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DEEPLEVEL IMPEDANCE SPECTROSCOPY OF ELECTRONIC MATERIALS By ANDREW NORBERT JANSEN DISSERTATION PRESENTED TO THE GRADUATE SCHOOL THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ACKNOWLED GEMENTS The author thanks the following individuals: to my wife Rosalie for her love and encouragement since we first met, to my parents Charles and Shirley for their love and guidance throughout the years, to Professor Mark E. Orazem for his advice and support during my grad uate studies, to Pankaj Agarwal for his Indian recipes and numerical regression exper tise, to Matthew Esteban, Oliver Moghissi, and Paul Wojcik for their assistance and friendship, to Professor ChihTang Sah for his discussions and reprints on semicon ductor physics and, to Professors Timothy J. Anderson, Oscar D. Crisalle, and Kevin S. Jones for their suggestions during the course of this work. The author thanks Mr. Darwin Thusius of Sula Technologies for the GaAs sample and discussions on the DLTS instrumentation. The author also thanks Dr. David Morton and Dr. Robert Miller of the U.S. Army Electronics Technology and De vice Laboratory for providing the ZnS:Mn TFEL panel and Dr. Raychem Corporation for providing the ZnO varistors. DARPA Gary Trost of the This work was supported by under the Optoelectronics program of the Florida Initiative in Advanced 11. snlan4.n: n a &a4 .. a1. TABLE OF CONTENTS CHAPTERS INTRODUCTION BACKGROUND a a a a a a a a a a S S 0 0 5 a 0 * a S . . . . . . . . * * * EXPERIMENTAL METHOD AND INSTRUMENTATION General Schematic Electrochemical Impedance Spectroscopy 3.2.1 3.2.2 3.2.3 Theory * * Signal Generation and Analysis Experimental Difficulties . Standard Characterization Techniques . a a a a .26 3.3.1 3.3.2 3.3.3 MottSchottky Profiling . . DeepLevel Transient Spectroscopy (DLTS) DLTS Equipment and Parameters . 32 Optical Spectroscopy DEVELOPMENT OF PROCESS MODEL 4.1 Shockley Read Hall Processes 4.1.1 4.1.2 4.1.3 4.1.4 Equilibrium . . Nonequilibrium . Small Signal Analysis Equivalent Circuit Surface States 4.2.1 4.2.2 S . a * 0 . .74 Single Energy Level Surface State Surface Band . . Complete Equivalent Circuit 4.3.1 j a a a a S S S S a a a a a a82 Simplifications of the Complete Circuit . . A 1 0 .1 13% 1 i a/l" < i l I ACE(NOWI;EDGEMENTS LIST OF TABLES ................... .............. LIST OF FIGUIEIES AIISTRACT Cuprous Oxide in AlkalineChloride Solutions 99 5.4.1 5.4.2 Background Experiment * S 9 S 9 9 S 9 . 9 9 9 9 9 * . 9 . S 9 9 9 9 S 9 9 9 9 RESULTS AND DISCUSSION 6.1 Gallium Arsenide 9 9 9 9 . 9 9 S 106 . 9 . 9 . 9 9 5 9 .106 6.1.1 6.1.2 Thermally Stimulated DeepLevel Impedance Spectroscopy Optically Stimulated DeepLevel Impedance Spectroscopy . 106 . 112 Importance of Weighting on Regression Results . . . 117 6.3 ZnS:Mn TFEL Panel .. .. .. .. 6.4 ZnO Varistors . . . . . 6.5 Cuprous Oxide in AlkalineChloride Solution 6.6 Comparison of Impedance and Admittance. 6.7 ThreeDimensional Representation . CONCLUSIONS . . . . . SUGGESTIONS FOR FUTURE WORK ..... S. .. 120 APPENDIX PROGRAM FOR DLTS INTERFACE REFERENCES S9 205 5 .9 .. .. 9. .. .. .9. *212 BIOGRAPHICAL SKETCH . 221 126 200 202 LIST OF TABLES A comparison of methods used to identify deeplevel states. Comparison of values from DLZS, DLTS, and literature [71]. . . 143 Model variance weighting parameters used in regression of impedance data. Values were determined by Pankaj Agarwal.. . . LIST OF FIGURES Energy band diagram of a ntype semiconductor with a deeplevel state. Experimental setup for DeepLevel Impedance Spectroscopy. Basic inverting operational amplifier circuit (a) and overly simplified equivalent circuit of an operational amplifier (b).. . . . Simplified block diagram for the Solartron 1286. . . . Simplified block diagram for the PAR 273. . . . . Equivalent circuit from regression of (a) Solartron 1286 and (b) PAR 273 impedance data. . . . . . . Comparison of Solartron cell data and equivalent circuit data for the Solartron and the PAR potentiostats in the form of an impedance plane plot. . S S S S S S S S S S S S S S S S S S S S jt Comparison of Solartron cell data and equivalent circuit data for the Solartron and the PAR potentiostats in the form of phase angle as a function of frequency. . S S S S S S S S S S S S S z 5 Comparison of PAR cell data and equivalent circuit data for the So lartron and the PAR potentiostats in the form of an impedance plane plot. .. . . . . . . Comparison of PAR cell data and equivalent circuit data for the So lartron and the PAR potentiostats in the form of phase angle as a function of frequency. Frequency dependence of gain and phase shift for control op amp [27]. Compensated (stable) uncompensated gain and phase shift for control op amp [27] . Energy levels as a function distance for a ntype semiconductor  t t  r . t    Intensity of the focused light as a function of the wavelength for the 410720 nm order sorting filter with 1200 and 600 grooves/mm diffrac tion gratings and the spectrometer exit/entrance slit width as param eters. . . . * * * * * Intensity of the focused light as a function of the wavelength for the 7201350 nm order sorting filter with the 600 grooves/mm diffraction grating and the spectrometer exit/entrance slit width as parameters. Intensity of the focused light as a function of the wavelength for the 12002000 nm order sorting filter with the 300 grooves/mm diffraction grating and the spectrometer exit/entrance slit width as parameters. 3.17 Intensity of the focused light as a function of the wavelength for the 18003000 nm order sorting filter with the 300 grooves/mm diffraction grating and the spectrometer exit/entrance slit width as parameters. Shockley Read Hall processes of a deeplevel state. Equivalent circuit corresponding to the small signal nonequilibrium case for a bulk deeplevel state. Equations and notation are given in the text. Equivalent circuit corresponding to the small signal nonequilibrium case for a surface state. Equations and notation are given in the text. The complete nonequilibrium equivalent circuit with surface and bulk deeplevel states for a Schottky barrier on the left side and an Ohmic contact on the right side with no DC leakage current. . . The reduced equilibrium equivalent circuit with surface and bulk deep level states for a Schottky barrier on the left side and an Ohmic contact on the right side. . Simplified equilibrium equivalent circuit with only surface states (a), and only bulk deeplevel states (b) for a Schottky barrier on the left side and an Ohmic contact on the right side. . . The reduced equilibrium equivalent circuit used in this work with a Schottky barrier on the left side and an Ohmic contact on the right side. Cell configuration for the copper electrode experiments. Imaginary component of the impedance as a function of the real com nonent of the imnedance for freauendes ranrine from 65000 to 1 Hz Capacitances from regression of simplified equivalent circuit to impe dance data as a function of inverse temperature for ntype GaAs with Schottky contact. .. .. . .. .. .. .. .. ... . Energy diagram of a shallow donor state (El), a midgap state (E2) slightly below the Fermi level (EF), and a shallow acceptor state (E3) for negative half cycle (upper) and positive half cycle (lower) of sinusoid.135 Inverse of resistances from regression of simplified equivalent circuit to impedance data as a function of inverse temperature for ntype GaAs with Schottky contact. . . . . 6.6 Equivalent circuit characteristic frequencies from regression of simpli fied equivalent circuit to impedance data as a function of inverse tem perature for ntype GaAs with Schottky contact. . . 137 6.7 Product of temperature and equivalent circuit resistance as a function of inverse temperature for ntype GaAs with Schottky contact.. 138 6.8 Product of temperature and equivalent circuit capacitance as a function of inverse temperature for ntype GaAs with Schottky contact.. 139 6.9 Equivalent circuit characteristic frequencies divided by the square of temperature as a function of inverse temperature for ntype GaAs with Schottky contact. .. . .. . . .. .. 140 6.10 DLTS emission rate divide by square of peak temperature as a function of inverse temperature for ntype GaAs with Schottky contact.. 141 MottSchottky plot for ntype GaAs with Schottky contact. . 142 Imaginary component of ponent of the impedance with optical energy as a contact . . the impedance as a function of the real com for frequencies ranging from 20000 to 4 Hz parameter for ntype GaAs with Schottky . . . 144 Dimensionless real component of the impedance as a function of optical energy with electrical frequency as a parameter for ntype GaAs with Schottky contact. . . . . . . Dimensionless imaginary component of the impedance as a function of optical energy with electrical frequency as a parameter for ntype GaAs with Schottky contact.. . . . . . Dimensionless real component of the imnedance as a function of lifht 6.16 6.17 6.18 6.19 6.20 6.21 6.22 Dimensionless imaginary component of the impedance as a function of light intensity at 1.1 eV with electrical frequency as a parameter for ntype GaAs with Schottky contact. Optical cross sections from DLOS experiments as a function of optical energy, from references [98, 99]. Dimensionless total impedance as a function of optical energy with electrical frequency as a parameter for ntype GaAs with Schottky contact. Dimensionless phase angle as a function of optical energy with electrical frequency as a parameter for ntype GaAs with Schottky contact. Resistances from regression of impedance data to equivalent circuit as a function of optical energy for ntype GaAs with Schottky contact. Capacitances from regression of impedance data to equivalent circuit as a function of optical energy for ntype GaAs with Schottky contact. Characteristic frequencies from regression of impedance data to equiva lent circuit as a function of optical energy for ntype GaAs with Schot tky contact. .. . Total system energy as a function of defect position.. Resistances from regression of impedance data to equivalent circuit with no weighting as a function of optical energy for ntype GaAs with Schottky contact. . . . . . . 156 Characteristic frequencies from regression of impedance data to equiva lent circuit with no weighting as a function of optical energy for ntype GaAs with Schottky contact. . . . . . 6.26 Resistances from regression of impedance data to equivalent circuit with proportional weighting as a function of optical energy for ntype GaAs with Schottky contact. .. . . Characteristic frequencies from regression of impedance data to equiv alent circuit with proportional weighting as a function of optical energy for ntype GaAs with Schottky contact. . . . . 6.28 Resistances from regression of impedance data to equivalent circuit with modulus weighting as a function of optical energy for ntype GaAs with Schottkv contact.. ... .... ..... 6.30 Imaginary component of the impedance as a function of the real com ponent of the impedance for frequencies ranging from 0.2 to 100 Hz and with optical energy as a parameter for the ZnS:Mn TFEL panel. 6.31 Dimensionless real component of the impedance as a function of optical energy with electrical frequency as a parameter for the ZnS:MN TFEL panel. . . * * * * * 6.32 Dimensionless imaginary component of the impedance as a function of optical energy with electrical frequency as a parameter for the ZnS:Mn TFEL panel. Change in photonreleased residual charge as a function of photon energy for a ZnS:Mn TFEL panel from reference [108]. . . 166 Total impedance as a function of frequency with optical energy as a parameter for the ZnS:Mn TFEL panel.. . . . 167 Phase angle as a function of frequency with optical energy as a param eter for the ZnS:Mn TFEL panel. 168 6.36 Resistances from regression of simplified equivalent circuit to impe dance data as a function of inverse temperature for the ZnS:Mn TFEL panel. 6.37 6.38 6.39 Capacitances from regression of simplified equivalent circuit to impe dance data as a function of inverse temperature for the ZnS:Mn TFEL panel. . . . . * * * Equivalent circuit characteristic frequencies from regression of simpli fied equivalent circuit to impedance data as a function of inverse tem perature for the ZnS:Mn TFEL panel. Imaginary component of the impedance as a function of the real com ponent of the impedance for frequencies ranging from 20000 to 0.05 Hz with optical energy as a parameter for the unaged ZnO varistor. 6.40 Imaginary component of the impedance as a function of the real com ponent of the impedance for frequencies ranging from 1500 to 0.2 Hz with optical energy as a parameter for the aged ZnO varistor. . 173 6.41 Dimensionless real component of the impedance as a function of optical energy with electrical frequency as a parameter for the aged ZnO varistor. 174 6.42 Dimensninmlpn s ima.oinarv mmnnnpnt nf t.h imp rlann ao a fmnnmn n4r 6.43 6.44 6.45 6.46 Dimensionless real component of the impedance as a function of optical energy with electrical frequency as a parameter for the unaged ZnO varistor Dimensionless imaginary component of the impedance as a function of optical energy with electrical frequency as a parameter for the unaged ZnO varistor. . Resistances from regression of simplified equivalent circuit to impe dance data as a function of inverse temperature for the unaged ZnO varistor. Capacitances from regression of simplified equivalent circuit to impe dance data as a function of inverse temperature for the unaged ZnO varistor. . Equivalent circuit characteristic frequencies from regression of simpli fled equivalent circuit to impedance data as a function of inverse tem perature for the unaged ZnO varistor.. . . . 180 Imaginary component of the impedance as a function of the real com ponent of the impedance for frequencies ranging from 1500 to 0.2 Hz with optical energy as a parameter for the unaged ZnO varistor an nealed at 600C.. . . . . 181 Total impedance as a function of frequency under no illumination and illumination of 1.77 eV for the aged and unaged ZnO varistors annealed at 600 C. * * . Phase angle as a function of frequency under no illumination and illu mination of 1.77 eV for the aged and unaged ZnO varistors annealed at 6000C. .. Imaginary component of the impedance ponent of the impedance for frequencies with optical energy near the bandgap as varistor. as a function of the real corn ranging from 1500 to 0.2 Hz a parameter for the aged ZnO S . 0184 Imaginary component of the impedance as a function of the real com ponent of the impedance for frequencies ranging from 65000 to 0.1 Hz with optical energy as a parameter for cuprous oxide (aged in light) in alkalinechloride solution. . . . . . 185 6.53 Phase angle as a function of frequency with optical energy as a param Dimensionless imaginary component of the impedance as a function of optical energy with electrical frequency as a parameter for cuprous oxide (aged in light) in alkalinechloride solution. . . 188 Black body radiation intensity per wavelength as a function of wave length with temperature as a parameter, from Planck distribution law. 189 Imaginary component of the impedance as a function of the real com ponent of the impedance for frequencies ranging from 10000 to 0.2 Hz with optical energy as a parameter for cuprous oxide (aged in dark) in alkalinechloride solution. . . . S .* 4 190 Phase angle as a function of frequency with optical energy as a param eter for cuprous oxide (aged in dark) in alkalinechloride solution. . 191 Dimensionless real component of the impedance as a function of optical energy with electrical frequency as a parameter for cuprous oxide (aged in dark) in alkalinechloride solution. .. .. .. Dimensionless imaginary component of the impedance as a function of optical energy with electrical frequency as a parameter for cuprous oxide (aged in dark) in alkalinechloride solution. . . . Density of states as a function of energy for a disordered semiconductor (left) and a crystalline semiconductor with a discrete state (right). . 194 Imaginary component of the impedance and of the admittance as a function of electrical frequency with optical energy as a parameter for the aged ZnO varistor. . . . . . . Dimensionless real component of the impedance as a function of optical energy and electrical frequency for ntype GaAs with Schottky contact. Dimensionless imaginary component of the impedance as a function of optical energy and electrical frequency for ntype GaAs with Schottky contact. Dimensionless real component of the impedance as a function of optical energy and electrical frequency for the ZnS:Mn electroluminescent panel.198 Dimensionless imaginary component of the impedance as a function of optical energy and electrical frequency for the ZnS:Mn electrolumines cent panel. .. 199 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DEEPLEVEL IMPEDANCE SPECTROSCOPY OF ELECTRONIC MATERIALS By ANDREW NORBERT JANSEN December, 1992 Chairman: Prof. Mark E. Orazem Major Department: Chemical Engineering DeepLevel Impedance Spectroscopy (DLZS) is suggested to be a new approach to analyze transitions of deeplevel states in electronic materials. The technique is based on interpretation of both the real and imaginary components of the impedance response over a continuous range of electrical frequencies. The two parameters that were varied, temperature and optical energy, directly influenced the transition rate or occupancy of deeplevel states. Transition rates of the deeplevel states were determined to have an Arrhenius relationship with temperature, indicating activation energy controlled reactions. Mea surement of the impedance response as a function of temperature is termed Thermally deeplevel state and a carrier band. This variant of the technique is termed Optically Stimulated DeepLevel Impedance Spectroscopy (OSDLZS). A mathematical model was developed from mass balance equations and Poisson's equation to aid the analysis of the impedance data. For convenience during the regres sion, the model was posed in terms of an equivalent circuit of resistors and capacitors whose values were related to system properties such as reaction rate constants and concentrations. The data were regressed to the equivalent circuit through complex nonlinear regression. The influence of temperature, monochromatic light, and elec trical frequency was evident in the impedance data and the regressed resistances and capacitances. The results from DLZS (data and regressed parameters) and the well accepted DeepLevel Transient Spectroscopy (DLTS) for the EL2 state in GaAs were found to be in close agreement. This technique was also used to analyze semicon ducting materials such as ZnS:Mn thin film electroluminescent panels, ZnO varistors, and cuprous oxide on copper in alkaline chloride solutions. CHAPTER 1 INTRODUCTION DeepLevel Impedance Spectroscopy (DLZS) is presented here as a new technique for characterization of electronic transitions in large bandgap semiconductor devices. This technique involves an analysis of the real and imaginary components of the impedance response over a continuous range of applied frequency under the influence of subbandgap illumination and/or temperature. A history of impedance techniques is presented in Chapter 2. This chapter deals only with techniques that are relevant to the work presented in this manuscript. There are a considerable number of techniques and their variations used to analyze semiconductors that are described elsewhere in literature. The experimental theory and application is discussed in Chapter 3. This discussion is designed to bring the next investigator up to speed on the experimentation and the details behind the impedance measurements. For those who have a good electronics background, much of this chapter may serve as a review. Chapter 4, some physical insight is provided as to the nature of electronic transitions between deeplevel states and the carrier bands. The governing equations are developed and transformed into a simplified equivalent circuit. A brief description of the samples used in this work is given in Chapter 5. samples described are: ntvue GaAs. a ZnS:Mn thin film pwlp.rtrnlmnonpFant+ annl 2 troscopy, and hopefully, provide new insight to the semiconducting nature of these materials. Chapter 7 is a summary of the key points in this work, and suggestions for future work are presented in Chapter 8 The computer program used to interface the DLTS equipment is included in the Appendix. CHAPTER 2 BACKGROUND In contrast to most spectroscopic techniques, Optically Stimulated DeepLevel Impedance Spectroscopy (OSDLZS) can be regarded as encompassing two frequency domains, one that is electrical and one that is optical. The potential utility of em playing a broad electrical frequency range is consistent with experimental observation that surface states have the largest influence on the impedance response at low fre quencies [1, 2] and that the space charge capacitance is obtained most easily from high frequency measurements. The application of subbandgap optical excitation of deeplevel states was sug gested body work describing Electrochemical Photocapacitance Spec troscopy (EPS) [36].The emphasis on interpretation of both the real and imaginary components of the impedance was driven by the results of a mathematical model [7 l0]that treated the influence of deeplevel states on the impedance response of a semiconductor by solving the equations which govern the physics of the system, e.g., Poisson's equation, conservation equations for electrons and holes, and homogeneous and heterogeneous rate expressions for generation and recombination. The modeling work suggested that, through use of monochromatic subbandgap optical excitation, the influence of even low concentrations of deeplevel electronic states could be seen on the real Dart of the imnedance measured at low electrical frenllnncies 4 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, deeplevel states are undesirable when they facilitate electronic tran sitions 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 en ergy level of the states caused by introduction of dopants determines the color of the emitted light. Interfacial states are believed to 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 deeplevel states. Deeplevel states also influence the performance of solidstate varistors. While this technology is more established, development of new processing recipes will be facili tated by understanding how fabrication techniques, composition, and aging processes influence the energy and location of deeplevel states. Copper in alkalinechloride solutions (marine environment) does not suffer from stress corrosion cracking if its oxide film forms under illumination[ll]. Deeplevel states may be involved because cuprous oxide is a large band gap semiconductor. The impact of deeplevel states can be significant, are very low by normal chemical standards. Several sta even in concentrations that Ltes 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 deeplevel electronic states. The techniques commonly , r.,, ,, +a 4, iAa/,.+ ,Aanlnral a+ fa +,,,A A ^ nk0 ,l,,4n*, 1 i ,n + +n.aa, 4.+ o 4a I ,l.n. el,, The electronic behavior of an electron donor deeplevel state with energy ET is illustrated in Figure 2.1. An electron can be emitted from the deeplevel state to the conduction band if it acquires energy of (Ec Er). Likewise, transfer of energy equal to (ET Ev) can excite an electron from the valence band to an ionized deeplevel state, (ED hole emission. An electron in the deeplevel state that acquires energy Er) can be excited to the donor state. Under equilibrium conditions, occupancy of the state is governed by the FermiDirac distribution function. In the absence of illumination, the deeplevel state will influence the impedance response of the semiconductor only if the probability of occupancy of the state is close to 1/2, i.e., if the state energy is close to the Fermi energy. If, for example, the probability of occupancy is essentially zero, the unfavorable energetic of the transition prevents the electrical excitation from moving electrons into the state. On the other hand, if the probability of occupancy is essentially unity, the need to satisfy Pauli exclusion prohibits transfer of electrons into the state. Similar arguments apply for transferring electrons out of the state. At an interface with a dissimilar material, a redistribution of charged species oc curs which results in the formation of a space charge region in the semiconductor. A space charge region can be formed at an interface with a metal, another semiconduc tor, or an electrolyte. In polycrystalline materials, a space charge region can also be formed at a grain boundary. Within the space charge region, the electron energy for the valence band, the conduction band, and the deeplevel state can be described as varying with reference to a fixed Fermi energy. Thus the probability of occupancy of the deeplevel state is a function of position, and, since the degree of band bending is determined by the Dotential annlied to the svstemn the der, nf nnnrmanmrv u a level states. The energy needed to excite deeplevel electronic transitions can also be provided through illumination. Monochromatic illumination with energy less than the bandgap energy is preferred because bandtoband transitions can easily obscure the transitions involving deeplevel states. The distinguishing features of some electrical techniques used to characterize deep level states are presented in Table 1. Anomalous changes in the space charge ca pacitance with applied potential have been attributed to the influence of deeplevel states. In the absence of deeplevel states, MottSchottky theory (see for example references [12, 13]) suggests that the inverse of the square of the space charge capac itance be a linear function of applied potential. The slope in this plot is proportional the concentration of carriers and the intercept is related by a constant to the flat band potential. Changes in the slope of the MottSchottky plot can be related to nonuniform distribution of dopants or to potentialdependent ionization of deep level states [8]. The presence of deeplevel states is suggested if the capacitance is a function of the electrical frequency at which the capacitance is measured. Development electrochemical photocapacitance spectroscopy (EPS) 6]extended analyses based on MottSchottky theory by using monochromatic sub bandgap illumination to change the occupancy of deeplevel states. Changes in the space charge capacitance, measured at a fixed electrical frequency (usually around 1000 Hz) and applied potential, were observed as a function of photon energy. energy at which a change in capacitance was observed was the energy of an allowed electronic transition. This approach can be used to map out the energy structure of an interface because transitions from valence band to the deeplevel state reduced the snace charge canacitance. and transitions from the deenlevel state to 7 be very small as compared to that associated with doping species, the sensitivity of capacitancebased techniques requires precise measurement of the imaginary part of the impedance response. Admittance Spectroscopy (AS) is a relatively new technique used to analyze deep level states [1418].In AS, the conductance (and the capacitance) of semiconductor samples are determined as a function of temperature at selected electrical frequencies. A plot of the conductance as a function of temperature is observed to have a peak. This peak temperature corresponds to a condition in which the emission rate of the trap equals the frequency of the perturbing sinusoid. An Arrhenius curve results if the frequency divided by the square of the peak temperature is plotted as a function of inverse temperature. The slope of this curve is proportional to the trap energy level from one of the carrier bands and the intercept is related to the capture cross section and trap concentration. This technique is similar to DLTS (described below) in that the emission rate is varied via temperature sweeps. conductance is monitored in AS, The difference is that the whereas, a capacitance difference is monitored in DLTS. Results from AS and DLTS were found to be in close agreement [1416]. DeepLevel Transient Spectroscopy (DLTS) perhaps, dominant capacitancebased technique used to detect deeplevel electronic states. In DLTS, one monitors the space charge capacitance of the semiconductor in response to a pulse in potential. In addition, the temperature is changed in the experiment to vary the coefficient of the emission rate. The capacitance is measured at two different times following a pulse (in the forward direction); therefore, the transient change in occupation of a state in response to a step change in potential is observed as a change 4n .n .*%t.. .a ,.. ..4 +n .t .*^i r, T..f + a.t+ rrr& +4h ,..; r^ Of ^^l K 0. +..0 4a 8 The experimental part of the technique presented here bears closest resemblance to Optical Admittance Spectroscopy (OAS) [20, The major difference is that the complex impedance response is analyzed in DLZS; whereas, the complex admittance response is treated in OAS. The distinction between the two experimental techniques is, perhaps, subtle since impedance is simply the reciprocal of admittance. difference lies in the data analysis. The main As shown later, analysis in terms of impedance may be more sensitive to deeplevel states at low frequencies. This work addresses a lightenhanced form of electrochemical impedance spec troscopy. 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 illumination 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 capacitance at a single high frequency. The use of a broad frequency range is the essential distinction between this approach and photocapacitance spectroscopy. The effect of temperature on the impedance response is also analyzed in this work; this technique is referred to as Thermally Stimulated DeepLevel Impedance Spec troscopy (TSDLZS). The rate of electronic transitions is dependent on the temper nfII.. in a n A' .rr ni, o nrnnnaarl n o n fTT TQ ,ar .n.ll7 0+i;,,,1 ,,1 TAflT AW. ,.f;f,, ,,,.,,.., hi Figure 2.1: Energy band diagram of a ntype semiconductor with a deeplevel state. t Table 2.1: A comparison of methods used to identify deeplevel states. Technique Optical Excitation Thermal Excitation Electrical Frequency of DeepLevel States of DeepLevel States Range single value about 1 kHz single value about 1 kHz rate windows (1 Hz to 5 kHz) DLTS3 capacitance measured at 1 MHz OAS5 single values (1 Hz to 1 MHz) 0.1 Hz to 1 MHz 0.05 Hz to 65 kHz 1 Hz to 65 kHz OSDLZS6 TSDLZS7 1. MS: MottSchottky Theory. Anomalous changes in differential capacitance are ob served as a function of applied potential. 2. PS: Photocapacitance or Electrochemical Photocapacitance Spectroscopy (EPS). Changes in differential capacitance are observed as a function of photon energy. DLTS: DeepLevel Transient Spectroscopy. capacitance are made Transient measurements of differential as a function of temperature. 4. AS: Admittance Spectroscopy. Changes in the admittance are observed as a function of frequency and temperature. OAS: Optical Admittance Spectroscopy. observed as a function of photon energy. Changes in the admittance spectrum are OSDLZS: Optically Stimulated DeepLevel Impedance Spectroscopy: Changes in the impedance spectrum are observed as a function of photon energy and applied potential. 7. TSDLZS: Thermally Stimulated DeepLevel Impedance Spectroscopy: Changes in CHAPTER 3 EXPERIMENTAL METHOD AND INSTRUMENTATION This chapter provides a detailed description of the equipment used throughout this work. The theoretical principles behind the techniques used are presented to help the reader understand the functions and limitations of the equipment. It is intended that this chapter supplement the operation manuals provided by the equipment distribu tors rather than replace them. A general description of the overall setup is given in the first section to provide an outline for the remainder of the chapter. The following sections describe the overall setup in greater detail. General Schematic A schematic of the experimental setup is given in Figure 3.1. The setup can be divided into two sections, one responsible for the optical frequency and the other for the electrical frequency. The light source was a 450 W Xenon lamp able to emit wavelengths of light greater than 300 nm. This light was diffracted in a SPEX Model 1681B Spectrometer to produce monochromatic light in accordance with Bragg's Law. The diffraction grating rulings were 1200, 600, 300 grooves/mm. The bandpass of the monochromatic light was set by means of an entrance and exit slit; the bandpass used was 18 nm unless otherwise noted. Harmonic frequencies were eliminated with order sorting filters and the lirht intensity was fixed with the use nf nuntlral rPnaitv filtir 12 A potential bias was maintained across the sample by a PAR 273 potentiostat or a Solartron 1286 Electrochemical Interface (potentiostat). This was coupled with a Solartron 1250 Frequency Response Analyzer (FRA) which applied a sinusoidal voltage at select frequencies across the sample. The current was measured by the potentiostat and the phase relationship between the applied voltage and the measured current was analyzed by the FRA. The output of the FRA was the real and imaginary components of the impedance at each measured electrical frequency. A Sula Technologies DLTS setup was used to perform the capacitancevoltage measurements and the DLTS experiments. The cryostatic chamber and tempera ture controller from this setup were also used to perform the temperature controlled impedance experiments. Electro chemical Impedance Spectroscopy The underlying principles behind impedance spectroscopy is presented first to better the understanding of the impedance equipment. basic schematic of the electronics is then given. 3.2.1 Theory The best single reference on the theory and application of impedance spectroscopy is a technical report written by Gabrielli [22]. This section provides a brief overview of some of the key concepts. The principle behind impedance spectroscopy is to relate a physical model to ex perimental data in the form of real and imaginary components of the impedance. This 13 an equivalent circuit with real and imaginary components, is in a form that can be directly compared to the impedance data. The electrochemical cell, or sample, can be thought of as a black box with two terminals where the black box consists of a circuit of electronic components men tioned above. The impedance of this circuit is Z(w) = a(w) + jb(w) where is the complex impedance that can be related to a real component, a(w), and an imaginary component, b(w), both being real numbers and having units of Ohms. The electrical frequency is given by w with units of radians per second, and  ,C All three impedance values are a function of potential. The potential dependence is not written explicitly because in typical impedance analysis the potential is fixed as a parameter while the frequency is varied. Let the voltage perturbation applied across the terminals be represented by a sinusoid, V(t) = Vo sinwt where Vo is the amplitude of the voltage perturbation. (3.1) The above equation assumes that the voltage perturbation is referenced to the steady state potential value. measured current that flows through the cell as a result of the applied voltage per turbation can be represented by a sinusoid, V0 I(t)= sin(wt + {(w)) + YmAn, sin(mwt ekm(w)) + n(t) (3.2) where Z(w) is the total impedance of the cell and (w) is the phase shift between the applied voltage and the measured current. If the electrochemical cell was a resistor, the phase shift would be zero. If the cell was a capacitor, the phase shift would ,, .. 14 The current expression above contains more information than is desired; only the impedance as a function of w is sought, not mw, and noise decreases the confidence in the impedance data. The digital frequency response analyzer minimizes harmonic responses and noise through integration of equation (3.2) multiplied by either sin wt or cos wt, i.e., 1 I, T () sin I(t) sinutdt= o cos 4(W) (3.3) 1 J T V0 I(t) cos wtdt = sin (w) z(tlt as T  00 (3.4) where T is the integration time in integer multiples of the sinusoid period, 1. The integrals that contain the harmonic response are zero when the integration is over a complete cycle(s) because sinwt, cost, and sin mwt form an orthogonal set of functions (see Chapter 2 of Haberman [24]). zero as the integration time increases ( 0). The noise integral will tend towards A trade off has to be made between "noise free" data (T S oo) and a practical time length for the experiment (T is finite), particularly if the system is corroding. Nonstationary effects would not be eliminated in the noise integral even as T + 00. The long integration time or short integration time in the FRA parameter menu refers to the integral time T. The impedance is the voltage divided by the current which can be represented as a complex number, i.e., Zcell = Zr,cell + jZ,cell = +31j'FI +14ir (3.5) where the subscript "cell" has been added to denote that this is the impedance of the cell (sample). Because the voltage is the applied signal, phase shift set to zero as a reference, the imaginary component of the voltage is zero (V. = 0. The real Dart of as II" The voltages and currents discussed so far occur across the sample. these values are modified by various gains in the potentiostat However, before reaching the frequency response analyzer. For instance, the voltage amplitude, Vo, from the signal generator will become KVo where K is the transfer function of the potentiostat. measured voltage passes through an amplifier of gain Gv before it reaches the FRA. The amplitude of the voltage signal at the FRA input becomes GvKVo. Likewise, the current signal becomes GIKVo/Zceu which is converted to a voltage signal by means of the measuring resistor, R. The current signal at the FRA input, in the form of a voltage, becomes GCRKVo/Zeeru. The impedance that is actually measured by the FRA is Zme aa r,menas Zjmeas  Gv = GI Zceli R GvKV. GIj R cos + jGx zV R sin (3.6) cos + j sin 6 The relationship between the measured real and imaginary impedance and the cell real and imaginary impedance is Z  r,meaa  GVl SR rZ,cell (3.7) j,meas Gvl G Rcel* CI RZ,1 (3.8) The ratio of the voltage and current gains in the potentiostat is designed to be unity over the frequency range used. Hence, the measuring resistor is the proportionality constant between the measured impedance and the cell impedance. 16 without first reading the ZPLOT manual and the operating manuals for the poten tiostats [27 28] and the frequency response analyzer [29]. The default parameters selected by ZPLOT are rarely appropriate to the system being studied. 3.2.2.1 Potentiostat There are several parameters involved with each potentiostat (Solartron 1286 or PAR 273) that must be tailored to each experiment. The first is the desired cell configuration. All potentiostats allow at least three electrodes to be monitored or controlled; these are the working electrode, the counter electrode, and the reference electrode. The working electrode is the electrode of interest (i.e., the sample elec trode) and its potential is controlled relative to the reference electrode via the counter electrode. For instance, suppose that the working electrode is desired to be 0.5 volts more positive than the reference electrode, the potentiostat would adjust the potential of the counter electrode to maintain this desired potential drop. The reference electrode potential versus the working electrode potential is mea sured with an operational amplifier, also referred to as an "op amp" It is worthwhile here to describe briefly the function of the operational amplifier (see for instance Malvino [30]). A basic inverting operational amplifier circuit is given in Figure 3.2(a) and the equivalent circuit for an operational amplifier is given in Figure 3.2(b). The minus sign is used to designate that the input is connected to the minus terminal of the op amp which means that the output has a phase shift of 180 degrees. The overall voltage gain of the op amp circuit, G, is defined as the ratio of the output voltage, V,, to the input voltage, V1g, fain. A. is defined as the ratio of V^.. /(V  = Vt/ ) n *'D U. A is larre ( The differential voltage 100. noon and V.., is 17 The op amp varies the output voltage Vo, in response to the input voltage VI,. The output voltage is feedback through the resistor, Ry, and thus affects the potential at the input terminal such that V1 V2 is small. Rin is the input impedance of the op amp and is typically the op amp. MO, large enough that essentially no current flows through There are two main features of an ideal op amp; no potential drop or current flow across the input terminals. Thus, a potential or current measurement can be made without affecting the system being measured. The overall voltage gain can be determined from the differential gain equation (AfVt V21  Vo,) and the fact that the currents through the two external resistors, Rf and R,, must be equal because no current flows through the op amp (I, = If = I). With these two assumptions, the voltage across the op amp can be expressed as V2 V =Vn  IR, (3.9) V2 = V + IRf. (3.10) Equation (3.9) can be related to the differential voltage gain, A, in order to derive an explicit current expression, i.e., A(MV  IR,) = vou=~I= Vot + AV1, (3.11) AR. This current expression is substituted into equation (3.10) to eliminate I, i.e., V011 + AV  AV,t ARvut + A R',t AR, (3.12) The last expression is rearranged to yield the overall voltage gain, TVot r _ ARf/R, (3.13) 1 + A + R/R, 18 This information will be used later in analyzing the current measurement capabilities of each potentiostat and in determining the proper bandwidth selection to ensure stability in potentiostatic control of the system. The PAR 273 potentiostat is capable of applying 100 volts between the counter and working electrodes just to maintain a fraction of a volt difference between the reference and working electrodes. As a safety precaution, the PAR 273 is installed with a cell enable/disable switch so that the electrode leads can be adjusted without fear of electrocution. The Solartron 1286 is capable of applying 20 volts between the counter and working electrodes and does not have such a safety switch. Three electrode configurations are used primarily in electrochemical experiments. For example, one of the systems studied here had a 0.5 M NaC1 (pH 10) electrolytic solution with a cuprous oxide working electrode, a saturated calomel reference elec trode (SCE), and a platinum counter electrode. The two cell configuration is used primarily in solid state systems such as semiconductor devices with either metal Schottky barriers or Ohmic contacts. It is extremely important to be mindful of the correct lead connections when using the two electrode cell configuration; the reference electrode lead is connected (shorted) to the counter electrode lead. This permits the measurement of the working electrode versus the reference electrode and control of this potential difference via the counter electrode. Never use just the reference or counter electrode alone versus the working electrode. This would not allow poten tiostatic control of the system because the reference electrode can only measure the potential between the reference and the working electrodes and the counter electrode can only control the potential between the reference and the working electrodes by n\ 0 a vi nr ak iT rAn + 1m a .n, v + r0 ar+ tunYo an A + 1. 4 M v i nnT al I n> alff*/ A 19 Simplified block diagrams that include only the options relevant to the impedance work done in this work are presented for the Solartron 1286 (Figure 3.3) and the PAR 273 (Figure 3.4). Not surprisingly, both block diagrams have the same gen eral features, with the exception of the current measurement circuit and the second reference electrode capability of the Solartron 1286. The second reference electrode permits measurement of the potential difference between two sections of the elec trolytic solution. The counter electrode maintains this potential difference. Note that the potential of the working electrode is not measured in this four electrode configuration. Thus, all two and three electrode cell configurations require that the second reference electrode lead be shorted to the working electrode. 3.2.2.2 Current measurement There are differences in the current measurement circuits of the two potentiostats that are worth discussing here. Both potentiostats determine the current passing through the working electrode with the aid of a high quality measuring resistor in se ries with the working electrode. The current flows through the resistor which creates a potential drop across the resistor equal to IP. This potential drop is measured with an op amp as shown in the block diagrams. Two differences in the current measuring circuits become evident from a closer look at the block diagrams. The first is that the resistors in the Solartron range from 0.1 f to 100 AkL and those of the PAR range from 1 f to 10 MSl. The second difference is that the Solartron uses an additional op amp to boost the low current signal (1, 10, and 100 kQl resistors). A possible disadvantage of the latter point is the introduction of the additional on amn circuit in the Solartron. This circuit micht 20 The larger measuring resistors in the PAR 273 permit low current measurements with less noise. This statement is based on the simple relation V = IR; small I re quires a large R to maintain the same potential drop and, hence, potential sensitivity. Currents in the nanoamp range are observed when the impedance of the cell is large (108lO), as was the case for most of the devices analyzed in this work. This brings up an important pointhow to be certain that the signal being measured actually reflects the cell alone or the electronic system of the potentiostat as well, especially if the impedance of the cell is near the limits of detection. The majority of systems analyzed can be represented as passive circuit elements such as resistors, capacitors, and inductors; some of the very same components used to apply and measure the voltage and current. A solution to this dilemma was suggested by Dr. Ron Haak of the ALZA Corpora tion. He proposed that the impedance data of the cell being analyzed be regressed to a suitable equivalent circuit and to build this circuit with resistors and capacitors on a breadboard. The same impedance scan would be performed on this equivalent circuit and the result compared with the original cell data. This test proved to be very fruit ful in the case of the ZnS:Mn thin film electroluminescent panel. Both potentiostats gave reproducible results; however, these results depended on the potentiostat used. The Solartron showed a change in phase angle from around 90 degrees to around 20 degrees at low frequencies (a 0.2 Hz); whereas, the PAR indicated that the phase angle remained relatively constant around 90 degrees in the same frequency range. The Solartron cell data was regressed with ZFIT [31] to a resistor in series with a resistor and capacitor parallel to each other as shown in Figure 3.5(a). This equivalent rcuit wa 21 only predict the qualitative features of the equivalent circuit, even though the original cell data was obtained with the Solartron. The PAR 273 cell data was regressed in a like manner to yield the equivalent circuit shown in Figure 3.5(b). Note that the capacitor is roughly the same as for the equivalent circuit above but that the parallel resistor is about 40 times greater. This resistor was considered to be so large that it could be assumed to have an in finite resistance, thus reducing the circuit to a resistor and capacitor in series. circuit would have a 90 degree phase shift at low frequencies. This It was not expected that the resistor and capacitor in series was a good fit to the original data; however, it was adequate to test the behavior of each potentiostat to such a highly capaci tive system such as the ZnS:Mn panel. The results from both potentiostats on the resistorcapacitor series are presented in Figures 3.8 and 3.9. The PAR was able to predict the capacitive nature much better than was the Solartron at low frequencies. Based on these series of experiments, it was decided that the PAR 273 was the ap propriate choice in potentiostats to analyze the ZnS:Mn panel. Similar checks were performed for the other systems. 3.2.2.3 Stability and bandwidth This subsection is a brief review of Chapter 9 of the Solartron 1286 operating manual A system is considered to be stable if its output is a direct response to an applied input signal. An unstable system can exhibit spontaneous undriven output or an output due to a previous input signal. Such a system could result in oscillations in the potentiostatic control of the system. In terms of operational amplifier theory, 22 a system becomes unstable when the loop gain is greater than or equal to unity and the phase shift is 0 or 360 degrees. The loop gain is defined as the product of all the gains that incorporate the loop. There are four gains in Figures 3.3 and 3.4: 1) GC differential voltage gain of potential control amplifier, A1, 2) GC  cell gain, 3) G2 differential voltage gain of potential measurement amplifier, A2, and 4) G,  gain due to resistors R1 and R2 (not shown in Figure 3.4). The last gain, Gr, is composed of resistors and hence will not impart a phase shift. It will only attenuate the signal. G2 has been designed to be frequency independent with a gain of unity except at very high frequencies. cell and the gain of the control op amp, GC is the The cell gain varies from cell to parameter that can be varied to ensure stability. The input signal to amplifier A1 is connected to the minus terminal which means that the output will experience a phase shift of 180 degrees. If the cell being analyzed is capacitive, it will also impart a phase shift of 90 degrees for a total of 270 degrees. The differential voltage gain, G1, is constant at low frequencies. However, the gain decreases at some frequency with an additional phase shift of 90 degrees as shown in Figure 3.10. The total phase shift becomes 360 degrees which means that the feedback signal is in phase with the input drive signal. at the minus terminal The two signals add together to form a larger amplitude input signal which the op amp amplifies with gain A1 to form the output signal. The output signal is feedback to the minus terminal where it is added to the in phase input signal and again amplified. This cyclic behavior will lead to inntahilitv if thp lnnn anin in wrap tha win r nMnil In The differential voltage gain of A1 occurs away from the 360 degree phase can be varied such that the unity loop gain . shift. This point can be graphically demon strated with the aid of Figure 3.11 where two G1 gains are shown. The uncompensated loop gain has unity gain at 360 degrees, and hence, is unstable. With the selection of a lower bandwidth gain (compensated) for the amplifier A1, the phase shift at unity loop gain becomes 270 degrees and the system is stable. The Solartron 1286 has ten ranges (labelled A to J) of bandwidths to choose from and the PAR 273 has two (labelled "high speed" and "high stability"). Bandwidths C, D, and "high stability" have high and low pass frequency filters and, due to the extra electronic circuits, should only be selected if none of the other bandwidths are appropriate. An oscilloscope is used to determine the proper bandwidth with a po tential square wave form applied across the cell. If the bandwidth is too wide, the square wave will exhibit 'ringing' which indicates that the differential voltage gain is too large. Too narrow of a bandwidth means that the gain is too small and the resulting wave form will be overdamped. correct bandwidth choice will yield a square wave form on the oscilloscope with perhaps a slight overshoot. 3.2.2.4 Frequency response analyzer The Solartron 1250 Frequency Response Analyzer (FRA) performs two duties: 1) generates an output wave form and 2) analyzes up to two input wave forms. The gen erated wave form can be a sine, square, or triangle wave at frequencies from 10 p Hz to 65,500 Hz. These wave forms are synthesized with a 10 MHz crystal oscillator along with digital processing to provide a frequency accuracy of 1 part in 10,000 [32]. method is reported to be better than the nrevinms nhaslnrlk mpthnr.l This ThP vR A 24 The analyzer section of the FRA can simultaneously measure two input channels which in the work presented here is the voltage and current (converted to a voltage signal) from the potentiostat. The analyzer displays the ratio of the two channels in either rectangular (a+ jb), polar (r,0), or log polar (log r, 0) coordinates. Integration time on the analyzer menu specifies the maximum number of cycles to be integrated if the convergence criteria is not meet. At least 1 cycle is integrated below frequencies of 655 Hz and 61 cycles above 655 Hz regardless of the number of cycles specified in the ZPLOT parameter menu. Only one integration over a cycle is necessary to eliminate the D.C. and harmonic frequencies of the fundamental frequency desired. However, integration over more cycles reduces the amount of noise in each channel. Two convergence criteria are available in the auto integration mode. The first is labelled "short integration time" which means that the channel measurement is continued and averaged until the standard deviation of the average does not vary by more than 10 percent of the average or until the maximum integration time has been reached as specified above. "Error 82" message will occur when the latter case has occurred; this does not imply that the data point should be discarded, only that the convergence criteria has not been met. "long integration time" The second option is labelled and is similar to the short integration time except that the convergence criteria is 1 percent. input channel. The convergence criteria can be applied to either The current channel (Channel 1) should be chosen since this channel is the response of the voltage signal from the generator and is expected to have the most noise. The voltage channel is the controlled channel in the potentiostat and hence is not expected to be as noisy. There are five voltage range selections for the channel innmit (f mV .nn mV RV should be measured on the oscilloscope to determine the peak voltage. Note that the voltage channel (Channel 2) can be fixed because the applied sine wave remains relatively constant when the Solartron 1286 or the PAR 273 is used as a potentiostat. However, the current channel (Channel 1) is best left in the Auto selection because the current varies several orders of magnitude as the impedance of the cell changes with frequency. 3.2.3 The reverse is true when in galvanostatic (current) control. Experimental Difficulties The large impedance of the materials used in this study made collection of accurate impedance data difficult at low frequencies. The experimental frequency range was constrained by an unfavorable signal to noise ratio. The currents measured at low frequencies were extremely small (in the nanoamp range), and currents smaller than this were beyond the range of the potentiostat used. This problem was addressed in this work by increasing the amplitude for the potential perturbation from the 10 mV characteristic of electrochemical studies to ZnO varistors. to 3 V for the ZnS:Mn panel and the Reproducibility of the impedance response for different magnitudes suggested that the system response was linear at even these large amplitudes. The signal to noise ratio was further improved, when possible, by increasing the area of the device tested. For example, the current output of the electroluminescent panel was increased by 2 orders of magnitude by electrically coupling 100 pixels. electronic transitions involving deeplevel states have time constants that are large in comparison to the inverse of the lowest measured frequency. In other words imaginary impedance did not tend toward zero at the lowest frequency used. lb .. a S a 26 required between runs for the ZnS:Mn TFEL panel and the unaged varistor; whereas, a wait time of 1 hour was sufficient for the aged varistor, and a few minutes for the GaAs semiconductor and the copper/cuprous oxide electrode. The problem of memory effects in electroluminescent panels is discussed in references [33, 34, 35]. Standard Characterization Techniques Two standard techniques were used in this work to provide a comparison to elec trochemical impedance spectroscopy. These are MottSchottky or CV profiling and DeepLevel Transient Spectroscopy (DLTS). 3.3.1 MottSchottky Profiling The MottSchottky relationship is used to determine the concentration of shallow level states in the bulk of a semiconductor. This relationship can be represented as [36] 1 2  C2 A2 where C is the measured capacitance, ((V qKeo(ND  hi) (3.15) NA) Vbi is the built in potential, V is the applied potential, A is the surface area, Keo is the permittivity of the semiconductor, ND is the donor concentration, and NA is the acceptor concentration. can be incorporated into this expression Deeplevel states by considering them to be either donor like or acceptorlike in nature. Assume, for example, that the deeplevel state is acceptorlike in nature in that it becomes negative when ionized. Equation (3.15) then becomes 1 2(Vi V  kT) P (2 nfl\ ~~~~~I A  t IA 27 The above equation can be expressed explicitly in terms of the net donor concen tration through the derivative of equation (3.16) with respect to the applied potential. The result is ND NANT=  Equations (3.16) (3.17) A2qKeo d(i)/dV (3.17) are implemented easily by plotting the inverse of the square of the measured capacitance ( ) versus the applied voltage, V The slope of this curve is referred to as the MottSchottky slope and is used in equation (3.17) to calculate the net donor or acceptor concentration. If the semiconductor is ntype the slope will be negative, and for ptype semiconductors the slope is positive. The capacitance is measured by applying a high frequency sinusoidal voltage per turbation about the set applied voltage bias. This frequency is typically in the MHz range, values too high to change the occupation of most deeplevel states. Changes in the MottSchottky slope with frequency as a parameter have been attributed to interface states or bulk deeplevel states in the literature [1,2,3739]. 3.3.2 DeepLevel Transient Spectroscopy (DLTS) Yau and Sah [40] first demonstrated the influence of deeplevel states on the capacitance decay of a pulsed semiconductor as a function of temperature in 1971. This concept was further refined by Lang [19] in 1974 into the present day technique known as DeepLevel Transient Spectroscopy (DLTS). DLTS has become the most widely used technique to determine the energy and concentration of deeplevel states. The theory behind DLTS is briefly outlined in this section for a ntype Schottky barrier, a more involved presentation for other situations is given by Lang [19]. 28 charge region of width W, which is a region void of majority carriers (electrons). All deeplevel states with energy greater than Fermi energy level, EFp, will emit their electron into the conduction band. all such states are empty. Steady state equilibrium is reached when This is shown in an energy versus distance diagram in Figure 3.12 (a). The semiconductor is forward biased for approximately 1 msec. The majority carriers are swept into the space charge region at a rate that can be considered to be instantaneous. The width of the space charge region decreases to Wf and the energy level of the deeplevel state falls below the Fermi energy level in a portion of the region Wr Wf. All the states in this portion capture electrons to reach a forward biased equilibrium. This situation is shown in Figure 3.12 (b). These states will not have time to capture electrons, reach equilibrium, if the pulse is too short in duration. The semiconductor is again reverse biased which creates a transient space charge region of width W7, that decays to the reverse biased equilibrium value, Wr. >w, because the trapped electrons are neutral species, and hence, more conduction band electrons vacate in order to maintain the same reverse bias level as before. deeplevel state energy level is above the Fermi energy level in a portion of the newly acquired space charge region. The trapped electrons in this region will be emitted into the conduction band to reach the reverse biased equilibrium. The space charge width will decrease towards the equilibrium width W, as these trapped electrons are emitted into the conduction band. This transient case is shown in Figure 3.12 (c). It is this transient decay of the space charge region that is monitored in DLTS experiments by observing the time dependence of capacitance. The emission rate constant can he extracted from the transient canacitance from Equation (3.16) can be rearranged into a more appropriate form, C(t) = A2qKeo(ND 2(Vu  NA  aT(t)) (3.18) k )  ' where it is noted that the capacitance and deeplevel state concentration are a func tion of time. It is convenient here to define a capacitance difference, i.e., AC(t) = C(t) C,, A2qK 2(Vb kT\   ND NA n(t) NDNA nT (3.19) where Cu, and nT, are the steady state reverse bias capacitance and deeplevel con centration, respectively. This equation is made dimensionless upon division by Co,, AC,) ( Ca. NDNA  nT(t) /ND  /WN NA nT. NA rT. ND NA NA  nT(t)  RT (3.20) Equation (3.20) relates the measured transient capacitance to the transient con centration of ionized deeplevel states. This is not of the form typically used in DLTS analysis to estimate the total concentration of states. Two additional assumptions are needed; the steady state (reverse bias) concentration of deeplevel states is sev eral orders of magnitude smaller than the net donor concentration, and the forward bias pulse is long enough that all of the deeplevel states are occupied. assumption can be expressed as The former AC(t) / Ca. nr(t) (3.21) NA A Taylor's series approximation of equation (3.21) about " NDNA AC(t) 1 n C,, 2 ND H 0 yields r(t) (3.22)  a  NA where NrT is the total trap concentration. This is the relationship used in most DLTS analysis. The kinetic expression for the rate of change of trap occupancy is used to link the concentration,. and hence, the measured capacitance to the desired emission rate constant. The transient concentration of trapped electrons can be expressed as dnT  = eJNr + e(NTT NT) + cnN(NTT Nr) + CpPNT dt (3.24) where en is the emission rate constant for electrons, ep is the emission rate constant for holes, c, is the capture rate constant for electrons, and cp is the capture rate constant for holes. The concentration of electrons and holes in the space charge region can be neglected when the reverse bias is applied. Hence, there are no electrons or holes to be captured by the deeplevel state and equation (3.24) is simplified to dnT dnT NT(e, dt + e,) + eNTT. (3.25) Under steady state reverse bias conditions, NT. NTT p (3.26) en + ep , An electron trap under reverse bias with energy greater than the Fermi energy will tend to be empty of electrons at steady state, or in other words, NT, implies from equation (3.26) that e, 0. This e,. Equation (3.25) can be solved easily with the neglect of the hole emission rate constant, the result being nT(t) = NTTexp(et). (3.27) Equations (3.22) and (3.27) are combined to provide the necessary link between 31 As can be seen from this relation, the capacitance decay following the forward bias to reverse bias step change is a simple exponential in time. Lang [41], in a preceding paper, also addresses the case in which nonexponential transient decays are present. He indicated that this case is likely for intermediate and shallow level states. DLTS can still be applied to nonexponential transients, however, the concentration of states cannot be determined via equation (3.23) because an extrapolation to AC(t = 0) is required [41]. The emission rate constant is given by 1 ErT e,= a1, T2exp( kT (3.29) where an is the capture crosssection of the trap, g is the trap spin degeneracy, and ET Ec is the energy difference between the deeplevel state and the conduction band. 7, is a temperature independent quantity that is derived in Section 4.3.2 along with equation (3.29). The procedure of DLTS can be best illustrated with the aid of Figure 3.13 where capacitance is shown as a function of both temperature and time. In the actual DLTS experiment, the complete capacitance decay is not monitored but rather, it is only determined at two points in time as shown in Figure 3.13. The difference between these two capacitance readings is the output signal that is typically connected to the yterminal of an plotter (the xterminal is connected to the temperature output signal). Time tl is referred to as the initial delay, i.e., the time between the start of the reverse bias (end of the forward bias) and the time of the first capacitance measurement. The ratio of tl and t2 is designed to be a constant. It is convenient to consider the emission rate. e.. as the inverse of a time constant. r. 32 out each DLTS experiment, and hence, the peak capacitance difference is found by varying the emission rate via the temperature (see equation (3.29)). The relationship between the emission rate and the two correlation times can be easily derived. From equation (3.28) we have c(t,) c(t2) oc exp(entl) exp(et2). (3.30) The derivative of equation (3.30) with respect to e, is zero at the peak capacitance difference. Setting the derivative of equation (3.30) to zero yields (3.31) eT=I( )/( 2). t2 The goal is to find the temperature at which equation (3.31) is true, temperature at which C(tl) C(t2) has its maximum value. A plot of the natural log of e,/T2 versus the inverse of the peak temperature will be a straight line of slope (ET Ec)/k and the intercept is proportional to an. 3.3.3 DLTS Equipment and Parameters The DLTS equipment was purchased through Sula Technologies (Palo Alto, CA) and consisted of a cryostatic chamber, a Sula Technologies Deep Level Spectrometer, a Lake Shore Model 805 Temperature Controller (Lake Shore Cryotronics, Inc. of Westerville, OH), a 7liter Dewar flask to hold liquid nitrogen, and a variable speed circulation pump to draw the liquid nitrogen through the cryostatic chamber, along with various tubing and electrical connectors. A roughing pump was supplied by Pete Axson, a technician for Dr. Timothy J. Anderson's research group located in Itt.i 21L .1 'I a ii 4 Ltf j __ _L _~ 33 Inside the cryostatic chamber was a thermal block connected to a heater element, controlled by the temperature controller, and a stainless steel tube through which liquid nitrogen is drawn. peratures. The combination provides a wide range of controllable tern The lowest temperature possible is 78K to 84K and the highest practical temperature is around 420K. A rather large brass block (0.25 inch thick x 3 inch di ameter) was attached to the thermal block by teflon screws with a thin teflon spacer in between the thermal block and the brass block to provide electrical insulation. The anode and cathode leads were loose wires; there was no sample mount provided. It was left to the purchaser to design his or her own sample mount. Pat Watson, while at Cornell University, suggested that the brass block be replaced because of a severe temperature lag between the thermal block and the sample due to the block's disk (1mm thick large mass. He recommended replacing the brass block with a sapphire x 2 cm diameter) that has a thin layer of gold deposited on it. sapphire has good thermal conductivity and is also an electrical insulator. One of the leads was connected to the gold with silver paste and the other lead was soldered to a needle. Silver paste was used to improve the connection between the gold layer and the semiconductor metal backing. However, the gold layer became damaged after repeated pasting and removal of the samples. It was decided to replace the sapphire disk with a 304 stainless steel plate of similar dimensions. block. The original teflon sheet was placed between the steel and the thermal Stainless steel was chosen because it does not form an electrically insulating oxide layer as do iron and most brasses particularly at higher temperatures. However, it is near impossible to solder to stainless steel. It was necessary to attach one of the leads to the steel late by drilling a small hole at one corner nf the nlats. hPnrm;n needle. The needle was blunted with a honing stone to prevent it from gouging into the semiconductor. Vacuum grease was used on both sides of the teflon spacer to improve the thermal contact. The needle was mechanically fastened to the thermal support with a cut and heatformed plexiglass sheet in the shape of a .The stainless steel plate was also fixed in place with a plexiglass block screwed into the thermal block. A small copper shell was "riveted" to the stainless steel plate to house the 1000 platinum temperature sensor used to measure the sample temperature, the other temperature sensor (sensor A) is in the thermal block and is used as the control sensor by the temperature controller. With this configuration, the thermal block sensor temperature reached 78K at its lowest and the sample sensor temperature reached 84K at its lowest. Although the temperature controller can maintain a temperature as high as 800K, it was found from experience that at temperatures above 420K problems arose due to the coaxial cables shorting out near the stainless steel plate. This problem was minimized by keeping the cables as far from the thermal block as possible. The Sula Technologies Deep Level Spectrometer did not have an IEEE488 com puter interface although the temperature controller did. Initially, a Metrabyte Corpo ration (Taunton, MA) analogtodigital converter was used but was later overloaded in a nonrelated experiment. However, a HewlettPackard 7090A xy plotter was avail able in our lab capable of analogtodigital data acquisition with 12 bit resolution. The interface programs for both the Metrabyte converter and the HP 7090A [42, 43] were written in Microsoft QuickBASIC 4.5 computer language. The program for the HP 7090A is included in the annendir thorough understanding of Lang's [19] paper before using the spectrometer. Several important relationships will be stressed here. The manual lists two heuristics that should be obeyed in the choice of front panel parameters. These are: The period (time between pulses) is greater than or equal to the length of the pulse plus ten times the initial delay (time tl). This guarantees that enough time has been allowed for completion of the correlation process, i.e., measurement of C(tl) and C(t2). The period is less than or equal to one hundred times the initial de This guarantees that the electronics of the spectrometer have time to respond to the pulsing in order to make an accurate capacitance mea surement. The ratio of times tl and t2 is designed to be constant in the spectrometer as recommended by Lang [19]. For this spectrometer, the rate window, or time constant, at the peak capacitance difference is "r " 1 = 4.3 where tl is also referred to as the initial delay. (3.32) Another important relationship used to estimated the state concentration is the capacitance difference at the start of the reverse bias, AC(t = 0) = 3[C(tl) C(t2)] evaluated at its peak value. (3.33) The steady state capacitance or background capacitance, C,,, is determined with a 150 mV peak to peak voltage sinusoid at a frequency of 1 MHz applied across the 36 in the MottSchottky plots described in an earlier section, however, the forward bias pulse should be off during these measurements. Optical Spectroscopy A SPEX Model 1681B Spectrometer was used to create the monochromatic light. The spectrometer was chosen over lasers because of the greater range of wavelengths available (the bandgap of the semiconductors analyzed ranged from 1.4 to 3.6 eV). Three diffraction gratings were used with rulings of 1200, 600, and 300 grooves/mm for visible light, near infrared, and infrared, respectively. The light source was a 450 W Xenon lamp that emits wavelengths of light greater than 300 nm. The light beam exiting the spectrometer passed through: (1) a order sorting filter to eliminate higher energy harmonics of the fundamental wavelength, (2) a collimating lens to convert the divergent rays of light to parallel rays, (3) neutral density filters to control the light intensity, and (4) a focusing lens to focus the beam onto the sample. The intensity of the focused light spot was typically 5 to 20 mW/cm2 and the dimensions were 8 mm. The intensity of the monochromatic beam was dependent on the wavelength of the light. This was due to the Xenon light source and the diffraction gratings. intensity of the focused light as a function of the wavelength are shown for each order sorting filter in Figures 3.14 to 3.17 with the diffraction grating and the spectrometer exit/entrance slit width as parameters. Curves are shown for two diffraction gratings in Figure 3.14 because the intensity with the 600 grooves/mm grating was larger than that of the 1200 grooves/mm grating for wavelengths greater than 660 nm. density filters and, in some cases, small variations of the exit slit width. Checks were made periodically on the accuracy of this method. There were some indications that the intensity spectrum is changing as the Xenon lamp ages. An estimate was made during the last month of experiments of the error in the intensity value by comparing the desired intensity with the intensity measured after the appropriate neutral density filters. It was found that the intensity varied by approximately 22 percent of the desired value. The error should be less for experiments that were conducted earlier. The order of experiments was: cuprous oxide in alkalinechloride solution, ZnS:Mn electroluminescent panel, ntype GaAs, and ZnO varistors. For the GaAs sample, the intensity was measured before the impedance experiments for each wavelength to be used and, thus, should have a much lower error in intensity. The bandpass of the spectrometer is the band of wavelengths that pass through the exit slit and is determined from the dispersion (wavelength in nanometers divided by the slit width in millimeters) and the exit slit width in millimeters. For instance, 1200 grooves/mm grating (dispersion of 3.7 nm/mm) set at a wavelength of 500 nm and an exist slit width of 5 mm will have a bandpass of 18.5 nm, or in other words, light of wavelength 490.75 to 509.25 nm will pass through the exit slit. The dispersion value is considered constant for each diffraction grating and harmonic order of the wavelength that is selected The dispersion is 3.7 nm/mm for the 1200 grooves/mm grating, 7.2 nm/mm for the 600 grooves/mm grating, and 14.4 nm/mm for the 300 grooves/mm grating all at the fundamental wavelength of the grating. The bandpass is one disadvantage of the spectrometer over a laser. The exact location of an optical transition is obscured by the range of wavelengths that exit & on mf&I ar an afarI A"^ Io ia n rrtn'rn wi 41,hia a eat4 W,4 r k* .1a a ~hP PnPr~mmcrt~cr* ~.C ~11A *Aj ~nn the spectrometer is set at 720 nm, light of wavelength 730 nm and its harmonic wavelength at 365 nm will also pass if the bandpass is large. The harmonic at 365 nm is the wavelength of concern because the 410720 nm order sorting filter begins to pass light of this wavelength as can be seen from an inspection of Figure 3.14. COUNTER AND REFERENCE ELECTRODE LEAD X" nIcRA IIC SPRINGLOADED CLAMP / / U.SO WORKING ELECTRODE LEAD .. . 0**t ..... S S..... *5~*** *5*St* S..... S.... 9..... 9*9e* *5* Light Source Spectrometer Black Order Sorting Filter Lens Neutral Density Filter Lens Sample IiI Computer Compudrive Power Supply Potentiostat v out out Figure 3.2: Basic inverting operational amplifier circuit overly simplified ~I~ Bias External Input Voltage Output Current Output High Measuring Resistors Low Measuring Resistors Bias External Input Current Output 7 Voltage Output IR Comp CeLL 121 MOhms 835 Ohms 9.23E9 farads 4000 MOhms 1620 Ohms b 9.23E9 farads 14.0 7.0 14.0 n ~~107 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 1 1 111 I I Original Circuit data data S U * Fi11T~ (Solartron) (PAR) o Circuit data (Solartron) 10.0 0.0 210 1O 100 10' Frequency . i. , Hz 11 1 1 1 i 1 1i A r i 102 103 4.0 3.0 2.0 1.0 2.0 4.0 6.0 Il 3e 107 90.0 87.0 84.0 81.0 78.0 2 100 100 101 102 i0a Frequency, log gain gain: =xl A1 pole JJ GAIN PLOT PHASE PLOT   cell pole phase 180 270'  360' log frequency IogigainI phase Al pole \\UNCOMPENSATED 180    270O COMPENSATED S 3603 gain= rl  log frequency cell pole Metal Metal Semiconductor S EC ^ E pI  E, ET Semiconductor  E  Er, E, Ev Metal E V Semiconductor SMor cap  Er ET lutor this aocltance as a transient a  I L t, ~4~ EC Tpeak  l I I I I I I I I I I I I Time C(tl)  C(t2) tO 0+ 300 400 500 600 700 800 WAVELENGTH, nm 1:.._ 1 A. T t r  1r :.l. s ..I. , .. .nl4. .. ',s *, rTloans4 in, +1, 600 700 800 900 1000 1100 1200 1300 1400 WAVELENGTH, nm r 91ct, T4na,r, at 4tn ,.1 nn in k .%.. vta pn .4 +h nnroant few a a *l Slit Width Entrance/Exit 9/7 mm 7/5 mm 4/2 mm 12002000 OSF 300 gr/mm 5 1000 1000 I ~ I I I I I I I I 1200 1400 I II 1 I II I I I 1600 1800 2000 WAVELENGTH, nm  r U  '3 IC. T......:2.. nL t tl n C...I AF i: n ..,. .tm An nC 4l\ ,utann*I tr II   18003000 OSF 300 gr/mm Slit Width Entrance/Exit 9/7 mm 7/5 mm 4/2 mm 1500 1700 1900 I 1 I 2100 I2300 1 2300 2500 2700 WAVELENGTH, nm S( r i y T I n hA ...P r P 1 r  v . Sr /f +r .rlralonn+h trw +h CHAPTER DEVELOPMENT OF PROCESS MODEL The objective of this chapter is to develop a mathematical model that describes transport and reaction processes involving deeplevel states. Some physical insight into the electronic processes of these states is provided in the first section. In the following sections, the governing equations for both bulk and surface states are devel oped, and the model is posed in terms of an equivalent circuit composed of resistors, capacitors, and current generators. Finally, various assumptions are made to obtain a usable circuit. Shockley Read Hall Processes The influence of deeplevel states or traps on the statistics of electronhole re combination was first described by Shockley and Read [46] and Hall [47]. Deeplevel states, as their name implies, lie close to the middle of the energy band gap of the semiconductor. Due to the large energy separation from the valence band and con duction band edges, deeplevel states are not fully ionized at room temperature. In contrast, shallowlevel states are those considered to be fully ionized at room tem perature due to thermal excitation. The interaction between a deeplevel state and electrons and holes can be de 57 to (Ec Er) by either radiative (photon) or nonradiative (phonon) processes, or a combination of both. Process (b) involves a trapped electron being emitted to the conduction band after receiving an amount of energy equal to (Ec Er) from either optical or thermal excitation for instance. Process (c) is the capture of a deeplevel state electron by the valence band. Valence band electrons are more tightly held to the crystal atom than are the con duction band electrons constants). , which have a much larger radius of travel (of several lattice This allows the analogy that conduction band electrons can be considered to be negative charges floating in a sea of fixed positive nuclei (lattice sites) as in a metal. Valence band electrons are held in a tight sphere about a nucleus and can move only if the neighboring nucleus has an electron vacant site that it can jump into. Once the electron moves into this vacant site, it leaves a net positive charge behind. This positive charge will appear to be an entity, a hole, moving opposite that of the valence band electron. Thus , it is feasible to consider process (c) as being the capture of a valence band hole. For a deeplevel state electron to be captured by the valence band, it must lose an amount of energy equal to (Er Ev), or likewise, for a valence band hole to be captured by the trap, it must lose the same amount of energy. Process (d) involves the emission of a valence band electron to the deeplevel state after receiving energy (Er Ev). This can also be thought of as hole emission from the deeplevel state to the valence band since an electron vacant deeplevel state has been filled by a valence band electron, thus leaving a hole in the valence band. The four processes involving deeplevel states can be described in terms of chemi cal reactions between two species. holes and electrons. Elementary reactions proceed 58 concentration of conduction band electrons and the concentration of electronvacant deeplevel states. The concentration of conduction band electrons per energy incre ment can be expressed as at energy energy f(E)D(E)dE where f(E) is the probability that a state E will be occupied by an electron and D(E) is the density of states at The concentration of electronvacant deeplevel states can be expressed as [1  fTr(r,t)]Nrr(r) where fr(r,t) is the probability that a deeplevel state will be occupied by an electron and is a function of position and time.  fTr(r,t)] is the probability that this state will be vacant or occupied by a hole. The total concentration of deeplevel states is given by NTT(r) which is a function of position only. Process (a) becomes dra = c (E)[1  fT(r, t)]NT(r)f(E)D(E)dE where c,(E) is called the capture coefficient and is a function of energy. c,(E) can be thought of as a reaction rate constant. This expression is integrated over the energy range of the conduction band (Ec ' oo), [1 fr(r, t)]NTrr(r) c4(E)f(E)D(E)dE. (4.1) Here, it is convenient to use a mean capture coefficient defined as c,(E)f(E)D(E)dE (4.2) ;f f(E)D(E)dE Process (a) becomes cn > [1 fT(r, t)lNr(r)n =< c, (4.3) where it is noted that the denominator of equation(4.2) is equal to n, the total concentration of conduction band electrons. and pr is the concentration of holes Ta = ro 1 Cn >= vUon Blakemore [48] suggests that this definition is a carry over from atomic physics where it was hoped that the capture crosssection would be propor tional to the crosssection of an atom (~ 1016 cm2 the capture crosssection can range anywhere from However, he points out that 10s2 to 1012cm2. Despite this apparent discrepancy, the term capture crosssection remains in common use. Grove [49] considers a,o to be a measure of how close a carrier has to come to a trap center in order to be captured. Hence, a deeplevel state can easily capture an elec tron if it has a large capture crosssection and/or if the mean thermal velocity of the electron is increased, by an increase in the temperature, for example. Electron emission from the deeplevel state, process (b), can be represented as a reaction between an electronoccupied deeplevel state and an electronvacant con duction band site. The concentration of electrons occupying a deeplevel state is given by fT(r, t)Nrr(r) and the concentration of vacant conduction band sites is given by [1 f(E)]D(E)dE at an energy increment dE. These symbols are defined above; note that [1 f(E)] is the probability that a conduction band site is vacant. An incremen tal expression for process (b) becomes: drb = en(E)fT(r,t)NTTrr(r)[1 f(E)]D(E)dE where e,(E) is the reaction rate constant or the emission constant. This expression is integrated over the conduction band energy range, rb = fT(r, t)NTT(r) e,(E)[1 f(E)]D(E)dE. (4.4) This rate expression is typically redefined as (4.5) where nT is the concentration of electrons occupying deeplevel states and e, is defined as rb = enfT(r, t)NTT(r)= enrT e, can be approximated by en(E)D(E)dE. (4.7) Note that as the concentration of free electrons increases, f(E) increases, and there fore, the emission rate constant decreases. Or in other words, the emission rate decreases as the number of free electrons increases because there are fewer vacant conduction band sites for the deeplevel state to emit its electron into. Hole capture by the deeplevel state can by treated as a reaction between an electron occupying a deeplevel state and a free hole in the valence band. centration of trapped electrons is given by The con fT(r,t)Nrr(r) and the concentration of holes in the valence band is given by [1  f(E)]D(E)dE. The incremental rate of process (c) is dr, = %(E)fT(r,t)NTT(r)[1  f(E)]D(E)dE which can be integrated over the energy range of the valence band to yield rc = fT(r, t)NTT(r) B~ K P(E)[1 f(E)]D(E)dE. (4.8) The hole capture coefficient is averaged similar to the electron capture coefficient, fE c,(E)[1  f_ [1 f(E)]D(E)dE (4.9) f(E)]D(E)dE in which case rc 4. > fr(r,t)NTT(r)p =< Cp (4.10) where it is noted that the denominator of equation(4.9) is the concentration of holes in the valence band. > is traditionally defined as the product of a hole capture crosssection, a,, and the mean thermal velocity of a hole, v,. Hole emission bv the deemlevel state is a reaction between an electronvacant e, X rate of hole emission is given as: drd = e,(E)[1 fTr(r,t)]NTT(r)f(E)D(E)dE where e,(E) is the hole emission constant. This expression is integrated over the valence band energy range to yield rd = [1 fT(r, t)]Nrr(r) ep(E)f(E)D(E)dE. (4.11) 00 This rate expression is redefined as rd = e[1 fT(r, t)]NTTrr(r) = eprP (4.12) where ep Ev La: e(E)f (E)D(E)dE IEv 00 e%(E)D(E)dE (4.13) because f(E) 4.1.1 1 in the valence band for the nondegenerate case. Equilibrium The above formulations are general in that no equilibrium constraints were placed on them. Here, relationships are derived based on the conditions of thermal equilib rium. "principle of detailed balance" will be used. Shockley [50] quotes John C. Slater's description of this principle: "When a system has reached thermal equi librium, it has run down and is no longer changing. Past and future are alike to Now imagine that a motion picture is made of the system, showing atoms and electrons in detail. This film can be projected backwards in time and since past and future are alike, the observer will not be able to tell the difference. Now suppose, for example, the forwardrunning picture shows on the average (Cap. n) electrons being captured per unit volume per unit time on traps giving up the energy in the form of he}at wavpn (nhnnnn s and llmrnn e it shnws (Eml. n') electrtnns hjinur emittAl nnr nnit to (Cap. n), forward and backwards running of the film can be distinguished, con trary to the assumption that the system is run down. Thus, the principle of detailed balance requires that each process and its reverse proceed at equal rates." With this in mind, process (a) must proceed at the same rate as its reverse, process (b). On average, a conduction band electron loses energy and is captured by a trap at the same time a trapped electron gains the same unit of energy and is emitted to the conduction band. Likewise, process (c) proceeds at the same rate as process valence band hole loses energy and is captured by a trap at the same time a trapped hole gains energy and is emitted to the valence band. These rate expressions are related as follows: ra = rb =r< r = rd '< where the subscript PTenc cut fll~e nTePe = Cpe lre e has been added to denote equilibrium conditions. (4.14) (4.15) These equa tions can be solved in terms of ratios by equating f of each relation, Pre nTe pT, The ratio of the third term to the second term is equal to unity, i.e., (4.16) (4.17) flepe The massaction law states that the product of the electron and hole concentration is equal to the intrinsic concentration squared (nepe = n?) under thermal equilibrium. II lIlk\ III U11 1 1 I~ epeene 63 The rate coefficients can be expressed in terms of an Arrhenius activation energy through the use of the principle of detailed balance. rates (a) and (b) at thermal equilibrium [46, 51], 1cPTe We have from the equality of [1 el2 (4.19) Under the assumption of nondegeneracy, the equilibrium concentration of electrons in the conduction band is given as (Boltzmann's approximation) EF Ei nE, = n exp( ) (4.20) where EF is the Fermi level of the semiconductor, E, is the intrinsic energy level, and k = 8.6173 x 10 s is the Boltzmann's constant. The probability of the deeplevel state at energy level ET being occupied by an electron can be described by a FermiDirac distribution function at equilibrium, fTe = 1+frexp( t") (4.21) where g is the impurity level spin degeneracy [48] which refers to the number of states having the same energy level Er. The subscript e has been added to the distribution function to indicate that this is an equilibrium expression. to electron occupancy in equation (4.19) becomes The ratio of hole occupancy  fTe] 1 Er = exp( g  E kT (4.22) Equations (4.20) and (4.22) are substituted into equation (4.19) to yield 1 ET ~ ~ T~ A = niexp( g (4.23) where nl is defined as the concentration of condnr.tinn hand wplrtimnn if thp TFnmi lnal The ratio pe/ > can be described in a similar manner when process (c) proceeds at the same rate as process (d) (thermal equilibrium), fTe [1 e [1 fr8] EF =gexp(  ErT (E  )ni exp( kT " (4.24) kT Ei = gniexp( 1 (4.25) pi is defined as the equilibrium concentration of valence band holes if the Fermi level coincides with the energy level of the deeplevel state, modified by g. 4.1.2 Nonequilibrium This section considers the carrier concentrations and the deeplevel states under nonequilibrium conditions as developed by Sah [51, The following transport equations are of the form Accumulation = Flux Gradient + Net Rate Production on a unit volume basis. The rate of change of the conduction band electrons can be represented as On(r,t) Cn > n(r, t)pTr(r, t) + e nT(r, t) + G.(r) + gn(r, t). (4.26) The rate of change of the valence band holes can be represented as ap(r, t) Q > p(r, t)nT(r, t) + e,,pr(r, t) + G,(r) + gp(r, t). (4.27) The rate of change of the concentration of deeplevel state electrons can be repre sented as 53]. ~,,I, r\ 65 The notation in parenthesis has been added to indicate the dependence on position and time. The terms G,(r),g,(r,t),G,(r), and gp(r,t) have been introduced to ac count for other net generation processes which are separated into steady state and dynamic terms. These net generation processes include such contributions as the in fluence of other deeplevel states, and conduction band to valence band generations and recombinations. GT(r) and gt(r, t) have been added to allow for optical excitation of the deeplevel states [51 The carrier flux can be attributed to contributions of drift (migration) and diffusion processes [52] i.e., jn = qpnnE + qD,Vn, and (4.29) jp = qpppE qDpVp. (4.30) These carrier flux expressions would have an additional term to treat any DC leakage current. For the present development, it is assumed that DC leakage current is not present. The above equations are related to the electrostatic potential via Poisson's equa tion: P = Q[(r, t)  n (r,t) + (ND(r, t) nD(r,t)) (NA(r,t) = KeoV  nA(r, t)) nT(r,t)] E = KeoV2 (4.31) where p is the concentration of charged species, q = 1.602 x 1019 C is the electronic charge, K is the dielectric constant, = 8.85419 x 1012  is the permittivity of vacuum, E is the electric field, 4 is the electrostatic potential, ND(r,t) is the donor concentration, n(r, t) is the concentration of unionized donors, NA (r, t) is the concentration of arrentorsn. nA (rt1 is thei ronrcfntratinn of nn innimd a.rrentnra 4.1.3 Small Signal Analysis The above nonequilibrium equations will be analyzed here with the small signal approach. are: There are several assumptions made in the present development which the perturbation from equilibrium (np n?) is small such that only the first two terms of the Taylor series are sufficient, the shallowlevel donors and acceptors are fully ionized (fD, nA = 0) at the temperatures analyzed, the semiconductor remains nondegenerate such that the nonequilibrium concentration of electrons, holes, and trapped electrons can be expressed with a quasiFermi energy level for each (F., F,, FT), and the system is onedimensional. The four rate constants are assumed to be equal to their equilibrium values (  aepc, >=< c, > and e/ e, This implies that the ratios > are constant and, in light of equations (4.23) and (4.25), ETEi= constant. (4.32) The quasiFermi energy level assumption, valid for a nondegenerate semicon ductor, permits the use of the Einstein relation in the flux equations. The Einstein relation is kT Di =  p. The onedimensional electron and hole flux equations become (4.33) OF, jn = pun ,x OFp jp = pp x (4.34) (4.35) The concentrations can be approximated by the first two terms of the Taylor >, >< where n(r,t) =F Ei = niexp( kT ). (4.37) The partial derivative with respect to time can be expressed as 9n(r,t) m It = exp( ___ Fn '9K Ot' N(r) (OFI LeT Ot Ot* (4.38) The hole concentration can be represented as p(r,t) Op(r, t) = P(r) + Op(r, t) (4.39) where (Ei = n"exp( (4.40) kT ' Op(r, t) OF, ~ ~) ni E; k^e xP( = nc exp( kfT =P(r)( e9Ei tm A OF ~ ) . (4.41) The trapped electron concentration can be represented as nT(r, t) = NT(r) + Onr(r,t) = NT(r)+ nt(r, t) (4.42) where nT(r, t) NTT(r) 1+lexp() 2 ItT (4.43) dnT(r,t) cu It NTT(r) [l+ lexp(F ) 1 ET T exp( gkT FTr aE, kT a Ot OF )a (4.44) PT(r) aE, NTT(r) at O)F,  The trapped hole concentration can be represented as Opr(r,t) , n' /_\ i  \ p(r,t) NT(r) P() +  li~p aEi o r\ In nr\ Opr(r, t) stIt Nrr(r)T exp(E T) 1 + exp(ZF) N_ (r) 1 1 ETF  1 F( ex+pT( ))T C1+~expQ~r&)J2 kT g ,DE, I' OF, , I = PT(r) tn Cl ~ PT(r) aEt OF NTT(]) t t Nnr(r)1 at 01 PTr(r) NT(r) (EO ) = (r) kT NTT(r) 9t Ft at (4.47) The results of the above derivations with the substitution of OEt produced here for clarity = E E;are re n(r,t) = N(r) + k(F, Ei), (4.48) p(r,t) = P(r) (r) IT" (4.49) nT(r, t) = NT(r) NT(r) Pr(r) kT NTT(r)' (4.50) pr(r,t) = PT(r) + PT(r) NT(r) ( . kT NTT(r) (4.51) Equations (4.48) to (4.51) are substituted in the nonequilibrium equation (4.26) to yield N(r) jF, kT Di OEI ai ' " [N(r) + q S(OF OE,)I[V(P, + OF)]) kT FEiy+F, cn > [N(r) + N(OF. 8Ei)][PT(r) + kT PT(r) NT(r) NTT(rDE OF,)] kT NTV(r) NT(r) PTr(r) +en[NT(r) (E (F,)] + Gn(r) + g(r,t) kT NTr(r) '("l q *(N(r)VF,)< c Sf" N(r)VOF, + q (4.52) > N(r)Pr(r) + eNT(r) + G,(r) N(r)VF. kT (OF, OE) + g(r,t) kTf r  d A . n at at v r NT(r) ,F. S> N(r)PT(r)Nr( NTT (r) 9E, OEOFt) Lr )r Im )* The terms in the curly bracket is the steady state equation and hence can be set equal to zero, and the last two terms are second order terms which can be dropped since the Taylor series included only first order terms. It is advantageous to divide the semiconductor into elements and to assume that in each element the static quasiFermi potentials and the static electron, hole, and trap concentrations are independent of position. The dynamic generation terms are also assumed to be independent of position within each element. However, these properties can vary from element to element but are constant within each element. Equation (4.52) reduces to kcT Ot attiN 0xL2+ cm>NPT T cn > NPT( T ) (4.53) [ NT Pr aF, > NPT + eNT]( le aI) Substitution of equations (4.48) to (4.51) into equation (4.27) yields P(r) (OE OF, kT t t)=  Cp(r)+ P(r)(BF S[P(r) + (dEi dF,)][V(F, + aF,)] > [P(r) + ((Ei OF,)][NT(r) N kT keT PT(r) Nr(r)(ES OF,)] NTT (r) P [P(r)(r) + e,[PT(r)+ ST NT(r) (E aFt)] + G,(r) + g,(r, t). This equation can be reduced with the subtraction of the steady state equation, the elimination of second order terms, and with the assumption of position independent static properties within each element as well as the dynamic generation term to yield P dI. REni2 82 F. ~Ar.~  E2 (4.54) Substitution of equations (4.48) to (4.51) into equation (4.28) yields NT(r)PT(r) (OF, kTNTT(r) 9t OEi) at N(r) + N(r)(F" (4.56) r + PT rN(r) NEU PTr + PT(]P) F > [P(r)+ P(r)(  aF,, rr 8F kT)][NT(r) kT  )F, PT(r) 9OE 9 F,,  NT(r) ( T ) NTr(r) kT S+ NT(r) OEi + e, [PT(r) + PrT (r) ( ( NT (r)  + GT(r) + Ft(rt). )] + GT(r) + g(rt). This equation is reduced by subtraction of the steady state equation, elimination of second order terms, with the assumption of position independent element properties to yield NTPT kTNTT OF, NT PT c > NPTN + eSNT NVTT N TT PT NT ] OF, > PNT +epPr ( NTT NTT O~i) leT > PNTOF > PNr(   'E*. kT )+ LeT F> N c, > NPT( " OE kT ) + t(t). (4.57) Poisson's equation becomes CV2(E, + OE,) EP(r)P(r)( P(r)+P(r)(a OF T 1')  N(r)  N(r)( E O) kT +ND(r) NA(r) ,PT(r) (Ei  NT(r) + NT(r)T(r) NTr(r)(  OF, kT (4.58) where it was assumed that the electrostatic potential is equal to the intrinsic energy level divided by the electronic charge, Ei. The derivative with respect to time of this equation is more useful here, 1 Keo q2 atV OE PO Ei kT 9F 9P) N OF, kT kCT OEi)+ at NTPT kTNTT OEi ( 9F) ). at (4.59) NT (r) ) (aEi [NT(r) NT (r)N( NTT(r) OE; a) , Cn >[ ,] e, aEi )] 4.1.4 Equivalent Circuit The small signal nonequilibrium equations derived above can be related to an equivalent circuit composed of common circuit elements such as resistors and capac itors. To do this , the partial quasiFermi and the partial intrinsic energy levels have to be related to a voltage [51 This can be done by dividing the energy levels by the electronic charge, OF  OFt   (4.60) Kirchhoff's current law is applied to each small signal equation. This law states that the sum of all the currents entering and exiting a circuit node must be zero. At this point it is useful to remember that the current through a capacitor is equal to the capacitance multiplied by the time derivative of the voltage across the capacitor and the current through a conductor is the conductance multiplied by the voltage drop across the conductor. The small signal equation derived above are reproduced below for clarity, N OF,, E aEi aaF, kT( at t ) q kT~ Di Ot q Ozx2 OF, Ei cI > NPT( kT ) k:T (4.61) NT > NPTN Nrr PT OFt + eNT ]( OaEi lv)? = g, kT( kT 9t [< cP  )E Ot PT > PNT  Nrr + NT + ePtr OF,9E, > PNT( kT ) kT (Oft 3( DaEi lv)? (4.62) = 9p, NrPT kTNrr NT PT c. > NPr_ + e.Nr _+ SF, Ot  'a BEi) at PT NT 1 (OF > PN + ePrI ( Clp 020Fp  P '" a 89 [< Cn aEi 1 qK^ Vtd SP(OF, kT kT  9Ei+ N8aF, at IcT' oEi 9E)+ at NrPr kTNTT aOF (4.64) The next step is to replace the quasiFermi energy level perturbations, with the corresponding voltages, qv, and multiply the above equations by the electronic charge, q. This changes the voltage coefficients into capacitances and conductances. There are four voltage parameters in equations (4.61) to (4.64), , Vp, vt, and vi. These equations are rearranged into voltage drops across circuit elements, i.  vi, vt V 2i,Vn1  vt, and vp  vt. It is assumed that direct band to band processes are included in g, and g, and hence, v,  v, need not be considered here. To find the coefficients of the these voltage drops, it is best to collect all the coefficients of like voltage and then regroup the voltages with common coefficient for each equation. The relationship, NTT = NT + PT is used several times. Upon doing the above, the equations can be expressed as q2 Na( V, lvi  vi) 22, +  qpN82 cn> NPT( T^ {n  t) 2 PT kTNTT [ > NPT  enNTr](vt = qgn, (4.65) q2P O(v,  vi)  qpP P  + > PNT r (" Vt) q2NT kTNTT ReC >PNT e,Pr](vt vi) = qgp, (4.66) q2 NTPT (vt kTNTr Vi) _ q [< kTN TT PNT, (vP enz'NP~ Vt)  ePrNT+ Cn> NPT( ^( n >PN4 Vi) epPrNT (Vt = qgt, vi) (4.67) e., t/v,vi, a~Eli at  vi) 73 At this point, the semiconductor is divided into equally spaced elements in order to treat the second order derivatives. The second order derivatives are discretized with the central finite difference method applied to three elements. The second order derivatives of each potential in the above governing equations can be expressed as 02Vrk BA: Vk+1 21vk + Vk_ (Az)2 Vk+l Vk (Aa)2 ktr (4.69) (Az)2 where the subscript k is used to indicate the element number. It is apparent that the second derivative in potential can be thought of as a potential drop across a circuit component at the beginning of the element minus the potential drop across a circuit component at the end of the element. We now let Cn= q2N kT' (4.70) C, = (4.71) q2NT PT kTNTT , (4.72) an= qp N =(Ax)2' = (Aaz)2' (4.73) (4.74) (4.75) Ke, (Ax)2' gr < Cn > NPT kT q2< Cp > PNT (4.76) (4.77) q2apT kfTNTT .2 AL. [ Ccc > NPT eNT], (4.78) Cr  CK  GnT = GpT = GnTi = (4.82)  qgt The above conductances have units of current per volt per volume and the capaci tances have units of charge per volt per volume. These conductances and capacitances are substituted in the above equations to yield Vi) Unk,+)+GnT(vn (4.83) vi) Vp,k)+Gp,k+i (vp,kVp,k+1 )+GpT(Vp = p, (4.84)  uv)  (GnTi + G,Ti)(v,  vi)  GpT(Vp  vt)  GnT(V,  t) = iT, (4.85) S(v, ^p  vi) +cn 8(v, vi) v.) +Cr  Vi,k) = CK,k  Vi,k+1) + CK,k+1 (4.86) The corresponding equivalent circuit is given in Figure 4 and Figure 4.2 are consistent with that derived by Sah [51, Equations (4.70) to (4.86) , 52]. Surface States The treatment of surface states presented here follows that of Sah Surface states can be a discrete single energy level defect or a distributed energy level defect (surface band). 4.2.1 The single energy level surface state is treated first. Single Energy Level Surface State cys S,,  Gnk (2t,,kl 2)nk)+ Gnk+l (Vnk vt)+GTi(vtvi) a(v, c,  Gpk (Vpk~l  ttt)+ GpTi (tlt t)i) d(Vik a(v;,kl is a boundary condition to Poisson's Equation in that it relates the first derivative of the potential to the charge located at the surface. The surface concentrations can be expressed in terms of quasiFermi energy levels that are either bulk values evaluated at the surface (F,, F,.) or newly defined levels associated with the surface state (FT.). The concentration of electrons evaluated at the surface is given by F= n exp( = niexp(  E kT ) (4.87) and of holes, p (t) =Ein = ni exp( FS) k:T (4.88) The concentration of electrons trapped by the surface state can be expressed as nTr(t) (4.89) 1+1 exp(&;E)F The concentration of surface states that are electron vacant is given by NTT, mi.t) exp( k) (4.90) l+ pexp( C ) where NTT is the total concentration of surface states, Ei, is the intrinsic energy level evaluated at the surface, ET, is the surface state energy level, and the subscript s denotes that these are either surface properties or parameters associated with the surface state. The Taylor series approximations of the surface concentrations are n,(t) = N, N.  9E;,), (4.91) p.(t) P + P P,+ (T(Ei,  OF,,), (4.92) nfT,(t) = NT. NTTB (8Ei8 (4.93) n.(t) NTTI  aF,,), 76 The nonequilibrium equations that describe the rate processes between the carrier bands and the surface state are the time rate of change of surface state occupancy 49fT8(t) (4.95) and Gauss' Law  Keo^ = qrT,(t) nf (4.96) where it is assumed that the potential gradient on the metal contact is zero. These equations apply to a twodimensional plane and hence, are based per unit of area, cm2 The surface state electron occupancy equation is twodimensional through out because nT, is per surface area, not volume. units of inverse time. The emission rate coefficients have The product of the capture rate coefficients and the respective carrier concentration have units of time. Both of these expressions are multiplied by a concentration of surface state, the product of such has units of per time per area. Upon substitution of the Taylor series approximations of the surface concentra tions, the above governing equations become NTPTr, (Ft, OE_ kTNTTr, Ot t pNa( OF P, NT,(  9E;, kT OF8 9E) > NPT,( ) kT + NT, PT.+ > NPTrTT + eNT, ~+ NTTs NTT Nrri (Oft. PT. > PNT, NTTs  9Ei, ,m ) = gt (t) (4.97) and the time derivative of Gauss' Law becomes 1 8 OOEi, s K Ox ao t : qNTPT. (dEi, kTNrT,r OF,t at (4.98) where the stePnAv state p1innatinn hna hen elimninatld as well as the srm ndnl nrdsr terms Cns > n~(t)pTo(t) ens 1ZTs(t) < Cpa > P.(t)nT, (t)+e~pr, (t)+ GT, +gt, (t) 77 can then be considered as simple resistors and capacitors with units of current and charge per volt per square area. The governing equations can be expressed as O(v~. v) vi,  (GnTis + GpT,) (vt,  v)i.)  GpTS(vps  Vt,) GnT,(vns  ve) == fl's (4.99)  t. 1 v^8) = CKs  Vil) (4.100) where q2NTPT. k TNTTs (4.101) (4.102) GnTs GuT. GyT. (4.103) q < Cn, > NPT, lkT qZ < C > PNT,. (4.104) GnTis q2PTs kTNTT. ke n. > N,PT.  ensNT.], (4.105) GpTiBr QNT.) k!TNTT, > PNT.  epPT,], (4.106)  nt The corresponding equivalent circuit is given in Figure 4.3 and Figure 4.3 are consistent with that derived by Sah [ 4.2.2 (4.107) !. Equations (4.99) to (4.107) 54]. Surface Band An equivalent circuit is developed in this section to incorporate a surface deep a(v,, a(vi, 78 before in that the capture and emission rate coefficients are a function of the carrier band energy level and also the energy between the carrier band and surface band. Each incremental process is developed first, then the small signal analysis is applied, and the equivalent circuit is deduced. Process (asb) is the rate of electron capture by the surface band and can be expressed as d2rab = c.,(E E,)f (E)D(E)[1  flb (Esb)]D,b(Ea)ddEdE. (4.108) where E8, is the energy level of the surface band increment dEg, and the subscript sb is used to indicate a surface band property. All other symbols are consistent with earlier definitions. This equation becomes upon integration over the conduction band dr.,b > n[1l  fab(Ea)]DD,(E,)dEb (4.109) where < Cnab(Eb) 1 o cnb >= J ri Ec Eb)f(E)D(E)dE (4.110) Process (bsb) is the rate of electron emission from the surface band and is given Pblbb  f(E)]D(E)f.b(E.,)D.b (E,b)dEdEb. = enb(E (4.111) This equation becomes upon integration over the conduction band = enb(Esb)fsb(E,b)Drb(E,a)dE,b (4.112) where enab (Esb) = nsb  f(E)]D(E)dE. (4.113) >=<    < C~,b (Elb) Cnbb (E Ea)[l drbsb enrb(E ,Eirb)[l 79 where the distribution function was assumed to have the same form as equation (4.20) and equation (4.22) was used to relate the equilibrium electron concentration to the intrinsic concentration. This relation will be used in the derivation of the equivalent circuit. Process (csb), the rate of hole capture can be expressed as (4.115) and upon integration over the valence band becomes > P.fSb(ESb)Dab(Eab)dEb (4.116) where < Cpb(Eb) >=< 1 Ev L E  f(E)]D(E)dE. (4.117) Process (dsb) is the rate of hole emission from the surface band and is given by d'~rd, = eps(E , Eb)f(E)D(E)[1 f,b(Eb)]D,a(Eb)dEdE,b. (4.118) This equation becomes upon integration over the valence band drdsb  eps(E,))[1  fab(E,b)]Dab(Ea )dE,sb (4.119) where eps (E,a)  ,.b Ev J1o , E,)f(E)D(E)dE. (4.120) The rate of process (csb) equals the rate of process (dsb) at each increment of surface band energy under thermal equilibrium. This yields the equilibrium relation hwptxwv rta~ wrnnntnn  f(E)]D(E)fsb(Eab)Dn6(Eb)dEdEb =< Cp,(Esh) ddr,~ Cprb(E, E~b)[l drc~b Edb )[1 Cpb (E %,a(E 80 The governing equations at the surface are similar to those of the single energy level surface state. The rate of surface band state concentration must be addressed at a particular energy level and not as an integrated sum of reactions because equilibrium conditions only apply to individual states at a particular energy level. Thus the rate of change of surface band concentration is given by anab(E.) = [dr.,b  drbsb  dr'es + dra, + g.b( Eb, t)J IlEs (4.122) Gauss' Law can be expressed as e08alt =fE (4.123) where E, and EI are the upper and lower energy bounds of the surface band, respec tively. The concentrations of electrons and holes, and the surface band distribution func tion can be approximated with the first two terms of the Taylor series expansion. These expressions are =N, N+ , + 8FT (4.124) P= + =P,+ (Eia, (4.125) f b(Esb) = Jfb  f8&)(OEi.  OF,,,), (4.126)  fAb(Eb) 1 f~b(1  9F  fb)(9Ei, (4.127) where fb (without the energy notation) denotes the static distribution function which is a function of E,. An additional assumption is necessary in the present case due to the distributed nature of the surface band; small perturbations in potential are independent of the f~a (Eb)Dsb(Eb)dEa   aF,,), 1 fdb(l LT 1 fb ~  8E. )][1 1  fb + fob(1 kTfa  fb9)(aEi. 1  f(1 ^fT  f.b)(9Ei.  OF,b)]Do,(Eb) (4.128) P, + (AE, < pb~lb  OFo)][fb 1 k f~a(l f.6)(8Ei, +epb(Eb) [1  fb)(OEi.  8F,6)]D,6(Eb) + g6(E.6, t)] dEs. The time derivative of Gauss' Law can be expressed as , all 1 Ot dr feb (1l  f8E 8 OF. )Da&(E~a)dE&. (4.129) The steady state equation is subtracted from the above equations and second order terms are neglected. The governing equations become, upon regrouping the dynamic energy terms, 1 Fb fa&(1 feb)D.b(Ea)dEb( 0g.) at (4.130) N,(1 fb)Dsb(Ea)  8Fb ) < cn (E,a) N,(1  fsDb)2Ds(Esb) dEoa(OFa OaEi.) < Cpb(E&b) Pfb Db (Esb)  epb (Eab) f(1  fb)D4b(Eb) dEb (OFb < Cpb(Eb)  OFB,) + P~f~ olatin b gsb (Ebs, t)dE,b e, 8 8OEx at Oax =qE. 1 Eb) kfa((l fab)( k:T at OFs )D )Db(EIb)dEsb. dt (4.131) These governing equations are related to common circuit components through the N  esh(Es}) [fs  fsb)Dab(Eb) 1 f.a + fbb(l k:[;r  ar~b)l Dbli(Elb)  dFb)]DI)(~Eb) < cnQ (Esb) f~b(i en~b(E~L) aEi~)  vi) = CK 8(vi.,  vil) (4.133) where the conductances have units of current per volt per area and the capacitances have units of charge per volt per area. These circuit components are defined as = q2 pfab(1 fsb)D,b(Eb )dEa (4.134) (4.135) N,(1 fb)Dsb (Eb) kT P.,f6 D,sb(Es) , > ~oEb (4.136) (4.137) < C ,an(Esb) N,(1  f, )2 Dsb (Es)  enaa(Eob)  (4.138) P, f,Dsb(Esba) < Cpsb(Eab) =Eq = q ,  fsb)D^s(Esb)  ep%(Eob) f,(1 dR,6 (4.139) (4.140) 9g, (E,, t)dEs. The corresponding equivalent circuit is identical to the equivalent circuit for the single energy level surface state given in Figure 4.3. are consistent with that of Sah 4.3 Co This circuit and its related equations implete Equivalent Circuit The surface state equivalent circuit can be coupled to the bulk state equivalent circuit by converting the bulk state capacitances and conductances from a h;aqi to a snrfarpe area ha.in volume This is dmnn hv mnltinlvintr thePQs terms hv the width of 9(Va~b dR.6 < Cnb (Esb) Gnsb < Cpb(E~b)  f~b) Dba(Eda) G,;,a Gpisb current. The Schottky contact can be treated as a capacitance, Co, connected only to the intrinsic energy line because electrons and holes cannot flow across a Schottky contact; only charging and discharging is allowed in an ideal case. The Ohmic contact on the right side of the circuit can be treated as a short of all three energy lines because a potential difference cannot be maintained in a metal contact. The circuit shown in Figure 4.4 represents the minimum number of elements needed to model a Schottky contact (one element) with surface states (one element). In the case of a pn junction, at least two bulk state elements are needed. It can be assumed that surface states at the Ohmic contacts can be neglected because the contacts would appear as a short across the surface states. One of the bulk state ele ments would represent a semiconductor with ptype behavior and the other element would represent ntype behavior. The use of only one element can be rationalized from an analysis of the deep level state terms at various locations in the semiconductor. The trap capacitance, Ct, is proportional to the product of the trapped electron concentration, NT, and the trapped hole concentration, PT. The trap can be assumed to be completely full or empty in the bulk of the semiconductor depending on the location of the trap energy level in relation to the Fermi level. Therefore the product NTPT is zero and, hence, the trap capacitance is zero. Likewise, band bending at the surface will make either NT or PT zero and the trap capacitance zero. The maximum in the trap capacitance will occur at the point where the trap energy level, ET, crosses the Fermi energy level at which point NT = PT. The product of the trapped electron probability and the trapped hole probability, fr[1 fT] is its maximum value of one fourth at this point. Tt in this rPwinn rnf the sPmircnrhdtrnr that thin n1nivalent cirrn1it aAIzdrP.aMPRP Th u nint 84 either a Schottky contact or pn junction, the equivalent circuit would be represented by a single conductance term, G, or G,. Figure 4.4 with a short between v,,, vi,, a (Replace the Schottky capacitance, Co in nd vt,). No space charge region would form , hence, the filling and emptying of traps would not take place. Although surface and/or bulk states may be present, neither would influence the electrical response. 4.3.1 Simplifications of the Complete Circuit It was assumed that the experiments performed in this work do not perturb the semiconductor far from equilibrium sucd equal the equilibrium Fermi level, EF. the steady state quasiFermi levels The steady state concentrations are equal to their equilibrium concentrations with this assumption. As a direct result of this equilibrium assumption, the transconductance terms, GnTi, GpTi, etc., are zero (rate of emission equals rate of capture) and can be removed from the circuit. Additional assumptions can be made if the semiconductor is ntype or ptype. All of the semiconductor devices studied in this work can be treated as ntype semi conductors with Schottky contacts. The concentration of electrons is many orders of magnitude greater than the concentration of electrons for this case (pe, e = , , 1017 a 010). This permits the removal of the hole conductance (Gp), hole capacitance (CG), ure 4.4. and the hole recombination conductance (GTi) terms in Fig This also has the effect of removing the hole surface state recombination term (GpTi,) from the circuit even though the concentration of holes at the surface might be of the same order of magnitude as the bulk concentration of electrons [57]. The resulting equivalent circuit is presented in Figure 4.5. r.1l* ..  "^J A i 4 4S1 #. 4 4 series is in parallel to a capacitance. The conductancecapacitance series can thought of as a recombination arm and the capacitance as a space charge capacitance. The space charge capacitance is composed of either the dielectric capacitance and the electron capacitance in series for the surface state case, or, capacitance for a bulk state. only the electron This space charge capacitance will remain constant in either case so long as the concentration of deeplevel states is several orders of magnitude smaller than the concentration of electrons due to shallowlevel states. It was assumed throughout the derivation of this circuit that the two dielectric capacitances were identical because in the finite difference approximation of the sec ond derivative of potential, the element width, was assumed to be constant. This is certainly true for the case of a large number of bulk elements. However, in the present case only one bulk state element was used and the assumption of equal sized element widths is not appropriate. The element width that comprises the space charge region is very small and of the same order of magnitude as the depletion width, Whereas, the element width near the Ohmic contact incorporates the bulk of the semiconductor and is on the order of a millimeter. capacitance terms, This implies that the dielectric which are inversely proportional to the element width, differ by several orders of magnitude. The dielectric capacitance near the Schottky contact is much larger than the dielectric capacitance near the Ohmic contact to the extent that the latter term can be treated as an open circuit. governing equations in this chapter have been developed for the case of one bulk deeplevel state and one surface state. There will be one additional gov ering equation that resembles the kinetic trap equation for each additional state nrefsentt TheSp will annear as rprrmhinati nn arms in naTallel tn thna glsir.tp l in 86 Based on the above discussion, the final simplified equivalent circuit used to regress the experimental data is given in Figure 4.7. This circuit has all the essential features that were mentioned above, namely, the equivalent circuit for a surface state is iden tical to that of a bulk state. It is possible to distinguish the two types of states by means of a MottSchottky plot (C2 versus applied bias) as discussed in Chapter A leakage resistor has been added to the equivalent circuit of Figure 4.7 to account for the nonideal behavior of the Schottky contact (DC leakage current). is proportional to the exponential of an activation energy. This term The activation energy can be the barrier height of the Schottky barrier or the energy difference between the conduction band and a surface state energy level. Leakage resistance was addressed for a pn junction in the literature [59, 60]. The leakage resistor may also be the in ternal resistor, R1,, of the measuring op amp, especially in cases in which the sample impedance is of the same order of magnitude as Rn[61] A similar circuit was derived by van der Ziel [62] in 1959 to account for generation recombination noise due to traps in semiconductors. Van der Ziel [63] also determined trap activation energies by measuring noise resistance as a function of frequency and temperature (similar to Admittance Spectroscopy). DareEdwards et al. [64] developed a similar circuit by eliminating terms from the governing equations be fore developing an equivalent circuit. Nicollian and Goetzberger [55] developed this circuit (without leakage resistance) to analyze interface states of metalinsulator semiconductor (MIS) capacitors. The impedance data was regressed to the simplified equivalent circuit by complex nonlinear regression [65, This regression method is based on the Levenberg MarrarUt nln,.mnthm ffi7l AnI altanatu' a tr iiea this nirtnInonr rnurpaarrocnn nnr lrsn 