Phys 48W
Physics and Chemistry at Surfaces An electronic course o↵ered at the Physics Department of the Bo˘gazi¸ci University to 4th-year and Masters-Degree students
Mehmet Erbudak
Physics Department, Bo˘gazi¸ci University and Laboratorium f¨ ur Festk¨orperphysik, ETHZ, CH-8093 Zurich
[email protected] [email protected]
September 2012
Phys 48W Physics and Chemistry at Surfaces 2012/2013 Fall Semester An electronic course o↵ered for the Bachelor- and Masters-Degree students Mehmet Erbudak Physics Department, Bo˘gazi¸ci University
[email protected] and Laboratorium f¨ ur Festk¨orperphysik, ETHZ, CH-8093 Zurich
[email protected] Several di↵erent phenomena are observed at surfaces that do not have a counterpart in bulk materials. Corrosion, epitaxial growth, heterogenous catalysis, or tribology are just few of these. While all bulk processes can be accounted for on equal footing owing to the universal description of electronic states, at surfaces symmetry is broken, and we need to redefine the electronic and crystal structure. In this course, we deal with the electronic structure of the bulk and surfaces, study the geometric structure and get acquainted with appropriate experimental tools to observe surfacespecific processes. A chemical analysis is part of the complete characterization of surfaces. We realize that the atomic structure, the electronic properties and chemistry of surfaces are all interrelated. Every week, students obtain the script for the week, exercises, and a short video clip. I will be present for a few lectures personally including the exam at the end of the semester. Prerequisite: Modern Physics.
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Preamble I will place the learning material as well as exercises to your disposal in internet in the pdf format every week. The learning material is planned to occupy your attention during about 3 hours per week to justify the 3 credit hours. I will mention to you some books as supporting material if needed, and will present relevant publications. With some basic knowledge on Quantum Mechanics and Solid State Physics, I assume you will appreciate the presented material as an introduction to several directions of Surface Physics and Chemistry as well as modern Materials Science. Similarly, the concepts you will be introduced correspond to those of low-dimensional phenomena. Thus, this course is thought to be as an introduction to your future research in many fields. I will mostly emphasize the experimental achievements. For any question please do consult me per mail. During the semester, you may use my Skype address erbudak if you wish a personal contact. The presented material may be too extensive. My intention is to trigger your interest on this subject. Interest and curiosity are required for innovative research and progress. In the following you will find a comprehensive introduction followed by a chapter on the electronic structure of the bulk and the surface as well as a chapter on photoemission. The next chapter is on the atomic geometry, likewise of the bulk and the surface. Following chapters are devoted on the chemical composition and adsorption of foreign atom on the surface as well as their behavior. I appreciate any suggestions or corrections on this material. As additional reading material, I suggest you follow some professional journals, such as Surface Science, Physical Review Letters, Science, or Nature. I recommend to you few books like: AZ - A. Zangwill, Physics at Surfaces, CUP D.P. Woodru↵ and T.A. Decker, Modern Techniques of Surface Science, CUP G. Ertl and J. K¨ uppers, Low-Energy Electrons and Surface Chemistry, Verlag Chemie, Weinheim JSB - J.S. Blakemore, Solid State Physics, W.B. Sounders Co., Philadelphia CK - C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, New York AM - N.W. Ashcroft and N.D. Mermin, Solid State Physics, Saunders College Publishing, Fort Worth I hope you will find the course useful and enjoy it!
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Contents Phys 48W Physics and Chemistry at Surfaces
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Preamble
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1 Introduction: Why surfaces? 1.1 Surface Processes . . . . . . . . . . . . . . . . 1.2 Surface-Induced Chemical Reactions . . . . . 1.3 Epitaxy . . . . . . . . . . . . . . . . . . . . . 1.4 Surface Melting . . . . . . . . . . . . . . . . . 1.5 Carbon-Based Structures . . . . . . . . . . . . 1.6 Scattering Cross Section and Mean Free Path 1.7 Vacuum Technique . . . . . . . . . . . . . . .
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2 Electronic Structure 2.1 Free Electrons in Metals . . . . . . . . . . . . . . 2.1.1 Free Electrons in 3DIM . . . . . . . . . . . 2.1.2 2DIM Electron Gas . . . . . . . . . . . . . 2.1.3 Fermi-Dirac Distribution Law . . . . . . . 2.2 Band Theory of Metals . . . . . . . . . . . . . . . 2.2.1 Periodic Lattice . . . . . . . . . . . . . . . 2.2.2 Motion of Electrons in a Periodic Potential 2.2.3 Band Gaps . . . . . . . . . . . . . . . . . 2.2.4 Bloch Functions . . . . . . . . . . . . . . . 2.2.5 Reduced-Zone Scheme . . . . . . . . . . . 2.3 Tight-Binding Approximation . . . . . . . . . . . 2.4 Surface Electronic Structure . . . . . . . . . . . . 3 Photoelectric Emission 3.1 Photoemission Process . . . . . . . . . . . . . 3.1.1 Optical Excitation: Conservation rules 3.1.2 Energy Conservation . . . . . . . . . . 3.1.3 Conservation of Momentum . . . . . . 3.1.4 Three-Step Model . . . . . . . . . . . . iii
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36 36 36 38 39 40
CONTENTS 3.2
Applications . . . . . . . . . . . 3.2.1 Band Structure . . . . . 3.2.2 Surface States . . . . . . 3.2.3 Spin Spectroscopy . . . . 3.2.4 Localized States . . . . . 3.2.5 Resonant Photoemission
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4 Crystal Structure 4.1 Chemical Bonding . . . . . . . . . . . . . 4.1.1 The van-der-Waals Bond . . . . . . 4.1.2 The Covalent Bond . . . . . . . . . 4.1.3 Covalent–van-der-Waals Structures 4.1.4 The Ionic Bond . . . . . . . . . . . 4.1.5 The Hydrogen Bond . . . . . . . . 4.1.6 The Metallic Bond . . . . . . . . . 4.2 Symmetry Operations . . . . . . . . . . . 4.2.1 Bulk Structure . . . . . . . . . . . 4.3 Determination of Bulk Structure . . . . . . 4.3.1 X-Ray Di↵raction . . . . . . . . . . 4.3.2 X-Ray Absorption Fine Structures
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5 Surface Structure 5.1 2DIM Structure . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Ideal Geometry . . . . . . . . . . . . . . . . . . . . . 5.1.2 Deviation from the Ideal Case . . . . . . . . . . . . . 5.1.3 Classification of Reconstructions . . . . . . . . . . . . 5.2 Determination of Surface Structure - Real-Space Techniques 5.2.1 Confocal Microscopy . . . . . . . . . . . . . . . . . . 5.2.2 Photoelectron Spectromicroscopy . . . . . . . . . . . 5.2.3 Scanning Electron Microscopy . . . . . . . . . . . . . 5.2.4 Field Emission Microscopy . . . . . . . . . . . . . . . 5.2.5 Field Ion Microscopy . . . . . . . . . . . . . . . . . . 5.2.6 Secondary-Electron Imaging . . . . . . . . . . . . . . 5.2.7 X-Ray Photoelectron Di↵raction . . . . . . . . . . . . 5.2.8 Scanning Microscopes . . . . . . . . . . . . . . . . . 5.2.8.1 Scanning Tunnelling Microscope . . . . . . 5.2.8.2 Atomic Force Microscopy . . . . . . . . . . 5.3 Determination of Surface Structure Reciprocal-Space Techniques . . . . . . . . . . . . . . . . . . 5.3.1 Low-Energy Electron Di↵raction . . . . . . . . . . . . 5.3.1.1 Intensity of Di↵racted Beams . . . . . . . . 5.3.1.2 Deviation from the Ideal Case . . . . . . . . 5.3.1.3 Calculation of Di↵racted Intensities . . . . .
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CONTENTS 6 Inner Shells 6.1 Introduction . . . . . . . . . . . . . . . . . 6.1.1 Filling the shells with electrons . . 6.1.2 Occupation of the shells . . . . . . 6.1.3 Shell Binding Energy . . . . . . . . 6.2 Ionization of Inner Shells . . . . . . . . . . 6.3 X-ray Absorption Spectroscopy . . . . . . 6.4 Excitation of an Atom by Electrons . . . . 6.4.1 Electron Energy-Loss Spectroscopy 6.5 X-Ray Photoelectron Spectroscopy . . . . 6.6 Relaxation Processes . . . . . . . . . . . . 6.6.1 Dipole Radiation . . . . . . . . . . 6.6.2 Auger Electron Emission . . . . . .
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7 Vibrational Spectroscopies 7.1 Infrared Absorption Spectroscopy . . . . . . . . . . . . . . . 7.2 Inelastic Electron Scattering . . . . . . . . . . . . . . . . . . 7.3 Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 A Classical Consideration for the Raman Process . . 7.3.1.1 An Isolated Molecule . . . . . . . . . . . . . 7.3.1.2 Selection Rules . . . . . . . . . . . . . . . . 7.3.1.3 An Example: CO2 Molecule . . . . . . . . . 7.3.1.4 Extended Systems . . . . . . . . . . . . . . 7.3.1.5 Raman Tensor . . . . . . . . . . . . . . . . 7.3.2 Quantum Mechanical Features of the Raman Process 7.3.3 Variants of the Raman Process . . . . . . . . . . . .
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135 135 135 136 137 138 139 141 145 148 153 153 156
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8 Surface Reactions 183 8.1 Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.2 Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 8.3 Desorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Index
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Chapter 1 Introduction: Why surfaces? The majority of processes, that played a crucial role in the development of our technological society, is based on physical and chemical properties of surfaces. Catalytical reactions and semiconductor (SC) structures are the most important examples. A pertinent question is whether we can describe the elementary surface processes of model catalysts on atomic scale and during the chemical reaction. Can we understand the basics of SC technology on microscopic scale? In all these issues, our goal is the insight into the connection between the microscopic properties of matter and its macroscopic behavior. What are the relevant concepts that help us reach this goal? The last few atomic layers of a solid constitute the interface with its environment. On this interface, there is a multiple of atomic and molecular processes that take place in the quasi 2 dimensional (2DIM) stage. These processes are the basis of our present-day technology. For example, without a detailed knowledge in the production of SC devices, no progress could have been achieved in the information and telecommunication technology. We also have access to nanostructured materials with extraordinary functional properties, such as SC quantum dots and carbon nanotubes. We have a growing understanding of how these structural features control the electronic properties. Throughout the years we have learned and mastered the crystal growth. A real revolution was the invention of epitaxy. It allows the fabrication of almost any material at will and makes possible the creation of any alloy in 2DIM which otherwise does not exist according to the 3DIM phase diagram. Most of the chemical reactions take place at the surface and heterogeneous catalysis is a surface reaction, while the catalytic substance does not take part in the reaction. A Ni-Fe-Cr alloy is called stainless steel because of its resistance to oxidation and corrosion. In fact, owing to adsorption-induced segregation, Cr di↵uses to the surface and binds to oxygen forming a thin oxide layer. The cromiumoxide cover at the surface of the alloy acts as a protection and prevents further oxidation. Similar surface passivation processes are successfully used in SC devices, like an atomic layer of Gd2 O3 on GaAs surface. Internal di↵usion of impurities to the surface results in segregation. In the worst case the grain boundary segregation of sulfur in 1
CHAPTER 1. INTRODUCTION: WHY SURFACES?
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stainless steel is responsible for its brittleness. We lose so much energy due to the friction, yet without friction we cannot even walk. Tribology deals with this surface e↵ect. The atoms of the bulk material are arranged in a symmetrical way. This symmetry allows many simplifications. The ion cores constitute a periodic potential. Under its influence the electrons of the material can be described as Bloch waves. There is a universal behavior of the bulk owing to the symmetry in all the phase transitions such that we can speak of universality. At the surface the symmetry is broken, no regularities can be found analogous to the bulk. The 3DIM phase diagram cannot be applied. Any observation has to be dealt with separately. These points are in fact responsible why Surface Physics or Surface Chemistry have advanced not before the last 50 years. I will now mention some processes specific of the surface.
1.1
Surface Processes
At a general surface, physical and chemical modifications can take place not known of for the bulk.
Figure 1.1: When a SC crystal is cleaved the top atomic layer may relax by x in either direction, as seen in the panel on the right-hand side.
One can cleave a SC perpendicular to a crystallographic direction and expose a surface for which the least amount of bonds are broken. No charge separation takes place as a result of cleavage. The surface atoms thereby have a reduced coordination and may react to this change in order to reduce the total energy. The shift normal to the surface of the top atomic layer is called relaxation as depicted in Fig. 1.1. If atoms are shifted pairwise lateral to the surface to form surface dimers the atomic symmetry of the surface will change and the periodicity is doubled [Fig. 1.2(left)]. This is a simple example for reconstruction. Buckling is illustrated in the Fig. 1.2(right).
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Figure 1.2: The top atomic layer may show reconstruction (left) or buckling (right).
There may also occur chemical modifications at a general surface. Foreign atoms may arrive at the surface and stick to it. Consider the potential formed between the foreign atom, the adsorbate, and the surface, the substrate, shown in (Fig. 1.3). In the case of physisorption there is a weak binding between the adsorbate and the substrate, as is in the noble-gas adsorption on surfaces or adsorption of gases on noble-metal surfaces. By a moderate heating the foreign atoms will be desorbed . The binding is strong for chemisorption, while the adsorbates are trapped by the strong attractive potential well (see Fig. 1.3).
Figure 1.3: The potential formed by the adsorbate and the substrate. In chemisorption there is an electron transfer between the substrate and the adsorbate, while physisorption is typically formed by some van-der-Waals forces. If chemisorption proceeds, a new compound, an oxide, can be formed that di↵ers from the bulk by chemical composition and atomic structure. Also surface segregation leads to a similar situation. Thus, we need to fully characterize the surface from scratch for its chemical composition, atomic geometry, and electronic properties.
CHAPTER 1. INTRODUCTION: WHY SURFACES?
1.2
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Surface-Induced Chemical Reactions
In a heterogenous catalysis, the catalyzer does not take part in the reaction, it just triggers the reaction. It provides the electron wave functions with the required symmetry in order to combine the reaction components. An extremely important example is the Haber-Bosch reaction for the synthesis of ammonia which dates back to a time prior to the advent of surface physics or chemistry. It is an exothermal reaction: 3H2 + N2 ! 2NH3 + 22.1 kcal (1.1) We need small Fe crystals for the reaction to proceed at 200 atu and 475 600 C. This reaction is used to produce the artificial fertilizer without which a great proportion of mankind would have starved during the 20. century. Fritz Haber received the Nobel Prize in 1918 for his achievement and Carl Bosch in 1931. The microscopic description of the reaction came as late as in 1975 by Gerhard Ertl. He is a surface physicist from Munich and could explain the process with proper wave functions upon which he received a professorship in Berlin at the Fritz-HaberInstitute. Later he was awarded with the Nobel Prize 2007 in Chemistry. Similarly, Fischer-Tropsch synthesis, is a collection of chemical reactions that converts a mixture of carbon monoxide and hydrogen into liquid hydrocarbons. The process produces synthetic fuel typically by burning low-cost coal, natural gas, or biomass. (2n + 1)H2 + nCO ! Cn H(2n+2) + nH2 O (1.2) The Fischer-Tropsch process operates in the temperature range of 150 uses Ni or Co catalysts.
1.3
300 C and
Epitaxy
The thermodynamics of 3DIM structures are governed by their phase diagram. This limitation does not apply to 2DIM systems, and therefore a wealth of di↵erent materials can be fabricated at the surface with tailored properties. The growth method is called epitaxy (epi = ‘top’ and taxis = ‘order’ in Greek).
Figure 1.4: Wetting of the surface by the adsorbate and island formation. In epitaxy1 the surface is exposed to a gas, e.g., metal vapor, which condenses on the surface. This way the surface becomes a contact place between two solids 1
J.R. Arthur, Surface Sci. 500, 189 (2002).
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which is called an interface. The fundamental question in epitaxy is whether the gas atoms adsorbed on the surface will wet the surface or form islands. Figure 1.4(left) schematically shows a mono atomic layer of adsorbate. This case occurs as a result of strong forces between adsorbate and surface atoms at T = 0. This is a typical case of adhesion. If, on the other hand, the adsorbate-adsorbate interaction is stronger than adsorbate-surface interactions, island form on the surface which are termed clusters. Hence, the wetting properties of a gas upon a specify surface are the necessary conditions for the epitaxial growth.
Figure 1.5: Schematic illustration of the three epitaxial growth modes. From R. Kern et al., In Current Topics in Materials Science, ed. E. Kaldis, Vol. 3, Chapter 3, North-Holland, Amsterdam, 1979.
Epitaxial growth can basically be classifies in three modes, illustrated in Fig. 1.5. In the simplest case, we may assume that the growth proceeds in a 2DIM fashion, one layer after the next, up to some required film thickness. This is called layer-bylayer growth. This is Frank-Van der Merwe (FV) growth, named after the investigators first described the process. However, this is not always the case. One often finds that the deposited material coagulates into clusters which at a stage may form a polycrystalline layer. This is Volmer-Weber type growth; 3DIM crystallites form upon deposition and some surface area remains uncovered at the initial stages of deposition. Stranski-Krastanov growth is in-between, few layers may grow in FV fashion before 3DIM clusters begin to form. We can in fact estimate in advance which growth mode is more probably for a given adsorbate-substrate system. We need to know three macroscopic quantities, namely the three surface tensions: a , i , s , the free energy per unit area at the adsorbate-vacuum interface, the adsorbate-substrate interface, and substrate-
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vacuum interface, respectively. We expect an ideal wetting of the substrate for = a+ i > 0, and SK growth for s < 0, FV growth, VW growth for = 0. For thicker films, i contains the contribution of the strained adsorbate layer and therefore depends on the film thickness. This is the case for pseudomorphic growth. In epitaxy, atoms or molecules are deposited on the substrate and some structures evolve as a result of a multitude of processes. This is a non-equilibrium phenomenon and any growth scenario is governed by the competition between kinetics and thermodynamics. Self assembly and self organization are modes through which desired nanometer-size structures grow on the surface. Microscopically, the primary mechanism in the growth of surface nanostructures from adsorbate species is the transport of these species on a flat terrace, involving random hopping processes at the substrate atomic lattice. This surface di↵usion is thermally activated. This means that di↵usion barriers need to be surmounted when moving from one stable (or metastable) adsorption site to another. The di↵usivity D, which is the mean square distance travelled by an adsorbate per unit time, obeys an Arrhenius law. If the deposition rate F of atoms in a growth experiment is kept constant, then the ratio D/F determines the average distance that an adsorbate species has to travel to meet another adsorbate for nucleation. Thus, the ratio of D/F is a key parameter characterizing the growth kinetics. If the deposition is slow (large D/F at the high-temperature limit), growth occurs close to equilibrium conditions: the adsorbates have sufficient time to explore the potential energy surface so that the system reaches a minimum energy configuration. If the deposition is fast (small D/F ), then the pattern of growth is essentially determined by kinetics; individual processes leading to metastable structures are important2 . SC nanostructures are usually grown at intermediate D/F values and their morphology is determined by the complex interplay between kinetics and thermodynamics. Strain e↵ects are particularly important and can be used to active mesoscopic ordering. Low-temperature growth of metal nanostructures on metal surfaces is the prototype of kinetically controlled growth methods. Metal bonds have essentially no directionality that can be used to direct interatomic interactions. Indeed, kinetic control provides an elegant way to manipulate the structure and morphology of metallic nanostructures. On homogenous surfaces, their shape and size are largely determined by the competition between di↵erent displacements the atoms can make along the surface, such as di↵usion on terraces, over and along step edges. Each of these displacement modes has a characteristic energy barrier, related to the local coordination of the di↵using atom. It is the natural hierarchy of di↵usion barriers that determines the details of the growth process. Terrace smoothening by step-flow growth is one example. Supermolecular self assembly is achieved at the high-temperature limit close to equilibrium. 2
J.V. Barth et al., Nature 437, 671 (2005).
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Figure 1.6: Schematic energy band of a SC. The conduction band edge is Ec and valence band edge is Ev . Analogous to metals, is defined as E1 EF , where EF is the Fermi level.
Figure 1.7: Energy bands of (a) narrow gap GaAs and (b) large band gap (AlGa)As. If GaAs is placed between two (AlGa)As layers by molecular beam epitaxy, a quantum-well structure is developed. Ref. [3].
Epitaxy is based on the revolutionary ideas of Leo Esaki and Raphael Tsu back in 1960’s and today it is extensively used in research and development as well as
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in technology. The fabrication of superlattice diodes is a prominent example. First let us look at the energy-level diagram of a SC shown in Fig. 1.6.
Figure 1.8: An electron micrograph of a superlattice structure. Ref. [4].
It is essential to note that the vertical axis is the energy of electrons. E1 Ev is called the ionization potential , and E1 Ec the electron affinity. For an intrinsic SC, EF is in the middle of the gap. Superlattices are manufactured by alternate epitaxial deposition of GaAs and (AlGa)As layers.3 GaAs quantum wells with small band gaps are found successively between (AlGa)As layers that possess electrons confined in 2DIM, as displayed in Fig. 1.7. Figure 1.8 shows a cross section observed in transmission electron microscope, TEM, across a laser diode consisting of a superlattice structure.4 Observe the precision in the production of numerous layers.
1.4
Surface Melting
Surface melting is a classical example for a surface-specific phenomenon. The slipperiness of ice is widely referred to as premelting, which is the existence of liquid at temperatures and pressures below the normal phase boundary5 . The atoms at the surface are loosely bound compared to those in the bulk. As a result, the amplitude 3
L.J. Challis, Contemp. Phys. 33, 111 (1992). D.D. Vvedensky, In Low-Dimensional Semiconductor Structures, ed. K. Barnham and D.D. Vvedensky, CUP, Cambridge, 2008. 5 J.G. Dash et al., Rep. Prog. Phys. 58, 115 (1995). 4
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of surface-atom vibrations is larger and hence the surface softer. There are supercooled liquids (like glass), but no superheated solids, possibly because the surface melts at a lower temperature than the bulk does. In the experimental terminology we may speak of lower Debye temperature. We characterize a phase with an appropriate order parameter (OP). In a phase transition, e.g., solid/liquid, OP is best chosen in such a way that it is zero in one phase and finite in the other. An abrupt change in OP at the critical temperature, Tc , is characteristic of a first-order phase transition. In this case the two independent curves of free energy cross each other and the system jumps from one state to the other one, like in the case of nucleation and growth. In this type of the transition, a seed is required to trigger the transition. In a continuous phase transition, two equivalent phases coexist and become indistinguishable. OP changes continuously with temperature, and near Tc it behaves like (T Tc ) . Figure 1.9 illustrates these two kinds of phase transitions schematically. OP
OP
t Tc
β
Tc T
T
Figure 1.9: A typical first-order (left) and second-order (right) phase transition. OP is plotted as a function of temperature, where t is the reduced temperature and the critical exponent.
The numerical value of , the critical exponent, only depends on some physical properties, like symmetry of the system or dimensionality of the order parameter. The property that the phase transition behaves similarly for all systems with the same dimensionality is called universality and suggests that unexpected phenomena might take place at the surface (2DIM) in contrast to the bulk (3DIM). Melting is a first-order phase transition for which the surface acts as a 2DIM seed. In all investigations so far, the OP for the surface behaves like that in a second-order phase transition so that we may say that the surface at temperatures much lower than Tc anticipates the bulk melting. Depending on crystallographic orientation, di↵erent melting temperatures have been observed for some metals6 . As yet, there is no universal microscopic theory for surface melting. 6
J.F. van der Veen et al., Phys. Rev. Lett 59, 2678 (1987).
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Figure 1.10: Conventional cubic cell of the diamond lattice. A sixfoldsymmetric planar honey-comb unit is highlighted.
1.5
Carbon-Based Structures
Graphene7 is a flat monolayer of carbon atoms tightly packed into a 2DIM honeycomb lattice. It is the basic building block for graphitic materials of any DIM. Carbon is the first element of the 4th group. It has the 1s2 2s2 2p2 electronic configuration. In the case of 3DIM diamond structure, the outer 2s2 2p2 electrons form an sp3 hybrid which has a tetrahedral symmetry with the extremely stable 109.47 the bond angle. It is a compact structure, macroscopically the hardest lattices. Diamond is an insulator with a large energy band gap. The diamond structure is shown in Fig. 1.10. Nearest-neighbor bonds are drawn in. The four nearest neighbors of each point form the vertices of a regular tetrahedron. In the 3DIM structure, a (111) plane is highlighted in order to emphasize the relation to graphene. Hence, diamond structure can be thought of a special way of stacking graphene layers along the [111] direction under consideration of the tetrahedral symmetry.
Figure 1.11: Graphite is formed by periodic stacking of individual graphene layers. 7
A.K. Geim and K.S. Novoselov, Nature Mat. 6, 183 (2007).
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11
Figure 1.12: Fullerenes with 60, 70, 76, and 78 carbon atoms. Ref. [8].
Another 3DIM carbon-atom morphology is the graphite. In graphite, carbon atoms occupy a 2DIM hexagonal lattice with 120 bond angles formed by the sp2 hybridization. The additional p electron fixes the hexagonal layers within a loose ⇡ bonding. The distance between the planes is almost 4.8 times the nearest-neighbor distance within planes. The graphite material is soft, black, and shows a metallic conduction along the graphene sheets. The graphite structure is schematically illustrated in Fig. 1.11. On the right-hand side, the sp2 electron orbitals are shown where the orbitals are 120 apart on a plane, while the ⇡ orbitals are perpendicular to this plane. Fullerene is a 0DIM cluster of carbon atoms arranged on the vertices of its typical dome-like structure in icosahedral symmetry, as shown in Fig. 1.12. Fullerenes are found as a by-product of carbon burning. They show spectacular properties when doped with metallic species from being magnetic to superconducting.8 Nanotubes9 are 1DIM cylindrical structures based on the hexagonal lattice of carbon atoms that forms crystalline graphite. By rolling up the graphene sheet a chiral vector C is defined by C = na1 + ma2 (chiral, cheir = ‘hand’ in Greek). The chiral angle is defined between C and a1 . For the chiral nanotube, is between 0 and 30 . The nanotube is termed armchair if n = m and = 30 , zigzag for m or n = 0 and = 0. This situation is illustrated in Fig. 1.13. The electronic properties of nanotubes are determined by their diameter and the chiral angle. For the motion of electrons, a nanotube is metallic if n m = 3q with q an integer. Thus, all armchair nanotubes are metallic, so are 1/3 of zigzag nanotubes; the rest is semiconducting. The conductivity does nor depend on the length L of the nanotube. 8 9
G. Sun and M. Kertesz, J. Chem. Phys. A 104, 7398 (2000). http://physicsweb.org/articles/world/11/1/9
CHAPTER 1. INTRODUCTION: WHY SURFACES?
12
Figure 1.13: Nanotube is a graphene sheet rolled into a 1DIM tube. Ref. [9].
1.6
Scattering Cross Section and Mean Free Path
According to the classical description, atoms consist of a positively charged nucleus which is enclosed by an electron cloud. This model is also referred to as the Rutherford model. Around 1910 experiments have been conducted with ↵ particles to investigate the classical ideas about the electron cloud. Not much was known about electron orbits microscopically, wave mechanics had not emerged yet. Nevertheless, some ideas about scattering were developed which are still used successfully today, as is done for scattering. So we first deal with elastic cross section in scattering. A scattering event is elastic if the energy of the system is not changed. So if a small particle with a mass m and initial velocity ~vo and initial momentum p~o collides with a larger mass M initially at rest, we have after the collision ~v1 and p~1 for the small mass and V~2 and p~2 for the larger mass. Consider central collision for simplicity. In elastic scattering momentum and energy are both conserved. Hence p~o = p~1 + p~2 and Eo = p2o /(2m) = p21 /(2m) + p22 /(2M ). For an energy transfer E during the collision we obtain 4mM E= Eo (1.3) (m + M )2
CHAPTER 1. INTRODUCTION: WHY SURFACES?
13
This expression is smaller for a noncentral collision. For M m we can write E ' 4m/M Eo . For scattering of electrons at isolated atoms (m/M 10 4 ), we obtain E 10 4 Eo . For scattering at a solid with 1023 atoms/cm3 the energy transfer is even smaller E ' 10 27 Eo . We realize that there is practically no energy transfer if a small particle collides with a larger one. The di↵erential cross section @ /@⌦ is defined as the e↵ective area per atom scattering into the solid angle ⌦. The number of scattered particles is given by Ns (⌦, ⌦) = Io
@ ⌦N, @⌦
(1.4)
where Io is the intensity of incoming particles, N number of target particles. The target area is R2 ⌦. Then the intensity of the scattered beam is given by Is (⌦) · R2 ⌦ = which leads to
Ns (⌦, ⌦).
(1.5)
@ Is (⌦) R2 = @⌦ Io N
(1.6)
@W @ N @ = = nd @⌦ @⌦ Ao @⌦
(1.7)
R2 and N are known quantities. The scattering probability is
with Ao the area, n the density of target, and d the path length of scattering particles. These ideas are valid for dilute targets where multiple scattering can be neglected. Integration over ⌦ results in W = nd. For W = 1 we obtain d = 1/n . This quantity is called the mean free path or escape depth, ⇤. Actually, this derivation is valid for thin targets with d < ⇤. Otherwise multiple scattering will dominate. For an infinitesimally thin layer we may write @W = n@d. If we scale with intensity, we obtain @I(d) = Io (d) n. @d
(1.8)
Considering the conservation of particles, i.e., Io (0) = I(d) + Io (d), leads to Io (d) = Io (0)e
nd
= Io (0)e
d/⇤
(1.9)
In all experiments we use in our investigations, there are electrons, photons, atoms, or ions involved. These particles interact with the solid with di↵erent intensities. As a result, the mean free path is limited, and particles are either strongly attenuated when they enter the solid or during escape. In any case, experiments are more surface sensitive the shorter the mean free path is. Generally, ⇤ for photons is quite long, while for electrons in the energy range 30 < E < 300 eV as short as few-atomic distances. Thus, experiments involving such electrons are the basic tools in Surface Science.
CHAPTER 1. INTRODUCTION: WHY SURFACES?
1.7
14
Vacuum Technique
There are several reasons why Surface Chemistry and Surface Physics have developed relatively late. The most important one is the question how to prepare a clean surface and how to keep it clean during the measuring time. In general, a SC surface is exposed by cleavage. Unfortunately, only a few crystallographically defined surfaces are accessible. Others, like all the metals, are first oriented along the desired direction by x-ray methods and subsequently cut by spark erosion to expose the net plane. Prior to introducing into the vacuum chamber, the surface is polished with appropriate powders with decreasing grain size down to 0.3 0.1 µm. In vacuum, surfaces are cleaned by bombardment with Ar+ ions of 500 2000 eV and heated to elevated temperatures to restore the crystalline structure. Once one has an appropriate surface, several systems, metals, alloys, SC’s, can be generated by epitaxy. A thus cleaned surface does not remain clean during long periods. The time a one-monolayer (ML) contaminant reaches the surface is used as a measure for the quality of the experiment and depends on the vacuum conditions. So the question is, for given vacuum conditions, how long does a surface remain clean? For an ideal gas, N = pV /kT is the number of particles in a volume V (l), at a pressure p (Torr) and Temperature T (K) with the Boltzmannqconstant k. Consider that the Maxwell distribution relates the average speed v ¯ = 8kT /⇡m of the particles with their mass m and the temperature T . Air molecules, CO or CO2 , at 20 C have an average speed of v ¯ ⇡ 500 m/s. p Mean free path ⇤ of the particles is given by ⇤ = 1/( 2⇡d2 )(N/V ) 1 , while d is the particle diameter. Thus, for air at p = 1 Torr, we have ⇤ = 4.5 µm at room temperature. These particles move in the experimental chamber and hit all exposed surfaces. The number of particles n that hit a surface of area F in a time t is given by n/ F / t = 1/4 v ¯ (N/V ) = v ¯/4 (p/kT ), where the number 4 considers di↵erent directions. Now we define the contamination time ⌧ (s) as an interval during which an originally clean surface is covered by 1 ML of adsorbed particles: 1015 /( n/ F / t) ⇡ 3x10 6 /p (Torr). (1 ML ⇡ 1015 atoms/cm2 ; 1 Pa = 10
5
bar; 1 Torr = 1.33 mbar = 1.33 x 10
2
Pa)
Now we have a stringent condition that we have to perform the experiments in vacuum with the best possible conditions. For a pump, one defines the pumping power Q = kT N˙ (Torr·l/s) and pumping speed S = Q/p (l/s). For the evacuation of an experimental chamber, we can write: ˙ N = dN/dt = (V /kT )dp/dt and N˙ = Q/kT = pS/kT , which results in: p(t) = po exp ( S/V t). Hence, the decrease of pressure is limited. Consider a chamber of cubic volume of V = 1 m3 at a pressure of p = 10 5 Torr. It contains N = 1017 atoms. The same chamber has ⇡1020 atoms (6 x 104 cm2 x
CHAPTER 1. INTRODUCTION: WHY SURFACES?
15
1015 atoms/cm2 ) sticking at its inner walls if only 1 ML of adsorbates is present. Hence, the number of atoms and molecules adsorbed at surfaces is much higher than those present in the volume. Therefore, we have to get rid of the adsorbates by making them desorb at elevated temperatures. So, we bake out the chamber to attain good vacuo. This fact limits the choice of materials used to construct the experimental chamber to those with high vapor pressure. In this course we will deal with spectroscopic experiments and results. Every spectroscopy has three ingredients. First, there is an initial disturbance, an excitation. As a result, the system undergoes a transition from the ground state to an excited state. This transition costs some energy and has a probability to occur. We measure both, considering that the probability of the transition is the intensity which is experimentally accessible. The excited state has some limited life time that determines the accuracy of the observations. In the last step, the system relaxes back to the ground state by emitting some energy. Also this time, we observe the process. Our hope is that the measured quantities constitute the dominant part of the transition. The major lesson is that we cannot ever measure the ground state – observable are the excitations.
Chapter 2 Electronic Structure The electronic properties of matter determines its macroscopic behavior. The magnetic phenomenon or the superconducting behavior of a metal has its roots in the electronic structure. With the expression electronic structure we mean the energetic and spatial distribution of electrons and how their energy is related to their motion in a solid. The electronic structure of each element is di↵erent. Yet, there are close resemblance between elements of the same group, i.e., that lie in the same column of the periodic table. The reason is given by the fact that the electronic shells are filled by one electron as we go from left to right in the periodic table, and elements with the same shell occupancy show similar behavior. Instead of dealing with di↵erent elements1 one considers elements in some characteristic groups. Here, we first analyze the behavior of free electrons. In this free-electron model the weekly bound electrons of the atoms move about freely in the metal without the influence of the attractive potential of ion cores. Thus the potential energy is neglected, the total energy is in the form of kinetic energy. The free-electron model gives us considerable information about several electronic properties of the so-called simple metals. The alkali metals or nobel metals can be regarded as simple metals. The free-electron model cannot explain the reasons why some elements are metals and others insulators. Besides, in the free-electron model electrons can travel long distances without scattering. In the next step, we place the electrons in the periodic potential of the ions and see the formation of energy bands, band gaps, e↵ective masses, etc. At the end, we mention the other extreme, the localized states. We need this approach to account for the nature of the 4f electrons of the Rare-Earth elements. These subjects will be a repetition for you with your Solid State Physics background. Subsequently, we apply these ideas to surfaces. Since the symmetry is broken in one direction (it is customary to define this as the z-direction) we expect a severe response in the electronic behavior because the electronic motion is restricted and 1
V.L. Moruzzi, J.F. Janak, and A.R. Williams, Calculated electronic properties of metals, Pergamon, New York, 1978.
16
CHAPTER 2. ELECTRONIC STRUCTURE
17
new states may be formed that do not exist in the bulk. In the following chapter, we will learn some experimental techniques that are currently used to investigate the electronic properties of the bulk and the surface.
2.1 2.1.1
Free Electrons in Metals Free Electrons in 3DIM
An unconfined electron in free space is described by the Schr¨odinger equation h ¯2 2 r '= 2m
✓
◆
h ¯ 2 @ 2' @ 2' @ 2' + 2 + 2 = E', 2m @x2 @y @z
(2.1)
where m is the free-electron mass. The solutions of this equation, 'k (r) =
1 eik·r 3 (2⇡)
(2.2)
are plane waves labelled by the wave vector k = (kx , ky , kz )
(2.3)
and correspond to the energy E=
h ¯ 2k2 h ¯2 2 = (k + ky2 + kz2 ). 2m 2m x
(2.4)
The vector components of k are the quantum numbers for the free motion of the electron, one for each of the classical degrees of freedom. The number of states in a volume dk = dkx dky dkz of k-space is N (k) dk =
2 dk (2⇡)3
(2.5)
with the factor of two accounting for the spin-degeneracy of the electrons. To express this density of states in terms of energy states, we use the fact that the energy (Eq. 2.4) depends only on the magnitude of k. Thus, by using spherical polar coordinates in k-space, dk = k 2 sin ✓ dk d✓ d ,
(2.6)
where the variables have their usual ranges (0 k < 1, 0 < 2⇡, and 0 ✓ ⇡) and integrating over the polar and azimuthal angles, we are left with an expression that depends only on the magnitude k: N (k) dk =
2 1 dk = 2 k 2 dk 3 (2⇡) ⇡
(2.7)
CHAPTER 2. ELECTRONIC STRUCTURE
18
By invoking Eq. 2.4, we can perform a change of variables to bring the right-hand side of this equation into a form involving the di↵erential of the energy: ✓
◆
✓
1 2 1 2mE dk 1 2m k dk = 2 dE = 2 2 2 ⇡ ⇡ dE 2⇡ h h ¯ ¯2
◆3/2 p
E dE
(2.8)
From this equation, we deduce the well-known density of states g(E) of a free electron gas in three dimensions: N (E) =
✓
1 2m 2⇡ 2 h ¯2
◆3/2 p
E
(2.9)
Notice the characteristic square-root dependence on the energy (Fig. 2.1). This results from the fact p that in 3DIM, the surfaces of constant energy in k-space are spheres of radius 2mE/¯h.
Figure 2.1: The densities of states g(E) of an ideal 3DIM electron gas. The density of states weighted by the amplitude square of the wave function is denoted as the local density of states. N (E, r) =
X i
|'i (r)|2 (E
Ei ),
where density of states is obtained by integration: N (E) =
2.1.2
(2.10) R
g(E, r)dr.
2DIM Electron Gas
If we restrict electron motion in one dimension, for instance in the z-direction, and let the electrons move freely in the other two directions, we encounter a quantum
CHAPTER 2. ELECTRONIC STRUCTURE
19
well in the z-direction. The Schr¨odinger equation is the same as that in Eq. 2.1 for free electrons in 3DIM h ¯2 2 r ' = E', (2.11) 2m but the boundary conditions on ' are di↵erent. In the direction of confinement along the z-direction we locate the confining planes with infinite potentials at z = 0 and z = L. The boundary conditions for ' are '(x, y, 0) = '(x, y, L) = 0
(2.12)
Thus, a natural way to solve the Schr¨odinger equation is by the method of separation of variables. By writing the wave function ' as '(x, y, z) = (x, y) (z)
(2.13)
and substituting into Eq. 2.2 we obtain the Schr¨odinger equations for ✓
h ¯2 @2 @2 + 2m @x2 @y 2 and for
,
◆
= Ek
(2.14)
h ¯ 2 d2 = En 2m dz 2
(2.15)
(0) = (L) = 0
(2.16)
with the boundary condition
Notice that the equation and boundary condition for are precisely those for a particle in a 1DIM box with infinite barriers at the edges. The solution for the not-normalized eigenfunction is n (z)
= sin(kn z),
(2.17)
where kn = n⇡/L and n a positive integer. The energy associated with these functions are h ¯ 2 kn2 En = (2.18) 2m By solving the Schr¨odinger equations in Eq. 2.14 and Eq. 2.15, we find that the energy eigenvalues for the quantum well in all three directions are given by En,k =
h ¯2 2 (k + kx2 + ky2 ), 2m n
n = 1, 2, . . .
(2.19)
This shows that this energy dispersion has features of both the 2DIM electron gas (associated with the motion in the x-y plane) and the 1DIM electron gas (associated with the confinement along the z direction). In the ground state of a system of N free electrons with an electron concentration of N/V , where V is the volume, the occupied states may be represented as points
CHAPTER 2. ELECTRONIC STRUCTURE
20
Figure 2.2: First three energy levels and wave functions of electron confined to z-direction within a sheet of thickness L. The wavelengths are indicated on the wave functions. From CK.
inside a sphere in k space. The energy at the surface of the sphere is the Fermi energy EF with the wave vector kF such that EF =
h ¯2 2 (k ). 2m F
(2.20)
Considering that there is one allowed wave vector for the volume element in k space and two allowed values of ms , the spin quantum number, for each allowed value of k, we find that EF depends on the mass and electron concentration: EF =
2.1.3
h ¯2 (3⇡ 2 N/V )2/3 . 2m
(2.21)
Fermi-Dirac Distribution Law
The ground state is the state of the system at absolute zero. The situation that happens as the temperature is increased is given by the Fermi-Dirac distribution function. The kinetic energy of the electron gas increases as the temperature is increased, and some energy levels are occupied which were vacant at absolute zero, and some levels are vacant which were occupied at absolute zero. The situation is illustrated in Fig. 2.3, where the plotted curve is of the function f (E) = (e(E
µ)/kB T
+ 1) 1 .
(2.22)
CHAPTER 2. ELECTRONIC STRUCTURE
21
Figure 2.3: Plot of Fermi-Dirac distribution function f (E) versus E/µ for zero temperature and for a temperature kB T = 15 µ. The value of f (E) gives the fraction of levels at a given energy which are occupied when the system is in thermal equilibrium. From CK.
The Fermi-Dirac distribution function gives the probability that a state at energy E will be occupied in thermal equilibrium. The quantity µ is also a function of temperature. The quantity µ is called the chemical potential, which is equal to EF at absolute zero. At all temperatures f (E) is equal to 12 when E = µ. As usual, EF is defined as the topmost filled energy state at absolute zero.
2.2
Band Theory of Metals
Every solid contains electrons which are arranged in energy bands separated by regions for which no electron energy states are allowed. Such forbidden regions are called energy gaps. If the number of electrons in a crystal is such that the allowed energy bands are either filled or empty, then no electrons can move in an electric field and the crystal will behave as an insulator. If the bands are partially filled, the crystal will act as a metal. To understand the di↵erence between insulators and conductors the free-electron model must be extended to take account of the periodic lattice of the solid.
2.2.1
Periodic Lattice
A crystal consists of a periodic repetition of a set of atoms in space. Periodic structures have long-range translational order and can be described by a lattice and
CHAPTER 2. ELECTRONIC STRUCTURE
22
a basis (unit cell). A unit cell is a collection of atoms at each point; this collection is identical at each lattice point. In simple solids, the basis consists of one single atom, complicated organic structures may have thousands of atoms. We need three basis vectors a. b, and c, which define the unit cell. With the basis and the lattice, all structures are uniquely defined. 3DIM structures can be described by symmetry operations which map the structure in itself. Translation is a parallel shift of the structure by T = pa + qb + r c, which is called translational invariance. p, q, and r are integers, analogous to the Miller indices. Any point in the crystal can be reached from the origin using T. The basis vectors a, b, c span the unit cell. The sum of all translations for any p. q, and r is called the translational group of the structure. The translational group defines the 3D periodicity of the structure. Rotation and mirror reflection are called point operations. They leave 1 point or 1 line unchanged. Point group consists of operations which leave 1 point unchanged and the structure invariant. In periodic structures there are two possible operations: mirror reflection around a line (mirror plane) and rotation through 2⇡/n (n = 1, 2, 3, 4, 6) around a point. The numbers give the n-fold rotation around a point. Only these are compatible with translational properties. In general, there are only some selected translation symmetries that are compatible with a given point group. There are fourteen di↵erent lattice types, Bravais lattices, which fulfill this requirement. More precisely, we can state that the choice of the unit cell in a crystal is not unique, the basis has to be defined accordingly. If the smallest possible unit cell contains just one atom, such a lattice is called a Bravais lattice. It is functional and appropriate, that any periodic function f (r) is represented by its Fourier transform. Then, we deal with Fourier components of the function rather than with the function itself in real space. Any function defined for a crystal, such as the electron density, is periodic with the same translation vector T as the basis vectors. We can write f (r + T) = f (r). (2.23) The crystal itself transforms into the reciprocal lattice with the vectors r⇤ , such that G = ha⇤ + k b⇤ + l c⇤ .
(2.24)
The function f ⇤ (r⇤ ) in the reciprocal space has the property f ⇤ (r⇤ + G) = f ⇤ (r⇤ )
(2.25)
We define a unit cell in the reciprocal space beyond which f (r) repeats itself. The smallest such unit cell is called a Brillouin zone. It is defined by the area surrounded by the planes that are perpendicular bisectors of the vectors from the origin to the reciprocal lattice points.
CHAPTER 2. ELECTRONIC STRUCTURE
23
It is remarkable that the value of k enters into the conservation laws for collision processes of electrons in crystals. For this reason k is called the crystal momentum of the electron. In a process involving a momentum transfer K, we can write k + K = k0 + G
(2.26)
with G being the reciprocal lattice vector.
Figure 2.4: Plot of energy E versus wave vector k (a) for a free electron and (b) for an electron in a monatomic linear chain with a lattice constant a. The energy gap Eg is associated with the first Bragg reflection at k = ±⇡/a. From CK.
2.2.2
Motion of Electrons in a Periodic Potential
In many situations the band structure of a crystal can be accounted for by the nearly-free electron model for which the band electrons are treated as perturbed only weakly by the periodic potential of the ion cores. Often the gross overall aspects of the band structure can be explained on this model. We consider, for the sake of simplicity, a linear chain of atoms with a lattice constant a. In Fig. 2.4 the band structure is shown (a) for free electrons and (b) nearly-free electrons with an energy gap at k = ±⇡/a. The Bragg condition (k + G)2 = k2 for di↵raction of a wave with the wave vector k becomes in 1DIM 1 k = ± G = ±n⇡/a, 2
(2.27)
where G = ±2n⇡/a and n is an integer. The first reflection and the first energy gap occur at k = ±⇡/a. The reflections at these particular values of k arise because the wave reflected from one atom in the linear lattices interferes constructively with the wave reflected from a nearest-neighbor atom. The di↵erence in phase between the two reflected waves is just ±2⇡ for these two values of k. The region in k space between ⇡/a and ⇡/a is called the first Brillouin zone of this lattice.
CHAPTER 2. ELECTRONIC STRUCTURE
24
Figure 2.5: (a) Variation of potential energy of a conduction electron in the field of the ion cores in a linear lattice. (b) Distribution of probability density in the linear lattice for | (+)|2 / cos2 (⇡x/a), | ( )|2 / sin2 (⇡x/a), and for a traveling wave. The wave function (+) piles up electronic charge on the cores of positive ions, thereby lowering the potential energy, while (+) piles up charge in the region between the ions removing it from the ion cores and thereby raising the potential energy. From CK.
2.2.3
Band Gaps
At k = ±⇡/a, when the Bragg condition is satisfied, a wave traveling in one direction is Bragg-reflected and then travels in the opposite direction. Thus, at k = ±⇡/a, the wave functions are made up equally of waves traveling to the right and to the left and form two di↵erent standing waves. If the traveling waves are in the form ei⇡x/a and e i⇡x/a , the standing waves are: (+) / (ei⇡x/a + e ( ) / (ei⇡x/a
e
i⇡x/a
) = 2 cos(⇡x/a);
i⇡x/a
) = 2 cos(⇡x/a).
(2.28)
The standing waves (+) and ( ) are even and odd, respectively, when x is substituted for x. Figure 2.5(a) indicates schematically the variation of potential energy of a conduction electron in the field of the positive ion cores of a monatomic linear chain. The potential energy of an electron in the field of a positive ion is negative. In Fig. 2.5(b) the distribution of electron density corresponding to the standing waves (+), ( ), and to a traveling wave is sketched. The traveling wave / eikx distributes electrons uniformly with | |2 = 1, while standing waves distribute electrons
CHAPTER 2. ELECTRONIC STRUCTURE
25
preferentially either midway between ion cores [ ( )] or on the ion cores [ (+)]. The potential energy of the three charge distributions is di↵erent in such a way that it is higher for ( ) than for a traveling wave and lower for (+) compared to a traveling wave. If the potential energies of ( ) and (+) di↵er by Eg , there is an energy gap of width Eg between the two solutions at k = ⇡/a in Fig. 2.4 or between the two solutions at k = ⇡/a. The wave functions at points A in Fig. 2.4 will be (+), and the wave functions above the energy gap at points B will be ( ).
2.2.4
Bloch Functions
Bloch2 has proved that the solutions of the Schr¨odinger equation with a periodic potential are of the form = uk (r)eik·r , (2.29) where u is a function, depending in general on k,which is periodic in x, y, z with the periodicity of the potential (that is, with the period a of the lattice). This amounts to that the plane wave eik·r is modulated with the period of the lattice. In order to justify the Bloch theorem, we can follow some simple arguments. Consider N lattice points on a ring of length N a, and suppose that the potential is periodic in a, so that V (x) = V (x + ga), (2.30) where g is an integer. Because of the symmetry of the ring we look for eigenfunctions such that (x + a) = C (x), (2.31) where C is a constant. Then (x + ga) = C g (x);
(2.32)
and, if eigenfunction is to be single-valued. (x + N a) = (x) = C N (x),
(2.33)
so that C is one of the N roots of unity, or C = ei2⇡g/N ;
g = 0, 1, 2, . . . , N
1.
(2.34)
We have then (x) = ei2⇡g/N a ug (x)
(2.35)
as a satisfactory solution, where ug (x) has periodicity a. Letting k = 2⇡g/N a, 2
F. Bloch, Z. Physik 52, 555 (1928).
(2.36)
CHAPTER 2. ELECTRONIC STRUCTURE
26
we have = eikx uk (x),
(2.37)
which is the Bloch result. A function in the form of the form Eq. 2.37 is known as a Bloch function. All one-electron wave functions in an ideal crystal are of the Bloch form.
2.2.5
Reduced-Zone Scheme
It is often convenient to select the wave vector k of the Bloch function so that it always lies within the first Brillouin zone. This procedure is known as the reducedzone scheme. If we encounter a Bloch function written as k0 (r)
0
= eik ·r uk0 (r),
(2.38)
with k0 outside the first zone, we may always find a suitable reciprocal lattice vector G0 such that k = k 0 G0 (2.39) lies within the first Brillouin zone. Then Eq. 2.38 may be written as k0 (r)
0
⇣
⌘
0
= eik ·r uk0 (r) = eik·r eiG ·r uk0 (r) ⌘ eik·r uk (r) =
k (r),
(2.40)
where we have defined 0
uk (r) ⌘ eiG ·r uk0 (r).
(2.41)
Figure 2.6: (a) The result of the perturbation associated with the nuclear potentials on the free-electron levels. Gaps are opened up at ±n⇡/a. (b) The bands using the extended-zone scheme may be folded into the first zone of (a). From J.K. Burdett, Chemical Bonding in Solids, Oxford University Press, Oxford, 1995.
CHAPTER 2. ELECTRONIC STRUCTURE 0
27
Both eiG ·r and uk0 (r) are periodic in the crystal lattice so uk (r) is also, hence k (r) is of the Bloch form. Even with free electrons it may be useful to work in the reduced-zone scheme, as is seen in Fig. 2.6. It follows also that any energy Ek0 for k 0 outside the first zone is equal to an Ek in the first zone, where k is related to k 0 by Eq. 2.39. Thus we need solve for the energy only in the first Brillouin zone, for each band. The opening of the band gap is illustrated in Fig. 2.7 for the reduced-zone scheme. Near the top or bottom of a band the energy is generally a quadratic function of the wave number, so that by analogy with the expression E = (h2 /2m)k 2 for free electrons we may define an e↵ective mass m⇤ such that @ 2 E/@k 2 = h ¯ 2 /m⇤ . The motion of the electron is characterized by this e↵ective mass, m⇤ .
Figure 2.7: (a) Ek relationship drawn in the reduced-zone scheme (a) for free electrons. The branch AC is reflected in the vertical line at k = ⇡/a gives the usual free-electron parabola for Ek vs k for positive k. (b) A crystal potential U introduces band gaps at the edges of the zone, but the overall features of the band structure remain. The dashed portions are due to free-electron parabola. Two energy bands ↵ and are shown separated by Eg at k = ±⇡a. From CK.
CHAPTER 2. ELECTRONIC STRUCTURE
2.3
28
Tight-Binding Approximation
A wide question relates to the relative importance of band structure and atomic correlation e↵ects in solids. The conduction electrons of free-electron like metals, such as alkali metals or Al, are shared between atoms for conduction, and may be treated by the above methods. The potential in which they move is rather smooth, and they can be well represented by plane waves. In band calculations, available electrons are successively filled into the calculated bands with di↵erent `components and the Fermi level is obtained by the state filled by the last electron. In contrast, if electrons, or pairs of electrons, are localized in covalent bonds, they are in a state associated with a specific atom. In the extreme case of the innershell electrons that are localized at atomic sites, the dominant process of conduction is the motion of electron from one atomic site to other. This motion is governed by correlation e↵ects came about by the interplay between repulsive electron-electron interaction and wave function hybridization. Usually, this interplay is in favor of charge interaction, and in the extreme case of heavy Fermions the system shows Fermi-liquid behavior with a large m⇤ . The 4f -electrons of most rare-earth metals and narrow-band d-electrons of transition metals behave as if they were unfilled core levels and atomic interactions dominate their behavior. In this case, the binding energy of electrons is a strong function of the band occupancy. Such bands show predominantly occupations by an integer number of electrons. For these localized-electron systems, interatomic type charge fluctuations may be responsible for electronic conduction in the form dnA dnB ! dnA 1 dn+1 B , where A and B are atomic sites and n the electron occupancies. Roughly, for the conduction to occur, the energy required for this process should be less than the d-band width w for transition metals. If the d d Coulomb and exchange interaction U is larger than the dispersional band width w than it is impossible for the above charge fluctuation to occur and the material becomes an insulator in spite of its unfilled d shell. These are the so-called Mott-Hubbard insulators. If U < w, the system will be a metal as predicted in the elementary band theory. The localized-electron behavior is well accounted for by the approximation which starts out from the wave functions of the free atoms which is known as the tightbinding approximation. It is quite good for the inner electrons of atoms and used to describe the d-bands of some of the transition metals. Consider an atomic orbital (r ~`) with a well-defined character, e.g., 1s, 2s, 2p orbitals centered on an atom at position ~` in the crystal. In this approximation, the wave functions are constructed using such orbitals and obey Bloch theorem k (r)
=
X
~
eik·` (r
~`).
(2.42)
~ `
In the orthogonalized plane waves method the localized and extended characters of the wave function are combined. An atomic region around each atom is defined
CHAPTER 2. ELECTRONIC STRUCTURE
29
where the wave function is described in terms of atomic orbitals. Outside this region, the relatively smooth parts of wave functions are expanded in terms of plane waves. Also in this method, one constructs a set of Bloch functions using occupied core states of the ions in each atom.
2.4
Surface Electronic Structure
Fundamental aspects of Surface Science have their roots in electronic properties. These include the charge density in the neighborhood of the vacuum interface, the di↵erence of the electron states near the surface compared to those in the bulk, chemical bonding states in the first few atomic planes, the electrostatic potential felt by surface atoms. Surface states that are present at the surface and not in the bulk give us mostly the clue about the behavior of surfaces. The macroscopic behavior of surfaces and interfaces, like oxidation, heterogeneous catalysis, or crystal growth, strongly depend on its electronic properties.
Figure 2.8: Electron density profile with a positively charged uniform background n ¯ for z 0. From N.D. Lang and W. Kohn, Phys. Rev. B1, 4555 (1970).
In order to study the electronic structure of surfaces, one starts with a semiinfinite crystal which has total number of electrons N at positions R. We have to consider the kinetic energy of all the electrons, the electron-ion interaction for all N and R, and the electron-electron repulsion for all electrons. The final term is not straightforward to handle, it is namely the exchange-correlation term, and since 1940’s several approaches have been used to satisfactorily solve the problem. In the
CHAPTER 2. ELECTRONIC STRUCTURE
30
so-called jellium model the discrete ion potential of a semiinfinite lattice is approximated by an averaged uniform positive charge density, n ¯ . Inverse charge density of the background is often related to a spherical volume: (4⇡/3)rs3 = 1/¯ n. Figure 2.8 displays the electron density profile n(z) for two choices of the background density rs . As a result of uncertainty principle electron density may not abruptly change from zero to its finite value n ¯ as we enter the solid from the vacuum side, this means that there is no sharp edge to the electron distribution. As a consequence, there is an exponentially decaying probability to find electrons outside the solid. In other words, electrons spill out into the vacuum region for z > 0 and thereby create an electrostatic dipole layer at the surface. We also notice that n(z) oscillates as it approaches an asymptotic value that exactly compensates the uniform bulk background charge. The wavelength of these Friedel oscillations is ⇡/kFr , where kFr = (3⇡ 2 n ¯ )1/3 . The oscillations arise because the electrons try to screen out the positive background charge distribution which includes a step at z = 0. The formation of a surface dipole layer is a result that the electrostatic potential in vacuum is larger than the mean value in the crystal. This potential step keeps the electrons within the crystal. The exchange and correlation is a bulk e↵ect which makes neighboring electrons stay away from each other and thus lowers the potential energy of each electron. The work function, , which is the minimum energy required to remove an electron from the bulk to a point away from crystal, is given by the dipole layer. Whenever the atomic density at the surface is large, the spilling out is similarly large, and the work function has a smaller value. Because of this surface contribution, depends sensitively on the exposed crystalline plane as well as on the impurity e↵ects at the surface. Surface geometric e↵ects, like reconstruction or relaxation, also modify surface dipole and consequently the work function, as expected. In calculations a reasonably accurate polycrystalline work function of some metal surface can be obtained using a uniform positive background jellium model. The jellium model of a metal surface neglects the electron-ion interaction and emphasizes the smooth surface potential barrier. Tamm3 has investigated a linear chain of 1DIM atoms possessing delta-function like positive potentials. For the bulk, he obtained solutions for the Schr¨odinger equation in the form of (z) = uk (z)eikz , where uk (z + na) = uk (z) reflects the periodicity of the linear chain. For the real values of k(z) the solutions for (z) are the common Bloch waves, extended over the entire chain. There are also complex values of k(z) which are associated with surface states, present at the surface and decay exponentially inside the surface for the z < 0. Shockley4 has similarly considered atomic potentials arranged in a linear chain with the interatomic distance a. For large a the energy values resemble that of free atoms, and there are no surface states. As a is deceased, energy values come together 3 4
J.E. Tamm, Z. Physik 76, 849 (1932). W. Shockley, Phys. Rev. 56, 317 (1939).
CHAPTER 2. ELECTRONIC STRUCTURE
31
and broaden to form bands. For adequately small nearest-neighbor distances, two discrete states move away from the bands to form surface states. In contrast to the jellium model, these models emphasize the lattice aspects of a linear chain of atoms and simplify the surface barrier. In fact, in any proper model, it is the surface barrier that makes the electrons reflect at z = 0 and leads to formation of surface states. Lets now consider a linear 1DIM chain of atoms periodically spaced in the z-direction starting from the surface. We make the assumption that the formation of the surface has no e↵ect on the interatomic distance. In the light of formation of dimmers at the surface as a result of reduced coordination and surface reconstruction, we know that this assumption is unrealistic. The second simplification is the modelling of the potential. We consider the step function at z = 0, but for the chain of ion cores, we consider a weak and smoothened periodic potential: V (z) = Vo + 2Vg cos gz, where g = 2⇡/a. Hence g is the reciprocal lattice vector of the chain. The solution of the Schr¨odinger equation ✓
◆
h ¯ 2 d2 · + V (z) ' = E', (2.43) 2m dz 2 using the screened ion-core potentials V (z) and neglecting the electron-electron interactions, leads to 'vac,1 = e+kz and 'vac,2 = e kz on the vacuum side with k = q 2m(V0 E)/¯h. The solutions must be finite, and we remain with 'vac,2 = e kz , because otherwise ' ! 1 for large values of z.
Figure 2.9: The periodic potential V (z) for 1DIM semi infinite lattice with a step at the surface, z = 0. The dashed curve is more realistic. In the bulk the solutions have the Bloch form 'k (z) = uk eikz because the potential is periodic: V (z + a) = V (z). We may assume for metals with nearly-free electrons that the potential is weak, and we expand V (z) around its average value V0 : X V (z) Vo = Vg eigz (2.44) g
Away from the zone boundaries we have in units of h ¯ /2m: 'k = eikz +
Vg ei(k k 2 (k
g)z
g)2
(2.45)
CHAPTER 2. ELECTRONIC STRUCTURE and Ek = k 2 +
32
|Vg |2 , (k g)2
k2
(2.46)
where Ek is measured relative to Vo . Near the zone boundary, k ⇠ |k g|, and above equations are not valid any more. One has to use degenerate perturbation theory with 'k = ↵eikz + ei(k
g)z
.
(2.47)
The coefficients can be found as (k 2
E)↵ + Vg = 0
⇣
and
Eq. 2.48 leads to 1 2 Ek = k + (k 2
2
g)2
Vg ↵ + (k
g) ±
r
⇣
k2
(k
g)2
⌘2
E
⌘
+ 4|Vg
|2
= 0.
(2.48)
!
(2.49)
.
with the wave functions near the band gap: 'k = eikx +
k2
E Vg
ei(k
g)z
.
(2.50)
At the zone boundary, k = 12 g, thus, 1 E± = ( g)2 ± |V | 2
and
g
'± = e i 2 z ±
V e |V |
i g2 z
.
(2.51)
The wave functions are standing waves determined by the sign of V : Energy
V >0
V 0, U + U0 > U . After considering all the details, e.g., the inner potential, mean free path of the electrons involved, the Debye temperature of the surface and bulk, the crystal
CHAPTER 5. SURFACE STRUCTURE
133
θ' θ Figure 5.69: The apparent scattering angle due to the refraction of electrons.
structure and the reconstruction at the surface, one uses realistic potentials and calculates the measured intensity curves using multiple scattering. In the case of adsorbates on the surface, one additionally has to take into account where exactly these species sit on the surface. The computed curves are compared with the experimental ones, and the input parameters to computations are modified until the results converge to an acceptable agreement. The input to calculations are then taken as most probable values representing the real case.
CHAPTER 5. SURFACE STRUCTURE
134
Figure 5.70: (Left) Measured di↵raction intensities of the 00 beam at a W(100) surface as a function of energy. (Right) Comparison of calculated and experimental di↵raction intensities for a Ni(100) surface covered with oxygen (right). Ref. [23].
Chapter 6 Inner Shells 6.1
Introduction
The electronic states near the Fermi level, EF , are responsible for the chemical and electronic, i.e., macroscopic, behavior of elements and compounds, while the binding energy EB of inner-shell states, the core levels, is a finger print of the identity and chemical state of the atom. The Koopmans’ theorem postulates that the binding energy of the core levels of the atoms corresponds to eigenvalues of oneelectron Hamiltonian ✏i in the ground state.1 Because each atom has a di↵erent nuclear charge Z and a di↵erent number of electrons, the resulting binging energy of a particular shell is also di↵erent for each atom. An assessment of EB provides a chemical analysis of the material, it is a finger print of the element. We can additionally determine the chemical state of the atom by considering the shift in EB relative to value in the neutral atom. In most of the cases, valence electrons are also involved in these measurements and thus we obtain simultaneously information on the electronic structure. This is the basis of the spectroscopic methods which investigate the transition between the inner shells of the atoms, while atomic species are identified by the binding energy of the inner shells. We may call such experiments with the title Inner-Shell Spectroscopies. We will deal with 1. the order in which the shells are filled with electrons, 2. the occupation of the shells with electrons, 3. the binding energy of the shells.
6.1.1
Filling the shells with electrons
The filling order of the electronic states is determined in principle by relativistic e↵ects. Figure 6.1 illustrates a classical recipe for the filling order of the shells that delivers acceptable results. Following this recipe, cesium with its 55 electrons has the configuration: 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s1 1
T. Koopmans, Physica 1, 104 (1934).
135
CHAPTER 6. INNER SHELLS
136
Figure 6.1: The gradual filling of the electronic states with an additional electron as the nuclear charge Z increases. The rows represent the shells with the main quantum number n and the columns stand for the orbital quantum number `. The additional electron occupies shells along the diagonal (constant n+`) with the smallest possible n + `. The numbers at the opposite axes represent the occupation number of each shell.
6.1.2
Occupation of the shells
The number of electrons in a shell is called the multiplicity, viz. degeneracy, of each shell. A state with |n, ` > has the total multiplicity 2(2` + 1). The factor 2 outside (. . . ) takes care of the spins. `=0 `=1
s p
2 degenerate states 6 degenerate states
`=2 `=3
d f
10 degenerate states 14 degenerate states
Here, s and ` are good quantum numbers. There are two mechanisms that partially lift the degeneracy and lead to a splitting: 1. Spin-orbit coupling which is a relativistic e↵ect. For the ith electron, it is: Zie↵ (ri ) ↵ (` · s), ri3 2
(6.1)
where ↵ is the fine structure constant, ` and s are orbital and spin momentum operators, Zie↵ is the e↵ective nuclear charge felt by the ith electron. Spin-orbit
CHAPTER 6. INNER SHELLS
137
coupling is large for large Z, large `, and small r. Now, ` und s are no longer good quantum numbers, but the total orbital momentum j = ` + s is. For this case, the multiplicity is given by (2j + 1). `=0 `=1 `=1 `=2 `=2 `=3 `=3
j j j j j j j
= 1/2 = 1/2 = 3/2 = 3/2 = 5/2 = 5/2 = 7/2
K-shell (K, LI , MI ) LII , MII -shell LIII , MIII -shell MIV , NIV -shell MV , NV -shell NV I , OV I -shell NV II , OV II -shell
s p1/2 p3/2 d3/2 d5/2 f5/2 f7/2
multiplicity multiplicity multiplicity multiplicity multiplicity multiplicity multiplicity
2 2 4 4 6 6 8
Let us now recall Hund’s rules that put the energies into correct order: a. The state with the maximum total spin has the smallest energy. b. The state with the maximum orbital momentum has the smallest energy. c. If a shell is less than half full, J = |L S| and J = |L + S|, if it is more than half full. 2S+1 It is customary to denote a state with its term symbol in the form of: LJ 2 3 Carbon atom with its p electrons in the unfilled outer shell has the P0 ground state, while the ground state for oxygen with its p4 electrons is 3 P2 . 2. Exchange splitting between a singlet and a triplet configuration. This splitting does not originate from magnetic dipole forces between the spins, but depends on the charge distribution that is modified by the Pauli principle according to the spin direction. In summary, the nuclear charge Z is responsible for the chemical identity. Di↵erent Z values change the binding energy of electrons and the structure of the electronic shells. The L S-coupling determines the fine structure of the shells.
6.1.3
Shell Binding Energy
The Koopmans’ theorem provides the binding energy of the core levels of the atoms as a result of one-electron eigenvalues in the ground state. This theorem is difficult to test in an experiment because only excitations are accessible in experiments not the ground state. At most we can examine how close the measured value agrees with the calculations. The discrepancy between these values is a measure of the response function of the system to our perturbation. Generally, this is referred to as relaxation energy and its magnitude depends on electron correlation in the excited state. There is no general recipe, but related materials can roughly be grouped into similar categories. The easiest case is the nearly-free electron systems which follow the Koopmans’ theorem where the so-called one-electron approximation prevails. Correlated electron systems with d or f electrons fall into the opposite category.
CHAPTER 6. INNER SHELLS
6.2
138
Ionization of Inner Shells
We can produce a hole in an inner shell by exciting an electron into a state above EF using energetic electrons, ions, or photons. We add energy to the system and promote it to an excited state.
Figure 6.2: Experimental electron binding energies. See e.g. Photoemission in Solids I, ed. M. Cardona and L. Ley, Springer, New York, 1978, p. 265.
CHAPTER 6. INNER SHELLS
6.3
139
X-ray Absorption Spectroscopy
An atom is excited by x-rays, but not ionized, e.g., 3p6 3dN ! 3p5 3dN+1 . In a typical absorption experiment, photons with the intensity Io traverse the specimen of thickness d. Some of the photons are absorbed, the rest, IT , is detected, as shown in the Fig. 5.39. The absorption coefficient µ is a function of h⌫ and depends on the density of the particles, nc , in the specimen and the ionization cross section, (h⌫). The optical excitation is selective with respect to the energy ⇤ (h⌫ = EN EN ), momentum ( ` = ±1), and the selection rules (symmetry of the wave functions, polarization of light, etc.). The method is referred to as x-ray absorption spectroscopy (XAS).
Figure 6.3: The dependence of the x-ray absorption coefficient on photon energy, and a typical experiment.
For low energies, µ has a monotonous dependence on photon energy h⌫. Whenever h⌫ is sufficiently high to excite an electron from an occupied shell to a state above EF , µ shows an abrupt increase. This is called an absorption edge. For still higher energies, µ drops monotonously with h⌫ until EB of the next shell is reached where µ shows another abrupt increase. Thus, the absorption coefficient displays a saw-tooth like energy dependence. The experiment, absorption process and a typical result are illustrated schematically in Fig. 6.3. Herewith we learn that: 1. The absorption edges correspond to EB of the shells. This quantity is defined with respect to EF , illustrated in Fig. 3.2. 2. Depending on the chemical environment of the absorbing atom, EB can be shifted up to a few eV due to the change of the screening of the nuclear charge as a result of electron transfer at the atomic site – this is called the chemical shift.
CHAPTER 6. INNER SHELLS
140
Figure 6.4: Experimental setup and data-acquisition method in XAS. Ref. [2].
Figure 6.4 shows on the top panel an experimental setup for XAS with a spatial resolution on the specimen in the nanometer range.2 X-rays are energy selected and focussed using a Fresnel plate onto the sample surface. The sample (lower left) is mounted on a 50-µm thin SiN holder that allows heating in a selected gas ambient. The sample holder is mounted on piezo-electric drivers that allow accurate positioning of the sample in the photon beam. The photomultiplier detects the transmitted light IT . In the experiment, for a preset photon energy, IT is determined at di↵erent areas A, B, or C of the sample. Finally, complete XAS spectra in the measured photon energy range is assembled using the acquired data for the areas A, B, or C. This method allows us to observe catalytic processes on di↵erent regions 2
E. de Smit et al., Nature 456, 222 (2008).
CHAPTER 6. INNER SHELLS
141
of the specimen with an unprecedented spatial resolution.
Figure 6.5: Chemical environment of Fe and O during the FischerTropsch process. Ref. [2]. Figure 6.5 depicts some results obtained on an iron oxide surface. On the lefthand side the photon absorption around the iron site is analyzed by observing the L2 and L3 core-level regions. On the right-hand side XAS around the oxygen site is observed in order to identify the chemical processes. Finally the maps are assembled in the middle panel. Thus, nonoscale chemical imaging of a working catalyst is achieved. Here, Fischer-Tropsch synthesis (Eq. 1.2) is observed on nanometer-size, active ion-oxide components.
6.4
Excitation of an Atom by Electrons
An excitation experiment performed with electrons displays several advantages. Electrons are easy to focus and their energy can be changed readily. Experiments are cheaper and can easily be combined with others. Data analysis can be ambiguous because there are two electrons in the final state that share the excess energy arbitrarily. It is a further problem that we do not exactly know whether the momentum-selection rules apply. Still, electron-excitation experiments present the advantage that they can be performed in the “home” laboratory.
CHAPTER 6. INNER SHELLS
142
An electron with the momentum p (= h ¯ k) loses energy Ee = (p2i p2f )/2m in an atomic collision. The atom gains the energy Ea = Ef Ei .3 We apply the 1st Born Approximation where V (r) is a week perturbation and i and f are plane waves. The probability for the transition to an excited state is given by the Fermi Golden Rule d! = with M = h
f |V
f
(r)|
i
2⇡ Ei ,pi 2 |MEf ,pf | (Ee h ¯ i i.
Ea )dpfx dpfy dpfz .
(6.2)
f,
are the wave functions of the atom before and after the scattering. iq m iki ·r e and f = (2⇡¯h1)3/2 eikf ·r are wave functions of the incident and i = pi
electrons, while the scattering angle
reflected
is between ki and kf .
Coulomb interaction between the incident electrons and the atom is given by: Z X
Ze2 V (r) = r
a=1
e2 |r
(6.3)
ra |
The first term is the attraction between the e and the nucleus, while the second term standsZ for the e-e repulsion. d We use d! = to obtain the scattering cross section. dE p2f dpf d⌦ = 12 pf d(p2f )d⌦ (in spherical coordinates). !
p2i p2f d ⇡Z i 2 = |MEEi,p | f ,pf dE h ¯ 2m 2m⇡ Ei,pi 2 |MEf ,pf | pf d! = h ¯
Ea pf d(p2f )d⌦ (6.4)
after integration over p2f i with MEEi,p = f ,pf
1 (2⇡¯ h)3/2
q
m pi
h
R
where M 0 = h f | e iq·r transfer. Using Eq. 6.3 0
M =h
f
|
Z
e
fe
|
ikf ·r
|V (r)| eiki ·r i i {z
}
M0
m2 pf d = 2 4 |M 0 |2 d⌦ (6.5) dE pi 4⇡ h ¯ V (r)dV | i i with q = ki kf with q the momentum
iq·r Ze
2
r
dV |
ii
h
f
|
Z
e
iq·r
Z X
a=1
e2 |r
ra |
dV |
i i.
(6.6)
The first expression is the Fourier transform of the Coulomb potential 1/r. Owing to the orthogonality of f and i it is zero. The second expression is the Fourier transform of 1/r with a delay of ra . FT 3
1 |r
ra |
!
=e
iq·ra
FT
H.A. Bethe, Ann. Physik (Leipzig) 5, 325 (1930).
✓ ◆
1 4⇡ = 2e r q
iq·ra
CHAPTER 6. INNER SHELLS
Then M 0 =
4⇡ 2 eh f q2
|
PZ
a=1
q = ki
iq·ra
e
e2 m h ¯2
d = dE
143
!2
| i i and therefore
4ps |h q 4 pinc
kf results in q 2 = ki2 + kf2
f
|
X
iq·ra
e
|
i i|
2
d⌦.
(6.7)
and qdq = ki kf sin d = ki kf d⌦ . 2⇡
2ki kf cos
Transformation d⌦ ! dq allows us to write for one electron d2 e 2 m2 = 8⇡ dEdq h ¯2 e2 = 8⇡ h ¯ 2 vi
!2
!2
1 |h q 3 ki2
1 |h q3
f
f
|e
|e
iq·ra
iq·ra
|
|
i i|
i i|
2
2
(6.8) (6.9)
with vi = h ¯ ki /m. If q · ra ⌧ 1, we can expand e iq·ra = 1 iq · ra 12 (q · ra )2 + ... Since i and f are orthogonal, the first expression has no contribution. Using q = ✏ˆ · |q|, the second expression reads d2 e2 = 8⇡ dEdq h ¯ 2 vi
!2
1 |h q
f
|ˆ✏ · ra |
i i|
2
.
(6.10)
This expression is proportional to Y1 ( , ) and describes the electrical dipole transitions (`0 = ` ± 1). 2
d Using the wave functions f and i , dEdq can be calculated. If the experiment is performed in a particular interval [qmin , qmax ]:
⇣
⌘
d min N (E) / dE / log qqmax |h f |ˆ✏ · ra | i i|2 . In analogy to the excitation with photons (ˆ e corresponds to the direction of light polarization) the absorption coefficient is given as µ(E) / ↵ (E) / |h f |ˆ e·ra | i i|2 .
The third term of the series expansion 1 (q 2
· ra ) 2 =
1 2 2 q z 2
=
1 2 2 q r 6
q2 (3z 2 6
r2 )
contains terms that are proportional to Y0 ( , ) and Y2 ( , ) and hence describe monopole (`0 = `) and quadrupole (`0 = ` ± 2) transitions. These are generally forbidden in optics. Higher-order terms describe multipole transitions. We can define a generalized oscillator strength, GOS , as d2 GOS = q12 |h f |e iq·ra | i i|2 and dEdq / 1q GOS
If Ei and Ef are known, ki and kf are determined, while q changes with the scattering angle . In fact, q varies between two extreme values: For min = 0 ! qmin ⇣p ⌘ q p p p qmin = ki kf = c Ei c Ef = c Ei Ei Ee with c = k/ E. For
max
= cos
1 kf ki
! qmax
CHAPTER 6. INNER SHELLS
144
Figure 6.6: Minimum and maximum values of momentum transfer q.
Figure 6.7: Maximum value of the momentum transfer qmax is only related to the energy transfer Ee during a collision, while the minimum value qmin is limited also by the primary energy Ei . q q p qmax = ki2 kf2 = c Ei Ef = c Ee . As summarized in Fig. 6.7, qmax only depends on the absolute value of the energy loss, while qmin is a↵ected by the primary energy as well. GOS describes to a great extend those transitions that occur at the smallest values of q. Therefore, results of experiments performed at higher primary energies Ei are expected to compare agreeably with those obtained in XAS.
CHAPTER 6. INNER SHELLS
6.4.1
145
Electron Energy-Loss Spectroscopy
GOS is the basis of electron energy-loss spectroscopy, EELS , that contains several characteristics analogous to XAS in the energy region of the core states.
Figure 6.8: On the left, x-rays excite a core level to states above EF . For constant matrix elements, the absorption coefficient µ maps the unoccupied density of states. On the right, an electron with a kinetic energy Ei loses E by promoting corelevel electrons into states above EF , just into the same states as in XAS. Hence the measured GOS is closely related to µ, shown as shaded area. Ref. [4].
Figure 6.9: EELS from selected compounds. The similarity at the nitrogen edge for ZrN and NbN is due to the local density of unoccupied states at the nitrogen site. The comparison between NbC and NbN reveals details of Nb-derived density of states above EF . Ref. [5].
CHAPTER 6. INNER SHELLS
146
The close analogy between XAS and EELS is indicated schematically in Fig. 6.8. Roughly, both experiments carry information about the density of unoccupied states at the site of the absorbing atom.4 While the absorption coefficient is clearly defined by the `-projected density of states, no clear evidence is presently available for the electron excitations. At sufficiently high electron energies (Fig. 6.7), electron excitations can be described within the dipole approximation, yet some mixing from non-dipole channels cannot absolutely be excluded. Nevertheless, for the sake of a rough inspection of the density of states, EELS is a very versatile tool. In Fig. 6.9 the measured EELS K-edge spectra of the nonmetals are compared with the total density of states above EF .5
Figure 6.10: XAS spectra at Fe 2p- and O 1s-core levels of di↵erent Fe-compounds. Ref. [6].
The fingerprinting of atoms in di↵erent coordinations is illustrated in Fig. 6.10. XAS spectra at Fe 2p- and O 1s-core levels show salient features that allow us to 4 5
M. De Crescenzi and G. Chiarello, J. Phys. C 18, 3595 (1985). J. Pfl¨ uger et al., Solid State Commun. 55, 675 (1985)
CHAPTER 6. INNER SHELLS
147
uniquely identify the Fe compound. On the left-hand panel we observe that both 2p3/2 - and 2p1/2 -components are measured. With additional consideration of O 1s-core levels even the ↵- and - phases of Fe(OH)O become distinguishable.6
Figure 6.11: The cubic spinel structure of Fe3 O4 with the two di↵erent sublattice sites. EELS data obtained in TEM from either sublattice use the density of states above EF as a fingerprint for di↵erentiating between two Fe ions. Ref. [7].
As a further example for EELS, Fig. 6.11 illustrates the structure of magnetite in the cubic spinel structure. It consists of tetrahedral and octahedral sublattices. The tetrahedral sites are occupied with Fe3+ ions and the octahedral sites with Fe2+ ions. In TEM standing-wave fields are created that have maxima either on tetrahedral or octahedral sites, depending on the tilt angle of the sample. Thus, di↵raction geometries are used to set up standing-wave fields with maxima at di↵erent sites. In Fig. 6.11(a) octahedral sites are selected and in Fig. 6.11(b) tetrahedral sites. With this site selection, the EELS data, shown on the right-hand side, originate either from (a) Fe2+ ions or (b) Fe3+ ions.7 Inclusion of di↵erent metals in carbon structures results in a wealth of intriguing phenomena. These include intercalated graphite, when metal ions are introduced between the carbon sheets. Similarly, metals in nanotube structures show ballistic conduction. Superconductivity is observed in metallofulleranes. 6 7
S.-Y. Chen et al., Phys. Rev. B 79, 104103 (2009). J. Taftø and O.L. Krivanek, Phys. Rev. Lett. 48, 560 (1982).
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Figure 6.12: (Left) A high-resolution TEM image and schematic illustration of the Gd-metallofullerane in a nanotube. The scale bars are 3 nm. (Right) An EELS spectrum of the same specimen measured in tandem in TEM within 35 ms. Ref. [8].
Figure 6.12 shows on the left-hand side (A) a conventional high-resolution TEM image and (B) a schematic representation of the structure. A chain of Gdmetallofulleranes is encapsulated in a single-walled nanotube. On the right-hand side of the figure, a typical EELS spectrum is shown acquired within 35 ms.8 We observe the Gd N -edge and C K-edge. This experiment proves that in a modern TEM one can detect single atoms within ms and identify them uniquely.
6.5
X-Ray Photoelectron Spectroscopy
We have introduced photoemission spectroscopy in Chapter 3 in detail in order to extract electronic information on the sample. In short, the process involves the excitation of electron states to a final state using photons with a known photon energy h⌫ and measuring the kinetic energy of emitted electrons outside the sample. Figure 6.13 shows schematically on the left-hand side the energy diagram of band states and some core levels in a metal in the initial state. According to the one-electron approximation, the excitation involves the promotion of every electron state by h⌫, while the response of the rest of the electron gas is neglected. The schematic energy distribution of the photoelectrons on the right-hand side just implies some broadening of the core levels and smoothening of the distribution at EF due to Fermi-Dirac function (Section 2.1.3). The electronic states near EF are responsible for the macroscopic properties of the sample, while the binding energy EB is taken as a fingerprint for the chemical identity of the atom. In the following we will observe how band states become core levels when we move from left to right in the periodic table by introducing each time one electron into the system. Subsequently, we will give examples for the utilization of core levels in chemical analysis. This is the reason why photoemission experiments are referred to as Electron Spectroscopy for Chemical Analysis, ESCA, 8
K. Suenaga et al., Science 290, 2280 (2000).
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Figure 6.13: The photoexcited electrons escape into vacuum, and their energy distribution is taken as a replica of electron states in the metal prior to excitation. This method is also referred to as X-ray Photoelectron Spectroscopy, XPS .
in some laboratories. Palladium atom has 4d10 configuration. In the metallic form it has the highest density of states at EF , so far known, as seen in Fig. 6.14 on the top spectrum. Hence, it can be used to easily determine the position of EF and calibrate the experimental energy scale. As one additional electron is introduced to the system, we obtain Ag with a completely filled 4d band, which appears at about 5 eV below EF . A splitting starts to already develop in the band. Ag is a good conductor owing to its 5s electrons at EF . As we go gradually to the right in the periodic table,9 we observe (a) how the 4d band further moves to higher EB , (b) that the spin-orbit splitting in the 4d band increases in energy making the splitting more pronounced, (c) how the sp states develop at and near EF . Figure 6.15 illustrates Al 2p spectra on the left as a function of oxygen exposure on the (100) crystal face measured at an excitation energy of 130 eV. Similar ex9
R.A. Pollak et al., Phys. Rev. Lett. 29, 274 (1972).
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Figure 6.14: The behavior of electron states in the transition metals Pd to Te. Ref. [9].
periments on the Al(111) surface is shown in the middle. The choice of the photon energy results in a kinetic energy of the photoexcited electrons with a very shallow escape depth. Hence, the spectra are extremely surface sensitive. Note the evolution of the 2.6-eV feature on both surfaces due to the partial oxidation of the surface. The additional chemical shift by 1.4 eV corresponds to the chemisorbed phase on the (111) surface as shown in (a) on the right-hand side in contrast to the (b) chemisorbed phase.10 This example shows that, besides determining the chemical identity of surface atoms, XPS reveals their chemical state. The spectral intensities are used as a measure for the elemental concentration. Hence, XPS is well suited for rapid chemical fingerprinting of surface species. Ytterbium is the last element of the rear-earth series. The completely occupied 4f states are located few eV below EF and show a typical spin-orbit splitting into 5/2 and 7/2 components. A Yb film, grown on a clean and flat Mo substrate, consists of atoms which have di↵erent neighbors. Yb-atoms at the Mo substrate have both Yb and Mo neighbors, while those at the surface have missing neighbors. Other Yb atoms, the “bulk” atoms, have a complete Yb coordination. Figure 6.16 displays 10
P.S. Bagus et al., Phys. Rev. B 44, 9025 (1991).
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Figure 6.15: Oxygen in the chemisorbed and oxide state produces di↵erent core-level shifts on Al(100) and (111) faces, while it is di↵erently incorporated at the surface. Ref. [10].
XPS spectra from a Mo substrate with 2, 3, and 4 ML of Yb. Each spectrum is decomposed into three components, the “interface”, the “bulk”, and the “surface” components for both spin-orbit components.11 A 2-ML film has no “bulk” atoms and consists of “interface” and “surface” atoms. The amount of “surface” atoms does not change for thicker films, so the spectral intensity remains constant. The amount of “interface” atoms does also not change as the film gets thicker, but its intensity decreases due to electron attenuation through the thicker film. The number of “bulk” atoms increases linearly with the film thickness, so we observe a marked increase in the “bulk” component. This example shows that di↵erent atomic coordinations are well resolved in XPS. Zn is a 3d transition metal with completely occupied 3d band. The photoelectron spectrum shows the energy region near the 3s and 3p regions for ZnF2 , as depicted in Fig. 6.17. A similar XPS spectrum is shown on the right-hand side for FeF2 .12 We observe a clearly-resolved splitting of the 3s level of Fe. There is no mechanism that splits the s level in the initial state as documented by th spectra for ZnF2 . Hence the observed splitting of the 3s level is a final-state e↵ect as long as the atom possesses an unfilled d shell. The splitting is generated by the alignment of either spin direction of the s state left behind in the photoemission process. The spin of 11 12
N. M˚ artensson et al., Phys. Rev. Lett. 60, 1731 (1988). L. Ley et al., Phys. Rev. B 8, 2392 (1973).
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Figure 6.16: Di↵erent neighborhoods of Yb induce di↵erent corelevel shifts on the 4f levels. Ref. [11].
this electron can namely be aligned parallel or antiparallel with the spins of the 3d
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Figure 6.17: Splitting of the 3s state in Fe which is missing in Zn. Refs. [12,13].
electrons of iron.13 Thus we observe the s-d exchange coupling of the final state. The splitting is a measure of the local magnetic field of d electrons influencing the s level. This process is in fact the smallest magnetometer we can think of. The material does not have to be ferromagnetic, the only requirement is an unfilled d shell. Hence, XPS can be used to measure the local magnetic field.
6.6
Relaxation Processes
The ionized core level has a finite life time after which it relaxes back to its ground state. The life time ( = h ¯ /⌧ ) is determined by the sum of all mechanisms that makes the excited state relax. The life time is very long for atoms, and relatively short for systems with overlapping electron states. Essentially, there are two di↵erent relaxation processes, the dipole radiation and electronic recombination, called Auger electron emission.
6.6.1
Dipole Radiation
An electron from a higher state fills the hole, and the energy gained is emitted as a quantum of light. Since this process is a dipole radiation, appropriate selection rules are obeyed: h⌫ = EB1 EB2 ` = ±1 j = 0, ±1 The emission of x-rays is used in research as a spectroscopic tool, in electron microscopy for elemental analysis, and in crystallography for structural analysis. Figure 6.18 illustrates schematically di↵erent x-ray emission processes. For the K series of emission, the initial hole in the K shell is transferred to higher states 13
B.V. Veal and A.P. Paulikas, Phys. Rev. Lett. 51, 1995 (1983).
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Figure 6.18: Principal structure of x-ray K- and L-series emission. D.S. Urch, In Electron Spectroscopy: Theory, Techniques and Applications, ed. C.R. Brundle and A.D. Baker, Vol. 3, Academic, New York, 1979, p. 3.
observing the selection rules. The emitted photons are designated as K↵ and K series. Using the table for binding energies of elements in Fig. 6.2 we can estimate some frequently-used x-ray transitions. In laboratory XPS measurements mostly Al or Mg anodes are used. The corresponding photon energies can easily be calculated for AlK↵1,2 1560 74 = 1486 eV and for MgK↵1,2 1305 52 = 1253 eV. In XRD experiments mostly a Cu anode is used, because copper conducts the power load in high-intensity sources e↵ectively. The Cu lines are well separated by 20 eV: K↵1,2 8979 951 = 8028 eV and 8979 931 = 8048 eV. Using Eq. 3.1 we find that these energies correspond to 1.575 ˚ A and 1.541 ˚ A wavelength which are comparable with interatomic distances and hence well suited for XRD work. The dipole radiation is called the fluorescence. While most of the observed fluorescence lines are normal, certain lines may also occur in x-ray spectra that, at first sight, do not abide to the basic selection rules. These lines are called forbidden lines; they arise from outer orbital levels where there is no sharp energy distinction between orbitals. As an example, in the transition elements, where the 3d level is only partially filled and strongly interacting with the 3p levels, a weak forbidden transition (the 5) is observed. A third type are satellite lines
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arising from dual ionization. Following the ejection of the initial electron in the photoelectric process, a short, but finite, period of time elapses before the vacancy is filled. This time period is called the lifetime of the excited state. For the lower atomic number elements, this lifetime increases to such an extent that there is a significant probability that a second electron can be ejected from the atom before the first vacancy is filled. The loss of the second electron modifies the energies of the electrons in the surrounding subshells, and thus x-ray emission lines with other energies are produced. For example, instead of the K↵1,2 line pair, a double-ionized atom will give rise to the emission of satellite lines such as the K↵3,4 and the K↵5,6 pairs. Since they are relatively weak, neither forbidden transitions nor satellite lines have great analytical significance; however, they may cause some confusion in the qualitative interpretation of spectra and may sometimes be misinterpreted as being analytical lines of trace elements.
Figure 6.19: XRF spectra from an Al-Pd-Mn alloy reveals the local density of occupied states at the site of each alloy component.
The intensity of fluorescence lines depends on the number of holes in the initial state, number of electrons that contribute to the dipole transition, and the dipole matrix elements. For certain transitions, the electrons involved may originate from bands near EF , and their number is the density of states. Consequently, an xray fluorescence spectrum (XRF) reveals the `-projected density of states at the atomic site. In an alloy, by tuning the photon energy to the core level energy
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of the alloy constituents, we can determine the density of electronic states in the neighborhood of these atoms. This situation is presented for the quasicrystal AlPd-Mn in Fig. 6.19. Figure 6.19 illustrates schematically the emission of x-rays originating from Al, Pd, and Mn atoms in an Al-Pd-Mn alloy. A spectrum around h⌫ = 118 eV will reflect the p-projected density of states of Al atoms, around h⌫ = 531 and 559 eV the s- and d-projected density of states at Pd site, and around h⌫ = 641 and 652 eV s- and d-projected density of states at Mn site. X-rays are not e↵ectively absorbed at air. Hence XRF experiments do not require very high vacuum conditions especially so if the initial hole is created by x-rays. Therefore, XRF is well suited for elemental analysis of technical alloys and widely used in industrial applications. We have to bare in mind that the method is not surface sensitive. The emission of characteristic x-rays is also used in SEM as described in Section 5.2.3 in order to identify the elements in the specimen which is currently imaged. The method is widely referred to as EDX or EDAX .
6.6.2
Auger Electron Emission
The intensity of electromagnetic radiation depends on the dipole interaction between the hole state and the state of the electron that fills the hole. This quantity depends on the energy di↵erence between the two states and hence on the atomic number Z square. It follows that the probability !R for radiation depends on Z 4 .
Figure 6.20: Dependence of the fluorescence yield (highlighted area) and Auger yield (white area) on atomic number. Ref. [14].
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Another process following a core excitation is of purely electronic nature. It is a redistribution of electrons and holes resulting in emission of an electron. This radiationless process is called Auger electron emission after Pierre Auger (1899 1993). He observed electrons with constant kinetic energy emitted from Ar+ ions. This process occurs between electronic shells and depends on the overlap of the shells and hence governed purely by Coulomb interaction. This interaction is e↵ective at the atomic site where the hole is located and depends on e2 /r. The probability for Auger electron emission is !A . The Auger process transfers the holes from states with higher EB to those with lower energy. In the radiation process the initial and final states both are a hole state, while in the Auger electron emission the initial state is a hole state and the final state has two holes. The initial hole state is bound to relax and hence the sum of the probabilities !A and !R is unity. !R depends on the nuclear charge Z in the fourth power.14 Figure 6.20 shows the Z dependence of the fluorescence yield. It is obvious that for lighter elements like C or O, !R is negligibly small which makes detection using EDAX extremely challenging. For the light elements Auger electron emission dominates, as indicated by the white area.
Figure 6.21: Estimated lifetime broadening of the ionization energies of the K, L, and M core levels. J.C. Fuggle, In Electron Spectroscopy: Theory, Techniques and Applications, ed. C.R. Brundle and A.D. Baker, Vol. 4, Academic , New York, 1981, p. 85. 14
R.W. Fink et al., Rev. Mod. Phys. 38, 513 (1966).
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If the system survives in a quantum state for a time ⌧ , in this case the initial hole state, the energy of the quantum mechanical system in principle cannot be determined with accuracy better than the spontaneous decay ⇡ h ¯ /⌧ . This is fundamental uncertainty relation for energy. In principle, no excited state has infinite lifetime ⌧ , thus all excited states are subject of the lifetime broadening , and the shorter the lifetimes of the states involved in a transition, the broader the corresponding spectral lines. As seen in Fig. 6.21, K shell broadening of some heavy elements is so large that the ionization of these levels cannot be detected. In general, the lifetime of an ionized state is short if there are several electronic states with smaller EB which can fill the state. These channels are numerous for heavy elements. For slow Auger processes, i.e., long life time of the ionized state, we can treat the ionization and relaxation process separately. If, on the other hand, the hole state is filled by an electron of the same shell, with the same principal quantum number n, the probability !A becomes very high owing to the strong overlap of the wave functions. The relaxation process is then very fast and is called the CosterKronig (CK) transition. Thus, CK transition instantly transfers the hole state to subshells with lower EB and displays a very broad spectral width. If all three states involved in the relaxation process originate from the same shell, we have a superCoster-Kronig transition with a still higher probability, and we cannot sort out the ionization and relaxation processes from each other because of the fast time scale.
Figure 6.22: Auger and Coster-Kronig transition in phosphorus. Unknown report.
Figure 6.22 shows the intensity of the L2,3 relaxation as a function of the excitation energy in phosphorus. When the energy is sufficiently high to ionize the L1 shell around 190 eV, the L1 hole state is transferred to L2,3 shells via a CK process, as shown the energy diagram on the right-hand side, and the probability
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!A grows markedly. In order a CK process can take place, the energy di↵erence between the L1 and L2,3 shells must be larger than the energy necessary to excite an electron from the valence band to vacuum. Hence, the CK process is not always energetically possible. The kinetic energy of Auger electrons is characteristic of the emitting atom and is well suited for elemental analysis. Yet, the exact energy value of the transition depends on several relaxation mechanisms. The intraatomic relaxation is a term associated with the redistribution of charge before the Auger process takes place. So, it is the response of the system to the creation of the hole state. In cases where the transitions are so fast that the spectator electrons cannot react, sudden approximation (= frozen-orbital approximation) is applicable and Koopmans’ theorem may be used: EB = ✏i . Relaxation of the system is taken negligible. On the contrary, if the hole state lives long, the spectator electrons can reorganize in energy. This situation is referred to as equivalent-core approximation (= Z + 1 approximation). This is the adiabatic limit with EB = ✏i Erelax . The estimation of Erelax is not straightforward, it changes from element to element, compound to compound. One usually considers besides the intraatomic relaxation the extraatomic relaxation which describes the screening of the final-state holes. The Auger transition takes place between electron shells with well-defined binding energies. As a result the Auger electrons have a definite kinetic energy characteristic to the element in which the hole state is created. Thus, the measurement of the kinetic energy of Auger electrons serves as a tool for chemical analysis of surfaces. The method is surface sensitive owing to the limited mean free path of the emitted electrons, given in Fig. 5.7. The emission intensity, on the other hand, is a measure for the concentration of the element in a near-surface region.
Figure 6.23: Auger transitions in the quasicrystal Al-Pd-Mn. M. Erbudak, unpublished.
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Figure 6.23 shows an Auger spectrum from a clean Al70 Pd20 Mn10 quasicrystal surface. The sample is excited by 2.4-eV electrons, and the backscattered electrons are energy analyzed. The signal is electronically di↵erentiated in order to detect the Auger signal without excessive noise due to the strong background. The main signal around 70 eV reveals the Al component, while the structure around 360 eV is due to Pd. Mn signal is less intense and is distributed between 600 700 eV. Similar spectra are used during the cleaning process of samples that show around 280 eV carbon and 520 eV oxygen signals due to contamination. One continues the cleaning process until these signals disappear.
Figure 6.24: The film B is vacuum deposited on the substrate A. If the growth is homogenous, known as FV growth, we expect an exponential dependence on the thickness d of the Auger signal for both elements. The broken lines point to a layer-by-layer growth. The Auger signal increases linearly within a film until the film is completed. Epitaxial growth can be monitored using Auger electron spectroscopy, as illustrated schematically in Fig. 6.24. In homogenous growth of the layer B on the substrate A, the Auger signal from the substrate decreases exponentially, IA / e d/⇤B , and the signal from the film material grows as IB / (1 e d/⇤B ) as the film thickness d increases. ⇤B the mean free path of electrons in the film B. Because of the high energy and high brightness of the incident electrons, highquality Auger electron spectra can be acquired with extremely high signal-to-noise ratios. Figure 6.25(a) shows a high energy resolution Auger electron spectrum of clean silver nanoparticles supported on a small MgO crystal; the silver MNN doublet is clearly resolved. Figure 6.25(b) shows the corresponding oxygen KLL Auger peak from the same specimen area.15 Surface compositional analysis of individual nanoparticles is essential for understanding the activity and selectivity of industrial bimetallic or multi-component catalysts used in a variety of chemical processes. Because of the high-surface sensitivity of Auger electrons, it is possible to determine qualitatively and, in some cases, quantitatively, the surface composition of nanoparticles consisting of multiple components. High spatial resolution Auger 15
J. Liu et al., Surface Sci. 262, L111 (1992).
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electron spectra can provide information about the surface enrichment of specific elements and information about how this enrichment varies with the size of the nanoparticles.
Figure 6.25: Auger electron spectra of (a) Ag MNN and (b) O KLL peaks of an Ag/MgO model catalyst. Auger maps of silver and oxygen are shown in (c) and (d), respectively. Ref. [15].
The primary-electron beam used to induce Auger transitions can be scanned on the surface to produce spatially resolved chemical information. The resolution is determined by the size of the focused beam and to a certain extent by the interaction volume of electrons in a near-surface region of the sample. Yet, for some samples, an image resolution < 1 nm can be achieved in scanning Auger microscopy. Such Auger maps have received great popularity because the experiment is easily combined with several others. Silver nanoparticles < 1 nm in diameter and containing as few as 15 silver atoms have been detected. Figures 6.25(c) and (d) show, respectively, Ag and O maps of an Ag/MgO model catalyst, clearly revealing the high-spatial resolution of Auger elemental maps. The resolution in images depends on several sample- and instrument-related e↵ects. The sample-related e↵ects include: (i) surface topography, (ii) escape depth of the collected Auger electrons,
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(iii) contribution from backscattered electrons and (iv) localization of the Auger electron generation processes. The last factor sets the ultimate resolution limit that will be achievable in images. Since the primary inelastic scattering processes involve excitation of inner-shell electrons, the generation of Auger electrons is highly localized. With thin specimens and high-energy incident electrons, the contribution from backscattered electrons should be negligible; it may, however, degrade the image resolution and a↵ect the image contrast of bulk samples. The instrumentrelated e↵ects include: (i) the intensity distribution of high-energy electron probes, (ii) the collection efficiency of the emitted Auger electrons and (iii) the instability of the microscopes. In a modern instrument, the instrument-related factors set the limits of obtainable resolution to ⇠ 1 nm in Auger peak images of thin specimens. The minimum detectable mass in high spatial resolution images is < 3 ⇥ 10 21 g. The Auger transition is a local process, because the driving force is the hole state localized at the atom. The process leaves behind the atom in a two-hole state. The interaction of these holes determines the line shape of the emission spectrum. For quasi-free Bloch states participating in the Auger transition, the Coulomb repulsion can well be neglected. As a result, the line shape for such a CV V transition, where C stands for a core state with the initial hole and V for valence band, corresponds to a self-convolution of the density of states in the valence band. Further, the initial hole is well screened in metals with Bloch states and lowers the total energy of the excited state. This is not the case in atoms. Surfaces behave in between.
Figure 6.26: Progressive symmetry-breaking for the final L shell configurations in KLL transitions. Ref. [16].
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On the contrary, the line shape of a CCC Auger transition is determined by the type of interaction of the two-hole state. We di↵erentiate three principal cases:16 a. In the light elements the Russel-Sounders coupling (Coulomb) dominates. b. For the heavy elements, mainly spin-orbit (j-j) coupling is observed. c. Intermediate coupling is relevant for elements in between. The relative energy splitting of each multiplet caused by these mechanisms is illustrated in Fig. 6.26. Far left, for light elements, energies are completely degenerate with nuclear Coulomb potential. There is no electrostatic interaction between electrons and no spin-orbit interaction, n, l, and s are good quantum numbers. As elements become heavier, electrostatic interaction has to be considered between electrons, resulting in states of di↵erent total orbital angular momentum L. Splitting into triplets and singlets due to exchange interaction occurs, and the e↵ect of spin-orbit interaction results in five allowed final states in pure LS-coupling (the triplet 3 P0,1,2 belonging to the (2s)(2p)5 configuration is degenerate in this limit). In the intermediate coupling we have nine allowed final states. Far right, for the heavy elements in the limit of jj-coupling, six allowed final states are presented.
Figure 6.27: Spectrum of the M4,5 N N Auger transition in Kr gas. Ref. [17]. 16
K. Siegbahn et al., ESCA: Atomic, Molecular and Solid State Structure by mans of Electron Spectroscopy, Almqvist and Wiksells, Uppsala, 1967.
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Krypton atom falls into the intermediate region, as illustrated in Fig. 6.26. The Auger spectrum recorded from Kr gas is depicted in Fig. 6.27 for the relevant energy region. In gases the electronic interactions between the excited state and the rest of electrons are extremely week and the final state lives very long. Hence the lifetime broadening is negligible and each final-state multiplet is well resolved as seen in the figure.17
Figure 6.28: Experimental LM M Auger transition spectrum from Cu. F. Vanini, unpublished.
Some filled shells in common metals may also behave like core levels, such as the 3d10 electrons of copper located near EF . Fig. 6.28 shows the Auger LM M transitions in Cu metal. We observe that the Auger spectrum is dominated by final-state multiplets, this time the final 3d8 state. The L3 M4,5 M4,5 components are located near 920 eV, while the L2 M4,5 M4,5 multiplets are observed near 940 eV. By inspection, we can state that among the possible multiplets the 3 F4 is the Hund’s ground state. This means that the emitted electrons leave the Cu atom in the least excited state and hence they have the highest kinetic energy. This transition is denoted with an arrow for the L3 M4,5 M4,5 group in the figure.
17
L.O. Werme et al., Physica Scripta 6, 141 (1971).
Chapter 7 Vibrational Spectroscopies Characterization of chemisorption has been the major goal of several studies of solid surfaces because of its obvious relevance to catalysis and corrosion. The study of kinetics, e.g., thermal desorption and flash desorption, directly yields binding states, relative saturation densities, and desorption rate parameters. Condensation kinetics provides an indirect means of characterizing adsorption because sticking coefficient depends on the state being populated, and the coverage dependence in a state is controlled by the mechanism of condensation. Surface di↵usion yields information on the potential experienced by an adsorbate as it moves laterally along a surface. A surface di↵usion process governs annealing and order-disorder transitions and may be rate-limiting steps in desorption of a dissociated species. We seek information related to adsorbate density n, fractional saturation coverage ✓, work function change , activation energy of desorption Ed , sticking coefficient or probability of condensation s, the number of binding states and their saturation densities, and symmetry of surface structure. The adsorbate structure, i.e., the location of adsorbate atoms on a surface, requires the knowledge of the positions of the substrate atoms. Most of these data are available in experiments we have reviewed like LEED, XPS, Auger Electron Spectroscopy, etc. Chemical-specific adsorbate site-symmetry information is contained in vibration properties of the species at the surface. The geometry around an atom or molecule adsorbed on a surface defines a certain point-group symmetry. This symmetry determines the number of observable vibrations. A single atom on the (100) surface of a cubic crystal can be adsorbed at an ontop, bridge, or fourfold position each with di↵erent symmetries. The same symmetries apply to an adsorbed dimer. The number of observable vibrational modes depends on the adsorbed species and the site symmetry. The observation of a single mode after adsorption of a molecule of an element is strong evidence for dissociative adsorption. Further information about the chemical nature and the position of the adsorbates is contained in the energy of vibration itself. For a molecular adsorption, Prof. Dr. Ceyhun Bulutay (
[email protected]), Department of Physics, Bilkent University, Ankara, has generously contributed the Section 7.3 on Raman Scattering.
165
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a vibration energy close to the vibration energy of the free molecule should be observed. The energy of vibration also contains some information about the adsorption site: For an adsorbate atom in a position on top of a substrate atom, the bond is concentrated on a single atom and the force constant e↵ective for the vibration normal to the surface will be comparatively high. For an atom in the bridge position or in a fourfold-coordination site the bond is shared between two and four atoms, respectively. For a vibration normal to the surface the bond forces however are only partially e↵ective. If one assumes the sum of the bond forces to be roughly equivalent for the three sites (which to some extent is equivalent to the assumption of the same desorption energy for the di↵erent sites), one may argue that the relation !top > !bridge > !f ourf old should hold. This relation is especially useful if the same adsorbate can be observed in di↵erent sites. Essential to any investigation of the vibrational properties of an adsorbed molecule is a classification of the localized vibrational modes. An N -atomic molecule has 3N degrees of freedom, of which 3 are translational, 3 rotational, and 3N 6 vibrational. If this molecule is adsorbed onto a surface and is not mobile, the translational and rotational degrees of freedom become vibrational degrees of freedom. These six “frustrated” translations and rotations are termed external modes, while others are internal modes. Although there will always be 3N localized vibrational modes associated with an isolated adsorbate molecule, it is not necessarily the case that 3N bands will appear in a vibrational spectrum. This will depend on the symmetry that is dictated by the molecular orientation and the surface site, i.e., on the molecular point group of the adsorbate complex (the surface molecule group). We examine two simple cases. The adsorption of a hydrogen atom results in the three translational degrees of freedom being converted into three vibrational modes. If the H atom is adsorbed on a threefold hollow site only two bands could be observed in an experimental spectrum: the “frustrated” translation perpendicular to the surface and the degenerate (due to the symmetry of the triangle) pair of “frustrated” translations parallel to the surface.1 This is shown in Fig. 7.1. The degeneracy is lifted if the adatom occupies a bridge site since the bridging atoms define a preferred direction in the plane, giving rise to a possible three bands in the experimental spectrum. It should be emphasized that this discussion has been restricted to the number of observable bands in a hypothetical spectrum. The number of bands actually observed will depend on the appropriate selection rules. These arguments confirm that vibrational spectroscopy is well qualified for detailed analysis of adsorbate structures. For chemical identification, the characteristic internal vibrational mode frequencies of gas phase molecules serve as ideal “fingerprints” that can be found in the signal derived from a solid surface covered with unknown chemisorbed species. Vibration of adsorbates can be excited using infrared (IR) radiation.2 Well 1 2
A.M. Bradshaw, Appl. Surface Sci. 11/12, 712 (1982). C. Backx et al., Surface Sci. 68, 516 (1977).
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Figure 7.1: Vibrational modes of adsorbed hydrogen in a threefold hollow site and a bridging site on a tungsten surface. Ref. [1].
monochromatized electron beams can alternatively be used to induce adsorbate vibrations. In both, the sensitivity of the measurement is ⇠ 0.005 ML. Surfaceenhanced Raman spectroscopy3 or inelastic electron tunnelling spectroscopy4 as well as inelastic He scattering5 are similar techniques to investigate adsorbate vibrations. In the following, we will limit our discussion to the first three of these.
7.1
Infrared Absorption Spectroscopy
Electromagnetic radiation travels with the speed of light, c = 3 ⇥ 108 m/s. The photon energy E = h⌫ and c = ⌫. For the energy of visible and x-ray radiation, used for the investigation of electronic properties of matter, the units are given in eV such that soft x-rays with a wavelength = 100 ˚ A have an energy of 124 eV (Eq. 3.1). IR refers to that part of the electromagnetic spectrum between the visible and microwave regions. In the mid IR region, the wavelength is 2.5 25 µm, which corresponds to 500 50 meV. Researchers in the field of IR prefer to work with ‘spectroscopic’ wavenumber6 and the range becomes 4000 400 cm 1 . In the IR spectroscopy, the excitation of adsorbate vibrations are measured in the reflected beam. Energy is extracted from the radiation field when the frequency of the light matches the eigenfrequency of the dipole-active molecule on a metal surface. The interaction between the radiation and the vibrating dipole is produced by the electric field of the light exerting a force on the charge of the oscillator. The wavelength of light is long compared to atomic distances and the excitation will therefore be almost completely in phase for neighboring dipoles. For metals the dielectric constant is rather high. Therefore the tangential component of the electric field is practically zero, while a substantial field normal to the surface exists for light 3
M.S. Dresselhaus et al., Group Theory, Springer, New York, 2010. P.K. Hansma, Physics Rep. 30, 145 (1977). 5 A.M. Lahee et al., Surface Sci. 177, 371 (1986). 6 What is called the “wavenumber” is nothing but the inverse of the wavelength. Hence 1 meV corresponds to 8 cm 1 . 4
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reflected at grazing incidence and polarized in the plane of incidence. Thus in IR spectroscopy only the perpendicular component of a surface vibration is excited. An important feature of IR spectroscopy is its broad-band capability. If a suitable source, window material, and detector are available spectra over a wide range can be measured. The second feature is high resolution; even in most of the commercial devices a resolution better than 1 cm 1 is routinely attained. The third important feature is the ability to polarize the radiation. Vibrations parallel and perpendicular to the surface can easily be distinguished, thus making it possible to determine the symmetry of the normal modes.
Figure 7.2: The electric field vectors for the two light polarizations. Ref. [8].
Figure 7.2 shows the field distribution for the two di↵erent polarizations. While p-polarized radiation can probe modes both parallel (x direction) and perpendicular (z direction) to the surface, s-polarized radiation is only sensitive to modes parallel (y direction) to the surface. The golden rule expression for the intensity of a vibrational band in IR spectroscopy is ~ i i|2 , I ⇠ |h'f |~µi · E|' (7.1)
~ is the electric where 'i and 'f are vibrational wave functions of the i -th mode, E field acting on the oscillator and µ ~ i is the dipole moment operator associated with the i -th mode. µ ~ i is often referred to as the dynamic dipole moment, i.e., the dipole moment associated with the mode when the nuclei are displaced from their equilibrium position. The excitation normally involved is from the ground state to the first vibrationally excited state. Using Group Theory a simple selection rule can be derived which states that the integral in Eq. 7.1 is only non-zero when the i-th mode belongs to the same irreducible representation of the molecular point group as one or more of the Cartesian components of the dynamic dipole. If this condition is fulfilled one speaks of a dipole-active mode. It is clear from Eq. 7.1 that maximum absorption occurs when E is parallel to the dynamic dipole. Important for the IR experiment is thus the behavior of ~ 2 upon reflection at a typical metal surface. Additional selection rules can be |E|
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Figure 7.3: The image dipole model. From AZ.
~ In the same way “induced” if specific adsorption conditions fix the direction of E. that the response of the metal valence electrons to the incident electromagnetic wave screens out the electric field components parallel to the surface, the dynamic dipole of the vibrational mode itself does not remain una↵ected. This phenomenon is best discussed in terms of a “fictitious” image dipole. Figure 7.3 shows two instantaneous dipoles, lying perpendicular and parallel to a metal surface. The response of the metal valence electrons to the dipole fields may be represented in the terms of an induced image dipole, which in turn interacts with the real dipole. Since for the dipole oriented normal to the surface the image dipole is in the same sense, this interaction corresponds to a reinforcement. For the parallel dipole the interaction e↵ectively gives rise to an electric quadrupole, for which excitation probability in IR spectroscopy is usually a factor 1000 lower. Hence, the combination of the two e↵ects - the screening of both the electric vector and the oscillating dipole - means that the chances of observing a dipole-active mode where the change in dipole moment is exclusively parallel to the surface are practically zero. Hydrogen adsorption on a Si(100) surface is a classical example for the IR spectroscopy, where its strength is evident.7 Hydrogen molecules dissociate upon adsorption on Si(100)(2 ⇥ 1) and form the monohydride phase. There are two local vibration modes separated by 12 cm 1 , illustrated in Fig. 7.4. The inset shows the directions of hydrogen-atom motion responsible for these modes. The s-polarized light only excites the antisymmetric stretch vibration (2087 cm 1 ), while the ppolarized light excites both the strong symmetric stretch vibration (2099 cm 1 ) and the antisymmetric vibration. This example shows that the IR spectroscopy has an unprecedented resolution to distinguish between these modes and exploits the polarizability of the excitation in distinguishing between the modes. In any absorption spectroscopy we measure how e↵ective a sample absorbs light at each wavelength. The classical and most straightforward way to do this is the technique of dispersive spectroscopy where a monochromatic light beam is directed at a sample and measure the intensity of the absorbed light. We repeat this procedure for each di↵erent wavelength. The Fourier-transform spectroscopy is a less intuitive way to obtain the same information. Rather than shining a monochromatic beam of light at the sample, this technique shines a beam containing many frequencies of light at once, and measures how much of that beam is absorbed by the sample. Next, the beam is modified 7
Y.J. Chabal, Surface Sci. 168, 594 (1986).
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Figure 7.4: IR spectrum on a Si(100)(2 ⇥ 1)-H surface. The solid line corresponds to p-polarized spectrum and the dashed line to s-polarized spectrum. The inset shows the symmetric and antisymmetric vibrations of the adsorbed hydrogen atoms. Ref. [8].
to contain a di↵erent combination of frequencies, giving a second data point. This process is repeated many times. Afterwards, a computer takes all these data and works backwards to infer what the absorption is at each wavelength. The beam is generated by starting with a broadband light source, containing the full spectrum of wavelengths to be measured. The light shines into a Michelson interferometer. As this mirror moves, each wavelength of light in the beam is periodically blocked and transmitted by the interferometer due to the interference. Di↵erent wavelengths are modulated at di↵erent rates, so that at each moment, the beam coming out of the interferometer has a di↵erent spectrum. Subsequently, a computer processing is required to turn the raw data, i.e., light absorption for each mirror position, into the desired result, i.e., light absorption for each wavelength. The processing required turns out to be a common algorithm of the Fourier transform, hence the name, Fourier transform IR spectroscopy or FTIR.8 Since an FTIR spectrometer simultaneously collects spectral data in a wide spectral range, which confers a significant advantage over a dispersive spectrometer, which measures intensity over a narrow range of wavelengths at a time, FTIR has made dispersive IR spectrometers all but obsolete, opening up new applications of IR spectroscopy. 8
Y.J. Chabal, Surf. Sci. Rep. 8, 211 (1988).
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7.2
171
Inelastic Electron Scattering
When an electron approaches the surface the electric field of the electron exerts a force on dipole-active oscillators located at the surface. Just as in IR spectroscopy and for the same reason for metals the field is practically normal to the surface. Therefore, the same selection rule with respect to the orientation of the dipole oscillator applies. As a consequence of the long-range nature of the Coulomb field the most significant contributions to the total interaction arise during the time where the electron is still many lattice constants away from the surface. The field is then nearly homogeneous on the atomic scale and therefore mostly long-wavelength surface waves are excited. The lateral extension of the electric field, however, being a function of the distance of the electron shrinks up as the electron approaches the surface. This is the reason, why unlike to IR spectroscopy, a continuous distribution of surface wave vectors is excited with electrons. In IR spectroscopy the field is periodic in time. Therefore for harmonic oscillators only the fundamental frequency is excited. This is di↵erent with electrons. The total interaction time of the electron is of the order of an oscillator period. From the standpoint of the oscillator the external force therefore contains all frequencies. As a result, multiple excitations may occur. The operator which causes excitations in inelastic electron scattering (IES) is iq·r e where r is the position of the scattered electron and q the transferred momentum, as we had in Eq. 6.9. In the case of a localized initial state, this operator may be expanded and the linear term dominates for q sufficiently small and causes dipole transitions; the quadratic term, which gives rise to monopole, quadruple, and higher-order cross terms in the matrix element, becomes important for larger q. Thus, when q is small, we observe the same transitions as are seen in optical absorption studies; “forbidden” transitions may be observed at larger q. In the case of electron reflection from an ideal conducting metal surface, the conduction electrons respond to the incident radiation producing a dipole image charge potential normal to the vacuum-metal interface. The time response of this long-range Coulombic interaction between the incoming charged particle and the surface electric field extends out into the vacuum. This crude representation of a complex scattering process serves to illustrate the point that the electron beam acts as a source of wide-band radiation. The optical absorption properties of an adsorbate will determine at which frequencies ⌫ “photons” are adsorbed from this radiation source, corresponding to an energy loss h⌫ of an electron scattered in the specular direction, i.e., relatively small net momentum transfer q associated with the loss. In addition to this optical dipole selection rule, the electric field normal to the surface, associated with the induced image charge, provides a second selection rule, which is important in the interpretation of energy-loss spectra, as it is in the IR spectra. In this case of atomic and molecular adsorbate vibrational excitations at IR
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frequencies, only vibrations which give dipole changes perpendicular to the surface will absorb radiation strongly. This “surface-normal dipole selection rule” is equally true whether the incident radiation is an electron or an IR photon. It can therefore be readily seen in Fig. 7.3 that, for example, a bond-stretching vibration, which would give rise to a dipole change parallel to the surface, will produce an equal and opposite change in the induced “image” dipole in the metal substrate; the net dipole change is e↵ectively zero. On the other hand, the oscillating dipole perpendicular to the surface will be reinforced by the oscillating image dipole. Another important point is that the surface carries a permanent normal dipole field (see, Section 2.4, the origin of the work function) that e↵ectively polarizes otherwise IR-inactive species when adsorbed. For this reason, the normal vibration modes of chemisorbed hydrogen produce inelastic scattering of the incident electrons.
Figure 7.5: Electron energy-loss spectra of hydrogen chemisorbed on W(100) at an impact energy of E = 9.65 eV: (a) specular beam direction; (b) 17 o↵ the specular direction towards the surface. The fundamental vibrational modes in the inset correspond to bridge-site bonding. Ref. [9].
In the IES experiment low-energy electrons of typically 2 5 eV energy are monochromatized (ideally E < 10 meV) with an electrostatic energy analyzer and reflected from the surface. The low resolution of IES makes it inferior to IR spectroscopy. A second electrostatic energy analyzer then discriminates the scattered electrons according to their energy. Those electrons which have excited vibrational modes in the adsorbate will appear as discrete loss peaks in the spectrum. Two excitation mechanisms appear to be primarily involved: dipole scattering and impact scattering. Dipole scattering is a long-range interaction between the incident or departing electron via its associated electric field and the oscillating dipole. Because of the response of the metal electrons to the approaching electron, which can be thought of in terms of the production of an image electron in the substrate, the lines of electric field strength terminate perpendicularly at the surface. Further-
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more, at distances far enough from the surface electrons cannot distinguish between the dipole and its image and so the simple dipole screening model of Fig. 7.3 is maintained. The discussion of the IR excitation mechanism based on Eq. 7.1 can thus in a first approximation be taken over and applied to dipole scattering in IES. In particular, we only expect to observe those modes which have a component of the dynamic dipole normal to the surface. Figure 7.5(a) shows energy losses from H adsorbed on W(100) in the specular beam. Only the dipole-active vibration mode is observed at 130 meV the same way as in the IR spectra. The fact that there is only one single loss peak is safely interpreted as dissociative adsorption of hydrogen and the loss energy corresponds to the symmetric stretch vibration normal to the surface. IES possesses one significant advantage towards IR spectroscopy: short-range impact scattering of the incoming electron from the local adsorbate potential can excite non-dipole active modes. Here, we are dealing with a short-range interaction between the electron and the atomic potentials of the adsorbate when the electron is at a distance from the surface comparable to the molecular dimensions and where the picture of classical image screening breaks down.9 Figure 7.5(b) displays energyloss spectra recorded in an o↵-specular direction from the same W(100)-H system. The 260 eV loss peak is the overtone (⌫ = 0 ! 2) of the fundamental dipoleactive vibration, while 80 and 160 eV losses are non-dipole excitations. A simple spring model predicts that the 160 meV mode is the asymmetric stretch and the 80 meV loss is then associated with an out-of-plane bending. The main feature of Fig. 7.5(b) is that all of the fundamental modes of the surface-molecule complex can be excited and observed when the measurement is made o↵ the specular beam direction. Whereas in dipole scattering the loss electrons are to be found in a narrow cone around the specular beam, in impact scattering they form a more isotropic distribution. The loss peak intensities will also be considerably weaker due to cross sections which are several orders of magnitude lower. Also important is the fact that impact scattering can occur not only from perpendicular modes but also from parallel modes and from modes that are not dipole-active. Thus by measuring scattering intensities as a function of angle it is possible to distinguish the perpendicular modes from the rest and to perform a structural analysis. IES experiments require ultrahigh-vacuum conditions, while IR spectroscopy can be performed on real surfaces.
9
W. Ho et al., Phys. Rev. Lett. 40, 1463 (1978).
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7.3
174
Raman Scattering
When a monochromatic electromagnetic wave is incident on a surface, we get the reflected and refracted beams as governed by Snell’s law. Even though this picture is quantitatively quite accurate, qualitatively it misses some important phenomena. Real materials also contain static and dynamic scatterers. The former cause an elastic scattering (i.e., at the same wavelength as the incoming wave) in all directions, which is about 1/10,000 of the incident intensity, and is known as Rayleigh scattering. Moreover, about 1/1000 of the scattered wave gets inelastically scattered (i.e., at a shifted wavelength with respect to incident wave) that arises from dynamic processes which is known as the Raman scattering. This Raman signal, though usually a tiny fraction (⇠ 1/107 ) of the incident intensity, plays a vital role in identifying the atomic constituents of the sample through their vibrational and rotational fingerprints. Hence, both Raman and IR spectroscopy are tools primarily for exploring vibrational properties of molecules, surfaces, and solids (crystalline or amorphous). They usually act as complementary techniques, as some vibrational modes are only IR active, and some are only Raman active. There are also cases where a particular mode can be active or silent for either one. Among the other di↵erences between the two techniques, Raman spectroscopy has a simpler spectrum than IR as often only the fundamental (lowest-order) processes are visible. Furthermore, in the Raman spectroscopy a monochromatic (usually a laser) source is used, whereas IR utilizes broadband sources.
7.3.1
A Classical Consideration for the Raman Process
First, we shall see that some essential features of the Raman scattering can be captured by a basic classical treatment.10,11,12 7.3.1.1
An Isolated Molecule
We start with a single molecule that may in general have a permanent dipole moment µ ~ , and an induced dipole moment through a polarizability tensor ↵ij , where i, j are the Cartesian indices, so that under an incident monochromatic electromagnetic ~ =E ~ 0 cos(!t), the resultant dipole moment becomes field E pi = µi + ↵ij Ej , with the convention that repeated Cartesian indices are implicitly summed over. Note that both µ ~ and ↵ij are functions of the coordinates of the nuclei and electrons. 10
W. Demtr¨ oder, Atoms, Molecules and Photons, Springer, New York, 2006. P.Y. Yu and M. Cardona, Fundamentals of Semiconductors, 4th Ed., Springer, New York, 2010. 12 D.A. Long, The Raman E↵ect, John Wiley, New York, 2002. 11
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We shall assume that electronic cloud can instantly respond to any change in the nuclear vibrations. Therefore, essentially these quantities are controlled by the nuclear displacements, for which we shall use their so-called normal coordinates as ~ k , where k labels any one of the vibrational degrees of freedom. As denoted by Q mentioned earlier, for an N -atom molecule, these are in total 3N 6. If we further assume that the incident light is o↵-resonant with both electronic and vibrational transitions, then such small-amplitude oscillations for the nuclear normal modes can be governed by the first two terms of the Taylor expansion around equilibrium ~ k = 0 yielding coordinates Q µi ' µi (0) +
3N X6
↵ij ' ↵ij (0) +
k=1 3N X6 k=1
@µi @Qkj @↵ij ~k @Q
Qkj ,
(7.2)
~k . Q
(7.3)
0
0
Each normal mode of vibration k will be oscillating at its eigenfrequency !k , there~ k (t) = Q ~ k0 cos(!k t). With this form the dipole moment of the fore we can write Q molecule becomes pi = µi (0) +
3N X6 k=1
@µi @Qkj
|
3N X6 @↵ij 1 + E0j ~ 2 k=1 @ Qk
Qk0j cos(!k t) + ↵ij (0)E0j cos(!t) 0
{z
IR spectrum 8 > >
> :|
{z
anti-Stokes
}
|
{z
}
!k )t]
Stokes
9 > > =
}> > ;
.
(7.4)
According to this expression, there is a component at the same frequency as the incident field, called the Rayleigh scattering; additionally we have a red- and blueshifted frequency parts which constitute the Raman scattering, where the former (latter) is called the Stokes13 (anti-Stokes) scattered light. This derivation suggests us that Raman scattering can be seen as a frequency mixing process just like the amplitude modulation concept used in electronic communication: incident field, acting as the carrier signal is modulated by the nuclear vibrations, here playing the role of the information signal. Fig. 7.6 illustrates the basic spectrum and the transitions corresponding to Stokes and anti-Stokes lines. 7.3.1.2
Selection Rules
As a result of the symmetries of the medium and of the vibrational modes involved in the scattering, some requirements are imposed. These are generally termed as 13
This terminology historically originates from a somewhat similar observation by Stokes on fluorescence where the frequency of the fluorescent radiation is always less than that of the incident. Ref. [12].
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Figure 7.6: The spectrum showing Rayleigh and Raman signals, together with the associated transition schemes. Ref. [10].
selection rules, which determine whether a specific perturbation V yields a nonzero transition from an initial state |vi to a final state |v 0 i as quantified by the matrix element hv 0 |V|vi. The study of such selection rules falls in the realm of so-called Group Theory, from which we shall state a few useful remarks without any derivation.14 For the IR and Raman spectroscopy, if we refer to Eq. (7.4), the operators that govern these transitions are the dipole moment µi and polarizability ↵ij operators, respectively. The former has components which transform as x, y and z, whereas the latter transforms as the binary products of x2 , y 2 , z 2 , xy, xz, yz, or equally their linear combinations such as x2 y 2 . According to Group Theory, if the direct product [v] ⇥ [µ] ⇥ [v] contains the totally symmetric irreducible representation of the point group, the transition v ! v 0 is IR active. Likewise, if the direct product [v] ⇥ [↵] ⇥ [v] contains the totally symmetric irreducible representation of the point group, the transition v ! v 0 becomes Raman active. 7.3.1.3
An Example: CO2 Molecule
In some cases, IR and Raman activity of certain vibrational modes can be extracted simply by observation, without the use of rigorous Group Theory machinery. One such example is a linear symmetric molecule ABA, such as the CO2 molecule. It has four modes of vibration: a symmetric stretching mode Q1 , an antisymmetric stretching mode Q2 , and two bending modes Q3a and Q3b which form a degenerate pair and have the same frequency of vibration. Three of these vibrations are shown in Fig. 7.7. We observe that symmetric stretching mode (on the left panel) has nonzero polarizability derivative at the equilibrium position (@↵/@Q 6= 0) therefore this vibration will give rise to an induced polarizability contribution under an incident field, hence it is Raman-active. However, as the A-B and B-A dipoles are of equal strength but pointing in opposite directions, this mode of vibration produces no net 14
D.C. Harris and M.D. Bertolucci, Symmetry and Spectroscopy, Dover, Mineola, 1989.
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dipole moment derivative at the center (@µ/@Q 6= 0), therefore it will be IR-silent. The situation is just the opposite for the center and right panels, as they are both IR-active but Raman-inactive.
Figure 7.7: Polarizability and dipole moment variations in the neighborhood of the equilibrium position for a linear ABA molecule. Ref. [10].
7.3.1.4
Extended Systems
Next, we move from a single N -atom molecule to an extended system (such as a solid) having n molecules per unit volume. The relation between the molecular quantities µ ~ , ↵ij , and those of the extended system, the permanent i and induced ij electric susceptibilities are i = nµi /✏0 , and ij = n↵ij ✏0 , where ✏0 is the permittivity of free-space. Similarly the polarization field of the medium is related to molecular dipole moment via P~ = n~p. Hence, for an incoming electromagnetic wave, ~ r, t) = E ~ 0 cos(~k ·~r !t), this time including its wave vector dependence ~k as well, E(~ ~ k (~r, t) = and expanding into Taylor series under nuclear vibrations with a form Q ~ k0 (~q , !k ) cos(~q ·~r !k t) yields the following terms, where we only show the Raman Q contributions Pi =
1 @ ij E0j ~k 2 @Q + cos[(~k |
0
8 > >
> :|
~q ) · ~r
{z
Stokes
(!
9 > > =
!k )t]> . }> ;
{z
(! + !k )t]
anti-Stokes
}
(7.5)
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For an isolated molecule the Raman scattering occurs in all directions with no angular preference. However, in an extended system we see that the Stokes-shifted wave has ~kS = ~k ~q , while the anti-Stokes is ~kAS = ~k + ~q . This means that in this case, the observation direction for the scattered wave inherently selects the participating phonon mode through the above momentum-conservation relations. Furthermore, note that for the visible light which is commonly used in Raman spectroscopy, its spectroscopic wavenumber is of the order of 105 cm 1 which makes up only a tiny proportion with respect to the extend of a typical Brillouin zone ⇠ 107 cm 1 . Therefore, in the case of crystals, because of the above momentum conservation requirement only the zone center phonons q ! 0 participate in a Raman process. As the nuclear vibrations attain larger amplitudes one may have to retain higherorder terms in the Taylor series expansion. These bring cross terms which will give rise to Raman shifts as ±!n ± !m for the second-order. When these modes are identical, the resultant two-phonon peak is called an overtone. 7.3.1.5
Raman Tensor
The intensity for the Raman scattered wave polarized along the unit vector direction eˆs in response to an incident polarization along eˆi is proportional to IS / eˆs ·
@
2 ij
~k @Q
0
~ k0 (q = 0, !k ) · eˆi Q
.
(7.6)
~ k , where i, j run Therefore the scattering is governed by a third-rank tensor @ ij /@ Q over Cartesian directions and k labels any of the optical phonon modes at the zone center. By introducing a unit vector along every available normal mode phonon ˆ = Q/| ~ Q|, ~ we can essentially introduce a second-rank (with the displacements Q same components as ij ) for each specific phonon polarization, called the Raman $ ~ 0 Q(! ˆ k ) , so that IS becomes tensor as15 R= (@ /@ Q)
2 $ !S4 e ˆ · ·ˆ e . s R i c4 Note that, we explicitly show the k 4 = ! 4 /c4 wavelength dependence which is ubiquitous in any form of sub-wavelength scattering like the Rayleigh scattering as popularized by the explanation of the blueness of the sky. If we neglect the small frequency di↵erence between the incident and scattered waves, the Raman tensor can be accepted as symmetric with respect to its Cartesian indices, just like the electric susceptibility ij .
IS (!S ) /
7.3.2
Quantum Mechanical Features of the Raman Process
Next, we discuss features which necessitate a quantum mechanical treatment.11 First, we notice from Eq. (7.6) that the scattered intensity is proportional to the 15
Here, we suppress the phonon polarization subscript on the Raman tensor.
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~ squared. Hence, according to the classical treatment there vibration amplitude Q would be no Stokes scattering if no atomic vibration is present. However, once we quantize the vibrational modes into phonons, in Stokes scattering where a phonon is excited in the medium by the incident radiation, the intensity becomes proportional to Nq + 1, while the anti-Stokes will be proportional to Nq . Here, Nq is the phonon occupancy, which is given in the case of equilibrium by the Bose-Einstein distribution. For low temperatures Nq ⌧ 1, therefore the ‘+1’ contribution in Stokes scattering intensity dominates. As one can recall, it arises from zero-point oscillations, a hallmark of quantum mechanics, and gives rise to spontaneous phonon emission even at zero temperature having Nq ! 0. Based on these facts, we can write the intensity ratio, anti-Stokes over the Stokes as ✓ ◆ ! + !k 4 Nq , ! !k Nq + 1 where the first term is again from the k 4 dependence for each frequency, while the second term contains the temperature dependence which simplifies to e ¯h!q /kB T . A fringe benefit of this is that the intensity ratio of the two lines can be used to extract the lattice temperature. In the light of this discussion, Stokes lines will be more intense as they originate from the v = 0 zero-phonon level, whereas anti-Stokes lines originate from the v = 1 one-phonon level with much less population. Another important feature is that the Raman lines have non-zero widths which further increase with temperature. Primarily it is due to the fact that the optical phonons taking part in the Raman process themselves are not of infinite lifetime because of the inherent anharmonicity of lattice vibrations causing phonon-phonon interactions. Therefore, they decay into two longitudinal acoustic phonons with large and opposite momenta. This finite phonon lifetime broadens the Raman resonances into a Lorentzian or a Gaussian line shape. The thermal variation of the width of the Raman line is given by16 ✓
(!k , T ) = (!k , 0) 1 +
2 e¯h!k /2kB T
1
◆
.
(7.7)
A further important deficiency of the classical picture is that it does not reflect the microscopic mechanism of how Raman scattering actually takes place. As a matter of fact, the direct excitation of phonons via photons is extremely weak. Therefore, this scattering proceeds quite di↵erently in at least three steps. 1. An incident photon with an energy h ¯ !i interacts with the charges within the sample through Hrad X , exciting the medium into an intermediate state |ni by creating a so-called exciton which is the bound state of an electron and hole pair. 2. This exciton being composed of an electron and a hole interacts strongly with the environment (lattice) via the electron-phonon (or hole-phonon) interaction 16
H. Kuzmany, Solid-State Spectroscopy, Springer, New York, 1998.
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(He ion ) and gets scattered into another state by emitting a phonon (considering the Stokes case) of energy h ¯ !k . This intermediate excitonic state will 0 be denoted as |n i. 3. The exciton in state |n0 i spontaneously recombines radiatively via Hrad with the emission of a so-called scattered photon of energy h ¯ !S .
X
So, this reveals that the electrons mediate the Raman scattering of phonons although they remain unchanged after the process. Since the transitions involving the electrons are virtual they do not have to conserve energy, although they still have to conserve (crystal) momentum. The corresponding Feynman diagram describing this process is shown in Fig. 7.8. Note that there are higher-order virtual processes other than the one above which also contribute Raman scattering. However, if we limit ourselves to only this lowest-order mechanism, then the Raman scattering rate can be calculated using Fermi Golden Rule as 2⇡ X hi|Hrad X (!S )|n0 ihn0 |He ion (!k )|nihn|Hrad X (!i )|ii WR (!S ) = h ¯ n,n0 [¯h!i (En Ei )] [¯h!i h ¯ !k (En0 Ei )] ⇥ (¯h!i
h ¯ !k
2
h ¯ !S ) .
(7.8)
Figure 7.8: The Feynman diagram corresponding to the basic Raman process. The left and right wiggly lines represent incident and scattered photon propagators, solid and dashed lines correspond to electron and hole propagators, and the spiral line is that of the phonon.
7.3.3
Variants of the Raman Process
The basic Raman process discussed above in practice comes with quite a few variations, with each one having a di↵erent utility and/or novelty. Some of these are very briefly mentioned below.12,10,11 • Brillouin Scattering – Inelastic scattering of light by acoustic waves was first proposed by Brillouin. As a result, this kind of light scattering spec-
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troscopy is known as Brillouin scattering. There is very little di↵erence between Raman scattering and Brillouin scattering. In solids the main di↵erence between them arises from the di↵erence in dispersion between optical and acoustic phonons. • Hyper-Raman – The intensity of Raman scattering is directly proportional to the irradiance of the incident radiation and so such scattering can be described as a linear process. When the intensity of the incident light wave becomes sufficiently large, the induced oscillation of the electron cloud surpasses the linear regime. This implies that the induced dipole moment p of the molecules are no longer proportional to the electric field E, but the E 2 and E 3 terms in the Taylor expansion of p(E) need to be retained giving rise to hyper-polarizability and second-hyper-polarizabilities, respectively. Such nonlinear optical e↵ects are named as hyper-Raman processes having quite di↵erent selection rules compared to linear one. The overtones of the main Raman signal as mentioned previously, are nothing but the manifestations of the hyper-Raman processes. • Electronic Raman – If the frequency di↵erence !i !k corresponds to an electronic transition of the system, we speak of electronic Raman scattering, which gives complementary information to electronic-absorption spectroscopy. This is because the initial and final states must have the same parity, and therefore a direct dipole-allowed electronic transition |ii ! |f i is not possible. • Resonant Raman – The enhancement of the Raman cross section near an electronic resonance is also known as resonant Raman scattering. Obviously, resonant Brillouin scattering is defined similarly. The enhancement in the Raman cross section at resonance is only two orders of magnitude relative to the nonresonant background. On the other hand, both free excitons and bound excitons have been shown to enhance the Raman cross section by several orders of magnitude because of their small damping constants (with a typical width of 3 meV) at low temperatures. Such strong resonance e↵ects have made possible the observation of new phenomena, such as wavevectordependent electron–LO phonon interaction, electric-dipole forbidden transitions, higher-order Raman scattering involving more than three phonons, and the determination of exciton dispersions. • Fourier-transform Raman – The combination of Raman spectroscopy with Fourier-transform spectroscopy allows the simultaneous detection of larger spectral ranges in the Raman spectra. • Surface-Enhanced Raman Scattering – Raman scattering cross-section is typically 14-15 orders of magnitude smaller over that of the fluorescence of efficient dye molecules. The intensity of Raman scattered light may be
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enhanced by several orders of magnitude if the molecules are adsorbed on a surface. A number of mechanisms contribute to this enhancement. Since the amplitude of the scattered radiation is proportional to the induced dipole moment, pi = ↵ij Ej , the increase of the polarizability ↵ by the interaction of the molecule with the surface is one of the causes for the enhancement. In the case of metal surfaces due to the presence of plasmonic e↵ects, the electric field E at the surface may also be much larger than that of the incident radiation, which also leads to an increase of the induced dipole moment. Both e↵ects depend on the orientation of the molecule relative to the surface normal, on its distance from the surface, and on the morphology, in particular the roughness of the surface. Small metal clusters on the surface increase the intensity of the molecular Raman lines. The frequency of the exciting light also has a large influence on the enhancement factor. In the case of metal surfaces it becomes maximum if it is close to the plasma frequency of the metal. Giant enhancement factors as high as 1014 have been reported on single-molecule studies.17 Because of these dependencies, surface-enhanced Raman spectroscopy has been successfully applied for surface analysis and also for tracing small concentrations of adsorbed molecules. • Coherent anti-Stokes Raman Scattering – Ordinary Raman scattering is an incoherent spontaneous process and as a result the intensity of scattering from a material system of N non-interacting molecules is simply N times that from one molecule. There are also non-spontaneous Raman processes, the three important ones are the stimulated Stokes Raman scattering, stimulated anti-Stokes Raman scattering and coherent anti-Stokes Raman scattering (CARS). The most popular is the last one where two incident laser beams are used with an energy di↵erence deliberately chosen to match a Raman-active vibrational mode of the sample. This produces highly directional beams of scattered radiation with small divergences. In particular, the intensity of the anti-Stokes signal is by far larger than in spontaneous Raman spectroscopy. The scattered intensity is proportional (a) to the square of the number of scattering molecules and (b) to the square of the irradiance of the incident radiations. With pulsed lasers time-dependent processes in molecules and their influence on the change of level populations can be studied by CARS; two examples being photosynthesis and the visual processes in our eyes.
17
K. Kneipp et al., Phys. Rev. Lett. 78, 1667 (1997).
Chapter 8 Surface Reactions In most of the investigations of solid surfaces the emphasis is placed on the characterization of clean and adsorbate covered surfaces because of the evident relevance to catalysis and corrosion. A reaction on the surface, as we have in heterogenous catalysis, proceeds at least in three steps. Each step is controlled by the reaction rate determined by the temperature and gas pressure: 1. Adsorption or coadsorption of two or several gases at the surface. 2. Reaction at the surface. 3. Desorption of reaction products from the surface. We will deal with adsorption, reaction, and desorption phenomena without going into much detail. Several spectroscopies we have studied so far had obvious connections to these subjects so that we can say that most of the time chemistry is one of the driving forces of detailed investigations in surface science.
8.1
Adsorption
Adsorption takes place when an attractive interaction between a particle and a surface is sufficiently strong to overcome the disordering e↵ect of thermal motion. When the attractive interaction is essentially the result of van-der-Waals forces then physisorption takes place. Physisorptive bonds are characterized by binding energies below approximately 1 eV. Chemisorption occurs when the overlap between the molecular orbitals of the adsorbed particle and the surface atoms permit the formation of chemical bonds, which are characterized by binding energy typically exceeding 1 eV. The chemisorption is always an activated process, i.e., the formation of a chemisorptive bond requires that an activation barrier is overcome. A common feature of molecular chemisorption is the weakening of intramolecular bonds that often lead to the dissociation of the adsorbed molecule. An important example for activated, dissociative chemisorption is the adsorption of oxygen molecules on most metal surfaces at room temperature. A descriptive view of chemisorption considers an adsorbate atom with a filled level of energy Ea that approaches a metal surface. As the atom nears the metal Ea 183
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will broaden by interaction with the metal. The closer Ea to EF the larger is the broadening. In the extreme case the overlap may lead to bonding and antibonding states. This interaction lowers the energy of the metal states below EF thus forming a strong bond. As found in all bonding the essential feature is that two electrons can simultaneously be on the adsorbate. Then the intraatomic Coulomb repulsion U of these electrons presents a crucial aspect of adsorption. If the broadening of the levels or the separation of bonding and antibonding orbitals exceeds U the chemical bond is stable. Fortunately, the value of U is considerably reduced near a metal surface by screening e↵ects which can be understood in terms of image interaction. Thus, the level Ea is pushed up by the image force (= e2 /4z) and the resulting ion interacts attractively with the metal through its image charge. The simplest approach to quantitative analysis of adsorption is to follow that what is already postulated by Langmuir. He made several assumptions: 1. A surface contains enumerable sites on which an adsorbate can be bound. The fraction ✓ of the occupied sites also called surface coverage, is 0 ✓ 1. When all adsorbed sites are occupied, the substrate surface is saturated. 2. Adsorption can only take place at sites that are not yet occupied.
Figure 8.1: Adsorption is possible at a previously not occupied site.
Since the fraction of the sites that are not occupied is given by (1 ✓), this number gives the fraction of species that are adsorbed at the surface upon arrival and is called the sticking probability, S(✓).
Figure 8.2: In the simplest case, the sticking coefficient S(✓) is linear with surface coverage, while real surfaces follow the power law.
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3. There is no interaction between the adsorbates. This means that the adsorbates “wet” the substrate and there is no island formation. like in an idealized FV-type growth (see Figs. 1.4 and 1.5). This assumption requires that the adsorbed particles are immobile. 4. Adsorption proceeds in equilibrium. With the number of adsorption sites N , (
dN dN )ads = ( )desorb dt dt
(8.1)
Using the constants k 0 , k 00 , and b = k 0 /k 00 and the gas pressure p, we can write dN dN )ads = k 0 p(1 ✓) and ( )desorb = k 00 ✓. (8.2) dt dt This leads to adsorption isotherms describing the surface coverage as a function of gas pressure over the sample: bP ✓= (8.3) 1 + bP Actually, b = f (T ). (
Figure 8.3: Adsorption isotherm of Langmuir.
We recognize two limiting cases: a. Low-pressure regime leads to weak adsorption, where ✓ = bp b. High-pressure regime leads to saturation with ✓ = 1. This case is a first-order rate equation. There is also a second-order process, which corresponds to dissociative adsorption and recombinative desorption of diatomic molecules, relevant for catalysis. Note that the dissociation of molecules can often be suppressed by lowering the temperature of the surface. An example is the adsorption of O2 on Ag(110), which is dissociative at room temperature, but non-dissociative at 150 K. For the second-order process, the coverage is given by p bP p ✓= (8.4) 1 + bP
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A modern approach to investigate the adsorbate structure, which simple means the location of the adsorbate atoms on the surface, requires the knowledge of the positions of the substrate atoms. We have seen several real-space and reciprocalspace techniques that deliver information on the clean solid surfaces. Di↵erent crystallographic orientations of bcc metals, like W, Mo, Ta, Nb, are frequently studied in chemisorption because of ease of their preparation. Most of the clean surfaces give only (1 ⇥ 1) LEED patterns, implying one deals with bulk-terminated surfaces. The fcc metals include the best catalysts, like Pt, Ni, Pd, and Rh and the noble metals Cu, Ag, and Au. LEED indicates that (111) planes have (1 ⇥ 1) symmetry for all the metals. On Ni, Pd, and Cu the (100) planes also have (1 ⇥ 1) symmetry. For other orientations, the situation becomes complex. The clean (100) planes of Pt, Au, and Ir exhibit a complicated (5 ⇥ 1) symmetry. The (110) surfaces of Au and Pt show a (2 ⇥ 1) LEED pattern. Electronic information on clean and adsorbate covered surfaces is gained by photoemission experiments, while Auger electron spectroscopy is widely used for chemical characterization of the adsorption process at any stage. Our knowledge on di↵erent adsorption sites is based on desorption spectroscopy (Section 8.3).
Figure 8.4: Flash desorption spectra for hydrogen on W(110) and W(111). Ref. [1].
Hydrogen on tungsten is one of the classical examples; clean tungsten surfaces have routinely been prepared for chemisorption studies. Desorption spectra have revealed two major adsorption sites on the (110) and at least four sites on the (111) plane.1 Figure 8.4 shows the relevant desorption data for H2 on W(110) and W(111). The pressure caused by desorbing hydrogen is plotted as a function of sample temperature, while maxima are related to individual adsorption/desorption energies. LEED data on W(110) indicate that there are two atomic states of equal density, the 1 and 2 phases, with the existence of (1⇥1) and (2⇥1) periodicity, respectively. 1
P.W. Tamm and L.D. Schmidt, J. Chem. Phys. 51, 5352 (1969); 54, 4775 (1971).
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The saturation density is approximately one H atom per surface W atom, as shown in Fig. 8.5. The 2 state occupies every other surface site and the 1 state occupies the remaining sites.
Figure 8.5: Proposed adsorbate structures for H2 on the W(110) surface with two states of equal density which fill sequentially. Ref. [1].
8.2
Reaction
A single catalytic reaction consists of a closed sequence of elementary processes or steps. Summation of these steps, multiplied each by an appropriate stoichiometric number reproduces the stoichiometric equation for the reaction. In the first step, a species called the catalyst enters as a reactant whereas it appears as a product in the last step of the sequence. The catalyst may be a gaseous molecule or a grouping of atoms at the surface of a solid called the active site and denoted by an asterisk “⇤”. In the latter case, catalysis is called heterogeneous. It is only a special case of a general phenomenon. The great technological advantage of heterogeneous catalysis is the easy separation between the solid catalyst and the fluid reaction medium. The two basic mechaTable 8.1: Dissociative and associative mechanisms in ammonia synthesis. Ref. [2]. Dissociative * 2⇤N 2⇤ + N2 ) 2⇤ + H2 * ) 2⇤H ⇤N + ⇤H * ) ⇤NH + ⇤ ⇤NH + ⇤H * ) ⇤NH2 + ⇤ ⇤NH2 + ⇤H * ) ⇤NH3 + ⇤ ⇤NH3 * ) NH3 + ⇤ N2 + 3H2 * ) 2NH3
1 3 2 2 2 2
Associative * ⇤N2 ⇤ + N2 ) ⇤N2 + H2 * ) ⇤N2 H2 ⇤N2 H2 + H2 * ) ⇤N2 H4 ⇤N2 H4 + H2 * ) 2NH3 + ⇤ N2 + 3H2 * ) 2NH3
1 1 1 1
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nisms of catalysis are dissociative and associative. As an example, let us postulate two possible sequences of steps in ammonia synthesis (Table 8.1).2 Qualitative considerations suggest that the steps in a dissociative mechanism will have a higher probability (entropy factor) but also higher energy barrier (energy factor) than their associative counterparts.3 Hence a dissociative mechanism may be favored at high temperatures, while at low temperatures associate mechanisms might prevail. The first quantitative measure of the catalytic act is the rate of the single reaction called the activity of the catalyst. The best measure of the rate, which lends itself to comparison of the activity of various catalysts is the turnover number defined as the rate per mole of site, that rate itself being defined as the rate of change of the extent of reaction with time. In most cases, only an average or nominal number may be obtained, as the number and types of sites may not be known. The activity of a catalyst is rarely its most important characteristic. Selectivity, which is defined as a ratio of rates, that of a desired reaction over the sum of rates of other undesirable side reactions, is often more important to achieve and more subtile to understand. The habit of catalytic materials is dictated by technological usage and by the frequent need for a very large specific surface area. Thus, porous materials with average pore dimensions around 10 nm are often used as the catalyst itself or as a support or carrier for the catalytic material. Clearly, the texture of a porous catalyst, i.e., the shape of a replica of the material is of great importance in determining activity and selectivity because of the unavoidable gradients of temperature and concentration in the porous medium. Quantitative measures of the texture of catalytic materials, namely the total specific surface area and the pore size distribution, can be obtained from isotherms of physisorbed nitrogen, according to standard methods which have received universal acceptance. In catalysis there are promoters and poisons which must be considered together: they consist of additives introduced with the reactants or during the catalyst preparation and they a↵ect catalyst activity or selectivity. A first kind of promoter, said to be textural , prevents loss of surface area of the catalyst. A typical example is alumina Al2 O3 introduced in small quantities during the preparation of ammonia synthesis iron catalysts. After reduction, the catalyst consists of metallic iron particles with a mean diameter of about 35 nm. The alumina is unreduced and covers about half of the iron surface, preventing sintering of metallic particles.4 Another kind of promoter is structural or chemical . An example is chlorine in the selective oxidation of ethylene to ethylene oxide on silver catalysts. It ap2
M. Boudart, In Interactions on Metal Surfaces, ed. by R. Gomer, Springer Series Topics in Applied Physics , Vol. 4, New York, 1975, p. 275. 3 G.K. Boreskov, New Approach to Catalysis, Catalysis Society of Japan, Tokyo, 1973. 4 P.H. Emmet, Structure and Properties of Solid Surfaces, ed. R. Gomer and C.S. Smith, The University of Chicago Press, Chicago, 1955.
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pears that chemisorbed chlorine inhibits the activated or non-activated dissociative chemisorption of oxygen on the metal. Since oxygen adatoms are responsible for the non-selective oxidation of ethylene, the role of the promoter is to enhance selectivity. Since the promoter occupies part of the surface, its role may also be conceived as that of a selective poison. There are as many types of poisons as there are types of sites. Some are reversibly held, i.e., they can be removed under reaction conditions as is the case for surface oxygen during ammonia synthesis. Some are irreversibly held as are most types of carbonaceous residues accumulating on catalytic surfaces during reactions involving hydrocarbons. These residues must be removed in a special step of catalyst regeneration. In fact, catalyst deactivation during use is the rule rather than the exception: it is the central problem in the transfer of any catalyst from the laboratory to the plant.5 Reversible poisons are called also inhibitors especially when they are participants in the reaction. Thus reaction products are often inhibitors as is the case for ammonia in ammonia synthesis on iron. The fraction of sites which is active in a given reaction depends on the catalyst and on the reaction, which is called active centers and are the site or group of sites taking part in the reaction. The identification of the active centers and the structure of their complexes with the reactive intermediates under reaction conditions is the central goal of research in catalysis. These are some classical ideas of catalytic reactions.
8.3
Desorption
Breaking the bonds of surface species and their liberation from the surface is called desorption. This can be accomplished in di↵erent ways. If the temperature of the system is high enough that a sizeable fraction of the adsorbate complexes has energies above the desorption energy in the Maxwellian distribution and can therefore leave the surface, one speaks of thermal desorption. Electron impact can lead to transitions to excited states of the adsorbate whose potential energy at the equilibrium distance of the ground state is higher than that of the respective free particle, so that ions or neutrals can be desorbed. This process is called electron-impact desorption or electron-stimulated desorption. Similar processes can be brought about by excitation by light; we speak of photodesorption. The impact of ions or neutrals can cause the removal of adsorbates in di↵erent ways; these processes are called ionimpact desorption. Finally, a very high electric field can bend down the ionic curve of the adsorbate so far that rapid tunneling occurs from ground state to the ionic state, and the resulting adsorbate ion carries away from the surface immediately; these processes are termed field desorption. 5
J.B. Butt, Advan. Chem. Series 109, 259 (1972).
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Measurement of desorption process, especially that of thermal desorption, have widely been used to define adsorbate states (Fig. 8.4) and to measure their populations. Thermal desorption is an important step in heterogenous catalysis, and the detailed analysis of the process is the basis of thermal desorption spectroscopy which is still used with success. It was the Swedish scientist Arrhenius who established the necessary formalism to calculate the desorption rate rd around 1920. rd =
dNA = dt
NA
d✓ = NA Ce dt
Ed /kT
(8.5)
Here, NA is number of adsorption sites per unit area, k is the Boltzmann constant, Ed the desorption energy, and C a constant. The last part of the Eq. 8.5 is the ansatz of Polanyi-Wigner. If we let the temperature change linearly as T = To + t, then rd (T ) = with
=
dT . dt
Recall that
dN dt
=
dN dT
·
dT dt
NA C =
e
Ed /kT
(8.6)
dN . dT
Figure 8.6: The change of desorption rate rd with increasing temperature. The desorption rate has a maximum value at TM , which helps us find the desorption energy Ed . The area under the curve corresponds to the number of particles liberated from the surface. The maximum value for rd requires
drd dT
= 0, which leads to
Ed C = e 2 kTM
Ed /kT
.
(8.7)
In most cases there are more than one maximum value for rd , indicating more than one desorption energy and, consequently, more than one di↵erent adsorption site at the surface. We di↵erentiate three main regions depending on the heating rate, : 1. Rapid heating implies dT = ! 1 and called flash desorption. It gives us dt information on the total surface coverage.
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191
dT dt
= 0 is isothermal tion and gives us information about the average time an adsorbate spends on the surface.
3. 0
