We report on the mechanisms governing electron transport using a comprehensive set of ZnO layers heavily doped with Ga (GZO) grown by plasma-enhanced molecular-beam epitaxy on a-plane sapphire substrates with varying oxygen-to-metal ratios and Ga fluxes. The analyses were conducted by temperature dependent Hall measurements which were supported by microstructural investigations as well. Highly degenerate GZO layers with n > 5 Â 10 20 cm À3 grown under metalrich conditions (reactive oxygen-to-metal ratio <1) show relatively larger grains ($20-25 nm by x-ray diffraction) with low-angle boundaries parallel to the polar c-direction. For highly conductive GZO layers, ionized-impurity scattering with almost no compensation is the dominant mechanism limiting the mobility in the temperature range from 15 to 330 K and the grain-boundary scattering governed by quantum-mechanical tunnelling is negligible. However, due to the polar nature of ZnO having high crystalline quality, polar optical phonon scattering cannot be neglected for temperatures above 150 K, because it further reduces mobility although its effect is still substantially weaker than the ionized impurity scattering even at room temperature (RT). Analysis of transport measurements and sample microstructures by x-ray diffraction and transmission electron microscopy led to a correlation between the grain sizes in these layers and mobility even for samples with a carrier concentration in the upper 10 20 cm À3 range. In contrast, electron transport in GZO layers grown under oxygen-rich conditions (reactive oxygen-to-metal ratio >1), which have inclined grain boundaries and relatively smaller grain sizes of 10-20 nm by x-ray diffraction, is mainly limited by compensation caused by acceptor-type point-defect complexes, presumably (Ga Zn -V Zn ), and scattering on grain boundaries. The GZO layers with n <10 20 cm À3 grown under metal-rich conditions with reduced Ga fluxes show a clear signature of grain-boundary scattering governed by the thermionic effect in the temperature-dependent mobility but with much higher RT mobility values compared to the samples grown under oxygen-rich conditions [34 vs. 7.5 cm 2 =VÁs]. Properties of GZO layers grown under different conditions clearly indicate that to achieve highly conductive GZO, metal-rich conditions instead of oxygen-rich conditions have to be used. V C 2012 American Institute of Physics. [http://dx