Periodic wave functions and number of extended states in random dimer systems

The electronic properties for a one-dimensional random dimer model (RDM) with a- and b-type atoms are studied within a tight-binding on-site model. We carry out a perturbative calculation on the energy spectrum for two different situations (a) εₐ-εb=t and (b) εₐ-εb=2t, where εₐ and εb are the site energies of a- and b-type atoms, respectivley, t is a nearest-neighbor matrix element. Let △E(i)=|Eᵢ-εₛ|, where s=a or b, Eᵢ is the ith eigenenergy; we find that △E(i) for Eᵢ around εₐ or εb equals 1.3m/N and 8.5m²/N² for cases (a) and (b), respectively, where m is the period of wave functions and N is the number of total states. Interestingly, by using the results of △E(i), we find √N and 0.34√N extended electronic states in RDM for cases (a) and (b), respectively.