted that improper ferroelectric-ity could emerge in perovskite superlattices like(PbTiO ) /(SrTiO ) , (LaFeO ) n /(YFeO ) n , aswell as the layer perovskites A B O , due to themodulation of non-polar antiferrodistortive modes. Forthese ( AB O ) n /( A ′ B O ) n , the ferroelectric polarizationappears to be nonzero when n is an odd number, whileit is fully compensated between layers for even n ’s .Our superlattices, in the type of ( AB O ) n /( AB ′ O ) n ,is somewhat different. First, the crystal structures areanalyzed. The space group of n = 1 superlattice is P /c , but that of n = 2 is P mc . The point group of P /c is 2 /m , and mm P mc . 2 /m belongs to the non-polar point group while mm b -axis. Therefore, the n = 1superlattice should be non-ferroelectric while a finite po-larization is expectable in the n = 2 superlattice. Such aprediction based on symmetry analysis should be phys-ical robust and qualitatively independent on details ofnumerical calculations (e.g. U eff ).The standard Berry phase method implemented inVASP is employed to evaluate the ferroelectric polariza-tion. As expected, the calculated polarization is zerofor the n = 1 superlattice, but in the case of n = 2, anet polarization up to ∼ . µ C/cm is obtained alongthe b -axis. These results can be verified using piezoelec-tric force microscopy or optical second harmonic gener-ation, as done in the LaFeO /YFeO superlattices . Inaddition, the intuitive point charge model can also beemployed to estimate the approximate value of polar-ization, which gives ∼ . µ C/cm , in agreement withthe corresponding Berry phase values qualitatively. Thequantitative difference is reasonable, since the Born ef-fective charge of ions can be different from their nominalvalences.The ferroelectricity, together with the magnetic order-ing of Fe, makes the n = 2 YFeO /YTiO superlatticemultiferroic. It is noteworthy that the previously stud-ied n = 2 LaFeO /LaTiO superlattice is non-magneticdue to the low-spin state of Fe . In this sense, theYFeO /YTiO superlattices can provide more physicalfunctions.The origin of polarization can be visualized in Fig. 3.Due to the antiferrodistortive mode, both Y and O − will move away from their corresponding high-symmetricpositions in the high-temperature cubic structure. In the n = 1 superlattice, all displacements have their asymmet-ric opponents nearby (Fig. 3 (a)), thus can not generatea net polarization, as in the parent materials. In con-trast, in the n = 2 superlattice, the sequence of B-site FIG. 3. (Color online) Sketch of ferroelectric distortions. Thearrows denote the displacements of Y . (a) n = 1. The dis-placements are compensated between layers. The (b) positiveand (c) negative ferroelectric distortion for n = 2. cations forms ...-Ti-Ti-Fe-Fe-... along the c -axis. Thissequence is analogous to the ...- ↑ - ↑ - ↓ - ↓ -... spin struc-ture in some type-II multiferroics (e.g. Ca CoMnO , o-HoMnO , and BaFe Se ), which owns parity. Thisparity, together with the antiferrodistortive mode, breaksthe space inversion symmetry . The displacements ofY (and O − ) sandwiched between the Ti-Ti (or Fe-Fe)bilayers and Ti-Fe bilayers are no longer asymmetric, giv-ing rise to a net in-plane polarization. Such distortionscan be reversed by reversing the antiferrodistortive pat-tern, as illustrated in Fig. 3(b) and 3(c). Then the ferro-electric polarization is switched from positive to negative,as confirmed in the Berry phase calculation.Considering the similarity and difference, our resulton ( AB O ) n /( AB ’O ) n is complementary to the previ-ous studied ( AB O ) n /( A ’ B O ) n , completing the theoryof improper ferroelectricity in perovskite superlatticesdriven by the modulation of non-polar antiferrodistortivemode . In fact, the parity-related origin of ferroelec-tricity in ( AB O ) n /( AB ’O ) n superlattice is even moregeneral, which can be independent of the details of anti-ferrodistortive mode.In the point charge model, a semi-quantitative parti-tion can be applied to the total ferroelectric polarizationto extract various contributions. The ferroelectric con-tributions of Ti-O-octahedra, Fe-O-octahedra, and Y-O-icosahedra (Fig. 4(a)) can be estimated as : − is a proper ferroelectric activeion with a considerable large dielectric coefficient. In thissense, the ferroelectric polarization in our n = 2 superlat-tice is not a pure improper one as in the previous studiedcases , but a hybrid one with both improper andproper contributions. total Y-O Fe-O Ti-O-1.0-0.50.00.51.01.5 U Fe =2 eVU Ti =1.6 eV (c) -144% D i po l e ( e ¯ ) U Fe =4 eVU Ti =3.2 eV (b) total Y-O Fe-O Ti-O -1.0-0.50.00.51.01.5 -30%37%93% FIG. 4. (Color online) (a) Sketch of Y-O-icosahedron (left),Fe-O-octahedra (middle), Ti-O-octahedra (right) in the su-perlattices. (b-c) Point-charge model estimation of individualcontribution to total polarization at different U eff ’s. Fe -Ti is adopted. (b) Insulating. (c) Metallic. D. Possible metallic ferroelectricity
Very recently, a few studies found some peculiar ma-terials, in which the ferroelectric distortion persists de-spite its metallicity . Such a metallic ferroelectricitycan be characterized according to the structural infor-mation, e.g. via measuring convergent-beam electrondiffraction or neutron scattering, as done in Ref. 43.Physically, a hint for the metallic ferroelectrics is theweak coupling between the electrons at the Fermi leveland the (soft) phonon(s) responsible for removing inver-sion symmetry .According to the above analysis, the ferroelectric dis-tortion from Y-O-icosahedra should be robust in our n =2 superlattice, which in principle can also be expectedin other ( AB O ) /( AB ’O ) multilayers. Following thisargument, a promising way to pursuit metallic ferroelec-tricity is to find a metallic ( AB O ) /( AB ’O ) superlat-tices. Still taking (YFeO ) /(YTiO ) as a model sys-tem, most electron bands near the Fermi level are con-tributed by Fe and Ti , but the structural distortionsof Y-O-icosahedra are the first driving force for polariza-tion. Thus, it is expectable that the ferroelectric distor-tion can survive even if the system could become metal-lic, namely rending a metallic ferroelectricity.Despite many experimental approaches (e.g. doping,vacancies, or using other perovskites) to make metallic( AB O ) /( AB ’O ) , here we use smaller U eff ’s in theDFT calculations to obtain the metallicity. Althoughsmaller U eff ’s may be not realistic for real YFeO /YTiO ,the propose of this hypothesis is to illustrate the generalphysical mechanism that the improper ferroelectricity in( AB O ) /( AB ’O ) is robust against the metallicity, go-ing beyond a special property limited to a concrete ma-terial (e.g. stoichiometric YFeO /YTiO ). In the DFT calculation, weak U eff ’s give rise to finite density of statesat the Fermi level although the charge transfer remains(almost) complete. Although the Berry phase methodcan not work anymore to evaluate the polarization in themetallic state, the point charge model still works.With decreasing ( U Ti , U Fe ), the system turns to bemetallic but its ferroelectric distortion can be even moreprominent. This enhanced ferroelectricity in metallicstate is also understandable: the negative contributionfrom Ti-O-octahedra is significantly suppressed while theone from Y-O-icosahedra is more robust. Taking the( U Ti = 1 . U Fe = 2 eV) case for example, the con-tribution from Ti-O-octahedra is only 45% of the orig-inal value, while the Fe-O-octahedra contribution is al-most unchanged (97% of the original one), as shown inFig. 4(c). The contribution of Y-O-icosahedra is slightlyincreased by 25%, which is the largest contribution to thetotal net polarization. This semi-quantitative partitioncan help to understand the metallic ferroelectricity.Our DFT study supports the argument that the im-proper ferroelectricity due to the geometry factor is ro-bust against the finite density of states at the Fermi level,when the origins of ferroelectric displacement and metal-licity are different. Although the U eff ’s used here maybe a little lower for concrete YFeO /YTiO , the gen-eral physics raised in the present work is scientific sound.Even if the stoichiometric n = 2 YFeO /YTiO super-lattice is not metallic, other approaches can be employedto make the superlattice metallic or other systems can bedesigned to realize the metallic ferroelectricity followingour above argument. IV. CONCLUSION
In summary, the (YFeO ) n /(YTiO ) n superlatticeshave been studied using the standard DFT calculation.Since the two parent materials are both Mott insulators,unexpected charge transfer has been found, in oppositeto the intuitional band alignment scenario. In addition,the ferroelectricity is predicted in the n = 2 superlat-tice from the symmetry analysis and confirmed by cal-culations. Considering the magnetism of Fe, this n = 2superlattice is multiferroic. In addition, this ferroelec-tricity is robust against the metallicity. Even if the real(YFeO ) /(YTiO ) may be insulating, this design prin-ciple can be extended to other superlattices to search formetallic ferroelectrics. ACKNOWLEDGMENTS
Work was supported by the 973 Projects of China(Grant No. 2011CB922101), National Natural Sci-ence Foundation of China (Grant Nos. 11274060 and51322206), the Natural Science Foundation of JiangsuProvince of China (Grant No. BK20141329). ∗ [email protected] P. Zubko, S. Gariglio, M. Gabay, P. Ghosez, and J.-M.Triscone, Annu. Rev. Condens. Matter Phys. , 141 (2011). H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Na-gaosa, and Y. Tokura, Nat. Mater. , 103 (2012). S. Dong, R. Yu, S. Yunoki, G. Alvarez, J.-M. Liu, andE. Dagotto, Phys. Rev. B , 201102(R) (2008). S. J. May, P. J. Ryan, J. L. Robertson, J.-W. Kim, T. S.Santos, E. Karapetrova, J. L. Zarestky, X. Zhai, S. G. E.te Velthuis, J. N. Eckstein, S. D. Bader, and A. Bhat-tacharya, Nat. Mater. , 892 (2009). B. R. K. Nanda and S. Satpathy, Phys. Rev. B , 054427(2008). C. Adamo, C. A. Perroni, V. Cataudella, G. De Filippis,P. Orgiani, and L. Maritato, Phys. Rev. B , 045125(2009). A. Ohtomo and H. Y. Hwang, Nature (London) , 423(2004). N. Nakagawa, H. Y. Hwang, and D. A. Muller, Nat. Mater. , 204 (2006). M. Gibert, P. Zubko, R. Scherwitzl, J. ´I˜niguez, and J.-M.Triscone, Nat. Mater. , 195 (2012). S. Dong and E. Dagotto, Phys. Rev. B , 195116 (2013). K. Ueda, H. Tabata, and T. Kawai, Science , 1064(1998). Y. Zhu, S. Dong, Q. Zhang, S. Yunoki, Y. Wang, andJ.-M. Liu, J. Appl. Phys. , 053916 (2011). G. Kotliar and D. Vollhardt, Phys. Tod. , 53 (2004). Y. Kawasugi, H. M. Yamamoto, N. Tajima, T. Fukunaga,K. Tsukagoshi, and R. Kato, Phys. Rev. Lett. , 116801(2009). E. Dagotto, Science , 257 (2005). M. Mochizuki and M. Imada, New J. Phys. , 154 (2004). M. P. Pasternak, W. M. Xu, G. K. Rozenberg, and R. D.Taylor, Mat. Res. Soc. Symp. Proc. , D2.7 (2002). J. Alaria, P. Borisov, M. S. Dyer, T. D. Manning, S. Lep-adatu, M. G. Cain, E. D. Mishina, N. E. Sherstyuk, N. A.Ilyin, J. Hadermann, D. Lederman, J. B. Claridge, andM. J. Rosseinsky, Chem. Sci. , 1599 (2014). D. Treves, J. Appl. Phys. , 1033 (1965). J. R. Hester, K. Tomimoto, H. Noma, F. P. Okamura, andJ. Akimitsu, ACB , 739 (1997). G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996). S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.Humphreys, and A. P. Sutton, Phys. Rev. B , 1505(1998). J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. , 8207 (2003). J. Heyd and G. E. Scuseria, J. Chem. Phys. , 1187(2004). J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. , 219906 (2006). X. Huang, Y. Tang, and S. Dong, J. Appl. Phys. ,17E108 (2013). P. X. Zhou, H. M. Liu, Z. B. Yan, S. Dong, and J.-M. Liu,J. Appl. Phys. , 17D710 (2014). J. E. Kleibeuker, Z. Zhong, H. Nishikawa, J. Gabel,A. M¨uller, F. Pfaff, M. Sing, K. Held, R. Claessen,G. Koster, and G. Rijnders, Phys. Rev. Lett. , 237402(2014). R. J. Harrison, S. A. McEnroe, R. B. Hargraves, andP. Robinson, Nature (London) , 517 (2002). J. Velev, A. Bandyopadhyay, W. H. Butler, and S. Sarker,Phys. Rev. B , 205208 (2005). R. Pentcheva and H. S. Nabi, Phys. Rev. B , 172405(2008). H. Chen, A. J. Millis, and C. A. Marianetti, Phys. Rev.Lett. , 116403 (2013). E. Bousquet, M. Dawber, N. Stucki, C. Lichtensteiger,P. Hermet, S. Gariglio, J.-M. Triscone, and P. Ghosez,Nature (London) , 732 (2008). N. A. Benedek and C. J. Fennie, Phys. Rev. Lett. ,107204 (2011). A. T. Mulder, N. A. Benedek, J. M. Rondinelli, and C. J.Fennie, Adv. Funct. Mater. , 4810 (2013). J. M. Rondinelli and C. J. Fennie, Adv. Mater. , 1961C(2012). Y. J. Choi, H. T. Yi, S. Lee, Q. Huang, V. Kiryukhin, andS.-W. Cheong, Phys. Rev. Lett. , 047601 (2008). I. A. Sergienko, C. S¸en, and E. Dagotto, Phys. Rev. Lett. , 227204 (2006). S. Picozzi, K. Yamauchi, B. Sanyal, I. A. Sergienko, andE. Dagotto, Phys. Rev. Lett. , 227201 (2007). S. Dong, J.-M. Liu, and E. Dagotto, Phys. Rev. Lett. ,187204 (2014). To analyze the origin of ferroelectricity, the net dipolemoment is partitioned into three components from Fe-O-octahedra, Ti-O-octahedra, and Y-O-icosahedra usingthe point charge model. First, the centroid of each oxy-gen octahedron ~r O is calculated. Then the dipole moments ~P Fe − O and ~P Ti − O are calculated using eV X ( ~r X − ~r O ), where V X ( X =Fe and Ti respectively) denotes the valence ofcation and e is the elementary charge. In this calculation,the oxygen octahedra only contribute the equivalent neg-ative charge − eV X , while the rest negative charge formsdipole moments together with Y . This method can esti-mate the individual contribution to total polarization semi-quantitatively. Y. Shi, Y. Guo, X. Wang, A. J. Princep, D. Khalyavin,P. Manuel, Y. Michiue, A. Sato, K. Tsuda, S. Yu, M. Arai,Y. Shirako, M. Akaogi, N. Wang, K. Yamaura, and A. T.Boothroyd, Nat. Mater. , 1024 (2013). D. Puggioni and J. M. Rondinelli, Nat. Commun. , 3432(2014). H. J. Xiang, E. J. Kan, S.-H. Wei, M.-H. Whangbo, andX. G. Gong, Phys. Rev. B , 224429 (2011). H. M. Liu, Y. P. Du, . Y. L. Xie, J.-M. Liu, C.-G. Duan,and X. Wan, Phys. Rev. B91