Electron localization in the insulating state: Application to crystalline semiconductors

We measure electron localization in different materials by means of a "localization tensor", based on Berry phases and related quantities. We analyze its properties, and we actually compute such tensor from first principles for several tetrahedrally coordinated semiconductors. We discuss the trends in our calculated quantity, and we relate our findings to recent work by other authors, We also address the "hermaphrodite orbitals", which are localized (Wannier-like) in a given direction, and delocalized (Bloch-like) in the two orthogonal directions: our tensor is related to the optimal localization of these orbitals. We also prove numerically that the decay of the optimally localized hermaphrodite orbitals is exponential.