Surface Physics: Fundamentals and Methods 9783110636697, 9783110636680

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Thomas Fauster, Lutz Hammer, Klaus Heinz, and Alexander Schneider Surface Physics

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Thomas Fauster, Lutz Hammer, Klaus Heinz, and Alexander Schneider

Surface Physics |

Fundamentals and Methods

Authors Prof. Dr. Thomas Fauster Universität Erlangen–Nürnberg Lehrstuhl für Festkörperphysik Staudtstr. 7 91058 Erlangen Germany [email protected]

Prof. Dr. Klaus Heinz Universität Erlangen–Nürnberg Lehrstuhl für Festkörperphysik Staudtstr. 7 91058 Erlangen Germany [email protected]

Dr. Lutz Hammer Universität Erlangen–Nürnberg Lehrstuhl für Festkörperphysik Staudtstr. 7 91058 Erlangen Germany [email protected]

Prof. Dr. Alexander Schneider Universität Erlangen–Nürnberg Lehrstuhl für Festkörperphysik Staudtstr. 7 91058 Erlangen Germany [email protected]

ISBN 978-3-11-063668-0 e-ISBN (PDF) 978-3-11-063669-7 e-ISBN (EPUB) 978-3-11-063699-4 Library of Congress Control Number: 2020934847 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Cover image: Alexander Schneider Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Surfaces of solids play an important role in basic physics research and in technical applications. Due to the broken bonds at the surface and often drastically different atomic arrangements compared to the bulk, surfaces may exhibit new properties. By depositing other substances on surfaces novel materials may be created. Surfaces are always involved as boundary or “skin” of a solid, when it interacts with its environment. Surfaces govern important phenomena, such as corrosion (or protection against), friction, or catalysis. With continuing miniaturization of semiconductor devices, the surface or interface fraction increases and a thorough understanding of interface properties becomes important. Because of this interest in basic research and the importance of associated applications, surface physics has rapidly grown during the last 50 years, and is still expanding as a subdiscipline of solid-state physics on its own footing with strong connections to surface chemistry and materials or nanoscience. Surface physics is regularly taught in study programs of physics and also physical chemistry or materials sciences. The large variety of uncommon properties which are absent in bulk solids is unfamiliar at first glance. A wide spectrum of surface-specific preparation and analysis methods with associated acronyms is even more confusing. This situation was pictured by Nobel-prize winner Wolfgang Pauli in the quote “God made the bulk; the surface was invented by the devil”. In this textbook, we attempt to disentangle the topic of surface physics at the level of bachelor or master students, who have basic knowledge of solid-state physics. The authors take advantage of long-standing experience in teaching courses on the subject and in surface-physics research. The selection of topics and methods was guided by the experimental expertise of the authors, and was intentionally limited to offer a textbook for a one-semester course. Nevertheless, we believe that we cover the relevant topics of surface physics at the bachelor or master level. Theoretical descriptions were kept at the minimum, which is necessary for the understanding and interpretation of measurements. However, in many cases the close interconnection between theory and experiment is essential for scientific progress in surface physics. We focus on surfaces as interfaces between a solid and vacuum or atmosphere. For adsorbates or deposited films, the interface to the solid, i. e., the solid-solid interface is also relevant and can be described using many concepts of surfaces. To limit its volume, magnetic properties of surfaces, catalytic reactions at surfaces, and processes, such as corrosion and friction, are not covered in this textbook. This textbook deals almost exclusively with the surfaces of crystalline solids. These systems can be treated with relative ease due to the translational symmetry parallel to the surface. The first part of the book lays the important Fundamentals of surface physics. Chapter 1 describes the basics of the atomic and electronic structure of surfaces and their lattice vibrations. The most important processes occurring at surfaces are presented in Chapter 2. The second part of the textbook is devoted to the most https://doi.org/10.1515/9783110636697-201

VI | Preface widespread Methods of surface physics. After the methods for sample preparation (Chap. 3) various experimental measurement techniques at surfaces (Chapters 4– 7) are presented. The appendix provides some useful supplementary information and tables. Each chapter closes with a short summary, some exercise problems or comprehensive questions, and a list of references. Answers to the problems and questions are collected at the end of the book, which closes with a list of acronyms and an index. The textbook addresses bachelor and master students in physics, surface chemistry, materials science or nanotechnology. The scope is geared for a one-semester course with two hours of lecture per week. Sections which may be too advanced for bachelor students are marked accordingly, and may be skipped during the first reading. The book is based on the second edition of our German textbook Oberflächenphysik. In recent years we have been teaching the surface physics course for the master students in English and realized the need for an English translation, which will be useful for a broader audience. Many of the experimental results presented in the various chapters are taken from diploma or PhD theses performed at the chair of solid-state physics of the FriedrichAlexander-Universität Erlangen-Nürnberg (FAU). It is our sincere pleasure to thank the graduates for their great work. We are also indebted to our colleagues for kindly providing figures used in this textbook: Jan Knudsen, Sabine Maier, Elias Vlieg, and Wolf Widdra. We thank Kristin Berber-Nerlinger for pushing the English translation, and Vivien Schubert for managing the publishing process. Erlangen, January 2020

Th. Fauster, L. Hammer, K. Heinz, M. A. Schneider

Contents Preface | V Legend | X

Fundamentals 1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.3

Properties of surfaces | 5 Atomic structure of surfaces | 5 Surface crystallography | 6 Superlattices and superstructures | 10 Surface relaxation | 14 Surface reconstruction | 17 Electronic structure | 23 Bloch’s theorem | 23 Surface Brillouin zone | 25 Projected bulk band structure | 27 Surface states | 30 Work function | 35 Lattice vibrations at surfaces | 37

2 2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.3.4

Processes at surfaces | 42 Energy-loss processes of electrons | 42 Energy distribution of scattered electrons | 43 Inelastic mean free path | 46 Adsorption, desorption, and diffusion | 47 Adsorption | 48 Desorption | 51 Diffusion | 52 Film growth and epitaxy | 54 Growth modes | 54 Nucleation | 55 Epitaxy | 56 Nanostructures formed by self-organization | 58

Methods 3 3.1

Preparation of surfaces | 65 Sources of contamination | 65

VIII | Contents 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.3.2

Preparation of clean single-crystal surfaces | 66 Cleaving | 67 Annealing | 67 Chemical cleaning | 69 Sputtering by ions | 70 Modification of surfaces | 72 Adsorbate-covered surfaces | 73 Layers on surfaces | 73

4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.3 4.4 4.5 4.6

Diffraction methods | 77 Kinematic description of diffraction | 79 Diffraction of low-energy electrons | 84 Experiment | 84 Geometry of the diffraction pattern | 85 Quantitative structure analysis | 88 Diffraction of high-energy electrons | 93 Surface x-ray diffraction | 93 Helium diffraction | 95 Fine structure of x-ray absorption | 96

5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.4 5.4.1 5.4.2 5.4.3

Electron spectroscopies | 101 Instrumentation | 102 Electron sources | 102 Photon sources | 103 Energy analyzers | 105 Electron detection | 107 Modulation technique | 108 Element-specific spectroscopy | 109 X-ray photoelectron spectroscopy | 109 Core-level shifts | 113 Auger electron spectroscopy | 115 Lineshape of Auger electron spectra | 118 Qualitative and quantitative analysis of chemical elements | 119 Determination of surface band structures | 121 Angle-resolved photoemission | 122 Inverse photoemission | 127 Two-photon photoemission | 128 Selection rules for photoemission | 130 Spectroscopy of surface vibrations | 132 Fundamentals and experiment | 132 Interactions and selection rules | 133 Applications | 137

Contents | IX

6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3 6.3.1 6.3.2 6.3.3 6.3.4

Scanning probe microscopies | 143 Principle of scanning probe methods | 143 Scanning tunneling microscopy | 144 Topographic images | 146 Tunnel process | 148 Spectroscopy | 151 Applications and other interaction mechanisms | 154 Atomic force microscopy | 155 Contact AFM | 156 Dynamic AFM | 156 Forces and their effects in AFM | 158 Applications and other interaction mechanisms | 160

7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.2

Particle spectroscopies | 162 Thermal desorption spectroscopy | 162 Fundamentals and experimental setup | 162 Types of desorption spectra | 164 Evaluation of desorption spectra | 166 Desorption spectra with several maxima | 169 Ion scattering | 170

A A.1 A.2 A.3 A.4

Appendix | 174 Planes and surfaces of hcp crystals | 174 Identification of plane groups | 175 Brillouin zones and mirror planes | 176 Energies of Auger transitions | 178

Answers to exercise problems and questions | 179 Acronyms | 185 Index | 187

Legend This symbol marks paragraphs of advanced in-depth topics which may be skipped during the first reading.

At the end of each chapter the essentials are summarized and put into context to other chapters.

Exercise problems and comprehensive questions help to gain a deeper understanding of the various topics of each chapter. The answers are collected at the end of the textbook.

Each chapter contains a list of references.

Vectors are underlined v and not marked with an arrow.

The title page presents a view into an ultrahigh-vacuum chamber. A LEED optics with an opened shutter and gold-plated grids can be seen in the background at the top. In the middle of the picture the sample manipulator with heated sample (center) and quartz crystal balance (left) is positioned. At the bottom of the photo a stainless-steel tube serves as doser for the gas inlet (left). The front ends of two evaporators for metals are located at the center and right. https://doi.org/10.1515/9783110636697-202

Two surfaces can be obtained in a gedankenexperiment by dividing a crystalline solid with a cut parallel to an atomic plane. The cutting of chemical bonds leads to atoms with unsaturated bonds, which is energetically unfavorable. To minimize the energy, atoms may change their position and engage into new bonds or modify existing bonds. The cut may also alter the lateral arrangement parallel to the surface. The atomic rearrangement can affect several layers parallel to the intersection, and the surface is created as a corresponding multilayer with new atomic and electronic configuration, and thus new physical properties, i. e., basically a new material. For example, an insulator or semiconductor may be electrically conducting at the surface. Due to the new bonds and the ones missing above the uppermost atomic layer, the surface atoms feel a changed potential compared to the bulk, and new surface vibrational modes can exist. The richness of geometric and electronic structures, which may be formed by the creation of a surface, including the associated modified lattice vibrations, is described in Chap. 1. The surface is of course always and foremost involved when a solid interacts with its environment. In Chap. 2 we first look at the interaction processes when electrons hit and enter a surface. The resulting small mean free path makes methods using lowenergy electrons the most important techniques in surface physics. The interaction of atoms or molecules at a surface on the other hand comprises the basic processes of adsorption, diffusion, and desorption. Deposited atoms may desorb immediately, or after a chemical reaction with another atom or molecule. Diffusion of adsorbed atoms may lead to the formation of islands or a closed adsorbate layer, but also to multilayer films, which can exhibit various growth modes and associated morphologies. A new material with surprising properties may be formed when a film adopts the lattice structure or constant of the substrate. Substrate surfaces, which are structured on the nanometer scale by steps or a reconstruction, may serve as templates for creating a nanostructured adsorbate or film. Nanostructures can also form on unstructured, flat surfaces by self-organization.

https://doi.org/10.1515/9783110636697-001

1 Properties of surfaces In this chapter, we treat the properties of surfaces separately for the arrangement of the surface atoms, the electronic states, and the vibrations of the surface atoms. This separation is justified for surfaces in the ground state or thermal equilibrium. However, that state is the result of the forces between the atoms, which are mediated by the electrons. On the other hand, the arrangement of the atoms does influence the electronic states, and thus the forces between the atoms and their vibrations. Since this interplay occurs on very short time scales, one observes in most experiments the stationary equilibrium state, which can be separated into atomic and electronic properties.

1.1 Atomic structure of surfaces An important goal of surface physics is the determination of the atomic arrangement of well-defined planar surfaces of single crystals. The knowledge of the crystallographic structure is essential, because the structure has an impact on all physical properties of the surface. In the most simple picture, one might imagine that the structure of a surface coincides with the corresponding cut through the bulk without any change of the atomic positions (bulk-like termination). However, this occurs only in rare cases, because the creation of a surface opens up new structural degrees of freedom. The cut of the chemical bonds in the gedankenexperiment of dividing the bulk crystal leaves the surface in an energetically unfavorable state. Consequently, the surface atoms assume new positions and find an energetically more favorable state. Existing bonds might be strengthened, in some cases, broken, or new bonds may be formed. These effects are most pronounced for semiconductors with their covalent bonding, which is associated with directional bond orientation. But also many metals, which are characterized by isotropic bonding, change their structure at the surface. The new geometric arrangement of the surface can be classified in two categories: surface relaxation, where only the distances between atomic layers parallel to the surface are changed, and surface reconstruction, where local atomic displacements lead to new lateral unit cells. At surfaces of alloys (with two or more atomic species), the chemical composition and order at the surface may change in addition. If the path of the surface atoms toward a new state of lowest energy goes via intermediate states, which are energetically less favorable (i. e., an activation energy has to be surmounted), the surface might not be able to reach the state of lowest energy, and adopts a metastable state. In such cases, the state finally assumed by the surface depends on the preparation steps. Clean surfaces can be modified by the adsorption of atoms or molecules, which influences the extent of the relaxation, and may alter the type of reconstruction (adsorbate-induced relaxation and reconstruction). Relaxation and reconstruction will be described in detail after an introduction regarding the fundamentals of surface crystallography. https://doi.org/10.1515/9783110636697-002

6 | 1 Properties of surfaces

Figure 1.1: (a) Three low-index surface cuts of an fcc crystal. (b–d) The surface planes are indicated in the top row by shading in the perspective view of the cubic unit cell. The bottom row shows the top view of the atomic arrangement and the surface unit cell (shaded). The atomic positions in the 1., 2. and 3. layer are marked by dark-brown, brown, and light-brown circles, respectively.

1.1.1 Surface crystallography In this section, we will treat the crystallographic description of ordered surfaces. They are obtained by cutting bulk crystals along selected planes, and are therefore denoted like crystal planes in three dimensions by Miller indices (hkl). Crystallographic directions are denoted as [hkl], correspondingly. Many elements assume the familiar face-centered cubic (fcc), body-centered cubic (bcc), or hexagonal close-packed (hcp) structures. Often low-index surfaces are investigated, because they have rather simple structures in combination with high symmetry and small unit cells. Figure 1.1(a) shows some examples of low-index surfaces of an fcc crystal. The top row (b–d) illustrates the surface planes by shading in the cubic unit cell with lattice constant a. The bottom row presents the top view on the atomic positions of the top three atomic layers (for (001) and (110), the third layer is hidden by the first layer). One recognizes, that the primitive basis vectors of the surface unit cell do not have to be primitive basis vectors of the three-dimensional lattice. The shortest basis vectors of the fcc lattice in Fig. 1.1 are of type a2 [110]. The primitive unit cell of the (110) surface is formed by the basis vectors a[001] and a2 [110]. The basis vectors of the surface unit cell are always perpendicular to the normal vector of the surface. The surface planes of hcp crystals may alternatively be denoted by a special four-index nomenclature, which is explained in Appendix A.1. The creation of a surface breaks the translational symmetry perpendicular to the surface. However, the periodicity parallel to the surface is often conserved, or a new periodic arrangement forms. The corresponding basis vectors a1 and a2 span the lateral unit cell (Fig. 1.2(a)). Perpendicular to the surface, the periodicity is lost and the

1.1 Atomic structure of surfaces | 7

Figure 1.2: (a) Perspective view of a simple surface. (b) Layer and unit-cell arrangement without atoms. (c) Corresponding point lattice with basis vectors a1 and a2 . The additional dashed vectors illustrate that the choice of the basis vectors and unit cells (shaded) is not unique.

unit cell is extended into the bulk crystal (and into the vacuum in the other direction). The surface structure is composed by a laterally periodic arrangement of vertical columns. Alternatively, one can describe the solid with a surface by a vertical sequence of stacked atomic layers as illustrated in Fig. 1.2(a),(b), which follows from the original bulk-crystal arrangement. The expression “surface” is commonly used for the stack of layers, in which the physical properties are different from the bulk. The choice of the basis vectors and unit cells is ambiguous as illustrated in Fig. 1.2(c). Conventionally, one chooses the smallest possible (primitive) unit cell and shortest basis vectors. The shorter basis vector is usually named a1 . The other basis vector a2 is then chosen such that the enclosed angle between the basis vectors becomes ≥ 90∘ . This convention is often ignored, because any other choice is also valid, and leads to the same physics. After removing the atoms, the lateral translational invariance leads to a two-dimensional point lattice or Bravais lattice (as illustrated in Fig. 1.2(c)), which characterizes the structure by points at the positions r mn = ma1 + na2

m, n ∈ ℤ.

(1.1)

This leads us to the question regarding how many different types of unit cells (point lattices or Bravais lattices) exist in two dimensions, which can completely fill a plane without voids. In three dimensions, there are 14 types of such unit cells. In two dimensions, this number is reduced to 5 Bravais lattices, which are shown in Table 1.1. For the rhombus (a1 = a2 , γ ≠ 60∘ , 90∘ , 120∘ ), one often chooses a nonprimitive, centered rectangular unit cell, which illustrates the inherent symmetry more clearly. This is analogous to the three-dimensional fcc or bcc lattices, where the primitive unit cells do not immediately reveal that one deals with cubic lattices. Bravais lattices are abstract point lattices. A crystal structure is generated, when each lattice point is associated with an atomic basis. In the most simple case, it consists of just one atom (per layer parallel to the surface), but it can also contain several atoms or molecules. Depending on the symmetry of the basis, certain symmetry operations leave the lattice invariant and characterize in combination with the translational

8 | 1 Properties of surfaces Table 1.1: The five Bravais lattices in two dimensions. Bravais lattice

Conditions

Real space

Parallelogram

a1 ≠ a2 , γ ≠ 90

Rectangle

a1 ≠ a2 , γ = 90∘

Rhombus (centered rectangle)

a1 = a2 , γ ≠ 60∘ , 90∘ , 120∘

Square

a1 = a2 , γ = 90∘

120∘ -Rhombus

a1 = a2 , γ = 120∘ or 60∘

Reciprocal space

∘

symmetry of the lattice the crystal structure. The associated symmetry elements are rotation axes (only 1-, 2-, 3-, 4-, and 6-fold rotation axes perpendicular to the surface are compatible with the translational symmetry), mirror lines, and glide lines with the notation letters m and g, respectively. This yields in two dimensions 17 plane groups, a small number compared to the 230 space groups of three-dimensional crystals. Since surfaces consisting of several layers are not pure two-dimensional objects, often the terms mirror planes and glide planes are used. These planes are then oriented perpendicular to the surface. The notation of the plane groups includes these symmetry elements. If some of these are generated by combination of two existent symmetries, they may be omitted to shorten the notation. A new mirror line may originate by combining a rotation and a mirror line. For a centered Bravais lattice, the letter c is added and one speaks about a centered structure. For a noncentered, so-called primitive structure, one may add the letter p, which is not really required, because the missing c is sufficient for distinction. Note, that the symmetry of the point lattice (Bravais lattice) does not have to be preserved by the occupancy with a basis. This can already occur for the layer stacking at a simple metal surface (Fig. 1.3). In many cases, the symmetry is reduced if a surface reconstructs or is covered by adsorbates. For a systematic determination of the plane group of a surface structure, one may follow the scheme given in Appendix A.2 in Table A.1. Symmetry reduction by layer stacking and adsorption is illustrated in Fig. 1.3. (a) shows the top layer of a Pt(111) surface (fcc(111)). The atoms occupy a hexagonal point lattice and preserve the 6-fold rotation axis and two mirror lines due to the spherical symmetry of the atoms (plane group p6mm). After adding the second Pt layer, each lattice point is associated with a two-atom basis. This reduces the rotational symmetry to a three-fold rotation axis, and one of the two mirror lines is lost, as illustrated in Fig. 1.3(b). The resulting plane group is p3m1. Adding the third layer (c) (or more layers in ABC stacking) does not lead to further changes of the symmetry or

1.1 Atomic structure of surfaces | 9

Figure 1.3: (a–c) Symmetry reduction by ABC-layer stacking of the top three layers on an fcc Pt(111) surface (red, brown, and light-brown circles), and (d) by adsorption of CO (blue-bordered circles). In (d) two different unit cells are plotted. The basis vectors are indicated, and the plane-group notation is explained in Section 1.1.2.

plane group. When CO molecules are adsorbed on the surface, the adsorption phase illustrated in Fig. 1.3(d) forms. Since the CO layer has the symmetry c2mm, the threefold rotation axis is lost in the combined adsorbate-substrate system and the plane group of the structure retains only one mirror line, and becomes p1m1. If the molecular axis is not perpendicular to the surface, as assumed so far, a further symmetry reduction may occur. More examples for the reduction of the symmetry of the substrate by surface reconstruction or adsorption are given in Figs. 1.13 and 1.18(a),(b), respectively. If a crystal is not exactly cut parallel to a low-index surface plane, a vicinal surface is formed. On such a stepped surface, terraces corresponding to low-index surface orientations alternate with steps. Sometimes one intentionally cuts the surface close to a low-index orientation to obtain a surface with many steps, because these offer special adsorption sites. Figure 1.4 shows the example of an fcc(775) surface (for Cu(117), see Fig. 5.20(b)) with a ball model and unit cell (a) and the respective directions and surface planes (b). The notation fcc(775) indicates the macroscopic surface orientation. A more detailed notation for stepped surfaces is written as M(S) − [m(ht kt lt )] × [n(hs ks ls )]. M(S) indicates a stepped surface of material M. The variables with indices t and s denote the orientation of the terraces and planes at the steps, respectively. The number of terrace atoms along a unit vector is given by m (including the atom at the step edge and in the step), and n is the number of atom layers forming the step. For the fcc(775) surface in Fig. 1.4 with step height n = 1, the notation is 7(111) × (111)̄ or 6(111) × (110), because the step orientation is ambiguous in this case. The angle φ between the orientation of the macroscopic surface (775) and the terrace (111) is calculated with the inner product [775]⋅[111] of the involved vectors cos φ = |[775]||[111]| , as φ = 8.47∘ .

10 | 1 Properties of surfaces

Figure 1.4: Steps (dark colored atoms) at an fcc(775) surface. (a) Perspective view with unit cell and (b) side view.

1.1.2 Superlattices and superstructures Many surfaces do not have the lateral periodicity, which would be expected for the cut through the bulk crystal. This might be caused by the reorganization of the bonds at the surface (reconstruction), or by the adsorption of atoms or molecules as illustrated in Fig. 1.3(d). The latter occurs in particular when evenly distributed adsorbates do not cover every substrate atom. This case corresponds to a fractional relative coverage θ, defined as the ratio of adsorbate to substrate atoms in the top layer, which is commonly given in monolayers (ML). The term superlattice is used for the resulting translational symmetry, whereas superstructure denotes the full crystallographic structure. Often the terms are used interchangeably. In the following, we illustrate the description for the case of adsorption. A reconstruction of the top layers of a bulk crystal may be treated in the same way. The most common positions of adsorbate species (adsorption sites) during adsorption are hollow sites (h), bridge sites (b), and top sites (t) (Fig. 1.5). Depending on the substrate, one has to distinguish hcp-like (h) and fcc-like (h󸀠 ) hollow sites, and short (b) or long (b󸀠 ) bridge sites. The number of bonds of the adsorbate to the top layer of the substrate is 1 for the top site and 2 for bridge sites. For hollow sites it is 3 or 4, depending on the symmetry of the top layer of the substrate. Bonds to the second substrate layer may also play a role for hollow site adsorption. Adsorbates in bridge sites lower

Figure 1.5: High-symmetry adsorption sites on (100), (110), and (111) surfaces of fcc crystals. The top site (t) is common to all surfaces. On the (110) surface, one has to distinguish short (b) and long (b󸀠 ) bridge sites. The (111) surface offers hcp-like (h) and fcc-like (h󸀠 ) hollow sites with a substrate atom right below the adsorbate in the second and third layer, respectively.

1.1 Atomic structure of surfaces | 11

Figure 1.6: Simple superlattices in Wood’s and matrix notation. The oxygen phases (a,c) on Ni(100) are really found in experiments. (b,d,e) are only used to illustrate structural differences.

the rotational symmetry of the surface in most cases. Top and hollow sites correspond to lateral positions of substrate atoms, and the symmetry is usually preserved. For the description of a superlattice the basis vectors of the adsorbate layer b1 and b2 are expressed in relation to the basis vectors of the substrate a1 and a2 . In a simple superlattice, all adsorbate species occupy equivalent sites of the substrate. This is illustrated in Fig. 1.6 for the example of oxygen on Ni(100). In (a) is b1 = 2a1 and b2 = 2a2 , thus the superlattice is denoted as (2 × 2)-O. The oxygen coverage relative to the number of atoms in the top substrate layer is 1/4. For the adsorption structure in Fig. 1.6(b), where all oxygen atoms occupy bridge instead of hollow sites, the same notation for the superlattice applies, though the superstructure is different. The superlattice notation contains no information on the structure, since it only describes the translational properties of the point lattice. Figure 1.6(c) depicts an adsorption structure for twice the oxygen coverage. The primitive unit cell of the superlattice is rotated by 45∘ with respect to the unit cell of the substrate, and the length of the vectors relative to the substrate is m = |b1 |/|a1 | = √2 and n = |b2 |/|a2 | = √2. The resulting notation for the superlattice is (√2 × √2)R45∘ -O. In this case, one may also use a centered unit cell labeled b󸀠1 , b󸀠2 , which would denote the superlattice as c(2 × 2)-O. The example in Fig. 1.6(d) shows that m and n might be different. The general case is described by Wood’s notation, S(hkl) - i(m × n)RΦ - N Ad. The substrate and its surface orientation is given by S(hkl), and Ad indicates the N adsorbate species per unit cell. The common rotation angle of b1,2 with respect to a1,2 is denoted by Φ. A centered and primitive unit cell is denoted by i = c and p, respectively. The label p may be omitted as well as the rotation angle for Φ = 0∘ . For reconstructed surfaces without adsorbate, obviously no species is given. The notation of the symmetry group of the structure may optionally be appended to Wood’s notation. Note that for Φ = 0∘ , m and n must be integer numbers. In addition, we point out that for

12 | 1 Properties of surfaces the so far considered simple superlattices the unit cell of the adsorbate layer is also the unit cell of the combined system (adsorbate + substrate), which comprises m ⋅ n unit cells of the substrate. If the rotation angles of b1 relative to a1 and b2 relative to a2 are not equal (see, for example, Fig. 1.6(e)), Wood’s notation fails. One resorts to the description of the superlattice in matrix notation, b1 = C11 a1 + C12 a2

b2 = C21 a1 + C22 a2

or

C b ( 1 ) = ( 11 b2 C21

C12 a1 a ) ( ) = C ( 1) . C22 a2 a2

The combined system of the surface with the primitive unit cell of the superlattice is described as S(hkl) - C - N Ad. The matrix C defines the relation between the basis vectors of substrate and adsorbate and all other quantities are analogous to those of Wood’s notation. Note that the matrix may depend on the choice of the type and order of the basis vectors. In the case of a simple superlattice, all elements of Cij are integer numbers and the superlattice vectors bi are the basis vectors of the combined system. The absolute value of the determinant of C gives the number of substrate unit cells, which are in the superlattice unit cell. For the superlattices shown in Fig. 1.6(a)–(d) both notations are applicable. For the superlattice in (e), only the matrix notation is given, because the vectors bi are rotated relative to ai for i = 1, 2 by different angles. However, it is possible to find a larger rectangular unit cell, which contains several adsorbate atoms. The proper Wood’s notation is (5 × 10)-5O for the structure in Fig. 1.6(e), but it does not contain the information about the relative position of the oxygen atoms in the unit cell, which is conveyed in the matrix notation. For commensurate superstructures or coincidence lattice structures the points of the ideal flat adsorbate lattice do not fall on an equivalent point of the substrate lattice. The matrix, which describes the periodicity, has at least one fractional element Cij . However, one always can find a common larger unit cell, which describes the periodicity of the combined system. This is not possible if one matrix element is irrational. In this case, one refers to an incommensurate superstructure. This definition has only formal mathematical relevance, because of the finite extent of any surface phase and the limited measurement accuracy, it is impossible to distinguish between an incommensurate and a long-range commensurate surface structure (measurements always give rational numbers). A simple example for a coincidence structure is the hypothetical adsorption phase on an fcc(100) surface shown in Fig. 1.7(a). The primitive adsorbate lattice is described by half-integer matrix elements. The combined system exhibits a (2√2×√2)R45∘ unit cell with a basis of two atoms. Even though the atoms are positioned exactly on the corners and centers of the unit cell, it is not a centered structure, because the assumed sites are crystallographically inequivalent (vertical and horizontal bridge sites). Since both sites have identical local bonding configurations, the adsorbate layer does stay flat. Such cases occur rarely, more often locally different sites are occupied as illustrated in Fig. 1.3(d) for the CO-adsorption phase on Pt(111). For the involved top and bridge sites, the chemical interaction is different and leads to different adsorption heights. Strictly taken, the lattice should not be described

1.1 Atomic structure of surfaces | 13

Figure 1.7: Examples for commensurate superlattices: (a) Hypothetical (2√2 × √2)R45∘ superlattice on an fcc(100) surface with locally identical adsorption sites. (b) (4 × 4) superstructure of Pb on Cu(111) with different adsorption sites. (c) Moiré structure of graphene on Ir(111).

by the primitive lattice of the adsorbate molecules. The interaction between adsorbate and substrate ensures that the periodicity of the substrate is impressed in the adsorbate layer and vice versa. This is exemplified for a close-packed Pb layer on Cu(111) shown in Fig. 1.7(b). Pb atoms are about one third larger than Cu atoms, and thus a coincidence structure forms, where three Pb atoms fit on four Cu atoms along close-packed directions. Due to the rather different adsorption sites of the Pb atoms (t, h, h󸀠 , and pseudo-b, cf. Fig. 1.5), the Pb layer and the top Cu layers are strongly corrugated (up to 0.26 Å [1.1]). The proper notation for this surface would be (4 × 4)-9Pb, which does not convey the information on the lateral arrangement of the atoms within the unit cell. Therefore the local distortions induced by the interaction between Pb and Cu atoms in the top layers are ignored and the matrix of the ideal adsorbate lattice is used. The matrix elements are given as fractions or decimal numbers with a number of digits representing the experimental accuracy. In some cases, Wood’s notation with fractional indices may be used, which would read as Cu(111)-(4/3 × 4/3)-Pb for the structure of Fig. 1.7(b). A special, commonly occurring case of long-range commensurate (or incommensurate) superlattices forms if lattice constants of adsorbate and substrate are only slightly different. Then, the local binding configuration of the adsorbate changes gradually from substrate unit cell to the next. A largescale moiré pattern forms as illustrated in Fig. 1.7(c) for the example of graphene (a layer of carbon in a honey-comb pattern) on Ir(111). The interaction with the substrate leads to an associated corrugation of the adsorbate layer. For the presented case of untwisted lattices does the periodicity of the moiré structure correspond to the reciprocal value of the relative lattice mismatch (in units of the lattice constant of the substrate). In Fig. 1.7(c) it is 10.7 aIr . In general, the unit cell of the moiré structure is not the unit cell of the coincidence structure.

The final topic in this section concerns domains of adsorption phases on surfaces. They may occur as different, symmetrically equivalent regions of finite extension if the symmetry of the superlattice is lower than the one of the substrate. For the example

14 | 1 Properties of surfaces

Figure 1.8: Formation of antiphase domains (framed by dashed lines) for a (2×2) adsorption structure on a square substrate.

of CO adsorption on Pt(111) (Fig. 1.3(d)), three domains exist, where the CO adlayers are rotated by 120∘ with respect to each other (rotational domains). In this case, the rotational and translational symmetries of the superlattice are reduced compared to the substrate. Figure 1.8 illustrates antiphase domains for a (2 × 2) superstructure on a square substrate, which are out of phase with respect to the (2 × 2) periodicity. Here only the translational symmetry is reduced. The antiphase domains occur, because the adsorbate structure starts growing in different places on the surface, which finally meet at the domain boundaries.

1.1.3 Surface relaxation Surface relaxation or layer relaxation describes the change of layer distances perpendicular to the surface, which occurs at any surface to various extent. The lateral translational symmetry remains unchanged in contrast to the surface reconstruction (Section 1.1.4). Figure 1.9(a),(b) demonstrates the surface relaxation using a cut perpendicular to a simple surface, which has only one atom per layer in the unit cell (e. g., an fcc(100) metal surface). Part (a) presents the bulk-like terminated surface. The spacings d and lateral lattice constants a of the layers are the same as in the bulk. The relaxation does not have to be limited to the first layer spacing, but also the spacings of deeper layers might relax (illustrated for three spacings in Fig. 1.9(a)). The change of several layer spacings is called multilayer relaxation, and the sequence of contractions and expansions forms the relaxation profile. Figure 1.9(c) shows relaxation profiles for several metal surfaces. In many cases, the relaxation profile shows alternating contractions and expansions of the layer spacings. More complicated profiles exist, e. g., if the number of nearest neighbors deviates from the bulk value for more than one layer at the surface, as is the case for the relatively open Mo(111) surface (Fig. 1.9(c)). The first layer spacing shows a contraction for most metals. For layers with dense packing of atoms, the relaxation is usually small (e. g., Ir(111) in Fig. 1.9(c)). In rare cases, as for the Pt(111) surface, one observes an expansion of the first layer spacing. The layer relaxation depends on the material and the surface orientation. The latter is demonstrated in Fig. 1.10(a)–(e) for various orientations of copper surfaces. The relative layer relaxations (di − d)/d of the two top layers are given as percentages. The size of the unit cells increases from

1.1 Atomic structure of surfaces | 15

Figure 1.9: Surface relaxation: (a) shows the bulk-like terminated, unrelaxed surface, where lateral lattice constant a and vertical layer spacing d have bulk values. (b) illustrates the case of vertical layer relaxation, where the first three layer spacings have changed (strongly exaggerated scale). (c) presents experimental values for layer relaxations determined by LEED for various metal surfaces [1.2, 1.3].

(a) to (e). The unit cell contains one copper atom per layer in all cases, accordingly the surfaces become more open, and the layer spacing decreases. The relative (and, in general, also the absolute) values of the layer relaxation increases with increasing openness of the surface. This is attributed to the decreasing number of nearest neighbors of atoms in the outermost layer(s) (Fig. 1.10(a)–(e)). Figure 1.10(f) illustrates the Smoluchowski smoothing, which provides a simple explanation for the origin of the layer relaxation. The atomic corrugation of the surface causes a corrugation of the electron density. Within the model of a free-electron gas, this corrugation attempts to be reduced by a smoothing of the charge distribution moving the center of the electronic charge density closer to the surface. The electrostatic attraction between the electrons and positively charged atom cores leads to a concomitant shift of the surface atoms toward the surface, which means a reduction of the first layer spacing. The effect increases with the atomic corrugation of the bulk-like terminated surface, i. e., the openness of the surface (in agreement with experiment). The model obviously cannot explain the rare situation of an expansion of the first layer spacing. A refinement of the model of redistribution of electronic charges can explain also the relaxation of deeper layers and finds an alternating relaxation profile, as shown in Fig. 1.9(c) for the Ir and Rh surfaces.

Pure lateral shifts (registry shifts) are also possible without changing the translational symmetry. This occurs in particular for high-index surfaces with corresponding low symmetry (missing mirror planes). For surfaces with more than one atom per unit cell, e. g., alloys or, in general, compounds with layers formed by different atomic species, intralayer relaxations within the unit cell can occur without changing the lateral translational symmetry of the surface. Figure 1.10(g) shows the side view of the CoAl(110)

16 | 1 Properties of surfaces

Figure 1.10: (a–e) Top view on the atomic structure of various copper surfaces with increasing openness and unit cell area. The relative relaxation (as a percentage) for the first two layer spacings is given, and below, the corresponding numbers of nearest neighbors [1.4]. (f) illustrates the Smoluchowski smoothing. (g) shows a cut through the first three layers of the (110) surface of a CoAl alloy with layers containing Co and Al. (h) Co and Al show different (intra-)layer relaxations within the unit cell. The size of the unit cell remains unchanged, and the associated layer corrugation amounts to 0.18 Å.

surface. The bulk layers contain equal amounts of Co and Al atoms, which can be arranged into sublayers of the two elements. These sublayers can have different vertical layer relaxations at the surface. Their vertical separation is called buckling, and amounts to 0.18 Å in the top layer of CoAl (Fig. 1.10(h), [1.5]). Another possibility (which does not occur in CoAl) are lateral intralayer relaxations without changing the lateral periodicity. In general, both vertical and lateral intralayer relaxations can be found together. Layer relaxations are commonly reduced by adsorption of other atoms (adsorbateinduced relaxation), which can be qualitatively explained by the (partial) reconstitution of the cut bonds at the surface. This is illustrated in Fig. 1.11(a),(b). The clean Rh(110) surface has a relative contraction of the first layer spacing of 6.9 %, which is reduced to 1.3 % after adsorption of a full monolayer of hydrogen [1.6]. Electronegative adsorbates may even convert a layer contraction into an expansion. For an incomplete coverage of the surface by an adsorbate (Fig. 1.11(c)), the reduction of the layer spacing occurs only locally around the adsorption site. The layer buckles and one obtains an adsorbate-induced reconstruction of the substrate surface, i. e., the top layers of the substrate assume the lateral periodicity of the adsorbate layer. This phenomenon leads to the topic of surface reconstruction in the following section.

1.1 Atomic structure of surfaces | 17

Figure 1.11: (a) Layer relaxation of a clean surface, which is reduced by a uniform adsorbate layer (b). For an incomplete coverage (c) the influence of the adsorbate is only local, which also leads to a change of the lateral periodicity, i. e., a reconstruction of the substrate.

Figure 1.12: Various types of reconstructions: (a) displacive, (b) bond-breaking, and (c) bond-forming reconstruction. The lattice constants of the bulk layers and the reconstructed surface are denoted by a and b, respectively.

1.1.4 Surface reconstruction Surface reconstruction describes the situation of a change in the lateral translational symmetry of the surface compared the bulk-like termination. In contrast to a pure relaxation, the unit cell has a different size and in the general case also a different shape. The frequently occurring reconstructions of clean surfaces (without adsorbates) are often driven by the formation of new bonds (e. g., saturation of broken bonds at semiconductor surfaces), or the breakage of bonds accompanied by the formation of microfacets, which finally lead to a lowering of the surface energy. Surface reconstructions can be altered or completely lifted by adsorbates with their additional bonds and resulting new energy balance. Figure 1.12 presents some simple cases for surface reconstructions. In (a) atoms at the surface are only shifted, whereas the atomic density stays constant. This is called a displacive reconstruction. In the other cases, the atomic density in the top layer changes, when bonds are broken (b) or new bonds are formed (c). The different cases may also be combined, i. e., in a large unit cell atomic displacements, bond breaking and formation of new bonds can occur in different places (see Fig. 1.16). In Fig. 1.12, atomic displacements only in the top layer are sketched. In general, reconstructions extend also to deeper layers (see Fig. 4.10(b)). Figure 1.13 shows the displacive reconstruction of the W(100) surface, which is observed below room temperature. As indicated by arrows in (a) on the bulk-like terminated surface, the atoms of the top layer are shifted in a zig-zag pattern (by about 0.2 Å). This leads, as indicated in (b), to an enlarged c (2 × 2) surface unit cell [1.7].

18 | 1 Properties of surfaces

Figure 1.13: (a) Bulk-like terminated W(100) surface (top view). The arrows indicate the displacements of the atoms, which lead to the zig-zag reconstruction in (b). One mirror line remains (solid line), and a new glide line (dashed) appears.

Figure 1.14: (a) Bulk unit cell of silicon and (b) bulk-like terminated Si(100) surface (side view). (c) The formation of dimer bonds (green) leads to a doubling of the lattice constant in the [011]̄ direction, and a reconstructed Si(100)-(2 × 1) surface results.

The symmetry is reduced from p4mm of the unreconstructed surface to p2mg for the reconstructed surface. Precise structural investigations revealed that also the atoms in the second layer are shifted by a substantially smaller amount [1.8]. The driving force for the zig-zag reconstruction could be the formation of closer nearest-neighbor bonds in the top layer. Displacive reconstructions are more frequently found at surfaces of semiconductors than of metals. Figure 1.14 illustrates this for the example of the Si(100) surface. (a) shows the bulk unit cell of Si, and (b) presents the unreconstructed, i. e., bulk-like terminated (100) surface in perspective view. Each of the surface atoms has two unsaturated dangling bonds. (c) illustrates the resulting reconstruction, in which pairs of neighboring atoms have shifted to form dimers. Half of the dangling bonds are now saturated, and an energetically more favorable state results. Compared to a bulk layer, the unit cell has doubled in one direction, and one obtains a (2 × 1) reconstruction. A more precise structural analysis reveals that the dimers are slightly asymmetric and are arranged in alternating orientation (Fig. 1.24(a)), leading to a further doubling of the unit cell size. In Section 1.2.4, the electronic states associated with the reconstructed surface are presented. Examples for bond-breaking and bond-forming reconstructions, which lead to a change of the atom density in the surface layer, are presented in Fig. 1.15. The upper

1.1 Atomic structure of surfaces | 19

Figure 1.15: (a) Bulk-like terminated Pt(110) surface and (1 × 2)-reconstructed surface in perspective (b), and side view (c). The arrows in (c) indicate the shifts of the atoms in the second, third, and fourth layer. (d) Top view on the bulk-like terminated Ir(100) surface and (e) the reconstruction phase Ir(100)-(5 × 1), where the atoms in the top layer have arranged in a quasi-hexagonal close-packed structure. The layer is buckled in the vertical direction. The quasi-hexagonal unit cell corresponding to the hypothetical layer without buckling is marked by dashed lines.

row shows the bond-breaking reconstruction of the (110) surface of Pt. (a) depicts the bulk-like terminated surface, (b) the reconstructed phase. Every second close-packed atom row is removed, giving this reconstruction the name missing-row structure. Perpendicular to the close-packed rows, the periodicity has doubled, which results in the notation Pt(110)-(1 × 2). The bond-breaking leads to a reduced atom density in the top layer. The side view (c) illustrates the zig-zag microfaceting of the structure marked by the dashed line. The arrows indicate the shifts of the atoms in the deeper layers, which result from the removal of the missing rows. The (111)-oriented facets have a dense atomic packing, which is energetically favorable. A reconstruction with the formation of new bonds is presented in the lower row of Fig. 1.15. The bulk-like terminated surface of Ir(100) is shown in (d). On the reconstructed surface (e) a hexagonal close-packed arrangement of atoms has formed in the surface layer. The atom density of the hexagonal layer is increased by 20 % compared to the bulk layers. The hexagon is slightly distorted (quasi-hexagonal), so that the lateral positions of the surface atoms (registry) repeat after five lattice constants with respect to the underlying Ir(100) substrate (a typical coincidence structure). The fivefold periodicity in the respective direction leads to the notation Ir(100)-(5 × 1). The interaction of the surface layer with the layer underneath causes it to slightly reconstruct as well. This continues down to the fourth layer with decreasing amplitude (see Fig. 4.10(b)). The quasi-hexagonal arrangement of the surface atoms in Fig. 1.15(e) can be rotated by 90∘ . Thus, there exist two energetically equivalent, orthogonal rotational domains for this reconstruction (see Figs. 4.7(b) and 4.10(a)). Within the hexagonal layer, more mutual bonds exist, which is energetically favorable. On the other hand, it requires the binding to the square substrate layer via top and bridge sites instead of the preferred hollow sites, which costs energy, and leads to a buckling of the hexagonal layer.

20 | 1 Properties of surfaces The question remains where the atoms go to or come from in a reconstruction process, which changes the atom density in the surface layer. The atoms can diffuse on the surface (Section 2.2.3), and excess atoms bind at steps on the surface (increasing the number of bonds compared to the diffusing single atom), whereas extra atoms are removed from steps (reduced binding at steps, due to the lower number of bonds compared the atoms in the surface layer). Steps are unavoidable, because a perfect cut parallel to a (low-index) crystallographic plane cannot be achieved in most situations. The various types of reconstructions can occur concurrently, and in general are accompanied by layer relaxations. The presumably most complicated reconstruction, quantitatively analyzed by experiment, is the case of the Si(111) surface. The uppermost Si bilayer (a bilayer are two closely spaced atom layers) shows drastic structural modifications, as illustrated in Fig. 1.16, and the unit cell is increased (7 × 7)-fold compared to the bulk-like terminated surface. A characteristic of the structure are the large corner holes, where the atoms of the top bilayer are missing. In the right half of the unit cell, the top bilayer is stacked as in the bulk (unfaulted stacking), whereas in the other half, the top bilayer is laterally displaced (faulted stacking). The connection between faulted and unfaulted areas occurs via rows of dimers. Several additional silicon atoms (adatoms) saturate the dangling bonds of atoms underneath. This cannot be achieved for all atoms, and the remaining atoms are called rest atoms. In total, the number of unsaturated energetically unfavorable bonds in the unit cell is reduced from 49 (one for each surface atom in the (7 × 7) unit cell) to 19 by the reconstruction. The impact

Figure 1.16: (7 × 7) reconstruction of the Si(111) surface. The top view (upper part) presents only the top three atomic layers. The side view (lower part) shows an additional three layers to point out the stacking sequence emphasized by strong lines.

1.1 Atomic structure of surfaces | 21

of the reconstruction on the electronic structure at the surface is discussed in Section 1.2.4. An interesting question may be posed, why the bonds on the Si(111) surface cannot be saturated by dimer formation just as on the Si(100) surface? After all, each atom on the bulk-like terminated Si(111) surface has only one dangling bond compared to two on the Si(100) surface. Indeed, the cleaved (111) surface of Si shows a (2 × 1) reconstruction, which is attributed to π-bonded chains of surface atoms. However, the bond configuration of the chain atoms is markedly different from the tetrahedral bulk coordination. As a result, the (7 × 7) reconstruction is energetically more favorable in spite of the incomplete saturation of the dangling bonds. The (2 × 1)-reconstructed cleaved surface is metastable, and transforms after annealing above 200∘ C into the (7 × 7) reconstructed surface. At surfaces of alloys (or more general solid compounds) the chemical degree of freedom provides a further type of reconstruction. For disordered alloys with a statistical distribution of the various chemical elements on given crystallographic sites, only an averaged unit cell exists in each surface layer of the bulk-like terminated unit cell, which is given by the atomic positions disregarding the chemical identity. However, one of the components can segregate at the surface and reach a higher concentration than in the bulk, if that reduces the surface energy. This enrichment may lead to a chemical ordering at the surface in addition to the positional order. The unit cell is thus increased by this chemical reconstruction. Figure 1.17(a) exemplifies this effect for the case of the (100) surface of the disordered alloy Fe0.97 Al0.03 , which is formed by alloying an iron crystal with 3 % of aluminum. The average bulk unit cell is marked in the second layer by strong lines. In the reconstructed phase, Al atoms segregate from the bulk to the surface and form, together with Fe atoms, an ordered surface layer with an enlarged c(2 × 2) unit cell. At surfaces of ordered alloys, segregation can also lead to a chemical reconstruction. Figure 1.17(b),(c) illustrates this for the (100) surface of the alloy (or compound) NiAl, which crystallizes in the cesium chloride structure. (b) presents the bulk-like terminated surface, where pure Al and Ni layers are alternating. The Al termination shown is energetically favorable for the ideal 1:1 stoichiometry of the compound. However, such a perfect stoichiometry can never be achieved, and there always exists a tiny excess of one component, which depends on the preparation conditions. Excess Ni atoms

Figure 1.17: Chemical reconstruction at disordered (a) and ordered (b,c) alloy surfaces. (a) shows a section of the (100) surface the compositionally disordered alloy Fe0.97 Al0.03 . (b) and (c) illustrate the chemical reconstruction at the ordered NiAl(100) surface.

22 | 1 Properties of surfaces

have to occupy sites on the Al sublattice, where they have only Ni neighbors, which is energetically unfavorable compared to Al neighbors in the perfect structure. The energy is lowered if Ni atoms segregate to the surface, and thereby reduce the number of unfavorable bonds. For the case of the NiAl(100) surface [1.9], a chemical reconstruction with a c(√2 × 3√2)R45∘ surface periodicity is formed (Fig. 1.17(c)).

Reconstructions can also be induced by adsorbates (adsorbate-induced reconstruction). Figure 1.18 presents the example of the carbon-induced reconstruction of Ni(100) with (a) showing the unreconstructed surface. Upon the adsorption of half a monolayer, the carbon atoms occupy hollow sites. (b) illustrates how the C atoms induce a rotation of the four neighboring Ni atoms in diagonal direction, alternating for neighboring C atoms (clock-anticlock reconstruction). The Ni atoms are shifted within the top layer by 0.45 Å such that they are coordinated with 5 instead of 4 Ni neighbors, plus 2 C atoms. The symmetry changes from p4mm for Ni(100) to p4mg for the surface covered by carbon with the corresponding notation Ni(100)-(2 × 2)-2C(p4mg). Adsorbates can also change existing reconstructions (reconstruction switch), or lift them completely. An example is provided by the reconstruction of the W(100) surface, which was presented in Fig. 1.13(b). After the adsorption of half a monolayer of hydrogen atoms, the zig-zag reconstruction is transformed into a phase with the same periodicity. The W atoms in the top layer now form dimers, which are bound by hydrogen atoms (Fig. 1.18(c)). After saturation of the surface by H atoms, the reconstruction is completely lifted. One finds two H atoms per W surface atom, which occupy all available bridge sites between the W atoms (Fig. 1.18(d)). In summary, we have demonstrated that surfaces exploit all structural degrees of freedom, which are opened by breaking the bulk symmetry at the surface. This comprises simple relaxations up to drastic reconstructions, including positional and chemical reordering. Adsorbates often do not simply occupy the obvious sites offered by the substrate, but can induce, change, or lift reconstructions. A flat surface can be unstable and split into regions of another orientation (facets). Existing steps may

Figure 1.18: (a) Top view on the bulk-like terminated (100) surface of Ni. (b) Reconstruction induced by carbon adsorption. The remaining mirror line is drawn as solid line and the new glide line as dashed line. The modification of the reconstruction of the W(100) surface (Fig. 1.13) by adsorption of hydrogen is illustrated in (c) for one H atom per two W surface atoms, and in (d) for the saturation coverage of two H atoms per one W surface atom [1.8].

1.2 Electronic structure

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wander and join other steps (step bunching). A surface can also roughen by developing height fluctuations (surface roughening). If a surface is used as substrate for the growth of another material, it may assume for a few monolayers the lattice parameter of the substrate. For thicker films, the increasing tension can be reduced by the formation of nanostructures (Section 2.3.4). Thus, the appearance of surfaces is full of surprises.

1.2 Electronic structure The termination of a solid by a surface changes the binding configuration of the surface atoms compared to the atoms in the bulk. This is the cause for the reordering of the atoms discussed in Chap. 1.1, which expresses itself, e. g., in relaxation or reconstruction of the surface. The forces between the atoms are conveyed by the electrons, which have to be treated by quantum mechanics. The thermodynamically stable configuration is reached by minimization of the total energy of the system. It can be determined in theoretical calculations, e. g., based on density functional theory, by variation of the structural parameters. Experimentally, a surface commonly can be investigated only in an energetically favorable configuration. How the atoms reach these favorite sites is discussed in Chap. 2. The complete electronic structure of a surface comprises the knowledge of the energy of all eigenstates depending on the quantum numbers, determined by the symmetry of the system. For periodically ordered surfaces, the wave vector parallel to the surface k ‖ is the relevant quantum number, and thus the most important conserved quantity. The dependence of the energy of the states on the wave vector E(k ‖ ) is called surface band structure. At the Fermi energy EF , one obtains the Fermi contour E(k ‖ ) = EF (Fermi surface in three dimensions), which is relevant for all properties related to electron transport. But also the excited states (electron E > EF and hole excitations E < EF ) are important for the formation of chemical bonds of adsorbates. In the following, we first concentrate on the basic general consequences of the periodicity of the surface for the surface band structure E(k ‖ ). We will see that for certain values E(k ‖ ), no electronic states of the bulk crystal exist. However, in these energy ranges, surface states may exist, which have a probability density concentrated at the surface. After the discussion of various types of surface states, we turn at the end of this section to the electronic work function. It can yield essential information on the surface, in particular, depending on the preparation conditions.

1.2.1 Bloch’s theorem A periodically ordered surface is described by two basis vectors, a1 and a2 , of the lattice. The general translation t = r mn = ma1 + na2 (eq. (1.1)) with integer numbers m

24 | 1 Properties of surfaces and n connects each point of the crystal to an equivalent point. Consequently, the potential V(r), which the electrons experience in two dimensions, is also periodic: V(r) = V(r + t). This applies obviously to the surface region, but is also true for the substrate underneath (cf. Fig. 1.2) and the region far above the surface, where the potential has the constant value of the vacuum energy EV . The wave functions ψ(r) of the electrons with mass m have to obey the Schrödinger equation, {−

ℏ2 Δ + V(r)}ψ(r) = E ψ(r). 2m

Let ψ(r) be a solution of the Schrödinger equation with energy E. Because of the two-dimensional periodicity of the potential and the translational invariance of the Laplace operator Δ, all wave functions ψ(r + t) are also solutions to the same energy E. Since this holds for arbitrary translations t, all solutions for different t have to be linearly dependent. Disregarding degeneracies (accidental or due to additional symmetries), the solutions can differ only by a complex factor with absolute value 1, which may depend on t. These considerations lead to Bloch’s theorem in two dimensions, ψ(r + t) = ψ(r) eik‖ ⋅t .

(1.2)

It asserts that the solutions ψ(r +t) are linearly dependent and differ only by a complex phase factor eik‖ ⋅t with absolute value 1. The most important consequence of Bloch’s theorem is that it is sufficient to know the wave function in a single unit cell, since the value in any other unit cell is uniquely defined by the phase factor. The wave functions are labeled by two quantum numbers, the two components of the wave vectors k ‖ . To emphasize this property Bloch wave functions are often written as ψk‖ (r) or ψ(k ‖ , r). The wave vectors k ‖ are the quantum numbers associated with the translational symmetry parallel to the surface. We will see later that the wave vectors are also experimentally accessible (Chap. 5.3). The surface band structure is given by the dependence of the energy on the wave vector E(k ‖ ), which is a continuous function of k ‖ . For each value of the wave vector k ‖ , the Schrödinger equation yields several solutions for the energy E, which correspond to different bands. In Bloch’s theorem (eq. (1.2)), the wave vector k ‖ appears in the exponential function exp(ik ‖ ⋅ t). One can find reciprocal lattice vectors g, for which g ⋅ t is an integer multiple of 2π, and thus exp(ig ⋅ t) = 1. The reciprocal lattice vectors form a twodimensional lattice with the basis vectors g 1 and g 2 in reciprocal space; this will be discussed in more detail in Chap. 4. From exp(ig ⋅ t) = 1, one readily obtains that k ‖ and k ‖ + g are equivalent, i. e., they yield the same phase factor in Bloch’s theorem (eq. (1.2)). Consequently, E(k ‖ ) = E(k ‖ + g), and the energy is a periodic function with the periodicity of the reciprocal lattice, which is spanned by reciprocal lattice vectors. This property will be exploited in the following Section 1.2.2 for the introduction of the surface Brillouin zone.

1.2 Electronic structure

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To become more familiar with the Bloch wave functions and reciprocal lattice vectors, we rewrite ψ(r) in eq. (1.2) as ψ(r) = exp(ik ‖ ⋅ r ‖ ) χ(r). It is straightforward to verify by substitution that χ(r) is a periodic function χ(r) = χ(r + t). Since χ(r) is a periodic function parallel to the surface, we can make the following ansatz of a discrete Fourier series: χ(r) = ∑ χg (z) eig⋅r‖ . g

Here the z direction is chosen perpendicular to the surface, and r ‖ is thus a vector in the xy plane. The proof follows from replacing r by r + t and using exp(ig ⋅ t) = 1. This ansatz corresponds to a Fourier expansion with the two-dimensional, reciprocal lattice vectors g parallel to the surface. The z dependence of the wave function is contained in Fourier coefficients χg (z). The Bloch wave functions can now be written as k

ψk‖ (r) = ∑ χg ‖ (z)ei(k‖ +g)⋅r‖ . g

(1.3)

In this expansion of the wave function into plane waves parallel to the surface, the dependence on the wave vectors k ‖ is accentuated. In principle, we obtained an exact solution of the Schrödinger equation by expanding the wave function into the complete basis system of plane waves parallel to the surface. In practice, this ansatz is only of limited value, since one would have to employ a great many plane waves (or reciprocal lattice vectors). The surface constitutes the border between the solid and the vacuum. In the vacuum region, sufficiently far away from the surface, an expansion of the wave function into plane waves is best suited. For an expansion of the wave function in the bulk of the solid, the corresponding bulk Bloch wave functions are an appropriate basis. These are characterized by the three-dimensional wave vectors K. To find out which wave vectors K contribute to a given k ‖ , we have to consider the surface Brillouin zone and its relation to the bulk Brillouin zone. This connection leads to the projected bulk band structure, which yields clear-cut criteria for the occurrence of surface states.

1.2.2 Surface Brillouin zone In the previous section, we learned that k ‖ and k ‖ + g are equivalent, since g ⋅ t is an integer multiple of 2π. Therefore, for the surface band structure E(k ‖ ), one does not need to cover the whole reciprocal space (k space), but a restriction on a finite region (area in two-dimensional reciprocal space) is sufficient. It is convenient to consider the shortest k ‖ vectors. This unique area is called the surface Brillouin zone, analogous to the bulk Brillouin zone in three dimensions. In Fig. 1.19, the construction is

26 | 1 Properties of surfaces

Figure 1.19: Reciprocal lattice and construction of the surface Brillouin zone (shaded) for the five Bravais lattices. The unit cells spanned by the reciprocal basis vectors g and g are drawn in blue. 1

2

illustrated for the five two-dimensional reciprocal lattices (Table 1.1). One draws the perpendicular bisectors to the nearest points g of the reciprocal lattice. The smallest area enclosed by the bisector lines is shaded in Fig. 1.19 and forms the surface Brillouin zone. For rectangular or square lattices, the surface Brillouin zone is also rectangular or square, respectively. In the other cases, the surface Brillouin zone is delimited by six edges. The surface Brillouin zones clearly show the symmetry of the underlying crystal lattices, which is not necessarily the case for the unit cells spanned by the basis vectors. In an extended zone scheme do the surface Brillouin zones cover the whole two-dimensional plane in reciprocal space, and thus have the same area as a primitive unit cell of the reciprocal lattice. Figure 1.20 shows the surface Brillouin zones for low-index surfaces of the facecentered cubic lattice. For the (001) surface, the surface Brillouin zone is square, and for the (111) surface, hexagonal. Corresponding figures for the (110) surface and the body-centered cubic and the hexagonal close-packed lattices are collected in the appendix (Fig. A.2). Special points and lines in the surface Brillouin zone are usually marked with an overbar to distinguish them from the special points and lines of the bulk Brillouin zone. The origin k ‖ = 0 is always denoted by Γ, in analogy to the Γ point at the center of the bulk Brillouin zone. Each point of the surface Brillouin zone corresponds to a line of the threedimensional bulk Brillouin zone of the substrate crystal, which runs parallel to the surface normal. For special points of the surface Brillouin zone, this connection is illustrated by vertical lines in Fig. 1.20. This relation between surface Brillouin zone and bulk Brillouin zone makes it evident that all bulk states, with K vectors along a line parallel to the surface normal, are associated with states of the surface band

1.2 Electronic structure

| 27

Figure 1.20: Surface and bulk Brillouin zone for the (001) and (111) surfaces of the face-centered cubic lattice. The blue shaded areas indicate mirror planes.

structure at the wave vector k ‖ . The relevance of this relationship for the existence of surface states will be elucidated in Section 1.2.3. Note that for low-index surfaces, many high-symmetry points of the bulk band structure are associated with special points of one or more surface Brillouin zones. For example, the X point of the fcc-bulk Brillouin zone contributes to the Γ and M point of the (001) surface Brillouin zone, as well as to the M point of the (111) surface Brillouin zone. In Section 1.1.1, additional symmetry elements, such as mirror planes, were discussed. The symmetry in real space carries over to the reciprocal space. In Fig. 1.20, mirror planes are indicated by blue shading. The mirror symmetry only exists if the basis of the lattice also has the respective symmetry. This is obviously the case if each lattice point is associated with one atom only. States with a k ‖ vector in a mirror plane can be classified according to the additional symmetry (Section 5.3.4). The wave functions have then either even or odd symmetry with respect to reflection by the mirror plane. For ̄ mirror planes are shown in an the face-centered and body-centered cubic lattices, the (001) and (110) extended zone scheme in the appendix (Fig. A.3). For low-index surfaces (Fig. A.2) or stepped surfaces (Figs. 1.4 and 5.20(b)) with simple step configurations the normal vectors often lie in mirror planes.

1.2.3 Projected bulk band structure At the end of Section 1.2.1, we obtained a solution of the Schrödinger equation by an expansion of plane wave parallel to the surface. The ansatz, according to eq. (1.3), is in principle exact if the sum runs over an infinite number of reciprocal lattice vectors. The k

difficulty lies in the calculation of the functions χg ‖ (z), which depend on two continuous variables, z and k ‖ . The potential has for large z—far away from the surface—the constant value EV (vacuum energy), and the wave functions are plane waves or decay exponentially. The solutions for negative z—in the bulk—are the Bloch wave functions

28 | 1 Properties of surfaces

Figure 1.21: Matching of the wave functions in the crystal and the vacuum region at the surface. Four cases are illustrated with energies E within and outside the band gap of the projected bulk band structure, and above and below the vacuum energy EV .

of the bulk band structure, which propagate through the whole crystal. In certain energy regions (band gaps), no Bloch waves exist, but solutions, which are localized at the surface and decay exponentially into the bulk. These are surface states. In a one-dimensional model for a given k ‖ , four different alternatives exist for the wave functions in crystal and vacuum. These are sketched in Fig. 1.21: – Propagating wave functions in crystal and vacuum. In the three-dimensional case, this corresponds to the transmission and reflection of electrons, as observed for the diffraction of low-energy electrons (Chap. 4.2). – Exponentially decaying wave function in the crystal, propagating solution in vacuum. This corresponds to a high reflectivity for incident electron waves. – Exponentially decaying wave functions in crystal and vacuum, which constitute surface states. – Propagating wave function in crystal, exponentially decaying solution in vacuum. This corresponds to a bound state in the crystal. If the wave function is enhanced at the surface, it is termed a surface resonance. For the existence of surface states band gaps are required. Absolute band gaps at the Fermi energy exist for semiconductors or insulators. But also in metals, energy regions can exist for some surface orientations and certain k ‖ , in which no Bloch-wave solutions are found in the bulk band structure E(K) = E(k ‖ , Kz ) for any Kz (relative band gaps). The existence of the surface changes the boundary conditions and solutions are allowed, which decay exponentially into the bulk. These can be described by complex K̃ z vectors, as illustrated in Fig. 1.22(a). A model calculation for nearly-free electrons was performed with a periodic potential of amplitude 4|V|. At the zone boundary, Re(K̃ z ) = G/2, a band gap of size 2|V| opens. In the energy range of this band gap, complex solutions exist. The real part is Re(K̃ z ) = G/2, as for the solutions with real-

1.2 Electronic structure

| 29

Figure 1.22: (a) Complex band structure E(K̃z ) within a two-band model. For the particular energies the real part Re (K̃z ) is plotted in brown, the imaginary part Im (K̃z ) in blue. The red curves show the dispersion of free electrons. (b) Bulk band structure of copper along several directions, which correspond to special points of the surface Brillouin zone of low-index surfaces (Fig. 1.20). The bands in the shaded regions contribute to the projected bulk band structure at the corresponding points.

valued Kz at the edges of the band gap. The imaginary part Im(K̃ z ) has a maximum near the center of the band gap and vanishes outside band gap. To facilitate identifying the relative band gaps, the calculated bulk band structure of copper is shown in Fig. 1.22(b) for several directions in reciprocal space. The bands along the ΓX line contribute to the Γ point of the surface band structure of Cu(001). One recognizes the flat d bands in the energy range between −4 and −2 eV, as well as the strongly dispersing sp-bands. The latter have nearly-free electron character and their dispersion resembles the results of the model calculation shown in Fig. 1.22(a). In the energy range from 2 to 7 eV, no bands are found along the ΓX line independent of the wave vector. In this relative band gap, image-potential states (Fig. 1.25(b)) can exist, since the vacuum energy EV of Cu(001) is about 4.5 eV above the Fermi energy EF . The band gaps depend on the wave vector k ‖ , as illustrated in Fig. 1.22(b) for the Γ, X and M points of the (001) surface. Figure 1.23 shows this dependence on k‖ in the projected bulk band structure of Cu(001) along the MΓX path of the surface Brillouin zone (Fig. 1.20). In the shaded regions bulk bands exist, thus surface states can only occur in the white areas. The Γ point derives from the ΓX line, which is shown in Fig. 1.22(b), and has a relative band gap between 2 and 7 eV. The M point corresponds to the XW line of the bulk band structure, and the X point connects adjacent L points while crossing the ΓK line (Fig. 1.20). The relevant bands of the bulk band structure are

30 | 1 Properties of surfaces

Figure 1.23: Projected bulk band structure of the Cu(001) surface along the Γ M and Γ X directions of the surface Brillouin zone (Fig. 1.20). The green lines show surface and image-potential states in the band gaps and a surface resonance.

plotted in Fig. 1.22(b). In the shaded regions bulk bands exist, such that the white areas correspond to the band gaps of the projected bulk band structure at the respective points of the surface Brillouin zone. Knowing the bulk bands, which contribute to a certain point of the surface Brillouin zone, is decisive for determining the energy regions in which surface states may be located. If states are identified in these regions by experiment or theory, they must be surface states. Most of the properties of the bulk bands at the band edges of the projected bulk band structure are carried over to the energetically close-by surface states. This becomes perspicuous if one regards the surface as a perturbation of the crystal potential. In this picture, the surface states are split off from the energetically close-by bulk states.

1.2.4 Surface states The band gaps in the projected bulk band structure indicate the regions in which surface states may occur. The band gaps obviously depend on the orientation of the surface. The detailed properties of the surface states are determined by the type of substrate and possible adsorbates. Free electrons in metals lead to surface states with characteristics which are distinguished from those associated with more localized bonds at semiconductor surfaces. The long-range screening of electrons in front of the surface generates image-potential states. The states mentioned so far are called intrinsic surface states in contrast to extrinsic surface states, which are created by a modification of the surface by adsorbates. In thin metal films, quantum-well states may

1.2 Electronic structure

| 31

Figure 1.24: (a) Bonds between surface atoms at the c(4 × 2)-reconstructed Si(100) surface. The side view (top) shows, in addition, the occupied and unoccupied dangling bond orbitals Dup and Ddown at the upper and lower dimer atom, respectively. The top view (bottom) indicates the unit cell resulting from the alternating tilt of neighboring asymmetric dimers. (b) Corresponding surface band structure and projected bulk band structure (shaded) [1.11].

form. Adsorbates will on the one hand modify (or quench) the intrinsic surface states by bonding to the substrate or, on the other hand, contribute new extrinsic states derived from their own electronic (atomic or molecular) levels. 1.2.4.1 Metals For the description of metals, the nearly-free electron approximation (Fig. 1.22(a)) is particularly well suited. The boundary conditions of the potential at the surface determine if and how many surface states exist. For the Cu(001) surface, the surface states are plotted as green lines in the projected bulk band structure of Fig. 1.23. In the vicinity of the X point, two intrinsic surface states with positive effective electron mass occur, which disperse upward in energy with increasing distance of the wave vector k‖ from the X point. A surface resonance is formed at the Γ point. At the M point, another intrinsic surface state is found at an energy of −1.8 eV with negative effective electron mass, which is slightly above the d bands of copper. For historical reasons, the latter state is called a Tamm state, whereas the other states are named Shockley states. 1.2.4.2 Semiconductors The covalent, directional bonds of semiconductors are more appropriately described in the picture of localized orbitals. On cutting a semiconductor crystal in a gedankenexperiment in two halves, one finds on the cut faces a large number of unsaturated bonds (dangling bonds). The energy gain from pairing two unsaturated bonds leads

32 | 1 Properties of surfaces to a rearrangement of atoms, and to sometimes drastic reconstructions at semiconductor surfaces (Fig. 1.16). As examples, we consider the reconstructed (100) and (111) surfaces of silicon. How the reconstruction of the Si(100) surface comes about was illustrated in Fig. 1.14. Figure 1.24(a) shows the associated orbital structure. Two neighboring surface atoms join in a dimer bond, whose orbitals lie in the energy range of the bulk bands. This cuts the number of unsaturated bonds in half. An asymmetric tilt of the dimers forms a doubly occupied orbital Dup at the upper dimer atom and an unoccupied orbital Ddown at the lower dimer atom. The alternating tilt of the dimers leads to a further small energy gain, and a c(4 × 2) reconstruction develops. To understand the electronic structure, it is important to count the number of states per unit cell: There are two dimers per c(4 × 2) unit cell, whose dangling bonds lead to the formation of one band each. Correspondingly, one finds in the surface band structure of Fig. 1.24(b), two occupied and two unoccupied surface bands Dup and Ddown , respectively. The surface bands are separated by a band gap, which has about half the value of the bulk band gap of silicon. Note that the valence band maximum of Si lies at Γ, and the conduction band minimum on the ΓX line of the bulk band structure. Both contribute to the Γ point of the projected bulk band structure of the Si(100) surface. The ̄ direction compared to dispersion of the surface bands is more pronounced in the [011] the [011] direction. This is attributed to the smaller distance between the dimers and the resulting larger overlap of the atomic wave functions along the rows, in contrast to the situation between the rows (Fig. 1.24(a)). The reconstruction of the Si(111)-(7 × 7) surface (Fig. 1.16) reduces the number of unsaturated bonds from 49 to 19. Accordingly, 19 surface bands would form, which are to a good part rather close in energy. Since each band can be occupied by two electrons with opposite spin orientation, there must be at least one half-filled band, and the Si(111)-(7 × 7) surface is (in a single-electron model) metallic. 1.2.4.3 Image-potential states A special class of surface states is formed by the image-potential states, which are generated if one brings an electron in front of a metal surface. A redistribution of the charges at the metal surface occurs, which ensures that the electric field lines are perpendicular to the surface. An identical field in front of the surface would be obtained if one replaces the surface charge density of the metal by an image charge. The image charge has the opposite charge and is positioned in the metal at the same distance |z| from the surface as the electron. Parallel to the surface, the electron can move freely, since the image charge follows the electron. The attractive force between electron and image charge can be obtained from the derivative of the image potential, which approaches for large distances the value of the vacuum energy EV : V(z) = EV −

e2 1 . 4πε0 4z

1.2 Electronic structure

| 33

The Schrödinger equation for this one-dimensional Coulomb potential corresponds to the radial equation of the hydrogen atom with a length scale expanded by a factor of 4. The solutions yield the energies En = EV −

13.6 eV 0.85 eV = EV − 16n2 n2

n = 1, 2, . . .

The infinite series of image-potential states results from the long-range Coulomb-like potential in front of the surface, and is distinct from the previously discussed surface states. The latter have wave functions, which decay exponentially into the vacuum (Fig. 1.21), as obtained for an abrupt potential step at the surface. Since these (Shockley) surface states are close to the Fermi energy, the image potential increases steeply (Fig. 1.25(b)), which can be approximated by a step-like potential. The surface of the crystal with a band gap in the projected bulk band structure is described by a continuously differentiable match of the wave functions in the bulk to the solutions in the image potential, as illustrated for the n = 1 and 2 image-potential states on Cu(001) in Fig. 1.25(b). The resulting energies are En = EV −

2 2 0.85 eV ℏ k‖ + (n + a)2 2m∗

n = 1, 2, . . .

(1.4)

The matching of the wave function at the surface introduces the quantum defect a, which is 0 at the upper edge of the band gap and reaches at the lower edge the value 1/2. Binding energies EV − En larger than 0.85 eV are not found for single-crystal surfaces,

Figure 1.25: (a) Electric field lines for an electron in front of a metal surface, surface-charge density (gray shading), and image charge. (b) Image potential (red) and probability density (green) for the first two image-potential states, with energies in the band gap of the projected bulk band structure and below the vacuum energy EV .

34 | 1 Properties of surfaces but may be found for overlayer or adsorbate systems with a modified potential at the surface. The preceding equation includes the motion of the electron parallel to the surface, which is described by the kinetic energy of a particle with an effective mass m∗ . In most cases, it has a value close to the free electron mass m, because charge and image charge can move freely parallel to the surface without change in the potential energy. At semiconductor surfaces, the charge is incompletely screened by the valence electrons, and the field lines are not perpendicular at the surface with a relative dielectric constant εr . Force and potential are weakened by a factor (εr − 1)/(εr + 1). The binding energy is reduced by the square of this factor compared to metals. The image-potential states arise by adding an additional electron to the surface, and are thus unoccupied in the ground state. This is consistent with the maximum binding energy of 0.85 eV relative to EV , which is smaller than any known work function Φ = EV − EF . Thus, the energies of the image-potential states are always above EF . The maximum of the wave function for the n = 1 state is located about 0.4 nm in front of the surface (Fig. 1.25(b)), and is thus larger than bond lengths of adsorbates. If adsorbates are present on the surface, an external charge is still screened by the substrate and image-potential states can still exist. Image-potential states can therefore be used as sensitive probes for modifications of the surface with very little disturbance of the electronic properties of the adsorbate system. With two-photon photoemission (Section 5.3.3), the energies of image-potential states can be determined with an accuracy of a few meV. 1.2.4.4 Quantum-well states

A thin metal film epitaxially deposited on a substrate has different surface properties as the surface of a bulk crystal of the same orientation. The surface states sense the potential and the different boundary conditions at the substrate and have therefore altered energies and dispersions compared to the corresponding surface of the bulk crystal of the film material. Due to the finite penetration depth of the wave functions of surface states, their properties converge toward those of the bulk crystal for film thicknesses larger than about five atomic layers. Additional quantization effects arise from the restriction of the electrons in bulk states in a film of finite thickness. These can be described in a simple quantum-well model. For energies in a band gap of the projected bulk band structure of the substrate, the states are confined to the film of thickness d. As a result, the band structure is not continuous in Kz (Fig. 1.22(b)), but the energies assume discrete values. The quantization condition can be written as a phase relationship, 2Kz d + ϕ = 2πn, where stationary states are obtained only if the phase is an integer multiple of 2π. The matching of the wave function at the substrate and the vacuum barrier is taken into account by the energy-dependent phase shift ϕ = ϕ(E). The quantization condition leads to discrete values for Kz , and the associated energies are obtained from the bulk band structure E(k ‖ , Kz ) (at fixed k ‖ ). Note that a phase-shift model can also be used to determine the energies of surface states at clean surfaces (d = 0). The significance of the phase of the wave function at the interface can be seen in Fig. 1.21, where the phase of the Bloch wave function (decaying exponentially into the bulk) at the surface has to be right to match the exponentially decaying wave function on the vacuum side.

1.2 Electronic structure

| 35

1.2.4.5 Adsorbate states Adsorbates can cause large changes of the electronic structure at the surface, because they contribute additional states (orbitals), which modify the bonding at the surface, or lead to coverage-dependent superstructures. For strongly bound adsorbates, the adsorbate states are usually substantially broadened and shifted in energy compared to the orbitals of the free adsorbate atom or molecule (see Section 2.2.1 and Fig. 2.6). Since the energies of the adsorbate states do not have to lie in a band gap of the projected bulk band structure, a clear-cut assignment and distinction from substrate states is difficult. For weakly-bound adsorbates or molecules, the adsorbate orbitals are only slightly modified, and the properties and notation may be adopted for the adsorbed species. The previously discussed surface dimers on the Si(100) surface (Fig. 1.24) have significantly different binding configurations compared to the substrate atoms. Therefore, one could also consider them as adsorbate molecules on a bulk-like terminated Si(100) substrate. Accordingly, the statements about the dispersion properties depending on the distance of the dimers apply as well to other adsorbate systems.

1.2.5 Work function The work function is defined as the minimum amount of work, which has to be expended to remove an electron from the solid. It is assumed that the electron is at rest sufficiently far away from the surface, i. e., its kinetic energy is zero. The electronic states in a solid are occupied up to the Fermi energy EF . The potential energy of an electron far away from the surface is the vacuum energy EV . The work function is then obtained as Φ = EV − EF . Since the electron has to overcome a potential barrier at the surface, the work function is a surface-specific quantity. It depends on the material of the solid and on the crystallographic orientation. The work function changes when layers or adsorbates are put on the surface. Defects also influence the value of the work function. The work function can be measured with an accuracy of a few meV, e. g., by photoelectron spectroscopy (Section 5.3.1). Coverage-dependent changes of the work function as differences in the contact potential relative to a reference electrode (Kelvin probe, Section 6.3.4) can be obtained with even better resolution. Work function measurements are therefore a simple method to characterize the condition of a surface. Figure 1.26(a) shows the work function of stepped Cu(11l) surfaces depending on the step density, which is the number of steps per length unit. At steps, the charge density does not change step-like or abruptly. It rather gets smoothed according to the model of Smoluchowski smoothing (Fig. 1.10(f)). Since the step atoms do not relax to the same extent as the charge density (Section 1.1.3), a positive excess charge occurs at the step atoms, compensated by a negative excess charge at the bottom of the step (Fig. 1.26(a)). An electric dipole moment perpendicular to the surface develops, which is oriented oppositely to the surface dipole moment generated by the potential step

36 | 1 Properties of surfaces

Figure 1.26: Work function as a function of (a) the step density on a Cu(11l) surface (after [1.12]), and (b) the sodium coverage on a Cu(111) surface (after [1.13]).

at the terrace surface, and thus lowers the work function. The effect increases with the number of step atoms (per unit length), and one obtains a linear relationship, as shown in Fig. 1.26(a) for stepped copper surfaces. This simple model even holds for Cu(115), though the terrace width is only 2.5 atomic distances. Adsorbates also influence the charge distribution, and thus the work function by the resulting dipole moments. Adsorbate systems offer the advantage that the density of the dipoles can be varied continuously by changing the coverage. As example, the work function for the adsorption of sodium on Cu(111) is presented in Fig. 1.26(b). Alkali atoms cause large work function changes, since the valence electron can easily be transferred to the substrate. The positively charged alkali atoms repel each other and are weakly bound to the substrate, which suggests evenly distributed Na atoms across the surface. The work function decreases linearly up to a coverage of 0.1 layers, which corresponds to a mean separation of approximately three atomic distances. It is interesting to note that the linear dependence was found for the stepped surfaces up to the same limit. A minimum of the work function is reached at 0.3 layers, and a maximum indicates the completed layer. For higher coverages, the work function decreases slightly, until for 2 layers, the value of metallic sodium is attained. The dependence for small coverages of < 0.3 layers can be explained by the influence of dipole moments of neighboring Na atoms. This leads to a reduction of the dipole moment per atom (depolarization) and can be described by the Topping model [1.14], as shown by the curve in Fig. 1.26(b). In general, one has to consider that the assumption of evenly distributed atoms is questionable at higher coverages if (metallic) islands form (Chap. 2.3). Finally, we address the problem of the work function of an inhomogeneous surface as it can occur for an incomplete coverage: Parts of the surface are covered by islands, and in between bare terraces of the substrate are exposed. Large islands should have the work function of the adsorbate-covered system and similarly large terraces of the substrate. At the boundaries between islands and substrate,

1.3 Lattice vibrations at surfaces | 37

electric field components parallel to the surface exist. These fields equalize the potential distribution in front of the surface, in such a way that far above the surface the average value of the work function of islands and substrate (weighted by their proportions) is reached. The distance where the potential equilibrates corresponds roughly to the lateral dimension of islands and terraces. This consideration holds if only small external fields are applied to measure the work function and it is sampled at a sufficiently large distance. When these conditions are not fulfilled, the experiment does not determine the average work function, but the local work functions of islands and terraces are measured instead. The n = 1 image-potential states or the 5p1/2 valence orbitals of physisorbed xenon atoms are examples for probes of the local work function at the surface. In order to image the different local work functions, one may also apply a strong electric field (≈ 10 kV/cm) to a surface. If the distance of the maximum of the potential from the surface is comparable to the lateral extent of islands and terraces, the electrons overcome different barriers, depending on their origin at the surface. This work function contrast is used in a photoelectron emission microscope (PEEM), which uses a large extraction field close to the surface. In a scanning tunneling microscope (Chap. 6.2), relatively small voltages are applied over small distances. This leads to large electric fields, and areas of different work function can be resolved with lateral resolution of nm.

1.3 Lattice vibrations at surfaces So far we have considered a rigid lattice and treated the electron motion independently from the motion of the atoms in the lattice. As known from solid-state physics of bulk crystals, the atoms can vibrate around the equilibrium positions. Since the mass of electrons is much smaller than the one of atoms, one can treat the problem in good approximation, assuming that the electrons are always in equilibrium with the instantaneous atom positions (adiabatic or Born–Oppenheimer approximation). Restoring forces on the atoms are generated by the dependence of the binding potential on the instantaneous atom positions. For sufficiently small deviations of the atoms from the equilibrium positions, the energy is proportional to the square of the amplitude of the deviation. In this harmonic approximation, one can treat the lattice vibrations by classical mechanics and quantize the vibrational frequencies as quantum mechanical (quasi-)particles called phonons. Lattice vibrations (and their coupling to the electron system) play an important role in processes, where energy is dissipated, e. g., excitations by light, chemical reactions, etc. Relaxation processes within the electronic system (of the solid, at the surface, but also of adsorbed molecules) occur on the femtosecond timescale, and the total energy remains in the electronic system. Only the coupling to the atomic lattice converts the energy to heat. Due to the large mass of the atoms, such processes occur on the picosecond timescale. We want to describe the lattice vibrations at surfaces in an atomistic picture, though also in the framework of the elastic continuum theory (the long-wavelength limit of the atomic model) surface and interface vibrational modes are obtained. Such modes have significant (technical) relevance, e. g., for material inspection by ultra-

38 | 1 Properties of surfaces sound. Also filter elements in the GHz range and radio-polled passive sensors use surface (acoustic) waves. Similar to the electronic surface states of the preceding Section 1.2.4 can the presence of the surface generate new solutions of the equation of motion of the atoms. This may be illustrated for a linear chain of atoms, which are coupled by ideal springs. This model corresponds to the harmonic approximation for vibrational modes along high-symmetry directions of a three-dimensional crystal, when whole crystal planes oscillate relative to each other. Considering the case of a one-dimensional crystal with a two-atom basis cut to a semi-infinite chain with the surface atom labeled with index 0 (Fig. 1.27(a)), the following set of equations of motion is obtained: M0 ü 0 = −D0 (u0 − v0 )

mv̈0 = −D0 (v0 − u0 ) − D(v0 − u1 )

M ü n = −D(un − vn−1 ) − D(un − vn ) mv̈n = −D(vn − un ) − D(vn − un+1 )

(1.5) n≥1

n ≥ 1.

Here un and vn are the displacements of atoms with mass M and m from the equilibrium position. At the surface, the force constant D0 and mass M0 may deviate from the respective values in the bulk. Since the number of atoms in the chain (crystal) is semi-infinite, one may assume that the dispersion relation deviates only slightly from the one of a chain infinitely extended in both directions. Using the ansatz un (q, t) = uei(q⋅na−ωt) and vn (q, t) = vei(q⋅na−ωt) (layer spacing a), one obtains from the equation of motion for n ≥ 1 the dispersion relation ω(q)2± =

D [(M + m) ± √(M + m)2 − 2mM(1 − cos(qa))]. mM

(1.6)

An oscillatory solution in the infinitely extended chain requires a real wave vector q. In a semi-infinite crystal, a complex q̃ with positive imaginary part Im(q)̃ > 0 may also lead to solutions with real-valued frequencies ω. It is exponentially decaying into the crystal and represents a surface mode. The conditions for their existence may easily be identified by the extension to complex wave vectors q̃ = qR + iqI . For real q̃ = q = qR , the dispersion relation ω(q) has the known appearance with optical and acoustic branch, as shown in Fig. 1.27(a). Since qI < 0 would correspond to an amplitude, which increases exponentially with distance from the surface, one has to restrict ̃ = cos(qR a) cosh(qI a) + i sin(qR a) sinh(qI a), one the solutions to qI > 0. Using cos(qa) obtains a real-valued ω only if qR a = 0 or qR a = π (qR reduced to first Brillouin zone). Replacing in eq. (1.6) cos(qa) by ± cosh(qI a), solutions with real ω above the bulk band are found at ω > √2D/m + 2D/M for qR = 0, and in the range √2D/M < ω < √2D/m (assuming M > m) for qR a = π, which is inside the band gap between the acoustic and optical branch (Fig. 1.27(a)). The energy eigenvalue of the surface mode is determined by the equations of motion of the surface atoms (first two lines of eq. (1.5)). An important difference to the electronic band structure in the two-band model (Fig. 1.22) lies in the fact that a maximum energy of the bulk phonon modes exists.

1.3 Lattice vibrations at surfaces | 39

Figure 1.27: (a) Phonon dispersion ω(q)̃ of a two-atomic linear chain as a function of the complex wave vector q.̃ For each frequency, the real part Re (q)̃ is plotted in brown and the imaginary part Im (q)̃ in blue. (b) Calculated dispersion relation of the phonons of a finite Cu(111) crystal (31 layers) along the ΓM direction. Surface modes are shown by green, bulk modes by brown lines (after [1.15]).

In the one-dimensional model, vibrational modes, whose amplitudes are exponentially damped with increasing distance from the surface, only occur if the force constant D0 or the mass M0 at the surface differ from the corresponding values in the chain. This situation is realistic, since by formation of a surface, bonds are broken and remaining bonds are altered by electron transfer, which may be taken into account by a modified force constant D0 at the surface. The different mass M0 at the end of the chain in the model corresponds to the case of adsorbates, which change the boundary conditions and allow the development of surface vibrational modes. However, adsorbate vibrations are often better described in a local picture (Section 5.4.3). The transition to a three-dimensional crystal with surface can be modeled by a set of equations of motion for the atoms in a laterally periodic slab of a crystal consisting of N layers. In analogy to the electron motion in a periodic potential (eq. (1.2)) are the vibrational states characterized by a vector q‖ , which may be restricted to the surface Brillouin zone. The set of equations for a crystal of N layers consists of 3N × 3N equations for each wave vector q‖ . The factor 3 arises from the three spatial dimensions for the vibrations of the atoms. In band gaps, solutions of the equations of motion can exist, which are exponentially damped inside the crystal perpendicular to the surface, as demonstrated for the one-dimensional model. A sufficient condition is the existence of a relative band gap, in which for certain q‖ in the surface Brillouin zone, no solutions for the three-dimensional periodic crystal (without surface) exist (cf. Sections 1.2.3 and 1.2.4 for the equivalent criteria for electronic states). Figure 1.27(b) shows a calculated dispersion relation for the vibrational modes at the

40 | 1 Properties of surfaces Cu(111) surface. The calculation was performed for N = 31 layers of the crystal with wave vectors q‖ along the ΓM direction. The modes described by the model of the one-

dimensional, single-atomic chain, can be found at the Γ point (q‖ = 0). For each q‖ , the calculation gives 3N modes, which represent standing waves along the [111] direction. For N → ∞, the frequencies of these modes would be closely spaced, forming a continuum of the projected bulk modes (brown). In regions, in which no bulk-like solutions exist, modes appear, which are localized in layers close to the surface (green). Surface modes are the Rayleigh mode below the continuum of bulk modes, and a mode near the M point in a band gap. The situation is analogous to the electronic states at the surface (cf. Fig. 1.23). These vibrational modes are intrinsic surface modes at the surface of a three-dimensional crystal. In addition to the vibrational modes at a surface, which are due to altered binding configurations or broken bonds, special modes can occur, due to the reconstruction of a surface. An example is a rocking mode of the dimers at the reconstructed Si(100)-c(4 × 2) surface (Fig. 1.24(a)). Such modes do not have to be in band gaps, but can lie also in regions of the projected bulk band modes. If the symmetry of the mode is compatible, it may couple to a bulk mode and become a surface resonance. The deposition of thin films or adsorbate layers can also create vibrational modes, which are localized at the surface. These modes are denoted extrinsic surface modes. Vibrations of adsorbed molecules are presented in Chap. 5.4. In this chapter, we addressed the description of the basic properties of surfaces. The presence of the surface implies new binding configurations of the atoms. The minimization of the energy of the system modifies the layer spacings, and may cause a reconstruction of the surface with a symmetry different from that of the bulk crystal. The surface unit cell then deviates from the simple cut of the bulk crystal. Corresponding notations relate the size and symmetry of the new unit cell to the ones of the ideal surface cut. With these changes, new electronic and vibrational surface states appear, which localize electrons or phonon modes near the surface. They may have energies, for which no corresponding states exist in the infinitely extended crystal (e. g., because of band gaps). The jump of the electron density at the surface is connected to a surface dipole moment. This leads to the characteristic work function of the surface, which is influenced by steps, defects, or adsorbates.

Q1.1: For the (100) surface of a bcc crystal, give the numbers of nearest neighbors of an atom in the first and second layer. How many atoms per primitive unit cell are in each layer? Q1.2: Identify the symmetry elements of the fcc(110) surface (Fig. 1.1(c)) and determine the plane group. Q1.3: Explain the matrix and Wood’s notation for superstructures. Can all superstructures be described by Wood’s notation? Q1.4: Explain the terms surface relaxation and surface reconstruction and give reasons for their occurrence. Q1.5: Check that the fcc(110)-(2 × 1) missing-row reconstruction in Fig. 1.15(c) consists of (111) microfacets. Q1.6: Discuss the importance of Bloch’s theorem and the reciprocal lattice vectors for the surface band structure.

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41

Q1.7: Sketch the construction of the Brillouin zone for the (100) surface of an fcc crystal. What is the correspondence to the Brillouin zone of the bulk crystal? Q1.8: Give the dependence of the potential energy of an electron as function of the distance from a metallic surface. Q1.9: Give the necessary condition for the occurrence of surface states, and describe different types with their properties. Q1.10: Sketch the Smoluchowski smoothing at a surface step, and explain why this reduces the work function. Q1.11: Explain why the adsorption of alkali atoms causes a lowering of the work function. Q1.12: How can the propagation velocity of surface waves be determined from the phonon dispersion relation ω(q)? Q1.13: How can special vibrational modes occur at a surface? In which regions of the ω(q) diagram can these modes be found with respect to the vibrational modes of the bulk crystal?

[1.1] [1.2] [1.3] [1.4] [1.5] [1.6] [1.7] [1.8] [1.9] [1.11] [1.12] [1.13] [1.14] [1.15]

S. Müller, J. E. Prieto, C. Rath, L. Hammer, R. Miranda and K. Heinz, J. Phys. Condens. Matter 13, 1793 (2001). S. Liepold, N. Elbel, M. Michl, W. Nichtl-Pecher, K. Heinz and K. Müller, Surf. Sci. 240, 81 (1990). M. Arnold, A. Fahmi, W. Frie, L. Hammer and K. Heinz, J. Phys. Condens. Matter 11, 1873 (1999). K. Heinz and U. Starke, Surface Crystallography in Surface and Interface Science (Ed. K. Wandelt), Vol. 2, p. 489, Wiley-VCH (Berlin 2012). V. Blum, C. Rath, G. R. Castro, M. Kottcke, L. Hammer and K. Heinz, Surf. Rev. Lett. 3, 1409 (1996). W. Nichtl, N. Bickel, L. Hammer, K. Heinz and K. Müller, Surf. Sci. 188, L729 (1987). M. K. Debe and D. A. King, Surf. Sci. 81, 193 (1979). G. Schmidt, H. Zagel, H. Landskron, K. Heinz, K. Müller and J. B. Pendry, Surf. Sci. 271, 416 (1992). D. R. Mullins and S. H. Overbury, Surf. Sci. 199, 141 (1988). M. Weinelt, M. Kutschera, R. Schmidt, Ch. Orth, Th. Fauster and M. Rohlfing, Appl. Phys. A 80, 995 (2005). M. Roth, Th. Fauster and M. Weinelt, Appl. Phys. A 88, 497 (2007). N. Fischer, S. Schuppler, Th. Fauster and W. Steinmann, Surf. Sci. 314, 89 (1994). J. Topping, Proc. R. Soc. Lond. A 114, 67 (1927). A. Eiguren, B. Hellsing, F. Reinert, G. Nicolay, E. V. Chulkov, V. M. Silkin, S. Hüfner and P. M. Echenique, Phys. Rev. Lett. 88, 066805 (2002).

2 Processes at surfaces In the previous chapter the description of the geometric, electronic, and vibrational properties of surfaces was laid out. To investigate these properties, an interaction with the environment in particular with photons, electrons, atoms or molecules is required. Whereas the interaction with photons will be discussed in the context of photoelectron spectroscopy in Sections 5.2.1 and 5.3, this chapter is devoted to processes which are caused by electrons and atoms, or molecules hitting the surface. The interaction processes of electrons at surfaces are multifaceted, and a good part of the methods used for surface studies relies on the backscattering or emission of electrons (Chaps. 4 and 5). The reason for this is the small mean free path of low-energy electrons in solids. Accordingly, we discuss in Chap. 2.1 the inelastic scattering and energy-loss processes, which are involved in all methods using low-energy electrons. Chapters 2.2 and 2.3 deal with processes on an atomic scale, which occur when atoms or molecules from the gas phase impinge on a surface. The surface is the boundary of a solid to its environment, i. e., every interaction between a solid and its surrounding phase occurs (initially) at the surface: Growth of adsorbate layers or films, dissolution of a crystal, or corrosion and catalysis, to name a few phenomena. All these are based on fundamental processes at the atomic level, which can be characterized with advanced methods experimentally and theoretically. These processes are very important for the understanding and modeling of properties and production procedures of surface coatings in industry, e. g., for semiconductor devices and optical, or corrosion-resistant coatings.

2.1 Energy-loss processes of electrons The interaction of electrons with matter occurs via the Coulomb interaction with the electrons and atomic nuclei of the solid. By elastic scattering processes, only the direction of the electron is changed. This contrasts inelastic processes, during which the electron looses a part of its energy. In principle, an electron may also gain energy, however, the crystal can provide only thermal energies of the order of kB T. Elastic and inelastic mean free paths describe the distances, which an electron moves on average through the solid before it is scattered elastically or inelastically, respectively. Due to the mass ratio, the scattering of electrons by atomic nuclei leads to a negligible energy transfer, and the scattering process is (quasi-)elastic. In contrast, electron-electron scattering can lead to a large momentum and energy transfer. Therefore, the scattering process is usually inelastic. Since a solid contains more electrons than atomic nuclei the penetration range of the electrons and the associated information depth is determined by the inelastic mean free path. In the following, we will focus on the discussion of inelastic scattering processes. https://doi.org/10.1515/9783110636697-003

2.1 Energy-loss processes of electrons | 43

Figure 2.1: Schematic illustration of the energy distribution of backscattered electrons.

2.1.1 Energy distribution of scattered electrons Electrons which move through a solid may excite other electrons individually or collectively. This applies to electrons impinging from the outside as well as electrons in the solid, which were excited in preceding processes. An electron can transfer an arbitrary fraction of its energy to another electron. To investigate these processes in detail, we consider a solid surface irradiated by electrons of a fixed energy (primary energy EP ) and record the energy distribution of the backscattered electrons. Such a spectrum is shown schematically in Fig. 2.1. Besides the elastically reflected beam (primary electrons), one observes a more or less continuous background of electrons at lower energy. The spectrum comprises electrons, which have suffered one or more inelastic processes while traversing the solid (backscattered electrons), and electrons from the solid, which were excited by inelastic processes above the vacuum energy (secondary electrons). On top of the continuous background, some relatively sharp lines are seen, which either arise from energy losses of discrete energies or from Auger electrons. The discrete losses have a fixed energy separation from the line of the elastically scattered primary electrons. The Auger electrons (Section 5.2.3) have a kinetic energy, which is characteristic for each chemical element and is independent of the primary energy. The fraction of electrons impinging on a surface, which are backscattered without energy loss, i. e., elastically, lies in the percentage range. The signal of the elastically reflected electrons exhibits a narrow linewidth, and therefore a large spectral intensity compared to backscattered electrons. Since each incident electron can excite in inelastic scattering processes further low-energy secondary electrons, the sum of the electrons leaving the sample is comparable to the number of impinging electrons. The ratio of the total number of emitted to incident electrons is called secondary-electron emission coefficient. For metals it has typically a value of the order of one, and shows a maximum for primary energies around 1 keV. At insulators, it may reach values up to 20, which is used in secondary-electron multipliers (Fig. 5.4(a)). The secondaryelectron emission coefficient depends not only on primary energy and incidence angle, but also on the chemical composition and structure of the sample’s surface. The

44 | 2 Processes at surfaces latter property is used to obtain an image of the three-dimensional topography of a sample by measuring the signal of emitted secondary electrons in a scanning electron microscope. In most electron spectroscopies used in surface physics, secondary electrons are just an annoying background. A notable exception is the change of the secondary-electron emission at the band edge of the projected bulk band structure (Fig. 5.17(a)). Figure 2.1 shows that energy losses of different kind and magnitude can occur. With increasing primary energy, more energy-loss processes are possible, which results in a strong energy dependence of the mean free path. The various excitation processes are discussed separately before we turn to the inelastic mean free path, which makes electron spectroscopies surface sensitive. 2.1.1.1 Low-energy excitations The energy losses to lattice or molecular vibrations are typically below 0.5 eV. They will be presented in Chap. 5.4; they do not play an important role in connection to the inelastic mean free path. The electronic excitations with the lowest energy are the creation of electron-hole pairs by the primary electron. These electronic transitions from an occupied into an unoccupied band at higher energy lead to a shoulder and an asymmetric broadening on the low-energy side of the primary line in the energy loss spectrum (Fig. 2.1). The width of these loss structures is related to the width of the bands involved and typically smaller than 10 eV. For semiconductors, electronic losses smaller than the band gap are not possible, and the shape of the primary line becomes symmetric. The electrons excited in the loss process contribute to the secondary electrons, and may leave the surface if their energy is above the vacuum energy. 2.1.1.2 Plasmon excitations A simple description of the electronic states of the valence electrons in a solid is provided by the model of a free electron gas. In this model, the solid is put together by a rigid lattice of positively charged atom cores, and the space in between is filled by the evenly distributed valence electrons. The Coulomb repulsion caused by an incident primary electron displaces the quasi-freely movable electron distribution. The restoring force of the lattice of ion cores leads to collective oscillations of the electron distribution. The frequency of the oscillation of the electron plasma is called plasma frequency and can be calculated in a simple model, as ωP = √ne2 /ε0 m∗ . It depends only on the electron density n and the effective mass m∗ . In quantum mechanics, the energy of the oscillation is quantized and the associated quasi-particles are the plasmons with an energy ℏωP . The plasmon energies of valence-electron oscillations in the bulk are in the range 5–25 eV. The altered boundary conditions at a surface generate new oscillation modes, which are localized in the surface region and are associated with quantized quasi-particles named surface plasmons. Their energy is reduced

2.1 Energy-loss processes of electrons | 45

Figure 2.2: Differentiated spectrum of surface (SP) and bulk plasmon losses (BP) from an aluminum surface.

compared to the bulk plasmons approximately by a factor 1/√2. In particular, for transition metals, the plasmon excitation may couple to electronic interband transitions, such that a clear distinction between single-electron and collective excitations is not possible. In these cases, the spectral dielectric response becomes rather complex. The coupling also shifts the plasmon energy considerably from the value predicted by the free-electron model. The excitation of plasmons causes discrete energy losses for the incident primary electrons, which leads to clearly separated satellite lines at the low-energy side of the primary line. For bulk plasmons, multiple losses can occur along the path of the electron through the solid. These processes form a sequence of equidistant lines, whose intensities rapidly decrease with increasing order. Surface plasmons in contrast can be excited only on the electron’s way in and out of the solid, and double losses are usually not observed. They may be observed, however, in combination with bulk plasmon losses. A measurement of the plasmon losses for aluminum is presented in Fig. 2.2. To enhance the structures relative to the background of backscattered and secondary electrons, the first derivate of the energy-loss spectrum (Fig. 2.1) was measured using modulation technique (Section 5.1.5). In the region of multiple plasmon losses, the lines become broader such that the surface plasmon cannot be resolved. The broadening can be explained by the increased average path length the electrons travel through the solid, during which the electrons suffer also other low-energy losses as discussed before. Surface plasmons are also relevant in the field of nanooptics, because the electric field is confined to the nanometer range.

46 | 2 Processes at surfaces 2.1.1.3 High-energy excitations Energy losses with higher energies than plasmon energies are connected with the excitation of electrons in inner shells. The excitation probability is relatively small (Section 5.2.3). Therefore, these processes play only a minor role, and cannot be identified in the undifferentiated spectrum of Fig. 2.1 as losses with well-defined energy relative to the primary line. For core-hole excitation energies < 3 keV, the recombination of the excited core holes occurs predominantly via emission of Auger electrons (Section 5.2.3). This leads to further lines in the energy loss spectrum (Fig. 2.1), whose energies do not depend on the primary energy.

2.1.2 Inelastic mean free path The inelastic mean free path (IMFP) is defined as the distance an electron can travel on average in a solid before it suffers an inelastic process. In the special situation of electron spectroscopy, only those processes are considered, which impact the integrated intensity of a line and not the lineshape. This typically implies energy losses on the order of 1 eV or more. In Fig. 2.3 the mean free path λin of electrons is shown qualitatively as function of energy above the Fermi energy. Below 10 eV, the dependence is determined by the excitation of electron-hole pairs. The inelastic scattering probability for such energy losses is proportional to the product of the available unoccupied and occupied states. Both factors are ∝ E, and hence, with increased scattering probability, the mean free path decreases with energy ∝ E −2 for energies below 10 eV. In this simple picture, a constant electron-electron scattering potential and density of states is assumed. The latter assumption is a crude approximation, and certainly breaks down for energies larger than the width of the valence band. With increasing energy, the excitation of plasmons (Section 2.1.1.2) or the ionization of core electrons becomes possible, which reduces the mean free path of the electrons. In a simple model, the inelastic scattering

Figure 2.3: Universal curve of the inelastic mean free path for electrons in metals (after [2.1]).

2.2 Adsorption, desorption, and diffusion

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cross sections are proportional to the interaction time of the passing electron, which decreases ∝ 1/√E with increasing energy. The resulting increase of the mean free path with energy is proportional to the velocity ∝ √E for E > 100 eV, as can be seen in Fig. 2.3. The curve shown in Fig. 2.3 is an adaption to a large number of experimental data for various elements (see, e. g., [2.2]). It is often called universal curve, though measurements for different elements may differ by a factor 3–5 (which leads in the doublelogarithmic plot to a moderate scatter). The universal curve is thus only a rough estimate for the actual mean free path and its energy dependence. The minimum of the mean free path is typically in the energy range of 20–100 eV, with a value of 3–10 Å. This establishes the high surface sensitivity of experimental methods, which employ electrons in this energy range.

2.2 Adsorption, desorption, and diffusion To study the interaction of atoms and molecules with a surface, we consider the exchange of particles between the solid and the surrounding gas phase. After an atom (or molecule) hits the surface of a solid, various processes can take place, which are illustrated in Fig. 2.4: (a) The atom may stick (adsorb) and subsequently relocate to other sites on the surface (diffuse). (b) Diffusion may also proceed via a site exchange with a substrate atom. (c) Possibly, the atom may not stay on the surface and leave again (desorb). At a step edge (d), at a defect (e), or after an encounter with another atom on a terrace (f), the atom may be bound more strongly and further diffusion or desorption may be suppressed. This constitutes the formation of a nucleus, to which further atoms may attach and layer growth begins. Each of these processes has a specific probability to occur. In a thermally activated process, the probability is determined mainly by the required characteristic energy, such as diffusion barriers, binding energies etc. The probability for the occurrence of a specific process thus depends strongly on temperature.

Figure 2.4: Elementary processes following the impact of an atom from the gas phase onto a surface: (a) Adsorption and diffusion, (b) site exchange with a substrate atom, (c) desorption, (d) attachment to a step edge, or (e) to a defect, and (f) formation of a nucleus.

48 | 2 Processes at surfaces 2.2.1 Adsorption A solid surface in thermal equilibrium with its environment exchanges atoms with the gas phase so that the rates of adsorption and desorption are equal on average (detailed balance). For an increase of the number of atoms adsorbed on a surface, the chemical potential of the respective species A in the gas phase has to be larger than the equilibrium value. In kinetic gas theory, the chemical potential is proportional to the logarithm of the partial pressure of the species. Therefore, the partial pressure of species A in the gas phase has to be larger than the vapor pressure of species A adsorbed on the surface. The relative coverage of the available adsorption sites for atoms of species A on the surface is denoted by θ, and its temporal change is the adsorption rate pA dθ S(θ) e−EA /kB TG . = dt √2πmk na B TG

(2.1)

The areal density na of adsorption sites on the surface is specific for the investigated system, and may depend also on other parameters like the surface temperature. The remaining terms of the quotient describe the rate of the atoms hitting the surface obtained from kinetic gas theory for a gas with temperature TG and partial pressure pA of the species A. The exponential takes into account that an activation barrier EA may have to be surmounted before the atom can bind to the surface. This barrier occurs in particular if a molecule first has to dissociate (dissociative adsorption), or if other particles adsorbed on the surface have to free the adsorption site (cf. Fig. 2.5(b)). The sticking coefficient S(θ) describes the probability that an impinging atom actually adsorbs. If the impinging atomic species is equal to the one of the substrate, i. e., the crystal simply grows, then S is independent of the number of atoms on the surface. Otherwise the sticking coefficient depends on the current coverage, since the number of available adsorption sites is reduced by the previously adsorbed atoms. In the simplest case, S(θ) ∝ (1−θ) applies. If for the adsorption, e. g., of a dissociating molecule, q free sites are required, the dependency changes to S(θ) ∝ (1 − θ)q . The sticking coefficient can also depend on the kinetic energy (and incidence angle) of the impinging species A. If the atoms cannot transfer the energy to the lattice fast enough, they may desorb immediately and are reflected by the surface (Fig. 2.5(a)). The adsorption process usually takes place at temperatures T, for which desorption can be neglected. Under these conditions, the change of the coverage is appropriately described by eq. (2.1). The applicable temperature range depends on the binding energy of the adsorbate at the surface. Two basic states of the adsorbate may be distinguished: physisorbed and chemisorbed. All adsorbates can physisorb on a surface, which means that no chemical bonds are formed with surface atoms (in particular, no electron transfer takes place). Instead van-der-Waals forces hold the adsorbate at the surface. Typical binding energies are less than 10 meV per atom, and the distance of the adsorbate from the surface is 3–4 Å. Thus, smaller molecules are usually not physisorbed at room temper-

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Figure 2.5: Total energy of an atom (a) and a molecule (b) at a surface. In (a) the atom does not have to overcome an energy barrier to reach the energetically lowest chemisorbed state. The kinetic energy of the impinging atoms has to be dissipated by additional processes for the atom to stay on the surface (gray line). (b) For dissociative adsorption of a molecule, a barrier exists between the molecular, physisorbed state and the atomic, chemisorbed state. The barrier results from the superposition of the increasing total energy of the molecule near the surface and the lowering of the total energy of the atom pair relative to the dissociation energy.

ature, and some desorption from the physisorbed state takes place even at low temperatures. The physisorption energy changes only weakly with the position parallel to the surface plane or adsorption site. Diffusion barriers are therefore quite small. The physisorbed state is often a precursor to chemisorption. During the time that the adsorbate is trapped in the physisorption well it can loose energy to the substrate lattice and find an energetically favorable configuration in a stable chemisorbed adsorption state (Fig. 2.5). In chemisorption, electrons are exchanged between the surface and the adsorbate. In particular for dissociative adsorption, the transition from the physisorbed to the chemisorbed state involves overcoming an energy barrier EA (Fig. 2.5(b)). The energy barrier for desorption EDes can therefore be slightly larger than the binding energy EB . The energetically highest occupied and lowest unoccupied adsorbate orbitals (frontier orbitals) are relevant for the binding, because they can most easily exchange electrons with the surface. In the case of molecules, they are called highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). The binding energy EB of a molecule in the adsorbed state amounts to 100 meV up to several eV. The distance to the surface in the chemisorbed state is 1–2 Å. The binding energy released during the adsorption process is to some part available for subsequent diffusion processes. The formation of a chemical bond strongly depends on the adsorbate and its chemical environment. Special adsorption sites exist, at which the sum of the lateral forces on the adsorbate cancels, due to local symmetry (Fig. 1.5). Larger molecules often bind to the surface with several atoms of the molecule. This can lead to selected but not necessarily uniquely defined adsorption sites and several favored orientations of the molecule.

50 | 2 Processes at surfaces

Figure 2.6: Formation of a chemical bond of an atom at a surface: Interaction of an adsorbate orbital εa with (a) delocalized states at the surface (s, p electrons) and (b) more strongly localized states (e. g., d electrons). (a) shows the shift of the energy and the energetic width of the adsorbate orbital, due to the attractive electrostatic potential (dashed line) and the overlap of adsorbate and substrate electron density. In (b), an additional interaction occurs with the more strongly localized orbitals εd of the substrate and bonding (σ) and antibonding σ ∗ hybrid orbitals form between adsorbate and surface just as in a molecule.

In the adsorbed state, the electrochemical potential of the adsorbate electrons is balanced with the one of the substrate electrons by electron exchange, i. e., the adsorbate orbitals are shifted relative to the Fermi level EF of the surface. The interaction with the states of a wide electron band of the substrate leads to an energetic broadening of the adsorbate orbitals. This corresponds to the situation that electrons can easily move from the adsorbate orbital to the substrate, and thus the probability density in the adsorbate orbital is reduced. This type of binding is similar to a metallic bond in a solid: The energy gain results from the delocalization of the electrons in the adsorbate orbital (Fig. 2.6(a)). Depending on the position of the resulting adsorbate orbital εa󸀠 relative to the Fermi level of the substrate, the bond may have strong ionic character. If, for example, an adsorbate orbital, which was originally occupied with electrons, is shifted above EF , the adsorbate is positively charged in the adsorbed state (cf. Na on Cu(111) in Fig. 1.26(b)). Another bond type which is similar to covalent bonding occurs if the substrate has narrow bands, e. g., dangling bonds at semiconductor surfaces (Fig. 1.24), or d states at metal surfaces. Then a combined surface-adsorbate orbital can form just as in the binding of a diatomic molecule with bonding and antibonding hybrid orbitals (Fig. 2.6(b)). Most catalysts owe their chemical activity to the formation of such bonds. The interaction of molecules with a surface and the concomitant broadening and shift of the orbital energies can lead to originally unoccupied antibonding molecular orbitals getting occupied by electrons. This breaks the intramolecular bond, and the molecule dissociates at the surface.

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2.2.2 Desorption By investigating the desorption from an adsorbate-covered surface, the binding energies of adsorbates and the relative coverage of a surface can be determined. This will be discussed in detail in Chap. 7.1. For the thermal desorption rate, one obtains the Polanyi–Wigner equation, dθ = −νDes (T) θq e−EDes /kB T . dt

(2.2)

The attempt frequency νDes (T) describes the number of trials per second of an adsorbate to leave the surface at temperature T. This rate is typically of the order of phonon frequencies (1013 Hz). The coverage-dependent factor θq takes into account that in the desorption process adsorbates have to be on the surface, and possibly q adsorbates have to jointly desorb as a q-atomic molecule. The exponential factor allows for the fact that the adsorbate is bound at the surface, and thus (thermal) energy is needed to break the bond. In simple cases, the desorption energy is EDes = EB + EA (Fig. 2.5). Note that different temperatures enter eqs. (2.1) and (2.2): For adsorption the temperature TG of the gas of impinging particles of species A determines the adsorption rate, whereas for desorption, the temperature T of the surface is relevant for the rate of desorption events. Besides thermal desorption, there is also electronically stimulated desorption. It might be excited by electrons or photons and is called electron- or photon-stimulated desorption (ESD or PSD), respectively. For the latter process, the short term photodesorption (PD) is also used. The common concept is appropriately described by the general term desorption induced by electronic transitions (DIET). Figure 2.7 shows the potential energy as function of the distance of the adsorbate from the surface, which has a minimum in the ground state at the equilibrium distance (blue curve). In contrast, the energy of an antibonding excited state (red curve) decreases with increasing distance from the surface. An excitation can lift an electron to an antibonding state. This electronic process is ultrafast (few femtoseconds), and the distance does not change. In molecular physics, this vertical transition is called

Figure 2.7: Electronically stimulated desorption. Excitation and deexcitation of electrons of an adsorbate occur on a femtosecond timescale, whereas the picosecond timescale is relevant for the change of the atomic coordinates and motion of the adsorbate.

52 | 2 Processes at surfaces

the Franck–Condon principle. The electronically excited adsorbate can now lower its potential energy by increasing the distance and simultaneously gain kinetic energy. The electron may also return to the ground-state curve at any point. The thereby released energy leads to the creation of an electronhole pair in the substrate. During the electronic deexcitation (again on a femtosecond timescale), the distance and the kinetic energy of the adsorbate remain unchanged. If the kinetic energy is sufficient to overcome the rest of the potential barrier, the adsorbate can leave the surface and desorb. In the end, the desorption process depends on the distance at which the deexcitation process occurs. The energy distribution P(E) of the desorbing adsorbate atoms or molecules has its maximum typically at energies of a few eV. Especially for small distances, the lifetime of the excited electron is short due to the strong interaction with the substrate and lies in the femtosecond range (Section 5.3.3). Therefore, only in rare cases does the excited state live long enough that the adsorbate can gain sufficient energy to overcome the remaining potential barrier after deexcitation. Electronically stimulated desorption is therefore usually a rather inefficient process in terms of the number of desorbing particles per incident electron or photon.

2.2.3 Diffusion If an adsorbate is bound at the surface and the desorption probability is very low at the selected temperature T, it may still diffuse on the surface. For the adsorbate, the surface presents itself as a potential landscape with more or less favorable adsorption sites. To get from one adsorption site to another, energy barriers have to be overcome (Fig. 2.8). Accordingly, the hopping rate kD (T) to a neighboring adsorption site can be written as kD (T) = νD (T) e−ED /kB T

(2.3)

with the attempt frequency νD (T) (which is of comparable magnitude as the one for desorption) and the diffusion barrier ED < EB . Typical diffusion barriers range from 10 meV for weakly bound molecules and atoms up to about 1 eV for strongly covalent bound adsorbates on surfaces. The diffusion barrier does not only depend on the combination of adsorbate and substrate, but

Figure 2.8: Potential landscape for a diffusing adsorbate on a substrate. To reach a neighboring adsorption site, the diffusion barrier ED has to be surmounted. In the vicinity of defects, the barrier can be different (at steps this is known as the Ehrlich–Schwöbel barrier ES ), and the diffusion is affected.

2.2 Adsorption, desorption, and diffusion

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Figure 2.9: Evaluation of the hopping rate of Cu atoms on a Cu(111) surface in the temperature range from 12 K to 15 K. (a) Overlay of two subsequent scanning tunneling microscopy measurements taken at 14.5 K. The atoms moved from the dark to the colored positions. The concentric circles around the Cu atoms are density oscillations of the electrons in the Cu(111) surface state, which are scattered by adsorbed atoms (Section 6.2.1). (b) The analysis with an Arrhenius plot (abscissa ∝ 1/T ) yields a diffusion barrier ED ≈ 40 meV [2.4].

also on the orientation of the substrate surface. Close-packed surfaces (e. g., fcc(111)) generally have a lower diffusion barrier than more open surfaces (e. g., fcc(110)). Diffusion barriers are difficult to measure, even though they are important parameters for the modeling of growth processes on surfaces (cf. Chap. 2.3), which are required, e. g., for the production of semiconductor devices. To follow the position of an individual atom on a surface as a function of time first succeeded with field ion microscopy (see [2.3]), which however can only be applied to very few systems. Scanning tunneling microscopy (Chap. 6.2) can on the other hand be applied to a wide range of substrates and adsorbates. By choosing the proper temperature, for which the hopping rate lies in the range of 0.01–1 Hz, the diffusion barrier can be obtained from an Arrhenius plot of the mean hopping rate collected for an ensemble of atoms (Fig. 2.9). Other methods infer the diffusion barrier from an analysis of film growth (Chap. 2.3). The diffusion barrier at defects can be different from the one at an ideal, defect-free surface. This can lead to vacancies in a growing film not being filled by adsorbates at certain temperatures, or that the first layer will not be completed before the second layer starts growing, because the adsorbates cannot overcome the Ehrlich–Schwöbel barrier ES (Fig. 2.8) at the step to get from an island to the substrate below. From the hopping rate in eq. (2.3), one obtains on a surface with lattice constant a the diffusion coefficient DS (T) = 1/4 kD (T) a2 . Diffusion can be viewed as a twodimensional random walk, and the mean square ⟨(Δx)2 (t)⟩ of the distance Δx from a

54 | 2 Processes at surfaces starting point at time t = 0 is related to the characteristic diffusion length LD , LD (t) = √⟨(Δx)2 (t)⟩ = √4DS (T)t ∝ e−ED /2kB T .

(2.4)

It is important to note that the diffusion length limits the number of possible adsorption sites, which an adsorbate can reach within a certain time interval. In particular, this implies that an adsorbate system may never be in thermal equilibrium on a surface, because the energetically most favorable adsorption configuration cannot be reached in the available time at the given temperature. The processes occurring at the surface are then kinetically limited.

2.3 Film growth and epitaxy For many applications, in particular in microelectronics, it is desirable to deposit homogeneous films of material A on substrate B. These layers should ideally be singlecrystalline and of uniform thickness on an atomic scale. We consider here only the deposition of material A from the gas phase. In molecular-beam epitaxy (MBE), a directed beam of atoms or molecules hits the surface. Other methods for the deposition of layers are presented in Chap. 3.3. In practical applications, films are often produced electrochemically, which will not be considered here, even though many basic processes are similar on the atomic scale.

2.3.1 Growth modes In the preceding section, we showed that the diffusion processes of the atoms on a surface have a decisive impact on the morphology of a thin film. If the atoms arriving at the surface got stuck on the first available adsorption site, a very rough film would result with strong thickness fluctuations. To reach a morphology close to that of an ideal homogeneous film, the deposited film is usually annealed, which activates diffusion processes thermally and reduces the number of defects. If the necessary diffusion processes cannot be activated sufficiently at the achievable temperatures, the film remains in a kinetically limited metastable state. Experimentally observed film morphologies are illustrated in Fig. 2.10. The layerby-layer growth (Frank-van-der-Merwe growth) is ideal for the fabrication of homogeneous films. Employing the RHEED method (Chap. 4.3), the completion of a layer can be detected as the film is deposited and grows. This allows determining the film thickness precisely by counting the number of completed layers. A frequently used modification of the layer-by-layer growth is the step-flow growth, which improves the crystalline quality of the film. The substrate is cut along a vicinal orientation in such a way that the diffusion length (eq. (2.4)) is larger than the terrace width. The deposited

2.3 Film growth and epitaxy | 55

Figure 2.10: Morphologies of films grown on substrates: (a) Layer-by-layer growth. (b) Island growth. (c) Initial layer-by-layer growth followed by island growth.

atoms diffuse across the terraces until they reach an upward step and get attached in the minimum of the potential landscape (Fig. 2.8). The thermodynamical condition for layer-by-layer growth is that the interface energy of the uncovered substrate to vacuum is larger than the one of the growing film to the substrate plus vacuum. If this condition is not met, it is energetically more favorable to leave a substrate area as large as possible uncovered, which results in island growth (Volmer–Weber growth, Fig. 2.10(b)). The final film consists of many adjacent crystallites of various thickness and is neither single-crystalline nor of homogeneous thickness (Fig. 2.11). In some cases, a mixed growth mode is observed, where the films start growing layer-by-layer, and later islands begin to form (Stranski–Krastanov growth, Fig. 2.10(c)). This happens in particular if the atomic structure of the growing film depends on its thickness, e. g., because the film grows initially in an epitaxial, stressed structure (Section 2.3.3), or because at the interface to the substrate, an alloy of substrate and film material forms (cf. site exchange in Fig. 2.4(b)).

2.3.2 Nucleation The occurrence of film morphologies as shown in Fig. 2.10 can be based on thermodynamics and kinetics. At sufficiently high substrate temperatures, atoms can leave the site of incidence and find energetically favorable adsorption sites, since the diffusion length (eq. (2.4)) becomes large. This favors the attachment of atoms at step edges or existing islands, because the number of binding partners gets maximized. In addition, high substrate temperatures aid overcoming the Ehrlich–Schwöbel barrier (Fig. 2.8). Atoms which landed on an existing island can leave the island and bind to the substrate. Activation of these diffusion processes lead to a morphology, as shown in Fig. 2.10(a). At lower substrate temperature, the diffusion length is small and the nucleation process competes with the attachment at existing islands or steps. Nucleation describes the process when two or more atoms get close to each other, occupy neighbor-

56 | 2 Processes at surfaces ing adsorption sites and form a nucleus by forming a bond. A nucleus is considered to be stable, if the number of atoms exceeds a certain value. Then the probability for further atoms attaching to the nucleus is larger than for loosing atoms. The number of atoms in a stable nucleus depends on the particular system, in many cases, two atoms are enough. The nucleus can still diffuse with a diffusion barrier, which is significantly larger than for a single atom, since more bonds have to be broken during the process, or several atoms have to move in a concerted motion. If the diffusion length is smaller than the distance between two step edges, nuclei and islands form on the substrate terraces. Step-flow growth is not possible then, and the diffusion length determines the average distance between two neighboring islands. The smaller the diffusion length, the larger the island density of the grown film becomes. At low substrate temperatures, atoms which landed on top of an island cannot leave the island, due to the Ehrlich– Schwöbel barrier at the island edges. These atoms are trapped on the island and form additional nuclei, which initiate the growth of the second layer. This continues, and the film grows in the island-growth mode (Fig. 2.10(b)), even if it would be thermodynamically more favorable to cover the substrate completely in a layer-by-layer growth mode. The morphology of a film can thus be controlled by the diffusion rate, which depends strongly on the substrate temperature (eq. (2.3)). A more detailed look into nucleation and film growth reveals that at low coverage the average distance of two islands has a power-law dependence on the ratio of diffusion coefficient DS (T) (Eq. (2.4)) to flux F of the impinging atoms. A high flux provides a large density of deposited atoms, which diffuse only short distances before forming a nucleus. This leads to small average island separations. In contrast, a large diffusion coefficient DS leads to large distances between islands. These concepts are summarized in the deposition-diffusion-aggregation (DDA) model. According to the DDA model, the film growth passes through three stages as a function of time: Phase 1 describes the formation of stable nuclei; phase 2, the growth of islands via attachment of atoms (or molecules) to stable nuclei. In phase 3, the islands grow together and finally join to form a continuous film. Using the model on experimental data of island densities allows determining the diffusion rate of atoms (or molecules) and obtain quantitative values for diffusion barriers [2.5].

2.3.3 Epitaxy So far, we did not consider a particular aspect of film growth, the match of the crystalline structure between film and substrate. Epitaxy denotes the growth of a thin film on a single-crystalline substrate, in which the crystallographic orientation of the film is ascertained by the substrate. An epitaxial film ideally should also be singlecrystalline, which can be realized only in a limited way, due to the nucleation processes. The film is often polycrystalline with adjoining crystallites based on islands

2.3 Film growth and epitaxy | 57

Figure 2.11: Scanning-tunneling-microscopy images of thin gold films: (a) Epitaxial growth at 400∘ C on a mica substrate. (b) Polycrystalline growth at room temperature on a glass substrate. Note the different length scales of the two images.

formed during the initial growth. These joints are called grain boundaries. The difference between a thin layer of a polycrystalline bulk sample and an epitaxial film is that in the latter all crystallites coincide at least in one crystallographic direction. In most cases, this is the growth direction of the film. A film is called (111)-oriented if all crystallites exhibit a (111) facet at the film surface. Fig. 2.11(a) presents an example of the result of the growth of a gold film with 100 nm thickness on a single-crystalline mica substrate. All gold islands have their [111] direction perpendicular to the substrate’s surface. Figure 2.11(b) shows for comparison a nonepitaxially grown gold film with 50 nm thickness on a glass substrate. Here the [111] directions of the grains enclose angles up to ±10∘ with the surface normal of the substrate. Whether epitaxial growth may be achieved depends on the right choice of the growth parameters, such as temperature and flux. The match of the lattice constants of the unit cells of substrate (aS ) and film material (aF ) is equally important. Only if the substrate and adsorbate are of the same material (homoepitaxy), or the growing film adopts the crystalline order of the substrate (pseudomorphic growth), the lattice constant of the film will conform to the one of the substrate. In the pseudomorphic case, the film is usually strained, and thus contains elastic energy. The criterion for pseudomorphic growth is based on the misfit, which is defined as the relative difference between the lattice constants |aF − aS |/aS . If the misfit is small (typically < 0.02), the film can initially grow pseudomorphically (Fig. 2.12(a)). Due to the associated elastic strain in the film, the total strain energy increases with film thickness. By successive incorporation of lattice dislocations, the strain is relieved and the film assumes its proper lattice constant at thicknesses typically above a few nanometers (Fig. 2.12(b)). For a larger lattice, misfit pseudomorphic growth becomes energetically unfavorable. The film then grows from the first layer on with its own lattice constant. If n aF ≈ m aS for small numbers m, n a commensurate superstructure can form by fitting n unit cells of the film on m unit cells of the substrate (Fig. 2.12(c)), though the film might still be slightly strained. Commensurability may also be achieved by rotation of the crystallographic directions of the film layers with respect to those of the substrate. More complicated situations may be envisaged, which follow in character and notation the superlattices discussed in Section 1.1.1 for surface reconstructions or adsorbate structures. There exist situations, in which the film does not gain energy if it adopts the ideal adsorption sites of the substrate, and gets strained thereby. This is in particular the case if the corrugation of the interaction between sub-

58 | 2 Processes at surfaces

Figure 2.12: (a) Pseudomorphic growth. (b) Release of the elastic strain of the initial pseudomorphic growth by introducing a dislocation. (c) Superstructures with large unit cells (here (7 × 7) with respect to the substrate lattice) are found, if it is energetically favorable for the film to keep its own lattice constant right from the beginning.

strate and film is small, e. g., for van-der-Waals interaction. The gain in binding energy is then smaller than the expense of the strain energy. This can happen also for the case of a strong interaction between film and substrate: If the superlattice required to match the substrate lattice becomes rather large, small displacements of the film relative to the substrate lattice lead to a configuration with almost the same energy. The explanation lies in the observation that a large superlattice samples the interaction potential on a fine grid and all, favorable or unfavorable, adsorption sites are taken by the superstructure independent of the lateral film position. This cancels the driving force for straining the film. Such a film forms on the substrate lattice a structural moiré pattern, which corresponds to a beating pattern of slightly different lateral periodicities (see also Fig. 1.7(c)).

2.3.4 Nanostructures formed by self-organization In the preceding section, we discussed how the density and size of islands (nanostructures) on a surface can be controlled via diffusion processes. In this section, we present further possibilities to control the morphology of nanostructures. One-dimensional nanostructures (like chains of atoms or molecules) can be obtained on stepped substrates, if one ensures that atoms (or molecules) attach to the step edges and no stable nuclei form on the terraces. Alternatively, one can exploit the anisotropy of diffusion barriers, which depend on the crystallographic direction. This can be easily seen, e. g., for the (110) surface of an fcc crystal, which consists of rows of close-packed atoms in ̄ direction separated by a distance of one cubic lattice constant a in [001] direction [110] (Fig. 1.1). Diffusion along the rows proceeds more easily than across the rows, which ̄ makes it easy to grow atomic chains along the [110] direction on an fcc(110) substrate. Another example is shown in Fig. 2.13(a). The hydrogen-induced (5 × 1) reconstruction of the Ir(100) surface provides a lattice of monatomic iridium chains, to which other metal atoms can attach. The reconstruction of the surface serves as a template for the further growth just as steps do on a vicinal surface. Besides the attachment of adsorbates to special structures on a surface, the formation of nanostructures can be created also by specific interactions between adsorbates. A particular example is the growth of organic molecules on surfaces. By modification of the end groups of the molecule (e. g., replacing a nonpolar by a polar one) one can achieve that at particular sites of the molecular structure

2.3 Film growth and epitaxy | 59

Figure 2.13: Self-organized growth imaged by a scanning tunneling microscope: (a) Fe-Ir-Fe chains on an Ir(100) surface (after [2.6]). (b) Organic molecules on a Ag(111) surface. Double-stranded chains (sketched on the right) form, which are stabilized by hydrogen bonds (after [2.7]). weak bonds between molecules form, and not at other sites. This favors a characteristic order of the molecules on the surface. Such weak bonds can be, e. g., hydrogen bonds (between C-O or C-N groups and the hydrogen atoms of the molecules), or coordination bonds between molecular groups and a metal atom. An example for hydrogen bonds is illustrated in Fig. 2.13(b) for benzoic acid molecules on a Ag(111) surface: The molecule binds with its π system to the substrate. At room temperature the molecules can diffuse and use the possibility to form hydrogen bonds to arrange in linear, doublestranded molecular structures. At low temperatures, the diffusion is inhibited and a regular pattern of double chains is observed. On the basis of the occurring structures, the hierarchy of involved interaction energies can be estimated: The interaction leading to the formation of a molecular pair is larger than the one which arranges the molecular pairs in chains. In addition, a repulsive interaction must exist between the double chains, which keeps them apart. This could be a substrate-mediated interaction by the modulation of the electron density or the strain of the substrate. The image in Fig. 2.13(b) presents the situation of the molecules at very low temperatures, that is, when the diffusive motion is switched off. This implies that the diffusion barrier is the smallest energy on the hierarchical scale.

Nanostructures with certain morphologies can be fabricated by exploiting the hierarchy of the characteristic energies in the system: Binding energies, diffusion barriers, and adsorbate-adsorbate interaction energies. A large range of possibilities exist in particular for the field of layers from organic molecules with growing interest for applications. Using molecules with tailored properties, the hierarchy of the energies in the system can be modified. Since the resulting structures are controlled by the interactions within the system, one refers to the self-organized growth of nanostructures. In this chapter, we had a look at many processes, which can occur at surfaces. Depending on the particles involved, various interactions and associated energy scales are relevant. Electrons can transfer energy to localized or collective excitations in the solid, or at the surface. These interactions are rather strong, so the mean free path for electrons becomes rather short. This makes electrons ideal probes for surface properties, as the methods discussed in Chaps. 4 and 5 will prove. Atoms or molecules on surfaces can adsorb, diffuse, combine, dissociate, react, and desorb. The diversity of processes is of relevance for applications in catalysis or film growth. Some of these processes we will exploit for the preparation of well-defined crystal surfaces in the following Chap. 3. The thermal desorption will be presented in detail in Chap. 7.1 with the aim to experimentally determine microscopic parameters.

60 | 2 Processes at surfaces

Q2.1: Sketch the energy distribution of scattered electrons and discuss the various regions. Where are the plasmon losses observed? Q2.2: In Section 2.1.1, it was stated that insulators may have a secondary-electron emission coefficient of up to 20. Can this be correct under steady-state irradiation by electrons? What prevents the crystal from charging and exploding by Coulomb repulsion? Q2.3: Discuss the inelastic mean free path of electrons as function of energy and its importance for methods investigating surfaces. Q2.4: Explain the terms adsorption, desorption, and diffusion. Q2.5: Figure 2.5 schematically shows the total energy of an atom as function of its distance from the surface. Discuss the forces involved for the different distance ranges. Q2.6: Consider the energy scheme shown in Fig. 2.6(b) for the interaction of an adsorbate orbital with, e. g., the narrow d bands of a transition metal. What happens to the binding of the adsorbate if the center of the band εd moves further away from the Fermi energy of the metal? Q2.7: Extend the energy scheme shown in Fig. 2.6(b) to the adsorption of a diatomic molecule with its ∗ ) orbital. What is the requirement to make the dissociation bonding (σMO ) and antibonding (σMO of the molecules at the surface probable? Q2.8: Which conditions lead to an island-growth mode (Fig. 2.10(b)) for kinetic reasons?

[2.1] [2.2] [2.3] [2.4] [2.5] [2.6] [2.7]

M. P. Seah and W. A. Dench, Surf. Interface Anal. 1, 2 (1979). C. J. Powell and A. Jablonski, NIST electron inelastic-mean-free-path database (2010); https://dx.doi.org/10.18434/T48C78. G. L. Kellogg, Surf. Sci. Rep. 21, 1 (1994). N. Knorr, H. Brune, M. Epple, A. Hirstein, M. A. Schneider and K. Kern, Phys. Rev. B 65, 115420 (2002). H. Brune, Surf. Sci. Rep. 31, 1 (1998). A. Klein, A. Schmidt, L. Hammer and K. Heinz, Europhys. Lett. 65, 830 (2004). J. Weckesser, A. De Vita, J. V. Barth, C. Cai and K. Kern, Phys. Rev. Lett. 87, 096101 (2001).

The techniques applied in surface physics may be grouped into preparation and characterization methods. The first group brings the surface in a reproducible, welldefined state for further investigation (Chap. 3). This concerns the removal of contaminants or impurities, the preparation of an ordered state with few defects, and in an optional final step, the controlled deposition of an adsorbate or a film consisting of several layers. The surface has to be kept clean during the measurements, which requires mounting it in an ultrahigh-vacuum chamber. The sample preparation and subsequent investigations are performed (in situ) in the ultrahigh-vacuum apparatus, which thus has to contain all the required measurement accessories. Accordingly, the equipment can be quite large, even though the sample dimensions are typically less than a centimeter. Many methods to investigate surfaces exist. They have been developed over decades and are still being refined or extended. They have to be surface sensitive, i. e., the particles used as probes must interact with the solid only in the surface region of the solid, and then leave the surface carrying information on its properties. Alternatively, the incident particles can release other particles from the solid coming exclusively from the surface region before they reach the detector. For example, incident photons can penetrate relatively deeply into the solid, whereas photo-excited, low-energy photoelectrons only from a region close to the interface between solid and vacuum can leave the surface and reach the detector. The various methods can be classified by the type of incident and probing particles, e. g., as photon in–electron out. However, we chose to present the most relevant methods according to their applicability to answer specific questions about surfaces. Chapter 4 presents methods, which provide information on the atomic arrangement, i. e., the crystallographic structure of the surface. These are foremost diffraction experiments using (x-ray) photons, electrons, or atoms, which take advantage of the lateral translational symmetry of the surface. Alternatively, local interference methods, such as measuring the fine structure of x-ray absorption, can determine the bond lengths between surface atoms without relying on long-range crystallographic order. The electron-spectroscopic methods, described in Chap. 5, aim at determining the chemical composition at the surface using Auger-electron or x-ray-photoelectron spectroscopy. The latter method, using lower photon energies, yields information on the valence-band electronic structure at the surface, and can measure the surface band structure using angle-resolved photoemission. To the topics of this chapter also belongs the high-resolution spectroscopy of the energy losses of electrons scattered at surfaces. This technique is able to measure the surface-phonon band structure and vibrational modes of adsorbed molecules, which contain information on the adsorption sites and their local configuration. Chapter 6 first describes the scanning tunneling microscope, which scans the electronic structure of electrically conducting surfaces. It can provide topographic images of the atomic structure of the surface if the electronic structure correlates with the https://doi.org/10.1515/9783110636697-004

64 | Methods atomic positions in simple cases, such as in metals. In spectroscopy mode, scanning tunneling microscopy yields local information on the electronic density of states at the surface. The chapter ends with a discussion on atomic force microscopy, which can be used for electrically nonconducting surfaces as well, and is often used for the investigation of nanostructures. The final Chap. 7 presents selected topics on the interaction of atoms, molecules, and ions with surfaces: Thermal desorption spectroscopy yields quantitative information on the binding energy of adsorbed atoms and molecules. Ion scattering is a method to investigate the composition and structure with high surface sensitivity.

3 Preparation of surfaces To investigate a surface on an atomic scale, a well-defined preparation of the surface is essential. This implies the removal of foreign atoms, which thermodynamically segregate to the surface from the bulk of the sample, or which adsorb on the surface from the surrounding gas phase. To keep the contaminations from the outside under control, very good vacuum conditions (typically 10−8 Pa) are generally required. Nevertheless, cleaning and preparation procedures have to be repeated regularly (usually daily). Surface research extends beyond the investigation of clean, uncovered crystal surfaces, and many studies concern adsorbate layers and films or nanostructures grown on a substrate. All these modified, new surfaces have to be prepared in a welldefined, controlled way in most cases repeatedly. This requires an array of various preparation techniques, and also a profound understanding of the underlying mechanisms. For many systems preparation recipes exist, which have to be implemented in the respective apparatus and adapted if necessary. For new, unknown systems, the first and often tedious task is to find the best preparation procedure with regard to the optimum homogeneity and minimum of defects in the desired surface phase. Many (if not the most) new surface phases are discovered by variation of preparation steps. The necessity to repeatedly prepare the samples in the respective vacuum chamber gives the researcher in surface physics full control over the sample quality, which in turn is essential for successful experiments with significant results.

3.1 Sources of contamination Most materials are available with limited chemical purity and contain foreign atoms in a relative atomic concentration in the 1–100 ppm range. A sample with a thickness of 1 mm consists of about 5 million layers of atoms. Only two of them are exposed at the surface, which corresponds to a relative fraction of 4 ⋅ 10−7 . Thus, there are enough foreign atoms present in the bulk to cover the surface by many layers. In addition, the production of the surface (e. g., by sawing and polishing) does introduce contaminations into the outer layers of the crystal, often up to a depth of several microns. The most common contaminants in crystals are carbon and sulfur, but depending on the material, other foreign elements can also be found. Annealing of the sample is usually necessary for cleaning the sample surface and many other preparation methods. During annealing, foreign atoms in the crystal start to diffuse and will accumulate at the surface, if that is thermodynamically favorable (segregation). In general, it is not possible, but also not necessary, to completely remove the foreign atoms from the bulk crystal. It is sufficient to create a depletion zone of adequate thickness below the surface. One proceeds as follows: The crystal is heated up to high temperatures to allow a fast diffusion of the foreign atoms to the https://doi.org/10.1515/9783110636697-005

66 | 3 Preparation of surfaces surface, exploiting the exponential dependence of the diffusion coefficient on temperature (eq. (2.3)). The enrichment at the surface does not necessarily increase at even higher temperatures, because the solubility in the bulk increases as well, and the foreign atoms are driven back into the crystal by entropy. The foreign atoms accumulated at the surface can then be removed by suitable cleaning methods (Chap. 3.2). By repeating the heating–cleaning cycle, the depletion zone below the surface grows, because the segregating foreign atoms have to diffuse to the surface from increasingly larger depths, which becomes more and more kinetically limited during the finite time of the heating cycle. The outer region of the crystal is thus brought deliberately into nonequilibrium with the bulk. After a sufficiently depleted zone has been generated in this way, it can be preserved for longer periods of time. One only has to ensure that the subsequent preparation steps proceed at lower temperatures than those used during the depletion steps. The most common source of external contaminations on a surface, besides adsorbates or deposited films from previous preparations, is the atmosphere surrounding the crystal. This implies that sample surfaces, which were exposed to air, are not atomically clean and have to be cleaned in situ under ultrahigh-vacuum (UHV) conditions. The UHV range means pressures p ≤ 10−5 Pa. But even if samples are kept in UHV, they have to be cleaned regularly, since the adsorption of residual gas cannot be ignored. Residual gas is not the rest of atmospheric air which has not been pumped away, but consists mainly of gases, which are released from materials inside the UHV chamber, or diffuse through its walls. A typical mass spectrum of residual gas reveals the main components as H2 , CH4 , CO, and CO2 . According to eq. (2.1) for the impingement rate of gas particles, a surface atom, which takes up an area of about 6 ⋅ 10−20 m2 , is hit under normal atmospheric conditions (pressure p = 105 Pa, temperature TG = 293 K) by a gas particle (e. g., CO, mCO = 28 amu) on average 2 ⋅ 108 times per second. To bring the impingement rate into the range below 1 particle per hour, which is a typical time for data acquisition, the pressure has to be lowered by at least 12 orders of magnitude. This means below 10−7 Pa, which can be routinely reached in modern UHV chambers. Their ultimate pressure (base pressure) is typically around 10−9 Pa. Sources for photons, electrons, ions, atoms, or molecules and sample preparation procedures often introduce gas into the UHV chamber, which results in a working pressure that can be about one order of magnitude above the base pressure.

3.2 Preparation of clean single-crystal surfaces Bulk single crystals of good quality are available for most materials, from which samples with the selected surface orientation can be produced. The surfaces are then polished and cleaned. For surface-sensitive investigations, the final preparation has to be done under UHV conditions. Various methods are available, which often have to be

3.2 Preparation of clean single-crystal surfaces | 67

combined and repeated, and are discussed in what follows. Which method can successfully be applied to a specific material can be found out by literature search [3.1], asking experienced individuals, or by trial and error.

3.2.1 Cleaving Some semiconductors, insulators, and layered crystals can be cleaved under UHV conditions. One obtains atomically clean surfaces, however, in general, only of one orientation: perpendicular to the plane with the weakest interlayer bonds. During the cleavage process a large amount of energy per atom is dissipated, such that the kinetics of the crack plane running through the crystal determines the type and frequency of created defects. To obtain macroscopically flat surfaces by cleaving, the construction of the cleavage device and the cleaving speed have to be optimized by trial and error. The surface created by cleaving is usually close to the truncated bulk configuration. It does not have to be the thermodynamically most stable surface, in particular for reconstructing surfaces, which require a long-range transport of surface atoms. For example, silicon does cleave parallel to a (111) plane and exhibits a metastable (2 × 1)-surface structure instead of the thermodynamically stable (7 × 7) reconstruction (Fig. 1.16). Layered crystals, such as graphite or many chalcogenides, may be cleaved successfully by simply peeling off some layers using adhesive tape, which is possible even under UHV conditions. This method has gained popularity in recent years for the fabrication of graphene or the preparation of surfaces of topological insulators. Cleaved surfaces–like any other surface–do not stay clean for prolonged times even under UHV conditions, and have to be cleaved repeatedly. Usually, the sample has to be transferred out of and back into the vacuum chamber via a load lock for the preparation of a new cleavage process.

3.2.2 Annealing For many preparations and experiments, the sample has to be heated. Also some contaminants may be removed simply by heating. To this group belong molecules, such as H2 , O2 , H2 O, CO, CO2 , or hydrocarbons like CH4 , some of which recombine from the atomic constituents adsorbed on the surface before desorption (Sections 2.2.2 and 7.1). In contrast, other contaminating atoms like carbon, sulfur, and phosphorus and many deposited metals (Section 3.3.2), usually cannot be desorbed by heating the sample. These contaminants either remain at the surface or dissolve in the bulk at higher temperatures, which is unwanted, because these foreign atoms my segregate back to the surface during the cooling of the sample. In this context, the heating and cooling rates also have to be considered. The temperatures required for the desorption of gas

68 | 3 Preparation of surfaces molecules depend on the binding energy of the particles to the substrate’s surface. For example, oxygen can be removed from a Pt(100) surface at a temperature of about 450∘ C, whereas on a W(100) surface, temperatures around 1800∘ C are needed. To heat a sample, several methods are available, which are selected according to the type of sample and required temperature: 3.2.2.1 Contact heating The sample is mounted in thermal contact to a heating element. This method has rather low heating and cooling rates, due to the heat capacity of the heating element. The maximum achievable temperatures are around 1000∘ C, which is not high enough for the preparation of many samples. 3.2.2.2 Heating by current A sample can be heated by running a current directly through it by resistive heating (Fig. 3.1(a)). This method is often used for semiconductors. Because of the negative temperature coefficient of the resistivity of semiconductors, the power supply has to be operated in constant current mode. The resistivity of metals is usually too low for running a current directly through the sample, and currents ≫ 100 A would be required. Instead the current is run through wires made from materials with high melting point (W or Ta). These wires are in thermal contact via grooves in the sample, and hold it in place (Fig. 3.1(b)). The achievable temperatures are limited by the melting point of the mounting wires, the thermal contact to the sample, and its surface area (determining the power loss by thermal radiation). Typically, maximum temperatures around 1200∘ C can be reached. Because of the positive temperature coefficient of the resistivity of the metallic mounting wires, the power supply has to be operated in constant voltage mode. 3.2.2.3 Heating by electron impact Electrons, which are emitted by thermionic emission from a heating filament (usually made from tungsten), are accelerated by a high voltage onto the sample (Fig. 3.1(c)). At accelerating voltages up to 2 kV and emission currents up to 50 mA, the sample can be heated with a power up to 100 W. The power to heat the filament is about 10 W and contributes to the heating of the sample at lower temperatures via thermal radiation. This heating method can reach temperatures up to 2000∘ C, at which almost all the heating power is converted to thermal radiation, such that the sample size puts a limit on the achievable temperature. Since the sample is mounted in thermal contact to the sample holder, the latter will also get warm during heating. To reduce unwanted degassing of the sample holder, it is advisable to actively cool it. This simultaneously ensures that the sample

3.2 Preparation of clean single-crystal surfaces | 69

Figure 3.1: Schematic illustration of different methods for sample heating in front view (top) and top view (bottom). Heating by running a current through (a) the sample and (b) the mounting wires holding the sample. (c) Heating by electron impact.

rapidly cools down to the measurement temperature. For efficient cooling, a good thermal contact to the cooling reservoir is needed. This, however, requires more power; otherwise, the attainable sample temperature during heating is limited. To counter these problems, one often connects the sample to the cooling reservoir via sapphire wafers, which have a strongly decreasing coefficient of thermal conductivity with increasing temperature, and are electrically insulating. For the measurement of the sample temperature, thermocouples are usually used, which are preferably mounted in direct contact to the sample. Several combinations are available for the thermocouple materials. They have to be selected for the required temperature range (so that the wires do not melt), but also an unwanted alloy formation with the sample has to be considered. If for technical reasons thermocouples cannot be engaged, pyrometers can be used to measure the sample temperature during heating through a window from the outside of the UHV chamber. Modern infraredrange pyrometers can be used for temperatures down to 300∘ C. However, one has to ensure the transmissivity of the UHV window in the employed infrared range. The emissivity of the sample at the infrared wavelength employed by the pyrometer also enters the measurement result. The emissivity is often not known precisely (or not at all) for the investigated system, which can lead to systematic errors up to 100 K.

3.2.3 Chemical cleaning Contaminants, which cannot be removed from the surface just by heating may chemically react with suitable gases (let into the UHV chamber) and desorb as molecules. Carbon, in particular, reacts during oxygen exposure on many metal surfaces to CO or CO2 at temperatures above their desorption threshold. Since carbon is a common contaminant and form the reoccurring residual gas contamination, oxygen annealing is a standard procedure during the sample cleaning of many metal surfaces. Similarly,

70 | 3 Preparation of surfaces sulfur can be removed during annealing in hydrogen as H2 S, and also oxygen can react to H2 O and desorb at temperatures well below the desorption temperature of oxygen. Since the reaction rate for these processes is often rather small, the surface has to be exposed to large doses of the respective gas. In such cases, it is advisable to guide the gas directly to the surface through a nozzle (doser), which reduces the gas load for the UHV chamber by about three orders of magnitude.

3.2.4 Sputtering by ions A method which can be used to clean almost all samples is the removal of surface atoms by bombardment with low-energy ions, called sputtering. For kinetic energies larger than about 100 eV, the collision between ions and atoms can be described as a binary encounter (Fig. 3.2(a)) between the atomic nuclei, since only a negligible amount of momentum can be transferred to the electrons. From energy and momentum conservation, the kinetic energy E2 transferred to the atom scattered by an angle ϑ2 relative to the kinetic energy E0 of the incident ion is obtained as E2 4A cos2 ϑ2 = E0 (1 + A)2

with the mass ratio A =

M2 . M1

(3.1)

The maximum energy transfer during a collision is obtained for equal masses A = 1. The dependence on the mass ratio A is not very strong: A deviation by a factor of 4 (A = 4 or A = 1/4) reduces the energy transfer to 64 % of the maximum value for A = 1. For subsequent collisions between sample atoms, A = 1 applies if all surface atoms have the same mass. If a sufficient amount of energy is deposited in the first collision, a cascade of further collisions develops (Fig. 3.2(b)), which can lead to the ejection and removal of several surface atoms. The sputter yield is the number of sputtered atoms per incident ion. In a simple approximation, it is ∝ E2 /EB , where EB is the binding energy of the surface atoms. The nominator E2 describes the energy transferred during the first collision, depending on the mass ratio and primary energy E0 (eq. (3.1)). For the sputter yield, the scattering angle ϑ2 is irrelevant, because the experiment contains an average over all angles ϑ2 . The yield first increases with the ion energy (Fig. 3.2(c)). For higher energies > 3 keV, the ions penetrate more deeply into the solid, and the energy deposited close to the surface decreases accordingly. The dependence on the ion species is given by the energy transfer during the binary collision between the primary ion and the surface atoms (eq. (3.1)), and reduces the sputter yield for very light ions. Crucial for the ejection of surface atoms is to deposit a large amount of energy immediately at the surface. Therefore, the sputter yield increases with the incidence angle up to about 70∘ relative to the surface normal. For larger incidence angles, the first collision may already lead to

3.2 Preparation of clean single-crystal surfaces | 71

Figure 3.2: (a) Sputtering process in the binary collision model. (b) Sputtering cascade. (c) Sputtering yield of a copper target at normal incidence for various primary ions [3.2].

a direct ejection of a surface atom in contrast to normal incidence, where at least two collisions are needed. However, for grazing incidence, the probability for ions to be simply reflected (quasi-)elastically increases. This reduces the deposited energy and the sputter yield decreases for incidence angles above about 70∘ . At a typical ion energy of 1 keV, the sputter yield is of the order of 1. For an ion current density of 1 μA/mm2 , one can estimate that it would take less than 3 seconds to remove one surface layer. However, the collision cascade leads to an intermixing of the surface region on the nanometer scale, which can displace contaminants from the surface to deeper layers. Therefore, sputtering times of several minutes are required to remove a layer of contaminants completely. Since primary ions are also implanted in the intermixing region, chemically inert noble-gas ions are generally used for sputtering. Implanted noble-gas atoms diffuse back to the surface and desorb during the obligatory thermal annealing of the defects created by the sputter collisions. An ion source usually has a fairly simple design (Fig. 3.3). The gas is ionized either in a continuous plasma discharge or by thermionically emitted electrons accelerated to about 100 eV (which corresponds to the maximum of the ionization cross section). The positive potential of the ionization region with respect to the sample potential defines the kinetic energy of the ions hitting the sample. The rest of the assembly is similar to an electron gun (Section 5.1.1) with reversed electric potentials: The positively charged ions are accelerated to the extraction aperture, through which the ion beam leaves the ionization region. Optionally, the beam can be focused by electrostatic lenses and/or deflected by pairs of condenser plates. The latter opens the pos-

72 | 3 Preparation of surfaces

Figure 3.3: Schematic design of an ion source.

sibility to scan the ion beam across the surface. Ion currents are typically of the order of 1–100 μA at energies of 0.5–5 keV. The ion-induced ablation or erosion is the foundation of analysis methods used in technological applications, which are outside the focus of this textbook on surface physics. The detection of the ions (or neutral particles) ejected by ion bombardment from the surface by a mass spectrometer gives quantitative information on the surface composition (secondary-ion mass spectrometry, SIMS). The signal of the various masses as a function of time during continuous ion bombardment can be converted to a depth profile of the elements after a suitable calibration (sputter yields depend on target mass). This depth can extend into the μm range with the resolution determined by the thickness of the intermixing region. In combination with a focused ion beam scanned across the surface, a three-dimensional element distribution can be obtained. Instead of the ejected particles, it is also possible to monitor the surface composition using electron spectroscopies (Chap. 5.2) during sputtering (Auger or XPS depth profiling).

3.3 Modification of surfaces The preparation of atomically clean single-crystal surfaces is not just an end in itself, but also provides well-defined substrates for the deposition of other materials. In this way, films or layers can be produced with surfaces exhibiting novel and interesting properties. The adsorption of gases for the investigation of catalytic or corrosive processes on an atomic scale also demands well-defined surfaces as a starting point. The basics of the physical processes of adsorption and layer growth have been extensively discussed in Sections 2.2.1 and 2.3. In this section, we will focus on the experimental realization.

3.3 Modification of surfaces | 73

3.3.1 Adsorbate-covered surfaces The simplest way to modify a surface is to adsorb gases on it (Section 2.2.1). The gas dose is determined by pressure ⋅ time with the traditional unit 1 Langmuir = 1 L = 10−6 Torr ⋅ s = 1.33 ⋅ 10−4 Pa ⋅ s. A closed adsorbate layer is obtained by a gas dose of the order of 1 L, under the assumption that all gas particles stick to the surface (sticking coefficient S = 1). Gas pressures in the relevant pressure range are measured by hotcathode ionization gauges (Bayard–Alpert gauge). The composition and purity of the gas is analyzed by using quadrupole mass spectrometers. In both cases, the neutral gas particles are first ionized by electrons, just as in an ion source (Section 3.2.4). The ions are then extracted to a collector, and the current is measured. For mass spectrometry the ions are accelerated into the quadrupole rod system for mass selection. The detected current is proportional to the pressure (or partial pressure for the selected mass) in the range below 10−1 Pa. Due to variations in the ionization probability, the measured current depends on the gas species. Ionization gauges are calibrated for nitrogen N2 as standard. For other gases, sensitivity factors R are used, which range from 0.20 for helium to more than 5 for hydrocarbons [3.3]. The values read on the ionization gauge have to be divided by R to obtain the correct pressure for the gas species introduced into the vacuum chamber.

3.3.2 Layers on surfaces Surfaces with different properties are obtained by depositing layers of atoms or molecules on a substrate surface (Section 2.3.3). If the layer has a well-defined crystallographic relation to the substrate, one speaks of epitaxy. One distinguishes between homoepitaxy and heteroepitaxy, depending on whether the same or a different material as the substrate material is deposited. The deposited layer thickness is often given in monolayers (ML), which has an obvious meaning for epitaxial growth. If the growth mode is not layer-by-layer (Section 2.3.1), the monolayer value corresponds to the average layer thickness. For adsorbed atoms or molecules, the coverage may also be given in monolayers with fractional values < 1 in many cases. For incommensurate superstructures or nonepitaxial growth, the monolayer is sometimes defined as a close-packed layer of the adsorbate. Layer thicknesses in the monolayer range can be measured by the frequency shift of a quartz crystal. The frequency change of the thickness-shear vibration (≈ 6 MHz) is proportional to the deposited layer thickness, and typically amounts to 5–50 Hz/ML. Many of the methods described in the following chapters also provide information on the deposited layer thickness and the growth mode. For producing films with a thickness in the monolayer range, a variety of different experimental approaches are available. The following are important evaporation methods:

74 | 3 Preparation of surfaces 3.3.2.1 Chemical vapor deposition In chemical vapor deposition (CVD), the material to be deposited is offered to the surface via a carrier molecule in the gas phase. These molecules decompose on the usually heated surface into the deposition material and other volatile components, which immediately desorb. The concept corresponds to the reverse reaction of chemical cleaning (Section 3.2.3). For example, metal layers can be obtained by CVD from the decomposition of metal carbonyls (metal-carbon-monoxide complexes) or silicon layers from disilane (Si2 H6 ). Graphene layers are also often produced by decomposition of hydrocarbons, such as ethene (C2 H4 ), on metal substrates (Fig. 1.7(c)). 3.3.2.2 Thermal evaporation The material to be deposited is heated in direct line of sight to the surface. The atoms evaporating with thermal kinetic energies can then condense on the sample held at a low temperature. To grow layers with thickness in the monolayer range, the sublimation rate of most elements is sufficiently high, so evaporation from the solid phase is feasible. One heats a piece of the high-purity material to the required temperature, either by running a current through it or by electron bombardment of the tip of a wire biased at a positive accelerating voltage (electron-beam evaporator). If the material has to be melted to reach a sufficient evaporation rate, it is placed in a crucible, which has to be used also for substances available only as powder. A special variation of thermal evaporators is the Knudsen cell, which is a completely closed compartment with the exception of a small evaporation aperture. It is homogeneously heated, such that a constant vapor pressure is maintained. Due to the good thermal shielding, the temperature of a Knudsen cell can be controlled very precisely, and stable evaporation rates are achieved. 3.3.2.3 Pulsed laser deposition In pulsed laser deposition (PLD), the material is evaporated by pulsed intensive laser radiation from a target, which is usually rotated and/or translated to avoid burning a hole in it. Besides neutral atoms, ions or clusters of atoms are also released from the target with kinetic energies up to 100 eV. Though the arrival time of particles on the target gets spread in time, due to different kinetic energies and flight times, many particles are deposited in a short time on the sample surface for each laser pulse. This leads to a higher nucleation rate (Section 2.3.2) than for thermal evaporation and often smoother layers. The possibly high kinetic energy of the deposited particles can lead to sputter processes and an intermixing in the surface region (Section 3.2.4). Pulsed laser deposition is particularly suited for materials consisting of several components (complex compounds), or those having high melting points. It also offers the option to deposit the material in a reactive gas atmosphere.

3.3 Modification of surfaces | 75

3.3.2.4 Sputter deposition Material sputtered by ion bombardment from a target (Section 3.2.4) can also be deposited on a sample surface. The energy and type (ions, atoms, or clusters) of the sputtered particles are similar to the situation in PLD. Instead of a directed ion beam (Fig. 3.3), a gas discharge can be used as an ion source. This permits large-area deposition and is used, e. g., in the semiconductor and coating industry. 3.3.2.5 Reactive segregation In contrast to the deposition of the material onto the surface, layers of chemical compounds, such as oxides, nitrides, or carbides, can be produced by including components of the substrate material via reactive segregation. The heated sample is exposed to a reactive gas atmosphere. At sufficiently high temperatures, all atoms in the nearsurface region can interdiffuse and reach the top surface layer. The reactive gas preferably binds to the substrate component, with which it can form the strongest chemical bond. These bound components cannot diffuse back into the bulk, and thus get enriched at the surface (adsorbate-induced segregation). At the employed high temperatures, often a well-ordered layer of the respective compound forms. However, the growing layer can act as a diffusion barrier, which inhibits the reaction between the substrate and gas atoms, and puts a limit on the achievable layer thickness. This effect can be sidestepped if the reactive component of the gas, e. g., nitrogen or carbon, is first implanted below the sample surface and both components segregate during the reaction. The layer thickness can then be controlled by the implanted dose. A good sample preparation is essential for the successful application of the various experimental methods of surface physics presented in the following chapters. As we have seen in this chapter, many techniques for the preparation of surfaces under ultrahigh-vacuum conditions are available, but not one for all systems. It is always crucial to feed back the results of the measurements to improve the sample preparation. The characterization of the sample and the understanding of the processes paves the way for producing surfaces with new geometric, electronic, and chemical properties using knowledge-based design.

Q3.1: Why are ultrahigh-vacuum conditions necessary for the investigation of atomically clean surfaces in most cases? Q3.2: Discuss various methods, which can be used to obtain atomically clean and well-defined surfaces of single crystals in UHV. Which method is suitable or unsuitable for surfaces of metals, alloys, and semiconductors? Q3.3: How do you select the ion species and energy to obtain a high sputter yield in ion-induced sputtering of a surface? Q3.4: A surface is exposed to a certain gas dose (pressure ⋅ time). Which factors determine the amount of gas (per unit area) found afterwards on the surface? Q3.5: Describe several methods for the deposition of layers on surfaces.

76 | 3 Preparation of surfaces

[3.1] [3.2] [3.3]

R. G. Musket, W. McLean, C. A. Colmenares, D. M. Makowiecki and W. J. Siekhaus, Appl. Surf. Sci. 10, 143 (1982). M. Schmid, A simple sputter yield calculator; www.iap.tuwien.ac.at/www/surface/sputteryield; Last visit 17.01.2020. J. E. Bartmess and R. M Georgiadis, Vacuum 33, 149 (1983).

4 Diffraction methods The geometric structure of a surface, i. e., the spatial arrangement of the atoms, influences almost all surface properties. The surface, in this context, is established by the first few atomic layers, in which the positions and/or the chemical character are different from those of the bulk (Chap. 1). Adsorbed atoms, molecules, or deposited layers also belong to the surface. Structure determination insinuates the visualization of individual atoms using real-space methods. Electron microscopes with atomic resolution exist, but they require extremely thin and specially prepared samples, have a limited surface sensitivity, and in most cases, image chains of atoms perpendicular to the surface. Scanning tunneling microscopes are extremely surface sensitive; they will be described in detail in Chap. 6. The images often give the impression to look at individual atoms. However, this is the case only if the electronic density of states correlates with the atomic positions at the particular tunneling voltage. In general, the electronic structure is scanned a few Ångstrøm above the surface, and atoms in deeper layers usually have negligible influence. A crystallographic determination of the complete surface structure is therefore in most cases not possible. For laterally periodic surfaces, the geometric structure can be determined with high precision using diffraction methods. Often the methods schematically presented in the upper part of Fig. 4.1 are employed: Low-energy electron diffraction (LEED), helium atom diffraction (HAS), and surface x-ray diffraction (SXRD). Common to all these diffraction experiments is the coherent irradiation of all accessible atoms in the surface region. The scattered contributions add up outside the surface, and a diffraction pattern is formed by interference. For electrons, multiple scattering contributions are important, whereas for HAS and SXRD, a description by single scattering is sufficient. The methods shown in the lower part of Fig. 4.1 are called interference methods. Electrons of a particular atom excited by photons (photoelectrons) can reach the detector either directly or after being scattered by neighboring atoms. The various paths interfere as direct and indirect signals just as in a holographic process, making this a holographic interference method using electrons. More common is the term photoelectron diffraction (PED), regarding the method as diffraction using an internal electron source. It does not require a periodic long-range order. However, many atoms with an identical (local) environment must be present for the excitation of photoelectrons in order to obtain a measurable signal. The same applies to the measurement of the surface extended x-ray absorption fine structure (SEXAFS), which makes use of the dependence of the excitation probability for a photoelectron on the backscattering from its neighbors. For periodic structures, diffraction methods are most adequate, because the coherent superposition of the scattering contributions from all unit cells leads to large signals in well-defined directions. The wavelengths of the employed radiation lie in https://doi.org/10.1515/9783110636697-006

78 | 4 Diffraction methods

Figure 4.1: Schematic illustration of different surface-sensitive methods relying on diffraction (top row) or interference (bottom row). Straight lines indicate beams of particles with finite mass, wavy lines x-ray beams. Incident beams are shown in red, outbound beams in green. The scattering paths (blue) are restricted to single-scattering events.

the range of the atomic distances, which leads to large diffraction angles facilitating the measurements. This condition determines the energy range for the respective radiation: For a wavelength of 1 Å, the corresponding energy is for photons Ehν = 12.4 keV, for electrons Ee = 150 eV, and for atoms (e. g., He) EHe = 20.6 meV. The photons in the x-ray range can penetrate deeply into the solid, and the surface sensitivity is low. Using grazing incidence and strong x-ray radiation sources (synchrotron radiation sources), precise surface structure determinations are possible. In contrast, atoms (typically helium) with thermal energies do not penetrate the surface at all, and thus “see” only the outermost atomic layer. Ideally suited for surface investigations are low-energy electrons, because their mean free path in solids is only a few Ångstrøm (Section 2.1.2), and the surface region of the solid is probed. Quantitative LEED and SXRD were developed into reliable methods for structure determination in the 1970s and 1980s, respectively. The diffraction of electrons and their associated wave character was demonstrated already in 1927 (for low-energy electrons by Davisson and Germer, for high-energy electrons by Thomson; Davisson and Thomson were awarded the Nobel prize in 1937), but the first quantitative structure determination by LEED succeeded only in the 1970s. In contrast was the first xray diffraction experiment performed already in 1912 by Laue (Nobel prize 1914), and the first structure analysis for bulk crystals was done immediately afterwards, in 1913, by Bragg (Nobel prize 1915). However, the first structure determination on surfaces by SXRD succeeded finally in 1986. LEED and SXRD carry some inherent problems: For LEED they lie on the theoretical side with the proper description of the multiple scattering of electrons. For SXRD they lie on the experimental side with the necessity of sufficiently intense x-ray sources, which depended on the development of syn-

4.1 Kinematic description of diffraction

| 79

chrotron radiation sources. Also SEXAFS and HAS are experimentally complex and require quantum-mechanical methods for their theoretical analysis and interpretation. Therefore, they were developed for quantitative structure determination about the same time as LEED and SXRD. In the following, we start with a description of the kinematic model of diffraction, which can be used for all diffraction methods (SXRD, HAS, LEED). Specific aspects of the particular methods are presented afterwards with emphasis on LEED, which is an experimental technique available in most surface-science laboratories.

4.1 Kinematic description of diffraction In kinematic theory, one assumes that the wave incident from the outside of an atomic scatterer (primary wave) is scattered by it and leaves the surface without any further scattering event. At the detector, the coherent scattering contributions from all scatterers hit by the primary wave interfere. For x-ray and helium diffraction, this description is almost exactly valid. For electrons, it has to be drastically modified, due to multiple scattering (Section 4.2.3). The primary wave is described as a plane wave with wave vector k 0 , as indicated in Fig. 4.2(a). The scattered waves are composed, due to the lateral translational symmetry from a set of plane waves with wave vectors k s ; only one of them is represented in Fig. 4.2(a). The scattering processes leading to the diffraction pattern are assumed to be elastic, i. e., |k s | = |k 0 |. Since perpendicular to the surface, the translational symmetry is broken, each unit cell with the basis vectors a1 and a2 parallel to the surface shown in Fig. 4.2(a) extends into the surface toward the bulk. In all these columns, the atoms have an identical arrangement, as illustrated in Fig. 4.2(b) for a simple case. The origin of the cell with number (m, n) is denoted with respect to an arbitrary zero, as r mn = ma1 +na2 . The various atoms j in the unit cell are reached by the vectors ρ . Collecting equivalent j

atoms from all unit cells leads to a description by stacked layers (with lateral periodicity) instead of columns. Besides the position of the atoms, their chemical identity can also vary, which then have different atomic scattering factors fj (k 0 , k s ) for the scattering from direction k 0 to direction k s . These scattering factors depend, of course, also on the interaction of the probing particles (photons, electrons, atoms) with the surface atoms, which will be discussed in later sections after the formalism common to all methods. For a primary wave with amplitude A0 = 1, the scattering contributions of all atoms add up at the detector to a scattering amplitude A = ∑ fj (k 0 , k s )e

−iΔk⋅ρ

j

j

∑ e−iΔk⋅rmn mn

80 | 4 Diffraction methods

Figure 4.2: (a) Diffraction setting at a surface and (b) construction of a unit cell from layers.

with the wave vector change (= momentum transfer), due to the scattering at the atoms, Δk = k s − k 0 . The diffraction intensity is obtained as 󵄨󵄨2 󵄨󵄨 󵄨2 󵄨 −iΔk⋅ρ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 j 󵄨󵄨 ⋅ 󵄨󵄨∑ e−iΔk⋅r mn 󵄨󵄨 ≡ |ℱ |2 ⋅ |𝒢 |2 . I = |A|2 = 󵄨󵄨󵄨∑ fj (k 0 , k s )e 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 mn ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ j ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 2 |𝒢 | |ℱ |

(4.1)

The factor |ℱ |2 is called form factor or structure factor, because it contains the structure, i. e., the positions of the atoms in the unit cell. The factor |𝒢 |2 is the lattice factor, because it depends only on the point lattice. Using r mn = ma1 + na2 the lattice factor |𝒢 |2 factorizes in 󵄨󵄨2 󵄨󵄨 +∞ 󵄨󵄨2 󵄨󵄨 +∞ 󵄨󵄨 −inΔk⋅a2 󵄨󵄨󵄨 −imΔk⋅a1 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 |𝒢 | = 󵄨󵄨 ∑ e 󵄨󵄨 ⋅ 󵄨󵄨 ∑ e 󵄨󵄨 , 󵄨󵄨 󵄨󵄨n=−∞ 󵄨󵄨 󵄨󵄨m=−∞ 󵄨 󵄨 󵄨 󵄨 2

(4.2)

where the individual factors are to be evaluated for a laterally infinite lattice. The two factors in eq. (4.2) vanish unless Δk ⋅ ai = Δk ‖ ⋅ ai = 2πqi

(i = 1, 2; qi ∈ ℤ),

(4.3)

a condition already used in Section 1.2.1. The condition is fulfilled, if and only if the parallel component of the momentum transfer Δk ‖ concurs with a discrete vector g of the reciprocal lattice (Laue condition), i. e., Δk ‖ = g hk = hg 1 + kg 2 .

(4.4)

4.1 Kinematic description of diffraction

| 81

Because of eq. (4.3), the basis vectors g 1 and g 2 have to fulfill the condition g i ⋅aj = 2πδij for i, j = 1, 2 and are obtained from the basis vectors of the lattice in real-space as g 1 = (g1x , g1y ) =

2π (a , −a2x ) Aa 2y

and g 2 = (g2x , g2y ) =

2π (−a1y , a1x ) Aa

(4.5)

with Aa = a1x a2y − a2x a1y . Solving g i ⋅ aj = 2πδij for the real-space vectors aj yields equivalent expressions a1 =

2π (g , −g2x ) Ag 2y

and a2 =

2π (−g1y , g1x ), Ag

(4.6)

with Ag = g1x g2y − g2x g1y . |Aa | and |Ag | are the areas of the unit cells in real space and reciprocal space, respectively. The translational symmetry enforces that, according to eq. (4.4), diffraction occurs only in directions with k s‖ = k 0‖ + g hk .

(4.7)

This allows labeling these directions with the indices (h, k). Note that this is independent of the used radiation, since the atomic scattering factors enter only the structure factor, and not the lattice factor. Because perpendicular to the surface, no translational symmetry exists, any amount of momentum can be transferred in that direction. This leads to the crystal truncation rods, which stand perpendicular to the plane spanned by the discrete reciprocal lattice points g hk and contain these points, as illustrated in Fig. 4.3. Since the scattering at the atoms is assumed to be elastic, the condition |k s | = |k 0 | (energy conservation) determines the momentum transfer for a particular diffraction direction.

Figure 4.3: Construction of the Ewald sphere for (a) normal or nearly normal incidence typically used in electron diffraction (LEED) and (b) oblique incidence typically used for surface x-ray diffraction (SXRD).

82 | 4 Diffraction methods The perpendicular component of k s is then obtained using eq. (4.7) as ks⊥ = ±√k02 − (k 0‖ + g hk )2 , where the + (−) sign applies to diffraction from an atomic layer toward the bulk (vacuum). For (k 0‖ + g hk )2 > k02 , one obtains a pure imaginary ks⊥ , i. e., the solution is an evanescent wave, and no wave gets diffracted back from the surface. The perpendicular momentum transfer is simply calculated as Δk⊥ = ks⊥ − k0⊥ . The Ewald sphere (Fig. 4.3) allows determining, by simple construction, the diffraction directions for a given incident wave vector k 0 (with energies E = (ℏk0 )2 /2m for particles of mass m and E = hν = cℏk0 for photons). In the reciprocal lattice of crystal truncation rods, the incident wave vector k 0 is pointed to the origin g 00 = 0, and a sphere with radius |k0 | is plotted around the origin (Ewald sphere). Only the crystal truncation rods, which intersect with the Ewald sphere, define k s vectors and lead to diffracted waves. With increasing k0 and, accordingly, increasing energy, more and more diffraction orders— labeled by g hk or (hk)—become available. Figure 4.3(a) illustrates the situation typical for low-energy electron diffraction, where usually for normal or near-normal incidence is measured: The diffraction order, on which the associated crystal truncation rod k s ends, is sampled by variation of the electron energy, and thus k0 and the intensity of the diffraction spot is measured. In surface x-ray diffraction, illustrated in (b), oblique incidence is used and the crystal truncation rod is sampled by rotating the crystal at fixed photon energy and k0 . For each diffraction spot (h, k), the lattice factor |𝒢 |2 has the same value. The intensity of the spot, and thus the information on the atomic arrangement in the column of the unit cell, is contained in the structure factor |ℱ |2 . Put simply, |𝒢 |2 gives the position of the diffraction spots and determines in reciprocal space the size and shape of the unit cell. |ℱ |2 is determined by the positions of the atoms in the unit cell. For an arbitrary arrangement of the atoms, the sum in |ℱ |2 cannot be easily evaluated. The salient features can be illustrated for the simple case of a bulk-like terminated surface. We take the situation shown in Fig. 4.2(b) with the additional simplification, ρ = cj +ρ = (j−1)a3 +ρ . The vector a3 connecting equivalent atoms in neighboring layj

j‖

‖

ers is assumed to be constant (independent of j), and ρ is the value of ρ , also chosen ‖

j‖

to be independent of the layer index j. For identical scatterers (fj (k 0 , k s ) = f (k 0 , k s )), and using |e

−iΔk⋅ρ

‖

| = 1, one obtains the intensity as

󵄨󵄨+∞ 󵄨󵄨2 󵄨󵄨2 󵄨󵄨+∞ 󵄨󵄨 󵄨󵄨2 󵄨󵄨󵄨 󵄨󵄨2 󵄨󵄨󵄨 󵄨󵄨 −iΔk⋅a3 j 󵄨󵄨󵄨 −iΔk⊥ a3 j 󵄨󵄨󵄨 I = |ℱ | = 󵄨󵄨f (k 0 , k s )󵄨󵄨 ⋅ 󵄨󵄨 ∑ (e ) 󵄨󵄨 = 󵄨󵄨f (k 0 , k s )󵄨󵄨 ⋅ 󵄨󵄨 ∑ (e ) 󵄨󵄨 . 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 j=0 󵄨 j=0 󵄨 2

For the further calculation, the attenuation of the primary wave on the incident path in the surface, and the attenuation of the diffracted wave on the exit path, have to be taken into account. An elegant, successful way is the introduction of an imaginary part kI for the wave number k0,s inside the solid, i. e., k0,s → k0,s + ikI . In a simple

4.1 Kinematic description of diffraction

| 83

Figure 4.4: Intensity of the (00) spot for normal incidence (a) for electrons as function of energy and (c) for x-rays (on a logarithmic scale) as function of the continuous diffraction index ℓ. In (b) the reciprocal lattice with the crystal truncation rods is shown. The open circles on the rods mark integer values of ℓ corresponding to the diffraction from a periodic lattice in three dimensions.

situation, we consider the (0,0) spot (g hk = 0) and normal incidence of the primary wave (k 0‖ = 0, k0⊥ = k0 ). This yields Δk⊥ = −2k0 − 2ikI , and the intensity is obtained as I00

󵄨󵄨+∞ 󵄨󵄨2 |f (k 0 , k s )|2 󵄨󵄨2 󵄨󵄨󵄨 󵄨󵄨 2(ik0 −kI )a3 j 󵄨󵄨󵄨 . ) 󵄨󵄨 = = 󵄨󵄨f (k 0 , k s )󵄨󵄨 ⋅ 󵄨󵄨 ∑ (e 󵄨󵄨 (1 − e−2kI a3 )2 + 4e−2kI a3 sin2 (k0 a3 ) 󵄨󵄨󵄨 j=0 󵄨

(4.8)

In Fig. 4.4(a), the intensity of diffracted low-energy electrons is plotted as function of the electron energy E = (ℏk0 )2 /2m for the scattering factor |f (k 0 , k s )| = 1. For the attenuation of the electron waves, a representative constant value of kI a3 = 0.2 (a3 = 2.12 Å) was used. The energy dependence of the mean free path (Section 2.1) and the resulting variation of the attenuation parameter kI was neglected. Maxima are found for energies, at which the real part of the momentum transfer Δk⊥R = −2k0 fulfills the condition Δk⊥R a3 = 2πℓ (ℓ ∈ ℤ). This is the third Laue condition, which has to be obeyed in the infinite three-dimensional lattice. It results from the constant distance a3 of the atomic layers at the surface assumed in the simple illustrative example. In Chap. 4.2, we will see that the multiple scattering of the electrons leads to a lot more maxima. The positions calculated from the third Laue condition are marked on the crystal truncation rods by small circles in Fig. 4.4(b). For x-rays, the attenuation is much weaker and the penetration into the bulk much deeper than for electrons. The previously described behavior is much more distinct, as shown in Fig. 4.4(c) for kI a3 = 5 ⋅ 10−4 and |f (k 0 , k s )| = 1. Here the notation for the three-dimensional case is commonly used, where k0 a3 is replaced by πℓ in eq. (4.8). The diffraction index ℓ is, in this context, a continuous variable in contrast to the integer values in threedimensional diffraction. The low intensity between the maxima becomes visible only on the logarithmic scale in Fig. 4.4(c). In the opposite situation, of very strong attenuation with large values of kI , e. g., in helium diffraction, only the topmost layer of the surface contributes to the diffraction, and there is no influence from the equidistant layer spacings.

84 | 4 Diffraction methods

For the calculation of the lattice factor, we assumed a laterally periodic lattice. If antiphase domains (Fig. 1.8) exist, which are smaller than the coherence length (Section 4.2.2), the respective 𝒢 have to be summed before squaring, which leads to a splitting of spots. The symmetry elements of the structure enter the calculation of the structure factor. However, diffraction spots, which should be identical due to a rotation axis, have the same intensity only if the rotational symmetry of the complete system is preserved. This includes the primary beam, which has to have normal incidence to observe the rotational symmetry in the diffracted beams. This observation can be used for a precise alignment of the normal incidence of the primary beam. Similarly, a systematic cancellation of spots due to glide planes is found only if the primary beam is incident parallel to the glide plane. Another point to consider in the calculation of the structure factor is the fact that the atoms of the unit cell are not fixed at their positions ρ , but vibrate with amplitudes u(t) around these positions, j

even at zero absolute temperature (ground-state vibrations). The atoms are only momentarily at their ideal crystallographic position. This reduces the scattering factor, which is described by the Debye– Waller factor. For its calculation, one needs to recognize that the interaction time of an electron or photon with an atom is much shorter than the vibrational period of the atom. To simplify the calculation, one further assumes that the atoms vibrate independently of each other, i. e., no fixed phase relation exists between their displacements. The scattering thus takes place at a distorted lattice, but during the scattering process the lattice is static. Therefore, the particles do not loose energy in the scattering process, which is also called zero-phonon scattering (for the inelastic scattering contributions see Sections 2.1.2 and 5.4). Some of the particles are scattered into all available directions (thermal diffuse scattering), away from ideal diffraction directions, which reduces their diffraction amplitude. This can be described by a reduction of the scattering factor. Averaging over all displacements [4.1, 4.2] leads to a temperature-dependent reduction factor e−M(T ) for the atomic scattering factors f = f (k 0 , k s ), f (T ) = e−M(T ) f

with

M(T ) =

1 2 ⟨(Δk ⋅ u) ⟩ = 2

1 2 2 (Δk) ⟨u ⟩ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 6

,

(4.9)

for isotropic displacements

where ⟨u2 ⟩ = ⟨u2 (T )⟩ is the mean square amplitude of the atomic vibrations at temperature T . The reduction factor for the diffraction intensities (Debye–Waller factor) is obtained in the kinematic approximation as I ∝ |f (T )|2 ∝ e−2M(T ) .

4.2 Diffraction of low-energy electrons 4.2.1 Experiment The LEED experiment is illustrated in Fig. 4.5(a), showing the schematic of a LEED optics. It consists of an electron gun and a spherical fluorescent screen with several spherical grids mounted in front of it. The sample is positioned in the center of those. From the electron gun, the primary beam with an energy E = −eV reaches the sample. The accelerating voltage V applied to the electron gun (see Section 5.1.1) is negative, since the sample is grounded. Besides the elastically back-diffracted electrons (green arrows) inelastically scattered electrons also leave the sample (Fig. 2.1). All electrons traverse on straight trajectories the field-free space between the sample and the first grid, which is at ground potential like the sample. To the next grid, a negative voltage VB (retarding potential with −eVB ≲ E) is applied, such that a large part of the

4.2 Diffraction of low-energy electrons | 85

Figure 4.5: (a) Schematic of a LEED optics and (b) diffraction image of an Ir(111) surface at 180 eV electron energy with marked reciprocal unit cell, and indices (hk) for several diffraction spots. The (00) spot is for normal incidence of the primary beam hidden by the electron gun.

inelastically scattered electrons with energies less than e|VB | is held back. Between the last grounded grid and the fluorescent screen, a high voltage of about 5 kV is applied, which accelerates the diffracted electrons onto the fluorescent screen. Visible diffraction spots are produced by luminescence, which are presented in Fig. 4.5(b) in planar projection. If the fluorescent screen is made on glass, the spots are visible from the back of the LEED optics without interference by the sample holder (though the electron gun casts a shadow in the center). In the most simple implementation of a LEED optics, only two grids are needed, and the grounded grid in front of the fluorescent screen may be omitted. If the LEED optics is used as a retarding-field analyzer for Auger-electron spectroscopy (Section 5.2.3), the energy resolution is improved by employing double grids for retardation, which reduces the penetration of the electric field. The grounded grid in front of the fluorescent screen is also required for Augerelectron spectroscopy, because it reduces the capacitance between retarding grids and screen, and thus the capacitive coupling of the modulation voltage (Section 5.1.5) from the retarding grid to the fluorescent screen, where the signal current is collected.

4.2.2 Geometry of the diffraction pattern Using a spherical fluorescent screen puts the diffraction spots in positions directly matching the Ewald sphere. The planar projection of the diffraction image in Fig. 4.5(b) corresponds to a cut through the arrangement of crystal truncation rods. From Fig. 4.5(a) one obtains for normal incidence of the primary beam for a spot g hk , sin α =

|g hk | |k 0 |

=

distance on screen d = . radius of screen R

86 | 4 Diffraction methods This relation shows that with increasing electron energy (increasing k0 ), the diffraction angles become smaller, and thus the spots move toward the center. A quantitative evaluation allows determining the length of the basis vectors of the surface unit cell with an accuracy of about 1 % using eq. (4.6). In the diffraction image of the Ir(111) surface for normal incidence, the innermost spots clearly show the threefold symmetry of the fcc(111) surface (Fig. 1.3). Note the different notation for equivalent spots in the hexagonal structure: (10), (11), (01). In crystallography, this is avoided by the introduction of an additional index (Fig. A.1). From the orientation of the equilateral triangle spanned by the (e. g., innermost) equivalent spots, one immediately recognizes the orientation of the mirror planes (vertical and rotated by ±120∘ , according to the threefold symmetry). The next six spots [(11), (12) . . . ] are not in mirror planes and are therefore equivalent, showing the same intensity. Because of the lateral translational symmetry of the surface, the factorization of the diffraction intensity I = |ℱ |2 ⋅ |𝒢 |2 into a structure and a lattice factor stays valid, even with multiple scattering of the electrons. The multiple scattering is caused by the large scattering cross section of low-energy electrons at atoms, which corresponds roughly to the cross-sectional area of an atom in the solid, about 10 orders of magnitude larger than for x-ray photons. Whenever an electron encounters an atom, it will be scattered with high probability. Since the atoms are tightly packed in a solid, an electron will be scattered again and again along its way. An atom thus does not only scatter the incident primary wave (Fig. 4.6(a)), but also scatters waves from neighboring atoms (Fig. 4.6(b)). The atom also scatters waves toward its neighbors, i. e., the total wave incident on the atom depends on its own scattering cross section (Fig. 4.6(c)). This requires a self-consistent treatment of the problem. In the end, the total scattered wave originating from an atom can be described instead of a single-scattering factor fj by a dynamic scattering factor fjD , which takes all scattering processes into account. The factorization of the intensity I = |ℱ D |2 ⋅ |𝒢 |2 into a dynamical structure factor and the unchanged lattice factor remains valid. The description of the electrons at the surface as Bloch waves introduced in Section 1.2.1 includes all multiple-scattering

Figure 4.6: Schematic illustration of the scattering dynamics. An atom in the surface (side view) scatters not only the directly incident primary wave (a), but also the waves scattered by its neighbors (b). In addition, multiple scattering processes at the atom and its neighbors have to be taken into account. One scattering path is shown in (c) as an example.

4.2 Diffraction of low-energy electrons | 87

processes. However, this description is very elaborate and not practicable for electron energies far above the Fermi energy, which would be required for LEED. The analysis of the geometry of a LEED pattern yields the position of the spots, and thus the shape and size of the unit cell in real space. In practice, one has to consider that for certain electron energies, the intensity of a spot may disappear, due to the energy dependence of the structure factor. A systematic vanishing of spots can occur, due to symmetry properties (e. g., a glide plane). The diffraction spots of the substrate can be recognized as long as the mean free path is longer than the thickness of the adsorption or reconstruction layer, which is usually the case for energies ≳ 100 eV (cf. Fig. 2.3). Commensurate superstructures caused by adsorption or reconstruction, in most cases, have larger unit cells compared to the one of substrate. Then the LEED patterns always contain spots at the positions of those of the substrate, and additional spots in between. As discussed in Section 1.1.2, most superstructures possess a lower symmetry than the substrate, and domains of different orientation can form. The incoherent superposition of the diffraction patterns from different domains leads in general to LEED patterns, whose spot positions show the full symmetry of the substrate. Figure 4.7 presents a few examples. The validity of the incoherent superposition relies on the limited coherence of an electron beam. The reason for it is mainly the finite size of the cathode, which independently (i. e., with uncorrelated phase) emits from different spots on its surface. The emitting area of the cathode is imaged by lenses onto the fluorescent screen (Fig. 5.1), such that a spot of finite width forms (a diffraction spot is actually an image of the source). This implies that the area of the surface, which is illuminated by waves with the same phase (coherence region), has a finite diameter instead of an infinite extension as assumed in Chap. 4.1. This finite width of the coherence region is also called instrumental transfer width. The finite energy spread of the thermionic electron source (hot cathode) also reduces the transfer width. The coherence region of a standard LEED optics has a diameter of about LK = 100–200 Å. Only the amplitudes of waves originating from a region of this diameter interfere. Amplitudes from other neighboring regions are superimposed incoherently, and the intensities are added. Therefore, diffraction spots— as seen in Fig. 4.5(b)—have spot profiles of finite full width at half maximum, which is reciprocal to LK , i. e., Δg‖ ≈ 2π/LK . For LK ≈ 30 a (a = unit cell extension), one obtains Δg‖ ≈ g/30. The spot width increases further if the diffraction occurs from regions (domains) with diameter LD smaller than the one of the coherence region. In such a case, Δg‖ ≈ 2π/LD results. Statistical defects within a domain do not increase the spot width, neither do the thermal vibrations of the scatterers (as mentioned before they do enter only the structure factor via the Debye-Waller factor). However, both effects increase the background in the diffraction image. By special provisions, one can build electron sources with coherence regions with diameters of more than 1000 Å. Then the amplitudes (and not the intensities) of differently ordered small domains are superposed, and a spot profile corresponding to this order (or disorder) is generated. Its evaluation yields information about the average order or disorder at a surface; the method is called spot profile analysis by LEED (SPALEED). The analysis can be performed in the kinematic approximation by evaluation of the lattice factor only, because the structure factor (this applies also to the dynamic structure factor) does not vary significantly across the region of the spot width.

88 | 4 Diffraction methods

Figure 4.7: Adsorption structures and symmetrically equivalent domains with incoherent superposition of their diffraction patterns. Substrate spots are plotted by filled circles, superstructure spots by open circles. (a) On a square substrate, the superposition of (2 × 1) rotational domains can be distinguished from a (2 × 2) structure. (b) illustrates the formation of the LEED pattern of the Ir(100)-(5 × 1) surface (cf. Fig. 4.10(a)). (c) For a hexagonal substrate, the diffraction pattern from three (2 × 1) domains rotated by 120∘ (rotational domains) coincides with the LEED pattern from a (2 × 2) structure.

4.2.3 Quantitative structure analysis Whereas the identification of the shape and size of the two-dimensional unit cell is straightforward, the determination of the structure, i. e., the positions ρ (and chemij

cal species) of the atoms within the unit cell, is more difficult. Since the information

4.2 Diffraction of low-energy electrons | 89

about the structure is contained in the intensities of the spots, the first task is to measure them, and the second step their appropriate analysis, which is the real hurdle, due to the multiple scattering. In principle, it would be sufficient to restrict the data set to the intensities of the multitude of spots observable at one single energy, which are distinctly different as the LEED pattern in Fig. 4.5(b) shows. To obtain as much information as possible, one measures the spot intensities as a function of the energy (I(E) or I-V spectra). Instead of an elaborate direct measurement of diffracted electron currents, nowadays a video camera is usually employed. It records the optical image on the fluorescent screen through a window in the UHV chamber. The video signal can be digitized by a computer, and the spots tracked online while the energy is scanned [4.2, 4.3, 4.4]. The I(E) spectra of the individual spots can also later be extracted off line from the recorded images. The measurement time is a few minutes, so sample contamination or degradation is minimized. The normal incidence of the primary beam can be checked quickly with an accuracy of ≈ 0.1∘ by test runs of symmetry-equivalent spots. Since LEED intensities are very sensitive to the incidence angle, one usually measures at normal incidence, which can be aligned very accurately. The multiple scattering leads to I(E) spectra with rich structure, as illustrated by the examples shown in Fig. 4.8 in comparison to Fig. 4.4(a). The extraction of structural data from the I(E) spectra faces two difficulties: First the information of the phase gets lost—as in any diffraction experiment—when the absolute square of the total amplitude is calculated to obtain the intensity (eq. (4.1)). Second, the structure factor |ℱ |2 contains the structure-dependent dynamic scattering factors f D instead of the simple

Figure 4.8: (a) Experimental and fitted calculated I(E) spectra from an Ir(111) surface for three selected beams. In (b) calculated intensity curves of the (01) beam of the Ir(111) surface are shown for different relaxations of the first layer spacing from the bulk value.

90 | 4 Diffraction methods atomic scattering factors. The dynamic scattering factors have to be calculated, taking the multiple scattering into account. With dynamic scattering theory [4.2, 4.5, 4.6], I(E) spectra for various structural models have to be calculated, and the models with the associated parameters have to be varied until a satisfactory agreement between experiment and theory has been achieved. The rich structure of the spectra and their drastic changes upon tiny structural modifications (Fig. 4.8(b)) require a quantitative measure for the comparison of experimental and calculated spectra, which is called reliability factor or short R factor. Most commonly used is the R factor of Pendry, which takes values between 0 ≤ R ≤ 2. It emphasizes the agreement in the positions of maxima and minima in the spectra instead of absolute intensities, and allows estimating the statistical error in the determination of the model parameters [4.7]. The smaller R, the better the agreement (fit) between measurement and calculation. In practice, one believes that the basic structural model is correct if the R factor is better than 0.2, though details of the structure might still be unidentified at that level of agreement. The more atoms are in the unit cell, the smaller the scattering contribution of an individual atom to the total wave field becomes. This means that a complex structural model requires a high quality of the fit for its validation. An improved fit also reduces the error in the resulting structure parameters. For well-prepared systems, R factors of 0.1 or smaller can be achieved. A reliable fit of model parameters requires a sufficiently large data basis. The width ΔE of a peak in an I(E) spectrum of typically 20 eV may be taken as a quantum of information (data point). A reliable determination of N structure parameters requires a cumulative energy range of all inequivalent spectra of rNΔE. The redundancy factor r should be in the range of 5–10. Put simply, one needs at least 100–200 eV spectral range for each model parameter fitted. Spectra of low-index diffraction beams are shown in Fig. 4.8(a). For the structure analysis of Ir(111), the total data basis was 5882 eV. The six topmost layer spacings (Fig. 1.9) were fitted, and an R factor of 0.05 was achieved. Due to the multiple scattering, the kinematic approximation of Chap. 4.1 cannot be used to calculate the diffraction intensities. In principle, one could solve the Schrödinger equation for the complicated potential constructed from the arrangement of the surface atoms (the kinematic approximation is the first Born approximation of the solution). This would be quite elaborate and almost impossible for complex structures. Instead, one proceeds according to a hierarchical concept (Fig. 4.9). One exploits the composition of the surface by atomic layers put together from atomic scatterers. For the dimensions of the lateral unit cell, the values of the substrate are usually used, which are taken from the literature. As a first step, the atomic scattering factor is calculated from the solution of the Schrödinger equation for a plane wave approaching the atom. Scattering occurs at the electron cloud and the atomic core. For heavy elements, the electrons gain considerable kinetic energy by the Coulomb attraction of the core, which requires the solution of the Dirac equation. Since the valence electrons contribute little to the scattering, one can assume a spherical atomic potential, which facilitates the calculation in a partial-wave representation, using phase shifts for a limited set of angular momenta. Figure 4.9(a) shows the absolute value of the scattering factor |f (k 0 , k s )| = f (ks , ϑ) as function of the k ⋅k

polar angle ϑ = arccos |k 0||ks | (polar plot: The arrow length is proportional to f (ks , ϑ).) for a platinum 0 s atom at an electron energy of 100 eV. The second step is the calculation of the layer diffraction (b), taking the multiple scattering between the atoms in surface parallel layers into account in a selfconsistent scheme, which is usually the most intricate part of the calculation. The final third step calculates the total diffraction from the surface (c) by stacking the individual layers, including the

4.2 Diffraction of low-energy electrons | 91

Figure 4.9: Three-step route (a–c) for the calculation of LEED intensity spectra. The blue curves in (b) and (c) are calculated kinematically and demonstrate, in comparison to the dynamically calculated spectra (magenta lines), the drastic failure of the kinematic approximation. The left side of (c) illustrates the multiple diffraction between layers for some multiple diffraction processes. multiple diffraction between the layers. The computational cost for calculating the diffraction intensities incorporating multiple scattering is several orders of magnitude above the one for the kinematic approximation and increases steeply with the complexity of the structure. In all calculations, one has to consider that the wave number k0 of the electrons increases when they enter the solid. In the approximation of a free electron gas, this can be described by an inner potential V0R , which depends on the material, and has values of the order of 10 eV. To describe the attenuation by an imaginary part kI of the wave number (Chap. 4.1), an imaginary part of the potential iV0I (optical potential) is introduced, such that an electron of kinetic energy E in vacuum has in the solid the energy E + V0R + iV0I . Using the dispersion relation for free electrons (ℏk)2 /2m = E + V0R + iV0I , one obtains in the approximation |V0I | ≪ E the real and imaginary part of the wave number in the solid as kR ≈

√2m(E + V0R ) ℏ

and kI ≈

√mV0I

ℏ√2(E + V0R )

.

The imaginary part kI thus also depends on the energy. Values in the range V0I = 4–6 eV yield realistic values for the mean free path (Fig. 2.3) for energies ≳ 50 eV. The width of a particular peak in an intensity spectrum is determined by the number and weight of the layers contributing to the diffraction processes. These contributions depend on the penetration depth described by kI , and therefore

92 | 4 Diffraction methods

the width depends on V0I . A calculation yields a full width at half maximum of ΔE = 4V0I [4.7]. Note that often peaks overlap, due to the many interference maxima in a spectrum caused by the multiple scattering.

Figure 4.8(a) illustrates how close calculated I(E) spectra for the best structural model can come to the experimental data. All important spectral features are reproduced, in particular the maxima and minima, as constructive and destructive interference of scattering contributions. Structural parameters (e. g., layer spacings, bond lengths, etc.) can be determined with an accuracy of 0.1 Å, or better. Note that atomic distances lie in the range of 2–3 Å. LEED spectra are particularly sensitive to vertical structural parameters, since the momentum transfer occurs predominantly perpendicular to the surface in the back-diffraction geometry of a LEED experiment (Fig. 4.5(a)). This is exemplified in Fig. 4.8(b) for the (01) spot from an Ir(111) surface. A reduction of the first layer spacing by 0.02 Å leads to noticeable changes of the spectrum, for 0.1 Å the changes are pronounced. Thus, parameters like the first layer spacing can be determined with an accuracy of 0.01–0.02 Å. Figure 4.10 shows the diffraction pattern from the Ir(100)-(5 × 1) surface and the results of a LEED structure analysis. The surface is heavily reconstructed, due to the quasi-hexagonal close-packed arrangement of the surface atoms (Fig. 1.15(e)). This leads to a strong buckling of the surface layer, which does not fit well on the underlying fcc(100)-like (i. e., square) atom layer. The interaction of the hexagonal layer with the underlying layers induces a reconstruction of the latter, which can be seen up to the fourth layer underneath. For the description of the resulting complex structure, 17 parameters are required, which were determined with

Figure 4.10: (a) LEED pattern of the Ir(100)-(5 × 1) surface at an electron energy of 174 eV and normal electron incidence. (b) Result of a LEED structure analysis in a vertically expanded side view together with the structure parameters (in Å).

4.3 Diffraction of high-energy electrons | 93

an accuracy of < 0.03 Å (Fig. 4.10(b)) [4.8]. In LEED structure analyses of even more complex structures, more than 60 structure parameters were determined [4.9]. As discussed in the preceding paragraphs a LEED intensity analysis requires as input information a structural model, whose parameters are then determined computationally. In LEED so-called “direct methods” were successful only in a few cases. In particular for complex structures, preliminary information from other surface methods are valuable if not indispensable. Scanning tunneling microscopy (Chap. 6) plays here a prominent role. It provides an image of the surface (electronic structure), which can suggest a structural model, or exclude many others models compatible with the lattice structure of the diffraction pattern. Theoretical methods, in particular those based on density functional theory, can also provide clues for structural models.

4.3 Diffraction of high-energy electrons At higher electron energies, the diffraction angles become smaller and more and more plane waves contribute to the diffraction. Measurement and quantitative analysis of the diffraction intensities, accordingly, become more difficult. Therefore, reflection high-energy electron diffraction (RHEED) is usually used for qualitative studies only and not for quantitative structure analysis. At the employed electron energies in the range of 5–50 keV, the elastically scattered electrons do not have to be accelerated to the fluorescent screen. Shallow incidence and exit angles relative to the surface plane are chosen, which leads to a small momentum transfer to the surface, and thus the Debye–Waller factor has little influence on the intensity (eq. (4.9)). The high energy makes the radius of the Ewald sphere large compared to the reciprocal lattice vectors. Due to the small angles, the sphere crosses the crystal truncation rods at an oblique angle (Fig. 4.3(b)), and the defect-related width leads to elongated spots. The diffraction intensities are quite sensitive to the roughness of the surface. RHEED is often employed for monitoring the layer growth, e. g., in semiconductor epitaxy. Completed layers have a high reflectivity, whereas partial layers have a low reflectivity, due to their inherent roughness (cf. Section 2.3.1), which can be monitored during the growth. The oblique incidence and exit angles of the electron beam permits a sideways mount of the electron gun and the fluorescent screen. This makes the surface accessible for evaporators from below, which is essential for evaporation from crucibles.

4.4 Surface x-ray diffraction Surface x-ray diffraction (SXRD) requires, as a rule, high-intensity synchrotron radiation sources, because only about 1015 /cm2 atoms contribute to the scattering signal from the surface compared to about 1023 /cm3 atoms for the diffraction from a bulk

94 | 4 Diffraction methods crystal. The surface sensitivity is achieved by oblique incidence and/or exit of the xray radiation, which prolongs the path of the x-rays in the surface region. The necessity of the use of synchrotron radiation makes the experiment quite elaborate compared to LEED, which is a standard equipment in UHV systems for surface studies. The validity of the kinematic approximation for the analysis of the diffraction intensities, however, is a definite advantage of SXRD compared to LEED. The large penetration depth of xrays opens the possibility of SXRD to study buried interfaces. These are created during the layer growth on a substrate and are unaccessible for most other methods, including LEED, if they are beyond a depth of 5–10 layers. The incident x-ray radiation is scattered by the electrons, which oscillate in the electric field of the electromagnetic wave. Scattering at the core is negligible due to its large mass. X-ray diffraction is thus sensitive to the electron density distribution in the unit cell, from which the atomic positions can be obtained. In Chap. 4.1, the structure factor |ℱ |2 of a (00) beam was calculated for a bulklike terminated crystal using the common notation I00ℓ for a three-dimensional crystal with a continuous diffraction index ℓ (Fig. 4.4(c)). For a general diffraction order (h, k) Bulk 2 of the bulk-like terminated crystal, we use in the following the notation Ihkℓ = |ABulk hkℓ | Bulk for the intensity of a beam with amplitude Ahkℓ . A bulk-like termination is in most cases not observed, the atom arrangement in the top layers is different from that in the bulk, due to relaxation, reconstruction, and/or adsorption. Therefore, this part of the surface (Fig. 4.11(a)) is described separately from the bulk by 2πi(hxj +kyj +ℓzj ) ASurf , hkℓ = ∑ fj e j

where (xj , yj , zj ) are the dimensionless coordinates of the surface atoms relative to the basis vectors of the bulk unit cell. The interference of bulk and surface amplitudes yields the total intensity 󵄨 Surf 󵄨󵄨2 Ihkℓ = 󵄨󵄨󵄨ABulk hkℓ + Ahkℓ 󵄨󵄨 .

Figure 4.11: (a) Division of a sample in a surface region and an unaltered bulk region. (b) Diffraction amplitude with the surface and bulk contributions from a sample with a 10 % contraction of the first layer spacing (after [4.10]).

4.5 Helium diffraction |

95

Figure 4.11(b) shows an example for a surface with a 10 % contracted first-layer spacing [4.10]. Note that in Fig. 4.11(b) the absolute values of the amplitudes are plotted and not the intensity as in Fig. 4.4(c) to emphasize the superposition of the amplitudes in the preceding equation. A particular advantageous situation for the application of SXRD is the existence of a superstructure, because new beams occur by diffraction at the surface. The superposition with diffraction contributions from the bulk can be neglected, because multiple diffraction does not play a role.

4.5 Helium diffraction Also, particles with higher mass than electrons can be scattered or diffracted at surfaces. For atoms, the term scattering and diffraction are often used interchangeably. Scattering often refers to inelastic scattering, whereas elastic diffraction experiments are sensitive to the surface structure. By inelastic scattering, the dispersion of surface phonons can be measured benefiting from the fact that energies and momenta of particles and phonons are of the same order of magnitude. Experiments for atom diffraction or atom scattering have been performed using H, He, Ne (also molecules, such as H2 , HD, and D2 , have been employed). Helium is mostly used, because it is scattered more strongly compared to the heavier neon atoms, and because it exhibits no chemical reactivity in contrast to hydrogen atoms or molecules. Therefore, we focus the discussion on helium atom scattering (HAS). Helium atoms of thermal energies (10–100 meV) have de Broglie wavelengths comparable to the atomic distances, and thus are diffracted at large angles. The beam of helium atoms is formed by adiabatic expansion of helium gas at high pressure (100–200 bar) through a small orifice of 5–10 μm diameter (nozzle) as an ultrasonic jet. It passes through several apertures, which separate differential pump stages to ensure that the helium pressure in the UHV chamber remains low. The thermal energy spread is reduced by multiple collisions within the ultrasonic beam, selecting only the central part by one or more apertures (skimmers). The diffracted atoms are neutral and have to be ionized before being sent through a mass spectrometer for detection. Due to their low kinetic energy, the atoms cannot penetrate into the surface. They interact only with the top-most atomic layer, which makes the method extremely surface sensitive. The low energy also allows investigating weakly-bound adsorbates without causing desorption, diffusion, or dissociation. Such processes can be triggered by the electrons in LEED and x-rays in SXRD (see Section 2.2.2 and Fig. 2.7). Upon approaching the surface, the atoms experience a strong repulsive interaction when their electron shell overlaps with the one of a surface atom (Pauli repulsion). Here the valence electrons play the dominant role (in contrast to LEED), and one obtains information on the geometric structure as well as on the binding configuration at the surface. Due to the strong repulsion, almost all atoms are back diffracted, which results in high diffraction intensities. The allocation of the total intensity to

96 | 4 Diffraction methods

Figure 4.12: Beam intensities from helium diffraction at the reconstructed Au(110)-(1 × 2) surface. The wavelength of the primary helium beam is 1.09 Å, and the incidence angle 48∘ (after [4.11]).

the various diffraction orders strongly depends on the electronic corrugation of the surface. For example, ionic crystals with positive and negative ions in the surface layer have a large electronic corrugation, due to the localization of the charge and the different ionic radii, and therefore higher diffraction orders have high intensities. Not surprisingly, the first helium-diffraction experiment was performed at a LiF(100) surface (1930 by Estermann and Stern). On the other hand, close-packed metal surfaces show only negligible electronic corrugation due to the Smoluchowski smoothing (Section 1.1.3), and almost the total diffraction intensity is found in the (00) beam (specularly reflected beam). At the Ag(111) surface, the intensity of the (10) beam is less than one percent of that of the (00) beam. More open or reconstructed metal surfaces can have a significant corrugation, which leads to considerable intensities in higher diffraction orders. Figure 4.12 shows the angular intensity distribution for the Au(110)-(1 × 2) surface, which shows the same missing-row reconstruction as the Pt(110) surface illustrated in Fig. 1.15(b),(c). This causes an electronic corrugation of about 1.5 Å compared to a geometric corrugation of about 2.5 Å. The precise determination of the electronic corrugation and conclusion on the geometric structure requires quantum-mechanical calculations.

4.6 Fine structure of x-ray absorption An x-ray photon can excite an electron (photoelectron) from an atom on or in a surface, and is thereby absorbed (x-ray absorption). If the photon energy hν is varied, no absorption takes place for hν < EB with the binding energy of the electron EB . At hν = EB , the absorption steeply increases, which led to the coining of the term absorption edge. For hν > EB , the excited photoelectron has the energy E = hν − EB + V0R in the solid, where V0R is the real part of the inner potential of the solid (Section 4.2.3). The corresponding wave number k is in a free-electron approximation k = √2mE/ℏ. As illustrated in Fig. 4.1, the emitted photoelectron wave is partially backscattered. The scattered wave, which returns to the emitting atom, interferes with the primary wave function of the photoelectron at the emitter atom. This depends on the amplitude and phase, and thus the quantum-mechanical matrix element for the excitation of a photoelectron and, in turn, the absorption probability of the photon. Since the amplitude

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of the backscattered wave at the emitter atom depends on the electron energy, the xray absorption shows oscillations as function of photon energy hν. They are called the fine structure of the x-ray absorption, because their amplitude is usually less than 10 % of the jump at the absorption edge. In an extended energy range E ≳ 50 eV above the absorption edge of the x-ray absorption, the functional dependence can be estimated relatively easily, because the consideration of a single backscattering by the neighboring atoms is sufficient. Measurements in this range are called extended x-ray absorption fine structure (EXAFS). For the dependence closer to the absorption edge, the terms near edge x-ray absorption fine structure (NEXAFS) or x-ray absorption near-edge structure (XANES) are used, and the multiple scattering of the electrons has to be taken into account. Experimentally, in all cases, the x-ray absorption coefficient of the sample is recorded. For the application to surfaces (surface EXAFS: SEXAFS) the intensity of emitted photoelectrons, Auger electrons or secondary electrons is measured. Due to their short mean free path the intensity signal becomes surface sensitive. Tunable x-ray radiation of sufficiently high intensities is available only at synchrotron radiation sources. To focus on the results originating from the backscattering by the neighbor atoms to the emitter atom in EXAFS and SEXAFS, one considers as fine-structure function χ(k) the change of the absorption probability σ(k) relative to the same atom in the gas phase (σ0 ) as function of k, χ(k) =

σ(k) − σ0 (k) . σ0 (k)

For an atom j at the distance Rj from the emitter, which scatters the electron wave back with a phase shift ϕj , the phase shift upon return at the emitter atom is (2kRj +ϕj ). Since the emitted and scattered waves are spherical waves, the amplitude scales going and returning each with 1/Rj , and in total with 1/R2j . The waves are in addition attenuated by inelastic scattering processes, which is described using the imaginary part kI of the wave number by a factor e−2kI Rj (Section 4.2.3). The backscattering (ϑ = π) at atom j with the square of the momentum transfer (Δk)2 = (2k)2 = 4k 2 has the probability am2 2 plitude fj (k) = fj (k, π), which is further reduced by the Debye–Waller factor e−2k ⟨uj ⟩/3 . The thermal vibrations of the atoms are assumed to be isotropic with a mean square vibrational amplitude ⟨u2j ⟩ (Chap. 4.1 and eq. (4.9)). If there are Nj identical atoms at the same distance Rj and several such atom shells around the emitter atom, one has to sum over all shells and types of atoms. This finally yields for the fine-structure function 1 χ(k) ∝ ∑ Aj (k) sin(2kRi + ϕj (k)) k j with the amplitude factor of atom j Aj (k) =

Nj

2 2 fj (k) e−2k ⟨uj ⟩/3 e−2kI Rj . 2 Rj

98 | 4 Diffraction methods By Fourier transform of χ(k), one obtains the radial distribution function of the atoms around the emitter. Note that χ(k) depends only on the distances from the neighboring scatterers, and not from the directions in which they are oriented relative to the emitter. Thus, SEXAFS is only sensitive to the bond lengths, and not to the complete geometric structure. In many cases, it is possible to infer the adsorption site from the bond lengths. The polarization and incidence angle of the synchrotron radiation can be used to obtain some information on the orientation of the bonds. An advantage of the method is that no long-range order around the emitter atom is required. By choosing the absorption edge of the emitter, the neighborhood of specific chemical elements can be investigated selectively. This can also be used to gain surface sensitivity if an emitter only present in an adsorbate layer is selected. Figure 4.13 illustrates the SEXAFS method using the adsorption system Ni(100)c(2 × 2)-S as an example. In (b), the measured yield of Auger electrons from sulfur are shown, which are generated after the excitation by x-ray photons. These data are used to extract the fine-structure function χ plotted in (a) depending on the energy E of the excited electrons in the solid above the absorption edge. After converting the energy to wave numbers k = √2mE/ℏ, the Fourier transform yields the curve shown in (c). The two pronounced maxima SA and SB correspond to distances dSA = 2.23 Å and dSB = 4.15 Å of the marked atoms in the structural model (d). These values agree well

Figure 4.13: SEXAFS measurement and data evaluation for the adsorption system Ni(100)-c(2 × 2)-S (after [4.12]).

| 99

with the result 2.19 Å and 4.15 Å of a LEED structure analysis [4.13]. The LEED study unambiguously determined also the hollow site as adsorption site, and some minor adsorbate-induced modifications of the substrate. Diffraction experiments are generally the most important methods for the precise determination of the geometric structure of periodic solids. X-ray diffraction is extremely successful for bulk crystals in three dimensions, but difficult to apply at surfaces. The diffraction of helium atoms is, on the other hand, so surface sensitive that it samples only the electronic corrugation of the charge density in front of the surface. The diffraction of low-energy electrons is the method of choice and permits determining the positions of surface atoms with picometer accuracy. Other methods are available, which rely on the interference of electrons. Some exploit the local excitation of electrons from atoms. In the following chapter, we will see that these excitations are unique for each element, and thus the local neighborhood may be investigated with chemical specificity.

Q4.1: Discuss the fundamental equations for the diffraction at the surface of a two-dimensional periodic crystal. Q4.2: Describe the diffraction pattern of a superstructure in comparison to the one of the substrate. Q4.3: Describe the experimental setup of an experiment for the diffraction of low-energy electrons. Q4.4: Which information is immediately obtained from the diffraction pattern of low-energy electrons? Q4.5: How can the positions of the atoms in the unit cell be obtained from diffraction experiments using low-energy electrons? Q4.6: In an adsorption experiment, the adsorbate atoms A occupy all hollow sites of a Ni(100) surface. Give the notation of the adsorption phase. Do new diffraction spots appear upon adsorption, and do the intensities of the previously observed spots of the substrate change? Q4.7: Explain why the oblique incidence angles used for the diffraction of high-energy electrons are advantageous for monitoring the epitaxial growth. Q4.8: Explain why a surface with a superstructure is advantageous for surface x-ray diffraction. Q4.9: Explain why helium is diffracted at low-index metal surfaces with low intensity and at ionic crystals with high intensity. Q4.10: How is the surface sensitivity achieved in a surface extended x-ray absorption spectroscopy experiment?

[4.1] [4.2] [4.3] [4.4] [4.5] [4.6] [4.7] [4.8] [4.9]

E. E. Castellano and P. Main, Acta Crystallogr. A41, 156 (1985). K. Heinz, Low-energy electron diffraction (LEED) in Surface and interface science (Ed. K. Wandelt), Vol. 1, p. 93, Wiley-VCH (Berlin 2012). K. Heinz, Rep. Prog. Phys. 58, 637 (1995). K. Heinz and L. Hammer, Z. Kristallogr. 213, 615 (1998). M. A. Van Hove, W. H. Weinberg, and C.-M. Chan, Low-energy electron diffraction: Experiment, theory and surface structure determination, Springer (Berlin 1986). J. B. Pendry, Low energy electron diffraction: The theory and its application to determination of surface structure, Academic Press (London 1974). J. B. Pendry, J. Phys. C, Solid State Phys. 13, 937 (1980). A. Schmidt, W. Meier, L. Hammer and K. Heinz, J. Phys. Condens. Matter 14, 12353 (2002). R. Bliem, E. McDermott, P. Ferstl, M. Setvin, O. Gamba, J. Pavelec, M. A. Schneider, M. Schmid, U. Diebold, P. Blaha, L. Hammer and G. S. Parkinson, Science 346, 1215 (2014).

100 | 4 Diffraction methods

[4.10] [4.11] [4.12] [4.13]

E. Vlieg, X-ray diffraction from surfaces and interfaces in Surface and interface science (Ed. K. Wandelt), Vol. 1, p. 375, Wiley-VCH (Berlin 2012). T. Engel and K. H. Rieder, Springer Tracts in Modern Physics 91, 55 (1982). S. Brennan, J. Stöhr and R. Jäger, Phys. Rev. B 24, 4871 (1981). U. Starke, F. Bothe, W. Oed and K. Heinz, Surf. Sci. 232, 56 (1990).

5 Electron spectroscopies Electrons with energies in the range of 1–104 eV are excellent probes for studying excitations at surfaces of solids. As charged particles, they interact directly and with rather large cross section with the electronic system of the surface (Chap. 2.1), and in turn they can also excite atomic vibrations. Due to their short mean free path (Fig. 2.3), electrons probe the depth region relevant for surface physics. From the experimental side, free electrons can be produced easily at low cost with low energy spread, they can be well focused, and the energy can be chosen over a wide range. Electrons can be analyzed according to their energy, direction, and spin orientation and can be detected with high sensitivity. Employing electrons for surface studies demands a sufficiently low pressure, such that the mean free path of the electrons is larger than the dimensions of the vacuum chamber. This usually requires pressures < 0.1 Pa, though with elaborate differential pumping and very short distances (≈ 1 mm) between sample and electron analyzer measurements at pressures in the 100 Pa range have become possible. The most important member in the family of electron spectroscopies is photoelectron spectroscopy (PES), in which electrons are excited in the solid by the absorption of UV or soft x-ray radiation. The surface sensitivity is not determined by the excitation process, but by the inelastic mean free path of the emitted photoelectrons (escape depth). If the electrons originate from core levels they have characteristic energies, whose exact values also depend on the chemical binding configuration of the emitter atom. The x-ray-induced photoelectron spectroscopy is also called for that reason electron spectroscopy for chemical analysis (ESCA). For the development of that method, Siegbahn was awarded the Nobel prize in 1981. The energy distribution of photoelectrons excited from valence bands reflects the density of valence states. In angle-resolved photoelectron spectroscopy (ARPES), the electronic band structure E(k ‖ ) of the occupied valence bands can be derived from the angular distribution of the photoelectrons. The stepwise excitation of photoelectrons by two photons in twophoton photoemission (2PPE) allows to obtain the equivalent information on intermediate states, and thus the unoccupied conduction band structure. The unoccupied core levels in the inner atomic shells (core holes), which are created by excitation by photons or electrons have a very short lifetime. The core holes decay either radiatively by emission of x-rays photons (x-ray fluorescence), or nonradiatively by emission of electrons with characteristic energy (Auger process). The emitted Auger electrons are superimposed on the spectrum of photoelectrons or backscattered primary electrons (Fig. 2.1). They also carry information on the chemical identity of atoms close to the surface, and their chemical binding configuration. The electronbeam-induced Auger electron spectroscopy (AES) is experimentally relatively simple and a widely used method for the chemical analysis of solid surfaces, which can also be carried out with lateral resolution in a scanning electron microscope. https://doi.org/10.1515/9783110636697-007

102 | 5 Electron spectroscopies Excitations in solids or at surfaces can also be detected by the corresponding characteristic energy losses, which the exciting electron suffers. The method is called electron energy-loss spectroscopy (EELS). If the incident electrons are produced by a simple nonmonochromatized electron source with an energy spread up to 1 eV, only excitations of deeper-lying levels (core-hole excitations) and collective oscillations of the valence electrons (plasmon excitations; see Chap. 2.1) can be investigated. To study low-energy losses, such as inter- and intraband excitations or lattice vibrations, the high-resolution electron energy-loss spectroscopy (HREELS) has to be used, which employs monochromatized primary electrons with an energy spread down to 1 meV. This method is employed, in particular, for the identification of the characteristic vibrational modes of adsorbed molecular species or ultrathin films (vibrational spectroscopy).

5.1 Instrumentation 5.1.1 Electron sources An electron source (Fig. 5.1) consists of components to produce electrons (cathode, Wehnelt cylinder, anode), a lens system (usually an electrostatic einzel lens), and optionally a deflection unit (crossed pairs of parallel plate condensers). The electrons are usually generated by thermionic emission from, e. g., a tungsten wire (work function ΦK = 4.5 eV, T ≈ 2700 K) or a metal ribbon coated with a material of low work function (BaO or LaB6 , ΦK = 1.2–2.2 eV, T ≈ 1300–1700 K). The thermally induced energy distribution of the emitted electrons is in the range of 0.3–0.6 eV. Space-charge effects can lead to an energy spread of the emitted electrons of up to 1 eV. Typical electron currents are in the μA range. The cathode is enclosed by an electrode named Wehnelt cylinder with a small aperture toward the anode (Fig. 5.1(a)). The Wehnelt cylinder is held at a slightly negative potential, and ensures the formation of a space-charge cloud surrounding the cathode. The penetration of the anode field through the Wehnelt aperture extracts electrons from a small spatial region and leads to a focusing at the crossover point, which is the actual point source, and determines the spatial coherence of the electron beam. The electrostatic electron lens (einzel lens) consists of three cylinders or apertures with the two outer electrodes at the same potential. The center electrode is at a different potential and leads to a focusing of the electron beam independent of its sign relative to the outer electrodes (Fig. 5.1(b)). The exit aperture is kept at ground potential to guarantee a field-free region for the electron beam leaving the source. In the most simple construction (Fig. 5.1(c)), the anode A is also at ground potential, and is used in conjunction with the exit aperture B as end electrode of the einzel lens L. After the exit aperture, one frequently finds pairs of deflector plates for adjusting the beam direction. Figure 5.1(d) illustrates that the kinetic energy Ekin of the electrons reaching

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Figure 5.1: (a) Components for the production of an electron beam and (b) of an electrostatic einzel lens. (c) Schematic of a simple electron source with (d) the associated potential distribution between cathode and sample.

the sample does not depend on the potentials inside the electron source. It only depends on the difference of the potentials and work functions of cathode and sample: Ekin = eUK + ΦK − ΦS . 5.1.2 Photon sources Typically, x-ray sources for photoelectron spectroscopy make use of characteristic xray emission in the soft x-ray range (0.1–10 keV). For a good energy resolution, the intrinsic linewidth of the used emission line should be as narrow as possible, and at the same time the photon energy hν high enough to be able to excite core electrons at all elements. Common anode materials are aluminum and magnesium, which have intense Kα -emission lines (transition 2p → 1s) at 1486.6 eV and 1253.6 eV, with a linewidth of 0.85 eV and 0.70 eV, respectively. Often both anode materials are combined in a double-anode x-ray source as illustrated schematically in Fig. 5.2(a). Two separate filaments offer the choice to bombard one or the other water-cooled anode with highenergy electrons (typically 10–15 keV, 500 W), in order to generate core-holes in the K shell. The radiative recombination occurs mainly from the L shell (Kα ). However, some satellite lines are also found with higher photon energies (5–70 eV), and relative inten-

104 | 5 Electron spectroscopies

Figure 5.2: Schematic cross section of (a) a double-anode x-ray source and (b) an x-ray monochromator.

sities of up to a few percent. They originate from the recombination from other shells (e. g., the M shell: Kβ ) or from multiple-ionized atoms. The discrete lines are superimposed on the continuous background from the bremsstrahlung continuum of the x-ray source. Its spectral intensity is at least two orders of magnitude below the one of the Kα line, whereas the integrated intensity is comparable. The x-ray source is completely encased and differentially pumped in order to keep desorbing particles, scattered electrons, and thermal radiation away from the sample and the vacuum chamber. Toward the sample, which is positioned quite close to the x-ray source, this is ensured by a μm thick aluminum foil, which only weakly absorbs in the range of the Al- and Mg-Kα lines, because their energies are slightly below the K-absorption edge. At the same time, the bremsstrahlung background at higher energies is suppressed quite effectively. The emitted x-ray radiation can be monochromatized by Bragg reflection, and at the same time focused (Rowland circle mount, typical R = 25 cm, Fig. 5.2(b)) at one or several bent quartz single crystals. With such a monochromator, a linewidth of 0.2– 0.3 eV is achieved for the Al-Kα radiation with a concomitant suppression of satellite lines. However, a significant intensity reduction results, due to the limited acceptance angle of the crystal monochromator. For valence-band photoemission (Chap. 5.3), usually photon energies between 10 and 100 eV are used. The range above the absorption edge of LiF at 11.8 eV is called vacuum-ultraviolet range (VUV), because no windows with sufficiently high transmission are available, and the light source has to be connected directly to the UHV chamber. For the generation of light in the VUV range, gas-discharge tubes are commonly used. In the gas discharge, a noble gas, e. g., helium, is excited and partially ionized. In electronic transitions from the 2p to the 1s level of excited atoms or ions,

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monochromatic UV light is emitted. The photon energies are 21.21 eV (He I radiation) and 40.81 eV (He II radiation), respectively. The intensity ratio of both lines can be controlled by the gas pressure. Doppler broadening and self-absorption lead to an overall linewidth of about 2 meV. In addition, some satellite lines occur with intensities of up to 2 % of that of the main line. The pressure difference between the gas discharge (typically 10 Pa) and the UHV chamber is maintained by several pumping stages. The VUV light is directed inside a glass capillary tube to the sample. Light in the UV range can also be generated by pulsed lasers with subsequent frequency multiplication. Up to about 6.2 eV sum-frequency generation in nonlinear optical crystals can be used. Photon energies up to 100 eV can be generated in gases by high-intensity laser radiation. Pulsed lasers can achieve pulse durations in the 10-fs range, which offers the possibility to study the electron dynamics in time-resolved measurements (Section 5.3.3). All laboratory light sources discussed so far (with the exception of lasers) are surpassed in brilliance (intensity per solid angle and energy interval) by synchrotron radiation, which is generated in an electron-storage ring by the deflection of the highenergy electron beam (≈ 1 GeV) in the deflecting magnets. The deflection by periodic magnetic fields in undulators generates radiation of particular high brilliance. Synchrotrons emit polarized radiation over a vast range of photon energies (typically 10 eV–10 keV). Subsequent monochromators have a relative resolution ΔE/E ≈ 10−4 , and deliver radiation with high intensity (typically 1013 photons/s) and tunable photon energy. Beam times at synchrotron radiation sources are limited and have to be applied for and planned well in advance. The UHV chambers installed at synchrotron beam lines, in many cases, do not offer all the preparation and analysis methods of the home laboratory. Therefore, good laboratory light sources are still needed and used and are indispensable for preparing for beam times at synchrotron radiation sources.

5.1.3 Energy analyzers Every electric (or magnetic) field with a component perpendicular to the direction of motion of the electron leads to a deflection, which depends on the energy (or velocity) of the electron. Since, in general, the deflection of electrons is also affected by their flight direction, a spectrometer should detect all electrons of the same energy over a finite angular range to achieve a high sensitivity. Electrons in the energy range below 3 keV are most conveniently detected by electrostatic analyzers, in which the electrons are deflected in a bent condenser and focused on the exit plane at points depending on the electron energy. By continuous variation of the condenser voltage, electrons of different energy can be scanned across an exit aperture and an energy spectrum is obtained sequentially. The most common types of analyzers are the cylindrical mirror analyzer (Fig. 5.3(a)), the hemispherical

106 | 5 Electron spectroscopies

Figure 5.3: Schematic illustration of common spectrometer types: (a) Two-stage cylindrical mirror analyzer. (b) Hemispherical analyzer with electron trajectories for several energies.

analyzer (Fig. 5.3(b)), and the 127∘ cylindrical sector analyzer (Fig. 5.22). In a cylindrical mirror analyzer, the electrons travel on curved paths between two concentric cylinders. The sample is imaged directly onto the detector aperture. The spectrometer has a rather large acceptance angle, with 360∘ azimuthal angular integration, due to the cylindrical symmetry. Therefore, it is well suited for quantitative elemental analysis by Auger or photoelectron spectroscopy. In the other two spectrometers, the electrons travel on approximately circular paths and the entrance slit is imaged onto the exit slit. Since the sample cannot be placed at the position of the entrance slit, these spectrometers usually have an electron lens system (Fig. 5.1(b)), which images the emitting spot at the surface onto the entrance slit. The transmission of these spectrometers is about one order of magnitude smaller than for the cylindrical mirror analyzer. However, they permit angle-resolved measurements of the emitted electrons at high energy resolution. The relative energy resolution ΔE/E of electron spectrometers is given by the entrance and exit slit widths s1 and s2 and the full acceptance angle α (in radians). For the most widely used hemispherical analyzer, one obtains ΔE/E = (s1 + s2 )/4r + (α/2)2 [5.1]. A large value of the nominal circular path of the electrons with radius r is favorable for a good energy resolution and, accordingly, hemispherical analyzers can be quite large (up to r = 300 mm). The effective entrance slit width s1 might be determined by the lens system, depending on the imaging conditions of the electronemitting spot on the sample. Since ΔE/E has a constant value given by the geometry of the analyzer, the energy width ΔE and also the transmission are proportional to the energy E of the detected electrons. The measured intensity becomes I(E) ∝ E ⋅ N(E), where N(E) denotes the true energy distribution of the electrons. Many spectrometers provide mechanically selectable slit widths to change the resolution. A more convenient way to improve the resolution is the retardation of the electrons with energy E to a smaller energy EPass , which can, e. g., be achieved in the lens system. The spectrometer is then negatively biased relative to the sample. In this way, the resolution ΔE

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can be improved in principle without limitations. However, the intensity is reduced by a factor (EPass /E)2 , which can be quite large. For an energy resolution in the meV range, the noise of the voltage supplies of sample and spectrometer has to be carefully reduced by appropriate filtering. To obtain a constant energy resolution ΔE across a whole spectrum, the internal potentials of the spectrometer are kept at constant values, and the bias voltage is varied (retardation mode). For less demanding applications a retarding-field spectrometer can be used, in which the electrons emitted from the sample have to overcome a potential barrier generated by a negative retarding voltage VB . All electrons with an energy larger than −eVB ∞ are detected, i. e., the intensity is I = ∫−eV N(E)dE. The signal has to be differentiated B with respect to VB to obtain the spectral energy distribution N(E). The typical setup of a retarding-field spectrometer corresponds to the one of a LEED optics (Fig. 4.5), where the fluorescent screen is used as the collector electrode, providing a large acceptance angle. The resolution of such a spectrometer is determined by the field penetration of the retarding grid and is usually ≥ 1 eV. For pulsed excitation sources (e. g., lasers or synchrotron radiation sources in special operating modes) the energy analysis can be realized also by measuring the flight time of the electron for a given drift distance. For low kinetic energies (≪ 50 eV), a very high energy resolution can be achieved.

5.1.4 Electron detection The detection of electrons at the exit of the spectrometer is most commonly done by a secondary-electron multiplier with continuous dynode (channeltron, Fig. 5.4(a),(b)). The amplification of a channeltron first increases steeply with the applied voltage, and then reaches a saturation regime (plateau, Fig. 5.4(c)). Voltages beyond the plateau destroy the channeltron. To detect single electrons, the voltage is adjusted at the lower end of the plateau. The amplification is then about 107 , and maximum count rates of several 106 counts/s are possible. For larger input currents (up to 1 nA) the channeltron may be operated at about 80 % of the nominal voltage as continuous current amplifier with an amplification of ≈ 104 . In hemispherical analyzers, the energy distribution is spread along the radial dimension of the exit slit (see Fig. 5.3(b)) and optionally registered by several channeltrons. The emission direction (or position) on the sample can be imaged simultaneously along the azimuthal dimension by suitable imaging modes of the lens system. For the detection of energy and angle (or position), the exit slit and channeltron are replaced by an areal detector (channelplate, Fig. 5.4(d)). A channelplate consists of a large number of microchannels (≈ 106 cm−2 ) packed next to each other in a plate with a thickness of < 1 mm. To achieve an amplification of 107 for a signal sufficient for detection by a preamplifier or a fluorescent screen, two or more channelplates are placed in succession. The electrons at the exit of the channelplate stack are accelerated onto a fluorescent screen for two-dimensional detection

108 | 5 Electron spectroscopies

Figure 5.4: (a) Principle of secondary-electron multiplication. (b) Photograph and (c) amplification characteristics of a channeltron. (d) Schematic of a channelplate.

by a camera, and subsequent data acquisition and processing, as in the LEED experiment (Section 4.2.3). An example for an energy-angle intensity distribution is shown in Fig. 5.16(a). The detection chain channelplate-camera has a lower dynamical range and larger nonlinearities in comparison to the counting of individual electrons with a channeltron. Alternative two-dimensional detector concepts for measuring energy and angle (or position) intensity distributions exist and are under continuing development.

5.1.5 Modulation technique In electron-beam-excited spectroscopies, the signals of interest are usually superimposed on top of a huge background of backscattered and secondary electrons (Fig. 2.1). This background is rather smooth, varies only weakly with energy, and can be eliminated by differentiating the spectrum with respect to the detection energy. The modulation technique permits the direct measurement of the differentiated signal at background contributions, which exceed the maximum count rate of a channeltron. The deflection or retardation electrodes of the spectrometer are modulated with a small alternating voltage (amplitude ≈ 1 V, frequency ≈ 1 kHz), which changes the detection energy periodically. The oscillating component of the detected signal (from a channeltron in current amplification mode or the collector electrode) is coupled via a highpass filter into a lock-in amplifier. The lock-in amplifier selectively filters the Fourier component of the signal corresponding to the modulation frequency, which corresponds to the first derivative of the signal current with respect to the detection energy. The second derivative may be obtained by locking into the Fourier component of the signal at twice the modulation frequency, and is used for energy-integrating

5.2 Element-specific spectroscopy | 109

Figure 5.5: Auger electron spectra of a copper sample. Top: Excited by x-ray radiation and electrons counted. Center: Spectrum from top numerically differentiated. Bottom: Excited by electron beam and measured by modulation technique and lock-in amplifier.

retarding-field analyzers. The modulation amplitude of the detection energy and the time constant of the lock-in amplifier (compared to the speed of the energy scan in the spectrum) lead to an additional broadening of the spectrum, and have to be chosen adequately. Figure 5.5 shows the comparison between an x-ray-excited, counted and numerically differentiated Auger electron spectrum of a copper sample (with a relatively low background, due to the x-ray excitation) and an equivalent electron-beam-excited spectrum, which was measured using modulation technique and a lock-in amplifier. The differences in the relative intensities of the groups of spectral lines at 840 and 920 eV are not associated with the different excitation processes, but are caused by the different energy resolution of the two measurement setups: The somewhat worse resolution in the bottom spectrum of Fig. 5.5 suppresses the sharp line at 920 eV compared to the center spectrum. Thus, the energy resolution has to be carefully taken into account in the quantitative evaluation of intensities of spectral lines (Section 5.2.5).

5.2 Element-specific spectroscopy 5.2.1 X-ray photoelectron spectroscopy In x-ray photoelectron spectroscopy (XPS), the surface is irradiated by photons of energy hν in the soft x-ray range (0.1–10 keV), and the emitted photoelectrons are detected as function of their kinetic energy. Upon excitation of electrons from an initial state of energy Ei into a final state of energy Ef (Fig. 5.6(a)), energy conservation yields Ef = Ei +hν and Ekin = hν−EB −Φ or EB = hν−Ekin −Φ. The binding energy EB in a solid is usually referred to the Fermi energy EF and has the opposite sign of the common electron energy scale. Bound electrons have a positive binding energy! The kinetic energy of an electron outside the solid is measured relative to the vacuum energy EV , which differs from the Fermi energy by the work function Φ = EV −EF . The work function may vary locally by up to several eV (Section 1.2.5), and thus also the vacuum energy rela-

110 | 5 Electron spectroscopies

Figure 5.6: (a) Energy scheme of the photoemission process. (b) Binding energies of core levels as function of atomic number. All levels above the horizontal, dashed line can be excited with a conventional Al-Kα x-ray source.

tive to EF . Since the measurement of the kinetic energy occurs in the spectrometer, its work function ΦSp is the relevant reference. However, the measured binding energies relative to EF are independent of the work function of the sample (or spectrometer), and can be used as reference standard for the binding energy in a specific chemical configuration. The photoemission of an electron from a localized core level to a free-electron final state couples states described by different symmetries, i. e., rotational and translational symmetry, respectively. Therefore, general selection rules cannot be formulated, but the dipole matrix element can lead to a vanishing photoelectron intensity in certain directions. This can be used to get information about the symmetry of adsorbates, and thus their adsorption sites. The photoemission matrix element and dipoleselection rules are discussed in Section 5.3.4. The plot of the binding energies of the core levels over the atomic number Z (Fig. 5.6(b)) shows that all elements have core levels, which can be excited by laboratory x-ray sources (e. g., Al Kα radiation). In the double-logarithmic plot, one recognizes the Z 2 dependence for the strongly-bound core levels, according to Moseley’s law. For the nomenclature of the electronic energy levels, two different notations are used: The anterior x-ray notation, which labels the spectral sequence by capital letters (K, L, M, …), and the atomic notation with the principal quantum number and angular momentum (s, p, d, f). They are often used interchangeably and are contrasted in Table 5.1.

5.2 Element-specific spectroscopy | 111 Table 5.1: Comparison of the commonly used notations for electronic levels up to the third shell. In the x-ray notation, the subscripts are occasionally written as capital roman numerals. X-ray notation Xi Atomic notation nlj Degeneracy 2j + 1

K 1s1/2 2

L1 2s1/2 2

L2 2p1/2 2

L3 2p3/2 4

M1 3s1/2 2

M2 3p1/2 2

M3 3p3/2 4

M4 3d3/2 4

M5 3d5/2 6

Because of the spin-orbit coupling, all levels with angular momentum l > 0 (p, d, f) are split energetically into two components with total angular momentum j = l ± 1/2. The size of the splitting increases with the binding energy, and for comparable binding energy, it increases with the principal quantum number n. If no splitting exists or it cannot be resolved, the index j for the total angular momentum may be omitted, and in the x-ray notation, the indices are merged (e. g., M2,3 = ̂ 3p). In Fig. 5.6(b), the average energy of both j components is plotted for clarity. The 2j + 1 different orientations of the total angular momentum are associated with different degrees of degeneracy of the states (Table 5.1). This leads to characteristic intensity ratios of the spin-orbit doublets in the spectra, i. e., p3/2 : p1/2 = 2 : 1; d5/2 : d3/2 = 3 : 2; f7/2 : f5/2 = 4 : 3. The cross section for photoemission from a certain core level strongly increases with increasing atomic number (Fig. 5.7), until the binding energy EB of the electron exceeds the photon energy hν. The curves end, since for EB > hν, no excitation of the core level is possible (cf. Fig. 5.6). For a tunable x-ray source (synchrotron radiation source), the photoemission yield can be optimized for photon energies about 50 eV above the binding energy of the core level under investigation. The resulting kinetic

Figure 5.7: Cross section for photoemission from various core levels as a function of the atomic number Z for excitation by hν = 1.5 keV photons [5.2]. The curves end for binding energies larger than the photon energy EB > hν. The filled circles on the vertical line mark the cross sections of the XPS lines of copper (cf. Fig. 5.8).

112 | 5 Electron spectroscopies energy of the photoelectron, of about 50 eV, also leads to a maximum surface sensitivity with a moderate secondary-electron background. The energetic width of a photoelectron line is primarily determined by the lifetime τ of the hole in the final state (0.1–100 fs). It leads to an intrinsic linewidth ℏ/τ, which is for commonly used XPS lines between 0.1 and 1 eV (lifetime broadening). Low-energy losses, which the photoelectron experiences either directly in the excitation process or on its way through the solid to the surface (Chap. 2.1), also contribute to the linewidth. This lineshape has to be convoluted with the experimental contributions from the spectral energy spread of the x-ray source and the energy resolution of the spectrometer. Figure 5.8 presents the XPS spectrum of a copper sample, which was excited by an Al-Kα x-ray source. The energy scale is given as kinetic energy at the bottom, and converted to binding energy at the top. The electrons emitted with the highest kinetic energy originate from the Fermi energy, which corresponds to a binding energy EB = 0. True photoelectron signals are only the lines labeled with L2 , L3 , M1 , M2,3 , and the contribution of the valence bands VB. The different intensities of the 2p (L2 , L3 doublet), 3s (M1 ), and 3p (M2,3 ) peaks reflect the different excitation cross sections of the levels illustrated in Fig. 5.7. For the L2 /L3 doublet, one can estimate an intensity ratio of 1:2, which is in agreement with the expectation from Table 5.1. The M2 /M3 doublet is not resolved, and the line is labeled M2,3 . For each photoelectron line, an additional peak S is found at higher kinetic energy, which is due to a satellite line of the x-ray source (Section 5.1.2). Finally, the spectrum also presents some Auger electron peaks (LMM transitions, cf. Fig. 5.5), which originate from the de-excitation of core-level holes cre-

Figure 5.8: Photoelectron spectrum of copper (VB: valence band; S: satellite peaks).

5.2 Element-specific spectroscopy | 113

ated by x-ray absorption. These peaks occur at constant kinetic energy (Section 5.2.3), i. e., their energetic position changes relative to the photoelectron lines upon variation of the photon energy. Therefore, laboratory x-ray sources often have two anodes (Al/Mg, Fig. 5.2), which provide an easy and convenient way to distinguish XPS and Auger lines or to avoid an overlap of both types of lines. Upon variation of the photon energy, photoelectron lines show constant binding energy, Auger transitions constant kinetic energy! Toward lower kinetic energies the background increases stepwise after each line. This is attributed to electrons, which are excited deep in the bulk and loose energy on their way to the surface. Assuming that each photoelectron with a distinct energy has a constant loss function (Fig. 2.1) in the low-energy range, the background B(E) of a photoelectron line is obtained as the integral of the lineshape I(E) as B(E) ∝ ∞ ∫E I(E 󸀠 )dE 󸀠 (Shirley background). At low kinetic energies in the range of several 10 eV, one observes a large signal from secondary electrons (Fig. 2.1), which is not shown in Fig. 5.8.

5.2.2 Core-level shifts The binding energy of a core-level electron is defined as the energy difference between the energy of the single-charged final state and the neutral initial state. Both energy contributions depend on the chemical environment of the ionized atom. The energy of the photoelectrons is thus not only characteristic for a specific element, but also sensitive to its chemical binding. One often distinguishes between initial- and finalstate effects in photoemission. A typical initial-state effect is a charge transfer caused by a shift of the valence-electron density, due to the binding at the atom with the respective core level (including the reaction of the neighboring charges), which changes the local potential. Energy contributions from the final state arise from the screening of the final-state hole by additional charge redistribution in the surroundings, and by the correlation of the hole state with the free valence electrons. The total chemically induced energy shift (chemical shift) of the photoelectron lines can amount to several eV. For example, the oxidation state of elements in a compound can be determined using experimental reference data [5.3]. Similarly different molecules, which contain the same atoms, can be identified by their core-level energies. This allows to follow chemical reactions at surfaces. Figure 5.9 shows as example the metalation of 2H-tetraphenylporphyrin (2H-TPP) to Mg-tetraphenylporphyrin (Mg-TPP) on a MgO film [5.4] by following the nitrogen 1s line. On Ag(100), the N1s line of adsorbed 2Htetraphenylporphyrin is split into two components with a difference in binding energy of 2 eV. These are assigned to nitrogen atoms, to which hydrogen atoms are bound, and those which have open bonds. Adsorption of 2H-TPP on a MgO film releases the two hydrogen atoms to the MgO film in exchange for a magnesium atom, which gets bound to the center of the tetraphenylporphyrin (TPP) molecule. In Mg-TPP, all four nitrogen

114 | 5 Electron spectroscopies

Figure 5.9: Metalation of 2H-tetraphenylporphyrin to Mg-tetraphenylporphyrin on a MgO film (after [5.4]).

atoms have identical bonds to the central magnesium atom, and hence a single N1s line is observed. Not only the spectroscopy of adsorbates, but also the high-resolution spectroscopy of substrate atoms, offers interesting applications. As illustrated in Chap. 1.1, surfaces exhibit geometric relaxations and reconstructions, which alter the density and distribution of the valence electrons compared to the bulk. Therefore, photoelectrons emitted from surface atoms may have energies slightly deviating from those emitted from deeper layers. These surface core-level shifts (SCLS) are prominent in spectra taken with photon energies, for which the mean free path of the emitted photoelectrons is small, and thus only a few layers contribute to the XPS signal. This is illustrated in Fig. 5.10(a) for the unreconstructed Ir(100) surface, where the signal from the 4f photoelectrons emitted from the top surface layer (S) is shifted to 0.75 eV lower binding energy compared to the signal from the deeper bulk layers (V). If several inequivalent substrate atoms exist directly at the surface, e. g., in the hexagonal reconstruction shown in Figs. 1.15(e) and 4.10, they all have different energy shifts (A–D in Fig. 5.10(b)), and provide a local probe of the binding configuration at the surface. This applies also to adsorbate systems, such as H/Ir(100) (Fig. 5.10(c)), where one can distinguish between iridium atoms coordinated with one (1) and two (2) hydrogen atoms. If one strives to achieve a quantitative understanding of photoelectron spectra beyond a qualitative description of line shifts, the outlined model based on a fully relaxed core hole in the final state is not adequate. One has to consider, in particular, that the whole process of photoexcitation occurs on very short time scales, and that the actual final state is a superposition of many different excited states. All types of valence-electron excitations can be coupled intrinsically to the photoemission process. Also vibrational excitations can be involved if the (relaxed) bond distance of the atom is markedly different from that of the ion in the final state. This leads to an increased spatial overlap of the initial-state wave function with the wave functions of higher vibrational states in the final state (Franck–Condon principle). The combina-

5.2 Element-specific spectroscopy | 115

Figure 5.10: Top: High-resolution XPS spectra of the 4f7/2 line of iridium [5.5] together with the fit curves for volume and surface components (V and S). Bottom: Top view of surface-structure models with differently coordinated surface atoms labeled A–D. (a) Nonreconstructed Ir(100)-(1 × 1)-surface. (b) Reconstructed Ir(100)-(5 × 1)-surface. (c) Ir(100)-(1 × 1)-surface covered by hydrogen with components attributed to 1- and 2-fold hydrogen-coordinated Ir atoms.

tion of all secondary processes can result in photoelectron spectra of rather complex structure.

5.2.3 Auger electron spectroscopy During an Auger transition an unoccupied inner level (core hole) X gets filled by an electron from an energetically higher-lying level Y. The available energy is transferred to another electron, usually from the same atom, in level Z, which can leave as Auger electron the solid (Fig. 5.11(a)). According to the involved levels, the process is called an XYZ transition. The core levels are labeled using the x-ray notation (Table 5.1), and for the valence band the abbreviation V is used. Unless both levels Y and Z are in the valence band, the atom is—after the Auger transition—in an excited state (core holes in levels Y or Z), and additional Auger electrons with lower kinetic energy can be emitted. The energy transfer during the Auger process takes place via the Coulomb interaction between the electrons. It is the quantum-mechanical analogue to an inelastic collision between two electrons. The transition probability w depends mainly on the overlap of the wave functions of the involved electronic initial and final states: w ∝ |⟨Ψf1 Ψf2 | 1r |Ψi1 Ψi2 ⟩|2 , where 1r describes the distance dependence of the Coulomb interaction between the two electrons. The overlap of the wave functions is determined primarily by the relative spatial distribution, and to a lesser extent by the atomic

116 | 5 Electron spectroscopies

Figure 5.11: (a) Energy scheme of the Auger process. For the core levels, binding energies are used, as in XPS. (b) Yield of the competing x-ray fluorescence process as function of the binding energy of the level being filled (data from [5.6]).

number Z, because the relative extension of the orbitals involved scales similarly with Z. Therefore, core holes are filled with higher probability by electrons from the next higher shell, i. e., K holes from the L shell, L holes from the M shell, etc., which leads to intense Auger lines (Fig. A.4). If the filling electron comes from the same shell as the emitted Auger electron, the overlap gets maximized, i. e., transitions of the type XYY are usually quite intensive. The Auger process competes with x-ray fluorescence, where the energy released in a transition Y → X is radiated as an x-ray photon with photon energy hν. The probability for x-ray emission scales for small energies approximately as (hν)2 and large energy quanta EX − EY > 10 keV (which occurs only for Z ≥ 32, Fig. 5.6(b)) are radiated predominantly in the form of x-ray photons. For binding energies below 2 keV, the fluorescence yield is always less than 6 % (Fig. 5.11(b)), and the recombination of the core hole proceeds almost exclusively via the emission of Auger electrons. Since all elements have core levels in this low-energy range (Fig. 5.6(b)), Auger electron spectroscopy can be employed quite generally for elemental analysis (with the obvious exception of hydrogen and helium). In practice, lower kinetic energies (≤ 1 keV) of the Auger electrons are often favored to achieve a high surface sensitivity (Section 2.1.2). The Auger process requires the excitation of a core level, which can be achieved by x-rays (Section 5.2.1) or electrons. The cross section QX for excitation of level X with binding energy EX by electrons with primary energy EP essentially depends on the ratio of the energies: QX = f (EP /EX ). The cross section normalized at the maximum is plotted in Fig. 5.12 for the excitation of the K shell and the L subshells. Above the threshold (EP /EX = 1), the cross section increases steeply and reaches at EP /EX ≈ 3 a broad maximum. For larger values of EP /EX , the cross section QX decreases slightly, which can be explained by the decreasing interaction time of the faster electrons with

5.2 Element-specific spectroscopy | 117

Figure 5.12: Cross section for the core-hole excitation as a function of electron energy normalized to the energy of the ionized level [5.7]. The energy dependence for the K shell and the L subshell is similar when the cross section is normalized at the maximum. For comparison, experimental data for the excitation of the K shell of carbon are shown [5.8].

the core level. In the range 2 < EP /EX < 7, the cross section QX is larger than 80 % of its maximum value for all shells, which makes a precise selection of the electron energy EP uncritical. The cross section QX scales in a crude approximation as 1/EX2 , i. e., energetically lower-lying levels are excited with significantly lower probability than higher-lying ones [5.7]. A high surface sensitivity is obtained for electrons with kinetic energies below 1 keV (Section 2.1.2). Since the binding energies of the core holes to be excited are usually only slightly larger than the emitted Auger electrons, an efficient excitation of most Auger lines is achieved by simple 3 keV electron sources (Fig. 5.1). The kinetic energy of the Auger electrons is obtained in a simple model as the difference of the binding energies Ei of the involved levels i, as determined by photoelectron spectroscopy: Ekin = EX − EY − EZ − ΔU − Φ.

(5.1)

It is thus characteristic for the particular element. For the correct evaluation, the work function Φ (of the spectrometer) has to be subtracted, as in XPS (Section 5.2.1). The correction term ΔU is needed, because the Auger process has a double-ionized final state in contrast to the single-ionized final state in x-ray photoemission. It takes into account the intraatomic relaxation (the presence of the hole in level Y lowers the energy in level Z), the extraatomic relaxation (screening of the additional charge in the final state by the surrounding charge density), and the interaction energy between the two hole states Y and Z in the spectroscopic final state. Each of these terms contributes a sizable amount of energy (> 10 eV). Fortunately, these contributions have different signs. Therefore in most cases, the difference of the binding energies gives a

118 | 5 Electron spectroscopies good approximation for the Auger electron energy. The energies EY and EZ enter the equation with the same sign, which shows that no difference exists between a XYZ and a XZY transition (Fig. 5.11(a)), i. e., it is not possible to say whether the Auger electron was emitted from level Y or Z. The conventional notation chooses the sequence with EY ≥ EZ . 5.2.4 Lineshape of Auger electron spectra Auger electron spectra also show a lifetime broadening of the observed lines as in photoelectron spectroscopy (Section 5.2.1), because the XPS final state corresponds to the AES initial state. A particular large transition probability, and thus large linewidth (up to 10 eV), is found for Auger transitions, in which the initial core hole is filled by an electron of the same shell (e. g., L1 L3 V). These ultrafast recombination processes are called Coster–Kronig transitions. In most cases, the released energy is rather small, and the lines are observed in the low-energy range of the spectra. The electrons participating in the Auger process interact not only directly with each other, but also with the surrounding valence electrons. These many-particle interactions can lead to the excitation of electron-hole pairs and plasmons, which can occur immediately in the Auger transition and also on the subsequent path of the Auger electron to the surface (Section 2.1.1). This causes the characteristic asymmetric lineshape of Auger transitions. In a XY V transition, each level of the valence band V with bandwidth W can contribute to the Auger process, which leads to a contribution to the total Auger line with the energy width W . The lineshape then replicates the electronic density of states D(E) of the valence band, under the assumption of a constant transition probability. For X VV transitions, one obtains an Auger line of width 2W , and the lineshape is given by the autoconvolution of the density of E states ∫E F D(E 󸀠 )D(E − E 󸀠 )dE 󸀠 . The fact that the electronic structure of the valence band determines the lineshape of the pertinent Auger transitions gives the method a sensitivity to the chemical surrounding of the respective atom. In a chemical compound, the valence band is formed by the overlap of different orbitals, which makes the energetic position, the bandwidth, and the density of states characteristic of the compound. As example, Fig. 5.13 presents drastically different Auger lineshapes for various boron compounds.

Figure 5.13: Differentiated Auger electron spectra of the KVV transitions of various boron compounds (after [5.9]).

5.2 Element-specific spectroscopy | 119

5.2.5 Qualitative and quantitative analysis of chemical elements An analysis of the chemical elements in the surface region of a sample can be performed by XPS and by AES using the same concepts. The following discussion focuses on the example of Auger electron spectroscopy with electron-beam excitation, which is the experimentally simpler and more common method. Since the intensity of the Auger lines is in general small compared to the background of inelastically backscattered and secondary electrons (Fig. 2.1), Auger electron spectra are usually recorded using modulation technique in the derivative mode (Section 5.1.5), and the obtained curves are plotted in this form (Fig. 5.5). Accordingly, the energy of the Auger transition is defined as the position of the negative minimum of the differentiated signal. This energy can be determined more reliably than the zero crossing, which would correspond to the maximum in the direct, undifferentiated Auger spectrum. As a practicable measure for the intensity, the difference between the positive and negative peak of the differentiated signal is used. However, one has to keep in mind that the peakto-peak amplitude is not a measure of the integrated intensity of the Auger line. It depends on the intrinsic linewidth, the modulation amplitude, and the energy resolution of the spectrometer, contributions which reduce the amplitudes of narrow lines more strongly than for broader lines (cf. Fig. 5.5). If the background varies noticeably with energy as typically in the low-energy part of the spectra (Figs. 2.1 and 5.14), the associated skew has to be accounted for in the determination of the peak-to-peak amplitude. The participation of three atomic energy levels in the Auger process often leads to a multitude of observable transitions, in particular, for heavier elements with several shells and subshells. Each element thus has a characteristic signature in the Auger electron spectrum, which permits an unambiguous identification. If the investigated sample contains several elements, the Auger lines may overlap. This occurs, in particular, for neighboring elements in the periodic table, as exemplified in Fig. 5.14 for an Fe49 Co23 Ni28 alloy. The low-energy MVV transitions and the higher-energetic LMM transitions of the three elements overlap considerably. This leads to substantial changes in the shape and amplitude of individual Auger lines, which has to be considered for a quantitative evaluation. Lines from light elements, which have only a few or just one transition in the spectrum (e. g. the surface contaminants argon, carbon and oxygen in Fig. 5.14), can easily be hidden by lines of other elements and be overlooked. For these reasons, it is important to check not only the energies of the spectral lines, but also the relative intensities of different transitions of an element against reference spectra (e. g., [5.10]), which are characteristic for the respective element. This might not provide a clear identification for Auger transitions involving valence bands, as exemplified in the preceding section (Fig. 5.13). The choice of analyzer type and resolution (Section 5.1.3) and the acquisition mode (Section 5.1.5) also influences the measured spectral intensities.

120 | 5 Electron spectroscopies

Figure 5.14: Auger electron spectrum of an Fe49 Co23 Ni28 alloy together with the reference spectra of the pure metal constituents (from [5.10]). The elements Ar, C, and O are typical contaminants on sputtered and unannealed surfaces.

To determine the (average) concentration ci of an element i in the near-surface region of a homogeneous sample, one has to consider that the intensity Ii of the corresponding Auger transition XYZ of this element scales not only with the excitation intensity I0 , but also depends on many other variables. For the case of electron-beam excitation, it is the energy and the angle of incidence of the primary electrons and the backscattering factor of the sample, because backscattered electrons can also excite core holes. Additional parameters are the probabilities for the excitation of a core hole X, the subsequent Auger decay, and also the mean free path of the Auger electron, which determines the depth from which electrons are detected. Finally, measurement parameters, such as the detection and acceptance angle, the transmission of the spectrometer (Section 5.1.3), and the sensitivity of the detector (Section 5.1.4), are also decisive for the registered Auger intensity. For a quantitative analysis, most of these quantities are usually not known with sufficient accuracy, and they are conveniently lumped into element-specific sensitivity factors Si (Ii = ci ⋅ I0 ⋅ Si ). They are obtained experimentally from reference spectra (usually from pure elements), and have been published for several spectrometers and primary energies (e. g., [5.10]). A plot of the energies and relative intensities for (almost) all elements of the periodic table can be found in the Appendix A.4. Since the intensities of different transitions of a specific element have fixed ratios, it is usually sufficient to restrict the evaluation to one line in the spectrum conveniently chosen as the most intense one. The concentration averaged over

5.3 Determination of surface band structures | 121

the sampling volume of a specific element can be calculated as ci = (Ii /Si )/ ∑j (Ij /Sj ), where the sum contains all detected elements j. For a reliable quantitative analysis of a sample, it is important to use for excitation energy, spectrometer resolution, and detection (Fig. 5.5), the same experimental parameters as for the reference spectra if possible. Even then can the specific properties of the sample, such as backscattering factor and mean free path (and diffraction or forward-scattering effects at single-crystalline samples), lead to deviations, which can amount up to a factor 5 in unfavorable circumstances. A considerable improvement can be achieved by introducing correction terms (matrix factors) to account for the chemical environment of the matrix, in which the atom is embedded. The vastly different lineshapes of the boron compounds (Fig. 5.13) illustrate this effect. Only for well-characterized systems can the element concentrations ci be determined with an accuracy of < 10 %. For any quantitative method, the following statement has to be kept in mind: The closer the reference samples are to the investigated sample, the better the results! For samples containing several chemical components, the depth distribution ci (z) is often inhomogeneous, due to segregation (Fig. 1.17) or selective sputtering (Section 3.2.4). The total registered signal of element i relative to a reference sample Iiref of the pure element is obtained as Ii /Iiref =

∞

cos ϑ cos ϑ z) dz. ∫ ci (z) exp(− λ λ

(5.2)

0

Here z is the distance (depth) from the surface in the crystal, λ the mean free path of the detected Auger electron, and ϑ the detection angle relative to the surface normal. The attenuation of the incident radiation with increasing depth can usually be neglected. The element concentration determined by electron spectroscopy is therefore a mean concentration averaged with exponentially decreasing weight over the depth (Auger or XPS concentration). Additional information on the depth distribution ci (z) can be otained from the evaluation of several lines of different kinetic energy, and thus mean free path. This can be achieved by measuring different lines of the same element or, in the case of XPS, by measuring the same line with different photon energies. Alternatively, the detection angle ϑ may be varied, but forward-scattering and, at single-crystal samples, diffraction effects may distort the results. The limited, experimentally attainable information usually does not permit determining the depth distribution on an atomic scale. However, the experimental data can be checked for consistency with calculated intensity ratios (eq. (5.2)) for various model distributions.

5.3 Determination of surface band structures The electronic band structure of solids comprises the occupied valence bands and the unoccupied conduction bands close to the Fermi energy. Information from the surface

122 | 5 Electron spectroscopies

Figure 5.15: (a) Energy diagram of the photoemission process. (b) Determination of the parallel component of the wave vector k ‖ from the emission angle ϑ.

region is obtained using electron energies below 100 eV, because of the associated small mean free path (Fig. 2.3). In this way, one gets access to the surface states discussed in Section 1.2.4. The most important experimental technique is photoemission (Fig. 5.15(a)), where an electron is lifted after the absorption of a photon with energy hν from an initial state |i⟩ in the valence band to a final state |f⟩. From the energy diagram 5.15(a), the energy conservation Ef = Ei + hν is readily obtained. If the energy Ef of the final state is above the vacuum energy EV , the electron can leave the surface, and its kinetic energy Ekin = Ef − EV can be measured. For valence bands, usually the initial-state energy Ei relative to the Fermi energy EF is used instead of the binding energy EB , which was introduced in Section 5.2.1 for the photoemission from core levels. If in addition the direction of the emitted electrons is determined in angle-resolved photoemission (ARPES), one obtains the surface band structure E(k ‖ ) (Chap. 1.2). 5.3.1 Angle-resolved photoemission In an angle-resolved photoemission experiment, the surface is irradiated by light of photon energy hν. Typically photon energies in range of 10–100 eV are used. Therefore, the method is often called ultraviolet photoelectron spectroscopy (UPS). The emitted electrons are detected at an angle ϑ relative to the surface normal (usually taken as z direction) in a certain azimuthal orientation. This got the technique the name angle-resolved UPS (ARUPS) or, more general, ARPES. The kinetic energy Ekin is determined by an energy analyzer, for which mostly hemispherical analyzers are used (Section 5.1.3). The implications of the direction of incidence (described by a wave vector q) of the incident light and the polarization of the light are discussed in Section 5.3.4. From the energy conservation, we obtain for a measured kinetic energy Ekin the initialstate energy Ei = Ekin + Φ − hν

(5.3)

relative to the Fermi energy EF . As introduced in Sections 1.2.5 and 5.2.1, Φ is the work function of the sample. The absolute value of the wave vector of the emitted electron is obtained from the kinetic energy as |k| = √2mEkin /ℏ2 . The component parallel to

5.3 Determination of surface band structures | 123

the surface (Fig. 5.15(b)) is calculated from Ekin and the emission angle ϑ as |k ‖ | = √2mEkin /ℏ2 sin ϑ.

(5.4)

When the electron passes the surface, basically only the momentum perpendicular to the surface is changed, due to the potential step at the surface. Parallel to the surface, the potential is periodic, which restricts changes of k ‖ to reciprocal lattice vectors g of the surface (cf. diffraction at the surface lattice eq. (4.3)). The parallel momentum k ‖ of

the emitted electron in vacuum is equivalent to k f‖ , i. e., it might differ by a reciprocal lattice vector g of the surface. If all k ‖ vectors are reduced to the surface Brillouin zone,

we obtain the momentum conservation k i‖ = k f‖ = k ‖ . The omission of the photon momentum q is justified in Section 5.3.4. Together with the energy conservation, we can,

in principle, determine the complete surface band structure Ei (k i‖ ) for the occupied states below EF by angle-resolved photoemission experiments.

5.3.1.1 Surface states As an example for angle-resolved photoelectron spectroscopy at surfaces, Fig. 5.16(a) presents the intensity distribution of the emitted electrons as function of initial-state energy Ei and wave vector k i‖ . The sample is a Cu(111) surface, which was excited by He I radiation (21.2 eV). A parabola of maximum intensity can clearly be identified. For the intensity maxima, the dispersion Ei (k i‖ ) (Fig. 5.16(b)) was determined using eqs. (5.3) and (5.4). The observed state is a (Shockley) surface state, because it is located in a band gap of the projected bulk band structure. The band gap is indicated by the white area in Fig. 5.16(b) and can be seen in the experimental data of Fig. 5.16(a) with low intensity (dark blue). For k ‖ = 0, the band gap at the Γ point can also be identified in the copper band structure of Fig. 1.22(b) along the ΓL line. With ordinary energy analyzers, the photoelectron spectra are recorded as energy distribution curves (EDC) for fixed emission angle ϑ as function of the kinetic energy.

Figure 5.16: (a) Photoemission intensity distribution and (b) experimental dispersion of a surface state on Cu(111). (c) The intensity distribution at the Fermi energy shows a ring-shaped Fermi contour (after [5.11, 5.12]).

124 | 5 Electron spectroscopies An example for a series of spectra for different emission angles is shown in Fig. 5.20(a). In Fig. 5.16(a), an EDC corresponds roughly to a vertical cut through the intensity distribution, because in the presented angular range of ±6∘ [5.11, 5.12], the approximation sin ϑ ≈ ϑ in eq. (5.4) can be used. For bands with parabolic dispersion, an increasing width of the EDCs is obtained for larger emission angles. A horizontal cut at fixed energy, as function of emission angle, yields a momentum distribution curve (MDC). This mode of data analysis has a particular significance at the Fermi energy, because it delivers a direct image of the Fermi surface at the surface. The measured Fermi contour of the Cu(111) surface state is shown in Fig. 5.16(c). As expected for a state with freeelectron-like character and parabolic dispersion, one observes a ring-shaped Fermi contour. The assignment of the structures observed in photoelectron spectra to surface states is not always so easy as in the presented example, because—in general—besides surface states, other structures also occur, which are due to Bloch waves in the bulk (bulk transitions). To identify surface states, several criteria exist, all of which have to be satisfied: 1. The state is located in a band gap of the projected bulk band structure. 2. The initial-state energy Ei of the structure is independent of the photon energy. 3. The intensity of the structure clearly decreases with the adsorption of gases. The first criterion was introduced as the necessary condition for surface states. The situation is clearly fulfilled for the surface state on Cu(111), as indicated by the projected bulk bands in Fig. 5.16. The second criterion eliminates direct transitions in the bulk band structure; this will be discussed in more detail in the following paragraph. This test, however, can erroneously categorize weakly dispersing bands (e. g., d bands) as surface states. The third criterion is the easiest from the experimental side, because after a few hours some contaminations from the residual gas accumulate on the sample, even under very good vacuum conditions (Section 3.1). But even this simple test is not unambiguous, because also for bulk transitions, the intensity decreases upon adsorption, due to the small mean free path of the emitted electrons. In addition, the intensity is scattered in arbitrary directions by disordered adsorbates when the photoelectrons pass through the surface. 5.3.1.2 Bulk states The incident photons can also excite electrons in bulk bands, which are then emitted. In the bulk of the crystal, the momentum has to be conserved also in the Kz direction of the surface normal, and the transitions occur vertically in the bulk band structure. In Fig. 1.22 or 5.17(b), one can see that upon a variation of the photon energy, in general, the initial-state energy and the final-state energy change, because the transitions for different photon energies occur at different Kz , due to the three-dimensional K = (k ‖ , Kz ) momentum conservation in the bulk. For surface states, this is not the

5.3 Determination of surface band structures | 125

case, which leads to the second criterion above. The photoemission from bulk bands can be described in a simple three-step model: Step 1: The photon excites an electron in the bulk from an initial state to a final state. Step 2: The electron in the final state travels to the surface and can, at the same time, be scattered. The associated attenuation of the intensity is described by the mean free path. Step 3: The electron passes through the surface. The potential step changes the direction, and the periodic surface lattice may lead to diffraction. The parallel momentum k ‖ is conserved up to reciprocal lattice vectors g of the surface. The three-step model correctly describes the processes of photoemission from bulk states. Energy Ei and parallel momentum k‖ are obtained from corresponding conservation laws, as in the case of surface states. The wave vector Kz is perpendicular to the surface, and thus the dispersion of the initial and final states in Kz direction though remain undetermined. One possibility would be to take the final-state energies Ef (k ‖ , Kz ) from a bulk-band-structure calculation. However, this would be inconsistent, because photoelectron spectroscopy should determine the band structure on its own. In many cases, the description of the final-state bands by free-electron bands with the minimum of the valence bands as energy reference is a good approximation. The same approximation is also used in LEED (Section 4.2.3). The inner potential V0R roughly corresponds to the energy minimum of the valence bands referenced to the vacuum energy. The dispersion of the nearly-free-electron approximation deviates from the one of free electrons only close to the band gap (Fig. 1.22). In this energy range, the states assume more the characteristics of standing waves (exactly at the edges of the band gap with zero group velocity) instead of traveling waves, and cannot transport electrons toward the surface into the vacuum. Thus, the relevant final states are from the strongly dispersing regions of the bulk band structure. Bulk-band-structure measurements are usually performed in normal emission ϑ = 0, and thus at k ‖ = 0 employing various photon energies. For low-index surfaces, high-symmetry lines of the band structure (Fig. A.2) are then sampled. For a given k ‖ ≠ 0, an appropriate matching of the emission angle would be required, according to eq. (5.4), if the kinetic energy changes upon variation of the photon energy. An example for a photoelectron spectrum in normal emission (ϑ = 0) is presented in Fig. 5.17(a). A Cu(001) surface was excited using 21.2 eV photons from a helium gas discharge (Section 5.1.2). In the occupied part of the band structure of Cu(001), no band gaps exist in the projected bulk band structure at Γ (Figs. 1.23 and 5.17(b)), and thus no surface states are observed in the photoelectron spectrum of Fig. 5.17(a). The prominent features in the spectrum of Fig. 5.17(a) are the emission from the copper d bands around 14 eV kinetic energy. These result from direct transitions in the bulk band structure (Step 1 of the three-step model), as indicated by arrows in Fig. 5.17(b). Five direct transitions can be identified with the three at the highest energy from the weakly dispers-

126 | 5 Electron spectroscopies

Figure 5.17: (a) Photoelectron spectrum of Cu(001) in normal emission as function of kinetic energy (bottom scale) and initial-state energy (top scale). (b) Direct transitions in the bulk band structure along the corresponding Kz direction.

ing d bands showing the highest intensity. The other two transitions can be identified only as weak structures in the spectrum. The weak features at about 2 eV higher energies than the main peaks are replica of the intense d bands, which are excited by higher-energy satellite lines of the He light source (Section 5.1.2). As final-state band for the bulk transition, only the free-electron-like, totally symmetric band starting at the X1 point was considered. The other final-state bands in the relevant energy range are antisymmetric with respect to reflections or rotations of the space group of the Cu(001) surface (p4mm, according to Table A.1) and can, therefore, in normal emission not couple to a plane wave in the vacuum region. The relevant final-state band can, in good approximation, be described by a free-electron band (green parabola in Fig. 5.17(b)), which has its minimum at the bottom of the valence band. In the spectrum of Fig. 5.17(a), a background can be seen with linearly increasing intensity toward lower energies. It arises from scattered electrons (Step 2 of the three-step model) or secondary electrons (Chap. 2.1). At kinetic energies below 3 eV, the background decreases again. Here the band gap below the X1 point (Fig. 5.17(b)) begins, in which no electrons can travel in the direction perpendicular to the surface and then be emitted. The intensity does not drop to zero, because electrons from other directions may be scattered by defects or phonons in the perpendicular direction when passing through the surface (Step 3 of the three-step model). At the left edge of the spectrum, the intensity drops to zero. This vacuum edge corresponds to electrons, which just overcome the work function and leave the surface with zero kinetic energy. If the work function of the sample is smaller than the one of the analyzer (more precise its entrance side), an accelerating bias voltage has to be

5.3 Determination of surface band structures | 127

applied to the sample, which has to be accounted for in the calibration of the energy scale of the spectrum. Electrons with the maximum kinetic energy hν − Φ are excited from the highest occupied states at EF (Fermi edge). Consequently, the width of the spectrum between vacuum edge and Fermi edge is hν−Φ, which can be used to readily determine the work function of the sample for known photon energy. Note that this relation is independent of the work function of the spectrometer or an accelerating bias voltage between sample and analyzer.

5.3.2 Inverse photoemission Photoelectron spectroscopy observes transitions from occupied levels below EF to free-electron states above EV (Fig. 5.18 left). The region in between is not accessible, though the unoccupied states in this energy range are relevant for possible bonds at surfaces. The time-reversed process to regular photoemission offers an alternative, and is named inverse photoemission (IPE). In IPE, an electron hits the surface, and a photon is emitted (Fig. 5.18 right). Transitions are only possible into unoccupied states above EF . The relations for energies and momenta (eqs. (5.3) and (5.4)) are identical for both methods. The fundamental disadvantage of inverse photoemission is the significantly lower count rate in comparison to regular photoemission.

Figure 5.18: Spectroscopies of occupied and unoccupied electronic states.

The detection of the emitted photons in the ultraviolet range is experimentally achieved by detectors, which are either only sensitive to a narrow photon energy range, or by grating spectrographs. A detector with band-pass characteristic is realized by Geiger–Müller counters filled with iodine or acetone and earth-alkaline fluoride windows (CaF2 or SrF2 ) [5.13]. The band pass at around 9.6 ± 0.3 eV photon energy is formed by the ionization threshold of I2 or acetone and the absorption edge of the window. A spectrum is obtained for a detector with fixed photon energy by recording the number of photons as a function of electron energy. Alternatively, the photon spectrum can be recorded at fixed energy of the incident electrons. The light is diffracted by a grating and detected by a channelplate, where the front surface serves as photocathode with the sensitivity enhanced by evaporated cesium iodide [5.14]. The achievable intensities in inverse photoemission are limited by the current of the electron source (1–100 μA), which cannot be increased beyond the space-charge

128 | 5 Electron spectroscopies limit. For the energy resolution in inverse photoemission, the thermal distribution of the electrons emitted by the cathode has to be taken into account. Typical overall energy resolutions are 0.3–0.6 eV. Improved values are accompanied by significant reductions in intensity.

5.3.3 Two-photon photoemission The problems and limitations of the investigation of unoccupied electronic states by inverse photoemission discussed in the preceding section can be overcome by twophoton photoemission (2PPE or TPPE). As illustrated in the center of Fig. 5.18 (for an online tutorial see [5.15]), a first photon excites an electron from an occupied initial state into an unoccupied intermediate state. From there, the electron is lifted by the second photon above the vacuum level EV and can leave the surface. Using the wellestablished methods of photoelectron spectroscopy (Section 5.3.1), its energy and direction can be selected, and finally it can be detected. The two-photon signal is significantly less intense than the one in usual one-photon photoemission, because in the excited state, only a small population can be established, which typically decays within about 1–100 fs. The energies of the employed photons should not exceed the work function in order to avoid an intense background from one-photon photoemission. This signal can be substantial even from thermally excited states above EF , which makes it advisable to cool the sample with liquid nitrogen or helium. The high photon intensities required for the two-photon process are generated by pulsed lasers, which have pulse lengths between 10 and 100 fs. This offers the possibility to vary the time delay between the two laser pulses and, additionally, observe the temporal decay of the population in the intermediate state. For this kind of experiment, it is advantageous to use different photon energies for the two laser pulses. Which of the two pulses serves as a pump or probe pulse is not determined a priori. This leaves consequently also two different possibilities for the energetic separation between intermediate and final state. The observation of the temporal decay of the two-photon intensity can help to decide which pulse probes the population. Alternatively, the photon energy can be varied, which leads to different shifts for initial and intermediate states. The essential experimental difference of two-photon photoemission from regular photoelectron spectroscopy is the use of a (femtosecond) laser system as light source. The time delay between the two laser pulses is implemented by movable mirrors in the beam path of one of the laser pulses. Moving the mirrors by 1 μm corresponds to a change in the path length of 2 μm, and therewith a time delay of 6.7 fs. For pulsed laser systems with repetition rates of < 300 kHz, electron-energy analysis and detection can be done using time-of-flight spectrometers. The flight time of the electrons from the sample to the detector is measured, and the kinetic energy is determined using the length of the drift tube. Time-of-flight spectroscopy has the advantage that every

5.3 Determination of surface band structures | 129

Figure 5.19: (a) Two-photon photoemission spectrum in normal emission from image-potential states on Cu(001). (b) Time-resolved measurements of the image-potential states on Cu(001) (after [5.16]).

emitted electron can be detected independent of its kinetic energy. In contrast, only electrons of a preselected energy (or a small energy range) are detected by electrostatic energy analyzers. A two-photon photoemission spectrum from a Cu(001) surface in normal emission is shown in Fig. 5.19(a). The three peaks in the spectrum are due to image-potential states (Fig. 1.25(b)) with the binding energy referenced to the vacuum energy EV . The energies match the prediction of eq. (1.4) for a = 0.2. The intermediate states were probed by photons of energy 1.55 eV after the initial excitation (pump) by the frequency-tripled laser radiation of photon energy 4.65 eV. The decay of the population of the image-potential states is shown in Fig. 5.19(b). Here the intensity is measured as a function of the time delay with the analyzer tuned to the energy of one of the states from Fig. 5.19. After a rapid increase of the intensity, which is essentially determined by the duration of the laser pulses (60 fs), a slower exponential decrease ∝ e−t/τ is observed. A semi-logarithmic plot yields a linear decrease with the slope directly related to the lifetime τ of the intermediate state. The higher image-potential states n = 3, 4 have a small energy difference and are coherently excited, due to the large bandwidth of the laser pulses. The result are quantum beats, in which the electron oscillates between the two states. Figure 5.20(a) presents two-photon photoemission spectra from a Cu(117) surface for various emission angles ϑ in the (110) mirror plane, which is illustrated in (b) as a side view. In the spectra, two peaks can be clearly identified, whose energies vary with the angle. These structures are assigned according to their energy to image-potential states. In Fig. 5.20(c), the dispersion E(k‖ ) of the states is plotted, which is determined from kinetic energy Ekin and emission angle ϑ using eqs. (5.3) and (5.4). For small wave vectors, a parabolic dispersion is obtained as expected for a free electron (eq. (1.4)). The periodicity of the stepped Cu(117) surface leads to Brillouin zone boundaries at

130 | 5 Electron spectroscopies

Figure 5.20: (a) Two-photon photoemission spectra from Cu(117) for different emission angles. The energy of the probing photon is 1.53 eV. (b) Side-view of the (110) mirror plane of the Cu(117) surface. (c) Dispersion of the image-potential states determined from the spectra in (a) compared to the result of a model calculation (after [5.17]).

k‖ = ±0.35 Å . Outside of the first Brillouin zone, the dispersion is continued in a periodic fashion with E(k ‖ ) = E(k ‖ +g), as supported by the data. The curves in Fig. 5.20(c) are the result from a model calculation using a multiple-scattering model [5.17], which leads to band gaps at the Brillouin zone boundaries (cf. Fig. 1.22(a)). −1

5.3.4 Selection rules for photoemission Photoemission is the most simple, lowest-order process for the interaction of light with electrons. Accordingly, the theoretical treatment is rather basic and leads to simple matrix elements and strict selection rules. The coupling between the incident light and the electrons is carried out in Coulomb e A⋅p. Here the light is described by a plane wave with the vector potengauge by the dipole operator mc

tial A(r, t) = A0 ei(q⋅r−ωt) . The wave vector q indicates the direction of incident light (Fig. 5.15(b)). The orientation of A0 defines the polarization, because the electric field is proportional to the temporal derivative of the vector potential. The Coulomb gauge implies A0 ⋅ q = 0, i. e., we consider light with transversal polarization. The transition probability between the initial state |i⟩ and the final state |f ⟩ is proportional to the absolute square of the matrix element ⟨f |A ⋅ p|i⟩. Here we again used the Coulomb gauge, which entails the permutability of A and p. For the evaluation of the matrix elements in a realspace representation, one can use for the wave functions of the initial and final states the expansion into plane waves k ‖ + g parallel to the surface (eq. (1.3)). The integration of the matrix element over f

the spatial coordinates parallel to the surface has a nonvanishing result only for k ‖ + gf = q + k i‖ + gi . ‖ Since the wavelength of the light is, in most cases, large compared to the atomic distances, |q| is small compared to the typical dimensions of Brillouin zones of low-index surfaces. Due to the limited experimental angular resolution, the term q can be neglected in most cases. After reduction of the wave ‖

5.3 Determination of surface band structures | 131

vectors k ‖ +g to the first surface Brillouin zone, we obtain the relation already known from Section 5.3.1 f

k ‖ = k i‖ : The transitions occur vertically in the reduced surface band structure E(k ‖ ). The neglect of q allows putting the vector potential in front of the matrix element: A ⋅ ⟨f |p|i⟩ = Ax ⋅ ⟨f |px |i⟩ + Ay ⋅ ⟨f |py |i⟩ + Az ⋅ ⟨f |pz |i⟩.

(5.5)

A suitable choice of the polarization (given by the direction of A0 ) and light incidence q (accounting for A0 ⋅ q = 0) allows to selectively excite individual components of the matrix elements. We illustrate this for a crystal with a mirror plane chosen as the xz plane and the corresponding reflection reversing y → −y. All electronic states must be either even or odd upon reflection. A final state, which leaves the surface as plane wave exp(ik ⋅ r) in the mirror plane, then has ky = 0, and is therefore even upon reflection. For the momentum operator, the component py is odd, whereas px and pz are even. Since the final state |f ⟩ is even, px and pz couple to even, and py to odd, initial states |i⟩. If the light is incident in the mirror plane, the individual components of the matrix element in eq. (5.5) can be selected: Ay is oriented perpendicular to the mirror plane (s-polarized light from German senkrecht) and excites odd initial states. Ax and Az are oriented parallel to the mirror plane (p-polarized light) and couple to even initial states. The effect of the polarization is exemplified in Fig. 5.21(a) for photoemission from the Si(553)-Au surface [5.18]. The spectra are completely different for the two different polarizations of the light. The peaks observed with s-polarized light correspond to odd initial states and are found at other energies than the even initial states excited using p polarization. The excitation was done by a laser source providing 6.2 eV photon energy, which allows changing the polarization quite easily. Besides the polarization one can also change the direction of light incidence. The photoemission intensity is proportional to the absolute square of the matrix elements of eq. (5.5): |A ⋅ ⟨f |p|i⟩|2 . The inner product of the vectors A and ⟨f |p|i⟩ leads to the general angular dependence ∝ cos2 α, where α is the angle between the two vectors. The vector ⟨f |p|i⟩ has, for each transition, a well-defined orientation relative to the coordinate system of the surface. Upon variation of the direction of the light q, the orientation of A0 also changes, because of the condition A0 ⋅ q = 0 for transversal light. This

changes the photoemission intensity ∝ cos2 α. This dipole distribution is shown in Fig. 5.21 in a polar diagram for a transition observed by inverse photoemission (Section 5.3.2). The intensity of the emitted 9.6 eV photons was measured as function of the detection angle at fixed energy and fixed angle of the incident electrons for a specific transition at the Cu(001) surface [5.19]. The dipole axis for an

Figure 5.21: (a) Photoemission spectra from Si(553)-Au for different polarizations (after [5.18]). (b) Dipole distribution of a transition in inverse photoemission from Cu(001) (after [5.19]).

132 | 5 Electron spectroscopies

emitted electromagnetic wave is oriented along the direction of the acceleration of the charge and corresponds, in this example, closely to the direction of the incident (refracted) electron.

5.4 Spectroscopy of surface vibrations 5.4.1 Fundamentals and experiment An electron hitting the surface can excite the electron system (Chap. 2.1) and thereby induce vibrations of atoms. These may be vibrational phonon modes of the substrate or ordered adsorbate layers, but also localized oscillations of individual atoms or molecules relative to the substrate or relative to other atoms in the molecule. Phonons typically have oscillation frequencies up to 1013 Hz, corresponding to energies up to 40 meV (Fig. 1.27(b)). Molecular vibrations may have significantly higher energies up to 500 meV, in particular if light atoms, such as hydrogen, are involved. Phonon modes in a periodic system have a continuous dispersion ω(q‖ ) as function of the wave vector q‖ (Chap. 1.3). On the other hand, molecules have discrete vibrational frequencies ωi (normal modes), which amount to 3N modes for an N-atomic adsorbed molecule. In the gas phase, this number is reduced by six degrees of freedom, three for the translational and three for the rotational motion of the freely moving and rotating molecule. The additional modes of the adsorbed molecule are sometimes called frustrated translations and rotations. By a suitable adapted combination of the atomic coordinates within the molecule to 3N normal coordinates ui it can be achieved that the complex spatial oscillations of the atoms in each normal mode can be described as a harmonic vibration with frequency ωi : ui (t) = ui0 eiωi t . If an electron with energy E0 is scattered at a surface, it can excite or de-excite a vibration of frequency ω. The energy of the scattered electron is then Es = E0 ± ℏω(q‖ ). The minus sign describes an energy loss, whereas the plus sign indicates an energy gain, in which the electron takes energy out of an excited vibrational mode. For molecules, the frequency does not depend on q‖ , and ω is one of the normal modes ωi . To simplify the notation, we suppress the index i from here on. For elastic diffraction of electrons, the momentum conservation eq. (4.7) holds, which has to be augmented for the inelastic scattering by the momentum of the phonons, k s‖ = k 0‖ + g hk + q‖ . For localized modes, the wave vector q‖ is meaningless, and the momentum conservation does not apply. For small electron energies, only the reciprocal lattice vector

5.4 Spectroscopy of surface vibrations | 133

Figure 5.22: Schematic setup of a spectrometer for high-resolution electron-energy-loss spectroscopy.

g hk = 0 contributes, which corresponds to the situation in LEED when only the (00) spot is observed. The energy losses (or energy gains for higher sample temperatures T, at which vibrational modes are thermally excited: kB T ≳ ℏω) are smaller than the typical energy spread of an electron source (Section 5.1.1). Thus, they can be resolved in the energy-loss spectrum only if the electron beam has been appropriately monochromatized. An electron monochromator corresponds to an energy analyzer (Section 5.1.3), whose pass energy is tuned to the intensity maximum of the incoming electron beam. Instead of a detector, an exit lens is mounted, which focuses the monochromatized beam on the sample. Because the incoming electrons produced by an electron source (Section 5.1.1) have a small angular divergence, hemispherical or cylindrical sector analyzers are best suited for electron monochromators with, in practice, a preference for the latter. For very high energy resolution and filtering out stray electrons, a second monochromator is placed after the first one. The experimental setup is completed by an analyzer for the energy selection and detection of the scattered electrons; it is schematically illustrated in Fig. 5.22. For angle-dependent measurements, the analyzer can be rotated about an axis through the sample surface. A complete HREEL spectrometer can be built quite compact, and be mounted on a single flange at the vacuum chamber, which can house other surface characterization and preparation methods as well. The achievable energy resolution is about 1 meV at electron currents of the order of 10 pA on the sample. 5.4.2 Interactions and selection rules The following discussion will be pursued for the example of localized molecular vibrations, but can readily be applied to delocalized lattice vibrations. For the inelastic scattering of electrons, several mechanisms exist, through which the incident electron can interact with the oscillating molecule bound to the surface. Scattering can

134 | 5 Electron spectroscopies occur by long-range electric fields, which are generated by molecular dipole moments (dipole scattering). The electron is scattered at distances far away from the surface (typically ∼ 10 nm). The differential scattering cross section dσ/dΩ can be obtained (in first Born approximation) from the absolute square of the Fourier-transformed scattering potential. It is peaked in forward direction and leads to a neglible deviation from the specular reflection of the scattered electrons. Scattering can also proceed by direct interaction with the atoms (impact scattering). The elastic contribution from this process was discussed for electron diffraction in Section 4.2.3. Due to the atomic scale of the scattering potential and the rather low energy, the electrons are scattered, in this case, into the complete half space. Finally, an incident electron of suitable energy can also be trapped for a short time in an unoccupied molecular orbital and excite a vibration upon leaving (resonance scattering). This mechanism plays, in general, not an important role for the analysis of surface vibrations, and hence will not be discussed further. Since the interaction time of the electron with the molecule is much shorter than the molecular vibrational period, the wave functions of electron and oscillating molecule may be separated: Ψtot = ψel ϕvib (Born–Oppenheimer approximation). For the inelastic scattering process, the scattering potential becomes time-dependent V = V(t). This holds for any oscillatory system of atoms, because ground-state vibrations exist even at zero temperature. For atomic vibrations, the amplitudes are small compared to the spatial extent of the interaction potential, and the displacements can be expanded along the normal coordinates u: V(r, t) = V(r, u(t)) = V0 (r) + u(t) ⋅ (𝜕V/𝜕u)u=0 + ⋅ ⋅ ⋅ , where V0 (r) is the static and thus the only elastically scattering part of the potential. The inelastic interaction is described by the second term of the expansion in good approximation. 5.4.2.1 Dipole scattering In dipole scattering, the interaction potential is given by the inner product V = ℰ ⋅ μ of the electric field ℰ of the passing electron and the dipole moment μ of the vibrating molecule. Due to the large scattering distance, the ℰ field can be considered constant across the region of the molecule and independent of u. The whole process of the scattering of the electrons from the state i → s, and the transition of the molecule from the vibrational state ν → ν + 1 (in most cases 0 → 1), is described by a matrix element, which can be factorized into a vibrational and an electronic part, i→s Mν→ν+1 = Mν→ν+1 ⋅ Mi→s = ∫ ϕ∗ν+1 u(

𝜕μ 𝜕u

)

u=0

ϕν du ⋅ ∫ ψ∗s ℰ ψi d3 r.

(5.6)

The term γ = u0 ⋅ (𝜕μ/𝜕u)u=0 represents the oscillation amplitude of the dipole moment, called dynamic dipole moment. It can also exist for nonpolar (e. g., homonu-

5.4 Spectroscopy of surface vibrations | 135

clear) bonds. The evaluation of the matrix element (eq. (5.6)) leads after a lengthy calculation [5.20] to the differential cross section dσ/dΩ for dipole scattering. For the common case of a scattering plane perpendicular to the surface, one obtains the expression γ 2 (δ − δ0 tan ϑ)2 cos ϑ dσ ∼ ⋅ . dΩ E0 (δ2 + δ02 )2

(5.7)

E0 is the energy and ϑ the angle of the incident electron, δ the scattering angle, and δ0 = ℏω/2E0 a characteristic angular spread in radians. The scattering cross section scales inversely with the energy, which favors low primary energies (typically 2–5 eV). The scattering-angle distribution (eq. (5.7)) is plotted in Fig. 5.23(a) for ϑ = 60∘ and δ0 = 0.2 after reflection from the surface. The pronounced maximum points close to the direction of the reflected beam (deviation < δ0 ) and the width is also of the order of δ0 . Since δ0 is typically one order of magnitude smaller than that used in Fig. 5.23(a) for illustration purposes, the scattering distribution coincides with the reflected primary beam ((00) beam) within the experimental angular resolution. This facilitates the experimental alignment, and ensures that almost the whole dipole-scattering intensity is collected within the acceptance angle of a few degrees, and good signal intensities are obtained. In the previous discussion, the substrate just served to reflect the electrons. At metals, each charge in front of the surface induces in the substrate an image charge (Fig. 1.25(a)). The electric field ℰ is therefore always perpendicular to the surface, and can only interact with the perpendicular component of the molecular dipole moment μ⊥ and dynamic dipole moment γ⊥ . Viewing the process from the perspective of the electron, it sees the superposition of the fields of the oscillating dipole and its image dipole. In a far-field approximation, this leads to a doubling of the perpendicular and a cancellation of the parallel component (cf. Fig. 5.23(b)). For a nonmetallic substrate, e. g., a semiconductor, the cancellation is not complete, and the dipole moment is re-

Figure 5.23: (a) Angular distribution of dipole-scattered electrons after reflection from the surface. For the calculation a typical incidence angle of ϑ = 60∘ and (for illustration purposes) an unrealistic large value of δ0 = 0.2 was used. (b) Canceling and amplification of the effective dipole moment at a metal surface for an orientation of a dipole parallel and perpendicular to the surface, respectively.

136 | 5 Electron spectroscopies duced by a factor 1/ε. Since for many semiconductors ε > 10 (e. g., εSi = 11.2) and the dynamic dipole moment enters quadratically in the cross section (eq. (5.7)), one can formulate the following dipole-selection rule: With dipole scattering only vibrational modes can be excited, which have a dynamic dipole moment perpendicular to the surface. Vibrations of bonds parallel to the surface or nonpolar bonds, e. g., the C=C bond in a flat-lying ethene molecule, can nevertheless have a nonvanishing perpendicular dynamic dipole moment. This occurs, because the variation of the intramolecular bond length influences also the bonds of the molecule to the substrate, which is accompanied by a perpendicular shift of the charges. A different approach to the dipole-selection rules offers the analysis of the symmetry of the matrix element for the vibrational excitation from the ground state M0→1 = ∫ ϕ∗1 γ⊥ ϕ0 du. The integral must be invariant under all applicable symmetry operations of the surface, i. e., it has to be totally symmetric. Since rotation axes and mirror planes are always perpendicular to the surface, γ⊥ is totally symmetric. This also holds for the wave function ϕ0 of the ground state and, consequently, ϕ1 must be totally symmetric as well. The dipole-selection rule can thus be summarized in an alternative way: With dipole scattering, only totally symmetric vibrational modes at the surface can be excited. 5.4.2.2 Impact scattering

For impact scattering, the scattering potential V (r, u) is strictly coupled to the corresponding atom coordinates. When vibrating, its derivative (𝜕V /𝜕u)u=0 has therefore the same spatial symmetry as the vibrational mode u. The matrix element for impact scattering cannot be factorized as in the case of dipole scattering, but it can be rearranged as follows: ∗ ∗ Mi→s ν→ν+1 = ∫ ϕν+1 (∫ ψs (

𝜕V ) ψ d3 r)u ϕν du . 𝜕u u=0 i

The inner integral formally corresponds to the dipole matrix element for photoemission (Section 5.3.4), where the vector potential A is replaced by (𝜕V /𝜕u)u=0 . Analogous to eq. (5.5), symmetry selection rules can be formulated, which state that antisymmetric vibrations parallel to the surface cannot be observed under certain scattering conditions. Actually measured angular distributions do not correspond to the differential scattering cross section for impact scattering, because the electron undergoes–before as well as after the inelastic scattering process–additional elastic scattering processes with high probability. The elastic multiple scattering causes a strong modulation of the scattering intensities as function of the primary energy, which is due to the local geometric structure as for the I(E) spectra in LEED. For impact scattering, the measured intensities are always quite small (a few electrons/s), since the spectrometer can collect only a small fraction of the total number of scattered electrons in contrast to the specular reflection in dipole scattering.

5.4 Spectroscopy of surface vibrations | 137

5.4.3 Applications 5.4.3.1 Phonon dispersion To determine the dispersion of surface phonons, one measures the energy of the phonon losses as a function of the scattering angle in a scattering plane commonly oriented along a low-index crystallographic direction. To completely cover the range up to the border of the surface Brillouin zone, the energy of the incident electrons usually has to be chosen > 100 eV, which puts high demands on the resolution of the spectrometer (ΔE/E0 ≈ 10−5 , which can only be achieved with strong pre-retardation). As an example, a series of spectra with increasing scattering angle from top to bottom 󸀠 is shown in Fig. 5.24(a) for a Ag(100) surface. The measurement direction ΓXΓ and the scattering geometry is sketched in Fig. 5.24(b). The prominent loss at the right side of the elastic line (ΔE = 0, LEED reflexes appear only at the Γ points) corresponds to the Rayleigh mode discussed in Chap. 1.3. The dispersion of this mode can be clearly seen in the spectra and is plotted in Fig. 5.24(c) as ℏω(q‖ ). The same mode can be identi-

Figure 5.24: (a) High-resolution electron-energy-loss spectra from a Ag(100) surface as function of the wave vector q‖ . The spectra were measured at a fixed total scattering angle of 2ϑ + δ = 120∘ by rotation of the sample. (b) Illustration of the measurement direction in k space (top) and real space (bottom). (c) Dispersion of the Rayleigh phonon measured in (a) (after [5.21]).

138 | 5 Electron spectroscopies fied also at the left side of the elastic line as energy gain. The low-energy vibrations (ℏω < 9 meV) of the Raleigh mode are significantly excited at the sample temperature of T = 300 K (kB T ≃ 25 meV), and can thus be de-excited by transferring energy to the electrons in the inelastic scattering process. 5.4.3.2 Determination of adsorption sites Also for weakly interacting adsorbates with negligible dispersion, loss spectra are often measured as a function of angle. In this case, the dipole activity of the various modes is studied by the intensity maxima at specular reflection in order to elucidate the symmetry of the adsorbate complex. This is illustrated in Fig. 5.25(a) for the vibrational losses of adsorbed hydrogen on a W(100) surface at saturation coverage. Three different vibrational modes are identified, only one of which is dipole active. From a simple inspection of the symmetries (Fig. 5.25(b)), the number and dipole activity of the various modes can be deduced for the various adsorption sites. The bridge site can be unambiguously identified for hydrogen on W(100), as illustrated in Fig. 1.18(c),(d). For top or hollow sites, the two nondipole-active modes parallel to the surface would be degenerate. An asymmetric site (e. g., on a slope with only one mirror plane) would show at least two dipole-active modes. With a nearest-neighbor spring model, the frequencies of the various modes can be roughly estimated (Fig. 5.25(b)). From the frequency ratio of the two stretching modes, the bond angle α relative to the surface normal can be estimated for the bridge site, and then the distance of the hydrogen atoms above the surface can be calculated using the known substrate lattice constant. For hydrogen on W(100), this leads to a bond angle of α = 51∘ and a distance of 1.28 Å, which deviates by only 0.11 Å from the result of a quantitative LEED structure analysis [5.23].

Figure 5.25: (a) Angular distribution of the vibrational energy-loss intensities for H/W(100) (after [5.22]). (b) Schematic overview of the atomic vibration modes for different adsorption sites and corresponding frequencies.

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5.4.3.3 Configuration and reaction of adsorbed molecules For molecular adsorbates, the chemical identity of the adsorbed species is at least as important as the spatial configuration. Only for physisorbed molecules, e. g., in thick condensed films, one can assume that the structure closely corresponds to the one known from the gas phase. As soon as a chemical bond with the substrate is formed, an hybridization of the molecular orbitals takes place, which changes bond lengths and angles. Furthermore, a rearrangement or even a breakup of the molecule, can occur with both processes possibly having an activation barrier. For molecules consisting of more than two or three atoms, the energy-loss spectra become rather complicated, due to the large number of different vibrational modes, and, accordingly, the correct assignment becomes difficult. The problem may be approached by exploiting the fact that many (in particular organic) molecules consist of a basic skeleton with attached molecular groups (e. g., –CH3 , =CH2 , –NH2 , –C=O), which are coupled only weakly to the rest of the molecule. This implies that many normal modes have significant amplitudes only for atoms within a particular group. Their frequencies are, accordingly, determined mainly by the coordination within the group, and thus characteristic for these. The group frequencies are well known from IR-absorption spectroscopy of free molecules for many molecular groups and tabulated (e. g., [5.24]). Further support for the mode assignment can be obtained by (partial) isotope substitution (in particular 1 H ↔ 2 D), which leads to characteristic frequency shifts (isotope shifts) in the spectra. Figure 5.26(a) illustrates this for the example of isobuten ((CH3 –)2 C=CH2 ) on a Ni(111) surface. Comparing the spectrum of normal isobuten (D0) to the one for partially deuterated molecules, where alternatively only the hydrogen atoms of the CH2 group (D2) or of the two CH3 groups (D6) were substituted by deuterium, one can assign, in particular, the CH-stretch modes (for D0 in the energy range 300–400 meV) according to their isotope shift. Due to the doubled mass of deuterium compared to hydrogen, the frequencies of the CH vibrations are reduced by about a factor 1/√2, with the exact value depending on the coupling to the rest of the molecule. The CH2 modes are noticeably broadened and shifted to unusually low energies, which hints toward additional hydrogen bridge bonds to the substrate. A detailed analysis of all frequencies and their isotope shifts [5.25] provides a complete assignment of all modes indicated in Fig. 5.26(a) by different symbols. In combination with the dipole activity obtained from the angular intensity distributions, a structural model of the adsorbed molecule can be constructed (Fig. 5.26(c), left). However, such analyses exist only for a few cases, particularly since partially isotope-substituted molecules are not always available. Vibrational spectra provide unique “finger prints” of the adsorbed species on the surface, even if the chemical identity and configuration of the adsorbed molecule is not completely characterized. Every change in the spectra, such as energetic shifts or vanishing of particular modes, provides evidence for a breakup or a chemical re-

140 | 5 Electron spectroscopies

Figure 5.26: (a) Vibrational spectra of a monolayer of regular (D0) and partially deuterated (D2 and D6) isobuten molecules chemisorbed on Ni(111) at 80 K. The modes of the CH2 (CD2 ) and CH3 (CD3 ) groups are marked accordingly. The scattering angle is δ = 18∘ relative to the specular reflection. (b) Comparison of the vibrational spectra of isobuten on Ni(111) as adsorbed at 80 K, and after annealing to 180 K. (c) Models of the corresponding surface species before and after the annealing step.

action of the molecule at the surface. Of course, the final product has to be analyzed as described before. This effect can be illustrated also for the isobuten system on Ni(111). Figure 5.26(b) shows that short annealing of the sample to 180 K leads to significant changes in the spectrum. Particularly noticeable is the disappearance of the CH2 stretch modes, which indicates the split-off of the hydrogen atoms with bridge bonds to the substrate. Consequently, the internal C=C bond assumes more the character of a double bond indicated by the shift to significantly higher energies. Based on additional findings [5.25], the structural model shown at the right of Fig. 5.26(c) is obtained for the decomposed molecule. The importance of electron-spectroscopic methods for surface physics is based on the small mean free path of low-energy electrons. It is a favorable coincidence that in order to work with electrons and with surfaces, good vacuum conditions are an essential prerequisite. We have seen that, depending on the energy of the electrons, different properties of surfaces become accessible: elemental analysis by core-level electrons (XPS or AES), electronic band structure of valence bands (ARPES), and surface or adsorbate vibrations (HREELS). Also in the following chapter, electrons are the prominent performers: Tunneling between a tip and a surface allows recording images with atomic resolution and taking local spectra of the electronic states.

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141

Q5.1: Q5.2: Q5.3: Q5.4: Q5.5:

Describe the essential elements of an electron source and explain their function. Which photon sources are available, and for which surface spectroscopies are they used? How can the kinetic energy of electrons be measured? How can electrons be detected? Explain the principle of the x-ray photoelectron and the Auger electron spectroscopy. How can the energies of the Auger lines be estimated from the binding energies of the core levels obtained by x-ray photoelectron spectroscopy? Q5.6: Which effects have to be considered in a quantitative elemental analysis by x-ray photoelectron or Auger electron spectroscopy? Q5.7: Explain how the surface band structure E(k ‖ ) can be determined from the measured kinetic energy Ekin and the emission angle ϑ of an observed transition in an angle-resolved photoemission experiment. Q5.8: How can one distinguish between surface states and transitions in the bulk band structure in an angle-resolved photoemission experiment? Q5.9: States in a mirror plane have either even or odd symmetry with respect to the reflection. Explain how the symmetry can be determined by a suitable polarization of the light in a photoemission experiment. Q5.10: Discuss the basic equations for the inelastic scattering of electrons at surfaces by phonons. Q5.11: Describe the elements of an experiment for the high-resolution electron-energy-loss spectroscopy of surfaces vibrations. Q5.12: Discuss the interaction mechanisms and selection rules for the high-resolution spectroscopy of surface vibrations.

[5.1] [5.2] [5.3] [5.4] [5.5] [5.6] [5.7] [5.8] [5.9] [5.10]

[5.11] [5.12] [5.13] [5.14] [5.15]

C. Tusche, Y. J. Chen, C. M. Schneider and J. Kirschner, Ultramicroscopy, 206, 112815 (2019). J. J. Yeh and I. Lindau, Atom. Data Nucl. Data Tabl. 32, 1 (1985). A. V. Naumkin, A. Kraut-Vass, S. W. Gaarenstroom and C. J. Powell, NIST X-ray photoelectron spectroscopy database (2012); https://dx.doi.org/10.18434/T4T88K. G. Di Filippo, A. Classen, R. Pöschel and Th. Fauster, J. Chem. Phys. 146, 064702 (2017). M. A. Arman, A. Klein, P. Ferstl, A. Valookaran, J. Gustafson, K. Schulte, E. Lundgren, K. Heinz, M. A. Schneider, F. Mittendorfer, L. Hammer and J. Knudsen, Surf. Sci. 656, 66 (2017). X-Ray Data Booklet, Center for X-ray Optics, (Berkeley, CA, 2009), https://xdb.lbl.gov; Last visit 17.01.2020. C. S. Campos, M. A. Z. Vasconcellos, J. C. Trincavelli and S. Segui, J. Phys. B: At., Mol. Opt. Phys. 40, 3835 (2007). K. Tawara, K. Harrison and F. D. Heer, Physica, 63, 351 (1973). G. Hanke and K. Müller, J. Vac. Sci. Technol. A 2, 964 (1984). K. D. Childs, B. A. Carlson, L. A. LaVanier, J. F. Moulder, D. F. Paul, W. F. Stickle and D. G. Watson, Handbook of Auger electron spectroscopy, 3. Ed., Physical Electronics Industries (Eden Prairie, MN, 1995). F. Reinert and S. Hüfner, New J. Phys. 7, 97 (2005). F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, and S. Hüfner, Phys. Rev. B 63, 115415 (2001). M. Budke, V. Renken, H. Liebl, G. Rangelov and M. Donath, Rev. Sci. Instrum. 79, 083903, (2007). Th. Fauster, D. Straub, J. J. Donelon, D. Grimm, A. Marx and F. J. Himpsel, Rev. Sci. Instrum. 56, 1212 (1985). 2PPE-Tutor (Lehrstuhl für Festkörperphysik, Universität Erlangen-Nürnberg, 2005), www.fkp.physik.nat.fau.eu/research/methods/2ppetutor/start; Last visit 17.01.2020.

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[5.16] [5.17] [5.18] [5.19] [5.20] [5.21] [5.22] [5.23] [5.24] [5.25]

Th. Fauster, Dynamics at Solid State Surfaces and Interfaces, Eds. U. Bovensiepen, H. Petek and M. Wolf, Vol. 1, p. 53. Wiley-VCH (Weinheim 2010). M. Roth, M. Pickel, J. Wang, M. Weinelt and Th. Fauster, Appl. Phys. B 74, 661 (2002). K. Biedermann, S. Regensburger, Th. Fauster, F. J. Himpsel and S. C. Erwin, Phys. Rev. B 85, 245413 (2012). R. Schneider and V. Dose, Further Topics in Low-Energy Inverse Photoemission, Vol. 69, p. 277. Springer (Berlin 1992). D. M. Newns, Vibrational spectroscopy of adsorbates, Ed. R. F. Willis, Springer Series in Chemical Physics 15, Springer (Berlin 1980). K. L. Kostov, S. Polzin and W. Widdra, J. Phys.: Condens. Matter 23, 484006 (2011). R. F. Willis, Surf. Sci. 89, 457 (1979). M. A. Passler, B. W. Lee and A. Ignatiev, Surf. Sci. 150, 263 (1985). NIST Chemistry WebBook; https://doi.org/10.18434/T4D303. L. Hammer, B. Dötsch, F. Brandenstein, A. Fricke and K. Müller, J. Electr. Spectr. Rel. Phenom. 54/55, 687 (1990).

6 Scanning probe microscopies For the investigation of the structure of surfaces, the scanning tunneling microscope (STM) (developed by Binnig and Rohrer, Nobel prize 1986) is now an established technique besides the diffraction methods. Developed shortly afterwards, the atomic force microscope (AFM) has gained less importance in surface physics, but has revolutionized the field of nanotechnology. The fascination of STM images lies in the fact that for the first time it was possible to determine the positions of individual atoms on extended surface areas in real space. Since more than 15 years, this is also possible with an AFM. As fascinating as the direct view of atoms at surfaces is, one also has to realize that scanning probe microscopes alone are by no means sufficient to elucidate the atomic structure of a surface: The measurement accuracy is limited by nonlinearities of the scanning motion and, in addition, an STM topographic image depends not only on the atomic, but also on the electronic structure of the sample. However, the combination with other methods, such as electron diffraction (LEED, Chap. 4.2) or density functional theory (DFT), allows a comprehensive characterization of surfaces and individual adsorbates on the atomic scale.

6.1 Principle of scanning probe methods All scanning probe methods have in common that a “tip” is brought close to the surface of the sample under investigation, and is scanned across a chosen area. The distance between sample and tip depends on the interaction mechanism. For an STM, it is a few atomic distances (0.5–2.0 nm), and distances of up to 10 nm are used for AFM. To bring the tip so close to the surface in a controlled way, a positioning system is needed, which can quickly travel distances of the order of millimeters in 10–100 nm steps. During the scanning motion, the interaction between sample and tip is kept constant using a feedback control system. The control variable in an STM is usually the tunneling current, whereas it is the force in an AFM. The required movement of the tip perpendicular to the surface to keep the control variable at the set value is used to generate the image. In addition, other quantities, which result from the interaction between sample and tip, can be measured and displayed point by point. Due to the minute distance between sample and tip, disturbances (which might move the tip in an uncontrolled way) have to be reduced by many orders of magnitude. The strongest source of such disturbances are building vibrations, which can have amplitudes of 10 μm or more in the frequency range < 100 Hz, mainly due to moving people or running machines. Therefore, a scanning probe microscope has to be mounted vibrationally insulated, and depending on the circumstances, also shielding of sound has to be considered. The realization of scanning probe microscopes has led to the development of many https://doi.org/10.1515/9783110636697-008

144 | 6 Scanning probe microscopies techniques particularly in the field of microscopic positioning systems, which have found widespread use in research and industrial applications. The following section describes the setup of a scanning tunneling microscope in detail. Many of the elements are found as well in an atomic force microscope.

6.2 Scanning tunneling microscopy In a scanning tunneling microscope (STM) (for an on-line tutorial see also [6.1]), a tunneling current I flows between sample and tip, due to the small distance (< 1 nm) and an applied bias (or tunneling) voltage V. Most commonly, wires from tungsten or platinum-iridium alloys are used as materials for the tip. It is formed either by an electrochemical etching process or simply by mechanical tearing or cutting of the wire. The tunneling currents are in the range from 10 pA to 1 μA, and can be converted to a voltage by a simple operational-amplifier circuit shown in Fig. 6.1. The voltage signal can be processed in further electronic circuits or digitized by an analog-digital converter in a computer. To simplify the electrical circuit, the tip is commonly held at ground potential and the tunneling voltage V is applied to the sample. Voltages range from a few millivolts to several volts, and can be controlled by a computer via a digital-analog converter. By applying a sufficiently high positive voltage, electrons tunnel into the conduction band of the sample, and for the reversed polarity, out of the valence band of the sample to the tip. The conductivity of the sample material should not be too low to guarantee that injected charge carriers can flow out via an electrical contact. This is always the case for metals and doped semiconductors, whose surfaces can always be studied by an STM. For insulators, the conductivity is usually too low,

Figure 6.1: Schematic setup of a scanning tunneling microscope (HV = high-voltage amplifier).

6.2 Scanning tunneling microscopy | 145

so they cannot be investigated as bulk materials, but only as few-atomic-layer thick films on a conductive substrate. The measured tunneling current is compared in a control loop to a set value ISet . The control (output) signal of the controller is fed into the z motion, and contains the information about the topography (also “z signal”). The bandwidth of the control loop is usually less than 1 kHz, and allows using simple integrator circuits or digital control loops as controllers. The relative motion of sample and tip is accomplished by piezoelectric elements. In Fig. 6.1, the tip is moved by a tube scanner, whose axial length depends on the electric field strength in the radial direction. By a suitable segmentation of the electrodes, the scanning motion parallel to the sample surface can be realized in addition to the z motion (Fig. 6.2(a)). This is achieved by applying triangular voltages of opposite polarity to the two electrode pairs of the tube for the x and y motion. The piezo elements (short piezos) have typical sensitivities of 1–10 nm/V, which conveniently allows moving the tip in the picometer range. However, for distances of several 100 nm, the low-voltage signals provided by analog-digital converters, or the control circuit, have to be amplified by high-voltage amplifiers. One has to keep in mind that piezo materials exhibit a hysteresis between applied electric field and displacement (Fig. 6.2(b)). The actual motion thus depends on the previous voltage or motion history. This explains why a precise, absolute determination of lattice parameters should not be done using a scanning tunneling microscope, but rather by a diffraction method (Chap. 4). An indispensable element of a scanning probe microscope is a mechanism, which can decrease the distance between sample and tip down to atomic distances in a controlled way (“coarse motion” in Fig. 6.1). Pure mechanical gear or lever reductions can achieve this in principle. More commonly, an inertial drive (Fig. 6.2(c)) is used: A movable plate is moved by saw-tooth-like, fast and slow piezo motions relative to the fixed piezos. During the slow motion, the plate adheres to the piezos and moves along with

Figure 6.2: (a) Piezo-electric tube scanner for executing (x, y, z) motions. (b) Hysteresis of the piezo motion (red curves) compared to the ideal motion (green line). (c) Piezo-electric inertial drive with corresponding driver voltage.

146 | 6 Scanning probe microscopies them. During the fast motion time interval, the accelerating force produced by the piezos exceeds the adhesion force between piezo and plate. As a consequence, the plate is only accelerated by the smaller sliding friction force, and moves relative to the piezos. Each saw-tooth step moves the plate by several 10 nm. To guarantee the stability of the sample-tip distance in a normal laboratory environment, a multi-stage damping system for building vibrations is installed. The table or UHV chamber are damped by an air-friction damping system with an eigenfrequency as low as possible, whereas the STM is constructed to be very stiff with a high eigenfrequency. The combination of both (and possibly additional damping stages) leads to the suppression of disturbances in the range of 1–1000 Hz by at least five orders of magnitude, resulting in variations of the sample-tip distance smaller than 10 pm. The temperature stability of the setup also deserves special consideration: A typical coefficient of thermal expansion of piezo materials is 5 ⋅ 10−6 K−1 , which implies for a piezo scanner with a length of ≈ 1 cm that the temperature is allowed to vary only in the milli-Kelvin range. This requires a sufficiently large heat capacity of the setup, such that rapid temperature changes do not affect the immediate surrounding of the piezo scanner.

6.2.1 Topographic images A scanning tunneling microscopic topography is obtained by keeping the tunneling current at a constant value I = ISet for a given voltage V during the scanning motion. The required distance variation z(x, y) is recorded and displayed in a false-color representation. Usually, brighter colors or shades indicate higher areas, where the tip was retracted from the sample. Even in the most simple model for the tunnel process, the quantum-mechanical penetration of a potential barrier by an electron wave, one obtains an exponential dependence of the tunnel current from the sample-tip distance (see Section 6.2.2 and Fig. 6.5(a)). Accordingly even small height variations of the sample have strong effects on the tunneling current, with corresponding retractions of the z piezo by the control circuit. Topography images were already presented in Chap. 2 (Figs. 2.9, 2.11, and 2.13), and Fig. 6.3 shows two further examples: The Au(111) surface 22 0 ) superstructure, the so-called herringbone structure. It has a reconstructs in a ( −1 2 higher density of surface atoms in the top layer, which leads to height variations of 10 pm. The STM image of Fig. 6.3(a) not only proves the existence of this unit cell with different orientations relative to the crystal axes (domains), but also local structural modifications at the domain boundaries (“elbows” of the structure). The second example in Fig. 6.3(b) presents an STM image of the Si(111)-(7 × 7) surface, whose structural model was already shown in Fig. 1.16. For the final confirmation of that model, STM has played a key role. At the same time, this example illustrates that an STM is more sensitive to the electronic structure of the surface than the geometric structure. The STM topography of a semiconductor changes with the sign of the

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Figure 6.3: STM images and unit cells (marked white) of (a) a reconstructed Au(111) surface with a “herring-bone” structure, and (b) a Si(111)-(7 × 7) surface for different tunnel voltages. Left: Image of the occupied states (V = −1.8 V). Right: Image of the unoccupied states (V = +1.8 V).

tunneling voltage, i. e., it is different if electrons tunnel into the tip (left image) or into the sample (right image). In the latter case (Fig. 6.3(b), right), unoccupied states are imaged, and the spatial structure corresponds to the arrangement of the adatoms of the surface. In the first case (Fig. 6.3(b), left), the STM topography images the spatial structure of the occupied states. At the Si(111)-(7 × 7) surface, the deeper-lying atoms of the reconstruction also contribute to the density of occupied states, and the effect of the different stacking in the two halves of the unit cell becomes visible. At metal surfaces, an influence of the polarity and size of the tunnel voltage is, in most cases, not observed. However, an example in which the electronic properties of the surface play a role for the imaging of a metal surface is shown in Fig. 6.4. A step at the edge of a Cu(111) terrace is imaged, as expected for the quantum-mechanical tunneling: If the barrier width suddenly changes by 220 pm during scanning, the tip has to be moved by the same distance to keep the tunneling current constant. Step edges at metal surfaces are thus often used to calibrate the z motion of the piezos. On the Cu(111) surface, “waves” with ≈ 5 pm amplitude appear at step edges and defects, which are particularly well visible at low temperature. Their wavelengths depend on the tunnel voltage, and they are not visible for voltages V < −500 mV. These waves have pure electronic origin (the metal atoms are arranged geometrically flat) and are caused by interference of quasi-free electrons in the Cu(111) surface state (Fig. 5.16) scattered by steps or defects. The wavelength of the interferences is associated with the wave vector k ‖ . Its dependence on the tunnel voltage can be used to determine the dispersion E(k ‖ ) of the surface state in agreement with ARPES measurements (Fig. 5.16). For the “standing waves,” the electronic influence is easy to identify and understand. For the point defects visible in Fig. 6.4, one might ask the question, whether the depressions are really “holes” or have a different electronic origin. In the present case, they are impurity atoms embedded in the copper surface. The more extended dark spots on the right terrace in Fig. 6.4 are also not depressions in the surface, but the electronic influence of defects or impurity atoms in deeper layers.

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Figure 6.4: Topography image of a step with a height of 220 pm on a Cu(111) crystal at T = 5 K. The eye-catching waves with a modulation amplitude of 5 pm and the extended darker spots on the right, upper terrace clearly demonstrate the sensitivity of the STM-imaging mode to the electronic properties of the sample.

Consequently, one can never be sure whether a bump on the surface of a crystal in an STM image of an unknown structure is an additional atom or if a dimple is a missing atom. One often observes that the tip changes during recording of an topography image, and valleys turn into mountains (or vice versa), or an atomic resolution appears or disappears. Working with a scanning tunneling microscope requires the skills to characterize the properties of the tip sufficiently well and to check whether the conclusions drawn from the experimental results depend on the atomic structure of the tip. One also has to take into account measurement artifacts caused by the meso- or macroscopic structure of the tip. A double or multiple tip can lead to the appearance of multiple images of steps or islands if, depending on the position on the sample, different parts of the tip contribute to the tunnel current. This problem occurs, in particular, when imaging surfaces with many steps or high nanostructures etc. But also images of atomic structures can be falsified by a double tip in ways which are difficult to recognize. To modify the structure of a tip, one can either in tunneling contact raise shortly (such that the control loop does not react to the higher current) the tunnel voltage by a significant amount, or push the tip (more or less) carefully into the sample surface. Alternatively, the tip can be bombarded outside of the STM by ions and heated (Sections 3.2.4 and 3.2.2), or has to be exchanged as a last resort.

6.2.2 Tunnel process The basic description of the tunnel process in an STM considers the transmission of an electron through a potential barrier. In the simplest model, the transmission probability 𝒯 (E, V, Φ, z0 + Δz) through a rectangular potential barrier can be calculated. Here the trapezoidal potential (obtained by neglecting the image potential, Fig. 1.25) between the two electrodes (Fig. 6.5(a)) for |V| ≪ Φ is replaced by a constant barrier of height Φ + eV/2 + EF,S . One obtains for the transmission probability of an electron with energy E 𝒯 (E, V, Φ, z0 + Δz) = exp(−2√

2m eV (Φ + + EF,S − E) (z0 + Δz)). 2 2 ℏ

(6.1)

6.2 Scanning tunneling microscopy | 149

Figure 6.5: (a) Simplified potential diagram of a tunnel contact. (b) Conductivity of the tunnel contact as a function of the sample-tip distance in a scanning tunneling microscope [6.2].

Here m is the mass of a free electron, and V the voltage applied to the tunnel contact. Only electrons can contribute to the tunnel current, whose energies lie in the interval of width eV between the Fermi energies of tip EF,T and sample EF,S , because electrons can tunnel only from occupied states of one electrode to unoccupied states of the other electrode. In the simple model, the effective work function Φ = (ΦS + ΦT )/2 is obtained from the work functions ΦS and ΦT of sample and tip, respectively. For small voltages |V| ≪ Φ and E ≈ EF = 0, the dependence of the tunnel current on the sample-tip distance z = z0 + Δz is determined only by Φ in the following way: I(Δz) = I(z0 )e−κΔz with κ = 2√2mΦ/ℏ = 10.25 nm−1 √Φ/eV. Figure 6.5(b) presents these relationships in an STM experiment, where the distance of the tip from a Ag(111) and a Cu(111) surface was varied and the electrical conductivity G = I(Δz)/V was measured at a voltage of V = 100 mV [6.2]. From the measurement of the tunnel current I(Δz), the inverse decay length κ can be determined. For metal surfaces, a reduction of the barrier width by |Δz| = 0.1 nm leads to an increase of the tunnel current by a factor of 10. This results in typical values for the inverse decay length of κ ≈ 23 nm−1 = 1/(0.043 nm), corresponding to an effective barrier height Φ ≈ 5 eV comparable to the work function of a metal (e. g., Cu(111): Φ = 4.93 eV). If one manages in an experiment to keep the tunnel current I(z0 ) constant with deviations of 10 %, the sample surface is scanned with a precision of Δz ≈ ±4 pm. This illustrates the high sensitivity of the scanning tunneling microscope to height differences of the sample surface, such as step edges (Fig. 6.4) or the reconstruction of the Au(111) surface (Fig. 6.3(a)). The measurement of the inverse decay length κ can be used to identify different materials or dipole fields present on the surface. Though, in this case, κ reflects changes of the local work function (Section 1.2.5), the two quantities should clearly be distinguished from each other. In principle, I(Δz) depends on the electric potential of the sample on the atomic scale, but in addition it also depends on the energy interval eV (Fig. 6.5(a)), from which the tunneling electrons contribute. Therefore, one cannot use I(Δz) directly to draw conclusions on the charge distributions relevant for the work function.

150 | 6 Scanning probe microscopies When the last atom of the tip gets in contact with the surface, one expects for the conductance of such a one-dimensional contact a single conductivity quantum G0 = 2e2 /h = 77.5 μS. The transition from the tunneling regime to the quantized conductance can be identified in Fig. 6.5(b) as a jump and leveling off of I(Δz) at small distances. The measurement shows that at a conductivity of the tunnel contact of 1 nS, the end of the tip is at a distance of z0 ≈ 0.5 nm (corresponding to 2–3 atomic bond lengths) from the sample surface, which is defined here as the position of electrical contact. The most commonly used theoretical description of the tunnel current in a scanning tunneling microscope is based on the work of Tersoff and Hamann [6.3]. The calculation is based on the transition probability of an electron from a state of the one electrode to a state of the other electrode in time-dependent perturbation theory. The tunnel current is then determined by the overlap of the wave functions of sample and tip in an arbitrary plane within the barrier region of the contact. In the Tersoff– Hamann model, it is assumed that the end of the tip can be described by a spherical potential, and that the relevant states of the tip, which contribute to the tunnel current, have an s-like character. Their amplitude depends only on the distance from the center of the (hemi-)spherical tip and their energy. Figure 6.6(a) presents a schematic illustration of the overlap of the wave functions of sample and tip. The amplitude of the wave functions in the barrier region is given by the numbers in arbitrary units. The assumption of an s-like wave function is justified only for sufficiently large sample-tip distances. One recognizes that the lateral extension of the region of maximum overlap roughly corresponds to the “diameter” of the last atom of the tip, i. e., the tunnel current is negligible outside of an area 0.4–0.6 nm in diameter. The Tersoff–Hamann model yields the following expression for the tunnel current: ∞

I(x, y, z, V) ∝ ∫ DT (E − eV)ρ(x, y, z, E)(f (E − eV, TT ) − f (E, TS ))dE.

(6.2)

−∞

Figure 6.6: Illustration of the tunnel process after Tersoff and Hamann: (a) Schematic representation of the amplitude (in arbitrary units) of the wave function in the STM. (b) Partitioning of the tunnel current, according to energy ranges and density of states DT and ρ∗ .

6.2 Scanning tunneling microscopy | 151

Here f (E, T) is the Fermi function with the Fermi level at E = 0; TT and TS are the temperatures of tip and sample, respectively; V is the potential of the sample relative to the tip; DT (E) is the density of states of the tip, and ρ(x, y, z, E) is the local density of states of the sample at the center of the tip (x, y, z). The local density of states is defined as ρ(x, y, z, E) = ∑ν |ψν (x, y, z)|2 δ(Eν − E), where ψν (x, y, z) is the amplitude of the wave function of the sample at position (x, y, z) with energy Eν , and the probabilities |ψν (x, y, z)|2 are summed over all states ν of the sample. Very often the Tersoff–Hamann equation is used for low temperatures, and also the decay of the states of the sample (cf. Fig. 1.21) is approximated by the density of states at the surface and a transmission probability (eq. (6.1)): ρ(x, y, z, E) = ρ∗ (x, y)𝒯 (E, V, z), where the barrier width z is the distance of the surface to the center of the spherical potential representing the tip. One obtains the more commonly used form of the Tersoff–Hamann equation for TS = TT = 0: eV

I(x, y, z, V) ∝ ∫ DT (E − eV)ρ∗ (x, y, E)𝒯 (E, V, z)dE.

(6.3)

0

For finite tunnel voltages V an energy interval of width |eV| results, in which electrons can tunnel from one electrode to the other. Figure 6.6(b) illustrates, by using the width of the arrows between the electrodes, how the total tunnel current is partitioned into several energy ranges taking into account the transmission coefficient of eq. (6.1). The highest tunnel probability is always obtained for states at the Fermi energy of the negatively biased electrode, according to eq. (6.1). The current is modulated in each energy range by the density of states of both electrodes. Whereas the high vertical resolution of the STM can easily be understood from the transmission probability eq. (6.1), the lateral resolution is more difficult to explain. As stated before, the tunnel current is spatially strongly localized by the pointed end of the tip (“the last atom”), however, a resolution of better than 0.3–0.4 nm cannot be rationalized in this simple picture. The Tersoff–Hamann model yields a dependence of the modulation strength of the density of states at a distance of 0.6–0.8 nm from the surface and the radius of the end of the tip. At metal surfaces, that modulation is quite small and the lateral resolution would be ≈ 0.4 nm, since for smaller distances between the atoms of the tip and sample, the vertical tip motions required to keep the tunnel current constant would be too small to control. For metals, atomic resolution is often only achieved using small tunnel voltages V = 1–5 mV, and large currents I = 1–10 nA. Then the tip is positioned relatively close to the sample surface and is scanned via a chemical bond (possibly involving an adsorbate) between sample and tip. Alternatively, more localized orbitals pz or dz2 of the tip have to be considered for the interpretation [6.4]. Both aspects go beyond the Tersoff–Hamann model.

6.2.3 Spectroscopy The local electronic structure of surfaces can be investigated in the spectroscopic mode of scanning tunneling microscopy called scanning tunneling spectroscopy (STS).

152 | 6 Scanning probe microscopies This method of spectroscopy is based on the dependence of the tunnel current on the local density of states and is able to determine this quantity as a function of energy. In contrast to photoelectron spectroscopy (Chap. 5.3), STS does not allow determining the dispersion relation of the electronic band structure E(k ‖ ). However, the occupied and the unoccupied density of states can be investigated by choosing the appropriate sign of the tunnel voltage. In STS, the voltage-dependent tunnel current I(x, y, z, V) is measured as function of the position (x, y, z) of the tip. Typically, the 𝜕 I(x, y, z(I0 , V0 ), V) is recorded in STS, or just the derivative of the tunnel current 𝜕V current I(x, y, z(I0 , V0 ), V) in current imaging tunneling spectroscopy (CITS). In both variants, the tip-sample distance is established by the choice of the set current I0 and voltage V0 prior to the spectroscopy measurement. Then the control loop is opened, keeping z0 = z(x, y, I0 , V0 ) at fixed position, and the voltage V is varied. This procedure can be executed at a fixed point (x, y) or at each point of an image. In STS, the derivative of the characteristic I(V) of the tunnel contact is usually obtained using a modulation technique, where the voltage V is modulated by an AC 𝜕 voltage ΔV(t) ≪ V. The amplitude of the 𝜕V I(V) signal is determined from I(V +ΔV(t)) with a lock-in amplifier, as described in Section 5.1.5. The frequency of the modulation voltage has to be larger than the cutoff frequency of the control loop, so that the modulation of the current signal is not compensated by a variation of the tip-sample distance. The advantage of this method is an improved signal-to-noise ratio in comparison to a numerical differentiation of the I(V) characteristic. However, the amplitude of the modulation voltage also reduces the energetic resolution of the spectra. Since the resolution also depends on the temperature T of tip and sample, one should choose |eΔV(t)| ≤ kB T for optimal performance and resolution. In CITS, usually a few voltages Vi are selected, for which the tunnel current I(x, y, z0 , Vi ) is measured at each image point. This method is particularly well suited to image lateral variations of a state, which contributes to the tunnel current. For the differential conductivity of the tunnel contact, one obtains from eq. (6.3) with z0 = z(x, y, I0 , V0 ) the following expression: 𝜕 I(x, y, z0 , V) ∝ e𝒯 (eV, V, z0 )DT (0)ρ∗ (x, y, eV) 𝜕V eV

+ ∫ ρ∗ (x, y, E)𝒯 (E, V, z0 ) 0

𝜕 D (E − eV)dE 𝜕V T

eV

+ ∫ ρ∗ (x, y, E)DT (E − eV) 0

𝜕 𝒯 (E, V, z0 )dE. 𝜕V

(6.4)

The differential conductivity is thus proportional to the density of states of the sample at the energy E = EF + eV (first term). The remaining terms can be interpreted as a slowly varying background. The derivation of eq. (6.4), starting from eq. (6.2) for finite and equal temperatures T for sample and tip, would have led in the first term of

6.2 Scanning tunneling microscopy | 153

Figure 6.7: (a) Current-voltage characteristic I(V ) and (b) STS spectrum dI/dV at T = 6 K, measured on an extended terrace of a Cu(111) crystal. The strong increase of the dI/dV signal at −0.44 V indicates the band minimum of the surface state.

eq. (6.4) to a convolution of the density of states of the sample ρ∗ (x, y, z0 , E) with the derivative of the Fermi function 𝜕f (E −eV, T)/𝜕E. The latter is a Gaussian-shaped curve with a full width at half maximum of ΔE = 3.5 kB T, explaining the strong temperature dependence of the resolution in STS. Figure 6.7 presents an example, which supports the proportionality of the differential conductivity to the density of states. The I(V) characteristic in (a) essentially shows a linear dependence (Ohmic behavior), as expected for a tunnel contact between two metallic electrodes. At a voltage V = −0.44 V, the slope of the curve changes noticeably. Accordingly, the dI/dV signal obtained by modulation technique exhibits a step-like increase at this voltage (Fig. 6.7(b)). The surface state on Cu(111) forms a two-dimensional electron gas characterized by a parabolic dispersion, starting at an energy E − EF = −0.44 eV (cf. Fig. 5.16). Therefore, a constant density of states and dI/dV signal is expected for V > −0.44 V. This represents only one contribution to the actual spectrum. The detailed shape of the slowly varying background depends also on the configuration of the tip. The simple interpretation of eq. (6.4) has to be considered with care. The common assumption is that the second and third term can be neglected for small voltages V and constant density of states DT (E). In practice, one finds tip configurations, which do not have a large variation of DT (E). The background of the measurement and the variation of DT (E) can then be identified by the lack of a position dependence. But even under these conditions, sample and tip are two equivalent electrodes of the tunnel contact in eq. (6.3), which means they can be interchanged. Then dI/dV becomes proportional to the density of unoccupied states of the tip at E = EF − eV , and the remaining terms describe the background, which contains the actual position-dependent information in this case. Which of the two interpretations prevails in the dI/dV signal depends also on the voltage V . Due to the higher tunnel probability of electrons at the Fermi energy of the negative electrode, the more the density of states of the tip attains higher weight, the more negative the voltage V becomes, because mainly electrons from the Fermi energy of the sample tun-

154 | 6 Scanning probe microscopies

nel into unoccupied states of the tip. For large positive voltages, the situation is reversed; for small voltages, both contributions get intermixed. In that case, the variation of the dI/dV signal as function of the voltage is dominated by the electrode, which exhibits the strongest variation of the density of states with respect to energy [6.5]. Strong variations of the density of states of the sample (e. g., band edges in semiconductors or orbitals of adsorbed molecules) can be identified also for negative voltages (occupied states of the sample) regardless of the discussed problems.

6.2.4 Applications and other interaction mechanisms Scanning tunneling microscopes are usually operated in ultrahigh vacuum at various temperatures, most conveniently at room temperature. In cryostats (as low as 10 mK), scanning tunneling microscopes exhibit an astounding stability and allow studying experimental low-temperature physics at surfaces (electron dynamics, superconductivity, many-electron effects, etc.). At high temperatures (up to about 800 K) growth processes at surfaces may be studied. An important aspect of surface studies concerns the identification of different materials or species at surfaces (chemical contrast). In an STM, this can be achieved either by measuring the local decay length of the tunnel current or by identifying different electronic properties in STS. In the context of research and development of molecular electronics, the properties of organic molecules on surfaces has gained increasing interest. Through the direct access on the nanometer scale, an STM permits the spectroscopy of individual single molecules on a surface and the investigation of the current transport through the molecule after forming a point contact between tip and molecule. Using a tip made from a ferromagnetic material or with a ferromagnetic coating, measurements of the topography or spectra can exhibit a contrast, which depends on the relative magnetization of sample and tip. This is explained by the fact that the electrons keep their spin orientation during the tunneling process. If tip and sample are ferromagnetic, their density of states is spin dependent. The tunnel current (of minority charge carriers) becomes large if the magnetization orientations of both electrodes are parallel. Surprisingly, a scanning tunneling microscope can also be operated under normal atmospheric conditions on sufficiently inert surfaces (e. g., graphite or gold). However, the situation is markedly different from tunneling under vacuum conditions. The apparent barrier height is often found to be only 100 meV or less, which points to the presence of different conductivity mechanisms, e. g., the activated conduction through molecules. Under well-controlled conditions, such mechanisms and growth processes can be investigated during the galvanic precipitation in an electrochemical scanning tunneling microscope (EC-STM), where the STM is operated in an electrolyte.

6.3 Atomic force microscopy | 155

6.3 Atomic force microscopy In contrast to STM, the distance measurement between sample and tip in an atomic force microscope (AFM) relies on measuring a force. This permits working on nonconducting samples as well. One distinguishes between the contact AFM, in which the force is exerted by direct contact between sample and tip, and the noncontact AFM, in which the change of the force is measured at a chosen distance away from the sample surface. The force is measured by sensing the motion of a micron-size cantilever with a tip located at its end. Such cantilevers are produced by lithographic processes on silicon wafers inexpensively, with high precision, and in large numbers (about 400 cantilevers on a 4-inch wafer). Cantilever, tip, and mount are made from the wafer material, optionally the tips can be coated. The geometric dimension of the cantilevers determines their force constant D. For a contact AFM, soft cantilevers with D = 0.005–0.5 N/m are used, and for noncontact AFM D = 1–50 N/m is required. In contact AFM, the deflection of the cantilever is measured, due to an applied static force, whereas in noncontact AFM, the cantilever oscillates and its resonance frequency or oscillation amplitude is registered as a function of the sample-tip distance. In both cases, the motion of the cantilever has to be measured in the nanometer range. Commonly used methods are the beam-deflection detection (Fig. 6.8), the interferometric detection, or for specially coated cantilevers, the utilization of a piezoresistive or piezo-electric effect. Since AFMs are often used to record images of surfaces on a micron scale, the nonlinearities of the piezos mentioned in Chap. 6.2 play a larger role, and have to be taken into account. Artifacts occur when imaging nanostructures with a large aspect ratio (e. g., holes or grooves, which are deeper than their lateral width). Here the tip may not reach the deepest point of the structure as it makes contact with the edges of the structure, and also sharp edges may appear rounded, due to the tip curvature.

Figure 6.8: Beam-deflection detection of the motion of an AFM cantilever. Normal forces between sample and tip deflect the cantilever in perpendicular direction. The photodiode registers the change of light incidence on quadrants A and B compared to quadrants C and D. Lateral forces lead to a torsion of the cantilever, which is detected by comparing the signals from quadrants A and C with those of B and D.

156 | 6 Scanning probe microscopies

Figure 6.9: (a) Contact-AFM image of a nanostructure formed by a 25 nm thick bismuth film. The polycrystalline structure of the 20 nm wide nanowires can be seen clearly. (b) Noncontact AFM image of a NaCl(100) surface at 5 K in UHV. Only the Cl− ions are imaged [6.6].

6.3.1 Contact AFM Figure 6.9(a) shows the image of a metallic nanostructure, which was obtained in air by a contact AFM. The image was recorded for a fixed repulsive force of ≈ 5 nN at the cantilever. A control loop, comparable to the one used in STMs, keeps the deflection of the cantilever, and thus the force constant during the scan. In contact mode, the achievable lateral resolution is limited by the contact area, which can be estimated from the elastic properties of the tip and sample material. In the Hertz model of contact mechanics, the circular contact area between an elastic sphere (tip) and an infinitely extended half space (sample) has a radius a, a=√ 3

3 FR . 4 E∗

Here F is the force acting at the contact, R the radius of the sphere, and E ∗ an effective common elasticity modulus of the two materials. For a silicon dioxide tip with a typical tip radius R = 10 nm and a force F = 5 nN, one obtains a contact radius a ≥ 1 nm, which is clearly too large to image a sample on the atomic scale. The force F is chosen as small as possible to minimize the contact area and to avoid a plastic deformation of sample and tip. This requires soft cantilevers with small force constants.

6.3.2 Dynamic AFM With a soft cantilever, only the repulsive forces between sample and tip can be used for imaging, because the tip is always in contact (Section 6.3.3). To guide the tip at a larger distance over the surface, such that only weak attractive forces act and contact with the surface is avoided or at least minimized, one has to operate the AFM in a dynamic mode, in which the cantilever executes a forced oscillation. The cantilever is mounted on a piezo, which is driven in such a way that the cantilever oscillates close to its resonance frequency (f0 ≈ 150 kHz). The forces on the tip, acting when brought close to the surface, change the eigenfrequency, and—possibly—also the quality factor of the resonance. For fixed driving frequency and amplitude, a change of the amplitude

6.3 Atomic force microscopy | 157

of the forced oscillation is measured depending on the mean distance between tip and sample. The simplest operation mode of a dynamic AFM is the tapping mode or intermittent contact mode. The tapping mode is the most widespread operational mode for AFM measurements in air. Though atomic resolution cannot be achieved, larger molecules or structures on a nanometer scale can be imaged reliably. The cantilever oscillates with amplitudes up to 10 nm and touches the surface during a short fraction of the oscillation period, which leads to a reduction of the oscillation amplitude. The tapping mode has the advantage over the contact mode, that the repulsive force between sample and tip acts only shortly and can be relatively small to be detected as amplitude modulation of the cantilever oscillation. The control loop is set up in such a way that the oscillation amplitude of the cantilever is kept constant by controlling the tip-sample distance. The oscillation amplitude is usually measured using the beamdeflection detection method by a lock-in amplifier. The influence of the repulsive force on the oscillation amplitude becomes noticeable after a time τ = 2Q/f0 , where Q is the quality factor and f0 the eigenfrequency of the oscillating cantilever. Therefore, this mode is not well suited for studies under UHV conditions, because without dampening adsorbate layers (typically water), the quality factor of the cantilever becomes very large, and accordingly, the tip has to be moved rather slowly across the surface. More interesting for surface physics is an operation mode, which can achieve atomic resolution under UHV conditions, is the following: In a noncontact AFM or FM-AFM (frequency-modulated AFM), the resonance frequency of the cantilever oscillation is reduced by the attractive force between sample surface and tip. This change of the resonance frequency can be detected on the time scale of the oscillation period. As control parameter for the distance control, a chosen frequency shift Δf ≈ f0 /100 is used, which then yields an image of the scanned surface. Figure 6.9(b) shows an image of a NaCl(100) surface obtained under UHV conditions using the noncontact mode of a “tuning-fork” AFM [6.7]. The atomic resolution indicates that a short-range force is used to image the surface. This force has the property of imaging only the Cl− ions as protrusions. The frequency shift Δf is determined by a phase-locked loop as tracking filter. A tunable oscillator is tuned to the current resonance frequency of the cantilever, and the associated control parameter is proportional to Δf . A distance control loop, as in an STM, keeps Δf constant by controlling the distance between sample and tip. The dynamic mode is operated using a constant driver or oscillation amplitude of the cantilever. The latter option requires an additional control loop, which adjusts the oscillation amplitude to the set value by changing the driver amplitude, but has the advantage that conservative and dissipative forces are kept apart in the topography signal. A soft cantilever with a force constant D < 50 N/m may still be pulled into contact by the attractive forces near the surface (Section 6.3.3). To prevent that from happening, the amplitude of the oscillation in the noncontact mode has to be large enough that during the operation the elastic energy of the cantilever deflection is larger than the potential minimum at the surface. Due to the large oscillation amplitude, the cantilever is close to the surface only for very short times during the oscillation period, which complicates

158 | 6 Scanning probe microscopies

the physical interpretation of the contour Δf = const. as topography. For an AFM with a tuning-fork sensor with a force constant of several 1000 N/m, the situation is easier. These sensors can be operated at very small amplitudes (< 0.1 nm) of the cantilever, which are measured by piezo-resistive elements. For small oscillation amplitudes, the frequency shift Δf is proportional to the gradient of the force on the tip at the chosen sample-tip distance.

Since long-range forces are always acting as well between sample and tip of an AFM, this puts high demands on the “sharpness” of the tip. In contrast to an STM, not only the configuration of the atoms at the end of the tip, but also the mesoscopic form of the tip on a 10 nm scale is relevant for the measured interaction strength.

6.3.3 Forces and their effects in AFM The atomic force microscope exploits for imaging the forces that act between the tip attached to an oscillating cantilever and the surface. The total force has various contributions that are relevant at different ranges of the distance z from the surface: – Repulsive force, due to the Pauli principle (z < 0.1 nm). – Force due to the binding between atoms (z = 0.1–0.4 nm). – Attractive van-der-Waals force (long range, dominating for z ≫ 0.5 nm). – Electrostatic forces (long range, dominating for z ≫ 0.5 nm). – Attractive capillary forces (long range, larger than van-der-Waals force) additionally occur in non-UHV environments. The repulsive force due to the Pauli principle arises when the overlap of the electron shell of the atom of the tip overlaps with the ones of the sample. It corresponds to the force that exists in contact between surface and tip in vacuum. This force actually rises exponentially with decreasing distance, but is often approximated by a repulsive term VPauli (z) ∝ z −12 within the Lennard-Jones model, which has an attractive atomic van-der-Waals term VB (z) ∝ −z −6 . The binding potential between two atoms has a well-defined minimum at the distance of a bond length. The van-der-Waals interaction between a sphere of radius R at a distance z from a plane surface has for z ≪ R a dependence VvdW (z) ∝ R ⋅ z −1 . The resulting force F(z) is obtained from the derivative of the sum of all potential contributions, as shown in Fig. 6.10(a). A possible electrostatic interaction (e. g., the contact potential VKP = ΔΦ/e for different work functions 2 −1 of sample and tip) results in a potential VC (z) ∝ RVKP z , and can be larger than the van-der-Waals potential. When operating an AFM in a normal air environment, capillary forces, due to the adsorbed water film on the surface, have also to be taken into account.

6.3 Atomic force microscopy | 159

Figure 6.10: (a) Typical dependence of the force F (z) between AFM tip and sample as a function of distance z, resulting from the sum of the contributions of the atomic Lennard-Jones and long-range van-der-Waals forces. In addition, the negative of the force FC (z, z0 ) is plotted, which the cantilever exerts on the tip. At the tip distances z1 , z2 , z3 , the resulting force on the tip is zero. (b) Resulting force-distance curve of FC (z, z0 ) for changing distances z0 (D ≈ 25 N/m).

The tip is mounted on the cantilever, which ideally exerts a force on it according to Hooke’s law FC (z, z0 ) = −D(z − z0 ). Here z0 is the distance of the tip from the surface, which the tip would have in the absence of interaction forces. The actual distance z of the tip is obtained from the condition F (z) + FC (z, z0 ) = 0. Figure 6.10(a) illustrates that if the force constant D of the cantilever is smaller than the gradient of the force between tip and surface F (z), depending on z0 , one or more distances z1 , z2 , z3 exist for which the resulting total force on the tip vanishes. For F (zi + δz) < −FC (z + δz, z0 ) and F (zi − δz) > −FC (z − δz, z0 ), a stable state is obtained. If the distance z0 is reduced in Fig. 6.10(a), the points z1 and z2 get close to each other until the distance z3 becomes the only stable state. The tip is then immediately attracted to the surface (jump-to-contact or snap-in). If the distance z0 is increased again, the point z3 moves toward the unstable state z2 , and the tip jumps away from the surface to a distance corresponding to z1 . This behavior leads to a hysteresis of the force-distance curve. In the static AFM mode, this bistability does not allow holding the tip at a distance, which corresponds to the attractive part of the tip-sample interaction. This drawback could be avoided by using a cantilever with large D ≫ 50 N/m, but then small forces cause only small deflections which are difficult to measure. Figure 6.10(b) shows schematically a force-distance curve, which would be measured by a soft cantilever. Note that the value of the repulsive force cannot be measured in contrast to the attractive force of adhesion. If the cantilever oscillates with a sufficiently large amplitude (at a correspondingly large distance z0 ), the force between sample and tip is never large enough to cause a “snap-in”. If a cantilever with large D oscillates with a small amplitude, no “snap-in” occurs, due to the stiffness of the cantilever. However, the forced oscillation of the cantilever is influenced by the force between sample and tip. The gradient of the force F (z) effectively modifies the force constant of the oscillating cantilevers and shifts the resonance frequency by an amount Δf (z0 ). For a sufficiently small oscillation amplitude, the f0 𝜕F (z) can be assumed to be constant and one obtains Δf ≈ − 2D (z) for Δf ≪ f0 . Under gradient 𝜕F 𝜕z 𝜕z these conditions, the oscillation can be stabilized in the attractive and in the repulsive range of the interaction force.

160 | 6 Scanning probe microscopies 6.3.4 Applications and other interaction mechanisms Applying the AFM, in particular to nanostructures, has lead to the development of many related techniques, some of which will be mentioned in what follows. The strong interaction between sample and tip in the contact mode prevents imaging the sample on an atomic scale, but offers other possibilities. The torsion of the cantilever in contact mode (Fig. 6.8) allows, rather easily, measuring lateral forces, which act during the scanning motion on the tip. This provides information on the tribological properties of surfaces or coatings. If the tip is coated by an electrically conductive material, nanostructures or larger molecules can be contacted and electrical transport can be measured in the AFM contact mode. Through the analysis of forcedistance curves (Fig. 6.10(b)), additional properties of the surface can be investigated: The adhesion force between sample and tip is directly accessible. If molecules with internal structural degrees of freedom are adsorbed on the surface, they can lead to additional jumps in the deflection traces. In particular for large molecules (polymers, nucleic acids, etc.), the folding of molecular strands can be studied. For high-resolution images or the investigation of atomically clean surfaces in UHV, the method of choice is the noncontact mode of AFM. The oscillation mode offers the option to study also dissipative forces by measuring the energy that is needed to sustain the cantilever oscillation. Furthermore, the phase shift of the cantilever oscillation relative to the driving signal can be used to image the variation of elastic properties of the sample and adhesion and friction forces on a nanometer scale. In the noncontact mode of the tuning-fork AFM with a rather small oscillation amplitude of the tip, the sample-tip distance can be chosen in a range similar to an STM. If a metallic tip is used, the setup may be operated as an STM and as an AFM. This combination permits correlating short-range forces and tunnel currents, and to understand common mechanisms on the atomic scale. With modified AFM probe tips, other interaction forces can be measured. The two most prominent modifications are the magnetic force microscope (MFM) and the Kelvin-probe force microscope (KPFM). In an MFM, a ferromagnetic layer is deposited on the tip, which can sense the forces originating from the magnetic stray field of magnetic structures of the sample. This allows investigating the domain structure of magnetic layers, as used in hard disk drives. In an KPFM, an electrically conducting tip is used. Variations of the contact potential generated by different work functions of sample and tip can be compensated and measured by applying a bias voltage between sample and tip. MFM and KPFM benefit from employing the dynamic imaging mode, which allows disentangling short-range forces characterizing the topography from the long-range electric or magnetic dipole forces. If instead of a cantilever the end of an optical fiber is used, the distance to the surface becomes smaller than the wavelength λ of the light, and the electromagnetic near field can be exploited to investigate optical properties on lateral length scales smaller than λ in a so called scanning near-field optical microscope (SNOM). The same

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can be achieved by illuminating the region between sample and AFM tip by focused laser radiation. The tip causes an enhancement of the electric field, and the corresponding effects on the polarizability of the sample can be detected in the far field of the scattered light (aperture-less SNOM). This research makes connection to the field of plasmonics, which deals with the interactions between electromagnetic fields and nanostructures. The scanning tunneling microscope comes close to turning the dream of “seeing” atoms into reality. It gives important clues for the sample preparation (Chap. 3) and the structure determination (Chap. 4). At second glance, one realizes that the images are actually determined by the electronic structure. This fact is exploited in scanning tunneling spectroscopy to investigate the electronic states in detail on an atomic scale, and to compare with results from photoelectron spectroscopy (Chap. 5.3). The atomic force microscope offers many experimental variants. The relevant forces between tip and sample strongly depend on the distance, and the achievable lateral resolution varies accordingly. The atomic force microscope has become an important tool in nanosciences, and atomic resolution requires special operation modes and good experimental skills.

Q6.1: In Fig. 6.1, a simple operational-amplifier circuit is shown to measure the tunnel current. What value has to be chosen for the resistor R to convert a current of 1 nA to a voltage of 1 V? Which thermal noise has to be expected, and what are typical problems in the practical realization of the circuit? Q6.2: The piezo scanners used in scanning probe microscopes have responsitivities of 0.5 nm/V in z direction and 5 nm/V in xy direction. What is the maximum permitted value of the noise level of the high-voltage amplifiers (HV x, y, z in Fig. 6.1) to allow scanning a lateral range of 1 μm with 0.05 nm resolution? What is the perpendicular resolution? Q6.3: What happens on areas with a lower work function on a surface when they are imaged in constant-current mode by a scanning tunneling microscope? Q6.4: Show in a one-dimensional model of the electronic states at a potential barrier that the current in a scanning tunneling microscope has a sinusoidal variation at a step on the surface. Q6.5: Explain why the contact between sample and tip in Fig. 6.5(b) has the conductivity G0 .

[6.1] [6.2] [6.3] [6.4] [6.5] [6.6] [6.7]

STM-Tutor (Lehrstuhl für Festkörperphysik, Universität Erlangen-Nürnberg, 2004), www.fkp.physik.nat.fau.eu/research/methods/stmtutor/start; Last visit 17.01.2020. L. Limot, J. Kröger, R. Berndt, A. Garcia-Lekue and W. A. Hofer, Phys. Rev. Lett. 94, 126102 (2005). J. Tersoff and D. R. Hamann, Phys. Rev. B 31, 805 (1985). C. J. Chen, Phys. Rev. B 42, 8841 (1990). G. Hörmandinger, Phys. Rev. B 49, 13897 (1994). S. Maier, (Department of Physics, Universität Erlangen-Nürnberg, 2004), unpublished. F. J. Giessibl, Appl. Phys. Lett. 73, 3956 (1998).

7 Particle spectroscopies During many processes at surfaces, particles are exchanged between the surface and the surrounding gas phase. Catalytic processes at surfaces involve the adsorption, dissociation, diffusion, reaction, and desorption of the reactants and the reaction products. Even the simple release or desorption of atoms and molecules from a surface can proceed in different ways: Thermally by the excitation of phonons or molecular vibrations, electronically by excitation with photons or electrons, and directly by impact of atoms or ions. Accordingly, a multitude of investigation methods are available with different degrees of complexity. Here two simple, widely used examples are presented: Thermal desorption spectroscopy allows determining the binding energies of atom and molecules at surfaces. Low-energy ion scattering offers the possibility to identify the atoms of the topmost one or two layers of a surface, according to their mass.

7.1 Thermal desorption spectroscopy The atoms or molecules adsorbed on a surface are bound with an energy (Chap. 2.2), which is characteristic for the atom or molecule, the specific surface, and the adsorption site. If this binding energy is provided to the adsorbate, e. g., by electronic, optical, or thermal excitation, it may desorb from the surface. In this section, the thermally stimulated desorption is discussed, where the adsorbate gains the energy necessary for the desorption from the thermal energy budget of the surface. This can occur at constant temperature of the surface (isothermal desorption). For that the gas phase of the particles adsorbing on the surface is rapidly pumped away and subsequently the surface is flashed to a constant elevated temperature (flash desorption). More widespread is the method of a linear increase of the sample temperature over time, and the simultaneous recording of the number of desorbing particles as function of time and temperature (thermal desorption spectroscopy (TDS), or temperature-programmed desorption (TPD)). The adsorbed particles desorb sequentially with increasing binding energy, which leads to an associated pressure increase in the vacuum chamber. In what follows, we focus on this method and explain how the binding energy of the adsorbed particles can be determined by TDS.

7.1.1 Fundamentals and experimental setup According to the ideal gas law, the partial pressure p of a particular gas species with the volumetric (particle-number) density n at the absolute temperature T is given as p = nkB T. If the same particle species desorbs from a surface and is then pumped away https://doi.org/10.1515/9783110636697-009

7.1 Thermal desorption spectroscopy | 163

by a vacuum pump, the associated pressure change is Ak T dθ S dp = − B na − p. dt V dt V

(7.1)

Here V denotes the volume of the vacuum chamber, A the surface area of the sample, S the pumping speed of the pump (unit m3/s), na the areal density of the adsorption sites on the surface, and θ the relative occupation of these sites (coverage) by atoms or molecules. In this balance, it is assumed that all particles are pumped away and do not readsorb. The desorption rate is defined as RD (t) = −na dθ/dt and leads to a positive source term in eq. (7.1), because for desorption, dθ/dt < 0 is always guaranteed. The ratio V/S can be interpreted as the time constant describing how fast the pressure reaches a new stationary value after a change of the desorption rate. If the temporal changes of the desorption rate occur more slowly than this time constant, a quasi-stationary state with dp/dt ≈ 0 is always found, and the pressure follows the desorption rate. This approximation is fulfilled, in particular, for small heating rates and large pumping speeds near the sample surface. The desorption rate is then obtained as t

RD (t) = −na

S S dθ = p and integrated as na (θ0 − θ(t)) = ∫ p(t 󸀠 )dt 󸀠 , (7.2) dt AkB T AkB T 0

where θ0 is the initial relative occupation of the adsorption sites at time t = 0. For ∞ t → ∞, all particles have desorbed (θ = 0), i. e., na θ0 = (S/AkB T) ∫0 p(t 󸀠 )dt 󸀠 . The integral over the complete pressure curve is thus proportional to the initial adsorbate coverage. The applicability of eq. (7.2) assumes that during the desorption experiment no particles desorb from the sides of the sample, the sample holder, or the walls of the UHV chamber. To minimize these problems, one often chooses the experimental setup sketched in Fig. 7.1. The aperture of a quadrupole mass spectrometer is put as close as possible to the sample surface, and an additional surrounding shield prevents the unwanted entrance of particles not coming from the surface into the spectrometer. The mass spectrometer is tuned to the mass of the desorbing particles, and the signal IM (t) is proportional to the desorption rate IM (t) = −α na dθ/dt. The proportionality factor α can either be calibrated by an experiment with known initial coverage na θ0 ∞ (α = (1/na θ0 ) ∫0 IM (t)dt), or by using information on the coverage from supplementary surface-sensitive measurements. The relative desorption rate is described by the Polanyi–Wigner equation RD (T) dθ =− = θq ν e−E/kB T , na dt

(7.3)

which was introduced already in Chap. 2.2 (eq. (2.2)). Here the indices of E and ν are omitted to simplify the notation. The desorption order is denoted by q, and E is the activation energy for desorption. The rate constant ν is also called attempt frequency of

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Figure 7.1: Experimental arrangement for measuring temperature-programmed desorption spectra. The lower part illustrates the conversion of the temporal signals of temperature and mass spectrometer into a spectrum as function of temperature. The temperature is measured, in most cases, by a thermocouple attached to the sample.

desorption. The attempt frequency multiplied by the Boltzmann factor gives the average number of successful desorption events per unit of time. The rate constant ν is, in many cases, of the order of phonon frequencies (≈ 1013 Hz). However, ν can be much larger (e. g., ν ≈ 1016 Hz for the CO desorption from a Pd(100) surface [7.1]) and can also depend on temperature and coverage. The desorption energy E can also be coverage dependent, e. g., if the adsorbed particles interact with each other. The desorption energy E is not necessarily identical to the binding energy EB of the atoms or molecules: If during adsorption an activation energy has to be surmounted, it has to be added to EB to obtain E (Fig. 2.5). If particles have to diffuse on the surface (e. g., to find a reaction partner, cf. Section 7.1.2) before they can desorb, the activation energy ED of surface diffusion (Section 2.3) may have to be considered as well. For temperature-dependent ν, a possibly involved additional activation energy has also to be added to obtain E.

7.1.2 Types of desorption spectra Different types of spectra are obtained depending on the desorption order q. A possible temperature or coverage dependence of ν and E also influences the spectral shape. For the beginning, we assume that ν and E are constant. Then the spectral type depends only on the order of the desorption. The cases q = 1 and q = 2 occur most frequently and have an intuitively clear interpretation: The case q = 1 represents the direct desorption of an adsorbed particle from its binding site (also termed associative or molecular desorption), with a desorption rate proportional to the coverage θ. For a recombinative desorption, two (q = 2) or more (q > 2) particles first have to diffuse

7.1 Thermal desorption spectroscopy | 165

Figure 7.2: Calculated thermal desorption spectra for different desorption orders q at E = 1.5 eV, ν = 1013 Hz, β = 1 K/s, and five different relative initial coverages θ0 .

on the surface to encounter each other with a probability proportional to θq (eq. (7.3)). Then they form a molecule, which desorbs. This type of desorption is called diffusion limited. It is often found in adsorption phases of two-atomic molecules, which dissociate at adsorption. Before the thermally activated desorption, the atoms diffuse on the surface until they recombine. Figure 7.2 presents in the top row the relative desorption rates for q = 1, 2 and for various initial coverages θ0 . The temperature linearly increases with time T = T0 + βt, as commonly found in experiments. For the heating rate β, a constant value in the range of 1–20 K/s is usually chosen. For q = 1, the desorption rate first increases with a strength depending on the initial coverage θ0 , then reaches a maximum at the temperature Tm independent of θ0 , and finally decreases asymmetrically, i. e., more rapidly than the increase. The maxima Tm found for q = 2 decrease with increasing initial coverage in contrast to the case q = 1. For q = 2, increase and decrease are approximately symmetric with respect to the maximum. The desorption orders in the lower row of Fig. 7.2 are more rarely observed. The case q = 0 occurs during desorption from a reservoir of particles: If, e. g., many layers of noble-gas or metal atoms cover a surface, the relative layer coverage θ abruptly drops at the very end of the desorption. The desorption rate first increases with increasing temperature approximately exponentially (in a temperature range, which depends on the total initial coverage). When all the layers have evaporated, the desorption rate suddenly drops to zero (this drop is usually broadened for experimental rea-

166 | 7 Particle spectroscopies

Figure 7.3: (a) Calculated thermal desorption spectra for various initial coverages θ0 and direct desorption (q = 1). Here the desorption energy E decreases linearly from 1.5 eV at low coverage to 0.9 eV at maximum coverage (θ ≈ 1). The further parameters are ν = 1013 Hz and β = 1 K/s. (b) Experimental desorption spectrum (β = 1 K/s) from a reconstructed Ir(100) surface (Ir(100)-(5 × 1)-H) at the saturation coverage of hydrogen, which exhibits three different desorption maxima [5.5].

sons). The fractional desorption order q = 1/2 is found if the adsorbate forms islands on the surface and the particles at the island edges have a markedly lower binding energy than the ones in the middle of the islands. Then the former with a relative proportion 1 ∝ θ /2 are preferentially desorbed. Also, in this case, the maximum Tm depends on the initial coverage θ0 . In contrast to the case q = 2, here Tm does increase with increasing θ0 . If E depends on coverage, the shape of the spectra and the temperature of the maxima change with varying initial coverage. Figure 7.3(a) presents such a case, which is observed, e. g., for CO on Pd(100) [7.1]. Often spectra exhibit several maxima (which are labeled by lower-case Greek letters α, β, γ, . . . ). This situation occurs for adsorbed particles occupying for certain initial coverages several adsorption sites with different binding energies (Fig. 7.3(b)). With increasing temperature, the particles from the sites with lower binding energy desorb first, followed by the ones from sites with increasingly higher binding energy.

7.1.3 Evaluation of desorption spectra The main aim of desorption experiments is the determination of the desorption energy E and, in addition, also q and ν. If the spectrum exhibits only one maximum, the evaluation of the measured desorption spectra is relatively simple, under the assumption of a coverage-independent E and ν. Spectra with more than one maximum have to be divided into several ranges, as illustrated in Fig. 7.3(b), which have to be evaluated separately. The initial coverage is obtained in all cases (after a suitable calibration of the spec∞ trometer) by integration of the spectrum using T = T0 + βt as na θ0 = ∫0 RD (t)dt = ∞

(1/β) ∫T RD (T)dT. The order q can be determined from the dependence of the temper0 ature Tm of the maximum desorption signal on the initial coverage θ0 , if as assumed in

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the following, a coverage dependence of E and ν can be excluded. For q = 0, the determination of E is particularly simple, because in an Arrhenius diagram the desorption rate RD (T) plotted as function of (1/T) has the slope −E/kB . For q > 0, the evaluation is more elaborate and involves the position Tm of the maxima on the temperature. They are obtained from eq. (7.3), given by d2 θ/dt 2 = 0, which yields for the corresponding q−1 2 values θm and Tm the equation q θm ν exp(−E/kB Tm ) = β(E/kB )(1/Tm ). For q = 1, this relation is independent of θm (and thus also of θ0 ). For q = 1/2, the determination of θm requires the integration of the spectra up to Tm . For q = 2, the spectra are approximately symmetric relative to the maximum, i. e., θm ≈ θ0 /2, and one obtains βE = θ0q−1 ν exp−E/kB Tm , 2 kB Tm

(7.4)

which is valid for q = 1 and 2. The shift of Tm with θ0 for q = 2 can be used to determine E by a linear fit: The logarithm of eq. (7.4) for q = 2 is 2 ln(θ0 Tm ) = E/(kB Tm ) + ln(βE/νkB ), 2 i. e., the plot of ln(θ0 Tm ) over 1/Tm yields a linear function with slope E/kB under the previously made assumption that E does not depend on temperature and coverage. In the practically relevant range 1013 < νTm /β < 1018 , approximate relations to extract E or Tm from eq. (7.4) are available: For q = 1, 2 one can use E ≈ kB Tm [ln(θ0q−1 νTm /β) − 3.64] and Tm ≈ E/[kB (ln(θ0q−1 νE/βkB ) − 7.15)] with an accuracy in the percentage range. The methods outlined so far required some information on ν and θ0 to obtain the desorption energy E. For constant ν, E can be determined without knowing ν and θ0 by measuring thermal desorption spectra for various heating rates β. The logarithm 2 of eq. (7.4) yields the relation ln(Tm /β) = E/(kB Tm ) + Cq , where the constant is Cq = q−1 2 ln(E/(kB νθ0 )). The term ln(Tm /β) is thus a linear function on 1/Tm with slope E/kB for q = 1, 2. To determine the desorption energy E with sufficient accuracy, the heating rate β has to be varied by more than one order of magnitude, which is experimentally challenging. The methods described so far fail for coverage-dependent E and/or ν, and a more elaborate evaluation procedure of eq. (7.3) is needed. We follow the method of King [7.2], which starts from the logarithm of eq. (7.3) for an arbitrary coverage θn ,

ln(−

dθn E(θn ) ) = q ln(θn ) + ln(ν(θn )) − . dt kB T

(7.5)

Depending on 1/T, one obtains for fixed θn a linear function with slope −E(θn )/kB . The axis intercept q ln(θn ) + ln(ν(θn )) yields for known q the rate constant ν(θn ) (or vice versa). To illustrate this procedure, thermal desorption spectra are calculated in Fig. 7.4(a) for recombinative desorption (q = 2) and three different initial coverages assuming

168 | 7 Particle spectroscopies

Figure 7.4: (a) Calculated thermal desorption spectra for recombinative desorption (q = 2) with linearly decreasing E from 1.5 eV at full coverage (θ0 = 1), down to 1.2 eV at zero coverage (other parameters: ν = 1013 Hz, β = 1 K/s). (b) Relative coverage θ(T ) obtained from the integration of the desorption spectra. (c) Arrhenius plot for θn = 0.4 from points 1–3 and θn = 0.7 from points 4–5, which yield the desorption energies E(θn = 0.4) = 1.32 eV and E(θn = 0.7) = 1.41 eV.

that the desorption energy E decreases linearly from 1.5 eV at θ = 1 to 1.2 eV at θ → 0. Note that the typical downward shift of the desorption maxima with increasing coverage for q = 2 (Fig. 7.2) is turned into an upward shift by the increase of desorption energy E. Each spectrum is integrated, starting at high temperatures, which yields the relative coverage θ(T) as a function of the temperature T (Fig. 7.4(b)). To determine the desorption energy at a specific coverage, e. g., θn = 0.4 or θn = 0.7, one obtains the intersection points 1,2,3 and 4,5 with the θ(T) curves, respectively. The corresponding desorption rates can be read off in Fig. 7.4(a). The Arrhenius plot of eq. (7.5) gives the two straight lines in Fig. 7.4(c), from which the desorption energies for the two selected coverages are obtained. In view of this fairly general evaluation procedure, one should keep in mind that thermal desorption spectra contain information on the state of the adsorbate just before or during the desorption process. This state does not have to be the same as the one at lower temperatures. Adsorbates may rearrange, form new molecules, or dissociate during the temperature increase. Particles (e. g., hydrogen atoms) dissolved in the bulk may diffuse to the surface. This may not only falsify the coverage calibration, but also superimposes the time and temperature dependence of the desorption with the

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one of the diffusion of the particles from the bulk to the surface. The situation becomes even more complex if the structure of the surface and concomitant, also the adsorption sites, change with temperature and/or coverage (e. g., by adsorbate-induced reconstruction). If a surface provides many adsorption sites with similar binding energy (e. g., a rough surface), many desorption maxima overlap and a TDS analysis becomes practically impossible. Finally, it should be noted that the basic Polanyi–Wigner equation 7.3 discussed in this chapter is also only an approximation (though often appropriate), and more complex relations of the desorption rate on coverage and temperature exist, since, after all, desorption is a nonequilibrium process. All determinations of desorption parameters have in common that they a priori do not permit identifying the adsorption site. This information can come from calculations of E for different adsorption sites (e. g., by density functional theory), which allows identifying the site by comparing with the experimental value. Alternatively, further surface-sensitive techniques allowing a quantitative structure determination for different coverages can identify the adsorption sites. The latter case will be illustrated in the experimental example that follows, which reveals further desorption characteristics and problems not discussed so far. 7.1.4 Desorption spectra with several maxima Desorption spectra often exhibit several maxima, which may partially overlap. Their interpretation may prove to be more difficult than outlined in the previous examples. Figure 7.5(a) shows the temperaturedependent H2 partial pressure (∝ H2 desorption rate), which is measured for the desorption of hydrogen from a Rh(110) surface at a heating rate of β = 10 K/s. The surface was exposed at 90 K to various amounts of hydrogen, which led to different initial coverages of hydrogen na θ0 [7.3]. The gas dose D = pΔt at (partial) pressure p for a time Δt is given in Fig. 7.5 in Langmuir units (L) (see Section 3.3.1). With increasing initial coverage, three different desorption maxima develop, which are labeled as α, β, and γ. From the gas dose, one cannot infer the resulting coverage na θ0 , because it is not known what fraction of the impinging H2 molecules adsorb on the surface. At best, one can assume na θ0 ∝ D for sufficiently small coverages when most adsorption sites are still available. This either requires a calibration of the spectrometer or bringing in another surface method. The latter option can be realized in the present example by quantitative structure determinations for the adsorbate system by LEED (Chap. 4.2) or helium diffraction (Chap. 4.5). The LEED measurements were done concomitantly with TDS for the same sample preparation [7.3], i. e., both data sets were obtained on the same adsorption phases. In the coverage range, where only the γ maxima are detected, the (3 × 1)-H and (2 × 1)-H phases depicted in Fig. 7.5(b) are observed, in which the H atoms (from the dissociated H2 molecules) are adsorbed in threefold-coordinated sites. The β maximum corresponds to the (3 × 1)-2H and (2 × 1)-2H phases with H atoms arranged in double rows and occupying neighboring threefold-coordinated sites. For even higher coverage, the α maximum develops, which correlates with the (1 × 1)-2H phase. In all phases, the H atoms occupy only threefold-coordinated sites at both sides of the close-packed Rh-atom rows, leading to twice as many sites as Rh atoms in the top layer of the Rh(110) surface. The completely developed superstructures correspond to the relative coverages θ = 1/6 ((3 × 1)-H), 1/4 ((2 × 1)-H), 1/3 ((3 × 1)-2H), 1/2 ((2 × 1)-2H), and 1 ((1 × 1)-2H). Note that the relative coverage

170 | 7 Particle spectroscopies

Figure 7.5: (a) Desorption spectra from a Rh(110) surface covered by hydrogen for various gas doses at a heating rate β = 10 K/s. (b) Structural models (top view) of various adsorption phases (after [7.3]). is normalized to the saturation coverage, at which two hydrogen atoms are bound to each Rh atom at the surface. For the low-coverage (3 × 1)-H and (2 × 1)-H phases, it is assumed that the chain-like arrangement is lost for higher temperatures, and the H atoms are statistically distributed on threefoldcoordinated sites. Before desorption the H atoms have to diffuse to form a H2 molecule and desorb as such. The desorption order would correspond to q = 2 in agreement with the experimental observation that the maxima shift to lower temperature with increasing initial coverage (Fig. 7.2). However, the spectra are not symmetric with respect to the desorption maxima Tm , which shift with coverage more than expected for constant E. This suggests a decrease of E with increasing coverage. A complete quantitative analysis of the adsorption system H on Rh(110) was successfully performed using methods of nonequilibrium thermodynamics [7.4]. For the phases with hydrogen double chains, one can suppose that direct molecular desorption takes place, because the H atoms are close to each other and do not have to diffuse to find a reaction partner. This implies the desorption order q = 1, which is consistent with the observed constant position of the α and β maxima in the absence of overlap with other maxima. One might suspect that for higher coverages, the neighboring H atoms adsorb as quasi-molecules and do not dissociate. This can be clarified by an isotope-exchange experiment with the simultaneous exposure of hydrogen and deuterium molecules (H2 and D2 ) during adsorption. The subsequent TDS experiment shows that besides H2 and D2 molecules, HD molecules also desorb, i. e., all molecules are dissociated in the adsorption state. From the temperature of the maxima Tmβ = 210 K and Tmα = 135 K for the α and β-Phase, the desorption energies Eβ = 0.53 eV and Eα = 0.33 eV are obtained for q = 1. These values are lower than for the γ phase with its higher desorption temperatures. Apparently the binding energy is reduced the closer the hydrogen atoms come to each other laterally, which can be related to an indirect substrate-mediated interaction between them.

7.2 Ion scattering During adsorption and desorption, particles have thermal energies ≪ 1 eV. At higher kinetic energies, the interaction is fundamentally different. If an atom or ion with an

7.2 Ion scattering | 171

Figure 7.6: (a) Scattering process in the binary collision model. (b) Ion scattering spectrum.

energy > 100 eV hits a surface atom, the process can be described by a binary collision model analogous to the sputtering process (Section 3.2.4). In Fig. 7.6(a), an incident ion with energy E0 and mass M1 is scattered by an atom with mass M2 . Using energy and momentum conservation, one obtains for the kinetic energy of the ion scattered by an angle ϑ1 cos ϑ1 + √A2 − sin2 ϑ1 2 E1 =( ) E0 1+A

with A = M2 /M1 .

(7.6)

For A < 1, the scattering angle is restricted by the condition sin2 ϑ1 < A2 , and an additional solution with a minus sign in front of the square root exists, which usually does not play a role. Normally, the scattering angle is fixed by the experimental setup. For ϑ1 = 90∘ and A > 1, one obtains the simple expression E1 /E0 = (M2 − M1 )/(M2 + M1 ). The experimental requirements are the already presented components ion source (Fig. 3.3) and energy analyzer (Section 5.1.3). The energy of the scattered ions is a unique function of the mass of the surface atom M2 . Therefore, the method is named ion scattering spectroscopy (ISS) or lowenergy ion scattering (LEIS). Figure 7.6(b) presents, as example, a spectrum of He+ ions of 1 keV energy scattered by a scattering angle of 164∘ from a nickel surface covered by oxygen. The lines associated with the scattering by oxygen and nickel at the energies given by eq. (7.6) can be clearly identified (see also [7.5]). The mass resolution is best for large scattering angles. In most experiments, either noble-gas or alkali ions are employed. To detect the common impurities C or O, only He+ and Li+ ions are available for scattering angles ≥ 90∘ . These light ions cannot, however, resolve heavier atoms very well. At smaller scattering angles, subsequent scattering by two surface atoms can occur. These double-scattering events lead to additional structures in the ion scattering spectrum. In addition, directly hit surface atoms with an energy given by eq. (3.1) can be detected by the spectrometer if they are ionized in the ejection process. This offers

172 | 7 Particle spectroscopies the possibility to detect hydrogen, which cannot be detected by most other techniques. These additional processes can be identified by varying the scattering angle, and play no significant role for scattering angles ≥ 90∘ . The surface sensitivity of low-energy ion scattering (E0 < 5 keV) is determined by the large scattering cross section and the high neutralization probability for ions, which are scattered in deeper layers. The scattering cross section can be calculated quantitatively from the scattering potential. The Coulomb potential (∝ Z1 Z2 /r) multiplied by an exponentially decreasing screening function is a good approximation for the scattering potential. Here Z1 and Z2 denote the atomic numbers of ion and surface atom at the distance r. The energy transferred in the scattering process to the surface atoms can displace them from their ordered crystal site, or even remove them as exploited for sputtering (Section 3.2.4). The beam damage due to the ion bombardment can be minimized by using a large beam cross section, low ion energies (E0 < 5 keV), and light ions, which lead to a small energy transfer, due to the large mass ratio A (eq. (7.6)). The high neutralization probability of noble-gas ions can be obviated by the detection of neutral particles. The energy analysis is done by registering the time of flight of the scattered particles, which requires a pulsed ion source. An alternative is offered by alkali ions, which have a low neutralization probability, but can also adsorb in contrast to noblegas ions. A quantitative analysis of the chemical elements on a surface is usually not possible, because the neutralization probability of the ions depends on the electronic structure of the surface atoms. In addition, the scattering potential and, consequently, the scattering cross section are not exactly known. Finally, atoms in deeper layers can be shaded depending on the incidence and exit angles. These latter effects can, however, be exploited for the determination of the surface geometry [7.6]. For the quantitative analysis by ion scattering, the statement from Section 5.2.5 applies as well: The closer the reference samples are to the investigated sample, the better the results! A variant of ion scattering, which uses ion energies of > 100 keV for the elemental analysis in the bulk, is called Rutherford backscattering (RBS). For energies in the MeV range, the collision (and thus the momentum and energy transfer) occurs at very small distances r. Then the unscreened Coulomb potential is a good approximation for the scattering potential and the scattering cross section can easily be calculated. In addition, the neutralization probability is smaller than in the case of ISS. If the energy of the backscattered particles is registered by pulse-height analysis in a semiconductor detector, ions and atoms are detected with the same probability. The energy losses of the ions after penetrating many layers lead to a continuous backscattering spectrum up to the energy given by eq. (7.6). For the incidence of the ions along open crystallographic directions (channels), most of the ions penetrate deeply into the crystal, an effect called channeling. Then only the ions scattered at the surface reach the detector without an energy loss.

| 173

In the final chapter, we discussed some selected aspects of the interaction of atoms, molecules, and ions with surfaces. Thermal desorption spectroscopy is a powerful method to determine microscopic parameters of adsorption, such as binding energies. The complexity of desorption processes shows that the progression of chemical or catalytic processes at surfaces is extremely difficult to decipher and understand. The interaction between surfaces and ions (or atoms) with energies much higher than thermal energies has different characteristics: A classic binary collision model describes the energy and momentum transfer if the energy of the ions significantly exceeds the binding energy of the surface atoms. With ion scattering, the chemical elements in the top layers of a surface can be identified and their relative placement determined.

Q7.1: Describe the experimental setup for thermal desorption spectroscopy. Q7.2: What conclusion can be drawn if in a thermal desorption experiment the maxima of the spectra, as function of temperature, do not depend on the initial coverage? How can the desorption energy of the adsorbate be determined in this simple case? Q7.3: Explain the term “recombinative desorption” and the characteristic features of the corresponding thermal desorption spectra. Q7.4: What can be concluded if a thermal desorption spectrum exhibits several maxima? Q7.5: Discuss under which conditions the interaction of ions with surface atoms can be described in a classic model of a binary collision between two atomic nuclei. Q7.6: How would you choose scattering angle and ion mass to obtain the optimum resolution for the mass of the surface atoms?

[7.1] [7.2] [7.3] [7.4] [7.5] [7.6]

R. J. Behm, K. Christmann, G. Ertl and M. A. Van Hove, J. Chem. Phys. 73, 2984 (1980). D. A. King, Surf. Sci. 47, 384 (1975). W. Nichtl-Pecher, J. Gossmann, W. Stammler, G. Besold, L. Hammer, K. Heinz and K. Müller, Surf. Sci. 249, 61 (1991). H. J. Kreuzer, Zhang Jun, S. H. Payne, W. Nichtl-Pecher, L. Hammer and K. Müller, Surf. Sci. 303, 1 (1994). M. Schmid, LEIS Energy Calculator; www.iap.tuwien.ac.at/www/surface/leis; Last visit 17.01.2020. Th. Fauster, Vacuum 38, 129 (1988).

A Appendix A.1 Planes and surfaces of hcp crystals

Figure A.1: For hexagonal (close-packed) crystals planes and directions are often denoted by four base vectors (a1 , a2 , a3 , c) and indices (hkil), instead of three (hkl). The additional base vector a3 = −a1 − a2 leads to the condition i = −h − k. Note that the indices are given in multiples of the base vectors, and do not refer to a Cartesian coordinate system in the case of three indices. The notations with three (black) and four (red) indices are given for some low-index planes. Between the (0001) top and bottom faces (perpendicular to the c-axis), a laterally shifted layer completes the close-packed stacking.

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A.2 Identification of plane groups | 175

A.2 Identification of plane groups For a structure, the associated plane group in the notation of Hermann–Mauguin may be determined following the scheme of Table A.1. Since surfaces are objects with an extension in the vertical direction, we use the terms mirror plane and glide plane, instead of mirror line and glide line, respectively (see Section 1.1.1). To determine the plane group, one proceeds as follows: The first letter (p or c) indicates a primitive or centered structure, respectively. The following number gives the maximum order n of the rotational symmetry (n = 1, 2, 3, 4, 6). The remaining two symbols denote the symmetries (m = mirror plane, g = glide plane, 1 = no symmetry) relative to one of the two translation directions of the point lattice. For the first symbol, one chooses a direction (main translation direction), which is perpendicular to a symmetry plane (m or g if it exists, 1 otherwise). If this applies to both translation directions, the choice is arbitrary. The second of the two symbols (m or g if it exists, 1 otherwise) indicates a mirror or glide plane, which is rotated with respect to the main translation direction for n > 2 by 180∘ /n, or parallel to it. If both symbols are 1, they may be omitted (e. g., p211 = p2). Additional abbreviated notations are given in parentheses in Table A.1. Table A.1: Identification of plane groups of a surface structure. Maximum order n of rotation axes

Mirror plane? Yes

n=6

p6mm (p6m)

n=4

Further mirror plane under 45 ? Yes: p4mm (p4m)

n=3

Rotation axes, which are not in mirror planes? Yes: p31m No: p3m1

n=2

Mirror planes, which are mutually perpendicular? Glide planes? Rotation axis not in mirror plane? No: p2mg (pmg) Yes: p2gg (pgg) No: p2 Yes: c2mm (cmm) No: p2mm (pmm)

n=1

Glide planes between mirror planes? Yes: c1m1 (cm) No: p1m1 (pm)

No p6 ∘

No: p4mg

p4 p3

Glide planes? Yes: p1g1 (pg)

No: p1

176 | A Appendix

A.3 Brillouin zones and mirror planes

Figure A.2: Surface and bulk Brillouin zones for the face-centered cubic lattice (fcc), body-centered cubic lattice (bcc) and hexagonal closed-packed lattice (hcp). Selected high symmetry points are indicated. For the hcp lattice the notation with four indices from Section A.1 is used in reciprocal space. The shading indicates mirror planes of the bulk structure.

A.3 Brillouin zones and mirror planes | 177

Figure A.3: Mirror planes of the fcc, bcc and hcp lattices in the extended zone scheme. The (001) ̄ mirror planes contain mirror planes contain only directions (hk 0) with third index l = 0. The (110) only directions (hhl) where the first indices are identical h = k. Dashed lines mark the boundaries of the surface Brillouin zones for selected low-index surfaces. The shaded areas correspond to those of Fig. A.2.

178 | A Appendix

A.4 Energies of Auger transitions

Figure A.4: Overview of the dominant Auger transitions of the elements [5.10]. The size of the symbols represents the intensity of the corresponding Auger lines.

Answers to exercise problems and questions Chapter 1 Q1.1: In the top layer, a surface atom has four nearest neighbors and eight in all other layers underneath. In each layer one atom is found in the primitive unit cell. Q1.2: See Table A.1: Two mirror planes perpendicular to each other and two-fold rotation axes, which are all located in the mirror planes: p2mm. Q1.3: The matrix notation represents the linear transformation of the basis vectors of the substrate to those of the superstructure, therefore it can be always applied. Wood’s notation contains just the ratio of the lengths, and the angle of rotation between the two systems of basis vectors. It can only be used if the angles between the basis vectors of the substrate and the superstructure are the same. Q1.4: The surface relaxation denotes the change of the layer distances of the topmost layer(s) and is caused by the missing bonds on the vacuum side. For metals, it can be qualitatively explained by the Smoluchowski smoothing, which shifts the electronic charge toward the substrate. The surface reconstruction is a rearrangement of the surface atoms. It is often found for semiconductor surfaces, because the directional unsaturated bonds of the truncated bulk are energetically unfavorable. ̄ Q1.5: From inspection of Fig. 1.1(c) one sees that the microfacet contains the vectors [110] (direction of ̄ the nearest neighbor along the chain plotted in red in Fig. 1.15(b)) and [101] (vector connecting two atoms at the center of the planes). These two vectors are perpendicular to (111). Q1.6: Bloch’s theorem (eq. (1.2) gives the relation between the wave function in one unit cell to the one in a unit cell shifted by a translation vector. Therefore, it is sufficient to know the wave function in one unit cell. The energies depend on the quantum numbers k ‖ and constitute the surface band structure E(k ‖ ). The periodicity E(k ‖ + g) = E(k ‖ ) with the reciprocal lattice vectors g implies that it is sufficient to know the band structure in the surface Brillouin zone. Q1.7: The basis vectors of the reciprocal lattice form a square as in real space (Table 1.1). The corresponding surface Brillouin zone is also a square (Fig. 1.19). There exists no direct correspondence to the bulk Brillouin zone; in particular, the surface Brillouin zone is not a simple projection of the bulk Brillouin zone (see Figs. 1.20 and A.2). For low-index surfaces high-symmetry points of the bulk Brillouin zone are often found at high symmetry points of the surface Brillouin zone. Q1.8: The charge of the electron is screened by the metal electrons in such a way that the electric field lines are perpendicular to the surface. The electric field in front of the surface corresponds to the one of a dipole consisting of the electron and the image charge separated by a distance 2z 2

e 1 (Fig. 1.25(a)). The associated image potential is V (z) = EV − 4πε . 0 4z Q1.9: Surface states exist in band gaps of the projected bulk band structure. Shockley surface states originate from the matching of exponentially decaying wave functions in bulk and vacuum at the surface (see Fig. 1.21). Image-potential states exist in the long-range potential, which electrons experience in front of a metal surface (Fig. 1.25(b)). Unsaturated bonds at semiconductor surfaces lead to localized dangling-bond states. States of adsorbates are modified in the binding process to the surface and form (extrinsic) surface states. Q1.10: The Smoluchowski smoothing leads to a shift of the charge density at the step-edge atom toward the lower-lying terrace (see Fig. 1.26). A dipole moment perpendicular to the surface occurs, which lowers the work function. Q1.11: Alkali atoms donate their outermost (least bound) s electron to the metal substrate. The charge transfer leads to a dipole moment perpendicular to the surface, which lowers the work function.

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180 | Answers to exercise problems and questions

Q1.12: The group velocity vg is obtained as slope (gradient in two dimensions) of the dispersion rela-

tion ω(q): vg = 𝜕ω(q) . 𝜕q Q1.13: The altered boundary conditions at the surface (due to missing binding partners for the surface atoms) can lead to additional oscillatory solutions of the equations of motion, which do not exist in the bulk. Similar to the situation for electronic surface states, these surface vibrational modes (phonons) exist only in band gaps of the projected bulk phonon band structure of the vibrations of the bulk crystal. Chapter 2 Q2.1: See Fig. 2.1. Plasmons have energies between 5 and 25 eV. Including multiple losses, they occur in the energy range up to 100 eV below the primary energy (Fig. 2.2). Q2.2: The emission of secondary electrons leads to a charging of the insulator, which cannot be compensated, due to lacking conductivity to an electrical contact. The electrostatic potential of the sample (surface) adjusts itself to a value, for which the emission coefficient is one, i. e., the same number of electrons leave the sample as are incident. Q2.3: See Fig. 2.3: For E < 10 eV, the phase space for electron-hole pair creation decreases, and the electrons have a large mean free path. For high energies, the mean free path increases, because of the reduced interaction time of the fast electrons. In an energy range of 10–600 eV, the mean free path has a broad minimum. Therefore, electrons detected with an energy in this range provide information on a few atom layers near the surface. Q2.4: Adsorption: Binding of a molecule or atom from the gas phase to the surface. Desorption: Breaking the bond of an atom or molecule to the surface and release of the adsorbate to the gas phase. Diffusion: Moving of a bound adsorbate from one adsorption site to a neighboring site. Q2.5: The characteristic minimum in the binding potential arises from the balance of attractive forces (van-der-Waals force and the associated energy gain by the formation of chemical bond) and the repulsive forces close to the surface, when the electron shells of adsorbate and surface atoms overlap (Pauli repulsion). Q2.6: If the state with energy εd moves further away from the Fermi energy, the antibonding state σ ∗ can drop below the Fermi energy and become occupied by electrons from the substrate. As a result the adsorbate is not bound to the surface anymore. This explains why fewer atoms or molecules adsorb on gold (d bands > 2.5 eV below EF ) compared to iron (d bands at EF ). Q2.7: The two (instead of one) orbitals of the molecule each form bonding and antibonding combinations with the state of energy εd . If the orbital binding to the surface, which is derived from the ∗ antibonding molecular orbital σMO , becomes located below the Fermi energy and is occupied by electrons, the bond within the molecule is weakened and it may dissociate. Q2.8: Island growth can occur if the system has a large Ehrlich–Schwöbel barrier (Fig. 2.8). Then atoms impinging on an existing island cannot reach (for kinetic reasons) at low substrate temperatures the more favorable adsorption sites at the step edge on the layer below. Chapter 3 Q3.1: Even under ultrahigh-vacuum conditions, each surface atom is statistically hit by an atom or molecule from the residual gas, which adsorbs with a probability given by the corresponding sticking coefficient. Q3.2: Cleaving for insulators, semiconductors, or layer crystals along specific directions. Annealing as long as impurities desorb thermally and do not show increased segregation from the bulk. Annealing in a gas atmosphere if impurities react with the gas and subsequently desorb. Removal of the top layers containing the impurities by sputtering and subsequent annealing can

Answers to exercise problems and questions | 181

be used in most cases. For alloys and compounds, the different sputter yields of the various elements have to considered. Q3.3: The ion mass should match the mass of the surface atoms to be removed to obtain a large energy transfer. The kinetic energy of the ions should be large compared to the binding energy of the surface atoms and be deposited in the topmost surface layers. The latter requirement is fulfilled for energies < 3 keV, and can be improved by oblique incidence angles (as long as the ions are not reflected). Q3.4: According to eq. (2.1), the adsorption rate is proportional to the sticking coefficient S(θ) (which depends on coverage and material), depending on the adsorbate mass m and gas temperature TG . Q3.5: Thermal evaporation of the material in the form of rods or from crucibles. Chemical vapor deposition by exposure to reactive molecules. Pulsed laser deposition by rapid, local evaporation of the target material (can also be applied for compounds and in a gas atmosphere). Deposition of sputtered material on the surface. Chapter 4 Q4.1: According to eq. (4.4), the component of the wave vector parallel to the surface can change only by a reciprocal lattice vector of the surface. This corresponds to the (crystal-)momentum conservation parallel to the surface, which has to be fulfilled in a periodic system only modulo reciprocal lattice vectors. The component of the momentum perpendicular to the surface is not conserved, due to the potential difference between bulk and vacuum. For elastic diffraction energy is conserved, i. e., the absolute values of the wave vectors of incident and diffracted waves are equal. Q4.2: The superstructure has larger basis vectors than the substrate. This is correlated to smaller reciprocal lattice vectors, which lead to additional spots in the diffraction pattern between those of the substrate. Q4.3: The electron beam hits the sample, which is positioned at the center of spherical grids and a fluorescent screen (Fig. 4.5(a)). The diffracted beams traverse a field-free region up to the first grid. The second grid retards the electrons, such that in the end only elastically scattered electrons are accelerated to the fluorescent screen. Q4.4: The position of the spots yields the basis vectors of the reciprocal lattice, from which the basis vectors of the real-space lattice can be determined using eq. (4.6). For perpendicular incidence of the electron beam onto the sample, rotation axes and mirror planes can be identified. Q4.5: For the determination of the positions of the atoms in a unit cell, the intensity of the diffraction spots has to be recorded. This should be done over a wide range of energies in order to obtain a large data base. Because only the intensity (and not the amplitude and phase) of the diffracted electrons can be measured, and because of the multiple scattering, a direct evaluation of the data is not possible. Calculated spectra are fitted to the experimental data by varying the structural parameters. Q4.6: The adsorbate atoms assume an identical lateral arrangement as the atoms of a substrate layer, i. e., a (1×1) layer. The positions of the diffraction spots are the same as the one of the substrate, but the intensities change. The adsorbate phase is denoted as Ni(100)-(1 × 1)-A. Q4.7: The surface is not blocked by the diffraction experiment, and stays accessible for evaporators. Evaporators can be mounted perpendicular to the sample, which is advantageous to achieve a uniform layer thickness on the substrate. Oblique incidence and exit angles also enhance the surface sensitivity for the high-energy electrons. Q4.8: The diffracted intensity of fractional-order spots of a superstructure comes mainly from the surface. Therefore, these spots are particularly well suited to determine properties of the surface. Integer-order diffraction spots, on the other hand, carry large contributions from the bulk. To

182 | Answers to exercise problems and questions

enhance the sensitivity to the structure of the surface, grazing incidence and exit of the x-rays is advantageous. Q4.9: The electronic corrugation at a low-index metal surface is small (due to Smoluchowski smoothing) compared to the one of an ionic crystal. The ion radii of the positive and negative ions are quite different, so the electrons of the helium atoms experience the Pauli repulsion at different distances from the surface. Q4.10: For adsorbate systems, the choice of an absorption edge of an adsorbate atom ensures that the measured signal comes exclusively from surface atoms. The surface sensitivity can be further enhanced by the detection of low-energy secondary or Auger electrons. Chapter 5 Q5.1: A filament emits electrons, which are collimated by the Wehnelt cylinder, focused at the crossover point and accelerated to the anode to the desired energy (Fig. 5.1). Subsequently, the electron beam can be focused by an einzel lens and deflected by a crossed pair of parallel plates. Q5.2: For x-ray photoelectron spectroscopy, water-cooled x-ray tubes are employed with Mg or Al anode. For ultraviolet photoelectron spectroscopy windowless gas-discharge lamps are used, which are usually operated with helium. For two-photon photoemission pulsed laser sources are required to obtain high enough intensity. Synchrotron radiation sources cover a large energy range with a significant higher brilliance than laboratory light sources (with the exception of lasers). Q5.3: Electron energy analyzers are based on the deflection in electric fields. Common geometries are the cylindrical mirror and the hemispherical analyzer (Fig. 5.3), as well as the cylindrical sector analyzer (Fig. 5.22). For less-demanding applications retarding-field analyzers (LEED optics, Fig. 4.5(a)) can be used. Q5.4: Single electrons are detected after amplification via secondary-electron multiplication by channeltrons or channelplates. If the signal can be measured as electron current, modulation techniques are often used to separate the signal from the background. Q5.5: From the kinetic energy Ekin of the electrons emitted by an x-ray photon of energy hν, the binding energy EB = hν − Ekin − Φ can be determined. The work function Φ = EV − EF accounts for the reference of EB to the Fermi energy EF , and of Ekin to the vacuum energy EV . In the Auger effect, a hole in an inner shell is filled by an electron from an outer shell. The excess energy is transferred to a second electron. For the energy balance (eq. (5.1)), one can— in a first approximation— use the binding energies obtained by x-ray photoelectron spectroscopy with a correction term, taking into account that the Auger process leaves a double-ionized final state in contrast to the single-ionized final state in x-ray photoelectron spectroscopy. Q5.6: The measured intensities are modeled as product of the concentration with sensitivity factors, which can be most simply determined from spectra of the pure elements. If the elemental distribution is not homogeneous, the mean free path at the respective energy of the emitted electrons has to be taken into account (eq. (5.2)). In certain circumstances, the chemical environment of the individual elements has to be considered. Q5.7: When the electron is emitted from the surface, the parallel component k ‖ of the wave vector stays unchanged. The emitted free electron has the kinetic energy Ekin = (ℏk)2 /2m. With the

emission angle ϑ (Fig. 5.15b), one obtains |k ‖ | = √2mEkin /ℏ2 sin ϑ (Eq. (5.4)). Q5.8: Surface states are located in a band gap of the projected bulk band structure, the initial-state energy of the spectral feature is independent of the photon energy, and the intensity of the spectral feature clearly decreases upon the adsorption of gases.

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Q5.9: The final state of an electron emitted in a mirror plane has even symmetry. Since the matrix element has to be even under reflection, the symmetry of the electric field (polarization of the light) with respect to the mirror plane determines which states can be excited: s- and p-polarized light select odd and even initial states, respectively. Q5.10: The energy of the electrons is increased or decreased by the excitation energy of the vibrational mode. In the momentum balance, the parallel momentum of the phonon and possibly a reciprocal lattice vector have to be added. Q5.11: To achieve the required energy resolution, the incident electron beam has to be monochromatized, usually in two stages (Fig. 5.22). For the investigation of the angular dependence of the scattering processes, the analyzer and electron detector can be rotated around the sample. Q5.12: At metal surfaces dipole moments parallel to the surface are screened by induced image dipoles, and only vibrational modes with a dynamic dipole moment perpendicular to the surface can be excited. Based on symmetry considerations of the matrix element, these modes can also be identified as totally symmetric modes. Chapter 6

Q6.1: R = 109 Ω. The thermal voltage noise of the resistor increases as ∝ √kB TR, which leads to a high noise level due to the large resistance. A parasitic capacitance parallel to the feedback resistor acts as a low-pass filter and limits the bandwidth of the circuit. Q6.2: The voltage noise should be < 10 mV. For the given resolution, a dynamic range of 1000 : 0.05 = 20000 : 1 is needed, i. e., this voltage noise level has to be maintained at an output voltage of 200 V. The vertical resolution is then limited to 5 pm for a maximum movement of the piezos in z direction of 100 nm. Q6.3: According to eq. (6.1), this leads to a higher tunneling probability, which is compensated in constant-current mode by a retraction of the tip. The respective low-work-function areas of the sample appear higher than they are geometrically. Q6.4: In front of such a potential barrier standing waves form. These can be described by the superposition of an incident and reflected one-dimensional wave. The density of states, which determines the tunnel current, is proportional to |ψ|2 and shows a sinusoidal variation with the distance from the barrier. In two dimensions, one obtains a Bessel function J0 . A step on a surface presents such a potential barrier for the electrons of a solid. It is more pronounced if the electrons are localized at the surface, e. g., in a surface state. Q6.5: G0 corresponds to the conductivity I/V of an ideal one-dimensional, conductive channel between two contacts. Such a channel is realized by a contact formed by one or more atoms in a chain-like arrangement. Chapter 7 Q7.1: The sample is placed in front of a mass spectrometer, and its temperature is increased linearly by a suitable heating method (Section 3.2.2). A shield reaching close to the sample ensures that only particles desorbing from the sample surface enter the mass spectrometer (see Fig. 7.1). The signal of the mass spectrometer is proportional to the partial pressure of the desorbing species, and is plotted as a function of temperature to obtain the thermal desorption spectrum. Q7.2: The situation represents the direct desorption of individual particles from their adsorption site (desorption order q = 1). The desorption spectrum is measured for various heating rates β, and the temperatures Tm of the desorption maxima are determined. The plot ln(Tm2 /β), as function of 1/Tm (eq. (7.4)), gives a linear relation with slope E/kB . Approximate values can be obtained using the expression given in the paragraph following eq. (7.4).

184 | Answers to exercise problems and questions

Q7.3: During recombinative desorption two (or more) atoms on the surface form a molecule, which subsequently desorbs. The maximum of the associated thermal desorption spectra (desorption order q = 2) shifts with increasing coverage to lower temperatures, because the probability for the involved atoms to meet another one rises. For second-order desorption the desorption spectra are approximately symmetric with respect to the maximum, which allows the evaluation using eq. (7.4). Q7.4: The adsorbates desorb from sites with different binding energies. These can be geometrically different sites, but also identical sites with coverage-dependent interaction energy, e. g., if adsorbates on neighboring sites repel each other. In this case, the binding energy of the desorbing particles increases once the coverage is low enough to avoid the occupation of neighboring sites. Q7.5: The kinetic energy of the ions has to be significantly larger than the binding energy of the surface atoms. Since only a negligible amount of momentum can be transfered to electrons, the scattering is dominated by the interaction of the nuclei of ion and surface atom, which can come rather close for energies > 100 eV. Q7.6: According to eq. (7.6), the scattering angle ϑ1 should be as large as possible, and the ion mass close to the mass of the surface atoms. The second condition ensures a large energy transfer, which can most easily be verified for the special cases ϑ1 = 90∘ or 180∘ .

Acronyms Acronym

English expression

Section

2PPE AES AFM ARPES ARUPS bcc CITS CVD DDA DFT DIET EC-STM EDC EELS ESCA ESD EXAFS fcc FM-AFM HAS hcp IMFP IPE ISS HOMO HREELS KPFM LEED LEIS LUMO MDC MFM ML MBE NEXAFS PD PED PES PEEM PLD PSD RBS RHEED

2-photon photoemission Auger electron spectroscopy atomic force microscope angle-resolved PES angle-resolved UPS body-centered cubic current imaging tunneling spectroscopy chemical vapor deposition deposition-diffusion-aggregation density functional theory desorption induced by electronic transitions electrochemical STM energy distribution curve electron energy-loss spectroscopy electron spectroscopy for chemical analysis electron stimulated desorption extended x-ray absorption fine structure face-centered cubic frequency-modulated AFM helium atom scattering hexagonal close-packed inelastic mean free path inverse photoemission ion scattering spectroscopy highest occupied molecular orbital high-resolution EELS Kelvin probe force microscopy low-energy electron diffraction low-energy ion scattering lowest unoccupied molecular orbital momentum distribution curve magnetic force microscopy monolayer molecular-beam epitaxy near edge x-ray absorption fine structure photodesorption photoelectron diffraction photoelectron spectroscopy photoelectron emission microscope pulsed laser deposition photon-stimulated desorption Rutherford backscattering reflection high-energy electron diffraction

5.3.3 5.2.3 6.3 5.3.1 5.3.1 1.1.1 6.2.3 3.3.2.1 2.3.2 6 2.2.2 6.2.4 5.3.1 5 5 2.2.2 4.6 1.1.1 6.3.2 4.5 1.1.1 2.1.2 5.3.2 7.2 2.2.1 5.4 6.3.4 4.2 7.2 2.2.1 5.3.1 6.3.4 3.3.2 2.3 4.6 2.2.2 4 5 1.2.5 3.3.2.3 2.2.2 7.2 4.3

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186 | Acronyms Acronym

English expression

Section

SCLS SEXAFS SIMS SNOM SPALEED STM STS SXRD TDS TPD TPP TPPE UHV UPS VUV XANES XPS

surface core-level shift surface EXAFS secondary-ion mass spectrometry scanning near-field optical microscope spot profile analysis by LEED scanning tunneling microscope scanning tunneling spectroscopy surface x-ray diffraction thermal desorption spectroscopy temperature-programmed desorption tetraphenylporphyrin two-photon photoemission ultrahigh vacuum ultraviolet photoelectron spectroscopy vacuum ultraviolet x-ray absorption near edge structure x-ray photoelectron spectroscopy

5.2.2 4.6 3.2.4 6.3.4 4.2.2 6.2 6.2.3 4.4 7.1 7.1 5.2.2 5.3.3 3.1 5.3.1 5.1.2 4.6 5.2.1

Index adhesion 159 adsorbate state 35 adsorption 48 – dissociative 48 – experimental 73 adsorption rate 48 AFM – contact 156 – dynamic 156 – frequency-modulated 157 – intermittent-contact 157 – jump-to-contact 159 – non contact 157 – tapping-mode 157 analyzer 105 annealing of samples 67 antiphase domain 14, 84 associative desorption 164 atom diffraction 77, 95 atom scattering 95 atomic force microscope see AFM attempt frequency 51, 52, 163 Auger electron spectroscopy 115 – lineshape 118 band gap 28 – phonons 39 – semiconductor 32 band structure – bulk 29 – projected 27 – spectroscopy 124 – complex 29 – surface 23 barrier – activation 48 – desorption 49 – diffusion 49 – Ehrlich-Schwöbel 53, 55, 56 basis vector – primitive 6 – reciprocal 26, 81 beam-deflection detection 155 bilayer 20 binding energy – at chemisorption 49

– core level 110 – for physisorption 48 – of an adsorbate 51 Bloch’s theorem 23 Born-Oppenheimer approximation 37, 134 Bravais lattice 7 bridge site 10 Brillouin zone – bulk 27, 176 – surface 25, 176 bulk Brillouin zone 27, 176 bulk plasmon 45 catalyst 50 centered structure 8 channelplate 107 channeltron 107 chemisorption 49, 140 cleaning – chemical 69 cleaving of crystals 67 coherence region 87 coincidence structure 12, 13 conductivity – differential of tunnel contact 152 core-level shift 113 cross section – core-level excitation 116 – dipole scattering 135 – photoemission 111 crystal truncation rod 81 dangling bonds 18, 31, 50 Debye-Waller factor 84, 97 decay length 149 delocalization 50 density of states – local 151, 152 depth distribution 121 desorption – associative 164 – electron-stimulated 51 – isothermal 162 – molecular 164 – photon-stimulated 51 – recombinative 164

188 | Index

– temperature-programmed 162 – thermal 51, 162 desorption order 163 desorption rate 51, 163 diffraction 77 – atom 95 – helium 95 – kinematic 79 – of high-energy electrons see RHEED – of low-energy electrons see LEED – spot 85 diffusion 52 – barrier 52, 53 – coefficient 53 – length 54, 55 – rate 52 dimer 18, 20, 22, 32, 35 dipole operator 130 dipole scattering 134, 134 dispersion relation – bulk bands 28 – phonon 38 – surface states 31, 123 dissociation 50 domain – antiphase 14, 84 – rotational 14, 19, 88 Ehrlich-Schwöbel barrier 53, 55, 56 electron lens 102, 106, 133 electron source 102 electron spectrometer 105 electron spectroscopy 101 elemental analysis 119 energy analyzer 105 energy loss – of electrons 42 – to plasmons 45 – to vibrations 132 energy resolution – electron spectrometer 106 – gas discharge 105 – synchrotron radiation 105 – x-ray source 103 epitaxy 56 – hetero- 73 – homo- 57, 73 – molecular-beam 54 evanescent wave 82

evaporation – by lasers 74 – by sputtering 75 – thermal 74 Ewald sphere 82 film growth 54, 73 – DDA-model 56 – epitaxial 56 – polycrystalline 56 – pseudomorphic 57 film morphology 54 film orientation 57 fine structure of x-ray absorption see SEXAFS force – adhesion 160 – friction 160 – magnetic 160 – van-der-Waals 48, 158 form factor 80 Franck-Condon principle 52 frontier orbitals 49 gas dose 73 gas-discharge tube 104 glide line see glide plane glide plane 8, 175 grain boundary 57 growth – Frank-van-der-Merwe 54 – island 55 – layer-by-layer 54 – step-flow 54 – Stranski-Krastanov 55 – Volmer-Weber 55 growth parameter 57 helium diffraction 77, 95 heteroepitaxy 73 hollow site 10 homoepitaxy 57, 73 hopping rate 52 hydrogen bond 59 hydrogen bridge bond 140 hysteresis – of piezo material 145 image charge 32, 135

Index | 189

image potential 32 – state 32, 129 impact scattering 134, 136 incommensurate superstructure 12 inertial drive 145 inner potential 91, 125 instability – AFM 159 interference method 77 intralayer relaxation 15 inverse photoemission 127 ion scattering 170 ion source 71 island density 56 island growth 55 isothermal desorption 162 isotope-exchange experiment 170 Kelvin probe 35, 160 kinematic diffraction 79 kinetically limited 54 kinetics 54 lattice 7 – reciprocal 80 lattice factor 80 lattice vector – reciprocal 24, 80 Laue condition 80 layer growth see film growth layer relaxation 14 LEED 77, 84 – optics 84 – pattern 87 – structure determination 88 lineshape – Auger electron spectra 118 mean free path – inelastic 46 – universal curve 46 metastable state 5, 21, 54, 67 Miller indices 6 mirror line see mirror plane mirror plane 8, 27, 175 misfit 57 modulation technique 108 moiré structure 13, 58 molecular desorption 164

molecular orbital – (anti-)bonding 50 – HOMO 49 – LUMO 49 molecular vibration 132 molecular-beam epitaxy 54 monolayer 10, 73 morphology 5 multilayer relaxation 14 nanostructure 58 – self-organized 59 neutralization probability 172 nucleation 55 optical potential 91 phonon 37, 132 phonon dispersion 38, 137 photoelectron diffraction 77 photoelectron spectroscopy – ultraviolet 122 – x-ray 109 photoemission 122 – angle-resolved 121 – binding energy 110 – cross section 111 – inverse 127 – one-photon 121 – two-photon 128 physisorption 48 piezo element 145 plane group 8, 175 plasmon 44 – bulk 45 – surface 44 point lattice 7 Polanyi-Wigner equation 51, 163 potential – chemical 48 – electrochemical 50 – inner 91, 125 – optical 91 primary electron 43 primitive basis vector 6 primitive structure 8 projected bulk band structure 27 pulsed laser deposition 74 quantum-well state 34

190 | Index

random walk 53 rate constant 163 Rayleigh mode 40, 137 reciprocal lattice 80 recombinative desorption 164 reconstruction 17 – adsorbate-induced 5, 16, 22 – displacive 17 relaxation – adsorbate-induced 5, 16 – intralayer 15 – multilayer 14 relaxation profile 14 residual gas 66 resolution – contact AFM 156 – non contact AFM 157 – STM 151 – STS 153 resonance scattering 134 RHEED 54, 93 rotation axis 8, 175 rotational domain 14, 19, 88 Rutherford backscattering 172 satellite line – gas discharge 105, 126 – plasmon 45 – x-ray source 103, 112 scanning probe microscope 143 scanning tunneling microscope see STM scanning tunneling spectroscopy 151 secondary electrons 43 secondary-electron emission coefficient 43 secondary-electron multiplier 107 segregation 65, 75 – reactive 75 selection rules – photoemission 130 – vibrational spectroscopy 133 self-organization 59 SEXAFS 77, 96 Shockley state 31, 147, 153 site – bridge 10 – hollow 10 – top 10 Smoluchowski smoothing 15, 35, 96

spectrometer 105 – cylindrical mirror 105 – cylindrical sector 106, 133 – hemispherical 106 – retarding-field 85, 107 spot profile 87 – analysis 87 sputter yield 70 sputtering 70, 75, 172 step 9, 20, 22, 27, 148 step-flow growth 54 sticking coefficient 48, 73 STM 144 – topography 145, 146 structure – centered 8 – primitive 8 structure determination 77 structure factor 80 superlattice 10 – commensurate 12, 13 – simple 11 superstructure 10 – commensurate 12, 13, 57 – in LEED 87 – in x-ray diffraction 95 – incommensurate 12 – matrix notation 12 – Wood’s notation 11 surface – stepped 9, 27, 129 – vicinal 9, 54 surface band structure 23 surface Brillouin zone 25, 39, 176 surface crystallography 6 surface mode – extrinsic 40 – intrinsic 40 surface plasmon 44 surface reconstruction 5, 17 surface relaxation 5, 14 surface state 30 – extrinsic 30 – intrinsic 30 – phononic 40 – spectroscopy 123 surface unit cell 6 surface vibration 37 – spectroscopy 132

Index | 191

surface x-ray diffraction 77, 93 synchrotron radiation 105 Tamm state 31 temperature-programmed desorption 162 terrace 9, 36 Tersoff-Hamann model 150 thermal desorption spectroscopy 162 thickness measurement – by quartz crystal 73 – by RHEED 93 three-step model 125 top site 10 topography image – AFM 156 – STM 146 transfer width 87 tunnel contact 150 tunnel current 149 tunnel process 148

two-photon photoemission 128 ultrahigh vacuum 66 vacuum ultraviolet 104 vapor deposition – chemical 74 vicinal surface 9, 54 wave vector 24 Wood’s notation 11 work function 35, 102, 127, 149, 160 – local 37, 149 x-ray absorption see SEXAFS x-ray diffraction 77, 93 x-ray fluorescence 116 x-ray photoelectron spectroscopy 109 x-ray source 103