Design of a multifunctional polar metal via first-principles high-throughput structure screening

Intrinsic polar metals are rare, especially in oxides, because free electrons screen electric fields in a metal and eliminate the internal dipoles that are needed to break inversion symmetry. Here we use first-principles high-throughput structure screening to predict a new polar metal in bulk and thin film forms. After screening more than 1000 different crystal structures, we find that ordered BiPbTi2O6 can crystallize in three polar and metallic structures, which can be transformed between via pressure or strain. In a heterostructure of layered BiPbTi2O6 and PbTiO3, multiple states with different relative orientations of BiPbTi2O6 polar displacements, and PbTiO3 polarization, can be stabilized. At room temperature, the interfacial coupling enables electric fields to first switch PbTiO3 polarization and subsequently drive 180° change of BiPbTi2O6 polar displacements. At low temperatures, the heterostructure provides a tunable tunnelling barrier and might be used in multi-state memory devices. Polar metal oxides are not frequently observed, yet offer attractive properties for functional devices. Now, high-throughput structure screening of a thousand crystal structures reveals that BiPbTi2O6 can form both polar and metallic structures.

P olar metals-analogy of ferroelectrics in metals-are characterized by intrinsic conduction and inversion symmetry breaking. Polar metals are rare (especially in oxides) because mobile electrons screen electric fields in a metal and eliminate internal dipoles that are needed to break inversion symmetry. The discovery of LiOsO 3 1 , a metal that transforms from a centrosymmetric R 3c structure to a polar R3c structure at 140 K, has stimulated an active search for new polar metals in both theory and experiment [2][3][4][5][6][7][8][9] .
Density-functional-theory-based first-principles calculations have proven accurate in describing crystal structures and have been succesfully applied to predict new functional materials, such as ferroelectrics, piezoelectrics and multiferroics 10 . Since crystal structure is the essential property of polar metals, we need to scrutinize the prediction by not presupposing an a priori favorable crystal structure. First-principles high-throughput crystal structure screening method, which is based on the marriage between first-principles calculations and a multitude of techniques such as particle-swarm optimization algorithm 11 and evolutionary algorithm 12 , has demonstrated its superior power in effectively searching for the ground state structures and metastable structures of functional materials with only the given knowledge of chemical composition [13][14][15] .
In this work, we use ab initio high-throughput structure screening to predict a new polar metal BiPbTi 2 O 6 (BPTO for short). After screening over 1000 different crystal structures, we find that ordered BPTO can crystallize in three different polar metallic structures (post-perovskite Pmm2, perovskite Pmm2 and perovskite Pmn2 1 ), each of which can be transformed to another via external pressure or epitaxial strain. The mechanism is that 6s lone-pair electrons of Bi and Pb ions tend to favor off-center displacements 10 . On the other hand, in the perovskite structures, Bi 3+ and Pb 2+ enforce a fractional valence on Ti, which leads to conduction; in the post-perovskite structure, strong hybridization between Bi/Pb 6p and O 2p states induces a finite density of states at the Fermi level.
Next we demonstate potential applications of the new polar metal BPTO by studying a BPTO/PbTiO 3 heterostructure. We find that different states in which BPTO polar displacements are parallel, anti-parallel and perpendicular to PbTiO 3 polarization can be stabilized in the heterostructure. Also, 180°switching of BPTO polar displacements needs to surmount an energy barrier of about 58 meV per slab. This implies that at room temperature where thermal fluctuations can overcome the switching barrier, the interfacial coupling between the polarization and polar displacements enables an electric field to first switch PbTiO 3 polarization and subsequently drive BPTO to change its polar displacements by 180°; at low temperatures where the switching barrier dominates over thermal fluctuations, the BPTO polar displacements can not be switched but the direction of PbTiO 3 polarization can be controlled by an electric field. This can stabilize three distinct states with different tunnelling barriers.

Results and discussion
Most stable crystal structures of bulk BPTO. The key question in predicting a new polar metal is to determine its crystal structure. Since ordered BPTO has not been synthesized in experiment, we perform a first-principles high-throughput search for the ground state structure using CALYPSO 11,16 method, in combination with CrySPY 17 . In the search, we do not constrain ourselves in any a priori favorable crystal structure. We screen >1000 different crystal structures among which we consider different Bi/Pb ordering in perovskite structure: layered ordering, columnar ordering and rock-salt ordering; and we also consider many non-perovskite structures, including post-perovskite structure and hexagonal structure. The computational details of our first-principles calculations and high-throughput structure screening method are provided in Methods. Figure 1 shows ten lowest-energy crystal structures of BPTO from our calculations. The details of these ten crystal structures are available in Supplementary Table 1. The lowest energy structure is post-perovskite with a polar symmetry Pmm2 (space group No. 25). The crystal structure is explicitly shown in Fig. 1b. The TiO 6 octahedra are both corner-sharing and edge-sharing. The lack of inversion symmetry can be appreciated from Ti atoms which have strong polar displacements with respect to neighboring O atoms towards x-axis. The next two lowest-energy crystal Fig. 1 The lowest-energy crystal structures and phase transitions. a Ten lowest-energy crystal structures of BPTO predicted by CALYPSO and DFT calculations. b Post-perovskite Pmm2 structure; c Perovskite Pmn2 1 structure; d Perovskite Pmm2 structure. e Enthalpy of the three lowest-energy structures as a function of pressure. The enthalpy of the post-perovskite Pmm2 structure under each pressure is set as the zero point. f Total energy of the two lowest-energy perovskite structures as a function of epitaxial strain. The energy of the perovskite Pmn2 1 structure constrained by an in-plane lattice constant of 3.94 Å is chosen as the zero energy.
structures are both perovskite with Pmn2 1 symmetry (space group No. 31) and Pmm2 symmetry (space group No. 25). Both Pmn2 1 and Pmm2 symmetries are polar. The two perovskite structures have almost the same energy. Figure 1c shows the perovskite Pmn2 1 crystal structure. Bi and Pb atoms form a rocksalt ordering and their displacements with respect to O atoms in the xy plane make the crystal structure acentric. Figure 1d shows the perovskite Pmm2 crystal structure. Bi and Pb atoms have a layered ordering with a stacking direction along z-axis. It is clear that Bi, Pb, and Ti atoms all have strong polar displacements with respect to O atoms along x-axis, which breaks inversion symmetry. While post-perovskite oxides are interesting by themselves 18,19 , perovskite oxides have been widely studied and are more suitable for device applications because many perovskite oxide substrates are available 20 , which makes it feasible to grow perovskite oxide thin films. Therefore, we consider using external pressure or epitaxial strain to transform BPTO among different polar structures. Pressure is widely used in bulk synthesis to isolate metastable phases of matter 21,22 . Figure 1e shows that both perovskite Pmn2 1 and Pmm2 structures become more stable than the post-perovskite Pmm2 structure under a few GPa. The reason is that the post-perovskite Pmm2 structure is very hollow with a very large volume of 130 Å 3 f.u. À1 under ambient conditions, while the two perovskite structures are more closely packed (122 Å 3 f.u. À1 and 126 Å 3 f.u. À1 under ambient conditions, respectively). Applying pressure favors structures with smaller volumes. If we want to grow BPTO thin films on a perovskite oxide substrate, the post-perovskite structure does not form due to very large lattice mismatch (see Supplementary Table 1 for the cell parameters of post-perovskite structure). The pseudo-cubic lattice constant of the perovskite Pmn2 1 structure is 3.94 Å, while that of the perovskite Pmm2 structure is 3.98 Å. It is anticipated that as the substrate lattice constant varies from 3.94 Å to 3.98 Å, the energetically favored structure changes from the perovskite Pmn2 1 structure to the perovskite Pmm2 structure. This is indeed what Fig. 1f shows. Since the perovskite Pmm2 structure has a layered ordering of Bi/Pb atoms, it is highly suitable for thin film growth methods such as pulsed layer deposition and molecular beam epitaxy 23 . Substrates such as NdScO 3 and KTaO 3 have a proper lattice constant to stabilize the perovskite Pmm2 structure in BPTO thin films. The DFT calculated lattice constants of KTaO 3 and NdScO 3 can be found in Supplementary Table 2 with the comparison to the experimental data. We also enforce the post-perovskite Pmm2 structure to be stabilized on a perovskite oxide substrate and we expectedly find that its total energy is about 10 eV f.u. À1 higher than the two perovskite structures because of the large lattice mismatch ( Supplementary Fig. 1). In addition to the study of phase transitions under strains and pressures, the temperature effect on phase transitions can be found in Supplementary Fig. 2.
Electronic and magnetic properties of polar metal BPTO. The upper panels of Fig. 2 show the DFT-calculated densities of states of the post-perovskite Pmm2, perovskite Pmn2 1 and perovskite Pmm2 structures. We calculate both total density of states (DOS) and orbital-projected DOS (Ti-3d, Bi-6s, Pb-6s, and O-2p). In three optimized structures, we do not find any magnetization or charge disproportionation. Therefore spin-up and spin-down are summed in the DOS. The DOS projected on the two Ti atoms are identical and hence are also summed. The three DOS share similarities but also have important differences. All three structures have a non-zero DOS at the Fermi level; both Bi-6s and Pb-6s are well below the Fermi level and are fully occupied. The difference is that in both perovskite structures (Pmn2 1 and Pmm2), because nominally Bi 3þ , Pb 2þ and O 2À , due to charge neutrality Ti must have a formal valence of Ti 3:5þ , i.e., every Ti atom has 0.5 electron in the 3d conduction bands (no charge disproportionation is found in the calculations). Figure 2b, c shows that the Fermi level crosses Ti-3d states in the DOS of the perovskite Pmn2 1 and Pmm2 structures. However, in the postperovskite structure (Fig. 2a), Ti-3d states have negligible contribution around the Fermi level. Instead, Bi-6p and Pb-6p, as well as O-2p states make the largest contribution to the DOS around the Fermi level, which can also be seen in Supplementary Fig. 3 where the electronic states around the Fermi level are zoomed in.
The Bader (static) charge analysis in Table 1 shows that Bi and Pb have about 0.25 and 0.19 more electrons in the post-perovskite structure than in the perovskite structures, which indicates stronger hybridization between Bi/Pb and O atoms in the postperovskite structure. Therefore in the post-perovskite structure, Bi-6p and Pb-6p states are not fully empty and thus appear around the Fermi level.
Pb and Bi are heavy elements and their spin-orbit interactions (SOI) are not negligible. In the lower panels of Fig. 2, we take into account SOI and show the corresponding densities of states of BPTO of the three polar structures. Similar to the results without SOI, we do not find any magnization or charge disproportionation in the fully relaxed structures. By comparing the densities of states calculated by DFT without SOI (upper panels of Fig. 2) and DFT with SOI (lower panels of Fig. 2), SOI almost unaffects the electronic structure, similar to previous studies on other polar metals 3,24 .
While DFT with/without SOI calculations do not find any magnetization or charge disproportionation, correlation effects from Ti-3d orbitals may favor spin ordering and charge ordering.   26 . Therefore we consider a Hubbard U Ti ranging from 0 to 5 eV. Within this range of U Ti , we do not find charge disproportionation but find robust metallicity in all the three polar structures of BPTO. Furthermore, in this range of U Ti , we find itinerant ferromagnetism in the perovskite Pmn2 1 structure at U Ti ! 1 eV and in the perovskite Pmm2 structure at U Ti ! 2 eV (antiferromagnetic ordering is less stable than ferromagnetic ordering). We do not find any magnetism in the post-perovskite Pmm2 structure up to U Ti ¼ 5 eV. The magnetic phase diagram for the three polar structures as a function of Hubbard U Ti is shown in Fig. 3. The origin of itinerant ferromagnetism in BPTO is Stoner instability 27 . In DFT+U calculations, the Stoner criterion to induce itinerant ferromagnetism is 28 : were U and ρðE F Þ are Hubbard U parameter and density of states at the Fermi level of a non-magnetic state, respectively. The upper panels of Fig. 2 shows that the perovskite Pmn2 1 structure has a large density of state at the Fermi level ρðE F Þ in its non-magnetic state; the perovskite Pmm2 structure has a slightly smaller ρðE F Þ.
Post-perovskite Pmm2 structure, on the other hand, has a very small ρðE F Þ (9 times smaller than that of the perovskite Pmm2 structure and 15 times smaller than that of the perovskite Pmn2 1 structure). This explains that the critical U Ti to stabilize itinerant ferromagnetism in the perovskite Pmn2 1 structure is the smallest, while a much larger U Ti (larger than 5 eV) is needed to induce magnetism in the post-perovskite Pmm2 structure.
The role of lone-pair electrons. A local structural instability arising from lone-pair electrons has been reported in ferroelectric insulators and degenerately doped ferroelectrics 10,[29][30][31][32][33][34][35] . However, lone-pair electrons alone are not sufficient to stabilize a polar state in metals nor a ferroelectric state in insulators. For example, BiFeO 3 is ferroelectric 30 but BiMnO 3 is anti-ferroelectric 36,37 although lone-pair electrons are present in both of them. Therefore, high-throughput crystal structure prediction is essential in predicting new polar metals and ferroelectric insulators.
In our study, the crystal structure screening takes into account both polar and anti-polar states for different cation orderings. We now show that in the three lowest-energy metallic phases of BPTO, the lone-pair 6s electrons in Bi and Pb play an important role in breaking inversion symmetry. We use electron localization function (ELF, defined in Methods) to explicitly visualize how lone-pair electrons of Bi and Pb break inversion symmetry in metallic BPTO. Figure 4a-c shows an iso-surface of ELF of the three polar structures of BPTO. Only the ELF of Bi and Pb ions are displayed for clarity. Similar to insulating Bi-based and Pb-based perovskite oxides 29,30,38 , the ELF shows that a lobe-like lone-pair resides on one side of Bi and Pb ions in all three polar metallic structures, which is the driving force to break inversion symmetry. On the other hand, ELF in the corresponding centrosymmetric structures shows spherical-symmetric feature, which is implied in Fig. 4d-f. Furthermore, we calculate the total energy variation as a function of normalized polar displacement λ from the centrosymmetric structures to the polar structures (see Fig. 4g-i). In all three cases, the energy curve monotonically decreases from the centrosymmetric structure to the polar structure, which indicates a continuous and spontaneous phase transition below a critical temperature (satisfying Anderson's and Blount's criterion of a ferroelectric-like metal 39 ). The energy difference between the polar structure and the corresponding centrosymmetric structure of BPTO in all three cases is larger than that of LiOsO 3 (about 25 meV f.u. À1 ), implying that the structural transition temperature of BPTO is higher than that of LiOsO 3 40 . We note that the above second-order structural phase transition is a key property to distinguish instrinsic polar metals from degenerately doped ferroelectrics 32-35,41,42 , because realistic dopants (cation substitution or oxygen vacancies) make the  crystal symmetry of doped ferroelectrics ill-defined and correspondingly there is no well-defined continuous structural phase transition at finite temperatures.
Switching barrier of BPTO thin films in a heterostructure. Next we study BPTO thin films. The Pmm2 perovskite structure of BPTO, which has a layered Bi/Pb ordering, is highly suitable for thin film growth and can be stabilized on a perovskite oxide substrate having a lattice constant of 3.98 Å or larger. The Bi/Pb stacking direction in the Pmm2 perovskite structure is chosen as the z-axis, while the polar displacements are in the xy-plane. In addition, we find that constrained by an in-plane lattice constant of 3.98 Å or larger, ferroelectric PbTiO 3 is under tensile strain and favors an in-plane polarization over an out-of-plane polarization ( Supplementary Fig. 4). Therefore, we study a BPTO/PbTiO 3 heterostructure, in which both BPTO polar displacements and PbTiO 3 polarization are parallel to the interface. We will show that different from previously studied ferroelectric/polar-metal heterostructures 2,4,43,44 , multiple states with different relative orientations of BiPbTi 2 O 6 polar displacements and PbTiO 3 polarization can be stabilized. When an electric field is applied to switch the polarization of PbTiO 3 , a finite energy barrier exists for BPTO to change its polar displacements by 180°. If the temperature is high enough that thermal fluctuations can overcome the energy barrier, the polar displacements of BPTO will follow the change of PbTiO 3 polarization. Otherwise, the polar displacements of BPTO stay put and form different configurations when the external electric field changes the direction of PbTiO 3 polarization. Figure 5a shows a BPTO/PbTiO 3 heterostructure. The in-plane lattice constant is constrained to 4 Å, which stabilizes both perovskite Pmm2 structure of BPTO and an in-plane polarization of PbTiO 3 . Experimentally substrates such as KTaO 3 and NdScO 3 can provide such a lattice constant. Figure 5a shows two different configurations: on the left (right) is a "parallel state" ("anti-parallel state") in which PbTiO 3 polarization is parallel (anti-parallel) to BPTO polar displacements. Both configurations are stabilized after relaxation in our calculations. We first find that one-unit-cell thin film of BPTO is still polar and metallic. Figure 5b shows layer-resolved conduction electrons by integrating the partial density states of Ti-d orbitals (Supplementary Fig. 5). Conduction electrons are mainly confined in BPTO with some charge leakage into a few unit cells of PbTiO3. This charge leakage is due to the proximate effect that Ti-d states in PbTiO 3 are empty while Ti-d states in BPTO nominally have 0.5e per Ti atom. Such a charge leakage can be effectively prevented by replacing PbTiO 3 with PbTi 1Àx Zr x O 3 45 , which is supported by our calculations in Supplementary Fig. 6. Figure 5c shows the layer-resolved cation displacements of Ti and Pb/Bi with respect to O atoms along the x-axis. The polar displacements of Bi and Pb in BPTO are almost bulk-like in both configurations. We note that the polar property of BPTO is not due to the interfacial coupling with PbTiO 3 (Supplementary Fig. 7). With PbTiO 3 replaced by a paraelectric SrTiO 3 substrate, oneunit-cell BPTO layer still has polar displacements and is metallic (Supplementary Fig. 8).
Next we study thermodynamics and the energy barrier of switching between "parallel state" and "anti-parallel state". DFT calculations find that the energy of "parallel state" is 37 meV per slab lower than that of "anti-parallel state". This is because in "anti-parallel state", a 180°domain wall is formed in PbTiO 3 close to the interface, which is clearly seen from Fig. 5c. Forming such a 180°domain wall in ferroelectrics increases energy 46,47 . Figure 5c shows that in both configurations, the interface strongly favors a parallel coupling between BPTO polar displacements and PbTiO 3 polarization.
While "parallel state" is more stable than "anti-parallel state", "anti-parallel state" can be stabilized by itself because it is a local minimum. Therefore a finite energy barrier exists for BPTO to 180°change its polar displacements from "anti-parallel state" to "parallel state". To quantitatively calculate the energy barrier and identify a possible switching path for polar displacement, we perform the climbing image nudged elastic band (NEB) calculations 48 and use transition state theory 49 . Transition state theory has been widely used in understanding polarization switching in ferroelectric thin films 50,51 , as well as ferroelectric domain wall motion 52 . We choose "anti-parallel state" as the initial state and "parallel state" as the final state. We study a possible switching path in which BPTO polar displacements are 180°"rotated" in the xy plane. The NEB results are shown in Fig. 5d. Along the structural transition path from "anti-parallel state" (labelled as "1") to "parallel state" (labelled as "3"), there is another metastable state (labelled as "2") where BPTO polar displacements are perpendicular to PbTiO 3 polarization in the xy plane (Fig. 5e). Between the three stable states ("1", "2", "3"), there are two energy barriers. The larger one, i.e., the energy difference between the anti-parallel state and the highest saddle point, is 58 meV per slab.
Multifunctions of the BPTO/PTO heterostructure. In this section, we discuss potential functions of the BPTO/PTO heterostructure based on the calculated switching barrier in the previous section.
We first discuss room temperature applications. The switching barrier of BPTO is about 58 meV per slab. From transition state theory 49 , at a given temperature T, an energy barrier ΔE with a magnitude of a few k B T can be easily surmounted [At room temperature, 1 k B T % 26 meV. In transition state theory, the probability P of overcoming the energy barrier is proportional to e ÀΔE=k B T , i.e., P/ e ÀΔE=k B T . The energy barrier (ΔE = 58 meV) is about twice the k B T, hence the probability of overcoming this barrier is around 14%.] (k B is the Boltzmann constant). Room temperature T ¼ 300 K is about 26 meV. Our energy barrier is about twice room temperature and therefore room temperature is sufficient to overcome the barrier. This implies that the interfacial coupling at the BPTO/PbTiO 3 interface enables an electric field to first switch PbTiO 3 polarization and subsequently drive BPTO to 180°change its polar displacements. This realizes an electrically switchable bi-state in the new polar metal BPTOat room temperature. We note that the transition path chosen in the NEB calculation is only one possiblity. The actual transition path could be different from the one in our study and the resulting energy barrier should be even lower, which will make the switching of BPTO polar displacements more feasible. Figure 6a schematically shows how we can use PbTiO 3 polarization to control the polar displacements of BPTO at room temperature.
The above switching mechanism is also applicable to multi-layer BPTO thin films. The mechanism is as follows. Our calculations find that bulk BPTO is more stable in the polar Pmm2 perovskite structure than the anti-polar Pmma perovskite structure by 65 meV f.u. À1 . The anti-polar Pmma perovskite structure is shown in Fig. 7a. The polar Pmm2 perovskite structure is shown in Fig. 7b for comparison. Therefore, for multi-layer BPTO thin films, once the bottom layer of BPTO is 180°switched via the interfacial coupling, the remaining layers of BPTO will be driven by thermodynamics to change their polar displacements in a layer-by-layer manner to avoid an anti-polar state in the film. The above physical picture is computationally confirmed in Supplementary Fig. 9. However, for device applications (e.g. the model device illustrated in Supplementary Fig. 10), BPTO thin films of single-unit-cell thick are most desirable, in analogy to twodimensional Van der Waals materials 9 .
Next we discuss low temperature applications. At sufficiently low temperatures where the energy barrier is much larger than k B T, the interfacial coupling can not drive polar metals to change their polar displacements when an electric field switches the polarization of ferroelectrics. However, this has interesting implications: as we use the electric field to change the direction of PbTiO 3 polarization, we can individually stabilize multiple configurations in which the BPTO polar displacements are "parallel", "perpendicular" and "anti-parallel" to the PbTiO 3 polarization (shown in Fig. 6b). Each configuration has different tunnelling resistance across ferroelectric insulators, because BPTO polar displacements and PbTiO 3 polarization have different relative orientation. As we use an electric field to change the direction of ferroelectric polarization (polar displacements do not follow due to low temperatures), we can tune tunnelling barriers between different states and therefore the BPTO/PbTiO 3 heterostructure can be used in multi-state memory devices.
In conclusion, we demonstrate the power of first-principles high-throughput screening in designing new functional materials and in particular predict a new polar metal BPTO by utilizing the Bi/Pb lone-pair electrons. The three lowest-energy structures of BPTO are all polar and metallic (post-perovskite Pmm2, perovskite Pmm2 and perovskite Pmn2 1 ), which can be transformed among each other via pressure or strain. In the perovskite structures, Bi 3þ and Pb 2þ enforce a fractional valence 3:5þ on Ti, which leads to conduction. In the post-perovskite structure, strong hybridization between Pb/Bi 6p and O 2p states induces a finite density of states at the Fermi level. In a BPTO/ PbTiO 3 heterostructures, at room temperature the interfacial coupling can overcome the switching barrier, which enables an electric field to first switch PbTiO 3 polarization and subsequently drive BPTO to 180°flip its polar displacements. This realizes an electrically switchable bi-state in the new polar metal BPTO. The switching method is applicable to other layered polar metals 3 . At low temperature, an electric field can control the direction of  PbTiO 3 polarization and stabilize multi states in which PbTiO 3 polarization and BPTO polar displacements have different relative orientations, implying different tunnelling resistance. This property can be used in tunable multi-state memory devices. We hope this work will stimulate experimentalists to synthesize the new polar metal in both bulk and thin-film forms.

Methods
First-principles calculations. For bulk structures, density functional theory (DFT) calculations are performed using a plane wave basis set and projector-augmented wave method 53 , as implemented in the Vienna Ab-initio Simulation Package (VASP) 54,55 . PBEsol, a revised Perdew--Burke--Ernzerhof (PBE) generalized gradient approximation for improving equilibrium properties of densely-packed solids 56 , is used as the exchange correlation functional and has been applied successfully to interpreting the experimental observations of polar metal LiOsO 3 in our previous work 24 . The Brillouin zone integration is performed with a Gaussian smearing of 0.05 eV over a Γ-centered k-mesh up to 12 × 12 × 12 and a 600 eV plane-wave cutoff. The threshold of energy convergence is 10−6 eV. Hubbard U corrections are also considered in our calculations to model the effects of strong correlation on electronic and magnetic properties. The rotationally invariant approach of Hubbard U proposed by Dudarev et al. 57 is used in our DFT+U calculations. Spin-orbit coupling (SOC) is also considered to study electronic structure in our DFT+U+SOC calculations 58 .
For the calculations of BPTO/PbTiO 3 structures, a Γ-centered k-mesh of 10 × 10 × 1 is used. The periodic slabs are separated by vacuum of 20 Å thick to diminish the interaction between them. Since asymmetrical interface modelling is used in our calculations, we employ dipole correction to eliminate the artificial electric field in the vacuum 59,60 . In all the interface calculations, the in-plane lattice constant is fixed to be 4 Å and the bottom layer of PbTiO 3 is fixed to simulate the bulk-like interior that is under tensile strain. All the other atoms are fully relaxed along the three axes. We consider two possible terminations of the heterostructure, i.e., BaO-and BiO-terminations. The former one is less stable than the latter one bỹ 220 meV per slab. Hence, we only report the BiO-terminated BPTO/PbTiO 3 interface in our study.
The energy barriers between the parallel and anti-parallel states, as well as the saddle points along the transition path are found by the nudged elastic band (NEB) calculations through the climbing image NEB method 48 . In NEB calculations, a set of intermediate structures (i.e., images) between the initial state (anti-parallel state) and the final state (parallel state) are generated. They are iteratively adjusted so as to minimize the increase in energy along the transition path.
The electron localization function in our study, which is used to visualize lone-pair electrons in the real space is defined as 61 : and Here ρ is the electron density and ϕ i are the Kohn--Sham wave functions.
Crystal structure search. The crystal structure search for bulk BPTO is carried out using the particle swarm optimization algorithm implemented in CALYPSO code 11,16 , with the assistance of CrySPY 17 . More than 1000 structures (50% 10atom BiPbTi 2 O 6 and 50% 20-atom Bi 2 Pb 2 Ti 4 O 12 ) are created in 20 generations. The structural optimization and computation of total energy are performed using VASP. In the first step of high-throughput screening of these 1000 crystal structure, we used non-spin polarized calculations with the exchange-correlation functional of PBEsol. The cutoff energy of 450 eV and the k-mesh grid density is about 2000 per atom. In the second step, the lowest 50 structures are re-calculated by the spinpolarized calculations in which the cutoff energy is increased to 600 eV and the kmesh grid density is >2500 per atom. We consider ferromagnetic ordering and different types of antiferromagnetic orderings such as A-type, C-type and G-type 62 to examine possible magnetic properties. The global structure search is performed under 0 GPa. The five lowest energy structures after screening are also studied under pressure. The space groups of the predicted crystal structures are examined by the FINDSYM code 63 .
Visualization. We use software VESTA to show crystal structures and real-space electron localized functions 64 .

Data availability
The authors declare that all the data supporting the findings of this study are available within the paper and its Supplementary Information.

Code availability
The high-throughput crystal structural predictions were carried out using the proprietary code VASP 54,55 , with the combination of CALYPSO 11,16 and CrySPY 17