A method for fabricating a four-junction space-qualified solar cell (200, 400, 450, 500) designed for operation at AM0 and at a 1 MeV electron equivalent fluence of at least 1 x 10 14 e/cm 2, the solar cell comprising solar subcells (A,B,C,D), comprising the following steps: providing a semiconductor growth substrate; forming a fourth solar subcell over or in the growth substrate; growing a sequence of layers of semiconductor material using a deposition process to form a solar cell comprising a plurality of solar subcells including a third solar subcell disposed over the fourth solar subcell and being lattice mis-matched from the fourth solar subcell, a second solar subcell disposed over and lattice matched to the third solar subcell and having a band gap in the range of approximately 1.55 to 1.8 eV and an upper first solar subcell disposed over and lattice matched to the second solar subcell and having a band gap in the range of 2.0 to 2.2 eV, wherein the average band gap of all four solar subcells is greater than 1.35 eV; providing an intermediate graded layer disposed between the third solar subcell and the fourth solar subcell, wherein the intermediate graded layer is compositionally graded to lattice match the third solar subcell on one side and the fourth solar subcell on the other side and is composed of any of the As, P, N, Sb based III-V compound semiconductors subject to the constraints of having the in-plane lattice parameter greater than or equal to that of the fourth solar subcell, and having a band gap energy greater than that of the third solar subcell, wherein the intermediate graded layer is compositionally graded such that sublayers (505a,...505zz) are defined having step-graded lattice constants, wherein an uppermost sublayer of the intermediate graded layer has a larger lattice constant than an adjacent layer adjacent to and disposed above the uppermost sublayer and a back surface field layer adjacent to and disposed above the adjacent layer, characterized in that the adjacent layer has a different composition than the uppermost sublayer and the back surface field layer, and in that the adjacent layer has the same lattice constant as the back surface field layer. providing a semiconductor growth substrate; forming a fourth solar subcell over or in the growth substrate; growing a sequence of layers of semiconductor material using a deposition process to form a solar cell comprising a plurality of solar subcells including a third solar subcell disposed over the fourth solar subcell and being lattice mis-matched from the fourth solar subcell, a second solar subcell disposed over and lattice matched to the third solar subcell and having a band gap in the range of approximately 1.55 to 1.8 eV and an upper first solar subcell disposed over and lattice matched to the second solar subcell and having a band gap in the range of 2.0 to 2.2 eV, wherein the average band gap of all four solar subcells is greater than 1.35 eV; providing an intermediate graded layer disposed between the third solar subcell and the fourth solar subcell, wherein the intermediate graded layer is compositionally graded to lattice match the third solar subcell on one side and the fourth solar subcell on the other side and is composed of any of the As, P, N, Sb based III-V compound semiconductors subject to the constraints of having the in-plane lattice parameter greater than or equal to that of the fourth solar subcell, and having a band gap energy greater than that of the third solar subcell, wherein the intermediate graded layer is compositionally graded such that sublayers (505a,...505zz) are defined having step-graded lattice constants, wherein an uppermost sublayer of the intermediate graded layer has a larger lattice constant than an adjacent layer adjacent to and disposed above the uppermost sublayer and a back surface field layer adjacent to and disposed above the adjacent layer, characterized in that the adjacent layer has a different composition than the uppermost sublayer and the back surface field layer, and in that the adjacent layer has the same lattice constant as the back surface field layer.