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The local atomic structure of Cu(111): An extended energy loss fine structure investigation
Journal of Electron Spectroscopy and Related Phenomena 95 (1998) 193–202
The local atomic structure of Cu(111): An extended energy loss fine structure investigation D.V. Surnin*, A.N. Deev, D.E. Guy, Yu.V. Ruts Physical–Technical Institute UB RAS and Udmurt State University, Kirov St. 132, Izhevsk 426001, Russia Received 17 October 1997; revised 31 March 1998; accepted 15 June 1998
Abstract Extended energy loss fine structures spectra (EELFS) above the M 2,3 ionisation edge of a Cu(111) single crystal have been ˚ . For obtained at primary electron energies of 1500 and 560 eV, corresponding to a studied layer depth of ⬃15 and ⬃8 A experimental EELFS spectra the solution of an inverse problem has been performed and pair correlation functions (PCFs) have been obtained by Tikhonov’s regularisation method with allowance made for normalisation of the atomic intensity of ionisation losses. Parameters of the nearest atomic environment, namely, the bond length, the co-ordination number, the Debye–Waller factor and the parameter of asymmetry of the first peak of the pair correlation function, have been determined ˚ ) and small (⬃8 A ˚ ) depths of the studied layer. 䉷 1998 Elsevier Science B.V. All rights reserved. for both large (⬃15 A Keywords: EELFS; EXAFS; PCF
1. Introduction The experimental determination of the atomic structure of matter is a traditional problem in solid state physics. Methods of electron and X-ray diffraction have had a dominant role in these investigations over a long period. Though traditional diffraction methods are still in use, over the last few decades there has been interesting developments in spectroscopic methods for studying the local atomic structure that are based on analysis of extended oscillating structures of spectra. Among these we quote EXAFS (Extended X-ray Absorption Fine Structure) [1], EELFS (Electron Energy Loss Fine Structure) [2] and other EXAFS-like methods [3]. By extended fine structures we mean oscillating * Corresponding author. Fax: + 007 3412 237901; E-mail: [email protected]
spectral features (with a period of oscillation of tens of electronvolts) located, as a rule, above the core ionisation threshold (the ionisation loss edge) and with an extension of several hundred electronvolts in EELFS, and more than 1 keV in EXAFS. Spectroscopic methods turn out to be most useful in studying the atomic structure of disordered multi-component materials, since these methods make possible analysis of the local atomic environment around an atom of specific chemical species [4]. Besides, spectroscopic methods allow for the mathematical formalisation of the inverse problem to determine the pair correlation function (PCF) from the experimental data [5,6], with the form of the inverse problem and its solution being independent of both chemical composition and the atomic structure of the matter studied [7]. Among spectroscopic methods, the EXAFS method is the oldest known and best developed. It is currently used as a standard method for the analysis of local
0368-2048/98/$ - see front matter 䉷 1998 Elsevier Science B.V. All rights reserved. PII: S 03 68 - 20 4 8( 9 8) 0 02 1 4- X
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atomic structure. Among more recent alternative methods, the EELFS technique is the one best developed. Compared to EXAFS, the EELFS method has both advantages and disadvantages. Advantages of EELFS are: (1) availability—one can obtain spectra using a conventional Auger spectrometer; (2) high degree of spatial localisation of the region under study; (3) surface sensitivity—the possibility for variation of the layer depth studied. The main disadvantages of EELFS are: (1) low intensity of the structure compared to the total signal intensity and thus complexity of extracting the oscillating structure from the experimental data; (2) the small extent of the oscillations that is determined by damping of the atomic intensity of ionisation losses. The latter leads to great difficulties in the solution of the inverse problem. In the present work the local atomic structure of a Cu single crystal (111) oriented surface has been investigated by the EELFS technique for different depths of the studied layer. Appropriate normalised pair correlation functions and parameters of the nearest atomic environment have been obtained by Tikhonov’s regularisation method. Our results are compared with those obtained by other authors with the help of both experimental methods and model calculations. Section 2 describes the known EELFS theoretical formalism in the EXAFS-like form for the case of the Cu M 2,3 ionisation loss edge. The experimental method for obtaining electron energy loss spectra and the procedure for extracting the oscillating part from the experimental data are given in Section 3. The Fourier-transform and Tikhonov’s regularisation methods used to analyse the extended electron energy loss fine structure signal are presented in Section 4. Discussion of the experimental results and comparison between the derived nearest atomic environment parameters and those obtained by other researchers are presented in Section 5.
2. EELFS formalism The extended fine structure above the edge of the ionisation loss is formed as a result of coherent scattering of the secondary electron on atoms of the nearest environment of an ionised atom. For the
structure above the M 2,3 edge of the Cu ionisation, the expression determining the intensity of this process has the form: JM2, 3 (Ep ) = J0 (Ep )[1 + x(p)]
(1)
The intensity of the process in Eq. (1) is presented as a function of E p (secondary electron energy). The secondary electron energy is determined by the energy conservation law: Ep = Ew − Eu − EM2, 3 , where E w is the incident electron energy, E u is the energy of an inelastically scattered electron registered by experiment, EM2, 3 is the M 2,3 core level electron binding energy. Taking into account only single scattering of the secondary electron on the nearest environment of an ionised atom brings the structural term in Eq. (1) to the EXAFS-like form [2]: ( ) exp i2(p + ig)R j 2 0 x(p) = Re exp(i2d2 )W (p)fj (p, p) 3 ipR 2j (2) where R j is the distance between ionised and neighbouring jth atoms, p is the wave number of the secondary electron, f j(p,p) is the back scattering amplitude of the secondary electron on the neighbouring jth atom, d02 is the phase shift resulting from the electron scattering on an ionised atom (the 2 corresponds to the d-symmetry of the secondary electron wave function), W(p) is the factor taking account of atom thermal vibrations. Summation over j (for all atoms of the environment of an ionised atom) is implied in Eq. (2). In the structural term of Eq. (2) it is assumed that the symmetry of the wave function of the secondary electron is d-type (l = 2), since 3p → ed excitation should be more intense than 3p → es (l = 0) in M 2,3 core level excitation. In Eq. (1) the processes of coherent scattering of both incident and inelastically scattered electrons are neglected. This approximation is true due to the fact that kinetic energies of these electrons far exceed the energy of the secondary electron. Besides the process of the energy loss of an incident electron by ionisation of the M 2,3 core level (Eq. (1)), processes of the energy loss of an incident electron by multiple excitation of electrons of the valence band make a contribution to this region of the spectrum. The latter gives the most intense but non-oscillating (smooth) component of the spectrum. Thus, the total
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non-oscillating part of the EELFS spectrum, J bg is the sum of the intensity of the atomic process of the M 2,3 core level ionisation loss (J 0) and the intensity of all processes of multiple losses by excitation of valence electrons.
3. Experiment and pre-treatment of data Electron energy loss spectra from the single crystal Cu(111) surface at room temperature have been obtained in a back scattering geometry with a JAMP-10S (JEOL) Auger electron spectrometer. The vacuum in the analysing chamber was not worse than 10 −7 Pa. The sample surface was precleaned with pure alcohol and acetone in an ultrasonic bath. In the analysing chamber the surface was cleaned by Ar + ion beam bombardment (0.5 keV). Surface cleanliness was monitored by Auger electron spectroscopy. During the whole experiment the density of impurity atoms on the surface did not exceed 1 at.%. In spite of the fact that during the experiment the crystallinity of the surface was not controlled, the use of the low-energy ion beam should ensure that the flatness and long range order of the Cu(111) surface is not grossly changed as shown by a number of investigations of the effect of low-energy ion beams on single crystal surfaces (see, for example Ref. [8]). Besides, it should be remembered that in these experiments we gain information, averaged over the depth of the analysed layer, about the local atomic environment which is insensitive to the crystallographic long range order of the sample. From the above considerations, we have assumed that the Cu surface under study is an ideal Cu(111) surface from a local structure perspective. Electron energy loss spectra were obtained at incident beam electron energies of 1500 and 560 eV in the integral mode (DE/E = 0.4%). Spectra were taken with a 1 eV step. Each spectrum was a summation of 1000 scans where each scan was obtained with 1000 samplings at each energy point. It is possible to estimate the studied layer depth from the mean free path of electrons registered in the experiment. For the spectrum obtained at 1500 eV excitation energy, we estimate that the ˚ while for 560 eV studied layer depth is ⬃15 A ˚ excitation energy the studied layer depth is ⬃8 A
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[9]. The information in the electron spectrum represents an average over the studied layer depth. Keeping this in mind, the structural results obtained from an electron energy loss spectrum taken at the excitation energy of 1500 eV will be termed ‘‘bulk’’ and the results obtained from the spectrum at the excitation energy of 560 eV will be considered as ‘‘surface’’. Initial ‘‘bulk’’ and ‘‘surface’’ spectra and corresponding oscillating parts are plotted on the scale of incident electron energy losses in Figs 1 and 2, respectively. EELFS oscillating parts were obtained from the recorded spectra by subtracting a non-oscillating part, J bg, which was approximated by fitting a polynomial curve to the data. The extended fine structure signals plotted in Fig. 1(b) and Fig. 2(b) are normalised to the unit value of the Cu M 2,3 edge jump. The recorded spectra were deconvoluted with the apparatus function, which was approximated by a Gaussian function with half-width determined from
Fig. 1. Electron energy loss spectrum (a) and corresponding M 2,3 extended energy loss fine structure (b) of Cu(111) which was obtained at the incident beam electron energy of 1500 eV.
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4. Solution of the inverse problem and discussion of results 4.1. Fourier-transformation
Fig. 2. Electron energy loss spectrum (a) and corresponding M 2,3 extended energy loss fine structure (b) of Cu(111) which was obtained at the incident beam electron energy of 560 eV.
DE/E for the apparatus. The derived oscillating parts of the Cu EELFS spectra are in good agreement with experimental results obtained by other investigators [2,10–12]. To solve the inverse problem to determine the characteristics of the nearest atomic environment of an ionised atom, there is a need to convert the oscillating parts of the EELFS spectra to the space of the secondary electron wave numbers. The ˚ −1) of the secondary electron (stanwave number (in A dardq for all EXAFS-like techniques) is p = 2m(Ep − E0 )=ប2 , where Ep = Ew − Eu − EM2, 3 is the secondary electron energy determined from the energy conservation law, E 0 is a correction parameter, the so-called inner potential or muffin-tin zero potential. The oscillating parts of the ‘‘surface’’ and the ‘‘bulk’’ signal (Fig. 3) were isolated using EM2, 3 = 74 eV and E 0 = 16 eV. The choice of the magnitude E 0 = 16 eV will be discussed in detail in Section 4.
According to Eq. (2) the distance to the nearest neighbours with a phase shift error can be determined by taking the Fourier transform of the extended fine structure. The Fourier transform procedure is a standard method for analysis of both EXAFS [4] and EELFS structures [2]. The magnitude of the Fourier transform (FT) of the oscillating structures plotted in Fig. 3 are given in Fig. 4. The positions of the peak maximum in the FT of both ‘‘bulk’’ (Fig. 4b) and ‘‘surface’’ (Fig. 4a) signals coincide. On the basis of the shape and the width of these FT peaks, in our opinion it is difficult to assess a clear difference between them within the distinction between the ˚) depth of the layer studied under ‘‘bulk’’ (⬃15 A ˚ and ‘‘surface’’ (⬃8 A) experimental conditions. Exponential damping of the electron wave and the presence of other amplitude components in Eq. (2) are not taken into account in the standard Fourier analysis of the oscillating part. Besides, a small
Fig. 3. (a) ‘‘Surface’’ and (b) ‘‘bulk’’ oscillating parts versus secondary electron wave number.
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into consideration the intensity of the atomic ionisation process). Furthermore, the method for solving the inverse problem by Tikhonov’s regularisation method makes it possible to find pair correlation functions for a multi-component matter under study, if results of one-component investigations are available [7]. The local atomic structure will be described by the conventional pair correlation function g(r) [7]. According to Eqs. (1) and (2), g(r) is the solution to an integral equation of the following form: 2 ⬁ 4pr0 J0 (Ep ) drg(r) 3 0 " # exp i2(p + ig)r + i2d02 × Re f (p, p) ip = J (Ep ) − Jbg (Ep )
Fig. 4. Fourier transforms of the (a) ‘‘surface’’ and (b) ‘‘bulk’’ EELFS extended fine structure of Cu(111).
energy extension of oscillating EELFS structures brings additional features in the Fourier transform. These difficulties may be avoided by solving an integral equation determined by the form of the oscillating part of Eq. (2). 4.2. Solution of the inverse problem by Tikhonov’s regularisation method To obtain an improved solution of the inverse problem we have used the technique, well developed for EXAFS [4–7,13] and already applied to EELFS [14–17], for solving the inverse problem by Tikhonov’s regularisation method (for a brief description of the regularisation method see Appendix A). Compared to the Fourier transform method, the technique is less sensitive to restrictions of the experimental result (extent, noise, errors due to the background extraction, etc.) and allows all coefficients present in the oscillating function [5,6] in Eq. (2) to be taken into account (in part, we take
3
where r 0 is the atomic density of matter, J(E p) is the intensity of energy losses obtained experimentally, and the other terms correspond to those given in Eqs. (1) and (2). As opposed to the oscillating part of Eq. (2), in the kernel of the integral equation (Eq. (3)) there is no W(p) factor because the thermal disorder of atoms in a solid is taken into account directly in the pair correlation function. Since the intensity of the oscillating structure is damped out rapidly due to damping of the intensity of the ionisation loss J 0(E p), it is not appropriate to normalise the oscillating part by division by J 0(E p). Besides, it does not always happen that we succeed in making an appropriate distinction between contributions from the process of ionisation losses and those from other processes forming the spectrum in the EELFS range, as in the present case (see Section 3). These difficulties, which emerged with consideration of J 0(E p), are important in the solution of the inverse problem. They may be obviated by using the theoretical intensity of the atomic M 2,3 ionisation process. For this purpose, we can use, for example, the approximate expression from Ref. [18]: 2 16p2 a6 4pa 2 p2 − a2 6 2 J0 (Ep ) = A 2 p + a2 (p + a2 )4 p2 + a2 2 p 2a2 +B 4 a p2 + a 2 where a is the inverse radius of the localization of the
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M 2,3 core level electron wave function. In the present paper a, along with coefficients A and B, are fit parameters. As opposed to the expression for J 0(E p) [18], a second term is added in Eq. (4). This modification is caused by the presence of zero at a 2 = p 2 in the first term in Eq. (4) that is a consequence of the use of approximate model wave functions in the calculation carried out in Ref. [18]. The added second term makes it possible to improve the agreement between the intensity J 0(E p) from Ref. [18] and the results as reported by the ab initio calculations of Aebi et al. [19]. To solve the integral equation (Eq. (3)), one should preset all parameters that form the kernel of the integral operator. Scattering amplitudes and phases were taken from Ref. [20]. The rest of the parameters of Eq. (3) are considered as fit parameters and have been chosen as follows: mh g= 2 ប p ˚ −2, A = 0.43, B = 1.28. For where h = 6 eV, a 2 = 40 A
the choice of these parameters, we have assumed that the ‘‘bulk’’ oscillating structure is formed by scattering on the nearest atomic environment corresponding to that of bulk crystalline copper. Thus, in the solution of the inverse problem, the ‘‘bulk’’ oscillating structure was the test experimental result (which led to the choice E 0 = 16 eV). For the numerical solution of Eq. (3) we have used the algorithm of the solution of the inverse problem by Tikhonov’s regularisation method [4] for the single-component EXAFS problem. In accordance with the distinction between the kernel of the integral EXAFS equation and that of Eq. (3) we have made corresponding changes in the program we used. The result of the solution of Eq. (3), the calculated pair correlation function g(r) obtained by 5% regularisation, is presented in Fig. 5 for ‘‘bulk’’ (Fig. 5b) and ‘‘surface’’ (Fig. 5a) oscillating parts, respectively, in comparison with the bulk crystallographic pair correlation function (PCF) (Fig. 5d).
5. Results and discussion
Fig. 5. Calculated pair correlation functions: (a) ‘‘surface’’ and (b) ‘‘bulk’’. (c) Reconstructed pair correlation function obtained by solving the inverse problem for the oscillating part calculated from crystallographic data within the experimental EELFS energy interval. (d) Bulk crystallographic pair correlation function.
A comparison between the Cu bulk PCF known from crystallographic data (Fig. 5d) and the calculated g(r) functions of the ‘‘bulk’’ (Fig. 5b) and the ‘‘surface’’ (Fig. 5a) shows general agreement of the PCF calculated peaks with known bulk values. However, even though Tikhonov’s regularisation method is relatively tolerant to the limitations of energy range of the experimental data, its short length does adversely affect the obtained solution, namely, in ˚ . The distortion effect of the the region from 3 to 4 A small energetic extent of the processed signal is noticeable when the crystallographic PCF is compared with the reconstructed PCF (Fig. 5c). The latter was obtained by solving the inverse problem from the oscillating part calculated by Eq. (2) with crystallographic data over the experimental range of values of the wave number. As can be seen from Fig. 5, the ˚ (in the range information in the range from 3 to 4 A of the second PCF peak position) is an artefact of the ˚) data processing. Displacement to the left (by 0.13 A of the third peak of the calculated ‘‘bulk’’ and ‘‘surface’’ PCFs compared to the crystallographic position is a result of a systematic error which arises because we have ignored the oscillating terms in the
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Fig. 6. First peak of the pair correlation functions for ‘‘surface’’ (dashed-dotted), ‘‘bulk’’ (dashed) and bulk crystallographic (solid) data.
kernel of integral Eq. (3) which are connected with consideration of monopole (3p → p) and quadrupole (3p → f) transitions from the M 2,3 core level. Nonetheless, the relative change of the maximum peak position of the ‘‘bulk’’ and ‘‘surface’’ PCF is unambiguous within the precision of Tikhonov’s solution of the inverse problem. Taking into account that 5% error in the regularisation parameters leads to an error in Tikhonov’s solution of the inverse problem ˚ , displacement of the maximum position of ⫾ 0.01 A of the third peak of the ‘‘surface’’ PCF to the left by ˚ as relative to the ‘‘bulk’’ PCF is within the 0.01 A limits of the calculation error.
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In the experimentally derived PCF the first co-ordination peak is the most reliable, as in other experimental techniques for obtaining PCF. Fig. 6 shows first co-ordination peaks of calculated ‘‘bulk’’ and ‘‘surface’’ PCFs and the crystallographic one. A comparison of results given in Fig. 6 shows that the calculated ‘‘bulk’’ result is practically in complete agreement with the crystallographic data. By contrast, the calculated ‘‘surface’’ peak has a clearly defined right-hand asymmetry and the maximum position is ˚. displaced to the left by 0.03 ⫾ 0.01 A With these results one can determine the nearest atomic environment of the surface layer of the sample averaged over the studied layer depth (⬃15 and ˚ ). Calculated structural parameters for the ⬃8 A ‘‘bulk’’ and the ‘‘surface’’, namely, the mean interatomic distance R 1 (the maximum position of the first PCF peak), the coordination number N 1 (proportional to the peak area) and the mean-square-displacement p hDR 2 i, proportional to the ratio of area to peak intensity, are given in comparison with crystallographic values in Table 1. In addition, Table 1 includes the quantity LS/RS—the numerical characteristic of asymmetry of the first PCF peak that is determined as the ratio of the area of the left (LS) to that of the right (RS) part of the peak. Obtained parameters of the nearest atomic environment of the ‘‘bulk’’ and the ‘‘surface’’ and their differences as a function of the studied layer depth ˚ ) of the single crystal Cu(111) (from ⬃15 to ⬃8 A surface decreases are in agreement with current concepts and experimental results of other investigations in the structure of the Cu(111) surface. Jona [21,22] studied the single crystal Cu(111) surface by Low Energy Electron Diffraction (LEED) and reported a ˚) decrease in the interatomic distance of 4% (⬃0.1 A compared to the bulk crystallographic value. In line with this result, the lack of change in the value of the ˚ ) interatomic ‘‘exchange’’ (averaged over ⬃15 A distance and the decrease in the ‘‘surface’’ (averaged
Table 1 Parameters of the nearest atomic environment obtained from bulk crystallographic data, ‘‘bulk’’ and ‘‘surface’’ EELFS experimental data p ˚) ˚) PCF R 1 (A hDR 2 i (A N1 LS/RS Crystallographic ‘‘Bulk’’ ‘‘Surface’’
2.556 2.56 ⫾ 0.01 2.53 ⫾ 0.01
12 12.0 ⫾ 0.1 11.2 ⫾ 0.5
0.085 0.095 ⫾ 0.010 —
1 1.0 ⫾ 0.1 0.8 ⫾ 0.1
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˚ ) result by 0.03 ⫾ 0.01 A ˚ appears to be over ⬃8 A quite reasonable. A small decrease (0.8 ⫾ 0.5) of the coordination number for the ‘‘surface’’ result compared with the ‘‘bulk’’ result is also quite reasonable. The most significant and interesting change that occurs with changes in the studied layer depth is the change in the shape of the first PCF peak. With ˚ ) studied layer depth the first PCF peak is (⬃15 A symmetric and pclose to a Gaussian function with a half-width of hDR 2 i⬃0.095. We therefore conclude that the average thermal oscillation in the outer ˚ is nearly harmonic which makes the first ⬃15 A PCF peak nearly a symmetric Gaussian. By contrast, ˚ is the first PCF peak averaged over the outer ⬃8 A asymmetric and it cannot be described by a single Gaussian function. Hence the approximation of harmonic thermal atom oscillations in thin surface layers is not valid and the use of the classic Debye– Waller factor is not appropriate in the case of describing and processing the corresponding extended fine structure. Experimentally obtained asymmetry of the ˚) first PCF peak at the small studied layer depth (⬃8 A is in good agreement with the result of the mathematical simulation of the corresponding peak derived by Clausen et al. [23] from molecular dynamic calculations on small Cu clusters with free boundary at room temperature. For a cluster consisting of 256 Cu atoms ˚ ) corresponding to a surface to (cluster radius of 16 A bulk ratio of ⬃0.2, the asymmetry of the first PCF peak was LS/RS⬃0.7. For the studied layer depth ˚ , we estimate the ratio of the number of ‘‘sur⬃8 A face’’ to ‘‘bulk’’ atoms is ⬃0.25. Thus the result of the model calculation (LS/RS⬃0.7) [23] correlates well with the experimental result (LS/RS⬃0.8 ⫾ 0.1).
˚ ). The asymmetry of the at the small thickness (⬃8 A first PCF peak obtained for the small studied layer ˚ ) may be due both to the effect of depth (⬃8 A averaging over the subsurface layers and to the influence of a greater anharmonic contribution of thermal atomic vibrations.
6. Conclusions
Appendix A.1 The Tikhonov regularisation
An EELFS experimental investigation of the nearest atomic environment in the near surface region of a (111)-oriented Cu single crystal has shown that the interatomic distance and the co-ordination number vary only slightly (⬃1%) with a decrease in the studied layer depth. By contrast, the shape of the first peak of the pair correlation function varies significantly (⬃20%) from symmetric at the large ˚ ) to highly asymmetric studied layer thickness (⬃15 A
An integral equation (Eq. (3)) for finding the atomic pair correlation function can be presented in the operator form
Acknowledgements We extend sincere thanks to Dr G.N. Konygin (PhTI UB RAS, Russia) for certifying the sample and Dr V.A. Shamin (PhTI UB RAS, Russia) for providing us with the program on the deconvolution of the experimental spectrum with the apparatus function. We are indebted also to Prof. Yu.A. Babanov (IMPh UB RAS, Russia) for providing the algorithm of the solution of the inverse problem by Tikhonov’s regularisation method and to Dr T. Tyliszczak (McMaster University, Canada) for the program of Fourier treatment of EXAFS spectra.
Appendix A Solving the inverse problem The method for solving the inverse problem (solving Eq. (3)) by the Tikhonov regularisation method is briefly described in this appendix. Although this method was described elsewhere, both purely mathematically and as applied to EXAFS (see, for example, Refs. [5–7,24,25]), this description is presented for a fundamental understanding of the method used for processing experimental data.
Ag = u
(A1)
where u is an experimental oscillating part, g is the required atomic pair correlation function. The integral operator A is determined by the following
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expression: Ag = 4pr0 J0
2 lfn (p, p)l 3 p
⬁
drexp( − 2gr)
0
× sin(2pr + vn + 2d02 )g(r)
(A2)
Eq. (3) is the Fredholm integral equation of the first kind and its solution minimises the functional of the form: kAg − uk2
(A3)
where k:::k denotes a norm of the function in space L 2. The problem involved is related to the class of inverse ill-posed problems characterised by their instability, i.e. changes of the right part of Eq. (A1) (experimental data), as small as one likes, can cause arbitrarily great changes of the required solution (an atomic pair correlation function). To solve problems of this kind the Tikhonov regularisation method can be used. The regularisation of the problem [25] consists actually in going from the functional of the form given in Eq. (A3) to that of the form: d ˜ 2 ˜ 2 + bk (g − g)k (A4) kAg − uk2 + akg − gk dr where g˜ is a trial function or zero. As a rule g˜ = 0 is chosen in the EXAFS method, as in the present paper. In the functional Eq. (A4) a and b are small positive regularisation parameters. In this case the parameter a provides continuity of the required solution, and the parameter b provides continuity of its first derivative. These requirements are natural when it is remembered that the required solution is the atomic pair correlation function. So, regularisation implies that prior information about the required solution is being taken into account. Appendix A.2 Numerical solution For the inverse problem to be solved, the collocation method is used. Then after minimisation and discretisation of the functional Eq. (A4), the required solution is the solution of the matrix equation of the following form [25]: (ATp, r Ap, r + Br, r )gr = ATp, r up
(A5)
where ATp, r is a transposed matrix, Ap, r are elements of the matrix, Ap, r are determined from Eq. (A2). Lower matrix indexes denote dimensions of this
201
matrix. The matrix Br, r being in fact the regularising matrix, takes the form: 1 0 a⬘ + 2b⬘ − b⬘ 0 ::: 0 C B C B − b⬘ a⬘ + 2b⬘ − b⬘ ::: 0 C B C B B ::: ::: ::: ::: C Br, r = B ::: C C B C B ::: ::: ::: ::: ::: A @ ::: ::: ::: ::: a⬘ + 2b⬘ (A6) b Dr ; b⬘ = DrDk ; Dr and Dk are step values where a⬘ = a Dk in r- and k-space, respectively. In the main text, regularisation parameters a and b are relative to the mean value of the diagonal element of the matrix A TA. The total matrix operator in Eq. (A5) is symmetrical, positive, definite and regular. This means that standard methods of linear algebra may be applied to solve the matrix equation Eq. (A5).
Appendix A.3 Procedure of iteration refinement The solution of Eq. (A5) has, as a rule, both positive and negative (essentially less intense) values. Negative values in the required solution result first of all from the fact that the kernel of integral equation Eq. (A2) does not fully correlate with the physical process that forms the experimental result. Refinement of the solution obtained, namely fulfilment of the condition of non-negativity required by the nature of the atomic pair correlation function, is performed by an iteration procedure of the following form [25]: ÿ − 1ÿ T (A7) Ap, r up + Br, r gr(n) gr(n + 1) = P ATp, r AP, r + Br, r where P is the projection operator, determined in the following way: ( gr , if gr ⱖ 0 (A8) Pgr = 0, if gr ⬍ 0 From experience of solving equations of the kind given in Eq. (A7) it follows that the iteration procedure allows the required solution to be improved by carrying out a few iterations. The main advantage of the Tikhonov regularisation method over other methods for obtaining parameters of the nearest atomic environment from experimental extended fine structures consists of the fact that in
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this case the inverse problem is solved directly. Moreover, unlike the conventional Fourier method, in the regularisation method all parameters of the physical process which form the oscillating structure may be directly taken into account in the kernel of an integral operator. This makes it possible to do away with the curve fitting procedure that is commonly used in the inverse Fourier method. Besides, the regularisation method allows us to formalise and obtain the correct solution of the integral equation mathematically. In closing it should be pointed out that at the present time the level of the mathematical progress of these methods [25–28] is much ahead of the level of their practical implementation, specifically in EXAFS and EXAFS-like methods. This is connected with the fact that these methods are complicated and difficult in practice. However it seems likely that the development of the extended fine structure methods and the expansion of the fields of their application to complex multi-component compounds and surface analysis will inevitably lead to the necessity of using regularisation algorithms for analysis of the experimental data. References [1] P.A. Lee, H.Y. Citrin, P. Eisenberger et al., Rev. Mod. Phys. 53 (1981) 769. [2] M. DeCrescenzi, Surf. Sci. Reports 21 (1995) 89. [3] E.A. Stern, Journal de Physique 12 (1986) C8–3. [4] F.W. Lytle, D.E. Sayers, E.A. Stern, Phys. Rev. B 11 (1975) 4825. [5] Yu.A. Babanov, V.V. Vasin, A.L. Ageev, N.V. Ershov, Phys. Stat. Sol. (b) 105 (1981) 747.
[6] N.V. Ershov, A.L. Ageev, V.V. Vasin, Yu.A. Babanov, Phys. Stat. Sol. (b) 111 (1981) 103. [7] Yu.A. Babanov, N.V. Ershov, V.R. Shvetsov et al., J. NonCryst. Solids 79 (1986) 1. [8] K.N. Eltsov, A.N. Klimov, S.L. Priadkin, V.M. Shevlyuga, V.Yu. Yurov, Phys. Low-Dim. Struct. 7/8 (1996) 115. [9] D.F. Stein, A. Joshi, Ann. Rev. Mater. Sci. 11 (1981) 485. [10] B. Luo, J. Urban, Surf. Sci. 239 (1990) 235. [11] M. DeCrescenzi, L. Papagno, G. Chiarello et al., Solid State Commun. 40 (1981) 613. [12] A.P. Hitchcock, C.H. Teng, Surf. Sci. 149 (1985) 558. [13] N.V. Ershov, Yu.A. Babanov, V.R. Galakhov, Phys. Stat. Sol. (b) 117 (1983) 749. [14] A.H. Kadikova, Yu.V. Ruts, N.V. Ershov et al., Solid State Commun. 82 (1992) 871. [15] A.H. Kadikova, N.V. Ershov, A.L. Ageev et al., Phys. Stat. Sol. (a) 122 (1990) K143. [16] D.V. Surnin, D.E. Denisov, Yu.V. Ruts, J. de Physique IV 7 (1997) C2–577. [17] D.V. Surnin, D.E. Denisov, Yu.V. Ruts, Surf. Rev. and Lett. 4 (1997) 556. [18] D.E. Guy, Y.V. Ruts, D.V. Surnin et al., Surf. Rev. Lett. 4 (1997) 223. [19] P. Aebi, M. Erbudak, F. Vanini et al., Phys. Rev. B 41 (1990) 11760. [20] B.K. Teo, P.A. Lee, J. Am. Chem. Soc. 101 (1979) 2815. [21] F. Jona, J. Phys. C: Solid State Phys. 11 (1978) 4271. [22] F. Jona, J.A.Stozier Jr, W.S. Yang, Rep. Prog. Phys. 45 (1982) 527. [23] B.S. Clausen, H. Topsoe, L.B. Hansen, et al., Jpn. J. Appl. Phys. 32, Suppl. 32-2 (1993) 95. [24] A.N. Tikhonov, V.Ya. Arsenin, in: J. Filtz (Eds), Methods of Solution of Ill-posed Problem, John Wiley and Sons, New York–Toronto–London, 1977, 287 pp. [25] V.V. Vasin, A.L. Ageev, Ill-posed Problems with A Priori Information, VSP, Utrecht, 1995, 255 pp. [26] A.B. Bakushinsky, A.V. Goncharsky, Ill-posed Problems: Theory and Applications, Kluver, Amsterdam, 1994, 256 pp. [27] A. Neubauer, O.A. Scherzer, J. Anal. Appl. 2 (1995) 369. [28] A.L. Ageev, E.V. Voronina, Nucl. Instr. Meth. 108 (1996) 417.
The local atomic structure of Cu(111): An extended energy loss fine structure investigation D.V. Surnin*, A.N. Deev, D.E. Guy, Yu.V. Ruts Physical–Technical Institute UB RAS and Udmurt State University, Kirov St. 132, Izhevsk 426001, Russia Received 17 October 1997; revised 31 March 1998; accepted 15 June 1998
Abstract Extended energy loss fine structures spectra (EELFS) above the M 2,3 ionisation edge of a Cu(111) single crystal have been ˚ . For obtained at primary electron energies of 1500 and 560 eV, corresponding to a studied layer depth of ⬃15 and ⬃8 A experimental EELFS spectra the solution of an inverse problem has been performed and pair correlation functions (PCFs) have been obtained by Tikhonov’s regularisation method with allowance made for normalisation of the atomic intensity of ionisation losses. Parameters of the nearest atomic environment, namely, the bond length, the co-ordination number, the Debye–Waller factor and the parameter of asymmetry of the first peak of the pair correlation function, have been determined ˚ ) and small (⬃8 A ˚ ) depths of the studied layer. 䉷 1998 Elsevier Science B.V. All rights reserved. for both large (⬃15 A Keywords: EELFS; EXAFS; PCF
1. Introduction The experimental determination of the atomic structure of matter is a traditional problem in solid state physics. Methods of electron and X-ray diffraction have had a dominant role in these investigations over a long period. Though traditional diffraction methods are still in use, over the last few decades there has been interesting developments in spectroscopic methods for studying the local atomic structure that are based on analysis of extended oscillating structures of spectra. Among these we quote EXAFS (Extended X-ray Absorption Fine Structure) [1], EELFS (Electron Energy Loss Fine Structure) [2] and other EXAFS-like methods [3]. By extended fine structures we mean oscillating * Corresponding author. Fax: + 007 3412 237901; E-mail: [email protected]
spectral features (with a period of oscillation of tens of electronvolts) located, as a rule, above the core ionisation threshold (the ionisation loss edge) and with an extension of several hundred electronvolts in EELFS, and more than 1 keV in EXAFS. Spectroscopic methods turn out to be most useful in studying the atomic structure of disordered multi-component materials, since these methods make possible analysis of the local atomic environment around an atom of specific chemical species [4]. Besides, spectroscopic methods allow for the mathematical formalisation of the inverse problem to determine the pair correlation function (PCF) from the experimental data [5,6], with the form of the inverse problem and its solution being independent of both chemical composition and the atomic structure of the matter studied [7]. Among spectroscopic methods, the EXAFS method is the oldest known and best developed. It is currently used as a standard method for the analysis of local
0368-2048/98/$ - see front matter 䉷 1998 Elsevier Science B.V. All rights reserved. PII: S 03 68 - 20 4 8( 9 8) 0 02 1 4- X
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atomic structure. Among more recent alternative methods, the EELFS technique is the one best developed. Compared to EXAFS, the EELFS method has both advantages and disadvantages. Advantages of EELFS are: (1) availability—one can obtain spectra using a conventional Auger spectrometer; (2) high degree of spatial localisation of the region under study; (3) surface sensitivity—the possibility for variation of the layer depth studied. The main disadvantages of EELFS are: (1) low intensity of the structure compared to the total signal intensity and thus complexity of extracting the oscillating structure from the experimental data; (2) the small extent of the oscillations that is determined by damping of the atomic intensity of ionisation losses. The latter leads to great difficulties in the solution of the inverse problem. In the present work the local atomic structure of a Cu single crystal (111) oriented surface has been investigated by the EELFS technique for different depths of the studied layer. Appropriate normalised pair correlation functions and parameters of the nearest atomic environment have been obtained by Tikhonov’s regularisation method. Our results are compared with those obtained by other authors with the help of both experimental methods and model calculations. Section 2 describes the known EELFS theoretical formalism in the EXAFS-like form for the case of the Cu M 2,3 ionisation loss edge. The experimental method for obtaining electron energy loss spectra and the procedure for extracting the oscillating part from the experimental data are given in Section 3. The Fourier-transform and Tikhonov’s regularisation methods used to analyse the extended electron energy loss fine structure signal are presented in Section 4. Discussion of the experimental results and comparison between the derived nearest atomic environment parameters and those obtained by other researchers are presented in Section 5.
2. EELFS formalism The extended fine structure above the edge of the ionisation loss is formed as a result of coherent scattering of the secondary electron on atoms of the nearest environment of an ionised atom. For the
structure above the M 2,3 edge of the Cu ionisation, the expression determining the intensity of this process has the form: JM2, 3 (Ep ) = J0 (Ep )[1 + x(p)]
(1)
The intensity of the process in Eq. (1) is presented as a function of E p (secondary electron energy). The secondary electron energy is determined by the energy conservation law: Ep = Ew − Eu − EM2, 3 , where E w is the incident electron energy, E u is the energy of an inelastically scattered electron registered by experiment, EM2, 3 is the M 2,3 core level electron binding energy. Taking into account only single scattering of the secondary electron on the nearest environment of an ionised atom brings the structural term in Eq. (1) to the EXAFS-like form [2]: ( ) exp i2(p + ig)R j 2 0 x(p) = Re exp(i2d2 )W (p)fj (p, p) 3 ipR 2j (2) where R j is the distance between ionised and neighbouring jth atoms, p is the wave number of the secondary electron, f j(p,p) is the back scattering amplitude of the secondary electron on the neighbouring jth atom, d02 is the phase shift resulting from the electron scattering on an ionised atom (the 2 corresponds to the d-symmetry of the secondary electron wave function), W(p) is the factor taking account of atom thermal vibrations. Summation over j (for all atoms of the environment of an ionised atom) is implied in Eq. (2). In the structural term of Eq. (2) it is assumed that the symmetry of the wave function of the secondary electron is d-type (l = 2), since 3p → ed excitation should be more intense than 3p → es (l = 0) in M 2,3 core level excitation. In Eq. (1) the processes of coherent scattering of both incident and inelastically scattered electrons are neglected. This approximation is true due to the fact that kinetic energies of these electrons far exceed the energy of the secondary electron. Besides the process of the energy loss of an incident electron by ionisation of the M 2,3 core level (Eq. (1)), processes of the energy loss of an incident electron by multiple excitation of electrons of the valence band make a contribution to this region of the spectrum. The latter gives the most intense but non-oscillating (smooth) component of the spectrum. Thus, the total
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non-oscillating part of the EELFS spectrum, J bg is the sum of the intensity of the atomic process of the M 2,3 core level ionisation loss (J 0) and the intensity of all processes of multiple losses by excitation of valence electrons.
3. Experiment and pre-treatment of data Electron energy loss spectra from the single crystal Cu(111) surface at room temperature have been obtained in a back scattering geometry with a JAMP-10S (JEOL) Auger electron spectrometer. The vacuum in the analysing chamber was not worse than 10 −7 Pa. The sample surface was precleaned with pure alcohol and acetone in an ultrasonic bath. In the analysing chamber the surface was cleaned by Ar + ion beam bombardment (0.5 keV). Surface cleanliness was monitored by Auger electron spectroscopy. During the whole experiment the density of impurity atoms on the surface did not exceed 1 at.%. In spite of the fact that during the experiment the crystallinity of the surface was not controlled, the use of the low-energy ion beam should ensure that the flatness and long range order of the Cu(111) surface is not grossly changed as shown by a number of investigations of the effect of low-energy ion beams on single crystal surfaces (see, for example Ref. [8]). Besides, it should be remembered that in these experiments we gain information, averaged over the depth of the analysed layer, about the local atomic environment which is insensitive to the crystallographic long range order of the sample. From the above considerations, we have assumed that the Cu surface under study is an ideal Cu(111) surface from a local structure perspective. Electron energy loss spectra were obtained at incident beam electron energies of 1500 and 560 eV in the integral mode (DE/E = 0.4%). Spectra were taken with a 1 eV step. Each spectrum was a summation of 1000 scans where each scan was obtained with 1000 samplings at each energy point. It is possible to estimate the studied layer depth from the mean free path of electrons registered in the experiment. For the spectrum obtained at 1500 eV excitation energy, we estimate that the ˚ while for 560 eV studied layer depth is ⬃15 A ˚ excitation energy the studied layer depth is ⬃8 A
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[9]. The information in the electron spectrum represents an average over the studied layer depth. Keeping this in mind, the structural results obtained from an electron energy loss spectrum taken at the excitation energy of 1500 eV will be termed ‘‘bulk’’ and the results obtained from the spectrum at the excitation energy of 560 eV will be considered as ‘‘surface’’. Initial ‘‘bulk’’ and ‘‘surface’’ spectra and corresponding oscillating parts are plotted on the scale of incident electron energy losses in Figs 1 and 2, respectively. EELFS oscillating parts were obtained from the recorded spectra by subtracting a non-oscillating part, J bg, which was approximated by fitting a polynomial curve to the data. The extended fine structure signals plotted in Fig. 1(b) and Fig. 2(b) are normalised to the unit value of the Cu M 2,3 edge jump. The recorded spectra were deconvoluted with the apparatus function, which was approximated by a Gaussian function with half-width determined from
Fig. 1. Electron energy loss spectrum (a) and corresponding M 2,3 extended energy loss fine structure (b) of Cu(111) which was obtained at the incident beam electron energy of 1500 eV.
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4. Solution of the inverse problem and discussion of results 4.1. Fourier-transformation
Fig. 2. Electron energy loss spectrum (a) and corresponding M 2,3 extended energy loss fine structure (b) of Cu(111) which was obtained at the incident beam electron energy of 560 eV.
DE/E for the apparatus. The derived oscillating parts of the Cu EELFS spectra are in good agreement with experimental results obtained by other investigators [2,10–12]. To solve the inverse problem to determine the characteristics of the nearest atomic environment of an ionised atom, there is a need to convert the oscillating parts of the EELFS spectra to the space of the secondary electron wave numbers. The ˚ −1) of the secondary electron (stanwave number (in A dardq for all EXAFS-like techniques) is p = 2m(Ep − E0 )=ប2 , where Ep = Ew − Eu − EM2, 3 is the secondary electron energy determined from the energy conservation law, E 0 is a correction parameter, the so-called inner potential or muffin-tin zero potential. The oscillating parts of the ‘‘surface’’ and the ‘‘bulk’’ signal (Fig. 3) were isolated using EM2, 3 = 74 eV and E 0 = 16 eV. The choice of the magnitude E 0 = 16 eV will be discussed in detail in Section 4.
According to Eq. (2) the distance to the nearest neighbours with a phase shift error can be determined by taking the Fourier transform of the extended fine structure. The Fourier transform procedure is a standard method for analysis of both EXAFS [4] and EELFS structures [2]. The magnitude of the Fourier transform (FT) of the oscillating structures plotted in Fig. 3 are given in Fig. 4. The positions of the peak maximum in the FT of both ‘‘bulk’’ (Fig. 4b) and ‘‘surface’’ (Fig. 4a) signals coincide. On the basis of the shape and the width of these FT peaks, in our opinion it is difficult to assess a clear difference between them within the distinction between the ˚) depth of the layer studied under ‘‘bulk’’ (⬃15 A ˚ and ‘‘surface’’ (⬃8 A) experimental conditions. Exponential damping of the electron wave and the presence of other amplitude components in Eq. (2) are not taken into account in the standard Fourier analysis of the oscillating part. Besides, a small
Fig. 3. (a) ‘‘Surface’’ and (b) ‘‘bulk’’ oscillating parts versus secondary electron wave number.
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into consideration the intensity of the atomic ionisation process). Furthermore, the method for solving the inverse problem by Tikhonov’s regularisation method makes it possible to find pair correlation functions for a multi-component matter under study, if results of one-component investigations are available [7]. The local atomic structure will be described by the conventional pair correlation function g(r) [7]. According to Eqs. (1) and (2), g(r) is the solution to an integral equation of the following form: 2 ⬁ 4pr0 J0 (Ep ) drg(r) 3 0 " # exp i2(p + ig)r + i2d02 × Re f (p, p) ip = J (Ep ) − Jbg (Ep )
Fig. 4. Fourier transforms of the (a) ‘‘surface’’ and (b) ‘‘bulk’’ EELFS extended fine structure of Cu(111).
energy extension of oscillating EELFS structures brings additional features in the Fourier transform. These difficulties may be avoided by solving an integral equation determined by the form of the oscillating part of Eq. (2). 4.2. Solution of the inverse problem by Tikhonov’s regularisation method To obtain an improved solution of the inverse problem we have used the technique, well developed for EXAFS [4–7,13] and already applied to EELFS [14–17], for solving the inverse problem by Tikhonov’s regularisation method (for a brief description of the regularisation method see Appendix A). Compared to the Fourier transform method, the technique is less sensitive to restrictions of the experimental result (extent, noise, errors due to the background extraction, etc.) and allows all coefficients present in the oscillating function [5,6] in Eq. (2) to be taken into account (in part, we take
3
where r 0 is the atomic density of matter, J(E p) is the intensity of energy losses obtained experimentally, and the other terms correspond to those given in Eqs. (1) and (2). As opposed to the oscillating part of Eq. (2), in the kernel of the integral equation (Eq. (3)) there is no W(p) factor because the thermal disorder of atoms in a solid is taken into account directly in the pair correlation function. Since the intensity of the oscillating structure is damped out rapidly due to damping of the intensity of the ionisation loss J 0(E p), it is not appropriate to normalise the oscillating part by division by J 0(E p). Besides, it does not always happen that we succeed in making an appropriate distinction between contributions from the process of ionisation losses and those from other processes forming the spectrum in the EELFS range, as in the present case (see Section 3). These difficulties, which emerged with consideration of J 0(E p), are important in the solution of the inverse problem. They may be obviated by using the theoretical intensity of the atomic M 2,3 ionisation process. For this purpose, we can use, for example, the approximate expression from Ref. [18]: 2 16p2 a6 4pa 2 p2 − a2 6 2 J0 (Ep ) = A 2 p + a2 (p + a2 )4 p2 + a2 2 p 2a2 +B 4 a p2 + a 2 where a is the inverse radius of the localization of the
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M 2,3 core level electron wave function. In the present paper a, along with coefficients A and B, are fit parameters. As opposed to the expression for J 0(E p) [18], a second term is added in Eq. (4). This modification is caused by the presence of zero at a 2 = p 2 in the first term in Eq. (4) that is a consequence of the use of approximate model wave functions in the calculation carried out in Ref. [18]. The added second term makes it possible to improve the agreement between the intensity J 0(E p) from Ref. [18] and the results as reported by the ab initio calculations of Aebi et al. [19]. To solve the integral equation (Eq. (3)), one should preset all parameters that form the kernel of the integral operator. Scattering amplitudes and phases were taken from Ref. [20]. The rest of the parameters of Eq. (3) are considered as fit parameters and have been chosen as follows: mh g= 2 ប p ˚ −2, A = 0.43, B = 1.28. For where h = 6 eV, a 2 = 40 A
the choice of these parameters, we have assumed that the ‘‘bulk’’ oscillating structure is formed by scattering on the nearest atomic environment corresponding to that of bulk crystalline copper. Thus, in the solution of the inverse problem, the ‘‘bulk’’ oscillating structure was the test experimental result (which led to the choice E 0 = 16 eV). For the numerical solution of Eq. (3) we have used the algorithm of the solution of the inverse problem by Tikhonov’s regularisation method [4] for the single-component EXAFS problem. In accordance with the distinction between the kernel of the integral EXAFS equation and that of Eq. (3) we have made corresponding changes in the program we used. The result of the solution of Eq. (3), the calculated pair correlation function g(r) obtained by 5% regularisation, is presented in Fig. 5 for ‘‘bulk’’ (Fig. 5b) and ‘‘surface’’ (Fig. 5a) oscillating parts, respectively, in comparison with the bulk crystallographic pair correlation function (PCF) (Fig. 5d).
5. Results and discussion
Fig. 5. Calculated pair correlation functions: (a) ‘‘surface’’ and (b) ‘‘bulk’’. (c) Reconstructed pair correlation function obtained by solving the inverse problem for the oscillating part calculated from crystallographic data within the experimental EELFS energy interval. (d) Bulk crystallographic pair correlation function.
A comparison between the Cu bulk PCF known from crystallographic data (Fig. 5d) and the calculated g(r) functions of the ‘‘bulk’’ (Fig. 5b) and the ‘‘surface’’ (Fig. 5a) shows general agreement of the PCF calculated peaks with known bulk values. However, even though Tikhonov’s regularisation method is relatively tolerant to the limitations of energy range of the experimental data, its short length does adversely affect the obtained solution, namely, in ˚ . The distortion effect of the the region from 3 to 4 A small energetic extent of the processed signal is noticeable when the crystallographic PCF is compared with the reconstructed PCF (Fig. 5c). The latter was obtained by solving the inverse problem from the oscillating part calculated by Eq. (2) with crystallographic data over the experimental range of values of the wave number. As can be seen from Fig. 5, the ˚ (in the range information in the range from 3 to 4 A of the second PCF peak position) is an artefact of the ˚) data processing. Displacement to the left (by 0.13 A of the third peak of the calculated ‘‘bulk’’ and ‘‘surface’’ PCFs compared to the crystallographic position is a result of a systematic error which arises because we have ignored the oscillating terms in the
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Fig. 6. First peak of the pair correlation functions for ‘‘surface’’ (dashed-dotted), ‘‘bulk’’ (dashed) and bulk crystallographic (solid) data.
kernel of integral Eq. (3) which are connected with consideration of monopole (3p → p) and quadrupole (3p → f) transitions from the M 2,3 core level. Nonetheless, the relative change of the maximum peak position of the ‘‘bulk’’ and ‘‘surface’’ PCF is unambiguous within the precision of Tikhonov’s solution of the inverse problem. Taking into account that 5% error in the regularisation parameters leads to an error in Tikhonov’s solution of the inverse problem ˚ , displacement of the maximum position of ⫾ 0.01 A of the third peak of the ‘‘surface’’ PCF to the left by ˚ as relative to the ‘‘bulk’’ PCF is within the 0.01 A limits of the calculation error.
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In the experimentally derived PCF the first co-ordination peak is the most reliable, as in other experimental techniques for obtaining PCF. Fig. 6 shows first co-ordination peaks of calculated ‘‘bulk’’ and ‘‘surface’’ PCFs and the crystallographic one. A comparison of results given in Fig. 6 shows that the calculated ‘‘bulk’’ result is practically in complete agreement with the crystallographic data. By contrast, the calculated ‘‘surface’’ peak has a clearly defined right-hand asymmetry and the maximum position is ˚. displaced to the left by 0.03 ⫾ 0.01 A With these results one can determine the nearest atomic environment of the surface layer of the sample averaged over the studied layer depth (⬃15 and ˚ ). Calculated structural parameters for the ⬃8 A ‘‘bulk’’ and the ‘‘surface’’, namely, the mean interatomic distance R 1 (the maximum position of the first PCF peak), the coordination number N 1 (proportional to the peak area) and the mean-square-displacement p hDR 2 i, proportional to the ratio of area to peak intensity, are given in comparison with crystallographic values in Table 1. In addition, Table 1 includes the quantity LS/RS—the numerical characteristic of asymmetry of the first PCF peak that is determined as the ratio of the area of the left (LS) to that of the right (RS) part of the peak. Obtained parameters of the nearest atomic environment of the ‘‘bulk’’ and the ‘‘surface’’ and their differences as a function of the studied layer depth ˚ ) of the single crystal Cu(111) (from ⬃15 to ⬃8 A surface decreases are in agreement with current concepts and experimental results of other investigations in the structure of the Cu(111) surface. Jona [21,22] studied the single crystal Cu(111) surface by Low Energy Electron Diffraction (LEED) and reported a ˚) decrease in the interatomic distance of 4% (⬃0.1 A compared to the bulk crystallographic value. In line with this result, the lack of change in the value of the ˚ ) interatomic ‘‘exchange’’ (averaged over ⬃15 A distance and the decrease in the ‘‘surface’’ (averaged
Table 1 Parameters of the nearest atomic environment obtained from bulk crystallographic data, ‘‘bulk’’ and ‘‘surface’’ EELFS experimental data p ˚) ˚) PCF R 1 (A hDR 2 i (A N1 LS/RS Crystallographic ‘‘Bulk’’ ‘‘Surface’’
2.556 2.56 ⫾ 0.01 2.53 ⫾ 0.01
12 12.0 ⫾ 0.1 11.2 ⫾ 0.5
0.085 0.095 ⫾ 0.010 —
1 1.0 ⫾ 0.1 0.8 ⫾ 0.1
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˚ ) result by 0.03 ⫾ 0.01 A ˚ appears to be over ⬃8 A quite reasonable. A small decrease (0.8 ⫾ 0.5) of the coordination number for the ‘‘surface’’ result compared with the ‘‘bulk’’ result is also quite reasonable. The most significant and interesting change that occurs with changes in the studied layer depth is the change in the shape of the first PCF peak. With ˚ ) studied layer depth the first PCF peak is (⬃15 A symmetric and pclose to a Gaussian function with a half-width of hDR 2 i⬃0.095. We therefore conclude that the average thermal oscillation in the outer ˚ is nearly harmonic which makes the first ⬃15 A PCF peak nearly a symmetric Gaussian. By contrast, ˚ is the first PCF peak averaged over the outer ⬃8 A asymmetric and it cannot be described by a single Gaussian function. Hence the approximation of harmonic thermal atom oscillations in thin surface layers is not valid and the use of the classic Debye– Waller factor is not appropriate in the case of describing and processing the corresponding extended fine structure. Experimentally obtained asymmetry of the ˚) first PCF peak at the small studied layer depth (⬃8 A is in good agreement with the result of the mathematical simulation of the corresponding peak derived by Clausen et al. [23] from molecular dynamic calculations on small Cu clusters with free boundary at room temperature. For a cluster consisting of 256 Cu atoms ˚ ) corresponding to a surface to (cluster radius of 16 A bulk ratio of ⬃0.2, the asymmetry of the first PCF peak was LS/RS⬃0.7. For the studied layer depth ˚ , we estimate the ratio of the number of ‘‘sur⬃8 A face’’ to ‘‘bulk’’ atoms is ⬃0.25. Thus the result of the model calculation (LS/RS⬃0.7) [23] correlates well with the experimental result (LS/RS⬃0.8 ⫾ 0.1).
˚ ). The asymmetry of the at the small thickness (⬃8 A first PCF peak obtained for the small studied layer ˚ ) may be due both to the effect of depth (⬃8 A averaging over the subsurface layers and to the influence of a greater anharmonic contribution of thermal atomic vibrations.
6. Conclusions
Appendix A.1 The Tikhonov regularisation
An EELFS experimental investigation of the nearest atomic environment in the near surface region of a (111)-oriented Cu single crystal has shown that the interatomic distance and the co-ordination number vary only slightly (⬃1%) with a decrease in the studied layer depth. By contrast, the shape of the first peak of the pair correlation function varies significantly (⬃20%) from symmetric at the large ˚ ) to highly asymmetric studied layer thickness (⬃15 A
An integral equation (Eq. (3)) for finding the atomic pair correlation function can be presented in the operator form
Acknowledgements We extend sincere thanks to Dr G.N. Konygin (PhTI UB RAS, Russia) for certifying the sample and Dr V.A. Shamin (PhTI UB RAS, Russia) for providing us with the program on the deconvolution of the experimental spectrum with the apparatus function. We are indebted also to Prof. Yu.A. Babanov (IMPh UB RAS, Russia) for providing the algorithm of the solution of the inverse problem by Tikhonov’s regularisation method and to Dr T. Tyliszczak (McMaster University, Canada) for the program of Fourier treatment of EXAFS spectra.
Appendix A Solving the inverse problem The method for solving the inverse problem (solving Eq. (3)) by the Tikhonov regularisation method is briefly described in this appendix. Although this method was described elsewhere, both purely mathematically and as applied to EXAFS (see, for example, Refs. [5–7,24,25]), this description is presented for a fundamental understanding of the method used for processing experimental data.
Ag = u
(A1)
where u is an experimental oscillating part, g is the required atomic pair correlation function. The integral operator A is determined by the following
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expression: Ag = 4pr0 J0
2 lfn (p, p)l 3 p
⬁
drexp( − 2gr)
0
× sin(2pr + vn + 2d02 )g(r)
(A2)
Eq. (3) is the Fredholm integral equation of the first kind and its solution minimises the functional of the form: kAg − uk2
(A3)
where k:::k denotes a norm of the function in space L 2. The problem involved is related to the class of inverse ill-posed problems characterised by their instability, i.e. changes of the right part of Eq. (A1) (experimental data), as small as one likes, can cause arbitrarily great changes of the required solution (an atomic pair correlation function). To solve problems of this kind the Tikhonov regularisation method can be used. The regularisation of the problem [25] consists actually in going from the functional of the form given in Eq. (A3) to that of the form: d ˜ 2 ˜ 2 + bk (g − g)k (A4) kAg − uk2 + akg − gk dr where g˜ is a trial function or zero. As a rule g˜ = 0 is chosen in the EXAFS method, as in the present paper. In the functional Eq. (A4) a and b are small positive regularisation parameters. In this case the parameter a provides continuity of the required solution, and the parameter b provides continuity of its first derivative. These requirements are natural when it is remembered that the required solution is the atomic pair correlation function. So, regularisation implies that prior information about the required solution is being taken into account. Appendix A.2 Numerical solution For the inverse problem to be solved, the collocation method is used. Then after minimisation and discretisation of the functional Eq. (A4), the required solution is the solution of the matrix equation of the following form [25]: (ATp, r Ap, r + Br, r )gr = ATp, r up
(A5)
where ATp, r is a transposed matrix, Ap, r are elements of the matrix, Ap, r are determined from Eq. (A2). Lower matrix indexes denote dimensions of this
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matrix. The matrix Br, r being in fact the regularising matrix, takes the form: 1 0 a⬘ + 2b⬘ − b⬘ 0 ::: 0 C B C B − b⬘ a⬘ + 2b⬘ − b⬘ ::: 0 C B C B B ::: ::: ::: ::: C Br, r = B ::: C C B C B ::: ::: ::: ::: ::: A @ ::: ::: ::: ::: a⬘ + 2b⬘ (A6) b Dr ; b⬘ = DrDk ; Dr and Dk are step values where a⬘ = a Dk in r- and k-space, respectively. In the main text, regularisation parameters a and b are relative to the mean value of the diagonal element of the matrix A TA. The total matrix operator in Eq. (A5) is symmetrical, positive, definite and regular. This means that standard methods of linear algebra may be applied to solve the matrix equation Eq. (A5).
Appendix A.3 Procedure of iteration refinement The solution of Eq. (A5) has, as a rule, both positive and negative (essentially less intense) values. Negative values in the required solution result first of all from the fact that the kernel of integral equation Eq. (A2) does not fully correlate with the physical process that forms the experimental result. Refinement of the solution obtained, namely fulfilment of the condition of non-negativity required by the nature of the atomic pair correlation function, is performed by an iteration procedure of the following form [25]: ÿ − 1ÿ T (A7) Ap, r up + Br, r gr(n) gr(n + 1) = P ATp, r AP, r + Br, r where P is the projection operator, determined in the following way: ( gr , if gr ⱖ 0 (A8) Pgr = 0, if gr ⬍ 0 From experience of solving equations of the kind given in Eq. (A7) it follows that the iteration procedure allows the required solution to be improved by carrying out a few iterations. The main advantage of the Tikhonov regularisation method over other methods for obtaining parameters of the nearest atomic environment from experimental extended fine structures consists of the fact that in
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