A latest fervent topic in condensed matter physics is the so-called topological insulators. An essential feature of these compounds is their having unique surface states. Surface states are states that decay into the vacuum as well as the bulk, thus qualitatively different from bulk states. These states always show up in the bulk forbidden gap, much the way as impurity states. In a sense they are indeed impurity states, because they are created by breaking periodic boundary conditions. Suppose you have a ring of N identical atoms. Now you break it, and you obtain two end atoms which have different surrounding than other bulk atoms. The breaking bond can be expressed as a perturbation, and thus leading to two impurity states. In history, they are called Shockley-Tamm states.
Introduction to surface states:
[1]http://philiphofmann.net/surflec3/surflec015.html
[2]http://en.wikipedia.org/wiki/Surface_states
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