Recent progress in the growth techniques makes it possible to fabricate low-dimensional structures, like thin films (planar multilayers), mesoscopic structures and nanostructures (lateral surface and multilayer gratings, quantum wires and quantum dots). These elements find applications in electronic and optical devices.
Optimization of the fabrication process and the physical understanding of the samples requires non-destructive structural studies of the samples produced. Complementary to the direct local probing methods (i.e., AFM microscopy), the X-ray elastic scattering methods probe locally the reciprocal space, thus providing information about the statistical properties of the structural parameters averaged over a large volume of the sample.
Nowadays, the development of X-ray scattering methods is encouraged by advanced technical equipment being widely available, like the multiple crystal arrangements enabling good precision in reciprocal space to be obtained. Further, complementary to the conventional or rotating anode laboratory sources, new high-intensity synchrotron radiation sources are advantageously involved in studies of low-dimensional objects whose diffraction power is small, as well as in the studies of low-intensity diffuse scattering. In addition to the improved dynamical range of measured intensity and highly parallel beam, synchrotrons also provide the possibility of wavelength tuning, thus changing the contrast of the sample constituents.
X-ray reflectivity (XRR), including both the specular X-ray reflection (SXR) and non- specular X-ray reflection (NSXR), is conveniently applied for the structural studies of both crystalline and amorphous multilayer samples. It is sensitive to the distribution of the refractive index in the sample. It studies the scattering at small angles around the critical angle of total external reflection, and it maps the distribution of the scattered intensity around the origin of the reciprocal space. Thus XRR is complementary to miscellaneous X-ray diffraction methods, namely the conventional symmetric and asymmetric X-ray diffraction, the more elaborate highly asymmetric diffraction, and the grazing incidence diffraction, which study the crystalline properties.
The present work is devoted to the X-ray reflection studies of multilayered samples, either planar or laterally structured.
In a rough approximation, the intensity scattered by a sample is proportional to the square of modulus of the Fourier transform of the electron density. Thus the electron density profile can be deduced from the measured intensity pattern, and subsequently the vertical properties (layer thicknesses) as well as the lateral properties (roughnesses and correlation properties of interfaces or lateral layer structure) characterizing multilayers can be determined. Therefore XRR is now being applied not only to the usual planar multilayers, but also to reveal the properties of various kinds of laterally structured samples, for instance gratings, multilayers grown on inclined substrate surfaces or layers with random island structures.
Several approaches for treating the theories of X-ray reflectivity are developed, solving the wave equation by various methods. The main approach comes from the usual optics of visible light, which is modified for X-rays since the index of refraction is near unity. The other treatments proceed from X-ray diffraction from crystals (the kinematical theory of scattering) and from quantum mechanics (the distorted-wave Born approximation).
The dynamical theories give rigorous solutions to the studied problems. However, they are usually not well suited for a qualitative physical understanding of the calculated results. Therefore I develop several approximative theories for XRR that explain easily the scattering phenomena and provide rapid algorithms for numerical calculation. On the other hand, the regions of validity of the approximations involved have to be determined.
In particular, the X-ray scattering theories studied in this work comprise the kinematical theory, the distorted-wave Born approximation, the dynamical theory and various approximations of the dynamical theory, like the single-reflection approximation, the multiple-beam approximation and the two-beam approximation.
Apart from the use of different approximations, also different notation is used by various authors, making a direct comparison of the theoretical approaches difficult. Therefore, one particular aim of the present work is to treat different theories using a uniform formalism that will enable the regions of validity of all treated theories to be studied, compared and discussed in a consistent and methodological way. For example, I show that the Fresnel coefficients, well-known from the optical reflectivity from planar multilayers, have their counterparts in the kinematical theory, and that they can be generalized in the case of reflectivity from gratings. I also show the advantage of the matrix formalism for the dynamical theories. In addition, I introduce into XRR the concept of the graphical representation of the scattering phenomena by means of the Ewald construction.
Now I briefly outline the structure of the presented thesis. In the first part, I deal with the basic characteristics of the reciprocal space. Relations providing the connection between the angular rotations during an experiment and the appropriate scans in the reciprocal space are provided.
Further, the specular reflectivity from planar multilayers is discussed. The usual dynamical theory is developed, from which the single-reflection approximation is derived. The kinematical theory treated by the stationary-phase method is discussed afterwards and the derived kinematical Fresnel reflection coefficients compared to the dynamical ones. The theories are applied to explain the specular reflectivity curve of a quasiperiodic Fibonacci multilayer.
Specular and non-specular X-ray reflectivity from rough planar multilayers are studied in the following chapter. Both the dynamical and kinematical theories for flat interfaces treated in the previous chapter are reformulated in order to cover the statistical properties of randomly rough interfaces. The application to fitting the experimental curves is demonstrated. The diffuse scattering is dealt with as well, for which the distorted-wave Born approximation is employed. The main features of the incoherent diffuse scattering are briefly presented from the measurement.
The main part of the work is devoted to the study of X-ray reflectivity from multilayer gratings. Following the general formalism and notation introduced in the previous chapters, the problem is solved by the kinematical theory, distorted-wave Born approximation and rigorous dynamical theory. Multiple-beam approximations, mainly the two-beam approximation, are derived from the dynamical theory. These theories are compared analytically and numerically, addressing mainly the short period gratings. The aim is to show the conditions under which the approximate single-scattering theories give correct results, and which are the regions where dynamical effects of multiple scattering prevail. Therefore the main interest of such a methodological study is given to the scattering theories themselves, and not to the optimization of the sample structure for particular applications.
The scattering from rough gratings is treated afterwards. The real structural imperfections of multilayer gratings are taken into account, comprising the roughnesses of the interfaces and of the side walls of the grating shapes. The proposed unified formalism enables these effects to be easily incorporated into both the dynamical and the kinematical theories.