Flexible 1.4-mu m-thick Cu(In,Ga)Se-2 (CIGS) films on stainless steel (SUS) substrates are fabricated by the process named "multi-layer precursor method" as the absorbers of the thin-film solar cells. It is shown that the [Ga]/([Ga] + [In]) (GGI) or bandgap (E-g) profiles were easily controlled to enhance photovoltaic performances, especially short-circuit current density (J(SC)) for the increased conversion efficiency (eta). It is determined that the double-grading GGI profile is most pronounced with the presence of the CuInSe2 (CIS) phase in the CIGS film, when annealing time and temperature during the fabrication are reduced to 1 min and 610 degrees C, respectively. Consequently, the minimum (mini) E-g value in the E-g profile is reduced to approximately 1.05 eV close to the Eg of the CIS material, resulting in the enhancement of external quantum efficiency in the long wavelength region. The J(SC) of the flexible CIGS solar cell is ultimately increased up to 40 mA/cm(2), thus enhancing the eta to 17.5% despite the reduction of CIGS thickness to 1.4 mu m. It is disclosed that the reduced CIGS thickness to 1.4 mu m with very short annealing time (1 min) leads to the possible reduction of the production cost of the flexible thin-film solar cell with the high eta.