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The drift velocity does not always saturate in the way shown in figure 4.7. In particular, it may not show a monotonic behaviour but it may reach a maximum and then saturate at a value below the maximum. The phenomenon does not occur in all semiconductors but only in the ones that have a direct gap with a higher subsidiary conduction band minimum as in GaAs see figure 2.11. The variation of the drift velocity in GaAs is shown schematically in figure 4.8. The analysis can be performed using a simple particle approach and does not need the Boltzmann equation which has, so far, verified any particle picture that we have used.

At small electric fields £ all the electrons in the conduction band reside in the conduction band minimum at к = 0. As the electric field £ increases, the average or drift velocity of these electrons increases in accordance with what we have described in section 4.4 (or chapter 3). But when £ reaches the order of magnitude of a few kV/cm, some electrons have gained enough energy to make the jump to the subsidiary conduction band minimum at к 0 which, however, has a heavier effective mass than the one at к = 0. As £ is further increased, more and more electrons jump to this heavier effective mass minimum and hence the average velocity falls. Any further increase of £ simply leads to saturation. This non-monotonic dependence of velocity on the electric field gives rise to the Gunn effect (discovered by Gunn at IBM), where the current density / as a function the electric field £ follows a variation with £ similar to the vdr (£) of figure 4.8. This is shown in figure 4.9. As can be seen from this figure, the Gunn effect displays a region of negative differential resistance for some £ which can be exploited for engineering microwave oscillators.

A very simple particle-like model can give a mathematical description of the Gunn effect. Let the number of electrons per unit volume n in the conduction band of GaAs, at any time, be divided into those in the light effective mass band-edge (i.e. minimum) и, and those in the heavy mass band-edge nh. Then

Obviously for the derivatives with respect to £ we will have

Differentiating the basic equation J = o£ we have

since о is £ dependent because n, and nh are functions of £.


where p; and xh are the mobilities of the light mass minimum and of the heavy mass

minimum respectively. Therefore, the RHS of 4.50 is negative (р/, < p/) and can make —-

negative in 4.49. This simple model gives the decreasing with £ part of the current density versus electric field characteristic. It should be clear that a more complicated empirical relation than 4.47 is needed to describe the velocity-field relation of this non-monotonic case.

So far what we have presented pertains to steady state conditions, i.e. independent of time. However, an interesting case arises in transient conditions called velocity overshoot. If the path of an electron is short or alternatively scattering mechanisms are absent during a short section of the electrons path, the electron may reach a velocity above its steady state value before it relaxes to its steady state velocity. Such effects are very important for modern state of the art transistors.

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