Surface spin characterization of Cr$_{\mathrm{2}}$O$_{\mathrm{3}}$ films epitaxially grown on (001) TiO$_{\mathrm{2}}$ and (0001) Al$_{\mathrm{2}}$O$_{\mathrm{3}}$*

Cr$_{\mathrm{2}}$O$_{\mathrm{3}}$ is an interesting antiferromagnetic material, which has been widely investigated because of its high Neel temperature and the electric field control of its exchange bias at room temperature. We have epitaxially grown the Cr$_{\mathrm{2}}$O$_{\mathrm{3}}$ films on (001) TiO$_{\mathrm{2}}$ and (0001) Al$_{\mathrm{2}}$O$_{\mathrm{3}}$. For the growth on TiO$_{\mathrm{2}}$, we find that the Cr$_{\mathrm{2}}$O$_{\mathrm{3}}$ film is (10-10) oriented, confirmed by XRD and high resolution TEM. As the spins of the Cr atoms are parallel to the $c$ axis, the surface spins of the (10-10) Cr$_{\mathrm{2}}$O$_{\mathrm{3}}$ film are manipulated to align in-plane, as indicated from the exchange bias measurement of the Py/(10-10) Cr$_{\mathrm{2}}$O$_{\mathrm{3\thinspace }}$interface. Furthermore, we observe a positive exchange bias that depends on the direction of the cooling and measuring magnetic fields. On the other hand, (0001) Cr$_{\mathrm{2}}$O$_{\mathrm{3}}$ film is epitaxially grown on (0001) Al$_{\mathrm{2}}$O$_{\mathrm{3\thinspace }}$substrate. And the crystal structure is confirmed by XRD. For this (0001) film, the exchange bias is found to be perpendicular to the plane, indicating that the spins of the (0001) Cr$_{\mathrm{2}}$O$_{\mathrm{3}}$ film are out-of-plane. Our results demonstrate crystal structure manipulations of the exchange bias and the collinear exchange coupling between the surface spins of the Cr atoms and the adjacent FM layer.