Abstract
With the fundamental work of Hofstadter on the combined effects of band structure and magnetic field on the electronic states in two dimensions (2D) as a starting point, we numerically study the effects on the Hofstadter butterfly of including a binary distribution of on-site potentials on a 2D lattice in the tight-binding picture. The effects of the external magnetic field are included through the so-called Peierls substitution. The problem is reduced to a one-dimensional set of difference equations when the binary distribution is constrained to be in one direction only. Besides a periodic structure, a number of aperiodically ordered distributions like the Fibonacci, Thue-Morse, and the Rudin-Shapiro sequences are considered, and the band structures presented and discussed. Also, 2D chessboard and Sierpinski carpet distributions are dealt with in some detail.
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