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Temperature dependent pulsed and ramped substrate bias measurements are used to develop a detailed understanding of the vertical carrier transport in the buffer layers in a carbon doped GaN power heterojunction field effect transistor. Carbon doped GaN and multiple layers of AlGaN alloy are used in these devices to deliver an insulating and strain relieved buffer with high breakdown voltage capability. However, understanding of the detailed physical mechanism for its operation is still lacking. At the lowest electric fields (< 10MV/m), charge redistribution within the C doped layer is shown to occur by hole conduction in the valence band with activation energy 0.86eV. At higher fields, leakage between the two-dimensional electron gas and the buffer dominates occurring by a Poole-Frenkel mechanism with activation energy ~0.65eV, presumably along threading dislocations. At higher fields still, the strain relief buffer starts to conduct by a field dependent process. Balancing the onset of these leakage mechanisms is essential to allow the build-up of positive rather than negative space charge, and thus minimize bulk-related current-collapse in these devices.