arXiv:1710.02381v2 [cond-mat.str-el] 6 Sep 2019

Origin of Ferroelectricity in Orthorhombic LuFeO3

Ujjal Chowdhury,1 Sudipta Goswami,2 Amritendu Roy,3 S.S. Rajput,4 A.K. Mall,5 R.

Gupta,6 S.D. Kaushik,7 V. Siruguri,7 S. Saravanakumar,8 S. Israel,9 R. Saravanan,10

A. Senyshyn,11 T. Chatterji,12 J.F. Scott,13 A. Garg,4, ∗ and Dipten Bhattacharya1, †

1Nanostructured Materials Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata 700032, India2Department of Solid State Physics, Indian Association for the Cultivation of Science, Kolkata 700032, India

3School of Minerals, Metallurgical and Materials Engineering,Indian Institute of Technology, Bhubaneswar 752050, India

4Department of Materials Science and Engineering,Indian Institute of Technology, Kanpur 208016, India

5Materials Science Programme, Indian Institute of Technology, Kanpur 208016, India6Department of Physics, Indian Institute of Technology, Kanpur 208016, India

7UGC-DAE Consortium for Scientific Research, Bhabha Atomic Research Centre, Mumbai 400085, India8Department of Physics, Kalasalingam University, Krishnakoil 626126, India

9Department of Physics, The American College, Madurai, India10Department of Physics, The Madura College, Madurai 625011, India

11Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II),Technische Universitat Munchen, D-85747 Garching b. Munchen, Germany

12Science Division, Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, France13School of Chemistry, University of St Andrews, St Andrews, Fife, KY16 9ST, United Kingdom

(Dated: September 9, 2019)

We demonstrate that small but finite ferroelectric polarization (∼0.01 µC/cm2) emerges in or-thorhombic LuFeO3 (Pnma) at TN (∼600 K) because of commensurate (k = 0) and collinearmagnetic structure. The synchrotron x-ray and neutron diffraction data suggest that the polariza-tion could originate from enhanced bond covalency together with subtle contribution from lattice.The theoretical calculations indicate enhancement of bond covalency as well as the possibility ofstructural transition to the polar Pna21 phase below TN . The Pna21 phase, in fact, is found tobe energetically favorable below TN in orthorhombic LuFeO3 (albeit with very small energy differ-ence) than in isostructural and nonferroelectric LaFeO3 or NdFeO3. Application of electric fieldinduces finite piezostriction in LuFeO3 via electrostriction resulting in clear domain contrast imagesin piezoresponse force microscopy.

PACS numbers: 75.70.Cn, 75.75.-c

I. INTRODUCTION

During the last few years, work on ferroelectricity inrare-earth orthoferrites RFeO3 (R = Sm, Dy, Tb, Y,Lu) poses quite a few puzzles. In SmFeO3, YFeO3, andLuFeO3 [1–3], the ferroelectric order is reported to setin right at the antiferromagnetic transition temperatureTN (∼600 K). On the other hand, in DyFeO3 [4], theferroelectric transition takes place at a much lower tem-

perature (TDyN ∼4 K) only when application of magneticfield H ‖ c induces a ferromagnetic component to theFe sublattice. While observation of ferroelectricity inDyFeO3 still remains unchallenged, the ferroelectricity inSmFeO3 below TN has been disputed from direct electri-cal measurement of polarization and capacitance-voltagecharacteristics [5] as well as from crystallography [6]. Ithas been pointed out that the rare-earth orthoferrites, ingeneral, are paraelectric down to T = 0 and could only ex-hibit ferroelectricity in thin film form upon introduction

∗Electronic address: [email protected]†Electronic address: [email protected]

of appropriate lattice strain [7]. If ferroelectricity at allemerges at TN in the bulk form of the sample, it should bedue to the inversion-symmetry-breaking magnetic struc-ture. The structure could either be noncollinear aris-ing from antisymmetric Dzyloshinskii-Moriya or p-d ex-change or collinear arising from exchange striction [8].The collinear magnetic structure in SmFeO3 seems toyield nonpolar Pbnm although possibility of polar m2mpoint group was also hinted [9]. The controversy sur-rounding the emergence of ferroelectricity in orthorhom-bic SmFeO3, therefore, calls for a thorough examinationof the issue in other such rare-earth orthoferrites. Ob-servation of finite ferroelectricity at TN in this class ofcompounds has got another important implication. Ifferroelectricity is indeed observed in them at TN , theycan form a new class of room temperature Type-II mul-tiferroics.

In this work, we examined the occurrence of ferroelec-tricity in orthorhombic LuFeO3 at TN (∼600 K). We em-ployed a special protocol within the modified Sawyer-Tower circuit to extract the intrinsic remanent ferroelec-tric polarization. This is complemented by piezoresponseforce microscopy. We investigated the electronic, crystal-lographic, and magnetic structures in the material using

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1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 01 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

P R (nC/

cm2 )

T ( K )

T N

- 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0- 9

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Rem

anen

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ariza

tion

(nC/

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F i e l d ( V / c m )

3 0 0 K 2 5 0 K 2 0 0 K 1 0 0 K

FIG. 1: (color online) The variation of the remanent polariza-tion (diamond) with temperature measured in orthorhombicLuFeO3; the Pel (circle) calculated from x-ray diffraction arealso shown; inset shows the remanent hysteresis loops mea-sured at different temperatures.

synchrotron x-ray and high resolution powder neutrondiffraction experiments while Raman spectrometry wasemployed to track the phonon modes across the ferroelec-tric transition. The results collectively suggest that thetiny ferroelectric polarization (∼0.01 µC/cm2), emerg-ing at the magnetic transition in orthorhombic LuFeO3,could originate from enhanced bond covalency thoughsubtle role of underlying lattice cannot be ruled out. Thetheoretical calculations conducted to investigate the ori-gin of ferroelectricity show the contribution of both elec-tronic and lattice structures to the observed polarization.In addition, possibility of a structural transition at TN(from Pnma → Pna21) could also be observed which,however, because of small energy difference between thesetwo phases (smaller than the room temperature thermalenergy), could not be detected experimentally.

II. EXPERIMENTS

Experiments have been carried out on phase purehigh quality bulk polycrystalline samples. The detailsof the sample preparation have been given elsewhere [3].The synchrotron x-ray data were recorded at the MCXbeamline of Elettra, Trieste (λ = 0.61992 A) and theneutron diffraction experiments were carried out at theSPODI FRM-II diffractometer of Technische UniversitatMunchen, Munchen (λ = 2.536 A), and also at the PD-3 diffractometer of NFNBR, Bhabha Atomic ResearchCentre, Mumbai (λ = 1.48 A). The synchrotron x-raydata have been refined by JANA 2006 and the struc-ture factors were used to construct the charge densitydistribution map within a unit cell by employing Maxi-mum Entropy Method (MEM). The neutron data have

been refined by FullProf for determining the magneticand crystallographic lattices. The remanent ferroelec-tric hysteresis loops were measured by PC Loop Tracerof Radiant Inc., and the piezoresponse force microscopy(PFM) were carried out by MFP-3D scanning probe mi-croscope of Asylum. In addition, Raman spectra havebeen recorded across 300-700 K in order to track thephonon modes around the ferroelectric transition.

III. COMPUTATIONAL DETAILS

The first-principles calculations were performed usingdensity functional theory as implemented in Vienna ab-initio Simulation Package (VASP). The generalized gra-dient approximations (GGA) by Perdew-Bruke and Ernz-erhof [10] (PBEsol), optimized for solids, has been used.In order to verify the robustness, some of the calculationswere tested using a different functional, PW-91 [11]. Thestrongly correlated electrons of the transition metal ionswithin the optimized structure have been taken care of bythe Hubbard potential (Ueff = 3-5 eV) within (GGA+U)for a separate set of calculations wherein a rotationallyinvariant approach by Dudarev et al. [12] was adapted.We used projected augmented wave potentials and con-sidered 9 valence electrons for Lu, 14 for Fe (includ-ing the semi-core states) and 6 for O ions. We used aMonkhorst-pack k-mesh of size 3×4×6 for all our calcu-lations. A paramagnetic state is the outcome of coun-teracting long-range ordering of magnetic moments andthermal energy to disrupt the above ordering. Therefore,it can be presumed that at zero temperature (no thermalenergy), the long-range ordering of the spins would as-sume the configuration that minimizes the total energyof the system. In view of the above, a 2×1×1 supercellwas constructed and different spin configurations [ferro-magnetic, A-antiferromagnetic (AFM), C-AFM, E-AFMand G-AFM] were enforced. Total energy correspond-ing to the above spin configurations was computed todetermine the most favored magnetic ordering. To ex-plore the possibility of a structural phase transition inpresence of antiferromagnetic order, the experimentallyobserved structure with Pnma symmetry at 298 K wastransformed to Pna21 (one of the subgroups of Pnma)using TRANSTRU within the Bilbao Crystallographicserver. The supercells with magnetic ordering and as-suming either Pnma, distorted Pnma or Pna21 struc-ture were fully relaxed such that the Hellman-Feynmannforces on the ions are less than 0.001 eV/Aand the to-tal pressure on the cell is close to zero. Total energyof each of the cases has been computed and compared.The electronic density of states and the band structurewere computed on the lowest energy structure. Polariza-tion within the insulating state of the system has beencomputed by Berry phase method [13]. The result wasfurther corroborated by the polarization obtained fromBorn effective charges computed using density functionalperturbation theory.

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FIG. 2: (color online) The (a) AFM and (b), (c) amplitude contrast and (d), (e) phase contrast PFM images under +100Vand -100V bias voltage, respectively; (f) strain-field and (g) phase switching hysteresis loops.

IV. RESULTS AND DISCUSSION

Figure 1 shows the typical remanent hysteresis loops ofthe sample measured at different temperatures and alsothe variation of the remanent polarization with temper-ature. The electronic ferroelectric polarization Pel (dis-cussed later), estimated from the x-ray diffraction data,are also shown. The measurement of remanent hystere-sis loops employs a specially designed protocol whicheliminates the contribution from non-remanent and non-hysteretic polarization components [14]. This protocolconsists of sending out a train of fourteen voltage pulseswhich measure the hysteresis loops formed from the con-tribution of remanent, nonremanent, hysteretic and non-hysteretic polarizations as well as from the nonrema-nent and nonhysteretic polarizations only. Subtractionof the latter loop from the former one yields the intrin-sic remanent hysteresis loop. The salient features of themeasurement protocol including the voltage pulse trainsent out for measuring the remanent polarization aregiven in the supplementary document [15]. The observa-tion of small yet finite remanent ferroelectric polarization(∼0.01 µC/cm2) ensures emergence of ferroelectricity atTN . The evolution of time scale along the hysteresis loopis counterclockwise which is consistent with true ferro-electric behavior and rules out charge injection. It isimportant to note that application of this specific proto-col on several compounds, either improper ferroelectricswith tiny remanent polarization or nonferroelectrics withno remanent polarization, is found to be effective in ex-tracting the characteristic P − E loop to determine the

magnitude of remanent polarization [14]. A rather smallnonlinearity in the left and right arms of the P −E loopcould possibly manifest the role of ferroelastic switchingas well. The loops have been blown-up in the supple-mentary document to show the extent of nonlinearity ofthese side arms clearly [15]. Of course, in general, thenonlinearity in the side arms is expected to be small inremanent hysteresis loops [15]. Square-looking ferroelec-tric hysteresis loops have previously been observed in afew cases such as in electrets, in orthoferrite SmFeO3

as well as in thin films of PbTiO3 of thickness 129 nmgrown on 0.7 wt% Nb-doped SrTiO3 with Pt top elec-trode. Electrets exhibit polarization which diminisheswith time [16]. This is not the case with our samples. Ina recent work [17], quantitative analysis of the hystere-sis loop shape using dielectric portraits is shown to offermore accurate information about the thickness of ferro-electric dead layer and its nature - Schottky barrier typeor other - and, therefore, may have, at least, peripheralrelevance to the loop shape observed here. The obser-vations ealier made for orthoferrites such as SmFeO3 areattributed to improper polarization, believed to originatefrom exchange striction giving rise to polar displacementof the oxygen ions at the magnetic domain walls [18].On the other hand, observations of similar square loopsin PbTiO3 thin films as well as in Pb(Mg1/3Nb2/3)O3-PbTiO3 composite (typical results shown in the supple-mentary document [15]) originate from the switching of90o domains [19] at higher frequency instead of com-plete 180o domain reversal which could possibly requirelower frequencies. In the present case, of course, we ob-

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serve complete 180o reversal of domains as witnessed bypiezoresponse force microscopy (PFM), as discussed inthe next paragraph. We also observe nearly frequencyindependent remanent hysteresis loops [14].

The temperature dependence of remanent polarizationsuggests a sharp drop in the polarization in the vicinityof TN (∼600 K) which is indicative of coupling betweenmagnetic and electrical ordering and perhaps a structuraltransition at this temperature which we further exploreusing temperature dependent x-ray and Raman studies.We conducted piezoresponse force microscopy (PFM) toexplore the ferroelectric switching in the samples. PFMwas used in a spectroscopic mode in which a dc bias volt-age is applied in a cyclic manner with tip remaining fixed.This yields a local piezoelectric loop which is basicallythe manifestation of the local piezoelectric vibration onthe voltage sweep. To observe the polarization switching,a sequence of dc voltage in a triangular sawtooth formwas applied with simultaneous application of 2 V ac volt-age to record the corresponding piezoresponse, measuredduring the off state at each step to minimize the effectof electrostatic interactions, resulting in a phase-voltagehysteresis loop. PFM amplitude and phase images ac-quired in PFM dual ac resonance tracking imaging mode,using a cantilever of stiffness 2 N/m and Ti/Ir tip. Figure2 shows the amplitude and phase contrast PFM imagesrecorded under +100V and -100V dc bias. Two typesof sub-micron-sized domains with dark purple and whitecolors could be seen in the phase contrast image cap-tured under -100V. These are antiparallel domains, alsocalled 180o domains, where polarization vector is orientedin phase with the applied voltage for purple and out ofphase for white. The orientation changes upon switch-ing the electric field. In principle, orthorhombic structurecan also exhibit 60o, 90o, and 120o domains [20] and pres-ence of multiple colors indeed points toward existence ofthese domains, albeit, in small proportions. The 180o do-mains, of course, are the dominant ones. The completeswitching spectroscopy PFM was also carried out andthe strain and phase switching angle versus field loopsare shown. The 180o switching of the domains under±100 V bias also indicates that during the measurementof remanent hysteresis loop saturation of polarization isachieved as identical bias voltage was applied in that casetoo. The butterfly shape of the strain versus electric fieldloop is indicative of the presence of piezoelectric activityin the sample. The distortion of the strain-field loop pos-sibly originates from difference in electrode-sample inter-face charge structure between top and bottom electrodes.It is important to point out here that though the PFMmeasurements have been carried out on polycrystallinesamples where the conductivity might have finite varia-tion across the grain-grain boundary network, the imagesrecorded indeed show the ferroelectric domains and theirswitching. The comparison of the topological and phase-contrast PFM images shows that the pattern observedin the topological image is quite different from the pat-tern observed in the phase-contrast PFM image. More-

over, we have carried out the measurements at differentplaces of the sample to ensure presence of finite intrinsicpiezoresponse in the sample. Therefore, influence of con-ductivity fluctuation on the PFM images is ruled out. Itis further mentioned that since the PFM data have beenrecorded on a polycrystalline sample, one does not knowthe orientation of individual grains. Therefore, it is notpossible to identify the crystallographic direction or planeof the measurement. While polarization vector could beoriented along a-axis (described later), if the grains haveorientations that are not perpendicular to the a-axis, onewould still observe the ferroelectric switching. Hence,since one serves the switching, one is witnessing the con-tribution of the component of the polarization along thedirection of the applied bias field under the PFM tip froma randomly oriented grain.

Using the synchrotron x-ray and neutron data, we nowexamine the contribution of lattice and electronic struc-ture to the overall polarization. Within the limit of theresolution of diffraction data and accuracy of the Rietveldrefinement, it appears (reliability factors and goodness offit vary within 1.0%-2.5% [15]) that the crystallographicstructure of the sample remains nonpolar orthorhombicPnma throughout the entire temperature range. There-fore, if at all there is any structural transition around TN ,it is of isostructural type. The isostructural transitionis rare and has implications for phonon dynamics. Thephonon symmetry does not change across the transition.The calorimetric trace and dc resistivity versus tempera-ture measurements reveal finite latent heat and resistivityhysteresis associated with the transition. The results areincluded in the supplementary document [15]. Althoughthe physics behind isostructural transition is not quitewell understood, there are suggestions that this could bedue to interaction of electrons with lattice vibrations [21].Both the x-ray and neutron scattering offer evidence ofprevalence of nonpolar Pnma structure even below TN .Of course, it is quite possible that extremely small non-centrosymmetry (of the order ∼0.16 mA mentioned inRef. 9), if present, remains undetected in these mea-surements. The small nonlinearity of the left and rightsides of the remanent hysteresis loops together with PFMdata also suggest a structural transition from nonpolarPnma to nonpolar yet ferroelastic P212121 at TN . InFig. 3, we show the variation of lattice parameters, vol-ume, ion positions etc, determined from the refinement ofx-ray data, as a function of temperature across 400-727K. The estimated standard deviation, obtained duringrefinement, varies within 0.2%-0.4% for all the param-eters. It represents the corresponding error bar. Clearanomaly could be observed in almost all the parametersaround TN signifying presence of strong spin-lattice cou-pling. The nature of the anomaly is similar in all thethree lattice parameters and hence the volume; they ex-hibit anomalous expansion at the onset of magnetic orderat TN . Temperature dependent evolution of the latticeparameters, bond lengths/angles together with in- andout-of-phase octahedral tilt and A-site ion displacement

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3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 0 7 5 05 . 5 65 . 5 75 . 5 85 . 5 97 . 5 87 . 5 97 . 6 07 . 6 17 . 6 27 . 6 35 . 2 35 . 2 45 . 2 55 . 2 62 2 02 2 12 2 22 2 32 2 4

b (A)

a (A)

T ( K )

o

o

o

v (A3 )

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T N

o

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3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 0 7 5 00 . 0 6 90 . 0 7 00 . 0 7 1- 0 . 0 1 9 5

- 0 . 0 1 9 0- 0 . 0 1 8 5

0 . 4 5 00 . 4 5 20 . 4 5 40 . 1 1

0 . 1 2- 0 . 3 0 8- 0 . 3 0 4- 0 . 0 6 0- 0 . 0 5 60 . 3 0 80 . 3 1 20 . 3 1 6

T ( K )

L u _ x

O 2 _ zO 2 _ y

O 2 _ x

O 1 _ z

O 1 _ x

L u _ z

T N

(b)

FIG. 3: (color online) Variation of the (a) lattice parame-ters and (b) ion positions with temperature obtained fromRietveld refinement of the x-ray diffraction data.

has earlier been tracked [22] across 25-1285 K for othermembers of the rare-earth orthoferrite family with largetolerance factor such as LaFeO3. Studies have also been

FIG. 4: (color online) The spin structure of LuFeO3 as perirrep Γ2. For clarity, Lu and O ions are not shown.

1 0 61 0 81 1 01 1 2

( f )

( e )

( d )

( c )

( b )

( a )4681 01 2

1 3 01 3 21 3 41 3 6

681 01 21 4

1 5 41 5 61 5 81 6 01 6 2

91 21 51 8

2 7 02 7 32 7 62 7 92 8 22 8 5

71 42 12 83 54 2

T e m p e r a t u r e ( K )

Line width (cm -1)Peak

Posit

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3 0 0 4 0 0 5 0 0 6 0 0 7 0 03 3 93 4 23 4 53 4 83 5 1

1 21 82 43 0

3 0 0 4 0 0 5 0 0 6 0 0 7 0 05 0 45 0 75 1 05 1 35 1 65 1 9

1 22 43 64 86 0

FIG. 5: (color online) The temperature dependence of Ramanmode frequency and linewidth.

done on orthorhombic PrFeO3 and NdFeO3 [23] and onR0.5R0.5’FeO3 (R = Sm, R’ = Pr, Nd) [24] and evidenceof spin-lattice coupling could be gathered from anoma-lies around TN (650-750 K). In the present case, the Lu,O1, and O2 ions exhibit anomalous displacement belowTN ; the position of Fe ion is fixed at (0,0,0.5). Within thelimit of accuracy with which the positions of the ions havebeen determined (error bar varies within 0.2%-0.4%), itappears that the anomalous displacement of Lu, O1, andO2 ions is consistent with the irreducible representationτ1 [25]. This signifies occurrence of isostructural transi-

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tion at TN . The allowed irreducible representations cor-responding to the anomalous ion displacements at TN ,obtained from the group theoretical analysis, as well asthe basis vectors for the τ1 mode are given in the sup-plementary document [15]. The comparison of crystallo-graphic parameters determined from x-ray and neutrondiffraction [15] shows that though there exists some dif-ference between the numerical values of the parametersthe overall trend is quite similar. It is important to pointout here that because of poorer scatteing of x-ray by thelighter ions such as oxygen, it is difficult to determine theposition of oxygen ions accurately from x-ray diffractiondata. On the other hand, for commensurate magneticstructure with k = 0, determination of ion positions fromneutron diffraction poses problem as both the magneticand nuclear peaks appear at the same point in reciprocallattice space. In this case, it is necessary to collect theneutron data at a spallation source across much largerQ range in order to eliminate the influence of magneticpeaks. The consistency in the structural parameters de-termined from both x-ray and neutron data reflects theaccuracy of the results obtained for the present case.The commensurate k = 0 magnetic lattice determinedfrom the neutron diffraction experiments is found to becollinear (Fig. 4), which corroborates the observationsmade in orthorhombic SmFeO3 [5, 6]. It is found, how-ever, that the magnetic lattice for LuFeO3 across 400-700K could be described by the single irrep Γ2 [15]; corre-sponding spin configuration is FxCyGz (Fig. 4). Thisis consistent with nonpolar structure. Below ∼400 K,spin-reorientation transition could be observed. We, ofcourse, concentrate here on the data across 400-700 K aswe are concerned about ferroelectricity right below TN .

We further investigate the role of lattice by carryingout Raman spectrometry across 300-700 K within theRaman shift range 90 to 1000 cm−1 [15]. The Ag andB1g modes [26] could be seen and their frequency shiftand linewidth are shown in Fig. 5 as a function of tem-perature. Distinct anomaly in both frequency shift andlinewidth could be observed at TN . However, unlike thephonon softening observed in systems with displacive fer-roelectricity arising out of even orthorhombic Pnma toorthorhombic P21ma phase transition [27], the phononmodes here exhibit a pattern expected in isosymmetricphase transition where the phonon modes are symmetricboth at above and below the transition [28] and the fre-quency changes only slightly (Cochran’s exponent < 0.1).Though reports exist [29] for other compounds where de-viation from the expected mode softening features couldbe observed even when the lattice ferroelectricity is finite,in the present case, we could not observe similar feature.Therefore, at best, clear evidence of lattice ferroelectric-ity appears to be undetectable in the experiments. Thetheoretical calculations (described later) show that thenonpolar to polar phase transition at TN could remainundetectable experimentally because of very small energydifference between these phases at below TN .

We then determined the electron charge density distri-

FIG. 6: (color online) The two-dimensional map of electroncharge density in (a) (100) and (010) planes showing Fe-O1and Fe-O2 bonds respectively and (b) (301) and (101) planesshowing Lu-O1 and Lu-O2 bonds respectively; the left panelsshow the data at 399 K (below TN ) while the right ones showthe data at 727 K (above TN ).

bution over the unit cell. Application of MEM/Rietveldrefinement to the high energy synchrotron x-ray diffrac-tion data yields the charge density distribution. The ac-curacy of MEM in determining the charge density dis-tribution and thus covalency of the bonds has alreadybeen established for different compounds including com-pounds containing combination of heavy elements suchas Pb together with light elements such as O [30]. TheMEM has also been used to observe the Mn3dx2−y2 or-bital order [31]. In the present case, MEM analysis hasbeen carried out by dividing the unit cell into 48×72×48pixels for all the temperatures. The details of the refine-ment and fit statistics are given in the supplementarydocument [15]. In Fig. 6, we show the two-dimensionalmaps of charge density distribution in (100), (010), (301),and (101) planes at 399 (i.e., below TN ) and 727 K (i.e.,above TN ) in order to show the charge density across Fe-O1, Fe-O2, Lu-O1, and Lu-O2 bonds, respectively. The

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0 . 5 1 . 0 1 . 5 2 . 00 . 40 . 81 . 21 . 62 . 0

o

L u - O 1 ( 3 9 9 K ) L u - O 1 ( 7 2 7 K ) L u - O 2 ( 3 9 9 K ) L u - O 2 ( 7 2 7 K )

Electr

on De

nsity

(e/A3 )

D i s t a n c e ( A )

o

(a)

0 . 5 1 . 0 1 . 50 . 00 . 51 . 01 . 52 . 0

F e - O 1 ( 3 9 9 K ) F e - O 1 ( 7 2 7 K ) F e - O 2 ( 3 9 9 K ) F e - O 2 ( 7 2 7 K )

Electr

on De

nsity

(e/A3 )

D i s t a n c e ( A )o

o

(b)

FIG. 7: (color online) The one-dimensional map of variationof electron density across (a) Lu-O and (b) Fe-O bonds at 399and 727 K.

background charge density is ∼0.2 e/A3 and the con-tours lines are mapped across 0 to 1 e/A3 at an intervalof 0.1 e/A3. The charge density around the midpointsof Lu-O1, Lu-O2, Fe-O1, and Fe-O2 bonds at 727 K are0.636 e/A3, 0.452 e/A3, 0.754 e/A3, 0.235 e/A3, respec-tively. The corresponding figures at 399 K are 0.792e/A3, 0.512 e/A3, 0.863 e/A3, and 0.401 e/A3, respec-tively. The covalency of all the Lu-O1, Lu-O2, Fe-O1,and Fe-O2 bonds has increased below TN by differentextent. The one-dimensional map of charge density dis-tribution across Lu-O and Fe-O bonds both at 399 and727 K are shown in Fig. 7. Different extent of chargetransfer below TN leads to asymmetric distribution ofcharges. The electrons are counted within the minimumcharge density surface around each ion of the cell. Thedifference between the electron count and atomic numbergives the charge state (n) of the ion. The charge statesfor Lu, Fe, and O ions turn out to be +3.93, +3.96, -3.53 (for four O1 ions), and -2.20 (for eight O2 ions)repectively at 727 K. At 399 K, the corresponding fig-ures for Lu and Fe ions are +3.80 and +3.50, respec-tively; O1 ions appear to be of -3.20 charge state while

O2 are of -2.0. It appears then that charge dispropor-tionation by nearly 10%-12% has taken place among Lu,Fe, and O ions as a result of magnetic transition. Com-parable extent of charge disproportionation has earlierbeen observed [32] in Fe3O4 below its charge order (Ver-wey) transition (TCO ∼120 K). In order to calculate theferroelectric polarization below TN , if any, as a result ofoff-centric charge density distribution, we first find outthe center of charge density distribution contour (c) foreach ion [15]. Using this result for all the cations andanions of the unit cell, the net off-centred shift (∆c) inthe charge density distribution contours or charge cloudhas been determined. Remarkably, ∆c turns out to befinite, though small (≈0.003-0.004 A), below the TN andusing the relation Pel = n.e.∆c/V where e = charge onan electron and V = volume of unit cell, we determinethe ferroelectric polarization resulting from off-centredcharge density distribution within a unit cell (Pel). In-terestingly, as against the observation made earlier [1],the Pel turns out to be oriented along a-axis. We plotthe values of Pel as a function of temperature in Fig. 1.The order of the magnitude of Pel appears to be compa-rable to what has been found from direct measurement(of the order of ∼0.01 µC/cm2 at 300 K). This result in-dicates that the tiny ferroelectric polarization measuredin orthorhombic LuFeO3 could possibily have electronicorigin. In spite of lattice centrosymmetry, consistent withΓ2 irrep, redistribution of charges below TN could yield afinite global ferroelectric polarization. In LaMn3Cr4O12

too, finite electronic ferroelectricity was claimed to resultfrom collinear spin ordering [33] within a cubic lattice.Of course, as pointed out earlier, a theoretical work [18]has recently suggested that polar displacement of ionsat the antiferromagnetic domain walls could induce fi-nite ferroelectric polarization even at TN in orthorhombicSmFeO3. This is proposed to be true for other rare-earthorthoferrites as well.

Interestingly, we also observe finite piezostriction inorthorhombic LuFeO3. Application of electric field in-duces detectable piezostriction (Fig. 2) as a result ofreasonably large dielectric constant [3] and electrostric-tive coefficient [34]. The ferroelectric domains observedin PFM actually represent those for the lattice. Howthey are related to the electronic ferroelectric domains,if present, is not yet understood. Of course, as pointedout above, presence of subtle noncentrosymmetry belowTN cannot be completely ruled out and it needs furtherinvestigation. Observation of lattice noncentrosymmetryonly under an electric field in presence of electronic fer-roelectricity has earlier been noted in a charge-transfercomplex tetrathiafulvalene-chloranil [35]. It will be in-teresting to search for similar result in other purely elec-tronic ferroelectric systems.

Since the origin of ferroelectricity in orthorhombicLuFeO3 is not quite conclusively understood, we em-ployed first-principles density functional theory (DFT)based calculations to investigate two distinct possibilities:(i) long-range magnetic order mediated hybridization of

8

FIG. 8: (color online) (a) Comparison of total energy of LuFeO3 for different spin configurations; (b) schematic of LuFeO3

supercell wherein G-AFM is the most favoured spin structure; (c) Electronic band structure and density of states of G-AFMstructure; and (d) differential charge density plot of distorted Pnma structure in presence of G-AFM order with and withoutHubbard parameter Ueff .

electronic oribitals leading to the asymmetric charge den-sity distribution and hence finite electronic ferroelectric-ity and (ii) breaking of spatial inversion symmetry of thecrystallographic structure in presence of magnetic order.The third possibility of finite ferroelectric polarizationbelow TN as a result of exchange-striction driven lat-tice ferroelectricity at the magnetic domain boundarieshas earlier been explored by others [18]. In fact, thispossibility has started gaining ground in the context ofrare-earth orthoferrites as it shows that a single-domainbulk sample cannot support ferroelectricity which is con-sistent with the experimental data [5]. However, in a realmultidomain sample, ferroelectricity emerges at the anti-ferromagnetic domain boundary. The magnitude of po-larization observed experimentally is also consistent withthe theoretical prediction. The relevance of this mecha-nism in the context of describing ferroelectricity in differ-ent magnetic ferroelectric systems has been highlightedby Scott and Gardner [19]. In a separate theoreticalwork [36], it has also been pointed out that orthoferrites(RFeO3) or orthochromites (RCrO3) with two magneticsublattices R and Fe/Cr, could exhibit ferroelectric polar-ization because of non-relativistic exchange striction with

TABLE I: Calculated structural parameters, magnetic mo-ment, and electronic band gap within Pnma structure.

GGA+U with PBE GGAwith PW

Ueff = Ueff =0 eV 3.0 eV 4.0 eV 5.0 eV 0 eV

a (A) 5.480 5.501 5.504 5.506 5.555b (A) 7.472 7.501 7.504 7.508 7.574c (A) 5.152 5.172 5.175 5.177 5.223µFe (µB) 3.63 4.02 4.09 4.15 3.66Eg (eV) 0.4 1.66 2.05 2.40 0.35

large magnetostructural effect. This mechanism could

explain the ferroelectricity in DyFeO3 below TDyN under amagnetic field [4]. Interestingly, a morphotropic mixtureof hexagonal and orthorhombic LuFeO3 too, has recentlybeen synthesized in thin film form and ferroelectric po-larization along with magnetoelectric coupling have beeninvestigated [37].

Figure 8(a) plots the total energy per formula unit ina 2×1×1 supercell relative to the lowest energy spin con-

9

figuration. It is found that the G-type antiferromagneticorder of the Fe ions corresponds to the lowest energywithin the orthorhombic Pnma symmetry. Experimen-tally determined spin-ordering also conforms to the abovecalculation. The closest competing spin structure is C-AFM which has ∼78 meV/f.u. higher energy over theG-AFM structure. The other spin structures correspondto still higher energies. Therefore, it is concluded thatthe G-AFM is the most favored spin structure within themagnetically ordered structure of orthorhombic LuFeO3

and all the calculations were performed assuming theabove structure of LuFeO3. To study the structural sta-bility of the centrosymmetric Pnma phase of LuFeO3,especially in presence of G-AFM order, as well as to un-derstand the evolution of ferroelectric polarization withinan apparent centrosymmetric phase, we performed first-principles density functional theory based calculations.Since G-AFM ordering of the Fe ions has been found tobe the most favored magnetic ordering, we used GGAand GGA+U methods to relax the experimentally ob-tained structure within G-AFM ordering. Table-I liststhe optimized structural details and electronic band-gapof the compound within Pnma symmetry obtained fromusing different functionals and different Ueff .

It is observed that the optimized lattice parameters,obtained from GGA and GGA+U calculations, are un-derestimation of the lattice parameters obtained experi-mentally at 298 K (a = 5.574 A, b = 7.600 A, and c =5.241 A). Such underestimation is not unusual consider-ing the large temperature difference between the struc-ture obtained experimentally (at 298 K) and the one ob-tained theoretically (at 0 K). In order to avoid bias to-ward ferroelectric instability, if any, we, however, usedthe optimized structure (instead of experimentally ob-served structure) for the calculation of the ferroelectricpolarization. Figure 8(b) schematically shows the G-typeantiferromagnetic ordering of the Fe-ions in the 2×1×1supercell of LuFeO3. It is interesting to note that the ar-rangement of the Lu ions along a-axis is chiral. The elec-tronic band gap within the G-type antiferromagnetic or-dering for GGA is estimated to be ∼0.4 eV, much smallerthan the experimental observation. Such underestima-tion of band gap by DFT is well known, in particular forstrongly correlated systems. We demonstrate that appli-cation of DFT+U method is helpful to increase the bandgap of the system. Electronic band structure, total andsite-projected density of states with GGA+U (Ueff = 4.0eV), is presented in Fig. 8(c). It is observed that the up-per part of the valence band is occupied predominantlyby O 2p states whereas the lower part of the conduc-tion band is dominated by Fe 3d states. The estimatedband gap is ∼2.05 eV. This is comparable to the obser-vations made (∼2.07 eV) in rare-earth orthoferrites [38].Further, the electrical polarization calculations were alsodone using GGA and GGA+U method with Ueff = 4.0eV. Small variation of Ueff was found not to affect thestructural stability.

We first explored the possibility of symmetry lower-

TABLE II: Comparison of the lattice parameters, magneticmoment, band gap, and difference in total energy betweenPnma and Pna21 structures

Pnma Pna21

a (A) 5.480 5.480b (A) 7.472 5.152c (A) 5.152 7.472∆E (meV/f.u.) 0.00 -0.23µFe (µB) 3.63 3.62Eg (eV) 0.4 0.36

ing structural phase transition to a polar group wherespontaneous polarization is realized. The calculation ofphonon density of states for Pnma structure in presenceof G-antiferromagnetic order, of course, confirms the sta-bility of the phonon modes [15]. However, as suggested inthe experiment, one of the subgroups of Pnma is Pna21

which is one of the possible polymorphs. Any subtlestructural phase transition which was not detected bythe experimental studies could be further explored by to-tal energy computation using density functional studies.Using the experimental structural parameters within theantiferromagnetic phase at 298 K, we transformed the ex-perimentally determined Pnma structure to Pna21 usinga program, TRANSTRU, within Bilbao crystallographicserver [39]. Comparison of the total energies of the fullyrelaxed Pnma and Pna21 structures, shown in Table-II, highlights similar values. The extent of difference inenergy between the two polymorphs is smaller than theroom temperature thermal energy and therefore couldnot be distinguished experimentally. Nonetheless, ourcalculations predict a polar phase (Pna21) for LuFeO3

in presence of G-type antiferromagnetic spin order al-though the difference in energy is of the order of tem-perature fluctuation and, therefore, one cannot be reallysure about the phase stability. Since such a small dis-tortion might involve movement of oxygen ions, neutrondiffraction experiment at a spallation source is necessaryto track the movement accurately. Interestingly, similarcalculations performed on relaxed and optimized struc-tures of isostructural yet nonferroelectric LaFeO3 andNdFeO3 show that the centrosymmetric Pnma structureis more stable even in presence of G-AFM order. Giventhis result, it will be interesting to examine (i) whetherrare-earth orthoferrites with tolerance factor (t) smallerthan a critical value (tC) could exhibit possibility of non-polar to polar phase transition in presence of magneticorder and (ii) whether engineering of lattice strain couldstabilize the polar phase below TN in epitaxial thin filmsor nanostructures. They may be addressed in subsequentworks.

We then considered the distorted Pnma in presenceof G-AFM structure. Calculation of polarization usingmodern theory of polarization (Berry phase formalism)predicts a small spontaneous polarization, ∼4.6 nC/cm2

at Ueff = 0. At Ueff = 4.0 eV, the polarization turns outto be ∼1.2 nC/cm2. The polarization (P ) is given by [40]

10

P = Z∗λuλ =Z∗2

λ Eωλ(T )2 +

∆Z∗λ

ωλ(T )2M(T )

where Z∗λ, ωλ(T ), and uλ are the effective charge, fre-quency of the phonon mode, and the ionic displacementassociated with λ (an order parameter which assumes thevalue 0 at the paraelectric phase and 1 at the ferroelec-tric phase) which, in the present case, is coupled with thespin. The first term is related to the dielectric contribu-tion while the second term arises from the spin-latticecoupling effect. Effect of magnetization vis-a-vis degreeof ordering of the spins can be further qualitatively as-sessed by the comparing the differential charge densityplot with and without the application of Hubbard param-eter Ueff , as shown in Fig. 8(d). Figure 8(d) shows thatupon application of the Ueff , the charge distribution overthe oxygen ions is modified. Under such condition, it hasbeen found that nearly 30% of the total polarization Pis originated from noncentrosymmetric electronic chargedensity distribtion while the rest 70% is from the lat-tice effect (distorted Pnma). Observation of finite elec-tronic ferroelectricity within distorted Pnma structure,in presence of G-AFM order, supports the experimen-tal observation of finite charge disproportionation at TN .However, the contribution of lattice too, turns out to befinite. This subtle contribution from lattice could not beclearly detected experimentally as neutron diffraction ata spallation source needs to be carried out. Of course, itis worth mentioning, in this context, that the orthorhom-bic distortion enhances by more than 2.5% below TN[15]. This could be the reflection of subtle distortionwithin the orthorhombic Pnma lattice which eventuallycontributes to the polarization too. Although, theoret-ical calculations, in the present case, do reveal contri-bution of both electronic and lattice structures to theferroelectricity, in strongly correlated electron systems,decoupling of electronic and lattice structural transitionis not rare [41, 42]. The collinear magnetic structure hasearlier been predicted to exhibit finite ferroelectricity atthe onset of magnetic order because of exchange stric-tion effect [8]. In many cases of rare-earth orthoferrites,concomitant structural transition to a polar phase couldnot be observed [6]. Our theoretical results point outthat, for LuFeO3, this is due to the comparable energyscales of the polar and nonpolar phases. Of course, insharp contrast to the observations made in orthorhombicLuFeO3, isostructural yet nonferroelectric LaFeO3 andNdFeO3 do not exhibit electronic and/or lattice ferroelec-tricity in presence of G-AFM. They also do not exhibitany instability towards Pna21 phase at the onset of G-AFM order. This result highlights the bias toward ferro-electricity in LuFeO3 in presence of G-AFM. As pointedout earlier, finite ferroelectricity could possibly emergein rare-earth orthoferrites in presence of magnetic orderonly if their tolerance factor (t) is smaller than a criticalvalue tC . In order to estimate the spontaneous polariza-tion for LuFeO3 using Born effective charges and corrob-orate the results obtained from Berry phase formalism,

TABLE III: Elements of Born Effective Charge tensors of ionsin presence of G-AFM structure.

Ueff = 0.0 eV Ueff = 4.0 eVIon Zxx Zyy Zzz Zxx Zyy Zzz

Lu 4.09 3.50 3.79 3.95 3.48 3.87Fe 6.20 5.59 5.16 4.08 3.82 3.84O1 -2.46 -4.06 -2.85 -2.11 -3.05 -2.54O2 -3.91 -2.51 -3.05 -2.96 -2.12 -2.59

we further employed ∆P = [P (u) − P (0)] = 1Ω

∫Z∗ijdu,

where Z∗ij represents Born effective charge tensor and uis displacement vector of the ions in the ferroelectric withrespect to the paraelectric phase. We calculated the Borneffective charge tensor using density functional perturba-tion theory with GGA+U (Ueff = 4.0 eV). Table-III liststhe principal elements of Born effective charge tensors ofthe ions within the distorted G-AFM structure. Interest-ingly, for Ueff = 4.0 eV, all the ions - Lu, Fe, and O -exhibit a maximum ∼33% rise in their effective charges -+3, +3, and -2, respectively. Such anomalous change inthe effective charges for Fe and O ions is indicative of asizeable covalent character of Fe-O bonds in LuFeO3. Infact, our electron localization function (ELF) calculationdemonstrates asymmetric distribution signifying prefer-ential accumulation of charge at one end of a bond which,in turn, is indicative of covalency. Application of Ueffappears to be reducing the effective charge of Fe andO while that of Lu remains nearly the same. The po-larization P , estimated from the Born effective charges,turns out to be comparable to what has been found fromBerry phase formalism. We further mention here thateven though Ueff influences the band gap and Born ef-fective charge of the ions, no clear Ueff dependence of Pcould be noticed.

It is true that the theoretically calculated polarization(P ), for distorted Pnma structure in presence of G-AFMand Ueff = 4.0 eV, is smaller than that observed exper-imentally by a factor of nearly six. This could be be-cause the calculations do not take care of the small lat-tice strain, domain boundaries, defect network etc whichcould always be present in as-prepared samples unlessspecial care is taken to remove them. The strain-fieldin a real sample (even in bulk form) could couple withferroelectric instability and offer higher polarization as aconsequence. However, from the theoretical calculationscarried out in this work, it is clear that this orthorhom-bic LuFeO3 compound (even in its strain-free most pris-tine form), in presence of G-antiferromagnetic order, isan electronic ferroelectric with lattice structure residingvery close to the ferroelectric instability.

V. SUMMARY

In summary, we show by using synchrotron x-ray, neu-tron, piezoresponse force, and remanent hysteresis datathat small but finite ferroelectricity indeed emerges below

11

TN even in the bulk sample of orthorhombic LuFeO3. Anearlier work [Phys. Rev. B 96, 104431 (2017)] showedthat this could result from lattice ferroelectricity dueto exchange striction at the antiferromagnetic domainboundaries within a multidomain system. The presentwork highlights that the ferroelectricity may have elec-tronic origin as charge disproportionation takes place andbond covalency enhances below TN . The first-principlescalculations show that within the distorted Pnma struc-ture, in presence of G-antiferromagnetic order, ferroelec-tricity has small but finite contribution from both elec-tronic and lattice structures. Theoretical calculationsalso highlight the possibility of structural transition fromPnma to Pna21 at TN . However, because of tiny energydifference between the phases (smaller than the roomtemperature thermal energy), the structural transitioncould not be detected experimentally. In presence ofG-antiferromagnetic order, this compound, therefore, inpristine and single domain form, is possibly, primarily, anelectronic ferroelectric with lattice structure residing veryclose to ferroelectric instability. Under an electric field,

tiny yet detectable piezostriction, of course, could be no-ticed. Small lattice strain, present even in ‘as-prepared’bulk sample, could also yield lattice ferroelectricity.

ACKNOWLEDGMENTS

Two of the authors - UC and SG - acknowledge Re-search Associateship and Senior Research Associateshipof CSIR, respectively. The authors UC, SG, and ARhave equal contribution to this work. The authors DBand AG acknowledge support from SERB, Govt. of India(project no. EMR/2016/001472). The author DB alsoacknowledges a collaborative research program (CRS-M-201) with Bhabha Atomic Research Centre, Mumbai, In-dia. The author AR acknowledges support from SERB,Govt. of India (project no. YSS/2014/000287). The au-thor TC gratefully acknowledges the financial supportprovided by FRM-II to perform the neutron scatter-ing measurements at the Heinz Meier-Leibnitz Zentrum(MLZ), Garching, Germany.

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