Numerical study of the localization-delocalization transition for vibrations in amorphous silicon

Numerical studies of amorphous Si in harmonic approximation show that the h ighest 3.5% of vibrational normal modes are localized. As the vibrational f requency increases through the boundary separating localized from delocaliz ed modes, near omega (c)=70 meV (the 'mobility edge') there is a localizati on-delocalization 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 to te st whether or not these diverge at omega (c). Results are consistent with a localization length xi which diverges above omega (c) as (omega - omega (c ))(-p) where the exponent is p approximate to 1.3 +/- 0.5. Below the mobili ty edge we find no evidence for a diverging correlation length. Such an asy mmetry would contradict scaling ideas, and we suppose it is a finite-size a rtefact. If the scaling regime is narrower than our (approximately 1 meV) r esolution, then it cannot be seen directly on our finite system.