Applied Surface Science 141 Z . 1999 357­365 Surface roughness of thin layers-a comparison of XRR and SFM measurements O. Filies a, O. Boling a ¨ , K. Grewer a, J. Lekki b,), M. Lekka b, Z. Stachura b, B. Cleff a a Institute of Nuclear Physics, UniÕersity of Munster, ¨ Munster, ¨ Germany b Institute of Nuclear Physics, Radzikowskiego 152, 31-342 Cracow, Poland Received 29 May 1998; accepted 12 August 1998 Abstract X-ray reflectivity Z . XRR studies of thin layers Z3 to 120 nm . thick were performed for the determination of layer thickness, density and roughness. The simulations of X-ray reflectivity measurements were performed using Parrat's recursive algorithm, while those of the reflection of X-rays from interfaces were performed using Fresnel formulae. Using this approach, the roughness of the interface was described by intensity damping by gaussian type functions. This allowed for the determination of layer thickness and density and average interface roughness. As an extension of this simple model, an enhanced theoretical description of rough interfaces proposed by Sinha was applied, where the X-ray reflection from interfaces was separated into a direct fraction and a diffuse scattered one with the use of the first Born approximation. A simulation procedure, calculating both fractions of the reflection was developed, that enabled the detailed characterisation of layers and inner layers. The complementary information required for proper adjusting of input simulation parameters was obtained from SFM measurements of the investigated surfaces. Surface roughness was described using fractal surface functions instead of simple gaussian peaks. A comparison between this method and SFM measurement shows a reasonable agreement, particularly in the estimation of shapes of interface structures. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 68.35Bs; 78.20Ci; 07.79 Keywords: X-ray reflectivity Z . XRR ; SFM; Thin layers; Surface roughness; Fractal surface scaling 1. Theoretical background ness. It allows also nondestructive studies of inner layers. Standard simulation methods of such X-ray X-ray reflectivity technique is a relatively simple, spectra use a simple, but functional term for the but powerful method for the determination of thin description of the interface roughness in a form of layer density, thickness and layer interface rough- X-ray intensity damping. X-ray reflections from multilayer systems of ny1 layers and n interfaces may be calculated using ) Corresponding author. Tel.: q48-12-637-0222 ext. 271; Fax: Parrat's recursive formulae w1,2x. The idea of this q48-12-637-1881; E-mail: lekki@alf.ifj.edu.pl approach is presented in Fig. 1. 0169-4332r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S0169-4332Z98.00524-8 358 O. Filies et al.rApplied Surface Science 141 ( ) 1999 357­365 The final recursion formula constructed with the use of Eqs. Z . 1 ­Z . 3 has the form: Rn,nq1qFny1,n 4 Rny1,nsany1 , Z4. Rn,nq1Fny1,nq1 where: fnyfny1 Fny1,ns , Z5. fnqfny1 The X-ray reflection calculations should be started at the lowest significant layer n Zi.e., for the layer number nq1 the factor Rnq1s0 and anq1s . and performed upwards until the topmost layer is Fig. 1. The idea of the recursive approach: X-rays coming from reached. medium 0 Zair or . vacuum are scattered and deflected at the first interface Zmedium . 1 . The scattered fraction is again scattered and deflected at the next interface. This mechanism is reproduced until the last significant layer is reached. Every reflected fraction interferes with fractions reflected at previous interfaces, producing 2. Interface roughness-Nevot's ´ model finally a measurable pattern. The simple model presented above describes only ideal interfaces: flat, homogeneous and isotropic. In For the ith interface the Fresnel factors f real measurements, such conditions are not met: in i are defined by Eq. Z . 1 , while Fresnel coefficients F general, the real surfaces are rough, inhomogeneous i j Zwhere jsiq . 1 are calculated using Eq. Z . 2 . and anisotropic, while the most significant role is played by interface roughness. 2 f The first simple description of interface roughness i s (u y 2 di y 2 i bi , Z1. was presented by Nevot ´ et al. w x 3 , where Gaussian where u is an incident angle, d is the dispersion functions were used for roughness modeling. Ac- factor and b represents absorption. cording to their model, the rough interface may be n approximated by a gaussian distribution of peaks and i sin ui y n j sin uj fiyfj Fijs s . Z2. valleys with respect to the mean surface ZFig. . n 2 . i sin ui q n j sin uj fiqfj The distance dn between interfaces Zand therefore also the thickness of layer n. may be described using the exponential thickness factor an defined as fol- lows: anseyik1 fngnr2 seZip rl.fndn, Z3. where kn is the wave vector in z-direction Zper- pendicular to the . surface and l is the X-ray wave- length. Therefore, the thickness factor an represents the attenuation of X-rays passing twice through layer n Fig. 2. Interface roughness according to Nevot. ´ The distribution of of thickness dn. The thickness factor of medium 0 is peaks and valleys on a mean interface level is described using the neglected, i.e., a0s1. gaussian function and its s parameter. O. Filies et al.rApplied Surface Science 141 ( ) 1999 357­365 359 Thus, the roughness coefficient may be calculated the type of surface: low h values represent sharply w x 3 as: formed surfaces, while higher h values correspond to s 2 mild curvatures. n z Self-affine property may be introduced into X-ray n s exp y8p f z l / 2 n fny1 , Z6. reflectivity formulas through the definition of the and the recursion from Eq. Z . 4 should be corrected mean-square height-deviation function GZ R.: w x 4 to the form: 2 GZ R. s² zZ r . yzZ rqR. :, Z10. RXny1,nsRny1,n zn. Z7. closely related to a height­height correlation func- tion CZ R.: 3. Fractal approach-Sinha's model CZ R. s² zZ r . =zZ rqR. :, Z11. by: The simulation of real interfaces with the use of roughness coefficients z 2 GZ R. s2 s yCZ R. . Z12. n is only a rough approxima- tion. Its weakness is particularly significant for non- In reality, for large separations RTM , it is desir- gaussian type of roughness, for example, for surfaces able to introduce the physical distance limitation, the with steps, periodical changes, etc. w x 5 . The simple model does not include the diffuse scattering from a rough surface, which may be significant. The model proposed by Sinha w6,7x and Palasantzas w x 8 brings a vast improvement of the simple model by separating the reflected X-ray beam into two fractions: specular and diffuse: RsRspecqRdiff . Z8. The specular part corresponds to Fresnel equa- tions for smooth surface, while the roughness influ- encing the diffuse fraction is described by a separate equation and treated as perturbation. As a calculation tool serves the Born approxima- tion Zfor large reflection . angles or Distorted Wave Born Approximation ZDWBA, for small . angles . In this way, it becomes possible to introduce into the theoretical model the interfaces described by growth functions Zobtained, for example, by simulations of molecular beam epitaxy . processes or by dedicated interface functions used in the Kardar­Parisi­Zhang model w9,10x. In particular, it is possible to apply periodical w5,7x or fractal functions w x 11,12 . Fractal type surfaces may be characterized by their scaling property. Surface zZr. is considered to be self-affine if the following simple equation is fulfilled: s Z L. ;Lh, Z9. where L represents the length scale Zsystem . size and s is a well-known root-mean-square value of Fig. 3. TOF-RBS spectra of CorAg thin layers deposited on a Si substrate at 108C Z . left and 1308C Z . 2 right . The measured mass the surface height variable s s² 2 z Zr.:. The Hurst density values are shown in figure insets Znumbers in parentheses exponent h Z0FhF . 1 contains information about represent values expected from deposition . conditions . 360 O.Filies etal.rApplied Surface Science 141 ()1999 357 ­365 Fig. 4. XRR spectra and simulations performed according to Nevot's ´ and Sinha's algorithms for the following layers deposited on a Si substrate: Z . a 7.3 nm Cor4.4 nm Ag double layer ZTable 1, sample . 1 , Z . b 96 nm Alr16 nm Ti double layer Zsample . 9 , Z . c 19 nm single In layer Zsample . 5 , Z . d 21 nm single Sn layer Zsample . 8 . Layers' thickness were obtained from simulation following either Nevot's ´ or Sinha's model. O. Filies et al.rApplied Surface Science 141 ( ) 1999 357­365 361 Table 1 The comparison of roughness parameters obtained for several selected samples from XRR and SFM measurements Sample Layer Thickness Roughness Z . nm Roughness Z . nm ZNsNevot, ´ Ss . Sinha Z . nm Z . SFM 1. CorAg on Si, evaporation at 108C Co 7.4"0.7 0.29"0.04 Z . NS 2.06"0.4 Ag 4.44"0.4 2.11"0.32 Z . NS Si ­ 0.85"0.13 Z . NS 2. CorAg on Si, evaporation at 108C Co 8.1"0.8 0.33"0.05 Z . NS 2.57"0.5 Ag 4.98"0.5 2.01"0.30 Z . NS Si ­ 0.93"0.14 Z . NS 3. CorAg on Si, evaporation at 1308C Co 4.2"0.4 1.36"0.20 Z . S 1.67"0.6 Ag 3.96"0.4 0.30"0.05 Z . S Si ­ 0.5"0.07 Z . S 4. Al2O3rSn on Si, magnetron sputtering Al2O3 99.4"14.9 2.79"0.56 Z . S 3.71"0.74 Sn 19.9"3.0 2.63"0.53 Z . S Si ­ 1.19"0.24 Z . S 5. In on Si, magnetron sputtering In 19.2"1.9 0.79"0.12 Z . S 3.91"0.78 Si ­ 2.29"0.34 Z . S 6. Al2O3 on Si, magnetron sputtering Al2O3 16.4"1.6 1.17"0.18 Z . N 1.32"0.27 Si ­ 0.71"0.11 Z . N 7. Ti on Si, magnetron sputtering Ti 17.1"1.7 1.40"0.21 Z . S 1.88"0.38 Si ­ 0.47"0.07 Z . S 8. Sn on Si, magnetron sputtering Sn 21.06"2.1 2.63"0.53 Z . S 1.93"0.39 Si ­ 1.38"0.28 Z . S 9. AlrTi on Si, magnetron sputtering Al 96.5"14.5 2.54"0.5 Z . NS 35"3 Z . ! Ti 15.8"2.4 0.75"0.15 Z . NS Si ­ 1.04"0.21 Z . NS Layer thickness values were calculated from the XRR data analysis. cutoff length j. In the simplest model, GZ R. and performed over the full sample area. Eq. Z . 14 was CZ R. functions may be represented by: used as the computational basis for the X-ray reflec- tivity simulation and analysis software, DiffTool 2 h R w x 1 2 13 . The examples of its application are presented g Z R. s2s 1yexp y , z zj/ / in the experimental part of the actual paper. The height­height correlation function CZ R. or mean- 2 h R square height-deviation function GZ R. may be ob- 2 CZ R. ss exp y , Z13. z z tained from complementary measurements, and ap- j / / plied to X-ray reflectivity calculations. The most Then, using the previously mentioned mixture of straightforward method for this purpose seems to be the Born approximation and DWBA, the reflection described by Eq. Z . 7 should be modified to the form: X 2 Rn,ny1(Rn,ny1 zn,ny1qRn,ny1 zn,ny1Z1yFn,ny1. 1 DiffTool is a complete application software for X-ray reflec- =HZexp Z f 2nCZ R.. y1. J0Z R. R dR, tion Z . XRR analysis and simulation, with the use of different Z14. simulation algorithms. The package contains also the correction and analysis tools for X-ray diffraction Z . XRD , small angle diffraction and rocking curve scans. DiffTool 1.01beta is freeware where J Z 0 R. is the Bessel function and integration is and can be downloaded from http:rrpikp15.uni-muenster.der. 362 O. Filies et al.rApplied Surface Science 141 ( ) 1999 357­365 Fig. 5. Topography of four samples presented in Fig. 4 measured using SFM in air. the scanning force microscopy Z . SFM , providing the Most of the samples were characterized by the real, three-dimensional topographical data. In this time-of-flight Rutherford backscattering technique way both, CZ R. andror GZ R. functions may be Z . q TOF-RBS using 300 keV He ions. The RBS computed directly from the SFM data. The calcu- technique provided independent measurements of lated functions may serve as a check of X-ray spec- layers mass density Z 2 in mgrcm . and profile. Fig. 3 tra simulation correctness and as a source of inde- shows the TOF-RBS spectra of two double layer pendently measured input values for the simulation cobaltrsilver structures evaporated on silicon sub- procedure. Similar comparison was performed in a strate: the first one evaporated at 108C and the recently published work of Wang w x 14 . second one at 1308C. Both measurements yield quite close values of cobalt and silver mass densities. Fig. 4 presents XRR measurements Zlogarithm of counts 4. Experimental vs. detection . angle and simulations performed using Nevot's ´ and Sinha's algorithms of four selected sam- The above approach was used for the characteri- ples. zation of several tens of metal and metal oxide XRR spectra deliver more information than the layers, deposited on silicon substrates with the use of TOF-RBS measurements: from simulations it is pos- physical vapor deposition Z . PVD and magnetron sible to gain both the layer thickness and the inter- sputtering techniques. In order to assure diversified face roughness, as is presented in Table 1. It is deposition conditions, deposition was performed at desirable however, that simulation should be justified different temperatures, ranging from 10 to 1308C. using complementary measurements. Therefore, as Full results of all measurements and analyses are the next step of experimental procedure, the SFM presented in Ref. w x 4 . In the present paper, several images of the same samples were collected ZFig. . 5 , example cases will be presented. using a home built scanning force microscope w x 15 . O.Filies etal.rApplied Surface Science 141 ()1999 357 ­365 Fig. 6. A comparison of surface fractal scaling properties of four example surfaces ZFigs. 4 and . 5 measured using SFM Z . directly and XRR Zsimulation according to Sinha's . model . 363 364 O. Filies et al.rApplied Surface Science 141 ( ) 1999 357­365 All SFM data were obtained in air. The surfaces of may be obtained by simple fitting. For XRR mea- samples show quite different character, depending surements the definition of the GZ R. function from not only on layer composition but also on deposition Eq. Z . 13 must be used and the values of j and h are conditions. a result of fitting the XRR spectrum by Eq. Z . 14 . Table 1 presents several comparisons between the root-mean-square surface roughness obtained from XRR spectra fitting and from SFM measurements. Both methods produce usually roughness values of Z . 15 the same range, where XRR data are usually smaller. Very often, for double layers with a relatively thin However, a reasonable agreement between SFM topmost layer Znot exceeding 5­10 . nm , estimated and XRR results was obtained in almost all cases, by XRR as very smooth, the SFM roughness corre- independent of deposition method and conditions. sponds to XRR simulation for the second, more One can also notice that the range of mean lateral rough layer. In several cases one can notice, how- distance between the surface structures observed in ever, that the SFM results are significantly higher SFM images corresponds to the cutoff distance j. than the corresponding XRR data Zcf. position 9 in Table . 1 . This situation occurs always when a rela- tively flat surface is covered with more or less 5. Conclusion prominent islands Zcf. Fig. 5b corresponding to posi- tion . 9 . In such case, X-rays are partially reflected XRR and SFM measurements may be performed from the surface Zislands . base and partially from the nondestructively in a very short time Z;5 . min and islands upper regions Zflat . peaks , just like in case of do not require any special sample preparation. By existence of an additional thin layer, corresponding obtaining surface roughness and scaling parameters to peaks area. As a result, the surface roughness from SFM measurements and applying them to simu- determined by XRR method is significantly dimin- lations using the models of Nevot ´ and Sinha, it is ished. This effect influences the range of applicabil- possible to determine the following properties of ity of XRR technique for surface roughness determi- layers and their interfaces: nation. Ø The average layer thickness for single- and As it was shown in Eqs. Z . 13 and Z . 14 , using multi-layer systems, taking into account sample XRR simulations performed according to Sinha's porosity Zwhich influences the simulation through model and three dimensional topographic data ob- lowering the film . density . The range of applicability tained by SFM, it is possible to compare directly the covers thicknesses from single nanometers to 200 fractal scaling properties of the surface studied by nm, while the resolution of the fitting procedure is both methods. Fig. 6 shows a comparison of GZ R. not worse than 10%. Best fits are obtained in the function calculated from XRR and SFM data. range of 10­100 nm. It is convenient to display the GZ R. function in a Ø The average global interface roughness may be log­log scale. Then, the linear part of the plot obtained, also for inner interfaces and buried layers. represents the region of applicability of the surface However, as the interface roughness influences scaling property Zit is also the mean lateral structure strongly the interference conditions, it should not size as illustrated in Fig. . 5 and the linear slope is the exceed the limit of ;10 nm. measure of the Hurst exponent h. At some distances, linearity breaks down and the GZ R. function reaches slowly saturation. This length is a measure of the References cutoff distance j and gives an estimate of the mean lateral extension of the surface structures. w x 1 L.G. Parrat, Phys. Rev. 95 Z . 1954 359­369. In the case of SFM measurements it is possible to w x 2 C. Rhan et al., J. Appl. Phys. 74 Z . 1 Z . 1993 146­152. w x 3 L. Nevot ´ et al., Rev. Phys. Appl. 15 Z . 1980 761­779. compute the GZ R. directly, according to Eq. Z . 12 . w4x O. Filies, Ront ¨ genreflektometrie zur Analyse von Then, the cutoff distance j and the Hurst exponent h Du¨nnschichtsystemen-Charakterisierung ultradu¨nner O. Filies et al.rApplied Surface Science 141 ( ) 1999 357­365 365 Schichten, PhD thesis, Part I, Institute of Nuclear Physics, w x 12 A. Bunde, S. Havlin, Fractals in Science, Springer-Verlag, Munster, ¨ 1997. 1995. w x 5 P. Doig et al., J. Appl. Cryst. 14 Z . 1981 321­325. w x 13 O. Filies, DiffTool-Program zur Analyse von Rontgens ¨ - w x 6 S.K. Sinha, Acta Phys. Pol. A 89 Z . 2 Z . 1996 219­234. pektren, PhD thesis, Part II, Institute of Nuclear Physics, w x 7 S.K. Sinha et al., Phys. Rev. B 38 Z . 4 Z . 1988 2297­2311. Munster, ¨ 1997. w x 8 G. Palasantzas, Phys. Rev. E 49 Z . 2 Z . 1994 1740­1742. w x 14 J. Wang, Europhys. Lett. 42 Z . 3 Z . 1998 283­288. w x 9 M. Kardar et al., Phys. Rev. Lett. 56 Z . 1986 889­892. w x 15 J. Lekki, Scanning Force Microscopy of Implanted Silicon, w x 10 G. Palasantzas et al., Phys. Rev. B 48 Z . 5 Z . 1993 2873­2877. PhD thesis, Institute of Nuclear Physics, Cracow, 1996. w x 11 A.-L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, 1995.