{'fraction_non_alphanumeric': 0.03263133533904189, 'fraction_numerical': 0.020431778358184282, 'mean_word_length': 4.345113977734582, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Oxygen vacancies at titanate interfaces: two-dimensional magnetism and orbital reconstruction

We show that oxygen vacancies at titanate interfaces induce a complex multiorbital reconstruction which involves a lowering of the local symmetry and an inversion of t2g and eg orbitals resulting in the occupation of the eg orbitals of Ti atoms neighboring the O vacancy. The orbital reconstruction depends strongly on the clustering of O vacancies and can be accompanied by a magnetic splitting between the local eg orbitals with lobes directed towards the vacancy and interface dxy orbitals. The reconstruction generates a two-dimensional interface magnetic state not observed in bulk SrTiO3. Using generalized gradient approximation (LSDA) with intra-atomic Coulomb repulsion (GGA+U), we find that this magnetic state is common for titanate surfaces and interfaces.

Two distinct doping mechanisms, controlled by the growth conditions of the samples, appear to be responsible for the formation of interface electron carriers [19][20][21][22][23] . The intrinsic polar charge doping in the heterostructures prepared at high oxygen pressures (10 −4 mbar) suggests doping with an upper carrier density limit of 3 · 10 14 cm −2 which corresponds to 0.5 electrons per interface unit cell (uc) and compensates the interface polar discontinuity. The resulting two-dimensional electron liquid forms in a nm-thick interface layer 20,21,24-28 and can be a possible source for superconducting properties tuned by gate electrostatic fields 20,29 . The extrinsic mechanism of charge doping due to oxygen vacancies dominates in the LAO/STO samples grown at lower oxygen pressures (10 −6 mbar) reported in Refs. 2 and 4. Remarkably, the oxygen reduced heterostructures are characterized by higher charge densities of 10 16 − 10 17 cm −2 with the charge spatially extended into the µm-thick interfacial layer inside the STO 21 . Recent cathode and photoluminescence experiments 4 provide direct evidence for oxygen vacancies in the STO-substrate. The oxygen vacancies can diffuse in STO with activation energy of 0.75 eV, which is comparable to the activation energy 1 eV of the vacancy migration through the LAO layers 30 . At each oxygen vacancy in the TiO 2 layer, two electrons are weakly bound to retain charge neutrality of a configuration in which the two titanium ions in the dimer share two electrons. Due to these moderately bound electrons the oxygen vacancies act as n-type dopants. We note though that in STO, these O-vacancy induced, weakly bound electrons do not contribute to resolve the polar discontinuity problem since they simply are there to produce charge neutrality upon removing the oxygens. The O-vacancy induced electrons are bound to the effective 2+ charge of the vacancy. Below we argue that these electrons enhance the interface conductivity considerably.
A critical issue related to the electron concentration is the electric conductivity of LAO/STO heterointerfaces. The experimental measurements show that the conductivity of n-type LAO/STO bilayers increases with lowering the oxygen partial pressure p O2 and exhibits an abrupt jump from 10 −2 to 10 2 Ω −1 / when p O2 changes from 10 −5 to 10 −6 mbar 2,4 . The high conductivity shown by samples grown at low oxygen pressures is caused by a large amount of oxygen vacancies which act as electrondonors in such samples 2,13 .
The local electronic state of the vacancy-containing transition metal oxides can exhibit extraordinary properties not observed in chemically stoichiometric materials. Recent first principle studies demonstrate a vacancyrelated magnetic exchange splitting in nonmagnetic materials like CaO or LAO/STO 10,31 . For LAO/STO, calculations of the vacancy formation energies 10,32 suggest a predominant positioning of O-vacancies in the top AlO 2 surfaces. The energy barrier for transport of O-vacancy from the interface TiO 2 to the surface AlO 2 layer strongly depends on the vacancy concentration 10,33 .
Recent STM, cathode luminescence studies and conductivity measurements provide strong support for clustering of the oxygen vacancies in STO 4,13 . Moreover, the STM experiments and DFT+U studies suggest the possibility that oxygen vacancies form pieces of linear chains, which appear to be more energetically favourable 34 as compared to the isolated vacancies [35][36][37] .
In this work, we explore the electronic structure of the first titanate surface/interface layers considering different configurations and concentrations of oxygen vacancies. In bulk SrTiO 3 , the octahedral crystal ligand field, covalency and Coulomb repulsion result in higher energies of Ti e g states as compared to t 2g states with a splitting of about 2.5 eV [38][39][40] . Here we find that in contrast to the bulk stoichiometric SrTiO 3 , each Ti-O v -Ti dimer contains missing hybridization links which strongly lower the energy of the d orbitals with lobes pointing to the vacancy. As a result, we obtain a splitting of the doubly degenerate e g orbitals and the triply degenerate t 2g orbitals into orbital doublets and singlets. In the pure material, these e g -like antibonding orbitals are strongly pushed up due to the strong hybrization with the lower-energy O 2p orbitals. In the oxygen-reduced LAO/STO, the missing local covalent bonding and the local symmetry lowering results in a new kind of orbital reconstruction at titanate interfaces: a shift and partial occupation of the vacancy-directed Ti e g -orbitals accompanied by their magnetic splitting and mixing with the 3d xy states. Our calculations of SrTiO 3 -surfaces and LAO/STO interfaces demonstrate the universal character of the orbital reconstruction of the titanates due to surface/interface oxygen vacancies. In the linear clusters of vacancy stripes, in which each Ti is accompanied by two O-vacancies the e g orbital state pointing along the Ti vacancy direction, is further strongly lowered. In these e g states, both spin states are occupied, although a small amount of spin polarization is also present. The high spin polarization occurs predominantly in the t 2g -states.
We also show that the electronic states of the oxygenreduced STO and LAO layers are physically different. The vacancies in TiO 2 layers release extra 3d electrons which are confined to the quasi-two-dimensional layer of the electron liquid at the LAO/STO interface. In contrast, the oxygen vacancies in the LAO overlayer produce local electronic states of a mixed sp character which are redistributed between the surface AlO 2 and interface TiO 2 layers resulting in the confinement of electrons in the parallel, surface and interface 2D layers with different effective masses, a concept which has been recently discussed in Refs. 41 and 42. We find that for small concentrations of oxygen vacancies c V ≤ 1/8 at the AlO 2 surface, the vacancy released electrons are transferred to the LAO/STO interface and compensate the interface polardiscontinuity, which also leads to an insulating surface state.

II. OXYGEN VACANCIES AT LAO/STO INTERFACES
We use supercells in density functional theory to explore the charge and spin state around oxygen vacancies at LAO/STO heterointerfaces. The supercells contain a 4 uc thick LAO overlayer deposited on a STO layer of a thickness between 1 uc and 6 uc. The LAO-STO-LAO structures are separated by a 13Å thick vacuum sheet. In the supercell, oxygen vacancies can be located in the interface TiO 2 layer or in one of the AlO 2 -planes of LAO. Cells with three types of oxygen vacancy configurations in MO 2 (M=Ti, Al) are sketched in Fig. 1 Fig. 1, which as in configuration (a) has Ti-O v -Ti dimers but the density of vacancies is a factor of two smaller.
The density functional calculations were performed using the Generalized Gradient Approximation (GGA) in the Perdew-Burke-Ernzerhof pseudopotential implementation 43 in the Quantum Espresso (QE) package 44 . We use a kinetic energy cutoff of 640 eV, and the Brillouin zone of the 106-to 166-atom supercells is sampled with 5×5×1 to 9×9×1 k-point grids. An increase of the k-point mesh from (5x5x1) to (7x7x1) leads to a negligibly small change of the total energy by 0.005 Ry and to an increase of the Ti magnetic moments by small values of about 0.05 µ B for the case that O-vacancies are present. In our calculations we account for a local Coulomb repulsion of Ti 3d electrons by employing a GGA+U approach with U Ti = 2 eV 45 . The supercells have been structurally relaxed by a combination of the optimization procedures of the full potential WIEN2k-package and the pseudopotential QE package 44,46 . The in-plane lattice constants have been fixed to their room-temperature bulk-STO cubic values (a STO = b STO = 3.905Å). Although low-temperature SrTiO 3 is tetragonal with the lattice parameters a = 3.896Å and c = 3.899Å, in our studies we assume that the small difference between the lattice constants should not seriously affect the state of oxygen vacancies.

A. Vacancy stripe configurations
We start with the confiuration with the largest degree of clustering of O-vacancies in our models which is the vacancy stripes of type (b). Fig. 2 presents the spin polarized projected densities of 3d states for the Ti(0.5a,0) atom nearest to the O-vacancies of type (b) and for the Ti(0,0) at the corners of the Ti 2 O 4 plaquette in Fig. 1(b) located still in the interface layer but furthest away from the O vacancy stripe. For the Ti(0.5a,0) atom ( Fig. 2(a)), we find a strong "splitting" of the 3d 3y 2 −r 2 and 3d z 2 −x 2 states with the 3d 3y 2 −r 2 being almost completely occupied with close to 2 electrons and the 3d z 2 −x 2 almost completely empty and at very high energies above E f . The extremely narrow e g DOS peaks are located in the energy window (E F − 2.3 eV;E F + 0.5 eV) relative to the Fermi level. This is in contrast to the DOS of the Ti(0,0) which is more distant to the oxygen vacancy ( Fig. 2(c)). At Ti(0,0) the empty e g states are located about 3.5 eV above E f and 2 eV above the t 2g states, the latter being partially occupied by the electrons generated due to the polar interface.
To better understand the nature of the sharp e g peaks, we plot in Fig. 2(b) the 3d 3y 2 −r 2 density of states and see that actually the DOS in the sharp peaks is almost solely in an e g 3d 3y 2 −r 2 orbital with the lobes pointing along the Ti-O v direction, which implies a strong confinement of the vacancy-released electrons between the nearest Ti atoms. This strong preference of the 3d 3y 2 −r 2 orbital occupation is clearly seen also in the spatial charge density distribution in Fig 4(b). The very sharp peaks in the e g density of states is a result of the quasi-one-dimensional character of the band structure of these mainly 3d 3y 2 −r 2 composed bands. For charge neutrality we need to accomodate two electrons per Ti in the dimerized chain which aside from the polar induced charge indicates that the Ti ions in the chains are formally divalent with nearly two electrons each. We see indeed from Fig. 2(b) that both the spin up and down 3d 3y 2 −r 2 states are nearly fully occupied.These states are therefore rather magnetically inert. The empty 3d 3y 2 −r 2 stripe projected DOS, also clearly visible in Fig. 2(b), results from the other Ti ions which do not neighbor oxygen vacancies.
The extreme splitting of the e g states 3d 3y 2 −r 2 directed along the Ti-vacancy direction reflects the interface orbital reconstruction, a phenomenon recently observed at the interfaces between (Y,Ca)Ba  To estimate the chemical valence of the Ti atoms, we have calculated the 2D electron charge density in two different energy windows below the Fermi level corresponding to occupied e g and t 2g orbitals, and performed the integration in (x,y) planes across the interface. The results are presented in Fig. 3. The integration of the e g charge profile along the z-axis gives a total of about 1.9 electrons, which implies that the excess vacancy-produced charge is almost fully transferred to the e g orbitals and induces a change of the chemical valence of the nearest Ti atoms by −1.9. In contrast to the stoichiometric STO, the local e g -states originating from the vacancies in the interface TiO 2 layers of the STO are placed well below the Fermi level. The clustering of the vacancies into stripes at the TiO 2 /LaO interface leads to a strong concentration of the vacancy-released electrons due to the shift of the filled e g subbands by more than 1 eV below the Fermi level. Integration of the conducting (t 2g ) charge profile shown in Fig. 3 reveals that the charge of the 0.3 t 2g electrons in the TiO 2 layers near the interface is compensated by the charge of the holes present at the AlO 2 surface of the LAO film.
In contrast to the strong spin polarization of the t 2g states observed in Fig. 2, the occupied e g states are almost unpolarized. The integration of the sharp e g peaks below the Fermi level gives a negligibly small value of the spin polarization of about 0.001 µ B , which implies the existence of the orbital separation of the spin and charge degrees of freedom in the vacancy stripe configurations: the vacancy-released charge is localized in the spin-nonpolar e g -orbitals, whereas only the t 2g polarity-induced intrinsic charge contributes to the magnetization. The contribution from different orbitals to the magnetization of various vacancy configurations is presented in Table I. Fig. 4 shows the spatial distribution of the charge density generated in the energy windows below the Fermi level for the filled t 2g (Fig. 4(a)) and e g states ( Fig. 4(b)). One can see that the polarization-generated t 2g electrons mostly occupy the Ti 3d xy and O 2p y orbitals which are hybridized and do not contain any contribution from the oxygen vacancy state. In contrast, the vacancy-generated electronic charge in Fig. 4(b) has a mixed 3d x 2 −y 2 /3d 3z 2 −r 2 character (or 3d 3y 2 −r 2 character in the projection along the Ti-O v -Ti).
Due to Coulomb attraction with the effective 2+ charge of the nearest oxygen vacancy, the excess electron charge is trapped in the in neighboring Ti sites and because of the change in local e g -O 2p hybridization the charge is mainly found in Ti orbitals of 3d 3y 2 −r 2 character. These are placed around the center of the O-vacancy, similar to the bulk STO with point defects 37 .
The strong localization of the excess electrons is already noted in the oxygen-reduced bulk STO, where the tendency for the formation of local Ti states is connected to the degree of clustering of the oxygen vacancies 36 . Similar to bulk STO, a weaker clustering in LAO/STO heterointerfaces reduces the energy gap between the filled and empty Ti e g states to less than 1 eV, which is demon-  gap between the electronically occupied and the empty spin-up (majority) e g states. The occupied e g states are located in the energy window (E F − 1.5 eV; E F ) and hybridize with the t 2g states. As a consequence, the electronic state of the Ti-O v -Ti dimer is half-metallic with conducting electrons of mixed e g -t 2g character released in part by the O-vacancy and from the polar induced electronic reconstruction involving mainly t 2g states. The mixed e g -d xy character of the electrons in the Ti-O v -Ti dimer reflects itself in the interface charge-density plot in Fig. 6. Due to the additional magnetic splitting of the e g states, the magnetic moment of Ti in the relaxed struc-  Table I).

C. Ti dimer-type O-vacancies in (2 × 2)-plaquettes
To analyze lower concentrations of oxygen vacancies, we studied larger supercells which contain (2 × 2) M 4 O 8plaquettes in the (x,y) planes ( Fig. 1(c)). The elim-  ination of one oxygen atom in the interface (TiO 2 ) 4plaquette corresponds to c V = 1/8-concentration or 1.5 · 10 14 cm −2 density of vacancies homogeneously distributed in the interface TiO 2 layer. The orbital projected Ti 3d DOS for the vacancy configuration of type (c) is displayed in Fig. 7. As shown, even the smaller concentration of oxygen vacancies still causes a splitting of the e g orbitals of Ti atoms in the close proximity of the vacancy. Similar to the other dimerized configurations, the occupied e g and t 2g states are located in the same energy window (E F − 1.5 eV; E F ) which implies a mixed e g -t 2g character of the vacancy-generated electron states. Fig. 8(a) presents the conducting electron den- sity profile across the interface calculated by integrating the planar charge density generated in the energy range (E F − 1.5 eV; E F ). The location of the oxygen vacancies in STO leads to an insulating state of the AlO 2 -surface. In Fig. 8, the density plots are restricted to the interface region and are zero beyond 3Å from the interface due to insulating character of the top AlO 2 layer. Moreover, the integration of the charge profile in Fig. 8(a) gives exactly two electronic charges, in accordance with the charge released by the oxygen vacancy, which implies the absence of the polarity-induced interface electrons and a suppression of the polar character of the LAO/STO interface. The reason for this at first glance surprising effect is discussed in the next section.

D. Vacancy-enhanced LAO-critical thickness
An examination of the atomic positions in the structures with interface O-vacancies reveals a considerable increase of the atomic distortions near the vacancy in both the interface TiO 2 and in the LaO layers, presented in Table II. These distortions originate from the repulsion between La ionic charges and the nearest positive charged O-vacancy. Table II  vacancies in the interface TiO2-layer. The macroscopic potential is calculated by the macroscopic averaging procedure proposed in Ref. 51. As compared to the stoichiometric system with the macroscopic electric field E st = 0.17 eV/Å screened by the polar charge, the macroscopic electric field in the structure with the interface vacancies is screened by the field ∆P vac e generated by antipolar distortions and approaches the value E vc = 0.13 eV/Å . For the obtained internal field 0.13 eV/Å , the critical LAO-thickess sufficient for the dielectric breakdown of LAO/STO with the STO band gap 3.2 eV 27 in the structure with the interface O-vacancies is enhanced up to 24.6Å which corresponds to a 7 u.c. thick LAO layer.

III. A MODEL FOR THE eg LEVEL SPLITTING
The vacancy-induced Ti 3d orbital reconstruction can be understood from the analysis of the local bonding for a two-electron state of a Ti-O v -Ti-cluster (Fig. 10).
We focus on the contribution of the covalent Ti3d-O2p bonding to the diagonal and exchange contributions to the electron energy levels which determine the magnetic state (paramagnetic singlet or ferromagnetic triplet) of the two-electron cluster.
In the stoichiometric SrTiO 3 , the two-electron energy contains a pd-hybridization term due to the overlap between the 3d-states ot Ti atoms and 2p-states of the central oxygen atom. In a TiO 6 -octahedron, such a covalent contribution V pd produces shifts of the one-electron 3d energy levels 38 Here ε t = −6.258 eV and ε e = −5.638 eV are the Ti 3d ionization energies plus Madelung potential, renormalized by the electrostatic shifts due to the cubic field. The corresponding energies for the O 2p states are: ε || = −10.519 eV and ε ⊥ = −10.039 eV. In Eq. (1), s α = cos k α a (α = x, y, z) and s 2 = α s 4 α − α =β s 2 α s 2 β . With the pd-covalency parameters V pdσ ≈ 2.1eV and V pdπ ≈ 0.84eV defined in Refs. 39 and 40, we can estimate the splitting between the anti-bonding e g -and t 2g -states in the Γ-point of the Brillouin zone [38][39][40] : whereṼ pdσ = √ 6V pdσ ,Ṽ pdπ = 2 √ 2V pdπ , and ∆ 0 ≈ 4.5 eV is the bare gap between the 2p and 3d energy levels, unrenormalized by the covalent overlap.
The elimination of the oxygens in a · · · -O-Ti-O-Ti-· · · stripe along the y-direction is equivalent to the condition V pdσ s y = V pdπ s y = 0. Consequently, the absence of the local covalency lowers the local symmetry and leads to a splitting of the e g states with the lower energy state corresponding to a 3d 3y 2 −r 2 orbital energy with lobes along the bond direction and the one at an almost unchanged energy of 3d x 2 −z 2 i.e. with lobes in a plane perpendicular to the Ti-O v direction (see scheme of Ti-O v -Ti cluster in Fig. 10) with the splitting energies These estimates are for the stripe configuration which two O-vacancy neighbors for each Ti. The negative splitting ∆ 2 gt implies a strong negative shift of the e g -orbital with lobes pointing to the vacancy. The e g -orbital is shifted below the t 2g energy levels, the corresponding orbital inversion is observed in the orbital density of states in Fig. 2(b). In the dimerized Ti-O v -Ti configuration (a) shown in Fig. 10, the smaller splitting energy ∆ 2 gt ≈ −1 eV results from the missing of only half of oxygens in the vacancy stripe configuration.
The vacancy-induced orbital reconstruction is not restricted to the LAO/STO interfaces, but has a generic character for the titanate surfaces and interfaces. It is related to the specific electronic structure of Ti, with the covalence-induced splitting between the high-energy  Fig. 1(a). Panels (a), (c) and panels (b), (d) display the up-spin and down-spin energies, respectively. e g and low-energy t 2g states. For example, in a TiO 2terminated (001)-surface layer of a SrTiO 3 slab, the absence of the top covalent-bonded oxygens results in the five-fold coordination of surface Ti atoms with consequent d-orbital splitting and reduction of the energy gap by about 0.6 eV 38,52 . As a consequence, a surface Ovacancy in a TiO 2 -terminated (001)-layer is equivalent to the additional condition V pdσ s z = 0 in Eq. (1), which leads to a negative shift of the energy of one of the surface e g -states similarly to ∆ 2 gt of Eq. (4) for an interface layer. Our calculations of a one-uc thick STO-slab indeed confirm the formation of a surface 3d 3y 2 −r 2 state in the gap shifted by 0.6 eV below the 3d xz band (see Fig. 11).
In the bulk stoichiometric SrTiO 3 , the nonmagnetic 3d-electron state is associated with a small exchange energy splitting of about 0.06 eV separating singlet and triplet states 53 . As follows from the GGA+U electrondensity contours (Figs. 4(b) and 6), the oxygen vacancy induces a strong spatial shift of the 3d electron density between Ti atoms of about 0.5Å towards the center of Ti-O v -Ti-dimer, which increases the overlap exchange integrals by about 0.36 eV at the LAO/STO-interfaces of type (b) (Fig. 12). For the dimerized vacancy configuration (a), we obtain from the comparison of the total energy of ferromagnetic and antiferromagnetic configurations the value of the exchange of J = 0.28 eV, however we note that the exchange energy is typically overestimated in the DFT calculations.
The structural relaxation of the vacancy surroundings leads to alternating rotations of the z-directed Ti 1 O and Ti 2 O bonds by ∆Θ = ±8 • and to a tilting of the reduced TiO 5 -octahedra towards the vacancy center in the (x,z)and (y,z)-planes (Fig. 11), which increases the overlap. Due to the large J > 0, the two-electron local state on a Ti-O v -Ti forms a triplet with a triplet energy E 2 − J where E 2 is the two-electron energy in the Ti-O v -Ti cluster. Due to the planar electron transfer between the nearest Ti 2 O 4 -plaquettes, the magnetic triplet state originating from the localized O-vacancies, is spread within the interface TiO 2 -plane and is to be considered as twodimensional magnetic ordering stabilized by the exchange splitting of the surface 3d bands.
The notion that the magnetic ordering at the interface has a two-dimensional character is confirmed by calculations of the supercells in which the oxygen vacancy is residing in more distant layers from the interface TiO 2layer. In spite of the occurrence of the local orbital reconstruction in the vicinity of the O-vacancy (Fig. 13 For this calculation, a supercell LAO(4uc)/STO(4uc) has been considered.

IV. OXYGEN VACANCIES AT THE LAO SURFACE
Another possible source of conducting interfacial charge are electrons generated by oxygen vacancies in the LAO. Such vacancies are expected to occur predominantly in the top AlO 2 surface layer 10,32 . To explore the electronic state of the oxygen vacancies in the AlO 2 surface, we consider first a vacancy placed in the center of the (2 × 1) Al 2 O 4 -plaquette with c V = 1/4 . In this case, the surface atomic configuration is described as Al 2 O 3 , and the charged LAO/STO supercell is expected to be doped by the two excess electrons to preserve the overall electrostatic neutrality. To study the supercell charging, we have determined the two-dimensional charge densities in the energy window (E F − 1 eV; E F ) across the interface which corresponds to the charge occupation of the 3d-conducting bands. Fig. 14 presents the charge profile obtained by planar integration of the calculated electron densities. The integration yields two electron charges per interface unit cell, with 1.25 of electron charges distributed in the surface and the remaining 0.75 in the STO/LaO-interface layer. We note the absence of polar charge produced by polar discontinuities in the vacancy-free LAO/STO. Consequently, the surface oxygen vacancies suppress the polar field of LAO due to the supply of the polaritycompensating excess electrons between the surface and interface layers. As a result, the interface charging in LAO/STO with surface vacancies has a self-doping character. Fig. 15 shows the projected densities of states for the surface Al and O atoms and for the interface Ti. At the AlO 2 surface, the local states of a mixed Al3sp-O2p character form in the window (E F − 1 eV; E F + 1 eV), with the Fermi level located below the top of the local band. This implies that the vacancy-doped electrons are mobile. In the interface TiO 2 layer, the vacancy-doped charge occupies the bottom part of the Ti 3d xy conducting band, with the distinct ferromagnetic half-metallic character reported in Ref. 10. A spin polarization is also observed in the AlO 2 surface states, the surface magnetic moments being antiparallel to the magnetic moments of the interface TiO 2 . Fig. 16 shows the spatial distribution of the conducting charge, both for the charge which arises from the Al 3sp states and for the charge in Ti 3d xy states. The spatial distribution of the mobile charges at the AlO 2 surface is presented in Fig. 16(a). The conducting charge in the Al 3sp-orbitals and O 2p orbitals is shifted towards the oxygen vacancies which form stripes along the y-direction (for the configuration of Fig. 1(b)). In the interface TiO 2 layer, the conducting electrons reside mostly in the 3d xy orbitals of Ti atoms ( Fig. 16(b)) which are hybridized with 2p states of oxygen atoms.
For a surface vacancy concentration of c V = 1/8, the compensation of the interface polarity by the vacancyreleased electrons leads to a transition of the AlO 2 surface to the insulating state as can be seen in Fig. 17. The insulating surface state occurs due to the transfer of the total vacancy-released charge to the interface TiO 2 layer. In the interface TiO 2 , the transferred charges occupy primarily the 3d states of Ti atoms nearest to the vacancy. The vacancy-generated electron states at the interface are highly spin polarized with Ti magnetic moment M T i = 0.56 µ B , enhanced due to a mixed e g -t 2g character of spin polarization indicated in the orbitalprojected DOS in Fig. 17(b). The electron charge profile for this case is shown in Fig. 8(b), which demonstrates the interface character of the electrons. They are confined mostly to the TiO 2 layer near the LAO/STO interface. The integration of the charge profile in Fig. 8(b) reveals a total of two electrons per oxygen vacancy, which implies a complete suppression of the interface polarity by the surface oxygen vacancies, a fact supported also by recent calculations reported in Ref. 33. Fig. 18 demonstrates the calculated xy-averaged electrostatic potential along the [001]-direction. In contrast to the biased macroscopic potential in stoichiometric structure (Fig. 9), the macroscopically averaged potential in the structure with the surface vacancies is flat in the LAO-layer, which indeed supports the complete suppression of the LAO-polarity due to the surface oxygen vacancies.

V. CONCLUSIONS
Within DFT-calculations, we considered oxygen vacancies in LAO/STO heterostructures and performed studies of the orbital states at the LAO/STO interface allowing for several types of vacancy arrangements. Using generalized gradient approximation (LSDA) with intraatomic Coulomb repulsion (GGA+U), we have shown that the oxygen vacancies at the titanate interfaces produce a complex multiorbital reconstruction which involves a change of the occupancy of the e g states rather than of t 2g orbitals of the Ti atoms nearest to the oxygen vacancies.
The orbital reconstruction is accompanied by a magnetic splitting of the local e g and interface d xy orbitals. This reconstruction generates a two-dimensional mag- netic state not observed in bulk SrTiO 3 . Moreover, oxygen vacancies placed in the TiO 2 layer farther away from the interface induce a sizable magnetic moment only in the TiO 2 interface layer. Also, oxygen vacancies in the AlO 2 surface where they are expected to be in the most stable configuration, generate a magnetic moment only in the interface titanate layer. In this latter case, the electronic reconstruction mechanism due to polar catastrophe is suppressed for a vacancy concentration of c V ≥ 1/8 by the charge introduced by vacancies. The surface is then free of charge carriers which implies the formation of an insulating state at the AlO 2 surface.
In configurations with vacancy stripes, we have found an orbital separation of the charge and spin degrees of freedom with the vacancy-released charge carriers localized in e g -orbitals, and the spin polarization occurring predominantly in t 2g -states. Moreover, we have provided evidence for the generic character of the two-dimensional magnetic state at titanate surfaces and interfaces.