Metal-semiconductor contacts are key components of nano-electronics and atomic-scale integrated circuits. In these components Schottky diodes provide a low forward voltage and a very fast switching rate but suffer the drawback of a high reverse leakage current. Improvement of the reverse bias characteristics without degrading performance of the diode at positive voltages is deemed physically impossible for conventional silicon-based Schottky diodes. However, in this work we propose that this design challenge can be overcome in the organic-based diodes by utilizing reversible transitions between distinct adsorption states of organic molecules on metal surfaces. Motivated by previous experimental observations of controllable adsorption conformations of anthradithiophene on Cu(111), herein we use density functional theory simulations to demonstrate the distinct Schottky barrier heights of the two adsorption states. The higher Schottky barrier of the reverse bias induced by a chemisorbed state results in low leakage current, while the lower barrier of the forward bias induced by a physisorbed state yields a larger output current. The rectifying behaviors are further supported by nonequilibrium Green's function transport calculations.