Ferroelectric atomic displacement in multiferroic tetragonal perovskite Sr$_{1/2}$Ba$_{1/2}$MnO$_3$

We investigate the crystal structure in multiferroic tetragonal perovskite Sr$_{1/2}$Ba$_{1/2}$MnO$_3$ with high accuracy of the order of 10$^{-3}$ Angstrom for an atomic displacement. The large atomic displacement for Mn ion from the centerosymmetric position, comparable with the off-centering distortion in the tetragonal ferroelectric BaTiO$_3$, is observed in the ferroelectric phase ($T_\mathrm{N}$ $\leq$ $T$ $\leq$ $T_\mathrm{C}$). In stark contrast, in the multiferroic phase ($T$ $\leq$ $T_\mathrm{N}$), the atomic displacement for Mn ion is suppressed, but those for O ions are enlarged. The atomic displacements in the polar crystal structures are also analyzed in terms of the ferroelectric modes. In the ferroelectric phase, the atomic displacements are decomposed into dominant positive Slater, negative Last, and small positive Axe modes. The suppression of Slater and Last modes, the sign change of Last mode, and the enlargement of Axe mode are found in the multiferroic phase. The ferroelectric distortion is well reproduced by a first-principles calculation based on Berry phase method, providing an additional information on competing mechanisms to induce the polarization, electronic $p$-$d$ hybridization vs. magnetic exchange-striction.


I. INTRODUCTION
Since a large nonlinear magnetoelectric effect was found in perovskite TbMnO 3 , multiferroic materials have been extensively investigated [1][2][3][4] .It is well known that the electric polarization of the most multiferroic materials is farc smaller than those of conventional ferroelectric materials such as BaTiO 3 .Nevertheless, rather large electric polarization among the multiferroic materials are theoretically shown in tetragonal perovskite BaMnO 3 [5][6][7] .In the paramagnetic phase, it is prospected that the ferroelectricity is induced by the off-centering distortion of the Mn and O ions.Since the magnetic Mn 4+ ion directly contributes to the emergence of the ferroelectricity, the large magnetoelectric effect is expected.However, in fact, it was known that the hexagonal structural phase is stable in BaMnO 3 .Sakai et al. found that Sr 1/2 Ba 1/2 MnO 3 with the smaller tolerance factor is crystallized in the tetragonal perovskite structure and shows ferroelectricity 8 .Below the magnetic phase transition temperature, the change of the crystal lattice with large reduction of the electric polarization is observed in tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 8 .
Perovskite Sr 1/2 Ba 1/2 MnO 3 exhibits two phase transitions, ferroelectric and antiferromagnetic, at T C ∼ 400 K and T N ∼ 185 K 8 , respectively.Here, we call the phases for T N ≤ T ≤ T C as ferroelectric and for T ≤ T N as multiferroic, respectively.Below T C , it is reported that the crystal system changes from the centrosymmetric cubic to the polar tetragonal, determined from the temperature variation of the c/a-lattice-constant ratio.The c/a increases with decreasing temperature below T C and sat-urates near 250 K.In contrast, below T N , the c/a turns to decrease with decreasing temperature and saturates below 150 K. From the reduction of c/a, it was speculated that the electric polarization is also reduced.The crystal structure analyses with use of the twin-free single crystal was also performed in the ferroelectric phase 8 .The result indicates that the origin of the ferroelectricity is the off-centering distortion of O1-Mn-O1 bond angle, same with the tetragonal ferroelectric BaTiO 3 [See Fig. 1 (a)] 9-11 .In the multiferroic phase, the distortion of O1-Mn-O1 at 50 K is smaller than that at 225 K in the ferroelectric phase.In view of the phase continuity from the antiferromagnetic ordered phase of SrMnO 3 12,13 , the antifferomagnetic structure in the multiferroic phase is inferred to be G-type (G-AFM), in which nearest neighbor magnetic moments are aligned antiparallel as shown in Fig. 1 (b).In the earlier study, it was speculated that the ferroelectric polarization is suppressed to obtain the gain of the magnetic exchange energy 8,14 .To our best knowledge, however, the quantitative comparison between the experimental result and the theoretical calculation for the electric polarization in the multiferroic phase has not yet been done.To quantitatively discuss the suppression mechanism of the ferroelectric polarization, information of the atomic displacements and the frozen ferroelectric modes in the ferroelectric and multiferroic phases is necessary.Nonetheless, in the earlier crystal structure analysis, the obtained atomic displacements of the ions in multiferroic phase is smaller than the experimental uncertainties 8 .Thus, the accurate crystal structure analysis in the multiferroic phase of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 has been desired.In a multiferroic system, an essential contribution of the quantum Berry phase of valence electrons can be revealed by the combined study of accurate structural analysis and first-principles calculation 15 .For tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 , Giovannetti et al. performed the first-principles density functional theory (DFT) band simulation and claimed that the ferroelectric polarization caused by Mn-O2 hybridization is suppressed by Mn-O1-Mn superexchange interaction in the G-AFM ordering 14 .In their study, the crystal structure in the multiferroics phase was theoretically optimized with generalized gradient approximation (GGA) potential whereas the comparison with the experimental structure was missing.It is also noteworthy here that a simulation study can provide an ideal magnetic structure that enhances ferroelectricity.In the multiferroic materials, the change of the magnetic structure may induce much larger ferroelectric polarization 16,17 .Thus, for the further understanding of the multiferroic properties in tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 , it is important as well to evaluate the ferroelectric polarization in hypothetical magnetic structures.
In this paper, we report the atomic displacements in the ferroelectric and multiferroic phases of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 determined by the crystal structure analyses using the twin-free single crystal and higher-Q diffraction data than earlier work 8 .
By the ferroelectric mode analyses, the polar crystal structures in the ferroelectric and multiferroic phases for Sr 1/2 Ba 1/2 MnO 3 and other tetragonal perovskite (AB O 3 ) were classified.By a first-principle calculation based on the accurate-crystal-structural parameters, we quantitatively clarify the suppression mechanism of the ferroelectric polarization in multiferroic phase and discuss the possible magnetic structure that enhances the electric polarization.

II. EXPERIMENTAL AND COMPUTATIONAL PROCEDURES
A single crystal of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 was synthesized by a high-pressure treatment on the precursor sample of oxygen-deficient single crystals 8 .A synchrotron x-ray diffraction experiment was performed on BL02B1 at SPring-8, Japan 18 .The photon energy of the incident x rays was tuned at 35.04 keV.Using the high energy x-ray, we can access diffraction peaks with high spatial resolution up to Q ∼ 30 Å−1 .The single crystal was crushed into cubes with a typical dimension of about 20 µm.The absorption coefficient µ is calculated to be 37.85 cm −1 .The empirical absorption correction was carried out 19 .Rapid-Auto program (Rigaku Corp.) was used to obtain an F-table.CRYSTAL STRUCTURE (Rigaku Corp.) program was used for analyzing the crystal structure from the F-table.In the crystal structure analysis in the multiferroic phase at 50 K, the isotropic atomic displacement parameter B iso was used for the Ba/Sr site.
First-principles calculations were performed using the VASP code 20 within the GGA + U 21 formalism with various U values.In addition, we employed the Heyd-Scuseria-Ernzerhof (HSE06) screened hybrid functional method 22 , which mixes the exact non-local Fock exchange and the density-functional parametrized exchange.The HSE06 is known to improve the evaluation of the band gap energy and the structural distortion, with respect to GGA + U approaches 23 .The cut-off energy for the plane-wave expansion of the wavefunctions was set to 400 eV and a k-point shell of (4, 4, 3) was used for the Brillouin zone integration according to Monkhorst-Pack special point mesh.The crystal structure was optimized with respect to internal atomic coordinates until the remaining forces were less than 1 meV/ Å while the lattice parameters were kept at the experimental values.

A. Synchrotron x-ray diffraction and crystal structure analysis
The synchrotron x-ray diffraction experiments have been carried out in the ferroelectric (T = 225 K) and in the multiferroic (T = 50 K) phases of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 .All observed diffraction spots can be indexed by those of the P 4mm space group.By using these data sets, we performed crystal structure analyses.Here, the A-site ion is fixed at the centrosymmetric position.The comparisons between observed and calculated structure factors are shown in Fig. 2. The structural parameters at 225 K and 50 K are summarized in Tables I and II, respectively.Schematic views of the atomic displacements in the ferroelectric and multiferroic phases of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 are shown in Figs.3(a) and 3(b).As a reference, the O1-Mn-O1 bond angles of tetragonal BaTiO 3 and cubic SrMnO 3 are also shown in Fig. 1(a).In the ferroelectric phase of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 , the atomic displacements along the c-axis at 225 K are respectively 0.0711(5) Å for Mn ion, 0.0218( 16) Å for O1 (2c site), and 0.0350 (19) Å for O2 (1b site), which are of the same order of magnitude as those reported by the earlier study 8 .The off-centering distortion can be estimated by the O1-Mn-O1 bond angle, as 174.45(13) • at 225 K in the ferroelectric phase, comparable with that of tetragonal BaTiO 3 24 .The atomic displacements and O1-Mn-O1 distortion at 50 K in the multiferroic phase are respectively changed as 0.0177(15) Å for Mn ion, 0.040(5) Å for O1, 0.062(4) Å for O2, and 176.5(2) • .In this study, since the atomic displacements are determined with the accuracy of the 10 −3 Å order, we observed that the atomic displacements of O ions are larger than that of Mn ion in the multiferroic phase.This enlarged atomic displacements of the O ions can not be explained only by the suppression of the offcentering distortion.The reason for this enlargement will be discussed later.
Next, we analyzed the observed atomic displacements by the ferroelectric modes to compare the ferroelectric and multiferroic phases of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 with other ferroelectric perovskite materials.At the structural phase transition from cubic P m3m to tetragonal P 4mm in the perovskite oxide, the polar vibrational motion is decomposed by three modes, so-called Slater, Last, and Axe modes [See Fig. 3(c)].The analysis by these ferroelectric modes is commonly performed to classify the soft phonon mode obtained from the optical, x-ray, and neutron spectroscopy experiments [25][26][27][28][29][30] .In this paper, we used polar atomic displacements from centrosymmetric positions to estimate frozen ferroelectric modes.The ratio of the frozen ferroelectric modes in tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 are compared with those in other ferroelectrics in Table III.The frozen ferroelectric modes can be quantified from the masses and the atomic displacements from the centrosymmetric positions of ions, as Harada et al. did using the inelastic structure factor of the soft phonon modes 31 .Here, the polar atomic displacements of B -site, O1, and O2 sites are represented by ξ B , ξ O1 , and ξ O2 , respectively.The coefficients of the ferroelectric modes, S Slater , S Last , and S Axe , can be defined as Here, s Slater = (−k, 1, 1),  A-site, B -site, and O ions, respectively.Thus, In Table III, the coefficients of the ferroelectric modes are summarized for tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 , in comparison for other perovskite ferroelectrics, tetragonal BaTiO 3 , KNbO 3 , PbTiO 3 , and BiCoO 3 9,24,32,33 .In the ferroelectric phase of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 , the dominant positive S Slater , relatively large negative S Last , and small positive S Axe are obtained.The contributions from the |S Slater |, |S Last |, and |S Axe | are approximately 70 %, 18 %, and 12 %, compatible with the result (71 %, 24 %, and 5 %) obtained by the optical and inelastic x-ray spectroscopies 30 .In tetragonal BaTiO 3 , the dominant positive S Slater , small negative S Last , and relatively large positive S Axe are observed.In tetragonal KNbO 3 , the positive S Slater is dominant but the error bars for the other modes are too large.As a commonality, they share two characteristics, the dominant positive S Slater and negative S Last .
To explain the origin of the commonality, we refer the earlier first-principles calculations for perovskite oxides, which have pointed out the importance of the covalency between the B -site and the apical O2 ions for the emergence of ferroelectricity [34][35][36][37] .This is the reason why the dominant parameter is the Slater mode with contracting the distance between the B -site and O2 ions as shown in Fig. 3(c).The negative S Last and positive S Axe play a role in reducing the extra atomic displacements of the O1 ions generated by the Slater mode.
In PbTiO 3 , the earlier first-principles calculation also pointed out that the hybridization between the 6p band of Pb and 2p band of O1 induces additional component of the electric polarization 36 .In that case, the distance be- tween the Pb and O1 ions also decreases.This atomic displacement induces the combined S Slater and S Last mode, which can be actually seen in PbTiO 3 , as listed in Table III.In BiCoO 3 , since the ratio of the ferroelectric modes is similar with that of PbTiO 3 , we speculate that the origin of the ferroelectricity for BiCoO 3 is the same for PbTiO 3 .
In the multiferroic phase of Sr 1/2 Ba 1/2 MnO 3 , S Slater and S Last are suppressed, while S Axe is enlarged.In addition, the sign of S Last changes to positive.The G-AFM exchange interaction prefers 180 • O1-Mn-O1 bond angle, being contradictory with off-centering distortion of O1-Mn-O1 bond angle.Therefore, the displacement for Mn ion is suppressed, and consequently gives rise to the decrease of S Slater and S Last modes.In stark contrast, the apical O2 is relatively free from the restriction of the magnetic exchange interaction.Thus, we speculate that the atomic displacement for apical O2 is enlarged to obtain the gain of the covalency between Mn and O2, resulting in the enlarged S Axe parameter.To eliminate the extra atomic displacements of the O1 from S Axe , the sign of S Last mode changes to positive in the multiferroic phase.From the experimentally determined crystal structure information and the results of the mode analyses, we discussed and speculated the qualitative suppression mechanism of the ferroelectricity.To support this speculation and provide more quantitative discussion, we performed the first-principles calculation.

B. First-principles calculation
For the discussion of the atomic displacements and the resulting ferroelectricity in this system, we performed first-principles calculations for tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 .To understand the effect of the magnetic order upon the ferroelectricity in the multiferroic phase, here we also simulate the ferroelectric polar-ization in the hypothetical A-type antiferromagnetic (A-AFM) structure (the magnetic moments are aligned parallel in a-b plane with antiparallel coupling with neighbor planes as shown in Fig. 1(b)) as well as the ground-state G-AFM structure in Sr 1/2 Ba 1/2 MnO 3 .
Figures 4(a) and 4(b) show the density of states from GGA + U calculations.When we set U = 3 eV and J = 1 eV as consistent with the previous DFT study 14 , the system is insulator while the energy gap is significantly underestimated as E gap ∼ 0.5 eV, being inconsistent with the experimentally estimated energy gap ∼ 2 eV for SrMnO 3 38 .The underestimation of the energy gap was not improved when the U value was increased up to 6 eV [see Fig. 4(b)]; on the contrary, the band gap was reduced to be ∼0.3 eV.This result might seem counterintuitive but this is due to a property of GGA + U method that adds effective Coulomb potential only to the localized orbital states (such as 3d and 4f orbital states).Indeed, GGA + U Coulomb potential shifts down the occupied Mn 3d states but keeps delocalized O 2p states at the original energy levels around the valence top state.When the O 2p states are located at shallow energy level, Mn ion favors to show trivalent instead of quadrivalent ionic state.This is the reason why the band gap tends to be closed as increasing the U value.To make matters worse, this narrow energy gap is closed when hypothetical ferromagnetic phase or A-AFM phase is calculated.Therefore, we conclude that GGA + U approach is not appropriate to describe the wide gapped insulating state and evaluate the ferroelectric distortion in Sr 1/2 Ba 1/2 MnO 3 .
Figure 4(c) shows the density of states from HSE06 calculation, leading to the wider energy gap (E gap ∼ 2 eV, consistent with the experimental data in SrMnO 3 38 ) with Mn quadrivalent state.In this case, the fraction of exact Hartree-Fock exchange in HSE06 scheme shifts down both the occupied Mn 3d levels and O 2p levels.Hereinafter, we will focus on HSE06 results and discuss the Density of states (DOS) in G-AFM of Sr 1/2 Ba 1/2 MnO3 calculated from GGA + U method with (a) U = 3 eV and J = 1 eV, (b) U = 6 eV and J = 1 eV, and from (c) HSE06 method.Upper and lower panels show majorityand minority-spin states, respectively.Projected DOS for Mn t2g and eg orbital states are highlighted with green and orange colors, respectively.ferroelectric property.By using the experimental and DFT-optimized crystal structures, the ferroelectric polarization was calculated as listed in Table IV.In order to investigate the influence of the magnetic ordering to the ferroelectric polarization, we consider the ground-state G-AFM and the hypothetical A-AFM configurations.
It is noted that the calculated electric polarization by the optimized structure based on point-charge model with nominal ionic charges (Ba and Sr; 2+, Mn; 4+, O; 2-), i.e., the ionic displacement contribution to the electric polarization, shows good agreement with that estimated by the experimental crystal structure: P PCM ∼ 10.1 µC/cm 2 with G-AFM in both experimental and optimized structures at T = 50 K.This result supports the advantage of use of HSE06 functional for the polar structural distortion with a high accuracy.The total electric polarization P Berry , i.e., the summation of ionic and electronic contributions, is almost double of the P PCM , as often seen in other ferroelectric manganites 39 , and is of the same order of magnitude with the electric polariza-  5: Schematic illustration of ionic distortion (narrow gray and red arrows) and induced electric polarization P (wide black arrows) for (a) G-type (G-AFM) and (b) A-type (A-AFM) antiferromagnetic orderings.Narrow gray and red arrow stand for the atomic displacement originating from the hybridization between Mn 3d and apical O2 2p bands and the in-plane Mn-O1-Mn magnetic magnetic exchange striction, respectively.P hyb and Pexstr stand for the electric polarization from the hybridization and the exchange striction, respectively.U and D denote up-and down-spins Mn sites, respectively.The detailed crystal and magnetic structures are shown in Fig. 1(b).tion (13.5 µC/cm 2 ) experimentally obtained in the earlier study 8 .
Next, we focus on the the suppression mechanism of the ferroelectricity in the multiferroic phase.Basically, we consider two mechanisms to induce the polarization: hybridization between Mn 3d and apical O2 2p states (P hyb and in-plane Mn-O1-Mn magnetic exchange striction (P extr ) as shown in Fig. 5.The p − d hybridization drives polar ionic distortion of Slater mode, by which Mn and O2 ions are respectively shifted downward and upward.The magnetic exchange striction modulates the in-plane Mn-O1-Mn bond angle (φ), resulting in the suppression of the S Slater and the sign change of S Last .Since the driving mechanism of the change of the ferroelectricity upon the ferroelectric to multiferroic phase transition is the magnetic exchange interaction, the atomic displacement should depend on the Mn spin configuration.In G-AFM, the magnetic exchange striction favors φ = 180 • so that Goodenough-Kanamori rule is satisfied for Mn 4+ ion [40][41][42] .This magnetic exchange striction prevents the atomic displacement of the side O1 ion so that total electric polarization is reduced.In contrast, in case of the hypothetical A-AFM, the magnetic exchange striction favors φ = 90 • and enhances the hybridizationinduced polarization as shown in Fig. 5.The calculated polarization with G-AFM and A-AFM is P Berry = 20.17 and 30.24 µC/cm 2 , respectively, as being consistent with the above discussed mechanism.The difference of the P values allows us to decompose the polarization into two contributions, P hyb ∼ 25 and P extr ∼ 5 µC/cm 2 .The former is comparable to the conventional ferroelectric polarization in BaTiO 3 (P ∼ 26 µC/cm 2 ) and the latter is comparable to the magnetically-driven polarization in multiferroic HoMnO 3 (P ∼ 6 µC/cm 2 ) 39 .Thus, we conclude that since only positive P hyb contributes the ferroelectric polarization in the paramagnetic phase, negative P extr causes the suppression of the polarization observed in the multiferroic phase.If one succeeded in stabilizing A-AFM in tetragonal AMnO 3 system, it might be a milestone multiferroic demonstrating the polarization larger than representative ferroelectric BaTiO 3 .Nonetheless, since A-AFM is energetically unfavored by 1.2 eV/f.u. with respect to G-AFM, a study to stabilize the A-AFM AMnO 3 is left as a topic for future work.

IV. SUMMARY
In summary, we have performed the synchrotron x-ray diffraction experiment to investigate the accurate crystal structures in the ferroelectric and multiferroic phases of tetragonal perovskite Sr 1/2 Ba 1/2 MnO 3 using twin-freesingle-crystalline sample.The large atomic displacement for Mn ion was observed in the ferroelectric phase.In the multiferroic phase, by contrast the atomic displacement for Mn ion is suppressed, but those for O ions are enlarged.From the obtained crystal structural parameters, the ferroelectric mode analyses were carried out.In the ferroelectric phase, the atomic displacements can be decomposed as dominant positive Slater, negative Last, and small positive Axe modes.The suppression of Slater and Last modes, the sign change of Last mode, and the enlargement of Axe mode are found in the multiferroic phase.The first-principles calculation using HSE06 functional successfully described the wide-gap insulating electronic states and quantitatively reproduces the experimentally observed ferroelectric polarization.The calculated ferroelectric polarization is further decomposed into two parts relevant to the hybridization and exchange striction mechanisms.

FIG. 1 :
FIG. 1: (Color online) (a) Schematic illustrations of the perovskite AB O3 structure.The O1-B -O1 bond angles of the BaTiO3 and SrMnO3 are also shown.(b) Schematics to explain the G-type antiferromagnetic (G-AFM) and A-type antiferromagnetic (A-AFM) structures on AB O3, drawn by VESTA 43 .Here, the A-site ions are not shown.Blue and brown circles stand for the up and down spins on B -site, respectively.O1 and O2 sites are indicated by the red circles.In G-AFM, the neighbor magnetic moments are aligned antiparallel.In A-AFM, the magnetic moments are aligned parallel in a-b plane with antiparallel coupling with neighbor planes.
FIG. 4:Density of states (DOS) in G-AFM of Sr 1/2 Ba 1/2 MnO3 calculated from GGA + U method with (a) U = 3 eV and J = 1 eV, (b) U = 6 eV and J = 1 eV, and from (c) HSE06 method.Upper and lower panels show majorityand minority-spin states, respectively.Projected DOS for Mn t2g and eg orbital states are highlighted with green and orange colors, respectively.
and M O stand for the mass of

TABLE II :
Structure parameters of Sr 1/2 Ba 1/2 MnO3 at 50 K in the multiferroic phase (Space group P 4mm (No. 99)).The 23321 reflections were observed, and 3409 of them are independent.The 14 variables were used for the refinement.The lattice parameters are a = 3.84300(10) Å and c = 3.8549(3) Å.The reliability factors are R = 2.99%, Rw = 2.35%, GOF(Goodness of fit) = 0.99.In Ba/Sr site, the isotropic atomic displacement parameter is used for the crystal structure analysis.

TABLE III :
Frozen ferroelectric modes estimated from the atomic displacements for B -site and O ions in the tetragonal ferroelectric and multiferroic phases.S Slater , SLast, and SAxe stand for the coefficients of Slater, Last, and Axe ferroelectric modes, respectively.The contribution ratio from the |S Slater |, |SLast|, and |SAxe| are also shown in brackets.ξB, ξO1, ξO2 are respectively the atomic displacements for B, O1, and O2 sites.Here, we selected the sign of the atomic displacement ξB so that the S Slater is positive.∠O1BO1 stands for the distortion of O1-B -O1 bond angle.

TABLE IV :
Calculated ferroelectric polarization for the experimental (E) and optimized (O) crystal structure at T = 50 K for the G-AFM and A-AFM antiferromagnetic ordering as based on HSE06-exchange-correlation functional.Both the net polarization obtained by Berry phase method (PBerry) and the ionic contribution based on point charge model (PPCM) are shown in unit of µC/cm 2 . FIG.