Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1994-894083.pdf [ 2.11MB ]
Metadata
JSON: 831-1.0085021.json
JSON-LD: 831-1.0085021-ld.json
RDF/XML (Pretty): 831-1.0085021-rdf.xml
RDF/JSON: 831-1.0085021-rdf.json
Turtle: 831-1.0085021-turtle.txt
N-Triples: 831-1.0085021-rdf-ntriples.txt
Original Record: 831-1.0085021-source.json
Full Text
831-1.0085021-fulltext.txt
Citation
831-1.0085021.ris

Full Text

PHOTOELECTRON EXPERIMENTS AND STUDIES OF X-RAY ABSORPTION NEAR EDGE STRUCTURE IN ALKALINE-EARTH AND RARE-EARTH FLUORIDES By Yuan Gao B. Sc., The University of Science and Technology of China, 1985 M. Sc., The University of British Columbia, 1990 A THESIS SUBMITThD IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 1994 © Yuan Gao, 1994  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the IJbrary shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or pubhcation of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  F H IS I C S  The University of British Columbia Vancouver, Canada  Date APRIL ZI,  DE-6 (2188)  i994  Abstract  Alkaline-earth fluorides and rare-earth trifluorides possess technological importance for applications in multi-layer electronic device structures and opto-electronic devices. Interfaces between thin films of YbF3 and Si(l 11) substrates were studied by photoelectron spectroscopy and x-ray absorption spectroscopy using synchrotron radiation. Results of 3 YbF / Si(l 11) were compared with those of TmFgSi(l 11). While electrons in the Si valence band are prevented from occupying the empty 4f levels in 3 at the interface by the on-site Coulomb repulsion energy, the charge transfer from TmF 3 is possible because the totally filled 4f states in Yb still lie below the Si Si to YbF valence band maximum. The theory of x-ray absorption near edge structure (XANES) is incomplete except for a few particularly simple special cases. A Bragg reflection model was developed to qualitatively explain the oscillations in XANES, in terms of the scattering of the photoelectron wave between families of lattice planes as set out by the Bragg condition for backscattering. The model was found to represent the data for systems with nearly free electron like conduction bands reasonably well. High resolution CaF2 fluorine K edge XANES was used as a prototype to understand XANES in more depth on systems with strong core hole effects. Unlike previous work which involved multiple scattering cluster calculations that include only short range order effects, both the long range order and the symmetry breaking core holes are included in a new bandstructure approach in which the core hole is treated with a supercell technique. A first principles calculation with the use of pseudopotentials successfully reproduced all the main features of the first 15 eV of the fluorine K edge in CaF2 which had not been explained with the cluster calculations. A comparison of the 11  theoretical and experimental fluorine K edges in CaF 2 and BaF2 was used to identify the structure related features. The possibility of multi-electron excitations being responsible for higher energy features in the XANES was investigated by comparing the energy loss satellites in the fluorine is x-ray photoelectron spectra with features at corresponding energies in the fluorine K edge absorption spectra. Finally the fluorine K edges in the rare-earth trifluorides LaP , CeF3, NdF 3 , SmF3, EuF 3 , DyF3 and YbF 3 3 were explored for the first time with the high resolution x-ray absorption spectroscopy. The near edge part of the fluorine K edges in all seven rare-earth trifluorides was found not to be dominated by the Bragg peaks because of the short life time of the photoelectron and the low crystal symmetry.  111  Table of Contents  Abstract  ii  Table of Contents  iv  List of Tables  vi  List of Figures  vii  Acknowledgements  xii  Chapter 1 Introduction  1  Crystal Structures of the Alkaline-earth Fluorides and the Rare-earth  4  Trifluorides Chapter 2 Apparatus  10  2.1 The Ui Beamline at NSLS  10  2.2 Thin Film Deposition  14  Chapter 3 Photoelectron Spectroscopy  17  3.1 Experimental Arrangement of Photoelectron Spectroscopy  17  3.2 Photoelectron Spectroscopy of 4f Levels at the 3 YbF / Si(1 11) and  22  TmF I 3 Si(1 11) Interfaces Chapter 4 X-ray Absorption Spectroscopy  37 iv  4.1 Experimental Arrangement of X-ray Absorption Spectroscopy  37  4.2 Fluorine K edges of Alkaline-Earth Fluorides  39  4.3 Bragg Reflection Model for XANES in Crystalline Solids  43  Case Study  59  Copper  60  Iron  67  Silicon  69  Neon  73  Calcium Fluoride  76  4.4 Bandstructure Calculation of XANES at the Fluorine K Edge in  82  CaF2 and BaF2 4.5 Multi-electron Excitations in XANES  105  4.6 Fluorine K edges of Rare-earth Trifluorides  110  Chapter 5 Summary and Conclusions  135  Bibliography  138  V  List of Tables  Table  Page  4.1  The reciprocal lattice vectors for f.c.c. copper.  61  4.2  The reciprocal lattice vectors for b.c.t. copper.  67  4.3  The reciprocal lattice vectors for b.c.c. iron.  68  4.4  The reciprocal lattice vectors for silicon with non-zero structure  70  factors. 4.5  The reciprocal lattice vectors for solid neon.  74  4.6  The reciprocal lattice vectors for CaF2.  78  4.7  The Fourier coefficients of the crystal potential U used in the  90  calculation of the fluorine K edge absorption in CaF . 2 4.8  The Fourier coefficients of the crystal potential U used in the  100  calculation of the fluorine K edge absorption in BaF2. 4.9  The bias potentials applied on the front grid of the detector in  110  obtaining the fluorine K edges of the seven rare-earth trifluorides. 4.10  The photon energies of the features labeled in Figures 4.27-33.  123  4.11  The loss energies of the satellites of the F is main photoemission  132  peak in the XPS spectra in Figures 4.35-41.  vi  List of Figures  Figure  Page  1.1  The fluorite structure.  7  1.2  Two projections of the hexagonal cell of LaF3,  8  1.3  The orthorhombic YF 3 structure.  9  2.1  The theoretical resolution curves and the measured resolution data  11  points of the ERG monochromator with gratings Gi, G2, and G3. 2.2  Top view of the experiment station of the Ui beamline at NSLS,  12  Brookhaven National Laboratory. 2.3  The schematic diagram showing the configuration of the  15  evaporator used for thin film deposition. 3.1  The schematic diagram illustrating the relative energy levels in  21  equation (3.1.6). 3.2  Valence band region of TmF3 on Si taken at two different photon  24  energies A and B. 3.3  Valence band region of YbF 3 on Si taken at two different photon  25  energies A and B. 3.4  The schematic diagram showing the binding energy distribution of 12 the 4f  —  4f  transition.  vii  28  3.5  The schematic diagram showing the binding energy distribution of the 4f 13  3.6  —*  29  1 transition. 4f  The schematic diagram showing the positions of the cation 4f  30  levels relative to the Si valence band and the F 2p valence band for 3 and YbF TmF 3 on Si(1 11) substrates. 3.7  Photoemission spectra of YbF 3 films of two different thicknesses  35  as deposited on room temperature Si(1 11) substrates. 3.8  Photoemission spectra with the resonant photon energy of 182 eV taken on a 10  36  A YbF 3 film deposited on Si(1 11) before and after a  1 minute anneal at about 400 °C. 4.1  The experimental absorption spectrum of the fluorine K edge in  40  . 2 CaF 4.2  The experimental absorption spectrum of the fluorine K edge in  41  . 2 SrF 4.3  The experimental absorption spectrum of the fluorine K edge in  42  . 2 BaF 4.4  The free electron wave length as a function of the free electron  45  kinetic energy. 4.5  The calculated electron inelastic mean free path (IMFP) in LiF by  46  Tanuma et al using various algorithms. 4.6  Constant energy surfaces in k-space with one Bragg plane. viii  51  4.7  One-dimensional schematic diagram showing examples of the  54  Fourier coefficients UG being complex. 4.8  The ratio of the absorption coefficient for the model potential with  56  various values to the absorption coefficient for the unperturbed plane waves. 4.9  The absorption ratio with complex U.  57  4.10  The absorption ratio with various magnitudes of U, assumed to be  58  real and positive. 4.11  X-ray absorption at the K edge of f.c.c. copper.  63  4.12  X-ray absorption at the K edge of b.c.t. copper.  64  4.13  X-ray absorption at the K edge of b.c.e. iron.  66  4.14  X-ray absorption at the K edge of crystalline silicon.  71  4.15  X-ray absorption at the K edge of solid neon.  75  4.16  . 2 X-ray absorption at the fluorine K edge in CaF  79  4.17  The schematic diagram showing the super unit cell.  87  4.18  2 is The experimental fluorine K edge absorption spectrum for CaF  94  shown at the top and the calculated absorption spectrum is shown at the bottom. 4.19  2 with Calculated fluorine K edge absorption spectra for CaF different binding energies of the initial is state. ix  95  4.20  Calculated fluorine K edge absorption spectra for CaF 2 with  96  different F- radii as indicated. 4.21  Calculated fluorine K edge absorption spectra for CaF 2 with  97  different Ca 2 radii as indicated. 4.22  The calculated fluorine K edge absorption spectrum for CaF2  98  including the core hole potential is shown at the top. 4.23  The experimental fluorine K edge absorption spectrum in BaF2 is  103  shown in the top spectrum, and the calculated spectrum is shown at the bottom. 4.24  A plot of the position of the peaks a-d for BaF 2 from Figure 4.23 as  104  2 from a function of the position of the corresponding peaks in CaF Figure 4.18. 4.25  The x-ray photoemission spectrum of the F is core level and its  106  satellites in CaF 2 measured with the Mg K a line. 4.26  The x-ray photoemission spectrum of the F is core level and its  107  satellites in BaF2 measured with the Mg K a line. 4.27  . 3 The fluorine K edge absorption spectrum of LaF  iii  4.28  The fluorine K edge absorption spectrum of CeF3.  112  4.29  . 3 The fluorine K edge absorption spectrum of NdF  113  4.30  . 3 The fluorine K edge absorption spectrum of SmF  114  x  4.31  The fluorine K edge absorption spectrum of EuF . 3  115  4.32  The fluorine K edge absorption spectrum of DyF ; 3  116  4.33  The fluorine K edge absorption spectrum of YbF . 3  117  4.34  The energies of peak b and c relative to peak a in Figures 4.27-33.  121  4.35  The x-ray photoemission spectrum of the F is core level and its  125  satellites in LaF 3 measured with the Mg K a line. 4.36  The x-ray photoemission spectrum of the F is core level and its  126  satellites in CeF 3 measured with the Mg K a line. 4.37  The x-ray photoemission spectrum of the F is core level and its  127  satellites in NdF 3 measured with the Mg K a line. 4.38  The x-ray photoemission spectrum of the F is core level and its  128  satellites in SmF3 measured with the Mg K a line. 4.39  The x-ray photoemission spectrum of the F is core level and its  129  satellites in EuF 3 measured with the Mg K a line. 4.40  The x-ray photoemission spectrum of the F is core level and its  130  satellites in DyF3 measured with the Mg K a line. 4.41  The x-ray photoemission spectrum of the F is core level and its satellites in YbF3 measured with the Mg K a line.  xi  131  Acknowledgments  I would like to first thank Dr. Tom Tiedje, my supervisor, for his continued guidance, 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. Wolfgang Eberhardt, Tony van Buuren, Brian M. Way, Dr. Jan N. Reimers, and Dr. Stefan Cramm for the help on the data acquisition at the Ui beamline, NSLS, Brookhaven National Laboratory. I am grateful to Dr. P. C. Wong and Dr. K. A. R. Mitchell for providing the XPS 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 few years. Useful discussions with Dr. Birger Bergersen and Dr. Ian Affleck are also acknowledged. I am very grateful for the warm Canadian hospitality I have experienced ever since I stepped on this beautiful land. Finally, no words can fully express my gratitude to my wife Ginny, who has generously supported me in every possible way.  xii  Chapter 1  1  Chapter 1 Introduction  Most chemical and physical properties of a solid depend on the behavior of the electrons, which can be described by electronic states, both occupied and unoccupied. The understanding of occupied and unoccupied states complements each other since any dynamic view of electrons will involve them both. The photoelectron process which involves exciting an electron from an occupied state into an unoccupied state is a powerful probe for studying these states. Two techniques involving photoelectrons were used in the work of this thesis, namely, photoelectron spectroscopy and x-ray absorption spectroscopy.  In photoelectron spectroscopy (sometimes also called photoemission  spectroscopy), the incident photon energy is fixed while the kinetic energies of the emitted electrons are analyzed. In x-ray absorption spectroscopy, the absorption of a sample is measured with a varying incident photon energy that is enough to excite a core electron. However, while the study of occupied states has matured with photoelectron spectroscopy during the past 20 years, an equivalent understanding of low energy unoccupied states with x-ray absorption spectroscopy has lagged behind [1]. In this thesis, it is attempted to push forward the understanding of low energy unoccupied states with an extended study on x-ray absorption near edge structure (XANES).  The  knowledge on occupied states energy alignments of the rare-earth trifluorides and silicon interfaces is also extended.  Chapter 1  2  Alkaline-earth fluorides CaF , SrF 2 , BaF2 and their alloys have been found to 2 grow epitaxially on a number of semiconductor surfaces [2-15].  There have been  extensive 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-layer device structures. Epitaxial growth of rare-earth trifluorides on Si(1 11) have also been investigated [16]. Laser action of the systems LaF :Nd and LaF3:Er 3 3 has been known for a long time [17], because of the numerous excited 4f states [18]. Direct doping of rare-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 Tm trifluorides on Si( 111) with photoelectron spectroscopy and x-ray absorption spectroscopy [22, 23] are extended in this thesis and the substantially different results of /Si(1 11) will be compared with those of TmF 3 YbF ISi(1 11). 3 X-ray absorption spectroscopy can be used for electronic states studies together with photoelectron spectroscopy. The x-ray absorption process involves exciting an electron to an empty state with a photon. A typical x-ray absorption spectrum is usually divided into two regions: the first 50 eV or so above the absorption threshold is called the x-ray absorption near edge structure (XANES) and the higher energy part of the spectrum is called the extended x-ray absorption fme structure (EXAFS) [24]. The EXAFS region usually has weaker but longer period modulations and the distinction between the XANES region and the EXAFS region is a loose one [25]. A division with more physical meaning was suggested by Bianconi, namely that it should be roughly at the energy where the wavelength of the excited electron is equal to the distance between the  3  Chapter 1  absorbing atom and its nearest neighbors [26]. EXAFS was explained by Sayers, Stern and Lytle in terms of short range scattering of the photoelectron wave by the neighboring atoms around the photoexcitation site [27, 28] and rapidly developed into a technique to obtain local interatomic distances in solids, both crystalline and noncrystalline [29-31]. XANES is known to be rich in chemical and structural information. It is more complex 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 Bragg reflection model is developed to qualitatively explain the oscillations in XANES, in terms of the scattering of the photoelectron wave between families of lattice planes as set out by the Bragg condition for backscattering.  The model is tested on a number of  representative systems. Because of the technological importance of CaF2 mentioned earlier, also in order to understand XANES in more depth, high resolution CaF2 fluorine K edge XANES is used as a prototype to develop a model of XANES calculations for wide band gap insulators with strong core hole effects. Unlike previous work which involved multiple scattering cluster calculations that included only short range order effects [32, 33], both the long range order and the symmetry breaking core holes are included in a new bandstructure 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 the first time.  4  Chapter 1  Crystal Structures of the Alkaline-earth Fluorides and the Rare-earth Trifluorides The alkaline-earth fluorides CaF2, SrF2 and BaF2 all have the same fluorite structure. 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 cube  lattice constant at room temperature is 5.463  A, 5.800 A, and 6.200 A for CaF2, SrF2 and  BaF2 respectively [34]. In the CaF2 structure, the fluorine lattice consists of a simple cubic lattice of F- ions packed together with a separation determined by their ionic radius. 2 ion expands the fluorine lattice so that the F-F In the BaF2 structure, the large Ba distance is larger than the sum of the ionic radii. The rare-earth trifluorides discussed in this thesis have two different crystal structures. Most of the light rare-earth trifluorides have the tysonite structure (also called 3 structure) [35]. According to Wyckoff, this crystal structure has a hexagonal LaF structure 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 where u  =  ±(2/3, 1/3, u+l/2)  0.57. Figure 1.2 shows the proposed hexagonal tysonite structure. The lattice  constants of the relevant rare-earth trifluorides are listed below [35].  LaF3  a(A)  c(A)  4.148  7.354  Chapter 1  5  CeF3  4.107  7.273  3 NdF  4.054  7.196  3 -TmF  3.905  6.927  However, more recent results suggest that the tysonite structure is actually a trigonal structure [86]. The other rare-earth trifluorides discussed in this thesis have the orthorhombic 3 structure [35]. It has the orthorhombic symmetry with the atoms at the following YF positions [35]: 4 cations at:  ±(u, 1/4, v)  4 fluorines at: ±(u’, 1/4, v ) t  and  ±(u+1/2, 1/4, 1/2-v)  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?  v’  =  0.528,  x=0.165,  =  0.60 1,  y=0.060,  z=0.363.  The orthorhombic YF 3 structure is also shown in Figure 1.3. The lattice constants of the relevant rare-earth trifluorides with the orthorhombic YF 3 structure are listed below [35]:  Chapter 1  6  a(A)  b(A)  c(A)  3 SmF  6.669  7.059  4.405  EuF3  6.622  7.019  4.396  DyF3  6.460  6.906  4.376  3 TmF  6.283  6.811  4.408  3 YbF  6.216  6.786  4.434  Chapter 1  7  Figure 1.1: The fluorite structure (after Wyckoff [34]). The left hand side shows the 2 projected on a cube face. Lettered positions of the atoms within the unit cell of CaF circles refer to the corresponding spheres at the right. The right hand side shows a . The dark spheres are the 2 perspective drawing of the atoms within the unit cube of CaF 2 ions and the light spheres are the F- ions. Ca  Chapter 1  8  Figure 1.2: Two projections of the hexagonal cell of LaF 3 (after Wyckoff [35]). In the upper part the black spheres are the La 3 ions and the lighter spheres are the F- ions. In the lower part the numbers show the displacement of the atoms in the c direction.  Chapter 1  9  a  Figure 1.3:  The orthorhombic YF 3 structure (after Wyckoff [35]).  On the left a  3 structure is shown. projection along the c axis of the orthorhombic YF  The  displacement of the atoms in the c direction is shown by the numbers in the circles. The unit of the numbers is one hundredth of c. Shown on the right is a packing drawing viewed along the c axis. The black spheres are the cations and the dotted spheres are the F- ions.  Chapter 2  10  Chapter 2 Apparatus  §2.1 The Ui Beamline at NSLS The photoelectron experiments in this thesis including photoemission and x-ray absorption experiments were conducted at the Ui beamline on the VUV storage ring at the National Synchrotron Light Source (NSLS) of Brookhaven National Laboratory. Accelerating charged particles lose energy by radiation of electromagnetic waves. The highly relativistic electrons in the storage ring are magnetically constrained to travel around a closed path and the consequent centripetal acceleration results in the emission of the synchrotron radiation. The radiation is peaked in the forward direction and is polarized 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 the resultant synchrotron radiation has a characteristic energy of 486 eV [36]. The U 1 beamline is an ultra high vacuum beamline designed to cover the photon energy range from 25 to 1300 eV continuously using an extended range grasshopper (ERG) monochromator [37]. The heart of the ERG monochromator consists of an entrance 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 and focus it at the sample position and a set of premonochromator slits to aperture the beam vertically and horizontally in order to reduce stray light in the forward monochromator  Chapter 2  11  0.750 I  lOli. SLITS 0.500  -, 0  D _I  0  0  0.250 -  •  Gi ,  I  0.000 0  ...-  -—-  I 250  500  PHOTON ENERGY (eV)  Figure 2.1: The theoretical resolution curves and the measured resolution data points of the ERG monochromator with gratings Gi, G2, and G3 (figure after Sansone et a! [37]).  Chapter 2  12  DEPOSITION  ANALYSIS  LEPJ( VALVE  EVAPORATOR SPU1TERING aN  TRANSFER ARM  ANALYZER  SAMPLE STORN3E  LOAD LOCK  VALVES THICKNESS MONIfOR  MASS SPEC.  MCP DETECTOR OPTICAL PYROMETER BEAMLINE TO SYNCHROTRON  Figure 2.2: Top view of the experiment station of the Ui beamline at NSLS, Brookhaven National Laboratory. Picture after Colbow [38].  Chapter 2  13  section. The combination of the flat mirror and the entrance slit are optically equivalent to a bilateral slit with a range from 10 to 300 tm while the exit slit is a bilateral device with a range from 10 to 1000 tm. The resolution and the photon throughput of the monochromator 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 eV  respectively. The minimum linewidth (FWHM) of the zero order peak with the smallest slits setting (10 0.02  A  i)  is measured to be 0.09  A for grating G1,  0.03  A for grating  G2, and  for grating 03 [37]. Figure 2.1 shows the theoretical resolution curves and  measured 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. A transfer arm can transport samples between storage shelves, thin film deposition chamber, and the main analysis chamber under UHV conditions. Once in the main analysis chamber, the sample is mounted on a rotary manipulator arm that can also move in all three directions.  The sample can be rotated facing the appropriate directions for  photoemission measurement with the hemispherical electron energy analyzer, or for x-ray absorption measurement with the microchannel plate detector. In addition, there is sample cleaning capability in the main analysis chamber. The sample can be sputter cleaned with a sputtering gun in the main chamber.  The sample holder on the  manipulator arm also serves as a heating stage. With the sample mounted in a Ta foil basket 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 red pyrometer. The temperature reading of the pyrometer was calibrated by oxide desorption on GaAs wafers and also by a type-K (chromel-alumel) foil thermocouple [38]. The  Chapter 2  14  agreement between the sample temperatures determined by the thermocouple and by the desorption of the oxides were within 10 C over the temperature range of 450-600 C and within 20 °C for temperatures down to 350 °C [38].  §2.2 Thin Film Deposition Deposition of thin films of rare-earth trifluorides on clean silicon wafers were required to study the interfaces between them and Si(1 11). This was achieved in the deposition chamber connected to the main analysis chamber. With this arrangement samples with freshly made films can be transported directly to the main analysis chamber under UHV conditions, so that the possibility of surface contamination is minimized. The evaporation source material was contained in a tube shaped vessel spot welded 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 diameter hole in the middle facing the direction of the substrate and the thickness monitor. It was resistively heated to evaporate the powdered material enclosed in it. A typical current of 25 35 A and 1 Volt of voltage was needed to sustain the evaporation. The background -  pressure was kept below 10-8 Torr during the evaporation and the substrates were kept at near room temperature, heated only by the radiation from the evaporator. A shutter was placed between the evaporation tube and the substrate. Once the deposition rate was stabilized at 0.2 0.5 A/s and about 20 30 A of the film was deposited on to the shutter, -  -  Chapter 2  15  E—4cm--)  .\  thickness monitor  10cm  substrate  shutter  Ta evaporator tube  electrodes  Figure 2.3: The schematic diagram showing the configuration of the evaporator used for thin film deposition.  Chapter 2  16  the shutter was opened to commence the deposition on to the substrate. Figure 2.3 schematically shows the evaporation configuration. The deposition rate and the film thickness was monitored with an Inficon quartz crystal thickness monitor during the deposition. The geometry of the thickness monitor and the substrate relative to the evaporator was kept as symmetric as possible. In that case, the theoretical uncertainty of the crystal thickness monitor should be less than 3% [39]. However, the directionality of the evaporation source could introduce much greater uncertainty in the film thickness determination. Therefore, the following complement method [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 film and the intensity I of the photoemission signal from a core level of the substrate can be approximated by, -d/2sinO R(d)=K[1,SG  where K  1° =  ]  is a constant expressing the relative sensitivity of the spectral features of  the film to the substrate which can be obtained experimentally, and 8 is the angle between the sample surface and the detector. I is the photoemssion intensity of the same feature as I, measured on a thick film (>50  A or thick enough that no signal from  the substrate is seen) and normalized by the incident photon intensity.  I is the  normalized photoemssion intensity of the same feature as Is’ measured on a bare substrate. With the use of X  =  6  A,  the film thickness estimation on the rare-earth  trifluorides was evaluated by Colbow to be accurate to a factor of two [38].  Chapter 3  17  Chapter 3 Photoelectron Spectroscopy  §3.1 Experimental Arrangement of Photoelectron Spectroscopy Photoelectron spectroscopy (PES) probes the occupied electronic states in materials by analyzing the kinetic energies of the photoelectrons emitted during the photoionization 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 is emitted after it absorbs a photon. If the radiation has a monochromatic photon energy  ho), 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 of the 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 occupied  states from the spectra of the kinetic energy distribution of the direct photoelectrons. Strictly from energy conservation, the energy of the absorbed photon equals the energy change of the whole system including the photoelectron and all other electrons in the system,  Chapter 3  ho)  18  =  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 binding energy EB 1 of the electron i is conventionally defined as the one-electron energy separation between the state concerned and the Fermi level of the sample, 1 EB  Ci-EF.  (3.1.3)  The one-electron energy in solids may not be a good quantum number in general because of the electron-electron interaction. In other words, the energy of an electron could be a function of many electrons. It can be described, however, in terms of the total energy of the system without considering the details of the electron-electron interaction. The oneelectron energy of the state i is thus defined as the difference of the total energy of the sample 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 the Fermi level. Since the zero of the energy is the vacuum level, we have •sEF. Combining eqn(3.1.2 5) and omitting the subscript i, we have eqn(3.1.1). -  (3.1.5)  Chapter 3  19  Eqn(3.l.1) gives the impression that the mapping of the binding energy EB to the kinetic energy EK depends on the sample work function  .  This is not desirable  because the work function is sensitive to preparation conditions and must be measured independently. However, it turns out not to be a concern. The kinetic energy of the photoelectron has to be measured by an electron energy analyzer, and the important quantity is the retarding potential V between the analyzer and the sample which is required to bring the photoelectron to rest which the analyzer takes as the measured kinetic energy of the photoelectron.  From the definition of V. EK=V+( a  øa is the work function of the electron energy analyzer and (Øa  -  -  ) where  ) is the contact  potential. 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 binding energy distribution of the occupied states by the measurement of V and it is independent of the sample work function •.  This is desirable because one does not need to know  the sample work function. The spectrometer work function a’ which can be assumed constant, is sufficient. The photoelectron spectroscopy experiments in this work were carried out at the Ui beamline of the National Synchrotron Light Source of Brookhaven National Laboratory.  The beamline provides a high intensity photon source from 25 eV to  1300 eV continuously through an extended range grasshopper (ERG) monochromator. A hemispherical electron energy analyzer (VSW-HAC 100) was used to detect the  Chapter 3  20  photoelectrons 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 potential Vret and the analyzer pass energy F.: V  Vret + E. The work function of the analyzer  was assumed to be 4.5 eV. The combined resolution of the monochromator and the electron energy analyzer for the photoemission experiments was variable depending on the photon energy, the setting of the monochromator slits and the pass energy setting on the analyzer. The photon energy used in each experiment was calibrated by measuring the Fermi edge on a freshly sputter cleaned copper surface on the sample manipulator which was always available in the analysis chamber.  Chapter3  21  e Vacuum Level  —-  Fermi Level  EK V Vacuum Level  —  Fermi Level  S  EB  Core Level  Sample  Figure 3.1:  Spectrometer  The schematic diagram illustrating the relative energy levels in  equation (3.1.6).  Chapter 3  22  §3.2 Photoelectron Spectroscopy of 4f Levels at the 3 /YbF Si(111) and TmF /Si(111) 3 Interfaces Epitaxial layers of LaF 3 and related rare-earth trifluorides grown on semiconductor substrates offer interesting possibilities for opto-electronic devices including integrated diode-laser pumped solid state lasers [16, 21] In addition the possibility exists of direct electrical excitation of rare earth ions at interfaces between the semiconductor and a rare-earth trifluoride. Accordingly the positions of the 4f levels of the rare-earth ions at the interface between silicon and various rare-earth trifluorides have been investigated [22, 23] Most of the light rare earth trifluorides have the hexagonal tysonite 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 orthorhombic 3 structure. In this section we report on a photoemission study of the 4f levels at the YF interface between YbF 3 and Si(1 11) and compare these results with 3 TmF / Si(1 11). The experiments were carried out at the Ui beamline at the NSLS, Brookhaven National Laboratory. The substrates were 1 cm 2 p-type Si wafers mounted in Ta foil baskets spot-welded to Ta wires which could be heated resistively. The substrates were cleaned by a repetitive, low energy (500 eV) Ar sputter-anneal process, until no residual carbon or oxygen was detectable in photoemission. The samples were annealed at approximately 600 °C for one minute after each sputtering cycle.  The sample  temperature was measured with a pyrometer calibrated with a thermocouple. The clean substrates were transfered under UHV to a separate chamber, where powders of the rareearth trifluorides were evaporated at 0.2  -  0.5  A/s  from tube-shaped Ta boats in a  background pressure of 8x10 9 Torr. A quartz crystal thickness monitor was used for  Chapter 3  23  relative thickness measurements during the depositions.  The samples were then  transfered in UHV to the analysis chamber for photoemission and X-ray absorption studies. The determination of the position of the 4f levels in the rare-earth cations is complicated by the fact that the photoemission signal from the partially filled 4f orbitals overlaps with the F 2p valence band. However the 4f orbitals can be distinguished by resonant excitation of the giant 4d—. 4f transition in the rare-earth cations [221. The broad feature above the 4d absorption threshold in the rare-earths was attributed to a collective excitation of the outer shell (4d4f) electrons of the ion during the 4d 4f—’ 1 10 4f 9 4d transition [401.  A favored decay mode of this excitation is through — 1 4 9 4d f ’  1 4d’O4 + photoelectron f’ leaving the ion in the same final state as reached by direct 4f photoemission [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  A  film of TmF 3 on Si(1 11). The photoemission spectra were taken at two different photon energies corresponding to resonant (A) and non-resonant (B) excitations and are normalized by the incident photon flux on the sample. Spectrum B (the non-resonant one) contains only the signal from the F 2p valence band and the non-resonant photoemission from the Tm 4f levels. Spectrum A (the resonant one) contains not only the above mentioned signal, but a large enhancement of the photoemission signal from 3 4f 12 levels due to excitation of the giant resonance in the outer shell electrons the Tm of the rare-earth ion. Spectra A and B are normalized by the current in the synchrotron storage ring during the measurement. The spectrum (A-B) was obtained by subtracting the non-resonant spectrum from the resonant one, which therefore corresponds to the  Chapter 3  24  Cl) 0 D  0  C)  >-  F— C/)  z w  I  z  -20  -15 -10 -5 BINDING ENERGY (eV)  0  Figure 3.2: Valence band region of TmF 3 on Si taken at two different photon energies A and B. The difference spectrum (A-B) is also shown. The inset shows the absorption spectrum of TmF 3 in the region corresponding to the Tm 4d absorption edge. The count 3 at the maximum of spectrum A and is about zero at the high energy number is 1.4X10 ends of all three spectra.  Chapter 3  25  U) C’, 4C  0 C)  >-  I— C’)  z  w I  z  -20  -15 -10 -5 0 BINDING ENERGY (eV)  5  Figure 3.3: Valence band region of YbF 3 on Si taken at two different photon energies A and B. The difference spectrum (A-B) is also shown. The inset shows the absorption spectrum of YbF 3 in the region corresponding to the Yb absorption 4d edge. The count number is about 800 at the maximum of spectrum A and is about zero at the high energy ends of all three spectra.  Chapter 3  26  photoemission signal from the 4f 12 orbftals on the Tm 3 ion only, provided we neglect the small difference (less than 15% [42]) in the cross section of the F 2p orbital for the two photon energies used. The inset in Figure 3.2 shows the absorption spectrum of the TmF film in the vicinity of the Tm 4d edge; the photon energies used to obtain same 3 spectra A and B are indicated by vertical lines. Photoemission spectra of the valence-band region for a 10  A thick film  of YbF 3  annealed at about 400 °C are shown in Figure 3.3. As in Figure 3.2 the photon energies used to obtain Spectra A and B are shown in the inset. The difference spectrum (A-B) was obtained in the same manner as for TmF , and the first two peaks at 15 eV and 10 eV 3 binding energy correspond to the partially filled 4f levels in the Yb 3 ions. We attribute the peak at 2.5 eV binding energy in spectra A and B to the regular (non-resonant) photoemission signal from the completely filled 4f 14 levels in the Yb 2 ions. The filled 14 configuration of Yb 4f 2 has no 4d-4f transition, so that there is no extra enhanced signal from these levels with photon energy corresponding to the 4d 13 transition as -4f 10 in the Yb 3 case. Therefore the difference spectrum (A-B) only maps out the 4f 13 levels 3 ions. We attribute the feature near 3 eV binding energy in the difference in the Yb spectrum to incomplete cancellation of the 4f 14 photoemission due to systematic errors in the normalization of the spectra. The prominent splitting of about 5 eV in the 4f 12 levels of Tm 3 and 4f 13 levels of Yb 3 is due to the exchange splitting associated with whether the photoelectron comes from a state with a spin parallel or anti-parallel to the majority spin of the 4f electrons [43].  Chapter 3  27  Let  t  indicate the direction of the majority spin and  -I-  indicate the direction of  the minority spin.  In the Tm 3 case, the ground state configuration of 4f 12 is  5 ( 7 (4ft) 4fL)  =  either  with S  5 ( 6 (4ft) 4f.L)  configuration with S  1. The final states after removal of one 4f electron will then be  with S =  1/2 or 4 (4f..L) with S =3/2 [43]. 7 (4ft)  =  3/2 has a lower energy than the configuration with S  Since the =  1/2 due to  the exchange interaction [43], the binding energy of the 4ft electron is larger than the binding energy of the 4f .I- electron. The area under the two exchange splitting peaks should have a ratio of 7:5 in TmF 3 because there are seven electrons to choose from for the transition 5 (4fi-) 7 (4ft) (4f..L) 7 (4ft) transition 5  —*  —  (4ft) ( 6 4fJ) and there are only five choices for the  4 ( 7 (4ft) . 4f..L) This is consistent with what we observe in the  difference spectrum (A-B) in Figure 3.2 by visual inspection. The angular momentum multiplet splitting due to the spin-orbit interaction in the initial state and the final state further broadens the binding energy distribution in each transition [441. Since the final state  4 ( 7 (4ft) 4fI-)  has a larger total spin of S  which has a total spin of S  =  =  3/2 than the final state 5 (4fJ-) 6 (4ft)  1/2, the feature associated with the transition  7 (4f) (4ft) 5  —>  7 (4f-L) (4ft) 4 is wider than the feature associated with the transition  (4f..L) 7 (4ft) 5  —  5 ( 6 (4ft) . 4fL)  of the 4f —’ 4f 12  The above discussion of the binding energy distribution  transition is shown schematically in Figure 3.4.  Yb case, the 4f 13 ground state has the configuration of (4ft) In the 3 7 (4f.i-) 6 with S  =  1/2.  The final states after removal of one 4f electron will either be  (4f..L) with S (4ft) 6  =  0 or  5 ( 7 (4ft) 4f.L)  with S  =  1. Similarly, the binding energy of  the 4ft electron is larger than the binding energy of the 4f.i- electron. The area under the two exchange-split peaks have a ratio of 7:6 because there are seven electrons to choose  Chapter 3  28  3 Tm  Binding Energy spin-orbit splitting spin-orbit splitting  Figure 3.4: The schematic diagram showing the binding energy distribution of the . 4f 12 4f 11 transition. The transition ( 5 ( 7 (4ft) 4f-) S = 1) the ground-to-ground transiton.  —*  4 ( 7 (4ft) ( 4f.J.-) S = 3/2) is  Chapter 3  29  3 or Tm Yb 2  Binding Energy  spin-orbit splitting  >1  Figure 3.5: The schematic diagram showing the binding energy distribution of the 12 transition. The transition ( —’ 4f 13 4f 6 ( 7 (4ft) 4f.i-) S = 1/ 2) the ground-to-ground transition.  —  5 ( 7 (4ft) ( 4fJ.-) S = 1) is  Chapter 3  Si  30  Si  3 TmF  3 YbF  CB  CB  VB  VB  14 4f  13 4f  F2p  F2p  13 4f  12 4f  Figure 3.6: The schematic diagram showing the positions of the cation 4f levels relative to the Si valence band and the F 2p valence band for TmF 3 and YbF 3 on Si( 111) substrates.  Possible electron occupation is illustrated with the shaded areas.  unshaded area shows where the 4f 13 levels of Tm would be.  The  Chapter 3  31  from for the transition (4ft) 6 ( 7 4f..L) transition 7 (4ft) ( 4f J.,)6  —*  (4f.L) 6 (4ft)  and there are six choices for the  5 ( 7 (4ft) , 4f..L) which is consistent with our observation in the  difference spectrum (A-B) in Figure 3.3. The broadening of the two exchange-split features is due to the angular momentum multiplet splitting in the initial and the final states. Since the total spin S  =  0 for the final state 6 (4fJ,-) and the total spin S (4ft)  =  1  for the final state , 5 ( 7 (4ft) 4fL) the feature associated with the transition 7 (4f..L) (4ft) 6  —  (4f..L) is narrower than the feature associated with the transition 6 (4ft)  7 (4f.L) (4ft) 6  —  7 (4f..I-) (4ft) . The discussion of the binding energy distribution of the 5  4fl . 3 ..  12 transition is again schematically shown in Figure 3.5. 4f There is no exchange splitting in the 4f 14 configuration of Yb 2 since only one  spin state exists in the photoemission final state: 4f (S 14  =  0)—. 4f (S 13  =  1/2). The width  of about 2.5 eV of this feature can be attributed to the angular momentum multiplet splitting in the 4f 13 final state which has S=1/2. The relative positions of the energy levels are summarized in Figure 3.6. The valence-band offset between Si and TmF , 3 namely the energy separation between the top of the silicon valence band and the top of the F 2p level, has been measured to be 7.0 eV [22]. We were unable to determine the valence-band offset between Si and YbF3 because the 4f 14 levels of the Yb 2 ions obscure the overlapping Si valence band. We assume the offset between Si and the F 2p valence band of YbF 3 is the same as that between Si and TmF . 3 As it was reported earlier [22], the on-site Coulomb repulsion energy U prevents the Si valence-band electrons from occupying the empty 4f levels in both NdF 3 and TmF3. U is defined as the difference between the ground state energies of 4f’ 1 1 and 4f configurations [45]. It can be determined in photoemission from the difference between  Chapter 3  32  the binding energies associated with the ground-to-ground transitions of 4f’ 1 4ffl  4ffl4  -.  4f” and  of the same atom [46]. For example, U for Tm can be taken as the binding  energy difference between centers of gravity of the feature corresponding to the transition 5 ( 7 (4ft) 4f..L) —+ 7 (4ft) ( 4f..L)’ in the Tm 3 ion and the feature corresponding to the transition  6 ( 7 (4ft) 4f1-) —* (4ftY(4ü) in the Tm 2 ion [44]. Because the 4f levels are  very localized, this energy is large. For Tm, U is about 6.5eV [44, 45]. An earlier study of Tm 3 and Tm 2 ions in TmSe and TmTe alloys also shows that the width of the 13 4f  12 feature associated with the final state being the ground state 4f  5 ( 7 ((4ft) ) 4f..L-) ,  is about 5 eV for Tm 2 ions [44]. This compares well with the observed width of the 4f 13 12 feature in the Yb 4f 3 ions associated with the final state being the ground state (see Figure 3.3) which has the same  5 ( 7 (4ft) 4f..L) configuration. The on-site Coulomb  repulsion energy U together with the width of the 4f 13 ground state levels prevents electrons from transferring from the Si valence band into the empty levels in the 4f 12 configuration of the Tm 3 ions at the interface. In other words the highest occupied state 13 configuration is above the top of the Si valence band as shown schematically of the 4f in Figure 3.6. On the other hand, the 4f 14 levels of Yb 2 ions are apparently below the Si valence-band maximum which is consistent with the on-site Coulomb repulsion energy U of 6.5 eV [45, 46] and the small width of 4f 14 levels. This makes charge transfer from the Si valence band to YbF 3 possible. Figure 3.6 illustrates the above mentioned distinct phenomena at the TmF /Si(1 11) and 3 3 YbF / Si(1 11) interfaces. Figure 3.7 shows photoemission spectra of YbF 3 films for two different thicknesses deposited at room temperature on Si(l 11) substrates. The photon energy was 151.4 eV. Peaks from the Si substrate are clearly visible in the spectrum for the 1 oA  Chapter 3  33  film 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 4f 13 levels and peak B is from 14 levels. The ratio of peaks B/A for the 10 the Yb 4f corresponding ratio for the 70  A film.  A film  is 3 times larger than the  This suggests that there is a higher concentration  2 ions at the interface. But the very appearance of peak B in the thick film of the Yb while no Si peaks can be observed also suggests that the Yb 2 ions are not only at the interface but also in the film. Figure 3.8 shows photoemission spectra for the 10  A YbF 3  film before and after  an 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 weaker while the signal from Yb 4f’ 4 stays approximately the same after the anneal. The smaller photoernission signal from YbF 3 after annealing can be explained by a change in the deposited film as described below. One possibility is that the initially uniform film forms a monolayer coverage together with islanding of the YbF 3 overlayer (Stranski-Krastanov growth) after annealing.  Alternatively, it is possible that some loosely bound as-  3 re-evaporates on crytallization during the anneal. In either interpretation deposited YbF the concentration of the Yb 2 ions is higher at the interface which is consistent with the finding that the 4f 14 levels in the Yb 2 ions lie below the top of the Si valence band. In conclusion, we have studied and compared the energy alignment of TmF 3 and 3 on Si(1 11) substrates using photoemission spectroscopy and x-ray absorption YbF spectroscopy. By resonantly exciting the 4d-4f transition in the rare-earth, we were able to enhance the photoemission signal from the 4f electrons and thereby distinguish it from the overlapping signal from the F 2p valence band of the rare-earth trifluorides. While  Chapter 3  34  the electrons in the Si valence band are prevented from occupying the empty 4f levels in 3 at the interface by the on-site Coulumb repulsion energy U, the charge transfer TmF from Si to YbF 3 is possible because the totally filled 4f states of Yb still lie below the Si valence band maximum.  Chapter 3  35  Cl) Cl) 4-’ C  0  C)  >-  I C’)  z  w H  z  -100  -80  -60  -40  -20  0  BINDING ENERGY (eV)  Figure 3.7:  Photoemission spectra of YbF 3 films of two different thicknesses as deposited on room temperature Si(l 11) substrates. The photon energy used was 151.4 eV. The count number is l.4X10 5 at peak A in the top spectrum and is about zero at the high energy ends of all two spectra.  Chapter 3  36  •II1IIII111  IlIllIllIll  I  I  Cl) Cl) D  0  C.)  >F  As deposited  (I)  z  w H  z  1111111  -20  I  11111  I  III  I  111111  -15 -10 -5 0 BINDING ENERGY (eV)  5  Figure 3.8: Photoemission spectra with the resonant photon energy of 182 eV taken on a 10 A YbF 3 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 at the high energy ends of all two spectra.  Chapter 4  37  Chapter 4 X-ray Absorption Spectroscopy  4.1 Experimental Arrangement of X-ray Absorption Spectroscopy The experiments of x-ray absorption studies in this work were carried out at the Ui beamline of the National Sychrotron Light Source of Brookhaven National Laboratory. As described in Chapter 2, this beamline provides a high intensity photon source from 25 eV to 1300 eV continuously through an extended range grasshopper (ERG) monochromator. The highest resolution (FWHM) achieved on this beamline is 0.02  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 total electron yield method in which all electrons emitted from the sample are collected regardless of their kinetic energies. As it has been shown [47], total electron yield is to a good approximation, proportional to the absorption coefficient. However, it is not exactly proportional to the absorption as we show below. In a photoelectric process, for every core hole created by a photon, there are 1-ö photoelectrons directly emitted from the sample, o3 secondary electrons emitted from the sample due to inelastic scattering of the primary photoelectrons, xA Auger electrons, and czM(0) electrons emitted due to multi-electron excitation processes. Therefore, in an energy region around the threshold  of a core excitation, the total number of electrons emitted from the sample per incident photon is  Chapter 4  Total Yield  38  =  po){(l-6)  +  x6  + aA + aM((o)  },  where ji(co) is the absorption coefficient. Although 6 and as depend on the photon energy 0)  indirectly through the photoelectron kinetic energy,  they are structureless  monotonically increasing functions of 0 due to the averaging over a series of statistical scattering processes [47]. While aA is independent of w, the only factor that introduces structure is the multi-electron coefficient aM(co). Therefore, in energy regions where there is no multi-electron excitation or multi-electron excitation is not strong, the total electron yield is a very good measure of the x-ray absorption spectrum. Although seemingly more precise x-ray absorption measurements can be made by the partial electron yield method which involves collecting only Auger electrons, the total electron yield method gives a better signal to noise ratio since only a small fraction of the direct photoelectrons and Auger electrons escape into vacuum compared with the total electron yield which is dominated by inelastically scattered electrons. A microchannel plate detector with two microchannel plates in series was used to count the emitted electrons. Each microchannel plate has an output of lO electrons for a single input event [38]. In addition, either the sample or the detector or both can be biased to optimize the signal acquisition depending on the specific experimental conditions. The whole chamber where the experiments took place was under UHV conditions ( iO Torr) throughout the experiment.  Chapter 4  39  §4.2 Fluorine K edges of Alkaline-Earth Fluorides Fluorine K edges of the three alkaline-earth fluorides CaF , SrF 2 2 were , and BaF 2 measured on powdered samples pressed on indium foils. A +80 V bias was applied to the front grid of the detector in the case of CaF 2 to achieve the best signal to noise. The bias applied in the case of SrF 2 was -80 V. In the case of BaF , a -20 V bias was applied to 2 the front grid of the detector and a -80 V bias was applied to the sample.  The  experimental data of the fluorine K edges of all three alkaline-earth fluorides studied in this work are shown in Figure 4.1, Figure 4.2 and Figure 4.3 respectively. A resolution of about 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]. The first sharp peak in the CaF 2 spectrum has been identified as a transition from the F is state to a bound core exciton state, through deexcitation studies by Tiedje et al [48]. We assume that the first peaks in the SrF 2 spectra are also core excition transitions. 2 and BaF The discussion about the features at higher photon energy will be deferred to §4.4 at which point we will have a better understanding of XANES.  Chapter 4  9.0 1  40  III liii..  II 11111  111111  III liii  liii liii 111111111111 11111111  FKedgein CaF 2  8.0 1  7.0 1 0 0  .4-  D 0  6.0 1  C.)  5.0 1 4.0 1 3.0  11111111 I 1111111111  680  690  liii  700  111111  liii  710  iii IIIII11I1I1II1I  720  730  Photon Energy (eV)  Figure 4.1:  I liii III  The experimental spectrum of the fluorine K edge in CaF . 2  740  Chapter 4  41  1.2 10 1.1 10 9.3 Cl) Cl)  D 0  8.0 1  C)  6.7 1 5.3 1 4.0 680  690  700  710  720  730  Photon Energy (eV)  Figure 4.2:  The experimental spectrum of the fluorine K edge in SrF . 2  740  Chapter 4  42  2.0 1.8  Cl) C’,  1.6  —S  D  0  1.4  C.)  1.2 1.0 8.0 680  690  700  710  720  730  Photon Energy (eV)  Figure 4.3:  The experimental spectrum of the fluorine K edge in BaF . 2  740  Chapter 4  43  §4.3 Bragg Reflection Model for XANES in Crystalline Solids As described in chapter 1, x-ray absorption extending less than about 50 eV above the 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 as extended x-ray absorption fine structure (EXAFS). EXAFS is well understood in terms of the interference of the photoelectron wave with backscattered electron waves from the surrounding atoms [27, 28]. XANES is more difficult to model quantitatively than EXAFS because of the very strong interaction between the photoexcited electron and the neighboring atoms, including atoms beyond the nearest neighbors [24, 25]. There is no general method for calculating XANES spectra and XANES spectra for a large number of solids have not been calculated [23]. As early as in 1932, using the nearly free electron approximation, Kronig first proposed that EXAFS oscillations were due to the Van Hove singularities in the density of states at special points in the k-space, namely, at the points where the wave vector k=G12, where G is a reciprocal lattice vector [49].  (4.3.1) This explanation was finally shown  conclusively to be incorrect by Stern in 1974 [28]. Stern pointed out that Kronig ignored the absorption matrix element effect, which is more important than the density of states contribution. Also it has been shown that even the absorption matrix element requiring long range order can not account for the EXAFS oscillations which are sinusoidal with periods generally in the order of 100 eV. It has been shown that the short range order  44  Chapter 4  contribution, namely, the scattering of the photoelectron by the nearest neighbor atoms is the most important mechanism of EXAFS which has been supported experimentally with the observation of EXAFS also in amorphous systems. However, XANES and EXAFS are very different in terms of the nature of the photoelectron scattering even excluding the core hole effect and the multi-electron effect that are often associated with XANES but normally ignored in EXAFS. In EXAFS, the wavelength of the photoelectron is smaller than an interatomic distance and the important scattering 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 than typical interatomic distances. (The wavelength of a free electron verses its kinetic energy is shown in Figure 4.4.) Also, the electron inelastic mean free path increases rapidly from a few inter-atomic distances to tens or hundreds of lattice constants below 50 eV kinetic energy 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 fluoride versus the electron kinetic energy as an example [52].) Therefore, the long range order effect that is not important in EXAFS might be important in the XANES case. In light of this difference, it is appropriate to explore the role of long range order scattering, namely the scattering of photoelectron waves by crystal planes, in XANES. We have previously proposed the Bragg reflection model relating positions of 2 G” h 2 peaks 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 the copper K edge for the usual f.c.c. form of copper and for strained thin films of copper  Chapter 4  45  15  I  I  I  11111  I  I  I  11111  I  I  I  11111  I  I  I  iiiiiL  I  I  I  IIIiiI  I  I  I  11111  € 4-  c,) C a) -J 10 >  C  0 4-  C) 0)  5  uJ 0) 0) U-  0 100  101  2 1o  Free Electron Kinetic Energy (eV)  Figure 4.4: The free electron wave length as a function of the free electron kinetic energy.  46  Chapter 4  LITHIUM FLUORIDE • IMFP Values Mod. Bethe Eq. TPP-2  50  -  -  -  0140  30 /  1 3Q  20  .  20  -  .  10  10•  .—  0  0  I  .  0  500  •  100  200.  .1  1000  1500  2000  Electron Energy (eV)  Figure 4.5: The calculated electron inelastic mean free path (IMFP) in LiF by Tanuma et al using various algorithms [52].  47  Chapter 4  with the body centered tetragonal structure [53]. The principle of the model is that the condition k  on the electron wave vector k gives the strongest constructive  =  interference between the outgoing photoelectron wave and the backscattered wave. It is not clear however, why the condition of the maximum constructive interference of the electron final states should necessarily enhance the absorption so as to give a peak at the corresponding energy. Destructive interference or some other feature could be present instead. In the following, we will reintroduce the Bragg reflection model in a more quantitative way and examine the limits of the model. The wave equation that the electrons in a periodic system have to satisfy is Schrodinger’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 G  where  0  k G  =  —(k 2m  —  2 is the free electron energy, e is the energy eigenvalue, G)  (k GI iy) -  =  .  $ei(r)dr unit  (4.3.3)  cell  is the Fourier coefficient of the electron wave function i(r) where V is the volume of the unit cell, and UG  =  -L  Je’U(r)dr  (4.3.4)  unit cell  is the Fourier coefficient of the potential energy. Its matrix form is simply üiy = where  Chapter 4  48  UG  •..  U,  :“  UG UGI  Ek_G  ••  UGG  (4.3.5)  UG_G  It is worth mentioning that the diffraction of electron waves by a periodic system is analogous to x-ray diffraction, with UG playing a similiar role in electron diffraction as the Fourier coefficient of the lattice electric charge distribution PG does in x-ray diffraction. That is to say, electrons are diffracted by the periodic crystal potential while x-rays (electromagnetic waves) are diffracted by the periodic charge distribution. Hence, some concepts associated with the x-ray diffraction are also useful here. Taking the example of monatomic lattices for simplicity, we borrow the concepts of structure factor and form factor from x-ray diffraction. The periodic crystal potential can be expressed as the sum of potentials of individual atoms or ions U(r)  u(r  —  r ) 1 ,  (4.3.6)  then  u G =! =  IerU(r)dr3  unit cell fe’u(r)dr whole crystal  = SGf(G)  (4.3.7)  49  Chapter 4  with 2 denoting the volume of the whole crystal, where SG  =  !Ee1Gri is the structure factor and f(G)  =  ._Jeru(r)dr3 is the form  factor, which have a similar meaning as in x-ray diffraction [62]. N is the number of 2/N. The structure factor SG depends on the geometrical 2 = atoms in the crystal and 0 configuration of the atoms in the crystal while the form factor f(G) represents only the contribution 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 of constructive interference between the incident wave and the reflected wave, also applies to the case of electron waves in a crystal. In the limit of a weak crystal potential, i.e., when  UI << e°  the solutions of  Eqn(4.3.2) are not very different from the free electron wave function except for the states very close to Bragg planes in k-space. One can expect that only those states that are close to Bragg planes contribute to the structure in the XANES spectrum significantly. 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 electron wave and have different absorption cross sections from the free electron wave. In a qualitative 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 energy 2 /G.2 t —  1  2m ‘ 2)  2IUGI  and a valley just above the same energy with a spacing of approximately  [28].  centered at  Stern further concluded that the absorption would have only a valley  2 /G2 h  I —I 2m2) —  corresponding to the valley in the density of states [28]. Stern  50  Chapter 4  concluded that these variations were too small to account for the large EXAFS oscillations. Now in order to understand the role of Bragg planes in XANES more accurately, we are going to quantitatively examine the absorption changes associated with a Bragg plane. We begin by showing how constant energy surfaces in k-space are affected by a Bragg plane. Figure 4.6 shows the constant energy surfaces in k-space with only one Bragg plane shown for clarity. It is equivalent to having only two reciprocal lattice (k2 uG simply is I vectors F and G in k-space. Thus the Hamiltonian matrix 2 un  UG Ik-GI  )  which we set all the physical constants equal to unity. In the picture we have set = 9. The dotted line indicates the sole Bragg plane T=(0, 0, 0), G=(l0, 0, 0), and defined by G/2. The numbers on each curve indicate the energy of that particular copstant energy surface. The constant energy surfaces are determined by the magnitude of 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 for the wave exp(Gr) while a negative UG means that the crystal potential is attractive for the wave exp(Gr). We proceed with our discussion assuming that UG is positive for definitiveness. 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 hand side are cosine-like [28, 54]. In the case of K edge absorption where the initial state is a is state, the absorption mathx element with a sine-like final state is  -.J  times the one  with a plane wave while the absorption matrix element with a cosine-like state is zero.  Chapter 4  51  k 12 10 8  6 4 2  G -2  -4 -6  -  -  -  -12 I  I  0  I  I  I  5  I  I  I  10  Figure 4.6: Constant energy surfaces in k-space with one Bragg plane. With all the physical constants equal to unity, the free electron energy  =  k. The dotted line  indicates the Bragg plane defined by G12. The number on each curve indicates the energy of that particular constant energy surface.  52  Chapter 4  Since the absorption coefficient is proportional to the square of the matrix element, states close to the Bragg plane on the left hand side have twice as large a contribution to the absorption coefficient as the the states that are far from the Bragg plane, and states that are close to the Bragg plane from the right hand side do not contribute to the absorption coefficient. The states with energies smaller than e  =  =25 are all below the  = 2 k  Bragg plane, while the states with energies larger than  (_)  +  IUGI = 34  are distributed  on both sides of the Bragg plane. When the energy is much smaller than 25, the constant energy surfaces are not very different from spheres (or circles in two dimensions), and all the states on the sphere have about the same contribution to the absorption coefficient as the 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 states , the absorption matrix elements are increased, and therefore, the total tt on the “neck absorption at this particular energy is enhanced. When the energy exceeds 34, the constant energy surface crosses the Bragg plane. As some states start to appear on the right hand side of the Bragg plane, their suppressed absorption contribution causes the overall absorption to decrease. When the energy is much larger than 25, the constant energy surface is again not very different from a sphere except it is broken at the Bragg plane. The number of states that are close to the Bragg plane and on its left hand side is about the same as the number of states that are close to the Bragg plane and on its right hand side. Therefore, the deficient contribution to the absorption coefficient from the states on the right hand side of the Bragg plane is made up by the enhanced contribution from the states on the left hand side of the Bragg plane, so that the total absorption  Chapter 4  53  contribution from all the states with that particular energy is about the same as that by the unperturbed plane wave states with the same energy. From this qualitative analysis, one can see that an oscillation occurs in the absorption coefficient when the energy sweeps through the special value defined by the =  Bragg plane e  =  ()2.  2 k  =  In order to know what exactly happens at e  =  2 k  however, a quantitative analysis is needed. For this purpose, we calculate numerically the absorption coefficient with a artificial potential UG, this time in three dimensions. To reflect 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 and  the 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 k space, 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 are interested in XANES which involves deep core states as initial states. For simplicity we make 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, if the crystal forms a Bravais lattice which has inversion symmetry about every atom, UG must be real. When a crystal does not form a lattice with inversion symmetry (e.g., a Bravais lattice with a basis), UG can be complex. An example is crystalline silicon which . There is 2 has no inversion symmetry about a silicon atom. Another example is CaF  Chapter 4  54  >1 wave  eI9X  (a)  wave  eI9X  (b)  wave  e9X  (c)  • species 1  o species 2 Figure 4.7:  One-dimensional schematic diagram showing examples of the Fourier  coefficients UG being complex. All three waves above have g = 2., where a is the lattice constant or the periodicity of the lattice. (a) U isreal. 9 (b) U 9  =  (1  (c) Ug  =  1 with the origin at species 1; U --f 9  1)  f, where f denotes the form factor. =  1 with the origin at species 2. jf  55  Chapter 4  Ca sites, but no inversion symmetry about F- sites. + inversion symmetry about 2 2 site and UG is imaginary for Consequently, UG is real when the origin is chosen at a Ca some values of G when the origin is chosen at a F- site.  Figure 4.7 schematically  illustrates in one dimension examples of UG being complex due to lack of inversion symmetry. Figure 4.8 shows the ratio of the absorption coefficient for the model potential described above to the absorption coefficient for the unperturbed plane waves. The ratio of the density of states with the model potential to the density of states without the potential is also shown in the figure for comparison. The light solid line is the density of states ratio with  =  3. It is independent of the phase of U. The heavy solid line shows  the absorption ratio with U=3, the dotted line shows the absorption ratio with U=-3 and the dashed line shows the absorption ratio with U=±i3. The arrow indicates the energy of 2 (G)  peak at  As we expected, the absorption ratio with the real positive U has a prominent below  (—)  The absorption ratio with the real negative U has a prominent  .  ()2. The absorption ratio with an imaginary U is very similar to  valley at UI above  the density of states ratio which has a less prominent peak at prominent valley at U  =  IUI  above  (_)  .  IUI  and a less  below  The absorption ratios with U  =  .=(1 ± i) and  =f1 ± i) respectively are shown in Figure 4.9. The absorption ratio with the real  part of U being positive has a similar behavior as the one with the real positive U and the absorption ratio with the real part of U being negative has a similar behavior as the one  Chapter 4  56  2  1  0 10  15  20 25 30 Energy (Dimensionless)  35  40  Figure 4.8: The ratio of the absorption coefficient for the model potential with various values, to the absorption coefficient for the unperturbed plane waves (shown by the heavy solid line, the dotted line, and the dashed line). The thin solid line shows the ratio of the density of states with IUI = 3 to the density of states with U = 0. The arrow indicates the energy defined by (G/2) . 2  Chapter 4  57  2  1  0  10  15  20 25 30 Energy (Dimensionless)  35  40  Figure 4.9: The absorption ratio with complex U. The arrow indicates the energy defined by (0/2)2.  Chapter 4  58  2  1  0 10  15  20 25 30 Energy (Dimensionless)  35  40  Figure 4.10: The absorption ratio with various magnitudes of U, assumed to be real and positive. The arrow indicates the energy defined by (0/2)2.  Chapter 4  59  with the real negative U. Figure 4.10 shows the effect on the absorption ratio of different magnitudes of U for U being real and positive. As the magnitude of U decreases, the absorption ratio approaches one, as expected. We have illustrated how Bragg planes affect the absorption cross section through their 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 of the photoelectron wave by a family of lattice planes generates both sine and cosine-like standing waves relative to the site of the photoabsorption. As the sine-like standing waves have an enhanced K edge absorption and the cosine-like standing waves have a suppressed K edge absorption, they collectively create a peak or valley in the absorption spectrum in the vicinity of the energy defined by k=G/2.  Case Study In the following, we illustrate and test the model with K edges in a number of materials. Copper is the classic test case for new models of x-ray absorption. It is a good test material because the core hole is well screened so that core hole effects are minimal in the absorption spectra and it has a simple crystal structure. More importantly, for us the copper K edge has been measured on two forms of copper with different crystal structures (f.c.c. and b.c.t.) [53, 55] which is an ideal situation for identifying the crystal structural effect in XANES. We look at iron next in order to compare the XANES in two materials with similar structures but different composition, namely, b.c.t. copper (almost  Chapter 4  60  b.c.e.) and b.c.c. iron. Then we test the model on crystalline silicon as an example of a covalent solid and solid neon as an example of a molecular solid. Finally, we test the Bragg reflection model on the fluorine K edge in CaF2 which is an insulating ionic compound.  Copper We know UG is real for systems that have inversion symmetry which is the case for Cu (f.c.c. structure) and Fe (b.c.e. structure). We only need to determine the sign of UG in these cases. Harrison has shown, taking Al as an example of a simple metal and Cu as an example of a transition metal [56, 57], that the pseudopotential form factors of these metals are negative for small G and positive for large G. The form factor crosses from negative to positive at about G I kF  1.5 for Al and G / kF  slowly approach to zero again when G / kF —+  oo•  1.8 for Cu and they  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 crystal  forms a Bravais lattice. As we have shown earlier, a positive UG causes a peak in K-edge XANES in the 2 “G” h 2 vicinity of the final state kinetic energy —I — I Thus, there should be peaks at these 2m2) .  energies in the Cu K edge XANES spectrum. The allowed reciprocal lattice vectors and their corresponding degeneracies NG and energies 8 G are tabulated in Table 4.1 for copper.  61  Chapter 4  (eV)  eG  (eV)  G  NG  8.7  331  24  54.9  6  11.6  420  24  57.8  220  12  23.1  422  24  68.8  311  24  31.8  511  24  77.4  222  8  34.7  333  8  77.4  400  6  46.3  440  12  92.3  G  NG  111  8  200  G 8  Table 4.1: The reciprocal lattice vectors for f.c.c. copper. NG is the degeneracy or the number of planes associated with the reciprocal lattice vector and h (G’ 2 2 2m’2  =  —  Figure 4.11 shows the comparison between the Bragg reflection model and the measured x-ray absorption data for the K-edge of f.c.c. Cu [53]. The known structure of copper defines the energy positions of the Bragg reflection resonances in the XANES spectra. In the weak potential limit the amplitude of the Bragg reflections will be proportional to IUG. For simplicity we assume that these Fourier coefficients have constant magnitudes in the energy range of interest. The total scattering amplitude, due to a given lattice plane, will also depend on the number of equivalent planes of different orientation and on the atom and plane density for that particular set of lattice planes. The product of the last two factors will be a constant for all the Bragg peaks, since any set of planes defined by a reciprocal lattice vector must include all the atoms in the crystal. We  Chapter 4  62  also neglect effects due to polarization of incident photons relative to the crystal. Thus the x-ray absorption spectrum can be modelled by,  (4.3.8)  where CF is the Fermi energy (7 eV for Cu [54]) and s G  2 G” h 2  =  —I 2m2} —  .  The function c(e)  describes the energy dependence of the absorption cross-section in the absence of the backscattering from the neighboring atoms. The oscillatory part is in the second term which is the sum of Lorentzian functions centered at each  —  EF  simulating the peaks  produced by each G. The constant ex is a parameter which describes the fraction of the outgoing electron wave which is Bragg reflected. The linewidth factor  is equal to the  sum 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 obtained from RPA (Random Phase Approximation) calculations of the imaginary part of the electron self-energy [59] which are in reasonable agreement ( 50%) with experimental measurements at selected energies [60, 61]. For an f.c.c. crystal the allowed reciprocal 2,r lattice vectors are of the form G = —(h,k,1) where h k 1 are all even or all odd. For the a energy dependent cross-section c(E), we have used the smooth-curve fit to the experimental data shown by the thin dashed line in Figure 4.11. The model spectrum has been convolved with a Gaussian broadening (FWHM 1 eV) to simulate the experimental resolution.  Chapter 4  63  U)  4-’  C  D  C 0 4-’  0 0 U) .  -10  0  10  20  30  40  50  60  Energy (eV)  Figure 4.11: X-ray absorption at the K edge of f.c.c. copper. The experimental spectrum is shown as the solid line [55] and the calculated spectrum from the Bragg reflection model is shown as the dotted line. The thin dashed line is a smooth-curve fit to the data that is used to model the absorption cross section c(e) in equation (4.3.8).  Chapter 4  64  Cl) C  D  C 0 4-  0  0 Cl) -  -10  0  10  20  30  40  50  60  Energy (eV)  Figure 4.12: X-ray absorption at the K edge of b.c.t. copper. The experimental spectrum  is shown as the solid line [55] and the calculated spectrum from the Bragg reflection model is shown as the dotted line. The thin dashed line is the absorption cross section used in equation (4.3.8).  Chapter 4  65  As shown in Figure 4.11, the model reproduces all the main features of the experimental data with the exception of the structure near the origin where the model shows a shoulder near the zero of energy corresponding to the (111) reciprocal lattice vector and a peak corresponding to the (200) reciprocal lattice vector, while the experimental data shows only one peak. This is because the Fermi surface of copper forms a neck connecting to the (111) Bragg plane due to slight deviations from free electron 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 is observed. In order to see the role of the crystal structure in XANES, we test the model with absorption data for a new body-centered tetragonal (b.c.t.) form of copper that is grown as an epitaxial thin film on a (100) single crystal silver substrate [55]. This b.c.t. copper film is found to have lattice constants a=2.88  A  and c=3.l0  A,  which is 7.6% expanded  vertically from a perfect b.c.c. silver crystal structure [55]. The reciprocal lattice vectors and their correponding energies are listed in Table 4.2. The model spectrum is obtained using 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. The absorption spectra in Figures 4.11 and 4.12 clearly illustrate the role of the crystal structure in the XANES spectra and show that the model is able to predict the change in the positions of the absorption oscillations, when the structure of the copper changes.  Chapter 4  66  Cl)  .1-’  C  D  C  0 0 0 Cl)  -10  0  10  20  30  40  50  60  Energy (eV)  Figure 4.13: X-ray absorption at the K edge of b.c.e. iron. The solid line is the measured spectrum [55]. and the dotted line is the calculated spectrum from the Bragg reflection model. The thin dashed line is the absorption cross section used in equation (4.3.8).  Chapter 4  67  G  NG  101  8  110  (eV)  (eV)  G  NG  8.4  123  16  58.0  4  9.0  312  16  60.8  002  2  15.7  321  16  62.5  200  4  18.0  004  2  63.0  112  8  24.8  400  4  72.1  211  16  26.5  114  8  72.0  202  8  33.8  411  16  80.6  220  4  36.1  303  8  76.0  013  8  39.9  330  4  81.1  301  8  44.5  024  8  81.0  310  8  45.1  402  8  87.9  222  8  51.8  420  8  90.2  CG  CG  Table 4.2: The reciprocal lattice vectors for b.c.t. copper. NG is the degeneracy or the number of planes associated with the reciprocal lattice vector and h (G 2 2 2m2  Iron The experimental absorption data for a b.c.c. iron crystal [55] is shown in Figure 4.13.  The parameters needed for modelling are as follows: the Fermi  Chapter 4  energye  68  =  11.1 eV, and the lattice constant a=2.87A [54]. For a b.c.e. crystal the  allowed reciprocal lattice vectors are of the form G = indices h+k+l  must be even.  where the sum of the  The reciprocal lattice vectors for iron and their  corresponding energies are listed in Table 4.3. The Fermi momentum of iron is 1.71 so that (1,1,0) = l. kF and 8  =  A-i,  kF. We need to determine the sign of 6 . 2  UG just as in the copper case. It is uncertain where the form factor changes sign for Fe although 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 than 2,r 2r —(2,0,0). The sign of UG for G = —(1,1,0) as not significant, because is below a a .  .  the Fermi energy in this case. The model spectrum shown in Figure 4.13 is obtained by the same procedure that  I  e (eV)  h  (eV)  G  NG  9.1  222  8  54.8  6  18.3  123  48  63.9  211  24  27.4  400  6  73.0  220  12  36.5  411  24  82.2  310  24  45.7  330  12  82.2  G  NG  110  12  200  G 8  Table 4.3: The reciprocal lattice vectors for b.c.e. iron. NG is the degeneracy or the number of planes associated with the reciprocal lattice vector and 1i2 (G\2 C G =—I—  2m2  Chapter 4  69  was used in the case of the f.c.c. copper. Once again, the model reproduces all the main features of the experimental data, except at the absorption edge where the experimental data shows an extra shoulder. This extra shoulder at the edge of the experimental spectrum may be associated with d bands or other breakdown in the nearly free electron picture, close to the Fermi energy. The 0.05 eV exchange interaction in iron  [771 has  been neglected in this analysis.  Silicon XANES spectra for semiconductors and insulators are more complicated than for metals in the vicinity of the absorption edge because of core hole effects. The core hole potential becomes more important as the screening decreases. In cases where the core hole 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 for silicon in addition to the core hole effects. Silicon is a covalent solid in which the bottom of the conduction band is made up by the antibonding nearly s-like sp hybrid. The nearly free electron approach on which the Bragg reflection model is based is likely to be a poor approximation for the bottom part of the conduction band, which has a tight binding character. Thus, the Bragg reflection model is not expected to work as well near the absorption edge in silicon as it does at higher energies. As is well known silicon has the diamond structure consisting of a f.c.c. Bravais lattice with a basis of 000 and  The conventional cube lattice constant of silicon is  Chapter 4  70  G  NG  SG  111  8  (1±i)/2  220  12  311  (eV)  (eV)  G  NG  SG  3.8  620  24  1  51.0  1  10.2  533  24  (1±i)/2  54.8  24  (1±i)/2  14.0  444  8  1  61.2  400  6  1  20.4  551  24  (l±i)12  65.0  331  24  (l±i)12  24.2  711  24  (1±i)/2  65.0  422  24  1  30.6  642  48  1  71.4  333  8  (1±i)/2  34.4  553  24  (1±i)/2  75.2  511  24  (1±i)12  34.4  731  48  (1±i)/2  75.2  440  12  1  40.8  800  6  1  81.6  531  48  (1±i)/2  44.6  733  24  (1±i)12  85.4  CG  G 8  Table 4.4: The reciprocal lattice vectors for silicon with non-zero structure factors. NG is the degeneracy or the number of planes associated with the reciprocal lattice vector G. SG is the structure factor, and CG  5.430  A [54].  h G 2 2 I 2m’\2 —  Table 4.4 lists all the reciprocal lattice vectors that yield non-zero structure  factors along with the corresponding free electron energies EG  =  The pseudopotential form factor of silicon for small k values (k/kp <2) is readily available. For example, Harrison gives: fs (111) 1  =  -1.84 eV, f (220) 5 1  form factor crosses from negative to positive at kfkF  =  1.5 where kF  =  =  0.61eV. The  1.8  A-i  and is  defined by the valence band width [62]. We assume that the form factors for reciprocal  Chapter 4  71  U) C  D .c C 0 0 U) .0  -10  10  30 50 Energy (eV)  70  Figure 4.14: X-ray absorption at the K edge of crystalline silicon. The experimental spectrum is shown as the solid line [641 and the calculated spectrum from the Bragg reflection model is shown as the dotted line. The vertical bar indicates the origin of the free electron energy CG.  Chapter 4  72  lattice vectors larger than (220) are all positive. Therefore, these reciprocal lattice vectors should all generate peaks in the Si K edge spectrum. A comparison between the experimental data [64] and the Bragg reflection model is shown in Figure 4.14. The absorption intensity near the absorption edge is enhanced due to the core hole effect as expected [63]. No effort was made to account for this effect in the model. The model spectrum was obtained by summing Lorentzians centered at each energy eG  2 (G h 2 —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 Bragg reflections, as was done for the metals. Each Lorentzian was weighted by the scattering strength of each G which was approximated by the product of the number of planes NG as a function of orientation and the square of the structure factor SG 12. The scattering strength dependence on the potential form factor is ignored for simplicity since the form factor is more slowly varying than the other two factors. The vertical bar in Figure 4.14 indicates the origin of the free electron energy 8 G The best fit (by eye) to the experiment was obtained with the bar at 8 eV below the absorption edge. This suggests that the valence electrons should occupy the bottom part of 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 electrons intermediate between core electrons and free electrons. As a result, the valence band can be interpreted as being evolved from the bottom of a parabolic free electron band. In this picture, the origin of the free electron energy e =  h k 2 should be in the vicinity of the 2m  73  Chapter 4  bottom 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 to the bottom of the conduction band in silicon. As the silicon valence electrons are not free electrons, the discrepancy between the 8 eV separation suggested by the Bragg reflection model and the 13.6 eV separation is not surprising. Overall, the Bragg reflection model gives at best a qualitative fit to the silicon K edge spectrum.  Neon Solid neon forms a f.c.c. lattice with the lattice constant a  4.426  A.  Table 4.5  lists all the allowed reciprocal lattice vectors with their corresponding energies of 2 (G h 2 =—I 2m2 —  The 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 holder fixed on a cold end of a liquid He cryostat [66]. The first two sharp features at 868.3 eV and 869.6 eV respectively have been interpreted as excitonic transitions [66]. The model spectrum is shown in the top part of Figure 4.15 which was obtained in the same manner as before except the (000) peak was added to the spectrum to indicate the origin of eG.  Chapter 4  74  G  NG  111  8  200  (eV)  (eV)  G  NG  5.8  440  12  61.4  6  7.7  531  48  67.2  220  12  15.4  600  6  69.1  311  24  21.1  442  24  69.1  222  8  23.0  620  24  76.8  400  6  30.7  533  24  82.6  331  24  36.5  622  24  84.5  420  24  38.4  444  8  92.2  422  24  46.1  551  24  97.9  511  24  51.8  711  24  97.9  333  8  51.8  640  24  99.8  G 8  G 8  Table 4.5: The reciprocal lattice vectors for solid neon. NG is the degeneracy or the number of planes associated with the reciprocal lattice vector and eG  2 (G” h 2 =—I 2m2 —  We can think of the solid neon as being held together by the weak Van der Waals interaction 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 the neon conduction band and the origin of EG should be close to the bottom of the conduction band, which we indeed observe in Figure 4.15. Since solid neon is insulating,  Chapter 4  75  Solid Ne K edge  -10  10  30  50  70  90  Energy (eV)  1.5 >4 H U)  z  1 U  H  0  .5  0 860  880  900  920  PROTON ENERGY  940  /  960  980  eV  Figure 4.15: X-ray absorption at the K edge of solid neon. The experimental spectrum is shown in the lower part [66] and the calculated spectrum from the Bragg reflection model is shown in the upper part.  Chapter 4  76  the less screened core hole effect dominates the first 5 eV of the absorption edge which is not expected to be accounted for by the Bragg model. Neither did we attempt to reproduce the shift of weight from the higher energy part to the lower energy part due to the attractive core hole potential. Setting this aside, a rather good agreement is observed between 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 the ones in the model spectrum starting from 40 eV above the edge. This can be expected because we have assigned equal weight to all the Fourier coefficients UG as an approximation while in fact one would expect UG to decrease at large G. Also this is already in the EXAFS region where the photoelectron wavelength is much smaller than the nearest interatomic distance to produce strong scattering by the crystal planes [251.  Calcium Fluoride Finally, we attempt on a more complex solid in which there are two different atoms and the core hole effect is strong at the same time, namely the fluorine K edge in . CaF2 is a wide band gap (Eg 2 CaF  =  12.1 eV [67]) ionic insulator with the fluorite  structure. The crystal forms a f.c.c. lattice with a basis consisting of a Ca 2 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 for both Ca 2 and F- ions. We have, with the origin at a Ca 2 site, 2+n j = j{fCa(G)+ 2fF(G)cos[(nl + n 2 U(Ca )]} 3  Chapter 4  77  and with the origin at a F- site, UG(F)  1 =  3 1 i—(n + 2 ) n 2  IC  1 2f(G)cos[-(n  +  2 n  where n 1, n2, and n3 are indices of the reciprocal lattice vector G as expressed by , n 1 , n 2 ). As shown in Table 4.6 UG(Fj for (111), (311) and so on is 3 G =-(n imaginary and UG(F) for (200), (220) and so on is real. We have shown earlier that an imaginary UG or a positive UG will generate a peak in K edge XANES. We do not know whether UG(Fj for the (200) series is positive or not. Nevertheless, we assign peaks for each G and make peak intensities only dependent on the degeneracy number NG for simplicity.  Chapter 4  78  G  NG  111  8  200  6  220 311  (eV)  H  (eV)  G  NG  UG(F)  3.8  422  24  (2fF+fCa)  FCa) 2  5.0  511  24  Ca  34.1  12  (2fF+fCa)  10.1  333  8  Ca  34.1  24  Ca  13.9  440  12  (2fF+fCa)  40.4  UG(F) Ca  CG  if  8  400  6  331  24  420  24  30.3  i  i  1/  222  eG  FCa) 2  15.1  531  48  F Ca)  20.2  600  6  24.0  442  25.2  620  Ca  if  if  Ca)  45.4  24  FCa) 2  45.4  24  V’F’Ca)  50.5  F  i  Ca  44.1  if  i(  F’Ca)  Table 4.6: The reciprocal lattice vectors for CaF . NG is the degeneracy or the 2 number of planes associated with the reciprocal lattice vector G. UG is the Fourier coefficient of the crystal potential where Ca and fF are the form factors of 2 G h 2 the Ca 2 ion and the F- ion respectively, and e G = —1 2m’\2 —  Since the band gap of CaF 2 is large, we assume that the origin of 8 G would be close to the bottom of the conduction band (like the case of neon and unlike the case of silicon) for the parabolic nearly free electron band would be a good approximation for the CaF2 conduction band. Figure 4.16 shows the comparison between the experimental spectrum of the fluorine K edge in CaF2 and the model spectrum. The experiment has been described in §4.2. The first sharp peak (689.3 eV) is due to core exciton transition and the bottom of the conduction band is believed to be at the minimum between the first  two high intensity peaks (690.7 eV) [48].  Chapter 4  79  liii,..  111111 111111 IlIlilIll IlillIllIll IllilIllIl  Fluorine K edge (1) C  D  c)  C 0  0 0  o  o  —---  I  0  :•: :  Cl)  •  I.—  C’,  I  I  ‘4  ,__  ‘-0 ‘-0  R.’ •_  C)  :  :  0 C’, .  C’J  C’)  .  !_  .‘  •——.•  •.___•  1111111111111111111111111111111111111111111111111  680  690  700  710  720  730  Photon Energy (eV)  Figure 4.16: X-ray absorption at the fluorine K edge in CaF2. The solid line shows the experimental spectrum and the dotted line shows the calculated spectrum from the Bragg reflection model.  Chapter 4  80  The model spectrum was obtained with the same manner as before. The (000) peak was added to the spectrum to indicate the origin of  G 8  As shown in Figure 4.16, a  good agreement between the experimental spectrum and the model spectrum can be observed in the non-excitonic part although there is some discrepancy in the energies particularly for the two low energy peaks indicated by (111) and (200). This is not surprising considering the deviations from free electron behavior one might expect in 2 near the bottom of the conduction band and also the fact that the details of the CaF atomic form factor are ignored in the Bragg model. There is another notable feature that the 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 is in 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 the Bragg 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 gives the peak positions approximately. We know that peak positions are energy of  2 ,G.2 h  —f 1 2m’\2) —  below the  and that UG varies with G. However, the basis of this model is the 2 h  nearly free electron approximation that requires IUG <<  —I I 2m2)  deviation of the peak positions away from the energy of  —I I 2m\2I  —  2 G” h 2 —  ,  in which case, the is small. Therefore,  we expect the Bragg reflection model to be valid for materials with nearly free electron like conduction bands. For metals the bottom part of the parabolic nearly free electron  Chapter 4  81  band is occupied up to the Fermi level so that the origin of 8 F below the bottom of G is 8 the conduction band. For semiconductors or systems with small band gaps the valence band occupies the bottom part of the parabolic nearly free electron band and the origin of G 8  is close to the bottom of the valence band. For insulators or systems with large band  gaps, the origin of 8 G is close to the bottom of the conduction band since the valence electrons in these systems are too deeply bound to be approximated as nearly free electrons. The model cannot account for features caused by core hole potentials. The core hole potential is most important when there is little screening such as in insulators. In the case of the fluorine K edge in CaF , for example, the excitonic peak is prominent in the 2 measured absorption edge, and not present in the Bragg reflection model. The model also does not reproduce the enhancement of the absorption intensity at the edge due to the attractive core hole potential. After all, the Bragg reflection model is a semi-quantitative model in which the cross section between the initial core state and the final state modulated by the scattering is not actually calculated.  In order to have a more  quantitative understanding of XANES spectra including not only positions but also shapes and intensities of the features, and especially the influence from the core hole potential, one must actually calculate the final states resulting from both the long range scattering by the crystal potential and the attraction by the core hole potential. This is what we intend to do in the next section.  Chapter 4  82  §4.4 Bandstructure Calculation of XANES at the Fluorine K Edge in CaF 2 and 2 BaF Separate approaches are often used to calculate the excitonic part and the higher energy part of the XANES spectrum for insulators like CaF . For example, in the case of 2 the calcium L edge in CaF 2 which is dominated by the localized excitonic features at the edge, 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 —> 3d transitions [681. The relatively weak features in the higher energy region were studied separately. They were shown to agree qualitatively with the Bragg peaks in a synthetic spectrum, formed by adding two experimental fluorine K edge spectra with one shifted down in energy by 3.5 eV to simulate the spin-orbit splitting of Ca 2p, suggesting the common Bragg scattering origin of these features in both calcium L edge and fluorine K edge [53]. The best quantitative model so far for the fluorine K edge absorption in CaF2 is a multiple scattering calculation on a 23 atom cluster [32]. However this model does not show as many peaks as the experimental spectrum. Presumably the cluster is not large enough to describe the crystal potential over a sufficiently long range to give an accurate description of the scattering of the photoexcited electron in the final state. The fluorine K edge absorption is difficult to calculate accurately because of the need to include both the localized potential of the core hole and the long range periodic potential of the surrounding crystal lattice at the same time.  Conventional electronic structure  calculations are typically designed to address either the molecular limit or the long range  Chapter 4  83  periodic potential (band structure) limit [33]. In the fluorine K edge problem, optical transitions must be calculated between a strongly localized F is core electron and a final state which depends on the localized core hole potential as well as the long range periodic potential of the crystal. Interactions with a large number of neighboring atoms must be included 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 interband transitions, which is 12.1 eV for CaF . This is an unfavorable situation for multiple 2 scattering calculations because many scattering paths must be considered if the inelastic mean free path is long. In order to improve our understanding of the fluorine K edge in 2 and BaF CaF , in this section we adopt a complementary approach to the earlier multiple 2 scattering calculations and use a one-electron band structure technique with a pseudopotential approximation for both the crystal potential and the core hole potential. We calculate the absorption coefficient from the transition rate given by standard time-dependent perturbation theory, —  6(ElS 2 i(EfIpIElS)  where ha, is the photon energy,  —  Ef  —  ha,)  (4.4.1)  ) is the initial is state, lEf) is the final state in the 1 IE  conduction band, and p is the momentum operator. A one electron Hamiltonian is used /2m 2 to calculate the final states in the conduction band: p  +  U  +  Uc. This Hamiltonian  includes the potential U of the ions in the absence of the core hole as well as the potential Uc of the photoexcited core hole. With a plane wave basis, the one electron Schrodinger equation is,  Chapter 4  (e_G  84  —e)(k—GIy)+ (uGP_G +UG_G)(k—G j)— 0  (4.4.2)  G’G  where k is the wave-vector; G and G’ are reciprocal lattice vectors defined by the crystal  structure including the artificial supercells used to model the core hole potential, as described below. UG and U are the Fourier coefficients of the crystal potential and the core hole potential respectively. The crystal potential U is approximated by a pseudopotential in which each ion is represented by a truncated Coulomb potential. We express U in terms of the lattice sum of the potential due to each ion in the crystal, U  Ua(r aia  =  —  rj)  (4.4.3)  where the potential due to the ex ion at na is expressed as 2 Ze —  4,rse0 r—r.la  /  Ua(r  —  r)  r—r.la >r (4.4.4)  =  2 Ze —  2reeora 4  r—r.la  That is, more than a pseudopotential ionic radius ra from the ion the potential of the ion is represented by the Coulomb potential associated with a Za charged ion in a dielectric medium. Inside the ion the potential is assumed to be a constant equal to the potential at the pseudopotential ionic radius ra. Thus in addition to the crystal structure three parameters are needed to determine the crystal potential U: the pseudopotential ionic radius of the Ca 2 (or Ba ) ion, the radius of the F- ion, and the dielectric constant e. 2  Chapter 4  85  The value of the pseudopotential core radius for Ca in Harrison’s solid state table [62],  O.9A, was taken for the Ca 2 ionic radius.  While an F- ionic radius of 1.3A is often used  for fluoride compounds [62] we found it necessary to treat the ionic radius of F- as a parameter in the calculation; accordingly we found that a F- radius of  i.oA gave the best  results when compared with the experimental data. The appropriate value for the dielectric 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 the linewidth of the features observed in the low energy part of the experimental spectra. We attribute the observed linewidth to experimental resolution and broadening associated with the lifetime of the photo-excited electron. In any case the lifetime broadening observed in the experimental data is less than about 5 eV for all the peaks less than 30 eV above the edge. The corresponding excited state lifetime is sufficiently long that we can use the optical dielectric constant e of 2 appropriate for CaF2 in the photon energy range 0.1  -  10 eV [70], in the expression for the pseudopotential. Any wave vector dependence  of the dielectric constant is neglected. The potential energy U’ due to the core hole is approximated by leaving out the potential 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 an approximation to the true non-periodic potential, we use a super unit cell which contains a single core hole but several standard unit cells and then periodically repeat the super unit cell to obtain an artificial crystal with a periodically repeated core hole. This method has been used earlier for including the core hole effect in x-ray absorption spectra  86  Chapter 4  [63,71]. In our case the super unit cell is a simple cube with sides four times the nearest neighbor fluorine-fluorine distance; it contains 32 Ca and 64 F atoms and its size is limited by our available computing power. The super unit cell is schematically shown in Figure 4.17. Since U has the same periodicity as the crystal and Uc has the periodicity imposed by the supercells, the Fourier coefficient UG is non-zero for G is non-zero for G  =  =  (nl,n2,n3)21t/a and UCG  (ml,m2,m3)1r/a, where the integers n, fl2 and fl3 must either be all  even or all odd as required by the fcc symmetry, while the integers mi, m and m are unrestricted. , 9 2 The Fourier coefficients are obtained from the Poisson equation V  =  —.  We  show in the following, the advantage of determining the Fourier coefficients of the crystal potential U from the Fourier coefficients of the charge distribution. The potential is more difficult to evaluate directly than the charge density since the tails of the Coulomb potential originating from every ion in the crystal contribute to the total potential at any given point in the space. On the other hand, the charge distribution of each ion is localized with no long range contribution. From the Poisson equation, the definition of U=G 2 the potential energy U = —ep, and V UGexp(—iG. r), it is clear that 2 G  UG=—-Ge  (4.4.5)  where the Fourier coefficient of the charge distribution PG  =  r. 3 Jp(r)exp(—iG. r)d one cell  (4.4.6)  Chapter 4  87  0 0  Fluorine Ion  Fluorine with Core Hole  Calcium Ion Figure 4.17: The schematic diagram showing the super unit cell. Each super unit cell contains one core hole and 96 atoms.  Chapter 4  88  Let pa(r) denote the charge distribution of a single ion a. With the approximation on the potential made in eqn(4.4.4) which is equivalent to having the charge Za uniformly distributed on the surface of a sphere of radius ra, we have, Pa(r) 2 4,rra  (4.4.7)  6(r—ra),  and !  JPa(r)exp(_iG. r)d r 3 all space  =  Zaesin(Gra) Gr V  (4.4.8)  Thus, =pexp(iG.r ) PG 1 a 1 a Zaesin(Gra)(.G \ V Gra a 1 a  ‘I  1/  (4.4.9)  where the summation goes over all the ions within a single unit cell, and na denotes the position of the ith ion of the a species. For CaF2, we set the unit cell to consist of one 3 the other F- at !!!. The volume of the unit cell V = 2 at 000, one F- at U! Ca and 444 444 4 Thus, with Zat+ 2 and ZF=- 1, we have, UG —  G e 3 e a 3  {-sin(Grca)  —  _2cos[(ni +  +  )n 3 ]sin(GrF)}  (4.4.10)  Chapter 4  where i, n and 2,r G  89  fl3  are indices of the reciprocal lattice vector G as expressed by  One thing worth pointing out is that, unlike many other commonly used approximations such as the muffin-tin approximation in which the potential is arbitrarily set to zero at the interstitial space between atoms that introduces discontinuity in the potential between two different kinds of atoms, the Fourier coefficients obtained with eqn(4.4. 10) imply no discontinuity in the potential. The best of all, the model potential used here is closer to the reality since it includes the Madelung potential which could be important. Table 4.7 shows the numeric values of the Fourier coefficients of the crystal potential U with the origin 000 chosen at the Ca site. Since the Ca site has the inversion symmetry, this choice of the origin yields real values for all the Fourier coefficients of the potential U. One should note, however, it is more convenient to have the origin at the F site for the fluorine K edge absorption calculations since the initial is state involved in the absorption process is at a F site. One can simply translate the Fourier coefficients of U with the origin at the Ca site to the Fourier coefficients of U with the origin at the F site 2 +n )] in the fluorine K edge absorption 3 by UG(Fsite)= UG(Casite)exP[i-(nl +n calculation. The Fourier coefficients of the wave function are truncated at (3,1,1)2icla; this reduces the matrix defined by eqn(4.4.2) to 1029x 1029. Eigenvalues and eigenvectors of the matrix are obtained at each k point in the first Brillouin zone and are then used in eqn(4.4. 1) to calculate the absorption. Because of the crystal symmetry, only the section of the zone defined by k,  k  k, which is 1/48 of the first Brillouin zone, needs to be  Chapter 4  90  G  UG (eV)  111  -0.609  200  -0.628  220  -0.045  222  0.086  311  0.026  400  -0.003  331  0.038  420  0.066  422  0.011  333  0.018  511  0.018  Table 4.7: The Fourier coefficients of the crystal potential U used in the . The dielectric constant e of 2 calculation of the fluorine K edge absorption in CaF 2 is used. The values of the Fourier coefficients are obtained with the Ca site as the origin. included. It is scanned in steps of irl4a in k-space. The absorption coefficient is averaged with the momentum operator p oriented in the x, y, and z directions respectively, which takes into account the random crystal orientation relative to the photon polarization.  Chapter 4  91  The pseudopotential we use in the final states calculation has less amplitude at large wave numbers than the real potential since the pseudopotential approximates the rapidly varying potential near the ion core with a slowly varying (constant) potential. This makes it possible to calculate the electron energy eigenvalues accurately with a relatively small number of plane waves while truncating the Fourier expansions of the wave functions in k-space. While the pseudopotential can give a good representation of the energy band structure it does not correctly reproduce the shape of the wave functions in the vicinity of the ion core. This can cause problems with the calculation of the optical matrix element between the conduction band state and the F is core level which is strongly localized at the ion core. In order to obtain an accurate value for this matrix element we need to know the detailed shape of the conduction band wave functions at the ion core position which is not possible with the pseudopotential approximation. A consequence of this loss of high spatial frequency infonnation is that the size of the features in the x-ray absorption are reduced because of the small overlap matrix element between the high spatial frequency is wave function and the slowly varying wave functions 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. The pseudo-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 the eigenfunctions of the pseudopotential in the empty states. The calculated 2 CaF fluorine K edge absorption spectrum is shown in Figure 4.18. The spectrum has been convoluted with an energy dependent Lorentzian to take into account the final state lifetime, and a 1.0 eV wide Gaussian to reflect the experimental  Chapter 4  92  resolution. The energy dependence of the width parameter f in the Lorentzian is approximated with an empirical expression designed to take into account the increase in the electron inelastic scattering at the threshold for interband transitions. The energy dependence 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 the absorption edge (686.6 eV) in the model spectrum which is the s-like ground state of the pseudopotential representing the core hole. This feature is too small to see in the calculated spectra for the vertical scales shown in Figures 4.18-23. Since the core level is also an s-state the absorption cross section for this state is weak because the pure s to s transition is dipole forbidden. The first large peak in the model absorption edge is an overlapping combination of 2s and 2p-like states. Both the is and 2 si peaks are p excitonic in the sense that they lie below the bulk crystal conduction band edge. In the real fluorine ion potential we expect the first large peak in the absorption to be a combination of 3s/3p-like states and not 2s/2p.  The next lower states in the real  photoexcited fluorine ion potential are the filled 2p and 2s states and the half-filled F is core level. There is no state comparable to the is-like state that is present with our pseudopotential. Thus the 2s/2p-like wave functions in the pseudopotential represent the 3s/3p-like wave functions of the real potential. The purpose of the pseudopotential model is to replace the real potential with a model potential that has the same energy eigenvalues but is mathematically easier to solve. One accomplishes this by replacing the rapidly varying potential in the vicinity of the ion core with a slowly varying average potential. The wave functions of the slowly varying potential will approximate the real wave functions between the ion cores but will have a smaller number of nodes in the  Chapter 4  93  vicinity of the ion core. The is level in the pseudopotential will be ignored as an artifact that has no counterpart in the real potential. The effect of changing the binding energy of the initial is state is shown in Figure 4.19. The calculated spectra in Figure 4.19 were obtained with the best fit values of the Ca 2 radius of 0.9  A and the F- radius of 1.0 A.  As shown in Figure 4.19, the use  of the pseudo-core level with a different binding energy in place of the true F is core level only affects the amplitude of the absorption edge features and does not add additional structure or change their energy position since the empty state wave functions are still more slowly varying than the pseudo-core wave function. To illustrate the effect of changing the pseudopotential core radii of F- and Ca 2 in the absorption calculation, we show in Figure 4.20 the calculated spectra with fixed 2 radius of 0.9 Ca  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 and different Ca 2 radii, namely 1.1  A, 0.9 A and 0.7 A respectively.  A,  The binding energy  of the initial is state used in obtaining the spectra in Figure 4.20 and Figure 4.21 is the best fit value of 150 eV. The peak spacings are not very sensitive to the change of the pseudopotential core radii, while all the peaks move towards smaller photon energy as the potential strength is increased by reducing the pseudopotential core radii.  This is  consistent with an earlier finding that peak spacings are more or less determined by crystal structures [53]. The change of the pseudopotential core radii is found to have more effect on the amplitudes and shapes of the peaks.  Chapter 4  94  C,)  D  C 0 ci 0 0  680  700 720 Photon Energy (eV)  740  Figure 4.18: The experimental fluorine K edge absorption spectrum for CaF 2 is shown at the top and the calculated absorption spectrum is shown at the bottom. The peaks labeled a, b, c, and d are used in Figure 4.24.  95  Chapter 4  Cl) C  D  C 0 0 I— 0 U)  680  690  700  710  720  730  Photon Energy (eV)  2 with different Figure 4.19: Calculated fluorine K edge absorption spectra for CaF 2 radius of 0.9 A and the F- radius of binding energies of the initial is state. The Ca 1.0  A were fixed.  96  Chapter 4  I  I  IllillIllIl  11111111111  3 rF=l.  A  U) C  D  C 0  i.oA  r=  I  0  U)  .0  7 rF=O. I  680  I  I  I  I  i  i  i  690  i  I  i  700  i  i.i  I  i  710  i  i  A  i  I  I  720  I  I  I  730  Photon Energy (eV)  Figure 4.20: Calculated fluorine K edge absorption spectra for CaF2 with different F Ca radius (0.9 radii as indicated. The 2  (150 eV) were fixed.  A) and the binding energy of the F is initial state  Chapter 4  97  11111  I  111111111111111111  Cl)  rCa= uA  D  S 0  rCa= O.9A  0 C,)  0  IIIIIIIIIIIIIIIIII I 11111  680  690  700  710  720  730  Photon Energy (eV)  Figure 4.21: Calculated fluorine K edge absorption spectra for CaF 2 with different Ca 2 radii as indicated. The F- radius (1.0 (150 eV) were fixed.  A)  and the binding energy of the F is initial state  Chapter 4  98  huh  IhIlIhuIlIhI  111111  Absorption with Core Hole  Density of States 5  0 I  680  I  11111111111 I  690  700  Ii  710  I  111,111  720  730  Photon Energy (eV) Figure 4.22: The calculated fluorine K edge absorption spectrum for CaF 2 including the core hole potential is shown at the top. The second spectrum is the same calculation except the core hole potential has been omitted. The third spectrum is the calculated global density of states with the crystal potential and the core hole potential left out as in the second spectrum. The dotted line shows the broadening function r used in the Lorentzian convolution of all the spectra.  Chapter 4  99  The core hole potential has the effect of increasing the absorption cross section in the vicinity of the absorption threshold, at the expense of the absorption cross section at higher energy, as expected [71]. To illustrate this we show in Figure 4.22 the absorption spectrum calculated with and without the core hole potential. The spectra are convoluted with the same broadening functions as in Figure 4.18 above. The first large peak in the model absorption spectrum with the core hole lies 1.8 eV below the edge jump in the spectrum without the core hole. This is consistent with experimental observations which show that the first peak in the CaF 2 fluorine K edge absorption spectrum is excitonic with a binding energy of about 1.0 eV [48]. The global density of states for CaF 2 calculated with the model crystal potential is also shown in Figure 4.22. Except for the gap in the density of states near 696 eV produced by U 111 and , 200 the structure in the density of states is generally less U pronounced than the structure in the absorption. This shows that the optical matrix element 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 in the 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 core level and final state standing waves scattered by the crystal planes. Their spacing should be sensitive to the crystal lattice constant. As pointed out earlier the energy spacing of these peaks is consistent with successively higher order Bragg backscattering reflections from the surrounding crystal lattice [53]. The high intensity of the first peak after the  Chapter 4  100  G  UG (eV)  111  -0.478  200  -0.590  220  0.036  222  0.067  311  0.038  400  0.004  331  0.032  420  0.061  422  -0.008  333  0.009  511  0.009  Table 4.8: The Fourier coefficients of the crystal potential U used in the calculation of the fluorine K edge absorption in BaF . The dielectric 2 constant c of 2 is used. The values of the Fourier coefficients are obtained with 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 the bottom of the conduction band are more strongly affected by the core hole potential and have a larger amplitude on the excited atom and hence a larger absorption cross section.  101  Chapter 4  As a test of the structural interpretation of the peaks a-d we calculate the fluorine 2 but a larger lattice constant. 2 which has the same crystal structure as CaF K edge of BaF 2 case (1.0 We use the same value for the F ionic radius as in the CaF 2 radius to be Ba  i.iA.  A)  and take the  We take the dielectric constant to be 2. Table 4.8 shows the  numeric values of the Fourier coefficients of the crystal potential U with the origin , the binding energy of the pseudo-is core state was 2 chosen at a Ba 2 site. As in CaF reduced to 150 eV to better couple the core wave function to the final state wave functions obtained from the pseudopotential approximation and improve the quality of the fit to the relative amplitude of the experimental peaks. A comparison between the experimental absorption spectrum and the calculated absorption spectrum is shown in Figure 4.23.  The calculated spectrum was convoluted with the same broadening  2 case discussed above. functions as in the CaP 2 has a large excitonic peak at the threshold similar to The fluorine K edge in BaF . The next higher energy peaks are closer together and smaller in amplitude than in 2 CaF 2 for both the experimental data and for the model. As pointed out above we attribute CaP the peaks a-d to the effect of electron standing waves reflecting from crystal lattice planes. Assuming that the final states are free electron-like, and that the Bragg condition describes the electron reflections from the crystal lattice, then the product aIE, where E is the kinetic energy of the photoelectron and a is the lattice constant, should be constant 2 [72]. (This relation for corresponding peaks in the absorption spectra of CaF2 and BaF follows from the condition for Bragg backscattering reflections, k idea, in Figure 4.24 we plot  E aJ 4  =  G12.) To test this  2 against for the first four non-excitonic peaks for BaF  , for both the experimental 2 the corresponding quantity for the same four peaks in CaF  Chapter 4  102  spectra and for the models. The origin in energy is taken at the minimum in the absorption just above the excitonic line in both cases. This approximates the bottom of the conduction band. The lattice constants are 6.20A and  2 and CaF2 5.46A for BaF  respectively [34J. If the nearly free electron interpretation is correct, then we would expect the points in Figure 4.24 to fall on a line through the origin with unity slope. As illustrated in Figure 4.24 both the theoretical and the experimental spectra agree reasonably well with this prediction; the agreement is somewhat better for the calculated spectra than for the experimental ones.  Chapter 4  103  C,)  D £  C 0 0 0 Cl)  680  700  720  740  Photon Energy (eV)  Figure 4.23: The experimental fluorine K edge absorption spectrum in BaF 2 is shown in the 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  104  25  20  15  10  5  5  10 112 a(E.)  15  20  25  for CaF 2  Figure 4.24: A plot of the position of the peaks a-d for BaF2 from Figure 4.23 as a function of the position of the corresponding peaks in CaF2 from Figure 4.18. The peak positions are plotted as the square root of the energy of the peak above the bottom of the band multiplied by the appropriate lattice constant. The bottom of the conduction band is taken to be the minimum in the absorption between the first two large peaks in the absorption spectrum. The peak positions from both the experimental and the calculated spectra are plotted as indicated.  Chapter 4  105  §4.5 Multi-electron Excitations in XANES In the experimental absorption data for CaF 2 (Figure 4.18), there are broad and prominent features in the absorption in the region 15 eV-35 eV above the edge. Similar broad features are even more prominent in the BaF 2 edge, although at a smaller energy (Figure 4.23). These features are in a region of the spectrum where the band structure approach is no longer valid due to the computational limitations on the number of plane waves we are able to include in the calculation. Because the broad features have a qualitatively different shape and larger oscillator strength than the lower energy peaks a , one is tempted to conclude that they have a different physical 2 d, particularly in BaF origin. For example they may be due to resonant transitions to higher angular momentum states such as d and f-like states associated with the cations in the CaF 2 and BaF 2 conduction bands respectively [73]. Since the d and f like states are on the cations and the 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 create multi-electron excitations such as plasmons, in addition to the is core hole. In x-ray absorption experiments multi-electron transitions are difficult to distinguish from crystal structure related peaks in single electron transitions because the structure from the two types of transitions overlaps. In order to isolate the spectrum of the multi-electron excitations from the single electron excitations we have measured the loss 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 together with the main photoemission peaks of the F is core levels for CaF 2 are shown 2 and BaF in Figure 4.25 and Figure 4.26 respectively. The energy losses associated with the major  Chapter 4  106  2 CaF 1.0 35.6 eV  N .—  C  O.&  17.2 eV  0.6  X4.7  > CD  27.4 eV  0.4  730  720  710  700  690  680  Binding Energy (eV)  Figure 4.25: The x-ray photoemission spectrum of the F is core level and its satellites in CaF2 measured with the Mg K x line (1253.6 eV). The labels on the satellite peaks indicate the energy separation from the main peak.  Chapter 4  107  2 BaF 1 24.6 eV N  0  I 730  720  710  700  690  680  Binding Energy (eV)  Figure 4.26: The x-ray photoemission spectrum of the F is core level and its satellites in BaF2 measured with the Mg K z line (1253.6 eV). The label on the satellite peak indicates the energy separation from the main peak.  Chapter 4  108  satellite features are labeled in the figure. The satellites are similar to those observed in electron energy loss (EELS) measurements [74, 75] except that the feature with 27.4 eV energy loss is significantly stronger in the XPS than in EELS. The larger intensity in XPS compared with EELS suggests that there is an intrinsic process in XPS in which plasmons are created by the sudden appearance of the core hole in addition to excitation by the final state electrons. We interpret the feature at 17.2 eV loss as an interband transition in which a valence electron is excited to the conduction band, and the feature at 35.6 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 the large features in the x-ray absorption discussed above, they are too small in photoemission to account for the rather large features observed in the x-ray absorption spectra. (The ratio of the integrated area of the loss peaks to the main peak is about 0.2 in the XPS spectra.) Nevertheless it is possible that the intrinsic plasmon generation will have a significantly higher cross section at the absorption threshold where the core exciton is created. The plasmon is a collective excitation of the F 2p electrons. The electric field experienced by the F 2p electrons on the photoexcited ion, due to the creation of the is core hole, will be augmented in the case of the core exciton by the electric field due to the bound photoelectron. This might increase the strength of the coupling to the plasmon enough to make a plasmon satellite observable in the x-ray absorption spectrum. In this case one would expect to observe a replica of the large excitonic features in the absorption edge, shifted up in energy by the plasmon energy.  Chapter 4  109  There has been a recent x-ray fluorescence study of the fluorine K edge in CaF , 2 that involves exciting a F is electron with the x-ray and detecting the emitted photon when the F is core hole is filled by a F 2p electron [76]. This study shows that when the incident photon energy is 23 eV above the fluorine K edge absorption threshold, the usual F 2p  —*  F is fluorescence peak starts to develop a satellite that is 4 eV higher in photon  energy. The ratio of the satellite to the main fluorescence peak increases to 0.13 when the incident photon energy is 29 eV above the fluorine K edge threshold [76], or in other words at the center of the large and broad feature in the x-ray absorption spectrum. This is a clear indication that F 2p valence electrons are excited when the x-ray absorption energy exceeds 23 eV above the fluorine K edge, and is consistent with the intrinsic plasmon interpretation of the broad feature in the absorption spectrum in this energy range. In this interpretation, the satellite in the fluorescence spectrum results from the presence of the plasmon during the x-ray fluorescence. The multi-electron excitation will screen both the F is core hole in the initial state and the F 2p valence hole in the final state. Since the x-ray satellite is on the high energy side, we can conclude that the final state is more strongly screened than the initial state by the plasmon, thus causing a net increase in the energy difference between the F is and F 2p levels. The relative strength of the intrinsic plasmon creation process is still uncertain because the plasmon can propagate away or decay before the fluorescence photon is emitted. Nevertheless, the ratio of the satellite to the main fluorescence peak (0.i3) gives a lower limit to the strength of the intrinsic plasmon creation during the x-ray absorption process.  Chapter4  110  §4.6 Fluorine K edges of Rare-Earth Trifluorides Fluorine K edges in seven rare-earth trifluorides: LaF3, CeF3, NdF , SmF3, EuF3, 3 DyF3, and YbF 3 were also measured. Powdered samples pressed on indium foils were used in the measurements. The bias potentials applied on the grid of the detector in each case 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 are very different from those of alkaline-earth fluorides. The main features are labeled in Figures 4.27-33. The photon energies of these features and the energy separations of each feature relative to the first high intensity peak at the absorption edge are tabulated in Table 4.10. We have not been able to observe any appreciable feature that can be explained by  Vand 3 LaF  120V  3 CeF  140V  3 NdF  140V  3 SmF  160V  3 EuF  240 V  3 DyF  140V  3 YbF  100V  Table 4.9: The bias potentials applied to the front grid of the detector in obtaining the fluorine K edges of the seven rare-earth trifluorides.  Chapter 4  111  1.3 io6 1.1 io6 U) U) .4C D  9.0 1  0  C-)  7.0 1 O  5.0 1 O 3.0 1 680  690  700  710  720  730  Photon Energy (eV)  Figure 4.27: The fluorine K edge absorption spectrum of LaF . The main features are 3 labeled with letters a-e. Marks A, B and C indicate the energies relative to peak a corresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.35).  Chapter 4  112  5.0 1 0  4.5 1 4.0 1 0 3.510 3.01O 2.5 1 2.0 1 1.5 1O 680  690  700  710  720  730  Photon Energy (eV)  Figure 4.28: The fluorine K edge absorption spectrum of CeF . The main features are 3 labeled with letters a-e. Marks A, B and C indicate the energies relative to peak a corresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.36).  Chapter 4  113  6.5 1 O  5.5 1 O U) Cl)  .-  4.5 1  3.5 1 O  2.5 1 O 680  690  700  710  720  730  Photon Energy (eV)  Figure 4.29: The fluorine K edge absorption spectrum of NdF . The main features are 3 labeled with letters a-e. Marks A, B and C indicate the energies relative to peak a corresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.37).  Chapter 4  114  3.2 1 0 2.8 1 0 Cl) Cl)  2.4 1 O  •1-  D  0  C-)  2.0 1 O  1.6 1.2 1O 680  690  700  710  720  730  Photon Energy (eV)  Figure 4.30: The fluorine K edge absorption spectrum of SmF3. The main features are labeled with letters a-e. Marks A, B and C indicate the energies relative to peak a corresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.38).  Chapter 4  115  3.5 1 O  3.0 1 O Cl) Cl,  2.5 1  2.0 1  1.5 10 680  690  700  710  720  730  Photon Energy (eV)  Figure 4.31: The fluorine K edge absorption spectrum of EuF . The main features are 3 labeled with letters a-e. Marks A, B and C indicate the energies relative to peak a corresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.39).  Chapter 4  116  7.0 1 6.5 1 0 6.0 1 0 Cl) U) C D  0  5.5 1 0  C-)  5.0 1 4.5 1 4.0 1o 680  690  700 710 Photon Energy (eV)  720  730  Figure 4.32: The fluorine K edge absorption spectrum of DyF3. The main features are labeled with letters a-e. Marks A, B and C indicate the energies relative to peak a corresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.40).  Chapter 4  117  8.0 1O  7.0 1  0 0  6.0 1O  .4-  D  0  C)  5.0 1  4.0 3.0 1 680  690  700 710 Photon Energy (eV)  720  730  Figure 4.33: The fluorine K edge absorption spectrum of YbF . The main features are 3 labeled with letters a-e. Marks A, B and C indicate the energies relative to peak a corresponding to the loss energies A, B and C in the XPS spectrum (Figure 4.41).  Chapter4  118  the scattering of the photoelectron by the crystal lattice in the seven x-ray absorption spectra of the rare-earth trifluorides. The lack of Bragg peaks observed can be attributed to the short lifetime of the photoelectron, and the small energy separation of the Bragg peaks. The inelastic scattering length 1 can be estimated from the spectral width T of the peaks in the absorption spectrum. From the free electron kinetic energy EK have F 1= x  =  =  AE  =  2 h —2kk.  g E 2 K .  From the uncertainty relation Axtk  Since the free electron wave length  =  =  2 h —k 2 we 2m ,  1, we find  equals h I g2mE, we obtain  the ratio of the electron inelastic scattering length to its wavelength, in terms of the ratio of the electron kinetic energy to the spectral width,  .  ,rF  (4.6.1)  The lifetime broadening is significantly larger in the rare-earth trifluorides spectra than in the CaF 2 and BaF 2 spectra, presumably because there are more excitation levels available in the rare-earth ions [18]. For example, the spectral width changes from 1 eV at the absorption edge, which reflects mostly the experimental resolution, to 1.5 eV at 15 eV above the edge in the CaF 2 fluorine K edge spectrum. The spectral width in the fluorine K edge spectra of the seven rare-earth trifluorides, however, is 3-4 eV in the same energy region. This suggests that the electron inelastic scattering length is only about one wavelength in the rare-earth trifluorides in this energy region. Therefore, the Bragg peaks would be too weak to be detected since the back scattered photoelectron wave loses its coherency so quickly.  Chapter4  119  In the higher energy region (15 eV above the absorption edge and higher), the detection of Bragg peaks will be even more difficult. Even if the ratio of EKIT’ makes the coherent interference between the back scattered and outgoing photoelectron wave possible, the larger broadening in this energy region would smear all the possible Bragg peaks. The Bragg peak positions are determined by, for the hexagonal LaF 3 structure and the orthorhombic YF3 structure, 2 2 3 fl 1 h 2 2 —-(n n 1 1 + +n ) 2 +—n 8m a c  —  G  2mk2) h  2  2  1 n  —  8 m a  2  n  +  b  hexagonal  ,  2  +  2 n  orthorhombic  ,  c  2 fl 212 2 37.6 eVA 1 1 + +n ) 2 +4 -(n n 2 — n  37.6 eV  2 A  [; +4÷4].  ,  hexagonal  orthorhombic  where n 2 and n ,n 1 3 are indices of the reciprocal lattice vector G, and a, b and c are the lattice constants.  Because of the low crystal symmetry of these trifluorides (see  Chapter 1) [35], any integer is allowed for n 2 and n ,n 1 , except the Bragg peak is weak 3 for (0, 0, n =odd) in the case of the LaF 3 3 structure and (0, n =odd, 0) in the case of the 2 3 structure. Taking a typcal lattice constant of 6 YF  A,  one can see that the Bragg peak  spacing is in the order of 1 eV, which could well be surpassed by the large spectral width in the rare-earth trifluorides spectra in the high energy region.  Chapter 4  120  The first peak (peak a) in all seven spectra is due to a transition to a state close to the bottom of the conduction band, which is enhanced by the attractive core hole potential. It is uncertain, however, whether the final state is a bound exciton or a localized quasi-bound state at the bottom of the conduction band. Peak b shifts towards the absorption edge from LaF 3 to YbF 3 with increasing atomic number of the cation (see Table 4.10). If it is a Bragg peak associated with the scattering of the photoelectron by the crystal, the energy of the peak relative to the bottom ,-,-,2 2 h 1 of the conduction band should be related to where a is the lattice 1 oc 2mk2} 2 a —  —,  constant. Since the lattice constant decreases from LaF 3 to NdF 3 which have the LaF 3 tysonite structure, and from SmF 3 to YbF 3 which have the orthorhombic YF 3 structure (see Chapter 1), peak b shifts in the opposite direction to what we would expect for a Bragg peak. Accordingly, we attribute peak b to a final state in the bottom of the conduction band which is an atomic-like high angular momentum state localized on the cation. For example, there are empty rare-earth 5d and 4f levels in this energy region.  The  centrifugal barriers experienced by the high angular momentum states trap the otherwise nearly-free electron in a quasi-bound state in the continuum, and thus the absorption is enhanced with this quasi-bound final state [78]. The energy of this atomic-like state decreases with increasing atomic number of the rare-earth ion because of the increasing Coulomb attraction of the nucleus. This fact is reflected in the decreasing atomic radius with increasing atomic number. The atomic radii of the seven rare-earths are shown in the following table [79]:  Chapter4  121  (eV) 11  E-E C a  10  9  8  fittoE-E a 0  E-E D a 7  6  5  5758 LaCe  60 Nd  6263 SmEu  66 Dy  70 Yb  z  Figure 4.34: The energies of peak b and c relative to peak a in Figures 4.27-33. The filled circles are the observed values. The triangles linked with the solid line are the best fit values to Eb A  =  -  18.52 eV and B  Ea in the form Eb 30.62 eV.A.  —  Ea  =  A  —  where the best fit parameters  Chapter 4  122  atomic 1 radius r  (A)  La  Ce  Nd  Sm  Eu  Dy  Yb  2.74  2.70  2.64  2.59  2.56  2.49  2.40  As a first order approximation, the energy of an atomic orbital due to the Coulomb Z 5 Pj potential should be where R = 13.6 eV is the Rydberg constant, rB = 0.529 A is —  ,  the Bohr radius, Z’’ is the effective charge felt by the orbital, and r is the radius determined from the expectation value of hr of the orbital [80]. Therefore, one may expect that the energy separation between peak b and peak a to follow the relation B The energy separation between peak b and peak a in all seven rare Eb Ea = A r —  —  —.  earth trifluorides is plotted in Figure 4.34 and is shown to fit a relation of the form =  A  Taking Z’  =  Eb  —  Ea  —  -p-, where the best fit parameters are A =  3 for the 3+ cation, one obtains RrBZ  reasonably well with the fit parameter B  =  =  18.25 eV and B 21.58 eV•A.  =  30.62 eV.A.  This compares  30.62 eV.A if one considers the fact that the  atomic radius r 1 might be larger than the expectation radius of the atomic-like quasibound final state, since the atomic radius r 1 is mainly determined by the 6s orbital of the free 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 the penetration of the final state wavefunction into the cation core. The energy separation of peak c relative to peak a in the seven rare-earth trifluorides is also shown in Figure 4.34. The position of peak c is relatively stable, but the energy separation between peak c and peak b increases slightly from LaF 3 to NdF 3  Chapter 4  123  a (eV)  b (eV)  c (eV)  3 LaF  683.5  690.6 (7.1)  692.9 (9.4)  704.7 (21.2) 707.6 (24.1) 715.5 (32.0)  3 CeF  686.5  693.3 (6.8)  695.6 (9.1)  706.5 (20.0) 709.4 (22.9) 717.5 (31.0)  3 NdF  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)  3 EuF  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)  3 YbF  686.5  691.9 (5.4)  696.4 (9.9)  717.9 (31.4)  d (eV)  d’ (eV)  705.1 (18.6)  e (eV)  Table 4.10: The photon energies of the features labeled in Figures 4.27-33. The uncertainty is ±0.3 eV. The numbers in brackets are energy separations relative to peak a.  and from SmF 3 to YbF 3 as the the atoms in the crystal are getting closer. This seems to suggest 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 compared to peak b except for LaF 3 and CeF 3 which have an additional structure at d. The insensitivity of these peak positions in relation to the change from LaF 3 to YbF 3 suggests that these features are not crystal structure related as would be expected in the case of Bragg scattering of the photoelectron by the crystal lattice. As in the case of CaF 2 and BaF2, there is the possibility of multi-electron excitations contributing to the x-ray absorption spectra when there exists multi-electron  Chapter 4  124  excitation channels within reach of the photon energy. Accordingly, we have measured the 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 the main photoemission peaks of the F is core levels for the seven rare-earth trifluorides are shown in Figures 4.35-41. Three main loss features observed in the XPS spectra for all seven rare-earth trifluorides are labeled with A, B and C. The energy separations of the three loss features relative to the main F is photoemission peak are summarized in Table 4.11. We interpret the three loss features just as in the CaF2 case as follows: loss peak A is due to an interband transition in which a valence electron is excited to the conduction band, loss peak B is due to the creation of plasmons by the sudden appearance of the core hole (intrinsic process) in addition to excitation of plasmons by the final state electrons (extrinsic process), and loss peak C is due to excitations of a shallow core electron (F 2s or rare-earth 5p) to an empty state. The energy separations relative to peak a in the absorption spectra, corresponding to the loss energies of A, B and C in the XPS spectra are marked in Figures 4.27-33 for comparison.  Chapter 4  125  3 LaF C 1.0  B  N  E 0  I  A  0.8  0.6  730  720  710  700  690  680  Binding Energy (eV)  Figure 4.35: The x-ray photoemission spectrum of the F is core level and its satellites in 3 measured with the Mg K x line (1253.6 eV). The main satellite features are labeled LaF with A, B and C and the corresponding loss energies are listed in Table 4.11.  Chapter 4  126  3 CeF  1.0 N  .—  E C  Z0.8  0.6  730  720  710  700  690  680  Binding Energy (eV)  Figure 4.36: The x-ray photoemission spectrum of the F is core level and its satellites in 3 measured with the Mg K cx line (1253.6 eV). The main satellite features are labeled CeF with A, B and C and the corresponding loss energies are listed in Table 4.11.  Chapter 4  127  3 NdF  1.0 C)  B  N  .,-  E  A  0.8  C  I  X3.4 0.6  0.4 730  720  710  700  690  680  Binding Energy (eV)  Figure 4.37: The x-ray photoemission spectrum of the F is core level and its satellites in 3 measured with the Mg K *x line (1253.6 eV). The main satellite features are labeled NdF with A, B and C and the corresponding loss energies are listed in Table 4.11.  Chapter 4  128  3 SmF  1.o  C  N  B A  C  Z0.8  If  0.6  730  720  710  700  690  680  Binding Energy (eV)  Figure 4.38: The x-ray photoemission spectrum of the F is core level and its satellites in 3 measured with the Mg K x line (1253.6 eV). The main satellite features are SmF labeled with A, B and C and the corresponding loss energies are listed in Table 4.11.  Chapter 4  129  3 EuF  1.0 N  C  B A  C  I  0.8  X19  0.6  730  720  710  700  690  680  Binding Energy (eV)  Figure 4.39: The x-ray photoemission spectrum of the F is core level and its satellites in 3 measured with the Mg K a line (1253.6 eV). The main satellite features are labeled EuF with A, B and C and the corresponding loss energies are listed in Table 4.11.  Chapter 4  130  3 DyF  1.0 C) N  C  .—  E C  B A  0.8  X2.3 > 0.6 C)  0.4 730  720  710  700  690  680  Binding Energy (eV)  Figure 4.40: The x-ray photoemission spectrum of the F is core level and its satellites in DyF3 measured with the Mg K cs line (1253.6 eV). The main satellite features are labeled with A, B and C and the corresponding loss energies are listed in Table 4.11.  Chapter 4  131  3 YbF  1.0  C  N  E C  B A  0.8  X2.3  I  0.6  0.4 730  720  710  700  690  680  Binding Energy (eV)  Figure 4.41: The x-ray photoemission spectrum of the F is core level and its satellites in 3 measured with the Mg K a line (1253.6 eV). The main satellite features are labeled YbF with A, B and C and the corresponding loss energies are listed in Table 4.11.  Chapter 4  132  A (eV)  B (eV)  C (eV)  3 LaF  15.0±1.0  27.0±1.5  41±4  3 CeF  16.0±1.0  27.0±1.5  35±4  3 NdF  15.0±1.0  26.0±1.5  32±2  SmF3  15.0±1.0  26.0±1.5  32±3  EuF3  15.0±1.0  26.0±1.5  35±3  DyF3  16.0 ±1.0  27,0 ±1.5  37 ±3  3 YbF  17.5±1.0  27.0±1.5  41±4  Table 4.11: The loss energies of the satellites of the F is main photoemission peak in the XPS spectra in Figures 4.35-41.  There is seemingly no direct match between the energies of the loss features in the XPS spectra and the features in the corresponding x-ray absorption spectra. However, the presence of the bound final state electron responsible for the high intensity peak at the x ray absorption threshold could raise the final state energy of the valence electrons at the photoionization site relative to the one hole XPS final state. In that case, the energy separation between the high intensity peak at the absorption threshold and its replica due to multi-electron excitations would be smaller than the loss energy of the corresponding satellite in the XPS spectrum. This is due to the fact that the high energy photoelectron in the XPS final state is far from the core hole and does not influence the valence electrons at that site. We do not know the magnitude of this interaction between the fluorine  Chapter 4  133  valence electrons at the photoionization site and the bound photoelectron plus the core hole. Nevertheless, we cannot rule out the possibility that the multi-electron excitation involving an interband transition may contribute to peak c and the excitation involving intrinsic plasmon creation may contribute to peak d in the x-ray absorption spectra. As in the CaF2 case, although the strength of multi-electron excitations in XPS is relatively small, the strength could be larger at the absorption threshold because the electric field experienced by the valence electrons on the photoexcited ion, due to the creation of the core 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 absorption threshold in the absorption spectra of all seven rare-earth trifluorides. It appears to be insensitive to both the crystal structure and the atomic number of the cation. Interestingly 2 and BaF a similar feature also appears in the CaF 2 spectra. The energy separation between the feature and the first peak at the absorption threshold is 31 eV for CaF2 and . A similar feature has also been observed in the fluorine K edge in SF 2 30.5 eV for BaF 6 gas molecules with a energy separation of 33 eV relative to the first peak at the absorption threshold [82].  The insensitivity of this feature to the environment of  fluorines suggests that it is an atomic feature related to fluorine atoms. One possibility is that it is due to a multi-electron excitation with a F 2s electron excited. The energy separation between F 2s and F 2p is 19 eV [42]. Assuming all the rare-earth trifluorides to have the same 10 eV band gap [83], we have 29 eV as the energy separation between F 2s and the bottom of the conduction band. Therefore, it is possible that feature e in the x ray absorption spectra includes a contribution from a two-electron excitation in which a F 2s electron is excited into the conduction band while a F is electron is photoexcited.  Chapter 4  134  We have shown in this section, that the near edge part of the fluorine K edge absorption spectra of the rare-earth trifluorides is not dominated by features associated with the scattering of the photoelectron wave by the crystal lattice. The detection of appreciable Bragg peaks requires the inelastic scattering length of the final state photoelectron to be long enough for coherent interference from several lattice planes. Consistently with the lack of Bragg peaks which are long range order effects, features that can be explained with an atomic model are observed. The fluorine K edges of the rare-earth trifluorides are more complex than the alkaline-earth fluorides because of more excitation channels available in the rare-earths [181, that still need further studies. The complexity of XANES is demonstrated in particular by the possibility of multi-electron excitations contributing to the absorption spectra in the XANES region.  Chapter 5  135  Chapter 5 Summary and Conclusions  The energy alignment of the interfaces between thin films of YbF 3 and Si(1 11) substrates have been studied under UHV conditions by photoelectron spectroscopy and x ray absorption spectroscopy using synchrotron radiation. Results of YbF ISi( 111) were 3 compared with those of TmF ISi(1 11). The determination of the position of the 4f levels 3 in the rare-earth cations is complicated by the fact that the photoemission signal from the partially filled 4f orbitals overlaps with the F 2p valence band of the rare-earth trifluorides. However the signal from the 4f electrons was distinguished by resonantly exciting the giant 4d-4f transition in the rare-earth. While only the 4f 12 configuration of 3 ions were observed in 3 the Tm TmF I Si(1 11), both the 4f 13 configuration of the Yb 3 ions and the 4f 14 configuration of the Yb 2 ions were observed in 3 YbF / Si(1 11). While the Yb 2 ions were also observed in the film, the concentration of the Yb 2 ions was found to be higher at the interface than in the film. This is due to the fact that electrons in the Si valence band are prevented from occupying the empty 4f levels in TmF 3 at the interface by the on-site Coulomb repulsion energy U, whereas the charge transfer from Si to YbF 3 is possible because the totally filled 4f states of Yb still lie below the Si valence band maximum. The results suggest that the 4f levels of TmF 3 might be excited by direct electrical injection of electrons from the semiconductor [841.  Chapter 5  136  A simple Bragg reflection model has been developed to qualitatively explain the oscillations in XANES in terms of the scattering of the photoelectron wave between families of lattice planes. It was shown that the positions of the Bragg peaks could be determined by the simple relation of  =  For metals, the zero of energy £G  is at the bottom of the free electron conduction band or eF below the absorption threshold; for semiconductors, the origin of CG is in the vicinity of the bottom of the valence band; and for wide band gap insulators, the origin of 8 G is close to the bottom of the 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 in good agreement with experiment for the elemental metals. The model cannot account for features caused by the core hole potential which is important when there is little screening such 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 to calculate the fluorine K edges in CaF 2 and BaF2. The effects of both the long range scattering by the periodic crystal potential and the attraction by the localized core hole potential were included. The symmetry breaking core holes were treated with a supercell technique. This is the first calculation of the fluorine K edge in BaF2 [33, 85]. The model successfully reproduced all the main features in the first 15 eV of the absorption edge which had not been explained previously with the cluster calculations [85]. The excitonic peak at the absorption threshold was found to be followed by a series of peaks whose spacing changes in going from CaF 2 to BaF 2 by an amount consistent with electron diffraction from crystal lattice planes. The peak spacings reproduced by the  Chapter 5  137  model are found to be insensitive to changes of the pseudopotential core radii of the ions in the crystal. The model breaks down at higher energies due to the limited k-space volume that can be included in the calculation. A model for the absorption edge at higher energies is complicated by the possibility of multi-electron excitations. This was studied by comparing the energy loss satellites in the fluorine is x-ray photoelectron spectra with features at corresponding energies in the fluorine K edge absorption spectra. Finally the fluorine K edges in the rare-earth trifluorides LaF , CeF 3 , NdF 3 , 3 , EuF 3 SmF , DyF 3 3 and YbF 3 were explored for the first time with high resolution x-ray absorption spectroscopy and x-ray photoelectron spectroscopy. The XANES part of the fluorine K edges in all seven rare-earth trifluorides was found not to be dominated by features associated with the scattering of the photoelectron wave by the crystal lattice. The absence of these effects which are dominant in CaF 2 can be attributed to the short life time of the photoelectron and the. small energy separation of the Bragg peaks. The larger lifetime broadening is attributed to the high density of low level excitations of the 4f shell that are available in the rare-earth ions [181. The considerably smaller energy separation of the Bragg peaks is due to the low crystal symmetry of the rare-earth trifluorides compared with the alkaline-earth fluorides. The detection of appreciable Bragg peaks requires the inelastic scattering length of the final state photoelectron to be long enough for coherent interference from several lattice planes. The features that were observed are interpreted as being due to absorption by atomic-like high angular momentum states of the rare earth cations. The possible contribution from multi-electron excitations was investigated for the rare-earth trifluorides by a study of loss satellites in xPs.  138  Bibliography  [1] J. C. Fuggle and J. E. Inglesfield, in Unoccupied Electronic States. Fundamentals for XANES, EELS, IPS and BIS, edited by 3. C. Fuggle and 3. E. Inglesfield (Springer Verlag, Berlin, 1992). [2] R. F. C. Fanow, P. W. Sullivan, G. M. Williams, G. R. Jones, and D. 0. Cameron, J. Vac. Sci. Technol. 19, 415 (1981). [3] R. A. Hoffman, S. Sinharoy, and R. F. C. Farrow, App!. Phys. Lett. 47, 1068 (1985). [4] S. Sinharoy, P. G. McMu!lin, J. Greggi Jr., and Y. F. Lin, J. Appl. Phys. 62, 875 (1987). [5] S. Sinharoy, F. A. Hoffman, 3. H. Rieger, R. F. C. Farrow, and A. 3. Noreika, J. Vac. Sci. Technol. A 3, 842 (1985). [6] D. Mao, K. Young, A. Kahn, R. Zanoni, J. Mckinley, and 0. Margaritondo, Phys. Rev. B 39, 12735 (1986). [7] Y. Yamada Maruo, M. Oshima, T. Waho, T. Kawamura, S. Maeyama, and T. Miyahara, Appi. Surf. Sci. 41/42, 647 (1989). [8] K. M. Colbow, T. Tiedje, D. Rogers, and W. Eberhardt, Phys. Rev. B 43, 9672 (1991). [9] D. Rieger, F. J. Himpsel, U. 0. Karisson, F. R. McFeely, J. F. Morar, and J. A. Yarmoff, Phys. Rev. B 34,7295 (1986).  139 [10] M. A. Olmstead, R. I. G. Uhrberg, R. D. Bringans, and R. Z. Bachrach, Phys. Rev. B 35, 7526 (1987). [11] M. A. Olmstead, and R. D. Bringans, Phys. Rev. B 41,8420(1990). [12] J. D. Denlinger, E. Rotenberg, U. Hessinger, M. Leskovar, and M. A. Olmstead, Appi. Phys. Lett. 62, 2057 (1993). [13] C. A. Lucas, G. C. L. Wong, D. Loretto, Phys. Rev. Lett. 70, 1826 (1993). [14] J. M. Philips, L. C. Feldman, J. M. Gibson, and M. L. MacDonald, Thin Solid Films 104, 101 (1983). [15] P. W. Sullivan, R. F. C. Farrow, and 3. R. Jones, J. Cryst. Growth 60, 403 (1982). [16] P. G. McMullin and S. Sinharoy, 3. Vac. Sci. Technol. A 6, 1367 (1988). [17] A. A. Kaminski, Laser Crystals, Springer Series in Optical Sciences Vol. 14 (Springer-Verlag, Berlin, 1981). [18] G. H. Dieke, Spectra and Energy Levels ofRare Earth Ions in Crystals (Wiley, New York, 1968). [19] V. F. Masterov and L. F. Zakharenkov, Soy. Phys. Semicond. 24, 383 (1990). [20] A. Taguchi, M. Taniguchi, and K. Takahei, Appl. Phys. Lett. 60, 965 (1992). [211 H. D. MUller, J. Schneider, H. LUth, and R. StrUmpler, Appl. Phys. Lett. 57, 2422 (1990). [22] K. M. Colbow, Y. Gao, Tiedje, J. R. Dahn, J. N. Reimers, and S. Cramm, 3. Vac. Sci. Technol. A 10, 765 (1992).  140 [23] T. Tiedje, K. M. Colbow, Y. Gao, J. R. Dahn, 3. N. Reimers, and D. C. Houghton, Appi. Phys. Lett. 61, 1296 (1992). [24] P. J. Durham, in X-ray Absorption, edited by D. C. Koningsberger and R. Prins (Wiley, New York, 1988). [25] J. B. Pendry, in EXAFS and Near Edge Structures, edited by A. Bianconi, L. Incoccia and S. Stipcich (Springer-Verlag, Berlin, 1983). [26] A. Bianconi, Appi. Surf. Sci., 6, 392 (1980). [27] D. E. Sayers, E. A. Stern, and F. W. Lytle, Phys. Rev. Lett. 27, 1204 (1971). [28] E. A. Stern, Phys. Rev. B 10, 3027 (1974). [29] F. W. Lytle, D. E. Sayers and E. A. Stern, Phys. Rev. B 11,4825 (1975). [30] E. A. Stern, D. E. Sayers and F. W. Lytle, Phys. Rev. B 11, 4836 (1975). [31] E. A. Stern, Contemp. Phys. 19, 289 (1978). [32] H. Oizumi, T. Fujikawa, M. Ohashi, H. Maezawa, and S. Nakai, J. Phys. Soc. Japan 54, 4027 (1985). [33] P. Kizier, Phys. Lett. A 172, 66 (1992). [34] R. W. G. Wyckoff, Crystal Structures, 2nd Edition, Vol. I, (Wiley, New York, 1963). [35] R. W. G. Wyckoff, Crystal Structures, 2nd Edition, Vol. II, (Wiley, New York, 1964).  141 [36] NSLS Experimenter’s Handbook, (Brookhaven National Laboratory, 1988).  [371 M. Sansone, R. Hewitt, W. Eberhardt, and D. Sondericker, Nuci. Instrum. Methods A 266, 422 (1988). [38] K. M. Colbow, Ph.D. Thesis (University of British Columbia, Vancouver, 1992). [391 Y. Gao, M.Sc. Thesis (University of British Columbia, Vancouver, 1990). [40] H. P. Kelly and Z. Altun, in Giant Resonances in Atoms, Molecules, and Solids, edited by J. P. Connerade, J. M. Esteva, and R. C. Karnatak, (Plenum Press, New York, 1987). [41] U. Becker, in Giant Resonances in Atoms, Molecules, and Solids, edited by 3. P. Connerade, J. M. Esteva, and R. C. Karnatak, (Plenum Press, New York, 1987). [42] 3. 3. Yeh and I. Lindau, Atomic Data and Nuclear Data Tables 32, 1 (1985). [43] G. K. Wertheim, A. Rosencwaig, R. L. Cohen, and H. 3. Guggenheim, Phys. Rev. Lett. 27,505 (1971). [44] M. Campagna, E. Bucher, G. K. Wertheim, D. N. E. Buchanan, and L.D. Longinotti, Phys. Rev. Lett. 32, 885 (1974). [45] 1. F. Herbst, R. E. Watson, and 3. W. Wilkins, Phys. Rev. B 17, 3089 (1978). [46] S. Htifner and G. K. Wertheim, Phys. Rev. B 7, 5086 (1973). [47] W. Gudat and C. Kunz, Phys. Rev. Lett. 29, 169 91972). [48] T. Tiedje, K. M. Colbow, D. Rogers, and W. Eberhardt, Phys. Rev. Lett. 65, 1243 (1990).  142 [49] R. de L. Kronig, Z. Phys. 75, 191 (1932). [50] A. Zangwill, Physics at Surfaces, (Cambridge, New York, 1990). [51] D. R. Penn, Phys. Rev. B 13, 5248 (1976). [52] S. Tanuma, C. J. Powell, and D. R. Penn, Acta Physica Polonica A 81, 169 (1992). [53] T. Tiedje, J. R. Dahn, Y. Gao, K. M. Colbow, E. D. Crozier, and D. T. Jiang, Solid State Commun. 85, 161 (1993). [54] N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders College, Philadelphia, 1976). [55] D. T. hang, E. D. Crozier, and B. Heinrich, Phys. Rev. B 44, 6401 (1991). [56] W. A. Harrison, Phys. Rev. 181, 1036 (1969). [57] W. A. Harrison, Solid State Theory, (McGraw-Hill, New York, 1970). [58] M. Krause and J. H. Oliver, 3. Phys. Chem. Ref. Data 8, 329 (1979). [59] B. I. Lundquist, Phys. Status Solidi 32, 273 (1969). [60] F. 3. Himpsel and W. Eberhardt, Solid State Commun. 31, 747 (1979). [61] D. E. Eastman, J. A. Knapp, and F. 3. Himpsel, Phys. Rev. Lett. 41, 825 (1978). [62] W. A. Harrison, Electronic Structure and Properties of Solids, (Dover, New York, 1989). [63] U. von Barth and G. Grossmann, Physica Scripta 21, 580 (1980). [64] D. T. Jiang and E. D. Crozier, (private communication).  143 [65] 3. R. Chelikowsky and M. L. Cohen, Phys. Rev. B14, 556 (1976). [661 A. Hiraya, K. Fukui, P. K. Tseng, T. Murata, and M. Watanabe, J. Phys. Soc. of Japan. 60, 1824 (1991). [67] 0. W. Rubloff, Phys. Rev. B 5, 662 (1972). [681 F. M. F. de Groot, J. C. Fuggle, B. T. Thole, and 0. A. Sawatzky, Phys. Rev. B 41, 928 (1990). [69] M. 0. Krause, J. Phys. Chem. Ref. Data 8, 307 (1979). [701 D. F. Bezuidenhout, in Handbook of Optical Constants of Solids II, edited by E. D. Paik (Academic, Boston, 1991). [71] P. J. W. Weijs, M. T. Czyzyk, 3. F. van Acker, W. Speier, J. B. Goedkoop, H. van Leuken, H. 3. M. Hendrix, R. A. de Groot, G. van der Laan, K. H. 3. Buschow, G. Wiech, and 3. C. Fuggle, Phys. Rev. B 41, 11899 (1990). [72] M. Kasrai, M. E. Fleet, G. M. Bancroft, K. H. Tan, and I. M. Chen, Phys. Rev. B 43, 1763 (1991). [73] 3. P. Connerade, in Giant Resonances in Atoms, Molecules, and Solids, edited by J. P. Connerade, J. M. Esteva, and R. C. Karnatak, (Plenum Press, New York, 1987). [74] J. Frandon, B. Lahaye, and F. Pradal, Phys. Status Solidi B 53, 565 (1972). [75] K. Saiki, T. Tokoro, and A. Koma, Japan 3. Appi. Phys. 26, L974 (1987). [76] J.-E. Rubensson, S. Eisebitt, M. Nicodemus, T. BOske, and W. Eberhardt (submitted to Phys. Rev. B).  144 [77] C. Kittel, Introduction to Solid State Physics, 6th Edition, (Wiley, New York, 1986). [78] J. L. Dehmer, D. Dill, and A. C. Parr, in Photophysics and Photochemistry in the Vacuum Ultraviolet, edited by S. P. McGlynn, G. L. Findley, and R. H. Huebner, (Reidel, Dordrecht, 1985). [79] Periodic Tables of the Elements, (Sargent-Welch, Skokie, 1980). [80] E. U. Condon and G. H. Shortley, The Theory ofAtomic Spectra, (Cambridge, New York, 1964). [81] C. C. Lu, T. A. Carlson, F. B. Mailk, T. C. Tucker, and C. W. Nestor, Jr., Atomic Data 3, 1(1971). [82] E. Hudson, D. A. Shirley, M. Domke, G. Remmers, A. Puschmann, T. Mandel, C. Xue, and G. Kaindl, Phys. Rev. A 47, 361 (1993). [83] C. G. Olson, M. Piacentini, and D. W. Lynch, Phys. Rev. B 18, 5740 (1978). [84] Y. Gao, K. M. Colbow, T. Tiedje, T. van Buuren, J. R. Dahn, J. N. Reimers, and B. M. Way, in The Proceedings of the 21st International Conference on the Physics of Semiconductors, Vol. I, p534 (World Scientific, Singapore, 1993). [85] Y. Gao, T. Tiedje, P. C. Wong, and K. A. R. Mitchell, Phys. Rev. B 48, 15578 (1993). [86] B. Maximov and H. Schulz, Acta Cryst. B 41, 88 (1985); A. Zailcin and D. H. Templeton, Acta Cryst. B 41, 91(1985).  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085021/manifest

Comment

Related Items