A method for forming single crystal or large-crystal-grain thin-film layers deposits a thin-film amorphous, nanocrystalline, microcrystalline, or polycrystalline layer, and laser-heats a seed spot having size on the order of a critical nucleation size of the thin-film layer. The single-crystal seed spot is extended into a single-crystal seed line by laser-heating one or more crystallization zones adjacent to the seed spot and drawing the zone across the thin-film layer. The single-crystal seed line is extended across the thin-film material layer into a single-crystal layer by laser-heating an adjacent linear crystallization zone and drawing the crystallization zone across the thin-film layer. Photovoltaic cells may be formed in or on the single-crystal layer. Tandem photovoltaic devices may be formed using one or several iterations of the method. The method may also be used to form single-crystal semiconductor thin-film transistors, such as for display devices, or to form single-crystal superconductor layers.