The current-voltage (J-V) characteristics of ZnO/PbS quantum dot (QD) solar cells show a QD size-dependent behavior resulting from a Schottky junction that forms at the back metal electrode opposing the desirable diode formed between the ZnO and PbS QD layers. We study a QD size-dependent roll-over effect that refers to the saturation of photocurrent in forward bias and crossover effect which occurs when the light and dark J-V curves intersect. We model the J-V characteristics with a main diode formed between the n-type ZnO nanocrystal (NC) layer and p-type PbS QD layer in series with a leaky Schottky-diode formed between PbS QD layer and metal contact. We show how the characteristics of the two diodes depend on QD size, metal work function, and PbS QD layer thickness, and we discuss how the presence of the back diode complicates finding an optimal layer thickness. Finally, we present Kelvin probe measurements to determine the Fermi level of the QD layers and discuss band alignment, Fermi-level pinning, and the V(oc) within these devices.