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QUANTITATIVE XRAY PHOTOELECTRON SPECTROSCOPIC METHODS AND THEIR APPLICATION TO LEAD IONSELECTIVE ELECTRODE SURFACE STUDIES BY PAUL C. MCCASLIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 ACKNOWLEDGMENTS I would first like to thank Dr. V. Y. Young for her guidance, patience, and fruitful suggestions, all of which led to the present work. Her support has been invaluable during my graduate years. In addition, I wish to acknowledge the aid of my other committee members, J. D. Winefordner, A. BrajterToth, G. M. Schmid, and G. B. Hoflund. Their assistance along the way is greatly appreciated. Special thanks are due Gar Hoflund, who was kind enough to read an early version of this dissertation. Ken Matuszak, Rustom Kanga, Jack Davis, Linda Volk, and Mike Clay have helped me out along the way with their discussions, opinions, and friendship. I appreciate the many hours we have shared. Finally, I want to thank my wife, Ann, who has been uniformly loving, supportive, and tolerant. Without her, graduate school would have been much more trying and much less enjoyable. TABLE OF CONTENTS Page ACKNOWLEDGMENTS..... ............. .............. .... .. ....... .... i ABSTRACT............................................................ v CHAPTERS 1 INTRODUCTION..... ......................................... XRay Photoelectron Spectroscopy (XPS).....................1 Quantification in XPS..................................... .3 Measurement in XPS.........................................4 Instrumentation...........................................7 Lead IonSelective Electrodes..............................12 2 THEORETICAL BACKGROUND....................................16 3 MODELING OF SURFACE TOPOGRAPHY EFFECTS IN XPS.............24 Introduction..............................................24 Theory of Modeling......................................27 Methodology..............................................29 4 TESTING AND APPLICATIONS OF THE SURFACE ROUGHNESS MODEL...43 Testing of Roughness Program...............................43 Sinusoidal Function Results........................... 44 Grating Function Results.............................52 Particulate Surface Model Results and Applications........58 Experimental Surface Roughness Studies....................74 Results of Gold Studies..............................75 Results of Carbon Studies: Model Comparison.........84 The Effect of Pressure on the VAXPS Curve.............95 Peak Structure Studies on Abraded Surfaces...........99 5 THEORY AND TESTING OF A METHOD FOR NONDESTRUCTIVE DEPTH PROFILING..........................................119 Introduction.............................................119 Numerical Inverse Laplace Transform (NILT) Method Development........................... ................. 121 Theoretical Basis...................................121 Mathematical Model................................. 123 Testing with Theoretical Functions.......................133 Experimental Testing of the NILT Method..................143 6 QUANTITATIVE XPS OF LEAD IONSELECTIVE ELECTRODE MEMBRANES ......... ................................... 151 Introduction............................................. 151 Experimental Analysis of Pb ISE Membranes................152 Operating Conditions and Preliminary Data Treatment......... ............................... 152 Composition and Preparation Effects................156 Effects of HCO................................. 170 Effects of Fe ..................................175 7 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK............ 184 Summary and Conclusions................................. 184 Future Work...................... ...................... 186 APPENDICES A SURFACE ROUGHNESS PROGRAM SOURCE CODE...................189 B NILT PROGRAM SOURCE CODE. ................................203 REFERENCES.. ..................................................... 220 BIOGRAPHICAL SKETCH ............................................. 225 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 QUANTITATIVE XRAY PHOTOELECTRON SPECTROSCOPIC METHODS AND THEIR APPLICATION TO LEAD IONSELECTIVE ELECTRODE SURFACE STUDIES BY PAUL C. MCCASLIN April, 1988 Chairperson: Vaneica Y. Young Major Department: Chemistry The development of methods applicable to the Xray photoelectron spectroscopic (XPS) analysis of solid surfaces, in general, and solid state lead ionselective electrode (ISE) membranes, in particular, is the focus of the present dissertation. These methods fall into two categories. The potential effects of surface roughness on XPS analysis have been investigated. In addition, a method has been developed which allows the composition of the nearsurface region to be investigated in a nondestructive fashion. The study of surface roughness effects has been carried out from both a theoretical and an experimental perspective. A program has been developed to determine the nature and magnitude of effects attributable to surface topography. Its predictions have been compared to experimental data; the agreement obtained is quite good. The effects of the pressure used in sample preparation, which is related to the sample topography, on the XPS signal have been investigated. Finally, the effect of increasing surface abrasion (and thus increasing surface roughness) on the relative makeup of the signal from the various species present in the nearsurface region has been studied. In order to obtain depth profile information nondestructively from a solid sample, a technique known as variable angle XPS (VAXPS) can be utilized. In this technique, the photoelectrons are collected at a series of takeoff angles with respect to the mean plane of the surface. These VAXPS results contain the desired depth profile data. However, the data must be mathematically treated in order to obtain the depth profile. A method of performing this data transformation has been developed. Its description is presented, along with testing of the theory behind it. Finally, these methods, although developed for general usage, have been applied to the study of the surface composition of lead ISEs consisting of a mixture of lead sulfide and silver sulfide. Several studies have been carried out, investigating the effects of membrane composition, preparation, cleaning, and poisoning of the response on the surface composition. The results are presented, along with several conclusions related to the optimization of lead ISE behavior. i CHAPTER 1 INTRODUCTION XRay Photoelectron Spectroscopy The roots of Xray photoelectron spectroscopy (XPS) may be traced far back. One of the four fundamentally important papers published by Einstein in 1905 concerned an explanation for the photoelectric effect (1). This effect had long been known to occur; however, Einstein was the first person to correctly describe the nature of the photoelectric effect. The phenomenon, which is the physical basis for XPS, can be described simply as an excitation process, whereby a photon of energy hv impinges on a solid, interacts with an atom's electron density cloud, and causes the emission of a socalled photoelectron, whose kinetic energy Ek relative to the measuring spectrometer is given in the absence of surface electric double layers by the expression Ek = hv Eb f where Eb is the binding energy of the electron in the atomic or molecular system measured relative to the Fermi level and of is the instrumental work function, all typically measured in eV (2). Thus, the electron's kinetic energy is independent of the incident photon intensity. Early attempts to construct a highresolution electron spectrometer made use of magnetic focusing and thus were susceptible to stray magnetic fields. The theory behind such instrumentation can jl be traced back to 1956, when the first doublefocusing magnetic spectrometer was described in the literature by Siegbahn and co workers (3). However, the potential usages of such an instrument were vastly increased with the report by the same group that not only did the instrument yield elemental information about the surface, but chemical state information as well. It was discovered that Cu 2p electrons from copper and copper oxide samples had different kinetic energies and thus different binding energies. Combined with its sensitivity to species at the surface and in the nearsurface region (typically, down to approximately 10 nm), this factor has been responsible for the development of XPS as an analytical technique. The next major step in the evolution of XPS was the elimination of the problems caused by external stray magnetic fields. Carlson and Krause were the first researchers to design an instrument using electrostatic focusing. This development was the crucial step needed to induce several instrument manufacturers to construct and market photoelectron spectrometers. The field experienced a dramatic increase in popularity in the early 1970s, as the first generation of commercial instruments were introduced. Since then, such instrumental advances as Xray monochromators (to reduce the Xray linewidth and thus enhance resolution), multichannel detectors (to increase the collection efficiency and thus reduce run times), elaborate lens systems (to further improve resolution), various in situ sample treatment methods, and smallspot XPS instruments (to enhance the spatial resolution) have been introduced. li Quantification in XPS The achievement of quantitative XPS results may be approached from two directions. One may make use of a number of standards to construct a series of empirical sensitivity factors based on a comparative analysis of peak areas. This method was applied early on to collect a limited number of sensitivity factors (4). However, the difficulty in preparing suitable standards has limited the applicability of this technique primarily to the study of homologous series of compounds. The other approach involves a theoretical analysis of the chain of events leading from the excitation of the photoelectron to its energy determination and subsequent detection. A fundamental work by Fadley et al. (5) laid the foundation for quantification in XPS. This paper outlined for the first time the basic principles of angular distribution experiments in XPS, the methodology for analysis of thin surface films and their effect on quantitative analysis of the bulk sample, and the possible effects of surface topography on the XPS signal. Since then, much progress has been made in characterizing the various contributing factors in the XPS peak intensity equation. Reports of methods for quantitative analysis without standards began appearing around 1977 (6,7). A more comprehensive approach taking a larger number of possible factors into account was introduced in 1982 (8) and has been revised since then (9). The early models typically made the implicit assumptions that the sample composition was homogeneous throughout the volume of sample analyzed, and the surface topography had a negligible effect on the XPS signal. Later modifications have made an attempt to address these two factors in a simplistic fashion. They allow for the possibility of an impurity overlayer's interfering with the signal. The problem of surface roughness in quantification has not been addressed in detail; however, some experimental results have been reported (10) indicating its effect on the determination of overlayer thicknesses. Measurement in XPS The basic measurement process in XPS is outlined in Figure 11. The technique makes use of a photon source of constant energy, typically the Ka Xray lines of aluminum or magnesium, aimed at a surface from which photoelectrons are emitted. The commonly used electron energy analyzer for XPS, the hemispherical analyzer (HSA), consists essentially of two closely spaced hemispherical plates, across which a potential can be applied, so that only electrons of a specified energy may pass through to the detector. The entire measurement process must take place under higher vacuum, e.g., 109 Torr, in order to slow the buildup of ubiquitous hydrocarbon contamination layers, keeping the surface as clean as possible and allowing for greater sensitivity toward the true surface species. The surface sensitivity arises not from the depth of penetration of the incident Xrayswhich may travel 1001000 nm before being appreciably attenuatedbut from inelastic scattering of the photoelectrons. This process accounts for the exponential decay of the photoelectron population as a function of depth, with the characteristic inelastic mean free path (IMFP) defined as the straightline distance an electron population travels before its number drops to 1/e (37%) of 0 0 L O 0 4J c 0( 0 0 cuO COO i4 0. 4 P 0 C to A5 o a 0 0 oco or cu 0 a. 4 te CU 0Q) c) 0)3 ci 00 CO 0 n L 4 El4) ~ ' C LU m CL E 0 U) L. O O 0 <) s a) a) 1r~ to 0 I its original value due to inelastic collisions. For photoelectrons with typical kinetic energies of 5001500 eV, the IMFP ranges from 0.110 nm. Instrumentation The instrument used to collect the majority of the XPS data is a Kratos XSAM 800 Xray photoelectron spectrometer. A block diagram of the spectrometer is shown in Figure 12. This unit consists of an ultrahigh vacuum (UHV) spectrometer console, a sample introduction system, an electronics console, and a computer for instrument control and data manipulation. The spectrometer console comprises a spherical sample analysis chamber, with a water cooled dual anode Xray gun, electron flood gun, macrobeam ion gun, electron energy analyzer, detector system, and a chamber pumping system. The Xray gun furnishes either Al Ka (1486.6 eV) or Mg Ka (1253.6 eV) radiation, allowing differentiation between kinetic energyinvariant processes, such as Auger electron spectroscopy (AES), and binding energy invariant processes, such as XPS. For reasons mentioned below, the bulk of the experimental work was carried out using the Al anode. The Xray gun may be isolated from the analysis chamber and separately pumped with a small dedicated ion pump, to reduce main chamber vacuum degradation. The electron flood gun is used to neutralize the charge buildup on the surface of insulating samples due to electron emission. The ion gun is differentially pumped using a Leybold Heraeus TMP 150 turbomolecular pump, which has a nominal pumping speed of 150 L/s. The ion gun is typically used to sputter clean surfaces __ I V11, 10 for analysis using Ar+ ions, thus reducing the intensity of the ubiquitous hydrocarbon signal to a manageable level; it may also be used as an excitation source in performing ion scattering spectroscopy (ISS), or as an etching source when carrying out destructive depth profiling of a sample surface. The XSAM 800 uses a 1800 deflection double focusing hemispherical electrostatic analyzer with a mean radius of 127 mm and an aberration compensated input lens (ACIL) system. A choice of two analyzer modes is given. In the fixed retarding ratio (FRR) mode, the kinetic energy of each electron entering the ACIL is reduced by a fixed fraction of its initial kinetic energy as it moves through the HSA. In the fixed analyzer transmission mode (FAT), the electron's kinetic energy is reduced to a constant value for all electrons passed to the detector. The relative kinetic energy information necessary to distinguish the electrons energetically is thus contained in the ACIL focusing program to reduce electron energy. The FRR mode gives higher count rates for peaks with low binding energies (high kinetic energies); this region being predominantly the region of study, the FRR mode was used in nearly all experiments. The detector system consists of a fast single channel multiplier and a fast response head amplifier. It can be operated in either a pulse counting mode, which gives a true electron count rate, or a current mode, which acts as a pulse count integrator. The former mode is more sensitive; however, it leaves the electron multiplier susceptible to saturation. For this reason, the current mode was used universally in these studies. The chamber pumping system comprises a rotary vane roughing pump, the dedicated Xray ion pump, the turbomolecular pump (which can also be valved to pump on the main chamber), a titanium sublimation pump, a 200 L/s ion pump, associated pumping manifolds and valves, and several vacuum measuring devices. Pressure in the roughing vacuum lines is monitored by two thermocouple gauges. An ionization gauge is attached to the sample chamber; it 4 6 serves to monitor pressure predominantly in the 10 10 Torr range. For UHV pressure measurements, a cold cathode Penning trigger 6 11 gauge is provided, with an operating range of 10 10 Torr. In order to prevent inadvertent instrument operation, an interlock system has been provided. If the ionization gauge cannot be energized, i.e., the pressure is >10" Torr, the analysis and detection electronics cannot be turned on. Also, if the Xray gun ion pump is not on, or if the anode cooling water flow is not sufficient, the Xray source cannot be operated. The system has a base operating pressure of 11 9 5 x 1011 Torr; typical operating pressures are in the low 10 to 10 high 10 Torr range. The sample introduction system allows a sample to be entered into the UHV chamber from atmospheric pressure in a matter of minutes. It comprises a direct insertion lock with an isolation ball valve and a fast insertion sample probe. The insertion lock is pumped by the rotary vane roughing pump. The sample probe can accommodate a variety of probe tips for analyzing different types of samples. It can also be heated or cooled over a temperature range of 1500C to +6000C. The probe can be rotated on its axis, thus altering the orientation of the sample surface relative to the Xray gun and ACIL entrance slit. The electronics console contains the control units for pressure monitoring and sample temperature setting, Xray power supply, Xray filament emission current stabilizer, ion gun control and raster units, electron flood gun control unit, and control units for both ion pumps and the titanium sublimation pump. It also contains the scan control unit and a display control unit, both of whose functions have been largely usurped by the computer control and display system. The instrument is controlled by a Digital Equipment Corporation Micro PDP11 computer. A Tektronix 4105A color computer display terminal serves as a user interface, and a Tektronix 4695 sevencolor graphics copier provides hardcopy output. The computer has 256 Kbytes of random access memory, a dual 5 1/4" floppy diskette drive system, and a Winchester hard disk system with a formatted capacity of 11.059 Mbytes. The DS800 data system included with the system resides on the hard disk. Backup files can be stored on floppy diskettes, each having a capacity of 400 kbytes. The data system allows computer control of data acquisition, including spectrometer control and extensive data display and manipulation features, such as peak area determination, background subtraction, peak deconvolution, and curve fitting. Lead IonSelective Electrodes Electrodes selective to lead ions date from 1969 (11). The first such electrodes utilized a homogeneous membrane consisting of a co precipitate of lead sulfide (PbS) and silver sulfide (Ag2S) pressed into a pellet, the potential across which can be related to the lead concentration. Other lead ionselective electrodes (ISEs) were soon introduced, including a liquid membrane ISE (12); a homogeneous membrane using lead selenide (PbSe) or lead telluride (PbTe) in place of PbS (13); another homogeneous system consisting of PbS, Ag2S, and Cu2S (14); and several homogeneous membranes with PbS or PbS/Ag2S present at the electrode surface (15,16). Since then, other lead ISEs have been proposed, utilizing PbS/Ag2S on a PTFE/graphite support (Selectrode), PbS deposited on an ionic conducting support material (17), Pb02 on a Selectrode body (18), coated wire type Pb ISEs (19), and PbS/Ag2S/As2S3 glass membranes (20). However, by far the most popular designs have been the homogeneous PbS/Ag2S membrane electrode and, to a lesser extent, the Selectrode body with PbS/Ag2S impregnated on the surface. For this reason, we have concentrated our investigations on this powder mixture pressed into a homogeneous pellet. The number of studies carried out on such membranes is too great to fully document. Several detailed electrochemical studies have been carried out (2123). The electrodes typically show a linear response in a potential versus log apb plot, with a response sensitivity of 2730 mV/decade, close to the Nernstian value for Pb2+ of 29.5 mV/decade. Several common ions interfere with the response of the ISE, including Ag+, Cu2+, and Hg2+, all of which 2 cause surface species deposition. The ions MnO and Cr20 act as oxidizing agents, thus interfering irreversibly with the electrode response. The response tends to degrade over a period of several weeks' usage; however, the electrode can be rejuvenated by polishing its surface, or exposing it to either ammoniacal EDTA or HC104 solution. Relatively few studies have been carried out to investigate the surface behavior of Pb ISEs. Chaudhari et al. (24) were the first investigators to use surface sensitive techniques to analyze this system. They investigated the effects of various cleaning agents on the surface composition. Fixed angle XPS was also used to examine the effects of Cu2+ and Zn2+ on these membranes (25). Scanning electron microscopy (SEM) with an energydispersive Xray spectrometer has also been applied to the investigation of these ISEs (26). A more sophisticated XPS study involving a rudimentary approach to resolving the species concentrations as a function of depth below the surface has been carried out (27). Thus, precedent exists for using surface techniques for investigating Pb ISE membranes. The choice of XPS as the analytical tool of choice can be justified by the nature of such electrode membranes. Their response is governed chiefly by the interaction of the solution being monitored with the electrode surface. An understanding of the surface chemistry is critical to elucidating the mechanism of Pb ISE behavior, particularly with regard to the destruction of this behavior by various agents present in solution. As opposed to other surface techniques, XPS stands out as the method of choice because of its ability to resolve chemical state information of surface species. The PbS/Ag2S system is a complex one, with several different molecular species present at the surface. A mere knowledge of the relative amounts of Pb, Ag, and S is not sufficient to provide a description of the method of membrane operation; one must know the nature of the species which interact with the electrolyte ) I solution. However, the concentration gradients of the important species in the nearsurface region have not been investigated in detail. In addition, the possibility that the sample's surface topography may affect the XPS results must be considered in order for the results to be treated with confidence. These factors have led to the present study. A model for determining the possible effects of surface topography on the nature and magnitude of the XPS signal has been developed and applied to the study of Pb ISEs. In addition, several more general ramifications of the surface roughness effect have been elucidated by a combination of theoretical and experimental investigation. These results are reported in the following chapters. CHAPTER 2 THEORETICAL BACKGROUND In order to present an equation for the XPS intensity, certain assumptions must be made. We initially assume a surface in which elastic scattering and electron transport anisotropies are neglected. Elastic scattering refers to the process where an electron's directional vector is altered with a conservation of energy. Thus, the electron, if detected, will appear in the species peak of interest. More will be said about this factor below. The condition of negligible electron transport anisotropies implies that no preferences are observable for electron motion along a particular direction. This condition is generally met in practice when dealing with polycrystalline or amorphous solids; however, single crystals show evidence of anisotropies which can be used to obtain information about surface bond orientation (28). Also, Xray reflection and refraction are assumed to be negligible. This condition is met for all geometries except grazing photoelectron exit, which is not of interest in the present study. Finally, the Xrays and electrons are assumed to be attenuated exponentially, the decay being characterized by a mean free path (MFP). For the electrons, the characteristic decay length is the inelastic mean free path (IMFP or A). The Xray MFP is typically much larger than the electron IMFP for the electron kinetic energies of interest in XPS. Given these assumptions, the intensity I (E ,y) of photoelectrons from level y of species c in sample s is given as (29) I (E ,y) Io(hv) doY/dg(hv,*) F(E ,EA) T(E ,EA,Q ) SD(E ,E) G(Q ,e) f( N (z) exp(z/A (E )*sin 9) dz. c A o o0 s a Here, I (hv) corresponds to the incident Xray flux of energy hv; doa/dQ(hv,0) is the differential photoionization crosssection of sub shell y of species c, also dependent on Xray energy and the angle * between the incident Xrays and the emergent photoelectrons which can reach the analyzer. It is given by (30) do /do (ac/4w)C1 Bn(Ec)/4 (3 cos2 1)], where ao is the total crosssection of subshell y of species c, C typically measured at 1487 eV in units of the C is crosssection of 13600 barns, and 1(Ec ) is the angular asymmetry factor, a function of the electron energy Ec. The values of aY have been theoretically c determined and reported as a function of atomic number and core level (31), and they agree well with experimental results (3234). Although the photoelectron energy does not depend on the angle between the incident Xray excitation beam and the detected photoelectron stream, the intensity does. The angular asymmetry term takes this variation into account. On the Kratos XSAM 800, this angle is fixed at 800; thus, this factor becomes a constant for a given peak. Theoretical values for n1(Ec) have been reliably calculated for all core levels and tabulated as a function of atomic number (35), and the effects of elastic scattering on the angular distribution of photoelectrons have been studied as well (36,37). The electronoptical factor F(Ec,EA) allows for the effects of deceleration of the photoelectrons from energy Ec to analyzer energy EA. In order to explain why the electrons must be decelerated before being energy analyzed, it is necessary to examine the resolving power of the photoelectron energy analyzer. For a point source imaged into a point image, the resolving power is given by Resolving power AE/E = (W/2R) + Ka2 where W is the slit width, R is the mean analyzer radius, a is the semiangle in the dispersive plane, and K is a constant. For a slit width of 8 mm and a mean analyzer radius of 127 mm, the resolving power is >0.012. Thus, for an electron with 1000 eV kinetic energy, resolution is about 12 eV, far too poor to distinguish between chemical states from a particular core level. By removing energy from the electron before it is energy analyzed, one can improve the resolving power, at the expense of a loss in sensitivity. Two methods of energy removal can be applied. In fixed analyzer transmission (FAT) mode, the transmission energy of the analyzer is held constant, and the electrons are retarded by a varying proportion. In fixed retarding ratio (FRR) mode, electrons entering the analyzer have their energy retarded by a fixed ratio of their initial kinetic energy. The analyzer transmission function T(E ,EAo ) is dependent on electron kinetic energy, the analyzer energy, and the solid angle of acceptance o for the analyzer; D(E ,EA) is the detector efficiency; and G( o,9) is a geometric factor allowing for the change in effective area of sample analyzed. For the Kratos XSAM 800 instrument, the geometric factor is not a function of e when operating in the FRR mode. N (z) refers to the atom concentration (atoms/cm3) of species c at depth z; As(E ) is the electron IMFP, dependent on the sample and electron energy; and 8 is the photoelectron escape angle. The product A (Eo )sin 9 denotes the effective IMFP, the effective depth from which photoelectrons emerge (Figure 21). For a series of peaks collected at various values of 0, the average distance an electron travels before being inelastically scattered remains constant, even though the effective depth below the surface from which the electron emerged varies. The lower the angle of photoelectron exit from the surface with respect to the tangent, or slope, at the point of exit, the smaller will be the depth from which the electron emerged. As will be seen in later chapters, this effect has a profound impact on the nature of XPS depth profiling experiments and the determination of quantitative information from them. The actual determination of the IMFP is difficult to achieve from either an experimental or a theoretical approach. Experimentally, several researchers have published universal meanfree path curves as a function of electron kinetic energy (3841). A power law dependence with an exponent in the range of 0.50.75 is typically reported, although more sophisticated methods of determination exist (41). From a theoretical viewpoint, several workers (4244) have modeled the mean free path of a freeelectronlike material and shown agreement between . I c, 0 to 0. 4 *5 a' . S.C 0  0 . O* "4 S0) 4 cS 0.) CD 0 0 r4 So a' 0 00 Cu C.. r24 *C 11 C 6 0Cu o^( 21; ^0 3 L) (Q .3 4 * r ey E U > their results and experimental measurements. Neither approach yields results of sufficient accuracy to preclude further research in this area, which is ongoing. The uncertainty in the IMFP determination is a chief complicating factor in performing absolute quantitative XPS completely from basic principles. At least part of the uncertainty arises from the unaccountedfor contribution of the elastic mean free path to the overall mean free path, as well as its effect on the XPS signal intensity; the latter effect is related to the variation in the angular asymmetry term discussed above. The general consensus among researchers is that there is a significant elastic scattering effect, but it does not in general affect the exponential decrease of the signal. Rather, it acts to effectively increase the electron mean free path (45,46). The elastic scattering contribution can be reduced by using small solid angles of acceptance at the energy analyzer; it is also more pronounced at smaller values of the photoelectron escape angle (47). The equation shown above indicates that the intensity depends on many parameters and points out what are likely to be some major limitations of quantitative XPS. First of all, a reliable method of estimating the asymmetry parameter, photoemission crosssection, and IMFP must be found, in order to account for these factors' effects on the XPS intensity. As discussed above, extensive work has been directed toward this end. The first two factors are generally recognized to be well characterized; however, research is still ongoing into the nature of the IMFP. Provision has been made for the uncertainty in this parameter in the development of the methods to be discussed in this work. Secondly, surface roughness may have a significant effect on e and on G(Qo,e). Quantitative VAXPS will be generally useful only when the effect of surface roughness may be adequately taken into consideration. One of the purposes of this work is to clarify the surface topography effect, both from a theoretical and an experimental standpoint. The various instrumentdependent functions remain constant in a VAXPS experiment, where only the escape angle is changed. Thus, keeping in mind the possible surface topography effects, as well as the uncertainty in the IMFP, these factors can be collected for a VAXPS experiment into an equation of the form I (9) Kc(9) fo N (z) exp Ez/A (E0 )sin 9] dz. The angular dependence of K (9), if present, can typically be determined empirically. Results from VAXPS experiments may be used to characterize the nature of the intensity I (9) as a function of the c photoelectron takeoff angle e. The use of such VAXPS data in this equation provides an opportunity for obtaining information about the depth profile Nc(z) of the species under investigation. The methodology necessary for extracting depth profiles from VAXPS data is presented in a later chapter, as are several applications. CHAPTER 3 MODELING OF SURFACE TOPOGRAPHY EFFECTS IN XPS Introduction In order to achieve the desirable goal of quantitative analysis of surface species by electron microscopy, one important prerequisite is the characterization of the surface topography of the sample under study. Surface roughness may, depending on its magnitude, affect the relationship between the instrumental setting of the angle of photoelectron escape and the nature and magnitude of the resultant XPS signal. Previous studies of the effects of surface topography on the magnitude and nature of photoelectron signal almost unanimously involved attempts to model the surface roughness as a periodic function. Ebel et al. (48) made use of a model based on closepacked cubes to approximate the effects of surface roughness on the photoelectron signal magnitude of gold films on abraded surfaces. Their rudimentary model took account of photoelectron shading (discussed below), but did not examine the changes in sampling depth occasioned by the topography of a sample surface. A more comprehensive study by Fadley et al. (5) made use of a onedimensional sinusoidal model to predict changes in the average emission angle, photoelectron intensity reduction effects, and the alteration of the average photoelectron escape depth. Although several samples were I analyzed, no systematic experimental investigation of surface roughness effects was undertaken. The change in emission angle was focused upon. Baird et al. (49,50) experimentally verified the presence of surface roughness effects in the XPS analysis of triangularly periodic aluminum diffraction gratings and unidirectionally polished aluminum foil samples. On the basis of their findings, they stated that the effects of surface topography would be nonnegligible if the roughness was at least of the same magnitude as the inelastic mean free path A. More recently, Wagner and BrUmmer (51) investigated four one dimensional models of surface roughness, the two models discussed above as well as a closepacked semicircular model and a rectangular roughness model. No experimental comparison was made. Wu et al. (52) carried out a purely experimental analysis of surface roughness effects on gold films, concentrating on the reduction in photoelectron intensity with increasing surface area, as measured by the Brunauer, Emmett, and Teller (BET) technique. 'They made no attempt to investigate the variation of the XPS signal as a function of escape angle. De Bernardez et al. (53) constructed curves showing the effects of surface roughness on the number of photoelectrons as the escape angle was varied. These curves were constructed using a two dimensional conical surface roughness model, whose photoelectron emission was computed using a Monte Carlo electron trajectory simulation. They compared their theoretical results to the experimental work of Wu et al. (52). The effect of electron shading d on the determination of thin overlayer thicknesses has also received some attention (10,54). Surface roughness effects are important in Auger electron spectroscopy (AES) as well. The importance of induced surface roughness due to the use of Ar+ ion sputtering was recognized early on (55,56) and has generated extensive interest. Surface roughness affects Ar+ ion sputtering by decreasing depth resolution (57), reducing ion sputtering yield (58), and changing the evolution of topography during sputtering (59). Holloway (60) investigated the surface roughness effect on the AES signal, concluding that its presence tends to reduce signal magnitudes and affect quantification, regardless of whether absolute or relative intensities are used. In a recent paper (27), a sinusoidal model of surface roughness using data derived from profilometer tracings showed a negligible roughness effect for the particular samples studied at the experimental resolution. However, the results suggested that a more detailed analysis of the effects of surface topography on the XPS signal was necessary. It has been shown that the depth resolution is lowered by the presence of surface roughness (50). In addition,other effects related to the change in the angle of photoelectron escape are present, and they affect the makeup of the XPS signal. Also, the presence of photoelectron shading acts to reduce the overall area sampled by XPS; one must account for its effects. In this chapter, the theory and methodology of a technique developed for the analysis of surface roughness is presented. Applications of the technique for the analysis of variable angle Xray photoelectron spectral data, as well as a comparison to such experimental data and a discussion of the results, are presented in the next chapter. Theory of Modeling In order to present an equation for the XPS intensity, certain assumptions must be made. We initially assume a surface in which elastic scattering and electron transport anisotropies are neglected. Also, Xray reflection and refraction are assumed to be negligible. Finally, the Xrays and electrons are assumed to be attenuated exponentially, the decay being characterized by a mean free path (MFP). For the electrons, the characteristic decay length is the inelastic mean free path (IMFP or X) described above. The Xray MFP is typically to be much larger than the electron IMFP. In this case, the intensity Ic(E ,y) of photoelectrons from level y of species c in sample s has been shown to follow the form Ic(Ey) Io(hv) daY/dQ(hv,) F(Ec EA) T(Ec,EA' o) SD(E ,E ) G(R ,9) J N (z) exp(z/A (E )*sin 8) dz. Of interest in this chapter are the geometric factor G(Q ,9) and the integral term. A surface with appreciable topography possesses a surface area greater than the apparent area defined by the projection of the aperture onto the surface, due to the presence of peaks and valleys rather than a planar surface. However, only a fraction of this rough surface is typically capable of emitting photoelectrons which may reach the analyzer aperture, due to electron shading of portions of the surface (i.e., valleys) by other areas of the surface (i.e., peaks). This electron shading is an important factor to consider in quantitative analysis, since its magnitude is related to the actual amount of surface area "visible" to the analyzer. The method of determination of the shading fraction assumes that the electrons act as particles traveling in straight lines after exiting the surface. A geometrical approach to the shading phenomenon is utilized, wherein electrons free of the surface and travelling toward the analyzer entrance slits are inelastically scattered if their path causes them to reenter the surface. Within the integral term, the angle of photoelectron escape is affected by surface roughness as well (Figure 31). Note that the photoelectron escape angle is defined for a point on a surface relative to a tangent drawn at that point. Thus, at any point on a rough surface, the actual photoelectron escape angle differs from the instrumental settingwhich is based on the assumption of an atomically flat surfaceby a factor related to the tangent, or slope, of the surface at that point. As mentioned above, the actual value for the sine of the angle of photoelectron escape is directly related to the effective sampling depth. If the species c has a nonconstant depth profile N (z), then the surface roughness will affect the XPS signal by changing the depth from which the characteristic photoelectrons are emerging. Since the concentration of atoms may vary with depth, this change can affect the total number of photoelectrons generated. To summarize, among the effects which surface roughness may exert on the XPS signal intensity, the change in surface area, the presence of electron shading, and the change in the actual range of photoelectron escape depths are prominent. Methodology A program to estimate the magnitude of these factors has been developed. A flow chart is shown in Figure 32. As input, it relies on a digitized vector or, in the more general case, a matrix of values representing the height above or below some defined surface level as a function of displacement along the surface. From these data, a shading fraction Fs, which represents the amount of surface free from shading, is calculated by comparing the height of a point on the surface with the heights of the other successive points along the direction of photoelectron emission, as shown in Figure 33. Specifically, if hi + nb < hi+n n = 1,2,... then that particular point i will be shaded from emitting photo electrons capable of reaching the analyzer aperture. Thus, the program works under the assumption that adjacent, shading areas of the surface do so completely, with no probability of electron trans mission. Here, hi represents the height of point i, b Tx tane, where Tx is the distance between successive height values and 9 is the instrumental setting for the photoelectron escape angle with respect to the plane of the surface, and hi+n refers to the heights of points along the analyzer axis. A "length" of surface along the direction of analysis is computed as the vector distance between points hi and hi+n; this surface length value is used to properly weight the parameters associated with point i. The vector distance L is calculated using the equation I L = 2 + (h h )2 x i+1 This normalization process rests on the assumption that a larger surface area pixel (greater value of L) emits a proportionately larger number of electrons. All the points in the vector or matrix are scanned in this manner, and Es is found by carrying out a weighted sum over all unblocked points and dividing by the sum of surface length values, representing the total surface length along that particular vector. In order to account for the change in the angle of photoelectron escape from the instrumental setting e to the average actual value <9'>, the value of 8i' at each point is approximated (Figure 34). The slope at a point on the surface is estimated to be the difference in two successive points divided by the distance rx separating them. In this manner, 91' 9 + i' where i Arctan(hi hi+ /Tx) A weighted average over all points previously found to be unshaded is carried out for 9i' and sin 8'l, to yield <8'>, distribution of ratio O 0 O r_ 0 ioM OS 0. ,0 4 010 CO O( D L. OW0 0 C5 00 U 4 ) co Q.3 COD 4. rC *^ 0.0. o CO 4, CO O 0 c0 0 . (D Ja cB * 4>, S.g LOJ 41 * 0 > >! 0 C 0 C 4n C L 0 4) I. N 8. l 2 4 N A (D OD CU MO 0 1 Ca 0o W 0} 0 0 40 4)L . C CC o 0 O 0 .o '.aCU 0* (D 0 0C a) 4 0} 40 404 0 m 4 m \ \ % \ %C \ N 1 b b \ '.4 I I I I I N . 4 I ,I CM L **n N I I I I I .I a  ?_: ,ai fm Figure 34. Illustration of the relationship between 9, for the three possible cases. hi> h.i I 1+1 h.= h. I 1M h.< h. I 1.1 general equal to of the average depth analyzed, which has been shown to be directly related to sin 8 for an ideally flat surface, may be obtained by looking at entire rough surface. The program does output the distribution of sin 9( values as well, in order that a more complete analysis may be carried out if called for. In order to determine how the population of electrons which can reach the analyzer entrance slits varies as a function of takeoff angle and surface roughness, it is necessary to take two factors into account. First, the area of each pixel which is unshaded must be determined. Second, the depth below the surface from which the photoelectrons may emerge without being inelastically scattered must be found for each pixel. These two values may then be multiplied together for each pixel to yield its effective sampling volume. Finally, the volume values over all the unshaded pixels are summed. The surface roughness modeling program then reports these values normalized so that near normal exit at 8 890 yields an electron population of unity. The routine for determining the normalized electron population of a surface while taking surface topography into account has little precedent. It provides a valuable means of comparing the normalized peak areas obtained in XPS as a function of escape angle with expected results determined from the surface roughness model. A literature review indicates that little progress has been reported along these lines of investigation. However, in order to adequately account for the effects of surface topography, and thus come one step closer to a truly quantitative XPS methodology, such investigations are called for to provide critical information about electron populations and the factors which go into their makeup. A means of presenting an alternative method of characterizing the nature of the surface roughness is provided as well. The program has the capability of calculating the autocorrelation vector for each height vector taken along the axis of analysis. These vectors are then summed over the entire matrix to give an effective autocorrelation vector for the height matrix in the direction which the photoelectrons must travel to reach the analyzer slits. This approach has been applied to modeling the surface roughness effect on the AES signal (60), where it affects both the excitation and detection processes. Other researchers (61,62) have also made use of autocorrelation functions to characterize the surface roughness profile. If one were to input an experimental roughness vector, this function would provide information about any periodicities in the data. Also, the autocorrelation lengththe average distance over which the structure is correlatedcan be used as a measure of the effective magnitude of the topography; rough surfaces are in general less correlated over large distances than smooth surfaces. An IMSL subroutine is used in the program to perform the actual autocorrelation calculation. For a given height vector in the xdirection (i.e., along the analyzer axis), the average height first calculated. Then the autocovariance vector Axx(j) is calculated for a specified number of point offsets K, which is typically about 10% of the number of points in the height vector. The equation used to calculate Axx(j) is 1 n A x(J) (n) E (hi i1I The variance s2 of the vector is calculated by 2 1 n 2 s (n) Z (h ii The autocorrelation vector is then given by A xx(j)/s2. Finally, the program attempts to estimate the total surface area for a matrix of digitized height values. The approach is to divide the surface into a series of triangles joining three nearspaced points, calculating the area of these triangles, and summing. Better surface area measurement techniques are readily available in cases for which the quality of the data justifies a more quantitative measure. It is simply designed to give another, qualitative measure of the magnitude of surface roughness. The program thus allows an evaluation of the two major factors related to a sample's surface topography which are responsible for varying the XPS signal. The signal can be decreased in magnitude by a certain region of the sample blocking electrons emanating from another region in the direction of the analyzer entrance slits. In addition, electrons from unshaded regions may leave the surface at an angle quite different from the instrumental photoelectron escape angle. This phenomenon leads to a change in the effective sampling depth of the photoelectrons, which is related to these two factors to create the overall electron population as affected by the surface topography is determined by the electron 42 population modeling subroutine. The autocorrelation behavior of the surface matrix along the axis of analysis is also available as a means of characterizing the surface topography. Finally, provision is made for approximating the total unshaded surface area represented by the input matrix of height values. CHAPTER 4 TESTING AND APPLICATIONS OF THE SURFACE ROUGHNESS MODEL In this chapter, the various analytical roughness patterns used to evaluate the performance of the surface roughness model are discussed and the results obtained are compared to the expected values. In addition, several applications of the program output to experimental data are presented. Lastly, the experimental investigations into the magnitude and nature of surface topography effects are presented and discussed in light of the theoretical results. Testing of Roughness Program Two functions which have been applied several times previously to the analysis of possible surface topography effects are the sinusoidal function and a generalized grating function. Both functions are one dimensional in their surface height variation; in fact, all known previous studies have treated the surface roughness problem as effectively a unidirectional one, with one exception. De Bernardez et al. (53) were interested in modeling the effects of damage from a static electron or ion beam, and so they used an axially symmetric coneshaped surface. The rationale behind onedimensional modeling arises from the geometric relationship between the sample surface and the electron energy analyzer entrance slits. The technique of variable angle XPS (VAXPS) does not, and in fact cannot, make use of .I an attractive potential between the sample and the analyzer entrance slits. If such a potential were to exist, then photoelectrons from a large range of escape angles, and thus a large range of depths, would be energyanalyzed and detected in the species peak. In effect, depth and kinetic energy resolution would be severely degraded. Thus, since the axis of sample rotation in VAXPS is perpendicular to the direction which the emergent photoelectrons must possess to enter the energy analyzer, essentially only one dimension of the surface roughness that dimension perpendicular to the axis of rotation and in the plane of the sample surfacehas any effect on the resultant XPS signal in the limit of an infinitesimally narrow analyzer entrance slit. For both the sinusoidal and grating functions, the shading fraction, vector length, and values related to the photoelectron escape angle can be predicted algebraically as a function of the ratio of amplitude to wavelength for the periodic function. A definition of these parameters, as well as their method of calculation, was presented in the previous chapter. The ability to compare program results with the analytic solutions made these functions excellent candidates for testing the accuracy of the surface roughness model's output. Sinusoidal Function Results For the sinusoidal function S(x) given by S(x) A sin (2wx/A), where A is the amplitude and A is the wavelength, the shading fraction Fs is given by fx2 "1 + [(2wA/A) cos (2rx/A)]2 dx F  s 2 J1 + [(2wA/A) cos (2wx/A)]2 dx where x1 = (A/2w) cos1 [(A/2wA) tan e] and A cot 9 sin (2wx2/A) x2 = A cot 9 sin (2rx /A) x A. The instrumental setting for the photoelectron takeoff angle is given by 9. It can be deduced from the above equations that no photoelectron shading occurs if (A/2wA)tan 0 > 1.0. Once x1 is obtained from the first expression, it can be substituted into the second expression to obtain x2. These values can in turn be substituted into the integral equation to yield the shading fraction. Although this integral form does not possess an analytic solution, it can be numerically integrated to any desired precision to give values for Fs algebraically. Calculation of the parameters related to the photoelectron escape angle 9 from first principles is more complex. Essentially, the procedure involves the determination of the surface tangent for a vector element dx, multiplying it by the length of dx, summing over all the unshaded vector elements, and dividing by the overall unshaded length. Again, numerical integration techniques are required to obtain an algebraic solution to the problem. Comparison between the analytic results and the program output for the onedimensional sinusoidal function can be seen in Figures 41 and 42 for several different wavelengths (holding the amplitude constant) over the range of escape angles such that 100 9 < 89. The curves in Figure 41 represent the true, algebraically arrivedat values for F,, while the points represent the roughness program results for a sinusoidal vector consisting of 50 wavelengths digitized to 5000 points. As A is increased relative to the amplitude, the surface topography becomes less pronounced. Thus, A 0.1 represents an analogy to a very rough surface, while A 20.0 corresponds to a smooth, essentially flat surface. In general, the agreement is quite satisfactory; relative errors are greatest at very low values of Fs, where the program must make use of a small number of points in its calculation. This error can in principle be reduced by increasing the digitized resolution. However, constraints on the memory available for array storage during execution of the program make such error reduction difficult to achieve. The general trend followed by Fs as a function of escape angle can be seen. As 9 is increased, more of the surface has the direct lineofsight to the analyzer entrance slits necessary for emitted photoelectrons to be collected. Thus F. increases as 9 is increased. At some cutoff angle 9c, the surface is fully visible to the analyzer, with no shaded areas. At this and all higher escape angles, the shading fraction is unity. For the 0 4. )L. Oic (0' 4 41 , 00 C 0 010 60 4 C 3 LC 0a, 0 (0 c 0 CU 00 00 cal bC O 0 41 0 M> 5, .C3 Se SF O>i 48 + * S 4 C I **. e I I i c < III *D '** t. IS S 4.o i '0 .. C I. ' 0 1 c \ * ~h i, ii ~ \ S N l t S S CO 0 bO 0 90 O (a 4) 0 4 0 O a, 4 C, O 0 (, O 1. 1d f,. bO  CD 0 0 + to tD II i II 2 5 *.'. 1 'M N < '4 S * .5 SI c< *1 + 4. 1 (') Cd '4 SQ 'I 4. *2 4 I' I I I I I 3 S; & :r a * S i45 I: + I 1 1* * i +4 I I 4 sinusoidal function, the cutoff angle 89 increases as the wavelength decreases; for a rougher surface, the shading effect is apparent over a greater range of 9 values. Finally, the curves demonstrate that not only is there shading over a greater escape angle range for a rougher surface, but the shading is greater, apparently allowing less of the actual emitted photoelectrons to make their way to the analyzer entrance slits. In Figure 42 are several curves showing the variation in the ratio R photoelectron takeoff angle, which differs from the instrumental setting sin 9 for rough surfaces. Results for the same four sinusoidal vectors as those shown in Figure 41 are illustrated here; again, the Curves represent an analytic determination of R, while the points correspond to program output. Agreement is in general better between output and true values. This value is not quite as sensitive to the use of small numbers of points in its calculation as the shading fraction. For a perfectly smooth surface, R is equal to unity. Over much of the range of escape angles, R is close to unity for the analogous smooth vector where A 20.0. Only below the cutoff angle 9c does it begin to show an increase. For the lower values of A corresponding to rougher surfaces, a general decrease in R is observed as 9 is increased, with the curve leveling out beyond 9G. Note that at low values of 9, the ratio is in general greater than unity, while the reverse is true at high values of 9. This observation has ramifications for the determination of the overall sampling volume to be discussed below. Grating Function Results The second analytic test function, a grating function, was chosen for several reasons. It has been previously studied by several researchers. Also, its use addresses the ability of the surface roughness program to deal with discontinuitiespoints of indeterminate slopein the topography of a surface. Finally, this function is not in general isotropic in its roughness along the direction of measurable photoelectron exit. It has a different surface area facing toward the analyzer from the side facing away from the analyzer, a category into which certain real world surfaces may fall. For the onedimensional grating function, analytic solution can be accomplished as well. The variable parameters are the primary and secondary blaze angles p and Y, respectively; the baselinetopeak distance is fixed at 2.0. The wavelength A is then given by A 2.0 (sin p + sin Y). If the photoelectron escape angle is greater than the primary blaze angle p, then the surface is fully illuminated and Fs 1. Otherwise, one must use a geometric approach to the determination of Fs. Recognizing that the function has a peaktotrough amplitude of 2.0, the initial cutoff point x1 is given by x 2.0/tan p. By drawing a series of right triangles and using elementary analytic geometry, the second cutoff point x2 is determined as x + A + 2 cot e (tan Y/tan p) x2 1 + cot 9 tan Y The equation for Fs turns out to be F (1/L) (x x )2 + (h 2.0) , where the surface length L across one wavelength is given by L 2.0 (1/sin p + 1/sin Y) and the height hx at point x2 equals hx 2.0 + 2.0 (tan Y/tan p) x2 tan Y. The same basic procedure used for calculating the parameters related to the angle of photoelectron escape in the case of the sinusoidal function is again used for the grating function. Comparison between analytic results (curve) and program output (points) is shown for the grating function in Figures 43 and 44. For the program's input height vector, 20 wavelengths were digitized to give 1000 points. As the blaze angles increase, the effects associated with increased surface roughness increase as well. The same general trends observed for Fs (Figure 43) and R (Figure 44) as a function of photoelectron escape angle in the sinusoidal functional case are reproduced here as well, lending an air of generality to such behavior. In this case, the cutoff angle is easily seen to equal the primary blaze angle; only below this angle can shading occur. 4 (0 0O 0 0. ) 0 I 4 4C L 8 ) E4 to 0 L H 0.0 00 C OC 0 0 cd 9 o 4 4 Vo o mCCU L a44 0 00 L C OH 0 0 (B S0 0 0 4. a 0 .C5 4 cu (0 0. r L4 I. wd O^ D ~d 60aL cr4~c Ebl I s 4. S \ C \ N J i A^ '.4. 1* 4 I. *3 4. + im (I ) ! + N I J * t s .4. )i N C ci '9' U) 1n * +1 Id Nt 14 IL N' ^l" * k iL O N 0) L. 0 (D 4) t, 0 r' b4 I 0 C4 0 0 C* (D 03 A r 44 w m4 0 0 o 0 0 00 4 *Se 0) ct s + 4. t 4. I o" V. + k i N c J! m e I, oD r tt ci 4. + 4. + II 1b N 1i ('a S~C I I I I I i Agreement between the program output and the analytic values is better for the grating function than the sinusoidal function, primarily because the pixel slope, upon which several key calculations are based, does not vary continuously in this case, but can take at most only one of two discrete values. Note also that the slope of the curves near 9, is sharper in this function. This factor arises from the periodic point discontinuities in the roughness vector. The greater the blaze angles p or Y, the more the surface vector behavior deviates from that for a smooth surface; increasing p causes an increase in 9c, while increasing Y reduces Fs and increases R. Particulate Surface Model Results and Applications The surface roughness program was first used to test for any difference in the effects of surface topography on two different surfaces. As experimental data, powder size distributions derived from scanning electron micrographs were compiled. In order to transfer these data to a form suitable for input into the program, the particulate surface model (PSM) was developed. The PSM is a simple approximation to the actual surface topography vector. At present, it is strictly onedimensional in its approach. The inherent assumptions are as follows: 1) Surface roughness effects on the determination of the shading fraction and the effective escape angle parameters are predominant primarily along the analyzer axis. 2) Given a particle size distribution, it is assumed that the particles are closepacked along the analyzer axis. 59 3) The maximum height of each particle above a surface baseline is modeled by taking a random fraction of its radius. 4) The particles are assumed to be spherical, orto be more preciseto give a circular crosssection along the "slice" of surface which is analyzed by the program. The resulting model surface resembles that shown in Figure 45 (a). This surface is then digitized to give a vector of height values in Figure 45 (b). Given a vector representing the randomized fraction of each particle's radius, the height h as a function of surface displacement p for a particle with diameter R is given by h 0.5 /4d (R d) m, where d R/2 F q C/2 p C 4F (R F) m R/2 F F (R/2) RND (0,1), as seen in Figure 46. The function RND (0,1) represents a random number generator with output s such that 0 < s < 1. The displacement pn from the edge of particle n is related to the total displacement x by n1 p = x E F . i=1 Figure 45. Illustration of a portion of surface height vector obtained from the PSM: a) construction of the vector from a close packing of particles; b) result of digitization, yielding a suitable input vector. Particulate Surface Model Digitized Height Values .,rlnrll,,,,,,,.,,lll,,.lllliilllili.il co 4 0 bO CO 0 4) 4) (D C 0 0 0 4 0o 0 0 I 0 1) CU 0I 0 0 4 0} 0 0 0 4 L c0 b ao 0r 60a eeIp~ Ioo4 LL .  cr r The sample powders whose surface topography effects are to be compared are homogeneous Pb ISEs manufactured from a coprecipitation of PbS and Ag2S. Two slightly different powder preparation techniques have been used. The general experimental procedure is to dissolve Pb(NO3)2 (MCB Chemicals, Reagent grade) and AgNO3 (MCB Chemicals, Reagent grade) in deionized water in amounts such that 2np nAg. A 0.1 M Na2S solution is then prepared. In the first procedure, the Na2S solution is added in excess to the mixture of Pb2+ and Ag+ salts, while stirring at room temperature. The second procedure is similar; the mixture of Pb2+ and Ag+ salts is added to an excess of Na2S solution. The resultant precipitates are isolated and washed several times with 0.1 M HNO3 to remove any excess sulfide. The powders are finally washed with deionized water, filter dried overnight, dried in a 1100 oven for 24 hours, and stored in a dessicator. The scanning electron microscopy (SEM) photographs shown in Figure 47 were taken on a JEOL JSM35CF microscope operating at 25 kV accelerating potential and 20,000X magnification. These micrographs, which are representative of those for each powder, correspond to powders prepared using procedure 1 (Figure 47 (a)) and procedure 2 (Figure 47 (b)) described above. Certain general differences between the two powders may be noted. The particles from the first procedure appear to be larger as a rule and more widely scattered in size. These indications are borne out by an analysis of the particle size distributions, shown in Figure 48. The skewedness of the distribution in Figure 48 (b) may be due in part to the lower resolution of the SEM photo in Figure 47. Scanning electron photomicrographs of two PbS/Ag2S powders: a) preparation by procedure 1, and (b) preparation by procedure 2; both of which are described in the text. The bar at the bottom of each photograph corresponds to 1 um. (A) (B) Figure 48. Particle size histograms, showing the number of particles as a function of particle diameter, derived from the SEM photomicrographs shown in Figure 47: a) histogram of particle sizes from the SEM photo in Figure 47 (a); b) histogram of particle sizes from the SEM photo in Figure 47 (b). SEM PHOTO 969S 111 : ..II 8 12 15 ll. ll EI..  I ~ ~`~ ~ ~I. ~ 18 21 24 27 38 33 42 45 48 4 1 PARTICLE DIAMETER SEN PHOTO 0065 18 .1 PARTICLE I Il EI m m M3DETER 338 CRO X 2F86 DIRIETER MICHROH X 209) 8 12 15 atrurrcIL Figure 47 (b). The presence of any smaller particles would only strengthen the argument that the average particle size differs in the two photomicrographs. In the particle size distributions determined from the two micrographs, photo 0005 (procedure 2) shows a mean diameter of 0.13 um, while the particles in photo 0009 (procedure 1) possess a mean diameter of 0.15 um. If we make the assumption that each particle population follows a Gaussian distribution in particle diameter, then the hypothesis that the second mean diameter is not greater than the first mean diameter can be rejected at a 99.5% confidence level. The histogram shown for photo 0005 probably underestimates the sample standard deviation and overestimates the sample mean. However, this hypothesis rejection holds even if we use the larger of the two sample standard deviations as a measure of the population standard deviation, as opposed to pooling, or if we lower our estimate of the first sample mean. These powder distributions were then used as the input vector for the PSM and an analysis of the comparative surface roughness effects was carried out. The comparison results are shown for the shading fraction Fs in Figure 49, and for the ratio R Figure 410. It is immediately apparent that the difference in particle diameters is not sufficient to cause a noticeable difference in their XPS signals attributable to the differing surface topography. A statistical comparison between the results for both powders supports this statement. However, both powders are expected to show appreciable surface roughness effects in their respective VAXPS signals, especially at low values for the photoelectron escape Figure 49. Plots of the shading fraction F as a function of photoelectron takeoff angle 8 (deg), derived from the SEM photomicrographs shown in Figure 47: a) F versus 9 plot for the powder depicted in Figure 47 (a3; b) F versus 9 plot for the powder depicted in Figure 47 (bT. 71 S ) 75. **  .. .5 .' PHOTO e889 .25 30 THETA 6"' 90 I. (B) F. 5 .*' PHOTO 8885 .25 fl_1 "_ 3B THETA 68 90 Figure 410. Plots of the ratio R takeoff angle 9 (deg), derived from the SEM photomicrographs shown in Figure 47: a) plot of R versus e for the powder depicted in Figure 47 (a); b) plot of R versus e for the powder depicted in Figure 47 (b). I (A) PHOTO 8889 '.. 38 THETA 6 98 B>) PHOTO 8885 DJ T T 68 9 2.5 R 2 1.5 1 .5. 2.5 R 2 1* .5 + * *. Y r \ 36 THETA 6 98 angle. Also, note that no sharp transitions can be observed in the curves. As has been previously seen, such transitions arise from periodicities in the surface roughness vector; since the PSM produces an essentially periodic roughness vector, as most randomly rough surfaces would be expected to produce, this observation is expected. Thus, the PSM results provide several useful pieces of information. They more closely approximate the expected behavior of Fs and R of randomly rough surfaces as a function of e. In the present case, the results allow the conclusion that the two methods of Pb ISE preparation do not produce different surface topographical effects. This factor is of importance to the discussion of Pb ISEs in a later chapter. Finally, and of more general interest, the PSM provides a means of estimating the surface roughness effect of a powder sample using particle size distributions readily available from SEM photomicrographs. Experimental Surface Roughness Studies An analysis of the effects of varying surface roughness has been carried out on two surface chemical systems of relative simplicity. The first system consists of sputtercleaned gold surfaces. These surfaces are essentially free of significant levels of contamina tion. Therefore, their normalized peak areas as a function of photoelectron escape angle can be used to characterize the instru mental response of the XPS instrument. The Au 4f7/2 peak at 83.80 eV is also commonly used to standardize the instrumental binding energy response. The second system is a carbon sample with measurable surface oxidation. Such samples are more readily available than gold samples, and they provide easily observable evidence of the contribu tion of oxidized species to the overall carbon concentration. Gold is relatively inert, and the Au 4f7/2 peak does not show the presence of oxides. The value of being able to resolve oxidized and unoxidized carbon arises from the information about the electron sampling depth available from such measurements. If the sampling depth is relatively small, then the carbon peak envelope shows a significant contribution from oxidized carbon species. For greater sampling depths, the bulk graphitic carbon contribution is predominant. Results of Gold Studies Initial studies using gold samples focused on the determination of the response function of the XPS instrument. In order to minimize surface topography and vertical inhomogeneity effects, a gold foil sample (Alfa Chemicals, 99.9+% purity, 0.25 mm thick) was sputter cleaned until the residual carbon contamination peak was minimized in intensity. Source power was 300 W (15 kV, 20 mA), and a small solid angle of detection (LO magnification mode) was used. The peak area of the Au 4f7/2 and 4f5/2 peak doublet was determined after subtraction of the background signal (Figure 411) using an iterative technique developed by Shirley. Peak area is calculated using the equation AT (count) Energy step size (eV/ch) AN (counteV/sec) Dwell time (sec/ch) ' where AT represents the sum of counts in all the channels in the peaks after background subtraction, the energy step size is an instrumental setting defining the ultimate energy resolution of the 4 V 04 0 00) 0 0 0 O 4 0 0 0 * Cu, C0 c io 0 LH 0 lO 0 S 04) 40 0 ( P19J (0 L4g Q)J 0 ^ e o  o t > de j *P O O (B 05 ( L << ( 0} 0 r ST 3) 60 cW Cu 4. CK 8 r (f (S,4U On) RiA4;suuI L w 5, spectrum, and the dwell time is given by Dwell time = Scan time (sec) # of sweeps Step size (eV/ch) (sec/ch) Energy scan range (eV) Since the Xrays used are not strictly monochromatic, satellite lines are visible in this spectrum. The satellites do not interfere with the peak area determination in this instance. The area was determined for a series of photoelectron escape angle settings ranging from 900 to 50. This area was then normalized by dividing by the largest peak area value for the range of escape angle settings. The results of these variable angle plots are shown in Figure 412 for both Mg Ka and Al Ka radiation. The decrease in peak area at 9 900 when using the Mg anode results from the arrangement of the anode faces. The anode tip is wedgeshaped, with the upper side aluminumcoated and the lower side magnesiumcoated. For 9 900, the sample surface is approximately even with the tip of the wedge and, thus, does not receive the total Xray flux when using the Mg anode. Since data taken at 900 were to be used in the surface roughness studies, the Al anode was used throughout the investigation. In order to ascertain the basic effects of increasing surface topography on the results of VAXPS experiments, two other gold samples were prepared and analyzed. As a model for the behavior of a very rough surface, a gold powder (Alfa Chemicals, 99.995% purity, 20 mesh powder size) was sprinkled onto the surface of conducting copper tape, which was then affixed to the sample holder using colloidal silver adhesive. In this manner, a rough gold surface could be examined without any of the sample charging effects due to the use of 0. 0 4 .o Cd l.0 a) 0} C 0O rC 0 0 O) 90 * * o.o 4) C 0.00 Le 0. O) bo n < 0 W O. C S. W 0. V0. O^n xo, 4 C. 4 3O 0 snb U 4 L t .  S S . * * * * S 'S Sx * S S * * * . 55. S*. ^. .. S o .^ *5 I I I L : I  x + 'W~ON _ * a m common adhesive tape. In addition, this powder was pressed at 7 psi into a pellet to be analyzed as well. This pellet was analyzed to determine the general surface roughness behavior of a sample prepared using a common, simple hand press which is often used in sample preparation for routine XPS. Operating conditions were the same as above, and the VAXPS curvesplots of total doublet peak area as a function of photoelectron escape angle 9for the foil, pellet, and powder samples are compared in Figure 413. In this graph, all the peak areas were normalized only to account for any differences in the peak collection times. Thus, the overall peak areas as a function of escape angle are compared for the three samples. Several noteworthy changes are apparent as the surface roughness increases in the order foil < pellet < powder. First, the doublet peak area drops off significantly at high values of 9, then actually increases as 9 is decreased. As the surface roughness increases, the escape angle where the maximum peak area is obtained decreases. In addition, in the range of low e, the peak area is greater for rougher surfaces. These two factors combine to lower the range of peak area values over the range of escape angle settings. Whereas the peak area for the foil increases by 2.5 orders of magnitude over the range of angles, only an order of magnitude increase is seen for the pellet sample, followed by a decrease at higher takeoff angle settings. For the powder sample, the peak area increases only by a factor of 3 before decreasing at higher 9. Obviously, this observation has important ramifications when one is varying the angle setting in order to maximize the XPS signal. In addition, the tacit assumption that surface roughness C C 00 ol X4 e_ 0 ( Q 1 0 #.0 o M 8,. 0 LtO 60 00 Q. 0,> .3 0 C 4 0 0 010 c3 4) 0) W WO 0} 0 0 0 00 bO .,40 1. r. CO ^ 1 ^ (0 S < Vl 0 SC: Q. (0 0) + X + 0 4a I I I + X a ~D + 0 x + a x + a x + 0 x+ 0 X+ 0 X I "I I ci 1 6 1 ~ 1 1 1 Q b d) ( CD aG ( zCD O0 a) to CDU CD CD CD C OC> M I0 CD G U) co b) G N c % qW effects are negligible with pressed pellet samples is shown to be false, due to the difference in VAXPS signal between gold foil and compressed powder pellet. This point forms the basis of a study of the effect of pressure on the XPS signal, presented below. Finally, the effects of surface roughness may be quite dramatic, as shown in the VAXPS response curve for the gold powder sample. Results of Carbon Studies: Model Comparison The first series of experiments investigating a graphitic carbon surface were designed to test the predictions of the electron population model which was developed as a routine in the surface roughness program. As discussed above, this model outputs an approximation to the normalized VAXPS curve as a function of the topography of the input surface height vector or matrix. The predictions of the model about the changes in the VAXPS curve are compared here with some experimental studies of carbon surfaces of various roughness. Experimentally, the samples studied included a carbon foil (Goodfellow Metals, 99.8+% purity, 0.125 mm thickness) which, although smooth, is not as smooth as the gold foil sample. Also, studied were two carbon powder samples (Alfa Products, 99.5% total purity, 325 mesh and 20 +60 mesh). For the 325 mesh powder, all particles are smaller than 44 pm in diameter; for the 20 +60 mesh powder, particle diameters range from 250841 pm. However, for the latter powder, the particles flake together, forming smooth sections of surface. Both powders were sprinkled onto conductive copper tape, then affixed to the sample holder using colloidal silver. The carbon powder (325 mesh) sample forms a surface with appreciable topography; it serves to act as a very rough surface. The carbon flake (20 +60 mesh) sample consists of disjointed sections of smooth surface; its surface roughness is intermediate between the powder and the carbon foil. The instrumental operating conditions for data collection were the same as in the case of the gold samples. For all three samples, a series of VAXPS experiments was carried out by varying the instrumental setting of the photoelectron takeoff angle 0 in the range [50,90]0. The C 181/2 peak areas were normalized in each case so that at the angle where the peak area was maximum, the area had a value of unity. The electron population model results over the same range of 0 were compared for a series of wavelengths using the one dimensional sinusoidal function and matching the best approximations to the experimental results. The curves are compared for the three samples in Figure 414 (carbon foil), Figure 415 (carbon flakes), and Figure 416 (carbon powder). The average percent difference between model and experimental points is 2.34% for the carbon foil, 4.57% for the carbon flakes, and 1.81% for the carbon powder. The carbon foil shows a response very close to an ideally flat surface. In fact, the percent difference between the carbon foil and a model flat surface is 2.42%. Thus, the carbon foil serves to act as an experimental flat surface standard. Note the good agreement between model and experimental results. As is expected, the wavelength of the sinusoidal function which best matches the experimental curve increases as the roughness of the sample decreases. That is, a smoother model surface vector is required to fit a smoother Figure 414. Comparison of theoretical electron population and experimental VAXPS response curves for a ruffle function (A 20.0) and a carbon foil, respectively. + RUFFLE FUNCTION + A 1.8 A 2e. 0 5 38 THETA 6 9S I. 25 1 .75 1 .5o .25 4 + 6+++ THET ,, 98_= SFO L SAMPLE JO THETA 6S 9Q Figure 415. Comparison of theoretical electron population and experimental VAXPS response curves for a ruffle function (A 10.0) and a flaky carbon sample, respectively. i. RUFFLE FUNCTION A A * 1 . 10.0 a38 THETA 6 8 98 .+ FLAKE SAMPLE 4 4 t" _ I I .75. .5. .75 .5. .2S 14.4.4. 4. I 3S THETA 68 90 Figure 416. Comparison of theoretical electron population and experimental VAXPS response curves for a ruffle function (A 0.2) and a carbon powder sample, respectively. .RUFFLE FUNCTION + A 38 THETA 68 1' POWDER SAMPLE 39 THETAr 68 4. 4. i.e 0.2 25+  I 98 .75% .25+ Sn  90 experimental surface. The uneven nature of the theoretical curve in Figure 416 is attributable to the small shading fractions for this very "rough" surface. The results are obtained by averaging over a relatively small number of points and are, thus, more subject to random error. Even though the surface roughness varied dramatically among these samples, they all show somewhat similar VAXPS curves. This phenomenon is due to the offsetting nature of the two effects of the surface topography on the magnitude of the XPS signal. The first effect is the shading fraction F variation. For a rough surface, the shading fraction is low, and a smaller signal is expected. However, for this same rough surface, the electron population model predicts that the regions of the sample which are capable of emitting analyzable photoelectrons emit proportionately more photoelectrons. An examination of Figure 417 helps to explain this result. Here, the relative population of analyzable photoelectrons emitted by each surface pixel is plotted in bar chart form directly below that pixel for several escape angle settings. For this portion of a hypothetical surface roughness vector, the electron population is relatively evenly distributed for 8 89. As 8 is decreased, portions of the surface are shaded, but the unshaded portions emit a greater number of photoelectrons. At higher instrumental takeoff angles, the decrease in the signal due to a lower shading fraction is more important, and an overall lowering of the signal is observed. At lower escape angle settings, the increase in the sampling depth is more important than Figure 417. Plots of the relative populations of emitted photoelectrons from each pixel of a small portion of a surface height vector for several different values of the instrumental takeoff angle e. The vertical scale is in arbitrary units. SURFACE SLOPE EFFECT ON X SURFACE SLOPE EFFECT ON A SURFRCE SLOPE EFFECT ON X 