⇦Chinese Physics Letters, 2022, Vol. 39, No. 9, Article code 097701 Ferroelectricity in Charge-Ordering Crystals with Centrosymmetric Lattices Yali Yang (杨亚利)1,2, Laurent Bellaiche3, and Hongjun Xiang (向红军)1,2* Affiliations 1Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, China 2Shanghai Qizhi Institution, Shanghai 200232, China 3Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA Received 21 August 2022; accepted manuscript online 22 August 2022; published online 3 September 2022 *Corresponding author. Email: hxiang@fudan.edu.cn Citation Text: Yang Y L, Bellaiche L, and Xiang H J 2022 Chin. Phys. Lett. 39 097701    Abstract The switchability between the two ferroelectric (FE) states of an FE material makes FEs widely used in memories and other electronic devices. However, for conventional FEs, its FE switching only occurs between the two FE states whose spatial inversion symmetry is broken. The search for FE materials is therefore subject to certain limitations. We propose a new type of FEs whose FE states still contain spatial inversion centers. The change in polarization of this new type of FEs originates from electronic transfer between two centrosymmetric FE states under an external electric field. Taking BaBiO$_{3}$ as an example, we show that charge-ordering systems can be a typical representative of this new type of FEs. Moreover, unlike traditional ferroelectrics, the change in polarization in this new type of FEs is quantum in nature with the direction dependent on the specific FE transition path. Our work therefore not only extends the concept of FEs but may also open up a new way to find multiferroics.
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 DOI:10.1088/0256-307X/39/9/097701 © 2022 Chinese Physics Society Article Text Since the discovery of the Rochelle salt in the 1920s,[1] ferroelectric (FE) materials have been extensively studied in fundamental research and have been demonstrated to have huge application potential in diverse applications, such as memory devices, sensors, and actuators, which pertains to industry and our daily lives.[2-7] Furthermore, ferroelectricity is believed to be always associated with piezoelectric and pyroelectric properties, which make FEs even more appealing.[8,9] From a crystallographic point of view, conventional ferroelectricity is reported to only occur in one of the ten polar point groups: $C_{1}$, $C_{2}$, $C_{1h}$, $C_{2v}$, $C_{4}$, $C_{4v}$, $C_{3}$, $C_{3v}$, $C_{6}$, and $C_{6v}$, all of which are non-centrosymmetric. In these systems, the spatial inversion symmetry disappears as the temperature is lowered below the Curie temperature, accompanied by a structural phase transition from the paraelectric (PE) phase to the FE phase. For example, for the prototypical FE perovskite BaTiO$_{3}$, the spatial inversion symmetry breaks due to the off-center motions of Ti$^{4+}$ and O$^{2-}$ ions (with respect to that of Ba$^{2+}$ ions) when the crystal transforms from the cubic PE phase ($Pm\bar{3}m$) to the tetragonal FE phase ($P4mm$).[10-12] Here, the polarization itself is the primary order parameter. In recent years, improper (including hybrid improper) ferroelectricity has also attracted much attention, in which the polarization acts as a secondary order parameter coupled either to some primary nonpolar lattice distortions, e.g., the octahedral rotations in the superlattice PbTiO$_{3}$/SrTiO$_{3}$[13] and the Ruddlesden–Popper perovskite A$_{3}$B$_{2}$O$_{7}$ (A=Ca, Sr; B=Mn, Ti),[14,15] or to some specific spin orders, e.g., the cycloidal spiral spin order in TbMnO$_{3}$.[16-19] In these improper FE systems, the spatial inversion symmetry is also broken. On the other hand, as a necessary condition for conventional FE materials, ferroelectricity must be switchable in response to an applied electric field. As a consequence, this excludes some non-centrosymmetric polar materials (e.g., wurtzite ZnO) from belonging to the FE class of compounds.[12] For the conventional ferroelectrics mentioned above, the application of an external electric field results in transforming a non-centrosymmetric state to another symmetrically equivalent non-centrosymmetric state. However, one may wonder if it is possible also to use an external electric field but to transform a centrosymmetric state having a polarization to another symmetrically equivalent centrosymmetric state possessing a different polarization. If yes, this will expand the concept of ferroelectricity, enrich the number of FE materials, and may lead to unprecedented applications. To address such issue, in this work, we first propose a new type of FEs for which both FE states are centrosymmetric and the polarization difference between these two FE states are quantized. Then, we take the BaBiO$_{3}$ system, in which Bi$^{3+}$(6$s^{2}$) and Bi$^{5+}$(6$s^{0}$) cations coexist due to the disproportionation of the Bi$^{4+}$ ions as an example to demonstrate, via first-principles calculations, this FE-like behavior in a centrosymmetric charge-ordering system. In addition, we suggest that the FE polarizations in BaBiO$_{3}$ may be switched by electric fields along different directions. Results—General Idea. To have a clear understanding of ferroelectricity in centrosymmetric phases, let us start from a simple model: the one-dimensional (1D) binary charge-ordering system AB, where A and B are cations and anions, respectively. The A cation may be in two different valence states (A$^{m+}$ and A$^{n+}$ with the integers $m < n$). As shown in Figs. 1(a) and 1(b), we construct two periodic 1D charge-ordering states, namely, state I and state II, respectively. These two 1D chains are periodic along the $x$ axis. The A$^{m+}$ (in orange) and A$^{n+}$ (in purple) cations align alternatively with a distance of $a$/2, where $a$ (in units of Å) is the lattice constant, and the B$^{v-}$ anions (in green) sit at the middle of the adjacent cations. It is easy to realize that the two systems are centrosymmetric with all the cations locating at spatial inversion centers. Note that the B$^{v-}$ anions are not at the spatial inversion centers because the inequivalent A$^{m+}$ and A$^{n+}$ cations locate on the left and right sides of each B$^{v-}$ ions, respectively. Now, if an appropriate external electric field is applied to the periodic 1D chain of state I in Fig. 1(a) along the $-x$ direction, a specific number (i.e., $n-m$) of electrons on each A$^{m+}$ cation would transfer to the adjacent right A$^{n+}$ cations via the A$^{m+}$-B$^{v-}$-A$^{n+}$ bonds. Then, one can easily see that the resulted structure after the electronic transfer is exactly the periodic 1D chain of state II shown in Fig. 1(b). Consequently, the periodic 1D chains of state I and state II in Fig. 1 are symmetrically equivalent, but with the A$^{m+}$-B$^{v-}$-A$^{n+}$ bond changing to A$^{n+}$-B$^{v-}$-A$^{m+}$. Alternatively, one can take the view that the periodic 1D chain of state II is obtained by translating the periodic 1D chain of state I in Fig. 1(a) with $a$/2 along the $x$ axis.
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Fig. 1. Schematic of the two symmetrically equivalent ferroelectric states (a) state I and (b) state II of the one-dimensional (1D) binary charge-ordering system AB. The A$^{m+}$, A$^{n+}$, and B$^{v-}$ ions are indicated by orange, purple, and green balls, respectively, with $m < n$. Here $a$ is the lattice constant of the unit cell in which two A and two B atoms are included. The periodic and finite models of each states are both presented. The 1D chains are parallel to the $x$ axis. The black curved arrow indicates the electron transfer under the electric field along the $-x$ direction.
Let us now explicitly calculate the polarization difference between state I and state II shown in Figs. 1(a) and 1(b), respectively. According to the modern theory of polarization,[20] the polarization difference between two periodic systems is uniquely defined only if the transition path between the two states is given. Consider the electronic transfer path shown in Fig. 1(a), the change of polarization (in fact the dipole moment in units of $e\cdot $Å) can be computed by $\Delta P=P_{\rm I\!I}-P_{\rm I}=-\frac{(n-m)a}{2}$. Interestingly, the polarization difference is an integer times half of the lattice constant. In fact, one can easily prove that the polarization difference between any two centrosymmetric states with the same lattice vectors is quantized. According to the modern theory of polarization, the polarization of a periodic insulating system is multi-valued: ${\boldsymbol P}={{\boldsymbol P}^{0}}+\sum_{k=1,3} {q_{k}{{\boldsymbol a}_{k}}}$, where ${{\boldsymbol a}_{k}}$ are the lattice vectors and $q_{k}$ are arbitrary integers. If the system contains a spatial inversion center, ${\boldsymbol P}=-{\boldsymbol P}$, thus ${{\boldsymbol P}^{0}}=\sum_{k=1,3} {\delta_{k}{{\boldsymbol a}_{k}}}$ with $\delta_{k}=0$ or 1/2.[21] Therefore, the polarization difference between any two centrosymmetric states is $\Delta {\boldsymbol P}={\boldsymbol P}_{\rm I\!I}-{\boldsymbol P}_{\rm I}=\sum_{k=1,3} {(q_{k}+\delta_{k}){{\boldsymbol a}_{k}}}$. To better understand the counterintuitive fact that the polarization difference between two centrosymmetric states could be nonzero, let us consider the more realistic finite chain systems corresponding to the two periodic states (see Fig. 1). We note that, in order to maintain electroneutrality, the A$^{m+}$ and A$^{n+}$ ions will be paired, resulting in different cations at the two terminals of the finite-size chain. It is clear that both chains do not contain the spatial inversion centers and thus possess nonzero dipole moments. Using the definition of the dipole moment of a finite system, one can easily show that the dipole moment difference between the two chains is $-\frac{(n-m)a}{2}$ per unit cell, in agreement with the previous computed result for the infinite periodic systems. Realistic Example. In the above discussions, we consider the 1D model systems to demonstrate the novel concept that an FE state may display the spatial inversion symmetry. In fact, similar to the 1D charge-ordering chain, the new type of FEs can also appear in two-dimensional (2D) and three-dimensional (3D) system models [see Figs. 2(a) and 2(b)], in which the charge ordering is arranged in a checkboard pattern. We now consider the issue whether it is possible to find a realistic material to realize the new type of FEs. In recent years, the concept of electronic ferroelectrics has attracted attention, in which the polarization originates from the dipole moments produced by the ordered alignment of the mixed-valence cations.[22-28] As in conventional ferroelectrics, the spatial inversion symmetry is also broken in the previously examined electronic ferroelectrics, e.g., in Fe$_{3}$O$_{4}$,[29] LuFe$_{2}$O$_{4}$,[27,28,30] and some manganites (e.g., R$_{1-x}$Ca$_{x}$MnO$_{3}$, R=La, Pr).[31] Here to demonstrate the new concept of ferroelectricity without breaking the spatial inversion symmetry in real systems, we take the 3D perovskite oxide BaBiO$_{3}$ as an example. In such a compound, the Bi$^{3+}$ (6$s^{2}$) and Bi$^{5+}$ (6$s^{0}$) cations coexist and are arranged alternatively in three dimensions due to the charge disproportion of the Bi$^{4+}$ ions.[32] BaBiO$_{3}$ is reported to undergo several phase transitions in the temperature range of 4.2–973 K.[33] At room temperature, BaBiO$_{3}$ crystallizes in a monoclinic structure having centrosymmetric $I2/m$ symmetry. When the temperature is lowered below $\sim $132 K, it transforms to the ground state with $P2_{1}/n$ symmetry[33] which is also centrosymmetric. In the present work, this 3D $P2_{1}/n$ structure is adopted to demonstrate the new type of FEs we proposed above. Figure 2(c) shows the unit cell of the low-temperature $P2_{1}/n$ phase of BaBiO$_{3}$, which contains 20 atoms. In such a phase, the Bi$^{3+}$ (6$s^{2}$) and Bi$^{5+}$ (6$s^{0}$) cations arrange alternatively in three dimensions along the [110], [$\bar{1}$10] and [001] directions.[32] In addition to the $a^{-}$$a^{-}$$c^{0}$ tilting pattern, the BiO$_{6}$ octahedra also exhibit a breathing distortion mode with respect to the ideal 5-atom cubic structure. In Fig. S1(a) of the Supplemental Material (SM), we show a schematic diagram of the three-dimensional breathing distortion mode in BaBiO$_{3}$. The BiO$_{6}$ octahedron breathes in and out for Bi$^{5+}$ and Bi$^{3+}$, respectively, leading to nonequivalent Bi–O bond lengths, i.e., the shorter Bi$^{5+}$–O$^{2-}$ bond and the longer Bi$^{3+}$–O$^{2-}$ bond. In Fig. S1(b) of the SM, we show the length of the Bi$^{3+}$–O$^{2-}$–Bi$^{5+}$ bond along the $[\bar{1}10]$ direction of the relaxed $P2_{1}/n$ ground state of Fig. 2(c). The lengths of the longer Bi$^{3+}$–O$^{2-}$ and shorter Bi$^{5+}$–O$^{2-}$ bond are 2.289 and 2.141 Å, respectively. As a consequence of the charge ordering and the breathing mode, the low-temperature monoclinic structure becomes insulating with an indirect band gap of 0.2–1.1 eV.[34] We note that defect-free BaBiO$_{3}$ is not magnetic while magnetization measurement indicates the presence of possible local moments on the grain boundaries of BaBiO$_{3}$.[35] Now we start to demonstrate ferroelectricity in the centrosymmetric charge-ordering BaBiO$_{3}$ system. Firstly, it is reasonable to imagine that, similar to the 1D charge-ordering chain described above, if an appropriate electric field is applied to BaBiO$_{3}$ along one of the three charge-ordered directions [e.g., the [001] direction in Fig. 2(c)], the 6$s^{2}$ electrons of each Bi$^{3+}$ cation will transfer to the adjacent Bi$^{5+}$ cation through the Bi$^{3+}$–O$^{2-}$–Bi$^{5+}$ bond in the direction opposite to the electric field. The Bi$^{3+}$–O$^{2-}$–Bi$^{5+}$ bond will thus reverse into Bi$^{5+}$–O$^{2-}$–Bi$^{3+}$ after the charge transfer, and the entire resulting structure [see Fig. 2(d)] is symmetrically equivalent to the initial state. Experimental evidence of charge-hopping from Bi$^{3+}$ (6$s^{2}$) to Bi$^{5+}$ (6$s^{0}$) in the BaBiO$_{3}$ system was experimentally demonstrated.[36-39] Here, we define the two FE states in Figs. 2(c) and 2(d) as state I and state II, respectively. It is easy to find that the main difference between these two FE states is that the Bi$^{3+}$ and Bi$^{5+}$ cations in Fig. 2(c) become Bi$^{5+}$ and Bi$^{3+}$ in Fig. 2(d), respectively. The “breath-in” (“breath-out”) BiO$_{6}$ octahedra in Fig. 2(c) thus become the breath-out (breath-in) in Fig. 2(d), accompanied by the small motions of the O atoms. Thus, similar to the case of the 1D chain, the transition between state I and state II under electric field in BaBiO$_{3}$ can also be regarded as the ferroelectricity switching.
cpl-39-9-097701-fig2.png
Fig. 2. (a) Two-dimensional (2D) and (b) three-dimensional (3D) charge-ordering system models, in which the anions are omitted for clarity. (c) Ferroelectric state I and (d) ferroelectric state II of 3D charge-ordering system BaBiO$_{3}$. The orange and purple balls in (a) and (b) indicate the A$^{m+}$ and A$^{n+}$ cations, respectively, where $m < n$. The Ba$^{2+}$, Bi$^{3+}$, Bi$^{5+}$, and O$^{2-}$ ions in (c) and (d) are represented by green, orange, purple, and red balls, respectively. The straight light and dark blue arrows indicate the direction of the external electric field and polarization, respectively. The black curved arrows indicate the direction of electron transfer under the electric field. The grey curved arrows indicate the transition from state I to state II.
Secondly, let us evaluate the exact electric polarization of the two FE states, or be more precisely the change in polarization between the two FE states of BaBiO$_{3}$ during the ferroelectricity switching under an electric field. Since the breathing mode in BaBiO$_{3}$ happens in three dimensions, the ferroelectricity shows an interesting triaxial property such that it would be switchable along three axes, i.e., the in-plane [110], $[\bar{1}10]$ and out-of-plane [001] axes [see Fig. 2(c)]. For each of these three axes, the 6$s^{2}$ electrons will behave like those in the one-dimensional chain in the sense that the electrons transfer through the Bi$^{3+}$–O$^{2-}$–Bi$^{5+}$ bond in the direction antiparallel to the electric field, leading to a change of polarization parallel to the electric field [see Fig. 2(c)]. Thus, as shown in Figs. S2(a)–S2(c) of the SM, we present the three cases that the electric field directs along [001], [110] and $[\bar{1}10]$ axes, respectively. It is easy to see that no matter which axis the electric field is along, there are two chains in the unit cell of BaBiO$_{3}$ along the direction of the electric field. Therefore, the electric dipole moment produced by electron transfer in the unit cell will be twice that in each chain. For example, in Fig. 2(d) the electric field is along the [001] axis, and the change of polarization is evaluated to be $|\Delta {\boldsymbol P}|=2|\boldsymbol{c}|$ (in units of |$e\cdot$Å|), where $\boldsymbol{c}$ is the lattice vector along the [001] axis. Similarly, the change of polarization will be $|\Delta {\boldsymbol P}|=2|{\boldsymbol a}+{\boldsymbol b}|$ and $|\Delta {\boldsymbol P}|=2|{\boldsymbol a}-{\boldsymbol b}|$ for the cases in which the electric field is along the [110] and $[\bar{1}10]$ axes, respectively [see Fig. S2(b) and S2(c) of the SM], where ${\boldsymbol a}$ and ${\boldsymbol b}$ are the lattice vector along [100] and [010] axes, respectively. To be more specific, the exact values of polarization changes are 84.12, 84.54 and 84.54 µC/cm$^{2}$ when the electric field is directing along [001], [110], and $[\bar{1}10]$, respectively. Such polarization changes are larger in magnitude than that of BaTiO$_{3}$ ($\sim$$52\,µ$C/cm$^{2}$),[40] while slightly smaller than that of PbTiO$_{3}$ ($\sim$$120\,µ$C/cm$^{2}$)[41] and BiFeO$_{3}$ ($\sim$$180\,µ$C/cm$^{2}$).[42] One would thus find that due to the distortion of the lattice parameters of the FE state in low temperature relative to the ideal cubic structure, the change of polarization along the different axes will show some difference. However, in the conventional ferroelectric system, this phenomenon is unlikely to occur, because the transition path with the smallest polarization change generally has the lowest energy barrier. We note that the change of polarization in the 3D charge ordered BaBiO$_{3}$ is also quantized similar to the 1D model case. Moreover, the neutral BaBiO$_{3}$ crystals with a finite size may not contain the spatial inversion centers and thus possess nonzero dipole moments. As a result, ferroelectricity appears in such finite-size BaBiO$_{3}$ crystals, as similar to the 1D model case. To be more specific, the ferroelectricity discovered here is a bulk effect, since the change of total dipole moment between the two FE states is linearly related to the real size of the crystal. In our calculations, it should be noted that the contributions of O atoms to the charge distribution and thus the change of polarization are fully taken into account. However, we find that unlike the motions of ions in conventional FEs (e.g., BaTiO$_{3}$), the displacement of the oxygen atoms during the FE switching process does not contribute to the polarization in BaBiO$_{3}$, since the oxygen moves opposite to each other along the charge-ordering chain Bi$^{3+}$–O$^{2-}$–Bi$^{5+}$ and leads to the vanishing of net ionic movement (see details in Section IV of the SM). Therefore, the oxygen displacement induced by the electron transfer between the adjacent cations does not change the quantum nature of the change of the electric polarization. Finally, let us investigate the possible transition path between these two FE states of BaBiO$_{3}$. It is known that the energy barrier between the two FE states determines the ability of ferroelectricity to switch, thus, to get the related transition barrier and the transition state between state I and state II of BaBiO$_{3}$, the climbing nudged elastic band (cNEB) calculation[43,44] is performed. As shown in Fig. 3, the energy barrier between the initial state I and the final state II is about 0.025 eV/f.u., which is close to those of conventional FEs[45-49] and can thus be overcome under an external electric field. In addition, the transition state is found to have Pbnm symmetry, which is also centrosymmetric but has higher symmetry than that of the two FE states. As shown in the insets of Fig. 3, the four BiO$_{6}$ octahedra in the two FE states and in the transition state are presented from the side view perpendicularly to the ($\bar{1}$10) plane of the 20-atoms cells. One can see that during the phase transition, the oxygens atoms move away from Bi$^{5+}$ to the adjacent Bi$^{3+}$ cations. As a result, the breathing mode, as well as the charge disproportion, disappears in the Pbnm transition state. The nonequivalent bonds Bi$^{3+}$–O$^{2-}$ and Bi$^{5+}$–O$^{2-}$ become the equivalent Bi$^{4+}$–O$^{2-}$ bonds. The $\beta$ angle changes from 90.31$^{\circ}$ (89.69$^{\circ}$) in FE state I (state II) to 90$^{\circ}$ in the transition state. To verify the electron transfer between the two FE states during the ferroelectricity switching, we calculate the Wannier functions (WFs) for the Bi $6s$ electrons in the two FE states. As shown in the insets of Fig. 3, the Bi $6s^{2}$ electrons are localized around Bi$^{3+}$ cations in all of the initial and final states. Thus, it is seen that under the appropriate electric field the 6$s^{2}$ electrons indeed transfer from the Bi$^{3+}$ cations to the Bi$^{5+}$ cations in the FE states via the Bi$^{3+}$–O$^{2-}$–Bi$^{5+}$ bond. Here, we only show the WFs of 6$s^{2}$ electrons on one Bi$^{3+}$ cation in each FE state for clarity, while more detailed information on WFs is given in Section V of the SM.
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Fig. 3. Transition path between ferroelectric state I and state II (in $P2_{1}/n$ symmetry) of BaBiO$_{3}$. The insets represent the zoom-in of the four BiO$_{6}$ octahedra in the 20-atom unit cell of BaBiO$_{3}$. The balls in orange, purple, yellow, and red represent Bi$^{3+}$, Bi$^{5+}$, Bi$^{4+}$, and O$^{2-}$, respectively. Ba$^{2+}$ cations are not shown for clarity. The Wannier functions corresponding to the $6s$ electrons of a Bi$^{3+}$ ion of the two ferroelectric states are also shown in the insets.
As discussed above, it is now clear that one cannot just rely on the symmetry information of a bulk structure to reply whether it is FE or not. It was widely accepted that all ferroelectrics display piezoelectric and pyroelectric properties. Now, this traditional view may also need to be changed. In our new type of ferroelectrics without breaking spatial inversion symmetry, the systems may not display piezoelectricity, since the charge order shall not change under a small strain. However, the pyroelectricity may still exist in the system. For instance, if the charge ordering is multidomain at high temperature, it would transform into a single domain when the temperature decreases, and the electrons would transfer along one specific direction to generate pyroelectric current. In summary, a new type of ferroelectrics is proposed in this work, for which the ferroelectric state can exist in charge-ordering systems even when they are centrosymmetric. The FE states can be switched due to the electron transfer between ions with different valence states under an external electric field without breaking the spatial inversion symmetry. Compared to the usual FEs, the switch speed of electronic FEs we considered may be faster since the electron hopping is faster than ion displacements, which may be promising for the application in ultrafast electronic devices. The polarization of the new FEs is quantized as required by symmetry. Although BaBiO$_{3}$ is non-magnetic, many other charge-ordering systems (such as CaFeO$_{3}$ and RNiO$_{3}$ with R representing rare earth elements[50-52]) are magnetic, thus the new ferroelectricity proposed in this work would pave a new way to design high-performance multiferroic materials. Supplemental Material. Computational methods; three-dimensional breathing mode in BaBiO$_{3}$; ferroelectric switching in BaBiO$_{3}$ under different electric fields; zero contribution from the breathing mode to ferroelectricity in BaBiO$_{3}$; Wannier functions of Bi $6s$ electrons during ferroelectric switching. Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11825403 and 11804138), the Qing Nian Ba Jian Program. L. Bellaiche thanks the Office of Naval Research for the support (Grant Nos. N00014-17-1-2818 and N00014-21-1-2086). 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