S. Scandolo, ICTP

Lesson 15 (Crystals: band structure, optical Properties)

The lecture starts with comparing the band structure of the free electron model with the tight binding model. The latter assumes that the crystal is close to collection of independent atoms and it builds a perturbative expansion with atomic wavefunctions (Linear Combination of Atomic Orbits). The free electron model is a good choice for alkalies, while tight binding is for rare gases. These simple models are less accurate in the middle columns of the periodic table.

The professor discusses how these models could be made more accurate by leaving behind perturbation theory but keeping the two basis of eigenfunctions (plane waves and atomic orbits) to express a general solution for Schroedinger's equation.

The effect of finite temperature is also estimated and it is found that the transition from occupied to empty states is practically a step function.

The second part of the lecture is about the optical properties of a crystal. In a collision of a photon and an electron, both energy and momentum are conserved. Since the momentum of a photon is negligible compared to the size of the Brillouin zone, only "vertical" transitions are possible between wave bands. In other words, the k quantum number remains the same. With this, the absorption coefficient for an insulator can be sketched: there must be a minimum and maximum energy between which the coefficient is non zero. This determines whether the crystal is transparent or opaque to visible light or, if it only absorbs part of the visible spectrum, its color. The visible light has around 1.5eV - 3eV energy.