3B2v7:51c ED: CM = BG GML4:3:1 MAGMA : 8448 Prod:Type: com pp:123ðcol:fig::NILÞ PAGN: sandhya SCAN: Padma ARTICLE IN PRESS 1 3 Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 5 7 Method of the reflections function in the X-ray 9 reflectometry study of multilayers 11 N.V. Kourtina*, E.A. Kravtsov, V.V. Ustinov 13 Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, 18, S. Kovalevskaya St., GSP-170, Ekaterinburg, 620219, Russia 15 17 Abstract 19 A theory of specular X-ray reflectivity from a rough interface based on the reflection function method is proposed. By 21 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 23 obtain the degree of interface asymmetry. The model X-ray reflectometry profiles for Fe/Cr superlattice are used to illustrate the method. r 2001 Published by Elsevier Science B.V. 25 Keywords: Multilayers, metallic; Interface structure; X-ray reflectivity 27 29 57 31 The X-ray reflectometry is a useful tool for studying scattering matrix surface and interface structure in thin films and multi- ! 59 33 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 61 35 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 63 37 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 65 39 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 67 41 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 69 43 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: 71 45 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 73 47 one-dimensional Helmholtz equation particular, the first order approximation easily enables one to go beyond DWBA. We denote qðzÞ ¼ d2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffi 75 49 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: 77 51 Here EðzÞ is 79 53 UNCORRECTED PROOF 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 81 i z 55 *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 0304-8853/01/$ - see front matter r 2001 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 9 1 1 - 8 MAGMA : 8448 ARTICLE IN PRESS 2 N.V. Kourtina et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 1 where AðzÞ and CðzÞ are the amplitude functions. In and qc ¼ maxjqðzÞj: In the X-ray reflectometry studies a 57 addition, we apply the following condition: is of the order of mean-root-square interfacial roughness 3 Z s ¼ 228 (A [4] and q 59 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 5 dz 2 2 z0 (0oyo41). These estimates make it possible to find the 61 Z i z solution of Eq. (3) in the form 7 CðzÞ exp qðxÞ dx : ð2bÞ 2 BðzÞ ¼ B 63 z 0ðzÞ expðbðzÞÞ; ð4Þ 0 9 The reflection function BðzÞ is defined as BðzÞ ¼ where 65 CðzÞ=AðzÞ:Taking into account the continuity of EðzÞ B0ðzÞ ¼ ðqðzÞ q2Þ=ðqðzÞ þ q2Þ; 11 and Eq. (1) one can prove that BðzÞ satisfies the first 67 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q order non-linear differential equation q2ð1Þ ¼ 2k sin2 y þ w7 13 Z 69 d q0ðzÞ z BðzÞ ¼ exp i qðxÞ dx and 15 dz 2qðzÞ 71 z0 X N Z bðzÞ ¼ b z nðzÞen: 17 B2ðzÞ exp i qðxÞ dx ð3Þ n¼1 73 z0 The function B0ðzÞ corresponds to the boundary 19 75 Eqs. (1) and (3) should be supplemented by the condition BðþNÞ ¼ 0 and it gives the Fresnel reflection 21 boundary conditions. For example, the choice of coefficient rF11 ¼ ðq1 q2Þ=ðq1 þ q2Þ from an ideal sharp interface. The series bðzÞ yields the corrections due to the 77 BðþNÞ ¼ 0 corresponds to the X-ray beam, incident interfacial non-ideality. 23 from zo0; and in this case a reflection coefficient r11 is The use of ansatz (4) is the essential step in the 79 given by the relationr11 ¼ Bð NÞ: We also take into derivation. It enables us partially to sum up the 25 consideration the dimensionless functions g 81 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 bnðzÞ are associated with the 27 w 83 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 29 85 obey the relation g that, at each step n, one encounters the only linear þðzÞ þ g ðzÞ ¼ 1: One can regard g inhomogeneous differential equation for b 7ðzÞ as a ``shape'' of the interface, which reproduces nðzÞ: The 31 87 the gradual transition from the first layer to the second details of this derivation will be presented elsewhere. 33 one. We shall call interface ``symmetric'', if ðq=qzÞg As a result, up to the third order of e, the elements of the ðzÞ is matrix S 89 an even function of z; otherwise interface is ``asymmetric''. 12 can be written in the form 35 In case of grazing incidence angles, Eq. (3) can be r11 ¼ rF11 exp iq1d 12q1q2s2 91 solved in the approximation of the abruptly changing þiq ; ð5Þ 37 potential. The small parameter e of this expansion is 1½ðq21 þ 3q22Þm1 þ ðq21 q22Þm2 s3 93 defined as e ¼ aq c=2p; where a is the characteristic r 39 length corresponding to the variation of the potential 22 ¼ rF 22 exp iq2d 12q1q2s2 95 þ iq2½ ðq21þ3q22Þm1 þ ðq21 q22Þm2 s3 ; ð6Þ 41 97 t12 ¼ t21 ¼ tF12 expð12iðq1 q2Þd þ 18ðq1 q2Þ2s2 43 þ 1 99 2iðq1 q2Þ½ðq21 þ 3q22Þm1 þ ðq1 þ q2Þ2m2 s3Þ: ð7Þ 45 Here rF; tF are Fresnel's reflection and transmission 101 amplitudes and parameters d; s; and m1ð2Þ are expressed 47 via g7ðzÞ as follows 103 Z þN 49 d ¼ zg= 105 51 107 53 Fig. 1. The sponding to the UNCORRECTED PROOF ðzÞ dz; ð8Þ N s2 ¼ 2Ið2Þð ; þÞ Z þN Z þN linear segment form of the profile g ðzÞ; corre- ¼ 2 g 109 ðz1Þ dz1 gþðz2Þ dz2 ð9Þ N z1 interface of a width 2a: The ``symmetric'' case is 55 shown by the dashed line, and the ``asymmetric'' one is depicted 111 m by the shaded region. 1ð2Þ ¼ ½I ð3Þð ; þ; þÞ7I ð3Þð ; ; þÞ =4s3; ð10Þ MAGMA : 8448 ARTICLE IN PRESS N.V. Kourtina et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 3 1 where 108 Z þN Z þN 55 3 Ið3Þð ; 7; þÞ ¼ g 107 ðzÞ dz1 g7ðzÞ dz2 N z1 106 57 Z þN 5 g 105 þðzÞ dz3: 59 zN 7 104 Consider now the physical meaning of Eqs. (5)­(10). intensity 61 First of all, the phase shift d arises due to transmitting 103 9 electromagnetic wave in the non-uniform interface 102 63 region. This phase shift is equivalent to ``effective'' 11 increase in the thickness of layer 1 to value d: z0 ¼ z d 101 65 (see Fig. 1). In the process of the numerical treatment of 13 100 the X-ray reflectometry profiles this fact enables one to 0.0 0.1 0.2 0.3 0.4 0.5 0.6 67 adjust the ratio between the layers' thicknesses in the q, Å-1 15 periodical cell of the superlattice in order to obtain the Fig. 2. Model X-ray reflectivity profiles for multilayer structure 69 best fit to experimental data. The second order correc- 17 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 71 the well-known Nevot­Croce [5] approximation. The 19 correction (m1 ¼ 0:2; solid line). Wave length l ¼ 1:789 (A. magnitude s has the meaning of the root-mean-square 73 interfacial roughness and it is given by Eq. (9). 21 The phase correction corresponding to the third order numerical algorithm will be presented elsewhere. We 75 terms of the expansion is a new feature in the question. 23 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 77 m2: We found, that m2 is in general non-zero for a wide 25 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 79 the asymmetric interfaces only. Hence, this property 27 variety of possible symmetric as well as asymmetric may be used to define m1 as the measure of the interface interfaces. 81 asymmetry. The parameter m2 has no such evident 29 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 83 ðq 1 q2Þ=q1 {1 the contribution from m2 in Eqs. (5) 31 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 85 asymmetric term, proportional to m1: 33 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 87 symmetric Epstein profile gE ðzÞ ¼ ð1 þ e z=aÞ 1 for 35 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. 89 obtained s ¼ ðp= 3Þa and m2 ¼ ð3 3=2p3ÞBð3ÞD0:100: 37 We have also found that m2 has the same order of 91 magnitude regardless of the exact choice of g The research was partially supported by RFBR 7ðzÞ (one 39 possible form from many others is shown in Fig. 1). (Grants No. 01-02-17119 and 00-15-96745). 93 Assuming further m2 ¼ 0:1 the model X-ray profile 41 corresponding to the Al2O3/Cr(70 (A)/[Fe(20 (A)/ 95 Cr(9 (A)]8 multilayer has been calculated, taking into 43 account the possible asymmetry m1 in the interfacial References 97 structure. Provided the matrices Sk;kþ1 are known, the 45 solution of Eq. (1) and, hence, the scattering matrix S of [1] V. Holy, U. Pietsch, T. Baumbach, High-Resolution X-ray 99 the whole multilayer is found by means of recurrent Scattering from Thin Films and Multilayers, Springer, 47 scheme [6]. The results obtained are shown in Fig. 2. In Berlin, Heidelberg, 1999. 101 agreement with Eqs. (5)­(7) the phase correction be- [2] X.-H. Zhou, S.-L. Chen, Phys. Rep. 257 (1995) 223­348. 49 comes 103 51 105 53 UNCORRECTED PROOF 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. 107