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J. Phys. Chem. B 2006, 110, 15166-15171
µ-GISAXS Experiment and Simulation of a Highly Ordered Model Monolayer of PMMA-beads Andreas Fro1 msdorf* and Richard C ˇ apek Institut fu¨r Physikalische Chemie, UniVersita¨t Hamburg, Grindelallee 117, D-20146 Hamburg, Germany
Stephan Volkher Roth HASYLAB / DESY, Notkestr. 86, D-22603 Hamburg, Germany ReceiVed: April 7, 2006; In Final Form: June 21, 2006
Uniform sized PMMA-beads were deposited as a monolayer on silicon substrates using dip-coating techniques. High-resolution grazing incidence X-ray small angle scattering experiments were performed using a micrometer sized beam (µ-GISAXS) to determine the structure of a highly ordered monolayer with two-dimensional hexagonal arrays. A clear and strong interference pattern coming from the reflection and refraction effects of particles on flat surfaces with small uncorrelated roughnesses is shown. The quantitative analysis and simulations of the X-ray scattering pattern have been performed, and a detailed explanation of the analysis is reported. The results were directly compared and verified with atomic force microscopy (AFM) measurements and their resulting FFT spectra.
Introduction The structure analysis of deposited monolayers has become more and more important overthe years due to increasing interest in the field of nanotechnology and surface science. Direct imaging techniques are often used to characterize the surface morphology and the lateral spacings of such structures. The limitations of this method are the very small observable area and the inability to determine buried and inner structures of such layers. Here, the small-angle X-ray scattering under grazing incident angles (GISAXS) is a very powerful tool. With this method, one is able to determine sample surface structure as well as inner electron density fluctuations of the deposited material. But there is still a need for the complete understanding of the obtained scattering pattern and for the quantitative analysis of the structure. During the past years the number of GISAXS investigations increased owing to the ability to perform such experiments at many experimental stations at synchrotron radiation sources. The method is now well established, and the angular resolution of the scattering experiments is improved by using smaller beam sizes with low divergence.1 For the investigation of coatings, films, and particles on surfaces, the GISAXS method has several important advantages over transmission scattering techniques. A highly intense scattering pattern is always obtained, even for films of nanoscale thickness, because the X-ray beam path length through the film plane is sufficiently long. The detectable length scales from a few to hundreds of nanometers are not limited by a beamstop which covers the information near qy ) 0 due to the possibility to perform so-called out-of-plane scans in GISAXS geometry.2 There is a reduced bulk scattering from the substrate due to the limited penetration depth of the incoming beam at glancing angles near the critical angle of the substrate on which the film is deposited. The sample preparation methods, * Corresponding author e-mail: hamburg.de.
[email protected] Figure 1. GISAXS scattering geometry.
such as spin- or dip-coating, simple drop-casting, or sputtering techniques, are well established. GISAXS can be applied to determine internal morphologies of thin films3-7 as well as top surface morphologies of films, coatings, and substrates.8-12 Expressions of the scattering cross section of GISAXS patterns based on the distorted wave Born approximation (DWBA) have been developed to describe the complicated reflection and refraction effects, which are not found in conventional SAXS. This has been done for rough surfaces,13-17 buried particles,18 and for supported islands. Rauscher et al.19 and later also Leroy et al.20 give a description for the scattering effects of the DWBA of free-standing Ge islands. In this high-resolution scattering experiment on an almost perfect model system, we show clearly the form factor interference effects of the DWBA. It was possible to simulate the scattering pattern in full agreement with the experimental data. The model system used is a highly ordered monolayer of
10.1021/jp0621880 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/19/2006
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Figure 2. The four scattering events in the distorted wave Born approximation.18
PMMA-beads with two-dimensional hexagonal arrays. These domains have diameters of several tens of micrometers. The monolayers can be obtained by dip-coating techniques and because of the narrow size distribution of the manufactured PMMA-beads, they self-assemble in large ordered systems. For the characterization of such monolayers, GISAXS experiments are the first choice to get quantitative information about the quality of the layer structure. In our experiment, a strong interference pattern appears just from the fact that the beam is scattered in four different conditions at a single bead, and thus four different form factor waves interact. Together with the crystalline reflections of the lateral lattice, a complex scattering pattern is shown, and we were able to explain all reflection and refraction effects. By simulating the interference function and the form factor for (a) the simple Born approximation (BA) and (b) the distorted wave Born approximation (DWBA), it is shown that the BA is not sufficient for simulating GISAXS patterns. It is also shown that it is possible to analyze simple out-of-plane cuts, such as a conventional SAXS curve, with some restrictions.
30 µm (vertical) at the sample position. Piezo-driven slits were installed in front of the sample to reduce diffuse scattering from the collimation devices of the beamline. The sample was placed horizontally on the goniometer and tilted to a glancing angle of Ri ) 0.51° with respect to the incoming beam, which is well above the critical angle of the native oxide layer of the silicon substrate. This separated the specular and the Yoneda peak clearly and has enabled us to identify the form factor, structure factor, and the different terms of the DWBA. The small beamsize allowed us to scan the sample laterally to approve the surface homogeneity. The illuminated surface area was 3.4 mm × 60 µm. The accumulation time was 1 h. Out-of-plane cuts (Intensity vs qy at constant qz; ∆qz ) 0.01 nm-1) were made at the Yoneda maximum (Rf ) Rc ) 0.195°) and at the specular position (Rf ) Ri ) 0.51°). With this setup we were able to determine structures within a scale of some nanometers up to around 400 nm, which is the range of the coherence length (∼250 nm) due to the strongly focused, and thus divergent, beam. Theory
Experimental Section The polymerization of the PMMA-beads was performed via a modified emulsion polymerization called “surfactant-free emulsion polymerization” (SFEP).21 In a 250 mL three-necked flask, 100 mL of demineralized water were stirred for 30 min under a nitrogen flow in an oil bath. The nitrogen flow was stopped, and 3 mL of freshly distilled MMA were added. The oil bath temperature was increased to 90 °C and the mixture was stirred for 2 h. Afterward, an aqueous hot solution of 0.5 g potassiumperoxodisulfate (p.a.) was added and the mixture was left stirring for another 2 h. Then the flask was opened and stirred under heating another 1 h to remove unreacted MMA by evaporation. The crude product was cleaned by several centrifugation steps. It is possible to fabricate particles with a very small size distribution and diameters between 100 nm up to several hundreds of nanometers. The resulting aqueous solutions (15 wt-%) were diluted 1:1 with ethanol. Using dipcoating techniques (KSV DC, dragging velocity ) 3 mm/min) the PMMA-beads were deposited on polished Si(111) surfaces (Wacker Siltronic AG). The bare silicon wafers were cut in 2 cm × 2 cm pieces, washed in ethanol, dried, and then cleaned in oxygen plasma for 10 min before use. The resulting amount of layers was controlled by variation of the concentration of the aqueous solutions. The characterization of the obtained twodimensional (2D) nanoparticle-arrays was done by atomic force microscopy (AFM) and grazing incidence X-ray small angle scattering (GISAXS). The AFM investigations were made in intermittent contact mode, using a JPK Nanowizard microscope with a Si-tip (ν ) 280 kHz, r ) 10 nm) installed. For the GISAXS measurements we used the grazing incidence setup of the experimental station BW4 22,23 at HASYLAB (Hamburg/ Germany) equipped with a high-resolution 2D CCD detector (MAR research, 2048 × 2048 pixel, pixel size 79 µm) at a distance of 2.5 m (sample - detector). The wiggler beamline was set to a wavelength of λ ) 0.138 nm. The flight path was fully evacuated and the beam size was focused by an additional beryllium lens system24 to a size of only 60 µm (horizontal) x
Using the simple Born approximation (BA), the reflectionrefraction effects are not taken into account for simple GISAXS analysis; this leads to a wrong interplay between the interference function and the form factor. Furthermore, in most cases the interparticle distance D is roughly calculated from the maximum of the appearing reflection by using the simple equation D ) 2 π/qmax. However, the coupling between the interference function and the form factor greatly increases the complexity of the analysis, and only a direct modeling of the data will give sufficiently accurate results. For a quantitative analysis of X-ray scattering curves, methods have been developed for ordered lattices of spheres using particle distribution functions25-28 and paracrystal theory29,30 and for lamellae and bilayers using interfacial distribution functions 31 or the Caille´-model.32-34 These models consider the influence of particle size, volume fraction, and degree of order to calculate the scattering curves. By the general framework given by Ruland,35,36 allowing to calculate scattering curves of the most prevalent ordered system, combined with the approach developed for the calculation of particle form factors,37 a general approach based on the decoupling approximation (DA) was outlined. This yields analytical expressions for the scattering curves of ordered three-dimensional structures. For transmission SAXS experiments, this modeling can be done with the software Scatter,38 written by S. Fo¨rster. For simulations and the analysis of GISAXS experiments, the modeling software IsGISAXS,39 written by R. Lazzari, is a powerful tool. It takes the DWBA into account and combines it with the analytical expressions of the transmission SAXS using the local monodisperse approximation (LMA). It is possible to simulate the off-specular scattering from rough surfaces, layers, buried particles, and supported islands. Concerning the highly ordered monolayers investigated in this paper, the influence of the use of either DA or LMA in the simulations leads to the same results. The geometry of a GISAXS experiment is illustrated in Figure 1. A monochromatic X-ray beam with the wavevector ki is
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Figure 3. (a) Atomic force micrograph (6 µm × 6 µm), height profile cut along marked line and corresponding fast Fourier transformation (FFT) of a monolayer of PMMA-beads. (b) Atomic force micrograph (1 µm × 1 µm), height profile cut along marked line and phase image of a monolayer of PMMA-beads.
directed on a surface with a very small incident angle Ri with respect to the surface. The Cartesian z-axis is the normal-tothe-surface plane, the x-axis is the direction along the surface parallel to the beam, and the y-axis perpendicular to it. The X-rays are scattered along kf in the direction (2θf, Rf) by any type of electron density fluctuations at the illuminated portion of the surface. The appearing scattering wavevector q for the three spatial directions is defined as follows:
[
cos(Rf)cos(2θf) - cos(Ri) 2π cos(R )sin (2θ ) qx,y,z ) f f λ sin (Rf) + sin (Ri)
]
(1)
In general, a 2D detector records the scattered intensity at angles of a few degrees for the observation of lateral sizes ranging from a few up to hundreds of nanometers. The sample detector distance is normally in a range of 1-4 m for GISAXS, and up to 13 m for grazing incidence ultra small angle scattering (GIUSAXS) experiments,23,40 whereas the detectable lateral size increases to several micrometers. Gas-filled wire frame detectors,
CCD-detectors as well as imaging plates are in use. The direct and the reflected specular beam are often suppressed by two small beamstops to prevent damage or saturation of the detector. The scattering intensity I(q) for a lateral electron density fluctuation on the surface can be described as follows:
I(q b) ) 〈|F|2〉S(q|)
(2)
where F is the form factor and S(q) is the total interference function. The interference function describes the spatial arrangement of the objects on the surface and thus their lateral correlations. It is the Fourier transform of the island position autocorrelation function. In the simple Born approximation (BA), F is the Fourier transform of the shape function of the objects and is defined as follows:
F(q b) )
r 3r ∫V exp(iqb ‚b)d
(3)
If reflection-refraction effects at the surface of the substrate have to be accounted for, F has to be calculated within the
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distorted wave Born approximation (DWBA) and has a more complex expression. Figure 2 illustrates the physical picture of the full calculation13,18 for the scattering cross section in the DWBA, which is given by eq 4. The four terms involved are
dσ ∝ |F(q b|,kfz - kiz) + R(Ri)F(q b|,kfz + kiz) + F(q b|, - kfz dΩ b|, - kfz + kiz)|2 (4) kiz) + R(Ri)R(Rf)F(q associated to different scattering events, which involve or not a reflection of either the incident beam or the final beam collected on the detector. These waves interfere coherently, giving rise to the effective form factor (FDWBA), in which the classical form factor (eq 3) comes into play, but with respect to the specific wavevector transfers. Each term is weighted by the corresponding Fresnel reflection coefficient R, either in incidence or in reflection. The reduced reflectivity of the substrate by an uncorrelated roughness can be calculated in a classical way with a mean standard deviation.41 The scattered cross section is proportional to the scattered intensity and thus to the modulus square of the Fourier transform of the electron density. The polarization effect for X-rays can be dropped out safely as the scattering angles are small. The combination of the calculated form factor within the DWBA in eq 4 together with the total interference function leads to the expression for the total incoherent cross section:
dσ ) 〈|FDWBA|2〉S(q|) dΩ
Figure 4. (a) GISAXS experimental data and (b) the simulated scattering pattern performed with IsGISAXS19 (DWBA form factor for spheres with r ) 173 nm with 2% size distribution. 2D hexagonal lattice with a ) 444 nm, Gaussian pair correlation function with ω ) 60 nm, domain size 2000 nm with random orientation). The qz positions of the out-of-plane cuts for the specular beam (S) and the Yoneda maximum (Y) are marked by arrows.
(5)
Results and Discussion After dip-coating, the obtained monolayer was first characterized by means of AFM to check the quality of the order of the structure. Figure 3a shows an atomic force micrograph of one of the fabricated monolayers. As one can see, the 2D hexagonal structure is highly ordered in large domains, due to the small size distribution of the particles. This part of the sample shows a domain of some tens of micrometers with some small crystal imperfections, but without loss in long-range order. The typical monolayer domain size ranges between 2 and 40 µm2. As a result of the highly ordered system, the FFT of the AFM image shows strong hexagonal peaks along several orders. Figure 3b shows the AFM measurement of the same monolayer at a missing bead in the lattice at a higher magnification. The height profile and the phase contrast image illustrate clearly that the PMMA-coating is really a monolayer. The substrate has a harder surface, which leads to a higher phase angle in the phase contrast image. The high-resolution GISAXS experiment shows a very strong interference pattern. Figure 4 illustrates (a) the GISAXS measurement and (b) the simulation performed with the software IsGISAXS including the modeling parameters (DWBA form factor for spheres with r ) 173 nm with 2% size distribution. 2D hexagonal lattice with a ) 444 nm, Gaussian pair correlation function with ω ) 60 nm, domain size 2000 nm with random orientation) we used. The simulation using a monolayer and the DWBA approach fits very well with the observed scattering. One part of the interference is due to the lateral lattice function, which leads to repeating reflections along qy. Figure 5(a) shows the scattering part of the lattice function of the full simulation. At higher qz the scattering rods decrease to smaller values of qy and finally disappear due to the fact that they are no longer in a reflection position on the Ewald-sphere. At larger scattering angles, the curvature of the Ewald-sphere comes into play and
Figure 5. (a) Interference function and (b) form factor of the GISAXS simulation from Figure 4b (Software IsGISAXS19) by assuming the distorted wave Born approximation (DWBA).
is no longer negligible. The other interference has its maximum around the reflected (specular) beam and a radial decrease. For flat, smooth, and thus highly reflecting substrates and particles with small X-ray absorption, the reflection terms 2 and 3 of the DWBA (Figure 2) become very strong and interfere in direction of the reflected beam, this leads, together with the very narrow size distribution of the PMMA-beads, to a very strong interference pattern. This part of the simulation is shown in Figure 5b. To demonstrate that the simple Born approximation is not sufficient to explain such scattering phenomenon, we simulated a scattering pattern with the same parameters but within in BA instead of the DWBA. Figure 6a shows the part of the form factor, and Figure 6b shows the combination of the form factor
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Figure 6. (a) Form factor and (b) and the resulting scattering pattern for both interference function and form factor of the GISAXS simulation from Figure 4b (Software IsGISAXS19) by assuming the simple Born approximation (BA).
Figure 7. GISAXS experimental out of plane scattering at specular position, intensity profiles obtained from the simulation (Figure 4b) performed with IsGISAXS19 (DWBA and interference function) and a calculated form factor for spheres with a radius r ) 173 nm and a size distribution of 2% using the Software Scatter18 (simple BA approach).
and the interference function. Obviously, the scattering of the form factor has its center around the origin of the nonreflected primary beam, because only the term 1 in the DWBA was used. The radius of the PMMA-beads can be obtained by analyzing the intensity profile cut along qy at the specular beam position as it is shown in Figure 7. Assuming that the scattering around the specular beam is not affected by the scattering coming from the Yoneda maximum (this can be done by using an incident angle well above the critical angle of the substrate), the terms 2 and 3 of the DWBA form factor are playing the main role. The reflected beam itself can be interpreted as a second scattering source directed to the specular position and thus probes reciprocal space at Ri ) Rf. As one can see, the simulation with the BA, which is usually applied in transmission SAXS, leads here to the marginally better result: The full DWBA approach weights all the terms equally and thus shifts the minima slightly at higher qy-values. To get the lateral order, distances of the particles and their lateral radius one has to
Fro¨msdorf et al.
Figure 8. FFT frequency spectrum (AFM) and 2D hexagonal lattice function (a ) 435 nm, spherical form factor with radius r ) 173 nm) calculated with Scatter;18 GISAXS out of plane scattering at Yoneda maximum and intensity profile obtained from the simulation with the DWBA form factor (Figure 4b) and the same profile assuming the BA (Figure 6b) performed with IsGISAXS.19
analyze the scattering intensity along 2θf at the position of the critical angle Rc (Yoneda maximum) of the substrate. The native oxide layer of the substrate has a critical angle of Rc ) 0.195° at the given wavelength. Here the Yoneda maximum at Rc ) Rf probes reciprocal space as it can be interpreted as the scattering origin. In the lower part of the Figure 8 one can see that the BA and DWBA approach give identical results. The same information can be obtained by a radial integration of the fast Fourier transformation (FFT) of the AFM micrograph. Both curves are displayed in the upper part of Figure 8 and a calculated 2D hexagonal lattice function was fitted to the experimental data of the FFT. The resulting lattice constant is a ) 435 nm and the spherical particle radius r ) 173 nm which is identical with the results one can get by the scattering experiment. The GISAXS scattering curve below shows, that the resolution of this µ-GISAXS experiment was not high enough to precisely determine such large interference functions, but it shows a broad maximum, which lies in the same q-range as the 11-reflection of the lattice. The further analysis of the curve with a 2-D hexagonal lattice model gives a lattice constant of a ) 444 nm, which is also in good agreement with the data, calculated from the FFT function. Conclusions After synthesis of uniform-sized PMMA-beads via a modified emulsion polymerization, highly ordered monolayers of PMMAbeads on silicon substrates were fabricated using dip-coating techniques. This serves as a model system for fundamental GISAXS investigations. High-resolution grazing incidence X-ray small angle scattering experiments were performed using a micrometer-sized beam (µ-GISAXS) to determine the structure of a highly ordered monolayer with two-dimensional hexagonal arrays. They show strong interference patterns that belong to the reflection and refraction effects of the incident X-rays described in the DWBA. A strong experimental effect is clearly shown, and we were able to explain and simulate the full scattering patterns. The form factor and interference intensities are separately simulated and the results are corroborated by the comparison with atomic force micrographs and their translation in Fourier space.
µ-GISAXS on monolayers of PMMA-beads Acknowledgment. This research was supported by the DFG (SFB 508 “Quantum Materials”) and by the University of Hamburg. We thank R. Lazzari for the helpful discussions and the IsGISAXS software, the HASYLAB staff for the support at the beamline BW4, and especially S. Fo¨rster for scientific and financial support. References and Notes (1) Roth, S. V.; Walter, H.; Burghammer, M.; Riekel, C.; Lengeler, B.; Schroer, C.; Kuhlmann, M.; Walther, T.; Sehrbrock, A.; Domnick, R.; Mu¨ller-Buschbaum, P. Appl. Phys. Lett. 2006, 88, 021910. (2) Levine, J. R.; Cohen, J. B.; Chung, Y. W.; Georgopoulos, P. J. Appl. Crystallogr. 1989, 22, 528. (3) Lee, B.; Park, I.; Yoon, J.; Park, S.; Kim, J.; Kim, K.-W.; Chang, T.; Ree, M. Macromolecules 2005, 38, 4311. (4) Cavicchi, K. A.; Berthiaume, K. J.; Russell, T. P. Polymer 2005, 46, 11635. (5) Tokarev, I.; Krenek, R.; Burkov, Y.; Schmeisser, D.; Sidorenko, A.; Minko, S.; Stamm, M. Macromolecules 2005, 38, 507. (6) Papadakis, C. M.; Busch, P.; Posselt, D.; Smilgies, D.-M. AdV. Solid State Phys. 2004, 44, 327. (7) Dourdain, S.; Bardeau, J.-F.; Colas, M.; Smarsly, B.; Mehdi, A.; Ocko, B. M.; Gibaud, A. Appl. Phys. Lett. 2005, 86, 113108. (8) Leroy, F.; Renaud, G.; Letoublon, A.; Lazzari, R.; Mottet, C.; Goniakowski, J. Phys. ReV. Lett. 2005, 95, 185501. (9) Sun, Z.; Wolkenhauer, M.; Bumbu, G.-G.; Kim, D. H.; Gutmann, J. S. Physica B 2005, 357, 141. (10) Du, P.; Li, M.; Douki, K.; Li, X.; Garcia, C. B. W.; Jain, A.; Smilgies, D.-M.; Fetters, L. J.; Gruner, S. M.; Wiesner, U.; Ober, C. K. AdV. Mater. 2004, 16, 953. (11) Mu¨ller-Buschbaum, P.; Hermsdorf, N.; Roth, S. V.; Wiedersich, J.; Cunis, S.; Gehrke, R. Spectrochim. Acta B 2004, 59, 1789 (12) Tate, M. P.; Eggiman, B. W.; Kowalski, J. D.; Hillhouse, H. W. Langmuir 2005, 21, 10112. (13) Sinha, S. K.; Sirota, E. B.; Garoff, S.; Stanley, H. B. Phys. ReV. B 1988, 38, 2297. (14) Holy´, V.; Kubuena, J.; Ohlı´dal, I.; Lischka, K.; Plotz, W. Phys. ReV. B 1993, 47, 15896. (15) Holy´, V.; Baumbach, T. Phys. ReV. B 1994, 49, 10668.
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