Diffuse scattering of x rays from nonideal layered structures M. Kopeck$@ Institute of Physics, Czech Academy of Sciences, Na Slovance 2, I80 40 Prague, Czech Republic (Received 8 July 1994; accepted for publication 22 November 1994) A new theory of nonspecular x-ray scattering from layered systems with random rough interfaces based on the distorted-wave Born approximation is presented. Calculations of the diffuse scattering from a single gold layer and two W/Si multilayer mirrors has been carried out. The theory explains the existence of maxima and minima in the angular distribution of diffusely scattered intensity resulting from standing-wave-enhanced scattering and other dynamical effects. The influence of the mutual correlation between individual interface profiles on x-ray scattering is discussed. 0 1995 American Institute of Physics. I. INTRODUCTION The generalization of a scattering theory for layered structures with rough interfaces is a very complicated prob- Nowadays, thin layers and synthetic layered structures lem. The scattered field is a superposition of many plane are of an increasing significance in applied physics. The waves (modes) and it is impossible to take the multiple scat- physical properties of layered systems depend not only on tering of every mode within the system into account. There- the composition and thickness of sublayers but can be fore a simplified model of interaction between electromag- strongly affected by the quality of interfaces. netic radiation and stratified medium has to be considered. Recent progress in the controlled deposition of ultrathin The development of the BA for the case of x-ray scat- films has made it possible to fabricate nanometer-period tering from nonideal multilayer structures has been carried multilayer structures which may be used as optical elements out by Stearns.' His solution is based on the so-called for soft x rays.' It is well known that the specular reflectivity "specular field approximation." In this approach, the coher- can be considerably modified by boundary imperfections. ent (specular) field within the multilayer is treated dynami- Moreover, interfacial roughness can lead to a loss of contrast cally, including multiple reflection and extinction, and the between specularly reflected and diffusely scattered radiation incoherent (diffuse) field is treated kinematically, i.e., the which is detrimental for imaging systems. For these applica- total incoherent field is approximated by the sum of diffuse tions, both the root-mean-square (rms) value and the lateral scattering from each interface. characteristics of interfacial profiles are important. The de- More recently, the extension of the DWBA for layered termination of interfacial roughness is therefore a necessary systems with rough interfaces has been performed by Holy assumption for predicting scattering properties of multilayers et al.Y In their approach, the specular field within the layered and, prospectively, for improvement of deposition technol- system with smooth interfaces has been used for calculating the diffusely scattered intensity. ogy- X-ray methods provide tools for investigating surfaces, tn this paper, a new theory of x-ray diffuse scattering thin layers, and layered structures. There is a large body of from layered systems based on the DWBA is described. The real specular field and the real transmission coefficient of the work discussing the spectdarly reflected component of x-ray radiation diffusely scattered from inner interfaces are consid- scattering.2*3 The reflectivity curves yield information on ered. By means of this theory, the previously measured densities and thicknesses of sublayers and are frequently modulations in the diffusely scattered intensity are explained. used for a rms roughness evaluation. Considerably less atten- The influence of mutual correlation between individual inter- tion has been devoted to the diffuse component of the scat- face profiles on the x-ray diffuse scattering is discussed. tering, which depends on the rms value as well as on the correlation function.4-8 Several theories of nonspecular scattering from a single II. THEORY random rough surface have been developed. The Born ap- proximation (BA) and the distorted-wave Born approxima- Let us consider a plane wave ET with wave vector `kr tion (DWBA) are mainly used in the x-ray region.4 The BA incident on a system of N- 1 layers deposited on a thick is valid at glancing angles much greater than the critical substrate. This wave is specularly reflected and refracted (co- angle 8,. whereas it breaks down as we approach the total herent field) and simultaneously diffusely scattered (mcoher- external reflection region. On the contrary, the DWBA is ent field) from each interface. suitable for glancing angles close to tic. However, at larger A, The coherent field glancing angles, the expressions for the differential cross We start by determining the coherent field within the section derived in the DWBA reduce to the results in the layered structure which may be in each medium i expressed BA." as the superposition of two plane waves ET and Ef (Fig. 1). The continuity of the tangential components of the electric a)E-mail: kopecky@fzu.cz and magnetic vectors at the boundary i may be written as" 2380 J. Appl. Phys. 77 (6), 15 March 1995 0021-8979/95/77(6)/2380/8/$6.00 Q 1995 American Institute of Physics Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp Ri=EPIET is the reflection coefficient of the system below air (vacuum I 2 the boundary i. The reflection coefficient ~i,i+l of this rough medium interface interface is related to the Fresnel reflection coefficient Ti,i+, of a smooth interface byi ri,i+ 1 =ri,i+l Xi(J&ZTL). (5) ?+=ll `x xi denotes the one-dimensional characteristic function of the rough interface i. Vector q,(O,O,2'&) represents the differ- =2 ence of wave vectors of waves EF, ET. The set of equations (4) may be solved by starting at the bottom medium N+ 1, where RN+ i = 0 (since the thickness of the substrate is as- sumed to be infinite). Further, the coherent field may be determined using formulas13 E;(8A=Ri(8,W:(8,), (6) i+2 `i*l Eir+O,j= l+Ri~~f:$f$)l,e,, goi+1(4.WT(ed. N E;: v E:: \ N- (7) ZN N+l The calculation is now carried out from top to bottom. ' ET N+l substrate 8. The diffuse x-ray scattering from a single rough interface FIG. 1. Scheme of the coherent field within stratified medium. As the second step, the scattering of the coherent field from each rough interface has to be evaluated. Let us con- sider rough interface (surface) i between two homogeneous 1 media i and i+ 1. The deviation of this interface from the z$(el)+Ef(elj= @Au average reference plane z = zi is described by the profile 9i+l(ed function h,(x,y). (14 The electric field for x rays poIarized perpendicular to the plane of incidence satisfies the wave equation V2~(r>+k~~(r)-V(r)~,(r)=0, (8) where r is the position vector and the scattering potential V(r) is related to the refractive index n(r) by V(r) =k$l -n'(r)]. (91 where `ki is the wave vector of the wave ET, 4oi the ampli- The polarization of x rays in the plane of incidence is not tude factor corresponding to the perpendicular depth d, , discussed since at grazing incidence the results are the same for both the polarization components. qi(O1)=e-iLki~di for i=Z,...,N, (2) Let us assume that the real rough interface represents a and ~+++,=l. small perturbation from the smooth interface, for which the According to SnelI's law, the wave vectors `ki are deter- exact eigenfunctions are known. Then it is convenient to split mined by the refractive indices ni of individual layers, wave- the scattering potential into two parts: length A, and the glancing angle 0i of the incident wave: V(r) = VI (4 + V2(rL (10) Ikix=`kl, 9 (34 where V1 is the scattering potential of the system with a smooth interface and V, is regarded as the perturbation due `kiv=lkly, CW to the roughness: `ki,=-ko@-c0S2 Bl=-`Q, (11) where k,,= 27~0~ is the magnitude of the vacuum wave vec- for :Zir `*w= (14) & dy(e-iPi+lzktx~Y)- 1) ti,i+l(lkijei'kf+~`, for Zz~, ried out over the illuminated area SO of the size L,L, . "i&r) = * (19 In a similar way, the field diffusely scattered from the i t&+`( -`kijeiLki+lr, for ZCZi, interface i into the medium i+ 1 may be derived. Then the which is a time reversed state for the incident plane wave eigenstate `4 is chosen as the time reversed state for the with the wave vector --`ki (Fig. 2). wave incident on the interface i from the medium i+ 1 with The T matrix for scatteriilg between states `ki and `ki the wave vector `ki+l (Fig. 3): (proportional to the scattered electric field) has in the DWBA i'ki', for z>zi, the form'" 2$crj= ti*,l,i(-`ki+l)e e'2ki+~rfr~+l,i( -2ki+l)ei2k[+l', for z by .YFR, YT' (the scattering of the In the medium i, the sources Yi'_`, , YfT,, Y"TR, .55fR wave ET) and YPR -i"f' (the, scattering of the wave produce the electromagnetic field, which is a superposition EP+, (pi+:). It follows'from Eqs. (17) and (21) that of two specularly related waves 99:, @. The boundary con- ditions for the interfaces i- 1 and i yield ~~R(ql)=ET(Bl)k~(a~-n~+l) Xt i,i+l(e,)t,,i+l(e2)Fi(15i+l+2~i+1), (224 2 2 .Y;T(qJ=ET(e*)kO( I n. -4+,) -tYiT_T*(q& (25) Xti,*+l(b)l)ti+l,i (e2)Fi(-`~i+1+25i+l)r (22b) where Qi are the reflection coefficients of layered systems above the interface i. We can determine Ri( 0,) from Eq. (4) ~~~ql)=--EP+1(~l~cpi+l(e1)ko2~~~+l-~?) and Qi( 0,) from the similar recursive formula xti+*,i(~l)ti,i+1(B2)~i(15i-2~i), (224 Q,(e )= Qi-l(e2)(P~-l(e2)+Y"i,i-l(e2) ~~T~~l)=-~~+l~elj~i+l~eljk~(~~+l-~~~ 1 2 Qi-l(e2)cp~-l(e2)Ti,i--l(e2j+l. (26) xt~+l,i(e,>ti+l,i(e2jFi(-1~i-2~ij, (224 The values of Qj are calculated in the direction from top where the tangential components qX, q, of vectors qi , qi+ , (Q, = 0) to bottom. are omitted in the argument of functions Fi. The variables If we express the term ti from Eqs. (24) and (25), `ci are defined analogically to `& as multiply it by the transmission coefficient Ti( 0,) for the sys- "ci=ko tem above the interface i, and carry out the summation over (23) all interfaces, the final matrix element Z?& corresponding to If the multiple diffuse scattering is neglected, the vari- the total diffusely scattered field outside the layered structure ables Yi may be considered as "independent sources of x may be obtained: I @i(ql)=; Ti(B&Z9f(ql)=$ Ti( 0,) ~i(~2)~i(~2)~~_Tl(ql)+~~~~l(q~)l+~R(ql)+.~(ql) i=l i=l 1 -Ri(e2)Qi-l(e2j~o?(e2j (27) The transmission coefficients Ti( 0,) may be determined cients Ti( O,), jGi, has to be performed starting by the wave in the following way: Let the plane wave I$+ t (6,) in the not in the substrate but in a layer closer to the surface. substrate pass through the system of layers toward the sur- By substituting Eqs. (22a)-(22d) for .Y a1 fils the Bragg condition," the sharp maxima arise. However, satellite minima may sometimes occur in these positions. -g IO3 +02 = 1.50 deg. This somewhat surprising but already experimentally ob- -2. 6 IO2 served phenomenon13V16*21 can be explained within the 3z framework of the theory presented above. When one of the 2 IO' angles 0,) a, approaches the Bragg angle, not only are the G interference effects and the coherent field modulations im- 2 - IO0 portant but, in addition, the transmission of the incident wave or the waves scattered from individual boundaries exhibits a 10-l 4 local extreme. Whether the increase or the decrease of the -0.6 -0.3 0.0 0.3 0.6 transmission occurs it depends particularly on the ratio of W and Si in the bilayer and on the number of periods. That is Q [deg.] why no decision-which effect is dominant and whether the IO' -3 scattering in a given direction is enhanced or attenuated- (b) may be done without carrying out the numerical calculations. 91 +*2 = 3.00 deg. 2 IO0 For uncorrelated roughness, rZ-+O, CijiO if i#j and -is the sum in the relation (33) for the diffuse component of the I IO" differential cross section has only N nonzero terms. On the 32 other hand, if the roughness is fully conformal, r,--+m and $ lo-2 all terms contribute to the total diffuse scattering. Therefore the nonspecular intensity is usually in the former case much 8 smaller. That is why the vertical correlation is very undesir- ,o-3 able at layered systems used as image elements, because it causes the contrast degradation of the final image. But the 0.0 opposite situation may also occur and the incoherent scatter- 01 [deg.] ing from the layered structure having uncorrelated interfaces may be comparable or even higher [Fig. 6(c)] than for the FIG. 4. Theoretical transverse scans for the gold layer (dAu=20 nm) on the identical system. The reason why the multilayer with corre- silicon substrate. The differential cross sections calculated using the coher- lated interface profiles exhibits a very low diffusely scattered ent field within the stratified medium with rough (solid line) and smooth intensity at the third-order Bragg maxima is the fact that the (dashed line) interfaces are compared. The rms roughness ~~2.0 nm and the third order is nearly structure factor forbidden. Another in- correlation length ~0.1 ym are assumed at both interfaces. teresting feature is also that incoherent scattering from sys- tems with completely uncorrelated roughness may show fine lation of the coherent field is therefore negligible, and the structure but this is mostly less apparent. This can be ex- fringes disappear. Only the calculations starting from the co- plained by the effect of primary extinction in the vicinity of herent field within nonideal stratified medium give realistic Bragg angles22 (reduced or enhanced transmission if the results in this case. nodes of the standing wave pass through the W or Si layers). Figure 5 depicts the influence of the vertical correIation16~`s~`g on the theoretical 0, scan. If the roughness is correlated (identical layered systems), the maxima and minima periodically alternate in the angular distribution of IO4 6.. s ' ' s * - I * Q r s *. ' *. I * * * - - * = * - I * - = * *. * - `4 the scattered intensity. Similar to the Kiessig structure of reflectivity curves, these oscillations result from the interfer- ence between the waves diffusely scattered from the top and bottom interfaces. However, if the interface profiles are com- pletely uncorrelated, nonspecular scattering monotonously decreases with the increasing angle 0,. For the demonstration of the diffuse scattering from more complicated structures, two, W/Si multilayers have been chosen. These x-ray mirrors differ only in the thickness ratio of layers composed of heavy and light elements. The rms roughness rr=OS nm and the correlation length r= 1 ,um IO" F. 0.0 1.0 2.0 3.0 4.0 are considered at each interface. 0, Meg.1 The series of theoretical transverse scans at 28 values corresponding to the first-, second-, and third-order Bragg maxima are plotted in Figs. 6 and 7. The x-ray scattering FIG. 5. Theoretical Ci, scans for the gold layer (dA,=20 nmj with correlated from multilayers with correlated and uncorrelated roughness (solid line) and uncorrelated (dashed line) interface profiles. The rms rough- ness (r= 1.0 nm and the correlation length ~0.1 pm are considered at both is compared. These pIots show several interesting features. interfaces. J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. Kopeclj 2385 Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp IO5 b' ' ' ' * - * * ' I - ' 8 ' ' ' 1' * b ' ' `. * ' ' - - I ' * - * g ' *. `j F (a) 0, +e,=I88deg. 1 lo2'....,,...1......~~~~~~~~~~~~~~~,.*~~~~~' -0.6 -0.3 0.0 0.3 0.6 ' --0.6 -0.3 0.0 0.3 0.6 0 [deg.1 0 Wg.1 ~~.~,~~~`~~"~)~.`.~~"~J~.~. (b) 01 + 92 = 3.60 deg. $1 + 92 = 3.60 deg. 0.0 co [deg.] 0 Wg.1 ~"~.~`,"`~~.*","~`~`.`.I.`~~`~"`,~~ 0, + e2 = 5.36 deg. y IO3 -g IO2 g IO' iz IO0 3 2. IO" 1 o-2 -2.0 -1.0 0.0 1.0 2.0 -2.0 -2.0 -1.0 -1.0 0.0 0.0 1.0 1.0 2.0 2.0 w Peg.1 0 0 Wg.1 Wg.1 FIG. 6. Transverse scans calculated for the multilayer WEi @,=I.7 nm, FIG. 7. Transverse scans calculated for the multilayer W/S (d,=1.2 nm, dsi=3.3 nm, 40 periodsj at (a) the first-, (b) second-, and (c) third-order ~&~=3.8 nm, 40 periods) at (a) the first-, (bj second-, and (c) third-order Bragg maxima. Bragg maxima. in the specular reflection. For the parameter 7z = 0.2 pm, not The off (0,2 19) scan measurement appears to be a very only the profiles of adjoining boundaries are correlated but a promising method for determining vertical correlation. The certain degree of correlation also exists between the rough- numerical calculations were carried out for the muhilayer ness of the substrate and of the surface. This again results in consisting only of 10 pairs of Fe/C (Fig. 8). A smaller num- fast oscillations equivalent to the Kiessig ones. ber of periods was chosen in order to reach a better resolu- The presented results show that the structure of layered tion of subsidiary interference fringes. This scan exhibits no systems has a strong impact on their scattering properties. fine structure for completely uncorrelated interfaces. With increasing vertical correlation, the interference maxima start IV. SlJttlMARY to appear in the regions where the angle of scattering 20 is equal to twice the Bragg angle. These peaks correspond to In this paper, the theory of diffuse x-ray scattering from the constructive interference of waves diffusely scattered rough layered systems based on the distorted-wave Born ap- from individual bilayers in a full analogy to the Bragg peaks proximation has been presented. To author's knowledge, this 2366 J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. Kopeck$ Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp may be quantified. Different scattering properties of systems with correlated and uncorrelated roughness enables one to specify the level of vertical correlation. A paper devoted to the characterization of layered struc- tures by coherent and incoherent scattering of x rays is being prepared. ACKNOWLEDGMENTS This research has been supported by the Grant Agency of the Czech Republic under Grant No. 202/93/1023 and by the Grant Agency of Czech Academy of Sciences under Grant No. 110427. 0, Wg.1 `See. for example, P Chakraborty, Int. J. Mod. Phys. B 5, 2133 (1991). FIG. 8. Theoretical offset (0.28) scans for multilayers Fe/C (dFe= 1.77 nm, "A. Horn&up, F. E. Christensen, J. Polny, and H. W. Schnopper, Proc. &=3.28 nm, 10 periods) with different vertical correlation. The same val- SPIE 982,409 (1988). ues of u=OS MI and 7= 1.0 ym are considered at each interface. "0. Renner, M. Kopecky, E. Krousky, F. Schafers, B. R. Miiller, and N. I. Chkhalo, Rev. Sci. Instrum. 63, 1478 (1992). 4S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, Phys. Rev. B 38, is the first model valid also at small glancing angles, where 2297 (1988). `D. G. Stearns, J. Appl. Phys. 65, 491 (1989). the calculation of the diffusely scattered intensity respects 6R A. J. de Korte and R. Lain&, Appl. Opt. 18, 236 (1979). both the real coherent field within a nonideal stratified me- 7E. L. Church and P. Z. Takacs, Proc. SPIE 640, 126 (1986). dium and a real transmission of waves scattered from inner *W. Weber and B. Lengeler, Phys. Rev. B 46, 7953 (1992). boundaries. `V. Holi, J. Kub&ra, I. Ohlidal, K. Lischka, and W. Plotz, Phys. Rev. B 47, 15896 (1993). By using the above-mentioned approach, the Yoneda t"L. G. Parratt, Phys. Rev. 95, 359 (1954). anomalous scattering of x rays can be described. Further, the "J. H. Underwood and T. W. Barbee, Jr., Appl. Opt. 20, 3027 (1981). modulation in the nonspecularly scattered intensity can be `*L. Nevot and P. Croce, Rev. Phys. Appl. 15, 761 (1980). explained as the consequence of dynamical effects. The scat- 13M. Kopecky, Doctoral thesis, Technical University, Prague, 1993. 14L. I. Schiff, Qtrantum Mechanics (McGraw-Hill, New York, 1968). tering from periodical multilayers exhibits satellite maxima 15B. B. Mandelbrodt, The Frucral Geometry qf Nature (Freeman, New York, and minima when 8, or ~9, approaches the Bragg angle. 1982). These features depend on the mutual correlation between in- 16D E Savage, J. Kleiner, N. Schimke, Y.-H. Phang, T. Jankowski, J. Ja- dividual boundaries but they can be observed for completely cdbs,.R. Kariotis, and M. G. Lagally, J. Appl. Phys. 69, 1411 (1991). 17Y. Yoneda, Phys. Rev. 131, 2010 (1963). correlated as well as for uncorrelated interface profiles. "D E Savage, H. Schimke, Y.-H. Phang, and M. G. Lagally, J. Appl. Phys. This scattering theory may be also used for fitting ex- 7;, ;283 (1992). perimental data. In this way, basic statistical parameters 19E. Spiller, D. G. Stearns, and M. Kmmrey, J. Appl. Phys. 74, 107 (1993). (root-mean-square roughness, correlation function) of ran- "A. E. Rosenblutb and P Lee, Appl. Phys. Lett. 40, 466 (1982). "J. B. Kortright, J. Appl. Phys. 70, 3620 (1991). dom rough surfaces and interfaces within layered structures "B. W. Batterman and H. Cole, Rev. Mod. Phys. 36, 681 (1944). J. Appl. Phys., Vol. 77, No. 6, 15 March 1995 M. Kopeck$ 2387 Downloaded 10 Sep 2002 to 148.6.178.13. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp