Process for fabricating a thin single-crystalline layer n, including steps of: a) providing a support substrate n, b) placing a seed sample n, c) depositing a thin layer n so as to form an initial interface region n including a proportion of seed sample n and a proportion of thin layer n, the proportion of seed sample n decreasing from the first peripheral part n towards the second peripheral part n, e) providing an energy input to the initial interface region n contiguous to the first peripheral part n so as to liquefy a portion n of the thin layer and form a solid/liquid interface region n, and f) gradually moving the energy input away from the seed sample n so as to solidify the portion n so as to gradually move the solid/liquid interface region n.