A Mathematical Model for the Influence of Deep-Level Electronic States on Photoelectrochemical Impedance Spectroscopy: 1. Theoretical Development

A mathematical model is developed to calculate the impedance response of a semiconductor electrode to a sinusoidal current perturbation under subbandgap monochromatic illumination. The model accounts explicitly for electron and hole transport as well as generation and recombination through band-to-band mechanisms and through bulk deep-level electron acceptors of specified energy. The model results are compared to experiment and illustrate how the impedance response obtained under monochromatic subbandgap illumination can be used to identify the energy, density, distribution, and recombinat ion rate constants associated with deep-level electronic states. This may have application to in situ characterization of semiconductor-electrolyte interfaces and to characterization of solid-state materials and devices. Electrochemical impedance spectroscopy, coupled with monochromatic subbandgap illumination, may provide a room temperature approach for characterizing deep-level and interfacial electronic states in large bandgap semiconductors. The electronic states of interest lie within the bandgap of the semiconductor. The goal is to determine that these states exist as well as to determine their concentration distribution and the associated rate constants for electronic transitions. Knowledge of these parameters is essential for the engineering of many electronic devices. For example, deep-level states are undesirable when they facilitate electronic transitions which reduce the efficiency of photovoltaic cells. In other cases, the added reaction pathways for electrons result in desired effects. Electroluminescent panels, for example, rely on electronic transitions that result in emission of photons. The energy level of the states caused by introduction of dopants determines * Electrochemical Society Student Member. ** Electrochemical Society Active Member. 1Present address: Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218. 2 Present address: Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611. the color of the emitted light. Electroluminescent panels represent one approach for development of thin-film color television screens. Interfacial states play a key role in electroluminescence, and commercial development of this technology will hinge upon understanding the relationship between fabrication techniques and the formation of deep-level states. The impact of deep-level states can be significant, even in concentrations that are very low by normal chemical standards. Several states can be associated with a chemical species, and such states may also appear as a result of vacancies or other crystalline defects. Traditional chemical means of detection, therefore, do not provide complete identification of deep-level electronic states. The techniques commonly employed to detect deep-level states [such as application of Mott-Schottky theory, see e.g. (1, 2), deep-level transient spectroscopy (DLTS) (3), or photocapacitance spectroscopy (4-9)] tend to be electrical in nature since it is through their electronic behavior that these states influence device performance. These techniques rely on determining the change of space charge capacitance associated with ionization of deep-level states. The capacitance is usually determined from a single, high-freJ. Electrochem. Soc., Vol. 139, No. 1, January 1992 9 The Electrochemical Society, Inc. 119 quency measurement. Since the charge held in the states may be very small as compared to that associated with doping species, the sensitivity of capacitance-based techniques requires precise measurement of the imaginary part of the impedance response. The real part of the impedance response has been shown to be more sensitive to ionization of deep-level states, particularly at low frequencies. Nicollian and Goetzberger (10), DeClerck et al. (11), and Nagasubramanian et aI. (12) have attributed variations observed in the real component of the impedance with potential to midbandgap surface states. This paper addresses a light-enhanced form of electrochemical impedance spectroscopy. In this technique, the effect of photonic excitation of electronic transitions by light at selected wavelengths is detected by impedance spectroscopy applied over a broad frequency range. The photonic energy of the light used is less than the bandgap energy; therefore, any changes in the impedance spectrum with i l lumination can be attributed to transitions involving energy levels within the bandgap. This method differs from the more commonly used DLTS in that the wavelength of monochromatic subbandgap light is varied to excite electronic transitions at a fixed temperature (e.g., room temperature): whereas, in DLTS, temperature is varied to change the occupancy of the states. A broad range of frequency (with emphasis on lower frequencies) for the impedance measurements is used instead of measuring an effective capacity at a single high frequency. The use of a broad frequency range is the essential distinction between this approach and photocapacitance spectroscopy. In principle, physical parameters such as the energy at which deep-level states exist, the concentrations of these states, the rate constants for transitions involving these states, and the distribution of these states could be extracted from data obtained using light-enhanced impedance spectroscopy. Such detailed interpretation of impedance data required an extension to currently available models for this system. The overall goal of this work, therefore, was to develop a comprehensive mathematical model which treats the physical phenomena governing the response of the system without use of overly restrictive assumptions. This work was motivated by two goals: (i) to support the current application of Mott-Schottky theory and photocapacitance spectroscopy for identification of deep-level states and (ii) to provide a framework for identification of deep-level states through interpretation of both the real and the imaginary parts of the impedance spectrum. The theoretical development and validation of the model by qualitative and quantitative comparison of model results to experimental data is presented in this paper. The application of the model to assess MottSchottky methods of analysis is presented elsewhere (13).