PhD thesis by Petr Mikulík

X-ray Reflectivity From Planar and
Structured Multilayers

Title in French:

Réflectivité des rayons X par de multicouches planaires et structurées

Title in Czech:

Rentgenová reflektivita na rovinných a strukturovaných multivrstvách

Introduction     Abstract     Publications     Download     Table of Contents


My thesis has been worked out in the framework of the scientific collaboration between the Czech Republic and France under the sponsorship of the Masaryk University and la these en co-tutelle  ("thesis under double leadership", "dizertace pod dvojim vedenim") grant of the French government. I studied at the Department of Solid State Physics at Faculty of science of Masaryk University in Brno and in Laboratoire Louis Néel (CNRS) associated to the Université Joseph Fourier in Grenoble.

I defended my thesis on:

6. 3. 1997 at Université Joseph Fourier in Grenoble, France, and
29. 4. 1997 at Masaryk University in Brno, Czech Republic.

Na Masarykově univerzitě jsem promoval 27. 11. 1997 a zde je k vidění moje promoční kartička s různými obrázky.


The thesis has been printed in 90 copies and they are all gone. Because of the continous demand, I have put the printable electronical version on the web and give you the chance for download of my thesis for printing.

Bugs in the original (1997) paper version

Few mistakes (mostly in writing, nothing influences the physics) in the printed version have been found until 12.  10.  1999 and they are listed here. You are welcome to update your copy.


List of publications resulting directly from the thesis is presented here:

Abstract (English)

The study of the X-ray reflectivity from planar and structured multilayers is presented using different theoretical approaches. The studied scattering phenomena are the specular reflection from planar multilayers with different stacking sequences (single layer, periodic, quasiperiodic), diffuse scattering from rough multilayers, and the scattering from surface and multilayer gratings. The theories employed for the calculation are the kinematical theory, the distorted-wave Born approximation, the dynamical theory and various approximations of the dynamical theory (single-reflection approximation, the two-beam approximation and the multiple-beam approximation). The unified formalism enables to discuss and compare all these theories in a consistent and methodological way. Numerical calculations are applied to fit the experimental curves in order to reveal the structural parameters of various types of layered samples.

Abstract (francais)

La reflectivite des rayons X par des multicouches planaires et structurees est presentee en utilisant differentes approches theoriques. Les phenomenes de diffusion etudies sont: la reflectivite speculaire par des multicouches planaires ayant diverses sequences d'empilement (monocouche, periodique, quasiperiodique), la diffusion diffuse de multicouches rugueuses, et enfin la diffusion par des reseaux de surface et par des reseaux de multicouche. Les theories employees pour les calculs: la theorie cinematique, l'approximation de l'onde deformee de Born, la theorie dynamique, et plusieurs approximations de la theorie dynamique (approximation a une reflexion unique, approximation a deux ondes et approximation   diffusion multiple), sont developpees dans un formalisme unique. Ce formalisme permet de discuter et de comparer toutes les theories d'une maniere solide et methodologique. Les calcules numerique sont appliques pour l'ajustement des courbes experimentales pour mettre en evidence les parametres structuraux des divers systemes multicouches.

Abstrakt (česky)

Rentgenová reflektivita rovinných a strukturovaných multivrstev je prezentována za použití různých teoretických postupů. Diskutovány jsou následující typy rozptylu: spekulární reflektivita rovinnými multivrstvami s různou sekvencí vrstev (monovrstva, periodická či kvaziperiodická posloupnost vrstev), difuzní rozptyl od drsných multivrstev, a konečně rozptyl vrstevnatými mřížkami. Teorie použité při výpočtech --- kinematická teorie, Bornova aproximace porušených vln, dynamická teorie, a různé aproximace dynamické teorie (aproximace jediného odrazu, dvouvlnná aproximace a aproximace vícenásobného rozptylu) --- byly vyvinuty v jednotném formalismu. Tento formalismus umožňuje diskutovat a porovnat všechny tyto teorie v jednotném a metodologickém stylu. Numerické výpočty byly použity při simulování experimentálních křivek pro nafitování strukturních parametrů vzorků.

Table of Contents

Title page



1 Introduction

Introduction (Francais)
Introduction (English)

2 Exploring reciprocal space

2.1 Introduction
2.2 Reciprocal space and angular representations
2.3 Experimental scans in the reciprocal space

3 X-ray reflectivity from planar multilayers

3.1 Introduction
3.2 Wave equation
3.3 Refraction index
3.4 Dynamical theory of X-ray reflection
3.4.1 Boundary conditions
3.4.2 Penetration depth
3.4.3 Fresnel coefficients
3.4.4 Reflection from a multilayer
3.4.5 Single layer
3.5 Kinematical theory of X-ray reflection
3.6 Multilayers with the layer sequence given by a deterministic rule
3.6.1 Periodic multilayer
3.6.2 Fibonacci multilayer
3.7 Conclusion

4 X-ray reflectivity from rough multilayers

4.1 Introduction
4.2 Statistical properties of rough interfaces
4.2.1 Description of a single rough interface
4.2.2 Description of rough interfaces in a multilayer
4.2.3 Root mean square roughness and the thickness fluctuations
4.3 Specular reflectivity from a rough multilayer
4.3.1 Dynamical calculation
4.3.2 Kinematical calculation
4.4 Specular reflectivity measurements
4.4.1 Instrumental factors in the specular reflectivity measurements
4.4.2 TbFe2 sandwich multilayer
4.4.3 NbSi periodic multilayers
4.5 Incoherent scattering: Distorted-wave Born approximation calculation
4.5.1 Distorted-wave Born approximation
4.5.2 Calculation of the coherent reflectivity
4.5.3 Diffuse scattering
4.6 Reflectivity map of a periodic multilayer

5 X-ray reflectivity from multilayer gratings

5.1 Introduction
5.2 Review of the published work
5.3 General characteristics of multilayer gratings
5.4 Kinematical theory
5.5 Distorted-wave Born approximation
5.5.1 Specular scan
5.5.2 Non-specular scans
5.6 Dynamical theory
5.6.1 Wavefields and the dispersion relation
5.6.2 Boundary conditions
5.6.3 Relation to the Fresnel coefficients
5.6.4 Two-beam case and multiple-beam approximations of the dynamical theory
5.7 Discussion of the reflectivity from a perfect rectangular surface grating
5.7.1 Intensity of the diffracted waves
5.7.2 Specular reflectivity
5.7.3 Azimuthal angle dependence
5.8 Other grating structures
5.8.1 Quasiperiodic gratings
5.8.2 Trapezoidal gratings
5.9 Rough gratings
5.9.1 Rough side walls
5.9.2 Rough interfaces
5.10 Experiment
5.11 Conclusion

6 Conclusion

Conclusion (Francais)
Conclusion (English)


This web-ed page of my thesis has been observed [counter] times since the last crash of this counter. The page has been created on 17. 3. 1997.