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Photoelectron experiments and studies of X-ray absorption near edge structure in alkaline-earth and rare-earth… Gao, Yuan 1994

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PHOTOELECTRON EXPERIMENTS AND STUDIES OF X-RAYABSORPTION NEAR EDGE STRUCTURE IN ALKALINE-EARTH ANDRARE-EARTH FLUORIDESByYuan GaoB. Sc., The University of Science and Technology of China, 1985M. Sc., The University of British Columbia, 1990A THESIS SUBMITThD IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994© Yuan Gao, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the IJbrary shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpubhcation of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_________________________________Department of F H IS I C SThe University of British ColumbiaVancouver, CanadaDate APRIL ZI, i994DE-6 (2188)AbstractAlkaline-earth fluorides and rare-earth trifluorides possess technologicalimportance for applications in multi-layer electronic device structures and opto-electronicdevices. Interfaces between thin films of YbF3 and Si(l 11) substrates were studied byphotoelectron spectroscopy and x-ray absorption spectroscopy using synchrotronradiation. Results of YbF3/Si(l 11) were compared with those of TmFgSi(l 11). Whileelectrons in the Si valence band are prevented from occupying the empty 4f levels inTmF3 at the interface by the on-site Coulomb repulsion energy, the charge transfer fromSi to YbF3 is possible because the totally filled 4f states in Yb still lie below the Sivalence band maximum.The theory of x-ray absorption near edge structure (XANES) is incomplete exceptfor a few particularly simple special cases. A Bragg reflection model was developed toqualitatively explain the oscillations in XANES, in terms of the scattering of thephotoelectron wave between families of lattice planes as set out by the Bragg conditionfor backscattering. The model was found to represent the data for systems with nearlyfree electron like conduction bands reasonably well.High resolution CaF2 fluorine K edge XANES was used as a prototype tounderstand XANES in more depth on systems with strong core hole effects. Unlikeprevious work which involved multiple scattering cluster calculations that include onlyshort range order effects, both the long range order and the symmetry breaking coreholes are included in a new bandstructure approach in which the core hole is treated witha supercell technique. A first principles calculation with the use of pseudopotentialssuccessfully reproduced all the main features of the first 15 eV of the fluorine K edge inCaF2 which had not been explained with the cluster calculations. A comparison of the11theoretical and experimental fluorine K edges in CaF2 and BaF2 was used to identify thestructure related features. The possibility of multi-electron excitations being responsiblefor higher energy features in the XANES was investigated by comparing the energy losssatellites in the fluorine is x-ray photoelectron spectra with features at correspondingenergies in the fluorine K edge absorption spectra. Finally the fluorine K edges in therare-earth trifluorides LaP3,CeF3, NdF3 SmF3, EuF3 DyF3 and YbF3 were explored forthe first time with the high resolution x-ray absorption spectroscopy. The near edge partof the fluorine K edges in all seven rare-earth trifluorides was found not to be dominatedby the Bragg peaks because of the short life time of the photoelectron and the low crystalsymmetry.111Table of ContentsAbstract iiTable of Contents ivList of Tables viList of Figures viiAcknowledgements xiiChapter 1 Introduction 1Crystal Structures of the Alkaline-earth Fluorides and the Rare-earth 4TrifluoridesChapter 2 Apparatus 102.1 The Ui Beamline at NSLS 102.2 Thin Film Deposition 14Chapter 3 Photoelectron Spectroscopy 173.1 Experimental Arrangement of Photoelectron Spectroscopy 173.2 Photoelectron Spectroscopy of 4f Levels at the YbF3/Si(1 11) and 22TmF3ISi(1 11) InterfacesChapter 4 X-ray Absorption Spectroscopy 37iv4.1 Experimental Arrangement of X-ray Absorption Spectroscopy 374.2 Fluorine K edges of Alkaline-Earth Fluorides 394.3 Bragg Reflection Model for XANES in Crystalline Solids 43Case Study 59Copper 60Iron 67Silicon 69Neon 73Calcium Fluoride 764.4 Bandstructure Calculation of XANES at the Fluorine K Edge in 82CaF2 and BaF24.5 Multi-electron Excitations in XANES 1054.6 Fluorine K edges of Rare-earth Trifluorides 110Chapter 5 Summary and Conclusions 135Bibliography 138VList of TablesTable Page4.1 The reciprocal lattice vectors for f.c.c. copper. 614.2 The reciprocal lattice vectors for b.c.t. copper. 674.3 The reciprocal lattice vectors for b.c.c. iron. 684.4 The reciprocal lattice vectors for silicon with non-zero structure 70factors.4.5 The reciprocal lattice vectors for solid neon. 744.6 The reciprocal lattice vectors for CaF2. 784.7 The Fourier coefficients of the crystal potential U used in the 90calculation of the fluorine K edge absorption in CaF2.4.8 The Fourier coefficients of the crystal potential U used in the 100calculation of the fluorine K edge absorption in BaF2.4.9 The bias potentials applied on the front grid of the detector in 110obtaining the fluorine K edges of the seven rare-earth trifluorides.4.10 The photon energies of the features labeled in Figures 4.27-33. 1234.11 The loss energies of the satellites of the F is main photoemission 132peak in the XPS spectra in Figures 4.35-41.viList of FiguresFigure Page1.1 The fluorite structure. 71.2 Two projections of the hexagonal cell of LaF3, 81.3 The orthorhombic YF3 structure. 92.1 The theoretical resolution curves and the measured resolution data 11points of the ERG monochromator with gratings Gi, G2, and G3.2.2 Top view of the experiment station of the Ui beamline at NSLS, 12Brookhaven National Laboratory.2.3 The schematic diagram showing the configuration of the 15evaporator used for thin film deposition.3.1 The schematic diagram illustrating the relative energy levels in 21equation (3.1.6).3.2 Valence band region of TmF3 on Si taken at two different photon 24energies A and B.3.3 Valence band region of YbF3 on Si taken at two different photon 25energies A and B.3.4 The schematic diagram showing the binding energy distribution of 28the 4f12 — 4f transition.vii3.5 The schematic diagram showing the binding energy distribution of 29the 4f13 —* 4f1 transition.3.6 The schematic diagram showing the positions of the cation 4f 30levels relative to the Si valence band and the F 2p valence band forTmF3 and YbF3 on Si(1 11) substrates.3.7 Photoemission spectra of YbF3 films of two different thicknesses 35as deposited on room temperature Si(1 11) substrates.3.8 Photoemission spectra with the resonant photon energy of 182 eV 36taken on a 10 A YbF3 film deposited on Si(1 11) before and after a1 minute anneal at about 400 °C.4.1 The experimental absorption spectrum of the fluorine K edge in 40CaF2.4.2 The experimental absorption spectrum of the fluorine K edge in 41SrF2.4.3 The experimental absorption spectrum of the fluorine K edge in 42BaF2.4.4 The free electron wave length as a function of the free electron 45kinetic energy.4.5 The calculated electron inelastic mean free path (IMFP) in LiF by 46Tanuma et al using various algorithms.4.6 Constant energy surfaces in k-space with one Bragg plane. 51viii4.7 One-dimensional schematic diagram showing examples of the 54Fourier coefficients UG being complex.4.8 The ratio of the absorption coefficient for the model potential with 56various values to the absorption coefficient for the unperturbedplane waves.4.9 The absorption ratio with complex U. 574.10 The absorption ratio with various magnitudes of U, assumed to be 58real and positive.4.11 X-ray absorption at the K edge of f.c.c. copper. 634.12 X-ray absorption at the K edge of b.c.t. copper. 644.13 X-ray absorption at the K edge of b.c.e. iron. 664.14 X-ray absorption at the K edge of crystalline silicon. 714.15 X-ray absorption at the K edge of solid neon. 754.16 X-ray absorption at the fluorine K edge in CaF2. 794.17 The schematic diagram showing the super unit cell. 874.18 The experimental fluorine K edge absorption spectrum for CaF2 is 94shown at the top and the calculated absorption spectrum is shownat the bottom.4.19 Calculated fluorine K edge absorption spectra for CaF2with 95different binding energies of the initial is state.ix4.20 Calculated fluorine K edge absorption spectra for CaF2with 96different F- radii as indicated.4.21 Calculated fluorine K edge absorption spectra for CaF2with 97different Ca2 radii as indicated.4.22 The calculated fluorine K edge absorption spectrum for CaF2 98including the core hole potential is shown at the top.4.23 The experimental fluorine K edge absorption spectrum in BaF2 is 103shown in the top spectrum, and the calculated spectrum is shown atthe bottom.4.24 A plot of the position of the peaks a-d for BaF2 from Figure 4.23 as 104a function of the position of the corresponding peaks in CaF2 fromFigure 4.18.4.25 The x-ray photoemission spectrum of the F is core level and its 106satellites in CaF2measured with the Mg K a line.4.26 The x-ray photoemission spectrum of the F is core level and its 107satellites in BaF2 measured with the Mg K a line.4.27 The fluorine K edge absorption spectrum of LaF3. iii4.28 The fluorine K edge absorption spectrum of CeF3. 1124.29 The fluorine K edge absorption spectrum of NdF3. 1134.30 The fluorine K edge absorption spectrum of SmF3. 114x4.31 The fluorine K edge absorption spectrum of EuF3. 1154.32 The fluorine K edge absorption spectrum of DyF3; 1164.33 The fluorine K edge absorption spectrum of YbF3. 1174.34 The energies of peak b and c relative to peak a in Figures 4.27-33. 1214.35 The x-ray photoemission spectrum of the F is core level and its 125satellites in LaF3 measured with the Mg K a line.4.36 The x-ray photoemission spectrum of the F is core level and its 126satellites in CeF3 measured with the Mg K a line.4.37 The x-ray photoemission spectrum of the F is core level and its 127satellites in NdF3 measured with the Mg K a line.4.38 The x-ray photoemission spectrum of the F is core level and its 128satellites in SmF3 measured with the Mg K a line.4.39 The x-ray photoemission spectrum of the F is core level and its 129satellites in EuF3 measured with the Mg K a line.4.40 The x-ray photoemission spectrum of the F is core level and its 130satellites in DyF3 measured with the Mg K a line.4.41 The x-ray photoemission spectrum of the F is core level and its 131satellites in YbF3 measured with the Mg K a line.xiAcknowledgmentsI would like to first thank Dr. Tom Tiedje, my supervisor, for his continuedguidance, constant support, and inspiring feedback throughout the course of this study.I would also like to thank Dr. Kevin M. Colbow, Dr. Jeff R. Dahn, Dr. WolfgangEberhardt, Tony van Buuren, Brian M. Way, Dr. Jan N. Reimers, and Dr. Stefan Crammfor the help on the data acquisition at the Ui beamline, NSLS, Brookhaven NationalLaboratory. I am grateful to Dr. P. C. Wong and Dr. K. A. R. Mitchell for providing theXPS measurements. Acknowledgement should also be made to many members of Dr.Tiedje’s lab, especially, Duncan Rogers, Jim Mackenzie, Steve Patitsas, Chritian Lavoie,and Shane Johnson, who have provided me with help in various ways during the past fewyears. Useful discussions with Dr. Birger Bergersen and Dr. Ian Affleck are alsoacknowledged.I am very grateful for the warm Canadian hospitality I have experienced eversince I stepped on this beautiful land.Finally, no words can fully express my gratitude to my wife Ginny, who hasgenerously supported me in every possible way.xiiChapter 1 1Chapter 1IntroductionMost chemical and physical properties of a solid depend on the behavior of theelectrons, which can be described by electronic states, both occupied and unoccupied.The understanding of occupied and unoccupied states complements each other since anydynamic view of electrons will involve them both. The photoelectron process whichinvolves exciting an electron from an occupied state into an unoccupied state is apowerful probe for studying these states. Two techniques involving photoelectrons wereused in the work of this thesis, namely, photoelectron spectroscopy and x-ray absorptionspectroscopy. In photoelectron spectroscopy (sometimes also called photoemissionspectroscopy), the incident photon energy is fixed while the kinetic energies of theemitted electrons are analyzed. In x-ray absorption spectroscopy, the absorption of asample is measured with a varying incident photon energy that is enough to excite a coreelectron. However, while the study of occupied states has matured with photoelectronspectroscopy during the past 20 years, an equivalent understanding of low energyunoccupied states with x-ray absorption spectroscopy has lagged behind [1]. In thisthesis, it is attempted to push forward the understanding of low energy unoccupied stateswith an extended study on x-ray absorption near edge structure (XANES). Theknowledge on occupied states energy alignments of the rare-earth trifluorides and siliconinterfaces is also extended.Chapter 1 2Alkaline-earth fluorides CaF2, SrF2, BaF2 and their alloys have been found togrow epitaxially on a number of semiconductor surfaces [2-15]. There have beenextensive studies, for example, on CaF2IGaAs (3.5% lattice mismatch) [5-8] and CaF2ISi(0.6% lattice mismatch) [9-13] interfaces because of their potential use in multi-layerdevice structures. Epitaxial growth of rare-earth trifluorides on Si(1 11) have also beeninvestigated [16]. Laser action of the systems LaF3:Nd and LaF3:Er3has been knownfor a long time [17], because of the numerous excited 4f states [18]. Direct doping ofrare-earth ions into ITT-V semiconductors [19, 201 and into LaF3/Si(1 11) interface [21]have been explored for possible opto-electronic applications.Previous studies of the energy alignment at the interfaces of the La, Nd and Tmtrifluorides on Si( 111) with photoelectron spectroscopy and x-ray absorptionspectroscopy [22, 23] are extended in this thesis and the substantially different results ofYbF3/Si(1 11) will be compared with those of TmF3ISi(1 11).X-ray absorption spectroscopy can be used for electronic states studies togetherwith photoelectron spectroscopy. The x-ray absorption process involves exciting anelectron to an empty state with a photon. A typical x-ray absorption spectrum is usuallydivided into two regions: the first 50 eV or so above the absorption threshold is called thex-ray absorption near edge structure (XANES) and the higher energy part of the spectrumis called the extended x-ray absorption fme structure (EXAFS) [24]. The EXAFS regionusually has weaker but longer period modulations and the distinction between theXANES region and the EXAFS region is a loose one [25]. A division with more physicalmeaning was suggested by Bianconi, namely that it should be roughly at the energywhere the wavelength of the excited electron is equal to the distance between theChapter 1 3absorbing atom and its nearest neighbors [26]. EXAFS was explained by Sayers, Sternand Lytle in terms of short range scattering of the photoelectron wave by the neighboringatoms around the photoexcitation site [27, 28] and rapidly developed into a technique toobtain local interatomic distances in solids, both crystalline and noncrystalline [29-31].XANES is known to be rich in chemical and structural information. It is morecomplex because the low energy final state electron interacts more strongly with the solid[24]. Consequently, there is no general theory for XANES. In this thesis, a Braggreflection model is developed to qualitatively explain the oscillations in XANES, in termsof the scattering of the photoelectron wave between families of lattice planes as set out bythe Bragg condition for backscattering. The model is tested on a number ofrepresentative systems.Because of the technological importance of CaF2 mentioned earlier, also in orderto understand XANES in more depth, high resolution CaF2 fluorine K edge XANES isused as a prototype to develop a model of XANES calculations for wide band gapinsulators with strong core hole effects. Unlike previous work which involved multiplescattering cluster calculations that included only short range order effects [32, 33], boththe long range order and the symmetry breaking core holes are included in a newbandstructure approach in which the core hole is treated with a supercell technique.Finally, the fluorine K edges in a number of rare-earth trifluorides are explored for thefirst time.Chapter 1 4Crystal Structures of the Alkaline-earth Fluorides and the Rare-earth TrifluoridesThe alkaline-earth fluorides CaF2, SrF2 and BaF2 all have the same fluoritestructure. It can be described as a f.c.c. lattice with a basis consisting of a cation at 000,and two anions at and respectively (Figure 1.1) [34]. The conventional cubelattice constant at room temperature is 5.463 A, 5.800 A, and 6.200 A for CaF2, SrF2 andBaF2 respectively [34]. In the CaF2 structure, the fluorine lattice consists of a simplecubic lattice of F- ions packed together with a separation determined by their ionic radius.In the BaF2 structure, the large Ba2 ion expands the fluorine lattice so that the F-Fdistance is larger than the sum of the ionic radii.The rare-earth trifluorides discussed in this thesis have two different crystalstructures. Most of the light rare-earth trifluorides have the tysonite structure (also calledLaF3 structure) [35]. According to Wyckoff, this crystal structure has a hexagonalstructure with the basis consisting of the atoms at the following positions [35]:2 cations at: ±(l13, 2/3, 1/4)2 fluorines at: ±(0, 0, 1/4)4 fluorines at: ±(l/3, 2/3, u) and ±(2/3, 1/3, u+l/2)where u = 0.57. Figure 1.2 shows the proposed hexagonal tysonite structure. The latticeconstants of the relevant rare-earth trifluorides are listed below [35].a(A) c(A)LaF3 4.148 7.354Chapter 1 5CeF3 4.107 7.273NdF3 4.054 7.196-TmF3 3.905 6.927However, more recent results suggest that the tysonite structure is actually a trigonalstructure [86].The other rare-earth trifluorides discussed in this thesis have the orthorhombicYF3 structure [35]. It has the orthorhombic symmetry with the atoms at the followingpositions [35]:4 cations at: ±(u, 1/4, v) and ±(u+1/2, 1/4, 1/2-v)4 fluorines at: ±(u’, 1/4, vt) and ±(u’+112, 1/4, 1/2-v’)8 fluorines at: ±(x, y, z), ±(x, l/2-y, z), ±(x+1/2, y, 112-z), ±(x+1/2, 1/2-y, 1/2-z)where u=0.367, v=0.058,u?= 0.528, v’ = 0.60 1,x=0.165, y=0.060, z=0.363.The orthorhombic YF3 structure is also shown in Figure 1.3. The lattice constants of therelevant rare-earth trifluorides with the orthorhombic YF3 structure are listed below [35]:Chapter 1 6a(A) b(A) c(A)SmF3 6.669 7.059 4.405EuF3 6.622 7.019 4.396DyF3 6.460 6.906 4.376TmF3 6.283 6.811 4.408YbF3 6.216 6.786 4.434Chapter 1 7Figure 1.1: The fluorite structure (after Wyckoff [34]). The left hand side shows thepositions of the atoms within the unit cell of CaF2 projected on a cube face. Letteredcircles refer to the corresponding spheres at the right. The right hand side shows aperspective drawing of the atoms within the unit cube of CaF2. The dark spheres are theCa2 ions and the light spheres are the F- ions.Chapter 1 8Figure 1.2: Two projections of the hexagonal cell of LaF3 (after Wyckoff [35]). In theupper part the black spheres are the La3 ions and the lighter spheres are the F- ions. Inthe lower part the numbers show the displacement of the atoms in the c direction.Chapter 1 9Figure 1.3: The orthorhombic YF3 structure (after Wyckoff [35]). On the left aprojection along the c axis of the orthorhombic YF3 structure is shown. Thedisplacement of the atoms in the c direction is shown by the numbers in the circles. Theunit of the numbers is one hundredth of c. Shown on the right is a packing drawingviewed along the c axis. The black spheres are the cations and the dotted spheres are theF- ions.aChapter 2 10Chapter 2Apparatus§2.1 The Ui Beamline at NSLSThe photoelectron experiments in this thesis including photoemission and x-rayabsorption experiments were conducted at the Ui beamline on the VUV storage ring atthe National Synchrotron Light Source (NSLS) of Brookhaven National Laboratory.Accelerating charged particles lose energy by radiation of electromagnetic waves. Thehighly relativistic electrons in the storage ring are magnetically constrained to travelaround a closed path and the consequent centripetal acceleration results in the emission ofthe synchrotron radiation. The radiation is peaked in the forward direction and ispolarized in the plane of the electron orbit, and it has a well defined continuous spectrum.The electrons in the VUV ring of the NSLS travel at the energy of 750 MeV and theresultant synchrotron radiation has a characteristic energy of 486 eV [36].The U 1 beamline is an ultra high vacuum beamline designed to cover the photonenergy range from 25 to 1300 eV continuously using an extended range grasshopper(ERG) monochromator [37]. The heart of the ERG monochromator consists of anentrance slit Si, three interchangeable spherical gratings G 1, 02, G3, and an exit slit S2.Other components include a series of mirrors to collect the synchrotron radiation andfocus it at the sample position and a set of premonochromator slits to aperture the beamvertically and horizontally in order to reduce stray light in the forward monochromatorChapter 2 110.750IlOli. SLITS0.500-,0D_I 000.250- • ...-Gi, -—-0.000 I I0 250 500PHOTON ENERGY (eV)Figure 2.1: The theoretical resolution curves and the measured resolution data points ofthe ERG monochromator with gratings Gi, G2, and G3 (figure after Sansone et a! [37]).Chapter 2 12DEPOSITIONEVAPORATORANALYSISLEPJ(OPTICALPYROMETERFigure 2.2: Top view of the experiment station of the Ui beamline at NSLS, BrookhavenLOADTRANSFER ARM LOCKSPU1TERINGVALVEaNSAMPLESTORN3EANALYZERVALVESTHICKNESSMONIfORMCPDETECTORMASSSPEC.BEAMLINE TOSYNCHROTRONNational Laboratory. Picture after Colbow [38].Chapter 2 13section. The combination of the flat mirror and the entrance slit are optically equivalentto a bilateral slit with a range from 10 to 300 tm while the exit slit is a bilateral devicewith a range from 10 to 1000 tm. The resolution and the photon throughput of themonochromator are functions of the aperture of these slits. The three gratings 01, G2,and G3 cover the photon energy ranges of 25 - 190 eV, 150- 550 eV, and 270 - 1300 eVrespectively. The minimum linewidth (FWHM) of the zero order peak with the smallestslits setting (10 i) is measured to be 0.09 A for grating G1, 0.03 A for grating G2, and0.02 A for grating 03 [37]. Figure 2.1 shows the theoretical resolution curves andmeasured resolution data points for the three gratings used as given by Sansone et al [37].Figure 2.2 shows the geometry of the experiment station of the Ui beamline. Atransfer arm can transport samples between storage shelves, thin film deposition chamber,and the main analysis chamber under UHV conditions. Once in the main analysischamber, the sample is mounted on a rotary manipulator arm that can also move in allthree directions. The sample can be rotated facing the appropriate directions forphotoemission measurement with the hemispherical electron energy analyzer, or for x-rayabsorption measurement with the microchannel plate detector. In addition, there issample cleaning capability in the main analysis chamber. The sample can be sputtercleaned with a sputtering gun in the main chamber. The sample holder on themanipulator arm also serves as a heating stage. With the sample mounted in a Ta foilbasket spot welded to Ta wires, it can be heated resistively in the main chamber.The surface temperature of the sample is monitored with an IRCON infra redpyrometer. The temperature reading of the pyrometer was calibrated by oxide desorptionon GaAs wafers and also by a type-K (chromel-alumel) foil thermocouple [38]. TheChapter 2 14agreement between the sample temperatures determined by the thermocouple and by thedesorption of the oxides were within 10 C over the temperature range of 450-600 C andwithin 20 °C for temperatures down to 350 °C [38].§2.2 Thin Film DepositionDeposition of thin films of rare-earth trifluorides on clean silicon wafers wererequired to study the interfaces between them and Si(1 11). This was achieved in thedeposition chamber connected to the main analysis chamber. With this arrangementsamples with freshly made films can be transported directly to the main analysis chamberunder UHV conditions, so that the possibility of surface contamination is minimized.The evaporation source material was contained in a tube shaped vessel spotwelded from tantalum sheet. Powders were used for rare-earth trifluorides evaporation.Each evaporation tube was specifically made to hold each evaporation source material.The evaporation tube was about 2 cm long and 6-8 mm in diameter with a 3 mm diameterhole in the middle facing the direction of the substrate and the thickness monitor. It wasresistively heated to evaporate the powdered material enclosed in it. A typical current of25 - 35 A and 1 Volt of voltage was needed to sustain the evaporation. The backgroundpressure was kept below 10-8 Torr during the evaporation and the substrates were kept atnear room temperature, heated only by the radiation from the evaporator. A shutter wasplaced between the evaporation tube and the substrate. Once the deposition rate wasstabilized at 0.2 - 0.5 A/s and about 20 - 30 A of the film was deposited on to the shutter,Chapter 2 15E—4cm--)thicknessmonitorFigure 2.3: The schematic diagram showing the configuration of the evaporator used for.\substrate10cm shutterTa evaporator tubeelectrodesthin film deposition.Chapter 2 16the shutter was opened to commence the deposition on to the substrate. Figure 2.3schematically shows the evaporation configuration.The deposition rate and the film thickness was monitored with an Inficon quartzcrystal thickness monitor during the deposition. The geometry of the thickness monitorand the substrate relative to the evaporator was kept as symmetric as possible. In thatcase, the theoretical uncertainty of the crystal thickness monitor should be less than 3%[39]. However, the directionality of the evaporation source could introduce much greateruncertainty in the film thickness determination. Therefore, the following complementmethod [38] was also used to check the film thickness.Let X be the electron escape depth of the overlayer, d be the overlayer thickness.Then the ratio of the intensity Ij’ of the photoemission signal from a core level of the filmand the intensity I of the photoemission signal from a core level of the substrate can beapproximated by,-d/2sinO1°R(d)=K[1,SG ]where K = is a constant expressing the relative sensitivity of the spectral features ofthe film to the substrate which can be obtained experimentally, and 8 is the anglebetween the sample surface and the detector. I is the photoemssion intensity of thesame feature as I, measured on a thick film (>50 A or thick enough that no signal fromthe substrate is seen) and normalized by the incident photon intensity. I is thenormalized photoemssion intensity of the same feature as Is’ measured on a baresubstrate. With the use of X = 6 A, the film thickness estimation on the rare-earthtrifluorides was evaluated by Colbow to be accurate to a factor of two [38].Chapter 3 17Chapter 3Photoelectron Spectroscopy§3.1 Experimental Arrangement of Photoelectron SpectroscopyPhotoelectron spectroscopy (PES) probes the occupied electronic states inmaterials by analyzing the kinetic energies of the photoelectrons emitted during thephotoionization processes. Only solid materials were involved in the work of this thesis.When a sample is exposed to photo radiation, the sample is ionized when an electron isemitted after it absorbs a photon. If the radiation has a monochromatic photon energyho), it is shown in many standard text books from the conservation of energy,(3.1.1)where EB is the binding energy of the initial electronic state, EK is the kinetic energy ofthe direct photoelectron which suffers no energy loss when it escapes from the sample,and • is the sample work function. Thus, one gets the binding energies of the occupiedstates from the spectra of the kinetic energy distribution of the direct photoelectrons.Strictly from energy conservation, the energy of the absorbed photon equals theenergy change of the whole system including the photoelectron and all other electrons inthe system,Chapter 3 18ho) = Total energy of the system without the electron i + EK- Total energy of the system with the electron i. (3.1.2)It is easy to derive eqn(3.1.1) from eqn(3.1.2) with the following definitions. The bindingenergy EB1 of the electron i is conventionally defined as the one-electron energyseparation between the state concerned and the Fermi level of the sample,EB1 Ci-EF. (3.1.3)The one-electron energy in solids may not be a good quantum number in general becauseof the electron-electron interaction. In other words, the energy of an electron could be afunction of many electrons. It can be described, however, in terms of the total energy ofthe system without considering the details of the electron-electron interaction. The one-electron energy of the state i is thus defined as the difference of the total energy of thesample before and after an electron at the state i is taken away from the sample,Total energy of the system with the electron i-Total energy of the system without the electron i. (3.1.4)The work function is defined as the energy difference between the vacuum level and theFermi level. Since the zero of the energy is the vacuum level, we have•sEF. (3.1.5)Combining eqn(3.1.2- 5) and omitting the subscript i, we have eqn(3.1.1).Chapter 3 19Eqn(3.l.1) gives the impression that the mapping of the binding energy EB to thekinetic energy EK depends on the sample work function . This is not desirablebecause the work function is sensitive to preparation conditions and must be measuredindependently. However, it turns out not to be a concern. The kinetic energy of thephotoelectron has to be measured by an electron energy analyzer, and the importantquantity is the retarding potential V between the analyzer and the sample which isrequired to bring the photoelectron to rest which the analyzer takes as the measuredkinetic energy of the photoelectron. From the definition of V. EK=V+( a -) whereøa is the work function of the electron energy analyzer and (Øa - ) is the contactpotential. One gets= -EB + EK +EB+V+a. (3.1.6)This relation is shown schematically in Figure 3.1. Thus one can map out the bindingenergy distribution of the occupied states by the measurement of V and it is independentof the sample work function •. This is desirable because one does not need to knowthe sample work function. The spectrometer work function a’ which can be assumedconstant, is sufficient.The photoelectron spectroscopy experiments in this work were carried out at theUi beamline of the National Synchrotron Light Source of Brookhaven NationalLaboratory. The beamline provides a high intensity photon source from 25 eV to1300 eV continuously through an extended range grasshopper (ERG) monochromator. Ahemispherical electron energy analyzer (VSW-HAC 100) was used to detect theChapter 3 20photoelectrons with specific kinetic energies. In this case, the potential V in eqn(3.1.6)and Figure 3.1 is determined by the combination of the actual analyzer retarding potentialVret and the analyzer pass energy F.: V Vret + E. The work function of the analyzerwas assumed to be 4.5 eV. The combined resolution of the monochromator and theelectron energy analyzer for the photoemission experiments was variable depending onthe photon energy, the setting of the monochromator slits and the pass energy setting onthe analyzer. The photon energy used in each experiment was calibrated by measuringthe Fermi edge on a freshly sputter cleaned copper surface on the sample manipulatorwhich was always available in the analysis chamber.Chapter3 21e—- Vacuum LevelFermi LevelEKVVacuum Level—Fermi Level SEBCore LevelSample SpectrometerFigure 3.1: The schematic diagram illustrating the relative energy levels inequation (3.1.6).Chapter 3 22§3.2 Photoelectron Spectroscopy of 4f Levels at the YbF3/Si(111) and TmF3/Si(111)InterfacesEpitaxial layers of LaF3 and related rare-earth trifluorides grown onsemiconductor substrates offer interesting possibilities for opto-electronic devicesincluding integrated diode-laser pumped solid state lasers [16, 21] In addition thepossibility exists of direct electrical excitation of rare earth ions at interfaces between thesemiconductor and a rare-earth trifluoride. Accordingly the positions of the 4f levels ofthe rare-earth ions at the interface between silicon and various rare-earth trifluorides havebeen investigated [22, 23] Most of the light rare earth trifluorides have the hexagonaltysonite structure whose a axis is within 2-8% of the lattice constant of the Si( 111)surface unit cell [35]. The heavier rare earth (Sm-Lu) trifluorides have the orthorhombicYF3 structure. In this section we report on a photoemission study of the 4f levels at theinterface between YbF3 and Si(1 11) and compare these results with TmF3/Si(1 11).The experiments were carried out at the Ui beamline at the NSLS, BrookhavenNational Laboratory. The substrates were 1 cm2 p-type Si wafers mounted in Ta foilbaskets spot-welded to Ta wires which could be heated resistively. The substrates werecleaned by a repetitive, low energy (500 eV) Ar sputter-anneal process, until no residualcarbon or oxygen was detectable in photoemission. The samples were annealed atapproximately 600 °C for one minute after each sputtering cycle. The sampletemperature was measured with a pyrometer calibrated with a thermocouple. The cleansubstrates were transfered under UHV to a separate chamber, where powders of the rare-earth trifluorides were evaporated at 0.2- 0.5 A/s from tube-shaped Ta boats in abackground pressure of 8x109 Torr. A quartz crystal thickness monitor was used forChapter 3 23relative thickness measurements during the depositions. The samples were thentransfered in UHV to the analysis chamber for photoemission and X-ray absorptionstudies.The determination of the position of the 4f levels in the rare-earth cations iscomplicated by the fact that the photoemission signal from the partially filled 4f orbitalsoverlaps with the F 2p valence band. However the 4f orbitals can be distinguished byresonant excitation of the giant 4d—. 4f transition in the rare-earth cations [221. The broadfeature above the 4d absorption threshold in the rare-earths was attributed to a collectiveexcitation of the outer shell (4d4f) electrons of the ion during the 4d10f—’4d9f1transition [401. A favored decay mode of this excitation is through 4d9f1—’4d’O4f’1+photoelectron leaving the ion in the same final state as reached by direct 4fphotoemission [41]. Thus, the 4f photoemission is enhanced with the resonant excitation.Figure 3.2 shows photoemission spectra in the valence-band region for a 10 Afilm of TmF3 on Si(1 11). The photoemission spectra were taken at two different photonenergies corresponding to resonant (A) and non-resonant (B) excitations and arenormalized by the incident photon flux on the sample. Spectrum B (the non-resonantone) contains only the signal from the F 2p valence band and the non-resonantphotoemission from the Tm 4f levels. Spectrum A (the resonant one) contains not onlythe above mentioned signal, but a large enhancement of the photoemission signal fromthe Tm34f12 levels due to excitation of the giant resonance in the outer shell electronsof the rare-earth ion. Spectra A and B are normalized by the current in the synchrotronstorage ring during the measurement. The spectrum (A-B) was obtained by subtractingthe non-resonant spectrum from the resonant one, which therefore corresponds to theChapter 3 24Cl)0D0C)>-F—C/)zwIz-15 -10 -5 0BINDING ENERGY (eV)Figure 3.2: Valence band region of TmF3 on Si taken at two different photon energies Aand B. The difference spectrum (A-B) is also shown. The inset shows the absorptionspectrum of TmF3 in the region corresponding to the Tm 4d absorption edge. The countnumber is 1.4X103 at the maximum of spectrum A and is about zero at the high energyends of all three spectra.-20Chapter 3 25U)C’,4-C0C)>-I—C’)zwIz-20 -15 -10 -5 0 5BINDING ENERGY (eV)Figure 3.3: Valence band region of YbF3 on Si taken at two different photon energies Aand B. The difference spectrum (A-B) is also shown. The inset shows the absorptionspectrum of YbF3 in the region corresponding to the Yb absorption 4d edge. The countnumber is about 800 at the maximum of spectrum A and is about zero at the high energyends of all three spectra.Chapter 3 26photoemission signal from the 4f12 orbftals on the Tm3 ion only, provided we neglectthe small difference (less than 15% [42]) in the cross section of the F 2p orbital for thetwo photon energies used. The inset in Figure 3.2 shows the absorption spectrum of thesame TmF3 film in the vicinity of the Tm 4d edge; the photon energies used to obtainspectra A and B are indicated by vertical lines.Photoemission spectra of the valence-band region for a 10 A thick film of YbF3annealed at about 400 °C are shown in Figure 3.3. As in Figure 3.2 the photon energiesused to obtain Spectra A and B are shown in the inset. The difference spectrum (A-B)was obtained in the same manner as for TmF3,and the first two peaks at 15 eV and 10 eVbinding energy correspond to the partially filled 4f levels in the Yb3 ions. We attributethe peak at 2.5 eV binding energy in spectra A and B to the regular (non-resonant)photoemission signal from the completely filled 4f14 levels in the Yb2 ions. The filled4f14 configuration of Yb2 has no 4d-4f transition, so that there is no extra enhancedsignal from these levels with photon energy corresponding to the 4d10-4f3 transition asin the Yb3 case. Therefore the difference spectrum (A-B) only maps out the 4f13 levelsin the Yb3 ions. We attribute the feature near 3 eV binding energy in the differencespectrum to incomplete cancellation of the 4f14 photoemission due to systematic errors inthe normalization of the spectra.The prominent splitting of about 5 eV in the 4f12 levels of Tm3 and 4f13 levelsof Yb3 is due to the exchange splitting associated with whether the photoelectron comesfrom a state with a spin parallel or anti-parallel to the majority spin of the 4f electrons[43].Chapter 3 27Let t indicate the direction of the majority spin and -I- indicate the direction ofthe minority spin. In the Tm3 case, the ground state configuration of 4f12 is(4ft)7fL5with S = 1. The final states after removal of one 4f electron will then beeither (4ft)6f.L5 with S = 1/2 or (4ft)7f..L) with S =3/2 [43]. Since theconfiguration with S = 3/2 has a lower energy than the configuration with S = 1/2 due tothe exchange interaction [43], the binding energy of the 4ft electron is larger than thebinding energy of the 4f .I- electron. The area under the two exchange splitting peaksshould have a ratio of 7:5 in TmF3 because there are seven electrons to choose from forthe transition (4ft)7fi-5— (4ft)6J and there are only five choices for thetransition (4ft)7f..L)5—* (4ft)7f..L). This is consistent with what we observe in thedifference spectrum (A-B) in Figure 3.2 by visual inspection. The angular momentummultiplet splitting due to the spin-orbit interaction in the initial state and the final statefurther broadens the binding energy distribution in each transition [441. Since the finalstate (4ft)7fI- has a larger total spin of S = 3/2 than the final state (4ft)6fJ-5which has a total spin of S = 1/2, the feature associated with the transition(4ft)7(4f)5 —> (4ft)7(4f-L) is wider than the feature associated with the transition(4ft)7f..L)5— (4ft)6fL5.The above discussion of the binding energy distributionof the 4f12—’ 4f transition is shown schematically in Figure 3.4.In the Yb3 case, the 4f13 ground state has the configuration of (4ft)7(4f.i-)6with S = 1/2. The final states after removal of one 4f electron will either be(4ft)6f..L) with S = 0 or (4ft)7f.L5with S = 1. Similarly, the binding energy ofthe 4ft electron is larger than the binding energy of the 4f.i- electron. The area under thetwo exchange-split peaks have a ratio of 7:6 because there are seven electrons to chooseChapter 3 28Tm3Binding Energyspin-orbit splittingFigure 3.4: The schematic diagram showing the binding energy distribution of the4f12.4f11 transition. The transition (4ft)7-5S= 1) —* (4ft)74f.J.-)S= 3/2) isthe ground-to-ground transiton.spin-orbit splittingChapter 3 29Yb3 or Tm2Binding EnergyFigure 3.5: The schematic diagram showing the binding energy distribution of the4f13—’ 4f12 transition. The transition (4ft)7f.i-6S= 1/ 2)— (4ft)7fJ.-)5S= 1) is>1spin-orbit splittingthe ground-to-ground transition.Chapter 3Si TmF3CBVB304f13Figure 3.6: The schematic diagram showing the positions of the cation 4f levels relativeto the Si valence band and the F 2p valence band for TmF3 and YbF3 on Si( 111)substrates. Possible electron occupation is illustrated with the shaded areas. Theunshaded area shows where the 4f13 levels of Tm would be.Si YbF3CBVB4f134f14F2p F2p4f12Chapter 3 31from for the transition (4ft)7f..L)6—* (4ft)6f.L and there are six choices for thetransition (4ft)7fJ.,)6 (4ft)7f..L)5,which is consistent with our observation in thedifference spectrum (A-B) in Figure 3.3. The broadening of the two exchange-splitfeatures is due to the angular momentum multiplet splitting in the initial and the finalstates. Since the total spin S = 0 for the final state (4ft)6fJ,-) and the total spin S = 1for the final state (4ft)7fL5, the feature associated with the transition(4ft)7(4f..L)6 — (4ft)6f..L) is narrower than the feature associated with the transition(4ft)7(4f.L)6 — (4ft)7(4f..I-)5. The discussion of the binding energy distribution of the4fl3... 4f12 transition is again schematically shown in Figure 3.5.There is no exchange splitting in the 4f14 configuration of Yb2 since only onespin state exists in the photoemission final state: 4f14(S = 0)—. 4f13(S = 1/2). The widthof about 2.5 eV of this feature can be attributed to the angular momentum multipletsplitting in the 4f13 final state which has S=1/2. The relative positions of the energylevels are summarized in Figure 3.6. The valence-band offset between Si and TmF3,namely the energy separation between the top of the silicon valence band and the top ofthe F 2p level, has been measured to be 7.0 eV [22]. We were unable to determine thevalence-band offset between Si and YbF3 because the 4f14 levels of the Yb2 ionsobscure the overlapping Si valence band. We assume the offset between Si and the F 2pvalence band of YbF3 is the same as that between Si and TmF3.As it was reported earlier [22], the on-site Coulomb repulsion energy U preventsthe Si valence-band electrons from occupying the empty 4f levels in both NdF3 andTmF3. U is defined as the difference between the ground state energies of 4f’1 and 4f1configurations [45]. It can be determined in photoemission from the difference betweenChapter 3 32the binding energies associated with the ground-to-ground transitions of 4f’1 -. 4f” and4ffl 4ffl4 of the same atom [46]. For example, U for Tm can be taken as the bindingenergy difference between centers of gravity of the feature corresponding to the transition(4ft)7f..L)5—+ (4ft)74f..L)’ in the Tm3 ion and the feature corresponding to thetransition (4ft)7f1-6—* (4ftY(4ü) in the Tm2 ion [44]. Because the 4f levels arevery localized, this energy is large. For Tm, U is about 6.5eV [44, 45]. An earlier studyof Tm3 and Tm2 ions in TmSe and TmTe alloys also shows that the width of the4f13 4f12 feature associated with the final state being the ground state ((4ft)74f..L-)5),is about 5 eV for Tm2 ions [44]. This compares well with the observed width of the 4f134f12 feature in the Yb3 ions associated with the final state being the ground state (seeFigure 3.3) which has the same (4ft)7f..L)5 configuration. The on-site Coulombrepulsion energy U together with the width of the 4f13 ground state levels preventselectrons from transferring from the Si valence band into the empty levels in the 4f12configuration of the Tm3 ions at the interface. In other words the highest occupied stateof the 4f13 configuration is above the top of the Si valence band as shown schematicallyin Figure 3.6. On the other hand, the 4f14 levels of Yb2 ions are apparently below the Sivalence-band maximum which is consistent with the on-site Coulomb repulsion energy Uof 6.5 eV [45, 46] and the small width of 4f14 levels. This makes charge transfer from theSi valence band to YbF3 possible. Figure 3.6 illustrates the above mentioned distinctphenomena at the TmF3/Si(1 11) and YbF3/Si(1 11) interfaces.Figure 3.7 shows photoemission spectra of YbF3 films for two differentthicknesses deposited at room temperature on Si(l 11) substrates. The photon energywas 151.4 eV. Peaks from the Si substrate are clearly visible in the spectrum for the 1 oAChapter 3 33film while there are no features from the substrate in the spectrum for the 70 A film.Peak A is from the F 2p valence band and overlapping Yb 4f13 levels and peak B is fromthe Yb 4f14 levels. The ratio of peaks B/A for the 10 A film is 3 times larger than thecorresponding ratio for the 70 A film. This suggests that there is a higher concentrationof the Yb2 ions at the interface. But the very appearance of peak B in the thick filmwhile no Si peaks can be observed also suggests that the Yb2 ions are not only at theinterface but also in the film.Figure 3.8 shows photoemission spectra for the 10 A YbF3 film before and afteran anneal at about 400°C with the resonant photon energy of 182 eV (see Figure 3.3),normalized by the incident photon flux. The signals from Yb 4f’3 and F 2p are weakerwhile the signal from Yb 4f’ stays approximately the same after the anneal. The smallerphotoernission signal from YbF3 after annealing can be explained by a change in thedeposited film as described below. One possibility is that the initially uniform film formsa monolayer coverage together with islanding of the YbF3 overlayer (Stranski-Krastanovgrowth) after annealing. Alternatively, it is possible that some loosely bound as-deposited YbF3 re-evaporates on crytallization during the anneal. In either interpretationthe concentration of the Yb2 ions is higher at the interface which is consistent with thefinding that the 4f14 levels in the Yb2 ions lie below the top of the Si valence band.In conclusion, we have studied and compared the energy alignment of TmF3 andYbF3 on Si(1 11) substrates using photoemission spectroscopy and x-ray absorptionspectroscopy. By resonantly exciting the 4d-4f transition in the rare-earth, we were ableto enhance the photoemission signal from the 4f electrons and thereby distinguish it fromthe overlapping signal from the F 2p valence band of the rare-earth trifluorides. WhileChapter 3 34the electrons in the Si valence band are prevented from occupying the empty 4f levels inTmF3 at the interface by the on-site Coulumb repulsion energy U, the charge transferfrom Si to YbF3 is possible because the totally filled 4f states of Yb still lie below the Sivalence band maximum.Chapter 3 35Cl)Cl)4-’C0C)>-IC’)zwHz-100 -80 -60 -40BINDING ENERGYFigure 3.7: Photoemission spectra of YbF3 films of two different thicknesses asdeposited on room temperature Si(l 11) substrates. The photon energy used was151.4 eV. The count number is l.4X105 at peak A in the top spectrum and is about zeroat the high energy ends of all two spectra.-20 0(eV)Chapter 3 36Cl)Cl)D0C.)>-F(I)zwHz-20 -15 -10 -5 0BINDING ENERGY (eV)Figure 3.8: Photoemission spectra with the resonant photon energy of 182 eV taken on a10 A YbF3 film deposited on Si(1 11) before and after a 1 minute anneal at about 400 °C.The count number is 1.7X 10 at the highest peak in the top spectrum and is about zero atthe high energy ends of all two spectra.•II1IIII111 IlIllIllIll I IAs deposited1111111 I 11111 I III I 1111115Chapter 4 37Chapter 4X-ray Absorption Spectroscopy4.1 Experimental Arrangement of X-ray Absorption SpectroscopyThe experiments of x-ray absorption studies in this work were carried out at theUi beamline of the National Sychrotron Light Source of Brookhaven NationalLaboratory. As described in Chapter 2, this beamline provides a high intensity photonsource from 25 eV to 1300 eV continuously through an extended range grasshopper(ERG) monochromator. The highest resolution (FWHM) achieved on this beamline is0.02 A, which corresponds, for example, to 0.8 eV at 700 eV photon energy.The method of measuring the x-ray absorption spectra in this work is the totalelectron yield method in which all electrons emitted from the sample are collectedregardless of their kinetic energies. As it has been shown [47], total electron yield is to agood approximation, proportional to the absorption coefficient. However, it is notexactly proportional to the absorption as we show below. In a photoelectric process, forevery core hole created by a photon, there are 1-ö photoelectrons directly emitted fromthe sample, o3 secondary electrons emitted from the sample due to inelastic scattering ofthe primary photoelectrons, xA Auger electrons, and czM(0) electrons emitted due tomulti-electron excitation processes. Therefore, in an energy region around the thresholdof a core excitation, the total number of electrons emitted from the sample per incidentphoton isChapter 4 38Total Yield = po){(l-6) + x6 + aA + aM((o) },where ji(co) is the absorption coefficient. Although 6 and as depend on the photon energy0) indirectly through the photoelectron kinetic energy, they are structurelessmonotonically increasing functions of 0 due to the averaging over a series of statisticalscattering processes [47]. While aA is independent of w, the only factor that introducesstructure is the multi-electron coefficient aM(co). Therefore, in energy regions wherethere is no multi-electron excitation or multi-electron excitation is not strong, the totalelectron yield is a very good measure of the x-ray absorption spectrum. Althoughseemingly more precise x-ray absorption measurements can be made by the partialelectron yield method which involves collecting only Auger electrons, the total electronyield method gives a better signal to noise ratio since only a small fraction of the directphotoelectrons and Auger electrons escape into vacuum compared with the total electronyield which is dominated by inelastically scattered electrons.A microchannel plate detector with two microchannel plates in series was used tocount the emitted electrons. Each microchannel plate has an output of lO electrons for asingle input event [38]. In addition, either the sample or the detector or both can bebiased to optimize the signal acquisition depending on the specific experimentalconditions. The whole chamber where the experiments took place was under UHVconditions ( iO Torr) throughout the experiment.Chapter 4 39§4.2 Fluorine K edges of Alkaline-Earth FluoridesFluorine K edges of the three alkaline-earth fluorides CaF2,SrF2, and BaF2weremeasured on powdered samples pressed on indium foils. A +80 V bias was applied to thefront grid of the detector in the case of CaF2 to achieve the best signal to noise. The biasapplied in the case of SrF2 was -80 V. In the case of BaF2, a -20 V bias was applied tothe front grid of the detector and a -80 V bias was applied to the sample. Theexperimental data of the fluorine K edges of all three alkaline-earth fluorides studied inthis work are shown in Figure 4.1, Figure 4.2 and Figure 4.3 respectively. A resolution ofabout 0.8 eV was achieved as evidenced by the FWHM of the first peak in Figure 4.3.This resolution is comparable to or better than the ones in earlier work [9, 32, 48]. Thefirst sharp peak in the CaF2 spectrum has been identified as a transition from the F isstate to a bound core exciton state, through deexcitation studies by Tiedje et al [48]. Weassume that the first peaks in the SrF2 and BaF2 spectra are also core excition transitions.The discussion about the features at higher photon energy will be deferred to §4.4 atwhich point we will have a better understanding of XANES.Chapter 4 4000.4-D0C.)III liii.. II 11111 111111 III liii liii liii 111111111111 11111111FKedgein CaF29.0 18.0 17.0 16.0 15.0 14.0 13.068011111111 I 1111111111 liii 111111 liii iii IIIII11I1I1II1I I liii III690 730700 710 720Photon Energy (eV)740Figure 4.1: The experimental spectrum of the fluorine K edge in CaF2.Chapter 4 41Cl)Cl)D0C)1.2 101.1 109.38.0 16.7 15.3 14.0680 690 700 710 720 730 740Photon Energy (eV)Figure 4.2: The experimental spectrum of the fluorine K edge in SrF2.Chapter 4 42Cl)—SC’,D0C.)2.01.81.61.41.21.08.0680 690 700 710 720 730Photon Energy (eV)740Figure 4.3: The experimental spectrum of the fluorine K edge in BaF2.Chapter 4 43§4.3 Bragg Reflection Model for XANES in Crystalline SolidsAs described in chapter 1, x-ray absorption extending less than about 50 eV abovethe absorption threshold is normally referred to as x-ray absorption near edge structure(XANES) while absorption extending above about 50 eV from the edge is referred to asextended x-ray absorption fine structure (EXAFS). EXAFS is well understood in termsof the interference of the photoelectron wave with backscattered electron waves from thesurrounding atoms [27, 28]. XANES is more difficult to model quantitatively thanEXAFS because of the very strong interaction between the photoexcited electron and theneighboring atoms, including atoms beyond the nearest neighbors [24, 25]. There is nogeneral method for calculating XANES spectra and XANES spectra for a large number ofsolids have not been calculated [23].As early as in 1932, using the nearly free electron approximation, Kronig firstproposed that EXAFS oscillations were due to the Van Hove singularities in the densityof states at special points in the k-space, namely, at the points where the wave vectork=G12, (4.3.1)where G is a reciprocal lattice vector [49]. This explanation was finally shownconclusively to be incorrect by Stern in 1974 [28]. Stern pointed out that Kronig ignoredthe absorption matrix element effect, which is more important than the density of statescontribution. Also it has been shown that even the absorption matrix element requiringlong range order can not account for the EXAFS oscillations which are sinusoidal withperiods generally in the order of 100 eV. It has been shown that the short range orderChapter 4 44contribution, namely, the scattering of the photoelectron by the nearest neighbor atoms isthe most important mechanism of EXAFS which has been supported experimentally withthe observation of EXAFS also in amorphous systems.However, XANES and EXAFS are very different in terms of the nature of thephotoelectron scattering even excluding the core hole effect and the multi-electron effectthat are often associated with XANES but normally ignored in EXAFS. In EXAFS, thewavelength of the photoelectron is smaller than an interatomic distance and the importantscattering is from the nearest neighbors, for which multiple scattering can be neglected.On the other hand, in XANES the electron wavelength is comparable to or larger thantypical interatomic distances. (The wavelength of a free electron verses its kinetic energyis shown in Figure 4.4.) Also, the electron inelastic mean free path increases rapidly froma few inter-atomic distances to tens or hundreds of lattice constants below 50 eV kineticenergy as shown by the universal curve of electron inelastic mean free path [50, 51].(Figure 4.5 shows the calculated electron inelastic mean free path in lithium fluorideversus the electron kinetic energy as an example [52].) Therefore, the long range ordereffect that is not important in EXAFS might be important in the XANES case. In light ofthis difference, it is appropriate to explore the role of long range order scattering, namelythe scattering of photoelectron waves by crystal planes, in XANES.We have previously proposed the Bragg reflection model relating positions ofh2 G”2peaks in XANES with the special final state kinetic energy of e = —I — I [53].2m2)Remarkable agreement between the experiment and the model has been shown in thecopper K edge for the usual f.c.c. form of copper and for strained thin films of copperChapter 4 45€4-c,)Ca)-J>C04-C)0)uJ0)0)U-151050I I I 11111 I I I 11111 I I I 11111I I I iiiiiL I I I IIIiiI I I I 11111100 101 1o2Free Electron Kinetic Energy (eV)Figure 4.4: The free electron wave length as a function of the free electron kineticenergy.Chapter 4 465001403020100500 1000 1500 2000Electron Energy (eV)Figure 4.5: The calculated electron inelastic mean free path (IMFP) in LiF by Tanuma et• IMFP ValuesMod. Bethe Eq.- -- TPP-2LITHIUM FLUORIDE3Q1..0 100 200.. I • .1/20 -10•.—0al using various algorithms [52].Chapter 4 47with the body centered tetragonal structure [53]. The principle of the model is that thecondition k = on the electron wave vector k gives the strongest constructiveinterference between the outgoing photoelectron wave and the backscattered wave. It isnot clear however, why the condition of the maximum constructive interference of theelectron final states should necessarily enhance the absorption so as to give a peak at thecorresponding energy. Destructive interference or some other feature could be presentinstead. In the following, we will reintroduce the Bragg reflection model in a morequantitative way and examine the limits of the model.The wave equation that the electrons in a periodic system have to satisfy isSchrodinger’s equation with a periodic potential. Its k-space form is(eG — e)(k — Gji)+ UG,G(k — G’Ily) =0 (4.3.2)G Gwhere 0 G = —(k — G)2 is the free electron energy, e is the energy eigenvalue,k 2m(k- GI iy) = . $ei(r)dr (4.3.3)unit cellis the Fourier coefficient of the electron wave function i(r) where V is the volume ofthe unit cell, andUG = -L Je’U(r)dr (4.3.4)unit cellis the Fourier coefficient of the potential energy. Its matrix form is simply üiy =whereChapter 4 48:“UG •.. U,UG Ek_G UGG (4.3.5)UGI •• UG_GIt is worth mentioning that the diffraction of electron waves by a periodic system isanalogous to x-ray diffraction, with UG playing a similiar role in electron diffraction asthe Fourier coefficient of the lattice electric charge distribution PG does in x-raydiffraction. That is to say, electrons are diffracted by the periodic crystal potential whilex-rays (electromagnetic waves) are diffracted by the periodic charge distribution. Hence,some concepts associated with the x-ray diffraction are also useful here. Taking theexample of monatomic lattices for simplicity, we borrow the concepts of structure factorand form factor from x-ray diffraction. The periodic crystal potential can be expressedas the sum of potentials of individual atoms or ionsU(r) u(r —r1), (4.3.6)thenu =! IerU(r)dr3Gunit cell= fe’u(r)drwhole crystal= SGf(G) (4.3.7)Chapter 4 49with 2 denoting the volume of the whole crystal,where SG = !Ee1Gri is the structure factor and f(G) = ._Jeru(r)dr3 is the formfactor, which have a similar meaning as in x-ray diffraction [62]. N is the number ofatoms in the crystal and20=2/N. The structure factor SG depends on the geometricalconfiguration of the atoms in the crystal while the form factor f(G) represents only thecontribution to the total crystal potential from the potential of an individual ion (or atom).Another similarity is that the Bragg condition, which describes the condition ofconstructive interference between the incident wave and the reflected wave, also appliesto the case of electron waves in a crystal.In the limit of a weak crystal potential, i.e., when UI << e° the solutions ofEqn(4.3.2) are not very different from the free electron wave function except for thestates very close to Bragg planes in k-space. One can expect that only those states thatare close to Bragg planes contribute to the structure in the XANES spectrumsignificantly. Indeed, it causes oscillations in the density of states across a Bragg plane.More importantly, the states close to a Bragg plane are different from the free electronwave and have different absorption cross sections from the free electron wave. In aqualitative analysis of the states near a Bragg plane in the presence of a negative UG,Stern concluded that the density of states would have a peak just below the energyt2 /G.2— 1 and a valley just above the same energy with a spacing of approximately2m ‘ 2)2IUGI [28]. Stern further concluded that the absorption would have only a valleyh2 /G2centered at—I — I corresponding to the valley in the density of states [28]. Stern2m2)Chapter 4 50concluded that these variations were too small to account for the large EXAFSoscillations. Now in order to understand the role of Bragg planes in XANES moreaccurately, we are going to quantitatively examine the absorption changes associated witha Bragg plane.We begin by showing how constant energy surfaces in k-space are affected by aBragg plane. Figure 4.6 shows the constant energy surfaces in k-space with only oneBragg plane shown for clarity. It is equivalent to having only two reciprocal lattice(k2 uvectors F and G in k-space. Thus the Hamiltonian matrix is simply I G 2 unUG Ik-GI )which we set all the physical constants equal to unity. In the picture we have setT=(0, 0, 0), G=(l0, 0, 0), and = 9. The dotted line indicates the sole Bragg planedefined by G/2. The numbers on each curve indicate the energy of that particularcopstant energy surface. The constant energy surfaces are determined by the magnitudeof UG, and are independent of its phase. In a periodic system with inversion symmetry,UG must be real. A positive value for UG means that the crystal potential is repulsive forthe wave exp(Gr) while a negative UG means that the crystal potential is attractive forthe wave exp(Gr). We proceed with our discussion assuming that UG is positive fordefinitiveness.With a positive UG, the states close to the Bragg plane on the left hand side(lower energy) are sine-like while the states close to the Bragg plane on the right handside are cosine-like [28, 54]. In the case of K edge absorption where the initial state is ais state, the absorption mathx element with a sine-like final state is -.J times the onewith a plane wave while the absorption matrix element with a cosine-like state is zero.Chapter 4 51k12108642G-2 --4 --6 --12I I I I I I I I0 5 10Figure 4.6: Constant energy surfaces in k-space with one Bragg plane. With all thephysical constants equal to unity, the free electron energy = k. The dotted lineindicates the Bragg plane defined by G12. The number on each curve indicates theenergy of that particular constant energy surface.Chapter 4 52Since the absorption coefficient is proportional to the square of the matrix element, statesclose to the Bragg plane on the left hand side have twice as large a contribution to theabsorption coefficient as the the states that are far from the Bragg plane, and states thatare close to the Bragg plane from the right hand side do not contribute to the absorptioncoefficient. The states with energies smaller than e =k2= =25 are all below theBragg plane, while the states with energies larger than (_) + IUGI = 34 are distributedon both sides of the Bragg plane. When the energy is much smaller than 25, the constantenergy surfaces are not very different from spheres (or circles in two dimensions), and allthe states on the sphere have about the same contribution to the absorption coefficient asthe unperturbed plane wave states. When the energy is smaller than, but approaches 25,the constant energy surface forms a “neck” connected to the Bragg plane. For those stateson the “necktt, the absorption matrix elements are increased, and therefore, the totalabsorption at this particular energy is enhanced. When the energy exceeds 34, theconstant energy surface crosses the Bragg plane. As some states start to appear on theright hand side of the Bragg plane, their suppressed absorption contribution causes theoverall absorption to decrease. When the energy is much larger than 25, the constantenergy surface is again not very different from a sphere except it is broken at the Braggplane. The number of states that are close to the Bragg plane and on its left hand side isabout the same as the number of states that are close to the Bragg plane and on its righthand side. Therefore, the deficient contribution to the absorption coefficient from thestates on the right hand side of the Bragg plane is made up by the enhanced contributionfrom the states on the left hand side of the Bragg plane, so that the total absorptionChapter 4 53contribution from all the states with that particular energy is about the same as that by theunperturbed plane wave states with the same energy.From this qualitative analysis, one can see that an oscillation occurs in theabsorption coefficient when the energy sweeps through the special value defined by theBragg plane e = k2= ()2.In order to know what exactly happens at e = k2=however, a quantitative analysis is needed. For this purpose, we calculate numerically theabsorption coefficient with a artificial potential UG, this time in three dimensions. Toreflect the 3-dimensionality, we include the reciprocal lattice vectors G=(10, 0, 0),(0, 10, 0), and (0, 0, 10). Thus, there are only three Bragg planes at the same distance at:(5, 0, 0), (0, 5, 0), and (0, 0, 5). We cover only part of the Brillouin zone from the origin,defined by k > 0, k > 0, and k > 0 when we calculate the absorption coefficient andthe density of states. The other three equivalent Bragg planes (-5, 0, 0), (0, -5, 0), and(0, 0, -5) need not be included since they are always far from the region we cover in kspace, and we are only interested in the behavior near the Bragg plane. In the calculation,we treat the initial state involved in the absorption as a deep core state since we areinterested in XANES which involves deep core states as initial states. For simplicity wemake this initial state a &function which means it has an infinite binding energy.The phase of UG is also important in the calculation. As we have stated earlier, ifthe crystal forms a Bravais lattice which has inversion symmetry about every atom, UGmust be real. When a crystal does not form a lattice with inversion symmetry (e.g., aBravais lattice with a basis), UG can be complex. An example is crystalline silicon whichhas no inversion symmetry about a silicon atom. Another example is CaF2. There isChapter 4 54wave eI9X>1(a)wave eI9X(b)wave e9X(c)• species 1o species 2Figure 4.7: One-dimensional schematic diagram showing examples of the Fouriercoefficients UG being complex. All three waves above have g = 2., where a is thelattice constant or the periodicity of the lattice.(a) U9isreal.(b) U9 = (1 1) f, where f denotes the form factor.(c) Ug = --f1 with the origin at species 1; U9 = jf1 with the origin at species 2.Chapter 4 55inversion symmetry about Ca2+ sites, but no inversion symmetry about F- sites.Consequently, UG is real when the origin is chosen at a Ca2 site and UG is imaginary forsome values of G when the origin is chosen at a F- site. Figure 4.7 schematicallyillustrates in one dimension examples of UG being complex due to lack of inversionsymmetry.Figure 4.8 shows the ratio of the absorption coefficient for the model potentialdescribed above to the absorption coefficient for the unperturbed plane waves. The ratioof the density of states with the model potential to the density of states without thepotential is also shown in the figure for comparison. The light solid line is the density ofstates ratio with = 3. It is independent of the phase of U. The heavy solid line showsthe absorption ratio with U=3, the dotted line shows the absorption ratio with U=-3 andthe dashed line shows the absorption ratio with U=±i3. The arrow indicates the energy of(G)2 As we expected, the absorption ratio with the real positive U has a prominentpeak at below (—) . The absorption ratio with the real negative U has a prominentvalley at UI above()2.The absorption ratio with an imaginary U is very similar tothe density of states ratio which has a less prominent peak at IUI below and a lessprominent valley at IUI above (_) . The absorption ratios with U = .=(1 ± i) andU = =f1 ± i) respectively are shown in Figure 4.9. The absorption ratio with the realpart of U being positive has a similar behavior as the one with the real positive U and theabsorption ratio with the real part of U being negative has a similar behavior as the oneChapter 4 5621010 40Energy (Dimensionless)Figure 4.8: The ratio of the absorption coefficient for the model potential with variousvalues, to the absorption coefficient for the unperturbed plane waves (shown by the heavysolid line, the dotted line, and the dashed line). The thin solid line shows the ratio of thedensity of states with IUI = 3 to the density of states with U = 0. The arrow indicates theenergy defined by (G/2)2.15 20 25 30 35Chapter 4 57210Energy (Dimensionless)Figure 4.9: The absorption ratio with complex U. The arrow indicates the energy10 15 20 25 30 35 40defined by (0/2)2.Chapter 4 5821010 40Figure 4.10: The absorption ratio with various magnitudes of U, assumed to be real andpositive. The arrow indicates the energy defined by (0/2)2.15 20 25 30 35Energy (Dimensionless)Chapter 4 59with the real negative U. Figure 4.10 shows the effect on the absorption ratio of differentmagnitudes of U for U being real and positive. As the magnitude of U decreases, theabsorption ratio approaches one, as expected.We have illustrated how Bragg planes affect the absorption cross section throughtheir effects on the absorption final states in the presence of a crystal potential.Equivalently, it can be also described with the language of scattering. The scattering ofthe photoelectron wave by a family of lattice planes generates both sine and cosine-likestanding waves relative to the site of the photoabsorption. As the sine-like standingwaves have an enhanced K edge absorption and the cosine-like standing waves have asuppressed K edge absorption, they collectively create a peak or valley in the absorptionspectrum in the vicinity of the energy defined by k=G/2.Case StudyIn the following, we illustrate and test the model with K edges in a number ofmaterials. Copper is the classic test case for new models of x-ray absorption. It is a goodtest material because the core hole is well screened so that core hole effects are minimalin the absorption spectra and it has a simple crystal structure. More importantly, for usthe copper K edge has been measured on two forms of copper with different crystalstructures (f.c.c. and b.c.t.) [53, 55] which is an ideal situation for identifying the crystalstructural effect in XANES. We look at iron next in order to compare the XANES in twomaterials with similar structures but different composition, namely, b.c.t. copper (almostChapter 4 60b.c.e.) and b.c.c. iron. Then we test the model on crystalline silicon as an example of acovalent solid and solid neon as an example of a molecular solid. Finally, we test theBragg reflection model on the fluorine K edge in CaF2 which is an insulating ioniccompound.CopperWe know UG is real for systems that have inversion symmetry which is the casefor Cu (f.c.c. structure) and Fe (b.c.e. structure). We only need to determine the sign ofUG in these cases. Harrison has shown, taking Al as an example of a simple metal andCu as an example of a transition metal [56, 57], that the pseudopotential form factors ofthese metals are negative for small G and positive for large G. The form factor crossesfrom negative to positive at about G I kF 1.5 for Al and G / kF 1.8 for Cu and theyslowly approach to zero again when G / kF —+ oo• As !(1,1,1) =2.kF for Cu [54],the form factor of Cu for all the allowed reciprocal lattice vectors G are positive.Therefore, UG for Cu are all positive since the structure factor SG = 1 as the Cu crystalforms a Bravais lattice.As we have shown earlier, a positive UG causes a peak in K-edge XANES in theh2 “G”2vicinity of the final state kinetic energy—I — I . Thus, there should be peaks at these2m2)energies in the Cu K edge XANES spectrum. The allowed reciprocal lattice vectors andtheir corresponding degeneracies NG and energies 8G are tabulated in Table 4.1 forcopper.Chapter 4 61G NG 8G (eV) G NG eG (eV)111 8 8.7 331 24 54.9200 6 11.6 420 24 57.8220 12 23.1 422 24 68.8311 24 31.8 511 24 77.4222 8 34.7 333 8 77.4400 6 46.3 440 12 92.3Table 4.1: The reciprocal lattice vectors for f.c.c. copper. NG is the degeneracyor the number of planes associated with the reciprocal lattice vector andh2 (G’2= —2m’2Figure 4.11 shows the comparison between the Bragg reflection model and themeasured x-ray absorption data for the K-edge of f.c.c. Cu [53]. The known structure ofcopper defines the energy positions of the Bragg reflection resonances in the XANESspectra. In the weak potential limit the amplitude of the Bragg reflections will beproportional to IUG. For simplicity we assume that these Fourier coefficients haveconstant magnitudes in the energy range of interest. The total scattering amplitude, dueto a given lattice plane, will also depend on the number of equivalent planes of differentorientation and on the atom and plane density for that particular set of lattice planes. Theproduct of the last two factors will be a constant for all the Bragg peaks, since any set ofplanes defined by a reciprocal lattice vector must include all the atoms in the crystal. WeChapter 4 62also neglect effects due to polarization of incident photons relative to the crystal. Thusthe x-ray absorption spectrum can be modelled by,(4.3.8)h2 G”2where CF is the Fermi energy (7 eV for Cu [54]) and s = —I—. The function c(e)G 2m2}describes the energy dependence of the absorption cross-section in the absence of thebackscattering from the neighboring atoms. The oscillatory part is in the second termwhich is the sum of Lorentzian functions centered at each— EF simulating the peaksproduced by each G. The constant ex is a parameter which describes the fraction of theoutgoing electron wave which is Bragg reflected. The linewidth factor is equal to thesum of the broadening due to the lifetime of the core hole in the K-shell (1.5 eV FWHM[58]) and the lifetime of the excited electron. An estimate for the latter can be obtainedfrom RPA (Random Phase Approximation) calculations of the imaginary part of theelectron self-energy [59] which are in reasonable agreement ( 50%) with experimentalmeasurements at selected energies [60, 61]. For an f.c.c. crystal the allowed reciprocal2,rlattice vectors are of the form G = —(h,k,1) where h k 1 are all even or all odd. For theaenergy dependent cross-section c(E), we have used the smooth-curve fit to theexperimental data shown by the thin dashed line in Figure 4.11. The model spectrum hasbeen convolved with a Gaussian broadening (FWHM 1 eV) to simulate the experimentalresolution.Chapter 4U)4-’CDC04-’00U).6063Figure 4.11: X-ray absorption at the K edge of f.c.c. copper. The experimental spectrumis shown as the solid line [55] and the calculated spectrum from the Bragg reflectionmodel is shown as the dotted line. The thin dashed line is a smooth-curve fit to the datathat is used to model the absorption cross section c(e) in equation (4.3.8).-10 0 10 20 30 40 50Energy (eV)Figure 4.12: X-ray absorption at the K edge of b.c.t. copper. The experimental spectrumis shown as the solid line [55] and the calculated spectrum from the Bragg reflectionmodel is shown as the dotted line. The thin dashed line is the absorption cross sectionused in equation (4.3.8).Chapter 4 64Cl)CDC04-00Cl)--10 0 10 20 30Energy (eV)40 50 60Chapter 4 65As shown in Figure 4.11, the model reproduces all the main features of theexperimental data with the exception of the structure near the origin where the modelshows a shoulder near the zero of energy corresponding to the (111) reciprocal latticevector and a peak corresponding to the (200) reciprocal lattice vector, while theexperimental data shows only one peak. This is because the Fermi surface of copperforms a neck connecting to the (111) Bragg plane due to slight deviations from freeelectron behavior. Thus, there are no empty states that satisfy k=G/2 for G =and one would expect the (111) peak to be absent in the experimental data, as in fact isobserved.In order to see the role of the crystal structure in XANES, we test the model withabsorption data for a new body-centered tetragonal (b.c.t.) form of copper that is grownas an epitaxial thin film on a (100) single crystal silver substrate [55]. This b.c.t. copperfilm is found to have lattice constants a=2.88 A and c=3.l0 A, which is 7.6% expandedvertically from a perfect b.c.c. silver crystal structure [55]. The reciprocal lattice vectorsand their correponding energies are listed in Table 4.2. The model spectrum is obtainedusing exactly the same procedure as in the case of f.c.c. copper. As shown in Figure 4.12,the model also reproduces the positions of all the peaks for the b.c.t. copper. Theabsorption spectra in Figures 4.11 and 4.12 clearly illustrate the role of the crystalstructure in the XANES spectra and show that the model is able to predict the change inthe positions of the absorption oscillations, when the structure of the copper changes.Chapter 4 66Cl).1-’CDC000Cl)-10 0 10 20 30Energy (eV)40 50 60Figure 4.13: X-ray absorption at the K edge of b.c.e. iron. The solid line is the measuredspectrum [55]. and the dotted line is the calculated spectrum from the Bragg reflectionmodel. The thin dashed line is the absorption cross section used in equation (4.3.8).Chapter 4 67G NG CG (eV) G NG CG (eV)101 8 8.4 123 16 58.0110 4 9.0 312 16 60.8002 2 15.7 321 16 62.5200 4 18.0 004 2 63.0112 8 24.8 400 4 72.1211 16 26.5 114 8 72.0202 8 33.8 411 16 80.6220 4 36.1 303 8 76.0013 8 39.9 330 4 81.1301 8 44.5 024 8 81.0310 8 45.1 402 8 87.9222 8 51.8 420 8 90.2Table 4.2: The reciprocal lattice vectors for b.c.t. copper. NG is the degeneracyor the number of planes associated with the reciprocal lattice vector andh2 (G22m2IronThe experimental absorption data for a b.c.c. iron crystal [55] is shown inFigure 4.13. The parameters needed for modelling are as follows: the FermiChapter 4 68energye = 11.1 eV, and the lattice constant a=2.87A [54]. For a b.c.e. crystal theallowed reciprocal lattice vectors are of the form G = where the sum of theindices h+k+l must be even. The reciprocal lattice vectors for iron and theircorresponding energies are listed in Table 4.3. The Fermi momentum of iron is 1.71 A-i,so that (1,1,0)= l.8kF and =2.6kF. We need to determine the sign ofUG just as in the copper case. It is uncertain where the form factor changes sign for Fealthough we know it changes sign at G/ kF 1.5 for Al and at 0/ kF 1.8 for Cu [57,62]. Nevertheless we assume that UG is positive for G equal to or larger than2,r . 2r .—(2,0,0). The sign of UG for G = —(1,1,0) as not significant, because is belowa athe Fermi energy in this case.The model spectrum shown in Figure 4.13 is obtained by the same procedure thatG NG I e (eV) h G NG 8G (eV)110 12 9.1 222 8 54.8200 6 18.3 123 48 63.9211 24 27.4 400 6 73.0220 12 36.5 411 24 82.2310 24 45.7 330 12 82.2Table 4.3: The reciprocal lattice vectors for b.c.e. iron. NG is the degeneracy orthe number of planes associated with the reciprocal lattice vector and1i2 (G\2C =—I—G 2m2Chapter 4 69was used in the case of the f.c.c. copper. Once again, the model reproduces all the mainfeatures of the experimental data, except at the absorption edge where the experimentaldata shows an extra shoulder. This extra shoulder at the edge of the experimentalspectrum may be associated with d bands or other breakdown in the nearly free electronpicture, close to the Fermi energy. The 0.05 eV exchange interaction in iron [771 hasbeen neglected in this analysis.SiliconXANES spectra for semiconductors and insulators are more complicated than formetals in the vicinity of the absorption edge because of core hole effects. The core holepotential becomes more important as the screening decreases. In cases where the corehole potential is important, the core hole not only changes the shape of absorption spectra[63] but it may also generate new features in the absorption edge such as excitonic states,which are not included in the Bragg reflection model. There is another difficulty forsilicon in addition to the core hole effects. Silicon is a covalent solid in which the bottomof the conduction band is made up by the antibonding nearly s-like sp hybrid. The nearlyfree electron approach on which the Bragg reflection model is based is likely to be a poorapproximation for the bottom part of the conduction band, which has a tight bindingcharacter. Thus, the Bragg reflection model is not expected to work as well near theabsorption edge in silicon as it does at higher energies.As is well known silicon has the diamond structure consisting of a f.c.c. Bravaislattice with a basis of 000 and The conventional cube lattice constant of silicon isChapter 4 70G NG SG CG (eV) G NG SG 8G (eV)111 8 (1±i)/2 3.8 620 24 1 51.0220 12 1 10.2 533 24 (1±i)/2 54.8311 24 (1±i)/2 14.0 444 8 1 61.2400 6 1 20.4 551 24 (l±i)12 65.0331 24 (l±i)12 24.2 711 24 (1±i)/2 65.0422 24 1 30.6 642 48 1 71.4333 8 (1±i)/2 34.4 553 24 (1±i)/2 75.2511 24 (1±i)12 34.4 731 48 (1±i)/2 75.2440 12 1 40.8 800 6 1 81.6531 48 (1±i)/2 44.6 733 24 (1±i)12 85.4Table 4.4: The reciprocal lattice vectors for silicon with non-zero structurefactors. NG is the degeneracy or the number of planes associated with theh2 G2reciprocal lattice vector G. SG is the structure factor, and CG I —2m’\25.430 A [54]. Table 4.4 lists all the reciprocal lattice vectors that yield non-zero structurefactors along with the corresponding free electron energies EG=The pseudopotential form factor of silicon for small k values (k/kp <2) is readilyavailable. For example, Harrison gives: fs1(111) = -1.84 eV, f51(220) = 0.61eV. Theform factor crosses from negative to positive at kfkF = 1.5 where kF = 1.8 A-i and isdefined by the valence band width [62]. We assume that the form factors for reciprocalChapter 4 71U)CD.cC00U).0Figure 4.14: X-ray absorption at the K edge of crystalline silicon. The experimentalspectrum is shown as the solid line [641 and the calculated spectrum from the Braggreflection model is shown as the dotted line. The vertical bar indicates the origin of thefree electron energy CG.-10 10 30 50 70Energy (eV)Chapter 4 72lattice vectors larger than (220) are all positive. Therefore, these reciprocal lattice vectorsshould all generate peaks in the Si K edge spectrum.A comparison between the experimental data [64] and the Bragg reflection modelis shown in Figure 4.14. The absorption intensity near the absorption edge is enhanceddue to the core hole effect as expected [63]. No effort was made to account for this effectin the model. The model spectrum was obtained by summing Lorentzians centered ath2 (G2 .each energy eG = —I — I , which simulates the effect of the Bragg reflection planes.2m2)No effort was made to fit the energy dependent cross section in the absence of the Braggreflections, as was done for the metals. Each Lorentzian was weighted by the scatteringstrength of each G which was approximated by the product of the number of planes NGas a function of orientation and the square of the structure factor SG 12. The scatteringstrength dependence on the potential form factor is ignored for simplicity since the formfactor is more slowly varying than the other two factors.The vertical bar in Figure 4.14 indicates the origin of the free electron energy 8GThe best fit (by eye) to the experiment was obtained with the bar at 8 eV below theabsorption edge. This suggests that the valence electrons should occupy the bottom partof the free electron parabolic band. As the crystalline silicon has a small band gap(relative to insulators) of 1.1 eV, its valence electrons are tight-binding electronsintermediate between core electrons and free electrons. As a result, the valence band canbe interpreted as being evolved from the bottom of a parabolic free electron band. In thish2kpicture, the origin of the free electron energy e= 2mshould be in the vicinity of theChapter 4 73bottom of the valence band. From the 12.5 eV band width of the valence band of silicon[65], one obtains 13.6 eV as the energy separation from the bottom of the valence band tothe bottom of the conduction band in silicon. As the silicon valence electrons are not freeelectrons, the discrepancy between the 8 eV separation suggested by the Bragg reflectionmodel and the 13.6 eV separation is not surprising. Overall, the Bragg reflection modelgives at best a qualitative fit to the silicon K edge spectrum.NeonSolid neon forms a f.c.c. lattice with the lattice constant a 4.426 A. Table 4.5lists all the allowed reciprocal lattice vectors with their corresponding energies ofh2 (G2=—I —2m2The comparison between the model and the experiment is shown in Figure 4.15.Experimental data for the neon K edge is shown in the bottom part of the figure [66].Solid neon was formed on an Al foil at 6.3 K which was attached on a sample holderfixed on a cold end of a liquid He cryostat [66]. The first two sharp features at 868.3 eVand 869.6 eV respectively have been interpreted as excitonic transitions [66]. The modelspectrum is shown in the top part of Figure 4.15 which was obtained in the same manneras before except the (000) peak was added to the spectrum to indicate the origin of eG.Chapter 4 74G NG 8G (eV) G NG 8G (eV)111 8 5.8 440 12 61.4200 6 7.7 531 48 67.2220 12 15.4 600 6 69.1311 24 21.1 442 24 69.1222 8 23.0 620 24 76.8400 6 30.7 533 24 82.6331 24 36.5 622 24 84.5420 24 38.4 444 8 92.2422 24 46.1 551 24 97.9511 24 51.8 711 24 97.9333 8 51.8 640 24 99.8Table 4.5: The reciprocal lattice vectors for solid neon. NG is the degeneracy orthe number of planes associated with the reciprocal lattice vector andh2 (G”2eG =—I —2m2We can think of the solid neon as being held together by the weak Van der Waalsinteraction between neutral closed-shell atoms. The ionization energy of neon is 21.7 eV[42] which means the gap between the occupied and unoccupied states is rather large.Thus, the parabolic nearly free electron band should be a good approximation for theneon conduction band and the origin of EG should be close to the bottom of theconduction band, which we indeed observe in Figure 4.15. Since solid neon is insulating,Chapter 4 75>4HU)zUH0Solid Ne K edge-10 10 30 50 70Energy (eV)901.51.50Figure 4.15: X-ray absorption at the K edge of solid neon. The experimental spectrum isshown in the lower part [66] and the calculated spectrum from the Bragg reflection modelis shown in the upper part.860 880 900 920 940 960 980PROTON ENERGY / eVChapter 4 76the less screened core hole effect dominates the first 5 eV of the absorption edge which isnot expected to be accounted for by the Bragg model. Neither did we attempt toreproduce the shift of weight from the higher energy part to the lower energy part due tothe attractive core hole potential. Setting this aside, a rather good agreement is observedbetween the model and the experiment starting from 5 eV above the absorption edge.The features in the experimental spectrum become smaller and smaller compared to theones in the model spectrum starting from 40 eV above the edge. This can be expectedbecause we have assigned equal weight to all the Fourier coefficients UG as anapproximation while in fact one would expect UG to decrease at large G. Also this isalready in the EXAFS region where the photoelectron wavelength is much smaller thanthe nearest interatomic distance to produce strong scattering by the crystal planes [251.Calcium FluorideFinally, we attempt on a more complex solid in which there are two differentatoms and the core hole effect is strong at the same time, namely the fluorine K edge inCaF2. CaF2 is a wide band gap (Eg = 12.1 eV [67]) ionic insulator with the fluoritestructure. The crystal forms a f.c.c. lattice with a basis consisting of a Ca2 ion at 000,and two F- ions at-4 and --j--- respectively. The lattice constant a = 5.46 A [34].Consequently, the Fourier coefficient of the crystal potential includes form factors forboth Ca2 and F- ions. We have, with the origin at a Ca2 site,U(Ca2j= j{fCa(G)+ 2fF(G)cos[(nl + n2 +n3)]}Chapter 4 77and with the origin at a F- site,1 i—(n+n23) ICUG(F) = 2 2f(G)cos[-(n1+ n2where n 1, n2, and n3 are indices of the reciprocal lattice vector G as expressed byG =-(n1,n2, n3). As shown in Table 4.6 UG(Fj for (111), (311) and so on isimaginary and UG(F) for (200), (220) and so on is real. We have shown earlier that animaginary UG or a positive UG will generate a peak in K edge XANES. We do not knowwhether UG(Fj for the (200) series is positive or not. Nevertheless, we assign peaks foreach G and make peak intensities only dependent on the degeneracy number NG forsimplicity.Chapter 4 78G NG UG(F) CG (eV) H G NG UG(F) eG (eV)111 8 Ca 3.8 422 24 (2fF+fCa) 30.3if i200 6 2FCa) 5.0 511 24 Ca 34.1220 12 (2fF+fCa) 10.1 333 8 Ca 34.1311 24 Ca 13.9 440 12 (2fF+fCa) 40.41/ i222 8 2FCa) 15.1 531 48 Ca 44.1if if400 6 F Ca) 20.2 600 6 F Ca) 45.4i if331 24 Ca 24.0 442 24 2FCa) 45.4i(420 24 F’Ca) 25.2 620 24 V’F’Ca) 50.5Table 4.6: The reciprocal lattice vectors for CaF2. NG is the degeneracy or thenumber of planes associated with the reciprocal lattice vector G. UG is theFourier coefficient of the crystal potential where Ca and fF are the form factors ofh2 G2the Ca2 ion and the F- ion respectively, and e = —1 —G 2m’\2Since the band gap of CaF2 is large, we assume that the origin of 8G would beclose to the bottom of the conduction band (like the case of neon and unlike the case ofsilicon) for the parabolic nearly free electron band would be a good approximation for theCaF2 conduction band. Figure 4.16 shows the comparison between the experimentalspectrum of the fluorine K edge in CaF2 and the model spectrum. The experiment hasbeen described in §4.2. The first sharp peak (689.3 eV) is due to core exciton transitionand the bottom of the conduction band is believed to be at the minimum between the firsttwo high intensity peaks (690.7 eV) [48].(1)CDC0I0Cl)Photon Energy (eV)Chapter 4 79liii,.. 111111 111111 IlIlilIll IlillIllIll IllilIllIlFluorine K edgec)0o —---o ‘-0‘-0:•: R.’:• •_•.___•0I.—0C’, I I ‘4,__ C’,C) . C’J C’): : .!_ .‘ •——.•1111111111111111111111111111111111111111111111111680 690 700 710 720 730Figure 4.16: X-ray absorption at the fluorine K edge in CaF2. The solid line shows theexperimental spectrum and the dotted line shows the calculated spectrum from the Braggreflection model.Chapter 4 80The model spectrum was obtained with the same manner as before. The (000)peak was added to the spectrum to indicate the origin of 8G As shown in Figure 4.16, agood agreement between the experimental spectrum and the model spectrum can beobserved in the non-excitonic part although there is some discrepancy in the energiesparticularly for the two low energy peaks indicated by (111) and (200). This is notsurprising considering the deviations from free electron behavior one might expect inCaF2 near the bottom of the conduction band and also the fact that the details of theatomic form factor are ignored in the Bragg model. There is another notable feature thatthe model does not seem to describe well in the photon energy range of 710 - 725 eV.This prominent feature in the experimental spectrum has a large oscillator strength and isin the region where the photon energy is high enough to create multi-electron excitations.We will discuss this possibility later in §4.5 since it is beyond the framework of theBragg reflection model we are presenting here.Now that we have illustrated the Bragg reflection model on a number of systems,we summarize the model as we conclude this section. The Bragg reflection model givesthe peak positions approximately. We know that peak positions are below theh2 ,G.2energy of—f — 1 and that UG varies with G. However, the basis of this model is the2m’\2)h2nearly free electron approximation that requires IUG << —I — I , in which case, the2m2)h2 G”2deviation of the peak positions away from the energy of—I — I is small. Therefore,2m\2Iwe expect the Bragg reflection model to be valid for materials with nearly free electronlike conduction bands. For metals the bottom part of the parabolic nearly free electronChapter 4 81band is occupied up to the Fermi level so that the origin of 8G is 8F below the bottom ofthe conduction band. For semiconductors or systems with small band gaps the valenceband occupies the bottom part of the parabolic nearly free electron band and the origin of8G is close to the bottom of the valence band. For insulators or systems with large bandgaps, the origin of 8G is close to the bottom of the conduction band since the valenceelectrons in these systems are too deeply bound to be approximated as nearly freeelectrons.The model cannot account for features caused by core hole potentials. The corehole potential is most important when there is little screening such as in insulators. In thecase of the fluorine K edge in CaF2, for example, the excitonic peak is prominent in themeasured absorption edge, and not present in the Bragg reflection model. The model alsodoes not reproduce the enhancement of the absorption intensity at the edge due to theattractive core hole potential. After all, the Bragg reflection model is a semi-quantitativemodel in which the cross section between the initial core state and the final statemodulated by the scattering is not actually calculated. In order to have a morequantitative understanding of XANES spectra including not only positions but alsoshapes and intensities of the features, and especially the influence from the core holepotential, one must actually calculate the final states resulting from both the long rangescattering by the crystal potential and the attraction by the core hole potential. This iswhat we intend to do in the next section.Chapter 4 82§4.4 Bandstructure Calculation of XANES at the Fluorine K Edge in CaF2 andBaF2Separate approaches are often used to calculate the excitonic part and the higherenergy part of the XANES spectrum for insulators like CaF2. For example, in the case ofthe calcium L edge in CaF2 which is dominated by the localized excitonic features at theedge, the dominant features at the edge were explained by the atomic Ca 3d-like exciton,and the atomic spin-orbit splitting and the crystal field splitting in the Ca 2p —> 3dtransitions [681. The relatively weak features in the higher energy region were studiedseparately. They were shown to agree qualitatively with the Bragg peaks in a syntheticspectrum, formed by adding two experimental fluorine K edge spectra with one shifteddown in energy by 3.5 eV to simulate the spin-orbit splitting of Ca 2p, suggesting thecommon Bragg scattering origin of these features in both calcium L edge and fluorine Kedge [53].The best quantitative model so far for the fluorine K edge absorption in CaF2 is amultiple scattering calculation on a 23 atom cluster [32]. However this model does notshow as many peaks as the experimental spectrum. Presumably the cluster is not largeenough to describe the crystal potential over a sufficiently long range to give an accuratedescription of the scattering of the photoexcited electron in the final state. The fluorine Kedge absorption is difficult to calculate accurately because of the need to include both thelocalized potential of the core hole and the long range periodic potential of thesurrounding crystal lattice at the same time. Conventional electronic structurecalculations are typically designed to address either the molecular limit or the long rangeChapter 4 83periodic potential (band structure) limit [33]. In the fluorine K edge problem, opticaltransitions must be calculated between a strongly localized F is core electron and a finalstate which depends on the localized core hole potential as well as the long range periodicpotential of the crystal. Interactions with a large number of neighboring atoms must beincluded because the inelastic mean free path for the photo-excited electrons can be large,especially for electrons with a kinetic energy less than the threshold for interbandtransitions, which is 12.1 eV for CaF2. This is an unfavorable situation for multiplescattering calculations because many scattering paths must be considered if the inelasticmean free path is long. In order to improve our understanding of the fluorine K edge inCaF2 and BaF2,in this section we adopt a complementary approach to the earlier multiplescattering calculations and use a one-electron band structure technique with apseudopotential approximation for both the crystal potential and the core hole potential.We calculate the absorption coefficient from the transition rate given by standardtime-dependent perturbation theory,— i(EfIpIElS)26(El — Ef — ha,) (4.4.1)where ha, is the photon energy, IE1) is the initial is state, lEf) is the final state in theconduction band, and p is the momentum operator. A one electron Hamiltonian is usedto calculate the final states in the conduction band: p2/2m + U + Uc. This Hamiltonianincludes the potential U of the ions in the absence of the core hole as well as the potentialUc of the photoexcited core hole.With a plane wave basis, the one electron Schrodinger equation is,Chapter 4 84(e_G —e)(k—GIy)+ (uGP_G +UG_G)(k—G j)— 0 (4.4.2)G’Gwhere k is the wave-vector; G and G’ are reciprocal lattice vectors defined by the crystalstructure including the artificial supercells used to model the core hole potential, asdescribed below. UG and U are the Fourier coefficients of the crystal potential and thecore hole potential respectively.The crystal potential U is approximated by a pseudopotential in which each ion isrepresented by a truncated Coulomb potential. We express U in terms of the lattice sumof the potential due to each ion in the crystal,U = Ua(r—rj) (4.4.3)aiawhere the potential due to the ex ion at na is expressed asZe2— r—r. >rla4,rse r—r./0 laUa(r—r) = (4.4.4)Ze2— r—r.42reeora laThat is, more than a pseudopotential ionic radius ra from the ion the potential of the ionis represented by the Coulomb potential associated with a Za charged ion in a dielectricmedium. Inside the ion the potential is assumed to be a constant equal to the potential atthe pseudopotential ionic radius ra. Thus in addition to the crystal structure threeparameters are needed to determine the crystal potential U: the pseudopotential ionicradius of the Ca2 (or Ba2) ion, the radius of the F- ion, and the dielectric constant e.Chapter 4 85The value of the pseudopotential core radius for Ca in Harrison’s solid state table [62],O.9A, was taken for the Ca2 ionic radius. While an F- ionic radius of 1.3A is often usedfor fluoride compounds [62] we found it necessary to treat the ionic radius of F- as aparameter in the calculation; accordingly we found that a F- radius of i.oA gave the bestresults when compared with the experimental data. The appropriate value for thedielectric constant is determined from a consideration of the lifetime of the excited state.The natural width of the F is level is about 0.2 eV [69] which is small compared with thelinewidth of the features observed in the low energy part of the experimental spectra. Weattribute the observed linewidth to experimental resolution and broadening associatedwith the lifetime of the photo-excited electron. In any case the lifetime broadeningobserved in the experimental data is less than about 5 eV for all the peaks less than 30 eVabove the edge. The corresponding excited state lifetime is sufficiently long that we canuse the optical dielectric constant e of 2 appropriate for CaF2 in the photon energy range0.1- 10 eV [70], in the expression for the pseudopotential. Any wave vector dependenceof the dielectric constant is neglected.The potential energy U’ due to the core hole is approximated by leaving out thepotential energy included in U due to the F ion that is located at the site of the transition.This is consistent with the fact that a fluorine ion with a core hole is electrically neutral.Clearly a core hole at one site breaks the translational symmetry of the crystal. As anapproximation to the true non-periodic potential, we use a super unit cell which containsa single core hole but several standard unit cells and then periodically repeat the superunit cell to obtain an artificial crystal with a periodically repeated core hole. This methodhas been used earlier for including the core hole effect in x-ray absorption spectraChapter 4 86[63,71]. In our case the super unit cell is a simple cube with sides four times the nearestneighbor fluorine-fluorine distance; it contains 32 Ca and 64 F atoms and its size islimited by our available computing power. The super unit cell is schematically shown inFigure 4.17.Since U has the same periodicity as the crystal and Uc has the periodicity imposedby the supercells, the Fourier coefficient UG is non-zero for G = (nl,n2,n3)21t/a and UCGis non-zero for G = (ml,m2,m3)1r/a, where the integers n, fl2 and fl3 must either be alleven or all odd as required by the fcc symmetry, while the integers mi, m and m areunrestricted.The Fourier coefficients are obtained from the Poisson equation V29, = —. Weshow in the following, the advantage of determining the Fourier coefficients of the crystalpotential U from the Fourier coefficients of the charge distribution. The potential is moredifficult to evaluate directly than the charge density since the tails of the Coulombpotential originating from every ion in the crystal contribute to the total potential at anygiven point in the space. On the other hand, the charge distribution of each ion islocalized with no long range contribution. From the Poisson equation, the definition ofthe potential energy U = —ep, and V2U =G2UGexp(—iG. r), it is clear thatGUG=—-- (4.4.5)Gewhere the Fourier coefficient of the charge distributionPG = Jp(r)exp(—iG. r)d3. (4.4.6)one cellChapter 4 8700Fluorine IonFluorine with Core HoleCalcium IonFigure 4.17: The schematic diagram showing the super unit cell. Each super unit cellcontains one core hole and 96 atoms.Chapter 4 88Let pa(r) denote the charge distribution of a single ion a. With the approximation onthe potential made in eqn(4.4.4) which is equivalent to having the charge Za uniformlydistributed on the surface of a sphere of radius ra, we have,Pa(r)4,rra26(r—ra), (4.4.7)and! JPa(r)exp(_iG. r)d3all space= Zaesin(Gra) (4.4.8)V GrThus,PG =pexp(iG.r1)a 1aZaesin(Gra)(.G‘I (4.4.9)V Gra \ 1/a 1awhere the summation goes over all the ions within a single unit cell, and na denotes theposition of the ith ion of the a species. For CaF2, we set the unit cell to consist of one3Ca2 at 000, one F- at U! and the other F- at !!!. The volume of the unit cell V =444 444 4Thus, with Zat+2and ZF=- 1, we have,UG— G3ee a3 {-sin(Grca) — _2cos[(ni + +n3)]sin(GrF)} (4.4.10)Chapter 4 89where i, n and fl3 are indices of the reciprocal lattice vector G as expressed by2,rGOne thing worth pointing out is that, unlike many other commonly usedapproximations such as the muffin-tin approximation in which the potential is arbitrarilyset to zero at the interstitial space between atoms that introduces discontinuity in thepotential between two different kinds of atoms, the Fourier coefficients obtained witheqn(4.4. 10) imply no discontinuity in the potential. The best of all, the model potentialused here is closer to the reality since it includes the Madelung potential which could beimportant. Table 4.7 shows the numeric values of the Fourier coefficients of the crystalpotential U with the origin 000 chosen at the Ca site. Since the Ca site has the inversionsymmetry, this choice of the origin yields real values for all the Fourier coefficients of thepotential U. One should note, however, it is more convenient to have the origin at the Fsite for the fluorine K edge absorption calculations since the initial is state involved inthe absorption process is at a F site. One can simply translate the Fourier coefficients ofU with the origin at the Ca site to the Fourier coefficients of U with the origin at the F siteby UG(Fsite)= UG(Casite)exP[i-(nl +n2+n3)] in the fluorine K edge absorptioncalculation.The Fourier coefficients of the wave function are truncated at (3,1,1)2icla; thisreduces the matrix defined by eqn(4.4.2) to 1029x 1029. Eigenvalues and eigenvectors ofthe matrix are obtained at each k point in the first Brillouin zone and are then used ineqn(4.4. 1) to calculate the absorption. Because of the crystal symmetry, only the sectionof the zone defined by k, k k, which is 1/48 of the first Brillouin zone, needs to beChapter 4 90G UG (eV)111 -0.609200 -0.628220 -0.045222 0.086311 0.026400 -0.003331 0.038420 0.066422 0.011333 0.018511 0.018Table 4.7: The Fourier coefficients of the crystal potential U used in thecalculation of the fluorine K edge absorption in CaF2. The dielectric constant e of2 is used. The values of the Fourier coefficients are obtained with the Ca site asthe origin.included. It is scanned in steps of irl4a in k-space. The absorption coefficient is averagedwith the momentum operator p oriented in the x, y, and z directions respectively, whichtakes into account the random crystal orientation relative to the photon polarization.Chapter 4 91The pseudopotential we use in the final states calculation has less amplitude atlarge wave numbers than the real potential since the pseudopotential approximates therapidly varying potential near the ion core with a slowly varying (constant) potential.This makes it possible to calculate the electron energy eigenvalues accurately with arelatively small number of plane waves while truncating the Fourier expansions of thewave functions in k-space. While the pseudopotential can give a good representation ofthe energy band structure it does not correctly reproduce the shape of the wave functionsin the vicinity of the ion core. This can cause problems with the calculation of the opticalmatrix element between the conduction band state and the F is core level which isstrongly localized at the ion core. In order to obtain an accurate value for this matrixelement we need to know the detailed shape of the conduction band wave functions at theion core position which is not possible with the pseudopotential approximation. Aconsequence of this loss of high spatial frequency infonnation is that the size of thefeatures in the x-ray absorption are reduced because of the small overlap matrix elementbetween the high spatial frequency is wave function and the slowly varying wavefunctions of the pseudopotential in the final state. To compensate for this we use a“pseudo-core level” in the absorption calculations in place of the true is orbital. Thepseudo-core level that we use has a lower binding energy (150 eV rather than 697 eV)compared with the true is core level in order to increase the overlap with theeigenfunctions of the pseudopotential in the empty states.The calculated CaF2 fluorine K edge absorption spectrum is shown in Figure 4.18.The spectrum has been convoluted with an energy dependent Lorentzian to take intoaccount the final state lifetime, and a 1.0 eV wide Gaussian to reflect the experimentalChapter 4 92resolution. The energy dependence of the width parameter f in the Lorentzian isapproximated with an empirical expression designed to take into account the increase inthe electron inelastic scattering at the threshold for interband transitions. The energydependence of F used in the model spectrum is shown in Figure 4.22.There is a weak feature (about 0.5% of the highest peak) at the foot of theabsorption edge (686.6 eV) in the model spectrum which is the s-like ground state of thepseudopotential representing the core hole. This feature is too small to see in thecalculated spectra for the vertical scales shown in Figures 4.18-23. Since the core level isalso an s-state the absorption cross section for this state is weak because the pure s to stransition is dipole forbidden. The first large peak in the model absorption edge is anoverlapping combination of 2s and 2p-like states. Both the is and 2sip peaks areexcitonic in the sense that they lie below the bulk crystal conduction band edge. In thereal fluorine ion potential we expect the first large peak in the absorption to be acombination of 3s/3p-like states and not 2s/2p. The next lower states in the realphotoexcited fluorine ion potential are the filled 2p and 2s states and the half-filled F iscore level. There is no state comparable to the is-like state that is present with ourpseudopotential. Thus the 2s/2p-like wave functions in the pseudopotential represent the3s/3p-like wave functions of the real potential. The purpose of the pseudopotential modelis to replace the real potential with a model potential that has the same energyeigenvalues but is mathematically easier to solve. One accomplishes this by replacing therapidly varying potential in the vicinity of the ion core with a slowly varying averagepotential. The wave functions of the slowly varying potential will approximate the realwave functions between the ion cores but will have a smaller number of nodes in theChapter 4 93vicinity of the ion core. The is level in the pseudopotential will be ignored as an artifactthat has no counterpart in the real potential.The effect of changing the binding energy of the initial is state is shown inFigure 4.19. The calculated spectra in Figure 4.19 were obtained with the best fit valuesof the Ca2 radius of 0.9 A and the F- radius of 1.0 A. As shown in Figure 4.19, the useof the pseudo-core level with a different binding energy in place of the true F is corelevel only affects the amplitude of the absorption edge features and does not addadditional structure or change their energy position since the empty state wave functionsare still more slowly varying than the pseudo-core wave function.To illustrate the effect of changing the pseudopotential core radii of F- and Ca2in the absorption calculation, we show in Figure 4.20 the calculated spectra with fixedCa2 radius of 0.9 A, and different F- radii, namely 1.3 A, 1.0 A and 0.8 A respectively.Similarly, in Figure 4.21, we show the calculated spectra with the F radius fixed at 1.0 A,and different Ca2 radii, namely 1.1 A, 0.9 A and 0.7 A respectively. The binding energyof the initial is state used in obtaining the spectra in Figure 4.20 and Figure 4.21 is thebest fit value of 150 eV. The peak spacings are not very sensitive to the change of thepseudopotential core radii, while all the peaks move towards smaller photon energy as thepotential strength is increased by reducing the pseudopotential core radii. This isconsistent with an earlier finding that peak spacings are more or less determined bycrystal structures [53]. The change of the pseudopotential core radii is found to havemore effect on the amplitudes and shapes of the peaks.Chapter 4 94C,)DC0ci00680 700 720 740Photon Energy (eV)Figure 4.18: The experimental fluorine K edge absorption spectrum for CaF2 is shown atthe top and the calculated absorption spectrum is shown at the bottom. The peaks labeleda, b, c, and d are used in Figure 4.24.Chapter 4 95Cl)CDC00I—0U)680 690 700 710 720 730Photon Energy (eV)Figure 4.19: Calculated fluorine K edge absorption spectra for CaF2 with differentbinding energies of the initial is state. The Ca2 radius of 0.9 A and the F- radius of1.0 A were fixed.Chapter 4 96U)CDC0I0U).0680 690 700 710Photon Energy (eV)Figure 4.20: Calculated fluorine K edge absorption spectra for CaF2 with different Fradii as indicated. The Ca2 radius (0.9 A) and the binding energy of the F is initial state(150 eV) were fixed.I I IllillIllIl 11111111111rF=l.3 Ar= i.oArF=O.7 AI I I I I i i i i I i i i.i I i i i i I I I I I720 730Chapter 4 97Cl)DS00C,)0680 690 700 710 720 730Photon Energy (eV)Figure 4.21: Calculated fluorine K edge absorption spectra for CaF2 with different Ca2radii as indicated. The F- radius (1.0 A) and the binding energy of the F is initial state(150 eV) were fixed.11111 I 111111111111111111rCa= uArCa= O.9AIIIIIIIIIIIIIIIIII I 11111Chapter 4 98680 690 700 710Photon Energy (eV)Figure 4.22: The calculated fluorine K edge absorption spectrum for CaF2 including thecore hole potential is shown at the top. The second spectrum is the same calculationexcept the core hole potential has been omitted. The third spectrum is the calculatedglobal density of states with the crystal potential and the core hole potential left out as inthe second spectrum. The dotted line shows the broadening function r used in thehuh IhIlIhuIlIhI 111111Absorption with Core Hole5Density of StatesI I 11111111111 I Ii I 111,1110720 730Lorentzian convolution of all the spectra.Chapter 4 99The core hole potential has the effect of increasing the absorption cross section inthe vicinity of the absorption threshold, at the expense of the absorption cross section athigher energy, as expected [71]. To illustrate this we show in Figure 4.22 the absorptionspectrum calculated with and without the core hole potential. The spectra are convolutedwith the same broadening functions as in Figure 4.18 above. The first large peak in themodel absorption spectrum with the core hole lies 1.8 eV below the edge jump in thespectrum without the core hole. This is consistent with experimental observations whichshow that the first peak in the CaF2 fluorine K edge absorption spectrum is excitonic witha binding energy of about 1.0 eV [48].The global density of states for CaF2 calculated with the model crystal potential isalso shown in Figure 4.22. Except for the gap in the density of states near 696 eVproduced by U111 and U200, the structure in the density of states is generally lesspronounced than the structure in the absorption. This shows that the optical matrixelement is important in determining the structure in the absorption edge, as expected [28].The calculated absorption spectrum in Figure 4.18 reproduces the main features inthe absorption edge reasonably well up to about 15 eV above the absorption threshold.As mentioned above, the first peak is excitonic, due to the attractive core hole potential.The next four peaks (a-e in Figure 4.18) are due to selective coupling between the is corelevel and final state standing waves scattered by the crystal planes. Their spacing shouldbe sensitive to the crystal lattice constant. As pointed out earlier the energy spacing ofthese peaks is consistent with successively higher order Bragg backscattering reflectionsfrom the surrounding crystal lattice [53]. The high intensity of the first peak after theChapter 4 100G UG (eV)111 -0.478200-0.590220 0.036222 0.067311 0.038400 0.004331 0.032420 0.061422 -0.008333 0.009511 0.009Table 4.8: The Fourier coefficients of the crystal potential U used inthe calculation of the fluorine K edge absorption in BaF2. The dielectricconstant c of 2 is used. The values of the Fourier coefficients are obtainedwith the Ba site as the origin.exciton (peak a) relative to the ones at higher energy is due to the fact that the states at thebottom of the conduction band are more strongly affected by the core hole potential andhave a larger amplitude on the excited atom and hence a larger absorption cross section.Chapter 4 101As a test of the structural interpretation of the peaks a-d we calculate the fluorineK edge of BaF2which has the same crystal structure as CaF2 but a larger lattice constant.We use the same value for the F ionic radius as in the CaF2 case (1.0 A) and take theBa2 radius to be i.iA. We take the dielectric constant to be 2. Table 4.8 shows thenumeric values of the Fourier coefficients of the crystal potential U with the originchosen at a Ba2 site. As in CaF2, the binding energy of the pseudo-is core state wasreduced to 150 eV to better couple the core wave function to the final state wavefunctions obtained from the pseudopotential approximation and improve the quality ofthe fit to the relative amplitude of the experimental peaks. A comparison between theexperimental absorption spectrum and the calculated absorption spectrum is shown inFigure 4.23. The calculated spectrum was convoluted with the same broadeningfunctions as in the CaP2 case discussed above.The fluorine K edge in BaF2 has a large excitonic peak at the threshold similar toCaF2. The next higher energy peaks are closer together and smaller in amplitude than inCaP2 for both the experimental data and for the model. As pointed out above we attributethe peaks a-d to the effect of electron standing waves reflecting from crystal latticeplanes. Assuming that the final states are free electron-like, and that the Bragg conditiondescribes the electron reflections from the crystal lattice, then the product aIE, where Eis the kinetic energy of the photoelectron and a is the lattice constant, should be constantfor corresponding peaks in the absorption spectra of CaF2 and BaF2 [72]. (This relationfollows from the condition for Bragg backscattering reflections, k = G12.) To test thisidea, in Figure 4.24 we plot a-4JE for the first four non-excitonic peaks for BaF2 againstthe corresponding quantity for the same four peaks in CaF2, for both the experimentalChapter 4 102spectra and for the models. The origin in energy is taken at the minimum in theabsorption just above the excitonic line in both cases. This approximates the bottom ofthe conduction band. The lattice constants are 6.20A and 5.46A for BaF2 and CaF2respectively [34J. If the nearly free electron interpretation is correct, then we wouldexpect the points in Figure 4.24 to fall on a line through the origin with unity slope. Asillustrated in Figure 4.24 both the theoretical and the experimental spectra agreereasonably well with this prediction; the agreement is somewhat better for the calculatedspectra than for the experimental ones.Chapter 4 103C,)D£C000Cl)680 700 720 740Photon Energy (eV)Figure 4.23: The experimental fluorine K edge absorption spectrum in BaF2 is shown inthe top spectrum, and the calculated spectrum is shown at the bottom. The peaks a, b, c,and d used in Figure 4.24 are indicated.Chapter 4 10425201510510a(E.)112Figure 4.24: A plot of the position of the peaks a-d for BaF2 from Figure 4.23 as afunction of the position of the corresponding peaks in CaF2 from Figure 4.18. The peakpositions are plotted as the square root of the energy of the peak above the bottom of theband multiplied by the appropriate lattice constant. The bottom of the conduction band istaken to be the minimum in the absorption between the first two large peaks in theabsorption spectrum. The peak positions from both the experimental and the calculatedspectra are plotted as indicated.5 2015for CaF225Chapter 4 105§4.5 Multi-electron Excitations in XANESIn the experimental absorption data for CaF2 (Figure 4.18), there are broad andprominent features in the absorption in the region 15 eV-35 eV above the edge. Similarbroad features are even more prominent in the BaF2 edge, although at a smaller energy(Figure 4.23). These features are in a region of the spectrum where the band structureapproach is no longer valid due to the computational limitations on the number of planewaves we are able to include in the calculation. Because the broad features have aqualitatively different shape and larger oscillator strength than the lower energy peaks ad, particularly in BaF2, one is tempted to conclude that they have a different physicalorigin. For example they may be due to resonant transitions to higher angular momentumstates such as d and f-like states associated with the cations in the CaF2 and BaF2conduction bands respectively [73]. Since the d and f like states are on the cations andthe is core hole is on the anion, such transitions do not violate the dipole selection rule.A further complication is that in this region the photon energy is high enough to createmulti-electron excitations such as plasmons, in addition to the is core hole.In x-ray absorption experiments multi-electron transitions are difficult todistinguish from crystal structure related peaks in single electron transitions because thestructure from the two types of transitions overlaps. In order to isolate the spectrum ofthe multi-electron excitations from the single electron excitations we have measured theloss satellites on the F is photoemission peak by x-ray photoelectron spectroscopy (XPS)using a 1254 eV Mg Ka x-ray source. The multi-electron loss satellites in XPS togetherwith the main photoemission peaks of the F is core levels for CaF2 and BaF2 are shownin Figure 4.25 and Figure 4.26 respectively. The energy losses associated with the majorChapter 4 106N.—C>CDCaF2Binding Energy (eV)Figure 4.25: The x-ray photoemission spectrum of the F is core level and its satellites inCaF2 measured with the Mg K x line (1253.6 eV). The labels on the satellite peaksindicate the energy separation from the main peak.35.6 eV 27.4 eV1.0O.&0.60.417.2 eVX4.7730 720 710 700 690 680Chapter 4 107N0IBaF2Binding Energy (eV)Figure 4.26: The x-ray photoemission spectrum of the F is core level and its satellites inBaF2 measured with the Mg K z line (1253.6 eV). The label on the satellite peakindicates the energy separation from the main peak.124.6 eV730 720 710 700 690 680Chapter 4 108satellite features are labeled in the figure. The satellites are similar to those observed inelectron energy loss (EELS) measurements [74, 75] except that the feature with 27.4 eVenergy loss is significantly stronger in the XPS than in EELS. The larger intensity inXPS compared with EELS suggests that there is an intrinsic process in XPS in whichplasmons are created by the sudden appearance of the core hole in addition to excitationby the final state electrons. We interpret the feature at 17.2 eV loss as an interbandtransition in which a valence electron is excited to the conduction band, and the feature at35.6 eV as the excitation of a shallow core electron (F 2s or Ca 3p) to an empty state.Although the loss peaks are at approximately the right energy to account for thelarge features in the x-ray absorption discussed above, they are too small inphotoemission to account for the rather large features observed in the x-ray absorptionspectra. (The ratio of the integrated area of the loss peaks to the main peak is about 0.2 inthe XPS spectra.) Nevertheless it is possible that the intrinsic plasmon generation willhave a significantly higher cross section at the absorption threshold where the coreexciton is created. The plasmon is a collective excitation of the F 2p electrons. Theelectric field experienced by the F 2p electrons on the photoexcited ion, due to thecreation of the is core hole, will be augmented in the case of the core exciton by theelectric field due to the bound photoelectron. This might increase the strength of thecoupling to the plasmon enough to make a plasmon satellite observable in the x-rayabsorption spectrum. In this case one would expect to observe a replica of the largeexcitonic features in the absorption edge, shifted up in energy by the plasmon energy.Chapter 4 109There has been a recent x-ray fluorescence study of the fluorine K edge in CaF2,that involves exciting a F is electron with the x-ray and detecting the emitted photonwhen the F is core hole is filled by a F 2p electron [76]. This study shows that when theincident photon energy is 23 eV above the fluorine K edge absorption threshold, the usualF 2p —* F is fluorescence peak starts to develop a satellite that is 4 eV higher in photonenergy. The ratio of the satellite to the main fluorescence peak increases to 0.13 when theincident photon energy is 29 eV above the fluorine K edge threshold [76], or in otherwords at the center of the large and broad feature in the x-ray absorption spectrum. Thisis a clear indication that F 2p valence electrons are excited when the x-ray absorptionenergy exceeds 23 eV above the fluorine K edge, and is consistent with the intrinsicplasmon interpretation of the broad feature in the absorption spectrum in this energyrange.In this interpretation, the satellite in the fluorescence spectrum results from thepresence of the plasmon during the x-ray fluorescence. The multi-electron excitation willscreen both the F is core hole in the initial state and the F 2p valence hole in the finalstate. Since the x-ray satellite is on the high energy side, we can conclude that the finalstate is more strongly screened than the initial state by the plasmon, thus causing a netincrease in the energy difference between the F is and F 2p levels. The relative strengthof the intrinsic plasmon creation process is still uncertain because the plasmon canpropagate away or decay before the fluorescence photon is emitted. Nevertheless, theratio of the satellite to the main fluorescence peak (0.i3) gives a lower limit to thestrength of the intrinsic plasmon creation during the x-ray absorption process.Chapter4 110§4.6 Fluorine K edges of Rare-Earth TrifluoridesFluorine K edges in seven rare-earth trifluorides: LaF3, CeF3, NdF3 SmF3, EuF3,DyF3, and YbF3 were also measured. Powdered samples pressed on indium foils wereused in the measurements. The bias potentials applied on the grid of the detector in eachcase are tabulated in Table 4.9. The experimental data are shown in Figures 4.27-33.The spectra of these seven rare-earth trifluorides all have similar features and arevery different from those of alkaline-earth fluorides. The main features are labeled inFigures 4.27-33. The photon energies of these features and the energy separations ofeach feature relative to the first high intensity peak at the absorption edge are tabulated inTable 4.10.We have not been able to observe any appreciable feature that can be explained byVandLaF3 120VCeF3 140VNdF3 140VSmF3 160VEuF3 240 VDyF3 140VYbF3 100VTable 4.9: The bias potentials applied to the front grid of the detector in obtainingthe fluorine K edges of the seven rare-earth trifluorides.Chapter 4 111Photon Energy (eV)Figure 4.27: The fluorine K edge absorption spectrum of LaF3. The main features arelabeled with letters a-e. Marks A, B and C indicate the energies relative to peak acorresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.35).U)U).4-CD0C-)1.3 io61.1 io69.0 17.0 1 O5.0 1 O3.0 1680 690 700 710 720 730Chapter 4 1125.0 1 04.5 14.0 1 03.5103.01O2.5 12.0 11.5 1O680 730Figure 4.28: The fluorine K edge absorption spectrum of CeF3. The main features arelabeled with letters a-e. Marks A, B and C indicate the energies relative to peak acorresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.36).690 700 710 720Photon Energy (eV)Chapter 4 1136.5 1 O5.5 1 OU).-Cl)4.5 13.5 1 O2.5 1 O680Photon Energy (eV)730Figure 4.29: The fluorine K edge absorption spectrum of NdF3. The main features arelabeled with letters a-e. Marks A, B and C indicate the energies relative to peak acorresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.37).690 700 710 720Chapter 4 114Cl)Cl)•1-D0C-)Figure 4.30: The fluorine K edge absorption spectrum of SmF3. The main features arelabeled with letters a-e. Marks A, B and C indicate the energies relative to peak acorresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.38).3.2 1 02.8 1 02.4 1 O2.0 1 O1.61.2 1O680 690 700 710 720Photon Energy (eV)730Chapter 4 115Figure 4.31: The fluorine K edge absorption spectrum of EuF3. The main features arelabeled with letters a-e. Marks A, B and C indicate the energies relative to peak acorresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.39).3.5 1 O3.0 1 OCl)Cl,2.5 12.0 11.5 10680 690 700 710 720Photon Energy (eV)730Chapter 4 116Cl)U)CD0C-)7.0 16.5 1 0Figure 4.32: The fluorine K edge absorption spectrum of DyF3. The main features arelabeled with letters a-e. Marks A, B and C indicate the energies relative to peak acorresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.40).6.0 1 05.5 1 05.0 14.5 14.0 1o680 690 700 710 720Photon Energy (eV)730Chapter 4 11700.4-D0C)Figure 4.33: The fluorine K edge absorption spectrum of YbF3. The main features arelabeled with letters a-e. Marks A, B and C indicate the energies relative to peak acorresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.41).8.0 1O7.0 16.0 1O5.0 14.03.0 1680 690 700 710 720Photon Energy (eV)730Chapter4 118the scattering of the photoelectron by the crystal lattice in the seven x-ray absorptionspectra of the rare-earth trifluorides. The lack of Bragg peaks observed can be attributedto the short lifetime of the photoelectron, and the small energy separation of the Braggpeaks. The inelastic scattering length 1 can be estimated from the spectral width T of theh2 2peaks in the absorption spectrum. From the free electron kinetic energy EK = —k , we2mh2have F = AE = —2kk. From the uncertainty relation Axtk = 1, we find1= x =g2EK . Since the free electron wave length equals h I g2mE, we obtainthe ratio of the electron inelastic scattering length to its wavelength, in terms of the ratioof the electron kinetic energy to the spectral width,(4.6.1). ,rFThe lifetime broadening is significantly larger in the rare-earth trifluorides spectrathan in the CaF2 and BaF2 spectra, presumably because there are more excitation levelsavailable in the rare-earth ions [18]. For example, the spectral width changes from 1 eVat the absorption edge, which reflects mostly the experimental resolution, to 1.5 eV at15 eV above the edge in the CaF2 fluorine K edge spectrum. The spectral width in thefluorine K edge spectra of the seven rare-earth trifluorides, however, is 3-4 eV in thesame energy region. This suggests that the electron inelastic scattering length is onlyabout one wavelength in the rare-earth trifluorides in this energy region. Therefore, theBragg peaks would be too weak to be detected since the back scattered photoelectronwave loses its coherency so quickly.Chapter4 119In the higher energy region (15 eV above the absorption edge and higher), thedetection of Bragg peaks will be even more difficult. Even if the ratio of EKIT’ makes thecoherent interference between the back scattered and outgoing photoelectron wavepossible, the larger broadening in this energy region would smear all the possible Braggpeaks. The Bragg peak positions are determined by, for the hexagonal LaF3 structure andthe orthorhombic YF3 structure,2 2h 1 2 2 fl3— —-(n+n1)+—- , hexagonal8m a cG 2mk2) 2 2 2 2h n1 n n2— + + , orthorhombic8ma b c2212 2 fl37.6 eVA —2-(n+n1)+4 , hexagonal37.6 eV A2[;+4÷4]. orthorhombicwhere n1, n2 and n3 are indices of the reciprocal lattice vector G, and a, b and c are thelattice constants. Because of the low crystal symmetry of these trifluorides (seeChapter 1) [35], any integer is allowed for n1, n2 and n3, except the Bragg peak is weakfor (0, 0,n3=odd) in the case of the LaF3 structure and (0,n2=odd, 0) in the case of theYF3 structure. Taking a typcal lattice constant of 6 A, one can see that the Bragg peakspacing is in the order of 1 eV, which could well be surpassed by the large spectral widthin the rare-earth trifluorides spectra in the high energy region.Chapter 4 120The first peak (peak a) in all seven spectra is due to a transition to a state close tothe bottom of the conduction band, which is enhanced by the attractive core holepotential. It is uncertain, however, whether the final state is a bound exciton or alocalized quasi-bound state at the bottom of the conduction band.Peak b shifts towards the absorption edge from LaF3 to YbF3 with increasingatomic number of the cation (see Table 4.10). If it is a Bragg peak associated with thescattering of the photoelectron by the crystal, the energy of the peak relative to the bottomh2 ,-,-,2 1of the conduction band should be related to — 1 oc —, where a is the lattice2mk2} a2constant. Since the lattice constant decreases from LaF3 to NdF3 which have the LaF3tysonite structure, and from SmF3 to YbF3 which have the orthorhombic YF3 structure(see Chapter 1), peak b shifts in the opposite direction to what we would expect for aBragg peak.Accordingly, we attribute peak b to a final state in the bottom of the conductionband which is an atomic-like high angular momentum state localized on the cation. Forexample, there are empty rare-earth 5d and 4f levels in this energy region. Thecentrifugal barriers experienced by the high angular momentum states trap the otherwisenearly-free electron in a quasi-bound state in the continuum, and thus the absorption isenhanced with this quasi-bound final state [78]. The energy of this atomic-like statedecreases with increasing atomic number of the rare-earth ion because of the increasingCoulomb attraction of the nucleus. This fact is reflected in the decreasing atomic radiuswith increasing atomic number. The atomic radii of the seven rare-earths are shown inthe following table [79]:Chapter4 121(eV)111098765zFigure 4.34: The energies of peak b and c relative to peak a in Figures 4.27-33. Thefilled circles are the observed values. The triangles linked with the solid line are the bestfit values to Eb - Ea in the form Eb — Ea = A — where the best fit parametersE-EC aE-ED afittoE-E0 a5758 60 6263 66 70LaCe Nd SmEu Dy YbA = 18.52 eV and B 30.62 eV.A.Chapter 4 122La Ce Nd Sm Eu Dy Ybatomic 2.74 2.70 2.64 2.59 2.56 2.49 2.40radius r1 (A)As a first order approximation, the energy of an atomic orbital due to the Coulombpotential should be — Pj5Z , where R = 13.6 eV is the Rydberg constant, rB = 0.529 A isthe Bohr radius, Z’’ is the effective charge felt by the orbital, and r is the radiusdetermined from the expectation value of hr of the orbital [80]. Therefore, one mayexpect that the energy separation between peak b and peak a to follow the relationBEb — Ea = A — —. The energy separation between peak b and peak a in all seven rarerearth trifluorides is plotted in Figure 4.34 and is shown to fit a relation of the formEb — Ea = A — -p-, where the best fit parameters are A = 18.25 eV and B = 30.62 eV.A.Taking Z’ = 3 for the 3+ cation, one obtains RrBZ = 21.58 eV•A. This comparesreasonably well with the fit parameter B = 30.62 eV.A if one considers the fact that theatomic radius r1 might be larger than the expectation radius of the atomic-like quasi-bound final state, since the atomic radius r1 is mainly determined by the 6s orbital of thefree rare-earth atom which is rather large [79, 81]. Also the effective charge Z”experienced by the atomic-like final state could be larger than 3 because of thepenetration of the final state wavefunction into the cation core.The energy separation of peak c relative to peak a in the seven rare-earthtrifluorides is also shown in Figure 4.34. The position of peak c is relatively stable, butthe energy separation between peak c and peak b increases slightly from LaF3 to NdF3Chapter 4 123a b c d d’ e(eV) (eV) (eV) (eV) (eV) (eV)LaF3 683.5 690.6 (7.1) 692.9 (9.4) 704.7 (21.2) 707.6 (24.1) 715.5 (32.0)CeF3 686.5 693.3 (6.8) 695.6 (9.1) 706.5 (20.0) 709.4 (22.9) 717.5 (31.0)NdF3 686.3 692.9 (6.6) 695.4 (9.1) 709.4 (23.1) 717.3 (31.0)SmF3 686.5 693.0 (6.5) 696.2 (9.7) 705.8 (19.3) 717.5 (31.0)EuF3 686.5 692.9 (6.4) 696.4 (9.9) 705.8 (19.3) 717.5 (31.0)DyF3 687.3 693.3 (6.0) 697.2 (9.9) 705.4 (18.1) 717.7 (30.4)YbF3 686.5 691.9 (5.4) 696.4 (9.9) 705.1 (18.6) 717.9 (31.4)Table 4.10: The photon energies of the features labeled in Figures 4.27-33. Theuncertainty is ±0.3 eV. The numbers in brackets are energy separations relative topeak a.and from SmF3 to YbF3 as the the atoms in the crystal are getting closer. This seems tosuggest the possibility of peak c being a crystal field split of peak b.The positions of peak d and e relative to peak a are relatively constant comparedto peak b except for LaF3 and CeF3 which have an additional structure at d. Theinsensitivity of these peak positions in relation to the change from LaF3 to YbF3 suggeststhat these features are not crystal structure related as would be expected in the case ofBragg scattering of the photoelectron by the crystal lattice.As in the case of CaF2 and BaF2, there is the possibility of multi-electronexcitations contributing to the x-ray absorption spectra when there exists multi-electronChapter 4 124excitation channels within reach of the photon energy. Accordingly, we have measuredthe loss satellites on the F is photoemission peak in x-ray photoelectron spectroscopy(XPS) using a 1254 eV Mg Koc x-ray source. The loss satellites in XPS together with themain photoemission peaks of the F is core levels for the seven rare-earth trifluorides areshown in Figures 4.35-41. Three main loss features observed in the XPS spectra for allseven rare-earth trifluorides are labeled with A, B and C. The energy separations of thethree loss features relative to the main F is photoemission peak are summarized inTable 4.11. We interpret the three loss features just as in the CaF2 case as follows: losspeak A is due to an interband transition in which a valence electron is excited to theconduction band, loss peak B is due to the creation of plasmons by the sudden appearanceof the core hole (intrinsic process) in addition to excitation of plasmons by the final stateelectrons (extrinsic process), and loss peak C is due to excitations of a shallow coreelectron (F 2s or rare-earth 5p) to an empty state. The energy separations relative to peaka in the absorption spectra, corresponding to the loss energies of A, B and C in the XPSspectra are marked in Figures 4.27-33 for comparison.Chapter 4NE0I125LaF3CBA1.00.80.6680Binding Energy (eV)Figure 4.35: The x-ray photoemission spectrum of the F is core level and its satellites inLaF3 measured with the Mg K x line (1253.6 eV). The main satellite features are labeledwith A, B and C and the corresponding loss energies are listed in Table 4.11.730 720 710 700 690Chapter 4 126CeF31.0N.—ECZ0.80.6Binding Energy (eV)Figure 4.36: The x-ray photoemission spectrum of the F is core level and its satellites inCeF3 measured with the Mg K cx line (1253.6 eV). The main satellite features are labeledwith A, B and C and the corresponding loss energies are listed in Table 4.11.730 720 710 700 690 680Chapter 4 127C)N.,-ECINdF3BAX3.41.00.80.60.4Binding Energy (eV)Figure 4.37: The x-ray photoemission spectrum of the F is core level and its satellites inNdF3 measured with the Mg K *x line (1253.6 eV). The main satellite features are labeledwith A, B and C and the corresponding loss energies are listed in Table 4.11.730 720 710 700 690 680Chapter 4 128SmF31.oNCZ0.8If0.6Binding Energy (eV)Figure 4.38: The x-ray photoemission spectrum of the F is core level and its satellites inSmF3 measured with the Mg K x line (1253.6 eV). The main satellite features arelabeled with A, B and C and the corresponding loss energies are listed in Table 4.11.C BA730 720 710 700 690 680Chapter 4 129NCIEuF3C BAX191.00.80.6Binding Energy (eV)Figure 4.39: The x-ray photoemission spectrum of the F is core level and its satellites inEuF3 measured with the Mg K a line (1253.6 eV). The main satellite features are labeledwith A, B and C and the corresponding loss energies are listed in Table 4.11.730 720 710 700 690 680Chapter 4 130C)N.—EC>C)DyF3C BAX2.31.00.80.60.4Binding Energy (eV)Figure 4.40: The x-ray photoemission spectrum of the F is core level and its satellites inDyF3 measured with the Mg K cs line (1253.6 eV). The main satellite features are labeledwith A, B and C and the corresponding loss energies are listed in Table 4.11.730 720 710 700 690 680Chapter 4 131NECIYbF3C BAX2.31.00.80.60.4Binding Energy (eV)Figure 4.41: The x-ray photoemission spectrum of the F is core level and its satellites inYbF3measured with the Mg K a line (1253.6 eV). The main satellite features are labeledwith A, B and C and the corresponding loss energies are listed in Table 4.11.730 720 710 700 690 680Chapter 4 132A B C(eV) (eV) (eV)LaF3 15.0±1.0 27.0±1.5 41±4CeF3 16.0±1.0 27.0±1.5 35±4NdF3 15.0±1.0 26.0±1.5 32±2SmF3 15.0±1.0 26.0±1.5 32±3EuF3 15.0±1.0 26.0±1.5 35±3DyF3 16.0 ±1.0 27,0 ±1.5 37 ±3YbF3 17.5±1.0 27.0±1.5 41±4Table 4.11: The loss energies of the satellites of the F is main photoemissionpeak in the XPS spectra in Figures 4.35-41.There is seemingly no direct match between the energies of the loss features in theXPS spectra and the features in the corresponding x-ray absorption spectra. However, thepresence of the bound final state electron responsible for the high intensity peak at the xray absorption threshold could raise the final state energy of the valence electrons at thephotoionization site relative to the one hole XPS final state. In that case, the energyseparation between the high intensity peak at the absorption threshold and its replica dueto multi-electron excitations would be smaller than the loss energy of the correspondingsatellite in the XPS spectrum. This is due to the fact that the high energy photoelectron inthe XPS final state is far from the core hole and does not influence the valence electronsat that site. We do not know the magnitude of this interaction between the fluorineChapter 4 133valence electrons at the photoionization site and the bound photoelectron plus the corehole. Nevertheless, we cannot rule out the possibility that the multi-electron excitationinvolving an interband transition may contribute to peak c and the excitation involvingintrinsic plasmon creation may contribute to peak d in the x-ray absorption spectra. As inthe CaF2 case, although the strength of multi-electron excitations in XPS is relativelysmall, the strength could be larger at the absorption threshold because the electric fieldexperienced by the valence electrons on the photoexcited ion, due to the creation of thecore hole, is augmented by the presence of the bound final state photoelectron.The broad feature e is about 31 eV above the high intensity peak at the absorptionthreshold in the absorption spectra of all seven rare-earth trifluorides. It appears to beinsensitive to both the crystal structure and the atomic number of the cation. Interestinglya similar feature also appears in the CaF2 and BaF2 spectra. The energy separationbetween the feature and the first peak at the absorption threshold is 31 eV for CaF2 and30.5 eV for BaF2. A similar feature has also been observed in the fluorine K edge in SF6gas molecules with a energy separation of 33 eV relative to the first peak at theabsorption threshold [82]. The insensitivity of this feature to the environment offluorines suggests that it is an atomic feature related to fluorine atoms. One possibility isthat it is due to a multi-electron excitation with a F 2s electron excited. The energyseparation between F 2s and F 2p is 19 eV [42]. Assuming all the rare-earth trifluoridesto have the same 10 eV band gap [83], we have 29 eV as the energy separation between F2s and the bottom of the conduction band. Therefore, it is possible that feature e in the xray absorption spectra includes a contribution from a two-electron excitation in which a F2s electron is excited into the conduction band while a F is electron is photoexcited.Chapter 4 134We have shown in this section, that the near edge part of the fluorine K edgeabsorption spectra of the rare-earth trifluorides is not dominated by features associatedwith the scattering of the photoelectron wave by the crystal lattice. The detection ofappreciable Bragg peaks requires the inelastic scattering length of the final statephotoelectron to be long enough for coherent interference from several lattice planes.Consistently with the lack of Bragg peaks which are long range order effects, featuresthat can be explained with an atomic model are observed. The fluorine K edges of therare-earth trifluorides are more complex than the alkaline-earth fluorides because of moreexcitation channels available in the rare-earths [181, that still need further studies. Thecomplexity of XANES is demonstrated in particular by the possibility of multi-electronexcitations contributing to the absorption spectra in the XANES region.Chapter 5 135Chapter 5Summary and ConclusionsThe energy alignment of the interfaces between thin films of YbF3 and Si(1 11)substrates have been studied under UHV conditions by photoelectron spectroscopy and xray absorption spectroscopy using synchrotron radiation. Results of YbF3ISi(111) werecompared with those of TmF3ISi(1 11). The determination of the position of the 4f levelsin the rare-earth cations is complicated by the fact that the photoemission signal from thepartially filled 4f orbitals overlaps with the F 2p valence band of the rare-earthtrifluorides. However the signal from the 4f electrons was distinguished by resonantlyexciting the giant 4d-4f transition in the rare-earth. While only the 4f12 configuration ofthe Tm3 ions were observed in TmF3ISi(1 11), both the 4f13 configuration of the Yb3ions and the 4f14 configuration of the Yb2 ions were observed in YbF3/Si(1 11). Whilethe Yb2 ions were also observed in the film, the concentration of the Yb2 ions wasfound to be higher at the interface than in the film. This is due to the fact that electrons inthe Si valence band are prevented from occupying the empty 4f levels in TmF3 at theinterface by the on-site Coulomb repulsion energy U, whereas the charge transfer from Sito YbF3 is possible because the totally filled 4f states of Yb still lie below the Si valenceband maximum. The results suggest that the 4f levels of TmF3 might be excited by directelectrical injection of electrons from the semiconductor [841.Chapter 5 136A simple Bragg reflection model has been developed to qualitatively explain theoscillations in XANES in terms of the scattering of the photoelectron wave betweenfamilies of lattice planes. It was shown that the positions of the Bragg peaks could bedetermined by the simple relation of = For metals, the zero of energy £Gis at the bottom of the free electron conduction band or eF below the absorptionthreshold; for semiconductors, the origin of CG is in the vicinity of the bottom of thevalence band; and for wide band gap insulators, the origin of 8G is close to the bottom ofthe conduction band. The model was tested with K edges in a number of materials,namely, f.c.c. Cu, b.c.t. Cu, Fe, crystalline Si, solid Ne and CaF2, and was found to be ingood agreement with experiment for the elemental metals. The model cannot account forfeatures caused by the core hole potential which is important when there is little screeningsuch as in insulators.In order to have a deeper and more quantitative understanding of XANES spectra,a new bandstructure technique with a pseudopotential approximation was developed tocalculate the fluorine K edges in CaF2 and BaF2. The effects of both the long rangescattering by the periodic crystal potential and the attraction by the localized core holepotential were included. The symmetry breaking core holes were treated with a supercelltechnique. This is the first calculation of the fluorine K edge in BaF2 [33, 85]. Themodel successfully reproduced all the main features in the first 15 eV of the absorptionedge which had not been explained previously with the cluster calculations [85]. Theexcitonic peak at the absorption threshold was found to be followed by a series of peakswhose spacing changes in going from CaF2 to BaF2 by an amount consistent withelectron diffraction from crystal lattice planes. The peak spacings reproduced by theChapter 5 137model are found to be insensitive to changes of the pseudopotential core radii of the ionsin the crystal. The model breaks down at higher energies due to the limited k-spacevolume that can be included in the calculation. A model for the absorption edge at higherenergies is complicated by the possibility of multi-electron excitations. This was studiedby comparing the energy loss satellites in the fluorine is x-ray photoelectron spectra withfeatures at corresponding energies in the fluorine K edge absorption spectra.Finally the fluorine K edges in the rare-earth trifluorides LaF3, CeF3, NdF3,SmF3,EuF3,DyF3 and YbF3 were explored for the first time with high resolution x-rayabsorption spectroscopy and x-ray photoelectron spectroscopy. The XANES part of thefluorine K edges in all seven rare-earth trifluorides was found not to be dominated byfeatures associated with the scattering of the photoelectron wave by the crystal lattice.The absence of these effects which are dominant in CaF2 can be attributed to the shortlife time of the photoelectron and the. small energy separation of the Bragg peaks. Thelarger lifetime broadening is attributed to the high density of low level excitations of the4f shell that are available in the rare-earth ions [181. 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