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A polarity-induced defect mechanism for conductivity and magnetism at polar-nonpolar oxide interfaces

The discovery of conducting two-dimensional electron gas (2DEG) and magnetism at the interface between insulating nonmagnetic oxides, as exemplified by the polar LaAlO3 and nonpolar SrTiO3 has raised prospects for attaining interfacial functionalities absent in the component materials. Yet, the microscopic origin of such emergent phenomena remains unclear, posing obstacles to design of improved functionalities. Using first principles defect calculations, we reveal a unifying polarity-induced defect mechanism for both conductivity and magnetism at polar-nonpolar interfaces of nonmagnetic insulating oxides. We demonstrate that the polar-discontinuity across the interface triggers thermodynamically the spontaneous formation of certain defects that in turn cancel the polar field induced by the polar discontinuity. It turns out that the 2DEG originates from those spontaneously formed surface donor defects (oxygen vacancy), but the density of 2DEG is controlled by the interfacial anti-site acceptor defects (Al-on-Ti). The interface magnetism is found to originate from the un-ionized deep Ti-on-Al anti-site donor defects within the LaAlO3 side near the interface. Our results suggest practical design guidelines for inducing and controlling both 2DEG and magnetism at polar-nonpolar oxide interfaces.

disorder 32,33 , or interfacial oxygen vacancies 34 , which are difficult to reconcile why the magnetism appears at insulating p-type interfaces and at n-type interfaces with a critical thickness (L c ) similar to that for 2DEG.
Overarching unresolved questions: a crucial issue associated with the emergent conductivity and magnetism at polar-nonpolar interfaces is what mitigates the divergence of electrostatic potential as the thickness of the polar film increases 35 . Is it electronic reconstruction within polar catastrophe scenario, atomic reconstruction, or chemical defects? Different answer represents different mechanism, suggesting different controlling parameters. It has been purported that all these three effects exist to varying degrees in real materials and contribute to the emergent interface phenomena. However, whether and how these effects are physically connected, and what are their relative contributions, remain uncertain. Particularly, for defects, it is unclear which specific defects can be induced and are responsible for the emergent interface phenomena. Using first principles electronic and defect calculations, we identify the specific defects that can form spontaneously as a response to the built-in polar field and show how these defects lead to a unifying mechanism (Figure 1) that simultaneously explains the main features of both conductivity and magnetism at the interface.
The key physical quantities that feature in our explanation are (i) the formation energy ΔH (which controls the defect concentration) (Figure 2) of the defect in various charge states at thermodynamic equilibrium Fermi-energy E F , and (ii) the donor or acceptor electrical levels (Figure 3), i.e., the defect charge transition energy ε(q/qʹ′) defined as the E F where the ΔH of a defect at two different charge states q and qʹ′ equal. A donor can produce electrons and compensate holes, whereas an acceptor can produce holes and compensate electrons. These two quantities (i) and (ii) have not been previously calculated for charged defects in different layers across the interfaces and turn out to be crucial. The details of their first principles calculations are given in Methods section.
The central point of the proposed mechanism is that the polar-discontinuity induced built-in polar field triggers thermodynamically the spontaneous formation of the certain defects at 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 donor defect (here V O ) has its donor levels located energetically above the SrTiO 3 conduction band at the interface but below the LaAlO 3 conduction band. This donor level position is a prerequisite for 2DEG formation. Although the 2DEG owes its existence to the surface donors, the density of 2DEG is controlled by the interfacial deep acceptor defects (mainly Al-on-Ti antisite). It is also turn out that the interface magnetic moment is caused by the unionized deep Ti-on-Al antisite defects located within the LaAlO 3 side near the interface.
In what follows we address specifically 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 LaAlO 3 film, the density of 2DEG, the insulating p-type interfaces, and the origin of the local magnetism moments. During this process, we also distill the general rules that control the pertinent effects and allow future section of other polar-nonpolar interface materials with similar or improved interface properties.
The 2DEG is unlikely to originate from the defect-free scenarios: these include the ionization of the intrinsic LaAlO 3 valence bands (suggested by the polar catastrophe model 3,4 ) or the ionization of the LaO interface layer (suggested by the interfacial charge leaking model) 36 (Supplementary note 1). This is simply because that the creation of 2DEG in these defect-free scenarios requires the LaAlO 3 valence band maximum (VBM) to cross the SrTiO 3 conduction band minimum (CBM) or Fermi-energy E F . This is contrary to the experimentally observed weak field (negligible band-bending) [37][38][39][40] in the LaAlO 3 film, clearly showing that the LaAlO 3 VBM is far below the E F. The 2DEG also is unlikely to originate from interfacial point donor defects (La Sr , Ti Al , and V O ). Recall first that the defect formation energy ΔH depends on the E F (or chemical potential) and the defect charge transition energy ε(q/qʹ′) needs to be close to band edges in order to producing free carriers. In thermodynamic equilibrium, the E F of the system pins around the middle of SrTiO 3 band gap when n LAO < L c , and around the SrTiO 3 conduction band edge near the interface when n LAO ≥ L c (Supplementary note 2). In either case, Fig.2ab shows that the ΔH of the anti-site donor defects, La Sr 0 and Ti Al 0 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 E F is energetically favorable and would inevitably lead to interfacial cation mixing. However, at such E F , both La Sr 0 and Ti Al 0 defects are stable in their charge neutral states (as indicated by the superscript), contributing no free carriers. On the other hand, the interfacial V O defects are energetically stable in the charged states, i.e., V O 2+ (Fig.2ab). This means that if formed, the V O will donate electrons and thereby become positively charged. However, the ΔH of V O 2+ at such equilibrium E F is rather high (> 2.5eV), implying that V O 2+ have very low concentration under thermodynamic equilibrium conditions. The high ΔH also means that even if the V O defects are formed under non-equilibrium growth conditions, they can still be removed easily by the post O-rich annealing process 41 (Supplementary note 3). Thus, contrary to earlier postulations, these interfacial donor defects are not responsible for 2DEG, consistent with recent experiments 42 .

The oxygen vacancy, V O (S), at the free LaAlO 3 surface can explain the interfacial 2DEG.
For this to happen, three conditions ("design principles") need to be satisfied. First, V O (S) in the polar film material needs to have a sufficiently low formation energy ΔH so it could form. Fig.4a shows the ΔH of V O (S) decreases linearly as the film thickness n LAO increases, consistent with previous calculations 20,43 . When n LAO ≥ 3-4 uc, the ΔH becomes zero or negative, and V O (S) will form spontaneously and be stable even for high oxygen chemical potentials. (This suggests that even exposing the surface to air or post-annealing under O-rich environment cannot heal these vacancies). Second, the donor transition level of V O (S) in the polar film material should be higher in energy than the substrate (SrTiO 3 ) conduction band edge at the interface. Fig.3 shows that this condition is also satisfied. Third, the system needs to have a none-zero built-in polar field that would enable the electron to transfer from the surface to the interface. Such transfer sets up an opposite dipole (proportional to n LAO ), which in turn cancels the field and lowers the ΔH. The larger the n LAO , 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. These three conditions are satisfied in this LaAlO 3 /SrTiO 3 system.
It is important to note that the built-in polar field always exists during the layer-by-layer growth. This is because that the surface defects (here V O ) can cancel the built-in polar field only in the LaAlO 3 film between the interface and the surface, not the built-in polar field in the LaAlO 3 film to be grown on top of the surface. Such polar field can then always trigger V O formation at the surface of LaAlO 3 film to be grown, no matter whether the polar field has been cancelled in the LaAlO 3 substrate or not.
The emerging picture for the formation of the 2DEG is that the electrons ionized from V O (S) of the polar film material transfer to the nonpolar substrate material SrTiO 3 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 LaAlO 3 . After the built-in field has been cancelled, the ΔH of V O (S) return to a high value (> 3 eV) characteristic of the bulk, and V O (S) become again hard to form in thermodynamic equilibrium 20 . Thus, the maximum concentration of V O (S) is 0.25/S 2D (where S 2D is 2D unit cell area), i.e., one of eight oxygen missing at surface, donating 0.5 e/S 2D that cancels the polar field in LaAlO 3 completely. The compensation of polar field by V O (S) also means that the band bending in LaAlO 3 due to polar field is also removed. Thus, the LaAlO 3 valence bands fall well below the E F , contrary to what the polar catastrophe model would suggest. Consequently, no free holes can arise from depopulation of the LaAlO 3 valence bands at the surface, consistent with experiments 3,22 .

The emerging design principles for selecting materials that will likely form interface 2DEG:
(i) the nonpolar material has a conduction band minimum (CBM) sitting in the band gap of the polar material; (ii) The polar material has at least one deep donor defect with its donor level higher than the conduction band of the nonpolar material at the interface. The (i) and (ii) in together guarantee that the 2DEG forms only at the n-type interfaces. This picture suggests that the 2DEG at n-type LaAlO 3 /SrTiO 3 interfaces may also be induced and/or tuned by using some surface adsorbates (e.g., H 2 O, H) [44][45][46] or metallic contacts 47 provided that the ionization energy of the surface adsorbate or the metallic contact is not lower than the donor level of the V O (S).

What controls the critical thickness?
The linear decrease of ΔH of V O (S) with increasing polar film thickness n LAO naturally explains the critical thickness L c for the metal-insulator transition. The rate of decrease (i.e., the slope dΔH/dn LAO ) equals 0.19 eV/Å, which is same as the calculated built-in polar field in the defectfree LaAlO 3 film (Supplementary note 4). The V O (S) defects start to form spontaneously when the ΔH becomes zero at an L c of ~4 uc under a typical O-rich growth condition (Fig.3a). For the LaAlO 3 film that is one unit cell thinner than this L c , the calculated ΔH of V O (S) is 0.75 eV, which is too high to produce significant free carrier concentration. Thus, the appearance of V O (S) (and the ensuing metal-insulator transition) at L c is predicted to be sharp, distinct from the gradual appearance of 2DEG behavior as predicted from polar catastrophe model, but consistent with experiments 48 .  49 . This observation was originally explained by L c = ΔΦ!/4πeP 0 (where ΔΦ is the energy difference between LaAlO 3 VBM and SrTiO 3 CBM) within polar catastrophe model 49 . Since ΔΦ and ΔH o have similar value (~3-4 eV), it is not surprising that the L c predicted from these two models is also similar.

Implication on the design of carrier mobility: (i)
The relatively high 2DEG mobility may be enabled by a modulated doping effect 50 , whereby the source of carriers (here at the LaAlO 3 surface) is spatially separated from the place where the carriers reside (here at the LaAlO 3 /SrTiO 3 interface), thus minimizing carrier scattering by the defects. This minimal spatial separation is measured by the critical thickness L c . The relationship L c = ΔH o !/4πeP 0 suggests that a large L c (hence maintaining good mobility) could be achieved by selecting a polar materials with small polarization, large dielectric constant, and donor defects characterized by high formation energy ΔH at the interface or in the bulk. On the other hand, (ii) the concentration of interfacial defects should 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 achiever higher mobility.

What compensates the built-in polar field?
Experimentally, only very weak residual field has been observed in the LaAlO 3 film no matter whether its thickness is below or above the L c [37][38][39][40] . This observation cannot be explained within the defect-free interface scenario. In turn, whereas the V O (S) model explains the weak electric field in LaAlO 3 film above the L c , it does not explain it below the L c . This leads us to inspect the effects of all possible cation antisite defects across the interface. Fig.2ab shows that the La Sr , Sr La , Ti Al , and Al Ti antisite defects have lower ΔH than other point defects (e.g., vacancy) in the layer where they are located. This means that these four antisite defects are the dominating defects in their corresponding layers. Apparently, the interfacial La Sr donor in SrTiO 3 side cannot set up an opposite dipole across the LaAlO 3 film that can cancel the polar field inside the LaAlO 3 film. For the Ti Al donor in LaAlO 3 side, since its donor level is lower than the SrTiO 3 conduction band at the interface, the ionized electrons cannot be transferred to the latter either to cancel the polar field. For interfacial Al Ti and Sr La acceptors, the polar field compensation is similar to that in polar catastrophe model: before the LaAlO 3 VBM reaches the acceptor levels of Al Ti or Sr La , the polar field cannot be cancelled; After that, the electrons start to transfer from LaAlO 3 valence bands to the acceptor levels of these defects, and the internal field decreases gradually as n LAO increase, approaching zero at infinitely large n LAO .  Fig.3. Clearly, the electron-transfer from donor to acceptor in both pair  and  is unlikely since it will create a dipole in the same direction as the intrinsic dipole in LaAlO 3 , and thus increase the dipole moment (also the electrostatic energy) and destabilize the interface. In both pair  and , the charge transfer can cancel the polar field. However, the electron transfer in pair  is energetically much more favorable because (i) Al Ti has a lower acceptor level than Sr La and (ii) 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). Fig.4b (filled symbols) shows that the energy required to form such defect pair is negative (i.e., exothermic), and the largest energy gain is obtained when a Ti atom of TiO 2 -interface monolayer is exchanged with an Al of AlO 2 -surface monolayer, i.e., Al Ti (I)+Ti Al (S), which is consistent with previous first-principles calculations 51 . This means that Ti atom at the interface would hop to the AlO 2 surface layer and exchange with Al atom there. Similar to that of V O (S), the linear decrease of ΔH with increasing donor-acceptor separating distance (Fig.4b) is a sign of polar field compensation by electron transfer from Ti Al donor to Al Ti acceptor, which is expected since the donor level is higher than acceptor level (Fig.3a). Fig.4a also shows that the V O (S) has too high ΔH to form for n LAO < L c (Fig.4a). Therefore, the polar field is cancelled by those spontaneously formed [Al Ti (I)+Ti Al (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).

What controls the density of 2DEG?
Reinterpretation of the puzzle: According to Gauss' law, the experimentally observed weak electric field in LaAlO 3 film means that the total external charge density (mobile and/or immobile) at the interface must be ~0.5 e/S 2D (Supplementary note 6). For n LAO < L c , not conductivity at interfaces means that all (or almost all) interfacial charge does not contribute to the conductivity. For n LAO ≥ L c , only a fraction of 0.5 e/S 2D interfacial charge is seen in transport and so the majority of 0.5 e/S 2D charge does not contribute to the conductivity. The puzzle thus is why the ~0.5 e/S 2D charge exists at the interface with any n LAO but only a small part of it contribute to conducting 2DEG when n LAO ≥ L c . This puzzle cannot be explained within defectfree interface scenario by polar catastrophe model 3,4 or interfacial charge leaking model 36 , since both predict zero interfacial charge for n LAO < L c and an interfacial charge density much higher than the measured 2DEG density for n LAO ≥ L c (Supplementary Fig.S1). The possibility of multiple carrier types (i.e., the electrons occupying d xy and those occupying d xz /d yz sub-bands contribute differently in transport) at ideal interfaces has also been suggested to explain the measured 2DEG density above the L c [52][53][54][55] . Similarly, the existence of such multiple carrier types could not explain the 0.5 e/S 2D interface charge that indeed exists. Moreover, it is also difficult to reconcile why a full carrier density of 0.5 e/S 2D has been observed at GdTiO 3 /SrTiO 3 interfaces (where the same multiple carrier types exist) 56 .

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 ~ 0.5 e/S 2D , which corresponds to the (almost) complete polar field cancellation. In SrTiO 3 side (where the 2DEG is located), there are mainly three types of acceptor defects, namely, Al Ti , V Sr , and V Ti . At equilibrium E F , Fig.2ab shows that these acceptor defects all prefer to stay in negative charge states, i.e., Al Ti 1-, V Sr 2-, and V Ti 4-. (In other defect charge states, these defects have much higher ΔH and are not shown in Fig.2ab). It means that once these defects form they will trap free electrons from the system and get negatively charged. Among these acceptor defects, Al Ti 1acceptors have the lowest ΔH and thus they the most potent electron-trapper. For n LAO < L c , the Al Ti defects resulting from Ti⇔Al exchange trap all free electrons transferred from Ti Al (S) defects, and hence no free carrier can occur. For n LAO ≥ L c , due to V O (S), the ΔH of [Ti Al +Al Ti ] pair changes from negative to positive (Fig.4b), meaning that the concentration of Al Ti defect resulting from Ti⇔Al exchange is reduced, compare to that formed below the L c . Therefore, the Al Ti defect concentration is not sufficient to trap all 0.5e/S 2D electrons transferred from V O (S). Therefore, only a small fraction of 0.5 e/S 2D can contribute to interface 2DEG. 42 may also be understood within above picture. For Al-rich LaAlO 3 film, where both A-site and B-site sublattices are fully occupied (hence having no cation vacancies), the Al Ti anti-sites are the only electron-trapping defects and the incomplete trapping of 0.5e/S 2D interface charge by Al Ti defects leads to interface conductivity. However, for La-rich LaAlO 3 film, where B-site sublattice is not fully occupied, the cation vacancies (V Ti and V Al ) also become main electrontrappers, in addition to Al Ti (I). Though, the concentration of Al Ti is reduced, each cation vacancy induced in La-rich film traps more electrons than an Al Ti . The insulating character can be then attributed to the complete interfacial electron trapping by both interfacial cation vacancies and Al Ti (I).

The recently observed LaAlO 3 cation-stoichiometry effect on 2DEG formation
The picture of Al Ti (I) as main electron-trapping defects may be extended to SrTiO 3 /GdTiO 3 interfaces. The observed full carrier density of 0.5 e/S 2D there 56 can be ascribed to the fact that both SrTiO 3 and GdTiO 3 have the same Ti atom at B-site sublattice and have no Al Ti -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 Al Ti in stoichiometric or Al-rich film), which should be minimized for enhancing carrier density. Such Al Ti -like electron trapping defects may be completely removed by designing other oxide interfaces like GdTiO 3 /SrTiO 3 interfaces, whose bulk components have a common cation atom with multiple valence states.

Why are the p-type interfaces insulating?
The intriguing fact is that the so-called p-type interfaces are not p-type (hole) conducting and are actually insulating. The defect-free polar-catastrophe model for p-type interface predicts a hole conducting interface and an electron conducting surface when n LAO > ~7.3 uc (Supplementary Fig.1) in contradiction with the insulating behavior observed robustly in experiment. To explain this, defects thus be involved. The emerging defect picture below differs from the literature model based solely on interfacial hole-polaron 57 or interfacial holecompensating V O defects 23,43 , which assumes that the interface has holes arising from the depopulation of the intrinsic SrTiO 3 valence bands.
Interfacial point defects can neither cause conductivity nor cancel the polar field. Similar to that at n-type interfaces, the interfacial La Sr and Ti Ar are stable at their charge neutral states and have negligible or negative ΔH at equilibrium E F . It means that they cause inevitable interfacial cation intermixing but induce no free carriers. The V O and other defects at the interface have too high ΔH to form and thus they do not produce free carriers either. Also for similar reason, each leading antisite defect (La Sr , Sr La , Ti Al , or Al Ti ) 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.
In the p-type interfaces, the polar field points from surface to interface, which is opposite to that in the n-type interfaces. Among four donor-acceptor defect pairs as indicted in Fig.3b, the pair  (i.e., [La Sr +Sr La ]) is energetically most favorable in polar field cancellation. For n LAO < ~ 4uc, the [La Sr (I)+Sr La (S)] pairs have negative ΔH (Fig.4d) and can form spontaneously via La⇔Sr exchanges, whereas the [La Sr (I)+V La (S)] have too high ΔH to form (Fig.4c). So the polar field is cancelled by the charge transfer from La Sr (I) to Sr La (S), which can be expected from their relative defect levels (Fig. 3b) and their linear decreasing behavior in ΔH as a function of n LAO (Fig.4d). For n LAO ≥ ~4uc, the ΔH of [La Sr (I)+V La (S)] become negative (Fig.4c) and can also form spontaneously. Since V La has a lower acceptor level than Sr La (Fig.3b), the polar field is cancelled by the charge transfer from La Sr (I) to V La (S), rather than to Sr La (S). In absence of electric field, Fig.4d (open symbols) indicates that the La⇔Sr exchanges cannot occur anymore over a distance of ~ 4uc. Unlike that in n-type interfaces, the V O (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 E F according to those point defects turns out to stay always around the middle of SrTiO 3 band gap. It means that both VBM and CBM are far away from the E F 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 two-dimensional hole conductivity:
Clearly, the formation of interfacial free holes is prevented by these spontaneously formed deep La Sr defects that have donor level higher than the VBM at the interface. So to induce interfacial hole conductivity, one should search for the polar-nonpolar interfaces where all such donors have too 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.

What causes interface magnetism?
Distinct form previous models 31-34 that explain magnetism based on interfacial Ti(3+) within the SrTiO 3 (i.e, not a defect), we find below that the local magnetic moment originates from the un-ionized deep Ti Al anti site defect (i.e., Ti(3+)-on-Al(3+) within LaAlO 3 side near the interface. The interface magnetism depends on the concentration and spatial distribution of such Ti Al 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 interfaces 8 .

What causes local moment:
For n-type interfaces, when n LAO < L c , the polar field in LaAlO 3 is cancelled by the charge transfer from Ti Al (S) defects to the interface. These formed Ti Al defects are thus ionized, i.e., Ti Al 1+ (where superscript denotes the defect charge states). The Ti atom at this defect site has oxidation states of 4+, denoted as, Ti(4+) which has no local magnetic moment. When n LAO ≥ L c , the polar field in LaAlO 3 is cancelled by the charge transfer to the interface from V O (S) instead of Ti Al . All Ti Al (I) defects in LaAlO 3 film in absence of internal field are most stable in their charge neutral (or un-ionized) states, i.e., Ti Al 0 , where Ti appears as Ti(3+) oxidation state, having a finite local magnetic moment. Therefore, the interface magnetism at n-type interfaces due to those un-ionized Ti Al 0 defects should also have a critical thickness of ~4uc. For p-type interfaces, it is the charge transfer among the defects other than Ti Al defects that cancels the polar field in in LaAlO 3 . So all formed Ti Al defects there are not ionized, having local magnetic moments, and can cause interface magnetism.

The magnitude of local moment:
The local moment of a single Ti Al defect at the interface can be estimated from that in bulk LaAlO 3 , which is 0.84µ B from our hybrid functional calculation. For ferromagnetic order as observed in experiment, the total interface magnetic moment depends on the concentration of un-ionized Ti Al defects in LaAlO 3 and can be very small per Ti atom in average. The experimentally observed inhomogeneous landscape of magnetism that also varies from sample to sample 8,9 may be attributed to the various spatial distributions of Ti Al defects, which may be sensitive to sample preparation conditions (such as temperature and P O2 ) and local strain.

Comparing with V O (I) model:
Relative to V O (I) as the origin of local moment, the Ti Al (I) defects within LaAlO side are more reasonable in at least two aspects. First, the deep Ti Al defect is spatially localized and has an unambiguous local moment. While for V O , which is a shallow donor, it donates electrons to the lower-energy interfacial Ti d xy sub-bands that have light effective mass inside the interface plane 55 , and the resulting Ti(3+) may then be itinerant. Second, the Ti Al defects would form easily or even inevitable due to its small or negative ΔH, whereas the interfacial V O require significant energy to form and if formed they may be removed completely after annealing.

Discussion
This work establishes 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. This link reveals a unifying mechanism that simultaneously explains the experimental observations listed in Table 1. This mechanism leads to a set of design rules for both conductivity and magnetism at the LaAlO 3 /SrTiO 3 interfaces and enables the design of other polar-nonpolar interfaces (not limited to oxides) with similar or improved interface properties by first principles defect calculations.
Having ruled out the electronic reconstruction, interfacial V O , and interfacial cation intermixing mechanism as the possible origin of 2DEG in our calculations, we conclude that the 2DEG at ntype interfaces with n LAO ≥ L c originates from the spontaneously formed V O (S) defects. This conclusion stems from the finding that the donor level of deep V O in in LaAlO 3 side is higher than the SrTiO 3 conduction band edge at the interface. This finding explains why the formation energy of V O (S) decreases linearly as n LAO increases. The linear decreasing relation reveals a general formula and some new controlling parameters for the critical thickness of sharp metalinsulator transition in absence of the electric field in the polar LaAlO 3 film.
Instead of causing 2DEG, we find that the defects resulting from interfacial cation anti sites play key roles in other aspects. Specifically, (i) the cation intermixing cancels the polar field only below the critical thickness for both-type of interfaces. (ii) the interfacial Al Ti acceptor defects control the density of 2DEG. (iii) the un-ionized interfacial Ti Al defects within LaAlO 3 side cause the local magnetic moments. The (iii) suggests that all previous models on interface magnetism based on Ti(3+) ions in SrTiO 3 side should be revisited. Importantly, we find that the cation mixing is confined near the interface and does not extend in LaAlO 3 side over more than ~4 uc, above which the cation intermixing is prevented by spontaneous formation of some surface vacancy defects. This spatial confinement is critical for realizing nanoscale interface magnetism.
This mechanism provides three distinctive predictions to be tested in experiment as further validation. (i) For n-type interfaces, the AlO 2 -surface layer is dominated by Ti Al defects when n LAO < L c and by V O defect when n LAO ≥ L c . (ii) For p-type interfaces, the LaO-surface layer is dominated by Sr La and V La defects, respectively, below and above an L c of ~4 uc. (iii) Ti 4+ and Ti 3+ signals exist in both sides of the interface. The appearance of the Ti 3+ signals should not be taken as a sign of conductivity. Whether the Ti 3+ signals detected by photoemission below the L c 21,39,58,59 can be truly assigned to those Ti 3+ ions in SrTiO 3 side should be revisited carefully. How these Ti Al 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
All calculations were performed using density functional theory (DFT) and plane-wave projector augmented-wave (PAW) 60 method as implemented in the VASP code 61 . 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. The in-plane lattice constant was fixed to 3.943 Å (the relaxed lattice constant of SrTiO 3 by GGA 62 ). In slab calculations, the 4 uc (~16 Å) vacuum layer was used and the dipole correction was always applied to remove artificial dipole interactions 63 . The results in Fig.2 and Fig.3 were obtained by using HSE hybrid functional 64 on the top of the GGA relaxed structures.

The formation energy of a defect (D) calculated from
! and ! ! are the total energies of a supercell with and without defect, respectively, and D being in charge state q. n α is the number of atoms of specie α needed to create a defect. E F is the Fermi energy relative to VBM (E v ). Δµ α is the relative chemical potential of specie α with respect to its elemental solid (gas) (µ 0 ). The equilibrium Fermi-energy was calculated self-consistently according to charge neutrality condition 65 . 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 formation of other competing phases in thermodynamic equilibrium ( Supplementary Fig.S2). The details of theory and calculations can be found in Ref. [66]. The symbol of '✔' and '✖' mean that the mechanism agrees or disagrees, respectively with the experimental observation. The '?' symbol denotes uncertainty.   Figure. Other cation defects that have higher ΔH are shown in Supplementary Fig. 3. 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 P O2 = 6.1×10-6 Torr. where the ΔH of a given defect in two different charge states equal. Some defects may have multiple charge transition energy levels. For example, V Sr 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. Supplementary Information for "A polarity-induced defect mechanism for conductivity and magnetism at oxide interfaces" by Liping Yu and Alex Zunger Supplementary Note 1:

The failures of polar catastrophe model
The key argument in favor of the polar catastrophe mechanism is its predicted critical thickness (L c ) of metal-insulator transition. Both electrostatics and the first principles calculations based on density functional theory (DFT) with the generalized gradient approximation (GGA) 1 , shown in Fig. S1c lead to an L c between three and four unit cells 2,3 . However, these results may be clouded by specific uncertainties: the electrostatic model depends on the choice of the LAO film dielectric constant and LAO/STO band offsets, whereas the GGA calculation suffers from the well-known band gap underestimation problem 4 . To examine this point we have applied the HSE hybrid functional 5 , which predicts correctly the experimental bulk STO band gap of 3.2 eV. We find an L c of 4-5 uc (Fig. S2c). It is noteworthy that, for n LAO = 4 uc, the band gap of the system turns out to be 1.1 eV, suggesting an insulating behavior, contrasting with the experiment where the robust conductivity has been observed at this thickness. Hence, strictly speaking, the defect-free polar catastrophe model cannot explain the L c either 6 .
In a defect-free interface structure, both previous DFT-GGA calculations 7,8 and our HSE calculation (Fig.S2d) predict a large electric field of 0.19 V/Å in the LAO film of n LAO < L c , which decreases gradually as n LAO increases for n LAO ≥ L c , approaching zero only at infinitely thick n LAO . This prediction of the PC model conflicts with experiments, where only very weak residual field has been observed for all investigated n LAO 9-12 .

Thermodynamical equilibrium Fermi energy as a function of LaAlO 3 thickness
Under a given growth condition, the thermodynamic equilibrium E F is established as a balance of all electrons and holes concentrations contributed by thermal ionization of all considered defects. According to charge neutrality condition 13 , the E F can be calculated self-consistently from first principles using defect formation energies as inputs.
As seen in the main text, the charge transfer from surface defect to interface defect always cancels the built-in polar field. For the n-type interface below the critical thickness and p-type interfaces at any thickness, all involved defects are deep defects, the resulting equilibrium E F always sits near the middle of SrTiO 3 band gap. While for n-type interfaces above the critical thickness, the surface O vacancies (V O ) form spontaneously. This V O defect has a donor level higher than the SrTiO 3 conduction band at the interface. The electrons will be transferred from Vo donor level to the SrTiO 3 conduction bands and pins the E F near the SrTiO 3 valence band minimum.

Supplementary Note 3:
Justification for the thermodynamic equilibrium The thin film growth is an out of equilibrium process. Our thermodynamic calculations may be still applied. The more detailed justification can be found in Ref [14]. In brief, the calculated formation energies and defect levels (deep or shallow) in this work are physically meaningful for both equilibrium and non-equilibrium processes. Non-equilibrium implies that once certain high-energy defects form, kinetic barriers may preserve them, even if their concentration exceeds the nominal equilibrium value. However, it should be clear that defects with high formation energy would always be unlikely to form, since a lot of energy needs to be expended in their creation, and the driving force to lowering the energy is large.

Supplementary Note 4:
The In a supercell calculation, the formation energy of an oxygen vacancy at LAO surface (denoted as V O (s) hereafter) is where ℰ ! ! and ℰ ! ! are the total energies of the supercell structures in the absence of an electric field across the LAO film with and without the vacancy. The second term in each of the brackets corresponds to the electrostatic energy rise due to the presence of internal electric field ! ! and E in LAO with and without V O , respectively. In a 2×2 2D supercell considered here the creation of a single V O (S) leads to zero electric field in LAO (! ! = 0 for all n LAO ). 8,15,16 Thus, where the second term can now be viewed as an opposite dipole, which decreases linearly at a rate of eE as n LAO increases. Therefore, the larger the n LAO , the larger the opposite dipole moment that lowers the total energy of the system, and the more energy decrease in ΔH. For V O (I) defects, which can not create an opposite dipole that lowers the ΔH, the ΔH is thus independent of n LAO and remains high as n LAO increases 8 .
The slope dΔH/dn LAO ) remains same or almost same as V O (S) defects accumulate gradually. This is because that the built-in polar field is created by the polar charge at surface/interface. Microscopically, the electron transfer from "one V O (S)" to the "interface" mainly cancels the polar field caused by the "polar charge at this V O (S) site", not the "polar field caused by the polar charge at other V O (S) defect sites". It means that during gradual formation of V O (S), though the in-plane-averaged electric field is lowered, each V O (S) to be formed still face the (almost) same polar field to be cancelled by it. Hence, the decrease in ΔH of V O (S) should be almost same as VO(S) defects accumulates.

Critical thickness vs oxygen chemical potential
Eq.(1) suggests that the critical L c can be approximately determined by the ΔH of the interfacial V O in TiO 2 layer and the electric field E in the defect-free LAO film (from electrostatics), via which is a function of the oxygen chemical potential µ O and the built-in electric field in defectfree LAO. Supplementary Figure 1b shows the variation of resulting L c due to V O (S) with respect to Δµ O (i.e., the oxygen chemical potential relative to 1/2O 2 ) at T = 0K. The corresponding Δµ O of current growth conditions ranges from -1.73 to -2.25 eV, where the L c is found to be between 3.1 and 3.8 uc. The O-rich annealing conditions have Δµ O ranging from -0.61 to -1.44 eV, where the resulting L c varies from 4.2 to 5.3 uc. Considering the possible high kinetic barrier associated with the low-temperature annealing, the O vacancies induced during the growth generally may not be completely compensated, and thus it is expected that the actually L c in the annealed samples be ~ 4 uc, consistent with experiments.

Supplementary Note 6:
Interfacial charge vs. internal electric field in polar film In absence of externally applied electric field, the electric displacement D STO = 0 in SrTiO 3 side, and D LAO = E + 4πP in LaAlO 3 side. According to Gauss' law, the external interface charge Q = D LAO -D STO = E+4πP. For E = 0, Q = 4πP = 0.5 e per two dimensional unit cell area, which is the maximum. For E = -4πP, Q = 0, which is the minimum, i.e., the case below the L c within the polar catastrophe model. Figure S1: Electronic properties of abrupt defect-free interfaces within the polar catastrophe scenario. a,b, schematic band diagram for n-type and p-type interfaces respectively. c, GGA and HSE band gaps of 6STO/nLAO/vacuum interfaces. d, Calculated macroscopic planar-averaged electric field in the center of LAO film by HSE. The experimental data (*) is taken from ref. 10 . e, Calculated interfacial carrier density: HSE (black dots) vs. experiment (red stars) 17 . The orange lines in d,e are calculated from the simple electrostatic model using Φ=3.3 eV (see a) and the LAO film dielectric constant of 30, suggesting a critical thickness of 4.3 uc for n-type interfaces. Using the same Φ and dielectric constant for p-type interfaces, the resulting critical thickness is 7.3 uc, which is shown in b. Figure S2: Thermodynamic equilibrium growth conditions and critical thickness for STO/LAO interface. a, typical growth (circles) and annealing (diamonds) conditions vs. oxygen chemical potential ∆! ! defined relative to O 2 molecule. The ∆! ! that corresponds to the experimental (P O2 , T) growth conditions was calculated according to the thermodynamic model 18 . The red dashed line corresponds to T = 1050 K and P O2 = 6.1×10 -6 Torr, which is referred in Fig.2 Fig.3 refer to ∆! ! = -2.0 eV. c, the critical thickness (L c ) as a function of O chemical potential used during growth and annealing process. Figure S3: Properties of interfacial defects in the 6STO/2LAO heterostructure containing both n--type and p--type interfaces. a, n-type interface. b, p-type interface. Each panel in a,b shows various defects in a given atomic layer. Each line represents the ΔH of a donor (red) or acceptor (blue) defect. Different slops of line segments represent different charge states of a defect that are most stable at given E F . Open circles mark the defect charge transition energies, i.e., the E F where the formation energy of a defect in two different charge states equal. The shaded regions in each panel denote the variation range of the equilibrium E F . The chemical potentials 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 -6 Torr.