The Calculation of Transmission Coefficients at Heterogeneous Semiconductor Interfaces: A Case Study Based on the n-InP
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The Calculation of Transmission Coefficients at Heterogeneous Semiconductor Interfaces: A Case Study Based on the n-InP | poly(pyrrole) Interface Carrie Daniels-Hafer, Meehae Jang, Frank E. Jones, Shannon W. Boettcher, Rob Danner, and Mark C. Lonergan, Dept. of Chemistry and The Materials Science Institute, University of Oregon, Eugene, OR 97403-1253 ABSTRACT The n-InP | poly(pyrrole) interface is used as a case study to discuss the calculation of the transmission coefficient, describing the probability of majority carrier transfer, at a non-ideal semiconductor interface exhibiting anomalous behavior assumed to be due to a spatial distribution of barrier heights. The most notable anomaly is a weaker dependence of current on voltage than predicted by thermionic emission (i.e. quality or ideality factor greater than unity). Central to this discussion is the calculation of the equilibrium exchange current density Jo and barrier height φb in light of a heterogeneous and potentially voltage-dependent barrier distribution. Various approaches to the measurement of φb and Jo valid for semiconductor interfaces characterized by a uniform, voltage- and temperature-independent barrier are discussed when applied to a heterogeneous interface. In particular, the use of a capacitancevoltage measured barrier is demonstrated to result in an overestimation of κ whereas the use of a Richardson plot barrier is demonstrated to result in an underestimation. Depending on method, errors in excess of five orders-of-magnitude are observed for the n-InP | poly(pyrrole) interface under conditions where it exhibits only mildly anomalous behavior (ideality factor ≈ 1.2). 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. INTRODUCTION Interest in the use of molecules as active elements in microelectronic devices has stimulated a great deal of interest in charge transfer through molecules and at interfaces between delocalized and localized electronic systems. One area of interest particularly relevant for the integration of molecular systems with conventional microelectronics is the interface between an inorganic semiconductor and redox active molecules or polymers. Such semiconductor electrodes are also important platforms for the fundamental study of electron transfer processes. A primary quantity of interest in the study of charge transfer kinetics at semiconductor electrodes is the heterogeneous charge transfer rate constant. Depending on system, this rate constant can be cast in a variety of ways.[1] The determination of heterogeneous charge transfer rate constants relies on thermionic emission theory in which the equilibrium exchange current density, Jo, at the electrode is given by:[2] J o = κA*T 2 exp(− βφ b )
(1)
where A* is the Richardson constant, T is temperature, and β = q / (kBT) with q the elementary charge and kB the Boltzmann constant. The potential barrier φb arises from the region of ionized dopant atoms t
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