PHYSICAL REVIEW B VOLUME 57, NUMBER 15 15 APRIL 1998-I Nonspecular x-ray-reflectivity study of partially correlated interface roughness of a Mo/Si multilayer D. R. Lee Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea Y. J. Park Pohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang 790-784, Korea D. Kim and Y. H. Jeong Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea K.-B. Lee* Department of Physics and Pohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang 790-784, Korea Received 22 September 1997 Nonspecular x-ray-reflectivity intensities were measured to characterize the interface morphology of a Mo/Si multilayer. Longitudinal off-specular scans and transverse scans at several multilayer peaks and valleys were carried out. For the analysis of the experimental data, a height cross-correlation function between different interfaces was derived for a model multilayer whose interfaces are partially correlated. The parameters related to the interface morphology were obtained by fitting the measured intensities within the distorted-wave Born approximation. The intermixing widths of the graded interfaces, the correlated interface roughness amplitude, and a vertical correlation length were obtained by analyzing the off-specular intensities. S0163-1829 98 04615-3 Specular and nonspecular x-ray reflectivity in grazing in- interface, while uncorrelated random noise generates a devia- cidence angle have been effectively utilized to characterize tion from perfect correlated interfaces. For partially corre- interface morphology. While specular reflectivity yields a lated interfaces, the interface profile can be related to the density profile perpendicular to the sample surface, non- adjacent one by the following recursive equation:7 specular scattering yields lateral interface structures. Sinha et al. formulated a distorted-wave Born approximation h m 1 r Bm m 1 r Amh m r , 1 DWBA formalism for a single rough surface to analyze x-ray-reflectivity intensities near the critical angle.1 The ex- where r is the lateral coordinate in the average interface tension of the DWBA calculation to layered structures were plane. h m(r ) and m(r ) refer to the normalized profile of the done by Holy´ and Baumbach2 and others3,4 for epitaxially mth interface and normalized deviation from the perfect rep- grown heterostructures with relatively sharp interfaces. lication of the (m 1)th interface, respectively. The first Though the intermixing of elements between layers is an- term on the right-hand side of Eq. 1 represents the devia- other crucial factor determining interface characteristics, es- tion from a perfect correlated interface due to random noise pecially in magnetic metallic multilayers, a DWBA analysis at the mth interface. Am is a replication factor representing for nonspecular reflectivity intensities of multilayers with the degree of conformality, and satisfies the relation 2 2 graded interfaces has not been reported earlier, to our knowl- Am Bm 1. Then h m(r ) can be expressed as edge. m m 1 In this work we carried out nonspecular x-ray-reflectivity studies of a Mo/Si multilayer which has graded interfaces. h m r Bi 1 Aj i r . 2 i 1 j i The experimental data have been analyzed within the DWBA to extract the parameters related to its interface mor- Assuming that Ai's and Bi's are independent of i, the self- phology, including the intermixing widths. A simple model correlation function of the interface roughness is is presented to describe partially correlated interfaces in mul- m m tilayers. The cross-correlation function, which has been 2 widely used without derivation in order to characterize par- hm r hm 0 m B2Am iAm j i r j 0 i 1 j 1 tially correlated interfaces of epitaxially grown multilayers,4­6 is derived from the model, and has been ap- 2 2 m 1 A2m r 0 m r 0 , plied to analyze our nonspecular x-ray-reflectivity data. During deposition process of multilayers, a height profile 3 hm(r ) of the mth interface with respect to the average inter- where A2m is negligible and i(r ) j(0) vanishes for i j face plane z¯m is conformally transferred to the (m 1)th because they are uncorrelated to each other. Here m is the 0163-1829/98/57 15 /8786 4 /$15.00 57 8786 © 1998 The American Physical Society 57 BRIEF REPORTS 8787 rms amplitude of the correlated roughness of the mth inter- face. Then the morphologies of the interfaces are similar to each other, and can be expressed with the same parameters. If they can be described as self-affine, the self-correlation function is approximately proportional to 2me (r/ x)2H.1 Here r and H are the distance along the interface plane and the roughness exponent describing the jaggedness of the in- terface, respectively. x is the lateral cutoff length. The cross-correlation function between the mth and nth inter- faces (m n) can be expressed as Cmn r hm r hn 0 Am n hn r hn 0 m ne r/ x 2He z¯m z¯n / z 4 for Am n e z¯m z¯n / z. z reperesents the vertical correla- tion length along the layer growth direction. Here we note that roughness components of different wavelengths have the same replication factor A resulting in a constant correlation length z . If A depends on the wavelength of the interface roughness,8 the correlation functions in Eqs. 3 and 4 should be expressed as a convoluted function of the replica- tion factor and the random noise. Then z is a function of lateral component (q xy) of the momentum transfer of scat- tered x rays. FIG. 1. Specular scan and off-specular scans at two diffferent With the cross-correlation function given above, scattered offset angles: a the measured specular intensities circles and the x-ray intensities from a multilayer with N interfaces can be result of fitting solid line . Arrows indicate the values of qz , where expressed in the DWBA as2,3 rocking scans have been performed. b The measured off-specular intensities at an offset angle of 0.055° circles and the results of the 1 calculation for sharp interfaces without any intermixing dashed I q /I d 0 R qz 2 q xy l d , line . The solid line represents the best fit for the graded interfaces x ly d diff which have both correlated roughness and intermixing. c The measured off-specular intensities at an offset angle of 0.6° circles , d k4 N o 2 2 2 2 and the results of calculations with different values of z : 300 d ni ni 1 nj nj 1 * Å dashed line , 600 Å solid line , and 1200 Å dash-dotted line . diff 16 2i,j 1 3 Di 1Dj 1* cial width for the ith graded interface, containing both its S m n e 1/2[ 2 i 1 2 j 1* i,t qmz 2 j,t qnz 2] interface roughness amplitude and intermixing width.9 m,n 0 qi 1 j 1 mz qnz * The sample of this study is a Mo/Si 30 multilayer pre- pared by the rf magnetron sputtering method as described d2r i 1 j 1 eq * mz qnz Cij r 1 eiq xy*r , 5 elsewhere.10 X-ray-reflectivity measurements were carried out at a bending magnet beamline 3C2 at the Pohang Light where I(q ), I Source PLS .11 X rays of 1.608 Å monochromatized by 0, R (qz) , and (d /d )diff are the scattered intensity with momentum transfer of q , the incident inten- a Si 111 double-crystal monochromator were focused at the sity, the specular reflectivity, and the differential cross sec- sample position by a torroidal premirror. Three different tion of the diffuse scattering, respectively. l types of scans were conducted with a Huber four-circle dif- x ly is the beam cross section, while S represents the illuminated area by in- fractometer: 2 scans for specular reflectivity, offset cident x rays. k 2 scans for off-specular reflectivity, and transverse o is the wave-vector length in vacuum, and the n rocking scans. i's are refractive indices in the ith layer. Dm's and qmz's for each layer are denoted by3 Figure 1 shows the results of the x-ray-reflectivity mea- surements from the sample. Specular reflectivity intensities D show peaks corresponding to a bilayer thickness of 118.8 0 T1T2 , D1 T1R2 , D2 R1T2 , D3 R1R2 , Å in Fig. 1 a . Two off-specular scan intensities with offset q angles of 0.055° and 0.6°, as shown in Figs. 1 b and 1 c , 0z k1z k2z , q1z k1z k2z , q2z q1z , also show sharp resonant diffuse scattering RDS peaks in- q dicating strong vertical correlation in the interface 3z q0z , 6 roughness.2 The measured intensities for qx 0 in Fig. 1 a where the amplitudes T1,2 and R1,2 of transmitted and re- have been fitted, after subtracting diffuse intensities, using flected waves in a multilayer with ideally smooth interfaces Parratt's recursive relation12 modified for rough interfaces are defined for an incident wave vector k1 and for a scattered with error function interface profiles. Later the parameters wave vector k2, respectively. i,t represents a total interfa- were refined by fitting the measured intensities with the full 8788 BRIEF REPORTS 57 expression of Eq. 5 , as described below. The broadening of the peak widths at high qz's can be explained by incoherent random errors in each layer thickness,13 and the fluctuation in the thickness was estimated to be 1.2 Å. The solid lines represent the fit, while the circles represent the measured intensities. Fitting gives the total interfacial widths t's of 2.4 and 3.5 Å for Si-on-Mo interfaces and Mo-on-Si inter- faces, respectively. However, there is an ambiguity in the estimates of t's because the calculations give almost the same results when the values of two t's are exchanged. This appears to be a common difficulty in the analysis of multilayers.14 It has been revealed by TEM studies15 that amorphous intermixed layers are formed on Mo-on-Si inter- faces and these interfaces have thicker interfacial widths than Si-on-Mo interfaces, in both sputter deposited and evapo- rated multilayers. Therefore, we assign the larger value of t to the interfacial width of Mo-on-Si interfaces of our sample. Other parameters can be determined from an analysis of non- specular reflectivity intensities, as described below. If elements are intermixed in a certain range across the interfaces, the interfaces are graded ones which have both interface roughness and interfacial widths due to intermix- ing. For these graded interfaces, it is difficult to estimate interface roughness amplitudes c's and intermixing widths d's by analyzing the specular reflectivity data which pro- vides only t's. We note that Cij(r ) in Eq. 5 is the only term containing c's explicitly, and, therefore, an analysis of diffuse scattering intensities is necessary to estimate the pa- rameters related to the interface roughness. In order to calculate nonspecular scattering intensities, we have used the whole expression of Eq. 5 without a small-qz approximation, because it has been shown by others6 that there is a significant discrepancy in the region of qz 1 be- tween the results of calculations with the full expression and FIG. 2. Rocking curves at multilayer Bragg peaks a and at with the approximation. However, some parameters are more their valleys b . Solid lines represent fits using the correlation func- sensitive to specific scans. For example, a longitudinal off- tion and the parameters explained in the text. The intensities are specular scan is sensitive to correlated roughness amplitudes shown as a function of inc (2 /2) in order to illustrate the low c's, because it scans the intensities for different qz's while qz scans better while qx qz sin inc (2 /2) . q xy 0. Therefore, the parameters were roughly estimated first with data of the corresponding scans to avoid the dan- specular intensities were fitted simultaneously to minimize gers of becoming trapped in local minima in the fitting pro- possible errors due to the interplay between various param- cess. The lateral coherence length was roughly estimated eters used in the analysis.4 from the full width at half maximum of the diffuse part of Off-specular intensities were calculated for both models rocking curves at multilayer peaks. Then the coherence with and without graded interfaces, which is an extension of length and roughness exponent were evaluated together by the work of Wormington et al. for a single graded surface9 to fitting the shapes of diffuse parts of the rocking curves at a multilayer with graded interfaces. Figure 1 b presents the peaks with large-q measured off-specular scan intensities as well as the results z values. In this study we have used an averaged value of correlated roughness amplitudes for all of calculations. For a multilayer with sharp interfaces, the interfaces, with an assumption that the major features of cor- values of the total interfacial widths estimated above from relations between different interfaces in our sample can be the analysis of the specular scan intensities were used both described by averaged values for correlated roughness am- for i,t's and c's in Cij(r ) of Eq. 5 . The results of the plitudes and vertical correlation lengths. Since the shapes of calculation are presented as a dashed line. At high qz's the rocking curves at fixed qz values do not depend strongly on calculation gives higher intensities than the measurements, a correlated roughness amplitude, the amplitude was esti- and this is explained by the overestimated c's, resulting in mated from the slope of resonant diffuse peaks in longitudi- high diffuse scattering intensities. For a multilayer with nal off-specular intensities. The vertical correlation length graded interfaces, the total interfacial width t can be ex- was evaluated from the ratios bewteen the intensities of the pressed as 2 2 2 t c d , where c and d represent the cor- peaks and valleys of off-specular scans. To refine the param- related roughness amplitude and the intermixing width, re- eters, the whole set of the x-ray-reflectivity data including spectively. We have also calculated the diffuse scattering 57 BRIEF REPORTS 8789 intensities for these graded interfaces, and the best fit yields In this work we have assumed the same replication factor c of 1.8 Å. Then d's for Si-on-Mo and Mo-on-Si inter- for roughness components of different wavelengths, which faces were estimated as 1.6 and 3.0 Å, respectively. The results in constant z . Generally, it is expected that the com- result of the calculation is presented as a solid line, and it bined effects of random noises from deposition and thermo- shows good agreement with the measured intensities. We dynamic kinetics on growing surfaces result in a wavelength- have also simulated, for a model of accumulative interface dependent replication factor A(qx), and generate surfaces roughness, 2 2 whose correlation lengths and roughness amplitude change c, j 0 j 2, and found that it shows the same tendency. as functions of time or thickness before they saturate.16 In In off-specular scans of partially correlated systems, rela- multilayers the interfaces manifest such evolutions in the tive intensities at the valleys with respect to those of the RDS form of broadening in off-specular peaks, and observations peaks are sensitive to the value of of such evolutions have been reported recently.17­19 In this z , because the extent of the interference depends on the vertical correlation. For study, off-specular x-ray-reflectivity measurements have CoSi been performed, as shown in Figs. 1 b and 1 c . Off- 2/Si/CoSi 2 layers on a Si 111 substrate, z has been estimated from the amplitude of the oscillations at high q specular peaks become broader at higher q z by z's. However, the others.4 Figure 1 c shows the measured intensities with an broadening in off-specular peaks is also attributed mainly to offset angle of 0.6° with the results of the calculation with the incoherent fluctuations in layer thickness, because the three different values of peak widths at the same q z : 300, 600, and 1200 Å. The z do not show any significant qx vertical correlation length was estimated to be 600 Å, which dependence up to a value of qx of 0.03 Å 1, and RDS of is about one-sixth of the total thickness of the multilayer. broader specular peaks at higher qz's also gives broader off- Figure 2 shows the measured rocking curves at multilayer specular peaks of the same peak widths. These results imply peaks Fig. 2 a and at the valleys between them Fig. 2 b . that the replication factor for the roughness is constant for a The rocking curves around the valleys in Fig. 2 b show length scale longer than 200 Å, which is consistent with the different shapes of diffuse scattering from those of the estimate for lateral cutoff length 70 Å. Further studies in multilayer peaks in Fig. 2 a . The diffuse scattering intensi- different diffraction geometries are required for the extended ties at the valleys are relatively flat with respect to q range of the lateral momentum transfer18,19 to observe the x , and the ``Yoneda wings'' are more apparent compared to those transition from the conformal behavior. in the rocking curves in Fig. 2 a . This is because the trans- In summary, we presented a comprehensive x-ray- verse rocking scans at the peaks are the scans along the RDS reflectivity analysis for a Mo/Si multilayer sample with streak which have broad humplike peaks around specular graded interfaces. A simple model which describes partially peaks.2 The measured intensities have been compared with correlated interfaces was presented, and the correlation func- the calculation for I(q tion derived from the model was applied to analyze the mea- x), which is the integrated intensity of I(q ) along the y direction due to the wide slitwidth in the sured intensities with the DWBA calculations. The param- perpendicular direction to the diffraction plane. The best fit eters related to the morphology of the graded interfaces were gives 70 Å and 0.4 for extracted from the analysis. It was also demonstrated that the x and the H exponent, respectively. Solid lines in Figs. 2 a and 2 b represent the results of the analysis of the off-specular reflectivity intensities is effective fitting. Though the vertical correlation length to separate the correlated roughness amplitude and the inter- z is estimated as 600 Å, corresponding to one-sixth of the total film thick- mixing widths of the graded interfaces. ness, calculated intensities with the same set of parameters The authors thank V. Holy´ for valuable discussions. This show main features including Bragg-like peaks at corre- work was supported in part by the Korean Ministry of Edu- sponding angles in the rocking curves of many different or- cation through Research Fund and BSRI/special fund at der multilayer peaks and their valleys. This implies that our POSTECH, by KOSEF 96-0702-01-01-3 , and through estimates properly represent the average values of the param- ASSRC at Yonsei University. 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