Electronic Band Structures

Summary

A semiconductor sample contains a very large number of atoms. Hence a quantitative quantum mechanical calculation of its physical properties constitutes a rather formidable task. This task can be enormously simplified by bringing into play the symmetry properties of the crystal lattice, i. e., by using group theory. We have shown how wave functions of electrons and vibrational modes (phonons) can be classified according to their behavior under symmetry operations. These classifications involve irreducible representations of the group of symmetry operations. The translational symmetry of crystals led us to Bloch’s theorem and the introduction of Bloch functions for the electrons. We have learnt that their eigenfunctions can be indexed by wave vectors (Bloch vectors) which can be confined to a portion of the reciprocal space called the first Brillouin zone. Similarly, their energy eigenvalues can be represented as functions of wave vectors inside the first Brillouin zone, the so-called electron energy bands. We have reviewed the following main methods for calculating energy bands of semiconductors: the empirical pseudopotential method, the tight-binding or linear combination of atomic orbitals (LCAO) method and the k·p method. We have performed simplified versions of these calculations in order to illustrate the main features of the energy bands in diamond- and zinc-blende-type semiconductors.