Electronic states in one-dimensional self-similar alloy with ($3^1,3^2,\dots ,3^N$) periods

The one-dimensional $A_{1-x}{B}_x$ alloy is presented to study electron properties of binary alloys intermediate between periodicity and randomness. The system studied is represented by a tight-binding Hamiltonian with diagonal elements given by $\Sigma {n=1\mathrm{}}^N{\epsilon }_{B}f\left(cos\right(\frac{2\pi i}{3^n}+\frac{2\pi }{3}\left)\right)$, where if $x=1\mathrm{}$ then $f\left(x\right)=1\mathrm{}$, otherwise $f\left(x\right)=0\mathrm{}$, and ${\epsilon }_B$ indicates the site energy of the $B$ atom. The Hamiltonian is exactly renormalized with the use of a decimation technique. Densities of states and band structures are calculated by the recursion relations. It is shown that the bands break up into narrower subbands with increasing $N$. It appears that when $N$ becomes $N+1\mathrm{}$ each band breaks up into triplet subbands, and as $N\to \infty$ the density of states approaches a limit with an infinite number of band gaps.