Offspecular reflection from flat interfacesWenli Wu Citation: The Journal of Chemical Physics 101, 4198 (1994); doi: 10.1063/1.468464 View online: http://dx.doi.org/10.1063/1.468464 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/101/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A comparative study of Langmuir surfactant films: Grazing incidence x-ray off-specular scattering vs. x-rayspecular reflectivity J. Appl. Phys. 110, 102213 (2011); 10.1063/1.3661980 Time-resolved specular and off-specular neutron reflectivity measurements on deuterated polystyrene andpoly(vinyl methyl ether) blend thin films during dewetting process J. Chem. Phys. 131, 104907 (2009); 10.1063/1.3224125 Lateral length scales of latent image roughness as determined by off-specular neutron reflectivity Appl. Phys. Lett. 92, 064106 (2008); 10.1063/1.2841663 OffSpecular Xray and Neutron Reflectometry for the Structural Characterization of Buried Interfaces AIP Conf. Proc. 931, 185 (2007); 10.1063/1.2799367 Offspecular reflection from flat interfaces with density or compositional fluctuations J. Chem. Phys. 98, 1687 (1993); 10.1063/1.464284

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Off-specular reflection from flat interfaces Wen-Ii Wu Polymers Division, Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899

(Received 11 March 1994; accepted 9 May 1994)

The reflection of x. ray or neutron from a flat interface with density or compositional fluctuations was studied. An integral equation relating the reflection intensities to the interfacial fluctuations was derived in the Fraunhofer limit. At the specular condition this integral equation, although derived for the off-specular intensities, has an expression identical to a well-established one for the specular component. This integral equation was applied to a flat surface onto which gold spheres of uniform size were randomly adhered. The out-of-plane components for both the real wave and the time reversed wave were approximated by the solution from the specular component. Based on the calculated off-specular intensities near the specular region, an approximation scheme was proposed to estimate the in-plane correlation length.

INTRODUCTION

A variety of theoretical work has been developed to ad­dress the off-specular reflection from rough surfaces; meth­ods based on Born approximation, 1,2 the distorted wave Born approximation (DWBA) of Sinha et al. 2 or of pynn,3 the use of Green function4 or the method of Nevot and Croce5- 8

have been recently applied to this problem. The Green func­tion method and the DWBA yield virtually identical results. Applications of these theoretical results to liquid surfaces,9,lO polymer surfaces with well-defined domain structurell and thin soap film12 have also been reported. In this work the problem of reflection from interfaces with lateral composi­tional or density fluctuations was treated. In the particular example to be given in the later part of this work, both sur­face roughness and compositional fluctuations were intro­duced by depositing gold spheres on a silicon wafer. The resultant Yoneda peak position deviated considerably from that of the bare wafer. This observation is not expected from the DWBA or the Green function approaches if the bare wa­fer surface is chosen as the reference surface. This change in Yoneda peak position will be addressed in the present work.

In a previous workl3 the problem of reflection from flat interfaces with compositional or density fluctuations was treated. That treatment was limited to the two-dimensional problems applicable to most neutron and x-ray reflectivity measurements where slit collimations were used. In this work we extend the treatment to three-dimensional cases since the x-ray intensity from a synchrotron source is often intense enough for the measurement of off-specular intensi­ties with pin hole collimations. In such an arrangement the detector can often be off the plane of incidence defined by the incident beam and the normal of the interface. Although the algorithm used in this work was identical to the one used previously, attempts were made here to provide a clear defi­nition of the off-specular reflectivity. With the off-specular reflectivity clearly defined, we showed that under specular conditions the integral equation expression for the off­specular components was identical to the well-established results for specular reflectivity. 14

In the second part of this work, this integral equation was applied to a flat surface onto which spheres of uniform

J. Chern. Phys. 101 (5), 1 September 1994

size are attached. Random distribution of the spheres on the surface was assumed and the otf-specular intensities were calculated based on the following approximation; the wave function derived from the specular calculation was chosen as the z component for both the time reversed wave and the real wave. Based on this wave function approximation and an­other assumption that the potential is separable into a product of an in-plane component and an out-of-plane component,I3 a scheme was proposed to extract the in-plane correlation length. At low surface coverage of the gold spheres, the in­plane correlation length is just the diameter of the spheres. Therefore, comparison can be made between the estimated in-plane correlation length and the true one-the sphere di­ameter.

Some preliminary x-ray off-specular results from 260 A diameter gold spheres on silicon wafer are presented at the end of this work, the validity of various approximation schemes employed in this work can then be examined.

THEORY

The geometry of the system to be considered is given in Fig. 1, the interface is located at z=O, the space above is filled with medium I and the space below with medium 2. The wave vector k i of the incident beam propagates in the x-z plane with a grazing angle fJ i defined in its usual way as illustrated in the above figure. The detector is located along the scattering wave vector ks defined by a grazing angle Os and an azimuthal angle 4> •• The first objective of this work is to obtain an expression to relate the scattering amplitude fl (ki ,ks ) measured at a distance remote from the origin or the interface. The subscript 1 denotes the fact that ks is within medium 1. The derivation undertaken is limited to the situation where the distance r between the source (or the detector) and the illuminated area is much greater than the linear dimension l of the illuminated area. Furthermore, the quantity 1 is much greater than the wavelength A. This is a typical Fraunhofer condition. As in the previous work, the nomenclature of neutron reflectivity is adopted throughout the derivation, however, the results are applicable to x ray as welL We began our discussion with the time-independent Schrodinger equation

4198

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Wen-Ii Wu: Reflection from flat interfaces 4199

z

x

FIG. 1. Schematic illustration of the incident beam, the refleCted beam and the related angles.

2 2m _ - V if!+J;l [V(x,y,Z)- EJif!-O. (1)

if! stands for the wave function, m is the mass of a neutron, n is the Planck constant divided by 21T, E is the neutron kinetic energy in vacuum, and V(x,y,z) is the potential causing re­flection or scattering. All structural information is contained in the potential V(x,y ,z). We further assumed that the struc­ture remote from the interface was uniform for both medi­ums 1 and 2. In other words, the value ofV(x,y,z) reached an asymptotic, limit VI as z~O and reached V2 as z«O.

The potential of a neutron within a medium is defined as an average of the interaction between the neutron and the nuclei over a unit scattering volume, more explicitly

21Tn2 V= 2: -- hi, (2)

i m

where b i is the coherent scattering length of nucleus i. The magnitude of the neutron wave vector k i in potential Vi is defined as

(2m) 112 k[= ,h2 .~ .. [E- Vi(x,z)] 112. (3)

In the above two equatIons the subscript i labels a nucleus in the scattering volume. The asymptotic values of k in media 1 and 2 are denoted as k I and k2' respectively. At large dis~ tances from the interface, or r-';oo, the wave function qr in medium 1 can be approximated as ""

fl (k;.ks ) 'k if!1 = <Pi+-,- e' Ir, (4)

where the first term is simply the incident beam of unit in­tensity along vector k i and illuminates an area n on the interface. It is noteworthy that the incident beam <Pi is a pencil beam. fl (ki ,ks) represents the scattering length with incident angle OJ and the detector direction specified by angles Os and <Ps, the subscript 1 denotes that the incident and scattered rays are in medium 1. The factor 11 r in the second term arises from the geometrical considerations for the three-dimensional cases treated in this work. The" second

term of Eq. (4) is denoted as if!r where the subscript r stands for reflection. Within medium 2 at large distances from the illuminated area the wave function if! is simply

(5)

Ot refers to the grazing angle of the transmitted wave. The objective of this work is to obtain an expression for fl (k/ ,ks); its value at k;";;ks is related to, but not equal to, the specular reflectivity, and at Os* 0i it is the off-specular scattering length. It is noteworthy that both Eqs. (4) and (5) are approximations valid only as..r approaches infinity.

The difference between fl (ki=ks) and the reflectivity R, although minor in substance, is imperative to clarify before we can proceed further in the' derivation. Reflectivity R c;an best be defined for an interface without any in-plane fluctua­tions which includes a cutoff boundary, i.e., for a plane of infinite extent. However, both Eqs. (4) and (5), therefore all the fl (k; ,ks)' are defined in the Fraunhofer limit. To connect R with f 1 (ki = k s ) an aperture function A (ks ) is needed to relate a quantity defined in the Fraunhofer limit with that defined for an infinite plane. 15 In the Fraunhofer limit the amplitude, F(s,r) at r distance away from the source, defined as the origin, can be related to the intensity distribution at the source by the following equation:

__ ikl ;k1r (kl ) F(s,r) - 21Tr e far s,O . (6)

In the present case the source can be considered as the illu­minated area n at the origin. The parameter s within F{s,r) specifies the location on the spherical surface at a distance r from the origin. The function fa(k 1sir,0) is the Fourier transfotm of the function F at z = O. It is noteworthy that the position specified by the coordinate (s,r) is located along the scattering vector ks . One can relate the Fourier transform of the reflected amplitude along ks at the origin with the reflec­tivity and the aperture function as follows:

(7)

where the aperture function A (k .. ) is the Fourier transform of the illuminated area n. For the present case in which the linear dimension I of the area a is much greater than the wavelength l\., the aperture function can be approximated by a Kronecker delta function; therefore one has

fa(~ s,o) =R(k;,ks)A. (7')

By substituting Eq. (7') into Eq. (6) and comparing the result with the second term in Eq. (4) it becomes evident that

(8)

From the definition of an aperture function one has

(9)

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4200 Wen-Ii Wu: Reflection from flat interfaces

where z is the unit vector along the z axis and a is the mag­nitude of the illuminated area.

In the rest of the derivation, the scheme used in a previ­ous publication was followed closely. A virtual potential Vo(z) was introduced and its asymptotic values at z~ ±oo were identical to those of the real potential V(x,y,z). Noting that the virtual potential Vo depends only on z, i.e., no in­plane fluctuation was allowed. The equality of the asymp­totic values between the virtual and the real potential was the only restriction on the choice of Vo(z). Consider an incident beam of unit intensity directed along - ks interacting with this virtual potential. The wavelength of this virtual or time reversed beam was set identical to that of ki . Following the reasoning behind Eqs. (4) and (5), the virtual wave function "" has the following expressions: .

- - 11(-ks ,k)'k - -"'I=</J+ r e'lr=</J+"'n (10)

and

(11)

The above wave function lfJ is also a solution of the Schro­dinger equation

2 - 2m -- V "'+V (Vo- E)",=O. (12)

By multiplying Eq. (1) with lfJ and Eq. (12) with "'ins, the difference between these two products results in

_ _ 2m _

",V 2 "'- ",V

2 "'= T (V - Yo) ",,,,. (l3)

The left-hand side of Eq. (l3) can be rearranged as

_ _ 2m _

V . ( ",V "'- ",V",) = T (V - Yo) "''''. (14)

A volume integration of both sides of the above equation yields

J V·( ~V "'- ",V ~)du

= J ¥; (V- Vo)"'~ du. (14')

Based on Green's theorem, the left side integral can be con­verted into a surface integral, and one has

J 2m -

= V (V-' Vo)"'''' du. (15)

The subscript n denotes the components normal to the inte­gration surface. This surface, for convenience, is chosen to be a sphere of radius r centered on the illuminated area. Furthermore, the magnitude of r is chosen to be sufficiently large such that Eqs. (4)-(7) are valid approximations for '" and ;Po By substituting Eqs. (4)-(7) into the left side integral

of Eq. (15), the result is zero on the integration surface within medium 2. On the surface within medium 1, one has

(114) cos 8rs II </J cos 8ri ) 'k ] + e' l

r ds r r '

(16)

where 8rs and ()ri denote the angles between vector r and ks' r and k i , respectively. To obtain the above expression, all the terms with 11 r2 were neglected because the surface integra­tion was conducted at large r. The above integration can be simplified significantly by recognizing that </J exists only near k; and ;p exists only near ks' i.e., the concept of a wavepacket is implemented. By substituting Eq. (8) into the above integral it becomes

(17)

where () zi and () zs denote the angles between the z axis, ki and ks' respectively.

Referred to the definition of the grazing angles given in Fig. 1, we have cos 8zi =sin ()i and cos 0zs=sin Os' Equation (17) can then be written as

(18)

Now let us recall the definition of R introduced in Eq. (7); R is the ratio of the amplitUde of the reflected wave along ks and the incoming wave along ki . If one wants to compare the above definition of R to the one conventionally used in scattering, the following modification becomes necessary. In a scattering measurement, one measures the ratio of intensi­ties of the scattered beam and the incident beam. Note that the beam intensity is a product of the square of the amplitude and the beam cross sectional area, consequently a simple geometric factor ~sin 8/sin Oiis needed for converting RI to a conventional scattering quantity R,

- _ ~sin 0i R=R J -'--n'

SIn Us

(19)

The quantity R2 is then the intensity ratio of the reflected beam and the incident beam. By substituting the above equa­tion into Eq. (18), one has

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Wen-Ii Wu: Reflection from flat interfaces 4201

V(z>J T. V\ o ~ i L. ______ _

I I I I I I I I

o£11----, , o

FIG. 2. The xy-averaged potential of spheres adhered on a flat surface with a potential Vs (upper curve) and the potential of the silicon wafer alone (lower curve).

_ 1

R(ki,ks)=R(-ks,-ki ) 2'k ~. e . II l la SIll i SIll Us

x f (V- Vo)l/I~ du. (20)

This expression has the symmetry between k i and ks . By letting the virtual potential Vo equ~l to the real one V, the above equation becomes R(ki ,ks ) = R( - k i , - ks)' i.e., the op­tical reciprocal principle is recovered.

The difference between the above equation and the one previously obtained by the authorl3 is that the previous one is expressed in terms of reflectivity and the current one is in terms of the ratio of the intensity of the scattered beam and that of the incident beam.

In the rest of this article, Eq. (20) will be applied to a model system composed of gold spheres of a uniform size adhered on a flat surface. In all the calculations the surface coverage of the spheres is kept at 3%; no in-plane intersphere correlation is expected at such a low surface coverage. Therefore, the in-plane correlation length is proportional to the diameter of the spheres.

MONODISPERSE SPHERES ON A FLAT SURFACE

The off-specular intensities were calculated using Eq. (20). The incident beam was from the vacuum side and was directed onto the spheres. We further restricted the calcula­tion to the x-z plane where the surface normal was parallel to the z axis. The choice of the virtual potential Vo is the z-averaged potential profile V(z) shown in Fig. 2. By defini­tion

V= Lfy V(x,y,z)dx dy,

and is routinely deduced from specular reflectivity measure­ments. If the exact solution of the wave function 1/1 is used as the input in Eq. (20), the choice of virtual potential, includ­ing its position and its shape, is entirely arbitrary; it should not have any effect on the calculated reflectivity results. However, since the exact form of 1/1 is the unknown, an ap­proximation is the only choice to be used in Eq. (20). Con­sequently, the virtual potential should be chosen as close to

the real potential as possible. By letting Vo= V(z), the cor­responding wave function for the virtual beam is

~=eiklXl/lz' (21)

I/Iz represents the numeric solution of the wave function based on the z-average potential shown in Fig. 2. In practice its value can be deduced from the specular reflectivity data. Conversely, for any given potential profile such as the one in Fig. 2, its corresponding I/Iz can be calculated easily using the matrix method. 14 The product in the above equation was also used as the wave function 1/1 near the interface for Eq. (20). The choice of V(z) instead of a step function as the virtual potential is the major difference between this work and that of Sinha et al. 2 The advantage of our choice will be addressed later.

The procedure used to carry out the integration [Eq. (20)] can be summarized as follows. For any given sp!Iere size and number of spheres per unit area, the profile V(z) was calculated. Thereafter the matrix method was applied to obtain I/Iz. Since the xy averaged V(x,y,z) was chosen as Vo, the term V - Vo consists of two parts; one is the summa­tion of the potentials of individual spheres and the other is Vs - V(z). Vs stands for the potential of the flat substrate, hence the term Vs - V(z) does not depend on either x or y. Therefore, one can neglect this term without affecting the calculated off-specular intensities. That is to say, V - Vo = 2: Uk, the perturbation potential is a sum of all the spheres each with potential v k' The integral given in Eq. (20) can then be expressed as

f f f (v- Vo)~1/1 dx dy dz

= L J J J u k~1/I dx dy dz k

(22)

The integral in Eq. (22) can be further simplified by recog­nizing that for each sphere k, this integration can be sepa­rated as a product of a part related to the center coordinate (X k , Y k) of sphere k on the x - y plane and a part for a single sphere with its center located at (0,0)

z

x

FlG. 3. A potential profile where the separation of potential, 8 V = V(x,y) V(z), is applicable.

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4202 Wen-Ii Wu: Reflection from flat interfaces

=[ ~ I I eik[(cOS IIscf>.+cos 1I;)Xke ik[ cos lis sin cf>"Yk dx dY ]

X [I I J v(O,O)( eik[(cos lis cos cf>s+cos lIi)Xeik[ cos II. sin 4>sY) if!z(/Jz dx dy dZ] ==M·R. (23)

The first integral, denoted as M, is the sum of the Jwo­dimensional fourier transform of the center positions of all the spheres within the illuminated area; the second term, de­noted as B, is the three-dimensional integration of a single sphere with its center located at (O,O,dI2) while d is the diameter of the sphere. The· numeric calculation of B is straightforward. For M a ,<;imple formula to approximate the intersphere correlation of points with a hard core repulsive diameter d is used,

N

N 2'lT (24)

1+- "k2 (1- cos okxyd) a u xy

where N denotes the total number of spheres on the scatter­ing area n and okxy denotes the in-plane component of ks - k i . The off-specular intensity is then simply

(25)

From a practical point of view, approximation schemes for deducing these quantities important for characterizing the in-plane fluctuations from a combination of the specular and off-specular measurements are highly desirable. Toward this end, the following scheme was devised as the first step for estimating the lateral correlation length.

An additional assumption is now introduced, namely, the perturbation potential V-V 0 can be separated as

V- Vo= V(x,y)V'(z). (26)

As stated earlier, V-VO=LVko the quantity V'(z) is now approximated by V(z) - Vs' Based on this additional ap­proximation, the integral given in Eq. (23) can be simplified as

I I I eiok·~eiOkyY(V_ Vo)if!z;frz dx dy dz

= I I e-iokxXe-iOkyYV(x,y)dx dy

(27)

In the above equation Z, by definition, stands for the second integral and it can be calculated readily. All the information required for calculating Z can be obtained from specular measurements. The quantity L, the first integral of the above equation, holds all the in-plane information. Its values can only be reduced from a combination of the specular and the

off-specular data. Separation of the perturbation potential V - V 0 == A V to a product of two terms is legitimate in certain cases; for example, on a flat surface there exist islands of the same height and with steep banks. A two-dimensional illus­tration of such a case is given in Fig. 3 where the separation is strictly correct, i.e., Ll V= V(x,y) V(z), where V(z)=O ex­cept for Zj <Z<Z2 in which case V(z) = 1. For the present case of spheres of uniform size, the separation of the poten­tial differences can, at best, be regarded as an approximation.

For gold colloid spheres, the potential is 6.613 X 10-3

A -2 in terms of Q~ which is defined as the critical Q below which total internal reflection takes place, Q has its usual definition as 47T/"A. sin O. For a silicon wafer the correspond­ing value is l.04X 10-3 A -2. For any given sphere size and surface coverage, the integrals M· Band Z defined in Eqs. (23) and (27), respectively, can be calculated readily. Figure 4 shows the results of 1M. B 12 and Z2 as a function of graz­ing incident angle 0i in simulating a transverse scan with the angle between the incident beam and the detector fixed at

~.------.-------'-------r------.

-4

·10

-12L-____ ~ ______ ~ ______ ~ ______ ~

0.0 0.5 1.0 1.5 2.0

Grazing Incident Angle 0

FIG. 4. A calculated transverse scan using Eq. (23) (lower curve) and the Z2 term using Eq. (27). Gold spheres of 260 A diameter adhered on a'silicon wafer constituted the model system. The -specular condition is located at 0.960

• The ordinate is scattering intensity in arbitrary units.

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Wen-Ii Wu: Reflection from flat interfaces 4203

350

300

.g 250 Q) 200 N

(7)

~ 150 ttl ~ 100 (J)

w 50

0

-500

o

o

o

o o o

20 40 60 80 100 120 140 160 Particle Diameter (A)

FIG. 5. The estimated lateral correlation length from the calculation _of 1M BJ2~ZI2==ILi2 vs the diameter of the spheres used in the simulation.

1.92°. In the above example we have 500 spheres with 260 A diameter randomly distributed on a 8.8 X 103 X 105 A 2 sur­face. We further let 8ky= 0 and 8kx , by definition, is related to the grazing incident angle (Ji as 8kx =(2'Tr1 A)[cos(200-0i)-cos OJ and 00 was chosen to be 0.96°. The similarity between 1MB 12 and Z2 is striking; 1MB 12 is the calculated off-specular intensity and the only approxima­tion invoked is the wave function form [Eq. (21)], while Z2 is calculated based-on an additional approximation regarding the separation of the perturbation potential [Eq. (26)]. This observation suggests that the origin of maxima and other features observed in a rocking (transverse) scan is not nec­essarily the lateral correlation of the surface feature. In a transverse scan over an angular range of 1.92° the in-plane Q component sweeps through a domain about two orders of magnitude greater than that of the out-of-plane component. In other words, the scan shown in Fig. 4 is almost a pure in-plane scan. However, the calculated Z2 reveals the same type of feature as that from a more precise calculation where the only approximation involved is the form for the wave functions [Eq. (21)]. The origin of the maxima in Fig. 4 is mostly from the out-of-plane structure. According to Eq. (27), the lateral correlation can surely affect transverse scan results, but not all the features observed in a transverse scan can be attributed to lateral correlations. To extract lateral correlation from a transverse scan is the next topic to be addressed.

Based on Eqs. (23) and (27), we have IMBI2/Izj2=ILI2. ILf by definition is just the Fourier transform of lateral cor­relation. In the present case 8ky = 0 and L depends only on 8kx ' Near the region of 8kx approaching zero, which is in the region close to the specular condition or 8i -+ 80 , a Guinier-type approximation, namely,

ILI2_L~( 1-~2 8k;+"')

is expected to hold true. Calculated results of IMBI2/Z2 or ILI2 in the low 8q x region were fitted as a linear function of 8k;, and the results are given in Fig. 5. The abscissa is the sphere diameter used in the calculation and the ordinate is the fitted result based on the Guinier approximation for ILI2. There exists a relation between the input sphere diameter and

101 ~_l~~"-'--~~~-,,---,-~~

a ~ ~ ~ ~ 1~ 1M 1S 1_

8j (degrees).

FIG. 6. X-ray off-specular results of 260 A diameter gold spheres on silicon wafer. The amount of surface coverage is zero for the bottom curve and 3%, 7.5%, and 15%, respectively, for. the top three curves in the order of increas­ing reflected intensities. All the transverse scans were collected with a fixed angle of 1.92° between the incident beam and the detector.

the calculated one based on 1MB 121 Z2 . However, the relation is not a linear one, deviation from a linear relation increases as the sphere diameter gets larger. Nevertheless, the above scheme demonstrates some limited success in extracting lat­eral correlation from transverse scans. In cases where the surface roughness is of the type shown in Fig. 3, a linear relation is expected. It is noteworthy that in terms of experi­mental data 1MB 12 is the observed off-specular scattering intensities, and Z2, defined explicitly in Eq. (27), can be calculated using specular reflectivity data alone.

In the rest of this manuscript, we will report some pre­liminary results of x-ray off-specular measurements con­ducted on samples with monodisperse gold spheres adhered on silicon single crystal wafers. A thin film of 150 A thick­ness of poly(2-virlylpyridine) (PVP) was spin coated on a 7 cm (111) silicon wafer. Small (d=260 A) gold spheres were deposited onto the PVP coated silicon wafer by adsorption from an aqueous colloidal gold suspension. The sample preparation procedure used by Kunz et at. 15 was adopted without any changes. The rocking curves with detector fixed at 1.92° with respect to the incident beam were obtained (Fig. 6). The bottom curve is from the PVP coated silicon surface without any gold spheres, the results from 3%, 7.5%, and 15% surface coverages are also included as the top three curves. The off-specular intensities seem to increase linearly with the amount of gold coverage. The position of the Yoneda peak of the gold covered surfaces was significantly higher than 0.22°, that of the silicon. This observation of a shift in the Yoneda peak toward a high angle is duplicated by our calculation (Fig. 4). This shift of Yoneda peak in our calculation can be attributed to our choice of V(z) as the virtual potential. If Vs ' the potential of the underlying sili­con, had been chosen as the virtual potential, the calculated Yoneda peak would remain at 0.22°. The experimental con­dition is almost duplicated in the theoretical results of Fig. 4 except (1) line broadening due to a finite instrument resolu­tion is not included in the calculation, (2) the specular com-

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4204 Wen-Ii Wu: Reflection from flat interfaces

ponent, namely, R of Eq. (20) is not included in the calcula­tion. Consequently, the calculated result exhibits more pronounced maxima and minima in comparison with the data (Fig. 4 vs Fig. 6). Nevertheless, all the features in the experi­mental data are reproduced qualitatively in the calculation of IMBI2 vs (Ji (lower curve in Fig. 4) as well as in Z2. This observation demonstrates th~t the origin of the peaks in transverse scans or rocking curves is not necessarily the in­plane correlation. At least in the present case, monodisperse spheres on a flat surface, the Z component seems to be re­sponsible for all the peaks.

CONCLUSIONS

An integral equation was derived in the Fraunhofer limit for the off-specular reflectivity from a flat surface with in­plane structural fluctuations. Both the wave function I{I and the time reversed one, '(p, were modeled as a product of o/z and I{Ixy where I{Iz was taken from the results deduced from specular measurements, and I{IXY remained unperturbed from that of the incident wave. If one further assumed that the perturbation difference V(x,y,z) - Vo(z) could be expressed as a product of V(x,y)V'(z), the off-specular reflectivity itself can be factored into a product of the in-plane compo­nent and the out-of-plane component.

By invoking the first approximation, transverse scans of a model system-monodisperse spheres on a flat surface, were simulated. The results are in qualitative agreement with the x-ray data. By invoking both the first and the second approximations the transverse scans of the same model sys­tem were obtained. The result indicated that the peaks ob­served in the transverse scans could mostly be. attributed to the out-of-plane component. The formula developed herein for the off-specular reflectivity suggests that the in-plane cor­relation can also affect the transverse scans. However, the example given in this work provides strong evidence sug­gesting that not all the features observed in a transverse scan can be attributed to in-plane correlation. To remove the out-

of-plane contribution from transverse scans and to reduce the in-plane structure information, a scheme invoking the above two approximations and a Guinier type approximation was devised. To examine the validity of the above scheme, IMBI2/Z2 was calculated using computer simulated data and the lateral correlation length was then estimated. This simu­lated results using the model system of monodisperse gold spheres on silicon demonstrated the feasibility of estimating the lateral correlation using a combination of the specular and the off-specular data.

ACKNOWLEDGMENTS

The author is grateful for the support from the Office of Naval Research under a Grant No. N00014-92-F-0036. The author is also indebted to Dr. J. H. van Zanten for all the x-ray measurements.

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