TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Components: The second layer comprises many second-layer sublayers, where each of the second-layer sublayers has substantially the first-layer lattice parameter. In the solar cell (C1): the second buffer layer is of a different composition than the first buffer layer; the first layer is a non-photoactive germanium substrate, the second layer includes gallium indium arsenide alloy of formula (Ga(1-x)In(x)As), where x is 0-0.53, and the third layer includes gallium arsenide; and the first buffer layer includes graded gallium indium arsenide, and the second buffer layer includes transparent graded aluminum gallium indium arsenide. The solar cells (C1) and (C2) further include: a fourth layer containing a photoactive fourth-layer material contacting the third layer, where the fourth layer has a fourth-layer lattice parameter different from the third-layer lattice parameter, and further includes a transparent graded third buffer layer extending between and contacting the third layer and the fourth layer and having a third-buffer-layer lattice parameter that matches that of the third layer where the third buffer layer contacts the third layer and matches that of the fourth layer where the third buffer layer contacts the fourth layer; and a fifth layer containing a photoactive fifth-layer material contacting the fourth layer, where the fifth layer has a fifth-layer lattice parameter different from the fourth-layer lattice parameter, and further includes a transparent graded fourth buffer layer extending between and contacting the fourth layer and the fifth layer and having a fourth-buffer-layer lattice parameter that matches that of the fourth layer where the fourth buffer layer contacts the fourth layer and matches that of the fifth layer where the fourth buffer layer contacts the fifth layer. In the solar cell (C2): the first layer is a non-photoactive germanium substrate; the second layer includes silicon-germanium; the third layer includes gallium arsenide; the first buffer layer includes graded silicon-germanium; and the second buffer layer includes transparent graded gallium indium phosphide (arsenide). In the solar cell (C3): the first layer is transparent to wavelengths of light that are photoconverted by the second layer; and the first layer is a substrate that is not photoactive, where the solar cell comprises at least one additional photoactive layer made of a photoactive additional-layer material, the first layer is optionally between the additional photoactive layer and the second layer, and the buffer layer is made of a buffer-layer material having a minimum bandgap greater than a bandgap of the second layer by at least 50 milli-eV. In the solar cell (C4): the first layer is transparent to wavelengths of light that are photoconverted by the second layer; and the first layer is a substrate that is not photoactive, where the solar cell comprises at least one additional photoactive layer made of a photoactive additional-layer material, the first layer is optionally between the additional photoactive layer and the second layer, and the buffer layer is made of a buffer-layer material having a minimum bandgap greater than a bandgap of the second layer by at least 50 milli-eV. In the solar cell (C4): the first layer is transparent gallium arsenide, the second layer is gallium indium arsenide second layer 64 having 35% indium; and the graded buffer layer is aluminum gallium indium arsenide or gallium indium phosphide(arsenide), where the solar cell further includes a gallium arsenide additional solar subcell contacting the first layer on the opposite side from the second layer so that the first layer is between the gallium arsenide additional solar subcell and the second layer, and the gallium indium phosphide additional solar subcell contacting the gallium arsenide additional solar subcell.