Monocrystalline 1.7 eV Mg0.13Cd0.87Te/MgxCd1-xTe (x > 0.13) double heterostructure (DH) solar cells with varying Mg compositions in the barrier layers are grown by molecular beam epitaxy. A Mg0.13Cd0.87Te/Mg0.37Cd0.63Te DH solar cell featuring abrupt interfaces between barriers and absorber and the addition of a SiO2 anti-reflective coating demonstrate open-circuit voltage (V-OC), short-circuit current density (J(SC)), fill factor (FF), and device active-area efficiencies up to 1.129 V, 17.3 mA/cm(2), 77.7%, and 15.2%, respectively. The V-OC and FF vary oppositely with the MgxCd1-xTe barrier height, indicating an optimal design of the MgCdTe DHs as a trade-off between carrier confinement and carrier transport. Temperature-dependent V-OC measurements reveal that the majority of carrier recombination in the devices occurs outside the DHs, in the a-Si:H hole-contact layer, and at the interface between the a-Si:H layer and the MgxCd1-xTe top barrier at room temperature. Simulation results for the device with the highest efficiency show that the p-type a-Si:H layer and the Mg0.37Cd0.63Te top barrier contribute 1.3 and 2.4 mA/cm(2) J(SC) loss, respectively.