I synthesize the 2 at% Mn doped ZnO thin films using DC Co-sputtering method. I observed the shift in ZnO peaks about 0.6 degree after the doping of Mn.
XRD peak shift usually occurs because of strain due to planar stress (left shift or low 2theta for compressive stress and right shift or higher 2theta for tensile stress), change in stoichiometry, change in chemical composition among other reasons. Try to find out how the particular dopant in your case is contributing to these properties. left shift can indicate dopant is increasing compressive stress or altering compositional and stoichiometric properties. It is also suggested that lower materials correspond to lower angle xrd peak and heavy materials to higher angles. Again this has to do with composition change indicated earlier. comprehensive understanding of reasons leading to xrd peak shift is still an open question.
If you want to understand this completely, you will need to apply a method to determine the valence of your Mn ions (could be anything from +II to +VI) and another method to get information about the microstructure (lattice defects, segregation etc.).
Use XPS< XANES or EELS for the valence info.
Use SEM + (S)TEM for the microsctrutureal info.
Putting both together, you can then try to build a model based on how to incorporate cations of a given charge and hence size into the lattice structure.