JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 2 15 JANUARY 2001 Nonspecular x-ray reflectivity study of roughness scaling in SiÕMo multilayers J. M. Freitaga) and B. M. Clemens Department of Materials Science and Engineering, Stanford University, Stanford, California 94305-2205 Received 10 July 2000; accepted for publication 17 October 2000 The interfacial roughness and lateral correlation length of a series of Si/Mo multilayers with bilayer period 69 Å and number of bilayers ranging from 5 to 40 have been characterized by diffuse x-ray scattering. Superlattice peaks are preserved in offset radial scans indicating a high degree of conformality in the roughness. The lateral correlation length increases with total film thickness h as h0.55; however, the magnitude of the roughness is approximately 2 Å for all film thicknesses, in disagreement with scaling laws for self-affine growing surfaces. This observation suggests that interfaces retard the evolution of high-frequency roughness while replicating longer wavelength roughness from one layer to the next © 2001 American Institute of Physics. DOI: 10.1063/1.1332095 I. INTRODUCTION function h(r,t) at time t. According to the Family­Vicsek The continued shrinking of microelectronic device size, scaling theory,2 the time evolution of the interface width is which is nearing the diffraction limit of conventional photo- described by the relation lithography systems, necessitates the development of optical imaging systems operating in the soft x ray, or extreme ul- wL L f tLz , 1 traviolet EUV , regime. These next-generation EUV lithog- raphy systems will require reflective optics coated with where f (u) is the scaling function which has the following multilayer films. The Si/Mo multilayer coating is a leading form: candidate for this purpose due to the large electron contrast and smooth layering of its constituents. Yet, the nonideal f u u u 1 , 2 nature of the interfaces limits the reflectivity of these const. u 1 multilayer coatings. Imperfections arise during the growth and where L is the length scale over which roughness is process in part due to intermixing and reaction of Mo and Si. measured, and are the static and dynamic scaling expo- In general, roughness is defined as the standard deviation of nents, respectively, and z is / .3 In the limiting cases, Eq. the interface height and leads to nonspecular scattering. This 1 reduces to nonspecular, diffuse, scattering is problematic for EUV im- w aging systems because it decreases the useful throughput of L t t/Lz 1 , 3 the system and produces a background halo that reduces the and to contrast of the image.1 In this paper we will describe a series w of experiments using x-ray diffraction to characterize the L L t/Lz 1 . 4 evolution of roughness and lateral roughness correlations in For a self-affine surface, a plot of wL vs L on a log­log scale Si/Mo multilayers. The paper is organized as follows: Sec. II yields a straight line with slope 0 1; for a self-similar presents the theoretical background for scattering from a surface, 1. However, above some critical length , rough- single rough surface and its extension to multilayers. We ness saturates to the root mean square rms roughness describe the experiment in Sec. III, followed by results and h(r)2 1/2. This critical length for scaling can be viewed discussion in Sec. IV. We begin though by introducing some as the lateral correlation of a self-affine surface, which itself scaling concepts of surface growth theory. scales according to the Family­Vicsek relation. At the be- Surface morphology of growing films commonly shows ginning of growth, 0 because the entire surface is uncor- fractal behavior, i.e., film roughness appears similar over related. During growth increases with time. In a finite sys- many orders of magnification. Fractal objects that are pre- tem though, cannot grow indefinitely. When it reaches the served under an isotropic scale transformation are called self- size of the system L, the entire interface becomes correlated. similar; typically, surfaces must be rescaled anisotropically At this point, roughness saturates and we find and are known as self-affine. However, film morphology is L t/Lz 1 . 5 not self-affine at all length scales: at small enough magnifi- cation a growing film will appear smooth. Consider a grow- This saturation occurs, according to Eqs. 1 and 2 , when ing two-dimensional surface characterized by the height t Lz. Replacing L with , we obtain the scaling relation for the lateral correlation length a Electronic mail: jfreitag@us.ibm.com t / . 6 0021-8979/2001/89(2)/1101/7/$18.00 1101 © 2001 American Institute of Physics Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 1102 J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 J. M. Freitag and B. M. Clemens We also note that the scaling relationships involving time can be recast with film thickness when the film is de- posited at a constant rate. For instance, interface width there- fore scales as film thickness to the power . II. SCATTERING THEORY A. Scattering of x rays from a single rough surface The effect of roughness on the intensity of specular re- flections is well-known. For a single rough surface with rms roughness , the specular reflectivity Ispec is attenuated by an exponential FIG. 1. Schematics of multilayer structures: a ``ideal'' multilayer, b I 2 conformal roughness, c uncorrelated layer roughness. spec I0 exp qz 2 , 7 where I0 is the specular reflectivity of a perfect crystal and qz is the out-of-plane momentum transfer vector. The deri- that obey the scaling requirement have been proposed.6 In vation of this result is analogous to the Debye­Waller effect this study however, we defer to the most commonly used for a Gaussian distribution of thermally displaced atoms.4 In correlation function given in Eq. 11 . the case of a rough surface, to derive Eq. 7 , we still require We insert Eq. 11 into Eq. 9 and simplify the equation a Gaussian distribution of surface heights but we assume the to a one-dimensional problem by integrating both sides over disorder is frozen in. The intensity that is lost in the specular y. In practice, this integration is realised by detecting the direction is diffusely scattered by the arbitrarily rough sur- scattered intensity with a sufficiently long slit. For 1/2 or face; however, calculating the shape of this diffuse scattering 1, the equation can be solved analytically; however, we must is a complex problem. Sinha et al.5 have developed a scat- perform a numerical integration for arbitrary . With 1/2, tering theory for nonideal surfaces. In this model, the z di- the correlation function is rection is the average film normal and the surface heights are C R 2 exp R/ . 12 described by a single-valued function, h z(x,y), which is a Gaussian random variable. Starting from the first-order Born The term exp q2zC(x,0) in Eq. 9 can be expanded and each approximation of the differential cross section for scattering term Fourier transformed analytically. The integration yields d I 2 q 2 2 x ,qz 2 I0 e qz /qz d N2b2 dr dr e iq* r r , 8 V V 2 q2z 2 m 1 where b, for x rays, is the electronic scattering length 2 qx 2 m 1 m m! 1 q 2/m2 (e2/mc2) and N is the electron number density, it can be x shown that the scattered intensity per unit area is 13 which is a sum of a delta function the specular reflection I q I 2 2 2 0 exp qz 2 /qz dxdy exp qzC x,y being convolved, in practice, with a finite detector resolu- tion and a diffuse term. The diffuse term is a sum of Lorent- exp i qxx qyy . 9 zians whose width is inversely related to the correlation In this expression the height­height correlation function length . C(x,y), defined as B. Multilayer reflectivity C x,y z x x ,y y z x,y , 10 Up to this point we have only considered scattering from specifies roughness and provides quantitative information a single rough surface. In multilayers, the incident field scat- about both the height variations and the lateral correlation. A ters at each rough interface. Therefore, calculation of the simple form of the correlation function for a self-affine sur- scattering amplitude requires knowledge of the roughness face with a cutoff length , proposed by Sinha, is correlation function for each interface and their cross- C x,y C R 2e R/ 2 , 11 correlations, where the exponent is the roughness exponent introduced Cij hi r hj r , 14 in the scaling relation of Eq. 1 . Computer simulations of where i, j are interface labels. Figure 1 schematizes extreme surface dynamics with different roughness exponent give roughness distributions within a multilayer: a no rough- manifestly different interfaces: small values of produce ness, b each layer conformally replicating the underlying jagged surfaces, while values of approaching 1 appear roughness, and c uncorrelated roughness from layer to smooth. In Eq. 11 , we assume that roughness is isotropic so layer. The term ``correlated'' should not be confused with that C is only a function of the magnitude of R (x2 the lateral correlation length described previously. In the y2)1/2 for all points on the surface with coordinates sepa- present context it refers to how the profiles of consecutive rated by (x,y). Alternate forms of the correlation function interfaces map onto one another in the multilayer. For the Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 J. M. Freitag and B. M. Clemens 1103 case of uncorrelated roughness, i.e., Cij 0 for i j, the scattered intensity from each interface adds incoherently and we measure the average of the scattering cross sections of the top N interfaces, where N depends on the penetration depth of the x rays.7 Similarly, for perfect conformal roughness, we measure N statistically equivalent interfaces. It is well-known that the specular scattered fields of mul- tiple interfaces add coherently and produce peaks at the Bragg condition (qz m2 / , where is the bilayer pe- riod whose amplitudes are determined by the Fourier coef- ficients of the composition modulation.8 Rough multilayers can exhibit similar peaks in the diffuse scattering. These so- called Bragg sheets arise from interfacial defects that are to some degree replicated from interface to interface.9 Payne et al. show that conformal roughness is indeed a necessary FIG. 2. Transverse k-scans along the seventh order Bragg peak in Si/Mo condition for coherent nonspecular scattering.10 In practice, multilayers with N bilayers, where N, from top to bottom, is 5, 10, 20, and multilayer interfaces are only partially correlated. In this 40. case, the interface can be described by decomposing its pro- file function into a totally correlated component and an un- correct deposition times for Mo and Si. The technique, which correlated component representing the intrinsic roughness. takes into account the intermixing and associated contraction The correlated component scatters intensity in the Bragg of the bilayer period, is described elsewhere.11 sheets, whereas the intensity between two sheets results from uncorrelated fluctuations.7 We will see shortly that the Si/Mo multilayers presented B. Synchrotron x-ray diffraction in this paper exhibit strong coherent scattering features in- All small-angle reflectivity measurements were per- cluding finite thickness oscillations in the diffuse spectra. formed on the focused bending magnet powder diffraction Such features evidence a high degree of conformality in the beamline, BL 2-1, at the Stanford Synchrotron Radiation multilayer structure. In this regard, the scattering theory for a Laboratory SSRL . X rays of energy 8051 eV, monochro- single rough layer developed in the previous section can be mated by a Si 111 double-crystal, were focused to a spot justifiably applied to the case of the Si/Mo multilayers pre- size of 1.0 2.0 mm. The diffractometer is designed around sented here. Indeed, for perfect, conformally rough multilay- two large concentric Huber goniometers. Four types of scans ers, the scattered intensity is a convolution of the scattered were conducted: specular ­2 reflectivity scans, offset intensity of a single rough layer with the modulation factor scans with 2 /2 0 for off-specular reflectivity, trans- associated with the superlattice structure. verse or rocking scans, and, more useful, transverse k-scans in which only the transverse component of the scat- tering vector varies. III. EXPERIMENT A. Multilayer deposition IV. RESULTS AND DISCUSSION Si/Mo multilayers were deposited by dc magnetron sput- Before embarking on a detailed analysis using the theory tering in a system with a base pressure of 5 10 9 Torr presented in Sec. II, we note that an important conclusion and with an Ar sputtering pressure of 1.5 mT. The nominal can be drawn immediately by qualitative inspection of the deposition rate measured with a crystal rate monitor without diffuse scattering data. Consider in Fig. 2 the transverse tooling factors was 1 Å/s. All depositions were, nominally, at k-scans about the seventh order Bragg peak for a series of room temperature. A small temperature increase of less than Si/Mo multilayers with increasing number of bilayers N. One 4° had previously been measured under similar deposition observes quite clearly that the width of the diffuse scattering conditions. The individual layer thicknesses were controlled decreases with increasing N, indicating that the lateral with mechanical shutters over the elemental sputtering tar- length-scale of the roughness is increasing. As expected, the gets. scattered intensity, Eq. 13 , based on a self-affine roughness To study the effect of film thickness on roughness cor- model, also predicts that the width of the diffuse scattering is relations, we deposited a series of Si/Mo multilayers with inversely proportional to the correlation length. In the fol- nominal bilayer period of 69 Å, 0.4, and number of lowing analysis, we attempt to estimate the correlation length bilayers N 5.5, 10.5, 20.5, and 40.5, i.e., each stack began using this self-affine model. We begin by presenting specular and ended with Si. refers here to the volume fraction of and offset ,2 scans to establish the conformal nature of Mo in the multilayer. We also deposited a series of samples the roughness in Si/Mo multilayers. with 69 Å, N 20.5, and varying from 0.2 to 0.8. The Si 100 wafers used as substrates were cleaned with acetone A. Specular and diffuse reflectivity and ethanol. The native oxide was not stripped. We achieved An example of the small-angle specular and off-specular a bilayer period as close as possible to the desired 69 Å with reflectivities for a Si/Mo multilayer with 20 bilayers is the appropriate using x-ray diffraction to determine the shown in Fig. 3. The large number of superlattice Bragg Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 1104 J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 J. M. Freitag and B. M. Clemens FIG. 4. Reciprocal space map showing the paths of the scans from Fig. 3 as solid lines. The circles indicate the positions of the superlattice Bragg peaks. FIG. 3. Reflectivity spectra for a Si/Mo multilayer with 20 bilayers and The dotted line is the path a rocking curve about the third superlattice peak bilayer period 73.0 Å. From top to bottom, the ­2 scans are a specu- traces through reciprocal space. lar reflectivity 0 , b off-specular reflectivity with 0.25°, c 0.5°, d 1.0°, and e 1.5°. The inset is a detail of the second Bragg peak showing finite thickness oscillations in both the specular and off-specular 0.25° spectra. For clarity, off-specular scans are vertically sy /sin( i), where sy is the size of the beam defining slit. The offset as indicated in the figure. second factor is due to the increased absorption from the increased path length through the multilayer at low incident or exit angles. A correction factor for an N-bilayer mirror, peaks that extend above 16° in 2 (q estimated using kinematic diffraction theory,14 was applied z 1.13 Å 1) and high- reflectivity indicate very smooth layering. A sample with 40 to the rocking curve depicted in Fig. 5. A third factor, how- bilayers grown under the same conditions exhibited a peak ever, is more difficult to estimate. It arises from the geometry normal-incidence reflectivity of 65.7% at 136 Å wavelength of the path of a rocking curve through reciprocal space. As radiation. Qualitatively, from observation of the extinction of shown in Fig. 4, because the length of the scattering vector is higher-order Bragg peaks, the total roughness appears inde- constant in a rocking scan, its path dotted line deviates pendent of the number of bilayers. No theoretical modelling from the qx , qy plane at the superlattice diffraction condition was undertaken in this study, mainly due to the difficulty in qz when 0. The intensity of a rocking scan will conse- extracting a unique set of structural parameters from fitting quently fall off with increasing . For this reason we will to the data. Indeed, Mo and Si readily intermix at the inter- consider in our subsequent analysis only true transverse faces forming a compound with a stoichiometry close to scans in which the z-component of the scattering vector is MoSi kept constant and the diffraction condition is satisfied for all 2.12 The resulting interlayers are reported to be asym- metric with the Mo on Si interlayer roughly twice as thick as values of qx . Si on Mo.13 The complexity of this structure therefore neces- Of interest to note, rocking scans display satellite peaks sitates a large number of independent fitting parameters to at off-specular angles. The strongest of these peaks at the adequately simulate it. Instead, we will quantitatively deduce tails of the rocking curve in Fig. 5 occur when the incident or the correlated roughness from the diffuse scattering. exit angle satisfies the Bragg condition. The smaller peaks Offset ,2 scans are also depicted in Fig. 3 while Fig. 4 shows the paths through reciprocal space, as solid lines, that these offset scans trace. The circle symbols in the latter figure indicate the positions of the superlattice Bragg peaks for each scan. The presence in the diffuse spectra of these peaks due to coherent scattering, as well as finite thickness oscillations visible in the inset close-up of Fig. 3, clearly indicates a high degree of conformality and interface rough- ness replication. Recall that uncorrelated interface roughness would generate none of the coherent diffuse scattering evi- dent in Fig. 3. Transverse rocking scans provide information on the lateral length scale or in-plane correlation length of the film. An example of such a rocking scan about the third superlattice peak for the previous 20 bilayer multilayer is shown in Fig. 5. Several geometrical factors may affect the shape of these rocking curves. First, the footprint of the FIG. 5. rocking scan about the third superlattice peak in a 20 bilayer beam on the sample varies with the incident angle i as Si/Mo multilayer. Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 J. M. Freitag and B. M. Clemens 1105 TABLE I. Roughness and effective in-plane correlation length for Si/Mo multilayers with N bilayers, nominal Mo thickness ratio tMo /(tMo tSi), and measured bilayer period . N Å Å Å 5 0.4 74.8 2.5 19 10 0.4 77.5 2.6 24 20 0.4 73.0 2.2 33 40 0.4 70.8 2.1 52 20 0.2 68.0 2.4 31 20 0.6 67.4 3.2 51 20 0.8 69.3 4.1 111 appearing at intermediate angles corresponding to the second order Bragg reflection arise from double diffraction.14 Fur- thermore, Yoneda wings become manifest when the incident or exit angle equals the critical angle for total external reflection.15 None of these effects are accounted for in the fitting procedures used in this paper which we discuss in the following section. B. Fitting procedure for transverse scans In Sec. II A we developed an expression, Eq. 13 , for the diffuse scattering of x rays in the Born approximation using the Sinha correlation function, Eq. 11 , and 1/2. The validity of this choice of will be discussed shortly. Our diffuse spectra consist of transverse k-scans along the FIG. 6. a Transverse k-scan for a Si/Mo multilayer with 20 bilayers and Bragg sheets of each sample up to the eleventh order. We 0.8. The solid line is the best fit obtained by numerically integrating the model these profiles using Eq. 13 and vary the in-plane general expression for the scattered intensity with a roughness coefficient 0.5. b Close-up of high-q correlation length and roughness to obtain a best fit. x tail showing the effect on the shape of a best fit using different values of . The best-fit parameters for each were: Careful attention is paid to the convergence of the sum of 0.25, 22 Å, 1.8 Å; 0.5, 111 Å, 3.6 Å; 0.9, 216 Å, Lorentzians in Eq. 13 . For small product q 5.4 Å. z , the sum converges rapidly. With qz 2 up to 70 terms are required to achieve numerical convergence. For a given surface transverse k-scan along the fourth-order superlattice Bragg roughness and a limited number of terms in the fitting pro- sheet (qz 0.3718 Å 1) for a 20-bilayer Si/Mo multilayer cedure, this imposes an upper limit on q with 0.8. Correction factors for absorption at low incident z and the number of Bragg sheets that can be included in the analysis. or exit angles as well as the larger beam footprint at low incident angle have been applied. In contrast to rocking C. Correlated roughness results curves, no correction is necessary to account for any diver- gence from the x,y plane at the Bragg condition since the Table I summarizes the results of the diffuse scattering length of the scattering vector is automatically adjusted dur- fitting. The value quoted for roughness represents only the ing the scan with qz held constant. The solid line in Fig. 6 a correlated roughness since it is extracted from the diffuse is a fit obtained by direct numerical integration of the total scattering intensity of a single layer applied to multilayers scattering, Eq. 9 , with the correlation function of Eq. 11 under the assumption of perfect conformal roughness. In and 1/2. The fit parameters are consistent with the results general, the total roughness will have contributions from ver- obtained using the analytical expression, Eq. 13 . Figure tically uncorrelated roughness, 6 b is a close-up of the high-qx tail of the diffuse scattering 2 2 showing best fits with three different values of . Fitting the tot c u, 15 diffuse scattering with smaller values of , i.e., more jagged where c and u are the correlated and uncorrelated rough- surfaces, requires in general a smaller correlation length. The ness, respectively. As justified earlier, the roughness of the lateral correlation length decreases from 216 Å to 22 Å for mirrors in this study appear to be highly conformal. There- ranging from 0.9 to 0.25. The figure clearly shows that the fit fore we may presume that the correlated roughness will using a scaling coefficient of 1/2 best describes the shape of dominate the uncorrelated contribution. the diffuse scattering. The fitting procedure described in the Up to this point, Eq. 13 has been used exclusively to previous section is therefore justified. model the diffuse scattering without justification of the choice of the scaling coefficient 1/2. This assumption was D. Roughness scaling necessary to analytically solve the general expression for the Finally, we discuss the scaling laws for roughness pre- total scattering, Eq. 9 . Figure 6 shows an example of a sented earlier and upon which our particular choice of rough- Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 1106 J. Appl. Phys., Vol. 89, No. 2, 15 January 2001 J. M. Freitag and B. M. Clemens ent scattering effects in the diffuse spectra and on the high- reflectivity exhibited by the multilayers. Larger total rough- ness would suppress the high-order Bragg reflections that we observe. Furthermore, AFM measurements give rms rough- ness in agreement with the values reported in Table I. A question that remains unanswered is how roughness can be conformal while the in-plane correlation length in- creases. One possible reason, suggested by Savage et al.,16 is that the interfaces act to preferentially smooth the high- frequency components of roughness. The long wavelength components, on the other hand, are replicated through the multilayer stack and give rise to coherent diffuse scattering. Although the roughness values listed in Table I are, within experimental error, very similar, there appears to be a trend toward slightly smaller correlated roughness for samples FIG. 7. Lateral correlation length as a function of total thickness for a with more bilayers. The multilayers remain, nevertheless, series of Si/Mo multilayers with 0.4. The solid line is a power-law fit with the scaling parameter 1/z 0.55. highly conformal because growth attenuates only high- frequency roughness. The spatial filtering hypothesis is plau- sible in the context of surface growth models which incor- ness correlation function depended. Recall from Eq. 6 that porate a surface relaxation term since relaxation tends to the time evolution of the lateral correlation length should preferentially flatten sharp surface features see Ref. 17, for follow a power-law scaling relationship instance . Under such conditions, large surface defects would 0t1/z, where 0 is the correlation length for a given time t be propagated from the substrate to the top of the film de- 0 . For a constant deposition rate, time can be replaced by thickness. Figure 7 grading the reflective optics and masks used in extreme ul- shows the evolution of correlation length as a function of the traviolet lithography systems. total thickness h N for the series of multilayers with 0.4 and number of bilayers N ranging from 5 to 40. The V. CONCLUSIONS solid line is a power-law fit with a scaling coefficient 1/z The evolution of roughness and roughness correlations 0.55. Admittedly, the small data set presented in this graph were measured in sputtered Si/Mo multilayers by synchro- prevents drawing any strong conclusions. Nevertheless, tron x-ray diffraction. An expression for the total scattered given 1/2 and z / , our result implies a dynamic scal- intensity from a self-affine rough surface was presented. This ing coefficient 0.28. According to the scaling law, Eq. expression was extended to multilayers with correlated, con- 3 , roughness is expected to increase with thickness to the formal roughness. The conformal nature of the roughness in power ; however, Table I shows that roughness appears to Si/Mo multilayers was confirmed by the presence of coher- be independent of thickness within experimental error. Other ent scattering in the diffuse spectra. A static scaling coeffi- studies report similar findings for W/C multilayers.16 The cient 1/2 for roughness best describes this diffuse spectra. authors speculated there that the interfaces suppress the in- By fitting the shape and intensity of the diffuse spectra, we crease in roughness by providing periodic ``restarting lay- extracted the roughness and in-plane correlation lengths. A ers'' during growth. However, the mere presence of the in- series of multilayers grown with the same nominal bilayer terfaces is insufficient to completely explain the roughening structure and number of bilayers varying from 5 to 40 dem- process. For Si/Mo multilayers, both the lateral correlation onstrates that the correlation length increases, scaling with a length and the roughness increase with , as can be seen in coefficient / 0.55. The roughness of the multilayers, Table I. The Si layers, which are amorphous, appear to however, is independent of thickness, contrary to the scaling smooth the cumulative roughness intrinsic to polycrystalline law prediction. This discrepancy is ascribed to the smoothing Mo growth. effect of the amorphous interfaces. Conformal layer-to-layer In contrast, Stearns et al.1 report that roughness doubles roughness replication is retained assuming that interfaces from 0.9 Å at the substrate to 1.8 Å at the top surface in a preferentially flatten only the high-frequency components of 40-bilayer Mo/Si multilayer. Their observation is based on roughness. integrating the power spectral density PSD function which is the Fourier transform of the autocorrelation function ob- ACKNOWLEDGMENTS tained by atomic force microscopy AFM over the entire frequency range. Roughness at each layer is inferred from One of the authors J.M.F. gratefully acknowledges the PSDs of the bare substrate and top multilayer surface support from the Stanford Graduate Fellowship program. using a linear growth model to describe the growth dynam- Synchrotron work was done at SSRL, which is operated by ics. It is possible that the correlated roughness reported in the the Department of Energy, Office of Basic Energy Sciences. present paper greatly underestimates the total roughness which includes an uncorrelated contribution; uncorrelated 1 D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, J. roughness would, ostensibly, increase during growth. This Appl. Phys. 84, 1003 1998 . 2 F. Family and T. Vicsek, J. Phys. 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Cowley, Diffraction Physics North­Holland, Amsterdam, 1984 . 9 J. M. Elson, J. P. Rahn, and J. M. Bennett, Appl. Opt. 19, 669 1980 . 71, 3283 1992 . 10 17 A. P. Payne and B. M. Clemens, Phys. Rev. B 47, 2289 1993 . A.-L. Baraba´si and H. E. Stanley, Fractal Concepts in Surface Growth 11 J. M. Freitag and B. M. Clemens, Proc. Mater. Res. Soc. 562, 177 1999 . Cambridge University Press, Cambridge, 1995 . Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp