<Supplementary Information>
Designing New Ferroelectrics with a General Strategy
Ke Xu1,2,3, Xue-Zeng Lu4, and Hongjun Xiang1,3*
1Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, P. R. China
2Hubei Key Laboratory of Low Dimensional Optoelectronic Materials and Devices, Hubei University of Arts and Science, Xiangyang, 441053, P. R. China
3Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, P. R. China
4Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
*Email:
1. Design ferroelectrics based on the Pbnm perovskite structure
The most common structure in ABX3 perovskite’s family is the GdFeO3-type structure with the Pbnm space group. In this section, we will show how atom substitution can induce ferroelectricity in the Pbnm ABX3 perovskite. For clarity, we take LaMnO3 as a typical example system. Here, The A, B and X sites are La, Mn and O respectively. In Table S1, we list all the atomic positions of LaMnO3. All TICs and PICs in a 20-atom unit cell are shown in Fig S1 (The coordinates are listed in Table S2 and S3). The 4 A-site, 4 B-site and 12-X sites are labeled in Fig. S2.
For the 20-atom cell of LaMnO3, we find that there are 14 possible ferroelectrics via atom substitution. Among them, two and twelve ferroelectrics are obtained by A-site substitution (see Table S4) and X-site anion substitution (see Table S5), respectively. In addition, we consider the cases of larger cells, i.e., 40-atom cells. Since the number of atom-substitution induced ferroelectrics in 40-atom cells is too large, we only give the four ferroelectrics with a superlattice ordering (see Fig. S3) because the superlattice structures may be easily synthesized experimentally.
Table S1. The lattice parameters and atomic positions of the experimental structure of Pbnm LaMnO3.
Pbnm LaMnO3, a = 5.742 Å, b = 7.668 Å, c = 5.532 ÅAtom / x / y / z
La1 / 0.7990000 / 0.0000000 / 0.2600000
La2 / 0.2010000 / 0.5000000 / 0.7600000
La3 / 0.7010000 / 0.5000000 / 0.2400000
La4 / 0.2990000 / 0.0000000 / 0.7400000
Mn1 / 0.2500000 / 0.7500000 / 0.2500000
Mn2 / 0.7500000 / 0.2500000 / 0.7500000
Mn3 / 0.7500000 / 0.7500000 / 0.7500000
Mn4 / 0.2500000 / 0.2500000 / 0.2500000
O1 / 0.2360000 / 0.0000000 / 0.1800000
O2 / 0.7640000 / 0.5000000 / 0.6800000
O3 / 0.2640000 / 0.5000000 / 0.3200000
O4 / 0.7360000 / 0.0000000 / 0.8200000
O5 / 0.5590000 / 0.7890000 / 0.47400000
O6 / 0.4410000 / 0.2890000 / 0.9740000
O7 / 0.4410000 / 0.7110000 / 0.9740000
O8 / 0.9410000 / 0.2890000 / 0.0260000
O9 / 0.9410000 / 0.7110000 / 0.0260000
O10 / 0.0590000 / 0.2110000 / 0.5260000
O11 / 0.0590000 / 0.7890000 / 0.5260000
O12 / 0.5590000 / 0.2110000 / 0.4740000
Figure S1. The TICs (a) and PICs (b) in LaMnO3, as shown by the smallest black spheres. Note that some inversion centers are hidden by the real atoms.
Table S2: The coordinates of 8 TICs in Pbnm LaMnO3 perovskite structure, as shown in Fig. S1(a).
Number / True Inversion Centers1 / 0.2500000 / 0.2500000 / 0.2500000
2 / 0.2500000 / 0.2500000 / 0.7500000
3 / 0.2500000 / 0.7500000 / 0.2500000
4 / 0.2500000 / 0.7500000 / 0.7500000
5 / 0.7500000 / 0.2500000 / 0.2500000
6 / 0.7500000 / 0.2500000 / 0.7500000
7 / 0.2500000 / 0.2500000 / 0.2500000
8 / 0.2500000 / 0.2500000 / 0.7500000
Table S3: The coordinates of 24 PICs in Pbnm LaMnO3 perovskite structure, as shown in Fig. S1 (b).
Number / Pseudo Inversion Centers1 / 0.2500000 / 0.0000000 / 0.2600000
2 / 0.2500000 / 0.0000000 / 0.7600000
3 / 0.0490000 / 0.2500000 / 0.0100000
4 / 0.2500000 / 0.5000000 / 0.2600000
5 / 0.0490000 / 0.2500000 / 0.5100000
6 / 0.2500000 / 0.5000000 / 0.7600000
7 / 0.0490000 / 0.7500000 / 0.0100000
8 / 0.0490000 / 0.7500000 / 0.5100000
9 / 0.7500000 / 0.0000000 / 0.2600000
10 / 0.7500000 / 0.0000000 / 0.7600000
11 / 0.5490000 / 0.2500000 / 0.0100000
12 / 0.7500000 / 0.5000000 / 0.2600000
13 / 0.2990000 / 0.2500000 / 0.2500000
14 / 0.5490000 / 0.2500000 / 0.5100000
15 / 0.7500000 / 0.5000000 / 0.7600000
16 / 0.2990000 / 0.2500000 / 0.7500000
17 / 0.5490000 / 0.7500000 / 0.0100000
18 / 0.2990000 / 0.7500000 / 0.2500000
19 / 0.5490000 / 0.7500000 / 0.5100000
20 / 0.2990000 / 0.7500000 / 0.7500000
21 / 0.7990000 / 0.2500000 / 0.2500000
22 / 0.7990000 / 0.2500000 / 0.7500000
23 / 0.7990000 / 0.7500000 / 0.2500000
24 / 0.7990000 / 0.7500000 / 0.7500000
Table S4: All possible ways for inducing ferroelectricity by A-site substitution in Pbnm ABX3 with a 20-atom cell. “SG” represents “space group”.
Possible A-site substitution / SG of FE / SG of PE1 / A3, A4 / Pmn21 / Pmmn
2 / A2, A3 / Pmc21 / Pbam
Table S5: All possible ways for inducing ferroelectricity by anion X-site substitution in Pbnm ABX3 with a 20-atom cell. “SG” represents “space group”.
Possible anion substitution / SG of FE / SG of PE1 / X3, X4 / Pmn21 / Pmmn
2 / X2, X3 / Pmc21 / Pbam
3 / X9, X10, X11, X12 / P1 / P-1
4 / X8, X10, X11, X12 / P1 / P-1
5 / X8, X9, X11, X12 / P1 / P-1
6 / X8, X9, X10, X12 / P1 / P-1
7 / X8, X9, X10, X11 / Pmn21 / Pmmn
8 / X7, X10, X11, X12 / P1 / P-1
9 / X7, X9, X11, X12 / P1 / P-1
10 / X7, X9, X10, X12 / Pna21 / Pnna
11 / X6, X10, X11, X12 / P1 / P-1
12 / X6, X7, X8, X9 / Pmc21 / Pbcm
Figure S2. The atomic labels (A1-A4, B1-B4, X1-X12) for the 20 atoms in the Pbnm ABX3 structure. See Table S1 for the atomic coordinates.
Figure S3. The ferroelectric structures with a superlattice ordering induced by A-site or B-site substitutions in the Pbnm ABX3 with a 40-atom cell. (a) The A-site superlattice ordering along the c direction. (b) The B-site superlattice ordering along the c direction. This ferroelectric structure was proposed in by Zhang et al. [Phys. Rev. B 91, 195145 (2015)]. (c) The B-site substitution superlattice ordering along the a direction. (d) The B-site substitution superlattice ordering along the b direction. The space groups of the ferroelectric structures shown in (a), (b), (c) and (d) are Pmc21, Pmc21, Pmn21 and Pmc21, respectively.
2. Design ferroelectrics based on the R3c perovskite structure
Another family of ABX3 perovskite adopt the 10-atom cell R3c structure as the ground state. We find that atom substitution can also give rise to ferroelectricity in the R3c perovskite structure. We consider both the case of the 10-atom unit cell and 20-atom supercell. For the 10-atom unit cell case, there are 2 possible ways of anion X-site substitution to induce ferroelectricity (see Fig. S4). While for the case of 20-atom supercell, there are 3 possible ways to induce ferroelectricity through A-site, B-site and X-site substitution, respectively (see Fig. S5).
Figure S4. Two ferroelectric structures [(a) and (b)] induced by the anion X-site substitution in one unit cell R3c ABX3. The space groups of these ferroelectric structures are C2 (a) and P1 (b), respectively.
Figure S5. Three ferroelectric structures induced by the atom substitution in the 20-atom supercell of R3c ABX3. (a) One A atom is replaced by a A’ atom. (b) Two B atoms are replaced by two B’ atoms. (c) Three anion X atoms are replaced by X’ atoms. All the three ferroelectric structures adopt the C2 space group.
3. Other Supplementary Materials
Figure S6. The symmetry operations (21 screw axis) in Pbnm LaMnO3 and Pmc21 SrMnO2F. The A-sites are not shown here for clarity. For LaMnO3, there are three 21 screw axes in one unit cell. But in the SrMnO2F, only one 21 screw axis is kept and the direction of electric polarization is along this axis.
Figure S7. Phonon dispersion of La2CoAlO6 for the R32/m phase from the GGA+U calculations. There are 3 imaginary frequencies at the Γ point with the double degenerate Γ3- polar modes and non-polar single degenerate Γ1- mode, respectively. The phonon dispersion is calculated using the Phonopy code1 based on a supercell approach, where the force constants are obtained by GGA+U calculations.
Figure S8. Phonon dispersion of La2CoAlO6 for the C2 phase from the GGA+U calculations. There are no imaginary frequencies indicating that the C2 structure is stable.
Figure S9. Phonon dispersion of Sr4Zn4O6S2 for the Pmc21 phase from the GGA calculations. There are no imaginary frequencies indicating that the Pmc21 structure is stable.
Figure S10. The total energies of SrMnO2F as a function of the magnitudes of the polar mode and the rotation mode, respectively. The polar mode is not a soft mode and the rotation mode has a double well potential. These characteristics indicate that the SrMnO2F superlattice is an improper ferroelectrics. The method of the mode decomposition in SrMnO2F is the same as the LaCoAlO superlattice.
Reference
1. Togo, A., Oba, F. Tanaka, I. First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures. Phys. Rev. B 78, 134106 (2008).