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

In this communication, the band–gap structures of nn-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, Z.X. Wen, S.S. Jiang, W.T. Tong, R.W. Peng, D. Feng, Phys. Rev. B 48 (1993) 829], we find that the nn-CF sequences with 2≤n≤42≤n≤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=5n=5 sequence belongs to a critical case which lies at the border between quasiperiodic and non-quasiperiodic structures. The frequency range of the self-similarity pattern approaches 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.

Research highlights
► We study generalized nn-component Fibonacci (nn-CF) piezoelectric superlattices.
► 2, 3, 4-CF superlattices are quasiperiodic and 5-CF superlattice is a critical structure.
► Unique peak labeling works for 2, 3, 4-CF superlattices, but not for 5-CF superlattice.
► Self-similar pattern exists for 2, 3, 4-CF superlattices, but not found for 5-CF superlattice.
► Generalized nn-component Fibonacci lattice tunes the quasi-periodicity of structures.