Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154
An experimental study of the q>-dependence of X-ray resonant
diffuse scattering from multilayers
V.A. Chernova, V.I. Kondratievb, N.V. Kovalenkob, S.V. Mytnichenkoc,*
a Siberian SR Centre, Budker Institute of Nuclear Physics, 11 Lavrentyev Ave., 630090 Novosibirsk, Russia
b Budker Institute of Nuclear Physics, 11 Lavrentyev Ave., 630090 Novosibirsk, Russia
c Institute of Solid State Chemistry, 18 Kutateladze Str., 630128 Novosibirsk, Russia
Abstract
A study of X-ray resonant diffuse scattering from the W/Si multilayer was performed to examine its dependence on
the momentum transfer normally to the specular diffraction plane, q>. The data obtained show two evident
disagreements with the present theoretical approximations. Firstly, when the incident angle was approximately equal
to the Bragg angle, additional scattering concentrated in the specular diffraction plane was observed. Secondly, the
q>-dependence of the quasi-Bragg scattering intensity obtained from these experiments is not the same, at least at the
small momentum transfer, as can be obtained from the scans in the specular diffraction plane, having tendency
to accumulate near this plane. The possible reasons for these phenomena are discussed. r 2001 Elsevier Science B.V.
All rights reserved.
PACS: 68.55.@a; 61.10.Kw
Keywords: Multilayers; X-ray diffuse scattering
1. Introduction interfaces. Thus, in order to understand and control
physical behavior of multilayers, it is essential to
Thin films composed of synthetically-grown be able to determine the detailed structure of
multilayer structures represent a new class of layers and interfaces and to correlate this structure
materials having novel optical, electric, magnetic, with the measured physical properties.
and superconducting properties for a host of A promising technique for characterizing the
important applications. Since the properties of a roughness of surfaces and interfaces in multilayer
multilayer principally differ from those of bulk structures is X-ray diffuse scattering. This struc-
materials, it is not a surprise that these properties tural method has advantages that prove its
are often highly sensitive to the nature of layer usefulness. It is a non-invasive technique, well
suited for dynamic measurements, including in situ
growth studies. Large depth penetration of X-rays
*Corresponding author. Siberian SR Centre, Budker Insti- provides bulk structural information, including
tute of Nuclear Physics, 11 Lavrentyev Ave., 630090 Novosi- the cross-correlation effects. Structural informa-
birsk, Russia. Tel.: +7-3832-394013; fax: +7-3832-394163.
E-mail address: s.v.mytnichenko@inp.nsk.su tion can be obtained over a wide spatial range,
(S.V. Mytnichenko). from macroscopic (0.1100 mm) down to atomic
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 0 3 1 - 2
146 V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154
dimensions. The diffuse scattering data are aver- A standard diffuse scattering experiment repre-
aged over the sample area, which makes this sents a set of one-dimensional scans in the specular
technique complementary, for example, for atomic diffraction plane, with that another degree of
force microscopy. integration in the direction normal to it (see
In spite of these advantages, diffuse scattering as Fig. 1). It may be noted that the intensity
a technique for investigation of a multilayer distribution in this direction is automatically
interfacial structure is not widely used now. (In assumed to be independent of the incident and
this paper, only the case when the total integrated scattering angles. A typical image of a Bragg
diffuse scattering intensity is much smaller than the reflection fine structure obtained by such experi-
specular reflection one will be discussed. This ments is schematically shown in Fig. 2. The
condition is equivalent to soL=n, where s is the following distinguishing features of resonant
roughness dispersion, L is the multilayer period, diffuse scattering can be observed:
and n is Bragg order [1].) The reason is both the The incoming and outgoing Bragg-enhanced
experimental and data interpretation difficulties. diffuse scattering manifests itself as a resonant
Indeed, in contrast to specular diffraction, X-ray amplification of diffuse scattering in the case of
diffuse scattering occurs at any direction including y
the specular one due to the violation of multilayer 0EyB or y1EyB š1Ž
lateral translation symmetry by the roughness where yB is the Bragg angle. This dynamic
imperfections. At the same time, only a combina- phenomenon was observed experimentally by
tion of the specular and off-specular scattering several authors [1,2] and can be explained within
measurements provides full structural information the framework of the distorted-wave Born approx-
on the surfaces and interfaces. Thus, even though imation [35]. According to this approximation, in
multilayer interfacial roughness anisotropy is the case of complete correlation between rough-
unavailable, it is necessary to obtain a diffraction ness of different layers (cross-correlation) [6], the
space map instead of a standard one-dimensional contribution from this scattering to the total
scan. Roughness anisotropy of this sort will cause scattering amplitude has the form
the technique sophistication to further increase. AincomingBR0FšDcŽ and AoutgoingBR1F*šDcŽ
For a variety of reasons, diffuse scattering under-
goes resonant amplification and is brought into the š2Ž
relatively small angle regions near the specular where R0 and R1 are the specular reflection
Bragg reflections forming its setting. amplitude at the incident angles y0 and y1,
Fig. 1. The diffuse scattering geometry: y0 and y1 are the incident and diffracted angles, respectively; o ¼ y0-y1; f is the azimuthal
angle; qz is the momentum transfer normal to the lateral planes; q8 is the momentum transfer parallel to the lateral planes and specular
diffraction plane; q> is the momentum transfer parallel to the lateral planes and normal to the specular diffraction plane.
V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154 147
Fig. 2. The fine structure of the Bragg reflection from a multilayer as may be observed in any standard diffuse scattering experiment.
Here the sets of transverse scans (o-scans) are parallel to the ordinate axis, whereas the total diffraction angles are plotted along the
vertical axis.
respectively; Dc ¼ c0@c1; c is the phase shift of diffuse scattering, quasi-Bragg scattering is not
the wave as it passed through one multilayer caused by dynamic effects but only by diffraction
period; FšxŽ ¼ š1@eiNxŽ=š1@eixŽ is the usual of X-rays from the interfacial imperfections
diffraction function and N is the number of coherently repeating from one layer to another
multilayer periods. If the roughness cross-correla- (the interfacial cross-correlation) [6,7]. Indeed,
tion is absent, the mechanism of Bragg-enhanced due to the translation symmetry normally to
diffuse scattering is analogous to appearance of the multilayer surface, the momentum transfer
fluorescent Kossel lines. It should be noted that projection on this axis must be kept with an
not only the resonant amplification lines can be accuracy of 2p=L, where L is the translation
observed, but also the breaks in a diffuse scattering symmetry or multilayer period. It makes possible
background [1]. The smaller is the roughness cross to immediately obtain the quasi-Bragg condition
correlation, the better is to observe this effect, y
caused by the specular diffraction standing wave. 0 ž y1En2yB: š5Ž
Apart from condition (1), the scattering ampli- This condition means that the scattering occurs at
tudes A the total scattering angle n2y
incoming and Aoutgoing are subjected to B beyond any incident
resonant amplification if angle. The condition accuracy is determined by the
multilayer rocking curve.
y0Ey1 or DcE0: š3Ž The origin of quasi-Bragg scattering can be
Quasi-specular diffuse scattering is caused by this explained by a more illustrative method [8].
amplification. Condition (5) is neither more nor less than the
Besides, the additional term with the resonant reflection condition of diffraction from a grating
condition that is not coincidental. Fig. 3 demonstrates that
y quasi-Bragg diffuse scattering is caused by diffrac-
0Ey1EyB š4Ž tion from a grating placed normally to the
provides diffuse scattering near the Bragg point, interfaces and composed by interfacial imperfec-
forming the ``Bragg nimbus''. tions coherently repeating from one layer to
Finally, quasi-Bragg scattering was theoretically another. The multilayer and grating periods are
predicted in 1988 [7]. In contrast to other types of evidently the same.
148 V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154
Fig. 3. The nature of quasi-Bragg diffuse scattering.
These properties of quasi-Bragg diffuse scattering of roughness imperfections which do not obey
significantly simplify its theoretical descriptions, condition (6) is omitted from the consideration.
especially, when the incident angle is not equal to Nevertheless, it is reasonable to assume existence
the Bragg angle. In this case the nature of quasi- of another type of imperfections that are unlimited
Bragg scattering is clearly kinematic, and its in some lateral directions. The case of linear
intensity distribution versus the momentum trans- defects due to various physical and technological
fer is directly proportional to the power spectral reasons is of special interest. The diffuse scattering
density of interfacial roughness omitting the cross- from these roughness imperfections will be con-
correlation effects. Under these conditions, the centrated near the specular diffraction plane,
q>-dependence of the quasi-Bragg diffuse scatter- which is easy to show. In other words, in this case
ing reflects directly the roughness power spectral the diffuse scattering behavior in the specular
density as well as the q8-dependence [9]. diffraction plane will not strongly differ from the
All theories of resonant diffuse scattering based theoretically predicted one, but at the same time
on the distorted-wave Born or kinematic approx- the q>-dependence of resonant diffuse scattering
imations predict the same q>-dependence inde- will alternate, having the tendency to accumulate
pendently of the diffraction nature (kinematic or near the specular diffraction plane.
dynamic) and type of diffuse scattering, if the Besides, interfacial roughness was shown by
interfacial roughness is assumed to be isotropic in the atomic-force microscopy [11] to have not one
the lateral planes. Moreover, as it was mentioned but a few characteristic lengths in various spatial
above, q>- and q8-dependences must be equivalent ranges. The roughness imperfections of different
in the kinematic approximation. However, from spatial ranges are caused by different physical rea-
our point of view, the latter effect is not a universal sons and must have different correlation functions.
physical law and is caused by the use of the self- In the present work an experimental study of the
affine roughness model by Sinha et al. [10] with q>-dependence of resonant diffuse scattering was
heightheight self-correlation function of the form performed.
CšrŽ ¼ ozšr0Žzšr0 ž rŽ >¼ s2expš@šr=xŽ2hŽ š6Ž
where s is the rms roughness, x is the correlation 2. Experimental
length, and h is a coefficient connected with the
fractal dimension D¼ 32h. Indeed, the choosing The W/Si multilayer was deposited by magne-
of correlation function (6) means automatically tron sputtering on a flat silica wafer with a surface
that the roughness imperfections have a point roughness of 0.30.5 nm. The number of bilayers
nature, i.e. are spatially restricted in all lateral was 200. A least-squares fitting of the experi-
directions (if r-N then CšrŽ-0). So, a great class mental specular reflectivity data was performed to
V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154 149
calculate the multilayer optical parameters using and outgoing Bragg-enhanced diffuse scattering.
Parrat's recursive dynamical method [12]. Below According to the distorted-wave Born approxima-
are the multilayer parameters obtained. The tion as well as other theoretical calculations [14],
multilayer period, L, was 1.47 nm. The W-rich these intensities must be symmetrical relative to the
layer comprised approximately 0.5 of this period. specular vertical line, omitting the trivial geometry
The roughness parameter, s, was found to be factor. Moreover, this symmetry seems to be stated
equal to 0.6 nm. (This value reflects both macro- by the optics reciprocity theorem. Indeed, it seems
roughness and presence of mixed layers.) that the intensities must be equal if the infinitely
The X-ray diffuse scattering measurements were distant source and detector are replaced. Never-
performed using SR of the VEPP-3 storage ring of theless, from our point of view, the optics
the Siberian SR Centre at Budker INP, which reciprocity theorem cannot be true in the case of
operates at 2 GeV and with a maximum stored statistical systems, since this theorem is a direct
current of 165 mA. A triple-axis diffractometer consequence of symmetry of Maxwell's equations
(Fig. 4a) with a primary channel-cut single-crystal relative to the time reverse. Statistical averaging
Si(1 1 1) monochromator and a Ge(1 1 1) crystal- breaks this symmetry. Another important condi-
collimator of the ``anomalous scattering'' station tion is the fact that the incident beam does not
were used [13]. The measured angular broadening represent the ideal plane monochromatic wave but
of the diffractometer had a full-width at half- a wave packet having limited space sizes.
maximum (FWHM) of 1500. A scintillation detec- ``To be sure that the discussed phenomenon is
tor based on an FEU-130 photomultiplier with a not an experimental artifact we tried to replace the
NaI(Tl) scintillator was used. The dynamic range crystal-collimator with vertical slits and to use
of the detector system was about 5 104. To additional collimators, with no any success. It is
increase the dynamic range of the measurements necessary to mark that the anomalous high
upto about 107, calibrated copper fails were used intensity of incoming Bragg enhanced diffuse
to alternate the incident beam. In order to perform scattering was observed in Ref. [15], where the
the q>-experiments the additional horizontal slits, small-angle Bragg reflections from a semiconduc-
providing an azimuthal angle resolution about tor superlattice were studied. The experimental
0.051, were used. scheme of this work principally differs from the
Besides, for the study of quasi-Bragg scattering, scheme discussed here. In our previous work
an experimental scheme with an image plate was devoted to the study of diffuse scattering from
used (Fig. 4b). Ni/C multilayer mirrors at annealing [16] we also
The measurements were performed at the have observed this effect. The last study can be a
wavelength l ¼ 0:154 nm. conclusive proof that the discussed phenomenon is
not a sequence of the experimental setup. During
the annealing of a mirror in this experiment, the
3. Bragg-enhanced diffuse scattering incoming Bragg-enhanced diffuse scattering line
disappeared though the peak reflectivity of the
The diffraction space map of the first Bragg mirror stayed quite high (1030%). It is interest-
reflection obtained by the conventional method is ing that the degradation of incoming Bragg-
shown in Fig. 5a: the horizontal stripis the quasi- enhanced diffuse scattering was accompanied by
Bragg scattering, the inclined strips are the the mosaic-like spread of specular reflection. Also
incoming and outgoing Bragg-enhanced diffuse we have studied the effect asymmetry of incoming
scattering. Such a striprich pattern is caused by and outgoing Bragg-enhanced diffuse scattering
the fact that every Kiessig modulation has its own from various multilayers and superlattices. As a
quasi-Bragg sheet and Bragg-enhanced diffuse result the following rules were revealed: 1. This
scattering strips. effect is very important if the float glass or silica
The main difference between the maps in Fig. 5a wafers are used, whereas in cases when single
and 2 is the intensity asymmetry of the incoming crystals were used as substrates this effect is
150 V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154
Fig. 4. The experimental setup: (a) the conventional triple-axis geometry with additional horizontal slits; (b) the experimental scheme
with an image plate; (c) the slit geometry.
V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154 151
experimental condition difference between the data
in Figs. 5a and b is that the specular diffraction
plane was cut off by the horizontal slits in the latter
case. As it is well seen from Fig. 5, the incoming
Bragg-enhanced strip disappears, which means
that the major part of this scattering occurs in
the specular diffraction plane. The total disap-
pearance of the incoming and outgoing Bragg-
enhanced diffuse scattering strips can be expl-
ained by the limited dynamic range.
Our possible explanation of this phenomenon is
the distortions of the coherent wave front caused
by the macroscopic roughness imperfections with
large length (flatness). Indeed, the diffracted
X-rays do not ``sense'' these imperfections if their
lengths are more greater than the Bragg extinc-
tion length (a few mm). From this point of view,
the multilayer structure is ideal. Nevertheless, the
distortions of the coherent wave front cause the
off-specular scattering. In this case, the role of
the Bragg diffraction is very simple: there is a
reflection if the incident angle is equal to the Bragg
one, otherwise, there is no reflection.
Such diffuse scattering mechanism makes the
directions in and normal to the specular diffraction
plane non-equivalent. A small incident angle
results in different sizes of the irradiated sample
square in these directions. If the size of coherent
beam was about 2 mm in the direction normal to
the specular diffraction plane, then in the specular
diffraction plane the size of the coherently-irra-
diated sample square increased up to the value
about a few mm. The result is that the diffuse
scattering is located in the specular diffraction
plane. The suggested interpretation explains well
Fig. 5. The diffraction space maps of the first order Bragg the dependence of the asymmetry degree of the
reflection: (a) the conventional geometry; (b) the diffuse incoming and outgoing Bragg-enhanced diffuse
scattering in the specular diffraction plane was cut off by the
slits. scattering on the incident angle. Indeed, the greater
the incident angle, the finer is the effect discussed.
remarkably smaller. 2. The asymmetry degree is A situation paradox must be noted. If the
more clear if the incident angle is smaller, for incident wave would be completely plane and
example, this asymmetry is very fine near the coherent, then the incoming Bragg-enhanced
Bragg reflection satellites from a superlattice in the diffuse scattering would not be concentrated in
high-angle range. 3. The typical intensity depen- the specular diffraction plane. Nevertheless, under
dence of incoming Bragg-enhanced diffuse scatter- real conditions, if the transverse coherent length is
ing on standing wave localization is invisible.'' greater, then this effect can be observed clearly.
Another diffraction space map of the same It is interesting to mark that the incoming
Bragg reflection is shown in Fig. 5b. The main and outgoing Bragg-enhanced strips disappear
152 V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154
completely due to the insufficient dynamic range of same behavior as in the case of quasi-Bragg
the detector system, whereas the quasi-specular scattering.
scattering remains to be visible.
As an example of the importance of the
transverse coherent length, a diffraction space 4. Quasi-Bragg scattering
mapof the same sample and Bragg reflection
obtained at another experimental setup is shown in The diffraction space map from the W/Si
Fig. 6. In this case the additional horizontal slits multilayer mirror obtained by the use of an
were removed, and the secondary crystal-collima- image plate is presented in Fig. 7. The upper spot
tor was replaced by secondary vertical slits; in so in the figure is the specular reflection, and the
doing its thickness (100 mm) was six times smaller central halo is the quasi-Bragg scattering sheet.
than the thickness of the primary vertical slits In Fig. 8 the quasi-Bragg scattering intensity
(600 mm). It should be noted that this ``square versus the q>-momentum transfer obtained from
grid'' view can be observed near the Bragg refle- this experiment is compared to the quasi-Bragg
ction only. Away from the Bragg reflections, the intensity versus the momentum transfer in the
scattering profiles become usual. specular diffraction plane, q8. The latter data were
As for the q>-dependence of the outgoing obtained using usual scans in the specular diffrac-
Bragg-enhanced diffuse scattering, which is very tion plane. As in Ref. [9], at high azimuthal angles,
slightly visible in Fig. 5a, it was verified to have the the quasi-Bragg scattering intensity is in good
Fig. 6. The diffraction space map obtained by the use of the slit experimental setup (Fig. 4c).
V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154 153
Fig. 7. The diffraction space map of diffuse scattering from the W/Si multilayer obtained with the use of an image plate: the incident
beam sizes are 100 100 mm, the distance from the scattering point to the image plate is 520 mm, f is the azimuthal angle, 2y and 2yB
are the total and double Bragg angles, respectively.
agreement with the theory with heightheight self-
correlation function (6). Nevertheless, additional
diffuse scattering near the specular diffraction plane
is well observed. A similar effect was observed in
Ref. [9]. This feature was called ``resolution
dominated region'' in this work. In order to
remove any doubts, the scattering profiles of the
quasi-Bragg scattering and specular reflection,
whose width is determined by the angle resolution,
are shown in Fig. 9. This Figure demonstrates well
that the angle resolution is sufficiently better than
the width of the quasi-Bragg scattering profile.
Thus, the data obtained support the existence of
the roughness imperfections that are unlimited in
Fig. 8. The diffuse scattering intensities versus the momentum some lateral directions. Moreover, the scattering
transfer: the solid curve is normal to the specular diffraction plane,
q from these imperfections can dominate as well
>; the dashed curve is in the specular diffraction plane, with
integration over f, q8. The logarithmic intensity scale was used. observed from Fig. 9.
154 V.A. Chernov et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 145154
Fig. 9. The f-profiles of the quasi-Bragg scattering (points) and specular reflection (solid curve) were shown using a linear scale. The
width of specular reflection demonstrates the experimental angle resolution.
5. Conclusion References
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