CHAPTER 3. SURFACE STRUCTURE 84

CHAPTER 3. SURFACE STRUCTURE 84 3.3.1.2 Deviation from the Ideal Case 1. Lattice of finite size How does the interference function look like after summing up I G 2? Consider: M 1 x m = x m m=0 m=0 m=m I sin2 [ 1 2 M 1a (k R k o )] sin 2 [ 1 2 a (k R k o )] x m = 1 1 x xm 1 x sin2 [ 1 2 M 2b (k R k o )] sin 2 [ 1 2 b (k R k o )] (3.12) Maxima of the intensity is expected each time when the argument of the denominator is multiples of π. Thus 1a (k 2 R k o ) = hπ and 1b (k 2 R k o ) = kπ, since k R = 2πs λ R and k o = 2πs λ o (both s are unit vectors). We obtain a (s R s o ) = hλ and b (s R s o ) = kλ. These are the Laue conditions for constructive interference at a 2DIM lattice. Figure 3.49: Modulations in the diffraction intensity for finite M. From CK. The equation for I implies that the height of the interference maxima is proportional to (M 1 M 2 ) 2 and occurs whenever the denominator is zero, as depicted in Fig. 3.49. In order to find the half width of the diffraction spot we have to move away from the intensity maximum by ɛ to arrive at I = 0, i.e., the nominator becomes

CHAPTER 3. SURFACE STRUCTURE 85 zero. At this place sin[ 1M a k sin(α + ɛ)] = 0 or 1M a k sin(α + ɛ) = π. This 2 2 leads to, with ɛ the full-width at half-maximum (FWHM) of the peak. ɛ = λ/(m a cos α) (3.13) This relationship between the crystallite size, namely the domain size, and the width of the diffraction spot (see Fig. 3.50) is known as Scherrer equation for 3DIM case. It just follows from the relationship between the real space and reciprocal space. Figure 3.50: For a finite domain size M, reciprocal-lattice rods become dashed regions, which represent the FWHM of the intensity distribution 2π/M. There are (M 2) additional maxima between the main intensities. Ref. [26]. Hence, the width of a diffraction spot is proportional to 1/M and the area (= width hight) proportional to M. a M 1 or b M 2 is the radius of an area on the surface in which the periodicity prevails, it is called the region of coherence. Figure 3.50 shows the reciprocal-space rods broadened according to the limited region of coherence. The size of the diffraction spot is inversely proportional to the number of scatterers in a periodic region. Actually, 25 to 100 atoms already give a well-resolved diffraction pattern. Theoretically, the spot size goes to zero and the spot intensity diverges as the region of coherence grows. Actually, electrons used in the experiment have a coherence length as well. For the derivation of the diffraction intensity we have assumed electrons be plane waves, i.e., the coherence length is not limited. In practice, it is about 10 15 nm that puts an upper limit on the intensity of beams diffracted at a perfect surface. 26 26 M. Henzler, Appl. Surface Sci. 11/12, 450 (1982).

CHAPTER 3. SURFACE STRUCTURE 86 Coherence is a definite phase relationship. In reality, the incident wave bares an uncertainty in its wave vector, in magnitude and in direction. It is k U in magnitude and k (= divergence) in direction. One speaks about temporal and spatial coherence. The energy uncertainty U is due to the high temperature of the filament required for thermionic emission. Thus wave chains are produced which are only able to interfere within a short distance of the coherence length. Within this length scale, they keep their phase relationship. The path difference between two waves may not be larger than this distance for interference. The angular divergence (β) is related to the size of the emitting surface. If L is the width of a perfect, monoenergetic beam then the coherence length CL is given by λ CL = L 2β(1 + U/U) Example: Source width = 0.5 mm, distance source-to-sample = 10 cm, U = 0 und λ = 1 Å results in CL = 100 Å. U 0.4 ev for 2000 C ( 3 kt ) applies to W cathodes, 2 U 0.2 ev for 800 C applies to LaB 6 cathodes. CL is a measure for the deviation of electrons from plane waves. If the average coherence region (a M 1 b M 2 ) on the crystal surface is smaller than CL, then the form of the LEED spots is given by M 1 M 2, otherwise by CL. Hence the information is always limited. Figure 3.51: LEED patterns obtained from of an Au(110)-2 1 surface at different primary-electron energies. Ref. [27].

CHAPTER 3. SURFACE STRUCTURE 87 2. Phase Transitions The (110) surfaces of fcc metals, in particular, those of the noble metals show a 2 1 reconstruction. 27 Today, thanks to several STM investigations, a detailed insight into the reconstruction has been gained. However, LEED is still most helpful describing the reconstruction and, if applicable, the nature of structural phase transitions. 28 Figure 3.52: The bulk-derived (110) surface of Au is shown on the right-hand side top. The real-space views of the 2 1 reconstructed Au(110) is shown on the left-hans side. Lower panel presents the reciprocal space view, as observed in LEED, consisting of integer-order ( ) and half-order ( ) spots. Ref. [28]. Figure 3.51 shows LEED patterns from an Au(110) surface. We notice that the integer- and half-order spots are all elongated in the 00 1 direction along which the half-order spots appear. This elongation is a measure for some existing disorder in that direction. Figure 3.52 displays on the right-hand side the real-space view of the fcc lattice. The striking feature of the (110) surface is the presence of the atomic chains in the 1 10 direction. This surface would give a LEED pattern consisting of the integer-order spots ( ). In reality, however, we also observe half-order spots ( ). Such spots manifest the doubling of the periodicity in real space. One possible 27 D. Wolf et al., Surface Sci. 77, 265 (1978). 28 R. Feder et al., Z. Physik B 28, 265 (1977).

CHAPTER 3. SURFACE STRUCTURE 88 mechanism for the increased periodicity is referred to as the missing-row model, as seen on the left-hand side of the figure. Every second chain of atoms running in the 1 10 direction would be missing. The transformation of information from the Fourier space back to the real space is not unique, and several other real-space structures are possible. Figure 3.53: Some models of 2 1 reconstruction of the (110) surface of noble metals. Each structure is shown both from the side and from the top. In all the structures the nearest-neighbor distances are the same as in the bulk. Ref. [29]. Figure 3.53 shows some models that are capable to account for the observed 2 1 reconstruction at the (110) surface of Au. Above the critical temperature of 650 K both the half-order spots and the disorder disappear reversibly. Investigations of the spot intensity indicate a second-order-like transition at the surface with a certain critical exponent, as indicated on the right-hand side panel in Fig. 1.9. A fit of the temperature dependence of the order parameter allows the determination of critical exponents and hence the universality class of the transition. The Au(110) 2 1 case is unique because the 2 1 spots due to reconstruction originates from a very shallow surface sheet. 29 Here, there is no counterpart of the transition in the bulk. As another example we mention c(2 2)-to-1 1 order-disorder transition in Cu 3 Au single crystal. In the case of Cu 3 Au the structural phase transition takes place at the surface as well as in the bulk. In this crystal, below 390 C the basal plane contains both Cu and Au atoms, which occupy well-defined sites in the fcc lattice, depicted in Fig. 3.54(b). 30 Above 390 C, the unit cell is an average Cu/Au atom (one atom per cell), i.e., there is a statistical distribution of Au and Cu atoms among all the lattice sites, as illustrated in Fig. 3.54(a). In LEED the ( 1 1) spots disappear above 2 2 the transition temperature. We can assign the intensity of the fractional-order spots to to an order parameter such that t β describes the transition. 29 M.S. Daw and S.M. Foiles, Phys. Rev. Lett. 59, 2756 (1987) and references therein. 30 V.S. Sundaram et al., Surface Sci. 46, 653 (1974).

CHAPTER 3. SURFACE STRUCTURE 89 Figure 3.54: Cu 3 Au in its disordered and ordered phase. Ref. [30]. Detailed work has later shown that at temperatures moderately away from the critical point this approach may have its validity. Near the critical point, however, multiple-scattering contribution renders the diffraction intensity not conserved and modifies the values of critical exponents sufficiently to shift the assignment from one universality class to another. In short, a simple kinematic analysis of peak intensities may lead to errors in the determination of critical values and one has to resort to multiple scattering. 3. Influence of the Third Dimension In the simplest case, we consider surface diffraction by single scattering. This simplification of 2DIM case requires two Laue conditions: a (s R s o ) = hλ and b (s R s o ) = kλ, with k R = 2πs λ R and k o = 2πs λ o. We know that also the bulk has a non-zero contribution on diffraction. In 2DIM, the Laue condition for the third dimension c (s R s o ) = lλ is not satisfied simultaneously for all energies. Actually, due to strong inelastic scattering, only few surface layers take part in diffraction, and as a consequence, the influence of the third dimension actually appears as a modulation. In Fig. 3.55 the intensity of the (00) beam is shown as a function of electron energy. We observe strong modulation of intensity including secondary Bragg maxima owing to double scattering. Such results demonstrate that the kinematic theory is not sufficient to fully describe diffraction features from single crystal surfaces. 31 The unknown value of the inner potential is responsible for shift of the forecasted energy from the observed value. The influence of scattering properties can also be realized considering the timereversal invariance, i.e., the theorem of reciprocity (see Fig. 3.56). The reciprocity theorem for scattering processes states that the amplitude for scattering from k to k is equal to that for scattering from the reversed final direction k to the reversed 31 K. Christmann et al., Surface Sci. 40, 61 (1973).

CHAPTER 3. SURFACE STRUCTURE 90 Figure 3.55: The normalized intensity of the specular reflection at Ni(100) as a function of primary-electron energy. The leading zero in the indices for diffraction orders should be omitted. Ref. [31]. initial direction k. 32 This is analogous to the fact that the reciprocal space has an inversion symmetry. For LEED, this implies that the intensity of the diffracted beam does not change under time reversal I(k, k ) = I( k, k). As a consequence of the time-reversal invariance, the diffraction experiment doubles the symmetry of the (00) beam and we observe, owing to the time-reversal invariance, symmetries that do not exist in the crystal. The threefold-symmetric (111) surface of an fcc structure and the (0001) surface of a hcp lattice both give rise to a sixfold-symmetric rotation axis for the (00) beam. Similarly, the intensity of specular reflection from the ideal (111) surface, from a twinned surface, and from a faceted but not twinned surface all result in the same rotation diagrams. k ' k -k ' -k I k k' = I -k' -k Figure 3.56: Time-reversal invariance. 32 H.E. Farnsworth, Phys. Rev. 49, 598 (1936); Phys. Lett. 36 A, 56 (1971).

CHAPTER 3. SURFACE STRUCTURE 91 4. Influence of the Temperature When the temperature is raised, the atoms vibrate around their assumed positions r j. The atomic positions are then given by r(t) = r j + u j (t) (3.14) for a cell with one single atom. r(t) is the variation in time of r j, the position of the atom at T = 0. The interference factor and the intensity become F = e i k r j (3.15) I F 2 = ( e i k r j ) 2 e i k u(t) 2. (3.16) In the expression for I the first factor is for a rigid lattice, i.e., at T = 0, and is called I o. The second factor is a time average. We have then: I = I o e ( k)2 u 2 (t), (3.17) where u 2 = 3h2 T 4π 2 mk B θd 2 with θ D, the Debye temperature, and m mass of the atoms. (3.18) Figure 3.57: The change in θ D with the primary energy for diffraction at Ni(100). Ref. [33]. ( k) 2 u 2 (T ) is the Debye-Waller factor (DW). This consideration shows that the intensity of diffracted beams decreases exponentially with temperature, while the spot size does not change.

CHAPTER 3. SURFACE STRUCTURE 92 Already in 1962 Germer and McRae have shown that the θ D for the surface is lower than that of the bulk. 33 They have compared results of diffraction experiments from a Ni(100) surface at different electron energies (Fig. 3.57). Electrons at different energies have different penetration depths and therefore carry information from different depths below the surface of the sample. That way θ D can be studied as a function of sample depth. The bulk value is obtained from x-ray scattering. Fig. 3.57 clearly shows that the Debye temperature for the surface is appreciably lower than that of the bulk. We also note that the DW factor depends on k. This implies that it is directional, and we have only access on the component u that is parallel to k. Hence, by varying the experimental geometry we can measure different components of u. It has been determined that u 2 to the surface 2 u 2 for the bulk and u 2 to the surface 1, 2 to 1,5 u 2 for the bulk. This means that the atoms move more effectively normal to the surface compared to their lateral motion (Fig. 3.58). (Challenging exercise: anisotropic melting!) Figure 3.58: Intensity of electrons elastically scattered from a Ni(110) surface close to the [1 10] direction with ψ = 0 (open circles) and ψ = 70 (filled circles) as a function of temperature. Electron energy is 40 ev. Different components of u have different temperature dependences. Ref. [33]. 33 A.U. MacRae and L.H. Germer, Phys. Rev. Lett. 8, 489 (1962); Ann. N.Y. Acad. Sci. 101, 627 (1963).

CHAPTER 3. SURFACE STRUCTURE 93 5. Surface Melting LEED provides a very convenient and readily accessible tool for studying melting (Section 1.4) at crystal surfaces. It is extreme surface sensitive, and the information is available over diffracted intensities. Figure 3.59 illustrates schematically scattering of primary electrons with an intensity of I o at a clean surface on the left-hand side with the diffracted beam I G. The surface displayed on the right-hand side is additionally covered with a thin structureless film of thickness d which represents the already melted layers. The reflected intensity is I. The diffracted intensity I Go, which is I G at T = 0, is attenuated due to several factors: I 0 I 0 I G I Figure 3.59: Electron scattering at a clean and a homogeneously covered surface. d and I G = I Go e ( k)2 u 2 (T ) (3.19) I = I Go e ( k)2 u 2 (T ) e 2d/Λ. (3.20) We assume that melting is homogenous over the entire surface and proceeds in a layer-by-layer fashion. The logarithm of both sides yields ln I = ln I Go DW 2d/Λ. (3.21) Experiments with Pb are practical, because Pb melts at conveniently attainable temperatures. Frenken and van der Veen have observed, as shown in Fig. 3.60, that there is a linear behavior of the DW factor up to 500 K and a strong increase above this temperature. 34 I Go is independent of temperature. Their results include: a. The (111) surface is a bulk-like and closed surface. It melts at T M, the bulk melting point. b. The (110) surface is an open surface. This surface shows depending on the direction different melting behavior: (10) beam is observable as high as the bulk melting point T M = 600 K. Thus the [001] direction remains ordered up to T M. The (01) beam shows, however, that there is a complete disorder in the [110] direction at already 560 K (Fig. 3.60). Here we have anisotropic melting. 34 J.W.M. Frenken and J.F. van der Veen, Phys. Rev. Lett 54, 134 (1985).

CHAPTER 3. SURFACE STRUCTURE 94 Figure 3.60: Melting of surface layers of Pb(110). The inset is an expanded view of the highest 10-K interval. The shaded band corresponds to the calibration uncertainties in T M. The arrow indicates the surface melting point. Ref. [34]. Theories about melting imply that d diverges if the temperature goes towards T M. For long-range interactions d (T M T ) r (3.22) with a critical exponent r = 1 3.35 For short-range forces 36 d ln(t M T ). (3.23) 6. Stepped Surfaces We deal now with facetted or vicinal surfaces. These are surfaces with uniform steps. The chemical activity of atoms at step edges is strongly enhanced, that makes this subject interesting for catalysis. If these steps or terraces are periodic, as displayed in Fig. 3.61, we have an additional condition for the interference in form of modulation of the Bragg peak. 37 35 J.K. Kristensen and R.M.J. Cotterill, Phil. Mag. 36, 437 (1977). 36 J.Q. Broughton and G.H. Gilmer, Acta Metall. 31, 845 (1983). 37 B. Lang et al., Surface Sci. 30, 440 (1972).

CHAPTER 3. SURFACE STRUCTURE 95 Figure 3.61: Schematic representation of the atomic structure of a stepped Pt surface with (111) terraces. Ref. [37]. Imagine N steps perpendicular to a. Each terrace has a coherence region of M unit cells (Fig. 3.61). If the step hight is d, which is the basis vector for the steps: M N G = e ip k a e iq k d p=1 q=1 G 2 = sin2 [ 1 Ma k] 2 sin 2 [ 1a k] sin2 [ 1 N k d] 2 sin 2 [ 1a k] (3.24) 2 2 In Eq. 3.24 the first term represents the ideal surface, while the second term describes the modulation within the diffraction maxima as a function of electron energy. Figure 3.62: (a) Vicinal surface with monotonically increasing steps and constant terrace size. (b) The sharp rods represent a lattice inclined to the average surface with each terrace representing one lattice point. Their separation depends on the terrace size. The broad rods represent the FWHM 2π/N a of the terrace structure factor, which has (N 2) subsidiary maxima. Reflections occur where the product of these two reciprocal lattices is nonzero. The terrace size is five-atoms wide. Ref. [26].

CHAPTER 3. SURFACE STRUCTURE 96 Figure 3.62(a) is a side view of the stepped surface shown in Fig. 3.61. The diffraction condition in real and reciprocal space is displayed Fig. 3.62(b). A typical LEED pattern with split spots is shown in Fig. 3.63. Figure 3.63: Diffraction at a periodic array of steps on a Pt surface. Ref. [37]. 7. Domains A partial region on the surface with a well-defined structure is called a domain. If there is one single domain on the surface surrounded by the bare surface we speak of an island. Islands form if the interaction between adsorbates are larger than that between adsorbate and the surface as a result of adhesion forces. There are three distinct types of domains: a) If a domain is larger than the spot size of the electron beam, we obtain an individual diffraction pattern from each domain. b) If a domain is smaller than the beam diameter, but larger than the coherence length of electrons, the resulting LEED pattern is a superposition of individual pattern (not a coherent addition) from domains that are illuminated by the electron beam. c) If the typical domain sizes are smaller than the coherence length of electrons, we have interference of beams scattered from different domains located in a region as large as the coherence length. The resulting LEED pattern is a coherent addition the beams from individual domains. Any commensurately adsorbed monolayer that has a superlattice unit mesh larger than the substrate will form translational and possibly also rotational (depending on the symmetry of the unit mesh) antiphase domains. Figure 3.64 is a schematic representation of the former, while the latter is presented in Fig. 3.66, both schematically. In each region a 2 1 reconstruction prevails, while the superstructure of each adjacent region is shifted by half a unit in one direction. Thus, a fractional monolayer may exist in the form of antiphase islands at some finite

CHAPTER 3. SURFACE STRUCTURE 97 Figure 3.64: An antiphase domain structure formed by adjacent occupancy of the two equivalent adsorption sublattices of an ordered 2 1 overlayer. From AZ. temperature if there is a net attractive interaction between the adsorbate atoms. There may be several reasons why a fractional monolayer might exist as islands, the most important being kinetic limitations and point or line defects on the substrate that act as nucleation sites. Figure 3.65: (Left) clean Cu(100) surface. Observation of the antiphase domains due to successive oxygen adsorption. P. Aebi et al., unpublished. Figure 3.65 presents LEED patterns obtained from Cu(100) surface. The clean surface delivers the pattern on the left-hand side. With 1 ML oxygen, a reconstruction is observed typical of a c(2 2) structure. Further oxygen adsorption introduces still additional fractional-order spots. The new surface phase is accounted for by a rotational antiphase-domain structure, as illustrated in Fig. 3.66. Quasicrystals lack periodicity yet display characteristics of long-range orientational order. The ternary alloy Al-Pd-Mn has icosahedral symmetry in the bulk, and its pentagonal surface shows bulk-terminated behavior, namely it is not reconstructed and appears as expected from the bulk structure. Figure 3.67 shows on the left-hand side the diffraction pattern from the clean surface. Compared to crystals,

CHAPTER 3. SURFACE STRUCTURE 98 Figure 3.66: A model showing the c(2 2) antiphase domains of oxygen on Cu(100). where one observes diffraction spots due to only few Brillouin zones, quasicrystal surfaces display enumerable number of spots. In fact, the longer one chooses the recording time for diffraction intensities, the more spots appear on the collector screen. We notice that the azimuth distribution of the pattern has 2π/5 symmetry. The spot distribution on radial directions obeys the golden ratio, ( 5 + 1)/2, so does the intensity distribution of all the spots. Figure 3.67: LEED patterns obtained at 55 ev from (left) a clean pentagonal surface of icosahedral Al-Pd-Mn, (right) after exposure to O 2 at 700 K and subsequent annealing for 2 h. Ref. [38].

CHAPTER 3. SURFACE STRUCTURE 99 When the quasicrystal surface is exposed to oxygen at elevated temperatures, Al segregates from the bulk and enriches the surface. 38 In contact with oxygen, Al readily oxidizes and the LEED pattern transforms to one observed on the righthand side. This pattern mainly shows 30 equally bright and equally distributed spots. They represent five distinct groups of hexagonal spots, typical of the (111) surface of Al 2 O 3. This observation is interpreted in terms of five distinct crystalline Al 2 O 3 domains formed on the pentagonal surface of Al-Pd-Mn. Using Eq. 3.13 we estimate the Al 2 O 3 islands as large as 3 nm. 3.3.1.3 Calculation of Diffracted Intensities The symmetry of the atomic order at the surface is studied by observing the diffraction spots in 2DIM; we study the projection of the unit cell in the reciprocal space. Additional relevant information can be extracted investigating the scattered intensity, I(E), I(θ), I(φ). A comparison of these measurements with detailed scattering calculations on model structures facilitates detailed knowledge on interatomic distances, vibration amplitudes, and other structural features not accessible to simple inspection of the diffraction pattern. The process consists of the assumption of a model structure and calculation of the diffraction intensity in order to find some agreement. The structure parameters are varied and calculations are repeated until some agreement is found. This agreement is a necessary but not sufficient condition for the validity of the model structure and all the assumed ingredients of the computation. In the computations one solves the scattering problem. The steps include - using an effective potential usually from a band theory - slicing the sample in slabs parallel to the surface - calculating multiple scattering in each slab - adding up all the contributions considering multiple scattering between the slabs and attenuating the signal from deeper slabs by the mean free path Λ appropriate for the electron energy. One has thereby to consider that electrons are faster in the solid than in vacuum owing to their acceleration by the inner potential (Figs. 3.68 und 3.69). Consequently, the normal component of the electron velocity changes and the tangential component is conserved such that v o sin θ o = v 1 sin θ 1. Energy conservation dictates: Snell s law gives the index of refraction: e(u + U o ) = 1 2 mv2 1 (3.25) eu = 1 2 mv2 o (3.26) n = sin θ o sin θ 1 = v 1 v o = U + Uo U. (3.27) 38 J.-N. Longchamp et al., Phys. Rev. B 76, 094203 (2007).

CHAPTER 3. SURFACE STRUCTURE 100 θ θ Figure 3.68: Refraction of electrons while entering the solid. Note that U o > 0, U + U o > U. θ θ' Figure 3.69: The apparent scattering angle due to the refraction of electrons. After considering all the details, e.g., the inner potential, mean free path of the electrons involved, the Debye temperature of the surface and bulk, the crystal structure and the reconstruction at the surface, one uses realistic potentials and calculates the measured intensity curves using multiple scattering. In the case of adsorbates on the surface, one additionally has to take into account where exactly these species are located on the surface. The computed curves are compared with the experimental ones, and the input parameters to computations are modified until the results converge to an acceptable agreement. The input to calculations are then taken as most probable values representing the real case.

CHAPTER 3. SURFACE STRUCTURE 101 Figure 3.70: (Left) Measured diffraction intensities of the 00 beam at a W(100) surface as a function of energy. (Right) Comparison of calculated and experimental diffraction intensities for a Ni(100) surface covered with oxygen. Ref. [23].