We investigate the thermal stability of alumina supporting layers sputtered at different conditions and its effect on the growth of aligned single-walled carbon nanotube arrays. Radio frequency magnetron sputtering of alumina under oxygen-argon atmosphere produces a Si-rich alumina alloy film on a silicon substrate. Atomic force microscopy on the annealed catalysts reveals that Si-rich alumina films are more stable than alumina layers with low Si content at the elevated temperatures at which the growth of single-walled carbon nanotubes is initiated. The enhanced thermal stability of the Si-rich alumina layer results in a narrower (< 2.2 nm) diameter distribution of the single-walled carbon nanotubes. Thanks to the smaller diameters of their nanotube pores, membranes fabricated with vertically aligned nanotubes grown on the stable layers display improved ion selectivity.