The subject of the invention is the use, as catalyst support sublayer in a process for growing carbon nanotubes by chemical vapour deposition (CVD), of a multilayer stack formed of alternating layers of silica and of alumina, each of the layers having a thickness of less than or equal to 10 nm and consisting of one or more superposed atomic monolayer(s). It also relates to a multilayer structure comprising a substrate which has, on at least one of its faces, such a multilayer stack, and also to the use thereof for the growth of a mat of carbon nanotubes, which are in particular spinnable, by chemical vapour deposition, preferably hot-filament chemical vapour deposition.