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Leakage currents in TiN/high-κ-ZrO2/TiN capacitors were simulated by using a novel kinetic Monte Carlo algorithm specially designed to describe tunneling transport of charge carriers in high-κ dielectrics, including defect-assisted transport mechanisms. Comparing simulation results with experimental data, a model for electronic transport was established and validated. Transport was found to be dominated by Poole-Frenkel emission from positively charged bulk trap states at medium voltages and trap-assisted tunneling at high voltages. Information on the conduction band offset at the TiN/ZrO2 interface as well as on the trap depth was extracted. The model accurately describes the scaling of the leakage current with temperature and with thickness of the dielectric film, and it provides insight into the mutual interdependence of the competing transport mechanisms.