Effects of deterministic aperiodic and self-similar on-site potentials on the structure of the Hofstadter butterfly

With the fundamental work of Hofstadter on the combined effects of band str ucture and magnetic field on the electronic states in two dimensions (2D) a s a starting point, we numerically study the effects on the Hofstadter butt erfly of including a binary distribution of on-site potentials on a 2D latt ice in the tight-binding picture. The effects of the external magnetic fiel d are included through the so-called Peierls substitution. The problem is r educed to a one-dimensional set Of difference equations when the binary dis tribution is constrained to be in one direction only. Besides a periodic st ructure, a number of a periodically ordered distributions like the Fibonacc i, Thue-Morse, and the Rudin-Shapiro sequences are considered, and the band structures presented and discussed. Also, 2D chessboard and Sierpinski car pet distributions are dealt with in some detail. [S0163-1829(99)01038-3].