Observation of persistent centrosymmetricity in the hexagonal manganite family
The controversy regarding the ferroelectric behavior of hexagonal InMnO is resolved by using a combination of x-ray diffraction (XRD), piezoresponse force microscopy (PFM), second harmonic generation (SHG), and density functional theory (DFT). While XRD data show a symmetry-lowering unit-cell tripling, which is also found in the multiferroic hexagonal manganites of symmetry, PFM and SHG do not detect ferroelectricity at ambient or low temperature, in striking contrast to the behavior in the multiferroic counterparts. We propose instead a centrosymmetric phase as the ground state structure. Our DFT calculations reveal that the relative energy of the ferroelectric and nonferroelectric structures is determined by a competition between electrostatics and oxygen--site covalency, with an of covalency favoring the ferroelectric phase.
pacs:75.85.+t, 77.80.-e, 71.15.Nc, 71.20.-b
I Introduction: structure of the hexagonal manganites
Hexagonal h-MnO ( = Sc, Y, Dy–Lu) represents an established class of multiferroics in which ferroelectricity and antiferromagnetism exist simultaneously. Although their fundamental properties have been investigated for half a century, recent reports of intriguing characteristics such as interlocked antiphase (AP) and ferroelectric (FE) (AP+FE) domain wallsChoi et al. (2010); Jungk et al. (2010); Meier et al. (2012) are fueling continued interest. At the root of these behaviors is their unusual improper geometric ferroelectricity,Van Aken et al. (2004); Fennie and Rabe (2005) which is in turn related to their layered structure, in which planes of ions are interspaced by layers of corner-shared MnO trigonal polyhedra (Fig. 1).
InMnO crystallizes in the same hexagonal manganite structure as the h-MnO compounds, and might be expected to show analogous ferroelectric behavior. In fact, most previous x-ray and neutron powder diffraction refinements assigned InMnO to the polar structure adopted by the multiferroic hexagonal manganites.Greedan et al. (1995); Belik et al. (2009); Fabrèges et al. (2011) In the structure, the MnO trigonal bipyramids tilt and trimerize with a trimerization phase of , where is an integer. The ions on the sites displace up or down along the direction, depending on the tilting direction, and those on the sites in the opposite direction (Fig. 1). This tilt symmetry then enables an additional displacement of the sublattice relative to the Mn-O layers causing a net ferroelectric polarization. Rusakov and Belik pointed out that the nonpolar structure – in which the MnO polyhedra trimerize at intermediate angles and the inversion symmetry is retained (Fig. 1) – and polar structure have similar powder x-ray-diffraction values.Rusakov et al. (2011) They disregarded the model in their subsequent analysis, however, believing that all h-MnO compounds should be polar. Indeed, ferroelectricity has been reported in InMnO below 500 K based on the observation of polarization-electric field (P-E) hysteresis loops obtained by a ferroelectric test system. Such pyroelectric current measurements, especially if they are applied to amorphous samples or thin films, are notoriously sensitive to sample defects, however, and the P-E loops shown in Ref. Serrao et al., 2006 could indicate leaky dielectric behavior rather than ferroelectricity.Scott (2008) In agreement with this, Belik did not observe spontaneous polarization when they repeated the experiment.Belik et al. (2009); Rusakov et al. (2011) Therefore there is no clear evidence to date that InMnO has the structure or shows ferroelectric polarization.
In this study, we revisit the structure and polarization behavior of InMnO by using a combination of x-ray diffraction (XRD), piezoresponse force microscopy (PFM), optical second harmonic generation (SHG), and density functional theory (DFT) and show that InMnO is indeed centrosymmetric, with as the most likely space group (Fig. 1). We explain the difference between InMnO and the multiferroic h-MnO compounds using DFT analysis of the chemical bonding, and propose another candidate material TlMnO that should also show the nonferroelectric InMnO structure. Finally, we discuss the implications of this newly-identified structure for the multiferroicity in the h-MnO family in general.
First we use powder XRD to directly and quantitatively compare the refinements for the candidate polar and nonpolar structures. For sample preparation, a stoichiometric mixture of InO (99.9%) and MnO was placed in Au capsules and treated at 6 GPa in a belt-type high pressure apparatus at 1373 K for 30 min (heating rate 110 K/min). After heat treatment, the samples were quenched to room temperature, and the pressure was slowly released. The resultant samples were black dense pellets. Single-phase MnO was prepared from commercial MnO (99.99%) by heating in air at 923 K for 24 h. The synchrotron XRD data were obtained on powdered samples at the BL02B2 beamline of SPring-8.Nishibori et al. (2001) They were collected in a range from to with a step interval of 0.01 degrees and analyzed by the Rietveld method with RIETAN-2000.Izumi and Ikeda (2000)
An evaluation of the XRD data shown in Fig. 2 clearly reveals that the unit cell of InMnO holds six formula units. This indicates a deviation of the high-temperature phase due to unit-cell tripling with tilt-shear motions of the MnO bipyramids as in the ferroelectric MnO compounds. The centrosymmetric and the noncentrosymmetric subgroups with the highest possible symmetry that are compatible with a trimerization of the high-temperature phase are and , respectively. Here refinements clearly favor the latter structure which may contribute to former claims of the symmetry for InMnO.Rusakov et al. (2011) However, in contrast to the case of the ferroelectric manganites, the structure refinements of the XRD data reveals equally good fits for the centrosymmetric space group and the noncentrosymmetric space group not (a) consistent with a previous observation.Rusakov et al. (2011) In Table 1 we report our refined atomic coordinates and lattice parameters within the and space groups. In general, structures refined with the correct space group are lower in energy than those refined with incorrect space groups. Our DFT calculations for InMnO at the atomic positions and cell parameters obtained in the two competing best-fit experimental refinements (calculation details given later) indicate that the nonpolar structure is 200 meV per formula unit (f.u.) lower in energy than the structure. We therefore suggest the nonferroelectric phase as the ground state for InMnO.
To confirm this suggestion, we next used PFM to probe directly for the presence of FE domains; this technique avoids ambiguities caused by sample leakiness which might have occurred in previous macroscopic polarization measurements.Serrao et al. (2006) In order to calibrate the response from InMnO, PFM and simultaneous scanning force microscopy (SFM) measurements with a commercial SFM (Solaris, NT-MDT) were carried out on YMnO and InMnO. All compounds for PFM and SHG measurements were grown by the flux method as -oriented platelets.Yakel et al. (1963) For PFM an ac-voltage of 14 V at a frequency of 40 kHz was applied to a conductive Pt-Ir coated probe (NSC 35, Mikromasch). The out-of-plane component of the piezoelectric response was recorded by the in-phase output channel of an external lock-in amplifier (SR830, Stanford Research) with a typical sensitivity of 200 V and time constant of 10 ms. The PFM signal of each sample was normalized to a response of the face of PPLN (=7 pm/V) in order to maintain comparability of the PFM response, which was measured before and after each h-MnO measurement in order to exclude changes in the PFM sensitivity.not (b)
Our results are summarized in the equally scaled Figs. 3(a) and 3(b). YMnO reveals the familiar domain pattern of six intersecting AP+FE domains with alternating polarization .Choi et al. (2010); Jungk et al. (2010) Strikingly, the InMnO shows an almost homogeneous distribution of the PFM response with no sign of FE domains. The corresponding SFM data in Fig. 3(c) show that the slightly brighter speckles in the PFM image correspond to protrusions on the unpolished surface of the InMnO sample. If one relates the contrast obtained for opposite domains in YMnO to the spontaneous polarization of C/cm, any polarization in InMnO has to be at least two orders of magnitude smaller to avoid detection in our measurement. Our calculated polarizationKing-Smith and Vanderbilt (1993) for the DFT-optimized InMnO structure is 4.8 C/cm which would certainly be detectable. The absence of ferroelectricity (or of sub-resolution domains) in InMnO is further supported by poling experiments with the SFM tip which did not induce any lasting change of the PFM response.
Since PFM measurements could only be done under ambient conditions, we next used SHG to search for ferroelectric order at low temperature. As discussed in detail in Ref. Fiebig et al., 2005, the breaking of inversion symmetry by ferroelectric order leads to a characteristic SHG signal. The samples were mounted in a liquid-helium-operated cryostat and probed with 120 fs laser pulses in a standard transmission setup for SHG.Fiebig et al. (2005) For comparison, ErMnO was chosen for the SHG data because, unlike YMnO, it has the same magnetic SHG spectrum as InMnO. Figure 4 shows the anisotropy of the SHG signal taken under identical conditions at 5 K on ferroelectric ErMnO and on InMnO. The laser light was incident under 45 to the hexagonal crystal axes so that SHG components coupling to a spontaneous polarization along could be excited.Fiebig et al. (2005) ErMnO shows the double lobe characteristic of the ferroelectric order. In InMnO the double lobe is absent. Instead a SHG signal with the sixfold anisotropy characteristic of the antiferromagnetic order and 150 times weaker intensity is found, indicating that the only order parameter is the antiferromagnetism of the Mn ions. We therefore conclude that InMnO is not ferroelectric down to 5 K.
To resolve the origin of the difference between InMnO and the other hexagonal manganite h-MnO compounds, we used DFT calculations to evaluate the energy difference, , between the candidate and phases. Our spin-polarized first principles calculations were performed using the projector augmented-wave (PAW) methodBlchl (1994) as implemented in vasp.Kresse and Furthmller (1996) In this study, Sc 3, 3, 3, and 4, Y 4, 4, 4, and 5, In 5 and 5, Lu 5, 5, and 6, Tl 6 and 6, Mn 3 and 4, and O 2 and 2 were described as valence electrons. The PAW data set with radial cutoffs of 1.3, 1.4, 1.6, 1.6, 1.7, 1.2, and 0.8 , respectively, for Sc, Y, In, Lu, Tl, Mn and O was employed. The local density of states was also evaluated within the same spheres. Wave functions were expanded with plane waves up to an energy cutoff of 500 eV. All calculations were performed with 30-atom cells, which can describe unit cells of and phases. -points were sampled with a -centered 442 grid. In addition to InMnO, we also calculated for ScMnO, LuMnO, and YMnO, as well as for as-yet-unsynthesized TlMnO. To validate the results, we adopted four different exchange-correlation (XC) functionals: local density approximation (LDA), generalized gradient approximation (GGA), LDA+, and GGA+,Perdew and Zunger (1981); Perdew et al. (1997); Dudarev et al. (1998) with the value for on the Mn-3 orbitals set to 4 eV. In addition we tested two different spin configurations, so-called frustrated antiferromagnetic (FAFM)Medvedeva et al. (2000) and the noncollinear antiferromagnetic (NCAFM) adopted in Ref. Oak et al., 2011. The lattice constants and internal positions were fully optimized in each case until the residual stresses and forces converged to less than 0.1 GPa and 0.01 eV/Å respectively.
Figure 5 shows our calculated values for the various functionals and magnetic configurations. A positive indicates that the ferroelectric structure is stable. When the site is occupied with a b (Sc, Y, or Lu) ion, the phase is more stable than the phase consistent with the experimentally observed ferroelectricity. However, in the case of a a (In or Tl) ion, is close to zero. Note that in these calculations the lattice parameters and ionic positions for both structures were fully relaxed, within the constraint of the appropriate symmetry. [When we previously constrained our atomic positions and cell parameters to those obtained from the XRD analysis, the energy of the phase is much lower (200 meV/f.u.) than that of the phase as we reported in Sec. II.1.] We note that is quite insensitive to the magnetic configuration, but does show a dependence on the choice of XC functional, and for InMnO, the sign of depends on the functional, suggesting the possibility of competing low energy structures.not (c) [Previous calculations of the energy difference between and structures for InMnO derived from XRD data (Sec. II.1) and the polarization for InMnO with the symmetry (Sec. II.2), and subsequent calculations use the GGA+ and FAFM configuration.]
It is clear from Fig. 5 that is not the key factor in determining the phase stability. Instead we focus on the different chemistry of the group a ions, compared to the group b ions. It was previously suggested that the behavior of InMnO is dominated by high-lying occupied semicore (In) electrons;Oak et al. (2011) in contrast the valence states are formally unoccupied in the b ions. In fact it is well known that the presence or absence of semicore electrons can affect the structural stability as illustrated by the different structures of MgO (rock salt) and ZnO (wurtzite, with semicore s) in which the cations have very similar ionic radii ( and in four-coordination and and in six-coordination).
In Fig. 6(a) we show our calculated densities of states (DOS) for InMnO and YMnO (TlMnO, and ScMnO/LuMnO behave analogously to InMnO and YMnO, respectively), both calculated within the phase to allow a direct comparison. In both cases, the valence bands consist mainly of Mn- (up-spin and ) and O-2 states. The main differences occur in the DOSs on the ions. The In “semicore” states, however, form a narrow band that is around eV below the top of the valence band when the electrons are treated as valence (not shown). They do not directly contribute to covalent bonding with the oxygen anions, in contrast to the suggestion in Ref. Oak et al., 2011. The relevant difference is the substantially lower energy of the formally unoccupied and states in In compared with Y, caused by the well-known increase in nuclear charge without corresponding increase in screening across the series. As a result, in InMnO the (and to a lesser extent ) states, which would be completely empty in the ionic limit, develop significant occupation through In-O covalency, with occupied In states in fact forming the bottom of the valence band. (Similar behavior has been previously reported in other In oxides.Zhang and Wang (2011); Mryasov and Freeman (2001)) In YMnO, the Y and states are substantially higher in energy relative to the top of the valence band and so their hybridization with O and subsequent occupation is negligible. Instead there is a small hybridization with the formally empty Y states.
The difference in covalency between InMnO and YMnO manifests particularly strikingly in the calculated valence charge densities at the sites. In Fig. 7 we show the valence charge density differences between LuMnO and YMnO and between InMnO and YMnO in the structure. The charge density at the Mn sites is similar in all cases. Compared with YMnO and LuMnO, however, InMnO, has a decrease in charge density at the O sites adjacent to the In ions and an increase at the outer region of the In site indicating charge transfer from oxygen to In and stronger In-O than Y-O or Lu-O covalent bond formation.
Next we investigate how the additional covalency of the In-O and Tl-O bonds compared with those of Y-O and related compounds manifest in the spring constants. Our calculated -direction spring constants of the ions at the high-symmetry sites in the structure are 2.7, 3.2, and 3.1 eV/ for ScMnO, LuMnO, and YMnO, and are 4.4 and 4.2 eV/ for InMnO and TlMnO. As expected, the strong In-O and Tl-O hybridization results in larger spring constants in the In and Tl compounds. The larger spring constants make the In and Tl ions reluctant to shift from their high-symmetry sites, favoring instead equal -O bond distances. This in turn favors the structure, in which of the ions retain their fully 6-coordinated high-symmetry positions, over the ferroelectric phase, in which all ions are displaced from the high-symmetry positions.
We emphasize that the behavior here in which stronger covalency favors the nonferroelectric phase is completely different from that in conventional ferroelectrics such as BaTiO, in which stronger covalency favors the ferroelectric distortion through the second-order Jahn-Teller effect. In such conventional ferroelectrics, the Born effective charges which participate actively in the re-hybridization are anomalously larger than the formal ionic charges, reflecting the charge transfer that takes place during the ionic displacements to the ferroelectric phase; such anomalous Born effective charges are signatures of instability toward a ferroelectric phase transition.Ghosez et al. (1998) In both InMnO and YMnO the mechanism for the primary symmetry-lowering tilt distortion is geometric rather than due to a rehybridization, and the Born effective charges on all atoms are nonanomalous.not (d) In InMnO, the additional strong In-O covalency in the paraelectric phase resists the distortion of the In ions away from their high-symmetry positions favoring the space group, whereas the lower Y-O covalency provides less resistance, allowing the additional Y-O displacements required to reach the symmetry. In Ref. Cho et al., 2007, the hybridization between the Y- and O- orbitals was measured using polarization-dependent x-ray absorption spectroscopy (XAS) at the O -edge. Then the static charge occupancy in the Y orbitals was equated with an anomalous dynamical Born effective charge, which led to the claim that this hybridization is responsible for the ferroelectricity in YMnO. It is important to understand that the Born effective charge is the of the polarization with respect to ionic displacements, and is unrelated to the static orbital occupancy in a single structure: Partial hybridization of Y with O , while clearly present both in the experiments and in earlier and subsequent first-principles calculations, is not indicative of an anomalous Born effective charge and therefore does not indicate tendency toward ferroelectricity.
Experimentally, it is known that InMnO has an anomalously large lattice constant compared to the multiferroic hexagonal manganites.Greedan et al. (1995) In Fig. 8 we plot our calculated lattice constants (with experimental values where available for comparison) of the manganites series as a function of ionic radii . We point out first that this is not a consequence of the different space group that we have established here; our density functional calculations yield similar lattice constants for InMnO in the and phases. The calculated lattice constants are systematically overestimated compared with experiments as is typical of the GGA. We see that the in-plane lattice constants, , increase monotonically with , with InMnO showing only a small calculated anomaly. In contrast, the lattice constant of InMnO deviates strongly from the trend shown by the b manganites, both in our calculations and in experiment. Since this deviation is also identified in TlMnO, the anomalously large likely originates from covalency in a manganites as shown in Fig. 7.
Although the centrosymmetric phase that we propose in this work for InMnO may be seemingly less attractive compared with the ferroelectric structure, our results have implications for the multiferroic hexagonal manganites as a whole. Since the tilt pattern of the YMnO structure subsequently allows for the development of ferroelectricity, whereas that of the InMnO structure does not, the subtle chemical bonding differences identified here that favor one tilt pattern over another in turn determine whether the resulting structure can be multiferroic. Specifically, we have discussed here that an absence of -O hybridization is required to favor the YMnO tilt pattern over the InMnO tilt pattern; an absence of -O hybridization is therefore a requirement for ferroelectricity in the hexagonal manganites. Earlier theoretical papers correctly noted the electrostatic origin of the “geometric ferroelectricity” mechanism in the hexagonal manganites;Van Aken et al. (2004) we now understand that the relative stability of ferroelectric and nonferroelectric structures is determined by a competition between electrostatics (favoring the ferroelectric phase) and covalency (favoring the nonferroelectric phase).
In addition, since the polar and nonpolar phases are close in energy in InMnO as shown in Fig. 5, it might be expected that their relative stability could be changed using external perturbations such as epitaxial strain. Figure 9 shows the calculated energies of the and phases and their energy differences as a function of in-plane lattice constant , with the out-of-plane lattice constant and internal positions fully relaxed for each value. Because the critical strain of the phase boundary depends on the value, we performed the calculations using both the GGA and GGA methods. We see that the increases with the increasing in-plane lattice constant, indicating a larger in-plane lattice constant could develop the polar . Interestingly, the ferroelectric polarization develops in the out-of-plane direction, in striking contrast to the behavior in perovskites.Diéguez et al. (2005) Therefore we anticipate that InMnO could be tuned into the polar structure using tensile strain.
Finally we mention that a recent transmission electron microscopy study of the domain walls in ferroelectric hexagonal TmMnO and LuMnOZhang et al. (2012) revealed that a domain wall structure at the edges of the sample is similar to the nonpolar InMnO structure: The ion at the wall is at the centrosymmetric position, with one neighbor displaced in the up-direction and one in the down-direction. Detailed calculations of the domain-wall structure in MnO are ongoing.
In summary, we have proposed a different nonferroelectric ground state structure in the hexagonal manganite InMnO, and we predict its occurrence in as-yet-unsynthesized hexagonal TlMnO. The proposed phase has symmetry, and is closely related to the usual ferroelectric ground state but with a different pattern of polyhedral tilts that retains the center of inversion. The energy balance between the two related phases is determined by a competition between electrostatics and -O covalency, with -O covalency favoring the ferroelectric structure. Thus, the absence of ferroelectricity in InMnO reveals to us the reason for the presence of ferroelectricity (and therefore multiferroicity) in the other h-MnO compounds.
Acknowledgements.We thank M. Bieringer, Department of Chemistry, University of Manitoba for providing the InMnO single crystals. Y.K. acknowledges support by JSPS Postdoctoral Fellowships for Research Abroad. Y.K., M.L., N.L., M.F., and N.A.S. acknowledge support from ETH Zurich, and A.A.B. acknowledges support from MANA WPI Initiative (MEXT, Japan), FIRST Program (JSPS), and JSPS Grant No. 22246083. The SXRD was performed under Proposals No. 2009A1136 and No. 2010A1215. The visualization of crystal structures and charge density differences were performed with vesta.Momma and Izumi (2008)
- Choi et al. (2010) T. Choi, Y. Horibe, H. Yi, Y. Choi, W. Wu, and S.-W. Cheong, Nature Mater. 9, 253 (2010).
- Jungk et al. (2010) T. Jungk, A. Hoffmann, M. Fiebig, and E. Soergel, Appl. Phys. Lett. 97, 012904 (2010).
- Meier et al. (2012) D. Meier, J. Seidel, A. Cano, K. Delaney, Y. Kumagai, M. Mostovoy, N. Spaldin, R. Ramesh, and M. Fiebig, Nature Mater. 11, 284 (2012).
- Van Aken et al. (2004) B. Van Aken, T. Palstra, A. Filippetti, and N. Spaldin, Nature Mater. 3, 164 (2004).
- Fennie and Rabe (2005) C. J. Fennie and K. M. Rabe, Phys. Rev. B 72, 100103 (2005).
- Greedan et al. (1995) J. E. Greedan, M. Bieringer, J. F. Britten, D. M. Giaquinta, and H. C. zur Loye, J. Solid State Chem. 116, 118 (1995).
- Belik et al. (2009) A. A. Belik, S. Kamba, M. Savinov, D. Nuzhnyy, M. Tachibana, E. Takayama-Muromachi, and V. Goian, Phys. Rev. B 79, 054411 (2009).
- Fabrèges et al. (2011) X. Fabrèges, I. Mirebeau, S. Petit, P. Bonville, and A. A. Belik, Phys. Rev. B 84, 054455 (2011).
- Rusakov et al. (2011) D. A. Rusakov, A. A. Belik, S. Kamba, M. Savinov, D. Nuzhnyy, T. Kolodiazhnyi, K. Yamaura, E. Takayama-Muromachi, F. Borodavka, and J. Kroupa, Inorg. Chem. 50, 3559 (2011).
- Serrao et al. (2006) C. R. Serrao, S. B. Krupanidhi, J. Bhattacharjee, U. V. Waghmare, A. K. Kundu, and C. N. R. Rao, J. Appl. Phys. 100, 076104 (2006).
- Scott (2008) J. F. Scott, J. Phys.: Condens. Matter 20, 021001 (2008).
- Nishibori et al. (2001) E. Nishibori, M. Takata, K. Kato, M. Sakata, Y. Kubota, S. Aoyagi, Y. Kuroiwa, M. Yamakata, and N. M. Ikeda, N. Nucl. Instrum. Methods Phys. Res. Sect. A 467, 1045 (2001).
- Izumi and Ikeda (2000) F. Izumi and T. Ikeda, Mater. Sci. Forum 198, 321 (2000).
- not (a) The values for nonpolar are =6.61%, =4.90%, =2.48%, and =1.39%. Those for polar phase are =6.64%, =4.94%, =2.36%, and =1.33%. Those for nonpolar are =12.94%, =8.90%, =6.08%, and =3.34%.
- Yakel et al. (1963) H. L. Yakel, W. C. Koehler, E. F. Bertaut, and E. F. Forrat, Acta Cryst. 16, 957 (1963).
- not (b) Although the space group belongs to the trigonal group, in this study we use the term “hexagonal manganite” for simplicity to describe the structure class with and space groups for simplicity.
- King-Smith and Vanderbilt (1993) R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
- Birss (1966) R. Birss, Symmetry and Magnetism (North-Holland, Amsterdam, 1966).
- Fiebig et al. (2005) M. Fiebig, V. Pavlov, and R. Pisarev, J. Opt. Soc. Am. B 22, 96 (2005).
- Shannon (1976) R. D. Shannon, Acta Cryst. A32, 751 (1976).
- Blchl (1994) P. E. Blchl, Phys. Rev. B 50, 17953 (1994).
- Kresse and Furthmller (1996) G. Kresse and J. Furthmller, Phys. Rev. B 54, 11169 (1996).
- Perdew and Zunger (1981) J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
- Perdew et al. (1997) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78, 1396 (1997).
- Dudarev et al. (1998) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
- Medvedeva et al. (2000) J. Medvedeva, O. Mryasov, M. Korotin, V. Anisimov, and A. Freeman, J. Phys.: Condens. Matter 12, 4947 (2000).
- Oak et al. (2011) M.-A. Oak, J.-H. Lee, H. M. Jang, J. S. Goh, H. J. Choi, and J. F. Scott, Phys. Rev. Lett. 106, 047601 (2011).
- not (c) We also performed calculations with the HSE06 hybrid functional (Ref. Heyd et al., 2006) that is known to describe the electronic structure for 3 transition metal compounds more precisely (Ref. Kumagai et al., 2012; Akamatsu et al., 2011, 2012; Stroppa et al., 2010; Hong et al., 2012). The calculated energy differences with FAFM configuration and 332 -points are 0.6 meV/f.u. for InMnO and 19.9 meV/f.u. for YMnO, both of which are similar to the results in Fig. 5.
- Zhang and Wang (2011) Y. Zhang and Y. Wang, J. Electr. Mat. 40, 1501 (2011).
- Mryasov and Freeman (2001) O. N. Mryasov and A. J. Freeman, Phys. Rev. B 64, 233111 (2001).
- Ghosez et al. (1998) P. Ghosez, J.-P. Michenaud, and X. Gonze, Phys. Rev. B 58, 6224 (1998).
- not (d) We calculated by displacing cations in in the -direction. The obtained for InMnO and YMnO are 3.8 and 4.1. They are slightly higher than the formal charges, 3, but smaller than the s on cations in perovskites such as BaTiO () (Ref. Ghosez et al., 1998).
- Cho et al. (2007) D.-Y. Cho, J.-Y. Kim, B.-G. Park, K.-J. Rho, J.-H. Park, H.-J. Noh, B. Kim, S.-J. Oh, H.-M. Park, J.-S. Ahn, ., Phys. Rev. Lett. 98, 217601 (2007).
- Bieringer and Greedan (1999) M. Bieringer and J. Greedan, J. Solid State Chem. 143, 132 (1999).
- Gibbs et al. (2011) A. S. Gibbs, K. S. Knight, and P. Lightfoot, Phys. Rev. B 83, 094111 (2011).
- Diéguez et al. (2005) O. Diéguez, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B 72, 144101 (2005).
- Zhang et al. (2012) Q. H. Zhang, L. J. Wang, X. K. Wei, R. C. Yu, L. Gu, A. Hirata, M. W. Chen, C. Q. Jin, Y. Yao, Y. G. Wang, ., Phys. Rev. B 85, 020102 (2012).
- Momma and Izumi (2008) K. Momma and F. Izumi, J. Appl. Cryst. 41, 653 (2008).
- Heyd et al. (2006) J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 124, 219906 (2006).
- Kumagai et al. (2012) Y. Kumagai, Y. Soda, F. Oba, A. Seko, and I. Tanaka, Phys. Rev. B 85, 033203 (2012).
- Akamatsu et al. (2011) H. Akamatsu, Y. Kumagai, F. Oba, K. Fujita, H. Murakami, K. Tanaka, and I. Tanaka, Phys. Rev. B 83, 214421 (2011).
- Akamatsu et al. (2012) H. Akamatsu, K. Fujita, H. Hayashi, T. Kawamoto, Y. Kumagai, Y. Zong, K. Iwata, F. Oba, I. Tanaka, and K. Tanaka, Inorg. Chem. 51, 4560 (2012).
- Stroppa et al. (2010) A. Stroppa, M. Marsman, G. Kresse, and S. Picozzi, New J. Phys. 12, 093026 (2010).
- Hong et al. (2012) J. Hong, A. Stroppa, J. Íñiguez, S. Picozzi, and D. Vanderbilt, Phys. Rev. B 85, 054417 (2012).