\underline{143}

Conclusion

In my thesis I have dealt with the theory of the X-ray reflectivity. I have applied it to the study of the following types of multilayered samples:

1. planar multilayers with various stacking sequences (single layer, periodic, quasiperiodic),

2. rough multilayers, and

3. multilayer gratings (multilayers with a lateral structure).

My aim was to develop and present several theories together using one unified formalism while pointing out the links with the X-ray diffraction theories. The theories discussed are:

1. the kinematical theory,

2. the distorted-wave Born approximation,

3. the dynamical theory, and

4. various approximations of the dynamical theory (single-reflection approximation, twobeam and multiple-beam approximations).

These theories have been compared by their analytical expressions as well as by numerical simulations. I discussed their regions of good coincidence as well as their differences.

Further, I demonstrated the use of numerical simulations to fit measured data. This allowed me to reveal structural parameters of the samples we analyzed in our laboratories.

The first part of the thesis has been devoted to the representation of scattering in reciprocal space. Here, the relations between the angular movements during an experiment and the appropriate scans in reciprocal space have been described. These formulae have been applied by programming the motor movements of the goniometers for synchrotron measurements.

Further I discussed the specular reflectivity from planar multilayers. Firstly, the kinematical theory has been formulated. I calculated its diffraction integral by the stationary-phase method, whose validity is not restricted to the first Fresnel zone contrary to the Fraunhofer approximation mostly employed in calculating the kinematical treatments. Further, the usual dynamical theory of reflectivity has been formulated, from which I derived the single-reflection approximation. Dynamical and kinematical Fresnel coefficients have been compared.

The kinematical theory and the single-reflection approximations were successful especially for the calculation of the reflectivity pattern of a quasiperiodic Fibonacci multilayer. By applying the fundamental theorems from the physics of quasicrystals I have shown that the reflectivity curve exhibits a self-similarity and two integers are needed to describe the peak positions.

Furthermore, specular and non-specular X-ray reflection from rough multilayers has been discussed. The statistical properties of randomly rough interfaces have been employed in the specular reflectivity from both the kinematical and dynamical theories. I applied the simulations in fitting the experimental curves for sandwich multilayers and periodic multilayers. Further diffuse scattering from rough multilayers has been briefly discussed and the distorted-wave Born approximation (DWBA) employed for quantitative analysis. I presented the main features of incoherent diffuse scattering on a measured map from a periodic multilayer.

The main contribution of this work treats X-ray reflection from multilayer gratings. I solved this problem using the kinematical theory, the distorted-wave Born approximation and the rigorous dynamical theory. The dynamical theory has been treated in the framework of the matrix modal eigenvalue method. The multiple-beam approximations have been derived from the dynamical theory. As a limiting case of a single-scattering theory I formulated and thoroughly discussed the two-beam approximation. The kinematical theory was treated by the stationary phase method.

All three theories have been formulated within one general formalism. This made their discussion and comparison easier and transparent. It allowed me to generalize the Fresnel coefficients involved in conventional specular X-ray reflection from planar multilayers for the lateral diffraction case. In the kinematical theory, they were expressed by Fresnel reflection coefficients of kinematical diffraction, whereas in the dynamical theory the “interface” matrix of Fresnel coefficients has been generalized. Further I have shown that the formalism used in all the theories is suitable not only for periodic gratings, but also for calculating the reflection from more complicated quasiperiodic Fibonacci gratings.

My main interest has been devoted to short period gratings (period d ≈ 1 μm) and wavelengths around 1 Å, but also larger period gratings have been briefly discussed. A detailed discussion was performed for a short period surface grating (SG, d = 8000 Å) with the wire to period ratio one half and for a wavelength of 1.54 Å. The theories have been compared analytically as well as numerically bearing these values in mind. The proposed treatment enabled me to separate the single scattering and the multiple (dynamical) scattering effects.

• The single-scattering (“two-truncation rod” scattering) regions, where the calculation by

the dynamical theory, two-beam approximation and the DWBA coincide, were determined: – most of the profile of the strong truncation rods (kinematically non-forbidden) for angles of incidence outside the critical angles of the grating materials (the discussed SG),

– the whole profile of the strong truncation rods for a grating fulfilling the conditions of validity of the two-beam approximation (the measured multilayer grating),

– most of the profile of the strong truncation rods calculated by the DWBA.

• Strongly dynamical cases, where multiple-beam scattering among many truncation rods has to be involved and where the single-scattering theories fail:

– the intensity profile of the weak, kinematically forbidden truncation rods (non-zero even order truncation rods for the discussed SG),

– the intensity scattered in the grazing incidence geometry where the incident beam falls parallel to the wires.

• Multiple scattering regions, where the single-scattering theories give still approximate results and where the multiple interaction between several truncation rods has to be considered:

– near to the angle of incidence for which the wavefield of the TR +1 becomes real,

– in the region of the critical angles for the strong truncation rods and the calculation by the two-beam approximation.

The main advantage of the presented approach is the presentation of the regions of validity of the single-scattering approximations (kinematical theory, DWBA and the two-beam approximation). I demonstrated that within the two-beam approximation and the DWBA the scattered amplitude of the primary scattering process (the single scattering between the incidencetransmitted and the diffracted-reflected waves) is proportional to the Fourier transform of the susceptibility χ_{h}.

This proportionality was also the result of the treatment by the kinematical theory. However, this theory, equivalent to the first Born approximation, does not include the effect of refraction, which is of major importance in X-ray reflectivity.

The first order DWBA employed includes the refraction as well as the main features of the dynamical theory except for a small known region of strong interaction with TR +1. This DWBA has been found adequate for calculating the intensity of the measurable non-forbidden truncation rods, which confirms the legitimacy of the DWBA for gratings as an example of a “big roughness”. This confirms the potential usability of this method for the studies of scattering by randomly structured layers (e.g., island-layer structures). In addition, I discussed the regions where the dynamical effects of multiple scattering prevail and where the full dynamical theory or the DWBA of higher order have to be employed.

In order to include the structural imperfections of real multilayer gratings into the calculation, I studied scattering from rough multilayer gratings too. I have considered both the “side wall” roughness of the grating shape and “interface” roughness into the matrix dynamical formalisms as well as into the kinematical theory, from which the generalization of the roughness into the DWBA is straightforward.

In the dynamical theory, I have found that the elements of the “interface” matrix have to be multiplied by the characteristic function of the interface roughness probability distribution. This is similar to the “interface” matrices of Fresnel coefficients introduced for planar multilayers.

In the kinematical theory, the interface roughness acts as the kinematical Debye-Waller damping factor on the Fresnel coefficients of kinematical diffraction. The derived analytical formulae allowed the roughness influence on the scattered intensity to be predicted, which was verified by the numerical simulation. The side wall roughness was introduced under the approximation of single-scattering processes by averaging laterally the Fourier coefficients of the susceptibility.

Finally, the numerical simulations have been applied to fit the structural parameters of a partially etched InP/GaInAs multilayer grating.

The scientific perspectives of the X-ray reflection methods. X-ray reflection is nowadays frequently and successfully applied to the structural studies of different kinds of multilayered samples. Topics of recent investigations by this technique are:

• interface correlation functions of samples with different types of random interfaces,

• multilayers with terraced interfaces, e.g., multilayers grown on miscut substrates,

• laterally structured samples

– with nearly perfect structure (multilayer gratings, quantum wires and dots),

– with large imperfections, like the samples with non-continuous island-like layers.

In particular, we propose the following studies of the X-ray reflectivity from multilayer gratings, to which a great deal of this report has been devoted to.

• The discussion here has shown the degree of validity and the limits of the first order DWBA. In order to extend the application of the DWBA also for the regions of prevailing effects of multiple scattering (the kinematically forbidden truncation rods), this method could be expanded into the second order.

• The DWBA could be advantageously used to calculate the incoherent diffuse scattering from imperfect gratings, using either

– the eigenstates of a laterally averaged planar multilayer with the restriction to the regions of validity discussed in this work,

– the eigenstates given by the dynamical theory.

• The non-coplanar reflectivity technique can overcome the limitation of the accessible reciprocal space for the coplanar reflectivity. The presented matrix dynamical theory using the modal eigenvalue approach has been shown to cope with the non-coplanar scattering. However, the consistency of the eigenvalue and the point matching approaches for the strongly dynamical case when the incident wave falls parallel to the wires has not yet been theoretically confirmed.