Introduction

Oxide interfaces exhibit many spectacular phenomena not found in the respective bulk components or in conventional semiconductor interfaces1, providing new avenues for electronics2. The LaAlO3/SrTiO3 interface is a paradigm example, exhibiting conducting two-dimensional (2D) electron gas (2DEG)3,4 and magnetism5,6,7,8,9,10,11 between two insulating nonmagnetic metal oxides. In the [001] direction, two different interfaces can be formed between polar LaAlO3, which consists of alternating LaO)+–(AlO2)− layers, and nonpolar SrTiO3, which consists of alternating (SrO)0–(TiO2)0 layers. One is LaO/TiO2 stacking configuration (so-called n-type) and the other is AlO2/SrO configuration (so-called p-type). The remarkable feature is that the conductivity occurs only at n-type interfaces when the LaAlO3 film thickness (nLAO) is larger than three unit cells (uc)4,5, whereas the magnetism has been observed both at n-type interfaces with nLAO>~3 uc and at insulating p-type interfaces8. Table 1 lists some experimental observations representing the main puzzles12 that need to be resolved before the promised applications can be realized13.

Table 1 List of some important experimental observations at LaAlO3/SrTiO3 interfaces.
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For 2DEG at n-type interfaces, four main mechanisms have been suggested, yet no single one explains the full scope of these puzzles. The prevalent one is intrinsic electronic reconstruction (so-called polar catastrophe) involving ionization of the electrons from host valence band of LaAlO3 within the abrupt and defect-free interfaces (Supplementary Fig. 1)3,4. The other three mechanisms involve various defects, including the oxygen vacancies at the interface (denoted as VO(I), where ‘I’ means ‘Interface’)14,15,16, oxygen vacancies at LaAlO3 overlayer surface (denoted as VO(S), where S means ‘Surface’)17,18,19,20,21,22, and the La-on-Sr (LaSr) antisite donor defects induced by interfacial cation intermixing23,24,25,26,27,28,29. As Table 1 shows, each of these proposed mechanisms represents one aspect of the interface physics, explains some experimental findings, but conflicts with a few others2. None explains the insulating nature of p-type interfaces. Regarding interface magnetism, it was shown experimentally that the local magnetic moments are associated with Ti3+ ions5,6,7,8,9,10,11,30. However, it is yet unclear whether such Ti3+ ions reside in the interface within SrTiO3 side, or LaAlO3 side, or both sides. Theoretically, it has been argued that the Ti3+ ions arise in SrTiO3 side, owing to the occupation of the low-energy Ti-dxy-like sub-bands caused by the interfacial splitting of orbital degeneracy31, or interfacial disorder32,33, or interfacial oxygen vacancies34. However, these scenarios are difficult to explain the fact that magnetism occurs at p-type interfaces and n-type interfaces with a critical thickness (Lc) similar to that for 2DEG.

The centrosymmetric III–III–O3 perovskite has a non-zero formal polarization, as established by the modern theory of polarization35,36. The discontinuity in the formal polarization of LaAlO3 and SrTiO3 leads to a finite polar field that would cause the divergence of electrostatic potential as the nLAO increases. A crucial issue associated with the emergent conductivity and magnetism at polar–nonpolar interfaces is what mitigates such potential divergence. Is it electronic reconstruction within the polar catastrophe scenario, or the atomic reconstruction scenario with or without chemical defects? Different mechanisms suggest different experimental designs that would control conductivity, mobility and magnetism. Particularly, for defects, it is unclear which defects can be induced and are responsible for the emergent interface phenomena. Using first principles electronic and defect calculations, we find that the certain defects would form spontaneously in response to the built-in polar field. The ensuing polarity-induced defect mechanism (Fig. 1) simultaneously explains the main features of both conductivity and magnetism at the interface, as summarized in Table 2.

Figure 1: Schematic band diagram and change transfer among the defects at LaAlO3/SrTiO3 interfaces.
figure 1

(a) n-type interfaces with nLAO<Lc: all electrons transferred from TiAl(S) are trapped by deep AlTi(I), causing no 2DEG. (b) n-type interfaces with nLAO≥Lc: VO(S) defects donate ~0.5e S2D−1 to the interface. Part of ~0.5e S2D−1 is trapped by the AlTi(I) and the rest leads to interfacial 2DEG. The formed TiAl defects are ionized, i.e., Ti3+-on-Al3+, having local magnetic moments. (c,d) p-type interfaces with nLAO<Lc (~4uc) and nLAO≥Lc: all electrons transferred from LaSr(I) are trapped by SrLa(S) and VLa(S), respectively. All involved defects are deep and do not induce carriers. The un-ionized TiAl0 (not shown in c,d) also form and induce local moments. The superscripts (0,+,++,−) in the Figure denote the defect charge states, not the oxidation states of the ions there.

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Table 2 The specific defects and their charge transfer processes that explain the leading experimental observations at stoichiometric LaAlO3/SrTiO3 interfaces.
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The key defect-related physical quantities that feature in our explanation are (i) the formation energy ΔH of defects in various charge states (q) at the thermodynamic equilibrium Fermi energy EF (Fig. 2). This ΔH controls the equilibrium defect concentration; (ii) the defect charge transition energy levels (deep or shallow; Fig. 3), ε(q/q′) defined as the EF where the ΔH of a defect in two different charge states q and q′ equal. A donor can produce electrons and compensate holes, whereas an acceptor can produce holes and compensate electrons. These two quantities (i) and (ii) calculated for charged defects located in different layers across the interfaces turn out to be crucial. The details of their first principles calculations are given in the Methods section.

Figure 2: Formation energy of the interfacial point defects at thermodynamical equilibrium Fermi energy.
figure 2

(a,b) n-type interfaces with nLAO<Lc and nLAO≥Lc, respectively. (c) p-type interfaces. At a given EF, the defect in different charge states (for example, VSr0, VSr1−, VSr2−) usually has different ΔH and the only one with the lowest ΔH is shown in the Figure. The ΔH versus EF for these defects are shown in Supplementary Fig. 2, which also includes other high-ΔH defects not shown here. The chemical potentials used for Sr, Ti, La, Al and O are −4.36, −6.20, −6.10, −5.46 and −2.0 eV, respectively, relative to their corresponding elemental solid or gas phases, which corresponds to T=1050 K and PO2=6.1 × 10−6Torr (Supplementary Fig. 3).

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Figure 3: Charge transition energy levels of the interfacial point defects.
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(a) n-type interface. (b) p-type interface. The defect charge transition energy level is defined as the EF where the ΔH of a given defect in two different charge states equal. Some defects may have multiple charge transition energy levels. For example, VSr has the two transition energy levels (one is for the transition between neutral charge state and −1, and the other is between −1 and −2). In such case, if the defect is donor (red), only the lowest level is shown, and if the defect is acceptor (blue), the highest level is shown.

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The central point of the proposed mechanism is that the polar-discontinuity-induced built-in polar field triggers thermodynamically the spontaneous formation of certain defects at the surface and/or interface, which in turn compensate the built-in polar field and thus avoids the potential divergence. Thus, it is the polar-field-induced defects, rather than the electronic or atomic reconstruction, that are responsible for the conductivity and magnetism at the interface. Specifically, we find that the surface VO has its donor levels located energetically above the SrTiO3 conduction band at the interface but below the LaAlO3 conduction band. This donor level position is a prerequisite for 2DEG formation. Although the occurring of the 2DEG is because of the surface donors, the density of 2DEG is controlled by the interfacial deep acceptor defects (mainly Al-on-Ti antisite). It has also turned out that the interface magnetic moment is caused by the unionized deep Ti-on-Al antisite defects located within the LaAlO3 side near the interface.

We address below how this polar-field-induced defect mechanism resolves the long-standing puzzles on the origin of 2DEG, the critical thickness for 2DEG, the weak field in LaAlO3 film, the density of 2DEG, the insulating nature of p-type interfaces and the origin of the local magnetic moments. During this process, we also distil the general design principles that control the pertinent effects and could allow future section of better polar–nonpolar interface materials.

Results

The origin of the 2DEG

The 2DEG is unlikely to originate from the defect-free scenarios: these include the ionization of the intrinsic LaAlO3 valence bands (suggested by the polar catastrophe model3,4) or the ionization of the LaO interface layer (suggested by the interfacial charge-leaking model)37 (Supplementary Note 1). This conclusion stems from the fact that the creation of 2DEG in these defect-free scenarios requires the LaAlO3 valence band maximum (VBM) to cross the SrTiO3 conduction band minimum (CBM) or EF. However, this is contrary to the experimentally observed weak field (negligible band-bending)38,39,40,41 in the LaAlO3 film, clearly showing that the LaAlO3 VBM is located energetically far below the EF.

The 2DEG also is unlikely to originate from interfacial point donor defects (LaSr, TiAl and VO). Recall first that the defect formation energy (ΔH) depends on the EF (or chemical potential) and the defect charge transition energy ε(q/q′) needs to be close to band edges in order to produce free carriers. In thermodynamic equilibrium, the EF of the system pins around the middle of SrTiO3 band gap when nLAO<Lc and around the SrTiO3 conduction band edge near the interface when nLAO≥Lc (Supplementary Note 2). In either case, Fig. 2ab shows that the ΔH of the interfacial antisite donor defects, LaSr0 and TiAl0, is small positive or even negative (note: the superscript denotes the defect charge states, not the nominal oxidation state of the atom at the defect site). In other words, the formation of such antisite defects at the thermodynamic equilibrium EF is energetically favourable and would inevitably lead to interfacial cation mixing. However, at such EF, both LaSr0 and TiAl0 defects are stable in their charge neutral states (as indicated by the superscript), contributing no free carriers. On the other hand, the interfacial VO defects are energetically stable in the charged states, that is, VO2+ (Fig. 2a,b). This means that, if formed, the VO will donate electrons and thereby become positively charged. However, the ΔH of VO2+ at such equilibrium EF is rather high (>2.5 eV), implying that VO2+ have very low concentration under thermodynamic equilibrium conditions. The high ΔH also means that even if the VO defects are formed under nonequilibrium growth conditions, they can still be removed easily by the post O-rich annealing process42 (Supplementary Note 3). Thus, contrary to earlier postulations, these interfacial donor defects are not responsible for 2DEG, consistent with recent experiments43.

The oxygen vacancy, VO(S), at the free LaAlO3 surface can explain the interfacial 2DEG. For this to happen, three conditions (‘design principles’) need to be satisfied. First, VO(S) in the polar film material needs to have a sufficiently low formation energy ΔH; therefore, it could form in significant quantities. Figure 4a shows that the ΔH of VO(S) decreases linearly as the film thickness nLAO increases, consistent with previous calculations20,44. When nLAO≥3–4 uc, the ΔH becomes zero or negative, and VO(S) will form spontaneously. The large negative ΔH means that even exposing the surface to air or post annealing under O-rich environment cannot remove these vacancies. Second, the system needs to have a none-zero built-in polar field that would enable the electron to transfer from the surface of the polar material to the interface. Such transfer sets up an opposite dipole (proportional to nLAO), which in turn cancels the field and lowers the ΔH. The larger the nLAO, the lower the ΔH. Note that in the absence of such field, the surface-to-interface charge transfer would not occur since such a transfer would create a dipole that would increase the electrostatic energy (proportional to that dipole) and thus raise the total energy of the system. Third, the donor transition level of VO(S) in the polar film should be higher in energy than in the substrate (SrTiO3) conduction band edge at the interface (Fig. 3). These three conditions are satisfied in this LaAlO3/SrTiO3 system.

Figure 4: Properties of surface defects and defect complexes.
figure 4

(a) The GGA-calculated ΔH of VO(S) defect, under the O-rich growth condition (that is, ΔμO=−1.5 eV, Supplementary Fig. 3a). (b) the ΔH of [TiAl+AlTi] defect pair created from a Ti⇔Al exchange out of the ideal interface with and without a VO(S) in a 2 × 2 (SrTiO3)6/(LaAlO3)4/vacuum surpercell. (c) The GGA-calculated ΔH of [LaSr(I)+VLa(S)] defect complex as a function of nLAO, under ΔμSr=−4.36 eV (Supplementary Fig. 3b). (d) the ΔH of [LaSr+SrLa] defect pair created from a La⇔Sr exchange out of the ideal interface with and without a VLa(S) in a 2 × 2 (SrTiO3)6/(LaAlO3)4/vacuum surpercell, respectively. The dTi⇔Al and dLa⇔Sr in b,d are the distance between the components of corresponding defect pair. The orange lines are the guides to the eye.

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It is noteworthy that it is the built-in polar field that triggers the spontaneous formation of the VO(S) when nLAO≥Lc. Such built-in polar field always exists in the LaAlO3 to be grown during the layer-by-layer growth. This is because that the surface defects (here VO) can cancel the built-in polar field only in the LaAlO3 film between the interface and the surface, not the built-in polar field in the LaAlO3 film to be grown on top of the surface.

In the absence of interfacial defects, the emerging picture for creating 2DEG is that the electrons ionized from VO(S) of the polar film material transfer to the nonpolar substrate material SrTiO3 conduction bands at the interface via the built-in polar field, thus forming the 2DEG at that interface. This charge transfer in turn cancels the built-in polar field in LaAlO3, which caused the low ΔH of the surface vacancies in the first place. After the built-in field has been cancelled, the ΔH of VO(S) return to a high value (>3 eV) characteristic of the bulk, and VO(S) become again hard to form in thermodynamic equilibrium20. Thus, the theoretical maximum concentration of VO(S) is 0.25 S2D−1 (where S2D is 2D unit cell area), that is, one of eight oxygen missing at surface. These would donate maximally 0.5e S2D−1 that would completely cancel the polar field in LaAlO3. The compensation of polar field by VO(S) also means that the band bending in LaAlO3 because of polar field is removed. Thus, the LaAlO3 valence bands fall well below the EF, contrary to what the polar catastrophe model would suggest. Consequently, no free holes can arise from depopulation of the LaAlO3 valence bands at the surface, consistent with experiments3,22.

The emerging design principles for selecting materials that will form interfacial 2DEG are: (i) the nonpolar material needs to have a CBM positioned in the band gap of the polar material; (ii) the polar material needs to have at least one donor defect with its donor level higher in energy than the conduction band of the nonpolar material at the interface. This picture suggests that the 2DEG at n-type LaAlO3/SrTiO3 interfaces may also be induced and/or tuned by using certain surface adsorbates (for example, H2O and H)45,46,47 or metallic contacts48 provided that the ionization energy of the surface adsorbate or the metallic contact is not lower than the donor level of the VO(S).

The origin of the critical thickness

The linear decrease in ΔH of VO(S) with increasing polar film thickness nLAO naturally explains the critical thickness Lc for the metal–insulator transition. The calculated rate of decrease (that is, the slope dΔH/dnLAO) equals 0.19 eV Å−1, which is the same as the calculated built-in polar field in the defect-free LaAlO3 film (Supplementary Note 4). The VO(S) defects start to form spontaneously when the ΔH becomes zero at the Lc of ~4 uc under a typical O-rich growth condition (Fig. 4a). For the LaAlO3 film that is 1 uc thinner than this Lc, the calculated ΔH of VO(S) is 0.75 eV, which is too high to produce significant free carrier concentration. Thus, the appearance of VO(S) (and the ensuing metal–insulator transition) at Lc is predicted to be a sharp transition (Supplementary Note 5), distinct from the gradual appearance of 2DEG behaviour as predicted from polar catastrophe model, but consistent with experiments49.

Figure 3a suggests that the Lc resulting from VO(S) can be written as Lc=ΔHo/eEp, where ΔHo is the formation energy of the VO at interface (or the ΔH extrapolated at nLAO=0) and Ep is the built-in polar field. Using Ep=4πP0/ε (where ε and P0 are the dielectric constant and formal polarization of LaAlO3 film), this relation can be written as

which predicts an Lc of ~4 uc, depending slightly on the O-poor/rich growth conditions (Supplementary Note 6). The above formula provides an alternative explanation for the observed variation of the Lc with the fraction x in (LaAlO3)1−x(SrTiO3)x overlayer (where P0 is proportional to x)50. This observation was originally explained by Lc=ΔΦε/4πeP0 (where ΔΦ is the energy difference between LaAlO3 VBM and SrTiO3 CBM) within polar catastrophe model50. Since ΔΦ and ΔHo have accidentally similar value (~3–4 eV) in this system, it is not surprising that the Lc predicted from these two models is also similar. However, the VO(S) model clearly explains many other observations in which the polar catastrophe model fails (Table 1).

Implication on the design of carrier mobility: (i) the relatively high 2DEG mobility could be enabled by a modulated doping effect51, whereby the source of carriers (here at the LaAlO3 surface) is spatially separated from the location where the carriers reside (here at the LaAlO3/SrTiO3 interface), thus minimizing carrier scattering by the ionized defects. This minimal spatial separation is measured by the critical thickness Lc. The equation (1) suggests that a large Lc (hence maintaining good mobility) could be achieved by selecting a polar materials with small polarization, large dielectric constant and donor defects having high ΔH at the interface or in the bulk. On the other hand, (ii) the concentration of interfacial defects must be minimized in order to take advantage of (i). In addition, (iii) since the 2DEG is located at the conduction bands of the nonpolar material, it is advantageous to select the nonpolar material with low electron effective mass in order to achieve higher mobility.

Polar field compensation

Experimentally, only very weak residual field has been observed in the LaAlO3 film no matter whether its thickness is below or above the Lc(refs 38, 39, 40, 41, 52). This observation cannot be explained within the defect-free interface scenario, even including the ionic relaxations53. In turn, whereas the VO(S) model explains the weak electric field in LaAlO3 film above the Lc, it does not explain it below the Lc. This leads us to inspect the effects of all possible cation antisite defects across the interface.

Each individual interfacial antisite alone cannot cancel the polar field. Figure 2ab shows that the LaSr, SrLa, TiAl and AlTi antisite defects have lower ΔH than other point defects (for example, cation vacancies) in the layer where they are located. Therefore, the former are the dominant defects in their corresponding layers. The interfacial LaSr donor in the SrTiO3 side cannot set up an opposite dipole across the LaAlO3 film that can cancel the polar field inside the LaAlO3 film. Regarding the TiAl donor in the LaAlO3 side, the donor level is lower than the SrTiO3 conduction band at the interface. Therefore, the ionized electrons cannot be transferred to the latter so as to cancel the polar field. Regarding the interfacial AlTi and SrLa acceptors, the polar field compensation is similar to that in the polar catastrophe model: before the LaAlO3 VBM reaches the acceptor levels of AlTi or SrLa, the polar field cannot be cancelled.

The [TiAl+AlTi] defect pair is the most potent source of polar field cancellation among those donor–acceptor antisite defect pairs at n-type interfaces. The four leading antisite defects can form four types of donor–acceptor pairs: [TiAl+AlTi], [LaSr+SrLa], [LaSr+AlTi] and [TiAl+SrLa], denoted as , , and , respectively, in Fig. 3. Clearly, the electron transfer from donor to acceptor in both pairs and is unlikely since it will create a dipole in the same direction as the intrinsic dipole in LaAlO3, and thus increase the dipole moment (also the electrostatic energy) and destabilize the interface. In pairs and , the charge transfer can cancel the polar field. However, the electron transfer in pair is energetically much more favourable because, first, AlTi has a lower acceptor level than SrLa and, second, the donor–acceptor separation distance (also the associated opposite dipole moment that lowers the total energy of the system) is larger in pair (Fig. 3a). We thus next focus on [TiAl+AlTi] (that is, pair ).

For nLAO<Lc, the [AlTi+TiAl] antisite pair can form spontaneously via Ti⇔Al exchange across the interface and cancel the polar field. Figure 4b (filled symbols) shows that the energy required to form such defect pair is negative (that is, exothermic), and the largest energy gain is obtained when a Ti atom of TiO2-interface monolayer is exchanged with an Al of AlO2-surface monolayer, that is, AlTi(I)+TiAl(S), which is consistent with previous first principles calculations54. This means that Ti atom at the interface would hop to the AlO2-surface layer and exchange with Al atom there. Similar to the case of VO(S), the linear decrease in ΔH with increasing donor–acceptor separating distance (Fig. 4b) is a sign of polar field compensation. Indeed, the electron transfer from the TiAl donor to the AlTi acceptor is expected, since the donor level is higher in energy than the acceptor level (Fig. 3a). Figure 4a also shows that the VO(S) has too high ΔH to form for nLAO<Lc (Fig. 4a). Therefore, the polar field is cancelled by those spontaneously formed [AlTi(I)+TiAl(S)] pairs. On the other hand, since these defects are deep, they cannot cause free carriers in the both interface and surface regions (whence insulating).

For nLAO≥Lc, the polar field is cancelled by spontaneously formed VO(S), not by [AlTi(I)+TiAl(S)]. Recall that the polar field always exists in the LaAlO3 layers during the layer-by-layer growth. Such polar field can trigger the formation of VO(S) and/or TiAl(S) defects as nLAO increases. For nLAO≥Lc, both VO(S) (Fig. 4a) and [AlTi(I)+TiAl(S)] pair (Fig. 4b) have zero or negative ΔH, meaning that both could form in ideal interfaces. However, if both VO(S) and TiAl(S) are present, since VO has an energetically higher donor level than TiAl (Fig. 3a), the VO(S) would transfer electrons to the TiAl defects. The polar field that was initially cancelled by the electrons transferred from the TiAl defects are then released and get cancelled by the electrons transferred from VO(S) defects. Consequently, the polar field in the whole LaAlO3 film would be cancelled by the VO(S) defects (if present). The larger the nLAO, the lower the ΔH of the VO(S). After the polar field has been cancelled by VO(S), Fig. 4b (open symbols) shows that the ΔH of [AlTi(I)+TiAl(S)] pair becomes positive (0.4–0.7 eV), meaning that [TiAl+AlTi] pairs cannot be formed via Ti⇔Al exchange over a distance beyond Lc. In brief, the presence of the [TiAl+AlTi] defect pairs in the sample does not change the linear-decreasing behaviour in the ΔH of the VO(S) (Fig. 4a), suggesting that the metal–insulator transition still occurs at the Lc of ~4 uc. However, the presence of VO(S) would prevent [TiAl+AlTi] pairs forming further above the Lc, and reduce the concentration of these pairs formed below the Lc.

The density of the 2DEG

Reinterpretation of the puzzle: According to Gauss’ law, the experimentally observed weak electric field in LaAlO3 film means that the total external charge density (mobile and/or immobile) at the interface must be ~0.5e S2D−1 (Supplementary Note 7), as recently observed55. For nLAO<Lc, there is no interfacial conductivity and thus none of these interfacial charge contribute to the conductivity. For nLAO≥Lc, only a fraction of 0.5e S2D−1 interfacial charge is seen in transport, and so the majority of 0.5e S2D−1 charges do not contribute to the conductivity. The puzzle thus is why the ~0.5e S2D−1 charge exists at the interface with any nLAO, but only a small part of it contributes to conducting 2DEG when nLAO≥Lc.

This puzzle cannot be explained by defect-free polar catastrophe model3,4 or interfacial charge-leaking model37, since both predict zero interfacial charge for nLAO<Lc and an interfacial charge density much higher than the measured 2DEG density for nLAO≥Lc (Supplementary Fig. 1). The possibility of ‘multiple carrier types’ at defect-free interfaces (that is, those electrons occupying interfacial dxy sub-band and those occupying dxz/dyz sub-bands contribute differently in transport) has also been suggested to explain the measured 2DEG density above the Lc (refs 56, 57, 58, 59). However, this scenario could not explain the total 0.5e S2D−1 interface charge that is independent of nLAO. Moreover, it is also difficult to explain why a full carrier density of 0.5e S2D−1 has been observed at GdTiO3/SrTiO3 interfaces (where the same multiple carrier types exist)60.

The 2DEG density is controlled by the concentration of immobile acceptor defects that can trap itinerant electrons. Within the emerging defect picture, the total interfacial charge is always ~0.5e S2D−1, which corresponds to the (almost) complete polar field cancellation. In the SrTiO3 side (where the 2DEG is located), there are mainly three types of acceptor defects, namely, AlTi, VSr and VTi. At equilibrium EF, Fig. 2a,b shows that these acceptor defects all prefer to stay in negative charge states, that is, AlTi1−, VSr2− and VTi4−. (In other defect charge states, these defects have much higher ΔH and are not shown in Fig. 2a,b.). This means that once these defects form they will trap free electrons from the system and get negatively charged. Among these acceptor defects, the AlTi1− acceptors have the lowest ΔH and thus they the most potent electron-trapping agents. For nLAO<Lc, the AlTi defects resulting from Ti⇔Al exchange trap all free electrons transferred from TiAl(S) defects, and hence no free carrier can occur. For nLAO≥Lc, the ΔH of [TiAl+AlTi] pair changes from negative to positive because of VO(S) (Fig. 4b), meaning that the concentration of AlTi defect resulting from Ti⇔Al exchange is reduced, compared with that formed below the Lc. Therefore, the AlTi defect concentration is not sufficient to trap all 0.5e S2D−1 electrons transferred from VO(S). Therefore, only a small fraction of 0.5e S2D−1 can contribute to interface 2DEG.

The recently observed LaAlO3 cation-stoichiometry effect on 2DEG formation43 may also be understood within the above picture. For Al-rich LaAlO3 film, where both A-site and B-site sublattices are fully occupied (hence having no cation vacancies), the AlTi antisites are the only electron-trapping defects and the incomplete trapping of 0.5e S2D−1 interface charge by AlTi defects leads to interface conductivity. However, for La-rich LaAlO3 film, where B-site sublattice is not fully occupied, the cation vacancies (VTi and VAl) also become the main electron-trapping agents, in addition to AlTi(I). Although the concentration of AlTi is reduced, each cation vacancy induced in the La-rich film traps more electrons than an AlTi. The insulating character can thus be then attributed to the complete interfacial electron trapping by both interfacial cation vacancies and AlTi (I).

The picture of AlTi(I) as the main electron-trapping agent may be extended to SrTiO3/GdTiO3 interfaces. The observed full carrier density of 0.5e S2D−1 there60 can be ascribed to the fact that both SrTiO3 and GdTiO3 have the same Ti atom at B-site sublattice and thus have no AlTi-like antisite defects at the interface.

Implication on how to increase the density of 2DEG: The above picture suggests that the main controlling factor for the interface carrier density is the concentration of the acceptor defects (mainly AlTi in stoichiometric or Al-rich film), which should be minimized for enhancing carrier density. Such AlTi-like electron-trapping defects may be completely removed by designing other oxide interfaces such as GdTiO3/SrTiO3 interfaces, whose bulk components have a common cation atom with multiple valence states.

The origin of the insulting nature of p-type interfaces

An intriguing fact is that the so-called p-type interfaces are not p-type (hole) conducting but are actually insulating. The defect-free polar-catastrophe model for p-type interface predicts a hole-conducting interface and an electron-conducting surface when nLAO>~7.3 uc (Supplementary Fig. 1) in contradiction with the insulating behaviour observed robustly in experiment. To explain this, defects must be involved. The emerging defect picture below differs from the literature model based solely on interfacial hole–polaron61 or interfacial hole-compensating VO defects23,44, which assumes that the interface has holes arising from the depopulation of the intrinsic SrTiO3 valence bands.

Individual point defects alone at p-type interfaces can neither cause conductivity nor cancel the polar field. As was the case for n-type interfaces, Fig. 2b shows that the interfacial LaSr and TiAr are stable at their charge neutral states and have negligible or negative ΔH at equilibrium EF. This means that they cause inevitable interfacial cation intermixing but induce no free carriers. The VO and other defects at the interface require too high ΔH to form, and thus they do not produce free carriers either. For similar reason, each of the leading antisite defects (LaSr, SrLa, TiAl and AlTi) alone at p-type interfaces cannot cancel the polar field.

The spontaneously formed donor–acceptor defect pairs always cancel the polar field but do not induce free carriers. Among the four donor–acceptor defect pairs as indicted in Fig. 3b, the [LaSr+SrLa] (that is, pair ) is energetically most favourable to cause polar field cancellation. For nLAO<~ 4 uc, the [LaSr(I)+SrLa(S)] pairs have negative ΔH (Fig. 4d) and can form spontaneously via La⇔Sr exchanges, whereas the [LaSr(I)+VLa(S)] have too high ΔH to form (Fig. 4c). Therefore, the polar field is cancelled by the charge transfer from LaSr(I) to SrLa(S), which can be expected from their relative defect levels (Fig. 3b) and their linear decreasing behaviour in ΔH as a function of nLAO (Fig. 4d). For nLAO≥~4 uc, the ΔH of [LaSr(I)+VLa(S)] become negative (Fig. 4c) and can also form spontaneously. Since VLa has a lower acceptor level than SrLa (Fig. 3b), the polar field is cancelled by the charge transfer from LaSr(I) to VLa(S), rather than to SrLa(S). In absence of an electric field, Fig. 4d (open symbols) indicates that the La⇔Sr exchanges cannot occur anymore over a distance of ~4 uc. Unlike the case in n-type interfaces, the VO(S) defects in p-type interface always have too high ΔH to form. The defects involved in polar field cancellation are all deep. The calculated equilibrium EF according to those point defects turns out to be always around the middle of SrTiO3 band gap. This means that both VBM and CBM are far away from the EF, and there are no free carrier arising from the depopulation of VBM and CBM in both interface and surface regions (whence insulating).

Implication on the design of 2D hole conductivity: Clearly, the formation of interfacial free holes is prevented by these spontaneously formed deep LaSr defects that have donor level higher than the VBM at the interface. Therefore, to induce interfacial hole conductivity, one should search for the polar–nonpolar interfaces where all such donors have high enough formation energy to form or (ii) their donor levels below the VBM at the interface. Practically, the (ii) may be achieved more easily by searching for the polar material whose VBM is higher than the charge transition energy levels of those spontaneously formed interfacial donor defects.

The origin of interface magnetism

Distinct from previous models31,32,33,34 that explain magnetism based on the intrinsic interfacial Ti3+ ion in the SrTiO3 (that is, not a defect), we find below that the local magnetic moment originates from the unionized deep TiAl antisite defect (that is, Ti3+-on-Al3+ within LaAlO3 side near the interface. The interface magnetism depends on the concentration and spatial distribution of such TiAl defects. This picture explains not only why the magnetism appears at n-type interfaces with a similar critical thickness to that for 2DEG but also why the magnetism also appears at insulating p-type interfaces8.

What causes local moment? As discussed earlier, for n-type interfaces, when nLAO<Lc, the polar field in LaAlO3 is cancelled by the charge transfer from TiAl(S) defects to the interface. These formed TiAl defects are thus ionized, that is, TiAl1+ (where superscript denotes the defect charge states). The Ti ion at this defect site has the oxidation states of 4+, denoted as Ti4+, which has no local magnetic moment. Moreover, noted before, when nLAO≥Lc, the polar field in LaAlO3 is cancelled by the charge transfer to the interface from VO(S) instead of TiAl. In absence of internal field, all TiAl(I) defects in the LaAlO3 film are most stable in their charge neutral (or unionized) states, that is, TiAl0, where Ti appears as Ti3+ oxidation state, having a finite local magnetic moment. Therefore, the interface magnetism at n-type interfaces because of those unionized TiAl0 defects should also have a critical thickness of ~4 uc. For p-type interfaces, it is the charge transfer among the defects other than TiAl defects that cancels the polar field in LaAlO3. Thus, all TiAl defects formed there are not ionized, having local magnetic moments, and cause interface magnetism.

The magnitude of local magnetic moment: The local moment of a single TiAl defect at the interface can be estimated from that in bulk LaAlO3, which is 0.84μB from our hybrid functional calculation. For ferromagnetic order as observed in the experiment, the total interface magnetic moment depends on the concentration of unionized TiAl defects in LaAlO3 and can be very small per Ti atom in average. The experimentally observed inhomogeneous landscape of magnetism that also varies from sample to sample8,9 may be attributed to the various spatial distributions of TiAl defects, which may be sensitive to sample preparation conditions (such as temperature and PO2) and local strain.

The TiAl(I) defects within LaAlO side being the origin of the local moment are more reasonable than VO(I) in two aspects. First, the deep TiAl defect is spatially localized and has an unambiguous local moment. In contrast, VO is a shallow donor that transfer electrons to the lower-energy interfacial Ti dxy sub-bands that have light effective mass inside the interface plane59; therefore, the resulting Ti3+ may then be itinerant. Second, the TiAl defects would form readily because of the small or negative ΔH of TiAl, whereas the interfacial VO requires significant energy to form and if formed it may be removed completely after annealing.

Discussion

We establish a physical link between polar discontinuity and defect formation: the polar discontinuity triggers spontaneous formation of certain defects that in turn cancel the polar field induced by polar discontinuity. It is the subtle interplay of those spontaneously formed surface vacancy defects and interfacial cation antisite defects that control the physics of the system by their formation energies and relative defect levels. Table 2 summarizes how those defects shown in Fig. 1 explain the leading experimental observations and puzzles in Table 1. The explanation leads to a set of design principles for both conductivity and magnetism at LaAlO3/SrTiO3 and other polar–nonpolar interfaces and enables the design of better polar–nonpolar interfaces.

Having ruled out the electronic reconstruction, interfacial VO and interfacial cation intermixing mechanism as the possible origin of 2DEG in our calculations, we conclude that the 2DEG at n-type interfaces with nLAO≥Lc originates from the spontaneously formed VO(S) defects. This conclusion stems from the finding that the donor level of deep VO in the LaAlO3 side is higher than the SrTiO3 conduction band edge at the interface. This finding explains why the formation energy of VO(S) decreases linearly as nLAO increases. This linear decrease relation leads to some new controlling parameters for the critical thickness of sharp metal–insulator transition in absence of the electric field in the polar LaAlO3 film. Instead of causing the 2DEG, the anti site defect pair turns out to play a key role in canceling the polar field, controlling the density of the 2DEG, and inducing the local magnetic moments at the interface (Table 2).

The emerging mechanism provides three distinctive predictions to be tested in experiment as further validation. (i) For n-type interfaces, the AlO2-surface layer is dominated by TiAl defects when nLAO<Lc and by VO defect when nLAO≥Lc. (ii) For p-type interfaces, the LaO-surface layer is dominated by SrLa and VLa defects, respectively, below and above an Lc of ~4 uc. (iii) Ti4+ and Ti3+ signals exist in both sides of the interface. The appearance of the Ti3+ signals should not be taken as a sign of conductivity. Whether the Ti3+ signals detected by photoemission below the Lc (refs 21, 40, 62, 63) can be truly assigned to those Ti3+ ions in the SrTiO3 side should be revisited carefully. How these TiAl local moments are ordered (ferromagnetic, or antiferromagnetic, or else) and whether and how they interact with the itinerant 2DEG are still open questions that should be investigated further.

Methods

Computational techniques

All calculations were performed using density functional theory and plane-wave projector-augmented wave64 method as implemented in the VASP code65. An energy cutoff of 400 eV was used. The Brillouin zone was sampled by 8 × 8 × 1 and 4 × 4 × 1 k-point mesh for 1 × 1 and 2 × 2 in-plane supercell, respectively. The atomic forces were relaxed to be less than 0.03 eV Å−1. The in-plane lattice constant was fixed to 3.943 Å (the relaxed lattice constant of SrTiO3 by GGA66). In slab calculations, the 4-uc (~16 Å) vacuum layer was used and the dipole correction was always applied to remove artificial dipole interactions67. The results in Figs 2 and 3 were obtained by using HSE hybrid functional68 on top of the GGA-relaxed structures.

First principles defect theory

The formation energy of a defect (D) calculated from , where and are the total energies of a supercell with and without defect, respectively, and D is in charge state q. nα is the number of atoms of species α needed to create a defect. EF is the Fermi energy relative to VBM (Ev). Δμα is the relative chemical potential of species α with respect to its elemental solid (gas; μ0). The equilibrium Fermi energy was calculated self-consistently according to charge neutrality condition69. The chemical potentials relative to their elemental solid (or gas) phase are taken as variables and are bounded by the values that maintain a stable host compound and avoid the formation of other competing phases in thermodynamic equilibrium (Supplementary Fig. 3). The details of theory and calculations can be found in ref. 70.

Additional information

How to cite this article: Yu, L. et al. A polarity-induced defect mechanism for conductivity and magnetism at polar–nonpolar oxide interfaces. Nat. Commun. 5:5118 doi: 10.1038/ncomms6118 (2014).