Thin Solid Films 323 Z . 1998 1­5 Letter Analysis of X-ray reflectivity curves of non-Gaussian surfaces G. Vignaud a,), A. Gibaud a, F. Paris b, D. Ausserre b, G. Grubel c ´ ¨ a UniÕersite´ du Maine, URA 807 CNRS, Faculte´ des Sciences Le Mans Cedex 72017, France b UniÕersite´ du Maine, URA 509 CNRS, Faculte´ des Sciences, Le Mans Cedex 72017, France c ESRF, Experimental diÕision, AÕenue des Martyrs, BP 220, Grenoble Cedex F 38043, France Received 30 September 1997; accepted 30 October 1997 Abstract Surfaces of symmetric diblock copolymers thin films exhibiting non-Gaussian distribution of height are studied by X-ray reflectivity and atomic force microscopy Z . AFM . When deposited on a silicon substrate, the surface is essentially flat and its roughness may be described by a Gaussian distribution of height. Upon annealing, films operate a two-dimensional phase transformation and form islands at the free surface having height and size that evolve as a function of annealing time. The height probability function cannot be represented by a Gaussian distribution anymore, and the question that arises is how to take into account the morphology of such surfaces in the reflectivity calculations. In a first approach, we show that the height distribution function derived from AFM measurements is directly transferable to analyze X-ray reflectivity curves according to a formalism that we present. In a second part, we determine the height distribution function from a fit to the observed reflectivity. q 1998 Elsevier Science S.A. All rights reserved. Keywords: X-ray reflectometry; Thin film structure and morphology; Polymers; Elastomers and plastics 1. Introduction ble average. For any physical surface, GZ R. will saturate to a mean-square roughness s at sufficiently large hori- X-ray and neutron reflectivity have recently been known zontal lengths, i.e., when R is larger than the roughness as a veritable explosion of interest in the scientific commu- correlation length j w8,9x. For surfaces in which the nity working on the structural characterization of thin correlation length of the fluctuations of height is smaller films. Dating back to the work of Parratt w x 1 who initiated than the coherence length of the beam, the reflectivity the recursive technique, reflectivity curves are now mainly measurements are frequently analyzed by means of two analyzed via the matrix technique. More recently, the Born quantities such as: Z . 1 the mean-square roughness s of the approximation and the distorted Born approximation w2­ x 6 surface that produces a deviation of the intensity decay were used to model the diffuse scattering that is inevitably from the Fresnel reflectivity in the specular direction; and observed in off-specular directions as soon as the surface Z . 2 the height­height correlation function that is the rele- presents some kind of roughness with correlations between vant quantity intervening in the analysis of the diffuse height fluctuations. A wide variety of surfaces and inter- scattering. faces occurring in nature are well represented by a kind of This simple description is only possible if the height roughness associated with self-affine fractal scaling, de- distribution of the surface is Gaussian, or if the roughness fined by Mandelbrodt w x 7 in terms of fractional Brownian is sufficiently weak, i.e., qz s-1. In such a description, motion. An isotropic rough surface can be described by the only the second moment of the distribution ss²ZhZr.y mean-square height difference given by: Z .:.2:1r2 -h r , i.e., the roughness of the surface, is suffi- 2 cient to model the decay of the specular reflectivity curve. GZ R. s² hZ r . yhZ O. :, Z1. Although this assumption is frequently acceptable, it hap- where hZr. stands for the height of the surface at the pens that some surfaces can in no way be described by in-plane position r and the symbol ² : denotes an ensem- such a distribution. In such cases, the specular reflectivity can only be calculated by means of the probability pZ z. of finding some points of height z at the surface. Up to now, ) Corresponding author. little work on X-ray reflectivity has been presented on the 0040-6090r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved. PII S0040-6090Z97.01020-1 2 G. Vignaud et al.rThin Solid Films 323 ( ) 1998 1­5 case of non-Gaussian surfaces mainly because it was diffi- is possible to deduce pZ z. from a fit to the measured cult to access the height probability function pZ z.. With reflectivity. the recent development of the atomic force microscope The X-ray reflectivity experiments were made at the Z . AFM , it is now possible to extract statistical information Troika ¨ Beam line of the ESRF on diblock copolymer thin of the surface Zdistribution of height, correlation function, films of PS-PBMA, which were annealed at 1508C for 6 etc.. from microscopy images and to compare with the min and 4 h. The samples that we have studied were made results calculated from X-ray scattering measurements by spin-coating a toluene solution of the diblock copoly- w10,11x. In particular, one can precisely define pZ z. and mer on the surface of a flat silicon wafer. Upon annealing, transfer it in the reflectivity calculation of non-Gaussian such materials undergo a two-dimensional phase transfor- surfaces. It is the purpose of this work to present first, a mation in which the surface of the film presents either formalism to describe the reflectivity of non-Gaussian holes or islands depending on the initial value of its surfaces, and second, to examine the validity of such a thickness w12­1 x 4 . In this case, the initial thickness of the formalism in samples on which the probability pZ z. has film which was 400 A, led to the formation of islands, the been determined by AFM. In addition, we will show that it height and the size of which were evolving upon the Fig. 1. AFM image of relief domains at the free surface of PSrPBMA diblock copolymer films annealed for 6 min Z . a and 4 h Z . b at 1508C. G. Vignaud et al.rThin Solid Films 323 ( ) 1998 1­5 3 annealing time. Fig. 1 shows the morphology of the film film at the coordinates Z x, y. taking the origin of the z surface measured by AFM over a square surface of 5=5 axis on the surface of the substrate. ssi represents the mm2 and clearly evidences a huge difference in the statisti- roughness at the film­substrate interface assuming that the cal properties of the two surfaces. The height probability distribution of height is Gaussian at this interface. The function pZ z. of these two surfaces is presented in Fig. 2. roughness and the electronic density of the substrate have It is clear on this figure that, in both cases, the probability been determined on the non-annealed film ssis3 A, pZ z. cannot be represented by a Gaussian distribution. y y3 rsis0.73 e A and these values were kept fixed for the In the Born approximation, the differential scattering annealed films. Keeping only the specular part, the above cross-section is the Fourier transform of the correlation equation can be written as: function density­density. For a thin film of a diblock copolymer presenting a corrugated surface as shown in 2 ds rsiyr 1 cp 2 2 rcp Fig. 1, assuming that the material is homogenous, except 2 y qz ssi yiqz z sre e 2 q Hd zpZ z. e , dVspec. iqz iq for the presence of the surface, the differential cross-sec- z tion restricted to the specular direction becomes w x 14 : Z3. 2 ds rsiyr 1 cp 2 2 rcp 1 2 y q where pZ z. is the probability density function of the z s si yiqz zZ x, y. sre e 2 q HHd xd ye , dV iq surface heights of the film. It is clear from Eq. Z . 3 that the z iqz S S Z2. measured reflectivity is only dependent on the Fourier transform of the height probability pZ z.. However, note where re is the classical radius of the electron, S is the that the diffuse scattering integrated in the specular direc- coherently illuminated area of the sample, and r is the tion is not considered in this calculation. It appears from electron density of either the silicon wafer or the diblock the AFM images ZFig. . 1 that since the averaged distance copolymer. zZ x, y. defines the height of the surface of the between the islands and the size of the islands are small, we do not expect diffuse scattering to peak in the specular direction and it is possible to neglect this quantity Zsuch is not always the case w x 3 .. As shown by Eq. Z . 3 , the calcula- tion of the specular reflectivity can be made if one can measure the probability function pZ z., or if one has a sufficient knowledge of the surface properties, allowing a close guess of its analytical form. As presented above, the measurement of pZ z. is made possible with an AFM that measures the relative height of the points in a surface. From the heights measured by AFM, it is straightforward to determine pZ z.. Another alternative consists in model- ing pZ z. by an analytical form. This is easy for example, in the case of a grating, in which one knows that the surface presents two dominating heights Zthe heights of the grooves and of the . steps to adjust pZ z. to the X-ray reflectivity data w x 15 . It is, in principle, also possible to develop the characteristic function wZq . Z z which is the Fourier transform of pZ z.. as the following series expan- sion: q2z yiqz z 2 wZ qz. sHd zpZ z. e s1yiq ² z z: y ² z :q . . . 2! in n n q ² z :k . Z4. n! The coefficients of the polynomial ² n z :, which can be fitted to the data, are the moments of the distribution, and once these coefficients determined a Fourier inversion of wZq . z yields pZ z.. The major drawback of such a calcula- Fig. 2. Height probability function extracted from the AFM measure- ments on PSrPBMA copolymer films annealed for 6 min Z . a and for 4 h tion is that one generally needs many coefficients to Z . b at 1508C. Note the appearance of the peak at 600 A after 4 h of correctly describe a reflectivity curve, and that these coef- annealing. ficients are unknown. We show now on an example how 4 G. Vignaud et al.rThin Solid Films 323 ( ) 1998 1­5 we have analyzed our reflectivity curves with help of the height distribution function. To ascertain the validity of this method, we have first measured the height probability of the surface of the diblock copolymer annealed for 6 min. In such case, it is somewhat difficult to predict the functional shape of pZ z., since the surface of the film that was essentially flat before the annealing is undergoing deep modifications with the onset of the island growth. The reflectivity curve presented in Fig. 3a clearly shows some oscillations that would have been difficult to interpret in an ab initio calculation. With the help of the AFM measurements, pZ z. is introduced in the calculation, and a good agreement is immediately obtained between the calculated and the observed reflectiv- y1 ities. The calculation begins at Qzs0.05 A because of the limitation imposed by the Born approximation. Resolu- tion effects have been considered by convolving the calcu- Fig. 4. Z . a Height probabilities extracted from the AFM measurement compared to the functional form Zsolid . line used in the calculation. Z . b Calculated Zsolid . line and observed reflectivities Zdotted . line of a PSrPBMA copolymer film annealed for 4 h at 1508C Zcompare Fig. 4b and Fig. . 3b . lated intensity with a Gaussian resolution function. This result unambiguously shows that AFM profiles can give, under certain conditions that we will presume, the same statistical information as an X-ray reflectivity curve. There is indeed a restriction that clearly appears in the second example that we have chosen, i.e., the case of the surface of a diblock copolymer annealed for 4 h Zsee Fig. . 3b . In such case, the islands are well formed and have grown bigger. With our AFM, the size of the image was 5=5 mm2 so that only a few islands were present in this image. As a result, the shape of pZ z. was determined only on a limited number of islands so that the transfer of pZ z. in Eq. Z . 3 did not give a good agreement between the ob- served and calculated reflectivities. This shows that AFM and X-ray measurements give the same statistical informa- tion, provided that the size of the AFM image is large enough to produce a stationary condition on pZ z.. In this Fig. 3. Calculated Zsolid . line and observed reflectivities Zdotted . line of case, the main reason of the discrepancy between the two PSrPBMA copolymer film annealed for 6 min Z . a and for 4 h Z . b at curves was found in the poor determination of the bottom y1 1508C. The calculated curves begin at Qz s0.05 A because of the limitation fixed by the Born approximation. The calculations have been height of the film by AFM. As we know, with enough done by entering the height probability extract from the AFM measure- confidence, the morphology of our film, we have tried to ments shown in Fig. 2. include in the calculation a simple functional form pZ z. G. Vignaud et al.rThin Solid Films 323 ( ) 1998 1­5 5 shown in Fig. 4a, consisting of a truncated Lorentzian to case, the usual methodology is to assume Gaussian distri- describe the top part of the film, and of a Maxwellian to butions which, to our viewpoint, are too restrictive. describe the bottom part. The observed data are fit into the model, which leads to an excellent agreement between the observed and calculated reflectivities as shown in Fig. 4b. References w x 1 L.G. Parrat, Phys. Rev. 95 Z . 1954 359. w x 2 S.K. Sinha, E.B. Sirota, S. Garoff, H.B. Stanley, Phys. Rev. B 38 2. Conclusion Z . 1988 2297. w x 3 P.Z. Wong, A. Bray, Phys. Rev. B 37 Z . 1989 7751. In conclusion, we have shown in this letter that the w x 4 J. Daillant, K. Quinn, C. Gourier, F. Rieutord, J. Chem. Soc., Faraday Trans. 92 Z . 1996 505. statistical properties of a surface that are completely con- w x 5 A. Gibaud, N. Cowlam, G. Vignaud, T. Richardson, Phys. Rev. Lett. tained in the height probability distribution pZ z. are essen- 74 Z . 1995 3205. tial to describe X-ray reflectivity curves. As a general rule, w x 6 R. Pynn, Phys. Rev. B 45 Z . 1992 602. we believe that instead of assuming Gaussian distributions, w x 7 B.B. Mandelbrodt, The Fractal Geometry of Nature, Freeman, NY, as is the case in most of the literature, it would be more 1982. w x 8 P.P. Swaddling, D.F. McMorrow, R.A. Cowley, R.C.C. Ward, M.R. suitable to determine the height probability distribution, Wells, Phys. Rev. Lett. 68 Z . 1992 1575. either by AFM for simple surfaces, or by assuming some w x 9 G. Palasantzas, J. Krim, Phys. Rev. B 48 Z . 1993 2873. simple functional forms in more complex systems. We w x 10 Z. Cai, K. Huang, P.A. Montano, J.M. Bai, G.W. Zajac, J. Chem. have indeed evidenced in this study that AFM images that Phys. 98 Z . 1993 2376. verify the stationary condition are in full agreement with w x 11 J. Stettner, L. Schwalowsky, O.H. Seek, M. Tolan, W. Press, C. Schwarz, H.v. Kanel, ¨ Phys. Rev. B 53 Z . 1996 1398. X-ray reflectivity curves. However, except if one is inter- w x 12 G. Coulon, D. Ausserre, ´ T.P. Russell, J. Physiol. Z . Paris 51 Z . 1990 ested in only one interface for which AFM is the ideal 777. tool, it is frequent that interfaces are buried inside the w x 13 D. Ausserre, ´ D. Chatenay, G. Coulon, B. Collin, J. Physiol. Z . Paris material. Then pZ z. at each interface can only be deter- 51 Z . 1990 2557. mined via a model. This is how it is done when one tries to w x 14 G. Vignaud, A. Gibaud, J. Wang, S.K. Sinha, J. Daillant, G. Grubel, ¨ Y. Gallot, J. Phys. Condens. Matter 9 Z . 1997 L125. define the profile of electron density by using the represen- w x 15 M. Tolan, G. Vacca, S.K. Sinha, Z. Li, M. Rafailovich, J. Sokolov, tation of interface in terms of slabs. However, in such a H. Lorenz, J.P. Kotthaus, J. Phys. D 28 Z . 1995 A231.