## Introduction

Polar 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 LiOsO31, a metal that transforms from a centrosymmetric $$R{\bar{3}}c$$ structure to a polar $$R{3}c$$ structure at 140 K, has stimulated an active search for new polar metals in both theory and experiment2,3,4,5,6,7,8,9.

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 multiferroics10. 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 algorithm11 and evolutionary algorithm12, 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 composition13,14,15.

In this work, we use ab initio high-throughput structure screening to predict a new polar metal BiPbTi2O6 (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 $$Pmn{2}_{1}$$), 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 displacements10. On the other hand, in the perovskite structures, Bi3+ and Pb2+ 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/PbTiO3 heterostructure. We find that different states in which BPTO polar displacements are parallel, anti-parallel and perpendicular to PbTiO3 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 PbTiO3 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 PbTiO3 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 CALYPSO11,16 method, in combination with CrySPY17. 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 TiO6 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 structures are both perovskite with $$Pmn{2}_{1}$$ symmetry (space group No. 31) and $$Pmm2$$ symmetry (space group No. 25). Both $$Pmn{2}_{1}$$ and $$Pmm2$$ symmetries are polar. The two perovskite structures have almost the same energy. Figure 1c shows the perovskite $$Pmn{2}_{1}$$ crystal structure. Bi and Pb atoms form a rock-salt 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 themselves18,19, perovskite oxides have been widely studied and are more suitable for device applications because many perovskite oxide substrates are available20, 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 matter21,22. Figure 1e shows that both perovskite $$Pmn{2}_{1}$$ 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 $${\mathrm{\AA}}^{3}$$ f.u.$${}^{-1}$$ under ambient conditions, while the two perovskite structures are more closely packed (122 $${\mathrm{\AA}}^{3}$$ f.u.$${}^{-1}$$ and 126 $${\mathrm{\AA}}^{3}$$ f.u.$${}^{-1}$$ 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 $$Pmn{2}_{1}$$ 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 $$Pmn{2}_{1}$$ 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 epitaxy23. Substrates such as NdScO3 and KTaO3 have a proper lattice constant to stabilize the perovskite $$Pmm2$$ structure in BPTO thin films. The DFT calculated lattice constants of KTaO3 and NdScO3 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.$${}^{-1}$$ 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 $$Pmn{2}_{1}$$ 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 ($$Pmn{2}_{1}$$ and $$Pmm2$$), because nominally Bi$${}^{3+}$$, Pb$${}^{2+}$$ and O$${}^{2-}$$, due to charge neutrality Ti must have a formal valence of Ti$${}^{3.5+}$$, 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 $$Pmn{2}_{1}$$ and $$Pmm2$$ structures. However, in the post-perovskite 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 post-perovskite 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 metals3,24.

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. A long-range magnetic ordering with a charge disproportionation (Ti$${}^{3+}$$+Ti$${}^{4+}$$) can result in an insulating ground state25. To test the robustness of our prediction that BiPbTi2O6 is a polar metal, we apply an effective Hubbard $$U$$ correction on the Ti-3$$d$$ orbitals and calculate the densities of states for all three low-energy structures. The accurate value of correlation strength of BPTO is not known, but presumably it should not exceed that of Mott insulator LaTiO3, in which Hubbard $${U}_{{\rm{Ti}}}$$ is about 5 eV26. Therefore we consider a Hubbard $${U}_{{\rm{Ti}}}$$ ranging from 0 to 5 eV. Within this range of $${U}_{{\rm{Ti}}}$$, we do not find charge disproportionation but find robust metallicity in all the three polar structures of BPTO. Furthermore, in this range of $${U}_{{\rm{Ti}}}$$, we find itinerant ferromagnetism in the perovskite $$Pmn{2}_{1}$$ structure at $${U}_{{\rm{Ti}}}\ge 1$$ eV and in the perovskite $$Pmm2$$ structure at $${U}_{{\rm{Ti}}}\ge 2$$ eV (antiferromagnetic ordering is less stable than ferromagnetic ordering). We do not find any magnetism in the post-perovskite $$Pmm2$$ structure up to $${U}_{{\rm{Ti}}}=5$$ eV. The magnetic phase diagram for the three polar structures as a function of Hubbard $${U}_{{\rm{Ti}}}$$ is shown in Fig. 3. The origin of itinerant ferromagnetism in BPTO is Stoner instability27. In DFT+$$U$$ calculations, the Stoner criterion to induce itinerant ferromagnetism is28:

$$U\rho ({E}_{{\rm{F}}})\,> \, 1,$$
(1)

were $$U$$ and $$\rho ({E}_{{\rm{F}}})$$ 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 $$Pmn{2}_{1}$$ structure has a large density of state at the Fermi level $$\rho ({E}_{{\rm{F}}})$$ in its non-magnetic state; the perovskite $$Pmm2$$ structure has a slightly smaller $$\rho ({E}_{{\rm{F}}})$$. Post-perovskite $$Pmm2$$ structure, on the other hand, has a very small $$\rho ({E}_{{\rm{F}}})$$ (9 times smaller than that of the perovskite $$Pmm2$$ structure and 15 times smaller than that of the perovskite $$Pmn{2}_{1}$$ structure). This explains that the critical $${U}_{{\rm{Ti}}}$$ to stabilize itinerant ferromagnetism in the perovskite $$Pmn{2}_{1}$$ structure is the smallest, while a much larger $${U}_{{\rm{Ti}}}$$ (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 ferroelectrics10,29,30,31,32,33,34,35. However, lone-pair electrons alone are not sufficient to stabilize a polar state in metals nor a ferroelectric state in insulators. For example, BiFeO3 is ferroelectric30 but BiMnO3 is anti-ferroelectric36,37 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 oxides29,30,38, 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 $$\lambda$$ 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 metal39). The energy difference between the polar structure and the corresponding centrosymmetric structure of BPTO in all three cases is larger than that of LiOsO3 (about 25 meV f.u.$${}^{-1}$$), implying that the structural transition temperature of BPTO is higher than that of LiOsO340. We note that the above second-order structural phase transition is a key property to distinguish instrinsic polar metals from degenerately doped ferroelectrics32,33,34,35,41,42, 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 PbTiO3 is under tensile strain and favors an in-plane polarization over an out-of-plane polarization (Supplementary Fig. 4). Therefore, we study a BPTO/PbTiO3 heterostructure, in which both BPTO polar displacements and PbTiO3 polarization are parallel to the interface. We will show that different from previously studied ferroelectric/polar-metal heterostructures2,4,43,44, multiple states with different relative orientations of BiPbTi2O6 polar displacements and PbTiO3 polarization can be stabilized. When an electric field is applied to switch the polarization of PbTiO3, 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 PbTiO3 polarization. Otherwise, the polar displacements of BPTO stay put and form different configurations when the external electric field changes the direction of PbTiO3 polarization.

Figure 5a shows a BPTO/PbTiO3 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 PbTiO3. Experimentally substrates such as KTaO3 and NdScO3 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 PbTiO3 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 PbTiO3 are empty while Ti-$$d$$ states in BPTO nominally have 0.5$$e$$ per Ti atom. Such a charge leakage can be effectively prevented by replacing PbTiO3 with PbTi$${}_{1-x}$$Zr$${}_{x}$$O345, 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 PbTiO3 (Supplementary Fig. 7). With PbTiO3 replaced by a paraelectric SrTiO3 substrate, one-unit-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 PbTiO3 close to the interface, which is clearly seen from Fig. 5c. Forming such a 180° domain wall in ferroelectrics increases energy46,47. Figure 5c shows that in both configurations, the interface strongly favors a parallel coupling between BPTO polar displacements and PbTiO3 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) calculations48 and use transition state theory49. Transition state theory has been widely used in understanding polarization switching in ferroelectric thin films50,51, as well as ferroelectric domain wall motion52. 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 PbTiO3 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 theory49, at a given temperature $$T$$, an energy barrier $$\Delta E$$ with a magnitude of a few $${k}_{{\rm{B}}}T$$ can be easily surmounted [At room temperature, 1 $${k}_{{\rm{B}}}T\approx$$ 26 meV. In transition state theory, the probability $$P$$ of overcoming the energy barrier is proportional to e$${}^{-\Delta E/{k}_{{\rm{B}}}T}$$, i.e., $$P$$$$\propto$$ e$${}^{-\Delta E/{k}_{{\rm{B}}}T}$$. The energy barrier ($$\Delta E$$ = 58 meV) is about twice the $${k}_{{\rm{B}}}T$$, hence the probability of overcoming this barrier is around 14$$\%$$.] ($${k}_{{\rm{B}}}$$ is the Boltzmann constant). Room temperature $$T=300$$ 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/PbTiO3 interface enables an electric field to first switch PbTiO3 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 PbTiO3 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.$${}^{-1}$$. 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 two-dimensional Van der Waals materials9.

Next we discuss low temperature applications. At sufficiently low temperatures where the energy barrier is much larger than $${k}_{{\rm{B}}}T$$, 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 PbTiO3 polarization, we can individually stabilize multiple configurations in which the BPTO polar displacements are “parallel”, “perpendicular” and “anti-parallel” to the PbTiO3 polarization (shown in Fig. 6b). Each configuration has different tunnelling resistance across ferroelectric insulators, because BPTO polar displacements and PbTiO3 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/PbTiO3 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 $$Pmn{2}_{1}$$), which can be transformed among each other via pressure or strain. In the perovskite structures, Bi$${}^{3+}$$ and Pb$${}^{2+}$$ enforce a fractional valence $$3.5+$$ 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/PbTiO3 heterostructures, at room temperature the interfacial coupling can overcome the switching barrier, which enables an electric field to first switch PbTiO3 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 metals3. At low temperature, an electric field can control the direction of PbTiO3 polarization and stabilize multi states in which PbTiO3 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 method53, 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 solids56, is used as the exchange correlation functional and has been applied successfully to interpreting the experimental observations of polar metal LiOsO3 in our previous work24. The Brillouin zone integration is performed with a Gaussian smearing of 0.05 eV over a $$\Gamma$$-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 calculations58.

For the calculations of BPTO/PbTiO3 structures, a $$\Gamma$$-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 vacuum59,60. In all the interface calculations, the in-plane lattice constant is fixed to be 4 Å and the bottom layer of PbTiO3 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 by ~220  meV per slab. Hence, we only report the BiO-terminated BPTO/PbTiO3 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 method48. 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 as61:

$${\rm{ELF}}={\left[1+{\left(\frac{D}{{D}_{h}}\right)}^{2}\right]}^{-1}$$
(2)

where

$$D=\frac{1}{2}\sum _{i}| \nabla {\phi }_{i}{| }^{2}-\frac{1}{8}\frac{| \nabla \rho {| }^{2}}{\rho }$$
(3)

and

$${D}_{h}=\frac{3}{10}{(3{\pi }^{2})}^{\frac{5}{3}}{\rho }^{\frac{5}{3}}$$
(4)

Here $$\rho$$ is the electron density and $${\phi }_{i}$$ 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 code11,16, with the assistance of CrySPY17. More than 1000 structures (50% 10-atom BiPbTi2O6 and 50% 20-atom Bi2Pb2Ti4O12) 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 spin-polarized calculations in which the cutoff energy is increased to 600 eV and the k-mesh grid density is >2500 per atom. We consider ferromagnetic ordering and different types of antiferromagnetic orderings such as $$A$$-type, $$C$$-type and $$G$$-type62 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 code63.

### Visualization

We use software VESTA to show crystal structures and real-space electron localized functions64.