Relativistic electronic states in disordered systems

The energy spectrum of relativistic electrons in one-dimensional disordered systems is investigated. Using an approximate solution to the one-dimensional Dirac equation as given by Steslicka and Davison and following a method reported earlier by the author for nonrelativistic cases, an equation connecting the wave functions at three consecutive atomic sites is derived. This “connection equation” is then analysed with the help of (1) Worpitzky's theorem and (2) a perturbational approach due to Phariseau. With the help of these approaches, explicit equations are derived for electronic energies in disordered systems with small deviations from periodicity. These explicit equation indicate that, in general, a band structure of electronic energies exists in each region of the system; they reveal further that there occur energies in addition to those characterising the local bands and these extra energies constitute a disturbance to the (local) band structures.