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The band-gap structures and recovery rules of generalized n-component Fibonacci piezoelectric superlattices

The spectral evolution from periodic structure to random structure has always been an interesting topic in solid state physics, the generalized n-component Fibonacci sequences (n- CF) provide a convenient tool to investigate such process since its randomness can be controlled via the parameter n. In this letter, the band-gap structures of n-CF piezoelectric superlattices have been calculated using the transfer-matrix-method, the self-similarity behavior and recovery rule have been systematically analyzed. Consistent with the rigorous mathematical proof by Hu et al.[A. Hu et al. Phys. Rev. B. 48, 829 (1993)], we find that the n-CF sequences with 2 \leq n \leq 4 are identified as quasiperiodic. The imaginary wave numbers are characterized by the self-similar spectrum, their major peaks can all be properly indexed. In addition, we find that the n = 5 sequence belongs to a critical case which lies at the border between quasiperiodic to aperiodic structures. The frequency range of self-similarity pattern approaches to zero and a unique indexing of imaginary wave numbers becomes impossible. Our study offers the information on the critical 5-CF superlattice which was not available before. The classification of band-gap structures and the scaling laws around fixed points are also given.