Page 21 - Fall 2008

Fig. 14. Illustration of the propagation of a Bloch wave packet. The two plots (a) and (b) illustrate the propagation of the packet from one instant in time to a later instant in time. In the context of the tight binding model, it is interesting to see that the local oscillators give up their energy in just the right manner.
Another interesting plot that may be made involves the modulus of the modes. For standing wave modes that are not degenerate, the modulus is simply given by the absolute value of the mode: |ψk,n(x)|. For standing wave modes that are degenerate, then the modulus is given by the square-root of the sum of the squares of the two degenerate modes. It can be shown that the modulus of a mode gives a result that is pro- portional to the periodic part of the Bloch wavefunction: |Uk,n(x)|. The results for the moduli of the modes in Fig. 7 are shown in Fig. 9. The periodic nature of the modulus of the Bloch wavefunctions is apparent in the figure, and one may note a systematic pattern as the modes are followed upward, as the frequency is increased. Because the function |Uk,n(x)| is periodic, the modulus of the wavefunction has the same maximum value in every unit cell throughout the entire sys- tem; for this reason, the wavefunctions of a periodic system are said to be “extended.”
The presentation given above has used standing Bloch wavefunctions as graphical examples. However, traveling Bloch wavefunctions are more commonly used in other treat- ments—these are given by
(2)
A traveling Bloch wavefunction moving in the opposite direction is obtained by replacing k with –k. It can be shown that this is equivalent to taking the complex conjugate of Eq. 2. From Eq. 1 it can be seen that the Bloch standing wave- function is a linear combination of two Bloch traveling waves going in opposite directions, just as for waves on a plain string. However, it is important to note that the presence of the factor exp(ikx) or exp(-ikx) in a Bloch wavefunction does not mean that the motion of the system is that of a traveling wave with wavenumber k. The functions Uk,n(x) are complex and also involve a phase, so that the actual motion of the sys- tem is complicated. If the scatterers are strong (large m), then the actual motion of a Bloch wave is nearly that of standing waves between the scatterers, and the factor exp(±ikx) simply represents a phase change in proceeding from one local standing wave to another. Ordinary traveling waves have cer- tain relations with real force, velocity, momentum, energy, etc., and Bloch waves do not have straightforward relations with such quantities. For example, a Bloch wave traveling in one direction may have energy traveling in the opposite direction. The same caution in interpreting exp(ikx) may be found in textbooks on solid state physics, where it is noted that Plank’s constant times the Bloch wavenumber k is not real momentum, but is instead something referred to as “crystal momentum.”4
While Bloch waves may not be traveling waves in a sys- tem, it is possible to form wave packets that do behave like ordinary wave packets. As shown in modern physics texts,15 wave packets travel at the group velocity given by cg = 2πdf/dk. Most acoustic systems have linear dispersion, with f=ck/2π, so that cg = c. However, as discussed earlier, Bloch waves have non-trivial dispersion given by the curves in Fig.
8. In the second band (and every other band) the group velocity is negative, so that wave fronts in a wave packet may move in one direction while the wave packet itself moves in the opposite direction.
Given the tight-binding picture of waves in periodic systems, it is surprising that it is possible to have a traveling wave packet at all. With strong scatterers, the acoustic ener- gy is mostly stored in a standing wave for a local oscillator, with only the phase exp(ikx) changing from one local oscil- lator to the next. To have a traveling wave packet, a set of local oscillators must give up all of their energy to the next set of local oscillators in the direction of propagation. A computer calculation showing that this actually works is shown in Fig. 14. In this figure, the upper plot is the Bloch wave packet at one instant of time, and the lower plot is the packet at a later instant in time. The periodic scatterers are not shown in this plot, but there are about 64 unit cells across the figure.AT
References
1 L. Brillouin, Wave Propagation in Periodic Structures (McGraw- Hill, New York, 1946).
2 B. L. Altshuler, P. A. Lee, and R. A. Webb, Mesoscopic Phenomena in Solids, (North-Holland, Amsterdam, 1991).
3 C. Janot, Quasicrystals: A Primer (Clarendon, Oxford, 1992).
4 C. Kittel, Introduction to Solid State Physics (John Wiley, New
York, 1966).
5 G. Floquet, “On linear differential equations with periodic coef-
ficients,” Annales Scientifiques de l'École Normale Supérieure
12, 47–89 (1883).
6 F. Bloch, “About the quantum mechanics of electrons in crystal
lattices,” Zeitschrift für Physik, 52, 555–600 (1928).
7 A. D. Pierce, Acoustics, An Introduction to its Physical Principles and
Applications (Acoustical Society of America, Melville, NY, 1989).
8 Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical
Magazine 24, 145–149 (1887).
9 C. Elachi, “Waves in active and passive periodic structures: A
review,” Proceedings of the Institute of Electrical and Electronics
Engineers 64, 1666–1698 (1976).
10 P. W. Anderson, “Local moments and localized states,” Reviews
of Modern Physics 50, 191–201 (1978).
20 Acoustics Today, October 2008