Now that we have discussed the physics of infinite, perfectly ordered, bulk materials with essentially non-interacting electrons, it is time to consider what happens in systems that are less “ideal”.

Defects

No real material is ever perfect. The basic argument for the existence of some nonzero density of defects is an entropic one, and makes perfect sense from the perspective of free energy. Even if there is a moderate energetic cost for the creation of a defect, if there are a very large number of possible configurations for the defect (e.g., sites where the defect can reside), the system's total free energy can be lowered through defect formation. Common defects can be zero-dimensional (point defects), one-dimensional (line defects), or two-dimensional (grain boundaries or interfaces).

What are the effects of these defects, particularly on the electronic properties of the materials? The Bloch condition and our picture of Bloch waves as single-particle eigenstates of the lattice are predicated on the assumption of the perfect, infinite spatial periodicity of the lattice. That is, the single-particle Hamiltonian in crystalline solids is assumed to be invariant under discrete translational symmetry. Strictly speaking, once that symmetry is broken by defects, the Bloch states are no longer exact solutions to the single-particle problem.

If the defect density is low, then we generally don't care. It's hard to imagine that a handful of defects in a crystal could have a large impact on the electronic structure of a crystal containing 1022 atoms (though see Exercises). The vast majority of single-particle states in the crystal are still approximated very well as Bloch waves in this case. However, the defects do alter the spectrum of allowed states, and these changes can be significant if the number of atoms near defects becomes a substantial fraction of the total number of atoms. We discuss this further below.