Carriers in silicon race around at thermal speeds and scatter constantly. An electric field adds only a tiny bias between collisions, yet that bias is the drift current.
electronholetracked carrier and its collisions
Distribution of the velocity component along the field: the field shifts it only slightly (vertical lines: averages)
What to look for
Each free flight is a nearly straight line; the field bends it slightly. Watch the tracked carrier: its zig-zag barely changes when you switch the field on, yet the displacement arrow below grows steadily.
The drift velocity is a small fraction of the thermal speed at ordinary fields, so the velocity histogram shifts only slightly. Current comes from that small shift multiplied by an enormous number of carriers.
Electrons and holes move in opposite directions but carry current in the same direction.
The physics behind it
Between collisions a carrier accelerates at qℰ/m*. With a mean time τ between collisions the average extra velocity is vd = (qτ/m*)ℰ = μℰ (book eq. 1.3.8).
Mobility is set by scattering: more ionized impurities or stronger lattice vibrations mean a shorter τ and a lower μ (book Fig. 1.13).
At high fields (above about 3×104 V/cm for electrons) carriers lose energy to the lattice faster and vd saturates near 107 cm/s (Fig. 1.11). This simple picture does not include that, so the field slider stops at 10 kV/cm.