Numerical Study of the Localization-Delocalization Transition for Vibrations in Amorphous Silicon

Numerical studies of amorphous silicon in harmonic approximation show that the highest 3.5% of vibrational normal modes are localized. As vibrational frequency increases through the boundary separating localized from delocalized modes, near ωc=70meV, (the “mobility edge”) there is a localization-delocalization (LD) transition, similar to a second-order thermodynamic phase transition. By a numerical study on a system with 4096 atoms, we are able to see exponential decay lengths of exact vibrational eigenstates, and test whether or not these diverge at ωc. Results are consistent with a localization length ξ which diverges above ωc as (ω − ωc) −p where the exponent is p ∼ 1.3 ± 0.5. Below the mobility edge we find no evidence for a diverging correlation length. Such an asymmetry would contradict scaling ideas, and we suppose it is a finite-size artifact. If the scaling regime is narrower than our (≈ 1 meV) resolution, then it cannot be seen directly on our finite system.