## I. INTRODUCTION 

Thin films of transition metal oxides (ZnO, TiO $_{2}$, HfO$_{2}$) were found to be ferromagnetic a few years ago, with a high Curie temperature, when doped with magnetic ions 1–3 . The origin of this magnetic ordering is still debated, but meanwhile ferro-magnetism was also found doping with non magnetic ions, such as Cu$^{4}$, and also in undoped cases 5–7 . Trying to understand this, several experimental tests were performed in different laboratories: the magnetic ion, the substrate over which the films were grown, the substrate temperature an d the oxygen pressure during film growth were changed. The results are confusing, as some samples are magnetic while others are not$^{8}$. It now seems well established that structural defects, such as oxygen vacancies, and low dimensionality (thin films or nanoparticles) are necessaryfor the appearance of this type of ferro- magnetism$^{9}$. One can understand why it is difficult to obtain reproducibl e results from different experimental samples, as it is hard to asses quantitatively the influence of defects or disorder in any system. 

At the same time, several ab-initio calculations have been performed for these oxides in bulk 10–13 , with impurities and/or defects, and different models have been proposed to produce the spin ordering. However, few calculations have discussed if surfaces or interfaces have any influence in this type of magnetism $^{14}$. 

LaAlO$_{3}$ (LAO) single crystal is an excellent substrate for epitaxial growth of many oxides, such as SrTiO$_{3}$, BaTiO$_{3}$, PbTiO$_{3}$ and TiO$_{2}$. While the surface electronic structure and atomic relaxation of those oxides 15 and the SrTiO$_{3}$/LAO interface 16,17 have been widely studied, the TiO$_{2}$/LAO interface has not. In the present paper we focus on undoped films of TiO 2 grown on top of a LAO substrate. This system has been found to be ferromagnetic, with a Curie temperature of about 800K$^{6,7}$, grown either by pulsed laser deposition or by spin coating. It is also well suited for ab-initio calculations because LAO in its cubic phase and TiO$_{2}$ in anatase structure, both along the (001) direction, have a very small structural misfit. In f act, thin films of TiO 2 using LAO as a substrate develop anatase structure, indicating epitaxial growth. 

The aim of this work is to study the role played by interfacial oxygen vacancies on the magnetic properties of the interface. For this purpose, we consider a model system with a high concentration of vacancies. Under certain conditions we find a localized ma gnetic moment in the interfacial Ti atoms, depending not only on the presence of vacancies, but also on subtle structural details. We do not consider here the possible interaction between these localized moments. 

