I think m_e is the only thing that changes between materials, of course the electron mass is the *effective* electron mass, because the atomic lattice acts like a magnifying lens. And it has different values in different directions, which may be unexpected to those who think of mass as constant.
Anyway, in silicon, I recall it's on the order of 10^5 m/s. Pretty fast. An important quality of silicon is, when very pure, it has a low carrier density, so for a given current density, the carrier velocity is pretty high. It's possible to get it so high, it would otherwise exceed the thermal velocity. But this doesn't occur, and the resulting characteristic becomes constant-current as velocity saturates!
I don't actually know how easy it is to do this. The electric field must be very close to breakdown. Most devices are designed to avalanche before this occurs (or before other phenomenon, like punch-through).
Gallium arsenide does have a constant current characteristic, in fact negative resistance, which is due to an unrelated effect with the energy levels (a very quick effect, hence making Gunn "diodes" effective in the microwave range, which are not diodes but actually unodes!).
Tim