Mobility of each carrier species is normally computed through the dispersion (derivatives) of the associated band diagram (Energy vs k-points) from an ab-initio calculation. You can take the hole or electron bands and compute the mobility of each carrier species.
Great care however must be exercised when interpreting band diagrams from density functional theory. The H-K theorem guarantees charge density and total energy will match (with a perfect functional) but makes no warranty on the band diagrams. You have been warned!
Note that the effective mass that is used in mobility calculations can be different than the effective mass associated with the density of states. The former is given as m_c = 3/(1/m1 + 1/m2 + 1/m3) (see Sec 1.4.2 of Sze and Ng "Physics of Semiconductor Devices, 3rd ed.") instead of the usual geometric mean of m1,m2,m3. Here m1 is the effective mass along principle axis 1. The calculated effective mass using dense k-point lines and the second derivative is a low-temperature value. It can vary from this (and becomes less well-defined) at room temperature when the CB/VB are not well approximated by a paraboloid all the way up to Delta E = kT.
The effective mass is used in formulas for mobilities associated with phonon scattering and defect scattering, which assume that you know the elastic constant and defect concentration in the material respectively. The elastic constant is easily looked up for known materials (or can be estimated using a VASP calculation). Extrinsic defects densities come from doping defect densities, but intrinsic defect densities may be harder to look up (and they are a major project to calculate and the results can be off by more than an order of magnitude). Calculate the mobility for phonons first, and if it is doped, calculated the mobility from extrinsic defects. Then you can estimate the minimum intrinsic defect density at the intrinsic defect mobility becomes significant. Hopefully, someone can tell you "the intrinsic defects in that material should be well below that room-temperature significance threshold". I believe this is true in many cases. But for example, in CuInSe2, the intrinsic copper vacancies ~10^17 or 10^18 cm^{-3} (I believe) may be non-negligible at low temperatures, maybe even room temperature. I have not looked at numbers enough to know at what temperature a 10^17 concentration would become important.
In addition to the warning given by Buurma, there is the issue of the approximation of the functional you are using. Band structure comparisons that I have seen between GGA and GGA+GW (more accurate, but very expensive to converge) look like the second derivatives agree well. But I wouldn't expect a phonon-mediated mobility calculated this way to be more accurate than within a factor of, say, 5.
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