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Chapter 6: Q12CQ (page 223)

The plot below shows the variation of 蠅 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蠅 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.

Short Answer

Expert verified

The group velocity exceeds phase velocity in all regions except the regions near the jump.

Step by step solution

01

Definition of group and phase velocities

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.

02

Explanation

Only within the regions near the jump does the slope of a line from the origin, 蠅/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.

03

Conclusion

For values of 蠅 just above the gap, the second derivative is larger, and the effective mass is smaller. For 蠅 values slightly below the gap, the second derivative is negative.

Therefore, the effective mass is negative.

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Most popular questions from this chapter

Given the situation of exercise 25, show that

(a) as Uo鈫�鈭�, reflection probability approaches 1 and

(b) as L鈫�0, the reflection probability approaches 0.

(c) Consider the limit in which the well becomes infinitely deep and infinitesimally narrow--- that is Uo鈫�鈭�and data-custom-editor="chemistry" L鈫�0but the product U0L is constant. (This delta well model approximates the effect of a narrow but strong attractive potential, such as that experienced by a free electron encountering a positive ion.) Show that reflection probability becomes:

R=[1+2h2EmUoL2]-1

Exercise 39 gives the condition for resonant tunneling through two barriers separated by a space of width 2 s, expressed in terms of a factor 尾given in Exercise 30. (a) Suppose that in some system of units, k and 伪 are both2蟺. Find two values of 2s that give resonant tunneling. What are these distances in terms of wavelengths of蠄? Is the term resonant tunneling appropriate?(b) Show that the condition has no solution if s = 0 and explain why this must be so. (c) If a classical particle wants to surmount a barrier without gaining energy, is adding a second barrier a good solution?

How should you answer someone who asks, 鈥淚n tunneling through a simple barrier, which way are particles moving, in the three regions--before, inside, and after the barrier?鈥�

The diagram below plots 蠅(k) versus wave number for a particular phenomenon. How do the phase and group velocities compare, and do the answer depend on the central value of k under consideration? Explain.

Show that 蠄(x)=A'eikx+B'e-ikxis equivalent to 蠄(x)=Asinkx+Bcoskx, provided that A'=12(B-iA)B'=12(B+iA).
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