Direct Visualization of Irreducible Ferrielectricity in Crystals

In solids, charge polarity can one-to-one correspond to spin polarity phenomenologically, e.g. ferroelectricity/ferromagnetism, antiferroelectricity/antiferromagnetism, and even dipole-vortex/magnetic-vortex, but ferrielectricity/ferrimagnetism kept telling a disparate story in microscopic level. Since the definition of a charge dipole involves more than one ion, there may be multiple choices for a dipole unit, which makes most ferrielectric orders equivalent to ferroelectric ones, i.e. this ferrielectricity is not necessary to be a real independent branch of polarity. In this work, by using the spherical aberration-corrected scanning transmission electron microscope, we visualize a nontrivial ferrielectric structural evolution in BaFe2Se3, in which the development of two polar sub-lattices is out-of-sync, for which we term it as irreducible ferrielectricity. Such irreducible ferrielectricity leads to a non-monotonic behavior for the temperature-dependent polarization, and even a compensation point in the ordered state. Our finding unambiguously distinguishes ferrielectrics from ferroelectrics in solids.

iron ladders (labelled as A and B). Long-range block-type antiferromagnetism (block-AFM) appears below the Néel temperature TN~240-256 K [15][16][17][18][19]. The structural tetramerization due to the block-AFM leads to charge dipoles along the a-axis and the alignment of dipoles is almost antiparallel but with a tiny canting angle (~5.78 o at room temperature) between ladders A and B (schematic displacements are shown in Fig. 1c) [14]. It should be noted that all other members of the 123-series iron selenides (e.g. BaFe2S3) also own a similar (quasi-) one-dimensional (1D) ladder structure, but only BaFe2Se3 has the canting ladder characteristic.
According to theory [14], a residual polarization along the c-axis (Pc) is expected in BaFe2Se3, as a characteristic of a 'reducible' ferrielectric material. Each unit is composed of two iron-ladders. c, Room temperature structure with a tiny tilting angle between the ladders A and B. d, High temperature structure without tilting. The in-situ selected area electron diffraction patterns are shown. The lattice periodicity along the [100] direction changes from 11.90 Å to 5.96 Å as the temperature rises from 298 K to 623 K. e, DSC curves indicate two transitions at~610 K and~420 K. The~610 K transition is a second-order one with a step-like behavior, while the~420 K transition is a first-order one with a peak.
The tilting of the iron ladders gradually disappears with increasing temperature up to~600 K, leading to a high symmetric Bbmm phase, according to X-ray diffraction [16]. Our in-situ selected area electron diffraction (SAED) results give distinctive patterns (Fig. 1c vs Fig. 1d), confirming the disappearance of the ladder canting at high temperature. Differential Scanning Calorimetry (DSC) measurements (Fig. 1e) confirm the phase transition occurring at~610 K as a second-order one. Besides, there is a first-order phase transition at~420 K, which was also evidenced in Ref. 16 but its real origin remains a puzzle. In addition, the resistivity behavior also supports this first-order transition (Supplementary Figure 1). Since our neutron powder diffraction (NPD) data confirm that only one block-AFM transition appears at 250 K , obviously, the transition at~420 K can be excluded as a magnetic-ordering behavior Interestingly the structural tetramerization already exists at room temperature and, unexpectedly, the intensities for ladders A and B are not equivalent, creating a strong ladder and a weak ladder within a unit cell. The inequivalent features in strong ladders and weak ladders are embodied in magnitude of displacements, as shown schematically in Fig. 2a and in the high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images along the b ( Fig.   2b and 2d) and c axis ( Fig. 2c and 2e). The line profiles along these two ladders are shown in Supplementary Figure 4. A displacement vector-mapping algorithm was implemented on the cross-sectional HAADF-STEM images to measure the local displacement of the atoms. Based on the statistics of about 300 data for each length, the Fe atoms displacement in the strong ladders are stronger than that in the weak ladders (see Supplementary Figure 5 for an example).
Considering that the Fe-block tetramerization could induce Se ions displacement along the a-axis [14] (as clearly indicated in Fig. 2b and 2d), such inequivalence of Fe displacements will lead to a residual polarization along the a-axis (Pa), which is larger than the expected Pc. The arrow is added to make it more visible. The uncovered zoom-in images (raw data) have been shown in Fig. 2(b-c). Therefore, BaFe2Se3 is a room temperature ferrielectric with polarization mainly along the a-axis, rather than the expected low temperature ferrielectric with polarization along the c-axis [14]. The averaged Fe L2,3 edge spectrum from neighboring Fe chains is presented. A Fe-L3 peak shift of approximately 0.4 eV between neighboring Fe chain is observed.
To explore the origin of this unbalanced ladders, monochromated electron energy loss spectra (mono-EELS) were acquired to demonstrate the underlying charge modulation as shown in Fig. 2f. Using the monochromatror we reach an energy resolution of 0.3 eV, which is enough to detect subtle changes in the fine structure of the EELS excitation edges. The averaged Fe-L3 edges in strong chains and in weak chains show significant differences in their ELNES. Comparing with Fe reference spectra [20], the valence states of the strong and the weak chains in BaFe2Se3 are different, and the Fe-L3 peak shift between them is approximately 0.4 eV (shown as the distance between two dashed line in Fig. 2f). Note that the EELS have an ability to reflect a valance change, but the absolute valance identification is still challenging, since the absolute energy position always has several hundred meV uncertainty. Such charge disproportion is not unusual in correlated electron systems, especially in Fe-based oxides and fluorides, e.g. Fe3O4 and LiFe2F6 [21,22], although it has not been reported in selenides before.
Nominally, the valences of Fe can become +(2+) and +(2-) for the two sublattices, and a proper  (around 0.15 according to our fitting results) can lead to the structural tetramerization following the idea of Peierls transition [14]. It should be noted that the non-integer valences are possible in iron selenides, e.g. in . Therefore, charge-ordering, i.e. difference of local electron density, can be the key ingredient for the unbalanced structural tetramerization and affiliated polarization, which needs deeper investigation in future.

Variation of the BaFe2Se3 structure with various temperature
Then it is interesting to know its ferrielectric TC. To characterize the structural tetramerization, the difference () of nearest-neighbor Fe-Fe bond length (d) along the ladder direction is measured as a function of temperature (Fig. 3a). An in-situ heating experiment was performed using a DENSsolutions SH30 system to be able to measure over a wide temperature range. A lamella of BaFe2Se3 was transferred onto specialized chips using a probe-assistance method (Supplementary Figure 6), then the sample was heated to the set temperature by resistance heating. The displacement vector-mapping algorithm was implemented on the STEM images, and each is obtained by averaging around 300 measurements. The difference of Fe-Fe bond length as well as the tilting angle between the ladders at high temperature vanishes~600 K, and the in-ladder tetramerization almost drops to zero, implying a high symmetric nonpolar phase (in agreement with the DSC curve shown in Fig. 1e and the X-ray data in Ref. [16]). The discrepancy between strong and weak ladders disappears in this high symmetric phase, as also demonstrated by above SAED result (Fig. 1d). With decreasing temperature (e.g. at 473 K), unexpectedly, tetramerization emerges in one sublattice of ladders but not in the other, which is a unique characteristic of irreducible ferrielectricity, leading to an emergence of Pa (Fig. 3b) [calculated by density functional theory (DFT)] [26]. Meanwhile, the ladders become tilting ( Fig. 3c compared with Fig. 3d). With further decreasing temperature (e.g. at 423 K), tetramerization emerges in both ladders with different intensities and slopes, implying that the first-order transition occurring at~420 K corresponds to the starting of tetramerization of the weak ladders. At~373 K, the intensities of tetramerization are close to identical between both ladders, resulting in an (almost) canceled Pa, i.e. the unique compensation point (T0) of irreducible ferrielectricity. Below T0, the difference between the two ladders increases with decreasing temperature, leading to a reentrance of Pa. It is worth to emphasize that the direction of polarization (i.e. the roles of strong and weak ladders) may be probably reversed across Statistics of the deviation of Fe-Fe bond length in the strong and weak chain at different temperatures could be seen in Supplementary Table 1. Since the ferrielectric polarization can not be directly measured using electrical methods (and even pizeoelectric force microscopy) in the current stage due to the high leakage of the samples (considering the very small band gap~0.13-0.178 eV [17,18]), an optical second harmonic generation (SHG) experiment is employed to characterize the polarity of the materials (see Methods for experimental details). Our SHG signal (Fig. 3b) demonstrates its polarity below~600 K. Most importantly, the non-monotonic evolution of the SHG signal unambiguously matches the DFT calculated polarization, including the possible compensation point at~380-400 K, which provides a very strong evidence to support our STEM data. The irreducible ferrielectricity of BaFe2Se3 can be qualitatively described by Landau theory with a coupling between two ladders (PA & PB). The most simplified free energy formula can be written as: where the first to six items are the standard Laudau-Ginzburg-Devonshire type energy expression up to the sixth power for sub-lattices A and B, while the last item is the antiferroelectric coupling between two sub-lattices. All coefficients except 2 are positive and the small canting angle between PA and PB is neglected. Without fine tuning of the coefficients (see Methods for details), the simulated evolution of the polarization (Fig. 3e) is qualitatively reproducing the non-monotonic experimental behavior, implying the correct main physics captured in the model. Consistent with the DSC data, the high temperature transition is a second-order one, while the low temperature transition is a first-order one due to the negative 2. Previous experiments reported the space group Pnma for BaFe2Se3 at room temperature [15][16][17], which is nonpolar and does not allow the tetramerization. A recent work reported the space group Pmn21 (a subgroup of Pnma) at room temperature, allowing the tetramerization and polarization [27]. Another recent X-ray diffraction work reported the space group Pmn21 at 300 K but Pm (a subgroup of Pmn21 allowing the inequivalent ladders) at 150 K [28]. In fact, the patterns of NPD (or X-ray) are very subtle among these space groups (see Supplementary precisely. Instead, our STEM technique is more suitable to monitor these subtle distortions of inner coordinates. According to our studies, below TC1 the accurate space group should already be Pm, allowing the asymmetry between two ladders. The transition from Bbmm to Pm at TC1 is a second-order one while the first-order transition at TC2 does not change the symmetry. Other high-resolution technique, such as synchrotron X-ray diffraction, may be helpful to verify our STEM results in future.

Discussion
Finally, it should be noted that the irreducible ferrielectrics is not limited to BaFe2Se3 but with broader interests. For example, as an important branch of multiferroics, TbMn2O5 and other 125-type manganites showed strange 'ferroelectric' behavior of polarization as a function of temperature or magnetic field [29,30], including the compensation point of polarization. The real mechanism is that the ferroelectric contributions in TbMn2O5 are from three out-of-sync sources according to the SHG measurement [30].
Even though, our current work on BaFe2Se3 is not a marginal extension of TbMn2O5. The polarity in TbMn2O5 is magnetism-driven, i.e. it is a so-called type-II multiferroic material, instead of a proper ferroelectric material. It is not rare for a magnetic system to have sequential magnetic phase transitions. In this sense, the nontrivial evolution of polarization in TbMn2O5 is just a secondary effect of magnetic evolution, which occurs at very low temperature (<40 K) and gives a very weak signal of polarization (~<0.04 C/cm 2 and~<0.15 C/cm 2 ) [29,30]. In our case, the ferrielectricity is not magnetism-driven but a primary polar property, which occurs above room-temperature (15 times of TbMn2O5) and with a much larger polarization (5-15 times of TbMn2O5). In addition, limited by its very weak polarization signal, the experimental measurements of TbMn2O5 can only rely on the pyroelectric method, which can lead to a net polarization but the microscopic facts of different contributions were mostly by suspecting or indirect derivation from SHG signals. Instead, our current work, powered by the advanced in-situ STEM techniques and thanks to the strong signal of BaFe2Se3, the microscopic evolution of two contributions can be visualized directly, leading to a more decisive conclusion. In fact, although the nontrivial polarization of TbMn2O5 has been known for decades, it is more likely to be recognized as a type-II multiferroics with strange ferroelectric behavior. Our work will lead to a re-look at the irreducible ferrielectricity, including that in TbMn2O5.
The irreducible ferrielectricity combines both characteristics of ferroelectricity and antiferroelectricity, making these systems having more degrees of freedom to be controlled. For example, by tuning the amplitudes of sub-lattice polarizations near the compensation point, the macroscopic polarization can be easily switched, without the reversal process of dipole moments as required in ferroelectric cases. Moreover, complex ferroelectric+antiferroelectric domain structures may be expected in ferrielectrics [31], which deserve further studies.
In summary, employing spherical aberration-corrected STEM with sub-angstrom resolution, the structural evolution of BaFe2Se3 has been characterized in detail. Highly interesting phenomena, beyond previous experimental observations and theoretical predictions, have been detected and analyzed. First, BaFe2Se3 is a room temperature polar material. Second, combined with EELS analysis, the origin of its structural tetramerization is demonstrated to be driven by the local electron density, not the previously expected block-type antiferromagnetism. Third, most importantly, the evolution of the two ladders in BaFe2Se3 does not behave synchronously, leading to irreducible ferrielectricity. The compensation point, a unique fingerprint of irreducible ferrielectricity, is observed. The irreducible ferrielectricity reported here is conceptually different from previously reported reducible ferrielectricity which is actually equal to ferroelectricity. The irreducible ferrielectricity in BaFe2Se3 acts as the primary effect, leading to a stronger impact to the community to re-investigate this independent branch of polarity.
More functionalities are promisingly expected in future based on irreducible ferrielectricity, e.g.
the magnetic-field-tunable polarization as demonstrated in TbMn2O5. T field. Neutron powder diffraction confirms the appearance of long range magnetic ordering 34 below ~250 K, in agreement with previous report of (~255 K) [17,18] and (~256 K) [15]. Thus, 35 the peak of the magnetic susceptibility at ~415 K (appearing only under strong magnetic field, e.g.

121
Noting that our neutron diffraction pattern are generally in agreement with previous measurements 122 [15,17]. The two weak peaks which are allowed in the low symmetry space group but forbidden 123 in the high symmetry one are close to the precision limit of our neutron spectrometer. In fact, the 124 powder neutron diffraction data can be well refined with either the Pm, Pmn21 or Pnma space 125 group, as done in previous works. In short, for this material, it's hard to distinguish these space 126 groups with neutron data only, but the neutron data can be a supplementary support to our STEM 127 data.