Introduction

Two-dimensional electron gas (2DEG) can be generated at the interface of heterojunction constructed by two insulating perovskite materials. The 2DEG has many unique electronic properties, such as ferroelectric polarization enhancement1, high carrier mobility2, two-dimension superconductivity3, magnetism at the interface4, electronic spin polarization5,6, and so on. With the rapid development of thin film growth technology including molecular beam epitaxy7, chemical vapor deposition8, and pulsed laser deposition9, the nature and functionalities of 2DEG draw great attention from both experimental and theoretical points of view.

Generally, there are two primary mechanisms to be involved in the formation of 2DEG at the heterojunction interface: oxygen vacancies10 and electronic reconstruction11. A prominent experiment is LaAlO3/SrTiO3 system, in which the LaAlO3 slab is grown on the SrTiO3 substrate12. It is well known that a metallic phase is confined within several monolayers near the LaO/TiO2 interface and therefore formed a 2DEG. The LaAlO3 slab consists of alternating charged layers LaO and AlO2, while the SrTiO3 substrate can be considered a stack of alternating neutral layers SrO and TiO2 along the (001) direction. The polarization discontinuity at the LaO/TiO2 interface causes the electrostatic potential divergence. To avoid the polar discontinuity, half an electron is transferred from the charged layer LaO to the neutral layer TiO2 11, resulting in a metallic phase at the interface of the LaAlO3/SrTiO3 system.

It is found that the 2DEG may even become magnetic at low temperature by Brinkman et al.13, as generated at the LaO/TiO2 interface between the non-magnetic LaAlO3 and SrTiO3 materials. This interesting phenomenon has the origin of the induced electrons exchange splitting in the Ti-3d orbital14, which is confirmed by the first principles calculations with spin-polarization. Formation of the completely spin-polarized 2DEG is extremely promising for spintronics applications. From the previous work of Nazir et al.15, LaAlO3/BaTiO3 model shows higher interfacial charge carrier density, stronger quantum confinement, and larger magnetic moment than the previously reported SrTiO3-based model superlattices. In this work, we carry out first principles calculations of the LaAlO3/BaTiO3 system with a special emphasis of oxide thickness control. As a matter of fact, previous experiments16 of LaAlO3/BaTiO3 superlattices have been realized with 20 periodic numbers and varied stacking periodicity of 2/2 (2 unit cells LaAlO3/2 unit cells BaTiO3), 3/3, 4/4, 5/5 and 8/8.

BaTiO3 is a tetragonal perovskite-type structure at the room temperature, when the temperature is above 393 K, the bulk BaTiO3 changes from the tetragonal to cubic structure with a lattice constant of 4.012 Å17, band gap of 3.2 eV18, and Curie temperature of 415 K19. LaAlO3 is the other building block for the LaAlO3/BaTiO3 model and it can be used as substrate for growing many lattice-mismatched perovskite materials. At room temperature, LaAlO3 has a perovskite-like structure with a slight rhombohedral distortion, at high temperature LaAlO3 also has the cubic perovskite structure with a lattice constant of 3.789 Å, band gap of 5.6 eV12, and it has a lattice mismatch of 5.56% with BaTiO3 material.

In this work, for the first-principles electronic structure calculations, we first build two kinds of LaAlO3/BaTiO3 superlattices with different interfaces. For example, in the (LaAlO3)4.5/(BaTiO3)4.5 superlattice, we find that the LaO/TiO2 interface is the more preferential configuration due to its highest cleavage energy, compared with the BaO/AlO2 interface15,20,21,22,23. Furthermore, we show that the LaAlO3/BaTiO3 superlattices with LaO/TiO2 interface could produce the spin-polarized 2DEG, and the completely spin-polarized 2DEG is strongly confined at the interfacial TiO2 layer when the thicknesses of LaAlO3 and BaTiO3 layers are both larger than 4.5 unit cells.

By means of the analysis of the layer-resolved partial density of states (DOS), we discovered that the mechanism for the spin-polarized 2DEG formation is similar to the LaAlO3/SrTiO3 systems11. Half an electron transfers from the LaO layer to the TiO2 layer in the interface of non-stoichiometric LaAlO3/BaTiO3 superlattices, in which two symmetrical and identical LaO/TiO2 interfaces are considered in the computation24. The spin-polarized 2DEG contributed by the interfacial Ti 3dxy orbitals is found to be tightly confined at the interfacial TiO2 monolayer, corresponding to the theoretically computed charge carrier density of 2.25 × 1014 cm−2 in LaAlO3/BaTiO3 superlattice when the thicknesses of LaAlO3 and BaTiO3 are both greater than 4.5 unit cells. Whereas we constructed the superlattice (SrTiO3)4.5/(LaAlO3)4.5, the calculated two-dimensional electron gas density at the interface is 1.48 × 1014 cm−2 (see Fig. S7 in Supplemental Information) which is significantly lower than our LaAlO3/BaTiO3 superlattice. In order to qualitatively analyze the effect of the thicknesses of LaAlO3 and BaTiO3 layers on the properties of interfacial transport characteristics, we calculated the relative interfacial charge carrier mobility and conductivity by using the (LaAlO3)2.5/(BaTiO3)2.5 superlattice as reference. Our results suggest that, when the thicknesses of LaAlO3 and BaTiO3 layers become larger than 4.5 unit cells, the mobility and conductivity are converged to the above-mentioned value. Therefore, our work gives theoretical guidance for generating superior spin-polarized 2DEG in LaAlO3/BaTiO3 superlattice, and enhancing the interfacial transport properties of the charge carrier through adjusting the thicknesses of LaAlO3 and BaTiO3 layers.

We also consider carefully the effect of ferroelectric BaTiO3 on the BaTiO3/LaAlO3 superlattices. In detail, we constructed tetragonal BaTiO3 with lattice constants a = b = 3.9945 Å and c = 4.0335 Å for the different (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5 to 8.5). Note that after relaxation the initial uniaxial ferroelectric structure transforms to two opposite polar modes pointing towards the bulk inside BaTiO3 (see Fig. S1 in Supplemental Information), and a tiny residual polarization of about 2.9 µC/cm2 which points toward the direction of initial ferroelectric polarization we set in the perovskite BaTiO3. This means that our mirror symmetry is not exact and the superlattice presents a pseudo-ferroelectricity. In order to avoid the energy-consuming charged domain wall in the middle of BaTiO3 the polarization diminishes (see Fig. S1). Two separate bands are found near the conduction band minima which were actually almost degenerate in the exact mirror-symmetric BaTiO3 case (see Fig. S2 in Supplemental Information).

Results

Interface energetics

Periodic boundary conditions are applied when constructing (LaAlO3)n/(BaTiO3)n superlattices along (001) direction25, where n is the number of unit cells of LaAlO3 and BaTiO3 (n = 2.5 to 8.5)26,27. Both non-stoichiometric LaAlO3 and BaTiO3 slabs have two distinct kinds of terminal surfaces28, the LaAlO3 slab has LaO and AlO2 terminal surface, while the BaTiO3 substrate has BaO and TiO2 terminal surface. Thus, we build two kinds of symmetric interface for (LaAlO3)n/(BaTiO3)n superlattices, including LaO/TiO2 or BaO/AlO2 interfaces, as shown in Fig. 1. In the supercell we have two interfaces, namely the left/right side interface and the middle interface, respectively. The index n indicates the LaAlO3 (BaTiO3) unit cells counting from the middle interface location.

Figure 1
figure 1

Atomic structures of the supercells with the (a) LaO/TiO2, (b) BaO/AlO2 interfaces in (LaAlO3)8.5/(BaTiO3)8.5 superlattice.

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In order to compare the interface stability for these two types of interfaces, we introduced the cleavage energy15,20,21,22,23 Ecleav which is defined as:

$${{\rm{E}}}_{{\rm{cleav}}}=\,({{\rm{E}}}_{{\rm{LAO}}}+{{\rm{E}}}_{{\rm{BTO}}}-{{\rm{E}}}_{S})/{\rm{2A}}$$
(1)

where ES is the total energy for the superlattices containing two kinds of perovskite compositions that are LaAlO3 and BaTiO3; ELAO and EBTO represent the energy of LaAlO3 and BaTiO3 slab systems, which have the same superlattice lattice constant with LaAlO3/BaTiO3 supercell, but BaTiO3 or LaAlO3 slabs are replaced by the vacuum of same size. The area of an interface in the superlattices is defined as A, and the factor of 2 in the formula indicates two symmetric interfaces in the superlattice models. The physical meaning of cleavage energy is the energy needed to decompose the (LaAlO3)n/(BaTiO3)n superlattices into two sections, and thus, the numerical value of the cleavage energy determines the compactness of the interfacial cohesion between the LaAlO3 and BaTiO3 slabs, which can describe the thermodynamic stability of the interface21,22,29.

We have calculated cleavage energy Ecleav for (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5 to 8.5) with LaO/TiO2 and BaO/AlO2 interfaces, as listed in Table 1. Overall, it can be found that cleavage energy of LaO/TiO2 interface is higher than BaO/AlO2 interface for the same index n. This means that the superlattices with LaO/TiO2 interface are generally more stable than BaO/AlO2 interface. With the increase of the parameter n, it is clearly noted that cleavage energy decreased from 0.117 eV/Å2 to 0.066 eV/Å2 for BaO/AlO2 interface and decreased from 0.189 eV/Å2 to 0.127 eV/Å2 for LaO/TiO2 interface. This suggests that with the increase of the superlattice size, the interface becomes less stable.

Table 1 Calculated cleavage energy (eV/Å2) for the (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5 to 8.5) with the LaO/TiO2 and BaO/AlO2 interfaces.
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2DEG in LaAlO3/BaTiO3 superlattices

To investigate the interface type induced electronic properties modification for the (LaAlO3)n/(BaTiO3)n superlattices, we calculated the total DOS for the LaO/TiO2 and BaO/AlO2 interfaces for (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5, 4.5, 6.5, and 8.5), as shown in Fig. 2. It can be seen that spin-polarized 2DEG is generated in the superlattices with LaO/TiO2 interface. Like the classical model LaAlO3/SrTiO3, spin-polarized 2DEG is seen at the LaO/TiO2 interface, as reported by Nazir et al.30. Making 2DEG spin-polarized is a very important for spintronics applications, where the regulation the atomic layer types at the interface enlarges the spectrum of potential applications. Experimental scientists can control the interface types by inserting atomic layers, using feasible and widely applied techniques. For example, Yajima et al. successfully controlled the SrRuO3/Nb:SrTiO3 Schottky heterojunction by inserting an artificial interface dipole31, and Chen et al. enhanced the mobility of 2DEG through inserting a single-unit-cell layer of La1−xSrxMnO3 at the interface32. Inspection of Fig. 2 indicated that the electronic states in the model LaO/TiO2 are half-metallic and do not change much in the energy range from −1.4 eV to Fermi level when the index n ≥ 4.5.

Figure 2
figure 2

Calculated spin-resolved total density of states (DOS) for the (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5, 4.5, 6.5, and 8.5) with the LaO/TiO2 (left column) and BaO/AlO2 (right column) interfaces. The red and blue lines show the majority-spin and minority-spin, respectively. The vertical lines at 0 eV indicate the Fermi level in all DOS plots.

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It is important to reveal the origin of spin-polarized 2DEG in superlattices (n = 2.5 to 8.5), we thus plot the layer-resolved partial DOS along with the spin charge density for (LaAlO3)2.5/(BaTiO3)2.5 superlattice, as shown in Fig. 3(a). We use IF-1, IF-3, IF-5, and IF-7 to represent the first, third, fifth, and seventh TiO2 monolayers. The 2DEG originates from Ti 3d states in the IF-1 and IF-3 TiO2 monolayers, while the LaAlO3 slab does not contribute to the 2DEG in the (LaAlO3)2.5/(BaTiO3)2.5 superlattice, demonstrating that the electrons provided by LaO monolayer are shared by Ti atoms in all TiO2 layers. In other words, the spin-polarized 2DEG may diffuse from the interface to the inside of the BaTiO3. The orbital-resolved partial DOS figures, as shown in Fig. 3(b), suggest that the half-metallic states at IF-1 TiO2 monolayer are mostly from Ti 3dxy orbitals, Ti 3dyz orbitals and Ti 3dxz orbitals only give a little contribution. While at IF-3 TiO2 monolayer, different from IF-1, most of the half-metallic states come from Ti 3dyz orbitals, partially from Ti 3dxz orbital and none from Ti 3dxy orbital.

Figure 3
figure 3

(a) Calculated layer-resolved partial DOS for the (LaAlO3)2.5/(BaTiO3)2.5 superlattice in the range from −4.0 to 2.0 eV, with the spin density corresponding to each of the oxide monolayers. (b) The orbital-resolved partial DOS for Ti atom at the first (IF-1) and the third (IF-3) TiO2 monolayers.

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Now let us look at the distribution of the 2DEG and the contribution of Ti atomic orbitals as the thickness parameter n increases, hence the layer-resolved partial DOS along with the spin charge density for (LaAlO3)6.5/(BaTiO3)6.5 superlattice is shown in Fig. 4(a). It is noted that nearly all the half-metallic states are contributed by the interfacial TiO2 monolayer, and the contributions from the IF-3, IF-5, and IF-7 TiO2 layer are negligible. This clearly indicates that the spin-polarized 2DEG is tightly confined at the interfacial TiO2 layer.

Figure 4
figure 4

(a) Calculated layer-resolved partial DOS for (LaAlO3)6.5/(BaTiO3)6.5 model in the range from −4.0 to 2.0 eV, together with the spin density projected on to each of the oxide monolayers. (b) The orbital-resolved DOS for Ti atom at the first (IF-1), third (IF-3), fifth (IF-5), and seventh (IF-7) TiO2 monolayers, respectively.

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The mechanism for 2DEG formation is similar to the LaAlO3/SrTiO3 systems, in the (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5 to 8.5) with LaO/TiO2 interface, an “extra” electron is injected into the supperlattice as a result of an electron located at the additional LaO layer24. In order to analyze the contribution of the Ti 3d orbitals to the 2DEG, we plot the orbital-resolved partial DOS in Fig. 4(b). The spin-polarized 2DEG located at IF-1 TiO2 monolayer is entirely contributed by Ti 3dxy orbitals, and there is no obvious 2DEG in other TiO2 monolayers near the Fermi level. Briefly, we find that the spin-polarized 2DEG is tightly confined at the interfacial TiO2 monolayer in (LaAlO3)6.5/(BaTiO3)6.5 superlattice, and completely provided by the interfacial Ti 3dxy orbitals. Through a comparison of the superlattices with thickness n = 2.5 and n = 6.5, the results show that the electronic property has a strong dependence on the superlattice size. Such tightly confined 2DEG at the interfacial TiO2 monolayer, contributed by the Ti 3d states, is very consistent with the results of Nazir et al.15. Moreover, we have also find the difference of Ti 3d orbitals contribution to the DOS located near the interface when the period of superlattice is as short as n = 2.5.

Interfacial charge carrier density

Next, we calculated the interfacial charge carrier density21,28,33,34 for the superlattices via integrating the partial DOS near the Fermi level of the occupied Ti 3d orbitals located at the interfacial TiO2 monolayer21,29, and displayed the relation between the interfacial charge carrier density and the parameter n, as shown in Fig. 5 (black line). When n ≤ 4.5, the interfacial charge carrier density increased from 1.13 × 1014 cm−2 to 2.25 × 1014 cm−2 with the increase of the index n, when n ≥ 4.5, the interfacial charge carrier density was converged to 2.25 × 1014 cm−2. From the work of Meevasana et al.35, the 2DEG with the electron density was measured to be as large as 8 × 1013 cm−2, formed at the bare SrTiO3 surface in LaAlO3/SrTiO3 system. We also constructed the (SrTiO3)4.5/(LaAlO3)4.5 superlattice and found the 2DEG density at the interface is 1.48 × 1014 cm−2. We plot the layer-resolved partial DOS for (SrTiO3)4.5/(LaAlO3)4.5 superlattice, as shown in the Fig. S7. By comparison, the system we studied can form a 2DEG with even higher charge carrier density than the traditional LaAlO3/SrTiO3 systems15. For LaAlO3/SrTiO3 superlattice we found that there is more electron density penetrating into the SrTiO3 near the interface. This explains why we could obtain a higher electron density of LaAlO3/BaTiO3 superlattice than that of LaAlO3/SrTiO3 superlattice.

Figure 5
figure 5

Calculated interfacial charge carrier density for (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5 to 8.5) with LaO/TiO2 interface and the magnetic moment of the Ti atom at the interface as a function of the index n.

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In order to ensure the reliability of the relationship between interfacial charge carrier density and the parameter n, we also calculated local magnetic moment29,33,36 on Ti atoms of the interfacial TiO2 monolayer. From Fig. 5 (red line), we observe that the magnetic moment versus n shows a similar trend with the relationship between the interfacial charge carrier density and the parameter n. The Ti atomic magnetic moment increased from 0.24 µB to 0.42 µB with the increase of the parameter n, and then converged to 0.42 µB when the thicknesses of LaAlO3 and BaTiO3 layers are greater than 4.5 unit cells. Similar to LaAlO3/SrTiO3 systems, for the (LaAlO3)n/(BaTiO3)n superlattices, about half electron is transferred from addition LaO monolayer to the neighboring TiO2 monolayer in the interface as a result of the polar discontinuity, the half-electron will occupy partial Ti 3d orbitals37, and our saturation value result is very close to 0.46 µB reported by Nazir et al.15.

Calculation of electron effective mass

To further discuss the effect of superlattice size on electronic properties of the 2DEG, we calculated the band structure along the path M-Г-X of the Brillouin zone21,22,38, as shown in Fig. 6. We can see that there is only majority-spin electrons pass through the Fermi level, resulting in the spin-polarized 2DEG at the superlattice interface. Figure 6 also suggested that the bottom conduction band becomes deeper in energy as the parameter n increases. Moreover, such band consists two degenerate energy levels and splits into two bands when n = 8.5. The degenerate states are due to the fact that there are two identical interfaces in the superlattice, whereas a slight symmetry breaking in the superlattice structure may result in the degenerate level splitting. Through the above DOS analysis, we realize that the conduction band passing through the Fermi level is provided by the Ti 3d orbitals, mainly supplied by Ti 3dxy state.

Figure 6
figure 6

Electronic band structures for the (a) n = 2.5, (b) n = 4.5, (c) n = 6.5, and (d) n = 8.5 (LaAlO3)n/(BaTiO3)n superlattices, respectively. The red (blue) lines show the majority (minority)-spin sates, and the black horizontal dashed line at 0 eV indicates the Fermi level. The bottom conduction band is employed to calculate the relative electron effective mass.

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It is important to qualitatively compare the interfacial electron transport characteristics in the (LaAlO3)n/(BaTiO3)n superlattices. We calculated the electron relative effective masses (m*/me) along the Г-X and Г-M directions using the bottom conduction band for each superlattice, in which me is the electron mass, and the results are listed in Table 2. The calculated electron effective masses are obtained from the following formula20,21,22,38:

$$\frac{1}{{{\rm{m}}}^{\ast }}=\frac{1}{{{\rm{\hbar }}}^{2}}\,\frac{{\partial }^{2}{{\rm{E}}}_{{\rm{CB}}}}{\partial {{\rm{k}}}^{2}}$$
(2)

where ħ is the reduced Planck constant, ECB and k are the energy and wavevector of the bottom conduction band, respectively. From the Table 2 we can clearly find that the relative effective masses, whether along the Г-X or Г-M direction, decreased from 0.351 to 0.273 with the increase of the parameter n. When n ≥ 4.5, the m*/me is converged to 0.273, suggesting that the relative effective mass of the 2DEG becomes a constant. We can also find almost no difference between the two kinds of relative effective mass, indicating same characteristics along the two transport directions for the 2DEG. In the typical LaAlO3/SrTiO3 systems, the relative effective mass of the 2DEG confined at SrTiO3 surface is 0.5–0.6 measured by the angle-resolved photoemission spectroscopy from the report of Meevasana et al.35. Therefore our studied 2DEG has a smaller effective mass, and thus better carrier mobility than the LaAlO3/SrTiO3 systems.

Table 2 Calculated relative electron effective mass m*/me along the Г-X and Г-M paths for the (LaAlO3)n/(BaTiO3)n superlattices.
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When considering the ferroelectricity of BaTiO3 in (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5, 4.5, 6.5, and 8.5), by observing the change of its properties, two separate bands are found near the conduction band minima which were actually almost degenerate in the exact mirror-symmetric BaTiO3 case (see Figs S2–4 in Supplemental Information and Fig. 6). This shows that the two interfaces will exhibit different electronic band structure characteristics when BaTiO3 is in ferroelectric state.

Charge carrier mobility and conductivity

Now let us compare the interfacial electron mobility and conductivity in the (LaAlO3)n/(BaTiO3)n superlattices. For that, we calculated the relative interfacial electron mobility (µ/µ0) and electrical conductivity (σ/σ0)21,22,38,39 with respect to (LaAlO3)2.5/(BaTiO3)2.5 superlattice, and plotted the results as function of the index n, as shown in Fig. 7. We employed the following formulas:

$$\mu \,={\rm{e}}\langle{\rm{\tau }}\rangle/{{\rm{m}}}^{\ast }$$
(3)
$${\rm{\sigma }}\,=\mathrm{ne}\mu $$
(4)

where e, ‹τ›, m*, n, µ, and σ represent elemental charge, average scattering time, effective mass, charge carrier density, carrier mobility, and electrical conductivity, respectively. The scattering time ‹τ› is assumed to be a constant40, determined by several scattering factors such as phonon scattering, electron-electron scattering, impurity scattering, and so on41,42. Our results show that, when the thicknesses of LaAlO3 and BaTiO3 layers are greater than 4.5 unit cells, the superlattices have a high and saturated carrier mobility and conductivity. Interestingly, the carrier conductivity for superlattices with n ≥ 4.5 is about 2.6 times than the reference superlattice, which is consistent with the results of the interfacial charge carrier density and the relative charge carrier effective mass. These results show that we can obtain a spin-polarized 2DEG by regulating the thicknesses of LaAlO3 and BaTiO3 layers in the process of growing LaAlO3/BaTiO3 superlattices.

Figure 7
figure 7

Calculated relative interfacial electron mobility (µ/µ0) and electrical conductivity (σ/σ0) for the (LaAlO3)n/(BaTiO3)n superlattices, where the µ0 and σ0 representative the interfacial electron mobility and electrical conductivity for the (LaAlO3)2.5/(BaTiO3)2.5 reference system, respectively.

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Conclusion

In summary, spin-polarized density functional theory calculations are employed to investigate the possibility of enhancing the 2DEG quantum confinement at LaO/TiO2 interface and the interfacial charge carrier density, mobility, and conductivity in LaAlO3/BaTiO3 superlattice by using thickness control. We find that the superlattices with LaO/TiO2 interface are more stable than BaO/AlO2 from the cleavage energy calculations. When the critical thicknesses of LaAlO3 and BaTiO3 layers are over 4.5 unit cells, the spin-polarized 2DEG is mainly provided by the interfacial Ti 3dxy orbitals and completely confined at the interface. In such cases, through the study of the interfacial charge carrier density, mobility, and conductivity, we confirm that these properties become independent of the thickness. Our present work provides theoretical understanding to produce and enhance the interfacial quantum confinement for spin-polarized 2DEG and the interfacial transport properties by adjusting the thicknesses of LaAlO3 and BaTiO3 layers. In LaAlO3/BaTiO3 superlattice, electrons are almost bound at the interfaces, but for LaAlO3/SrTiO3 superlattice we found that there is electron density penetrating into the SrTiO3 near the interface.

Computational and Structural Details

In this study, for all density functional theory (DFT)43 calculations we used Vienna Ab initio Simulation Package(VASP)44. The electron exchange-correlation interaction was described by the generalized gradient approximation (GGA) parametrized by the Perdew-Burke-Ernzerhof (PBE) plus the on-site Coulomb interaction approach (GGA + U)45. The projected augmented method (PAW) was used to describe the electron-ion interaction. In order to more accurately describe the electronic states of strongly correlated electrons, effective U value of 5.8 eV and 7.5 eV were applied to Ti 3d and La 4 f orbitals46,47, respectively. Such U values are well tested and more results are in Figs S5–6 of the Supplemental Information. Our calculations were performed by employing a cutoff energy of 450 eV for the plane wave basis set and a Monkhorst-Pack k-point mesh of 6 × 6 × 1 (except for n = 2.5 we use 6 × 6 × 2). The convergence of electronic energy was set to 10−6 eV in the self-consistent calculations. The atomic positions were optimized to simulate the epitaxial superlattice until the interatomic forces were small than 0.03 eV/Å.

We used slab model to build (LaAlO3)n/(BaTiO3)n superlattices (n = 2.5 to 8.5) by staking the epitaxial slabs using BaTiO3 lattice parameters, in which n is a half-integer representing the number of LaAlO3 and BaTiO3 unit cells along (001) direction. We have constructed two types of superlattices interface, namely LaO/TiO2 and BaO/AlO2 interfaces. To simulate the epitaxial film growth in the experiment, we fixed the lattice constants in the ab-plane, and fully relaxed all atoms (unless otherwise stated) in the LaAlO3/BaTiO3 perovskite superlattices.