Introduction

Heterostructures of transparent conducting oxides (TCOs) have attracted the attention of researchers in view of both fundamental research as well as potential applications1,2. Among them, interfaces of the perovskite materials LaAlO3 and SrTiO3 are the most studied prototypes due to the emergence of fascinating physical phenomena including superconductivity, ferromagnetism3, and interfacial two-dimensional electron gases (2DEG)1. The latter is mainly a consequence of the electronic reconstruction to compensate the interfacial polar discontinuity induced by the deposition of the polar material LaAlO3 formed by alternatingly charged (LaO)+ and (AlO2)− planes on a neutral, TiO2-terminated SrTiO3 substrate. The 2DEG density confined within the SrTiO3 side of the interface can reach 0.5 electrons (e) per a2 (a being the lattice parameter of SrTiO3), corresponding to ~3.3 × 1014 cm−2 for a complete compensation of the formal polarization induced within LaAlO34. However, in real samples, the presence of defects impacts both polar discontinuity and electronic reconstruction, and thus carrier mobilities5. This includes cation intermixing5,6, oxygen vacancies7, edge dislocations8, and changes in surface stoichiometry9. Overall, the electron mobilities at such an interface are very sensitive to growth conditions5. Besides these extrinsic effects, the low interfacial mobility of this material system is also caused by the low-dispersion (large effective electron masses) of the partially occupied Ti-d states. Further, scattering of electrons within these bands induced by significant electron–phonon coupling (EPC) decreases the mobility from 104 cm2 V−1 s−1 at low temperature, to 1 cm2 V−1 s−1 at room temperature1,10.

According to the polar-catastrophe model, in a perfect LaAlO3/SrTiO3 interface, the formal polarization (\({P}_{{{{{{\rm{LAO}}}}}}}^{0}\)) allows for an insulator-to-metal transition at the interface beyond a certain thickness (tc) of LaAlO3. Estimating this quantity by considering the energy difference between the valence-band edge of LaAlO3 and the conduction-band edge of SrTiO34,11, one obtains a value of 3.5 unit cells. Higher tc values reported theoretically and experimentally are due to structural relaxations, i.e., polar distortions that induce a polarization opposite to \({P}_{{{{{{\rm{LAO}}}}}}}^{0}\). Such polar distortions that are not considered in the polar-catastrophe model maintain the insulating character of the interface above 3.5 unit cells as confirmed theoretically12,13 and later observed experimentally14. These distortions appear mainly in the LaAlO3 side of the interface and arise due to changes in inter-plane distances between La and Al planes upon interface formation. More recently, a competition between polar and nonpolar distortions through octahedra tilts has been observed15. Interestingly, the dependence of the octahdra tilts on the LaAlO3 thickness can be exploited to tune the functionality of such interface15. We expect fascinating characteristics to occur when involving a polar material with pristine octahedral distortions, such as a perovskite with an orthorhombic primitive cell.

Focusing first on nonpolar candidates as the substrate, cubic BaSnO3 (BSO) has emerged as a most attractive system to overcome the limitations of SrTiO3, as it exhibits extraordinary room-temperature mobilities, reaching 320 cm2 V−1 s−116,17,18,19,20,21. This value is the highest ever measured in a TCO and attributed to the low effective electron mass, as well as to the long relaxation time of the longitudinal optical phonon scattering compared to SrTiO316,22,23. In contrast to other suggested polar materials to be combined with BaSnO3, such as LaScO317, LaGaO318, or LaAlO324, LaInO3 (LIO) has the advantage of being nearly lattice-matched to BaSnO3 and exhibiting a favorable band offset to confine a 2DEG within the BaSnO3 side19,20,21,25,26. Interestingly, it has an orthorhombic structure with tilted InO6 octahedra, thus is ideal for exploring also the interfaces made of tilted and nontilted components.

In a previous work27, it was shown by transmission electron microscopy (TEM) that LaInO3 can be coherently grown on (001) BaSnO3, forming a sharp interface with negligible atomic disorder or misfit dislocations. This characteristic makes such an interface fascinating, since—as reported for LaAlO3/SrTiO38—the interface conductivity and the mobility of the electron gas are enhanced by minimizing the dislocation density. Later it was shown by HRTEM analysis28 that even if the BaO termination of the (001) BaSnO3 surface is most favorable, the LaInO3/BaSnO3 interface (termed LIO/BSO from now on) is characterized by the LaO/SnO2 termination, which is key for the formation of a 2DEG. Thereby, cation intermixing at the interface was rated to be negligible. Therefore, the combination of LaInO3 and BaSnO3 exhibits all the key features and appears superior to LaAlO3/SrTiO3 interfaces for reaching a high mobility 2DEG.

In this work, we explore the characteristics of ideal LIO/BSO interfaces, based on density functional theory (DFT), also employing many-body perturbation theory where needed. We focus only on intrinsic effects that may play a role in compensating the interfacial polar discontinuity, i.e., electronic reconstruction (formation of 2DEG) and possible structural distortions (formation of a depolarization field). Considering first periodic heterostructures, we address the competition between the polar distortions and the 2DEG charge density to compensate the interfacial polar discontinuity, upon increasing the thickness of the polar LaInO3 block. Second, we discuss the impact of the interface termination that may give rise to either a 2DEG or a 2D hole gas (2DHG). Finally, motivated by the advancements in synthesis techniques and in control of nanoscale structures29, we provide a detailed comparison between the characteristics of the 2DEG in a periodic heterostructure and a non-periodic LIO/BSO interface, discussing how one can exploit dimensionality for tailoring the properties of the 2DEG. Overall, our results demonstrate the potential of this material combination for tuning and achieving high electron density and mobility.

Results and discussion

Pristine materials

Before discussing the results for the periodic heterostructures LIO/BSO, we summarize the basic properties of the pristine materials. BaSnO3 crystallizes in the cubic space group Pm\(\overline{3}\)m and is built by alternating neutral (BaO)0 and (SnO2)0 planes along the Cartesian directions, making it a nonpolar material. Its static dielectric constant was estimated to be about 2016. The calculated lattice constant of 4.119 Å obtained with the PBEsol functional is in excellent agreement with experiment16. LaInO3 has an orthorhombic perovskite structure of space group Pbnm, containing four formula units per unit cell. The optimized structural parameters a = 5.70 Å, b = 5.94 Å, and c = 8.21 Å are in good agreement with experimental values30. The corresponding pseudocubic unit cell is defined as to have the same volume per LaInO3 formula as the orthorhombic structure (for more details, see Supplemental Information). Its calculated lattice parameter is about 4.116 Å. In LaInO3, the InO6 octahedra are tilted along the pseudocubic unit cell directions with an a−a−c+ pattern according to the Glazer notation31. Considering the formal ionic charges of La (+3), In (+3), and O (−2), the charged (LaO)+1 and (InO2)−1 planes along the pseudocubic unit cell [100], [010], and [001] directions make LaInO3 a polar material (see Supplementary Fig. 1).

The calculated lattice mismatch between the cubic BaSnO3 and the pseudocubic LaInO3 unit cells is about 0.07%, suggesting a coherent interface as confirmed experimentally27. In the latter work, it was shown that LaInO3 can be favorably grown on top of a (001) BaSnO3 substrate along the three pseudocubic unit cell directions, preserving the polar discontinuity at the interface, regardless of the orientation. Here, we consider the interface formed by [001]-oriented LaInO3 (i.e., c = 8.21 Å corresponds to the out-of-plane direction) on top of the (001) BaSnO3 surface (see Supplementary Fig. 1).

The electronic properties of both bulk BaSnO3 and LaInO3 have been reported by us in previous works32,33, revealing PBEsol band gaps of 0.9 and 2.87 eV, and quasiparticle band gaps of 3.5 and 5.0 eV, respectively, as obtained by the G0W0 correction to the results obtained by the exchange-correlation functional HSE06 (G0W0@HSE06). In both cases, the valence-band maximum is dominated by O-p states, while, the conduction-band minimum has Sn-s and In-s character, respectively.

Stoichiometric periodic heterostructures

In Fig. 1 we show a comprehensive compilation of the electronic properties of the LIO/BSO interface in a periodic heterostructure. Thereby, n-type (LaO/SnO2) and p-type (InO2/BaO) building blocks are periodically replicated in the out-of-plane direction (z). In this case, the system is stoichiometric and ideally suited for investigating the electronic reconstruction due to the polar discontinuity. The BaSnO3 block has a thickness of 10 unit cells which is enough to minimize the interaction with its replica. It is determined by making sure that the central part of the BaSnO3 block behaves like its bulk counterpart (see Supplementary Fig. 2). The LaInO3 block has a thickness of 12 pseudocubic unit cells. (We use the notation LIO12/BSO10 heterostructure in the following). Since the LaInO3 block has different terminations on the two sides, a formal polarization of P0 = e/2a2 = 0.47 C m−2, oriented from the (InO2)−1 to the (LaO)+1 plane, is induced (black arrow in the bottom panel of Fig. 1). As BaSnO3 is a nonpolar material, this gives rise to a polar discontinuity at the interface.

Fig. 1: Stoichiometric periodic heterostructure.
figure 1

The system is formed by LIO12/BSO10 in out-of-plane direction z (bottom), which is shared between panels (a–d) as their horizontal axis. P0 is the formal polarization oriented from the (InO2)−1 to the (LaO)+1 plane. a Local density of states per unit cell (LDOS) where the Fermi level is set to zero (also in panel (e)). The shaded gray area indicates the occupied valence band. The partially unoccupied valence (hole) and the occupied conduction (electron) states, resulting from electronic reconstruction, are highlighted by red and blue color, respectively. The dashed green and pink lines indicate the alignment of the conduction-band edges of BaSnO3 and LaInO3 at the interface. b In-plane averaged electrostatic potential along the z direction. The dashed lines are guides to the eye, highlighting the trend of the potential. c Total polarization per unit cell computed for different LaInO3 thicknesses (in units of the pseudocubic unit cell, PC). d Distribution of the electron (hole) charge densities obtained by integrating over the occupied conduction (unoccupied valence) states shown with blue (red) colored area in panel (a). e Electronic band structure along X-Γ-M. f Density of the 2DEG in electrons per BaSnO3 surface unit cell (e per a2). g Total polarization (\({P}_{{{{{{\rm{total}}}}}}}^{{{{{{\rm{LIO}}}}}}}\), violet) within the LaInO3 block and changes of polar distortions within the LaInO3 (ΔPLIO, magenta) and BaSnO3 (ΔPBSO, green) side as a function of the LaInO3 thickness. The respective orientations are shown in the structural model.

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The charge reconstruction is evident from the electronic band structure (Fig. 1e) showing that the combination of these two insulators has metallic character. From the local density of states (LDOS, per unit cell) depicted in Fig. 1a, along the z direction, we can clearly see that the dipole induced within the LaInO3 side causes an upward shift of the valence-band edge, evolving between the (LaO)+1 and (InO2)−1 terminations. This is also reflected in the in-plane averaged electrostatic potential shown in Fig. 1b (for more details see Supplementary Fig. 9). At the latter termination, the valence-band maximum crosses the Fermi level inside LaInO3, leading to a charge transfer to the BaSnO3 side in order to compensate the polar discontinuity. Consequently, the bottom of the conduction band of BaSnO3 becomes partially occupied, giving rise to a 2DEG confined within three unit cells (~10 Å). A 2DHG forms in the LaInO3 side localized within one pseudocubic unit cell (~4 Å). Integrating over these now partially occupied conduction states (see Supplementary methods), we find that the 2DEG density reaches a value of 2.7 × 1014 cm−2, i.e., ~0.46 e per a2 (a2 being the unit cell area of bulk BaSnO3). Obviously, the same value is obtained for the 2DHG when integrating over the now empty parts of the valence bands. The charge distribution shown in Fig. 1d reveals that the 2DEG is located mainly within the SnO2 plane and exhibits Sn-s character. This highly dispersive s-band suggests high mobility, unlike the situation in LaAlO3/SrTiO3. These results demonstrate the exciting potential of such a material combination as an ideal platform for achieving a high-density 2DEG. We also highlight the importance of the well-confined 2DHG hosted by O-p states in view of p-type conductivity. We note that in SrTiO3/LaAlO3/SrTiO3, another heterostructure34 formed by a polar and a nonpolar material, only recently the presence of a 2DHG has been confirmed experimentally29. The coexistence of high-density well-confined electron and hole gases within one system as shown here, appears as a promising platform for exploring intriguing phenomena such as long-lifetime bilayer excitons35 or Bose–Einstein condensation36.

Polar distortions in stoichiometric periodic heterostructures

The 2DEG density reached in the periodic heterostructure LIO12/BSO10, being slightly lower than 0.5 e per a2, indicates that polar distortions are involved to partially compensate for the polar discontinuity. Looking into its optimized geometry, we find that the tilt of the octahedra decreases gradually from the LaInO3 to the BaSnO3 side (see Fig. 1, bottom panel). Consequently, the out-of-plane lattice spacing increases close to the interface by about 3% (see Supplementary Fig. 2), in good agreement with an experimental observation26. We also find that the out-of-plane displacements of the inequivalent O atoms are not the same within all octahedra (see Supplementary Figs. 2 and 3). Moreover, the distances between AO and BO2 planes (A = Ba, La and B = Sn, In) across the interface are also unequal (see Supplementary Fig. 2). Using a simple linear approximation for the local polarization based on Born effective charges (Z*)37,38 of the atomic species (calculated for the pristine materials), we obtain a qualitative trend of the out-of-plane polarization induced by such structural distortions (see Supplementary discussion). We note that due to the tilt of the octahedra, calculating the polarization for such a heterostructure is less straightforward than for, e.g., LaAlO3/SrTiO3. We find that the structural distortions within the LaInO3 side induce a polarization ΔPLIO that counteracts the formal polarization P0. Moreover, the polar discontinuity at the interface is reduced by structural distortions within the BaSnO3 side, inducing ΔPBSO that is parallel to P0. The total polarization within the LaInO3 side, \({P}_{{{{{{\rm{total}}}}}}}^{{{{{{\rm{LIO}}}}}}}\), shown in Fig. 1g is the sum of −ΔPLIO and P0. As expected for the particular case of LIO12/BSO10, the average polarization within LaInO3 (\({P}_{{{{{{\rm{total}}}}}}}^{{{{{{\rm{LIO}}}}}}}\)) is smaller than P0 due to partial compensation by structural distortions. For this reason, the 2DEG density mentioned above is smaller than 0.5 e per a2. For better grasping of the polar discontinuity at the interface, we plot the total polarization along the z direction (see Fig. 1c). Focusing first on the particular case of LIO12/BSO10, we can clearly see that within the LaInO3 side, it is smaller than P0 and also non-negligible inside BaSnO3, making the polar discontinuity at the interface less pronounced. As we provide only a qualitative analysis of the polarization, we do not expect a full match between the values of the 2DEG and the corresponding polarization discontinuity. However, the obtained trend of the polarization strength is valuable to understand and explain the relationship between the calculated 2DEG density and the LaInO3 thickness.

Impact of the LaInO3 thickness in stoichiometric periodic heterostructures

Now, we fix the thickness of the BaSnO3 building block to 10 unit cells and vary that of LaInO3 between 2 and 12 pseudocubic unit cells, labeling the systems LIOm/BSO10 (m = 2, 4, 6, 8, 10, 12). Before discussing the results, we note that the critical thickness, tc, for an insulator-to-metal transition at the interface is about one pseudocubic LaInO3 unit cell, when ignoring effects from structural relaxation as given by the polar-catastrophe model4,11. This value is obtained as \({t}_{{{{{{\rm{c}}}}}}}={\epsilon }_{0}{\epsilon }_{{{{{{\rm{LIO}}}}}}}{{\Delta }}E/e{P}_{{{{{{\rm{LIO}}}}}}}^{0}\). Here, ϵLIO ~ 24 is the relative static dielectric constant of LaInO330,39, and ΔE = 1 eV represents the energy difference between the valence-band edge of LaInO3 and the conduction-band edge of BaSnO3 at the interface, obtained by the PBEsol functional (see Fig. 1a). In contrast to the unrelaxed LIO2/BSO10 system, where we find metallic character, with full electronic reconstruction of 0.5 e per a2 (see Supplementary Fig. 4), atomic relaxations lead to semiconducting character, i.e., no formation of 2DEG (see Fig. 1f). We observe that the polar distortions that counteract the formal polarization P0, hamper the electronic reconstruction at the interface and, thus, the formation of a 2DEG, up to a critical thickness of 4 pseudocubic LaInO3 unit cells (see Fig. 1f). This result highlights the importance of structural relaxations for compensating the polar discontinuity and stabilizing the interface.

Focusing on the electronic charge, we find that at a thickness of 4 pseudocubic LaInO3 unit cells, the density of the 2DEG is only about 0.03 e per a2 (see Fig. 1f), i.e., distinctively smaller than the nominal value of 0.5 e per a2. On increasing the LaInO3 thickness, the 2DEG density increases progressively and reaches a value of ~0.46 e per a2 with 12 pseudocubic LaInO3 unit cells. This means that the polar distortions are also non-negligible beyond the critical thickness. In Fig. 1g, we display the averages of ΔPLIO and ΔPtotal for the considered structures LIOm/BSO10, finding that the polar distortions (total polarization) is maximal (minimal) at two pseudocubic LaInO3 unit cells and decreases (increases) with LaInO3 thickness. ΔPBSO increases with LaInO3 thickness, but it is smaller than its counterpart in LaInO3. As a result, the polar discontinuity at the interface increases with increasing LaInO3 thickness (see Fig. 1c). Hence, the 2DEG density increases accordingly in order to compensate it (see Fig. 1f).

For a more reliable estimation of the critical thickness of LaInO3 for an insulator-to-metal transition, we evaluate ΔE by considering the quasiparticle band gaps of the constituents32,33. Applying a scissor shift to the PBEsol band offset leads to a quasiparticle value of ΔE = 3.4 eV, in agreement with the band offset reported in ref. 20 (see Supplementary discussion). Thus, we obtain a tc = 4 pseudocubic LaInO3 unit cell which is increased by 3 pseudocubic unit cells compared to that given by PBEsol offset (one pseudocubic unit cell). For the relaxed structures, we estimate tc to be about seven pseudocubic LaInO3 unit cells when considering quasiparticle band gaps, which is distinctively larger than the 4 pseudocubic unit cells derived from PBEsol (see Fig. 1f). This result indicates that a thick LaInO3 component is needed to reach high 2DEG densities in periodic heterostructures, as both sides contribute to the compensation of the polar discontinuity through atomic distortions. This result is in line with a previous theoretical discussion reported for oxide interfaces13.

Proceeding now to the nature of the atomic distortions, we find that, in contrast to the LaAlO3/SrTiO3 interface where the unequal distances between La and Al planes dominate14, the unequal displacements of the inequivalent oxygen atoms are most decisive for the polar distortions in the LaInO3 side of the interface (see Supplementary Fig. 2). This behavior is governed by the gradual tilts of octahedra across the interface which facilitates the compensation of the polar discontinuity. This indicates that below the critical thickness this compensation happens through such atomic distortions, rather than elimination by ionic intermixing or other defects, explaining the sharp interface and negligible intermixing observed experimentally27. The latter characteristic is crucial for achieving a high-density 2DEG beyond tc. The band offset at the interface shows that the conduction-band minimum of BaSnO3 is about 1.4 eV below that of LaInO3, confining the 2DEG at the BaSnO3 side (see Fig. 1a).

Non-stoichiometric nn-type periodic heterostructure

By adopting thicker polar building blocks, i.e., above 12 pseudocubic unit cells, the interaction between the n-type and the p-type interfaces is prevented. Due to the considerable computational costs, several models were proposed to predict the characteristics of such situations in oxide interfaces13. One of them is to consider non-stoichiometric structures, where the polar LaInO3 block is terminated by a (LaO)+1 plane on both sides. In this case, termed nn-type periodic heterostructure, the system is self-doped as the additional (LaO)+1 layer serves as a donor. As the LaInO3 building block is symmetric, the formal polarization P0 is induced from the middle of the slab outwards on both sides, which compensate each other. In this way, the built-in potential inside the LaInO3 is avoided, while the discontinuity at the LIO/BSO interface is preserved. To this end, we consider a periodic heterostructure formed by 10 BaSnO3 unit cells and 10 pseudocubic LaInO3 unit cells, which is large enough to minimize the interaction between the periodic n-type interfaces (see Fig. 2 (bottom) and Supplementary Fig. 6).

Fig. 2: Non-stoichiometric periodic nn-type heterostructure.
figure 2

The system is formed by LIO10/BSO10 in out-of-plane direction z (bottom), which is shared between panels (a–d) as their horizontal axis. The LaInO3 block is terminated by a (LaO)+1 plane on both sides. a Local density of states per unit cell (LDOS). The Fermi level is set to zero, and the electron population of the (original) conduction bands is shown as shaded blue area (also in panel (e)). The dashed green and pink lines indicate the alignment of the conduction-band edges of the two materials at the interface. b In-plane averaged electrostatic potential along the z direction. c Total polarization per unit cell. d Distribution of the electron charge density obtained by integrating over the occupied conduction states indicated by the blue area in panel (a). e Electronic band structure along X-Γ-M. f Band alignment as obtained by using the band gaps of the bulk systems (PBEsol, top and G0W0@HSE06, bottom), considering the energy difference between the potential of the bulk and the periodic heterostructure as shown in panel (b). The alignment at the interface obtained by PBEsol can be seen in panel (a).

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The electronic band structure, obtained by PBEsol, shows that this system has metallic character, where the partial occupation of the conduction band amounts to 1 e per a2 (see Fig. 2e). The corresponding effective electron mass, being 0.24 me, is quite low compared with that of LaAlO3/SrTiO3 interfaces (0.38 me34), and suggests a high electron mobility. This value is close to that found for pristine BaSnO3 (0.17 me obtained by PBEsol, 0.2 me by G0W0)32. In pristine SrTiO3, a transition from band-like conduction (scattering of renormalized quasiparticles) to a regime governed by incoherent contributions due to the interaction between the electrons and their phonon cloud has been reported upon increasing temperature40. In BaSnO3, the relaxation time for the longitudinal optical phonon scattering is found to be larger compared to SrTiO3, contributing to the high room-temperature mobility reported for the La-doped BaSnO3 single crystals23. Based on this, a high room-temperature mobility is also expected at the here-investigated interfaces as this material combination basically preserves the structure of the pristine BaSnO341. Overall, significant polaronic effects are not expected. In both constituents, we find typical EPC effects on the electronic properties32,33, i.e., a moderate renormalization of the band gap by zero point vibrations and temperature. Given the excellent agreement between theory and experiment that can explain all the features of the optical spectra32,33, we conclude that polaronic distortions do not play a significant role. Thus, we do not expect a dramatically different behavior at the interfaces.

Before analyzing the spatial charge distribution, we note that in a pristine symmetric LaInO3 slab, the electronic charge accumulates on its surfaces, accompanied by structural distortions that tend to screen the discontinuity of the polarization. Combined with BaSnO3, the polar distortions vanish at the LaInO3 side but appear in the BaSnO3 building block and are accompanied by a charge redistribution (see Fig. 2a, c). In Fig. 2c, we display the polarization induced by the structural distortions along the slab. We find that the gradual decrease of the octahedra tilt from the LaInO3 to the BaSnO3 side enlarges the out-of-plane lattice spacing at the interface and enhances the unequal oxygen-cation displacements within the BaSnO3 side (see Supplementary Fig. 6). This, in turn, induces a gradually decreasing polarization (ΔPBSO) oriented away from the interface. Due to this local dipole, the BaSnO3 conduction-band edge is shifted downwards near the interface, leading to partial charge redistribution (see Fig. 2a, d), and being also evident in the in-plane averaged electrostatic potential (Fig. 2b). Integrating the LDOS over the partially occupied conduction states, we find that the 2DEG is confined within three unit cells, amounting to 0.5 e per a2 at each side (see Fig. 2d). Thereby the 2DEG is mainly hosted by Sn-s states within the SnO2 planes. The valence and conduction-band edges at the LaInO3 side are almost the same in all pseudocubic unit cells as the polar distortions are negligible (only nonpolar tilts) (see Fig. 2a, c). These results are in line with the trend of the polarization and the 2DEG density obtained for the stoichiometric periodic system. While in the latter, we observed decreasing (increasing) structural distortions in the LaInO3 (BaSnO3) side as a function of LaInO3 thickness (see Fig. 1g), here the polar distortions vanish (are enhanced) at the LaInO3 (BaSnO3) side of the interface. Consequently, the conduction-band minimum of the BaSnO3 side is lowered by about 1.35 eV with respect to that of LaInO3 (see Fig. 2a). This value is also in line with that estimated for the stoichiometric periodic heterostructure LIO12/BSO10 (~1.4 eV). Such an offset is crucial for confining the 2DEG in order to reach a value of 0.5 e per a2 (see Fig. 2a, d).

Note that here we can determine the band offsets by considering the respective quasiparticle gaps of the pristine materials32,33, as well as an alternative approach based on the electrostatic potential42 (see Fig. 2b and Supplementary discussion). We find that the conduction-band offset between the middle of the BaSnO3 and LaInO3 blocks is almost the same when using quasiparticle or PBEsol band gaps (see Fig. 2b, f). Thus, we conclude that PBEsol is good enough to capture band offset and charge distribution at the interface. The latter conclusions are confirmed by calculations using HSE06 for a smaller (feasible) nn-type system (see Supplementary discussion).

Non-stoichiometric pp-type periodic heterostructure

Calculations for a periodic pp-type heterostructure (see Supplementary Fig. 5) with (InO2)−1 termination on both sides of LaInO3 reveal a 2DHG with a density of 0.5 e per a2. Interestingly, the hole stays at the LaInO3 side, confined within one pseudocubic unit cell. Like in the periodic nn-type heterostructure, the tilt of the octahedra decreases gradually from the LaInO3 to the BaSnO3 side. However, the BaO termination in the pp-type case favors nonpolar distortions within the BaSnO3 side, i.e., equal displacements of the inequivalent O atoms, while polar distortions appear only in the LaInO3 side. Consequently, local dipoles are induced in the latter, pushing up its valence-band edge above that of BaSnO3 (see Supplementary Fig. 5). Hence, the 2DHG exhibiting O-p character stays on the LaInO3 side of the interface. It has been shown recently by a combined theoretical and experimental investigation28 that the n-type interface is more favorable than the p-type interface. Since, at the BaO-terminated p-type interface, the 2DHG stays within the LaInO3 side, it contributes less to the compensation of the interfacial polar discontinuity. In contrast, the SnO2-terminated n-type interface allows for electronic charge transfer to the BaSnO3 side, forming a 2DEG that compensates the interfacial polar discontinuity more efficiently (see Supplementary discussion).

Stoichiometric non-periodic interface

Now, we investigate the case of a non-periodic LIO/BSO interface, consisting of a thin LaInO3 layer on top of a (001) BaSnO3 substrate. Considering stoichiometric systems, we only focus on the n-type interface as it is predicted to be more favorable28. For BaSnO3, we find that 11 unit cells are enough to capture the extension of the structural deformations and the 2DEG distribution away from the interface. We then vary the thickness of LaInO3 between one and eight pseudocubic unit cells, labeling the systems as LIOn/BSO11 (n = 1, 2, 4, 6, 8). In Fig. 3 (bottom panel), we show the optimized geometry of LIO8/BSO11. The BaSnO3 substrate terminates with a (SnO2)0 plane at the interface, and the surface termination of LaInO3 is a (InO2)−1 plane. Thus, LaInO3 is stoichiometric and has a formal polarization of P0 = 0.47 C m−2, oriented from the surface to the interface, as in the stoichiometric periodic heterostructures discussed above.

Fig. 3: Stoichiometric non-periodic interface (here termed IF).
figure 3

The system is formed by LIO8/BSO11 in out-of-plane direction z (bottom), which is shared between panels (a–d) as their horizontal axis. P0 is the formal polarization oriented from the (InO2)−1 plane towards the (LaO)+1 plane at the interface. a Local density of states per unit cell (LDOS) with the Fermi level being set to zero (also in panel (e)). The shaded gray area indicates the occupied valence states. The depleted valence-band region (holes) and the occupied conduction-band region (electrons) resulting from electronic reconstruction, are highlighted by red and blue color, respectively. The dashed green and pink lines indicate the alignment of the conduction-band edges of the two materials at the interface. b In-plane averaged electrostatic potential along the z direction. c Total polarization per unit cell computed for different LaInO3 thicknesses (in units of the pseudocubic unit cell, PC). d Distribution of the electron (hole) charge densities obtained by integrating the LDOS indicated by blue (red) color in panel (a). For comparison, the 2DEG distribution in the periodic heterostructure LIO12/BSO10 (here termed HS) is shown by the shaded gray area. e Electronic band structure along X-Γ-M. f Density of the 2DEG in e per a2 in the non-periodic (blue) and periodic (gray) heterostructures. g Total polarization (\({P}_{{{{{{\rm{total}}}}}}}^{{{{{{\rm{LIO}}}}}}}\), violet) within the LaInO3 side and changes of polar distortions within LaInO3 (ΔPLIO, magenta) and BaSnO3 (ΔPBSO, green) side as a function of LaInO3 thickness. The respective orientations are shown in the structural model.

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Electronic reconstruction, which leads to a metallic character, is evident from the resulting band structure, where the valence and conduction bands overlap at the Γ point (see Fig. 3e). In the LDOS, we clearly see that the dipole induced within LaInO3 causes an upward shift of the valence-band edge (most pronounced at the surface) which is also evident in the in-plane averaged electrostatic potential (see Fig. 3a, b). At the surface, the valence-band edge crosses the Fermi level, leading to a charge transfer across the interface that counteracts the polar discontinuity (see Fig. 3a). Consequently, the bottom of the conduction band becomes partially occupied within the BaSnO3 side, giving rise to a 2DEG that is confined within five unit cells (~20 Å) (see Fig. 3d). The 2DHG formed at the surface is localized within one pseudocubic LaInO3 unit cell (~4 Å). In this case, the conduction-band minimum of the BaSnO3 building block is about 1.7 eV below that of LaInO3 (Fig. 1a), in agreement with an experimental observation of 1.6 eV20. Integrating the LDOS over the region of these partially filled states, we find that the 2DEG density (and likewise the 2DHG density) reaches a value of 2.9 × 1014 cm−2, i.e., ~ 0.49 e per a2.

By increasing the LaInO3 thickness from one to eight pseudocubic unit cells, the polar distortions (ΔPLIO) decrease, being accompanied by an electronic charge transfer from the surface to the interface (see Fig. 3f, g). Compared to the periodic heterostructure (Fig. 1c), they are less pronounced at the BaSnO3 side (Fig. 3c). This enhances the polar discontinuity (see Supplementary Fig. 8), and thus the 2DEG density that reaches a higher value than in the periodic systems with similar LaInO3 thickness (see Fig. 3f). Focusing on the charge confinement, the conduction-band edge in the BaSnO3 side is gradually shifted up when moving away from the interface as ΔPBSO is less pronounced. This allows an extension of the 2DEG up to five unit cells (~20 Å) in the BaSnO3 (substrate) compared to three unit cells in the periodic heterostructure case (see Fig. 3d). The enhanced polar discontinuity reduces the critical thickness to only two pseudocubic LaInO3 unit cells compared to four found for the periodic heterostructure, when relying on PBEsol (see Fig. 3f). Based on the quasiparticle band gaps, however, we estimate tc to be five pseudocubic LaInO3 unit cells in the non-periodic system compared to seven in the periodic case. Overall tuning of the 2DEG charge density through the LaInO3 thickness is possible, and remarkably, also the type of heterostructure (i.e., periodic or non-periodic) impacts its spatial distribution. Both aspects can be exploited to tune the characteristic of the 2DEG.

In view of realistic applications, it is worth considering the results presented here in the context of existing experimental research on LIO/BSO interfaces. A main challenge here is the quality of the BaSnO3 substrate43. Previous experimental works20,26 investigated a field effect transistor, formed by a LaInO3 gate and a La-doped BaSnO3 channel on a SrTiO3 substrate. An enhancement of conductance with increasing LaInO3 thickness was observed, but there was no indication of a critical thickness for an insulator-to-metal transition at the interface. A maximal 2DEG density of only 3 × 1013 (0.05 e per a2) was reported for 4 pseudocubic LaInO3 unit cells, and a decrease beyond it. We assign the differences to our predictions for non-periodic interfaces mainly to the high density of structural defects (e.g., dislocations) due to the large mismatch (~5%) between the channel and the substrate. The La doping, needed to compensate the acceptors induced by such dislocations, may cause an alleviation of the polar discontinuity at the interface and, hence, limit the 2DEG density. On the other hand, it makes the system metallic without a clear critical thickness. With the recent advances in achieving high-quality BaSnO3 and LaInO3 single crystals30,33,43, as well as interfaces27,28, our predictions open up a perspective for exploring interfacial charge densities in combinations of these materials in view of potential electronic applications.

Conclusions

In summary, we have presented the potential of combining nonpolar BaSnO3 and polar LaInO3 for reaching a high interfacial carrier density. Our calculations show that, depending on the interface termination, both electron and hole gases can be formed that compensate the polar discontinuity. The gradual decrease of octahedra tilts from the orthorhombic LaInO3 to the cubic BaSnO3 side increases the out-of-plane lattice spacing at the interface and governs the unequal oxygen-cation displacements within the octahedra. The latter distortions induce a depolarization field, counteracting the formal polarization in the LaInO3 block and hampering electronic reconstruction, i.e., the formation of a 2DEG at the interface up to a critical LaInO3 thickness of seven (five) pseudocubic unit cells for periodic (non-periodic) heterostructures. While the PBSEsol functional provides a good description of the interfacial charge-density distributions as well as the type of band offset, it fails to determine tc reliably, as the knowledge of the quasiparticle gaps of the pristine materials is required. The polar distortions (polar discontinuity) decrease (increase) with LaInO3 thickness, leading to a progressive charge transfer until reaching a 2DEG density of 0.5 e per surface unit cell. The electronic charge density is hosted by a highly dispersive Sn-s-derived conduction band, suggesting a high carrier mobility. Overall, the 2DEG charge density can be tuned through the LaInO3 thickness. Interestingly, also the type of interface (i.e., periodic or non-periodic heterostructure) strongly impacts its density and spatial confinement. All these effects can be exploited in view of tailoring the characteristics of the 2DEG.

Methods

Theory

Ground-state properties are calculated using DFT, within the generalized gradient approximation (GGA) in the PBEsol parameterization44 for exchange-correlation effects. All calculations are performed using FHI-aims45, an all-electron full-potential package, employing numerical atom-centered orbitals. For all atomic species, we use a tight setting with a tier 2 basis set for oxygen, tier1 + fg for barium, tier1 + gpfd for tin, tier 1 + hfdg for lanthanum, and tier 1 + gpfhf for indium. The convergence criteria are 10−6 electrons for the density, 10−6 eV for the total energy, 10−4 eV Å−1 for the forces, and 10−4 eV for the eigenvalues. Lattice constants and internal coordinates are optimized for all systems until the residual forces on each atom are less than 0.001 eV Å−1. The sampling of the Brillouin zone is performed with an 8 × 8 × 8 k-grid for bulk BaSnO3 and a 6 × 6 × 4 k-grid for bulk LaInO3. These parameters ensure converged total energies and lattice constants of 8 meV per atom and 0.001 Å, respectively. The Born effective charges in pristine systems are computed using exciting code46 (see supplementary discussion).

For the heterostructures, a 6 × 6 × 1 k-grid is used. The in-plane lattice parameters are fixed to \(\sqrt{2}{a}_{{{{{{\rm{BSO}}}}}}}\) (aBSO being the bulk BaSnO3 lattice spacing) (see Supplementary Fig. 1). For the non-periodic systems, vacuum of about 150 Å is included and a dipole correction is applied in the [001] direction in order to prevent unphysical interactions between neighboring replica. In this case, we fix the first two BaSnO3 unit cells to the bulk structure to simulate the bulk-like interior of the substrate, and relax the other internal coordinates. For computing the electronic properties, a 20 × 20 × 1 k-grid is adopted for all systems. This parameter ensures converged densities of states and electron/hole charge densities up to 0.01 e per a2. Atomic structures are visualized using the software package VESTA47.