Pavel Exner, Ralf Gawlista
Band spectra of rectangular graph superlattices
(2735K, PostScript (Adobe, V. 2.0))
ABSTRACT. We consider rectangular graph superlattices of sides l1, l2
with the wavefunction coupling at the junctions either of the
delta type, when they are continuous and the sum of their
derivatives is proportional to the common value at the junction with
a coupling constant alpha, or the "delta-prime-S" 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 theta := l1/l2. If the latter
is an irrational badly approximable by rationals, delta lattices have
no gaps in the weak-coupling case. We show that there
is a quantization for the asymptotic critical values of alpha
at which new gap series open, and explain it in terms of
number-theoretic properties of theta. 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.