Designing new ferroelectrics with a general strategy

The presence of a switchable spontaneous electric polarization makes ferroelectrics ideal candidates for the use in many applications such as memory and sensors devices. Since known ferroelectrics are rather limited, finding new ferroelectric materials has become a flourishing field. One promising route is to design the improper ferroelectrics. However, previous approach based on the Landau theory is not easily adopted for systems that are unrelated to the Pbnm perovskite structure. To this end, we develop a general design rule that is applicable to any system. By combining this rule with the density functional theory calculations, we identify previously unrecognized classes of ferroelectric materials. It is shown that the R3̄c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\bar{3}c$$\end{document} perovskite structure can become ferroelectric by substituting half of the B-site cations. Compound ZnSrO2 with a non-perovskite layered structure can also be ferroelectric through the anion substitution. Moreover, our approach can be used to design new multiferroics as illustrated in the case of fluorine substituted LaMnO3. An innovative computational method could enable the design of new classes of ferroelectric materials. Ferroelectrics, whose spontaneous electric polarizations can be switched electrically, are useful for a range of applications, such as memory or sensing devices. However, relatively few naturally occurring materials are ferroelectric, and, while theoretical methods for designing new ferroelectrics are available, they are usually restricted to highly symmetric systems such as perovskite oxides. A team led by Hongjun Xiang from Fudan University have now developed a more general computational approach that can be applied to any system, and have used it to identify previously unrecognized classes of ferroelectrics. Such an approach could not only enable new ferroelectrics to be discovered, but it may also be suitable for designing multiferroic systems.


INTRODUCTION
Recently ferroelectrics (FEs) have attracted much attention due to their wide range of applications, especially in the electronic devices such as nonvolatile memory, 1,2 tunable capacitors, 3 solar cell, 4 and tunnel junction. 5 For traditional proper FEs, such as BaTiO 3 , the transition metal ion Ti 4+ with d 0 configuration can hybridize with the oxygen 2p states leading to the FE phase transition due to the pseudo Jahn-Teller effect. [6][7][8][9][10] However, FEs are very few in nature. 11 To find more high-performance FEs, improper (including hybrid improper) FEs become an intense research field. 12 The typical example of the conventional improper FEs is hexagonal manganite YMnO 3 , 13 where the FE buckling (P mode) of the Y-O planes is induced by the non-polar MnO 5 polyhedra tilt (Q mode). 13,14 The free energy expansion in this system contains the coupling term (PQ 3 ) between the Q and P, indicating that the non-polar distortion Q must be reversed. 15 The hybrid improper ferroelectricity (HIF) was recently discovered in the artificial superlattice PbTiO 3 /SrTiO 3 , 16 where the ferroelectricity is induced by a trilinear coupling (PQ 1 Q 2 ) between the FE mode (P) and two oxygen octahedral rotational modes (Q 1 and Q 2 , respectively). The HIF was also found in the double-layered Ruddlesden-Popper (RP) perovskite A 3 B 2 O 7 (A=Ca, Sr; B=Mn,Ti), [17][18][19] the 1:1 A-cation ordering perovskite-type superlattice, [20][21][22][23] A-cation ordering RP NaRTiO 4 (R = Y, La, Nd, Sm-Ho), 24 the 2:2 B-cation ordered superlattice, 25 and metal-organic perovskite material. 26 In the above-mentioned theoretically designed FEs, one usually starts from a high-symmetry structure (e.g., cubic perovskite structure), then the effect of atomic substitution and soft phonon modulations are examined to see whether the ferroelectricity can be induced or not. Finally, the trilinear coupling mechanism is discussed to understand the origin of improper ferroelectricity. This procedure is indeed informative. However, it is tedious and its applicability to other type of compounds is limited since even the high-symmetry structure may be unknown. Therefore, two key questions remain to be addressed: Is it possible to propose a general method to design improper FEs? Can improper ferroelectricity be obtained in systems with totally different structures from perovskite? The answers to these questions may widen the opportunities of finding the improper FE materials. In this work, we propose a general approach to design the FE materials. Our approach can not only reproduce the previous results (e.g., the 1:1 A-cation [001] ordered FE perovskite superlattices), but also predict new FEs. Our results show that the B-cation ordered½111-superlatticeLa 2 (Co,Al)O 6 with a R3c parent structure displays a spontaneous polarization. And compound ZnSrO 2 with a non-perovskite layered structure can also be FE by proper anion substitution. Moreover, compound LaMnO 3 with the fluorine substitution can become multiferroic of a sizable polarization with its direction perpendicular to the direction of magnetization. Hence, our study may pave a new route to find the FE materials.

General design guidelines
Firstly, the parent structure should have the true inversion centers (TICs) and pseudo inversion centers (PICs) simultaneously and also have a sizable band gap. The TIC is defined as a position on which there is inversion symmetry I of the structure. The PIC is a position which can become a TIC after small displacements of the atoms. With a given centrosymmetric structure (i.e., parent structure), we first find all the TICs and PICs. Then we find out all the possible atom substitutions within a given supercell of the parent structure, which lift all the TICs but still keep at least one PIC. These substituted structures can be candidates for FEs. After the slight displacements of the ions in a substituted structure where the PIC becomes a TIC, the corresponding paraelectric (PE) structures can be obtained. Figure 1a shows a chart for the general strategy of designing new FEs. Our method is illustrated clearly with a simple one-dimensional toy model (see Fig. 1b). Assuming that there is a one-dimensional chain with two atoms of the same type (represented by A) in a unit cell, the two atoms locate at positions 0.0 and 0.4, respectively. Then, one can find two TICs denoted by the crosses within a unit cell before any substitution. If replacing an A atom by a B atom in the unit cell, all the TICs disappear, resulting in a possible polar structure with polarization P along the left or right directions. However, there is a PIC at position 0.5 even after the substitution. When moving the B atom to the center of the unit cell, the PIC will become a TIC. From the above discussion, the key point in our method is lifting all the TICs by proper atomic substitution and maintaining at least one PIC in the system. To ensure the successful design of FEs, we need two additional conditions: (I) The FE phase should be locally stable; (II) Both the FE and PE phases should have large band gaps. Since the superlattice-type structures may be synthesized by means of the molecular beam epitaxy methods, we hereafter mainly focus on the superlattice-type structures.
Improper FEs based on the Pbnm ABO 3 perovskite structure Let us first concentrate on the design of FEs based on ABO 3 perovskites. The Pbnm structure with 20 atoms in the unit cell can be obtained if the cubic Pm3m structure undergoes an out-phase rotation and in-phase rotation of oxygen octahedral (a − a − c 0 and a 0 a 0 c + in Glazer notation, respectively). 11 By using our method, we find that there are 8 TICs (all B-sites are TICs) and 24 PICs in one unit cell of the Pbnm structure (see Fig. SI in the Supplementary Information (S1) for details). Within the 20-atom cell, 14 possible substitution-induced FEs in total are found, with 2 types of A-site substitutions and 12 types of anion substitutions (see Fig. S2 in the SI). One of the A-site substituted (A/A′)B 2 O 6 , is the [001] FE superlattice discovered in previous studies. [20][21][22][23] The A and A′-site ordering breaks all the TICs but maintains the 16 PICs. The appearance of net electric polarization is due to the fact that the A-site displacement and the A′-site displacement are not completely cancelled. We find that it is impossible to obtain improper FEs through B-site substitutions in a 20-atom cell, in agreement with previous result. 22 Nevertheless, considering a twofold supercell along the z direction, we can obtain FEs by any B-site substitution. For example, our result indicates that the B/B′ cation ordered 2/2 supercell adopts the FE Pmn2 1 symmetry (see Fig. S3 in the SI for the possible substitutions in the 2/2 B/B′' cation ordered superlattices). All these results demonstrate that our method can not only reproduce the earlier reported results, [20][21][22][23] but also predict previously unknown FE materials. In particular, we point out for the first time that the anion substitution may lead to improper ferroelectricity, to be discussed later in details.
Improper FEs based on the R3c ABO 3 perovskite structure Another family of perovskite oxides favor the 10-atom R3c structure as the lowest energy structure, 27 which has a rhombohedral a − a − a − tilt pattern around the [111] direction with respect to the Pm3m structure. There are eight TICs and eight PICs in one unit cell (the B-sites are TICs). Within the 10-atom cell, we find that there are two possible ways of the anion substitutions to induce improper FE (see Fig. S4 a, b in the SI). If considering a twofold 20-atom supercell, one has three types of FEs which result from the A-site, B-site and C-site substitutions, respectively (see Fig. S5 b, c and in the SI for the A-site, B-site, and C-site substitutions). Note that the A-site order in the 1:1 superlattice exhibits the non-FE structure with the R32 symmetry. 28 In the following, LaCoO 3 and LaAlO 3 are selected as the parent structures to demonstrate the B-site substitution induced ferroelectricity in the R3c ABO 3 perovskites. It was experimentally known that LaCoO 3 and LaAlO 3 take the R3c structure as the ground state, which is also confirmed by our test calculations. The mismatch in the lattice constants between these two compounds is only about 0.13%. LaCoO 3 with theR3c symmetry adopts a lowspin nonmagnetic ground state 29-31 (t 6 2g , S = 0) for Co 3+ . For all the perovskites with theR3c symmetry, we find that the ½111 B-site order in the 1:1 20-atom superlattice results in a polar structure with the C2 symmetry. The corresponding PE structure takes the R3m symmetry. Both the FE and PE structures have large band gaps about 2.0 eV, and the energy barrier between the FE and PE structures is~0.3 eV for a 20-atom cell. This spontaneous electric polarization is calculated to be 0.5 μC/cm 2 , aligning along the ½110 direction. The ferroelectricity is caused by the fact that the Bsite inversion symmetry is broken by the B-site substitution. For clarity, we show the local structures of LaAlO 3 and La 2 CoAlO 6 to understand the direction of polarization. A 10-atom La 3 Al 4 O 3 cluster in the R3c LaAlO 3 forms a tetrahedral structure. This cluster has one out-of-plane threefold-rotational axis on the central Al atom and three in-plane twofold-rotational axes along the La-La bond directions (see Fig. 2a). In La 2 CoAlO 6 with a ½111 B-site order, only one twofold rotation axis is kept due to the Co-atom substitution. This twofold rotation axis is exactly along the direction of polarization (see Fig. 2b). Interestingly, we find that the direction of polarization P½110 is parallel to the cross product of the B-site ordering vector D½111 and the octahedral rotational vector Ω [111] (see Fig. 2c The phonon dispersion also shows that the superlattice is stable (see Fig. S8 in the SI). In addition, our genetic algorithm (GA) 32 structure search confirms that the FE C2 structure is indeed the lowest energy structure. For proper FEs, the FE mode is the primary order parameter and there is a double well potential in the plot of energy versus polar displacement. But for the improper FEs, the FE mode is no longer the primary order parameter, i.e., the polarization is induced by one or two rotational modes. [13][14][15][16][17][18][19][20][21][22][23][24][25] To verify whether the ½111superlattice LaCoO 3 /LaAlO 3 is an improper FE or not, the stability of the FE mode will be examined. We adopt the ISOTROPY software 33 to obtain the symmetry-adapted phonon modes in the low-symmetry FE structure. To find out the appropriate FE mode and eliminate the effect of the symmetry breaking solely due to atomic substitution, we first replace the Co atoms in La 2 (Co,Al)O 6 back to Al atoms before allowing the mode decomposition with respect to the cubic Pm3m LaAlO 3 . We find two dominant modes, namely, FE mode and rotation mode. Hereafter, we will refer to these two modes as 'polar' and 'rotation' modes, respectively. Note that the PE structure is not R32=m here. After this mode decomposition, we replace the Al atoms back to Co, in order to obtain the PE reference structure La 2 (Co,Al)O 6 with the R32=m symmetry. The corresponding FE structure has the C2 symmetry. Second, we find that the earlier identified 'polar' and 'rotation' modes are both composed by the non-degenerate Γ À 1 and double degenerateΓ À 3 normal modes of the La 2 (Co,Al)O 6 R32=m PE structure. The Γ À 3 mode is the two-dimensional irreducible representation in active D 3d point group and can induce the inplane polarization but the Γ À 1 mode is non-polar. Here, we redefine the Γ À 3 components of the 'polar' and 'rotation' modes as P and Q modes, respectively. Our group theoretical analysis shows that the Landau free energy can be expanded in terms of P(P x ,P y ) and Q (Q x ,Q y ) around the reference R32=m structure: The last term of the free energy is the linear coupling between the P and Q modes, which is allowed since they belong to the same representation of the R32=m space group. For simplicity, we rescale the P and Q to the dimensionless quantities: P = 1 and Q = 1 mean that the total displacements of the P and Q modes are equal to 0.0496 and 0.8218 Å with respect to the 20-atom R32=m La 2 (Co,Al)O 6 PE structure, respectively. When applying the FE P mode to the PE structure, we find that the total energy increases with the mode magnitude, and thus the FE P mode is not a soft mode. For the Q mode, it has a double well potential in the plot of energy versus Q mode displacement (see Fig. 2d). These characteristics indicate that the layered (LaAlO 3 ) 1 /(LaCoO 3 ) 1 superlattice is an improper FE. The four parameters a, b, c and d of the Landau free energy can be obtained by a fitting of the density functional theory (DFT) results. We can see that the DFT is described rather well by the simple form of the Landau free energy (see Fig. 2d-f). The obtained parameters are a = 0.00612 eV, b = −0.49735 eV, c = 0.21836 eV, and d = −0.02829 eV, respectively. Since d is important, it is expected that the linear coupling between the P and Q modes will induce a finite FE P mode although the P mode itself is not unstable (a > 0).
New FEs based on the non-perovskite structures Previous works on designing improper FEs mainly focused on the perovskite-related structures. We note that if a crystal structure has the TIC, PIC, and band gap simultaneously, it is a possible parent candidate for designing new FEs through atom substitution. Here, we find that the non-perovskite structure ZnSrO 2 with the Pnma symmetry is suitable for such FEs design. We note that the experimentally synthesized ZnSrO 2 compound is a metastable structure since the formation energy of ZnSrO 2 is positive with respect to the rock salt SrO and wurtzite ZnO. The parent structure with eight TICs and eight PICs has a large direct band gap~2.2 eV at the Perdew-Burke-Ernzerhof level. The unit cell can be regarded as a layered structure along the z direction that contains two formula units of Zn 2 Sr 2 O 4 , in which there exist three twofold screw axes 2 1 as shown in Fig. 3a. By replacing the outer O atoms of the upper ZnSrO 2 layer with S atoms, we obtain a FE Pmc2 1 structure of Zn 4 Sr 4 O 6 S 2 . The corresponding PE structure has the Pmma symmetry. The energy barrier between the FE and PE states is 0.4 eV. The replacement of the oxygen atoms with the two sulfur atoms in a unit cell will break the screw axes along the a and c directions and induce a large spontaneous polarization (~20.0 μC/ cm 2 ) along the b direction (see Fig. 3b). The large polarization is due to the large atomic displacements induced by the sulfur replacement. Figure.   The corresponding structure of the La 3 Co 2 Al 2 O 3 cluster in C2 La 2 (Co, Al)O 6 is displayed in b, c The structure of the FE C2 La 2 (Co,Al)O 6 . The A-site La atoms are not shown for clarity. The direction of polarization P is perpendicular to both the superlattice direction D and the rotation direction Ω of the oxygen octahedron. d, e Show the total energies of La 2 (Co,Al)O 6 as a function of the magnitudes of the P mode and the Q mode, respectively. f Total energies of La 2 (Co, Al)O 6 as a function of the magnitude of the P mode with a fixed Q mode (Q = 1). For simplicity, we adopt the fractional unit for the P and Q modes, i.e., we rescale the P and Q to the dimensionless quantities: P = 1 and Q = 1 mean that the total displacements of P and Q modes are equal to 0.0496 and 0.8218 Å respect to the 20atom R32=m La 2 (Co,Al)O 6 PE structure. The black arrow indicates the lowest energy point Multiferroics based on the anion-substituted perovskite compound Our method is also suitable for designing new multiferroics. It is well-known that the ground state of LaMnO 3 is A-type antiferromagnetic (A-AFM) with Pbnm space group. 34 Thus, LaMnO 3 is selected as a parent structure for illustration. One can obtain a FE structure by replacing one-third of oxygen atoms with fluorine atoms to form a superlattice along the orthorhombic a direction (see Fig. 4a, b). We note that the replacement of oxygen ions in perovskite oxides by other anions such as fluorine ions has been achieved experimentally. [35][36][37] Since La and Mn ions in the parent LaMnO 3 structure are both trivalent, we also replace all the A-site La ions with bivalent Sr ions, whose radius is very similar to that of La 3+ . The chemical formula of the final substituted structure becomes SrMnO 2 F. The FE and PE SrMnO 2 F phases take the polar Pmc2 1 and non-polar Pbcm symmetries, respectively. The SrMnO 2 F is also an improper FE, see Fig. S10 in the SI for details. The electric polarization of the FE structure is 14.5 μC/cm 2 along the orthorhombic b direction. We find that the magnetic ground states for both FE and PE structures of SrMnO 2 F are the A-AFM, which are similar to the parent structure of LaMnO 3 . 34 When considering the spin-orbit coupling effect, SrMnO 2 F displays a magnetic anisotropy with the magnetic easy axis along the orthorhombic b direction. The Dzyaloshinskii-Moriya interaction 38,39 leads to a canted ferromagnetic magnetic moment of about 0.058 μ B along the orthorhombic c direction. In the parent structure of LaMnO 3 , the total canted magnetic moment is 0.027 μ B , which indicates that the anion substitution not only induces the ferroelectricity, but also increases the canted magnetic moment. Since the SrMnO 2 F with the Pmc2 1 structure has both weak ferromagnetism and ferroelectricity, it is a multiferroic. In addition, we find that the LaMnO 2 F with the FE Pmc2 1 structure has a G-type antiferromagnetic order with a very small magnetic anisotropy since the Mn 2+ ion is half-filled.

CONCLUSION
In conclusion, we have proposed a general method to design new FEs. We note that whether the designed FE is proper or improper is an open question. With this method, we can not only rediscover the HIF in A-site [001] ordered Pbnm perovskite oxides, but also discover new improper FEs. In particular, our results show that the B-site ordered½111 superlattice (LaCoO 3 ) 1 /(LaAlO 3 ) 1 can be an improper FE. We also demonstrate for the first time that the  anion substitution can be adopted to generate FEs as exemplified in the case of perovskite systems and ZnSrO 2 with a nonperovskite structure. Moreover, through the cation and anion substitution in LaMnO 3 , we find that SrMnO 2 F is a multiferroic, which indicates that our method is also useful for the design of multiferroic materials.

Strategy for finding PICs
The inversion center in a three-dimensional crystalline structure must locate at an atomic site or the center of two identical-element atoms. In our strategy for finding the PICs, we first consider all the possible inversion centers. For each possible inversion center, we first check whether it is a TIC or not. If yes, it is a TIC instead of a PIC. If not, we will consider whether it becomes a TIC after we displace the atoms in a given range (e.g., less than 2.0 Å). The element substitution may remove some of the TICs and PICs. It is also possible that some element substitution strategies will keep some of the TICs and PICs. For each substitution case, we check whether it breaks all the TICs but keeps at least one PIC. If yes, this substitution will lead to a possible realization of ferroelectricity.

Electronic calculations
In this work, the geometry optimization and electron structure calculation are performed by the state-of-the-art DFT 40 using the projector augmented-wave 41 potentials as implemented in the Vienna ab-initio Simulation Package. 42,43 In the DFT plane-wave calculations, the plane wave cutoff energy is set to 500 eV, and the exchange-correlation interactions are described by the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA). 44 In order to take into account for the proper orbital dependence of the on-site Coulomb and exchange interactions, we employ the GGA + U method 45 in treating the Co's and Mn's 3d orbitals and the values of U are set to 7.0 and 3.0 eV, respectively. For the relaxation of structures, the Hellmann-Feynman forces on each atoms are less than 0.001 eV/Å. The total electric polarization is calculated using the Berry phase method. 46,47 Global optimization We adopt a global optimization method based on the GA 32 to search for the ground state of layered compound La 2 CoAlO 6 . In our calculation, the basic lattice structure is fixed and our purpose is to find the distortions that leads to the lowest energy. It means that the basic framework (perovskitetype in current case) is fixed, but the structural distortions (FE displacements, oxygen octahedron rotations, cell deformation etc.) are allowed. In this GA searching method, the DFT calculation is adopted to relax the structure and the low-spin configuration is adopted for Co. The number of atoms in the supercell is fixed to 20. The population size and number of generations are set to 16 and 10, respectively.