We investigated a multiple charge-injection mechanism of a magneto-electric (ME) capacitor. Although the capacitor showed a multi-step charge-injection process with regard to its capacitancevoltage (CV) characteristics, the behaviors were found to be different depending on the tunneling layer thickness. Detailed analyses of currentvoltage (IV) characteristics revealed the presence of multi-step behavior in a sample exhibiting a FowlerNordheim (FN) tunneling mechanism. With decreases in the tunnel layer thickness, the FN tunneling behavior shifted to a high voltage region, at which point the multi-step behavior became unclear. These results indicate that the tunnel layer thickness plays an important role in this multi-step mechanism.