Because velocities are calculated by the slope of band dispersion while mobility is calculated from the scattering mechanism, can we calculate hole and electron velocities using band slopes in semiconductors similar to that of graphene?
The main question and the subsequent explanatory note seem to correspond to different issues. Be it as it may, it is not the band slopes (i.e. Fermi velocities) that are directly relevant to mobilities (of electrons and holes), but effective masses. The velocity that determines mobility (whether the mobility of electrons or that of holes) is the drift velocity, not the Fermi velocity, although the former depends on the latter. Slope of band enters in the expression for the effective mass. For this, assuming a parabolic band of the form ε(k) = k2/2m* (identifying h-bar with 1), where m* denotes effective mass, one has vF = kF/m*, or m* = kF/vF, where kF denotes the Fermi wave vector and vF the Fermi velocity. One sees that m* that enters in the expression for (electron) mobility, is not solely a function of vF, but also a function of kF. In general, for non-parabolic bands, the relationship is more complicated.
For some relevant discussions regarding conductivity and mobility, I refer you to the book Solid State Physics, by Ashcroft and Mermin (1976), pp. 563, 601-602.
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