Calculation of transmission coefficients at nonideal semiconductor interfaces characterized by a spatial distribution of barrier heights

Abstract

The use of temperature-dependent current–voltage and capacitance–voltage measurements in the determination of transmission coefficients, which can be related to various other heterogeneous charge-transfer rate constants, is analyzed for semiconductor interfaces characterized by a spatial distribution of barrier heights. The level of error introduced by the unknowing neglect of heterogeneity is analyzed using a discrete parallel network of regions with potentially voltage and temperature dependent effective areas and potential barriers. In general, the unknowing neglect of heterogeneity results in an overestimation of the transmission coefficient calculated from barrier heights based on capacitance–voltage measurements and an underestimation when based on barrier heights from temperature dependent current–voltage measurements (Richardson plots). Of particular focus is the calculation of transmission coefficients at semiconductor interfaces that exhibit anomalous behavior, most notably ideality or quality factors greater than unity, due to small-scale, “pinched-off,” barrier inhomogeneities characterized by voltage-dependent effective barriers. In general, the meanings of various empirical treatments of current–voltage data in light of a voltage-dependent barrier height distribution are clarified with a particular focus on the extraction of equilibrium exchange current densities and the meaning of Richardson plots. To be specific, the model of Tung for pinched-off barrier inhomogeneities is used to demonstrate that multiple orders-of-magnitude errors in the calculation of transmission coefficients are possible with systems exhibiting only mildly anomalous behavior (ideality factors less than 1.3) if heterogeneity is neglected. The conditions of applied bias, dopant density, and temperature where the error is minimized are discussed along with the criteria for the rigorous extraction of transmission coefficients. The greatest confidence in the transmission coefficients occurs when the ideality factor is unity and the capacitance–voltage barrier agrees with the Richardson plot barrier.