Abstract
We consider two topics in the theory of one-dimensional crystals. 1. Shockley states. These levels are ordinarily regarded as essentially different from states due to surface imperfections, for surface states of the Shockley type are possible, even in perfect crystals, provided the energy bands have crossed. We have noticed, however, that the analysis of Shockley levels and of levels originating from surface imperfections may be unified if the energy bands are ordered in a certain way. When the proper ordering of the energy bands is imposed, the unit cell shifts as a result of a band crossing, and an extra half-cell is left at each end of the crystal. These half-cells constitute “impurity layers,” and give rise to surface levels according to the usual theory. 2. When are energy band crossings possible? For a square well, it is known that the energy must be greater then the potential ( E > V) everywhere in the unit cell. We show that a similar result holds for more general potentials: if the potential is real and (for convenience) symmetric, then if E is not V everywhere in the unit cell, there are no band crossings, and hence no Shockley levels. This last result is based on the general behavior of the Bloch functions under time and space inversion, which is discussed in detail in the Appendix.
Published Version
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