963 J. Appl. Cryst. (1997). 30, 963-967 Non-Gaussian Roughness of Interfaces: Cumulant Expansion in X-ray and Neutron Reflectivity W. PRESS, J.-P. SCHLOMKA, M. TOLAN AND B. ASMUSSEN Institut fiir Experimentalphysik, Christian-Albrechts-Universit6t Kiel, 24098 Kiel, Germany. E-mail." pex51@rz.uni-kiel.de (Received 24 June 1996; accepted 20 March 1997) Abstract There are several ways of generating density profiles Within the Born approximation, a cumulant expansion is from reflectivity data, more precisely from true specular used for the formulation of X-ray or neutron reflectivity. reflectivities, where the diffuse scattering has been Odd- (third-) order cumulants indicate asymmetric pro- subtracted from the measured reflectivity. It is a matter files; they may only be detected in layer systems via a Q~ of fact that, in general, there is no way to reconstruct a dependence of the oscillation period of Kiessig fringes. density profile from a single reflectivity measurement. Fourth-order cumulants are also visible in the larger Q. The measurement of intensities introduces the well regime for single interface systems. As an example of an known phase problem and leads to ambiguities in the asymmetric surface, a triangular height distribution reconstruction of profiles. Several efforts have been function is discussed. made in the past to overcome this problem, both theo- retically (Clinton, 1993; Lipperheide, Reiss, Fiedeldey, Sofianos & Leeb, 1993) and experimentally (Sivia, 1. Introduction Hamilton, Smith, Rieker & Pynn, 1991; Felcher, Dozier, Huang & Zhou, 1992; Sanyal et al., 1993; Majkrzak & When X-rays or neutrons are scattered from surfaces and Berk, 1995). Anomalous dispersion (X-rays) or the interfaces, both a specular and a diffuse contribution are magnetic contribution from a reference layer (neutrons) observed. In the past few years the scattering theory has are used to give two prominent methods. been considerably advanced and now can deal with both We have looked into this problem again, with special contributions rather well. The major breakthrough for emphasis on the description of interfaces beyond scattering in the small-angle regime came with the for- Gaussian height distributions and uniqueness of the mulation of the scattering from rough surfaces within determined profiles. A cumulant expansion of the the distorted-wave Born approximation (Vineyard, amplitude of the field reflected from an interface with 1982) and the use of a fractal description of interfaces non-Gaussian roughness is used in kinematic calcula- (Sinha, Sirota, Garoff & Stanley, 1988). Recently, this tions. approach has been generalized for layer systems (Pynn, Simple model profiles which are non-Gaussian are 1992; Hol~,, Kub6na, Ohlidal, Lischka & Plotz, 1993; studied within a Fresnel-type description (without Hol~' & Baumbach, 1994). There are several applica- approximations) using matrix methods (Abel6s, 1950; tions which make use of this theory (Bahr, Press, Jeba- Parratt, 1954; Lekner, 1987). For angles of reflection sinski & Mantl, 1993; Schlomka et al., 1995; Schlomka ~i > 3~c (with the critical angle ~c), these calculations et al., 1996; Jenichen, Stepanov, Brat & Kroemer, 1996; agree with the kinematic approach. Several conclusions, Stettner et al., 1996). In many cases, the best results are pertaining both to surfaces and interfaces (we first achieved and a more stable set of surface parameters is restrict ourselves to a single layer), can be drawn. The obtained when true specular and diffuse scattering are loss of information when taking the modulus (squared) refined simultaneously. In this context, a problem has of Fourier-transformed distributions becomes quite arisen: sometimes, but not always, a sizable difference obvious, particularly for a single interface. between the r.m.s, roughness a obtained from the true specular reflectivity and that from the simultaneous fit (the a tends to be large) is found (Schlomka et al., 1995; Lfitt et al., 1997). This may be related to a premise in the 2. Cumulant expansion for describing rough surfaces scattering theories mentioned above: they all rely on a Gaussian height distribution p(z) of width a of a rough Cumulant expansions (Kendall, 1994) are rather fre- interface. An analysis of reflectivity data alone need not quently used in crystallography. They describe the follow this restriction. scattering from atoms and molecules performing large- ~.i 1997 International Union of Crystallography Journal of Applied Crystallography Printed in Great Britain - all rights reserved ISSN 0021-8898 ~ 1997 964 N O N - G A U S S I A N ROUGHNESS OF INTERFACES amplitude motions, both translational and orientational The relation between moments and cumulants up to the (Willis & Pryor, 1975; Johnson, 1969). Accounts of this tenth order is given by Kendall (1994). Unfortunately, can be found in the International Tables of Crystal- no unambiguous reconstruction of the distribution p(z) lography (and references therein) and in a recent review from a finite number of cumulants is possible. Realisti- (Kuhs, 1992). cally, one may hope to determine the cumulants up to Cumulant expansions can also be usefully applied to order l = 4 from a reflection experiment. interfaces with non-Gaussian roughness. The reflection Calculating the reflectivity makes it obvious that the from interfaces is only sensitive to the corresponding situation is even worse. From equations (1) and (2), it is projection of the height function z ( r ) [ r = (x,y) is a clear that only cumulants of even order can be deter- vector parallel to the surface] onto the interface normal mined. As le i~] = 1, odd-order terms do not affect the z. Therefore, only a one-dimensional description is reflectivity R and, hence, the asymmetry of an interface needed. The reflectivity within the kinematic theory can (more precisely, the asymmetry of its height distribution be written as (Als-Nielsen et al., 1994) function) cannot be determined from a simple reflection experiment. This is a demonstration of the phase pro- 1 ]" dp(z) 2. R ~- R F [~,~v. ~ e x p ( i Q z ) d z (1) blem already mentioned in the introduction. The resultant ambiguities are demonstrated with a triangle as model height distribution (Fig. 1), which Here, R F is the Fresnel reflectivity (o~ Q-4 for large Q), corresponds to a parabolic interface profile. In compar- p(z) is the density profile (with Pa,, the average density) ison to a Gaussian distribution function, which has only and p(z)cx dp(z)/dz the height distribution function; a nonzero second-order cumulant, the triangular dis- Q - Q. is the momentum transfer perpendicular to the surface. Apparently, the change in the density profile at an interface determines the contrast. To be more exact, 'density profile' means the profile of the refractive index n = 1 - 5 - i f l , where (5 is the dispersion and fl the absorption of the respective material. For X-rays, the \ dispersion is almost proportional to the electron density; I- I I \ ", -- mangle _ for neutrons it is proportional to the scattering length 0.06 / \ right density. / x, J l \ d - - trian)zle - ,, \ -] I / " F "~ 0.04 x, "]i \ left 2.1. Single surface Instead of modelling the interface profile in direct space, the Fourier-transformed quantity J'[dp(z)/ dz] exp(iQz)dz is expanded. We obtain 0.00 ~ -20 - I 0 0 10 20 1 I dp(z) Pay ~ e x p ( i Q z ) d z l0 o :(A) ' ' I ' ' ' r T r ~ ' ; ' ' ' ' 1 ' ' ' 1 ' ' ' ' 1 ' ' ' ' ~ "~ eiQze-l/2Q2~2e-l/6iQ~K(3)el/24Q4K(4'. (2) - - triangle right " I0 2 K (/) denotes the cumulant of order l. With restriction to terms of order l ~< 2 the usual (harmonic) Debye- Waller-like factor results. Higher-order terms (l >/3) 1 0 - 4 describe anharmonic motions in crystallography. In our e,,,' case, the term with l = 3 is related to the asymmetry of 10-6 the height distribution p(z) and l = 4 to symmetric deviations from the Gaussian shape [K (4) < 0 'flatter', K (4) > 0 'steeper' than a Gaussian]. A more quantitative 1() -g formulation is provided by the relation to the moments . . . . '. , , , , 1 : L ; .. [ ~ A ~ L I , . L . ~ ] , L , J I : ~ ~ ~ M (l) = .[zlp(z)dz of the distribution p(z). When taking 0.0 0-1 0-2 0.3 0-4 0.5 0.6 0.7 M(~)= 0 (then zo = 0 marks the mean interface posi- q=(, l) tion), one obtains Fig. 1. The two triangular probability functions (top) result in the same reflectivity curve (bottom) in the case of a single interface. The M (2) = K (2) = 0 .2 third-order cumulant is negative for the 'triangle right' profile, M (3) = K (3) (3) positive in the other case. For the calculation of the reflectivity, a silicon substrate (refractive index n = 1 - 7.56 x 10 -6- M (4) = K (4) + 3K (2)2 il.73 x 10 -7) with roughness a = 5 A and an X-ray wavelength of 1.54,~ (Cu K:O was assumed. W. PRESS et al. 965 tribution also has nonzero cumulants of higher order. tangent-shaped profile is very small. [See also Bahr, The two distributions 'triangle left' and 'triangle right' Press, Jebasinski & Mantl (1993).] In the calculations shown in Fig. 1 differ in the sign of the odd-order the latter is used. Because of the interference between cumulants. Independently of the sign, the same reflec- the refected waves from the two interfaces [equation tivity results. (6)] a Q-dependent oscillation period occurs and then Since one obviously cannot determine asymmetric the two parabolic density profiles shown in Fig. 2 components, it may be advisable to adopt the symmetric become distinguishable. For the 'triangle right' profile solution/5(z) = l/2[p(z)+ p(-z)] in the single interface [K (3) < 0], the period decreases with Q; for the 'triangle case. left' profile it increases. To estimate roughly the effect of higher-order None of the two Gaussian interface roughnesses cumulants, we calculate the even-order cumulants for should be too large, otherwise the contribution of one the profile in Fig. 1. [F is the full width at half-max- interface as well as that of the interference term decays imum (FWHM) of the distribution] rapidly and AK 13) cannot be determined. Here one may also note the following aspect. When K (2) = 2/9/- 2 (4) dealing with wetting problems, it is customary first to K 14) = - 4 / 1 3 5 F 4. characterize the dry substrate, in order to minimize the number of surface parameters in subsequent refinements. One calculates from equation (2) that intensities I(Q) As only even-order cumulants can be determined in a have to be measured up to Q ~_ 3 I F ~_ 1.5/a to obtain a measurement with a single surface, a problem may 10% contribution from the fourth-order term. For a result. A possible asymmetry of the substrate surface, roughness a = 5 A, this requires a dynamic range of at least six orders of magnitude, which can be obtained with good laboratory sources. O ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' I ' ' ' I ' ' ' 2.2. Two interjktces 6 ~ - - triangle right The reflectivity of a layer on a substrate (two inter- - - triangle left faces) reads as follows (always within the Born approximation) (Als-Nielsen et al., 1994) ~ 4 R ' ~ RFIF(Q)I 2 2 c~ RFIApl I-I exp[imOmK(lm)/m!] m + Ap2 ]--I at exp [flQtK~l)/l!)12 (5) 0 Here Apj refers to the change in electron density -20 0 20 40 60 80 100 120 (or neutron scattering length density) at the respective etA) I0 cl ~__,,,,, .... , .... ,,-,,, .... , .... , .... ! interface and K~I m), ~2~ l) are the cumulants of the two interface profiles. Differences of odd-order cumulants i l - - triangle right J now appear in the interference terms, proportional to 10 .-2 ~ ~,,,, - - triangle left - (neglecting cumulants of order >3) - \ cos[Q(z 2 - Zl) -- Q3(K~3) - K~13))/6]. (6) ~ 1 0 4 % The dominant contribution, at least at small Q, is the ,¢ modulation of the intensity with a period AQ--2re~d, 10 6 sometimes called Kiessig fringes, with the layer thickness d = z 2 - z 1. After introducing the difference AK (3) = K~ 3 ) - K/l 3), one can redefine an effective per- 10-8 iodicity AQ = 2rc/d(Q) with d(Q) -- d - Q2AKC3)/6. [ - ~ , , , I . . . . I . . . . . I . . . . . . , , , 0.0 0.1 0.2 0-3 0.4 0-5 0.6 0.7 Apparently, the modulation period becomes Q depen- q:(A l) dent in this case. An example is given in Fig. 2. The asymmetric triangular probability distributions (Fig. 1) Fig. 2. The density profiles of a 100 .~, film on a silicon substrate (top). now describe the surface of a 100 A, layer with half of The asymmetric probability distributions from Fig. 1 were used to the substrate electron density on top of a silicon sub- model the film surface; the substrate/film interface is tanh-shaped. The asymmetry leads to an increasing or decreasing 'effective per- strate. The substrate/film interface has the usual hyper- iodicity' (see text) for the calculated reflectivities (bottom). The film bolic tangent or error-function shape. The difference was assumed to have half of the electron density of the substrate; the between an error-function-shaped and a hyperbolic roughness of each interface is a = 5 ,~. 966 NON-GAUSSIAN ROUGHNESS OF INTERFACES which is invisible to the dry case, can indeed affect the That system has four free parameters (thickness, elec- measured reflectivity when the wetting film is present. tron density and roughness of the layer and substrate This may result in a wrong interpretation of the data, e.g. roughness) compared to only two parameters for the by introducing an additional layer in the refinement triangular profile [width ( = roughness) and asymmetry]. which is in fact only a manifestation of the asymmetry of the substrate. Using the cumulant expansion and non-Gaussian 3 . C o n c l u s i o n s density profiles may help to limit the number of free At present, we cannot provide an experiment in which parameters in the refinement procedure. For example, if odd-order cumulants (note: change of modulation period one measures the reflectivity of Fig. 1, which cannot be of Kiessig fringes) apparently contribute. There are, refined using the model of a single Gaussian-shaped however, examples where the fourth-order cumulant interface, the next assumption would be the presence of K (4) obviously plays a role. In these it is referred to as a thin surface layer. In fact, the 'data' can also be refined polymer interdiffusion in the near-surface regime, e.g. using a model of a single layer with tanh-shaped sym- Kunz & Stamm (1994, 1996). For modelling the inter- metric interfaces as shown in Fig. 3. (For a thin layer face profile it became necessary to introduce two with roughnesses of the order of the film thickness the Gaussians of different widths. The underlying reptation height distributions of the two interfaces overlap. The model for polymer diffusion requires at least two dif- refinement routine used in this example adds the two ferent diffusion constants. In this case, the introduction contributions numerically, obtains the density profile by of a cumulant K (4) would represent an alternative integration and calculates the reflectivity of that profile.) approach. It would also be extremely interesting to calculate the diffuse scattering of non-Gaussian profiles. Dietrich & 0.10 . . . . I . . . . Haase (1995) give solutions for the diffuse scattering I ' _' ' / \ * cross section of various profiles. Unfortunately, the 0.08 - - triangle right ///.~/i I given formulae require very time-consuming numeric / / ~ , calculations. Using the cumulant expansion may be a - - t a n h fit /' / t simple, alternative approach which could more easily be 0.06 /// l included into a z2-minimization algorithm. "0.04 ,,/~ 1 Note added in proof After acceptance of our manuscript we became aware of the fact that similar work has been f 0.02 / ' II published by Rieutord, Braslau, Simon, Lauter & Pasyuk (1996). 0.00 - - - i - We thank U. R6sler, J. Stettner, O. H. Seeck and M. 10 -20 - 10 0 20 Liitt for very helpful discussions. 10 o =(h) R e f e r e n c e s 10-2 © triangle right Abel,s, F. (1950). Ann. Phys. (Paris), 5, 596-640. Als-Nielsen, J., Jacquernain, D., Kjaer, K., Leveiller, F., "~ 10-4 Lahav, M. & Leiserowitz, L. (1994). Phys. Rep. 246, 251- 313. ,-¢ Bahr, D., Press, W., Jebasinski, R. & Mantl, S. (1993). Phys. 10-6 Rev. B, 47, 4385-4393. Clinton, W. L. (1993). Phys. Rev. B, 48, 1-5. Dietrich, S. & Haase, A. (1995). Phys. Rep. 260, 1-138. iO-S Felcher, G. P., Dozier, W. D., Huang, Y. Y. & Zhou, X. L. (1992). Surface X-ray and Neutron Scattering, edited by H. 0.0 0. I 0.2 0.3 0-4 0.5 0-6 0-7 Zabel & 1. K. Robinson, pp. 99-103. Berlin: Springer q:IA J) Verlag. Fig. 3. The reflectivity from Fig. 1 was refined using a model of a Holy, V. & Baumbach, T. (1994). Phys. Rev. B, 49, 10668- single layer with two tanh-shaped interfaces. The resulting prob- 10676. ability distribution (top) and the refined reflectivity are shown Holy, V., Kub~na, J., Ohlidal, I., Lischka, K. & Plotz, W. (bottom). The refined parameters of the film are: asub~t,~t e = 4.0,A,, (1993). Phys. Rev. B, 47, 15896-15903. cr~t m =02.9,~,, nfil m = 1 - 5.0 x 10 -6 - i1.14 x 10 -7, film thickness Jenichen, B., Stepanov, S. A., Brar, B. & Kroemer, H. (1996). = 8.0 A. The model has four free parameters in comparison to only J. Appl. Phys. 79, 120-124. two for the triangular profile (width and asymmetry). Johnson, C. K. (1969). Acta Cryst. A25, 187-194. W. PRESS et al. 967 Kendall, A. (1994). In Kendall's Advanced l'heo~ of Statis- Sanyai, M. K., Sinha, S. K., Gibaud, A., ttuang, K. G., Car- tics, Vol. 1, 6th ed., edited by A. Stuart & J. K. Ord. valho, B. L., Rafailovich, M., Sokolov, J., Zhao, X. & Zhao, London: Edward Arnold. W. (1993). Europhys'. Lett. 21,691~96. Kuhs, W. F. (1992). Acta C~st. A48, 80-98. Schlomka, J.-P., Fitzsimmons, M. R., Pynn, R., Stettner, J., Kunz, K. & Stamm, M. (1994). Macromol. Syrup. 78, 105- Seeck, O. H., Tolan, M. & Press, W. (1996). Physica B, 221, 114. 44-52. Kunz, K. & Stamm, M. (1996). Macromolecules, 29, 2548- Schlomka, J.-P., Tolan, M., Schwalowsky, L., Seeck, O. H., 2554. Stettner, J. & Press, W. (1995). Phys. Rev. B, 51, 2311- Lekner, J. (1987). Theory of Reflection. Dordrecht: Nijhoff. 2321. Lipperheide, R., Reiss, G., Fiedeldey, H., Sofianos, S. A. & Sinha, S. K., Sirota, E. B., Garoff, S. & Stanley, H. B. (1988). Leeb, H. (1993). Physica B, 190, 377-382. Phys. Rev. B, 38, 2297-2311. Ltitt, M., Schlomka, J.-P., Tolan, M., Stettner, J., Seeck, O. H. Sivia, D. S., Hamilton, W. A., Smith, G. S., Rieker, T. P. & & Press, W. (1997). Phys. Rev. B, 56, 4085 4091. Pynn, R. (1991). J. Appl. Phys. 70, 732-738. Majkrzak, C. F. & Berk, N. F. (1995). Phys. Rev. B, 52, Stettner, J., Schwalowsky, L., Seeck, O. H., Tolan, M., Press, 10827-10830. W., Schwartz, C. & v. K/inel, H. (1996). Phvs. Rev. B, 53, Parratt, L. G. (1954). Phys. Rev. 95, 359-369. 1398-1412. Pynn, R. (1992). Phys'. Rev. B, 45, 602~12. Vineyard, G. H. (1982). Phys'. Rev. B, 26, 4146-4159. Rieutord, F., Braslau, A., Simon, R., Lauter, H. J. & Pasyuk, V. Willis, B. T. M. & Pryor, A. W. (1975). Thermal Vibrations (1996). Physica B, 221,538-541. in C~stallography. Cambridge University Press.