PHYSICAL REVIEW B VOLUME 54, NUMBER 11 15 SEPTEMBER 1996-I Diffuse scattering from interface roughness in grazing-incidence x-ray diffraction S. A. Stepanov,* E. A. Kondrashkina,* M. Schmidbauer, R. Ko¨hler, and J.-U. Pfeiffer MPG-AG ``Ro¨ntgenbeugung,'' Hausvogteiplatz 5-7, Berlin D-10117, Germany T. Jach National Institute of Standards and Technology, Gaithersburg, Maryland 20899 A. Yu. Souvorov European Synchrotron Radiation Facility, Boi te Postale 220, Grenoble Cedex F-38043, France Received 12 March 1996 A theory of x-ray diffuse scattering from interface roughness in grazing-incidence diffraction GID is presented. The theory assumes dynamical diffraction of x rays from perfect multilayers with the diffuse scattering from roughness calculated in the distorted-wave Born approximation. This permits the calculation of scattering due to roughness at all points on the diffraction curves, including the vicinity of the Bragg peaks. It is shown that the measurements of diffuse scattering in GID can provide information on atomic ordering at crystal interfaces which is not accessible by usual x-ray specular reflection and nonspecular x-ray scattering. The theory is found to be in good agreement to the two GID experiments carried out with an etched Ge surface and an AlAs/GaAs superlattice at the Cornell High-Energy Synchrotron Source and European Synchrotron Radiation Facility, respectively. In the case of the etched Ge surface, an anti-Yoneda dip in the diffuse scattering pattern at the Bragg peak and two symmetrical shoulders on the Bragg curve wings have been found and explained. In the case of the AlAs/GaAs superlattice, the diffuse scattering has been separated from GID by means of high-resolution measurements. A comparison between diffuse scattering in GID and diffuse scattering in grazing incidence far from the diffraction conditions has shown that the atomic ordering was preserved in the interface roughness, while it was partially destroyed in the surface roughness. S0163-1829 96 05035-7 I. INTRODUCTION observed, while in Ref. 21 an attenuation of the tails with an exponent similar to that found by Ne´vot and Croce24,25 for The combination of Bragg diffraction and total external the tails of x-ray specular reflection curves was predicted. reflection TER effects in grazing-incidence x-ray diffrac- We have assumed that discrepancies between Refs. 15 tion GID opens up a wealth of possibilities in the study of and 21 were due to the diffuse scattering of x rays from thin surface layers of crystals.1­4 GID has been applied with surface roughness that was measured in Ref. 15 in combina- success to investigations of the surface treatment and surface tion with the diffracted intensity of GID. Therefore, the ef- oxidation of semiconductor wafers,5,6 to studies of structure fect of roughness on GID has become the subject of our transformations during ion implantation,7­9 and to analysis study. of strain relaxation in epitaxial layers1,10,11 and multilayers.12 In Sec. II a theory of diffuse scattering arising from sur- The measurements of diffuse scattering DS in GID can face roughness as well as correlated and uncorrelated inter- provide additional information on crystal lattice defects in face roughness in multilayers ML's is presented for GID. surface layers.13,14 However, along with diffuse scattering Our model is based on the dynamical theory of GID in caused by lattice defects, strong scattering due to surface and multilayers26­28 and the distorted-wave Born approxima- interface roughness can be expected in GID by analogy with tion.29­32 The results are general for GID and extremely x-ray TER studies, since the angle of incidence of the x rays asymmetric x-ray diffraction, and are applicable in the case is equally small in both these cases.15­19 Roughness can also of normal lattice strains in multilayers. give rise to a change in the diffracted intensities.20­22 Thus In Sec. III we derive expressions for the diffuse scattering one has to distinguish between the effects of roughness and observed in GID with different experimental setups. In Sec. crystal lattice defects on GID. IV the theory is applied to the explanation of Ref. 15, where The effect of roughness on x-ray Bragg diffraction has not GID curves were taken from a sample with strong surface been adequately explored. As shown in Ref. 23, in normal- roughness produced by etching. incidence diffraction geometries the effect is relatively weak. In Sec. V some numerical examples demonstrating the In GID, there are several theoretical predictions21,22 of a role of diffuse scattering in GID studies of semiconductor strong effect of roughness on the coherent reflection. How- multilayers are given. The effect of the correlation between ever, only a few experimental results20,15 are known, and roughness of different interfaces is discussed. Calculations they are not in good agreement with the theory. In particular, are carried out for different experimental geometries. in Ref. 15 an enhancement of the intensity of the tails of the In Sec. VI we describe a high-resolution GID experiment diffracted beam of GID measured from a rough surface was carried out with an AlAs/GaAs superlattice at the optical 0163-1829/96/54 11 /8150 13 /$10.00 54 8150 © 1996 The American Physical Society 54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8151 and 35 or dynamical3 diffraction theory. The kinematical theory is applicable i far from the Bragg peaks from perfect crystals, ii for imperfect mosaic crystal structures, or iii for GID from very thin layers 1-10 monolayers . The dif- fracted intensities in these cases are weak and given by the DWBA. Then the scattering from interface roughness can be calculated in the kinematical theory either simultaneously with the diffraction from layers17­19 or in the second-order DWBA. The dynamical theory is applicable at all points on GID curves corresponding to bulk reflections from perfect structures. In this case, a strong diffraction intensity at the Bragg peak and the effect of diffraction on the incident and specular waves are taken into account. That is just the case for the both experiments discussed in Secs. IV and VI, where the measurements are taken near the Bragg peaks of bulk reflections. Therefore, we use the dynamical diffraction theory in order to explain the peculiarities of diffuse scatter- ing near the Bragg peaks of GID. For the reader's conve- nience, a short outline of the dynamical theory is given be- low.For the sake of simplicity, we neglect the changes in x-ray FIG. 1. Schematic layouts of grazing-incidence x-ray diffraction polarization by diffuse scattering in GID. These changes can for a a real incident wave Ein0 and b for an ``imaginary'' incident be simply added to the model, but they are too weak of the wave Eout 0 which is inverted with respect to one of diffuse scattered order of noncoplanarity of GID, i.e., 10 3) to be of an waves shown in a by a dashed line. The cones in a schematically interest for present-day experiments. Our derivations below illustrate DS along the diffracted and reflected waves of GID. For are carried out for polarization. The equations are simply the explanation of other vectors, see the text. extended for polarization by incorporating cos(2 B) in the x-ray polarizabilities beamline of the European Synchrotron Radiation Facility h and h¯ . ESRF . These direct measurements of diffuse scattering in A. Some results from the dynamical theory GID are compared to calculations based on our model. We of GID in multilayers conclude with some possible uses of diffuse scattering in GID and their distinction from similar scattering in total ex- In the framework of the dynamical diffraction theory, the ternal reflection. wave fields Ein(r) and Eout(r) can be found as solutions to the system of the dynamical diffraction equations in II. THEORY multilayers.26­28 Let us consider x-ray Bragg diffraction in a multilayer see Fig. 1 . It is assumed to be a stack of N In GID, the diffuse scattering can be observed along the perfect crystalline layers with laterally matched lattice spac- directions of both diffracted and reflected beams see Fig. ing, and every layer can possess its own lattice spacing in the 1 a . We restrict our consideration to the scattering along direction normal to the surface: an n z az az , where the diffracted beam, since the majority of measurements are anz az , and index n denotes the number of the layer in made in this manner.1,5­14,20,22 The analysis of the scattering the stack counted from the surface. This model corresponds along the reflected beam of GID is analogous. to a so-called unrelaxed multilayer containing no misfit dis- The x-ray diffuse scattering is due to the deviations locations. (r) of the polarizability of the scatterer from an ``ideal'' At the first step, we assume flat interfaces between the distribution id(r). The most effective way for its calcula- layers. For Bragg diffraction from a reciprocal-lattice vector tion, in the lowest-order perturbation (r), is to apply the h approximately parallel to the surface, the polarizability can reciprocity theorem.32 The amplitude of DS can be repre- be presented as the following sum over the polarizabilities of sented as the layers: N f 2/4 Eout r r Ein r d3r, 1 flat r n r H z zn H zn 1 z , 2 n 1 where Ein(r) and Eout(r) are the wavefields in the ideal ob- ject with flat interfaces produced by the incident x-ray n r n n n 0 h ehn*r e hn*r. 3 h¯ wave and the wave backprojected to the object from the ob- n n servation point, respectively see Fig. 1 . The parameter is Here H(z) is the steplike Heaviside function, and zn are the the magnitude of the wave vector of these waves in vacuum. coordinates of interfaces. Reciprocal-lattice vectors hn in the The approximation 1 is commonly referred to as the nth layer differ slightly from the mean vector h because of distorted-wave Born approximation DWBA . the variations in the normal lattice spacing between layers: In case of GID, the wave fields Ein(r) and Eout(r) can be hn h hznZ, where hzn h, and Z is a unit vector found with the help of either kinematical33 see also Refs. 34 along the internal surface normal. 8152 S. A. STEPANOV et al. 54 In dynamical Bragg diffraction, the x-ray wave field in The substitution of Eq. 10 into Eq. 9 adds the phase each layer can be expanded over the sum of the transmitted factor to Dhnj corresponding to the expansion of the x-ray and diffracted Bloch waves with wave vectors k0n and wave field over h instead of hn in 4 . Thus we can transfer khn k0n hn , and amplitudes D0n and Dhn , this phase factor from 9 to 4 , and proceed to the expan- respectively:36­38 sion over h for the whole multilayer. At that, the dispersion equations 8 remain formally unchanged because the expo- En r D0neik0n*r Dhneikhn*r. 4 nents on the right side cancel each other. The amplitudes D The amplitudes D 0n j can be found with the help of the 0n and Dhn can be treated as constants boundary conditions for the x-ray waves and their deriva- satisfying the dynamical diffraction equations in each layer: tives at each interface.2­4 It is convenient to present the k2 2 boundary conditions in a (4 4) matrix form,9,26­28 0n 0 nD n D k2 D0n 0 0n h¯ hn , U 0n n SvEv S1F1 D1 , 5 k2 2 S L D U D hn h n 1F1 1 S2F2 2 , D nD k2 Dhn h 0n 0 hn . 11 hn n . . . Making use of the fact that the lateral components of all L U vectors k SN 1FN 1DN 1 SNFN DN . 0n and khn coincide because they remain unchanged at refraction and specular reflection, and substituting see Here Ev (E0 , 0, Es , Eh) and Dn (D0n1, D0n2, D0n3, Fig. 1 a : 0z sin 0 , hz sin h , k0zn un , and D0n4) are the four-component vectors composed by un- h n zn n (1 az/a), we can rewrite equations 5 as known x-ray amplitudes E0 , Es , and Eh are the amplitudes of the incident, specularly reflected and diffracted waves in u2 n n vacuum; see Fig. 1 a . Parameters S n sin2 0 0 D0n D h¯ hn , v , Sn , and Fn denote n (4 4) characteristic matrices of layers: 6 u n n n n 2 sin2 h 0 Dhn h D n 0n . The condition 2 2 1 0 1 0 0 1 0 1 h 0 presuming the elastic scattering of Sv , 12 x rays, gives sin 0 0 sin 0 0 0 sin sin2 h 0 sin h h sin 0 2 , 7 where (2 0h h2)/ 2 is the standard parameter for the angular deviation of the incident wave from the Bragg con- 1Vnj dition in the dynamical theory of diffraction. Snj , 13 u The values of u n j n are determined by the dispersion equa- tion which is the condition for the existence of a solution of Vnj unj Eq. 6 : and F(U,L) ni j ijexp(iunj z(U,L)), where indices (L) and (U) indicate that the exponents are evaluated at the lower upper u 2 n n n n n sin2 0 0 un n 2 sin2 h 0 . h¯ h boundary of layer, (L) n n n (U)n 1. 8 The solution to Eq. 11 is straightforward: Equation 8 is a fourth-order polynomial equation for un , E 1S 1S 1 S U D and has four solutions. As shown in Ref. 39, there are always v Sv 1Ft1S1 2Ft2 * * * SN 1 NFN N , 14 two solutions corresponding to x-ray waves damping with where Ftnij ijexp( iunj tn) and tn z(L) z(U). After cal- z Im(un) 0 , and two other solutions corresponding to the culating the matrix product in the right-hand side of Eq. 14 , waves growing with z Im(un) 0 . The latter waves are we obtain four linear equations for four amplitudes: Es , usually treated as being specularly reflected from the lower Eh , D0N1, and D0N2. The other amplitudes are given by interfaces of the layers. For each of the solutions, Eq. 6 Eq. 11 . For the details of the numerical procedure, the gives ( j 1, . . . ,4): reader is referred to Refs. 26­28. u2 sin2 n B. Amplitude of diffuse scattering from individual fluctuations D n j 0 0 hn j n D0nj VnjD0nj . 9 in interface positions h¯n Following the approach developed for x-ray reflection by The polarizabilities n rough surfaces,29­32 we describe rough interfaces between h correspond to vectors h n n which layers as local fluctuations z vary from layer to layer. Therefore, it is more convenient to n( ) in the interface positions ( is the coordinate vector along interfaces . This approach use the expansion over h see, e.g., Ref. 40 : is obviously valid at length scales much greater than the interatomic distances. Then instead of 2 the polarizability n n n h exp h z . 10 n h z of the layers is given by 54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8153 N rough r n r H z zn zn n 1 H zn 1 zn 1 z , 15 Respectively, the fluctuations of polarizability can be written as a sum over the interfaces, N r rough r flat r Pn r n r , 16 n 1 where P(r) is the steplike function introduced by analogy with Refs. 30 and 31: 1, z zn,zn zn , zn 0 Pn r 1, z zn zn ,zn , zn 0 0 elsewhere 17 and n r n n 1 n n 1 0 0 h h eih*r n n 1 e ih*r. 18 h¯ h¯ FIG. 2. Four different diagrams of x-ray scattering from a fluc- tuation of interface position shaded area in GID as given by the Thus the perturbation of the crystal polarizability due to four terms in Eq. 25 . Coherent and diffuse scattered waves are roughness possesses the form shown by thick and thin vectors, respectively. Horizontal lines present the Bragg planes in crystal. N n r P n n n n r 0 heih­r e ih­r . 4 4 n 1 h¯ 2 exp iQnijzn 19 f n 4 Enij i 1 j 1 iQnij Substituting Eq. 4 and Eq. 19 into Eq. 1 we also assume29­32 that the wave fields do not change considerably d2 eiQnij zn 1 eq* , 24 on the scale of the roughness. Then the wave fields of one of two layers forming an interface e.g., of the lower one can where it is denoted be used at both sides of the interface, and we obtain the scattering amplitude E in out n in out n ni j D0niD0n j h DhniDhn j h¯ N 2 N 4 4 z in out in out n n zn D D D D , 25 f f hni 0n j 0ni hn j 0 n d2 dz n 1 4 n 1 i 1 j 1 zn Q in out ni j uni un j . 26 ei in out in out 0 0 i uni un j z Din in 0ni Dhnieih*r The four terms in Eq. 25 can be treated as shown in out out Fig. 2. 0 heih*r h¯e ih*r D0n j Dhn jeih*r . i Term 1 corresponds to the Bragg diffraction of x rays 20 by the fluctuations of interfaces Fig. 2 a . We are interested in diffuse scattering about the direction ii Term 2 corresponds to the triple Bragg diffraction: of the grazing diffracted beam. Therefore, out in the incident wave is diffracted by ML, then diffracted at 0 ( 0 h), and it is convenient to write h in the fluctuations, and diffracted by the multilayer again Fig. 2 b . in in iii Term 3 corresponds to the Bragg diffraction of the 0 0Br qin, 21 incident wave by the multilayer, and the small-angle scatter- out in ing of the diffracted wave by the fluctuations Fig. 2 c . 0 0Br h qout, 22 iv Term 4 corresponds to the small-angle scattering of q qin qout, 23 the incident wave by the fluctuations and the Bragg diffrac- tion of the scattered wave by the multilayer Fig. 2 d . where in0Br is the vector in the incidence direction, exactly One can see that, in all four cases, x rays are scattered satisfying the Bragg condition. The terms in Eq. 20 con- through the large angle 2 B due to the diffraction from the taining exp( ih*r) oscillate with at an atomic scale and Bragg planes. That is, the momentum transfer due to scatter- can be neglected. Then, after carrying out the integration ing from roughness is small and the intensity of DS must be over z, Eq. 20 is transformed to much greater than that in Refs. 41 and 42. In those cases, 8154 S. A. STEPANOV et al. 54 x-ray scattering through large angles corresponding to large momentum transfers was measured from amorphous multi- layers with rough interfaces in a geometry formally similar to GID. C. Cross section of diffuse scattering from statistical roughness Proceeding from Eq. 24 to the cross section of diffuse scattering, we obtain d 4 N 4 4 * d f 2 16 2 EnijEn i j n,n 1 i,i 1 j,j 1 ei Q * ni j Qn i j zn Q * ni jQn i j 1 d2 d2 eiq eiQ * ni j zn iQ n i j zn 1 , 27 where . . . denotes averaging over random functions zn( ) assumed to be Gaussian. The application of the general formula43 exp( j jxj) exp( FIG. 3. Different degrees of collimation in the scattering plane jk j k xjxk /2), where j are constants and x j are Gaussian random variables, to Eq. 27 gives of grazing-incidence x-ray diffraction: a , b , and c show single-, double-, and triple-crystal experimental schemes, respectively. Co- herent and diffuse scattered waves are shown by thick and thin d 4 N 4 4 * * vectors, respectively. The incident and scattered beams are colli- d S 16 2 CnijEnijCn i j En i j mated and analyzed in a direction normal to the plane of the figure. n,n 1 i,i 1 j,j 1 d2 eQ * ( j 1 and 2 . This procedure can be called ``the specular- ni jQn i j Knn 1 eq* , 28 reflection approximation.'' Below, we show that it has more applications. where C 2 2 ni j exp(iQnijzn nQnij/2)/Qni j , n is the rms height of roughness, 2 2 n zn(0) , and Knn ( ) z III. INTEGRATED DIFFUSE SCATTERING n(0) zn ( ) is a correlation function. The same cor- relation functions proposed for diffuse scattering in x-ray IN DIFFERENT TYPES OF GID EXPERIMENT TER should be valid here.29,31,32,44,45 In the majority of cases the incidence and the exit angles Thus expression 28 for diffuse scattering in GID is for- of diffracted beam are controlled in GID experiments be- mally similar to that in TER compare Refs. 30­32 and 44­ cause these angles determine the penetration depth of GID 47 . However, there are two essential differences. The physi- inside the samples. However, the angles in the Bragg plane cal difference is that diffuse scattering in TER originates are not always controlled see Fig. 3 , and that provides from 0 the fluctuations of the mean target density , while some averaging of the pattern and simplification of Eq. 28 . in GID it is mainly due to h the crystal structure of the The general formula 28 is applicable in the case of fluctuations . That means that diffuse scattering in GID pro- triple-crystal measurements only, when the incident beam is vides information about the degree of crystal structure per- collimated and the scattered beam is analyzed in two planes fection at interfaces. The mathematical difference consists of Fig. 3 c . That would be the most informative case, but the the dependence of Enij and En i j on q, which is not the case low intensity of GID might impose serious experimental in TER. Hence the DS pattern is far more complicated in limitations. GID than in TER, and depends on four diffraction angles: the angles of the incident and the scattered x-ray waves with respect to the surface and their deviations from the Bragg A. Single-crystal scheme: No angular resolution angle in the surface plane. An analytical integration of Eq. in the Bragg plane 28 over one of the components of q, like that used to re- Some authors48,27,12 have performed measurements where duce the calculations in the case of diffuse scattering in the incident beam is collimated in 0 , but not in , and the specular reflection, is thus not possible. diffracted waves are separated over their exit angles by a slit One simplification occurs when DS is studied far from the or a position-sensitive detector PSD . This single-crystal direction of propagation of the coherent diffracted beam scheme49 is based on Eq. 7 , where 2sin(2 B)( B) where the Bragg diffraction of waves scattered by roughness 10 5 is proportional to the deviation of the in-plane angle can be neglected. Then Dout out out2 n hn j 0 and un j ( 0 0)1/2, from the kinematic Bragg angle B . The large width of giving Dout 0n j as solutions to the specular reflection problem Bragg peaks for 0 and h 10 2­10 3 makes these 54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8155 measurements very convenient. However, the coherent re- in the Bragg plane, but the acceptance of the scattered beam flection and diffuse scattered radiation are all counted to- is limited by an analyzer crystal. gether see Fig. 3 a . If neither the in-plane angle of the incident or diffracted IV. ANALYSIS OF A GID EXPERIMENT beam is collimated, then waves with large Bragg deviations dominate in the incident and scattered fans, and the solutions WITH AN ETCHED Ge SURFACE to the specular reflection problem can be used for both The theory given in Secs. II and III has been applied to Ein(r) and Eout(r). Thus the dynamical diffraction problem interpreting the results of the experiment in Ref. 15. In this of GID need not be considered for the calculations of DS at study, carried out at the Cornell High-Energy Synchrotron all. The experiment integrates over both the in-plane compo- Source CHESS , the grazing-incidence diffraction was mea- nents of q, providing (x) (y)/ 2 in integral 28 . Thus one sured from 220 Ge planes at 1.55 Å. A double-crystal finds scheme of measurements was used. The incident beam was collimated in the incidence angle d N 2 2 0 within 0.25 mrad, and 1 2 n in out in the diffraction angle within 0.0014 mrad. The sample was d S16 2 Cnij hD0niD0nj n,n 1 i,i 1 j,j 1 rocked through the Bragg angle B at fixed 0 4 mrad, and the entire scattered intensity was collected over the take- C n in out n i j h D0n i D0n j * off angle h and over the in-plane exit angle. The sample surface consisted of two different parts: a eQ * ni jQn i j Knn 0 1 . 29 high-quality polished part and an etched quadrant provided a As follows from Eq. 29 , the DS measured in the single- spectrum of surface roughness. The measured GID curves crystal scheme is completely determined by K for these two parts are presented in Figs. 4 a and 4 b . The nn (0). In the case where the roughness of different interfaces is com- curve taken from a smooth surface coincides well with the theoretical calculations for the perfect case. A peculiarity of pletely uncorrelated, K 2 nn (0) n nn , the measurements of both the experimental and theoretical curves in the perfect integral DS provide the rms roughness height. case is the zero reflection coefficient to the left of the Bragg In the case of small completely correlated roughness, peak, where the diffracted wave becomes surface trapped where Knn (0) n n and the exponent in 29 can be ex- the angle panded ( h becomes imaginary due to Eq. 7 . Contrary to nQni j 1), formula 29 for the diffuse scattering the smooth surface case, the experimental curve for a rough becomes very similar to that for the intensity of coherent surface exhibits two nearly symmetrical shoulders at both GID calculated in the DWBA see, e.g., Ref. 22 : sides of the Bragg peak. These shoulders are obviously due d N 2 to diffuse scattering at surface roughness, because the effect 1 S n in out of roughness on the coherent beam would appear to be a d 16 2 n hD0niD0nj n 1 i, j 1 Debye-Waller attenuation of the intensity on the wings.21 The diffuse scattering has been calculated with the help of ei uin out 2 in out ni un j zn n 2 uni un j 2 2. 30 Eq. 31 . In the case of only one interface and 0, it is greatly simplified: The expression for GID differs from 30 by the substitu- 2 2 Q*2 /2 tion of n d 2 3 e 2 Qi j h instead of n out in in h . The consequences of this 2 D * analogy are discussed in Sec. V, where we give some nu- d S 16 2 D0 j i, j 1 QiQj* Di merical examples. dx eQiQ*jK x 1 eqxx, 32 B. Double-crystal scheme: Partial angular resolution in the Bragg plane where Dout 2sin out/(sin out uout), Q in uout), In some more advanced double-crystal experiments, the 0 0 0 i (ui uout (sin2 out in are the two solutions to the incident beam is collimated in the Bragg plane, while the 0 0)1/2, and ui in entire in-plane spread of the scattered beam is accepted Fig. dispersion equation 8 with Im(ui ) 0. 3 b . Then the ``specular reflection'' approximation can be Equation 32 was integrated over h , and renormalized applied to Eout(r) and, additionally, one can average Eq. 28 to the reflectivity see Eq. 2.13 in Ref. 29 : over the components of q normal to h . This procedure re- 1 d duces the integral in 28 to a one-dimensional one, R 2 2 DS Ssin 0 0 d d h . 33 d N 4 2 2 3 in out d S 16 2 CnijDniD0nj The correlation function was chosen in the Gaussian form: n,n 1 i,i 1 j,j 1 K( ) 2exp( 2/ 2), where is the rms height and is C in out the lateral correlation length of roughness. The integration in n i j Dn i D0n j * 33 was carried out numerically. The normalized calculated flux of DS for different is dx eQ * ni jQn i j Knn x 1 eqxx, 31 presented in Fig. 4 c . The shape of DS curves strongly de- pends on the lateral correlation length of roughness: at where q n n x q* h / , and Dni D0ni h Dhni 0 The same greater correlation lengths DS is concentrated closer to the situation takes place when the incident beam is uncollimated coherent diffracted beam of GID. The dependence of calcu- 8156 S. A. STEPANOV et al. 54 lated DS on is presented on Fig. 4 d . The intensity of DS quickly grows with , while the shape of the curves is prac- tically independent of in a wide range up to 10 Å. At 10 Å, the DWBA starts to diverge at B 0, where the diffracted wave of GID is surface trapped. This diver- gence is due to a small penetration depth of surface-trapped x-rays see Fig. 4 e . The DWBA breaks down when the x-ray wave fields undergo large changes on the scale of the height of the roughness. The roughness height for the case in Fig. 4 b measured with a 3- m profilometer tip was 200 Å. This value was obviously beyond the applicability of the DWBA. In particu- lar, the DWBA fails to explain the attenuation of maximum reflectivity at the Bragg peak in Fig. 4 b . However, taking into account the weak dependence of the shape of DS curves on , we can fit the shape of the curves at small , and then extrapolate the data to higher where the DWBA diverges. The fitting procedure was carried out in two steps: first, the shape of the DS curve was fitted at B 0 where the coherent reflection is zero see Fig. 4 a . Then the calculated DS was added to the calculated coherent reflectivity attenu- ated by some empirical factor ch in order to fit the maximum of the reflection coefficient: R 2 2 2 total ch R coherent R DS . The factor ch was introduced to account for the relative con- tributions of coherent diffracted and scattered radiation when the roughness was great. The parameters of the fit presented in Fig. 4 b are 1600 Å and ch 0.4. The roughness rms height fitted at B 0, where the DWBA does not diverge is 43 Å. This value is consistent with the profilometer data, since the rms roughness at 1600 Å need not be as great as the rms roughness at 30 000 Å corresponding to the horizontal resolution limit of the profilometer. The long-wavelength roughness measured with the profilometer could not cause the DS on the tails of the curve Fig. 4 b because the DS corresponding to the long-wavelength roughness is strongly concentrated near the Bragg peak. This roughness was prob- ably responsible for the broadening of the experimental Bragg peak in Fig. 4 b . If, on the other hand, we consider the possibility that etching causes the crystal structure disor- dering, then, the parameter h in Eq. 32 would be re- duced by a static Debye-Waller factor, and the same intensity of DS would be achieved with a greater . Both the experiment and theory show a dip in the DS pattern at the Bragg peak. In Ref. 15 this dip was supposed to be due to a cutoff of the maximum roughness wavelength to which the experiment should be sensitive because of the limited coherence length of the diffracted beam ( 3 m . The theory presented here contains no assumptions on the coherence length of the source, and the interpretation of this FIG. 4. Fitting of GID data measured in Ref. 15 Ge, 220 effect is different. As follows from Eqs. 31 and 32 , the Bragg planes, 1.55 Å, and 0 4 mrad . Data are represented intensity of DS in the double-crystal scheme is approxi- by dots, and fits by solid lines. a Smooth surface fit assumes GID mately proportional to the total intensity of the x-ray wave with no diffuse scattering . b Rough surface fit is GID plus DS field illuminating the interface: calculated with 1600 Å and 43 Å . c Calculated shape of DS curves vs correlation length of roughness the dashed line indi- cates the total x-ray intensity at crystal surface . d Intensity of DS d 2 2 2 in in vs height of roughness the DWBA diverges at * in in B 0 for d Di Di 0 D0i Dhi 2 i,i 20, 30, and 40 Å . e Penetration depth inside the crystal cal- 1 i 1 culated for two wave fields Din1,2 of GID at 0 4 mrad the decrease in penetration depth at B 0 causes the divergence of in in in 0 E0 Es Eh 2. the DWBA in d . 34 54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8157 The right side of Eq. 34 exhibits a dip at 220 , pro- viding the minimum in DS see the dashed line in Fig. 4 c . Thus, the observed effect has the same origin as the Yoneda peaks50,29 in x-ray DS and the secondary emission yield ap- pearing near the critical angle for TER. Yoneda peaks are due to the enhancement of x-ray intensity at the surface near the critical angle for total external reflection. Here we have a dip in intensity at the atomic planes near the Bragg angle, which is matched in this instance with the threshold angle for total internal reflection with the angle , where h becomes an imaginary quantity due to Eq. 7 and the diffracted wave becomes surface trapped . This dip provides a minimum in DS in Fig. 4 b as well as a minimum in the fluorescence yield from lattice sited atoms which was observed in GID standing wave experiment.51,52 Therefore, our case can be referred to as an ``anti-Yoneda effect.'' At 0, the critical angle for total internal reflection given by Eq. 7 may not coincide with the Bragg angle. Then two dips in DS are predicted by our theory, the dip at the critical angle for total internal reflection being stronger than that at the Bragg angle. FIG. 5. Calculated x-ray reflectivity curves vs the takeoff angle for the single-crystal scheme of GID. The curves are for a 20-period AlAs/GaAs superlattice (t V. DIFFUSE SCATTERING IN MULTILAYERS AlAs 125 Å and tGaAs 95 Å on a 001 GaAs substrate. The parameters of the calculations are as follows: In the case of multiple and periodic rough interfaces, the 220 reflection, 1.5 Å; 0 0.3°, 5 Å, and 2000 Å. effects of roughness become much more prominent. We have Curves 1 and 2 show DS for uncorrelated ( z 0) and completely carried out calculations for a sample similar to that discussed correlated ( z ) roughnesses of interfaces, respectively. Curves 3 in Refs. 27 and 31: an AlAs/GaAs superlattice consisting of and 4 present the coherent GID reflection for flat and rough inter- 20 periods of 125 Å AlAs and 95 Å GaAs on a 001 - faces, respectively. oriented GaAs substrate. The calculations assumed a 220 Bragg reflection of polarized incident x rays with GID measurements is due to fluctuations in the crystal struc- 1.5 Å and ture h , while the parameters of roughness given by x-ray 0 0.3°. The correlation function was cho- sen in the form suggested by Ming et al.,44 reflectivity refer to the fluctuations of material density 0 . In the case where the crystal structure is destroyed by K roughness, the data of TER and GID may disagree: the latter nn n n e / 2e zn zn / z, 35 may exhibit a reduced intensity corresponding to a smaller with a rms roughness height n 5 Å and a lateral correla- . Then, to isolate the diffuse component of GID, one has to tion length 2000 Å. select scattered radiation in double- or triple-crystal measure- The calculated intensity of DS for the single-crystal case ments. of GID renormalized to a reflectivity as in Eq. 33 is pre- A comparison between the diffuse scattering in TER and sented in Fig. 5 as a function of exit angle h . Curves 1 and GID can be used for investigating the crystal structure of 2 correspond to the uncorrelated ( z 0) and completely cor- rough interfaces. In our model, one can add a Debye-Waller related ( z ) roughness of interfaces, respectively. One factor for h and h¯ in Eq. 25 describing the attenuation can see that in both the cases the curves exhibit multilayer of polarizabilities due to disordering of the crystal structure Bragg peaks at the same angular positions as the peaks of at rough interfaces. The value of this Debye-Waller factor coherent GID curves 3 and 4 . Therefore, it is difficult to can be found from the difference in given by TER and separate the diffracted and diffuse intensities. In the case of GID. Thus the diffuse scattering in GID can deliver a mea- correlated roughness, the DS about the multilayer Bragg sure of crystal structure ordering at rough interfaces. This peaks becomes much more prominent, and the intensity at information is not accessible by conventional x-ray scatter- large h is comparable to the intensity of coherent GID. The ing techniques. shape of the DS curve is very similar to that of coherent The calculations for the double-crystal scheme of GID are GID, as given by 30 and well seen in Fig. 5, but the de- presented in Fig. 6 for noncorrelated a and completely cor- crease in DS intensity with h is slower. These results may related b interface roughness in multilayers. Due to the explain the discrepancies between the theory and experiment in-plane angular collimation of the incident x rays, for each at high angles observed in the single-crystal measurements B the diffracted wave exits the crystal at the certain of GID in multilayers in Ref. 27. angle h , as given by Eq. 7 . The exit angle of the diffracted To distinguish the effect of DS on single-crystal GID wave at different B is traced on the maps by thick solid curves, one might measure the parameters of roughness by stripes. The intensity at all other points on the maps can be x-ray reflectivity and diffuse scattering in TER, substitute the attributed to DS. Thus a separation of coherent and diffuse roughness value into Eq. 29 , and subtract the calculated DS scattering is possible. from the measured GID curves. However, as follows from The maps in Fig. 6 clearly show the bunching of the DS Eq. 29 and Fig. 2, the DS contributing to the single-crystal into resonance diffraction sheets RDS's for correlated in- 8158 S. A. STEPANOV et al. 54 h can exit the crystal at each B . Diffuse and coherent scattering could be separated out in this case by means of the triple-crystal scheme. The calculations of DS for relaxed su- perlattices could be performed using Eq. 1 and the wave fields of GID found in Ref. 53. VI. MEASUREMENTS OF DIFFUSE SCATTERING FROM AlAs/GaAs MULTILAYER In order to provide a comprehensive test of our theory, we have undertaken high-resolution measurements of GID from a 20-period AlAs/GaAs superlattice SL . The GID experi- ment has been carried out at the optics beamline BL10 of ESRF. The superlattice grown by molecular-beam epitaxy on a 001 GaAs substrate was characterized in the laboratory by x-ray Bragg diffraction, x-ray specular reflection, and non- specular x-ray scattering. A -2 scan near the 004 Bragg peak did not reveal any strain relaxation, indicating that the superlattice possessed a laterally matched crystal structure. The thickness of the layers obtained from the fitting of this scan was tAlAs (154 1) Å, and tGaAs (73 1) Å. The x-ray-diffraction data also indicated a sample surface miscut of ( 0.38 0.02)° along 110 . The x-ray specular reflec- tion -2 scans confirmed the thickness of AlAs and GaAs layers and revealed a surface transition layer with a thickness FIG. 6. Calculated maps of diffuse scattering for the double- of (18 2) Å, probably due to natural oxidation. A fit to the crystal scheme of GID experiments. The parameters of the calcula- specular reflection curve gave the rms height of interface tions are the same as in Fig. 5. The equal-intensity map a is for roughness i (4 0.5) Å. Finally, the longitudinal and uncorrelated interface roughness, and b is the same for completely transverse scans of nonspecular x-ray scattering at grazing correlated roughness. Thick stripes marked by arrows show the po- incidence ( -2 scans with offsets of the sample and sition of the coherent wave of GID. scans at fixed 2 positions of the detector, respectively were carried out in order to determine the vertical and lateral cor- terface roughness. This effect is completely analogous to the relation lengths of roughness. The longitudinal scans exhib- formation of RDS's ``Holy bananas'' in DS during x-ray ited very sharp RDS maxima corresponding to a complete specular reflection.31 The vertical black fringe at 0 cor- correlation between the roughness of different interfaces. responds to the anti-Yoneda minimum in DS which we have Therefore, the data were fitted with the simple correlation discussed above for scattering from a rough surface. function 35 assuming z . The transverse scan along Another interesting peculiarity displayed by Fig. 6 b is RDS-8 (2 3.172°) and the corresponding fit are shown in the appearance of RDS's at negative B where the dif- Fig. 7. The fit consists of two independent contributions of fracted wave of GID is surface trapped and cannot exit the interface and surface roughness, which explain the central crystal. This effect is surprising because the surface-trapped part of the curve and the Yoneda peaks at the wings, respec- wave has a small penetration depth inside the crystal see tively. The fitted parameters are 4 Å, 3000 Å for the Fig. 4 e , and one could expect only a few interfaces con- interface roughness, and 9 Å, 500 Å for the surface tributing to DS at these angles. However, two types of x-ray roughness. The height of the surface roughness corresponds wave fields are generally excited in the crystal under GID: to half of the thickness of the surface transition layer found one wave can be roughly connected to the diffracted wave in by reflectometry. The lower-order RDS indicated a more vacuum, and the other one to the incident wave. The angle of complicated spectrum of roughness. For example, the trans- the latter wave to the surface is not small in our example, verse scan at RDS-5 (2 2.033°) was better explained by a thus providing a greater penetration depth. This effect can be combination of the same surface roughness and a sum of used for the experimental measurements of DS because the interface roughness with 4 Å, 4500 Å, 3 Å, separation of the diffracted beam is unnecessary. We note a 3000 Å and 2Å, 1500 Å. However, in the first small difference in the positions of RDS's at positive and approximation the parameters 4 Å, 3000 Å found at negative B . RDS-8 were applied to modeling the results of the GID ex- As long as the coherent and diffuse scattering can be dis- periment. criminated with the double-crystal scheme of the GID ex- The experimental configuration of the GID experiment is periment, the triple-crystal scheme Fig. 3 c is not of par- shown in Fig. 8. The grazing-incidence diffraction was mea- ticular interest. The situation might change in the case of sured from the 220 AlAs/GaAs planes in the double-crystal relaxed superlattices containing a distribution of lattice spac- scheme corresponding to Fig. 3 b , and similar to the experi- ings along the lateral direction. In this case, Eq. 7 becomes ment at CHESS.15 However, in contrast to Ref. 15, the an- inapplicable,53 and a fan of diffracted waves with different gular spectrum of the diffracted beam was analyzed with a 54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8159 FIG. 7. Transverse scan through the eighth-order resonance sheet (2 3.172°) of x-ray diffuse scattering taken from an AlAs/ GaAs superlattice far from diffraction conditions. Experimental FIG. 9. Double-crystal rocking curve of GID taken from the data are represented by circles, and the theoretical fit by the solid AlAs/GaAs superlattice at 0 0.5°. Dots represent experimental line. The fit is the sum of contributions of completely correlated data, and the solid line is a theoretical simulation convoluted with interface roughness ( 4 Å, 3000 Å and surface roughness the rocking curve of the five-reflection monochromator. The top ( 9Å, 500 Å . line is the magnified reflection coefficient for the specular x-ray beam. Vertical marks indicate the sample positions where PSD spectra of diffracted intensity were taken see Fig. 11 . linear PSD, and the experimental setup was optimized in order to provide a good separation of GID and DS. from a Ge wafer: the width of the Bragg peak corresponded A Si 111 double-crystal monochromator was tuned to to the calculated value. The angular resolution of the PSD 1.40 Å. The crystals in the monochromator were slightly over the takeoff angle was 15 sec of arc. offset in angle to suppress the third and higher harmonics.54 The front of the beam impinging on the sample was re- The divergence of the x-ray beam in the vertical plane over stricted by the output slits S3 with vertical and horizontal 0 was mainly determined by the monochromator since the sizes of 0.04 and 0.2 mm, respectively. This provided an vertical divergence of synchrotron radiation at ESRF is 1 illuminated area smaller than the sample size, and eliminated sec of arc. The collimation of the beam in the horizontal edge reflections. The primary and the secondary slits S1 and plane over was provided by five 220 reflections in a S2 restricted the beam front to 1 2 mm2 and 0.1 1 channel-cut Si crystal. The five-reflection collimator was mm2, respectively, and reduced the background in the ex- used to suppress the tails of the x-ray beam and provide a perimental hutch. better discrimination of GID and DS at the position-sensitive In the first step of the experiment, the PSD was replaced detector. by a scintillation counter. The diffracted beam was recorded The estimated parameters of the incident beam at the while scanning as in Ref. 15 with no separation of the sample were: / 1.2 10 4, 0 6 sec of arc, and coherent and diffuse components. A second counter Si pho- 4 sec of arc. The last parameter is given with account todiode simultaneously recorded the rocking curves of the of the dispersion effect caused by the difference in the Bragg specularly reflected x-ray beam. Figure 9 presents the mea- angles of Si and GaAs 220 reflections. The horizontal di- sured and calculated rocking curves for the incidence angle vergence was examined by recording a 220 rocking curve 0 0.5°. The parameters for the calculations are taken from the laboratory data presented at the beginning of this section and the algorithm is described in Sec. II and elsewhere.21 The experiment clearly demonstrates the high quality of the superlattice: i the half-widths of the Bragg peak and the superlattice peaks correspond to the calculated parameters; and ii the rocking curve of the specular beam exhibits a maximum at the first superlattice Bragg peak, thus proving that the diffraction is dynamical. At the same time, the reflectivity at the wings of measured GID curve is noticeably higher than expected and is probably due to DS. Figure 10 shows the map of DS calculated for the FIG. 8. The scheme of the high-resolution GID experiment car- ried out on the optical beamline of ESRF. The linear position- experimental conditions according to 31 . In contrast to the sensitive detector PSD provided an angular analysis of radiation example given in Fig. 6 b , the DS is concentrated along the scattered along the diffracted beam of GID. The intensity of the diffracted beam of GID because of a greater lateral correla- specular x-ray beam was recorded integrally using a Si photodiode. tion length. However, a characteristic pattern of DS with For the integral GID measurements presented in Fig. 9, the PSD periodic superlattice peaks in two directions is well devel- was replaced by a scintillation counter. oped. The SL peaks parallel to the -axis are especially 8160 S. A. STEPANOV et al. 54 FIG. 10. Calculated map of diffuse scattering in GID ( 0 0.5°) for the AlAs/GaAs superlattice studied at ESRF. The thick stripe marked by the arrow shows the position of the coherent wave of GID. The theoretical curves in Fig. 11 are the sections of this map drawn along the vertical axis at different B . strong, and should be clearly seen in PSD observations which cut the map along the h axis. The PSD spectra have been taken at different -positions of the sample as marked in Fig. 9. The results are shown in Fig. 11. The curves in the figure are vertically shifted with respect to each other as explained in the figure caption. The experiment is compared to the theoretical curves, which consist of four parts I IF Total h , IGID h , IDS h , ISurf DS h , IBackgr . 36 Here IGID( h , ) RGID ( h) R5 ( h) ( )] is the product of x-ray Bragg reflections from the sample and the five-reflection collimator, respectively; ( h) is the x-ray Bragg deviation for the sample given by Eq. 7 , and ( ) 2 sin(2 B) is the difference in for collima- tor and sample due their angular misalignment corre- sponding to a given PSD spectrum. The terms IIF Surf DS and IDS are the DS intensities for the interface and surface roughness, respectively, calculated according to Eq. 31 with the pa- rameters found from the laboratory measurements. Finally, IBackgr is a constant term equal to the experimental back- ground. Different terms in Eq. 36 explain different peculiarities of the experimental spectra in Fig. 11. The coherent reflec- tion provides the floating peaks marked by arrows. The po- sition of these peaks is given by Eq. 7 with ( ). The strongest effect of coherent reflection is observed at 3 arc sec, where two coherent peaks are found. At 0, the contribution of coherent reflection is small and invisible on the spectra. DS due to interface roughness forms regular superlattice FIG. 11. PSD spectra of GID intensity vs exit angle at peaks, with a maximum intensity around the position of the 0 0.5° and different B for the AlAs/GaAs superlattice. Dots coherent reflection, as expected from the map presented in present experimental data, and solid lines are calculations with pa- Fig. 10. At 0 and 65 arc sec, the integrated inten- rameters taken from the laboratory studies of the sample. Arrows sity of DS peaks exceeds the integrated intensity of the dif- mark coherent peaks. The curves are successively shifted by 103 for fraction peak itself, thus explaining the high-intensity wings clarity. The central curve is then reduced by a factor of 20 because of the experimental curve in Fig. 9. It is worth noting that su- of its high intensity. 54 DIFFUSE SCATTERING FROM INTERFACE ROUGHNESS . . . 8161 perlattice peaks are observed for 0. This proves that the ness in crystals can give rise to x-ray diffuse scattering about x-ray wave field in GID possesses greater penetration inside the diffraction beams of GID. The theoretical model is based the superlattice, even though the diffracted wave is surface on the distorted-wave Born approximation29 and the dynami- trapped. cal theory of GID by multilayers.26­28 The model can be Finally, the surface roughness provides a maximum in DS applied to asymmetric x-ray diffraction and to the diffraction near the critical angle of TER at of neutrons. Expressions have been obtained for GID experi- h 0.3°) for the data corresponding to 169 and 313 arc sec, and 0. ments in different geometries. The effect of interface- Here we have found that the intensity of DS calculated due interface roughness correlations on x-ray diffuse scattering in to the surface roughness with 9 Å was a factor of 4 GID has been taken into account. It has been demonstrated greater than observed in the experiment. The theoretical that in the case of periodic interfaces, interface-interface cor- curves presented in Fig. 11 correspond to 4.5 Å, half of relations give rise to the formation of resonance sheets in DS the value determined from the laboratory measurements. We similar to ``Holy bananas''31 in small-angle x-ray scattering. suggest that the difference in the observed values is due to When applied to the analysis of GID data taken from a Ge the distinction between diffuse scattering in TER and GID: crystal etched to provide a roughened surface, the theory the diffuse scattering in TER is proportional to 2 obtains the observed diffracted intensity and diffuse scatter- 0 , while ing. The intensity shoulder on the experimental curve at that in GID is mainly given by 2h , a measure of the atomic ordering in roughness. The crystal structure at the surface B 0, where the GID intensity must otherwise be zero, has been explained by diffuse scattering from the surface could be partially destroyed by the oxidation, giving rise to roughness. A dip in the diffuse scattering near the Bragg weaker DS in GID. Thus our experiment confirms the con- peak can be attributed to an ``anti-Yoneda'' effect. clusion derived in Sec. V that the measurements of DS in High-resolution measurements of GID have been re- GID may provide information on atomic ordering at inter- ported. This approach allowed the diffracted flux from an faces which is not accessible by DS in TER. AlAs/GaAs superlattice to be resolved into GID and diffuse While the agreement between our theory and experiment scattering due to correlated interface roughness. The experi- is reasonable, some discrepancies can be attributed to the mental results are in good agreement with the theory. simplified model of the correlation function used in the cal- Measurements of diffuse scattering in GID are sensitive to culations. It is worth noting that no diffuse scattering due to atomic ordering in roughness, thereby providing information crystal structure defects in the superlattice was observed. As which is not accessible by conventional small-angle x-ray shown in Refs. 55 and 56, a peak of DS due to pointlike scattering. defects could be expected at h 0 0.24°. Some discrep- ancies between the theory and experiment at small h near ACKNOWLEDGMENTS the critical angle of TER might be due to this kind of scat- tering, but the scattering from interface roughness was the One of us S.A.S. is pleased to thank S. Sinha Argonne major contribution. National Laboratory and V. Kaganer Institute of Crystal- lography, Moscow for stimulating discussions. We also VII. 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