## II. THE SYSTEM 

To simulate interfaces we use a super-lattice geometry, so that the system under study results periodic in 3D : ... LAO / TiO$_{2}$ / LAO / TiO$_{2}$ / ... 

Figure 1 shows the unit cells of the two component materials: LAO and TiO$_{2}$ anatase. LAO is a layered material and along the (001) direction. The planes LaO and AlO$_{2}$ have nominal charges of +1 and -1, respectively. TiO$_{2}$ anatase is almost layered: the oxygen atoms are slightly out of the Ti planes, as indicated by the small arrows in Fig. 1. Each plane has zero nominal charge, so that the interface with LAO will restructure due to electrostatics either changing the valence of Ti from Ti 4 + to Ti 3 + or Ti 2 + or by missing oxygen atoms. In this respect, our previous preliminary work 14 showed, using total energy calculations, that it is easier to remove an oxygen atom from the anatase side of the interface than from the LAO side. 

Figure 2 shows the two possible interfaces, namely the two ways of stacking to form the super- lattice, that maintain the octahedral environment of the Ti atoms. Without oxygen vacancies, our previous calculations 14 showed that neither one has a magnetic solution. With oxygen vacancies near the interface it is clear, due to electrostatics, that the AlO$_{2}$/TiO$_{2}$ interface will be preferred with respect to the LaO/TiO$_{2}$ one. While there is only one way of stacking in which LaO faces TiO$_{2}$, there are two ways with AlO$_{2}$ facing TiO$_{2}$, that we name A and B. They are shown in Fig. 3, that presents the super-lattice unit cells used in the present calculations. In order to have all interfaces in the super-lattice either of type A or type B, 7 anatase planes and 5 LAO planes are considered. If there are no oxygen vacancies, type A is clearly more stable, with lower total energy, as Al-O bonds are formed at the interface due to the off plane oxygens in the anatase structure. With oxygen vacancies, each distribution has to be studied separately. 

The size of the super-cells is obtained by minimizing the total energy in each case, and the atomic positions inside the cell are allowed to relax but only in the direction of the super-lattice. The reason for this last restriction is based on our previous results, apart from making the cal- culation more feasible. In fact, the calculations on pure bulk rutile TiO$_{2}$ 11 showed that oxygen vacancies can give rise to a local magnetic moment due to structural relaxation. However, this is only true for a fairly large concentration of vacancies and for a fixed cell size, which means there must be some restriction to the full structural relaxation. We could not find a similar result for bulk anatase TiO$_{2}$, but our assumption here is that the LAO substrate could impose such a structural restriction to the anatase film thus giving rise to magnetic s olutions. Therefore, the calculations in 

