The plot below shows the variation of ω with k for electrons in a simple crystal. Where, if anywhere, does the group velocity exceed the phase velocity? (Sketching straight lines from the origin may help.) The trend indicated by a dashed curve is parabolic, but it is interrupted by a curious discontinuity, known as a band gap (see Chapter 10), where there are no allowed frequencies/energies. It turns out that the second derivative of ω with respect to k is inversely proportional to the effective mass of the electron. Argue that in this crystal, the effective mass is the same for most values of k, but that it is different for some values and in one region in a very strange way.
The group velocity exceeds phase velocity in all regions except the regions near the jump.
Group velocity is defined as the velocity of the whole envelope of a wave in space.
Phase velocity is defined as the velocity of a phase or part of a wave in space.
Only within the regions near the jump does the slope of a line from the origin, ω/k, exceed the slope of the curve, or tangent line. Thus, only in these regions does the group velocity not exceed the phase velocity. Wherever the dispersion relation follows the parabolic plot, its second derivative is the same constant. Just for k values near the bandgap, it deviates.
For values of ω just above the gap, the second derivative is larger, and the effective mass is smaller. For ω values slightly below the gap, the second derivative is negative.
Therefore, the effective mass is negative.
The potential energy barrier in field emission is not rectangular, but resembles a ramp, as shown in Figure 6.16. Here we compare tunnelling probability calculated by the crudest approximation to that calculated by a better one. In method 1, calculate T by treating the barrier as an actual ramp in which U - E is initially , but falls off with a slop of M. Use the formula given in Exercise 37. In method 2, the cruder one, assume a barrier whose height exceeds E by a constant (the same as the average excess for the ramp) and whose width is the same as the distance the particle tunnels through the ramp. (a) Show that the ratio T1 /T2 is . (b) Do the methods differ more when tunnelling probability is relatively high or relatively low?
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