We used high-quality ZnO nanostructures/graphene substrates for understanding the mechanisms of charge transfer (CT) that take place under nonplasmonic conditions. As the optimal conditions for CT processes are found, we studied the range of CT normal to the ZnO surface that is coated with nanoscale HfO2 layers with different thicknesses. We could observe that CT decays over a few nanometers. In addition, we also observed a unique oscillation of the SERS intensity in the atomically thin oxide layers, which reflects the quantum tunneling effects of CT electrons across the oxide layers. To the best of our knowledge, this is the first direct observation of SERS-active charge transport and measurement of a CT span with atomic-scale accuracy.