Band-structural and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices

We study the electronic and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices generated by the stacking rule Sₗ₊₁=Sₗⁿ{S}_{l{{-}}1}ᵐ$ with positive integers n and m, where ${S}ₗ$ is the lth generational binary sequence. After showing that, in the limit of the large potential strength, the energy spectrum of a lattice with certain specific n and m can be determined by the associated characteristic value {τ}(n,m), we investigate the relation between the electronic band structure and the Fourier spectrum. When the lattice possesses the Pisot-Vijayaraghavan (PV) property (i.e., when n+1>m), the Fourier spectrum is closely related to the electronic band structure; the location and the relative strength of the Fourier spectral peak is in agreement with the location and the relative width of the energy spectral gap. On the other hand, when the lattice possesses no PV property (i.e., when n+1{≤}m), the Fourier spectrum is not directly related to the electronic band structure; the strength of the Fourier spectral peak is irrelevant to the width of the energy spectral gap, while the location of the peak corresponds to that of the gap. We also study the dependence of the electronic and Fourier-spectral properties on the initial conditions of the stacking rule through detailed study of the copper mean lattice (n=1,m=2) with initial conditions S₁={A} and S₂={ABᵖ}. It is found that the fractal structure of the energy spectrum is independent of the integer p, while some local electronic properties depend on p. It is also found that the global structure of the Fourier spectrum depends on p; it looks more blurred, and thus the aperiodic nature of the lattice becomes clearer with the increase of p.