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We discuss two types of defects in two-dimensional lattices, namely (1)translational dislocations and (2) defects produced by a rotation of thelattice in a half-space.  For Lipschitz-continuous and $\Z^2$-periodic potentials, we first show thattranslational dislocations produce spectrum inside the gaps of the periodicproblem; we also give estimates for the (integrated) density of the associatedsurface states. We then study lattices with a small angle defect where we findthat the gaps of the periodic problem fill with spectrum as the defect anglegoes to zero. To introduce our methods, we begin with the study of dislocationproblems on the real line and on an infinite strip. Finally, we considerexamples of muffin tin type. Our overview refers to results in [HK1, HK2].

Dislocation problems for periodic Schr\"odinger operators and mathematical aspects of small angle grain boundaries