Physica B 253 (1998) 278-289 Analysis of grazing incidence X-ray diffuse scatter from Co-Cu multilayers I. Pape , T.P.A. Hase , B.K. Tanner *, M. Wormington Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK Bede Scientific Inc, Suite G-104, 14, Inverness Drive East, Englewood, CO 80112, USA Department of Engineering, University of Warwick, Coventry, CV4 7AL, UK Received 15 July 1997 Abstract Grazing incidence diffuse X-ray scattering data from a Co-Cu multilayer with stepped interfaces grown by molecular beam epitaxy on a copper silicide buffer on a silicon substrate has been analysed using a computer code based on a fractal interface within the distorted wave Born approximation. We have extended the theory to include the scattering from a stepped interface and have shown that a single set of structural parameters can be used to obtain an excellent agreement between simulation and experimental data taken under very different X-ray optical conditions. The symmetry of the diffuse scatter on rotation about the surface normal can be explained if it arises from step bunching at the ends of extensive flat terraces. These steps have a self-affine nature, enabling the fractal model to be used successfully. 1998 Elsevier Science B.V. All rights reserved. Keywords: X-ray scattering; Magnetic multilayers; Terraced interfaces 1. Introduction associated with the crystallographic texture. In particular, it is still not clear why the GMR Although discovered in 1988 [1], the giant mag- in layers grown by molecular beam epitaxy (MBE) netoresistance (GMR) observed in multilayers of is usually lower than equivalent films grown by transition metals is still not understood. While the sputtering. concept of two spin-dependent scattering channels The copper-cobalt system, which exhibits ex- provides a very good qualitative description of the tremely high values of GMR [2,3], has received phenomenon, it is unclear whether the magnitude much attention and has the important advantage of the GMR is determined by the roughness of the that the systems are immiscible. Most studies of multilayer interfaces or by Fermi surface effects the correlation between interface roughness and GMR [4-6] have relied on specular reflectivity measurements that cannot distinguish between * Corresponding author. Tel.: #44 191 374 2137; fax: #44 roughness and compositional grading. We have 191 374 2111; e-mail: b.k.tanner@durham.ac.uk. recently shown that measurement of the diffuse 0921-4526/98/$ - see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 3 9 5 - 0 I. Pape et al. / Physica B 253 (1998) 278-289 279 X-ray scatter provides powerful insights into the with long lateral correlation. As a function of speci- interface structure of these systems. In collabora- men orientation with respect to the incident beam, tion with workers at the University of Leeds, we the relative positions of specular and diffuse peaks have measured the grazing incidence diffuse X-ray changed sinusoidally with angle of rotation about scatter and have shown that changes in spin-depen- an axis normal to the specimen surface. From this it dent scattering on annealing Co-Cu multilayers was concluded that the interface was stepped. This arise through metallurgical changes not associated paper presents a more detailed study of the scatter- with the multilayer itself [7]. The very substantial ing from such a stepped interface and compares the changes are associated with alloying of the top scattering observed under different X-ray optical copper layer with the gold anti-oxidation cap, and conditions with the predictions of a theoretical no changes in the roughness of the copper-cobalt model. interfaces were observed. In multilayer systems of 3d transition metals such as Co-Cu, the scattering from such a gold cap has a crucial effect on the 2. Experimental configuration grazing incidence X-ray scattering [8]. In order to distinguish the Co-Cu interface All experiments were performed on Co-Cu mul- scattering from that due to other interfaces, we tilayers grown on polished, single crystal, silicon have exploited techniques using anomalous dis- substrates oriented normal to the [1 1 1] direction. persion to change the scattering factor difference The silicon substrates were 1 mm-thick slabs, the between the 3d transition metal layers while films being grown on square 10 mm;10 mm faces. having little effect on the scattering from other All samples were grown at the University of Leeds interfaces. Measurements were made at two X-ray in the VG80M MBE facility with a base pressure of wavelengths, one far from the absorption edge 3;10\ mbar. During deposition, the pressure and the other, close to the absorption edge. remained below 1;10\ mbar and in some cases, In performing these measurements, we exploit the sample was rotated at a rate of 1 Hz. The 20 a unique property of synchrotron radiation, namely period multilayers, nominally 11.5 A> Co/7 A> Cu, that it has a continuous spectrum of slowly varying were grown on a 10 A> Cu layer, deposited directly intensity. Thus by examining the scatter close on the silicon substrate at 150°C. This then formed to and away from the edge, the scattering from a Cu the multilayer interfaces can be identified. Consis- Si compound on to which a 15 A> Au layer was deposited. Finally, a nominally 15 A> thick Au tency of wavelength was achieved between runs cap was laid to prevent oxidation of the top layer of by measurement of the fluorescence yield and copper. The data presented in this paper were taken calibrating against the near-edge absorption struc- on a single sample but is representative of all sam- ture of standard samples. During analysis, the data ples examined. were corrected for the changing area of sample Grazing incidence X-ray scattering measure- illuminated by the beam during the scan, the ments were made using the two circle powder dif- changing fraction of the scatter reaching the fractometer on station 2.3 at the Synchrotron detector, beam spill-off due to small sample size Radiation Source (SRS) at Daresbury Laboratory. and the decay of the electron beam current in the A water cooled, double bounce, Si (1 1 1) mono- storage ring. chromator was used to select the wavelength and A high GMR has recently been reported in Co- in the geometry selected this gave a dispersion of Cu multilayers grown by MBE on silicon with 1.5;10\ at a wavelength of 1.4 A>. With incident a cobalt silicide buffer. Grazing incidence diffuse beam slits before the monochromator of 100 m scatter studies showed that the diffuse scatter in height and 4 mm in width, this produced count through the first superlattice Bragg position was rates of typically 10 c.p.s. at the sample. The width strongly peaked close to the specular peak in trans- of the beam-defining slits as well as the slits before verse (specimen only) scans in reciprocal space [9] the detector were determined to an accuracy of and this was interpreted as arising from roughness 10 m. Analyser slits immediately in front of the 280 I. Pape et al. / Physica B 253 (1998) 278-289 detector were set to 100 m, giving a measured these, the transmitted wave, travels towards the instrumental resolution function in a detector scan substrate while the other, the reflected wave, travels which was Gaussian and of 40 arcsec at full width away from the substrate. From the continuity of the half height maximum (FWHM) [10]. Through use electric field and its normal derivative across each of a further set of anti-scatter slits, evacuation of the interface, Parratt's well-known recursion formula is air path to the high dynamic range scintillation obtained [12], detector [11], which was itself encased in lead, we obtained an experimental background of 1 c.p.s. r R J#RJ> J> limited by scatter of the main beam within the air J" . (1) 1#r path around the specimen. JRJ> J> Four types of X-ray measurements were made. Here, RJ"EJ /EJ denotes the reflection coefficient Specular ( "2 ) scans recorded the intensity as of the multilayer below the lth interface, where the detector is swept at twice the angular rate of the EJ and EJ are the amplitudes of the reflected and incidence angle and scanned parallel to the recipro- transmitted waves just above the lth interface. The cal space vector q Fresnel coefficients for reflection and transmission . Off specular longitudinal diffuse ( "2 # ) scans recorded the intensity of at each interface are given by rJ"(kX J!kX J> )/ the diffuse scatter just below the specular ridge by (kX J#kX J> ) and tJ"2kX J/(kX J#kX J> ), respec- scanning in the same way as in specular scans but tively. For the layer thickness dJ, the complex am- with a slight offset in the sample angle. In reciprocal plitude factor is given by the expression space, these trace out radial lines emanating from J"exp(ikX JdJ). The z component of the wave vec- the 000 reciprocal lattice point. Transverse (q tor in the lth layer is given, according to Snell's law, ) scans in reciprocal space were recorded by fixing by kX J"k(n J!cos ) , where nJ is the refractive the detector and scanning only the specimen. De- index of the layer and is the grazing angle of the tector-only scans were also performed, the effect of incident X-ray beam. which was to scan around the Ewald sphere in For a semi-infinite substrate, it follows that reciprocal space keeping the penetration depth of R,> "0. Thus, the reflectivity coefficient of the the X-rays into the sample constant. multilayer may be obtained by solving Eq. (1) re- Wavelengths were chosen close to (1.3801 A>) and cursively, starting at the substrate and working up. away from (1.48 A>) the Cu absorption edge in order Finally, the reflected intensity is given by to enhance the scattering from the Co/Cu interfaces IQ"I "R " , where I is the intensity of the incident and distinguish it from the scattering of top surface X-ray beam. and buffer layer interfaces. Although the contrast Real interfaces are not ideally sharp, as described enhancement at the Co edge was potentially better above, but possess both roughness and grading than the Cu edge [10], this was offset by a higher which tend to smooth the interface profile when intensity in the latter part of the spectrum. In addi- averaged over large areas. The electric susceptibil- tion, the presence of structure on the Cu absorp- ity across such an interface can be expressed as tion edge enabled us to calibrate wavelength accur- J[zJ! zJ(x, y)], where zJ(x, y) is a random vari- ately to one part in 10 , which was conveniently able that describes the deviation of the interface very well matched to the monochromator disper- from its mean position. Provided the total interface sion [10]. width is not too large, we may use the distorted- wave Born approximation to determine the re- placement, 1rJ2, for the Fresnel reflection coeffic- 3. Simulation technique ient in Eq. (1). After some mathematics we arrive at 3.1. Specular scatter 1rJ2+rJ exp(!2(kX JkX J> J> ) The specular, or coherent, field in each layer can ; dz g be expressed as the sum of two plane waves. One of J(z)exp(2i(kX JkX J> z), (2) \ I. Pape et al. / Physica B 253 (1998) 278-289 281 where we have assumed that zJ> (x, y) possesses where a Gaussian distribution with a standard deviation, J represents the difference in the electric susceptibility of layers l and l#1, calculated suffi- J> , and zero mean. Here, gJ> (z)" J> (z)/ ciently far from the interface so that the effects of J! J> ) is the normalised first derivative of the grading, described by the function g local electric susceptibility across the interface. The J(z), can be neglected. The electric field components directly exponential term following the ideal Fresnel reflec- below the lth interface, namely EM tion coefficient accounts for interface roughness J G and EM J G, are obtained by solving the specular problem (Eq. (2) and the Fourier integral takes into account the separately for the incident wave vector, k, and for effects of grading. For a rough interface possessing the inverted exit wave vector, !k , respectively. an abrupt local change in J> (z), the second term is The parameters q equal to unity. However, if the transition is smooth, X J G H denote the z-component of the scattering vectors q and can be approximated by an error function of J G H"k J G!kJ H. Finally, the summation is taken over all interface combinations width, E J> , the second correction term is equal to within the multilayer, and for both the transmitted exp [!2kJkJ> ) E J> ], which is identical to the (t) and reflected (r) components of the electric fields. roughness term. This is an important result as it Hence, the calculation of diffuse scatter from clearly demonstrates the inability of specular reflec- a multilayer is computationally very demanding as tivity to distinguish between roughness and grad- it generally involves a large number of terms ing at an interface; they both damp the reflection [13,14]. coefficient as the angle of the incident X-ray beam is The covariance functions C increased. J JY( ) contain all of the information about the morphology of indi- vidual interfaces and the way in which the morpho- logy propagates from layer to layer. In order to 3.2. Diffuse scattering calculate the diffuse scatter from a multilayer, par- ticular forms of the covariance functions have to be The distorted-wave Born approximation not assumed. The shape of the lth interface is described only allows us to calculate the effect of roughness by the auto-covariance function CJ( ) and is de- and grading on the specular reflectivity, but also fined by allows us to predict the intensity of diffuse scatter as C a function of the scattering vector, q. If we again J( )"1 zJ(0) zJ( )2, (5) assume Gaussian height distributions at each inter- where is a vector lying in the plane of the interface face, then after some lengthy mathematics, we ar- and 1 2 denotes a configurational average over all rive at the following expression for the scattering points in the interface. The random quantity zJ( ) cross-section, S(q), for a multilayer is the local centre of the grading profile at a lateral position with an RMS value A k , J "1CJ(0)2 and S(q)" P vanishing mean. In this work C 16 J G HP*JY GY HY J( ) is assumed to be J JY G GY P R H HY P R the isotropic auto-covariance function introduced by Sinha et al. [15] namely e\ O ; X J G H N J> O* X JY GY HY N JY q C X J G Hq*XY JY GY HY J( )" J exp[!( / J) &J], (6) where "" ". This describes the behaviour of ; d [eOX J G HO*X JYGYHY!J JY M !1]eGO M, (3) a self-affine fractal interface with a cut-off deter- mined by the correlation length J. Although this where A "A/sin is the area of the specimen illu- represents a particular class of morphology it is minated by an incident X-ray beam of area A. The quite general in that it can describe both jagged factors P and gently undulating interfaces depending on the J G H are given by Hurst parameter, HJ. This parameter is restricted to P the region 0(H J G H" JEMJ GEM J H dz gJ(z)exp iqX J G Hz), (4) J)1 and defines the fractal di- \ mension, DJ"3!HJ, of the interface. 282 I. Pape et al. / Physica B 253 (1998) 278-289 In general, we not only have to consider the number of computations scale linearly with the roughness of an individual interface, but also the number of layers, N, in the sample. Generally, the way in which the morphology propagates through computation time scales as N which is very slow the multilayer. If correlations exist between differ- for a large number of layers. ent layers, a non-zero covariance function The experimental geometry most widely used to measure the diffuse scatter defines the scattered CJ JY( )"1 zJ(0) zJY( )2 (7) wave vector by placing a long, narrow slit in front of the detector. Since the resolution of this arrange- has to be assumed. Here zJ and zJY represent the ment is poor out of the scattering plane we have to local centres of the grading profile at the lth and l th integrate Eq. (3) over the y-component of the scat- interfaces, located at zJ and zJY, respectively. tering vector. This procedure gives the following Two particular forms of CJ JY( ) used in this work, result, the first of which is given by A k , C S(q P J JY( ) V, qX)" 8 J G HP*JYGY HY J JY G GY P R H HY P R " J JY exp[!( / J JY) &] exp(!"zJ!zJY"/ ), (8) e\ OX J G H N J> O*X JY GY H N JY ; where the parameters J JY" J# JY and & J JY" q ( & X J G Hq*X JY GY HY J # & JY )/2 are defined by means of the RMS roughness J and correlation length J of individual ; interfaces. The paramater describes the tendency dx[eOX J G HO*X JY GYHY!J JY V !1]eGOVV, (10) of the individual layers to replicate the substrate roughness and is the distance over which correla- from which the diffusely scattered intensity, tions between the fluctuations at the lth and l th I(qV, qX), may be calculated using I (qV, qX)" interfaces are damped by a factor of 1/e. No cor- I A\ S(qV, qX) , where is the angular accept- relations are present in the case "0 and nearly ance of the detector slit in the scattering plane. perfect correlation exists when is much larger Thus, what is measured is related to a one-dimen- than the thickness of the multilayer. The computa- sional, rather than a two-dimensional, Fourier tion time for Eq. (8) scales as N ; however, it is transform of the covariance function. This is fine for extremely useful as the effects of partial correlation isotropic rough surfaces as it essentially yields the and frequency-dependent replication of interface same information, but may be misleading for an- morphology can be investigated. isotropic roughness. The second covariance function used can be writ- The Fourier transform in Eq. (10) does not in ten as general have an analytical solution and numerical techniques must be used to evaluate it. Hence, this CJ JY( )"( JY JY J JY# J Y JY) exp[!( / ) &] becomes the rate determining step when modelling (9) diffuse scatter. To optimise the calculation we have developed an efficient numerical approach. For the where particular covariance functions used in this work J JY is the Kronecker delta operator and is unity when l"l and zero otherwise. The RMS we may expand the Fourier transform as a series, roughness for example, J " J# J of the lth interface is de- fined in terms of an uncorrelated component J, that does not replicate from layer to layer, and dx[eOX J G HO*X JY GY HYN J JY \ M KJ JY & !1]e OVV a correlated component J, which replicates the morphology of the substrate perfectly. The correla- tion length and Hurst parameter are assumed to be q L "2 X J G H(q*X JY GYHY) L L J JY identical for all interfaces. By using this particular J JY F(qV J JY/n & , L n! form of CJ JY( ), we can rearrange Eq. (3) so that the (11) I. Pape et al. / Physica B 253 (1998) 278-289 283 where in practice only the first few terms of the summation have to be considered and F(q)" dx e\V &cos(qx). (12) Because the function F(q) may be accurately tabulated in advance for various values of H this method is computationally much faster than evalu- ating Eq. (10) directly [16]. All the simulations shown in this paper have been performed using this second covariance function. The model has been proven for the case of a single rough interface with an error-function grad- ing profile by fitting the experimental data from a polished surface of the glass ceramic Zerodur Fig. 1. Experimental and simulated reflectivity curves from [17]. We have been able to separate the grading a Cu-Co multilayer. and roughness by measurement of the specular scan and one of the diffuse scans; all other diffuse scans being excellently fitted by the parameters derived from the former data sets. 4. Results Fig. 1 shows the specular curve for one sample, corrected for the forward diffuse scatter measured in a !0.1° offset longitudinal diffuse scan ( " 2 # ), together with our best fit to a simulated curve. This latter curve has been simulated with a structure: Si (6 A> rms roughness), 15 A> Cu Si (6 A> rms roughness), 21.5 A> Au (6 A> rms roughness), +15.0 A> Co (6.5 A> rms roughness), 6.5 A> Cu (7.0 A> Fig. 2. Structure used to simulate the reflectivity curve in Fig. 3. rms roughness), ;19, 14.5 A> Co (6.5 A> rms rough- ness), 31 A> Au Cu (7 A> rms roughness) (Fig. 2). It proved essential to use an Au-Cu alloy for the cap, suggesting that the top Cu layer of the nom- positions are therefore determined by the electron inally 20 period superlattice had interdiffused with density at the surface. This change in surface den- the Au. When a pure Au cap was used in the sity was noted in annealed Cu-Co multilayers simulation, the interference fringe period was cor- grown on sapphire and high-angle X-ray diffrac- rect but the positions of the maxima and minima tion experiments confirmed the presence of the were displaced from those observed experimentally. alloyed surface layer [7]. This sensitivity of the interference fringe positions Fig. 3 shows transverse scans taken through the to the position of the critical angle is a general 1st-order Bragg peak for different azimuthal angles effect in specular reflectivity curves and the fringe where the specimen is rotated about the surface normal. We note the sharp peak in the diffuse scatter adjacent to the very narrow specular peak. Zerodur is a registered trademark of Schott Glaswerke, There is a clear displacement of the diffuse scatter Mainz. peak from the specular ridge and this is a strong 284 I. Pape et al. / Physica B 253 (1998) 278-289 Fig. 3. Transverse diffuse scan taken through the first-order Bragg peak on rotation about the sample's surface normal. function of the specimen orientation with respect to the incident beam. On rotation of the specimen about the surface normal, the relative position of the specular and diffuse scatter peaks reverses and the separation can be fitted well by a sine function (Fig. 4). Asymmetries in scattered X-rays which are a function of the direction of the input beam (#qW, !qW,#qV,!qV) have been observed previously [18-21]. In all cases they have been ascribed to a preferential direction on the surfaces and often, by implication terracing within the multilayers. How- ever, unlike these previous investigations, a single off-cut, giving rise to regular terraces, cannot be the case here, or we would expect to see two humps on Fig. 4. Separation of the diffuse and specular peaks as a func- either side of the specular peak, i.e. a blazed diffrac- tion of the azimuthal angle. The solid line is a sine function fitted tion grating [22]. to the data points. I. Pape et al. / Physica B 253 (1998) 278-289 285 It is interesting to note that no clear Yoneda model the variations in diffuse scatter. The same wings appear in either experimental or simulated strategy was adopted and we see from Fig. 3 that curves of Fig. 3. For the case of conformal rough- the agreement is again excellent. This provides fur- ness, the majority of the diffuse scatter through the ther support for the model of Cowley in this in- Bragg peak comes from the buried interfaces and as stance [24]. the Yoneda wings arise from the enhancement of However, it proved possible to obtain quite good the electric field amplitude in the surface at the fits to the transverse scans through the Bragg peak critical angle, these features are hidden when the for a range of correlation lengths between 1300 and majority of the scattering is from the conformal 400 A> and fractal parameters between 0.2 and 0.45, roughness of the multilayer. This effect has pre- there being significant coupling between the two viously been noted experimentally and the very in this region. In order to establish the uniqueness good agreement between the DWBA simulation of the simulated structure, the scattering at differ- and the experiment is noteworthy. ent angles and at different wavelengths was simu- With respect to the specular peak, the angle at lated. Transverse diffuse scans taken through a which the diffuse scatter rises to the mean value Kiessig maximum and minimum at a wavelength of far from the specular peak, i.e. the position of the 1.3801 A> are shown in Fig. 5 with the best-fit buried Yoneda wing, is constant for all four azi- simulations. Note that the Yoneda wings are well muthal settings and does not follow the sinusoidal dependence of the peak in the diffuse scatter. We also note that no change was observed in the specu- lar curve on rotation. The coupling of the Yoneda and specular positions can be understood in the following way. As the observed diffuse scatter in- tensity originates from roughness in the sample, if the roughness has a preferential direction, as in the case of terracing, this will be mirrored in the diffuse scatter. In contrast, the Yoneda wings arise as an increase in diffuse scatter at the critical angles, due to an enhanced electric field at the interface at these angles. This arises as a direct consequence of the boundary matching of electromagnetic waves at the interface. These same waves govern the direction from which the specular scatter emerges and so a coupling of the positions of the specular and Yoneda wing positions occurs. The theoretical treatment of the scattering of X-rays from terraced scatter systems has been given by several authors [21,23]. In some cases the specu- lar scatter is said to arise from the average surface [24], while others restrict the argument to regions that are able to scatter coherently [25]. We have observed in studies of large grain alumina ceramics that the measured position of the Yoneda wings relates to the average surface density of the mater- ial. From this observation it can be deduced that the electromagnetic waves matched across the aver- age surface within their coherence length [26]. Us- Fig. 5. Transverse scans through a Kiessig (a) maxima and ing this assumption, we were able to successfully (b) minimum, wavelength of 1.3801 A>. 286 I. Pape et al. / Physica B 253 (1998) 278-289 defined. Increasing the number of transverse diffuse scans dramatically reduced the number of possible solutions and a value of "450$50 A> and H" 0.28$0.05 was found to fit six independent data sets at wavelengths of 1.3801 and 1.48 A>. To obtain a good fit to the diffuse scatter, it was necessary to assume that the ratio of correlated to uncorrelated roughness on the Cu/Co interfaces was 60 : 40. A combination of scans in reciprocal space en- ables a contour map of the scattering to be pro- duced. Such a simulated map (Fig. 6) shows that the scattering is very substantially enhanced around the q value corresponding to the first Bragg reflection from the superlattice. This en- hancement of scattering is an interference effect that is characteristic of the presence of correlated roughness through the multilayer. When the lateral correlation length is short, these features appear much more extensive in reciprocal space [27]. Through the effect of the refractive index correction they are curved, leading to the name of "Holy´ bananas". In this case, however, the long correla- tion length restricts the scattering in the q direc- tion and the enhancement occurs only close to the specular Bragg peak. We also note that for correlated (conformal) roughness (Fig. 6a), Kiessig interference fringes oc- cur in longitudinal off-specular scans. They are not present when the roughness on the multilayer is uncorrelated through the stack. As indicated above, the presence of such fringes in the off-specular scans in Fig. 7 is a clear indication of significant corre- lated roughness and the fraction can be deduced from the fringe amplitude relative to that simu- lated. This is found to be in the ratio 60 : 40, in agreement with the fits to the transverse diffuse Fig. 6. Simulated reciprocal space maps of the scattering from scan data of Fig. 3. The interference fringes in the the Co/Cu multilayer (a) with vertically correlated roughness and (b) with vertically uncorrelated roughness. The asymmetry transverse scans of Fig. 5 do not relate to conform- about the origin is a result of the decreasing illuminated area as ality in the roughness but instead arise from the the specimen angle, , is increased. The neighbouring isointens- interference between the incident and specularly ity contours represent an intensity ratio of 10 . is the detector reflected waves. Such fringes are not predicted in angle. the Born-wave approximation but are very well simulated in the distorted-wave Born approxima- tion. In this case, the top surface showed more the off-specular, longitudinal scans (Fig. 7) is in correlated roughness, the ratio being 70 : 30. The good agreement. ratio of correlated to uncorrelated roughness in the However, it is still uncertain whether the sharply multilayer, determined from the amplitude of the peaked scattering is an artefact of a figured surface, Kiessig interference fringes and the Bragg peak in rather than genuine diffuse scatter. If the specimen I. Pape et al. / Physica B 253 (1998) 278-289 287 Fig. 7. Specular and off-specular longitudinal scans showing Fig. 8. Detector-only scan for three different incidence angles interference fringes in the diffuse scatter, characteristic of strong on the specimen. correlated roughness. contained a region with a moderate amount of curvature, which was misoriented with respect to the main part of the sample, a subsidiary peak would be observed that scaled with the intensity of the main specular peak and also had the angular dependence on specimen rotation seen in Fig. 4. To resolve this question, we performed scans of the detector only at three different angles of the speci- men with respect to the incident beam. These are shown in Fig. 8. For the specimen set to 1.2 and 1.25°, the sharp specular peak and the diffuse peak are well separated. However, for the specimen at 1.89°, corresponding to the Bragg position, the two peaks are almost coincident in the detector scan. As illustrated in Fig. 9, for a figuring artefact, the peak separation in a detector scan will increase with the scattering vector. If the peak arises from stepping of the interfaces, the centroid of the diffuse scatter will be displaced from the specular ridge. The detector angle between cutting the specular ridge and the broad peak from the conformal roughness de- creases with increasing specimen angle. As this be- haviour is indeed observed, we can be confident Fig. 9. Figuring and off-centred diffuse scatter displayed in re- that the peak corresponds to diffuse scatter and is ciprocal space. not figuring. Well developed interference fringes are shown in electric field amplitude in the specimen is Fig. 8. These are unlike those found in transverse modulated by the specular Kiessig fringes. In a de- scans, such as Fig. 5, where interference fringes can tector-only scan, however, the incident beam re- arise from uncorrelated roughness, as the overall mains at a fixed angle with respect to the specimen 288 I. Pape et al. / Physica B 253 (1998) 278-289 and the electric field inside the sample remains jumping down a step edge [29] can cause local constant throughout the scan. Thus the interference irregularities that develop on the surface to blow up fringes correspond to coherence in the scattered to create large mounds. These mounds are not wave and hence coherence in the diffuse scatter, self-affine in nature and dominate the surface pro- which can only originate from correlation in file. As a consequence, the self-affinity of the surface the roughness for the various interfaces in the is destroyed. Experimentally these mounds have sample. been observed in MBE systems [29,32,33]. How- ever, we feel that the evidence available at this time merely proves or disproves the existence of self- 5. The validity of a self-affine model affine scaling in very specific examples and this for MBE growth cannot be generalised to MBE grown surfaces as a whole. Although an excellent match to a single set of In the past there have been attempts to simulate parameters was found for most data, simulations of terraced surfaces by using a fractal model [19]. the Kiessig minimum showed a sharper peak in the However, in the last year or so, it has become diffuse scatter than that observed experimentally. widely believed that a terraced surface cannot be We have found that a suppression of this peak can described by such a model [34,35], due to such be obtained by increasing the degree of correla- a surface not scaling in a self-affine way. As a result, tion between the top and bottom of the layers. As new models are being devised such as roof, terrace, the penetration of the incident wave differs for the and castallation models to account for these effects Kiessig maxima and minima, a variation in the [34,35]. With regard to the Co/Cu multilayers degree of conformality of the roughness through grown on silicon, a simple off-cut cannot be present the stack will result in a different best-fit under for two reasons. Firstly, the diffuse scatter is sym- these two different conditions. Although our model metric on rotation of the sample by 90° about its enables us to specify the fraction of correlated to surface normal, and secondly, such a simple model uncorrelated roughness at each interface, it is ex- would give rise to a blazed grating effect [36]. In tremely difficult to perform such a multivariable addition, the local tilts within this structure must minimisation. be at an angle to the average sample surface. This is In order to keep computation times reasonable required so that the diffuse peak is offset from the for this quite complex system, it was assumed that specular position. Furthermore, there can be no vertical correlations extend equally over all fre- periodicities within this structure, since no grating quencies of roughness defined within the correla- effects are seen. tion length. Thus, ignoring the optical effects of the One possible structure that may give rise to the Yoneda wings, all of the diffuse scatter in q is observed diffuse scatter is one in which the atomic changed by the same amount on the introduction steps have bunched into regions separated by large of vertical correlations in the roughness. It may be flat terraces. Although such a surface is clearly not expected [28] that the shorter periods of roughness self-affine on a macroscopic scale, the fractal model are suppressed as they propagate up through the is seen to give a reasonable fit to the experimental stack. Thus, at the Kiessig minima, where the sensi- data. This apparent anomaly, may be resolved tivity to the near surface region is greater than at when consideration is given to where the bulk of the Kiessig maxima, the centre of the transverse the diffuse scatter is arising. It is clear that the diffuse scan would be lowered. surface is most rough in the region of the step Despite problems surrounding the exact math- bunching. As a result, although it is clear that the ematical description of the behaviour of MBE sys- macroscopic surface is not scaling in a self-affine tems, a number of researchers have found self- manner, it may be possible that on smaller length affine-type scaling properties of MBE grown surfa- scales, within the bunches themselves, such scaling ces [29-31]. However, the "Schwoebel barrier", is present. Thus, the self-affine model can provide a potential barrier which tends to deter atoms from a satisfactory fit to the observed diffuse scatter from I. Pape et al. / Physica B 253 (1998) 278-289 289 certain microscopic regions on the sample if they [2] S.S.P. Parkin, R. Bhadra, K.P. Roche, Phys. Rev. Lett. 66 dominate the diffuse scatter. (1991) 2152. [3] S.S.P. Parkin, Z.G. Li, D. Smith, Appl. Phys. Lett. 58 (1991) 2710. 6. Conclusions [4] E.E. Fullerton, D.M. Kelly, J. Guimpel, I.K. Schuller, Y. Bruynseraede, Phys. Rev. Lett. 68 (1992) 859. [5] M.E. Tomlinson, R.J. Pollard, D.G. Lord, P.J. The results described here confirm our previous Grundy, Zhao Chun, IEEE Trans Mag. MAG 28 (1992) conclusion [9] that the peak, displaced from the 2662. specular scatter, in transverse scans of Co-Cu [6] K.Y. Kok, J.A. Leake, Thin Solid Films 275 (1996) 210. multilayers grown on silicon is associated with long [7] H. Laidler, C.I. Gregory, I. Pape, B.J. Hickey, B.K. Tanner, correlation length roughness at stepped interfaces. J Magn. Magn. Mater. 154 (1996) 165. [8] M. Safa, B.K. Tanner, J. Magn. Magn. Mater. 150 (1995) The roughness is strongly correlated through the L290. multilayer, giving rise to characteristic interference [9] I. Pape, T.P.A. Hase, B.K. Tanner, H. Laidler, C. Emmer- fringes in the diffuse scatter. We have used a son, T. Shen, B.J. Hickey, J. Magn. Magn. Mater. 156 DWBA theoretical model to fit the experimental (1996) 373. data and have shown that the same set of para- [10] B.K. Tanner, D.E. Joyce, T.P.A. Hase, I. Pape, and P.J. Grundy, Adv. X-ray Anal. 40 (1998) in press. meters can be used for data taken under different [11] S. Cockerton, B.K. Tanner, Adv. X-ray Anal. 38 (1995) X-ray optical conditions. While it can never be 371. possible to have sufficiently good data to obtain [12] L.G. Parratt, Phys. Rev. 95 (1954) 359. a unique solution for the structural parameters by [13] V. Holy´, T. Baumbach, Phys. Rev. B 49 (1994) 10668. fitting to one data set, the probability of a local [14] S. Dietrich, A. Haase, Phys. Reports 260 (1995) 1. [15] S.K. Sinha, E.B. Sirota, S. Garoff, H.B. Stanley, Phys Rev. minimum in the difference between theory and ex- B 38 (1988) 2297. periment surviving a major change in experimental [16] D.K. Bowen, M. Wormington, Adv. X-ray Anal. 36 (1993) parameters is extremely low. We can thus be confi- 171. dent that the fractal model used to fit the data is [17] M. Wormington, I. Pape, T.P.A. Hase, B.K. Tanner, a good representation of the interface and layer D.K. Bowen, Phil. Mag. Letters 74 (1996) 211. [18] Q. Shen, Phys. Rev. Lett. 64 (4) (1990) 451. structure in these MBE-grown Co-Cu multilayers. [19] S.K. Sinha, Physica B 198 (1994) 72. [20] H. Zabel, Appl. Phys. Lett. 43 (1) (1983) 59. [21] D.K.G. de Bohr, X-ray Spec. 24 (1995) 91. Acknowledgements [22] P. Madakson, J. Appl. Phys. 68 (5) (1990) 2121. [23] V. Holy, Phys. Rev. B 55 (15) (1997) 9960. The work was supported financially by the En- [24] R.A. Cowley, Phys. Rev. B 48 (19) (1993) 14463. gineering and Physical Science Research Council [25] R. Pynn, Physica B 198, No. 1-3 (1994) 1. [26] I. Pape, Ph.D. Thesis, Durham University, 1997. through a CASE award with Daresbury Laborat- [27] V. Holy´, T. Baumbach, Phys. Rev. B 49 (1994) 10668. ory and with the UK Department of Trade and [28] D.E. Savage, J. Kleiner, N. Schimke, Y.-H. Phang, T. Industry through a LINK project. The Durham Jankowski, J. Jacobs, R. Kariotis, M.G. Lagally, J. Appl. work was undertaken within the European Com- Phys. 69 (3) (1991) 1411. munity Human and Capital Mobility Network [29] A.L. Barabasi, Fractal concepts in surface growth, Cam- bridge University Press, 1995. ERBCHRXCT 930320. [30] Y.P. Palasantzas, Phys. Rev. B 49 (1994) 4902. [31] B. Lengeler, Phys. Rev. B 46 (1992) 7953. [32] J.L. Whitehouse, Appl. Phys. Lett. 59 (1991) 3282. References [33] R.A. Cotta, Phys. Rev. Lett. 70 (1993) 4106. [34] S.K. Sinha, Acta Physica Polonica 89 (1996) 219. [1] M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, [35] V. Holy, Proc. X TOP'96, Palermo, Italy, Il Nuovo F. Petroff, P. Etienne, G. Creuzet, A. Fiedrich, J. Chazelas, Cimento (in press). Phys. Rev. Lett. 61 (1988) 2472. [36] M.G. Lagally, Phys. Rev. B 50 (19) (1994) 14435.