What factors cause the velocity saturation to occur at different electric fields for different materials? In semiconductors the velocity of carriers gets saturated after a certain value of electric field. In silicon it occurs at around $10^4 kV/cm$ and in GaAs at some other value. 


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*What factors are involved here? 

*Is it related to the value of band-gap alone or other parameters are at play?


Furthermore, suppose the value of saturated velocity of an electron in Si at room temperature is $10^7 cm/s$, for an electron in GaAs it is even higher. 


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*Why is that so?

 A: Maybe this will be helpful: Ridley–Watkins–Hilsum theory. It somehow explains why velocity of electron saturate. High speed electrons can change valley in band structure and gain higher mass, less mobility. Similar to scattering by optic phonon. 
A: Wikipedia takes a swing at explaining this phenomenon. There it says that the velocity saturation is caused by scattering from optical phonons, with
$$
\frac{1}{2}m*v_s^2 \approx \hbar \omega_o
$$
where $m*$ is the effective mass of the carrier (depends on conduction band), $v_s$ is the drift velocity in saturation and $\omega_o$ is the angular frequency of the optical phonon (depends on phonon band structure, i.e. crystal lattice). So, it is a complex interaction between electron and phonon band structure, which determines the saturation velocity.
The formula gives a heuristic understanding of the process: the carrier is accelerated in the electric field until it gets enough kinetic energy to excite an optical phonon, transferring the main part of its energy to it. Then the process starts again.
However, there is an unanswered question:
The charge carrier could always scatter from acoustic phonons, because they can have arbitrarily small energies. I think this can be answered by comparing cross sections of the scatter processes (though I am not sure).
