In this paper we present studies of the I-V characteristics of CdZnTe detectors with Pt contacts fabricated from high-resistivity single crystals grown by the high-pressure Brigman process. We have analyzed the experimental I-V curves using a model that approximates the CZT detector as a system consisting of a reversed Schottky contact in series with the bulk resistance. Least square fits to the experimental data yield 0.78-0.79 eV for the Pt-CZT Schottky barrier height, and <20 V for the voltage required to deplete a 2 mm thick CZT detector. We demonstrate that at high bias the thermionic current over the Schottky barrier, the height of which is reduced due to an interfacial layer between the contact and CZT material, controls the leakage current of the detectors. In many cases the dark current is not determined by the resistivity of the bulk material, but rather the properties of the contacts; namely by the interfacial layer between the contact and CZT material.