  We present a detailed low-energy muon spin rotation and x-ray magnetic
circular dichroism (XMCD) investigation of the magnetic structure in ultra-thin
tetragonal (T)-CuO films. The measured muon-spin polarization decay indicates
an antiferromagnetic (AFM) order with a transition temperature higher than
200K. The XMCD signal obtained around the Cu $L_{2,3}$ edges indicates the
presence of pinned Cu$^{2+}$ moments that are parallel to the sample surface,
and additionally, isotropic paramagnetic moments. The pinning of some of the Cu
moments is caused by an AFM ordering consisting of moments that lie most likely
in the plane of the film. Moreover, pinned moments show a larger orbital
magnetic moment contribution with an approximate ratio of $m_{orb}/m_{spin} =
2$, indicating that these spins are located at sites with reduced symmetry.
Some fractions of the pinned moments remain pinned from an AFM background even
at 360K, indicating that $T_N >$ 360K. A simple model could explain
qualitatively these experimental findings; however, it is in contrast to
theoretical predictions, showing that the magnetic properties of ultra-thin
T-CuO films differ from bulk expectations and is more complex.
