Spectrum of waves in stochastically modulated superlattices

A theory of the spectrum of waves in partially randomized multilayers is developed. A model of the superlattice in the form of a harmonic function with stochastically modulated period is considered. Correlation functions of the superlattice are deduced for different shapes of the modulation: plane one-dimensional defects, prolate along the superlattice axis, two-dimensional defects, and isotropic three-dimensional inhomogeneities. The law of the correlation decay depends essentially on the dimension of short-wave inhomogeneities and is Gaussian for long-wave inhomogeneities irrespective of their dimension. Dispersion laws and damping of waves are calculated at the boundary of the Brillouin zone of the superlattice in the weak-binding approximation. Both the damping and modification of the dispersion law depend on the dimension of the inhomogeneities and their correlation radii. In the absence of randomization a gap in the spectrum is determined by the change of some physical parameter at the interfaces. Randomization leads to the monotonic decrease of the gap for the case of short-wave inhomogeneities. For the case of long-wave inhomogeneities randomization leads at first to the increase of the gap and only then to its decrease and closing.