Does conductivity always depend on mobility, charge carriers (electrons or holes)? And does the size of the crystallites depend on the mobility and the concentration, ie does a high mobility have a large crystallite size?
Conductivity depends upon mobility as well as carrier concentration of the material.
Not at all, "high mobility has a large crystallite size", it depends upon the type of materials.
Even in the large crystal size material like c-Si, due to the high doping in the material, mobility reduces.
Mainly the mobility depends upon the scattering of the carriers. It may be grain boundary scattering, carrier-carrier scattering, carrier-phonon scattering, Carrier-impurity scattering, carrier-defect state scattering at surface or interface.
Large crystal size may reduce the grain boundary scattering, but another scattering may present in the material.
I think conductivity of Transparent conducting oxides (TCOs) depends on various factor such as Defect densities, carrier concentration and transparency of the Sample. Although few reports declined the fact that crystallite size have a direct relation with conductivity of the sample but It is a well established fact that high crystallite size facilitates the charge transfer and decreases the grain boundary scattering which subsequently increases the conductivity of TCOs. We have recently published a report in which we have observed similar type of phenomenon.