On the electronic energy spectrum of disordered systems

The present paper deals with the investigation of the electronic energy spectrum of a one-dimensional model of monatomic disordered systems with rectangular-well type atomic potentials. First, an equation describing the connection between the wavefunctions at three consecutive atomic sites is derived, the method followed for this purpose being based on the use of well-known “transfer matrix” together with an approach reported earlier by the author. This connection equation is further analysed with the help of continued fraction methods as well as a perturbational approach (first used by Phariseau), and explicit equations, giving allowed electronic energies for various one-dimensional models with small deviations from periodicity, are obtained thereby. It is found in general that, for the types of disorder considered here, a band structure of electronic energies exists; in addition, there appear extra localised energy levels. The treatments involving continued fractions show that such extra energies lie unambiguously within the forbidden regions of the band structure, while the extra energies obtained with the help of perturbational approach, are seen to have the possibility of lying both in the allowed and forbidden regions. With regard to localisation of electronic states, we have found that, for all tratments, the electronic states other than those giving the band structure are localised. As for the band states Phariseau's approach predicts them to be localised, while the continued fraction methods fail to predict whether the band states are localised or not.