Multiple scattering effects in strain and composition analysis ... - IEMN

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013112-2 Richard et al. Appl. Phys. Lett. 94, 013112 2009 FIG. 2. Color online a Simulated radial scan intensities around the Si400 reflection for Ge islands on Si001 whose strain has been calculated by FDM: a 124 nm wide and 20 nm high dome. The intensity is integrated between f 0° and f 1.5°. The total DWBA intensity is reported for different incident angles and compared to the BA first path intensity. The substrate scattering has been omitted. b Evolution of t i ,z 2 inaGe dome on Si001 as a function of height z and i / c for =1.097 Å. As the intensity scattered by each z-layer above the surface is proportional to t i ,z 2 , an x-ray scattering experiment will predominantly reflect the strain state of a given height depending on i rather than the full island strain state distribution. c Same as a but for a 70 nm wide and 7 nm high pyramid. To understand this effect, four paths according to the distorted wave Born approximation DWBA have to be taken into account, 8,9 F DWBA = f j e ik f −k i ·r j + r i f j e ik f +k i ·r j jQDs jQDs + r f f j e i−k f −k i ·r j jQDs + r i r f f j e i−k f +k i ·r j, jQDs where f j is the atomic scattering factor and r j is the position of an atom j in the island; k i and k f are the incoming and exit wave vectors and r i,f are the reflectivities of the incident and scattered waves. The first term in Eq. 1, which we call first path, corresponds to the kinematical scattering of the incident wave by the islands or Born approximation BA. The three other terms imply single or double reflections by the substrate. 8 To interpret these observations, the strain field in a coherent Ge/Si001 dome-shaped island with the experimental width and height was calculated with the finite difference method FDM, 10 assuming continuum theory of elasticity. For simplicity, a pure Ge dome was modeled. The scattered intensity see Fig. 2a was then calculated using Eq. 1 around the in-plane Si400 reflection for different values of i around c and summed up between f 0° and f 1.5° to account for the experimental conditions. The island form factor given by the BA shows two broad peaks that arise from two prevalent strain states in the dome-shaped island: the first peak close to the Si Bragg peak corresponds to the basis and also to the main part of the island, which is surrounded by the 113 and 15323 facets, while the second peak results from the strain state of the top part of the island, terminated by 105 facets. For i =0.62 c , only one 1 maximum is present in the simulated radial scan. It is positioned at h=3.944, i.e., displaced compared to the real maxima h=3.925 and 3.954 of the strain state given by the first path. Above the critical angle, the positions of the intensity maxima are close to the real positions of the maxima of the strain state given by the first path. To discuss these effects, the islands can be decomposed into layers of different heights z, each having its own lattice parameter az. With this hypothesis, a radial position h corresponds to the scattering by a given layer at z. We also suppose case of Ge islands on Si that the higher z, the larger the lattice parameter and hence the smaller h. Within the DWBA and as k z i,f = 2/ sin i,f , for small incident and exit angles, the amplitude scattered at a given h by the portion of the island at height z can be written as 11 F z DWBA i , f = F z BA h, i , f t i ,zt f ,z, where F z BA is the island form factor at height z and t,z =1+re −4iz/ is a generalized optical function. t i ,z varies as a function of i and z, as illustrated in Fig. 2b. Hence, the intensity distribution in Fig. 2a results from the product of the island form factor by the generalized optical function. Because z is directly correlated with h, the optical function varies along the radial direction. If the value of i is below or at the critical angle of the substrate, the optical function can be very strongly enhanced for certain heights z and thus for the corresponding radial positions h. Low incident angles maximize the generalized optical function for high z values and thus maximize the scattered intensity at lower h values. For i =0.62 c , this maximum corresponds to z20 nm see Fig. 2b, i.e., the intensity scattered by the island’s top, where it is the most relaxed, is strongly enhanced. Above the critical angle, the optical function does not display strong minima and maxima; the multiple scattering effects can be neglected, and the kinematical approximation is recovered. The condition for these DWBA effects to be significant is 11 = rzH2/ahd /dz 1, where H is the total height of the nanocrystal, z is the parallel strain of an isostrain area at height z, and rz is its radius. Hence, this phenomenon is observed only for large enough islands, with large enough vertical strain gradient. For the simulated Ge dome, 10. For smaller islands with smaller strain gradient, these multiple scattering effects are not observed. This was checked in the case of a small Ge pyramid on Si001 see Fig. 2c for which the intensities simulated for different i values can be superimposed. Indeed, in that case, the DWBA analysis for islands on a substrate is identical to that of islands buried below the surface: the total scattered amplitude see Eq. 2 is proportional to the form factor amplitude, F BA k f −k i , according to F DWBA i , f = F BA k f − k i t i t f , where t i,f is the usual amplitude transmission function as a function of incident exit angle. Having determined for which conditions the DWBA has to be used instead of the BA, the question arises whether the standard MAD analysis using the BA is still valid for large islands under grazing incidence. Equation 2 shows that the intensity scattered by a stripe at height z is simply the BA intensity multiplied by the incident and exit generalized op- 2 3 4
013112-3 Richard et al. Appl. Phys. Lett. 94, 013112 2009 Ge content 1 α 0.8 i =0.12° α i =0.2° 0.6 0.4 0.2 0 3.9 3.92 3.94 3.96 3.98 h (r.l.u.) FIG. 3. Color online Ge content as a function of h as extracted using grazing incidence MAD with 12 different energies around the Ge K-edge. The Ge content extracted for two different incident angles remains consistent in spite of the strong Ih dependence with i . tical functions, which are energy independent far from c in the case of a buffer containing nonanomalous atoms. Therefore during a MAD experiment the extracted structure factors are F Ge and F Si multiplied by these functions. Since the Ge composition is extracted by a ratio of these factors according to x Ge = f Si F Ge /f 0 0 Ge F Si + f Si F Ge , where f Ge and f Si are the Ge Thomson atomic form factor and the Si atomic form factor, respectively, the generalized optical functions disappear from the result, and the deduced composition is not affected by multiple scattering effects. This result was experimentally checked by comparing the results of grazing-incidence MAD analysis of the dome sample for different incident angles. Figure 3 shows the Ge composition calculated from the Ge and Si structure factors F Si and F Ge , which were extracted for two incident angles without any structural hypothesis, and refined through a least-squares minimization using a dedicated program. 12 This program evaluates the uncertainties of F Ge and F Si and hence the error bars on the Ge composition. The partial structure factors are affected by the generalized optical function as is the total scattering amplitude. However, only slight variations in the Ge composition are observed, with overlapping error bars. Hence, the correct Ge content is extracted, whatever the conditions of incidence. A BA treatment of anomalous diffraction can thus be performed to extract the composition of isostrain regions of the islands. The subcritical regime i c yields smaller scattering from the substrate and is thus preferred. To conclude, for large islands, multiple scattering effects strongly distort the intensity distribution in reciprocal space and need a complete DWBA analysis to extract the lattice parameter distribution. By contrast, the in-plane strain in small islands can be investigated using a BA treatment. Hence, scattering from small and large islands can be distinguished simply by varying the incident angle around the critical angle c , checking if the intensities can be superimposed to each other by rescaling. This may be most useful for bimodal island growth e.g., pyramids and domes in the Ge/Si case to discriminate the origin of the measured intensity. Most importantly, the standard MAD analysis can be applied to all islands as long as the x-ray energy is far from an absorption edge of the substrate and the substrate does not contain any elements violating this condition. Z.Z. acknowledges support from the FWF, Vienna. We acknowledge Hubert Renevier for fruitful discussions. 1 J. Stangl, V. Holý, and G. Bauer, Rev. Mod. Phys. 76, 7252004. 2 A. Létoublon, V. Favre-Nicolin, H. Renevier, M. G. Proietti, C. Monat, M. Gendry, O. Marty, and C. Priester, Phys. Rev. Lett. 92, 186101 2004. 3 I. Kegel, T.-H. Metzger, A. Lorke, J. Peisl, J. Stangl, G. Bauer, J. M. Garcia, and P. Petroff, Phys. Rev. Lett. 85, 1694 2000. 4 T. U. Schülli, J. Stangl, Z. Zhong, R. T. Lechner, M. Sztucki, T. H. Metzger, and G. Bauer, Phys. Rev. Lett. 90, 066105 2003; T. U. Schülli, M. Stoffel, J. Stangl, R. T. Lechner, E. Wintersberger, M. Sztucki, T. H. Metzger, O. G. Schmidt, and G. Bauer, Phys. Rev. B 71, 035326 2005. 5 A. Malachias, S. Kycia, G. Medeiros-Ribeiro, R. Magalhães-Paniago, T. I. Kamins, and R. Stanley Williams, Phys. Rev. Lett. 91, 176101 2003. 6 M. Schmidbauer, D. Grigoriev, M. Hanke, P. Schafer, T. Wiebach, and R. Kohler, Phys. Rev. B 71, 115324 2005. 7 I. N. Stranski and L. Krastanow Sitzungsber. Akad. Wiss. Wien, Math.- Naturwiss. Kl., Abt. 2B 146, 797 1938. 8 M. Rauscher, R. Paniago, H. Metzger, Z. Kovats, J. Domke, H. D. Pfannes, J. Schulze, and I. Eisele, J. Appl. Phys. 86, 6763 1999. 9 J. Coraux, V. Favre-Nicolin, M. G. Proietti, B. Daudin, and H. Renevier, Phys. Rev. B 75, 235312 2007. 10 V. Ranjan, G. Allan, C. Priester, and C. Delerue, Phys. Rev. B 68, 115305 2003. 11 I. Kegel, T.-H. Metzger, A. Lorke, J. Peisl, J. Stangl, G. Bauer, K. Nordlund, W. V. Schoenfeld, and P. Petroff, Phys. Rev. B 63, 035318 2001. 12 V. Favre-Nicolin, in preparation.
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