Band spectra of rectangular graph superlattices

We consider rectangular graph superlattices of sides ${\mathit{l}}_1$, ${\mathit{l}}_2$ with the wave-function coupling at the junctions either of the δ type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant α, or the ${\mathrm{\delta }}_{\mathit{s}}^{\prime }$ type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio θ:=${\mathit{l}}_1$/${\mathit{l}}_2$. If the latter is an irrational badly approximable by rationals, δ lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of α at which new gap series open, and explain it in terms of number-theoretic properties of θ. We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible localization properties under influence of an external electric field. © 1996 The American Physical Society.