  The magnetic properties of Zn$_{1-x}$Co$_x$O ($x=0.07$ and 0.10) thin films,
which were homo-epitaxially grown on a ZnO(0001) substrates with varying
relatively high oxygen pressure, have been investigated using x-ray magnetic
circular dichroism (XMCD) at Co $2p$ core-level absorption edge. The line
shapes of the absorption spectra are the same in all the films and indicate
that the Co$^{2+}$ ions substitute for the Zn sites. The magnetic-field and
temperature dependences of the XMCD intensity are consistent with the
magnetization measurements, indicating that except for Co there are no
additional sources for the magnetic moment, and demonstrate the coexistence of
paramagnetic and ferromagnetic components in the homo-epitaxial
Zn$_{1-x}$Co$_{x}$O thin films, in contrast to the ferromagnetism in the
hetero-epitaxial Zn$_{1-x}$Co$_{x}$O films studied previously. The analysis of
the XMCD intensities using the Curie-Weiss law reveals the presence of
antiferromagnetic interaction between the paramagnetic Co ions. Missing XMCD
intensities and magnetization signals indicate that most of Co ions are
non-magnetic probably because they are strongly coupled antiferromagnetically
with each other. Annealing in a high vacuum reduces both the paramagnetic and
ferromagnetic signals. We attribute the reductions to thermal diffusion and
aggregation of Co ions with antiferromagnetic nanoclusters in
Zn$_{1-x}$Co$_{x}$O.
