Combining self-consistently first-principles density functional theory(DFT) and non-perturbative scattering theory for transport calculations, we study leakage current through ultra-thin crystalline metal-oxides and SiO₂ barriers. In this talk we will introduce the method and describe its implementation. Because DFT is a ground state theory, transport in the conduction band can only be calculated approximately using this approach. We will discuss and illustrate this issue and show how to obtain valuable qualitative information not directly accessible to semiempirical approaches. Calculated leakage current through prototype (c,t,m)-ZrO₂ and m-HfO2 barriers will be presented for defect-free barriers and in the presence of impurities and crystal defects likely to be found in these materials. We will show that higher leakage currents in Zr(Hf)O2 barriers result from a vanishing Si/Zr(Hf)O₂ conduction band offset. The presence of a SiO₂ interfacial layer between Si and Zr(Hf)O₂, with its bigger conduction band offset, decreases the leakage current.