CdTe Schottky diodes for X- and γ-ray detection exhibit excellent spectroscopic performance, even though these are not stable under operative voltages. Improvements require the comprehension of the main carrier transport mechanisms, presently unclear. We address this issue by correlating the internal electric field and the flowing current. Depending on the temperature and applied voltage, different mechanisms become dominant where the deep levels always play a central role. Indeed, the partial ionization of deep levels directly controls the electric field distribution. Transient measurements show how, under high voltages, the field at the contacts controls the current flowing through the detector.