Journal of Magnetism and Magnetic Materials 240 (2002) 494­496 Method of the reflections function in the X-ray reflectometry study of multilayers N.V. Kourtina*, E.A. Kravtsov, V.V. Ustinov Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, 18, S. Kovalevskaya St., GSP-170, Ekaterinburg, 620219, Russia Abstract A theory of specular X-ray reflectivity from a rough interface based on the reflection function method is proposed.By using the approximation of the abruptly changing potential, we represent a reflectivity in the form of a series.Its first term reproduces the Nevot­Croce approximation and the second one gives the phase correction, which can be used to obtain the degree of interface asymmetry.The model X-ray reflectometry profiles for Fe/Cr superlattice are used to illustrate the method. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Multilayers, metallic; Interface structure; X-ray reflectivity The X-ray reflectometry is a useful tool for studying scattering matrix surface and interface structure in thin films and multi- ! layers.Usually, the rough surface X-ray reflection is r S 11 t12 analysed in the frame of the plane-wave Born approx- 12 ¼ t21 r22 imation (PWBA) or the distorted-wave Born approx- imation (DWBA) [1,2].In this work, we apply the related with the given interface between two subsequent reflection function method (RFM) [3] to the specular layers, which are denoted as 1 and 2. X-ray reflection from a rough surface or interface. The RFM starts from the transformation of the linear Let us consider the X-ray reflection on a non-ideal second order differential equation (1) for the wave interface structure.We assume that this structure is amplitude EðzÞ into a non-linear first order equation of homogeneous along the surface which is parallel to the Riccati type for the reflection function BðzÞ: This (xy) plane and the media can be characterized by its transformation is not unique and can be performed in dielectric susceptibility wðzÞ depending only on the a number of different ways.An advantage of the RFM normal coordinate z, where wðzÞ-w is that the perturbation expansion carried out in the 7 when z-7N: The change of the material occurs only in the z-direction framework of this scheme gives more rapid convergence perpendicular to the surface.Then one has to solve the in comparison with the conventional Born series.In one-dimensional Helmholtz equation particular, the first order approximation easily enables   one to go beyond DWBA.We denote qðzÞ ¼ d2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffi þ k2 sin2 y EðzÞ þ k2wðzÞEðzÞ ¼ 0: ð1Þ dz2 2k sin2 y þ wðzÞ; and represent the electric field EðzÞ in the form: Here EðzÞ is the electric field in the medium, y is the   Z  incident angle and k ¼ 2p=l; l being a wave length of i z EðzÞ ¼ q 1=2ðzÞ AðzÞ exp qðxÞ dx radiation.As the first step, we need to evaluate the 2 z0  Z  i z *Corresponding author.Fax: +7-343-92-327-37. þ CðzÞ exp qðxÞ dx ; ð2aÞ 2 E-mail address: hope@imp.uran.ru (N.V. Kourtina). z0 0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 9 1 1 - 8 N.V. Kourtina et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 494­496 495 where AðzÞ and CðzÞ are the amplitude functions.In and qc ¼ maxjqðzÞj: In the X-ray reflectometry studies a addition, we apply the following condition: is of the order of mean-root-square interfacial roughness   Z  s ¼ 228 (A [4] and q d i i z cBð4p=lÞ sin y: Therefore the EðzÞ ¼ q1=2ðzÞ AðzÞ exp qðxÞ dx condition e51 holds over the scattering angle region dz 2 2 z0 (0oyo41).These estimates make it possible to find the  Z  i z solution of Eq.(3) in the form CðzÞ exp qðxÞ dx : ð2bÞ 2 z BðzÞ ¼ B0ðzÞ expðbðzÞÞ; ð4Þ 0 The reflection function BðzÞ is defined as BðzÞ ¼ where CðzÞ=AðzÞ:Taking into account the continuity of EðzÞ B0ðzÞ ¼ ðqðzÞ q2Þ=ðqðzÞ þ q2Þ; and Eq.(1) one can prove that BðzÞ satisfies the first ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q order non-linear differential equation q sin2 y þ w   Z  2ð1Þ ¼ 2k 7 d q0ðzÞ z BðzÞ ¼ exp i qðxÞ dx and dz 2qðzÞ z0 X N  Z z  bðzÞ ¼ bnðzÞen: B2ðzÞ exp i qðxÞ dx ð3Þ n¼1 z0 The function B0ðzÞ corresponds to the boundary Eqs.(1) and (3) should be supplemented by the condition BðþNÞ ¼ 0 and it gives the Fresnel reflection boundary conditions.For example, the choice of coefficient rF11 ¼ ðq1 q2Þ=ðq1 þ q2Þ from an ideal sharp BðþNÞ ¼ 0 corresponds to the X-ray beam, incident interface.The series bðzÞ yields the corrections due to the from zo0; and in this case a reflection coefficient r interfacial non-ideality. 11 is given by the relation r The use of ansatz (4) is the essential step in the 11 ¼ Bð NÞ: We also take into consideration the dimensionless functions g derivation.It enables us partially to sum up the 7ðzÞ; which reducible parts of the expansion BðzÞ in powers of e; so are related to wðzÞ via equality wðzÞ w7 ¼ 7ðw that the coefficients b w nðzÞ are associated with the þÞg7ðzÞ: The function g ðzÞ-0; when z- N; and irreducible terms only.The series bðzÞ can be found by g ðzÞ-1; if z- þ N (See Fig.1).The functions g7ðzÞ means of subsequent iterations from Eq.(3).It turns out obey the relation gþðzÞ þ g ðzÞ ¼ 1: One can regard that, at each step n, one encounters the only linear g7ðzÞ as a ``shape'' of the interface, which reproduces inhomogeneous differential equation for bnðzÞ: The the gradual transition from the first layer to the second details of this derivation will be presented elsewhere. one. We shall call interface ``symmetric'', if ðq=qzÞg ðzÞ is As a result, up to the third order of e, the elements of the an even function of z; otherwise interface is ``asymmetric''. matrix S12 can be written in the form In case of grazing incidence angles, Eq.(3) can be r solved in the approximation of the abruptly changing 11 ¼ rF 11 exp iq1d 12q1q2s2 potential.The small parameter e of this expansion is þiq1½ðq21 þ 3q22Þm1 þ ðq21 q22Þm2Šs3 ; ð5Þ defined as e ¼ aqc=2p; where a is the characteristic length corresponding to the variation of the potential r22 ¼ rF22 exp iq2d 12q1q2s2 þ iq2½ ðq22þ3q21Þm1 þ ðq21 q22Þm2Šs3 ; ð6Þ t12 ¼ t21 ¼ tF12 expð12iðq1 q2Þd þ 18ðq1 q2Þ2s2 þ 12iðq1 q2Þ2½ðq1 q2Þm1 þ ðq1 þ q2Þm2Šs3Þ: ð7Þ Here rF; tF are Fresnel's reflection and transmission amplitudes and parameters d; s; and m1ð2Þ are expressed via g7ðzÞ as follows Z þN d ¼ zg= ðzÞ dz; ð8Þ N s2 ¼ 2Ið2Þð ; þÞ Z þN Z þN Fig.1. The linear segment form of the profile g ðzÞ; corre- ¼ 2 g ðz1Þ dz1 gþðz2Þ dz2 ð9Þ N z1 sponding to the interface of a width 2a: The ``symmetric'' case is shown by the dashed line, and the ``asymmetric'' one is depicted m by the shaded region. 1ð2Þ ¼ ½I ð3Þð ; þ; þÞ8I ð3Þð ; ; þÞŠ=4s3; ð10Þ 496 N.V. Kourtina et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 494­496 where 108 Z þN Z þN Ið3Þð ; 7; þÞ ¼ g 107 ðzÞ dz1 g7ðzÞ dz2 N z1 Z 106 þN gþðzÞdz3: 105 zN 104 Consider now the physical meaning of Eqs.(5)­(10). First of all, the phase shift d arises due to transmitting intensity 103 electromagnetic wave in the non-uniform interface 102 region.This phase shift is equivalent to ``effective'' increase in the thickness of layer 1 to value d: z0 ¼ z d 101 (see Fig.1).In the process of the numerical treatment of 100 the X-ray reflectometry profiles this fact enables one to 0.0 0.1 0.2 0.3 0.4 0.5 0.6 adjust the ratio between the layers' thicknesses in the q, Å-1 periodical cell of the superlattice in order to obtain the Fig.2. Model X-ray reflectivity profiles for multilayer structure best fit to experimental data.The second order correc- Al2O3/Cr(70 (A)/[Fe(20 (A)/Cr(9 (A)]8 calculated without asym- tion to the amplitudes rF; tF in Eq.(5)­(7) reproduces metric phase corrections (points), and with asymmetric phase the well-known Nevot­Croce [5] approximation.The correction (m1 ¼ 0:2; solid line).Wave length l ¼ 1:789 (A. magnitude s has the meaning of the root-mean-square interfacial roughness and it is given by Eq.(9). The phase correction corresponding to the third order numerical algorithm will be presented elsewhere.We terms of the expansion is a new feature in the question. would like to emphasize that the form of the scattering In addition to s2 it contains two extra parameters m1 and matrix as given in Eqs. (5)­(7) is rather general, i.e., it is m2: We found, that m2 is in general non-zero for a wide irrelevant to the precise form of a reflectivity profile. set of profiles wðzÞ; whereas m1 does not vanish in case of Thus it provides the unification description of a large the asymmetric interfaces only.Hence, this property variety of possible symmetric as well as asymmetric may be used to define m1 as the measure of the interface interfaces. asymmetry.The parameter m2 has no such evident Summing up, we have developed the theory of meaning as m 1: But we may note that due to inequality specular X-ray reflectivity from a rough interface based ðq 1 q2Þ=q1 {1 the contribution from m2 in Eqs.(5) upon the reflection function method.By using the and (6) turns out to be less essential, than the approximation of the abruptly changing potential we asymmetric term, proportional to m1: have found the phase correction to the reflectivity due to To test the obtained approximation we exploited the interface roughness and asymmetry, which is essential symmetric Epstein profile gE ðzÞ ¼ ð1 þ e z=aÞ 1 for for the description of the X-ray reflectivity spectra for which the exact solution is known.In this case we ffiffi p ffi ffiffi p ffi greater incident angles. obtained s ¼ ðp= 3Þa and m2 ¼ ð3 3=2p3ÞBð3ÞD0:100: We have also found that m2 has the same order of magnitude regardless of the exact choice of g The research was partially supported by RFBR 7ðzÞ (one possible form from many others is shown in Fig.1). (Grants No.01-02-17119 and 00-15-96745). Assuming further m2 ¼ 0:1 the model X-ray profile corresponding to the Al2O3/Cr(70 (A)/[Fe(20 (A)/ Cr(9 (A)]8 multilayer has been calculated, taking into account the possible asymmetry m1 in the interfacial References structure.Provided the matrices Sk;kþ1 are known, the solution of Eq.(1) and, hence, the scattering matrix S of [1] V.Holy, U.Pietsch, T.Baumbach, High-Resolution X-ray the whole multilayer is found by means of recurrent Scattering from Thin Films and Multilayers, Springer, scheme [6].The results obtained are shown in Fig.2.In Berlin, Heidelberg, 1999. agreement with Eqs.(5)­(7) the phase correction be- [2] X.-H. Zhou, S.-L. Chen, Phys. Rep. 257 (1995) 223­348. comes essential with the increase of the incident angle y [3] V.V. Babikov, Phase Function Method in Quantum Mechanics, Moscow, Nauka, 1976.(in Russian) and it provides a more adequate description of the [4] J.M. Bai, E.E. Fullerton, P.A. Montano, Phys. B 221 (1996) reflectometry spectrum for the scattering vectors in 411. the range from the first to the second Braggs' peaks. [5] L.Nevot, P.Croce, Rev.Phys.Appl.15 (1980) 761. The more exhaustive account and the details of our [6] L.G. Parratt, Phys. Rev. 45 (1954) 359.