A Mathematical Model for the Influence of Deep-Level Electronic States on Photoelectrochemical Impedance Spectroscopy: 2. Assessment of Characterization Methods Based on Mott-Schottky Theory

The impact of the assumptions inherent in using the Mott-Schottky theory to identify deep-level electronic states in semiconductors was assessed by comparison to the results of a less restrictive mathematical model. The model, developed in another paper, treated the transport and recombination reactions involving electrons, holes, and electronic states located within the bandgap. The capacitive component used in standard Mott-Schottky theory was found to be insensitive to bulk electronic states within the bandgap for concentrations significantly less than the doping level. The resistive component measured at low frequencies was much more sensitive to deep-level states and may be used to determine their distribution. For high concentrations of deep-level states, the model results were in agreement with the current practice of attr ibuting changes in the slope of the Mott-Schottky curve to partial ionization of single-energy deep-level states with applied potential. In the absence of frequency dispersion, these changes in slope could be attributed instead to nonuniform dopant distribution. Mott-Schottky theory is commonly used to evaluate the doping level and flatband potential of semiconductors [see, e.g., Ref. (1, 2)]. The technique is based on the equation 1 2(V + RT/F) [1] C~r eF(Nd -Na) which provides the relationship between the applied potential and the capacitance of the semiconductor spacecharge region for an evenly doped, ideally polarizable electrode under reverse bias potentials. Following this relationship, a plot of 1/C 2 for an evenly doped semiconductor in the absence of deep-level electronic states is linear over a range of potentials where a depletion layer is formed. A depletion layer is formed near the polarized interface at potentials such that the semiconductor is reverse-biased (V < 0 for an n-type semiconductor) and such that the electron and hole concentrations are significantly less than the dopant concentration. This requirement is violated at low and high reverse biases when the potential drop across the semiconductor reaches a value that allows either the electron or hole concentration to be of the same magnitude as the doping concentration. Mott-Schottky analysis of uniformly doped semiconductors is subject to the assumption that deep-level states are not present in significant concentration. Deviations from straight lines in Mott-Schottky plots, therefore, are frequently attributed to the influence of potential dependent charging of surface or bulk states. This interpretation is supported by analytic calculations of the contribution of deep-level states to the space charge as a function of applied potential (3-5). It should be noted that this approach treats only the high-frequency asymptotic limit to the imaginary part of the impedance. Deviations from linearity of Mott-Schottky plots could also be attributed to a nonuniform dopant distribution. The local dopant concentration is obtained through Eq. [1] from the slope of the Mott-Schottky plot at a given potential. This potential is related to a probing depth through an expression for the depletion layer width W (6) where w = [2] F(Na Na) * Electrochemical Society Student Member. ** Electrochemical Society Active Member. 1 Present address: Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Mm'yland 21218. Present address: Department of Chemical Engineering, University of Florida, GainesviiIe, Florida 32611. A further simplification can be made by noting that