- Email: [email protected]
X-ray diffraction study of the 5-fold plane surface of a Al70.4Pd21.4Mn8.2 single-grain
Physica B 283 (2000) 79}83
X-ray di!raction study of the 5-fold plane surface of a Al Pd Mn single-grain M.J. Capitan , Y. Calvayrac, D. Gratias, J. Alvarez *
Departmento Fn& sica de la Materia Condensada C-3, Universidad Autonoma de Madrid, 28049 Madrid, Spain CECM-CNRS, 15 rue Georges Urbain, 94407 Vitry s/Seine, France LEM-CNRS/ONERA, BP 72, 9232 Chatillon Cedex, France
Abstract The structure of a 5-fold symmetry quasicrystal surface of a thermodynamically stable icosahedral Al Pd Mn single-grain has been studied by means of the surface X-ray di!raction technique. The model used reproduces rather well both the Bragg peaks and the surface signal. The parameters obtained from the "t of the measured structure factor show a contraction of the outermost atomic layer. The surface also tends to be richer in A1 and Pd atoms forming a very dense outer-layer. 2000 Elsevier Science B.V. All rights reserved. PACS: 61.44.Br; 68.35.Bs; 61.10!i Keywords: Surface X-ray di!raction; Quasicrystals; Surface structure
1. Introduction After the discovery of the quasicrystal phases in 1984 [1] great e!ort has been made in order to determine the bulk structure of these quasiperiodic systems. Although the bulk structure is beginning to be well known [2,3], the surface structure of these alloys remains generally unknown. However, most of the properties of interest for practical uses of these materials (such as low friction coe$cient, good wear [4] and corrosion [5] resistances) are largely determined by phenomena related to the surface and interface regions. This is the main reason for the intensive studies on quasicrystal surface in the last few years. An important goal is to know if the properties mentioned have a relationship with the quasicrystalline structure, i.e. if the quasicrystalline structure is preserved up to the surface. Di!erent techniques have been applied to the study of the quasicrystal surface [6}14]. Although these
* Corresponding author. Tel.: #34-913975550. E-mail address: [email protected] (J. Alvarez)
techniques have helped to improve our knowledge of the quasicrystal surface a convincing model of the surface structure is still lacking. In this paper we propose the use of the surface X-ray di!raction technique for characterising the quasicrystal surface structure. We develop a theoretical formalism for the surface X-ray di!raction of quasiperiodic systems and we compare the model with the experimental results.
2. Surface di4raction amplitude calculation We are going to show in this paper that the quasicrystal surface structure can be obtained through the interpretation of the crystal truncation rod (CTR) intensities. CTRs are rods of intensity in reciprocal space parallel to the sample surface normal that result from the truncation of the in"nite bulk crystal [15]. Indexing the 5-fold symmetry axis perpendicular to the surface of our sample as [0 0 1], the CTRs intensities are then parallel to the (0 0 L) direction in the reciprocal space. The left-hand side of Fig. 1 presents an outline of the Bragg peak distribution characteristic of the icosahedral structure relative to this 5-fold plane surface.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 8 9 6 - 7
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Fig. 1. Outline of the reciprocal space corresponding to an icosahedral phase with respect to the 5-fold surface of the sample. It has been shadowed the measured region close to the (0 0 L) region which is represented on right side of "gure. It can be observed in the measured intensity the presence of an `extraa intensity which links the Bragg peaks corresponding to the quasicrystal phase.
The quasicrystal structure is characterised by the absence of a periodicity in the physical (3D) space. Thus, the quasicrystal structure can only be described within a periodic frame by increasing the dimension of the used space. It is then necessary to use a 6D space (the 3D real space#a 3D complementary space) for describing periodically an icosahedral structure. This quasiperiodicity has a direct consequence in the CTR calculation. The CTR intensity comes from the contribution of the atomic layers parallel to the surface plane [15,19]. In the case of a periodic structure, it is possible to factorise the layers contribution because of the existence of a lattice unit along the normal direction (even in the case of very complicated structures such as the untwinned YBa Cu O [16]). As it has been formerly said it does \d not exist in the quasicrystal as a lattice unit along any direction of the real space. That implies that, in the quasicrystals, it should be necessary to add the contribution of each individual atomic layer up to the in"nite. The CTR calculation as in"nite sum has two problems; "rst, it has associated with it a large computational error and, second, it is necessary to have an exact atomic layer
description in the real space. This last point is especially critical in the case of treating a lower symmetry direction, i.e. all the CTRs di!erent from the (0 0 L). Thus, we introduce a higher-dimensional space, used for describing the quasicrystal bulk [17,18], for calculating a CTR intensity within the Robinson [15] and Andrews and Cowley [19] frame [16]. To our knowledge, this is the "rst time that for the quasicrystals truncation is reported in the literature. Although in the X-ray surface di!raction experiments one generally adopts the convention of decomposing the momentum transfer vector into two components, parallel and perpendicular to the surface, this is in con#ict with the normal nomenclature used for describing quasicrystals within a higher-dimensional space (what is called the real or parallel 3D spaces plus the complementary or perpendicular 3D space). We decided, therefore, to name as Q the momentum transfer in the direction perpenX dicular to the surface. In the cut model of the quasicrystals given by Duneau and Katz [18], each node of the 6D periodic lattice (K) is occupied by an atomic surface (p). These objects are
M.J. Capitan et al. / Physica B 283 (2000) 79}83
projected in the physical space through k . The Fourier transform of the density function o is given by o( "(p( ) KH)*k( . (1) The quasicrystal truncation by a plane perpendicular to z at a c level modi"es only the characteristic measure k . Its Fourier transform is now given by i e pOX A k( "1 d (2) , V W 2p (q #ii) X instead of k( "1 d corresponding to the perfect , V W X in"nite quasicrystal. The di!raction amplitude characteristic function (o( ) di!ers only from the standard bulk spectrum by the continuous component along z that corresponds, in crystals, to the well-known rods of di!use intensity typical of surface di!raction. Eqs. (1) and (2) imply that each Bragg peak is convoluted by the rod function, the carrier of which is a line along the z-direction. If, as is the case, in practice, this direction is a quasicrystallographic direction, the convolution leads to a discrete sum of contribution terms issued from the wave vectors of the 9-module that are aligned along z. Let +qH, be the in"nite set of re#ections aligned along the z-direction passing through a given chosen re#ections and let q be a generic wavevectors along that direction. The di!racted amplitude is then given by: p( (qH)e\ pOHX A ie pEX A , (3) o( (q)" 2p (q !qH #ik) X H X where p( (qH) are the usual structure factors of the bulk quasicrystal which, for n atomic surfaces per unit cell, writes p( (qH)" , f (qH )g( (qH )e pqH RI . I I I , The cut level c (Eq. (2)) introduces a quasiperiodically varying phase in the sum that cannot be factorised in the calculation of the intensity. Indeed, contrary to the case of crystals, there are no possible c values for which the phase factors qH c are simultaneously integers for all wave X vectors of +qH,. This corresponds to the fact that, the termination planes being not periodically stacked, di!erent intensities can be obtained depending on where the quasicrystals is cut. All the possible terminations should be considered in order to take into account surface steps, di!erent domains, etc. Therefore, the experimental diffracted intensity is the superposition of all the possible terminations. Summarising, it can be observed that in the equation corresponding to the di!raction amplitude (Eq. (1)) appears a contribution that comes from the truncation of the in"nite array of atoms. This means that the Bragg peak intensities are convoluted by the Fourier transform of the step function that represent the surface [16]. This term produces an intensity decay of the Bragg peaks with a dependency of 1/*q which is the same type of dependence as obtained for periodic crystals [15,19]. It has
81
already been described in the literature [15] that for periodic structures, the minimum of the CTR between the two Bragg peaks is the point that is the most sensitive to the surface morphology.
3. Results and discussion There is some controversy about the preparation method of the surface of the quasicrystal. Therefore, the sample preparation was monitored from both a chemical point of view by the use of in situ auger electron spectroscopy (AES) and from a structural point of view by the use of in situ X-ray di!raction. The AES spectrum of the sample in the UHV conditions before any preparation process shows the presence of oxygen and aluminium displaying the characteristic AES peak shifting at lower kinetic energy of the oxidised Al. This spectrum evidences the presence at the surface of an oxidised aluminium layer with a small amount of Pd and surface contaminant like C. The AES spectrum of the sample annealed at 5503C for 60 min shows a decrease in the O peak, the appearance of an Al metallic peak and an increase in intensity of the Pd peak. The same trend is observed up to 7003C. After annealing at 7003C there are no traces of oxygen and only Pd and metallic Al can be seen. It is important to note that no traces of Mn can be detected. The quantitative analysis of the Al : Pd AES intensities suggest a surface chemical composition close to 3 : 1. These results can be interpreted as a `sublimationa process at sample temperatures above 5503C of a native aluminium oxide layer present at the quasicrystal surface. This process becomes very fast at around 7003C. It was observed by surface X-ray di!raction that the apparition of CTR intensity in the quasicrystal coincides with the total disappearance of the aluminium oxide signal as detected by AES. We can a$rm that we have prepared a `clean #ata surface on the quasicrystal by a simple annealing of the sample. Although CTR intensity was measured along di!erent ¸-directions (di!erent surface parallel momentum contribution) we are going to present and analyse only the specular CTR due to its higher symmetry (only perpendicular to the surface momentum contribution). Thus, this CTR has only information about the crystallographic order along the z-direction (5-fold axis) and, it is not perturbed by possible lateral atomic relaxation within the 5-fold plane (xy-plane). A 2D map of the measured intensity in the region around the (0 0 L) CTR is presented on the right-hand side of Fig. 1. This region corresponds to the shadowed zone drawn in the reciprocal space sketch on the left-hand side of Fig. 1. It can be observed that in between the measured Bragg peak intensities characteristic of an icosahedral structure there is an intensity remaining all along the (0 0 L) direction of the
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quasicrystal. This intensity links the Bragg peak. The FWHM along the lateral direction of this appeared intensity at the minimum between the two most intense Bragg peak indicates a `perfecta surface with 380 As terrace size. The CTR intensity is obtained by integrating the 2D measured reciprocal space intensity (right-hand side of Fig. 1) along the (0 0 L) direction. In order to obtain the surface structure factor, the integrated intensities [20] were subjected to several geometric corrections. Depending on the method used to record the di!racted intensity and on the position of the reciprocal lattice node, the time required for di!erent nodes to cross the Ewald sphere di!ers. In the present geometry and Lorentz correction that takes into account this factor is 1/[sin(2a)]. The polarisation factor depends on the state of polarisation of the incident X-ray beam and on the scattered angle of the di!racted beam. In the case of an horizontally polarised beam this correction is cos(k#c) (being the incidence angle and the surface scattering angle, respectively [20]). The illuminated sample area correction depends on the dimension and distance of the pre-sample slits and sample size which were set for this experiment to 0.1 mm;1 mm (perpendicular and parallel to the sample surface, respectively). This correction should be proportional to sin k. The CTR intercepted area correction is proportional to sin c. The measured specular (0 0 L) CTR integrated intensity for the presented clean sample (solid line) is shown in Fig. 2. Two main features should be noticed in this "gure. First, the large number of peaks of very di!erent intensity observed all along this CTR, thus the bulk peaks are extended in a large range into the complementary space (i.e. Q of the (23,19) is equal to 2.7 As \) characteristic of , the icosahedral phase. Second, the intensity between the most intense peaks (noted as (7,11) and (18,29)) decays as 1/(*q) which corresponds to the di!raction of a semiin"nite crystal [15,19]. As it has been shown in Section 2 of this article this q-dependent decay of the Bragg peaks indicate the presence of a well-ordered surface in the annealed sample. The quasi-periodic order characteristic of the icosahedral structure can be described by a periodic cubic lattice structure by using a higher (6D) dimensional space. For this we have used the model proposed by Katz and Gratias [2] for AlCuFe adapted to the case of AlPdMn which, as has been indicated by these authors, implies minor modi"cations. The square of the structure factor calculated (Eq. (3)) was "tted to the experimental data by the least-squares method (see Fig. 2). The quasicrystal reciprocal space is a very dense set of bulk Bragg peak. Due to this fact the (0 0 L) CTR contains bulk Bragg peaks with intensities which vary within a very large range. This large intensity variation determines the use of a de"nition for s based on the di!erence of the logarithm of the calculated and measured intensities
Fig. 2. Comparison of the square of the calculated structure factor (dashed lines) with the square of the measured structure factor (solid lines and circles) for the specular crystal truncation rod.
which is widely used in data analysis involving large dynamic range [21]. The free parameters of the "t were: the z-displacement of the last atomic layer, the atomic concentration changes of the last atomic layer, the surface roughness parameter (b) [15] and a global scale factor. Therefore, the "t gives information about the average state of the last atomic layer. The roughness parameter used here (b) corresponds to the covering percentage concept de"ned by Robinson [15] which is one of the most extensively used parameters for quantifying it. It should be noticed that the resulting "t describes rather well both bulk peaks and surface signal. The AlPdMn bulk was described by using the Debye}Waller factor (B) of 0.85 As [22] which is very close to the tabulated value for aluminium [23]. Within the limits given for the most used models [3,17] the calculated bulk di!raction pattern is in good agreement with the observed squared structure factor (even at high Q values). The s value obtained is 0.11 that is , noticeably better than the best "t obtained using the nominal bulk parameters (no contraction and bulk chemical composition for the last layer) giving a s value of 0.47. The surface signal "t gives a roughness value of almost zero (0.06$0.05), which corresponds to an
M.J. Capitan et al. / Physica B 283 (2000) 79}83
almost perfectly #at surface within the correlation length [15]. Taking into account that the present sample has a miscut of 43 and the calculated average mosaic block of 380 As giving a `perfecta surface within this distance, one can calculate an average step height of 28 As . It can be concluded that the surface presents step bunching. This justi"es the used approximation of an incoherent interaction between the di!erent surface terminations. Other results obtained from the "t are: the last layer tends to be contracted and richer in Al and Pd than in the bulk. The same trends were observed in LEED experiments of this surface [13]. In our case, the last interlayer distance su!ers a contraction of 0.83$0.01 As to be compared with the average bulk interatomic distance which is 3.8 As . The relative contraction factor (21%) is surprisingly identical to the value reported in literature for LEED experiment [13]. This contraction is rather important and could indicate that not the last layer alone but the last few layers may su!er a contraction. At present, the analysis of only one CTR makes it inconvenient to introduce additional free parameters in the "t. The "t reveals that the surface composition is depleted in Mn and enriched in Pd and Al by K110% and 36%, respectively, compared to the bulk concentration. That gives for the last layer a concentration close to Al : Pd/3 : 1 in agreement with the AES derived composition. The present results on roughness, surface contraction and concentration have shown to be independent of the model used for describing the bulk structure. Similar results have been obtained using the model described by Boudard et al. [3]. In this case, the best "t gives a s value of 0.16 showing the same chemical composition for the last layer and a bigger contraction of the last atomic layer (0.94$0.03 As ). In both cases the "t indicates that the quasicrystal has a highly dense packing at the surface. Gierer et al. [13] point out that in quasicrystals, as in metallic crystals, the closest-packed face has the lowest free energy. It can be observed in Fig. 2 that the surface signal is rather well-"tted except at low Q values where a signi"X cant di!erence is obtained. This di!erence could come from some problem in the data correction for the sample size. However, it can also be due to the existence of preferential terminations in the quasicrystal surface [16]. It has already been mentioned that the cut level produces unequivalent surface terminations and in this case we have considered a large enough number of possible terminations in order to obtain a representative average. However, the scanning tunnelling microscopy (STM) of the 5-fold plane surface made by T. Shaub et al. [10] shows the presence of an alternating Fibonacci series of short}long step height. The reported values of 6.78 As and 4.22 As could agree with the presence of a preferential type of terminations.
83
Acknowledgements The authors wish to thank Salvador Ferrer for his constant and inestimable help in the realization of this work. We thank the ESRF for facilitating the use of its installations for the completion of this work.
References [1] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [2] A. Katz, D. Gratias, J. Non-Cryst. Solids 153 (1993) 187. [3] M. Boudard, M. de Boissieu, C. Janot, G. Heger, C. Beeli, H.U. Nissen, H. Vincent, R. Ibberson, M. Audier, J.M. Dubois, J. Phys. C 4 (1992) 10149. [4] S.S. Kang, J.M. Dubois, J. von Stebut, J. Mater. Res. 8 (1993) 2471. [5] J.M. Dubois, S.S. Kang, Y. Massiani, J. Non-Cryst. Solids 153}154 (1993) 443. [6] M. Zurkirch, M. Erbudak, H.U. Nissen, M. Hochstrasser, E. Wetle, S. Ritsch, Philos. Mag. Lett. 72 (1995) 199. [7] M. Erbudak, H.U. Nissen, E. Welti, M. Hochstrasser, S. Ritsch, Phys. Rev. Lett. 72 (1994) 3037. [8] X. Wu, S.W. Kycia, C.G. Olson, P.J. Benning, A.I. Goldman, D.W. Lynch, Phys. Rev. Lett. 75 (1995) 4540. [9] S.L. Chang, J.W. Anderegg, P.A. Thiel, J. Non-Cryst. Solids 195 (1996) 95. [10] T.M. Schaub, D.E. BuK rgler, H.J. Guntherodt, J.B. Suck, Phys. Rev. Lett. 73 (1994) 1255. [11] A.R. Kortan, R.S. Becker, P.A. Thiel, H.S. Chen, Phys. Rev. Lett. 64 (1990) 200. [12] Ph. Ebert, M. Feuerbarcher, N. Tamura, M. Wollgarten, K. Urban, Phys. Rev. Lett. 77 (1996) 3827. [13] M. Gierer, M.A. Van Hove, A.I. Goldman, Z. Shen, S.L. Chang, C.J. Jenks, C.M. Zhang, P.A. Thiel, Phys. Rev. Lett. 78 (1997) 467. [14] E.G. McRae, R.A. Malic, T.H. Lalonde, P.A. Thiel, H.S. Chen, A.R. Kortan, Phys. Rev. Lett. 65 (1990) 883. [15] I.K. Robinson, Phys. Rev. B 33 (1986) 3830. [16] H. You, U. Welp, G.W. Crabtree, Y. Fang, S.K. Sinha, J.D. Axe, X. Jiang, S.C. Moss, Phys. Rev. B 45}49 (1992) 5107. [17] A. Katz, D. Gratias, in: F. Hippert, D. Gratias (Eds.), Lectures on Quasicrystals, 1994 (Chapter 3). [18] M. Duneau, A. Katz, Phys. Rev. Lett. 54 (1985) 2688. [19] S.R. Andrews, R.A. Cowley, J. Phys. C: Solid State Phys. 18 (1985) 6427. [20] E. Vlieg, J.F. van der Veen, S.J. Gurman, C. Norris, J.P. MacDonald, Surf. Sci. 210 (1989) 301. [21] E.E. Fullerton, I.K. Schuller, H. Vanderstraeten, Y. Bruynseraede, Phys. Rev. B 45 (1992) 9292. [22] M. Cornier-Quiquandon, A. Quivy, S. Lefebvre, E. Elkaim, G. Heger, A. Katz, D. Gratias, Phys. Rev. B 44 (1991) 2071. [23] C.H. MacGillavry, G.B. Rieck (Eds.), International tables for X-ray crystallography, Vol. 3, IUC, 1985, p. 232.
X-ray di!raction study of the 5-fold plane surface of a Al Pd Mn single-grain M.J. Capitan , Y. Calvayrac, D. Gratias, J. Alvarez *
Departmento Fn& sica de la Materia Condensada C-3, Universidad Autonoma de Madrid, 28049 Madrid, Spain CECM-CNRS, 15 rue Georges Urbain, 94407 Vitry s/Seine, France LEM-CNRS/ONERA, BP 72, 9232 Chatillon Cedex, France
Abstract The structure of a 5-fold symmetry quasicrystal surface of a thermodynamically stable icosahedral Al Pd Mn single-grain has been studied by means of the surface X-ray di!raction technique. The model used reproduces rather well both the Bragg peaks and the surface signal. The parameters obtained from the "t of the measured structure factor show a contraction of the outermost atomic layer. The surface also tends to be richer in A1 and Pd atoms forming a very dense outer-layer. 2000 Elsevier Science B.V. All rights reserved. PACS: 61.44.Br; 68.35.Bs; 61.10!i Keywords: Surface X-ray di!raction; Quasicrystals; Surface structure
1. Introduction After the discovery of the quasicrystal phases in 1984 [1] great e!ort has been made in order to determine the bulk structure of these quasiperiodic systems. Although the bulk structure is beginning to be well known [2,3], the surface structure of these alloys remains generally unknown. However, most of the properties of interest for practical uses of these materials (such as low friction coe$cient, good wear [4] and corrosion [5] resistances) are largely determined by phenomena related to the surface and interface regions. This is the main reason for the intensive studies on quasicrystal surface in the last few years. An important goal is to know if the properties mentioned have a relationship with the quasicrystalline structure, i.e. if the quasicrystalline structure is preserved up to the surface. Di!erent techniques have been applied to the study of the quasicrystal surface [6}14]. Although these
* Corresponding author. Tel.: #34-913975550. E-mail address: [email protected] (J. Alvarez)
techniques have helped to improve our knowledge of the quasicrystal surface a convincing model of the surface structure is still lacking. In this paper we propose the use of the surface X-ray di!raction technique for characterising the quasicrystal surface structure. We develop a theoretical formalism for the surface X-ray di!raction of quasiperiodic systems and we compare the model with the experimental results.
2. Surface di4raction amplitude calculation We are going to show in this paper that the quasicrystal surface structure can be obtained through the interpretation of the crystal truncation rod (CTR) intensities. CTRs are rods of intensity in reciprocal space parallel to the sample surface normal that result from the truncation of the in"nite bulk crystal [15]. Indexing the 5-fold symmetry axis perpendicular to the surface of our sample as [0 0 1], the CTRs intensities are then parallel to the (0 0 L) direction in the reciprocal space. The left-hand side of Fig. 1 presents an outline of the Bragg peak distribution characteristic of the icosahedral structure relative to this 5-fold plane surface.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 8 9 6 - 7
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Fig. 1. Outline of the reciprocal space corresponding to an icosahedral phase with respect to the 5-fold surface of the sample. It has been shadowed the measured region close to the (0 0 L) region which is represented on right side of "gure. It can be observed in the measured intensity the presence of an `extraa intensity which links the Bragg peaks corresponding to the quasicrystal phase.
The quasicrystal structure is characterised by the absence of a periodicity in the physical (3D) space. Thus, the quasicrystal structure can only be described within a periodic frame by increasing the dimension of the used space. It is then necessary to use a 6D space (the 3D real space#a 3D complementary space) for describing periodically an icosahedral structure. This quasiperiodicity has a direct consequence in the CTR calculation. The CTR intensity comes from the contribution of the atomic layers parallel to the surface plane [15,19]. In the case of a periodic structure, it is possible to factorise the layers contribution because of the existence of a lattice unit along the normal direction (even in the case of very complicated structures such as the untwinned YBa Cu O [16]). As it has been formerly said it does \d not exist in the quasicrystal as a lattice unit along any direction of the real space. That implies that, in the quasicrystals, it should be necessary to add the contribution of each individual atomic layer up to the in"nite. The CTR calculation as in"nite sum has two problems; "rst, it has associated with it a large computational error and, second, it is necessary to have an exact atomic layer
description in the real space. This last point is especially critical in the case of treating a lower symmetry direction, i.e. all the CTRs di!erent from the (0 0 L). Thus, we introduce a higher-dimensional space, used for describing the quasicrystal bulk [17,18], for calculating a CTR intensity within the Robinson [15] and Andrews and Cowley [19] frame [16]. To our knowledge, this is the "rst time that for the quasicrystals truncation is reported in the literature. Although in the X-ray surface di!raction experiments one generally adopts the convention of decomposing the momentum transfer vector into two components, parallel and perpendicular to the surface, this is in con#ict with the normal nomenclature used for describing quasicrystals within a higher-dimensional space (what is called the real or parallel 3D spaces plus the complementary or perpendicular 3D space). We decided, therefore, to name as Q the momentum transfer in the direction perpenX dicular to the surface. In the cut model of the quasicrystals given by Duneau and Katz [18], each node of the 6D periodic lattice (K) is occupied by an atomic surface (p). These objects are
M.J. Capitan et al. / Physica B 283 (2000) 79}83
projected in the physical space through k . The Fourier transform of the density function o is given by o( "(p( ) KH)*k( . (1) The quasicrystal truncation by a plane perpendicular to z at a c level modi"es only the characteristic measure k . Its Fourier transform is now given by i e pOX A k( "1 d (2) , V W 2p (q #ii) X instead of k( "1 d corresponding to the perfect , V W X in"nite quasicrystal. The di!raction amplitude characteristic function (o( ) di!ers only from the standard bulk spectrum by the continuous component along z that corresponds, in crystals, to the well-known rods of di!use intensity typical of surface di!raction. Eqs. (1) and (2) imply that each Bragg peak is convoluted by the rod function, the carrier of which is a line along the z-direction. If, as is the case, in practice, this direction is a quasicrystallographic direction, the convolution leads to a discrete sum of contribution terms issued from the wave vectors of the 9-module that are aligned along z. Let +qH, be the in"nite set of re#ections aligned along the z-direction passing through a given chosen re#ections and let q be a generic wavevectors along that direction. The di!racted amplitude is then given by: p( (qH)e\ pOHX A ie pEX A , (3) o( (q)" 2p (q !qH #ik) X H X where p( (qH) are the usual structure factors of the bulk quasicrystal which, for n atomic surfaces per unit cell, writes p( (qH)" , f (qH )g( (qH )e pqH RI . I I I , The cut level c (Eq. (2)) introduces a quasiperiodically varying phase in the sum that cannot be factorised in the calculation of the intensity. Indeed, contrary to the case of crystals, there are no possible c values for which the phase factors qH c are simultaneously integers for all wave X vectors of +qH,. This corresponds to the fact that, the termination planes being not periodically stacked, di!erent intensities can be obtained depending on where the quasicrystals is cut. All the possible terminations should be considered in order to take into account surface steps, di!erent domains, etc. Therefore, the experimental diffracted intensity is the superposition of all the possible terminations. Summarising, it can be observed that in the equation corresponding to the di!raction amplitude (Eq. (1)) appears a contribution that comes from the truncation of the in"nite array of atoms. This means that the Bragg peak intensities are convoluted by the Fourier transform of the step function that represent the surface [16]. This term produces an intensity decay of the Bragg peaks with a dependency of 1/*q which is the same type of dependence as obtained for periodic crystals [15,19]. It has
81
already been described in the literature [15] that for periodic structures, the minimum of the CTR between the two Bragg peaks is the point that is the most sensitive to the surface morphology.
3. Results and discussion There is some controversy about the preparation method of the surface of the quasicrystal. Therefore, the sample preparation was monitored from both a chemical point of view by the use of in situ auger electron spectroscopy (AES) and from a structural point of view by the use of in situ X-ray di!raction. The AES spectrum of the sample in the UHV conditions before any preparation process shows the presence of oxygen and aluminium displaying the characteristic AES peak shifting at lower kinetic energy of the oxidised Al. This spectrum evidences the presence at the surface of an oxidised aluminium layer with a small amount of Pd and surface contaminant like C. The AES spectrum of the sample annealed at 5503C for 60 min shows a decrease in the O peak, the appearance of an Al metallic peak and an increase in intensity of the Pd peak. The same trend is observed up to 7003C. After annealing at 7003C there are no traces of oxygen and only Pd and metallic Al can be seen. It is important to note that no traces of Mn can be detected. The quantitative analysis of the Al : Pd AES intensities suggest a surface chemical composition close to 3 : 1. These results can be interpreted as a `sublimationa process at sample temperatures above 5503C of a native aluminium oxide layer present at the quasicrystal surface. This process becomes very fast at around 7003C. It was observed by surface X-ray di!raction that the apparition of CTR intensity in the quasicrystal coincides with the total disappearance of the aluminium oxide signal as detected by AES. We can a$rm that we have prepared a `clean #ata surface on the quasicrystal by a simple annealing of the sample. Although CTR intensity was measured along di!erent ¸-directions (di!erent surface parallel momentum contribution) we are going to present and analyse only the specular CTR due to its higher symmetry (only perpendicular to the surface momentum contribution). Thus, this CTR has only information about the crystallographic order along the z-direction (5-fold axis) and, it is not perturbed by possible lateral atomic relaxation within the 5-fold plane (xy-plane). A 2D map of the measured intensity in the region around the (0 0 L) CTR is presented on the right-hand side of Fig. 1. This region corresponds to the shadowed zone drawn in the reciprocal space sketch on the left-hand side of Fig. 1. It can be observed that in between the measured Bragg peak intensities characteristic of an icosahedral structure there is an intensity remaining all along the (0 0 L) direction of the
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quasicrystal. This intensity links the Bragg peak. The FWHM along the lateral direction of this appeared intensity at the minimum between the two most intense Bragg peak indicates a `perfecta surface with 380 As terrace size. The CTR intensity is obtained by integrating the 2D measured reciprocal space intensity (right-hand side of Fig. 1) along the (0 0 L) direction. In order to obtain the surface structure factor, the integrated intensities [20] were subjected to several geometric corrections. Depending on the method used to record the di!racted intensity and on the position of the reciprocal lattice node, the time required for di!erent nodes to cross the Ewald sphere di!ers. In the present geometry and Lorentz correction that takes into account this factor is 1/[sin(2a)]. The polarisation factor depends on the state of polarisation of the incident X-ray beam and on the scattered angle of the di!racted beam. In the case of an horizontally polarised beam this correction is cos(k#c) (being the incidence angle and the surface scattering angle, respectively [20]). The illuminated sample area correction depends on the dimension and distance of the pre-sample slits and sample size which were set for this experiment to 0.1 mm;1 mm (perpendicular and parallel to the sample surface, respectively). This correction should be proportional to sin k. The CTR intercepted area correction is proportional to sin c. The measured specular (0 0 L) CTR integrated intensity for the presented clean sample (solid line) is shown in Fig. 2. Two main features should be noticed in this "gure. First, the large number of peaks of very di!erent intensity observed all along this CTR, thus the bulk peaks are extended in a large range into the complementary space (i.e. Q of the (23,19) is equal to 2.7 As \) characteristic of , the icosahedral phase. Second, the intensity between the most intense peaks (noted as (7,11) and (18,29)) decays as 1/(*q) which corresponds to the di!raction of a semiin"nite crystal [15,19]. As it has been shown in Section 2 of this article this q-dependent decay of the Bragg peaks indicate the presence of a well-ordered surface in the annealed sample. The quasi-periodic order characteristic of the icosahedral structure can be described by a periodic cubic lattice structure by using a higher (6D) dimensional space. For this we have used the model proposed by Katz and Gratias [2] for AlCuFe adapted to the case of AlPdMn which, as has been indicated by these authors, implies minor modi"cations. The square of the structure factor calculated (Eq. (3)) was "tted to the experimental data by the least-squares method (see Fig. 2). The quasicrystal reciprocal space is a very dense set of bulk Bragg peak. Due to this fact the (0 0 L) CTR contains bulk Bragg peaks with intensities which vary within a very large range. This large intensity variation determines the use of a de"nition for s based on the di!erence of the logarithm of the calculated and measured intensities
Fig. 2. Comparison of the square of the calculated structure factor (dashed lines) with the square of the measured structure factor (solid lines and circles) for the specular crystal truncation rod.
which is widely used in data analysis involving large dynamic range [21]. The free parameters of the "t were: the z-displacement of the last atomic layer, the atomic concentration changes of the last atomic layer, the surface roughness parameter (b) [15] and a global scale factor. Therefore, the "t gives information about the average state of the last atomic layer. The roughness parameter used here (b) corresponds to the covering percentage concept de"ned by Robinson [15] which is one of the most extensively used parameters for quantifying it. It should be noticed that the resulting "t describes rather well both bulk peaks and surface signal. The AlPdMn bulk was described by using the Debye}Waller factor (B) of 0.85 As [22] which is very close to the tabulated value for aluminium [23]. Within the limits given for the most used models [3,17] the calculated bulk di!raction pattern is in good agreement with the observed squared structure factor (even at high Q values). The s value obtained is 0.11 that is , noticeably better than the best "t obtained using the nominal bulk parameters (no contraction and bulk chemical composition for the last layer) giving a s value of 0.47. The surface signal "t gives a roughness value of almost zero (0.06$0.05), which corresponds to an
M.J. Capitan et al. / Physica B 283 (2000) 79}83
almost perfectly #at surface within the correlation length [15]. Taking into account that the present sample has a miscut of 43 and the calculated average mosaic block of 380 As giving a `perfecta surface within this distance, one can calculate an average step height of 28 As . It can be concluded that the surface presents step bunching. This justi"es the used approximation of an incoherent interaction between the di!erent surface terminations. Other results obtained from the "t are: the last layer tends to be contracted and richer in Al and Pd than in the bulk. The same trends were observed in LEED experiments of this surface [13]. In our case, the last interlayer distance su!ers a contraction of 0.83$0.01 As to be compared with the average bulk interatomic distance which is 3.8 As . The relative contraction factor (21%) is surprisingly identical to the value reported in literature for LEED experiment [13]. This contraction is rather important and could indicate that not the last layer alone but the last few layers may su!er a contraction. At present, the analysis of only one CTR makes it inconvenient to introduce additional free parameters in the "t. The "t reveals that the surface composition is depleted in Mn and enriched in Pd and Al by K110% and 36%, respectively, compared to the bulk concentration. That gives for the last layer a concentration close to Al : Pd/3 : 1 in agreement with the AES derived composition. The present results on roughness, surface contraction and concentration have shown to be independent of the model used for describing the bulk structure. Similar results have been obtained using the model described by Boudard et al. [3]. In this case, the best "t gives a s value of 0.16 showing the same chemical composition for the last layer and a bigger contraction of the last atomic layer (0.94$0.03 As ). In both cases the "t indicates that the quasicrystal has a highly dense packing at the surface. Gierer et al. [13] point out that in quasicrystals, as in metallic crystals, the closest-packed face has the lowest free energy. It can be observed in Fig. 2 that the surface signal is rather well-"tted except at low Q values where a signi"X cant di!erence is obtained. This di!erence could come from some problem in the data correction for the sample size. However, it can also be due to the existence of preferential terminations in the quasicrystal surface [16]. It has already been mentioned that the cut level produces unequivalent surface terminations and in this case we have considered a large enough number of possible terminations in order to obtain a representative average. However, the scanning tunnelling microscopy (STM) of the 5-fold plane surface made by T. Shaub et al. [10] shows the presence of an alternating Fibonacci series of short}long step height. The reported values of 6.78 As and 4.22 As could agree with the presence of a preferential type of terminations.
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Acknowledgements The authors wish to thank Salvador Ferrer for his constant and inestimable help in the realization of this work. We thank the ESRF for facilitating the use of its installations for the completion of this work.
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