Fractal Character of Eigenstates in Disordered Systems

Electronic eigenfunctions are studied on the tight-binding model of disordered systems at dimensionalities $d=1,2,3\mathrm{}$. It is found that the eigenfunctions have a self-similar (fractal) behavior up to length scales roughly equal to the localization length. For $d=3\mathrm{}$, above the mobility edge, the fractal character persists up to length scales about equal to the correlation length $\xi$. The dependence of the fractal dimensionality $D$ on disorder $W$ is presented. The fractal character of the wave function is suggested as a new method for finding mobility edges.