Copper deposition onto Cu 2 O thin films grown on Au(111) results in the formation of monolayer islands with hexagonal and rhombic shapes, as observed with scanning tunnelling microscopy. The differential conductance through the Cu islands is governed by distinct quantum well states (QWS), accompanied by pronounced electron standing wave patterns. Below the onset of the QWS, an extended region of negative differential conductance opens up, in which also the tunnelling current declines markedly with increasing bias voltage. The effect is assigned to the quantised electronic structure of the Cu islands in combination with the p-type conductance behaviour of the oxide film underneath. The latter promotes electron transport across the islands around the Fermi level, but leads to a closure of this transport channel at negative bias.