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.