Single crystal means the one crystal orientation.

Epitaxial thin film also exibits single crystal, but the epitaxial thin films are grown with single crystalline nature by lattice match between thin film and substrate.

To sumarize, epitaxial films are included in single crystals in view of crystalline nature, but the term of 'epitaxial' are used in special case of growth by lattice match between thin film and substrate.

Single crystal means the one crystal orientation.

Epitaxial thin film also exibits single crystal, but the epitaxial thin films are grown with single crystalline nature by lattice match between thin film and substrate.

To sumarize, epitaxial films are included in single crystals in view of crystalline nature, but the term of 'epitaxial' are used in special case of growth by lattice match between thin film and substrate.

Epitaxy is a greek word, and means "grown on". The film is coherent with the substrate, i.e. with the same crystallografic structure and orientation. For instance, we can grow silicon films on a silicon substrate, and this is an epitaxial film; in this case we also speak in terms of OMOEPITAXY.. We can also grow silicon single crystals on substrate of different nature but similar cristal structure (like sapphire), and we speak of HETEROEPITXY. Epitaxy is used in silicon technology, because is possible to have films with physical characteristics (like electrical resistivity) different from that of the substrate..

It should also be possible (if the growth process is dominated by thermodinamics and not by kinetics) to grow crystalline films on amorphous substrate.

To be more precise...epi- = above, upon taxis= ordering. So epitaxy means the formation of an epi-layer on a substrate in and "ordered" fashion, i.e. following the substrate crystallographic orientation. It does not requires to be lattice matched or coherent. In this case one refers to relaxed epitaxy (usually this happens in heteroepitaxy). Coming back to your question: in the case of homoepitaxy (epilayer and substrate are the same chemical species in the same crystalline phase) the result of the growth is a single crystal. A single crystal can be obtained in different manner but what you ultimately do is growing material on an ordered seed from different phases (solid, liquid, vapour)

The presence of lattice match between the thin film and the substrate induce the deformation of films. Thus, the epitaxial thin films can be considered as the deformed crystal. In the case of the temperature dependence measurements, these deformation can influence on the physical parametres of investigated materials.

The basic deference between epitaxial thin film and single crystal film is the roll of grain boundaries. In the single crystal there is no grain boundaries but this might be present in epitaxial film.

So would GaN-AlxGa1-xN-AlN on Si contain both homo and hetero epitaxy?

https://www.researchgate.net/post/How_do_I_figure_out_the_spatial_thickness_of_the_individual_layers_from_these_well_resolved_diffractograms

Thin film of single crystal means that this thin film has a single crystal structure or monocrystalline.

Epitaxial thin film means that there is a single crystal thin film had been grown on a substrate, where the matching between their lattice is very high