Recent Developments in the X-Ray Reflectivity Analysis for Rough Surfaces and Interfaces of Multilayered Thin Film Materials

X-rayrrflectometry is a powerful tool for investigations on rough surface and interface structuresof multilayered thin film materials. The X-ray reflectivity has been ca1cu lated based 011 the Parratt formalism, accounting for the effect of roughness by the theory of Nevot-Croceconventionally. However, in previousslUdies, theca1culalions ofthe X-ray rtflectivity often show a strange effect where interference effects would increase at a rough surface. And estimated surface and interface roughnesses from the X-ray reflectivity measurrments did not correspond to the TEM imagr observation results. The strange remit had its origin in a used equation due to a serious mistake in which the Fresnel transmission coefficient in the reflectivity equaticn is increased at a rough interface because of a lack of consideration of diffuse scattering. In this review, a new accurate formalism that corrects this mistake is presented. The new accurate formalism derives an accurate analysis of the X-ray reflecti\~ty from a multilayer surface of thin film materials, taking into account the effcrt of roughness-induced diffuse scattering. The calculated reflectivity by this accurate reflectivity equation should enable the structure of buried interfaces to be analyzed mOTe accurately,


Introduction
X-rays scattered from a material surface at a glancing angle of incidence provide a wealth of information on the structure of the surface layer of materials. X-ray scattering spectroscopy is a powerful tool for investigations on rough surface and interface Structures of multilayered th in film materials I l~ 23], and X-ray reflectometry is used for such investigations of various materials in many fields [14,15,20[. In many previous studies in X-ray reflectometry, th e X-ray reflectivity was calculated based on the Parrall formalism [I], coupled with the use of the theory of Nevot and Croce to include roughness [2). However, the calcu lated results of the X-ray reflectivity dOlle in this way often showed strange results where the amplitude of the oscillation due to the interference effects would increase for a rougher surface, Because the X-ray scattering vector in a specular reflectivity measurement is normal to the surface, it providesthe density profile solely in the direction perpendicular to surface. Specular reflectivity measurements can yield the magnitude of the average roughness perpendicular to surface and interfaces but cannot give information about the lateral extent of the roughness, In previous studies, the effect of roughness on the calculation of the X-ray reflectivity only took into account the effect of the density changes of the medium in a direction normal 10 the su rface and interface, On the other hand, diffuse scattering can provide information about the lateral extent of the roughness, In contrast to previous calculations of the X-ray reflectivity, in the present analYSiS, we consider the effect of a decrease in the intensity of pe netrated X-rays due to d iffuse scattering at a rough surface and rough interface.
In this review, we show that the strange result has its origin in a currently used equat ion due to a serious mistake in which the Fresnel tran~m i ssion coeffi cient in the reflectivity equation is increased at a rough interface, and the increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient because of a lack of consideration of diffuse scattering. The mistake in Neval and Croce's treatment originates in the fuct that the modified Fresnel coefficients were calculated based on the theory which contains the X-ray energy conservation rule at surface and interface. In their discussion, the transmission coefficients were replaced approximately by the renection coefficients by the ignoring diffuse scatteri ng term at the rough interface and according to the principle of conservation energy at the rough interface also. The errors oftransmitlance without the modificatio n cannot be ignored. It is meaningless to try to precisely match the numerical result based on a wrong calc ulating formula even to details of the reflectivity profile of the experimental result. Thus, because Nevot and Croce's treatment of the Parratt formalism contains a fundamenta l mistake regardless of tht' size of roughness, this approach needs to be corrected, In the present study, we present a new accurate formalism that corrects this mistake and thereby derive an accurate analysis ofthe X-ray reflectivity from a multilayer surface, taking into account the effect of roughness-induced diffuse scattering. The calculated reflectivity obtained by the use of this accurate reflectivity equation gives a physically reasonable result and should enable the structure of bu ried interfaces to be analyzed more accurately. This paper is the review article that is edited based on the two research articles of lOP Science [22, 23J and the later study.

Measurement for SUl'fa<:es and Interfaces of Multilayered Thin Film Materials
The surface and interfacial roughness of the same sample of multilayered thin film material was measured by transmission electron microscopy (TEM) and compared them with those from X-ray reflectivity me;lsurements. The surface sample for examination was prepared as follows; a GaAs layer was grown on Si(1lO) by molecular beam epitaxy (MB£). From TEM observations, the thickness of the GaAs layer was 48 nm, the root-mean-square (rms) roughness of the GaAssurface was about 2.8 nm, and the rillS roughness of the interface between GaAs and Si was about 0.7 nm. Figure 1 Journal of Materials 10' ., 10' , § shows a cross-section image of this GaAs/Si(lIO) sample observed by TEM. X-ray reflectivity measurements were performed using a Cu-Kal X-ray beam from an 18 kW rotat ing-anode source. Figure 2 shows the measured reflectivity of X-rays (wave length 0.154 nm) from a GaAs layer with a thickness of 48 nm on a silicon wafer. The dec rease in signal for angles larger than the total reflection critical angle shows oscillations. These oscillations are caused by interference between X-rays that reflect from the surface of GaAs layer and those that reflect from the interface of the GaAs layer and Si substrate. The characteristics of these oscillations re(lect the surface roughness and the interface roughness.

X-Ray Reflectivity Analysis
In Section 3.1, we consider the calculation of the X-ray reflectivity from a multilayer material by the Parralt formalism [l J, and in Section 3.2, the calculation of the X-ray retlectivity when roug h ness exists in the surface and the interface is considered.

X-ray Reflectivity from a MIJ/tilayer Material willi a Flat
Sur/ace and Flat Imerface. The intensity of X-rays propagating in the su rface layer; of a material, that is. the electric and magnetiC fields, can be obtained from Maxwell's equations [241. The effects of the material on the X-ray intensity are characterized by a complex refractive index 11 , which varies with depth. We divide a material in which the density changes continuously with depth into N layers with an index j. The complex refractive index of the jth layer is I'j" The vacuum is denoted as j = 0 and "0 = I. The thickness of the jth layer is il j , the thickness of the bottom layer being assumed to be infinite.
The reflectance of an N-layer multilayer system can be calcu lated using the recursive formalism given by Parran [I].
In the following, we show in detail the process of obtaining Parratt's expression and, further, show that this expression requires conservation of energy at the interface. We go on to show that the dispersion of the energy by interface roughness cannot be correctly accounted for Parratt's expression.
Following that approach, let IIj be the refractive index of Ihe jth layer. defined as (I) where 8 j and P j are the real and imaginary parts of the refractive index. These optical constants are related to the atomic scattering factor and eleclron density of the jth layer material.
For X-rays of wavelength;t, the optical constants of the jth larer material consisting of Nij atoms per unit volume can be expressed as (2) where re is the classical electron radius and iii and i2i are the real and imaginary parts of the atomic scattering factor of the ith element atom, respectively.
We take the vertical direction to the surface as the z-axis, with the positive direction pointing towards the bulk. The scattering plane is made the x-z-plane. The wave ve<:tor k j of the jth layer is related to the refractive index Ilj of the jth layer by and, as this necessitates that the x, y-direction components of the wave vector are constant, then the z-dire<:tion component of the wave vector of the jth layer is (4) In the Oth layer, that is, in vacuum, In the j lh layer, the components of the wave vector are The electric field of X-ray radiation at a glancing angle of incidence fJ is expressed as The incident radiation is usually decomposed into two geometries to simplify the analysis, one with the incident electric field E parallel to the plane of incidence (p-polarization) and one with E perpendicular to that plane (s-polarization ). An arbitrary incident wave can be represented in terms of these two polarizations. Thus, Eox and Eoz correspond to , , , k, p-polarization and Eo)" to s-polarization; those components of the amplitude's electric vector are expressed as The components of the wave vector of the incident X-rays are k Ox = kcosf}, koy = 0, ko: = ksin8. (9) The electric field of renected X-ray radiation of exit angle 8 is expressed as where (II) Because an X-ray is a transverse wave, the amplitude and the wave vector are orthogonal as follows: , , We consider the relation of the electric field Eo of X-rays incident at a nat surface from vacuum, the electric field EI of X-rays propagating in the first layer material, the electric field ~ of X-rays renected from the surface exit to vacuum, and the electric field E"; of X-rays propagating toward to the surface in the first layer material, as shown in Figure 3. The electric fields Ep E: in the first layer material below the surface are expressed as where kl,x = kcos8, k 1y = 0, k l .: = k~/Ji -cos 2 8. (14) 4 TIle relation of the amplitudes A I), A~, A I-and A') can be fou nd from the continuity eq uations of the electric fields for the interface bet\veen the Oth and llh layers as follo ws: , , Here, the Fresnel coefficient tensor $ for refraction at the interface between the Oth and Ith layers is given by

Journal of Materials
The Fresnel coefficient tensor "II for reflection from the interface between the Olh and lth layers is given by Here, we consider the reflection from a flat surface of a Single layer. The refl ection coefficient is defined as the rat io Ro.1 of the reflected electric field to the incident electric field at the surface of the material. The reflection coefficient Ro,I fro m a single-layer flat surface is equal to the Fresnel coefficientl.j'o.1 for reflection, as the fol lowing shows (20) In general, when X-rays that are linearly polarized at an For the reflectivity in the case of s-polarized X-rays inCident, Next, we consider the reflection from a fl at surface of a multilayer with flat interfaces. We consider the electric field E j _ 1 of X-rays propagating in the {jl)th layer material, the electric field E j of X-rays propagating in the jth layer material, and the electric field Ej_l of X-rays reflected from the jth layer material at Z = Zj_l.j of\he interface between the (jl)th layer and jth layers as shown in Figure 4.
The electric fields E j _ 1 , Ej_1 at the interface between the (j-I )th layer and j th layer and the electric fields E j , Ej below the interface betw'een the (j -J)th layer and jth layer are expressed as The electric fields of X-rays at the interface between the (jl)th layer and jth layer can be formally expressed as fo llows: where 'f' j _ 1.j is the Fresnel coefficient tensor for reflection from the interface between the j -1 and j layers, and lJ)j_l,j is the Fresnel coefficient te nsor for refraction at the interface between the j -I and j layers. In addition, the electric field with in the jth byer varies with depth II) as follows: The amplitudes A j and A~ at the jth layer are derived from the previous equations for the interface between the j -1 and j layers as follows: 6 Journal of Materials The Fresnel coefficient tensor '¥ for reflection from the interface between the j -1 and j layers is given by Here, the Fresnel coefficient tensor d> for refraction at the interface between the (j -1)1 hand j th layers is given by 2k)_ I ..: The amplitudes A j _ 1 and Aj_1 of the electric fields E j _ l • Then, the relations between the amplitudes A j -J> Aj_I' A j ' and Aj at the interface of the (j -J) th and jth layers are expressed as follows: The refl ection coefficient is defined as the ratio Ro.I of the reflected electric field to t he incident electric field at the surface of the material and is given by The reflection coefficient R j _ l . j of the elect ric field E; _1 to the electric fi eld E j _ 1 at the interface of (j -I)th layer and jth layer is (41) and the ratio R j _l,j is related to lhe ratio R j,j+ 1 as follows: Here, from the relation between the Fresnel coefficient for reflection and the Fresnel coefficient for refraction, We can fo rmulate the following relationship: (44) It is reasonable to ass ume that no wave will be reflected back from the substrate, so that Then, the X-ray reflectivi ty is simply

Previous CaiClliatiollS of X-Ray Reflectivity
WhCll Rough-lIess Exists at the Surface alld llllerface. When the surface and interface have roughness, the Fresnel coefficient for reflection is reduced by the roughness [8][9][10][11][12][13][14][15][16][17][18][19]. The effect of the roughness was previously put into the calculation based on the theory of Nevat and Croce [2] . The effect of such roughness was taken into account only through the effect of the changes in density of the medium in a vertical direction to the surface and interface. With the use o f relevant roughness parameters like the root-mean-square (n ns) roughness 0' j _I,j of the jth layer, the reduced Fresnel reflection coefficient 'l" for s·polarization is transfo rmed as fo llows: and the X-ray reflectivity is calculated using the following equation: ( 48) Figure 5 shows the result (dots) of a calculation based o n these expressions of the reflectivity of X-rays from a GaAs layer with a th ickness of 48 nm o n Si substrate. The rms rough ness of the interface of GaAs and Si was set to 0.7 nm, the value derived from the TEM observations. The rms roughness of the GaAs surface was set to 2.8nm, the value derived from the AFM measurements. The agreement of the calcu lated and experimental results in Figure 2 is not good . The calculated result suggests the following : if the value of the surface roughness and the interfacial roughness in the calcu lation would be made larger, the calculated result will more closely approach the experimental result. In the TEM observation and AFM measurements, on e half of the peak to peak value of the interface roughness equates to 1 nm, and that o f the GaAs surface is 4 n m. We then recalculated the reflectivity values of this order fo r the su rface roughness a nd the interface roughness in the calculation. Three calcu lated results for a roughness ofGaAs surface of 3.5 nm, 4 nm, and 4.5 nm, with an interface rough ness of I nm are shown in Figure 6.
A !though the calculated results did more closely approach those from experiment, they still showed poor ag reement. The ratio of the oscillation amplitude to the value of the reflectivity near an angle of incidence of 0.36· in the calculated reflectivity for theGaAs surface of 4 nm roughness in Figure 6 is larger than that of the reflectivity for a small roughness of 2.8 nm in Figu re 2, that is, near an angle of incidence of 0,36· interference effects appear to increase the reflectivity in the case of large roughness. [t seems very strange that interference effects would operate in this way. Figure 7 shows the reflectivity frolll a liaAs-covered silicon wafer, solid line shows the calculated result in the case of flat surface and flat interface, das hed line shows the calculated result in the case that the su rface has an rms roughness of 4 nm, and dOlled line shows the equivalent result when the surface and interface both have an rms roughness of 4 nm. In the latter case, the reflectivity curve (dots) decreases .q .:: more qUickly than that in Figure 3. However, the ratio of the oscillation amplitude to the value of the reflectivity does not decrease. It seems unnatural that the effect of interference does not also decrease at a rough surface and interface, because the amount of coherent X-rays should red uce due to d iffuse scatteri ng at a rough surface and interface. In the reflectivity curve (dashed line) for a surface roughness of 4 nm and with a flat interface, the ratio oftheoscillation amplitude to the size of the reflectivity near an angle of incidence of 0.36· is much larger than the reflectivity of the flat surface in Figure 3. It seems very strange that the interference effects would increase so much at a rough surface.
To probe these effects further, we then recalculated the reflectivity for surface roughnessof3.5 nm,4 nm, and 4.5 nm, and with a fl at interface. Those calculated reflectivity res u lts are shown in Figure 8. The ratio of the o scillation amplitude to the reflectivity near an angle of incidence of 0.36· in calculated reflectivity is larger in all cases than thaI of the reflectivity in the case of a flat surface in Figure 3.
For most angles of incidence within this range, the refl ectivityofthesurface with a ro ughness of 4 n m is near the mean value of the reflectivity of su rfaces with ro ughnesses of3.5 nm and 4.5 nm. However, near an angle of incidence of 0,36· , the reflectivity of the surface with a roughness of 4 nm is very much attenuated compared to that same average. [t seems very strange that the reflectivity o f the average roughness has a value quite different from the mean value of the reflectivity of each roughness, because the value of the roughness is not the value of the amplitude of a rough surface but the standard deviation value of vario us amplitudes of rough surface. Figure 9 shows the reflectivity from a tungsten-covered silicon wafer calculated by the theory in use prior to this work. The ratio of the oscillation amplitude to the value of the reflectivity from a surface with an rms su rface roughness of 0. incidence of 1.8· but increases. This result is strange and not reasonable.

Effect of Roughness 011 X-Ray Reflectivity of Multilayer
Surface. We now consider the previous Sl range result of the X-ray reflectivity wh ich was calculated based on the Parratt formalism [I J with the use of the Nevot and Croce approach to account for roughness [2]. In that calculation, the xray reflectivity is derived using the relation of the reflection coefficient R j-l.j and Rj,j +1 as follows:
However, the relationship between the reflection coefficients R j _l.j and Rj,j"l was originally derived as the following equation: (51) x exp(2ik j _ I ) l j _ I ).
Here, the following conditional relations between the Fresnel coefficient for reflection and refraction are relevant to the previous equation: , , 'fI j_ 1, j = -'¥j.j-I' then, that is, The Fresnel coefficients for refracl"ion at the rough interface are derived using the Fresnel reflection coefficient "I' as follows: , , lllj _ l .jlllj.j_l -lll j_l,jlll j,j_ l = '¥~,j-l (I -exp ( -2k j ,zk j _ I ,z(1}i_I)) > 0, Therefore, the Fresnel coefficients for refraction at the rough interface are necessarily larger than the Fresnel coefficient for refraction at the flat interface. The resulting increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient. These coefficients for refraction obviously comain a mistake because the penetration of X-rays should decrease at a rough interface because of diffuse scattering. We propose that the unnatu ral results in the previous calculation o f the X-ray reflectivity originate from the fact that diffuse scattering was not conSidered. In fac t (52) contains the X-ray energy conservation rule at the interface as the following identity equation for the Fresnel coefficient: Here, we consider the energy flow of the X-ray. In electromagnetic radiation, E, H, the energy fJow is equal to the Poynting vecto r where (60) and £ and p. are the dielectric and magnetic permea bility. The Poynting vector is therefore Then, the Poynting vector that crosses the interface is The amplitudes A j _ 1 and Aj_l of the electric fields Ei-I> E~_ l at the jth layer and amplitudes A j and A j of the electric fields E j , Ej at the (j + l)th layer are rela ted by the following equations: That is, the X-ray energy flow is conserved at the interface. When the Fresnel coefficients at the rough interface obeys the following equations, these coefficients fulfil X-ray energy flow conservation at the interface, and so d iffuse scattering was not considered at the rough interface. This conservation expression should not apply any longer when the Fresnel reflection coefficient is replaced by the reduced coefficient 0/ when there is roughening at the interface. Therefore, calculating the reflectivity using th is reduced Fresnel re fl ection coefficient 'f' in (50) will incorrec tl y increase the Fre;nel transmission coefficient $ '; that is, <D < <D'.
The penetration of X-ra ys should decrease at a rough interface because of diff'use scatteri ng. Therefore, the identity equation for the Fresnel coefficients become where D2 is a decrease due to d iffuse scattering. Then, in the calculation of X-ray reflectivity when there is roughening at the sur face or the interface, the Fresnel transmission Journal of Malerials coefficient $' should be used for the reduced coefficien t. Several theories exist to describe the influe nce of roughness on X-ray scattering [8][9][10][11][12][13][14][15][16][17][18][19]. When the surface and interface are both rough, the Fresnel coefficient for refraction has been derived in several theories [14 -19 ].  However. no one obtained the expression corresponding to <1>: .0' It is peculiar that <1>;.0 and <1>~. 1 are asymmetrical. It comes to cause a different result if Hh layer and Oth layers are replaced and calculated . Therefore this derived <1>' should not be used to calculate the reflectivity from rough surfaces and interfaces.

TIw Refractive Fresnel Coefficiellt of a Rouglllllterface
The derived Fresnel refraction coefficients <1>' increase. This increase in the transmission coefficient completely over· powers any d ecrease in the value of the reflection coefficient as the following: Moreover, if the deformat ion modulus of <1>;.0 is assumed to be <1>~, 1' the left side of (80) exceeds unity and therefore (78) is obviously wrong.
In Nevot and Croce's treatment of the Parratt formalism for the reflectivity calculation induding surface and inter· face roughness [2), the relations of the Fresnel coefficients between reflection and transmission as (52), (68), and (80) were not shown. Furthermore, the modification of the Fresnel coefficients according to Nevot and Croce has been used for only surface and interface reflection. However, the modification of the transmission coefficients has an important role when the roughness of the surface or interface is high, and the effect of diffuse scattering due to that roughness shou ld not be ignored, as shown in (69). The error in Nevol and Croce's treatment [2J originates in the fac t that the modified Fresnel coefficients was calculated based on the Parrall formalism which contains the X-ray energy conservation rule at the su rface and interface. In the d iscussion on pp.767-768 of Nevot and Croce's [2), their Fresnel coefficients at the rough interface fulfil X-ray energy flow conservation at the interface, and so diffuse scattering was ignored at the rough interface. In their discus.ion, the transmission coefficients tR and 1/ were replaced approximately by the reflection coefficients 'R and '/ by the ignoring diffuse scat· tering term, and according to the principle of conservation energy. The reflection coefficient 'R at the rough interface should be expressed as a function of the reflection coefficient 'I and transmission coefficient 'I. However, the reflection coefficient 'R at the rough interface was expressed only by the reflection coefficient 'I' while the transmission coefficient 'I had already been replaced by the reflection coefficient TI by the ignoring diffuse scattering term in the relationship based on the principle of the conservation of energy. Th us, the reflection coefficient' R at the rough interface as equation (I I) of p,77I in Nevot and Croce [2J had been expressed with the reflection coefficient '/ only, and this results in the equat ion was also sure to include the conservation of energy.
The resulting increase in the transmission coefficient completely overpowers any decrease in the value of the 13 reflection coefficient at the rough interface, Thus, because Nevot and Croce's treatment of the Parralt formalism can · tains a funda mental mistake regardless of the size of the roughness, results using this approach canl10l be correct. The size of the modification of the transmission coefficient is one-order smaller than that of reflection coefficient, but the size of transmission coefficient is one-order larger than the reflection coefficient at angles larger than critical angle. Thus, the errors of transmittance without the modification cannot be ignored.
Of course, there are cases where that Nevot and Croce's treatment can be applied. However, their method can be applied only to the case where there is no density d istribution change at all in the direction parallel to the surface on the surface field side, and only when the scattering vector is normal to the surface. A typical example of surface medium to which this model can be applied is one where o nly the denSity distribution change in the vertical direction to the surface exists, as caused by diffusion, and so forth. In such a speCial case, Nevot and Croce's treatment can be applied without any problem. However, because a general multilayer film always has structure in a direction parallel to the surface field side, Nevot and Croce's expression fails even when the roughness is extremely small. The use of only Fresnel reflection coefficients by Nevol and Croce is a fun damental mistake that does not depend on the size of the roughness.

71re
Refractive Fresliel Coefficient of a RDllgh IlIte'face Used ill New Reflectivity CalculatiolJs. To proceed, we therefore reconsider the derivation of the average value of the matrix as the same derivation of (70) when we consider the reduced Fresnel coeffi cient, which is known as the Croce-Nevot factor.
When the z'position of the interface of the Oth layer a nd 11h layer ZO.l fl uctuates vertically as a function of the lateral position because of the interface roughness, the relations between the electric fields are derived by the use of the Fresnel coefficient tensor <b for refraction and the Fresnel coefficient tensor'¥ for reflection as follows:   [20 J. TIle Fresnel refraction coefficients <1>' derived by this method are reduced and could be used to calculate the reflectivity from rough surface and interfaces. Acco rdingly, we calculated the reflectivity using these derived Fresnel refraction coefficients. However, the numerical results of this calculation did not agree with the experimental results when the angle of incidence smaller than the total reflection critical angle. In trying to account for the reason for th is disagreement, it should be noticed that our present approach to constructing the reduced reflection coefficient 'l'~ I term does not include any reference to the refractive index of the medium. Further, X-rays that penetrate an interface reflect from the interface below, and penetrate the former interface again without fail. Therefore, the refraction coefficient ¢~.l and 4>; ,0 should not be separately treated.

A New Formula for the Reflectivity for Rough Multilayer
Because X-rays that penetrate an interface reflec t from the interface below and penetrate former interface again without fail, it is necessary to treat the refraction coefficients <D~.1 and <1>;,0 collectively: The Fresnel refraction coefficients cP' derived by this method are reduced and can be used to calculate the reflectivity from rough surface and interface. Therefore, we calcu late the reflectivity using these newly derived Fresnel coefficient s in an accurate reflectivity equation of R j _ l . j and Rj.j-t l as follows: Based on the previous considerations. we again calculated the X-ray reflectivity for the GaAsJSi system but now considered the effect of attenuation in the refracted X-rays by diffuse scattering resulting from surface roughness. The results are shown as the dashed line in Figure JO for a surface roughness of 4 nm and flat interface. and the dotted line shows the calculated result in the case that the surface and interface both have an rms roughness of 4 nm. The ratio of the oscillation amplitude to the size of the reflectivity in the reflectivity curve (dot) in Figure \0 is smaller than that of the reflectivity curve Figure 7. In the reflectivity curve (dashed line), the ,'ery large amplitude of the oscillation near an angle of incidence of 0.36· in Figure 7 has disappeared in Figure 10. These results are now physicalJy reasonable. All the strange results seen in Figure 7 have disappeared in Figure 10. It seems natural that the effect of interference does decrease at a rough su rface and interface, because the amount of coherent X-rays should reduce due to diffuse scattering. Figure  the surface roughness of 4 nm is near the mean value of the reflectivity of the surface roughness of 3.5 nm and the reflectivity of the su rface roughness of 4.5nm. This resu lt is physically reasonable, because the value of the roughness is the standard deviation value of various amplitudes of rough surface. However, it was difficult to match the numerical result of X~ray refl ectivity to the results of TEM observation. Next, we again calculated the X-ray reflectivity for the WJSi system but now considered the effect of attenuation in the refracted X rays by diffuse scattering resulting from surface roughness. However, the reduced refraction coefficiellt in prior work varies [13-19[. Then about the reduced refraction coefficient, reduction as same as reflection coefficient was applied now. Figure 12 shows the calculated results with the use of improved X-ray reflectivity formalism. In the reflectivity curve from a surface with an rms surface roughness of 0.3 nlll (dashed line), the amplitude of the oscillation in Figure 9 has reduced in Figure 12. These results are now phYSically reasonable. The strange results seen in Figure 9 have disa ppeared in Figure 12. It seems natural that the effect of interference does decrease at a rough surface and interface, because the amount of coherent X rays should reduce due to diffuse scatteri ng.

Summary
In this review, we investigated the fact that the calcu lated result of the X-ray reflectivity based on Parrat! formal ism [I] with the effect of the roughness incorporated by the theory of Nevot-Croce [2] shows a strange phenomenon in which the amplitude of the oscillation due to the interference effects increases in the case of the rougher surface. The X~ray reflectivity calculation based on Parratt formalism [I] with the effect of the roughness incorporated by the theory of Nevot-Croce [2] shows as in (48), with the reduced Fresnel reflection coefficient ' 1" being as shown in (47). However, the relationship between the reflection coefficients R j _t ,j and Rri+ l was originally de rived as in (51) . Here, the fo llowing conditional relations between the Fresnel coefficient for reflection and refraction are relevant to (51); see (52) and (53). In these condition. the Fresnel coefficients for refraction at the rough in terface are derived using the Fresnel reflection coefficient,+, as shown in (57). Therefore, the Fresnel coefficients for refraction at the rough interface are necessarily larger than the Fresnel coefficient for refraction at the fl at interface. The resulting increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient. These coeffici ents for refraction obviously contain a mistake because the penetration of X-rays should decrease at a rough interface because of diffuse scattering. We propose that the unnatural results in the previous calculation of the X-ray reflectivity originate from the fact that diffuse scattering was not considered. We found that the strange result originates in the currently used equation due to a serious mistake where the Fresnel refraction coefficient in the reflectivity equation is increased at a rough interface. The increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient because of a lack of consideration of diffuse scattering. The mistake in Nevot and Croce's treatment originates in the fact that the modified Fresnel coefficients were calculated based on the theory, which contains the X-ray energy conservation rule at the surface and interface. In their discussion, the transmission coefficient was replaced by the reflection coefficient so as to conserve energy, and so diffuse scattering was ignored at the rough interface. It is meaningless 10 try to precisely match the numerical result based on a wrong calculating fo rmula even to details of the reflectivity profile of the experimental result. Thus, because Nevot and Croce's treatment of the Parratt formalism Journal of Materials contains a fundamental mistake regardless of the size of roughness, results based on this approach are not correct.
We have develo ped a new fo rmalism that corrects th is mistake, producing more acc urate estimates of the X-ray reflectivity for systems having surface and interfacial roughness, takin g into account the effect of roughness-induced diffuse scattering.
The new, accurate formalism is completely described in detail. The X-ray reflectivity R of a multilayer thin film material consisti ng of N layers is derived by the use of accurate reflectivity equations for R j _ Lj and R j .]+ l as the following: Here, the refractive index of the jth layer IIj = I -OJi/3], "0 = 1, the z-direction component of the wave vector of the jth layer k j • z = k~IIJcos 2 0, k = 2rr/).,).; wave length, 8; glancing angle of incidence, a N-Iayer multilayer system with a jth layer of thickness of I I] and j -I, jth interface roughness of a i-1 .i' k ],z is the z component of the wave vector in the jlh layer, and 'P j _ L · and <1>] -1.] are the Fresnel coefficients for reflection and retraction, respectively, at the interface between the (j -1)lh layer and the jth layer. Although formula for '+'j -1.] is well known where 0 j _1,] is the interface roughness between (j -I)th a nd jth layers, an accurate analytical formula fo r ¢l j -l .] indud ing the effect of the interface roughness is not available. 111ere are several approximations proposed so far and all these resu lts can be written as (107) where parameters C l , C 2 de pend on the proposed approximation. C 1 = 2 and C 2 = 0 is the most appropriate approximation [23]. The penetration of X-rays should decrease at a rough interface because of diffuse scattering. Therefore, the identity equation for the Fresnel coefficients become <' h' <' h' I ,,,I "'j-l.j"'j,j-l -'t'j _1.j T j,j_l I I 12 = <l>j_l./I>j,j_l + '+'j -l,j = 1 _ D 2 < 1, where D2 is a decrease due to diffuse scattering. Then, in the calculation of X-ray reflectivity when there is roughening at Ihe surface or the interface, the Fresnel transmission coefficient cI>' should be used for the reduced coefficient.
Figu re 13 shows the reflectivity from a tungsten-covered silicon wafer with an rms surface roughness of 0.3 nm. Dashed line shows the calculated result b)' (48) based on Parratt formalism with the effect of the roughness incorporated by the theory of Nevol-Croce in use prior to this work. The ratio of the oscillation amplitude to the value of the reflectivity from a su rface with an rms surface roughness ofO.3 nm does not decrease near an angle of incidence of 1.8· but increases than the reflectivity from a flat surface in Figure 9, This result is strange and not reasonable. Next, we again calculated the Xray reflectivity for the W/Si system, but now considered the effect of attetluation in the refracted X rays and the reduction in the sum intensity of reflective X-ray and refract ive xray by diffuse scattering. Solid line shows the calculated results with the use of improved X-ray reflectivity tormaJis/ll, In the reflectivity curve, the amplitude of the oscillation is smaller'than that of the reflectivity from a flat surface in Figure 12. These results are now phYSically reasonable. The st range results seen in the previous calculation have disappeared. It seems natura l that the effect of interference does decrease at a rough surface and interface, because the 19 amount of coherent X rays should reduce due to diffuse scattering. The reflectivity calculated with this new, accurate formalism (l05) gives a physically reasonable result. The use of this equation resolves the strange numerical results that occurred in the previous calculations that neglected diffuse scattering and is expected that buried interface structure can now be analyzed more accurately.