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Charge trapping properties of electrons and holes in copper-doped zinc oxide (ZnO:Cu) films have been studied by scanning probe microscopy. We investigated the surface potential dependence on the voltage and duration applied to the copper-doped ZnO films by Kelvin probe force microscopy. It is found that the Fermi Level of the 8 at.% Cu-doped ZnO films shifted by 0.53 eV comparing to undoped ZnO films. This shift indicates significant change in the electronic structure and energy balance in Cu-doped ZnO films. The Fermi Level (work function) of zinc oxide films can be tuned by Cu doping, which are important for developing this functional material. In addition, Kelvin probe force microscopy measurements demonstrate that the nature of contact at Pt-coated tip/ZnO:Cu interface is changed from Schottky contact to Ohmic contact by increasing sufficient amount of Cu ions. The charge trapping property of the ZnO films enhance greatly by Cu doping (~10 at.%). The improved stable bipolar charge trapping properties indicate that copper-doped ZnO films are promising for nonvolatile memory applications.