I'd think a diamond file/hone would be overkill for an axe, not to mention much more expensive than necessary. Steel in axe heads usually isn't very hard, nor does it contain a lot of carbides contributing to high abrasion-resistance. I'd bet a conventional file or a stone made for sharpening axes (such as a 'puck' style stone) should work. Anything in a coarse grit of aluminum oxide or silicon carbide should do. Many have also sharpened axes using wet/dry sandpaper (silicon carbide) on a firm/hard backing.
David