•RESET state of Ge-rich GST is not only resistive but also shows a capacitive component.•As good as trap-assisted, granular models explain heterogeneous materials electrical behavior.•SET state is not simply ohmic. Characteristics resemble those of a doped semiconductor.•Overall characteristics assigned to chemical and phases heterogeneities. In relation with the tendency of the Ge-rich alloys to decompose in pure Ge and GST phases. Ge-rich GeSbTe (GST) alloys are attracting Phase Change Materials for future memories as their higher crystallization temperature offers an extended range of applications. We have studied the electrical characteristics of PCM cells using such alloys as active layers. We show by impedance spectroscopy that the cells in the RESET (amorphous) state are not only resistive but also exhibit a capacitive component. Although trap-assisted conduction models are apparently able to describe the I(V) and I(T) characteristics of the devices in this state, their physical background is thus questionable. Alternatively, we show that granular models, describing electrical transport through conductive grains separated by insulating interfaces, are also able to simulate these characteristics, while fed by physically sound fitting parameters. Moreover, we show that the SET (crystalline) state is not simply ohmic but that its characteristics, as conductive as a metal but reacting as an insulator to temperature, resemble to those found in a semiconductor doped with a very low ionization energy defect. Finally, all these characteristics can be understood by considering that the electrical properties of cells made of Ge-rich GST layers are not those characteristic of some defective and homogeneous material but instead result from strong chemical heterogeneities found both in the amorphous and crystalline states of these Ge-rich alloys.