X-ray scattering from interfacial roughness in multilayer structures D. G. Stearns University of California, Lawrence Livermore National Laboratory P.O. Box 808, Livermore, California 94550 (Received 28 June 1991; accepted for publication 20 January 1992) A quantitative theory of the nonspecular scattering of x rays from multilayer structures having rough interfaces is presented. The results are valid for arbitrary polarization and angles of incidence (measured from the normal) less than the critical angle for total external reflection. A structural model is adopted wherein each interface is assumed to be described by a surface having statistically random roughness with a well-behaved power spectrum. In addition, the model accounts for arbitrary correlation of the roughness between different interfaces. Calcu- lations are presented for a variety of roughness configurations to investigate the dependence of the nonspecular scattering on the fundamental structural parameters. In particular, it is shown that the scattering from correlated roughness exhibits characteristic resonance behavior (quasi- Bragg diffraction). I. INTRODUCTION can be found in the literature.10-13 More recently there has Thin films composed of synthetically grown multilayer been increasing interest in using nonspecular x-ray scatter- (ML) structures represent a new class of materials having ing to study the roughness of multiple interfaces in ML novel optical, electrical, magnetic, mechanical, and super- structures. The first experimental results indicate that the conducting properties for a host of important applications. x-ray scattering can exhibit a rich variety of behavior as- Since the interesting and unique properties of ML derive sociated with the structural correlations between from the close proximity of different materials, it is not interfaces. 14-" However, the interpretation of these results surprising that these properties are often strongly sensitive has been limited by the lack of a quantitative theory that to the nature of the interfaces at the layer boundaries. In incorporates realistic models of the interface structure. The order to understand and control the physical behavior of goal of this paper is to present a simple theory that, within ML, it is essential to be able to determine the detailed the limitations imposed by certain simplifying approxima- structure of the layers and interfaces, and to correlate this tions, can provide a straightforward means of relating re- structure with the measured properties. alistic interface structures to measurements of nonspecular One important type of structural imperfection that can scattering. affect the properties of ML structures is interfacial rough- The scattering of radiation from multilayer optical ness. For example, in electronic and magnetic ML,' inter- coatings having rough boundaries has been considered pre- facial roughness increases the amount of electron scatter- viously. Early attempts'8"9 were directed at studying the ing by providing coupling to additional momentum states. effects of roughness on the specular scattering using scalar In x-ray optical ML,2 roughness both decreases the reflec- theory. Subsequently, Elson2' developed a vector theory tivity and introduces a background halo that can degrade for the scattering of radiation from surface roughness, and the resolution of imaging optics. There is also an increasing applied the theory to ML coatings in which the roughness interest in understanding roughness as an intrinsic dy- at the layer boundaries was either exactly reproduced from namic behavior of growing surfaces and interfaces. Recent layer to layer (complete correlation) or was completely theories3-7 predict that the roughening of a surface follows random at each layer (no correlation). Elson, Rahn, and simple scaling laws, and it has been shown* that ML struc- Bennett2' later extended the model to include a case where tures can be useful experimental systems for studying the the roughness accumulates from the bottom to the top of evolution of the surface roughness during film growth. the stack which corresponds to partial correlation. Bous- , A promising technique for characterizing the rough- quet, Flory, and Roche2* developed a comprehensive the- ness of surfaces and interfaces in ML structures is x-ray ory of scattering from ML coatings that accomodates ar- scattering. The use of x-ray scattering as a structural probe bitrary correlations between the roughness of the different has several important advantages. It is inherently a nonin- interfaces. Calculations based on this model have been vasive technique, well suited for dynamic measurements compared to measured nonspecular scattering from multi- including in situ growth studies. The penetration of x rays layer optical coatings for the special cases of no correlation allows both surfaces and buried interfaces as to be directly and complete correlation.23 It is also possible to induce probed. Furthermore, due to the short wavelength of x strongly correlated interface roughness in "ideal" ML rays, x-ray scattering can provide structural information coatings by applying surface acoustic waves. A purely ki- on spatial scales ranging down to atomic dimensions. The nematical description of x-ray scattering from such struc- roughness of single surfaces has been investigated using tures, assuming that the roughness is completely corre- x-ray scattering,' and corresponding theoretical treatments lated, has been compared to experimental results.24 applicable to a variety of conditions and approximations Most of this previous work was intended to describe 4286 J. Appl. Phys. 71 (9), 1 May 1992 0021-8979/92/094286-i 3$04.00 @ 1992 American institute of Physics 4286 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp scattering from optical ML coatings at visible and near- ultraviolet wavelengths. Although the theoretical formal- isms can in principle be applied to the scattering of x rays, we choose instead to apply the theory of the scattering of x rays from a nonideal ML structure developed in a previous paper" (referred to herein as Part I). This treatment uti- lizes the first Born approximation and hence is only valid under the condition that the scattering is weak and refrac- tion can be neglected; that is, for incident and scattering angles (measured from the normal) less than the critical angle for total external reflection. At x-ray wavelengths the dielectric function is close to unity for all materials, so that FIG. 1. The configuration of the radiation field scattered from a rough even the specular (zeroth order) scattering is weak. Then interface f(x,y) separating two uniform media. for many experimental configurations this approximation is acceptable and offers the advantage of significantly sim- E'. As a review, we consider the configuration shown in plifying the mathematical description of nonspecular scat- Fig. 1. A plane wave, Eo$ exp( L&x), propagating in di- tering, particularly the term that Elson and Bennett26 call rection $ with polarization $, is incident on the rough in- the "optical factor" which accounts for the geometric and terface from above and scatters into direction &, experi- polarization effects. encing a momentum transfer of In Sec. II we extend the results of Part I to treat the specific case of x-ray scattering from inter-facial roughness q = k(hi -i?), (1) in ML structures. A general model for inter-facial rough- where k is the vacuum wave number. The difference in the ness is introduced that can account for an arbitrary dielectric constants of the two media A = E - E' is always amount of correlation between layers. The model imposes small at x-ray wavelengths (Ag 1). Using the first Born no contraints on the power spectrum of the roughness, and approximation for the scattered field, the amplitude den- thus is compatible with both conventional correlation- sity of the field reflected into direction 6i with polarization length-type and fractal-type descriptions of surface &s given by roughness. 27 The correlation of the roughness between lay- ers is described by a frequency-dependent replication factor Wk~$ = - ,i9 rEo gz (i;*-i$k~,,~,). which can be used to either amplify or attenuate the rough- ness from layer to layer in a given frequency range. This Here yis the Fourier transform off, s = sxG + s,,? is the represents an important extention of the earlier models for projection of q in the x-y plane, and 2 represents the roughness propagation used in ML scattering theories, and complex conjugate to account for the the case of circular is more consistent with the models currently used in theo- polarization. It should be noted that the Born approxima- retical treatments of surface roughening during film tion is only valid when the scattering is weak and refrac- growth. tion of the transmitted field can be ignored, conditions that In Sec. III we present calculations of nonspecular scat- are generally satisfied for x rays when the incident and tering from a variety of different interface structures and scattering angles (measured from the normal) are less than configurations. In particular, we study the characteristic the critical angle for total external reflection. The Fourier dependence of the scattering on structural parameters such transform in Eq. (2) results from the assumption that the as the number of layers, the root-mean-square roughness, interface is "slightly rough," such that the correlation length of the roughness, and the degree of correlation of the roughness between layers. Systematic Iqzfky) I = k( m, - n,) If (x,-Y) 14 1 for all x,y. variations in the nonspecular scattering are observed, in- (3) cluding the appearance of resonance features (sometimes It is shown in Sec. II A that this condition is equivalent to called "quasi-Bragg diffraction") when the interfacial requiring that the total integrated nonspecular scattering is roughness is correlated from layer to layer. The purpose of small compared to the specular reflectance. We note that these calculations is to illuminate the close relationship Eq. (3) is always satisfied when the interfacial roughness between the nonspecular x-ray scattering and the detailed If(x,y) I <;1, the x-ray wavelength. structure of the interfaces. The differential power dP, reflected from an interface In Sec. IV we conclude by commenting on the imple- of area A into a solid angle da, per unit incident power, is mentation of the theory as it relates to the interpretation of given by experimental measurements. II. THEORY (4) In Part I we presented a description of x-ray scattering Similarly, for a plane wave, E$ exp( ik'G*x), incident from an interface of arbitrary structure. A special case is onto the rough interface from below, the field amplitude the scattering from a "slightly rough" boundary f(x,y) density transmitted into direction i;i with polarization 2 is between two media described by dielectric constants E and given by 4287 J. Appl. Phys., Vol. 71, No. 9, 1 May 1992 D. G. Stearns 4287 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp Once the specular field in each layer is known, the non- specular scattering is treated kinematically. The total non- specular field is approximated as the sum of the fields scat- tered from each interface, taking into account extinction of the field as it propagates towards the top of the ML stack. N-l We note that the kinematic treatment breaks down when the nonspecular scattering angle satisfies the ML Bragg i+l condition, such that the multiple scattering exhibits con- structive interference. This can either suppress or enhance i the nonspecular scattering, in the same way that the spec- i-l - ular transmission through a ML structure is modified at Et.1 . the Bragg condition.25 Such distinctive features arising . from multiple scattering of the nonspecular field have been E2 . I- experimentally observed.15p16 El Let us consider the field scattered from the ML stack 0m into the nonspecular direction i% with polarization 2. As before, the propagation of the field in the ith layer is altered FIG. 2. Schematic diagram of a ML structure having rough interfaces. by refraction to be in the direction &ii. There are two con- The inset shows the scattering at a particular interface, consisting of two tributions to the nonspecular field from each interface, cor- contributions. The specular field incident from above and below the in- responding to the scattering from the interface of the spec- terface is scattered into the mode (&a with amplitudes of ri and ti, ular fields incident on either side (see Fig. 2). In respectively. particular, the specular fields incident on the ith interface from above and below are, correspondingly, 3 Ei- (x) = ,E-$- exp(ikn,y)exp( - ik,n$) (8) t(iii,i,ii;G,3) = a() &. (ii+- E)~(s,,s,), (5) .? and and the power scattered into that direction is E+ (x) = Ei+$+ exp(ikn,y)exp(ik&), (9) dP, hhhh 4rPrn~ where j = i + 1 and the polarization $* corresponds to S z bww) = 2 lt12. (6) GA or P type in accordance with the polarization of the inci- Next consider the scattering from a multilayer stack dent field. The amplitudes of the specular fields, Ejm and consisting of a sequence i = 1,2,...,M of layers having Eii' , are determined using recursive or matrix methods as rough interfaces as illustrated in Fig. 2. The ith interface, mentioned above. Each of these specular fields scatters defined as the boundary between layers i and i + 1, is de- from the rough interface, generating a nonspecular field scribed by the surface fi(x,y). A plane wave of unit am- that propagates towards the top of the stack. The incident plitude 8 exp( ik%x) is incident at an angle 19, on the top of field Ei- scatters into mode % with an amplitude density the stack. For convenience, the propagation vector is cho- ri given by Eq. (2), sen to have components of n, = 0, nY = sin eo, and n, Aikj = cos eo, and the polarization 2 is either S or P type. The ri( i%,G;S,G- ) = - iEi m; e-2- >fi(S,JJ, propagation direction in the ith layer is altered by refrac- (10) tion to have the value 2, where where Ai = Ej - Ed Similarly, the incident field E+ scatters i n, = n,, into mode % with an amplitude density ti given by Eq. (5)) nf= n, (7) Aiki ti( iYp@$pS' ) = iEii j--J-& e-3+ )f,(s,Jy). (11) .z nf = & ,/kf - k2nz - k2nz. The nonspecular field scattered from the ith interface Here kj = \ E;`~ I k and ni is a complex quantity in the case accumulates phase as it propagates to the top of the ML of an absorbing medium. stack. The phase contributed by traversal of thepth layer is The x-ray scattering within the ML structure is calcu- defined as c#J~, given by lated using the specular field approximation, as discussed in Part I. In this approximation, the specular field in each &, = ktP ,im, (12) layer is determined from a complete dynamical treatment where tp is the thickness of the pth layer. The total phase of the specular scattering within the system of interfaces accumulated upon reaching the vacuum ihterface at the using well-known recursive28 or matrix methods.29 The top of the stack is pi = ZF= i + ,c$~ The total amplitude Fresnel reflection coefficients can be modified to account density of radiation scattered into the vacuum in the direc- for the interfacial roughness,25 although this is required by tion ;;i is Zy= ,bi( r, + ti) . Then, according to Eqs. (4) and condition (3) to be a small correction to the specular field. (6), the scattered power is given by 4288 J. Appl. Phys., Vol. 71, No. 9, 1 May 1992 D. G. Stearns 4288 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp gj( iii,i?;;s,$, = $pj 1 ii0 2vrj + fj) 12. ai decreases monotonically with 1 x ( , and the convolu- (13) tion serves to smooth the surface locally. This, of course, is the natural outcome of suppressing the replication of the The theoretical development presented thus far is higher-frequency components. mostly a recapitulation of the results of Part I. Equation It is interesting to relate the model for the propagation (13) describes the nonspecular scattering of x rays from a of interfacial roughness in a ML structure to the theory of series of rough interfaces having completely arbitrary the roughening of the surface of a single film during structures. To proceed we must adopt a specific model for growth. The difference equation (14), describing the the interfacial roughness in a ML stack. We postulate that roughness at a number of discrete interfaces, can be con- the roughness of an interface can be separated into two verted into a differential equation by considering the film, components which we call the "intrinsic" and "extrinsic" of total thickness t, as being composed of many thin I_ayers parts. The intrinsic foughness h(x), with the associated having thickness At. We define the functions q(s) rh(s)/ frequency spectrum h(s), corresponds to that part of the At and b(s) = [ 1 - Z( s)]/At, and take the limit of At-+0 to interface structure that is inherent to the formation of the obtain interface, and would be experimentally >bserved if the un- derlying interface was perfectly smooth. The extrinsic ah) roughness corresponds to the structure derived from the -= q(s) - b(s)f(s,t). at replication of the roughness of an underlying interface. Hence the extrinsic roughness accounts fc: Lhe propaga- If we let b(s) = 47?& then the Fourier transform of Eq. tion of roughness through the ML stack. It is reasonable (18) yields the well-known Langevin equation describing that the extent of propagation varies with spatial fre- the evolution of a growing surface derived by Edwards and quency; components of roughness having wavelengths Wilkinson,3 much longer than the layer thickness should be replicated, whereas the high-frequency components of roughness are a- (x,0 -= likely to be planarized. To keep the model general we de- at yV2f(xJ) + r](x). (19) fine a replication factor ai which describes the fraction This is a differential equation representing isotropic diffu- of the frequency component s in the (i - 1)th interface sion in two dimensions, where the thickness of the film that is replicated in the ith interface. Then the model for replaces the time variable. The first term on the right-hand inter-facial roughness in a ML stack can be written as side describes the relaxation of surface features due to a fi(S) = hi(S) + &(S)fi- 1(S)* (14) "diffusion coefficient" Y. The second term is a source term accounting for the introduction of random noise during The first and second terms on the right-hand side represent growth. Kardar and co-workers' have pointed out that it is the intrinsic and extrinsic roughness of the ith interface, necessary to include a nonlinear term proportional to respectively. The replication factor &(s) can have any (Vf)* in Eq. (19) when the direction of growth is locally functional form, but is physically constrained to have the normal to the surface of the film. However, the behavior of limiting values of unity and zero as 1 s 1 approaches 0 and the growth on sloped surfaces is likely to be strongly de- CO, respectively. By substituting recursively in Eq. ( 14) we pendent on the detailed characteristics of the deposition obtain process such as the collimation and energy of the incident adatoms, and it is not clear a priori whether the inclusion (15) of a nonlinear term is more physically relevant. In any case, Eqs. (14) and (17)-( 19) are linearized, lowest-order where descriptions of roughness propagation in film growth, and are certain to be good approximations when the interface Iii = (+im or surface slopes are small. CinEfl:,=gm* (16) Having established a structural model for interfacial roughness in a ML, we proceed to develop an expression Equation (15) explicitly shows that the roughness of th_e for the nonspecular x-ray scattering. We rewrite Eq. (13) ith interface is composed of its-own intrinsic roughness hi as and the intrinsic roughnesses h, of each of the underlying interfaces. The factor c/n represents the amount of intrinsic roughness inherited by layer i from the underlying layer n. (20) The physical significance of the replication factor is made clear by taking the Fourier transform of Eq. (14) to obtain where the quantity Wi is defined by a description of the interface structure in real space, WjzAj[Ej+ (@*$+) - Ei- (pm$-)]e'@i. (21) ft(X) =hi(x) +ai(x)*[fj- I(X)I* (17) From Eq. ( 15) we have The amount of roughness replicated from the underlying interface is determined by a convolution with the function Ti$ = ( nio cirzLn) ( i. ai( When &(s) decreases monotonically with 1 s 1 then cj&T)* (22) 4289 J. Appl. Phys., Vol. 71, No. 9, 1 May 1992 D. G. Stearns 4289 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp The intrinsic roughness hi(x) of each interface is statisti- the radiation fields scattered by interfaces that have corre- cally random, in the sense that it is completely uncorre- lated structure due to the replication of roughness from lated with the intrinsic roughness of any other interface. layer to layer. Hence the phase of the quantity i,,@ has a random value, If the configuration of the surface roughness or the and every term in Eq. (22) for which I#n vanishes when measurement geometry reduces to a one-dimensional scat- averaged over an ensemble of interface structures. This tering problem it is straightforward to show that Eq. (26) "random-phase approximation" is valid for measurements becomes, in which the spatial coherence length is much smaller than the dimensions of the x-ray beam, where the measurement c~,O~,G: Wi fi averages over an ensemble consisting of different areas of the ML sample. However, in the case where the incident beam is spatially coherent, all of the terms in Eq. (22) will + iil ( f: C+zcjndG!t) contribute to the scattered power, resulting in a varying n=O and complex angular distribution analogous to the speckle patterns produced by the scattering of laser light from X(Wiy+ flWj) 9 1 (27) rough surfaces. Applying the random-phase approximation reduces Eq. (22) to where G:(s) is now the one-dimensional power spectrum of the intrinsic roughness of the nth interface. Although Eq. (26) is a complicated result, it is quite f2 = n$o CjnCjnh",Lt, j -A=asA 10-i - n 1\.7OA ---`4=75A . . . . ..t.wA lo4 `.-/I 4 \ f 104 -1 --. `--\,y cx' P -__--- `..*-. u lo4 `_`-\->-\,' h normal incidanw 10d If&W=lWk 4rW=~ t lo'!.:.:.:.:.:.:.:.:`:.:' 10-r ! . : . : . : * : . : . : . : * : ' : . : * : ' 0 10 20 30 40 50 -60 -40 -20 0 20 40 60 Scattering Angle 6 (deg) Scattering Angle 6 (deg) Cc) (4 lo", I k,O MO A, A, 85 85" " incident inckmt angle 100 angle -- ;zy&, ;-y&, o(sub)=2 A o(=w=2A lo*! .:`:.:.:`:`:`:`:`:`:' I lo+ I I 0 10 20 30 40 50 0 18 18 36 36 54 54 72 72 ! 90 Scattering Angle 9 (deg) Scattering Scattering Angie Angie 9 9 (deg) (deg) FIG. 5. Calculations of nonspecular x-ray scattering from a Mo-Si ML having complete correlation, where the roughness of the substrate is replicated exactly at each interface. Variation of the differential scattered power with (a) the ML period, showing resvnant scattering at angles predicted by E.4. (45), (b) the angle of incidence, and (c) the number of ML periods. (d) Nonspecular scattering at A = 10 A from a ML having a period of A = 40 A, for two values of the layer-to-thickness ratio r. Four distinct orders of resonant scattering are labeled. fairly uniform with scattering angle. However, as L in- fields scattered from the interfaces, with a resulting pro- creases, the scattering becomes increasingly weighted in found effect on the nature of the scattered field. To illus- the specular direction (8 = 0'). Of course, in the limit trate this behavior we consider next the extreme case of L + 0~) , the ML interfaces become perfectly smooth and the complete correlation, where the replication factors are scattering is purely specular. The intensity of the scattering unity (&, = Zsi = 1). Furthermore, we choose the ML scales as L2 at small 8, consistent with the L dependence of interfaces to have no intrinsic roughness the power spectrum. ( aM, = asi = 0 A). This yields the condition of purely ex- The nonspecular scattering for different angles of inci- trinsic roughness considered in Sec. II C, where the surface dence ranging from f3e = V-40" is shown in Fig. 4(d) . The profile of the substrate is exactly reproduced at each ML interface parameters are LMo = Lsi = Lsub = 100 A and interface: or+&, = ffsi = aSUb = 2 A. It is evident that the scattering always peaks in the specular direction, due to the maxi- fySi(s) = i&,(S). (4) mum of the power spectrum at s = 0. The roughness of the substrate is assumed to be described statistically by Eqs. (36) and (37) with an autocorrelation 6. Correlated roughness length of &, = 100 A and rms roughness of o,,b = 2 A. An essential feature of any model for ML inter-facial The calculated nonspecular x-ray scattering from a roughness is the capability to account for the propagation MO-5 ML having completely correlated interfacial rough- of roughness through the ML stack. When propagation ness is presented in Fig. 5. The scattering predicted for x occurs, an interface is correlated with each of the underly- rays at normal incidence and a wavelength of/z = 130 A is ing interfaces. The correlation of the interface structure shown in Fig. 5 (a). The curves correspond to different provides a unique phase relationship between the radiation values of the ML period in the range of A = 65-80 A. In 4294 J. Appl. Phys., Vol. 71, No. 9, 1 May 1992 D. G. Stearns 4294 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 2am &z"~ f (47) qz=6nlA eff q,=4nlh where N,, is the number of ML periods over which the roughness is correlated. For completely correlated rough- qz=2nlA ness, Neff is simply the total number of ML periods, inde- pendent of frequency. In the case of partially correlated roughness, where the correlation decreases at higher fre- qY quencies s, the sheets of resonant scattering become broader and more diffuse with increasing distance from the qr axis. The x-ray scattering measurement at a particular wavelength /z interrogates points in reciprocal space that lie on a sphere of radius k = 2rrr//z having its center at the point q = - kir^ (the Ewald sphere). The intersection of FIG. 6. Geometrical construction depicting the location of resonant non- this sphere with the planes at qz = 2?rm/A are the set of specular x-ray scattering in reciprocal space. Resonant scattering occurs circles defined by Eq. (46), representing the peak in the wherever the sphere of radius k intersects the planes qr = Zrm/A, corre- sponding to the set of circles defined in J&J. (46). resonant scattering for a given experimental configuration (i.e., x-ray wavelength and incident angle). The angular position of the resonant scattering in real space is given by the direction of the circles with respect to the center of the each case the ratio I- of the MO layer thickness to the ML sphere. From this construction it is clear that, for a given period is kept at the constant value of 0.4. The scattering is ML period A, an infinite series of scattering resonances are characterized by a striking resonance surrounded by sec- available; the number of orders m of resonant scattering ondary oscillations. The angular position 0, of the peak of that are experimentally accessible is determined by the x- the resonance is seen to vary systematically with the ML ray wavelength (i.e., the radius of the Ewald sphere). The period. The origin of the resonance behavior can be under- width of the resonance is determined by the thickness of stood by noting that the ML structure has a series of re- the sheets as given by Eq. (47) and the trajectory of the ciprocal lattice vectors along the z direction with magni- scan in reciprocal space. Resonant nonspecular x-ray scat- tude 2n-m/A. The resonance in the nonspecular scattering tering of this type has been experimentally observed by occurs when the change in the momentum of the x ray several groups,`""' and some of the experimental issues are along the z direction matches a reciprocal lattice vector of discussed by Savage et ~1.`~ The calculations presented in the ML, such that this paper correspond to "28" scans, where the incident beam is fixed and the detector position is varied. (45) Further examples of resonant scattering from a ML of period A = 75 A are shown in Fig. 5 (b). The different where m = 1 and qr = k (cos 0, + cos 6JP). Equation (45) curves correspond to angles of incidence ranging from neglects refraction corrections that are usually small at 6, = V-40". Scattering resonances occur in both the for- x-ray wavelengths. This is identical to the Bragg condition ward and backward directions, which corresponds to view- for x-ray diffraction from a ML, except that Bragg scatter- ing a slice at qx = 0 through the m = 1 circle of resonant ing requires that qx = q,, = 0. The resonance behavior of scattering shown in Fig. 6. It is also apparent that the the nonspecular scattering can be considered as quasi- angular distribution of the scattering is weighted by the Bragg diffraction, where a finite momentum transfer in the uncorrelated scattering distribution from a single interface; x-y plane is permitted.t4 As a result, the introduction of the scattering is enhanced in the specular direction. correlated roughness to a ML structure has an interesting The scattering of normal incidence x rays from a ML effect in reciprocal space: The series of lattice points at of period A = 75 A having different numbers of layer pairs q = (2am/A)$ corresponding to scattering from an ideal is presented in Fig. 5(c). The enhancement of the reso- ML (i.e., having perfectly smooth interfaces) spread out nance behavior as the number of layer pairs increases is into diffuse sheets located at qr = 2rrm/A. Then, for a clearly evident. The intensity of the resonance peak scales given x-ray wave number k, the resonant nonspecular scat- as N2 (until saturation occurs due to extinction of the tering is described by a circle in reciprocal space according incident field) and the frequency of the secondary oscilla- to tions increases with the number of layers. This behavior is typical of coherent scattering from a periodic structure (s, + kr~,)~ + (s,, + knJ2rk2 - (27-rm/A + knJ2. (46) (e.g., diffraction from multiple slits). The peak of the res- onance corresponds to the condition where all of the fields The conditions for resonant scattering expressed by are in phase; the maxima in the secondary oscillations oc- Eq. (46) are illustrated in the geometrical construction cur at angles where the phase difference between the radi- presented in Fig. 6. Thecesonance scattering occurs along ation scattered from the top and bottom interfaces is an the sheets qz = 2rm/A. The sheets have finite thickness integral multiple of 2~. 6q, given approximately by Resonant scattering from correlated roughness can oc- 4295 J. Appl. Phys., Vol. 71, No. 9, 1 May 1992 D. G. Stearns 4295 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp cur whenever Eq. (46) is satisfied, resulting in a spectrum quency; the low-frequency components of roughness tend of harmonics associated with the different integer values of to propagate more effectively through the ML stack, while m. This behavior is illustrated in Fig. 5 (d), where we show the frequencies greater than - l/ fi are damped. Conse- calculations of the scattering of unpolarized x rays at an quently, the correlation of the interfacial roughness be- incident angle of 85" and a wavelength of ;1= 10 A from a tween the different interfaces is greatest for the lower fre- Mo-Si ML having N = 50 periods of A = 40 A. The quencies, and the x-ray scattering from the low-frequency atomic scattering factors used in the calculation are listed components of the roughness is expected to exhibit the in Table I. Four strong scattering resonances are clearly strongest resonant behavior. evident. The positions of the peaks are in good agreement With this particular model of roughness propagation, with Eq. (46) and the corresponding values of m are la- Eq. (14) becomes beled in the figure. The two curves represent different val- ues for the ratio of the MO layer thickness to the ML fyqs) = i&$(s) period: The solid and dashed curves correspond to values of I = 0.4 and 0.5, respectively. It is apparent that the ~yy(s). + ( 1 even orders are suppressed when I = 0.5. This is because l + 4~vMo,SitMo,SifMssIsi) the fields scattered from the Mo-on-Si and the Si-on-MO (50) interfaces in each layer pair are QT out of phase and hence destructively interfere. The same effect is observed for the The total of eight parameters, (TMo,si,sub, LMo,si,sub, and ditIi-action orders of a ruled grating, where the even orders YMo,si, in conjunction with the recursive relation (50), vanish when the line-to-space ratio is unity. completely define the structure of the interfaces in the ML stack. As an example, consider a ML$ucture in which the C. Partially correlated roughness layers have no intrinsic roughness [hMo,si(s) = 01. In this Although the cases of uncorrelated and completely case the roughness at each interface is due solely to the correlated interfacial roughness are of interest for showing replication of the roughness of the original substrate. If we the limiting behavior of the x-ray scattering, it is unlikely assume that all of the layers have the same thickness At that ML structures grown in the laboratory are well de- and value for the diffusion parameter v, then from Eq. scribed by either of these extremes. A more realistic model (50) the roughness of the ith interface can be written as for roughness propagation is the intermediate case of par- tial correlation, where the roughness of an interface in- Ij;l&(s). (51) cludes a partial replication of the structure of the underly- ing interfaces. In the model (14) for ML interfacial The inter-facial roughness damps exponentially with the roughness, the nature of the roughness propagation in the number of layers and the strength of the damping increases ML is dictated by the specific functional form of the rep- with increasing frequency. Taking the limit as At-0 and lication factors g(s). For the following calculations we use i- UJ yields an expression describing the roughening of a 1 continuous film as a function of thickness t, Z;(s) = 1 + 47f?vitisZ ' (48) T(s;t) = exp( -4??vt~)h,ub(s). (52) This form is chosen to be consistent with the linearized This is the solution of the differential equation J 18) de- Langevin equation ( 19) for surface roughness propagation scribing the diffusion of an initial perturbation h&,(d) in in the limit where the ML becomes a continuous film. In the case where the source term ij vanishes. particular, the Hankel transform of Eq. (48) is Calculations of nonspecular x-ray scattering from a 1 ML having partially correlated roughness (v = vMO ai(r) = -; K. zrrj, = Vsi) are presented in Fig. 7. The structure of the Mo-Si ML is similar to the previous example (A = 40 A, I = 0.4, where Kc(x) is the modified Bessel function having asymp- N = 20), as is the incident field (1 = 10 A, 19~ = 85"). Fig- toticvaluesof (-lnx) asx+Oand (x-1'2e-X) asx-+m. ure 7(a) shows the scattering for different values of the Equation (49) is the solution for isotropic diffusion from a diffusion parameter v in the case where the MO and Si point source in two dimensions, where the thickness of the layers have no intrinsic roughness ( (TM0 = osi = 0 A). In ith layer ti substitutes for the time variable and Vi is the this case, all of the interfacial roughness in the ML stack is diffusion coefficient. Hence for this specific choice of rep- due to replication of the substrate roughness characterized lication factor, the propagation of roughness behaves in a by ,&, = 100 A and a,,, = 1 A. The four scattering reso- manner analogous to surface diffusion: A spike introduced nances arising from the correlated roughness are clearly at the (i - 1)th interface propagates to the ith interface, evident. Increasing the diffusion parameter v causes the annealing into a surface feature having a radius of scattering intensity to decrease at all angles, as the inter- - &. Correspondingly, a surface fluctuation of fre- facial roughness is damped more effectively. The damping quency s is damped to one-half of its original amplitude is smallest at the scattering angles near the specular direc- when the overlayer has a thickness of ti = 1/(4?T2viS2). tion of 8 = 85", which correspond to low-frequency com- The replication of roughness is strongly dependent on fre- ponents (i.e., smaller values of s). 4296 J. Appl. Phys., Vol. 71, No. 9, 1 May 1992 D. G. Stearns 4296 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp G-9 ness from the PSD measured at any interface in the ML stack (e.g., the top surface). An example of the nonspecular x-ray scattering from a ML that includes intrinsic roughness is presented in Fig. 7(b). X-ray scattering (A = 10 A, 13, = 85") from the Mo- .,,.`i' `y `,`.hY.' Si ML considered previously (A = 40 A, r = 0.4, N = 20) `Y ,- ,I is calculated for the case in which the intrinsic roughness fi $\ ,' /' 0. ' ** . . :' of all the layers (and the substrate) is identical: LMO _I-- ,:' ;/ = Lsi = Lsub = 100 A, a~, = asi = U,,b = 1 A, and ,' ,' ./" vMO = Vsi. The curves in Fig. 7(b) correspond to different 10"' `. `;,/,,,.,. .A.' o(sub)=iA, L(sub)=l00A values of the diffusion parameter v. At small values of v the O(MMO = o(si)=oA .- HO h (85" incident angle scattering resonances due to correlated roughness are 1o.`7-1 . . . : * . * : * . * : . . . : . . . 1 clearly visible. However, the introduction of intrinsic 0 18 38 54 72 90 roughness entirely washes out the secondary oscillations. Scattering Angle 8 (deg) As v increases, the replication of roughness from layer to (W layer is damped, thereby diminishing the amount of corre- lation. Correspondingly, the intensity of the scattering res- onances decrease, and converge to a smooth background characteristic of the scattering from a ML having com- pletely uncorrelated interfacial roughness. The sensitivity of the resonance structure on the diffusion parameter var- ies with the order of the resonance. The highest orders are suppressed rapidly with increasing Y since these corre- spond to scattering from the higher spatial frequencies. IV. CONCLUSION c ASIOA, 850 incident angle The theory presented in this paper provides a general lo*- lo*- . . ...***: ...***:"`: "`:"`: "`:`" `" 0 0 18 18 38 38 54 54 72 72 framework for describing the nonspecular x-ray scattering Scattering Scattering Angle Angle 0 0 (deg) (deg) from interfacial roughness in ML structures, valid within the limitations imposed by the following simplifying as- sumptions: (i) The scattered field is weak and refraction FIG. 7. Calculations of nonspecular x-ray scattering (A = 10 A) from a can be neglected, (ii) multiple scattering of the nonspecu- Mo-Si ML (A = 40 A, N = 20, r = 0.4) having partially correlated lar field can be neglected, and (iii) the interfacial rough- roughness, for different values of the diffusion parameter Y = v,, = vsi. The differential scattered power is calculated for the cases of (a) no ness is slight as defined by condition (3). intrinsic roughness ( aMO = os, = 0 A) and (b) finite intrinsic roughness As in the case of x-ray diffraction from crystal lattices, ( u&=q,= 1 A,. the nonspecular scattering from roughness in a ML struc- ture is often sufficiently weak to justify the neglection of refraction and multiple scattering. Within this kinematic We next allow the ML interfaces to include a compo- approximation, the momentum transfer q of the scattered x nent of intrinsic roughness. In this case, the roughness of ray maps to a single point in reciprocal space, and the the underlying layers is continuously damped, while new intensity of the scattering at that point is directly related to random roughness is introduced at each layer. After a the power spectra of the interfacial roughness through Eq. number of layers has been deposited corresponding to a (26). The nonspecular x-ray scattering at measurable an- transient stage, the ML system converges to a steady-state gles derives from those frequency components of the inter- condition where the roughness of the interfaces is constant. facial roughness having spatial wavelengths in the vicinity If we assume for simplicity that each of the M = 2N inter- of il, the x-ray wavelength. The smallest spatial wavelength faces is characterized by-the same replication factor Z(s) that can be interrogated is -A/2, corresponding to back- and intrinsic roughness h(s), then it is straightforward to scattering. The largest spatial wavelength that contributes show that the power spectral density (PSD) of the inter- to the nonspecular scattering is the projection of the trans- facial roughness converges to verse coherence length of the x-ray beam onto the surface of the ML sample. Thus a specific frequency regime in the PSD= Is(s) I2 power spectrum of the interfacial roughness can be studied by choosing an appropriate wavelength and configuration o-%) = 11- ;;f CJ%) --) l _ 2(s) , as M--+c~, of the incident field. The theory predicts a wide range of interesting phe- (53) nomena, some aspects of which have been explored in the where (T and G(s) are the rms roughness and power spec- calculations presented in Sec. III. In particular, when trum associated with h(s). This relationship provides a roughness propagates through the ML, resulting in corre- direct method of determining the intrinsic interface rough- lation between the structure of the different interfaces, in- 4297 J. Appl. Phys., Vol. 71, No. 9, 1 May 1992 D. G. Stearns 4297 Downloaded 15 May 2003 to 148.6.178.88. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp terference in the x-ray scattering produces a characteristic per, Appl. 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