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

Abstract

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) ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{a}}$-${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{b}}$=t and (b) ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{a}}$-${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{b}}$=2t, where ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{a}}$ and ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{b}}$ are the site energies of a- and b-type atoms, respectivley, t is a nearest-neighbor matrix element. Let △E(i)=|${\mathrm{E}}_{\mathrm{i}}$-${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{s}}$|, where s=a or b, ${\mathrm{E}}_{\mathrm{i}}$ is the ith eigenenergy; we find that △E(i) for ${\mathrm{E}}_{\mathrm{i}}$ around ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{a}}$ or ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{b}}$ equals 1.3m/N and 8.${5\mathrm{m}}^{2}$/${\mathrm{N}}^{2}$ 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 $\sqrt{N}$ and 0.34$\sqrt{N}$ extended electronic states in RDM for cases (a) and (b), respectively.