Surface Crystallography by X-ray Diffraction

François Grey and Robert Feidenhans'l, Roskilde (Risø National Laboratory)

It has been said that all science is ventional Miller indices hkl of a crystal either physics or stamp collecting. If so, plane can most easily be understood as then surface science has consisted the shortest reciprocal lattice vector mainly of stamp collecting for the last normal to it.) 20 years. The collection is a vast and Without further precaution, however, complex one, because crystal surfaces the diffuse X-ray scattering from the display a bewildering variety of struc­ bulk will swamp all but the most intense tures which depend delicately on, for fractional-order reflections. The solution example, the crystal temperature or the to this problem is to perform the diffrac­ amount of any adsorbed atoms on the tion experiment under grazing angles of surface. To extract some physics from incidence, thus limiting the X-ray pene­ this collection, scientists require reliable Fig. 1 — The diffraction geometry. The tration depth considerably. The diffrac­ models of the surface atomic geometry. monochromatic X-ray beam is incident at an tion geometry is shown in Fig. 1. In par­ Such models have proved difficult to angle αi to the surface. After being diffrac­ ticular, below a certain critical angle of obtain experimentally. ted through an angle 29 by the reconstruc­ incidence, typically a fraction of a The ideal surface of a crystal — the ted surface, the X-rays are detected at an degree, the X-rays are totally reflected surface obtained by simply terminating exit angle αf, to the surface, αi, and αf are typically of the order 0.5°. The in-plane from the surface. This phenomenon is the bulk crystal structure at a particular component, q||, of the scattering vector q is the direct analogue of total internal crystallographic plane — is rarely an a reciprocal lattice vector of the surface reflection in optics, but occurs externally energetically favourable arrangement lattice; the component qz perpendicular to because the refractive index of materials for the surface atoms. To achieve a more the surface can be varied by altering αi or αf, for X-rays is slightly less than one : stable configuration, the surface will in order to probe along a rod in reciprocal n = 1 - , = 4.49 x 10-6NZλ2 often undergo a reconstruction, which space. N: number of atoms per Å3 can involve quite dramatic changes of Z: number of electrons per atom the surface atomic geometry. A classical fracted X-ray signal from a single atomic λ: wavelength in Å example, which illustrates the task layer is very weak. If the first layer of so that, in the small angle approxima­ facing the surface crystallographer, is atoms from a 1 mm2 region of the sur­ tion, the critical angle is: the 7x7 reconstruction of the (111) crys­ face were collected into a ball it would αc =  2 , tallographic surface of silicon. This sur­ measure only a few microns in diameter; typically of the order of 0.5°. face can be produced by cleaving Si in this effective crystal size is the lower Under the conditions of total external ultra-high vacuum and heating the crys­ limit of what can be studied with con­ reflection, the refracted wave is inhomo­ tal briefly to 400°C. The notation 7x7 ventional X-ray sources. That is why, to geneous, and exponentially damped in refers to the size of the reconstructed date, most surface X-ray diffraction the direction perpendicular to the sur­ unit cell relative to the 1x1 unit cell of studies have been made at synchrotron face. The penetration depth is then the ideal surface. Assuming that the X-ray facilities. typically only a few tens of Angstroms. reconstruction affects the first few Intense X-ray sources though, are not Although this grazing incidence geome­ layers of the crystal, the 7x7 unit cell the only requirement for surface crystal­ try is not essential for observing a contains well over 100 atoms. Such unit lographic studies. The technique must surface-diffracted signal, it has proved cells are more reminiscent of crystals of also be made surface sensitive. This is to be important for obtaining large crys­ large organic molecules than of simple because X-rays probe tens of microns tallographic data sets with sufficient elements. into the bulk of the crystal. Recons­ accuracy to perform detailed structural Given the long history of X-ray diffrac­ tructed surfaces provide a unique analysis. tion, and its prime role in the determina­ means of distinguishing the scattering The weakness of the diffracted X-ray tion of complex structures such as pro­ from the surface from the much more in­ signal from a surface has one very posi­ teins, the technique would seem an ob­ tense bulk scattering. The larger periodi­ tive aspect; the intensity is so low that vious candidate for surface structure city of the reconstructed surface layer multiple scattering can safely be neglec­ determination. It is therefore somewhat leads to extra Bragg reflections, well ted, and so analysis based on the Born surprising to find that the first attempt separated from the bulk peaks. In terms approximation is applicable. The simpli­ by Marra and Eisenberger [1] to solve a of the crystallographic Miller indices h,k city of the underlying scattering theory surface structure by X-ray diffraction of the ideal surface, these extra reflec­ is the crux of the method: it is possible dates from as recently as 1981. Part of tions have fractional indices, and are to analyse the results of surface X-ray the explanation for the tardy develop­ referred to as fractional-order reflec­ diffraction experiments in great detail, ment of this technique is that the dif- tions. (For the non-specialist, the con­ with very modest computational means. 94 This contrasts sharply with the standard problems is to use a miniature ultra-high will have peaks at positions relative to technique of surface crystallography, vacuum chamber which can be discon­ the origin which correspond to interato­ low energy electron diffraction (LEED). nected from a larger sample preparation mic distances in the unit cell, since the Low energy electrons are a surface sen­ chamber and mounted on a diffractome­ autocorrelation integral will have a maxi­ sitive probe because of their strong in­ ter. Recently, though, several instru­ mum when two atoms overlap. Such a teraction with matter, but this strong ments have been built where the sample function is plotted in Fig. 2a, based on interaction leads to multiple diffraction can be prepared and studied with X-ray data from a surface X-ray diffraction and refraction effects. Although LEED diffraction in the same chamber, which measurement of the Ge(001)2x1 re­ can reveal the symmetry of the surface permits a much wider range of experi­ constructed surface [3], structure, giving the size and orientation ments to be performed [2], In common with most surface X-ray of the surface unit cell, it is difficult to The steps from the surface X-ray dif­ structure determinations to date, the deduce reliable models for the atomic fraction measurements to the analysis measurements were made with grazing geometry from such measurements. of the data are straightforward: struc­ angles of incidence and exit, so that only On the other hand, the apparatus for ture factor intensities are obtained from a plane in reciprocal space was probed. surface X-ray diffraction is considerably the integrated, background-subtracted As a consequence, the Patterson func­ more complex and costly than the stan­ intensities of the Bragg reflections after tion is that of the 2x1 unit cell projected dard LEED equipment. The experiment correction for the polarization factor and onto the surface plane. Another impor­ must combine the full freedom and ac­ certain geometric factors connected tant point is that reflections with integer curacy of an X-ray diffractometer with with the scattering geometry. It must be Miller indices are not used in the analy­ the ultra-high vacuum technology ne­ borne in mind, though, that the data pro­ sis, because there is a coherent contri­ cessary for keeping the surfaces clean. vides only the amplitude of the structure bution to these reflections from the The simplest solution to these technical factors, and not their phases. Since the substrate. This systematic absence of beginnings of X-ray crystallography, a structure factor intensities tends to Fig. 2 — The Patterson function of Ge(001)- great variety of ingenious methods have distort the Patterson function slightly. 2x1 and dimer model. The Patterson P(r) is been devised to extract structural infor­ Despite these shortcomings, three inter­ the Fourier transform of the measured struc­ mation from diffraction data, and thus atomic distances are clearly visible in ture factor intensities: side-step the famous "phase-problem". Fig. 2a. It is possible to eliminate several P(r) ~  /Fhkl2cos[2 (hx + ky)] hk One of the strengths of the surface X-ray of the many structural models that have Positive peaks correspond to interatomic diffraction technique is that it can imme­ been proposed for this surface, simply vectors in the unit cell. Because integer- diately draw on this bank of knowledge. by comparison with the Patterson func­ order reflections are not included, only tion. Fig. 2b shows the projected struc­ layers significantly different from the bulk In-plane Structure ture of a model in which the surface layers are visible. The smallest non-repeat­ The first step in the data analysis is to form dimers in order to saturate the ing section of the Patterson is shown (a). plot the Fourier transform of the X-ray in­ broken bonds at the surface. The figure The main features can be identified with in­ tensities, familiarly called the Patterson indicates the three interatomic vectors teratomic vectors in a dimer model of the surface (b), slight distortion in the Patterson function. Because this is a transform of which are significantly distorted from (the vectors do not run through the middle of structure factor intensities, it does not the ideal surface geometry, correspon­ the rings) being due to the absence of the yield the electron density in the unit cell, ding to the features in the Patterson. integer-order reflections. The size of the unit but rather the autocorrelation function Once a model has been found which cell doubles in one direction when the top of that density. The Patterson function can account for the Patterson, a more layer atoms dimerize (c). EUROPEAN SYNCHROTRON RADIATION FACILITY GRENOBLE - FRANCE The ESRF, is an international synchrotron radiation source to be built in Grenoble, France. It will consist of a 6 GeV electon/positron storage ring surrounded by X-ray beam-lines. Our Experiments Division is looking for a SCIENTIST TO TAKE CHARGE OF HIGH X-RAY ENERGY SCATTERING Basic experience required: An international reputation for research involving high energy (> 50 keV) X-rays. Duties: To be responsible for the design and construction of beam-lines for high energy X-ray scattering and to provide scientific leadership in this area at ESRF. One of the characteristic properties of ESRF is that it will produce X-ray beams of high energy and low emittance. The scientist in charge of high energy X-ray scattering will be re­ quired to develop a relationship with users from countries participating in ESRF. He/She would liaise with the Optics Group and the Detector Group at ESRF in re­ search and development projects on the problems of constructing beam lines to match the quality of the planned ESRF beams. He/She should speak fluent English. Remuneration: Gross annual salary from 250to 300 kFF depending on qualifications and experience. In addition: monthly family supplement and expatriation allowance for non-French staff (calculated on the basis of family situation) and a settling in allowance and adaptation allowance (paid once). Applications should be sent, in English or French, with the names of three referees before 30 SEPT. 88, to: ESRF-Personnel Office -Ref. 17/2111 — BP 220 — F 38043 GRENOBLE CEDEX

95 these sites [4], The same pattern of twelve dots is familiar to surface scien­ tists from the scanning-tunnelling mi­ croscope pictures of the Si(111)7x7 unit cell obtained by Binnig and Rohrer. The methods used for the determina­ tion of the projected structure of the unit cell belong to the standard analysis techniques of X-ray diffraction. But Fig. 3 — Electron density difference map of Sn/Ge(1W7x 7. The structure is closely related there are two important features that to that of the Dimer-Adatom-Stacking fault (DAS) model of the Si (111) 7 x 7 surface. A map of distinguish surface X-ray diffraction electron density difference la) is given by from X-ray diffraction experiments on bulk crystals: the two-dimensional na­   (r) α hk (lFexpl - lFmodell )cos[2 (hx + kx) -  model] wherelFexphklare the experimental structure factors andlFmodelhklare the structure factors for ture of the surface, and the interference a DAS model containing only Ge atoms. The phases of the experimental structure factors are of surface and bulk scattering. Both fea­ unknown and therefore assumed to be  modelhk, the phases for the model structure factors. tures offer the possibility of obtaining The result shows missing electron density on the adatom positions (shaded in b). The ada­ additional information about the surface toms are thus identified to be Sn. structure. detailed fit of the structural parameters Fig. 4 — Schematic illustration of scattering along a Bragg rod of intensity from a can be performed, in which the atom reconstructed surface. If only one layer is different from the ideal bulk geometry, there is positions are varied to minimize a least- monotonic decay of intensity along the rod la). If several subsurface layers are perturbed by squares residual between the experi­ the reconstruction, they contribute to the Bragg rod too: the intensity has an oscillatory variation due to interference of the scattering from different layers (b). For Ge(OO1), the mental and calculated structure factor Bragg rods have pronounced oscillatory behaviour, which can be accounted for by elastic intensities. This fitting procedure is deformation of the subsurface layers due to the dimers at the surface. The data is well-fitted straightforward, but there is no guaranty by a model in which the first eight layers are relaxed (full line in c). of success. For simple unit cells, such as that of Ge(001), satisfactory agreement is obtained with the model suggested by the Patterson function. But for struc­ tures with large unit cells it may be necessary to include extra atoms that are not immediately apparent from the Patterson function. These extra atoms will appear in an electron density dif­ ference map. This is obtained by sub­ tracting the measured structure factor amplitudes from those of the best (but inadequate) model, assuming the struc­ ture factor phases of the model, and per­ forming a Fourier inversion. If the star­ ting model is not too different from the actual structure, then this procedure will reveal where there is too much or too little electron density in the model unit cell. An elegant example of the use of elec­ tron density difference maps comes from a study of a 7x7 reconstruction on the Ge(111) surface, induced by the ad­ sorption of a submonolayer amount of Sn. This study involved the collection of 269 different reflections, the largest sur­ face X-ray diffraction data set to date. The structure was found to be identical to the generally accepted Dimer- Adatom-Stacking fault (DAS) model for the Si(111) 7x7 surface. Since Sn has more electrons than Ge, it is possible to find which sites the Sn atoms occupy by performing an electron difference map between the data and a DAS model con­ taining only Ge atoms. The result is shown in Fig. 3a. The peaks in the map are the positions of the adatoms in the unit cell (the shaded atoms in Fig. 3b), indicating that Sn substitutes Ge at 96 Bragg Rods and Subsurface Relaxation the bulk Ge bond-length. The subsur­ large, and deeper layers which are The two-dimensional periodicity of a face relaxation occurs in order to mini­ essentially unperturbed. The scattering reconstructed surface translates into an mize the strain induced by dimerization at the integer-order reflections is then array of rods in reciprocal space, exten­ of the top Ge layer. The relaxation pat­ the coherent combination of a Bragg rod ding perpendicular to the surface plane. terns for this and other reconstructed from the surface region, and the crystal The reciprocal lattice rods are defined by semiconductor surfaces can be under­ truncation rod of the substrate. The sum three Miller indices, h and k in the plane stood from the elastic properties of Si of these two will depend sensitively on of the surface, and by a continuous and Ge: bond stretching is energetically the position of adsorbate atoms in the index / along the rod. There are no res­ more expensive than bond bending, unit cell, and measurements of the trictions on the third index simply be­ causing the atoms to relax so as to pre­ integer-order reflections can thus be cause the reconstructed unit cell does serve bond-length as far as possible. used to determine the registry of a mesh not repeat perpendicular to the surface. formed by atoms adsorbed on the crys­ If only a single atomic layer were in­ Crystal Truncation Rods and Adsorbate tal, and thus deduce the preferred ad­ volved in the reconstruction, then the Registry sorption sites. This technique has been rods would be essentially featureless, The other unique feature of surface applied to the  3x 3R30 reconstruc­ except for the monotonic dependence X-ray diffraction is the coherent combi­ tions of Pb chemisorbed on the Ge(111) on the atomic form-factor and thermal nation of scattering from the substrate surface [6]. A simple structure, called Debye-Waller factor. In practice, the re­ and from the surface reconstruction for the α-phase, occurs for a coverage of 1 construction perturbs several subsur­ Bragg reflections with integer Miller in­ Pb atom per unit cell. In the denser  - face layers. These layers are distorted dices h,k. The sharp interface between phase, the sites between the Pb atoms slightly from their bulk positions, acquir­ crystal and vacuum leads to tails of in­ at the corners of the unit cell are filled so ing the same periodicity as the surface. tensity about Bragg reflections, exten­ that there are 4 Pb atoms per unit cell, They therefore contribute to the fractio­ ding along the direction normal to the resulting in a dense-packed Pb layer (Fig. nal-order Bragg reflections. This con­ surface (see Fig. 5). Between reflec­ 6). Although these structural models for tribution is particularly apparent along tions, the intensity in these "crystal trun­ the two phases had been proposed over the Bragg rods, which become modula­ cation rods" is comparable to that due to twenty years ago on the basis of elec­ ted (see Fig. 4). a single atomic layer, and thus easily tron diffraction measurements, the Pb An example of a modulated Bragg rod measurable in a surface X-ray diffraction adsorption sites were not known. For for the Ge(001) surface is given in Fig. 4. experiment [5]. the simple α-phase, analysis of the To account for the experimental modula­ For a reconstructed surface, the integer-order reflections showed that tion it is necessary to include relaxations crystal can conveniently be divided into the Pb atoms sit in triangles of top layer in the first eight layers of the crystal. The two regions, a near-surface region of Ge atoms and above second layer relaxations of the subsurface layers are one or several layers for which the devia­ atoms. In the -phase, though, the ana­ of the order of 0.1 Å, or less than 5% of tions from the ideal bulk structure are lysis of the integer order reflections

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97 clearly indicates a shift of the unit cell so The technique is unique owing to both that the corner atoms sit on a new ad­ the simplicity of the analysis, and be­ sorption site, above 4th layer Ge atoms cause X-rays can probe the structure of (Fig. 6). The (5-phase is not, therefore, layers below the surface. Combining the produced by simply "filling the gaps" in in-plane structure with the information the α-phase, but by a significant chemi­ extracted from the reciprocal lattice rods cal change at the surface, involving a gives surface X-ray diffraction a very new bond configuration. wide scope : it should in future become a The few examples given here illustrate routine matter to determine the full the potential of X-ray diffraction for the three-dimensional structure of a recons­ investigation of surface crystallography. tructed surface. Fig. 5 — The principle of crystal truncation rods. The intensity along a Bragg rod passing through bulk Bragg reflections is shown. For a slab of N layers the scattered intensity has the form: I = Io[sin2(Nqza0/2)]/[sin2(qza0/2l] where l0 is the scattered intensity from a single Fig. 6 — The a and  phases of Pb/Ge(111) layer (a). The result is analogous to the multiple-slit diffraction pattern in optics. If the crystal  3x 3. The dilute α-phase has one atom is allowed to extend to infinity in one direction, the resulting scattered intensity has the form : per unit cell situated above the second layer I = lc/sin2lqza0/2l which is the envelope-function of the scattering from a finite crystal (b). Ge atom. In the p phase, Pb atoms form a Between the Bragg reflections, the intensity in the crystal truncation rod falls to l0, the scat­ dense-packed structure. From the analysis tered intensity from a single layer. of the in-plane integer-order reflections it can be deduced that the corner atoms of the unit cell are shifted to a new adsorption site in the p phase, in the holes of the honeycomb pattern formed by the first Ge bilayer (direct­ ly above the fourth layer Ge atoms). Note that the other atoms in the p phase are not on high-symmetry sites.

REFERENCES [1] Eisenberg P. and Marra W.C., Phys. Rev. Lett. 46 (1981) 1081. [2] Robinson I., in Handbook of Synchrotron Radiation, Eds. D.E. Moncton and G. Brown (North Holland) 1987. [3] Grey F. et al.. The Structure of Surfaces II, Eds. J.F. van der Veen and M.A. van Hove (Springer Verlag) 1988, p. 292. [4] Pedersen J. Skov et al., ibid., p. 352. [5] Robinson I., Phys. Rev. B 33 (1986) 3830. [6] Feidenhans'l R. et al., Surf. Sci. 178 (1986) 927.

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