Abstract
Magnetoelectrics offer tantalizing opportunities for devices coupling ferroelectricity and magnetism but remain difficult to realize. Breakthrough strategies could circumvent the mutually exclusive origins of magnetism and ferroelectricity by exploiting the interaction of multiple phonon modes in geometric improper and hybrid improper ferroelectrics. Yet, the proposed instability of a zoneboundary phonon mode, driving the emergence of ferroelectricity via coupling to a polar mode, remains to be directly observed. Here, we provide previously missing evidence for this scenario in the archetypal improper ferroelectric, yttrium manganite, through comprehensive scattering measurements of the atomic structure and phonons, supported with firstprinciples simulations. Our experiments and theoretical modeling resolve the origin of the unusual temperature dependence of the polarization and rule out a reported doublestep ferroelectric transition. These results emphasize the critical role of phonon anharmonicity in rationalizing lattice instabilities in improper ferroelectrics and show that including these effects in simulations could facilitate the design of magnetoelectrics.
Introduction
The prospect of controlling magnetic order via electric fields, and viceversa, is captivating broad interest in developing multiferroic materials, for applications ranging from nextgeneration data storage to sensing and energy conversion^{1,2,3,4,5,6,7,8,9,10,11,12}. Because magnetism and displacive ferroelectricity do not normally coexist (as they are normally associated with partially filled vs. empty transition metal orbitals^{13}, respectively), a quest for mechanisms enabling compatibility has flourished^{9,10,11,12,14}. A promising avenue was identified in socalled geometric improper and hybrid improper ferroelectricity^{7,15,16,17,18,19,20} coexisting with magnetism^{7,8,12,15,17,19,21,22}. It is well established that in “proper” displacive ferroelectric (FE) transitions, the polarization (P) emerges from the condensation of a soft polar mode at the zonecenter (wavevector q = 0), e.g., in BaTiO_{3}, KNbO_{3}, BiFeO_{3} ^{14,18,23}. However, the mechanism underlying “improper” FE transitions, exemplified by the multiferroic hexagonal manganite YMnO_{3}, still remains unconfirmed^{15,18,24}. It is proposed that such transitions arise from anharmonic interactions between an unstable nonpolar phonon mode at finite wavevector (q ≠ 0) and a soft but stable zonecenter polar mode^{15}. The improper FE transition is key to realizing topological (anti)vortices^{12,25}, and offers an opportunity to tune q ≠ 0 nonpolar distortions via electric fields to engineer multifunctional properties not accessible in proper FEs^{7,16,17,18,20}.
Despite extensive efforts, using a wide range of experimental techniques^{26,27,28,29,30,31,32,33,34,35} and theoretical simulations^{15,26,27,33,35,36}, no reported experiments or simulations, to our knowledge, have previously probed the q and Tdependence across \(T_{{\mathrm{FE}}} \simeq 1260\) K of the proposed q ≠ 0 phonon instability thought to drive the FE transition in YMnO_{3}. This is largely because of the experimental difficulty in observing the relevant phonon modes, and limitations of the harmonic approximation in firstprinciples simulations. Experimental studies with Raman and infrared (IR) spectroscopy^{26,29,34} were intrinsically limited to q = 0. Surprisingly, recent measurements of the momentumintegrated phonon density of states (DOS) using inelastic neutron scattering (INS) on polycrystalline YMnO_{3} showed little change across T _{FE} ^{35}. However, as we show here, the behavior of key phonon modes is obscured in the DOS, which averages over all wavevectors. The qresolved INS measurements of Petit et al.^{37}, on the other hand, focused only on the lowT antiferromagnetic transition (~75 K), leaving open questions about the mechanism of the phonon instability across T _{FE}.
Moreover, the structural evolution across the FE transition remains uncertain. It is known that YMnO_{3} crystallizes in the paraelectric (PE) P6_{3}/mmc (#194) at high temperature, and transforms to the FE P6_{3}cm space group (#185) on cooling below T _{FE} ^{31}. But conflicting reports of a doublestep^{27,28,31,34,38} vs. singlestep transition^{15,32,33} have impeded a coherent understanding of the transition sequence. Previous Xray and neutron powder diffraction studies have remained inconclusive^{28,31,39}, calling for singlecrystal measurements. Besides, diffraction experiments alone, lacking energy resolution, cannot resolve atomic dynamics critical in improper ferroelectrics.
Here, we report decisive measurements of momentumresolved phonon dispersions in YMnO_{3} with INS, including the behavior across the FE transition, as well as singlecrystal neutron and Xray diffraction (XRD) measurements of the structural distortion. INS is uniquely powerful to map phonon dispersions across the entire Brillouin zone, providing a stringent test of microscopic theoretical models. In addition, we performed firstprinciples phonon simulations with density functional theory (DFT), in both the FE and PE phases, explicitly including the anharmonic renormalization of phonon dispersions at high temperature. The excellent agreement between our experiments and simulations enables us to elucidate the mechanism of the improper FE transition, and determine the precursor instability in the PE phase. Our results establish the singlestep nature of the transition and directly reveal the unstable mode at the K_{3} zoneboundary point in the PE phase, which condenses at T _{FE}. We model the coupling between primary and secondaryorder parameters to rationalize the temperature dependence of P, which is strikingly different from that in proper ferroelectrics [P ∝ (T _{c} − T)^{0.5}]^{14}.
Results
Structural phase transition and lattice distortions
Our XRD and neutron diffraction measurements on single crystals reveal a clear, singlestep phase transition with trimerization of the unit cell at T _{FE} ~ 1260 K. Figure 1a–d illustrates the structure modulation settling across T _{FE}, with buckling of the yttrium plane along c and tilts of MnO_{5} polyhedra. Note the tripling of unit cell volume (trimerization) from PE to FE (Fig. 1e), leading to superlattice Bragg peaks in the FE phase (compare panels Fig. 1g, i). The transition temperature is confirmed by our calorimetry measurements (Supplementary Fig. 1) and is in good agreement with previous reports^{31,32,33}. Group theoretical analysis^{40} showed that the FE transition is enabled by four symmetryadapted modes of the PE phase: \({\mathrm{\Gamma }}_1^ +\) (identical representation), \({\mathrm{\Gamma }}_2^ \) (zonecenter polar mode), K_{1}, and K_{3} (phonon modes at q = 1/3, 1/3, 0 in the PE phase). The latter three are illustrated in Fig. 1b–d. Figure 1f, h, j show singlecrystal XRD and neutron diffraction patterns, respectively, measured in the (H0L) reciprocal plane across T _{FE}. All reciprocal space indexing is done with reference to the FE unit cell. As one can observe from Fig. 1f–j, H0Ltype superlattice reflections for H ≠ 3n (n integer) appear on cooling below 1300 K, revealing both K_{3} and K_{1} lattice distortions. Fennie and Rabe^{15} pointed out that K_{3} is the more potent distortion and should be the primary order parameter, inducing trimerization, which we confirm with INS (as shown below). Since the amplitude of the K_{1} mode (0.03 Å) is much smaller than that of K_{3} (0.93 Å)^{15,40}, the superlattice peak intensities are primarily determined by the K_{3} modulation. The stable zonecenter polar mode, \({\mathrm{\Gamma }}_2^ \), lowers the lattice symmetry from P6_{3}/mmc to P6_{3}mc (Space group: 186) but does not lead to new Bragg peaks below T _{FE}.
Dynamics of the nonpolar zoneboundary K_{3} distortion
Using INS data, we now investigate the dynamics of the primary K_{3} distortion across T _{FE} by following the outofplane (c) polarized transverse acoustic (TA) branch along the [H, 0, 10] reciprocal lattice direction, whose polarization overlaps with K_{3}. Figure 2a, b, d, e shows maps of the dynamical susceptibility from INS, \(\chi {\prime\prime}({\bf{Q}},E) = S({\bf{Q}},E){\mathrm{/}}\left( {n_s + \frac{1}{2} \pm \frac{1}{2}} \right)\) (Methods), at 300, 612, 1423, and 1753 K (see Supplementary Fig. 2 for additional temperatures). The INS data are compared with our 0 K and finite temperature DFT simulations in Fig. 2c, f. Here, Q denotes the wavevector transfer, E the energy transfer, and k _{B} is the Boltzmann’s constant. We write Q = τ _{ HKL } + q, with τ _{ HKL } the nearest reciprocal lattice vector, and q the reduced phonon wave vector. Note that Γ − K (τ + [HH0]) in the PE phase is equivalent to Γ − M (τ + [H00]) in the FE phase. Above T _{FE}, the TA branch shows pronounced dip at K points \(\left( {H \in {\Bbb Z},H \ne 3n} \right)\) but retains a finite gap (Fig. 2a, b). At the highest T measured in the PE phase, the TA branch rises from H = 0 up to ~10 meV at H = 0.5, then curves back down to E ~ 5 meV at (1, 0, 10). Phonon energies obtained by fitting Gaussians peaks to constantq cuts are shown with black markers in Fig. 2b. While a clear gap is seen at K points for T > T _{FE}, the excitation is also considerably broadened in energy, which indicates that the K_{3} oscillations are strongly damped in the PE phase, reflecting a large anharmonicity. The FE transition corresponds to condensation of the zoneboundary K_{3} modes at T _{FE} closing the gap and leading to the formation of the superlattice Bragg peaks at (1, 0, 10) and (2, 0, 10) as shown in Figs. 1j and 2d, e. The absence of superlattice Bragg peaks above T _{FE} indicates the absence of static, longrange correlations. However, precursor dynamical correlations exist above T _{FE} at K points, and these correlations are clearly seen in constantE maps at low energy, shown in Supplementary Fig. 3. We note that it is important to separate lowenergy dynamical correlations from static correlations, as was done here, and that this is harder to do in conventional diffraction measurements, which integrate over both signals.
The striking experimental behavior of the K_{3} soft mode is quantitatively captured in our DFT lattice dynamics simulations. As shown in Supplementary Fig. 4, phonon dispersions calculated using the harmonic approximation result in an imaginary (unstable) branch for the K_{3} distortion in the PE phase (represented as a negative frequency), which is most pronounced at K points. This instability is renormalized by anharmonicity in our ab initio molecular dynamics (AIMD) simulations^{41}, leading to a stabilized branch at 1500 K, as shown in Fig. 2c and Supplementary Fig. 5a. These are, to our knowledge, the first physically realistic firstprinciples simulations of the lattice dynamics of the PE phase in improper ferroelectrics, capturing renormalization of the unstable phonons by anharmonic effects at high T. The anharmonic DFT simulations correctly predict the finite energy gap opening (~4.8 meV) at [1, 0, 10] and [2, 0, 10] in the PE phase and are in quantitative agreement with our experimental values of 3.41 ± 0.76 and 5.44 ± 0.22 meV at 1423 and 1753 K, respectively. However the computed phonon DOS has only a small change across T _{FE} explaining the apparent lack of effect reported in previous measurements^{35} (Supplementary Fig. 5b, d). Although stabilized at 1500 K, the computed TA branch shows a pronounced dip at K points similar to INS measurements, reflecting the precursor instability at T > T _{FE}. This behavior is understood as a renormalization of the effective K_{3} potential by the phonon bath at high T, as illustrated in Supplementary Fig. 6a. We emphasize that Raman and IR spectroscopy cannot access the continuous condensation of the zone boundary q ≠ 0 instability in the parent PE phase, while INS provides the full dispersions across the Brillouin zone, in both phases. Even in the FE phase, at the superlattice Γ points, only one of the two optical modes (A _{1} and B _{2}) overlapping with the parent K_{3} distortion is accessible^{26,29,34}, since the B _{2} mode is silent in both Raman and IR. On the other hand, both optical phonon branches are accessible with INS (A _{1} is near ~20 meV at (0, 0, 10), B _{2} is near ~15 meV at (1, 0, 10), see Fig. 2e, f and Supplementary Fig. 2a).
We now follow the dynamics of the K_{3}derived modes on cooling in the FE phase. Figure 2d, e directly reveal lowq acoustic modes emanating from the new superlattice peaks whose intensity increases with the growing lattice distortion on cooling. The Tdependence of K_{3}like phonons can be determined from the constantQ cuts along [H, 0, 10]. For q away from the new Bragg peaks (e.g., (0.4, 0, 10)), the TA modes in FE phase show a recovery stiffening (increase) from 9.84 ± 0.07 meV at 1173 K to 12.75 ± 0.05 at 18 K on cooling as the K_{3} distortion increases (Supplementary Fig. 2h). This stiffening on cooling is also consistent with our DFT simulations of the temperature evolution of the K_{3} potential energy, where the curvature at the minimum increases on cooling below T _{FE}: \(\left( {\partial ^2E{\mathrm{/}}\partial Q_{{\mathrm{K}}_{\mathrm{3}}}^2} \right)^{1{\mathrm{/}}2} = K_{{\mathrm{K}}_{\mathrm{3}}}^{{\mathrm{eff}}^{1/2}} \propto \omega _{{\mathrm{K}}_{\mathrm{3}}}^{{\mathrm{eff}}}\), with \(K_{{\mathrm{K}}_3}^{{\mathrm{eff}}}\) and \(\omega _{{\mathrm{K}}_3}^{{\mathrm{eff}}}\) the effective stiffness and phonon energy of K_{3} lattice distortion, as shown in Supplementary Fig. 6a. We emphasize that by mapping the 4D S(Q, E), we were able to establish the absence of additional instabilities at any other q points. This is a critical point to benchmark theoretical predictions in this strongly anharmonic system.
Dynamics of the K_{1} distortion
Additional INS measurements were performed in the (HK0) scattering plane in order to investigate the dynamics of K_{1}like phonons, whose eigenvectors are parallel to the basal plane (Fig. 1d). The dynamics of the K_{1} distortion were measured along [H\(\overline {\mathrm{H}}\)0] around τ = (220). Figure 3a–c shows χ″(Q, E) along the [2 + H, 2 − H, 0] at 300 and 612 K, compared with our DFT simulations. The simulations and measurements are in good agreement, and both show that the K_{1}like TA branches emanating from (1, 3, 0) and (3, 1, 0) are exceedingly weak. While faint superlattice Bragg peaks are indeed observed at (1, 3, 0), (3, 1, 0), and (4, 0, 0) in the FE phase (Supplementary Fig. 7), these are much weaker than K_{3}modulation peaks at (1, 0, 10) and (2, 0, 10), and the intensity of TA phonons emanating from them was not observable. The Tdependence of a nearby TA mode at (2.4, 1.6, 0), of similar character as the K_{1} lattice distortion, is shown in Fig. 3d. We find no change in the energy of this mode between 300 (10.77 ± 0.07 meV) and 612 K (10.89 ± 0.06 meV), within the instrumental energy resolution. For comparison, in the same temperature range, a mode probing the K_{3} distortion in the FE phase shows a pronounced change of 2.07 ± 0.17 meV. This result is also explained by our firstprinciples simulations, which yield very similar phonon energies for both the PE and FE phases for \(\left {{\bf{Q}}  {\boldsymbol{\tau }}_{220}} \right < 0.5\) along [2 + H, 2 − H, 0], and compatible with group theoretical analyses that showed the amplitude of the K_{1} distortion, \(Q_{{\mathrm{K}}_1}\), to be significantly smaller than that of \(Q_{{\mathrm{K}}_3}\) and \(Q_{{\mathrm{\Gamma }}_2^  }\) ^{15,40}.
Dynamics of the zonecenter polar \({\mathrm{\Gamma }}_2^ \) distortion
Further, our firstprinciples lattice dynamics simulations, validated against INS measurements, are critical in clarifying the behavior of the \({\mathrm{\Gamma }}_2^ \) phonons. In the FE transition, the unstable K_{3} distortion creates a geometric field that couples to the polar \({\mathrm{\Gamma }}_2^ \) distortion to induce the finite P, as proposed in ref. ^{15} and also shown in Supplementary Fig. 6. However, previous simulations have not resolved the evolution of the \({\mathrm{\Gamma }}_2^ \) mode across the FE transition. Figure 4 shows χ″(Q, E) along [H, 0, 10] for H near zero, where the contribution from the \({\mathrm{\Gamma }}_2^ \) distortion is significant. The \({\mathrm{\Gamma }}_2^ \) modes in both phases are indicated with ellipses, and their phonon eigenvectors are illustrated in Fig. 4c–e. We note that, because the \({\mathrm{\Gamma }}_2^ \) intensity in the lowT phase is three orders of magnitude weaker than the acoustic modes, its signal is difficult to discriminate from the background in our INS measurements. The calculated \({\mathrm{\Gamma }}_2^ \) mode frequency in the PE phase is at E = 22.85 meV, and overlaps with two phonon modes in the FE phase (also predicted by Prikockyte et al.^{36}, but limited to the FE phase at 0 K). The energies of the resulting modes are 29.72 and 37.01 meV according to our 0 K DFT simulations, in good agreement with Raman spectroscopy: 26.04 meV at 15 K^{29} and 36.82 meV at room temperature^{34}, respectively.
Improper FE coupling of polar and nonpolar distortions
Based on measurements and simulations, we can now rationalize the experimental polarization data from Lilienblum et al.^{33} and clarify the controversy surrounding the single vs. doublestep character of the FE transition. In Fig. 5a, we plot the temperature dependence of P, and energy of \({\mathrm{\Gamma }}_2^ \) Raman phonons (from Bouyanfif et al.^{34}) and K_{3}like distortion from our INS data. As one can observe, although the distortion amplitude \(Q_{{\mathrm{K}}_3}\) increases quickly below T _{FE} (Fig. 5b), the phonon energy of K_{3}derived modes, \(E_{{\mathrm{K}}_{\mathrm{3}}  {\mathrm{like}}}\), shows a much more gradual rise, following the temperature dependence of P (Fig. 5a). The phonon energy of \({\mathrm{\Gamma }}_2^ \) distortion, \(E_{{\mathrm{\Gamma }}_2^  }\), also follows a trend similar to that of P, which is expected, given that \(E_{{\mathrm{\Gamma }}_2^  }\) is roughly linear in \(Q_{{\mathrm{\Gamma }}_2^  }\) (Supplementary Fig. 8c) and \(Q_{{\mathrm{\Gamma }}_2^  }\) is directly proportional to P. This unusual temperature dependence of \(Q_{{\mathrm{\Gamma }}_2^  }\), \(E_{{\mathrm{\Gamma }}_2^  }\), \(E_{{\mathrm{K}}_{\mathrm{3}}  {\mathrm{like}}}\), and P implies Tdependent coupling constants between \(Q_{{\mathrm{K}}_3}\) and \(Q_{{\mathrm{\Gamma }}_2^  }\), i.e., η(T) and ζ(T), with coupling increasing from small magnitude to DFT values calculated by freezing K_{3} and \({\mathrm{\Gamma }}_2^ \) distortions at 0 K (\(E \propto \zeta (T)Q_{{\mathrm{K}}_3}^2Q_{{\mathrm{\Gamma }}_2^  }^2 + \eta (T)Q_{{\mathrm{K}}_3}^3Q_{{\mathrm{\Gamma }}_2^  } + \gamma Q_{{\mathrm{\Gamma }}_2^  }^2 + \ldots\), see Supplementary Fig. 8b, d and Methods section for more details). On the other hand, as shown by Fennie and Rabe^{15}, Tindependent coupling constants would lead to \(Q_{{\mathrm{\Gamma }}_2^  } \propto Q_{{\mathrm{K}}_3}\forall T < T_{{\mathrm{FE}}}\) (since \(Q_{{\mathrm{K}}_3}\) increases suddenly below T _{FE}, and \(Q_{{\mathrm{\Gamma }}_2^  } \propto Q_{{\mathrm{K}}_3}\) for large \(Q_{{\mathrm{K}}_3}\)) and thus cannot account for the experimental trend of \(E_{{\mathrm{\Gamma }}_2^  }\) or P. The weak coupling between \(Q_{{\mathrm{K}}_3}\) and \(Q_{{\mathrm{\Gamma }}_2^  }\) at T _{FE} also confirms that the FE phase transition is indeed driven by the K_{3} instability as the primary order parameter, while \(Q_{{\mathrm{\Gamma }}_2^  }\) is a secondary order parameter. We note that our analysis of \(Q_{{\mathrm{\Gamma }}_2^  }(T)\) is markedly different from the doublestep model of Gibbs et al.^{31} based on powder diffraction, whose result for P(T) was contradicted by the measurements of Lilienblum et al.^{33} Rather, our singlecrystal results indicate that the apparent second phase transition between 900 and 1100 K^{28,31,34,38} is likely due to the gradual increase in coupling between \(Q_{{\mathrm{K}}_3}\) and \(Q_{{\mathrm{\Gamma }}_2^  }\), and associated changes in phonon dynamics. The nature of Tdependent coupling constants is entirely consistent with the topological nature of improper ferroelectricity, where topological defect vortices with continuously varying phase adopt a discrete Z _{6} symmetry at a critical value of coupling strength^{33,42}, and enable a coupled response of the K_{3} and \({\mathrm{\Gamma }}_2^ \) modes^{15}.
In summary, our comprehensive INS and XRD measurements on single crystals, combined with firstprinciples lattice dynamics simulations, including anharmonic effects, provide previously missing direct evidence of the mechanism of the geometric improper FE transition in YMnO_{3}. Our data and analysis resolve an ongoing controversy concerning the single vs. doublestep nature of the FE transition, and account for the unusual Tdependence of polarization. Our detailed study of atomic structure and dynamics across the FE transition not only validates but also refines the proposed transition path and quantitatively accounts for the origin of the polarization. Anharmonic DFT simulations of phonons at finite T, validated against INS measurements, provides a pathway to understand the unstable zoneboundary phonons central to geometric ferroelectrics in both “improper” and “hybrid improper” FE classes. These results and insights are general and applicable to a broad range of FE transitions resulting from anharmonic phonon–phonon interactions, and open a route toward the rational design of ferroelectrics and magnetoelectrics with desirable characteristics.
Methods
Sample growth
Single crystals of YMnO_{3} were grown by the floatingzone technique. The feed and seed rods for the crystal growth were prepared by solidstate reaction. Stoichiometric mixtures of Y_{2}O_{3} and Mn_{2}O_{3} were ground together and calcined in air at 1100 °C for 24 h. It was then reground again into a powder and pressed into a 6mmdiameter 60mm rod under 400 atm hydrostatic pressure, which were finally sintered at 1200 °C for 20 h. The crystal growth was carried out in air in an IRheated image furnace with a growth rate of 4 mm h^{−1}. Small pieces of the single crystals were ground into fine powder for XRD, which shows the pure phase of hexagonal YMnO_{3}.
Measurement procedures
Singlecrystal XRD and INS experiments were carried out to probe the structure and lattice dynamics across T _{FE}. XRD was performed on a 5 × 5 × 1 mm crystal with hard Xrays (E _{i} = 105.091 keV) at beamline 11IDC at the Advanced Photon Source. The crystal was mounted in a Linkam TS1500 stage, with the sample in air (Supplementary Fig. 9d). Singlecrystal INS experiments were performed using the timeofflight (TOF)wide angularrange chopper spectrometer (ARCS) and the hybrid spectrometer (HYSPEC) at the spallation neutron source (SNS), and tripleaxis spectrometer HB3 at the highflux isotope reactor (HFIR) at Oak Ridge National Laboratory^{43}. For TOF measurements on ARCS, we used a closedcycle helium refrigerator and a lowbackground resistive furnace for 18 ≤ T ≤ 300 K, and 300 < T ≤ 612 K, respectively, with an oscillating radial collimator. Two samples, each of mass ~3 gm, were coaligned on an Al mount. The mosaic of coaligned samples was <1.5° in H0L and 2° in HK0 scattering plane (Supplementary Fig. 9b, c). For highT (612 < T ≤ 1753 K) TOF INS measurements at HYSPEC, and all tripleaxis measurements, we used a hightemperature furnace with an air atmosphere to minimize oxygen vacancies at elevated temperature. A singlecrystal piece of mass ~3 gm was mounted in H0L scattering plane on an Al_{2}O_{3} post with Pt wires both of which are stable for our probing temperature range (Supplementary Fig. 9a). The sample mosaic was <0.75°. We did not observe any degradation or change in sample color after heating to 1753 K. We used E _{i} = 40 and 30 meV at ARCS (phonon creation) and HYSPEC (phonon annihilation, see Supplementary Fig. 10 for full coverage) with energy resolution of ~1.8 and ~2.0 meV at the elastic line (FWHM), respectively. The tripleaxis measurements at HB3 were performed using the PG002 monochromator and analyzer, with a constant final energy E _{f} = 14.7 meV, and collimation settings of 48′–60′–60′–120′. Furthermore, our thermogravimetric measurements using TGA Q5000IR (0.1 mg sensitivity) in atmospheric conditions (106.054 mg sample in ceramic pan for 1140 min at 1473 K) did not show any observable mass loss, while similar measurements with Argon flow (154.139 mg sample in ceramic pan for 1140 min at 1473 K) had 0.6% mass loss. Although our singlecrystal XRD measurements (not shown) with Argon flow showed similar results when compared to the results of atmospheric conditions.
Harmonic density functional theory simulations
Phonon simulations were performed in the framework of DFT as implemented in the Vienna Ab initio Simulation Package (VASP 5.3)^{44,45,46}. We used 4 × 4 × 2 gammacentered Monkhorst–Pack electronic kpoint mesh with a planewave cutoff energy of 500 eV in all of our simulations. The convergence criteria for electronic selfconsistent loop was set to 10^{−8} eV. The projectoraugmented wave potentials explicitly included 11 valence electrons for Y (4s ^{2}4p ^{6}5s ^{2}4d ^{1}), 13 for Mn (3p ^{6}4s ^{2}3d ^{5}), and six for O (2s ^{2}2p ^{4}). All our calculations were spin polarized (collinear) with Atype antiferromagnetic order. We used the local spindensity approximation (LSDA) with Atype AFM order and a Hubbard correction^{15}. To treat the localized delectron states of Mn in LSDA + U calculations, the total energy expression was described as introduced by Dudarev et al.^{47} with onsite Coulomb interaction U = 8.0 eV and onsite exchange interaction J = 0.88 eV^{15}. During the relaxation of the structure, the atomic positions were optimized until forces on all atoms were smaller than 1 meV Å^{−1}. We used lattice parameters a = 6.148 Å and c = 11.493 Å. Phonon dispersions were calculated in the harmonic approximation, using the finite displacement approach as implemented in Phonopy^{48}. The atomic forces were obtained with VASP from 20 independent atomic displacements. The phonon calculations used a 2 × 2 × 2 supercell of the hexagonal cell containing 240 atoms. The atomic displacement amplitude was 0.04 Å.
Finite temperature AIMD simulations
AIMD simulations were performed at 1500 K on a 3 × 3 × 1 supercell of highT hexagonal phase containing 90 atoms. We used experimental lattice parameters a = 3.619 Å, and c = 11.341 Å^{31}. AIMD simulations were performed using NVTensemble with Nosé–Hoover thermostat (MDALGO = 2, SMASS = 0.92). We used a planewave cutoff energy of 800 eV with the Γpoint mesh for Brillouin zone integration. The simulations ran for about 3000 fs with a time step of 2 fs. The remaining AIMD simulation parameters were kept identical to 0 K DFT simulations. The trajectories were subsequently postprocessed using TDEP code^{41,49,50} to obtain temperaturedependent effective potential surface and secondorder force constants at 1500 K. Secondorder force constants were used to obtain phonon dispersion along [H, 0, 10] direction (in the lowT FE unit cell notation) and phonon DOS as shown in Supplementary Fig. 5. The phonon energy of ~4.8 meV in AIMD simulations at 1500 K at Kpoint, i.e., (1, 0, 10) or (2, 0, 10) agree quite well with our experimental values of 3.41 ± 0.76 and 5.44 ± 0.22 meV at 1423 and 1753 K, respectively (Fig. 2). In addition, the shift in yttrium dominated ~15 meV and oxygen dominated ~78 meV phonon peak in 0 K DFT simulations to ~10 and ~70 meV, respectively, at 1500 K is consistent with experimental phonon DOS reported by Gupta et al.^{35}
Frozen phonon potential
Potential energy curves, as shown in Supplementary Fig. 6, were obtained by calculating energy for different amplitudes of K_{3} and \({\mathrm{\Gamma }}_2^ \) lattice distortions from DFT simulations in 30 atom lowT notation unit cell. Since K_{1} and \({\mathrm{\Gamma }}_1^ +\) lattice distortions are stable in highT phase and have comparatively smaller amplitude, we do not consider the coupling of K_{1} and \({\mathrm{\Gamma }}_1^ +\) with K_{3} lattice distortion. The energy of the crystal can now be written as a function of \(Q_{{\mathrm{K}}_3}\) and \(Q_{{\mathrm{\Gamma }}_2^  }\) ^{15},
Potential energy curves were fitted using the above energy expression, and the corresponding fit is shown in Supplementary Fig. 6. Our parameters, α = −1.278 eV, β = 0.804 eV, γ = 0.015 eV, δ = 5.01 × 10^{−4} eV, ζ = 0.080 eV, and η = −0.227 eV are in excellent quantitative agreement with results of Fennie and Rabe^{15}.
The Tdependent potential in the highT PE phase for K_{3} lattice distortion is calculated from secondorder force constants obtained from AIMD simulations at 1500 K. The potential curve at 1500 K can be expressed as \(E\left( {Q_{{\mathrm{K}}_3},T} \right) = \alpha (T)Q_{{\mathrm{K}}_3}^2\) for α(T = 1500 K) = 1.47 eV. The potential for K_{3} distortion can be interpolated between the highT PE and lowT FE phase to obtain the potential at the intermediate temperatures—one nearby T _{FE} where curvature in potential is close to zero and another for T < T _{FE}. Linear interpolation of the potential between 0 and 1500 K leads to T _{FE} ≃ 710 K, while for quadratic interpolation (by constraining \(\partial E{\mathrm{/}}\partial Q_{{\mathrm{K}}_3} = 0\) at 0 K) T _{FE} ≃ 1120 K, similar to the experimental value of ~1260 K. \(Q_{{\mathrm{K}}_3}\) obtained from quadratic interpolation of the potential curves follows the similar trend as experimental data shown in Supplementary Fig. 8a except that T _{FE} values have small difference. Moreover, for both interpolation schemes, square root of the curvature—\(\sqrt {\frac{{\partial ^2E}}{{\partial Q_{{\mathrm{K}}_3}^2}}} = \sqrt {K_{{\mathrm{K}}_3}^{{\mathrm{eff}}}} \propto \omega _{{\mathrm{K}}_3}^{{\mathrm{eff}}}\) at equilibrium position increases with decreasing T from T _{FE} to 0 K, thus leading to stiffening (increase) of phonon frequency as experimentally observed and shown in Supplementary Figs. 2h and 5a. Here \(K_{{\mathrm{K}}_3}^{{\mathrm{eff}}}\) and \(\omega _{{\mathrm{K}}_3}^{{\mathrm{eff}}}\) are effective stiffness and phonon energy of K_{3} lattice distortion.
Furthermore, for the \({\mathrm{\Gamma }}_2^ \) distortion, \(Q_{{\mathrm{\Gamma }}_2^  }\) at equilibrium position can be calculated by solving \(\partial E{\mathrm{/}}\partial Q_{{\mathrm{\Gamma }}_2^  } = 0\) to obtain \(Q_{{\mathrm{\Gamma }}_2^  } \simeq \eta (T)Q_{{\mathrm{K}}_3}^3{\mathrm{/}}\left( {2\gamma + 2\zeta (T)Q_{{\mathrm{K}}_3}^2} \right)\). To simplify the expression of \(Q_{{\mathrm{\Gamma }}_2^  }\) in terms of \(Q_{{\mathrm{K}}_3}\), the term involving δ have been left out owing to order of magnitude small value compared to other parameters. \(Q_{{\mathrm{\Gamma }}_2^  }(T)\) calculated from experimental values of \(Q_{{\mathrm{K}}_3}\) is shown in Supplementary Fig. 8a. The Tdependence of η(T) and ζ(T) has been expressed using c _{1} exp[−c _{2}(T/T _{FE})^{2}], where c _{1} is η and ζ values at 0 K calculated using frozen phonon DFT simulations, and c _{2} is found to be 3.5. Here we note that function c _{1} exp[−c _{2}(T/T _{FE})^{2}] is chosen for its simplicity to develop qualitative understanding, and may be of different form in other geometric ferroelectrics. Additionally, from the temperature dependent \(Q_{{\mathrm{K}}_3}\), \(Q_{{\mathrm{\Gamma }}_2^  }\), and ζ, square root of the curvature—\(\sqrt {\frac{{\partial ^2E}}{{\partial Q_{{\mathrm{\Gamma }}_2^  }^2}}} = \sqrt {K_{\Gamma _2^  }^{{\mathrm{eff}}}} \propto \omega _{{\mathrm{\Gamma }}_2^  }^{{\mathrm{eff}}}\) as a function of \(Q_{{\mathrm{\Gamma }}_2^  }\) and T is shown in Supplementary Fig. 8c, d, respectively. Here \(K_{{\mathrm{\Gamma }}_2^  }^{{\mathrm{eff}}}\) and \(\omega _{{\mathrm{\Gamma }}_2^  }^{{\mathrm{eff}}}\) are effective stiffness and phonon energy of \({\mathrm{\Gamma }}_2^ \) distortion.
Phonon intensity simulations
The simulated phonon intensity was calculated using the following expression:
where \(\overline {b_d}\) is neutron scattering length, Q = k − k′ is the wave vector transfer, and k′ and k are the final and incident wave vector of the scattered particle, q the phonon wave vector, ω _{ s } the eigenvalue of the phonon corresponding to the branch index s, τ is the reciprocal lattice vector, d the atom index in the unit cell, exp(−2W _{ d }) the corresponding DW factor, and \(n_s = \left[ {{\mathrm{exp}}\left( {\frac{{\hbar \omega _s}}{{k_{\mathrm{B}}T}}} \right)  1} \right]^{  1}\) is the mean Bose–Einstein occupation factor. The + and − sign in Eq. (3) correspond to phonon creation and phonon annihilation, respectively. The phonon eigenvalues and eigenvectors in Eq. (2) were obtained by solving dynamical matrix using Phonopy^{48}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank O. Hellman for providing and helping to use the TDEP software, C. Luo for help with XRD measurements, B.S. Lokitz for help with thermogravimetric analysis, W. Porter for initial testing of samples at highT under different environmental conditions in DSC, and R. Mills for technical support with the hightemperature furnace. Neutron and xray scattering measurements and analysis were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under the Early Career Award No. DESC0016166. H.Z. (sample synthesis) thanks the support from NSFDMR1350002. The use of Oak Ridge National Laboratory’s Spallation Neutron Source and High Flux Isotope Reactor was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. DOE. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DEAC0206CH11357. Thermogravimetric analysis was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. Theoretical calculations were performed using resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract no. DEAC0205CH11231. This manuscript has been coauthored by UTBattelle, LLC, under contract DEAC0500OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paidup, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doepublicaccessplan).
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D.B., J.L.N., and O.D. performed INS experiments with help from V.O.G., D.L.A, and S.C. D.B. performed singlecrystal XRD with help from Y.R. D.B. analyzed all the data, and performed DFT simulations. R.S. and H.Z. synthesized the samples. D.B. and O.D. wrote the manuscript and all authors commented on it. O.D. supervised the project.
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Bansal, D., Niedziela, J.L., Sinclair, R. et al. Momentumresolved observations of the phonon instability driving geometric improper ferroelectricity in yttrium manganite. Nat Commun 9, 15 (2018). https://doi.org/10.1038/s41467017023092
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DOI: https://doi.org/10.1038/s41467017023092
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