PHYSICAL REVIEW B VOLUME 57, NUMBER 8 15 FEBRUARY 1998-II Dynamical x-ray diffraction of multilayers and superlattices: Recursion matrix extension to grazing angles S. A. Stepanov,* E. A. Kondrashkina,* and R. Ko¨hler AG ``Ro¨ntgenbeugung,'' Humboldt University of Berlin, Hausvogteiplatz 5-7, Berlin 10117, Germany D. V. Novikov and G. Materlik HASYLAB at DESY, Notkestrasse 85, Hamburg 22603, Germany S. M. Durbin Department of Physics, Purdue University, West Lafayette, Indiana 47907 Received 7 February 1997 A generalized dynamical theory has been developed that extends previous models of x-ray diffraction from crystals and multilayers with vertical strains to the cases of grazing incidence and/or exit below the critical angle for total specular reflection. This provides a common description for extremely asymmetric diffraction, surface ``grazing-incidence'' , and grazing Bragg-Laue diffraction, thus providing opportunities for the appli- cations of grazing geometries to the studies of thin multilayers. The solution, obtained in the form of recursion formulas for 2 2 scattering matrices for each individual layer, eliminates possible divergences of the 4 4 transfer-matrix algorithm developed previously. For nongrazing x-ray diffraction in the Bragg geometry and for grazing-incidence x-ray specular reflection out of the Bragg diffraction conditions, the matrices are reduced to scalars and the recursion formulas become equivalent to the earlier recursion formulas by Bartels et al. Acta Cryst. A 42, 539 1986 and Parratt Phys. Rev. 95, 359 1954 , respectively. The theory has been confirmed by an extremely asymmetric x-ray-diffraction experiment with a strained AlAs/GaAs superlattice carried out at HASYLAB. A solution to the difficulties due to dispersion encountered in extremely asymmetric diffraction measurements has been demonstrated. Finally, the validity of Ewald's expansion for thin layers and the relation of the matrix method to the Darwin theory, as well as the structure of x-ray standing waves in multilayers are discussed. S0163-1829 98 05408-3 I. INTRODUCTION in studies of semiconductor crystal surface structures, includ- ing diffusion-induced, ion-implanted, and epitaxial layers In recent years x-ray diffraction schemes with grazing in- and multilayers, oxidized, etched, and corrugated surfaces, cidence and/or exit angles have attracted particular interest etc. see Refs. 26­37, Refs. 15,38­51, and Refs. 52­54 for because of their advantages in the studies of very thin surface EAD, GID, and GBL, respectively . However, these studies layers of crystals.1­4 Due to the total external reflection ef- also indicated the lack of a general diffraction model for the fect for grazing x rays, the x-ray penetration inside crystals is various grazing geometries. reduced from the micrometer range down to a few nano- X-ray diffraction at grazing incidence and/or exit can be meters, thus providing the possibility of studying surface treated with the help of either an extended kinematical theory structures with atomic depth resolution. often called the ``distorted wave Born Grazing geometries can be classified into three major approximation'' ,16,19,41,49,55 or extended dynamical types Fig. 1 : i The coplanar extremely asymmetric dif- theory.5,6,8,9,11,14,17,22,45,48,51,56­59 Both approaches take into fraction EAD is realized when the diffraction planes make account refraction and specular reflection effects for grazing the Bragg angle with the crystal surface and either the inci- x rays at crystal surfaces and interfaces. As with ordinary dent or exit x-ray wave is grazing.5­12 ii Surface or Bragg diffraction, the kinematical theory is applicable to mo- ``grazing-incidence'' diffraction13 GID is the geometry saic crystals, to the tails of the Bragg peaks, and to the dif- where the Bragg planes are perpendicular to the surface and fraction from layers thinner than the x-ray extinction depth. both the x-ray waves are grazing.14­23 iii Finally, grazing This depth decreases to about 10 ML under total external Bragg-Laue diffraction GBL is a combination of the EAD reflection conditions for grazing x rays. The application of and GID. It involves the diffraction from atomic planes in- the perturbation kinematical theory to the above-listed clined at a small angle to the crystal surface normal, so that cases is possible due to a small intensity of diffracted x rays. the reciprocal lattice vector points outside the crystal at few Otherwise the dynamical theory must be applied, which degrees to the surface. It is then possible to choose asymmet- takes into account comparable intensities of incident and dif- ric diffraction with either grazing incidence or grazing exit fracted waves and their multiple rescattering into each other. by a small variation in the incidence angle or even to switch Improvement in the dynamical theory is demanded by the between these two cases within one diffraction steadily improving quality and increasing complexity of experiment.24,25 semiconductor heterostructures. However, the majority of All three geometries have found widespread application theoretical studies have been concerned with perfect crystals 0163-1829/98/57 8 /4829 13 /$15.00 57 4829 © 1998 The American Physical Society 4830 S. A. STEPANOV et al. 57 x-ray wave fields in an infinite crystal, whose applicability to thin layers must be established. The aim of the present paper is to reformulate the matrix dynamical theory in a recursion matrix RM form that overcomes the numerical problems of the TM formulation, and to provide an experimental check to the Ewald expansion for thin strained multilayers. In Sec. II the (4 4) transfer matrix theory48,58,59 of x-ray diffraction from multilayers is outlined, and in Sec. III pos- sible numerical problems are demonstrated and explained. In Sec. IV the theory is reformulated in terms of recursion formulas for 2 2 matrices for individual layers. It is shown how this approach overcomes the numerical problems of the 4 4 TM algorithm. In Sec. V the reduction of the matrix recursion formulas to the scalar recursion formulas by Parratt63 and Bartels, Hornstra and Lobeek64 is demonstrated for x-ray grazing incidence far from the Bragg diffraction condition and for Bragg diffraction with no grazing waves. In Sec. VI the recursion matrix theory is verified with the help of the double-crystal EAD measurements taken from a strained AlAs/GaAs superlattice SL with known structure parameters. In Sec. VII the results of recursion matrix calculations are compared to those given by the Darwin theory applied to heterostructure diffraction.65 In contrast to our approach, the Darwin theory does not contain Ewald's expansion and di- rectly sums x-ray scattering of atomic planes. It is shown that in the case of ordinary Bragg diffraction the results of both theories coincide. Further comparisons require an extension of the Darwin theory to grazing angles, which is forthcoming.66 In Sec. IX the structure of x-ray standing FIG. 1. Diffraction geometries with x-ray grazing incidence waves in multilayers is discussed. and/or exit. a coplanar extremely asymmetric diffraction grazing- We conclude with some possible uses and further exten- incidence case , b grazing-incidence diffraction, and c grazing Bragg-Laue diffraction. Vectors sions of the recursion matrix theory. 0 , s , and h denote incident, specularly reflected, and diffracted waves, respectively; h is the reciprocal lattice vector corresponding to the Bragg planes, 0 , h , and are the angles of 0 , h , and h, respectively, with the II. 4 4... MATRIX THEORY OF MULTILAYER surface; B is the Bragg angle. DIFFRACTION or with greatly simplified models of defect crystals. The Let us consider x-ray Bragg diffraction in one of the ge- problem is that the diffraction from strained crystals in the ometries presented in Fig. 1. The crystal is assumed to be a case of grazing incidence/exit cannot be analyzed within the multilayer consisting of a stack of N perfect crystalline lay- standard Takagi-Taupin approach,60,61 which is based on the ers with laterally matched lattice spacing. We allow each assumption that the x-ray wave-field amplitudes vary slowly layer to possess its own lattice spacing anz in the direction at interatomic distances so that their second derivatives can normal to the surface: an n n z az az , where az az , and be neglected. A new general theory applicable to x-ray dif- n is the layer index numbered from the surface of the stack. fraction with grazing incidence and/or exit was constructed This model corresponds to a so-called unrelaxed multilayer in a (4 4) matrix form.45,48,51,56­59 In Refs. 56 and 57 the containing no misfit dislocations. Methods to extend this rank of matrices was 8 8 because - and -x-ray polariza- model to multilayers with misfits are briefly discussed in the tions were treated together. The two different formulations Conclusions this problem is not completely solved yet . A discussed in these papers, i.e., the differential matrix possibility of crystal curvature as a result of strains67 is also equations56,57 and the algebraic equations for transfer disregarded. That is, we assume that either the multilayer is matrices,45,51,48,58,59 are essentially equivalent. The former thin enough or the substrate is thick enough to ignore curva- approach is more convenient for continuous strain profiles in ture. crystals, while the latter one is superior for multilayers. This Those structures that contain additional amorphous layers transfer matrix TM technique is similar to the (2 2) ma- are not considered here for the reason of simplicity, although trix solution for grazing incidence x-ray specular reflection they can be readily included in the model.48,58 For the same of multilayers found by Abeles.62 reason, we neglect possible changes in x-ray polarization, Unfortunately, both the differential and algebraic versions which may occur due to the refraction effects in GID and of the (4 4) matrix technique may suffer from serious nu- GBL. The intermixing of and polarizations was taken merical problems in their computer implementation. In addi- into consideration, for example, in Refs. 56 and 57, but this tion, the TM theory makes use of Ewald's expansion for effect is shown to be small.68 Our derivations below are car- 57 DYNAMICAL X-RAY DIFFRACTION OF MULTILAYERS . . . 4831 ried out for polarization. The equations can be extended ray diffraction experiments, since 1 is usually not ac- for polarization by incorporating cos(2 B) in the x-ray cessed experimentally because of low reflectivity on the far susceptibilities h and h¯ . tails of Bragg peaks. Thus, the range 1 is outside the When Bragg diffraction from atomic planes in a ML with scope of our study. For those interested in possible exten- variable anz is considered, the reciprocal vector hn associated sions of the theory to 1 so-called asymptotic Bragg with the atomic planes in the nth layer slightly differs from diffraction,74 or crystal truncation rod scattering75 the solu- the mean vector h because of the deviation of the normal tion to Eq. 3 and the corrections to boundary conditions at lattice spacing in the layer: hn h hznZ, where large were obtained by Caticha.76 In addition, as shown by hzn h, and Z is a unit vector along the internal surface Colella77 the cases with 1 may require analysis in the normal. We assume the local crystal dielectric susceptibility framework of multiple Bragg diffraction theory, since the in each layer to have the same periodicity as the local atomic two-wave approximation 1 and 2 may become invalid. planes spacing in the layer: The condition 2 2 h 0 presuming the elastic scattering of x rays gives59 n r n n n 0 h eihn* r rn i n e ihn* r rn i n, 1 n h¯n 2h 0 2 , 5 where the beginning of coordinates rn is taken at the upper layer interface, and the initial phase Eq. 5 implies that the exit angle of a grazing diffracted n will be chosen later. For small strains the Fourier coefficients n n wave does not depend on vertical strains. h , can be set n h¯n The values of un are determined by the dispersion equa- equal to the Fourier coefficients n n h , of unstrained h¯ tion, which is the condition for the existence of a solution of crystal.60 Eqs. 4 : Following the standard Ewald approach to dynamical dif- fraction theory, we expand the x-ray wave field in each layer u 2 2 n 2 n n n n 0 0 un n 2 h 0 h . 6 over the sum of the transmitted and diffracted Bloch waves h¯n n with wave vectors k0n and khn k0n hn , and amplitudes Equation 6 is a fourth-degree polynomial equation for un D0n and Dhn , respectively:69­71 and has therefore four roots. As shown in Ref. 24, there are always two roots corresponding to x-ray waves with ampli- Dn r eik0n*r D0n Dhneihn* r rn i n . 2 tudes damping out with z Im(un) 0 , and two other roots Under the expansions 1 and 2 , the amplitudes D corresponding to the waves with amplitudes growing with z 0n and D Im(un) 0 . The latter waves are usually treated as being hn can be treated as constants satisfying the dynamical dif- fraction equations in each layer: specularly reflected from the lower interfaces of the layers. We shall assume that the roots uj are sorted over descending k2 2 n 0n 0 j D nD n D Im(un), so that j 3,4 correspond to the reflected waves. For k2 0n 0 0n h¯ hn , n each of the solutions Eqs. 4 give ( j 1,...,4) 0n 3 j j j j j 2 n n k2 2 D v D , v u 2 / . 7 hn h hn n 0n n n 0 0 h¯ D n D nD n k2 hn hn 0n 0 hn , hn Proceeding to the boundary conditions at multilayer inter- where 0 and h are the values of the incident and diffracted faces, one must choose the parameters n in a way that wave vectors in vacuum, respectively see Fig. 1 . provides a continuous phase of the waves in Eqs. 1 and 2 . The lateral components of all vectors k0n and khn coincide This is provided by the following choice which does not because they remain unchanged at refraction and specular affect Eqs. 3 ­ 7 : reflection. Then, Eqs. 3 can be expressed in terms of the normal wave-vector components, which are determined by n 1 n 1 the incidence and exit angles see Fig. 1 : L U n hzk zk zk hzktk , 8 k 1 k 1 0z sin 0 0 , hz sin h h . hzn n (1 an U,L z /a), where h­Z/ 2 sin sin B . Mak- where tk are the thicknesses of the layers and zk denote the ing these substitutions72 and introducing the dimensionless coordinates of the upper and lower interfaces, zU zL . complex parameters u k k 1 n k0zn / we arrive at17,73 With the substitution of Eq. 8 , the exponents in the expan- u2 2 n n z n 0 0 D0n D sions 1 and 2 become identical to h h¯ hn , 0 z(z)dz; they can n 4 also be presented in a more usual notation compare with u 2 n n Ref. 60 : n n 2 h 0 Dhn h D n 0n . The transition from Eq. 3 to 4 makes use of the assump- n 1 z U tion that the difference between k h h h 0n , khn , and 0 h is zn z zn zktk hzz z z dz k 1 0 small, so that the former two can be replaced by in the denominators at the left side of Eq. 3 . This is the typical z az z approximation used in most x-ray diffraction theories. It is hzz hz 0 a dz valid at small deviations from the Bragg condition, (2 0*h h2)/ 2 1, which is well justified for most x- hzz hzu z , 9 4832 S. A. STEPANOV et al. 57 where u(z) is the function representing the displacement of III. NUMERICAL PROBLEMS WITH THE DIRECT 4 4... atoms from their positions in the Takagi-Taupin theory. MATRIX SOLUTION Equation 1 with n given by Eq. 8 is equivalent to the expansions of the susceptibility (r) of strained crystals used One of the key steps of the direct 4 4 matrix solution by Takagi60 and, in fact, in the Darwin-like theory of hetero- 13 is the calculation of the matrix product at the right hand structure diffraction.65 It does not contain approximations of 13 . This product diverges for an infinitely thick about slow variations of (r) at interatomic distances. As to multilayer due to the accumulation of large exponents con- expansion 2 , its validity will be discussed in Secs. VI­VIII. tained in Fk . In practical computations, the loss of precision 1,2 The boundary conditions for grazing waves need to be in adding big numbers exp( iun tn) and small numbers formulated not only for x-ray amplitudes, but also for their exp( iu3,4 n tn) happens when the multilayer thickness is derivatives, which corresponds to accounting for x-ray re- greater than the x-ray extinction depth inside a crystal. As is fraction and specular reflection effects.14,17,22 The transmitted well known, the x-ray extinction depth at grazing angles can and diffracted wave field of each layer are matched to those be as small as a few nanometers, so numerical problems may of the upper and lower adjacent layers. The wave fields in the arise for quite thin structures. surface layer are matched at the surface to the following A previous solution to the numerical problems was as vacuum x-ray field which consists of incident, specularly re- follows.58,48 Successively calculating the matrix product in flected, and diffracted waves with the amplitudes E Eq. 13 from the left to the right, one is traveling from the 0 , Es , and E crystal surface towards the deeper layers. If the matrix prod- h , respectively see Fig. 1 : uct becomes great at some layer, it indicates that the x-ray E waves in this layer are very weak and the contribution to E v r ei 0 *r E0ei 0 0z Ese i 0 0z Ehe i 0 hz ih *r . v 10 coming from the layer's lower interface and from all the underlying layers can be neglected. Then, the overflow is The boundary conditions provide four equations for the overcome since the matrix product is truncated at the upper x-ray amplitudes at each interface, which can be formulated part of the multilayer. This idea is well understood for a in the (4 4) matrix form:45,48,58,59 perfect crystal formally subdivided into multilayers. Then, S 1 k Sk 1 1 and the matrix product is the inverted absorp- (L) S tion factor of x-ray waves: F 1. vEv S1D1 , 1F2 ...Fn Fn The above-described procedure is equivalent to the usual thick crystal approximation TCA widely used in the dy- S L U 1F1 D1 S2F2 D2 , 11 namical diffraction theory.69­71 That is, the solutions of the dispersion equation corresponding to the waves growing . . . . . . , with z Im(un) 0 are disregarded for thick crystal plate. However, as soon as the x-ray extinction strongly depends S L U on the grazing diffraction angles, the number of layers taken N 1FN 1DN 1 SNFN DN , into consideration may vary across a diffraction curve, so Here E 1 2 3 4 that the grazing-case TCA is dynamical. Essentially the same v (E0 1,0,Es ,Eh) and Dn (D0n , D0n , D0n , D0n) are the four-component vectors, and S idea to overcome the numerical problems was suggested by v , Sn , and Fn are the characteristic (4 4) matrices of the layers: Berreman and Macrander57 for their matrix differential dif- fraction equations of grazing-incidence diffraction. The application of the dynamical TCA to the transfer ma- 1 0 1 0 trix method provided a successful interpretation to the 0 1 0 1 1 1 1 1 v1 2 3 4 n vn vn vn grazing-incidence diffraction measurements of strained Sv ,Sn 1 2 3 4 , superlattices.48 However, we have found some cases where 0 0 0 0 un un un un TCA is unable to avoid numerical failures. 0 1 2 3 4 h 0 h wn wn wn wn The problem is that the four different x-ray wave modes 12 (D1 2 3 4 n , Dn , Dn , Dn) are characterized by different extinction lengths inside a crystal and may set different conditions for Fj(U,L) j (U,L) j j j n ij ij exp iun zn , and wn vn(un n). TCA. This is clearly seen in the case of grazing-incidence A direct formal solution to Eqs. 11 is diffraction Fig. 1 a . In the GID conditions, the dispersion equation 6 always gives a Borrmann wave field D1n with E 1 1 1 U v Sv S1F1S1 S2F2 . . . SN 1SNFN DN , 13 wave nodes between the diffraction planes and weak absorp- tion, and an anti-Borrmann wave field D2n with wave nodes where (F (U) (L) j n)i j Fn (Fn ) 1 i j i j exp( iun tn). After on the diffraction planes and strong absorption.78,22 Since the calculating the matrix product on the right hand of Eq. 13 Borrmann and anti-Borrmann modes are characterized by a and taking into account that the amplitudes of the waves strong and a weak interaction with crystal matter, they pos- reflected from the lower interface of a thick substrate layer sess different critical angles for total external reflection, are zero (D3 4 0N D0N 0), one arrives at four linear equations which are lower and higher, respectively, than the usual criti- for four unknown amplitudes: E 1 2 s , Es , D0N , and D0N . The cal angle c ( 0)1/2. At the exact Bragg position ( 0) other amplitudes are given by Eq. 11 . This is the transfer the critical angles are14 1,2 ( 0 h)1/2, and for the gen- matrix solution to the diffraction problem, as suggested in eral case the angular areas for total external reflection are Refs. 48, 58, and 59. shown by the hatched patterns I and II in Fig. 2 a . The wave 57 DYNAMICAL X-RAY DIFFRACTION OF MULTILAYERS . . . 4833 FIG. 3. An example of numerical problems that occur with the transfer matrix method. The calculations are for 220 grazing- incidence diffraction of Cu K 1 radiation and an AlAs/GaAs super- lattice 20 periods of 73 Å GaAs and 154 Å AlAs on 001 GaAs substrate . The scan is calculated at 0 and corresponds to the diagonal in Fig. 2 a . The dotted line and the thin solid line in a show the GID reflectivity calculated by the TM method with the 1014 and 1015 thresholds of the maximum matrix element, respec- tively the curves are shifted by 0.3 for clarity . Respective lines in b show the number of the top layers in the multilayer taken into account in the calculations. The calculations for different thresholds disagree with each other and with the recursion matrix calculations thick solid line in a in the gap between the two critical angles for FIG. 2. The angular areas of total external reflection for a total external reflection ( grazing-incidence x-ray diffraction and b grazing Bragg-Laue dif- 1 0 2). fraction ( 1.6°). The calculations are for 220 reflection of Cu K 1 radiation from GaAs crystal. Areas denoted 0, I, and II interfaces: the anti-Borrmann waves are excited in deep lay- correspond to the total reflection for none, one, and two wave fields ers by the Borrmann waves. in thick crystal, respectively. 0 is the incidence angle and h is An illustration to this problem is given in Fig. 3 for 220 the exit angle of diffracted wave, c ( 0)1/2 is the critical angle GID of an AlAs/GaAs superlattice consisting of 20 periods for total reflection in the absence of the Bragg diffraction, of 73 Å GaAs and 154 Å AlAs on 001 GaAs substrate. The 1,2 ( 0 h)1/2 are the critical angles for GID introduced in Ref. dotted and thin solid lines in Fig. 3 a show the reflectivity 14. curves of GID calculated by the TM method with the TCA applied when the maximum element of the matrix product is fields D3 4 n and Dn with Im(u) 0 are also Borrmann and 1014 and 1015, respectively. The same lines in part b of the anti-Borrmann modes, and the total reflection areas for these figure show how many layers out of a total of 41 are taken modes coincide with that of the modes 1 and 2, respectively. into account. The curves are plotted as a function of the The same consideration is applicable to the grazing Bragg- incidence angle at 0. This is the scan along the diagonal Laue diffraction see Fig. 2 b and to the EAD. in Fig. 2 a . The reflectivity curves with different TCA con- The TCA procedure works well for the areas 0 and II in ditions coincide at 0 1 , where all the x-ray waves are Fig. 2 where either none or all of the waves are strongly totally reflected and at 0 2 , where nothing is totally absorbed. In the area I in the gap between the two critical reflected. However, they differ in the gap between the two angles the extinction the penetration depth for the anti- critical angles where the anti-Borrmann waves are reflected Borrmann and Borrmann modes may be of the order of 102 and the Borrmann ones are not. This proves that the TCA Å and 105 Å, respectively. As a result of this great difference procedure is not applicable in this range. by three orders, the anti-Borrmann mode may give large ex- The TCA thresholds used in the above example are the ponents in Eq. 13 and require the TCA at a few layers, maximum ones achievable with a double-precision FORTRAN while the Borrmann mode would require taking into account program where the mantissa is 16 decimal digits. Performing diffraction in the whole multilayer. One cannot use separate computations with a longer mantissa may overcome the loss thick crystal approximations for different wave modes be- of precision in some cases, but cannot solve the problem in cause they are coupled via the boundary conditions at the principle. 4834 S. A. STEPANOV et al. 57 R0 (Es ,Eh), respectively. Also, the waves at the right hand of Eq. 15 can be viewed as two transmitted Im(u1,2) 0 and two incident waves Im(u3,4) 0 . The amplitudes of the latter waves coming to the surface from the crystal interior are zero in thick crystals, but we keep them for the general case where crystals have internal interfaces. Thus, we group the waves below the surface as the vectors T 1 2 1 (D0 ,D0) and R 3 4 1 (D0 ,D0), respectively. Splitting matrices X and F into four (2 2) blocks we obtain T0 Xtt Xtr F 0 T1 , 16 R0 Xrt Xrr 0 F R1 where F and F are diagonal matrices containing the in- creasing and decreasing exponential functions, respectively. Equation 16 enables the ``scattered'' waves R0 and T1 to be expressed via the ``incident'' waves T0 and R1: FIG. 4. On the derivation of matrix recursion equations for x-ray diffraction in cases of single heterostructure a and multilayer b . Tk and Rk denote the two-component vectors containing the ampli- T1 Mtt Mtr T0 , 17 R0 Mrt Mrr R1 tudes of transmitted and reflected waves, respectively. where Thus, the matrix technique must be reformulated in order to overcome the divergences. Mtt F 1 Xtt 1, Mtr MttXtrF , IV. RECURSION 2 2... MATRIX FORMULAS FOR MULTILAYER DIFFRACTION Mrt Xrt Xtt 1, 18 In the following consideration we make use of the ap- Mrr Xrr MrtXtr F . proach developed by Kohn79 for nongrazing x-ray diffraction with multiple Bragg- and Laue-case x-ray waves in multilay- Equations 18 have a clear physical interpretation. For ers. The Bragg- and Laue-case x rays in that problem can be example, the block Mrr is responsible for the scattering of viewed as being analogous to the transmitted and reflected R waves in our problem. The basic idea by Kohn is that Eq. 1 into R0 and the last line in Eq. 18 implies that the scattering may be a direct transmission R 13 diverges because the vacuum amplitudes E 1 R0 and may be v are sought a multiple scattering process R together with the substrate amplitudes D 1 T0 T1 R0. We note N . The former am- that Eqs. 17 and 18 do not cause any divergences because plitudes are of the order of 1, while the latter ones can be the increasing exponentials F are inverted. In the case of a evanescent in a thick crystal. A better way is to express the thick substrate vector R reflectivity of a multilayer containing n 1 interfaces via 1 approaches zero, and then R that of a multilayer with n interfaces. Such a recursion must 0 M rtT0. Proceeding to multilayers Fig. 4 b , the solutions of the converge because the effect of additional lower interfaces on scattering problem for multilayers incorporating n interfaces the reflectivity decreases with the distance of the interfaces and n 1 interfaces according to Eq. 13 can be presented from the surface. as We start with the following renormalization of the x-ray amplitudes:80 T tt tr n W Wn n T0 rt rr , 19 D L R W W R n Fn Dn , 14 0 n n n and denoting X 1 and n 1 Sn Sn 1 . Then, all equations 11 as- sume the universal form here and below the primes in Dn tt Wtr are left out : Tn 1 Wn 1 n 1 T0 , 20 R rt rr 0 Wn 1 Wn 1 Rn 1 Dn Xn 1Fn 1Dn 1 ,n 0,...,N 1. 15 respectively. Here Wn and Wn 1 are (2 2) matrices. At the The amplitudes D same time, according to Eq. 17 the scattering equations for n are constant within the layers and change at the interfaces. Therefore, the interfaces can be interface (n 1) are treated as ``scatterers'' for amplitudes. First, let us consider tt tr the scattering at a single interface. For clarity we discuss the Tn 1 Mn 1 Tn crystal surface Fig. 4 a , but our consideration is applicable Mn 1 . 21 R rt rr n Mn 1 Mn 1 Rn 1 to any internal interface as well. The waves at the left hand of Eq. 15 can be classified as two incident and two scat- The combination of Eqs. 19 ­ 21 results in the follow- tered waves. We group them in the vectors T0 (E0 ,0) and ing recursion formulas for Wn : 57 DYNAMICAL X-RAY DIFFRACTION OF MULTILAYERS . . . 4835 Wtt tt persion equation, the order of which is reduced to 2. The n 1 AnWn , boundary conditions Eq. 11 and Eq. 13 formally remain Wtr tr tr rr n 1 M n 1 AnWn M n 1 , 22 in the same form, but all the matrices are now (2 2). In particular, the scattering matrices S Wrt rt rt tt v and Sn are reduced to n 1 Wn BnM n 1Wn , Wrr rr n 1 BnM n 1 , Sv 1 1 , Sn 1 1 1 2 25 0 0 un un where it is denoted and for X xy we find A tt tr rt n 1 and M n 1 n Mn 1 1 Wn Mn 1 1, 23 a B rr rt tr n,n 1 n Wn 1 M n 1Wn 1. Xn 1 an,n 1 , 26 a n,n 1 an,n 1 Starting with the crystal surface and progressively apply- ing Eqs. 22 to lower interfaces, one arrives at the matrices Mttn 1 n,n 1 exp iun 1 tn 1 , Wxy tr N determining the reflectivity of the whole multilayer. The Mn 1 rn 1,n exp 2iun 1 tn 1 , recursion matrix RM solution does not cause any diver- 27 gences in the numerical calculations. As follows from Eq. Mrtn 1 rn,n 1 , 18 , the order of Mrt is about 1, while the other three blocks are small due to the factors F and (F ) 1. According to Mrr n 1,n exp iun 1 tn 1 . Eq. 22 , the same ratio of orders is preserved for the blocks n 1 Wxy. Thus, the block Wrt Here a (u N is the only one significant for a n,n 1 n un 1)/2un . The parameters n,n 1 thick multilayer and the solution to the diffraction problem is 2un /(un un 1) and rn,n 1 (un un 1)/(un un 1) R rt are the Fresnel transmission and reflection coefficients, re- 0 WN T0. The other blocks converge to zero at the recur- sions 22 . spectively, for the wave incident on the interface from layer The thick solid line on Fig. 3 a shows the GID reflectiv- n; n 1,n and rn 1,n are those for the wave incident on the ity calculated by the RM method for the example discussed interface from layer n 1. in the previous section. The RM calculation coincides with Thus, for specular reflection the recursion formulas 22 the transfer matrix results in angular areas II and 0. In area I, become scalar, but they do not have exactly the same form as there is a disagreement, because the TM method fails. How- Parratt's recursion formulas.63 The difference is that our ever, when the TCA threshold in the transfer matrix calcula- equations express the reflectivity of a multilayer consisting tions is increased, the mismatch between the two methods of n 1 interfaces via that of n interfaces and the reflectivity decreases. A complete coincidence would be achieved if one on (n 1)th layer, while the Parratt equations connect the had a computer with a hypothetically unlimited number of ratio Pn Rn /Tn with the respective ratio Pn 1 in the next significant digits. layer. The two types of equations are equivalent and can be Finally, let us find the x-ray wave-field amplitudes R reduced to each other. For example, the easiest way to obtain n and T the Parratt recursion formulas is to use Eq. 16 : n inside the layers. These are required for the interpretation of x-ray standing waves81 and diffuse scattering51 in diffrac- tion from multilayers. Equation 19 gives Xrt rr n 1 Xn 1Fn 1 Fn 1 1Pn 1 R rt rr Pn . 28 0 W tt tr n T0 Wn Rk . However, the direct solution Xn 1 Xn 1Fn 1 Fn 1 1Pn 1 R rr rt k (Wn ) 1(R0 Wn T0) leads to uncertainties like 0/0 for thick multilayers and one has to make use of recursions. A Substituting the explicit form of Xxy we arrive at combination of Eqs. 19 and 21 brings r R rt tr rr rt tt n,n 1 Pn 1e 2iun 1 tn 1 n 1 Mn 1Wn 1 Mn 1Rn 1 Mn 1WnT0 , Pn 1 r , 29 24 n,n 1Pn 1exp 2iun 1 tn 1 T tt tr n Wn T0 Wn Rn . where rn,n 1 is the Fresnel reflectivity defined above. Equa- Equations 24 must be progressively applied to all the layers tion 29 is the same as the Parratt recursion equation with starting at the crystal substrate where R the only difference that we define P N 0. n Rn /Tn at the lower layer interface, while Parratt used the definition at the middle of layers. In the general case where R V. REDUCTION TO SCALAR RECURSIONS n and Tn are not sca- lars, the Parratt method is not applicable, while the recursion IN PARTICULAR CASES equations 22 remain valid. A. Reduction to Parratt's formulas far from the Bragg diffraction B. Reduction to Bartels' formulas When the grazing x rays are far away from the Bragg for nongrazing Bragg diffraction conditions, the x-ray wave field above the surface is reduced When x rays satisfy the Bragg condition and the incident to the incident E0 and specular Es waves only, and the field and exit angles are not small, one can neglect the specular in each layer consists of one transmitted D1 j n and one reflected x-ray waves. Then, only those solutions un to the dispersion D2 1,2 n wave with the wave vectors kn ( un , ), respec- equation 6 are significant, for which the waves inside crys- tively. Here u 2 n j n ( 0 0)1/2 are the solutions to the dis- tal only slightly deviate from the waves in vacuum: un 0 , 4836 S. A. STEPANOV et al. 57 or uj In order to give an experimental verification to our theory, n n h . After discarding unimportant roots and proceeding from the large parameters u we have carried out symmetric and extremely asymmetric n to the small refrac- tion corrections s 2 n Bragg diffraction measurements of an AlAs/GaAs superlat- n un ( 0 0)1/2 in Bragg diffraction, Eq. 6 is reduced to the following second-degree tice. The sample was a 20-period AlAs/GaAs superlattice polynomial:25 grown on a 100 GaAs substrate by molecular-beam epi- taxy. The thickness of the layers was 154 Å AlAs and 73 Å n n GaAs, and the interface roughness was 4 Å, as found by h¯ h s n n n fitting grazing-incidence x-ray specular reflection data of the n sn 2 0. 30 0 4 0 h sample. The multilayer thickness was far below the critical thickness for the strain relaxation and formation of misfit Here n n n ( 0 2 0 h az/a)(1 )/ is the param- dislocations. The absence of relaxation was confirmed by the eter determining the deviation of x-rays from the Bragg con- measurements of symmetric 400 Bragg reflection see be- dition in layer n, and 0 / h is the asymmetry factor of low and asymmetric x-ray topographs the latter are not the reflection. The roots of Eq. 30 are the well-known so- shown here . Also, no noticeable sample curvature was lutions of the dynamical diffraction theory: found. The measurements of the superlattice can provide a good n n h¯ h 1/2 test for the assumption concerning the periodicity of x-ray s n n 2 n y 1 , 31 2 waves. First, the layers in the SL are as thin as a few mono- n y n 0 h 1/2 layers and, second, a possible deviation of the real wave field from the theory will be accumulated in a resonant way at the y n 1/2 SL peaks. n . 32 2 n n The symmetric 400 Bragg diffraction measurements were h¯ 1/2 n hn taken in the laboratory using a Philips materials research As soon as the refraction and specular reflection effects diffractometer MRD and CuK 1 radiation from a 2 kW are small, the boundary conditions can be formulated for x-ray tube monochromatized by a Ge 220 Bartels-type x-ray wave amplitudes only, and the solution to the monochromator. The extremely asymmetric diffraction ex- multilayer diffraction problem is obtained in the general periment was carried out at the CEMO beamline of HASY- form 13 with (2 2) scattering matrices. In this case the LAB, DESY. An (n, n) nondispersive setup was used with explicit form of S a Ge double-crystal monochromator symmetric 311 reflec- v and Sn is tion and coplanar asymmetric 311 reflection from the sample. The asymmetry of the sample Bragg reflection was S v 1 0 , S , 33 varied by changing the x-ray energy around 8.5 keV. 0 1 n 1 1 v1 2 n vn All the data were simulated with the help of the theory and the calculation of X xy presented in Sec. IV. The experimental angles are introduced n 1 and M n 1 is straightforward, but tedious. into the theory as follows:59 let a be a unit vector along the The important practical result is that the recursion formu- crystal scan axis. When the crystal is rotated round a through B las 22 become scalar. Again, as in the case of the specular an angle , the original wave vector 0 satisfying the exact reflection problem, our recursion formulas differ from that Bragg condition ( 0 is changed by a vector , which by Bartels, Hornstra, and Lobeek,64 who used the recursions can be expanded into the two mutually perpendicular vectors for P c and b, both lying in the plane of rotation: n Rn /Tn . However, the two types of equations are equivalent and the formulas by Bartels, Hornstra, and xc yb, 34 Lobeek can be obtained using Eq. 28 . In extremely asymmetric Bragg diffraction where only b B B*a a, c B a . 35 one x-ray wave is grazing, the dispersion equation 6 can be 0 0 0 reduced to a third-order polynomial with three roots.59 How- Then, the conditions: 2b sin( /2) and ( B0 )2 2 ever, some of the matrices in Eq. 22 then become (2 1) give rectangular. The way to handle rectangular matrices in recur- sion formulas was discussed by Kohn,79 who solved this x b/c sin , y 2 sin2 /2 . 36 problem for multiple Bragg diffraction with no grazing x rays in multilayers. As soon as is found, we can calculate 2( ­h)/ 2 and B 0 0 ( *Z)/ . The value of h is given by Eq. 5 . Unlike usual Bragg diffraction, accounting for the variations VI. EXPERIMENT in 0 and h during scans is absolutely necessary in grazing Ewald's expansion 2 , the starting point of our method, is geometries because these parameters may change signifi- obviously valid for thick layers mathematically-for an in- cantly. finite crystal , while for thin layers composed of a few For coplanar geometries (a B0 h ) and small scan atomic planes, a continuous expansion of Dn(r) in a Fourier angles ( 1) Eq. 34 is simplified to integral over h may be required. Thus one has to prove that B B 0 0 h /( hcos B), which brings the well- using the expansions over the local periodicity of atomic known expression for 2 sin(2 B) . planes gives results that are consistent with experiment and The data for the symmetric reflection are presented in Fig. with other theories containing no such assumption. 5. The experiment and the theory are shown by dotted and 57 DYNAMICAL X-RAY DIFFRACTION OF MULTILAYERS . . . 4837 FIG. 5. A comparison of the recursion matrix theory with ex- periment for 400 symmetric reflection of an AlAs/GaAs superlat- tice. S and the numbers 0, 1, ... mark the substrate Bragg peak and the different-order superlattice peaks, respectively. The inset shows the experimental setup with the 220 Bartels monochromator. solid lines, respectively. The theoretical curve is calculated using the normal lattice spacing mismatch ( a/a)AlAs 2.775 10 3 in the AlAs layers, as measured in the precise experiment by Bocchi et al.82 for fully strained AlAs on GaAs. The general match between the theory and the experiment is good, although the theory overestimates the reflectivity of the 2 SL peak. This can be explained by the effect of 4 Å interface roughness or by the presence of transition layers .83 In the symmetric case the RM calculations perfectly co- incided with the calculations provided by the commercial Philips software based on the Bartels, Hornstra, and Lobeek algorithm.64 This fact is not surprising because the Bartels recursion formulas are a particular case of the RM method. FIG. 6. Same as in Fig. 5 for 311 extremely asymmetric reflec- Figure 6 presents the experiment dotted lines and the tion. The insert shows the experimental setup with the double- theory solid lines for the 311 coplanar extremely asymmet- crystal monochromator in the nondispersive 311 Bragg position. ric Bragg reflection from the same sample. The successive The captions above each rocking curve indicate the asymmetry of curves correspond to increasing asymmetry of the Bragg re- the reflection at the respective wavelength of synchrotron radiation selected by the monochromator. flection which is determined by the difference B misc be- tween the kinematical Bragg angle and the miscut of the 311 planes. The actual difference deviates from this value grazing x rays interact with a greater number of atomic because of the refraction effects for incident x rays. The planes projected on their path, which results in a shorter reflection asymmetry was altered by small tilts of the double- extinction depth. crystal monochromator that caused small changes in the en- However, the general match between the theory and the ergy of incident synchrotron radiation. The upper and the experiment in the 311 case is worse than that in the 400 case. lower three energies correspond to the kinematical Bragg It cannot be due to the larger angular range of this scan, condition above and below the total-reflection critical angle because 0.02, so the approximation used in Eq. 4 is for the incident x rays, respectively. In the latter case the well justified. Possible explanations for the mismatch can be extinction length of x rays decreases and the reflectivity at a greater footprint of incident beam at the sample surface in the substrate peak S falls. asymmetric diffraction the reflectivity is averaged over a The theoretical curves are corrected for a geometrical fac- greater surface area , or a greater sensitivity of EAD to sur- tor the part of the diffracted intensity measured by the de- face defects due to a smaller extinction length. Also, it might tector was proportional to the incidence angle because of the be due to the sensitivity of EAD to the fluctuations of mate- large footprint of the incident x-ray beam at the sample sur- rial density and interface roughness,51 which affect the re- face and added to the experimental background. fraction of grazing x rays. As we see, the same theory with the same structure pa- We have found that an additional source of mismatch are rameters explains both the symmetric and asymmetric x-ray dispersion effects in the (n, n) scheme applied to EAD. diffraction experiments. It should be noted that the applica- The (n, n) scheme is dispersion free for conventional ge- bility of Ewald's expansion 2 to asymmetric diffraction is ometries, because the Bragg curves for different wavelengths justified even better than for symmetric diffraction, because possess the same shape and they are simultaneously mea- 4838 S. A. STEPANOV et al. 57 FIG. 8. Rocking curves of the 400 symmetric reflection for ideal Ge crystal and Ge crystal with linearly strained surface layer t 20 000a0, an (1 n )a0, where n 1,...20 000, 5 10 8, a0 1.44 Å is the interplanar spacing for 400 in Ge . The results of the calculations completely coincide with that calculated by the Darwin method and presented in Ref. 65. under Bragg diffraction, which might provide direct informa- tion on the structure of x-ray standing waves XSW . Recent XSW studies of short-period AlAs 3/ GaAs 7 superlattices81 constitute a step in this direction, but the fluo- rescence yield was not interface specific. VII. COMPARISON WITH THE DARWIN THEORY OF HETEROSTRUCTURE DIFFRACTION Since the assumption of the applicability of Ewald's ex- pansion to thin layers is the most critical point of our theory, FIG. 7. Same as in Fig. 6 for the experimental curves measured it is important to compare our results with the Darwin-type with an additional 333 double-reflection monochromator reducing theory of heterostructure diffraction,65 which contains no as- the wavelength dispersion. sumptions of that kind, but instead directly sums up x rays scattered from atoms in the individual atomic planes. sured at the same deviations from the Bragg angle. Thus, Figure 8 presents the results of the RM method for an dispersion effects in two crystals cancel each other. This ideal example discussed by Durbin and Follis.65 The reflec- does not remain true for extremely asymmetric diffraction. tivity of symmetric 400 Bragg diffraction is calculated for a One can see in Fig. 6 that when the incident energy is hypothetical structure where a Ge crystal has a surface layer changed by a small value E/E 10 3, the curves of asym- with a linearly increasing lattice parameter. The layer con- metric diffraction not only shift by the Bragg angle, but also sists of 20 000 atomic planes whose spacing an successively considerably change their shape. In our case the shape of the increases towards the surface as an (1 n )a0, where Bragg curves depends on the angular distance between the n 1,...20 000, 5 10 8, and a0 1.44 Å is the interpla- Bragg angle and the critical angle for total external reflec- nar spacing for 400 planes in Ge. In order to apply our tion. Therefore, the shape of the curves for different wave- method, we formally subdivided the strained layer into lengths is averaged in (n, n) measurements the effect is 20 000 sublayers and solved the dynamical diffraction prob- proportional to the wavelength spread of incident x rays . lem in each of them. Our result exactly coincides with the In order to avoid the dispersion effect, we have carried Darwin theory calculations, even though one cannot consider out an asymmetric diffraction experiment with an additional any periodicity at all in a layer consisting of just one atomic four-reflection Si 333 monochromator selecting a narrow plane. wavelength interval. The results are presented on Fig. 7. The coincidence of the two theories can be understood by Clearly, the experiment now tends to be in much better observing that the dynamical diffraction solution automati- agreement with the theory. cally reduces to the kinematical one for a very thin layer.71 Thus, we have shown that our theory gives a good expla- Our method thus gives the kinematical scattering of each nation for both symmetric and extremely asymmetric Bragg plane, and sums up the multiple scattering exactly in the reflections from a short-period superlattice containing 13 and same way as in the Darwin theory. 27 atomic layers of GaAs and AlAs, respectively. However, Retracing the calculation of superlattice diffraction dis- further experiments, especially with thinner layers and x-ray cussed in the experimental section, one can subdivide each standing waves measurements are welcome. These could be layer in the SL into sublayers corresponding to the atomic x-ray fluorescence measurements of interface-located atoms planes and apply both the Darwin and our theory. The result 57 DYNAMICAL X-RAY DIFFRACTION OF MULTILAYERS . . . 4839 of our method does not change if the perfect layers are for- mally subdivided into monolayers. On the other hand, it co- incides with the Darwin theory if the sublayers are atomic planes. Thus, our method gives the same reflectivity as the Darwin method with the advantage that with our approach the scattering from thick layers is summed up analytically. The above comparison is restricted by the symmetric Bragg case with ordinary incidence and exit angles. Recently it has been proposed to extend the Darwin theory to grazing incidence and/or exit by treating these cases as multiple Bragg diffraction.66 Then the Darwin theory will also require matrix recursion formulas, indicating the fundamental simi- larities of the two approaches. VIII. X-RAY STANDING WAVES IN MULTILAYERS The analysis carried out in previous sections has shown that the RM method gives correct reflectivities for x-ray dif- fraction of multilayers. The aim of this section is to prove the identity between x-ray standing waves in our method and other theories. The main point is to understand the seeming contradiction between x-ray waves ``hooked'' by atomic FIG. 9. The structure of x-ray standing waves at 400 symmetric planes in the Ewald expansion 2 used in our method and Bragg reflection of Cu K 1 radiation from a Ge crystal with 100 ``unhooked'' x-ray standing waves reported by both the Dar- surface monolayers stretched by a/a 10 3. Solid line presents XSW for heterostructure and dashed line shows the substrate XSW win and the Takagi-Taupin theories.65,84 That is, the wave extrapolated into the layer. a at the substrate Bragg peak; b at the field within a given layer as described by Eq. 2 must have overlayer Bragg peak. The depth is measured in Ge monolayers. SS the periodicity of the atomic planes in that layer, yet one and SW indicate the surface shift due to stretching and the XSW would not expect the total standing wave field to always shift, respectively. follow the periodicity of the individual layers, especially when it arises primarily from substrate diffraction, for ex- ample. XSW in the layer: Let us consider a symmetric Bragg reflection from a crys- I z ei u1z D1 2 0 DheiP z 2, tal with a strained overlayer of thickness t as an example. 38 Then, the intensity of XSW can be calculated as P z u2 u1 z h U zn z zn n , 4 I j U z ei unz Dj j i 0n Dhneihzn z zn n 2, 37 j 1 we find that the term ihznz is cancelled and the XSW has the periodicity of atomic planes in the substrate. However, if the where one uses (n 1, z(U) (U) deviation from the Bragg condition for the layer is small, the 1 0, 1 0 and (n 2, z2 t, 1 2 amplitudes D and D become considerable and the atomic 2 hz1t) inside the layer (z t) and the substrate (z t), h 0 respectively. Here the amplitudes Dj periodicity of layer may compete with that of the substrate. 0n are given by Eqs. 14 and 24 , and Dj To justify this conclusion, we have calculated the XSW hn are calculated according to Eq. 7 . Since for an overlayer consisting of 100 atomic planes on Ge 100 Dj j 0n and Dhn are constants within the layers and the phase substrate and the symmetric 400 reflection of CuK relation between them oscillates as 1/h 1 radia- zn , the XSW corre- tion. The overlayer was assumed to be stretched out by sponding to each wave mode has the periodicity of local a/a 10 3. Figures 9 a and 9 b present the calculated atomic planes. However, there is also an interference be- XSW as a function of the z coordinate for the two angles tween several wave modes with different ujn . ( Assume that we are interested in the XSW at the inci- B) 6 and ( B) 128 corresponding to the substrate and the overlayer Bragg conditions, respectively dence angle corresponding to the substrate Bragg peak, and ( the difference in lattice spacing of substrate and overlayer is B is the kinematical Bragg angle . At the substrate Bragg angle the XSW in the overlayer completely conforms with large enough to provide a splitting of their Bragg peaks. the XSW in the substrate, while at the overlayer Bragg con- Then, solving the dispersion equation 6 for the overlayer, dition the XSW starts with conformality near the substrate, we obtain two roots corresponding to a weak coupling be- and then it gradually shifts towards the expanded crystal lat- tween Dj j 0 and Dh . In the first approximation, one root tice of the overlayer. Near the surface the shift with respect u1 2 1 1 0 0 gives the pair of waves (D0 1, Dh 0) being to the XSW extended from the substrate is as large as about the continuation of incident wave in the layer, and the other 1/4 of the XSW period. However, we note that in the latter root u2 2 2 2 0 0 n gives the pair (D0 0, Dh 1) case the relative intensity of the XSW is rather small. Stron- corresponding to the continuation of the wave diffracted by ger XSW correspond to thicker layers, but then the effect of the substrate. Substituting these roots in the expression for ``hooking'' becomes evident. 4840 S. A. STEPANOV et al. 57 In summary, there is good agreement between our method theory can be applied independently for each of the Bragg and the results of other theories concerning x-ray standing peaks with the assumption of uniformly strained layers.85 waves. Then, scattering from strains around dislocations and other defects can be calculated as a perturbation86 using the wave IX. CONCLUSIONS fields given by the RM theory as a basis for the distorted wave Born approximation DWBA . We have presented a recursion matrix theory and experi- The combination of the wave fields provided by the RM mental results on x-ray diffraction from strained multilayer method and the DWBA can also be applied to the scattering crystals. It has been shown that the RM theory overcomes from surface gratings, like, e.g., in the recent analysis of the numerical problems of the former transfer matrix method roughness effects on GID.51 The same approach can be used and is generally applicable for ordinary as well as grazing as well for the scattering from point defects. angles of x rays. The RM method has been shown to reduce Finally, some grazing-incidence x-ray standing waves ex- to the scalar recursion formulas by Bartels, Hornstra, and periments from strained multilayers are in preparation, and Lobeek64 and by Parratt63 in cases of ordinary nongrazing will be useful tests of the theory. x-ray diffraction and grazing-incidence x-ray reflection far The results of this study are aimed at stimulating the ap- from the Bragg conditions, respectively. The results of the plication of x-ray diffraction schemes with grazing incidence Darwin theory for multilayer diffraction65 have also been and/or exit to semiconductor structure research and surface reproduced, and the behavior of x-ray standing waves has science. been demonstrated to be in agreement with the predictions of the Darwin and the Takagi-Taupin theories. The symmetric and extremely asymmetric Bragg diffrac- ACKNOWLEDGMENTS tion experiments with strained AlAs/GaAs superlattice have confirmed the RM theory. A dispersion effect has been found This work was supported by the Volkswagen Foundation, in the (n, n)diffraction scheme applied to the measure- Federal Republic of Germany Project No 1/72439 . One of ments of extremely asymmetric Bragg diffraction and the us S.A.S. is pleased to thank T. Jach National Institute of necessity of an additional monochromator has been demon- Standards, Gaithersburg , V. Kaganer Institute of Crystal- strated in order to suppress this effect. lography, Moscow , V. Kohn Kurchatov Institute, Mos- Extensions of the theory to relaxed multilayers containing cow , A. Macrander, and S. Sinha Argonne National Labo- misfit dislocations and lateral strains can be considered. The ratory for stimulating discussions. We are grateful to R. Hey relaxation is usually characterized by a considerable differ- Paul-Drude Institute, Berlin for the preparation of AlAs/ ence in lateral lattice parameters, so that the Bragg peaks GaAs superlattice, and to M. 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