Accuracy assessment of sheet-charge approximation for Fowler-Nordheim tunneling into charged insulators

Abstract

Various insulators are used as gate dielectrics and passivation layers in wide-bandgap (WBG) semiconductor devices as well as in advanced Si devices, and the understanding of their current conduction mechanism is essential to achieve their high performance and high reliability. Because these insulators are more or less charged, the conduction current is mostly caused by the Fowler-Nordheim (FN) tunneling into charged insulators, ruling out the conventional analytic FN formula. In order to facilitate the analysis of these currents, we focused on the method, named sheet-charge approximation (SCA), of approximating the charge distribution in the insulators by a charge sheet that has the same areal density and centroid as those of the original. Using, as references, the results obtained exactly calculating the tunneling current in the framework of the Wentzel-Kramers-Brillouin approximation, we confirmed the advantage of SCA over the previous method using a tunneling-endpoint field, the error of SCA-estimated areal charge densities being at most 30% for rectangular charge distributions of which charge centroids are known as in stacked films. In a more general case where the centroid is unknown, the SCA usually provides only a charge moment with reference to the insulator/anode interface, being unable to decompose the moment into the areal charge density and centroid. However, this demerit of SCA can be overcome through a reverse-biased current-voltage measurement using a capacitor formed on a heavily doped substrate or a capacitor with a diffusion layer attached, which measurement provides a charge moment with reference to the original cathode/insulator interface. Using these two kinds of charge moments, we can separately extract the areal charge density and centroid. Hence, the SCA has practical significance as a tool for analyzing conduction currents through charged insulators, especially through stacked films, and accordingly will play an important role in improving the performance and reliability of gate dielectrics and passivation layers for various WBG semiconductor devices as well as of high-k gate stacks for advanced Si devices.