Abstract
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 \(R\bar{3}c\) perovskite structure can become ferroelectric by substituting half of the Bsite cations. Compound ZnSrO_{2} with a nonperovskite 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 LaMnO_{3}.
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 JahnTeller effect.^{6,7,8,9,10} However, FEs are very few in nature.^{11} To find more highperformance 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 YO planes is induced by the nonpolar 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 nonpolar 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 doublelayered RuddlesdenPopper (RP) perovskite A_{3} B _{2}O_{7} (A=Ca, Sr; B=Mn,Ti),^{17,18,19} the 1:1 Acation ordering perovskitetype superlattice,^{20,21,22,23} Acation ordering RP NaRTiO_{4} (R = Y, La, Nd, SmHo),^{24} the 2:2 Bcation ordered superlattice,^{25} and metalorganic perovskite material.^{26}
In the abovementioned theoretically designed FEs, one usually starts from a highsymmetry 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 highsymmetry 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 Acation [001] ordered FE perovskite superlattices), but also predict new FEs. Our results show that the Bcation ordered\([11\bar{1}]\)superlatticeLa_{2}(Co,Al)O_{6} with a \(R\bar{3}c\) parent structure displays a spontaneous polarization. And compound ZnSrO_{2} with a nonperovskite 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.
Results
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 onedimensional toy model (see Fig. 1b). Assuming that there is a onedimensional 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 superlatticetype structures may be synthesized by means of the molecular beam epitaxy methods, we hereafter mainly focus on the superlatticetype 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 Pm \(\bar{3}\) m structure undergoes an outphase rotation and inphase 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 Bsites 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 20atom cell, 14 possible substitutioninduced FEs in total are found, with 2 types of Asite substitutions and 12 types of anion substitutions (see Fig. S2 in the SI). One of the Asite 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 Asite displacement and the A′site displacement are not completely cancelled. We find that it is impossible to obtain improper FEs through Bsite substitutions in a 20atom cell, in agreement with previous result.^{22} Nevertheless, considering a twofold supercell along the z direction, we can obtain FEs by any Bsite 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 \({\boldsymbol{R}}\bar{{\bf{3}}}{\boldsymbol{c}}\) ABO_{3} perovskite structure
Another family of perovskite oxides favor the 10atom \(R\bar{3}c\) structure as the lowest energy structure,^{27} which has a rhombohedral a^{−}a^{−}a^{−} tilt pattern around the [111] direction with respect to the Pm \(\bar{3}\) m structure. There are eight TICs and eight PICs in one unit cell (the Bsites are TICs). Within the 10atom 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 20atom supercell, one has three types of FEs which result from the Asite, Bsite and Csite substitutions, respectively (see Fig. S5 b, c and in the SI for the Asite, Bsite, and Csite substitutions). Note that the Asite order in the 1:1 superlattice exhibits the nonFE structure with the R32 symmetry.^{28}
In the following, LaCoO_{3} and LaAlO_{3} are selected as the parent structures to demonstrate the Bsite substitution induced ferroelectricity in the \(R\bar{3}c\) ABO_{3} perovskites. It was experimentally known that LaCoO_{3} and LaAlO_{3} take the \(R\bar{3}c\) 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 the\(R\bar{3}c\) symmetry adopts a lowspin nonmagnetic ground state^{29,30,31} (\({t}_{2g}^{6}\), S = 0) for Co^{3+}. For all the perovskites with the\(R\bar{3}c\) symmetry, we find that the \([11\overline{1}]\) Bsite order in the 1:1 20atom superlattice results in a polar structure with the C2 symmetry. The corresponding PE structure takes the \(R\bar{3}m\) 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 20atom cell. This spontaneous electric polarization is calculated to be 0.5 μC/cm^{2}, aligning along the \([1\bar{1}0]\) direction. The ferroelectricity is caused by the fact that the Bsite inversion symmetry is broken by the Bsite substitution. For clarity, we show the local structures of LaAlO_{3} and La_{2}CoAlO_{6} to understand the direction of polarization. A 10atom La_{3}Al_{4}O_{3} cluster in the \(R\bar{3}c\) LaAlO_{3} forms a tetrahedral structure. This cluster has one outofplane threefoldrotational axis on the central Al atom and three inplane twofoldrotational axes along the La–La bond directions (see Fig. 2a). In La_{2}CoAlO_{6} with a \([11\bar{1}]\) Bsite order, only one twofold rotation axis is kept due to the Coatom substitution. This twofold rotation axis is exactly along the direction of polarization (see Fig. 2b). Interestingly, we find that the direction of polarization P \([1\bar{1}0]\) is parallel to the cross product of the Bsite ordering vector D \([11\bar{1}]\) and the octahedral rotational vector Ω [111] (see Fig. 2c), i.e.,P ∥ D×Ω. This relation also holds for the other FE domains (There are six different domains since the octahedron rotation Ω may have six possible rotation directions: [111], [−1−1−1], [1−1−1], [−1,1,1], [1–11], [−11−1], and [−1−1−1]). 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\(\,[11\bar{1}]\)superlattice 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 symmetryadapted phonon modes in the lowsymmetry 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 \(Pm\bar{3}m\) 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 \(R\bar{3}2/m\) 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 \(R\bar{3}2/m\) 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 nondegenerate \({{\rm{\Gamma }}}_{1}^{}\) and double degenerate\({{\rm{\Gamma }}}_{3}^{}\) normal modes of the La_{2}(Co,Al)O_{6} \(R\bar{3}2/m\) PE structure. The \({{\rm{\Gamma }}}_{3}^{}\) mode is the twodimensional irreducible representation in active D_{3d} point group and can induce the inplane polarization but the \({{\rm{\Gamma }}}_{1}^{}\) mode is nonpolar. Here, we redefine the \({{\rm{\Gamma }}}_{3}^{}\)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 \(R\bar{3}2/m\) 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 \(R\bar{3}2/m\) 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 20atom \(R\bar{3}2/m\) 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 nonperovskite structures
Previous works on designing improper FEs mainly focused on the perovskiterelated 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 nonperovskite 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 PerdewBurkeErnzerhof 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. 3b shows the atomic direction and magnitude of the displacement (see red arrows) after the full relaxation.The S atom is nearly at the original O position. Because the bond length of ZnS is larger than that of ZnO, Zn_{1,} and Zn_{2} ions move along the –c or c directions, respectively. This subsequently leads to a large displacement of the O_{2} atom along the direction (i.e., approximately [011] direction) perpendicular to the Zn_{1}Zn_{2}bond direction. Overall, these ion displacements induce a large polarization along the –b axis in the relaxed structure of Zn_{4}Sr_{4}O_{6}S_{2}. Moreover, the new FEs Zn_{4}Sr_{4}O_{6}S_{2} is an environmentally friendly nontoxic material unlike PbTiO_{3}.
Multiferroics based on the anionsubstituted perovskite compound
Our method is also suitable for designing new multiferroics. It is wellknown that the ground state of LaMnO_{3} is Atype antiferromagnetic (AAFM) with Pbnm space group.^{34} Thus, LaMnO_{3} is selected as a parent structure for illustration. One can obtain a FE structure by replacing onethird 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 Asite 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 nonpolar 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 AAFM, which are similar to the parent structure of LaMnO_{3}.^{34} When considering the spinorbit coupling effect, SrMnO_{2}F displays a magnetic anisotropy with the magnetic easy axis along the orthorhombic b direction. The DzyaloshinskiiMoriya 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 Gtype antiferromagnetic order with a very small magnetic anisotropy since the Mn^{2+} ion is halffilled.
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 Asite [001] ordered Pbnm perovskite oxides, but also discover new improper FEs. In particular, our results show that the Bsite ordered\([11\bar{1}]\) 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.
Methods
Strategy for finding PICs
The inversion center in a threedimensional crystalline structure must locate at an atomic site or the center of two identicalelement 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 stateoftheart DFT^{40} using the projector augmentedwave^{41} potentials as implemented in the Vienna abinitio Simulation Package.^{42,43} In the DFT planewave calculations, the plane wave cutoff energy is set to 500 eV, and the exchangecorrelation 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 onsite 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 HellmannFeynman 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 lowspin 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.
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Acknowledgements
Work at Fudan was supported by NSFC, Research Program of Shanghai Municipality and MOE, the Special Funds for Major State Basic Research (2015CB921700), Qing Nian Ba Jian Program, Program for Professor of Special Appointment (Eastern Scholar), and Fok Ying Tung Education Foundation. K. X. was partially supported by NSFC 11404109. X. L. was supported in part by the National Science Foundation (NSF) through the Pennsylvania State University MRSEC under award number DMR1420620, which is supervised by Prof. James M. Rondinelli. We thank Panshuo Wang for useful comments on the manuscript.
Author Contributions
The study was proposed and planned by H.X. The calculations were carried out by K.X. All authors discussed the results and wrote the manuscript.
Competing Interests
The authors declare no conflict of interests.
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Affiliations
Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai, 200433, People’s Republic of China
 Ke Xu
 & Hongjun Xiang
Hubei Key Laboratory of Low Dimensional Optoelectronic Materials and Devices, Hubei University of Arts and Science, Xiangyang, 441053, People’s Republic of China
 Ke Xu
Department of Materials Science and Engineering, Northwestern University, Evanston, IL, 60208, USA
 XueZeng Lu
Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, People’s Republic of China
 Ke Xu
 & Hongjun Xiang
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Correspondence to Hongjun Xiang.
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