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Lecture Notes in Monographs Editorial Board
Beig, Vienna, Austria J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Zftrich, Switzerland R. L. Jaffe, Cambridge, MA, USA R.
Kippenhahn, G6ttingen, Germany Ojima, Kyoto, Japan H. A. WeidenmillIer, Heidelberg, Germany J. Wess, MiInchen, Germany J. Zittartz, K61n, Germany R.
I.
Managing Editor W.
Beiglb6ck
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technique
of
jean Daillant Alain Gibaud
X-Ray and Neutron Reflectivity: Principles and Applications
N
Q] -
-
04’
Springer
Author
Jean Daillant
Physique de I’Etat Saclay
Service de CEA
Condens6
F-9-ii9i Gif-sur-Yvette Cedex, France Alain Gibaud Laboratoire de
Physique de FEW Condens6,
UPRES A 6o87
Universit6 du Maine, Facult6 des sciences F-72o85 Le Mans Cedex 9, France
Library of Congress Cataloging-in- Publication Data. Die Deutsche Bibliothek
-
CIP-Einheitsaufnahme
and neutron reflectivity : principles and applications / Jean Daillant ; Alain Gibaud (ed.). Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris Singapore ; Tokyo Springer, 1999 (Lecture notes in physics : N.s. M, Monographs 58) ISBN 3-540-66195-6
X-ray
-
ISSN 0940-7677
(Lecture Notes in Physics. Monographs)
ISBN 3-540-66195-6
Springer-Verlag Berlin Heidelberg New York
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Foreword
The reflection of x-rays and neutrons from surfaces has existed as an experimental technique for almost fifty years. Nevertheless, it is only in the last decade that these methods have become
enormously popular
as
probes of
surfaces and interfaces. This appears to be due to the convergence of several different circumstances. These include the availability of more intense neutron and x-ray
of
magnitude
sources
(so
that
reflectivity
can
be measured
and the much weaker surface diffuse
detail);
over
scattering
many orders
can now
also be
growing importance of thin films and multilayers in both technology and basic research; the realization of the important role which roughness plays in the properties of surfaces and interfaces; and finally the development of statistical models to characterize the topology of roughness, its dependence on growth processes and its characterization from surface scattering experiments. The ability of x-rays and neutro4s to study surfaces over four to five orders of magnitude in length scale regardless of their environment, temperature, pressure, etc., and also their ability to probe buried interfaces often makes these probes the preferred choice for obtaining global statistical information about the microstructure of surfaces, often in a complementary manner to the local imaging microscopy techniques, This is witnessed by the veritable explosion of such studies in the published literature over the last few years. Thus these lectures will provide a useful resource for students and researchers alike, covering as they do in considerable detail most aspects of surface x-ray and neutron scattering from the basic interactions through the formal theories of scattering and finally to specific applications. studied in
some
the
weakly with surfaces the kinematic theories that so simple weakly enough general of scattering are good enough approximations to describe’the scattering. As most of us now appre, iate, this is not always true, e.g. when the reflection is close to being total, or in the neighborhood of strong Bragg reflections (e.g. from multilayers). This necessitates the need for the full dynamical. theory (which for specular reflectivity is fortunately available from the theory of optics) or for higher-order approximations, such as the distorted wave Born approximation to describe strong off-specular scattering. All these methods are discussed in detail in these lectures, as are also the ways in which the magnetic interaction between neutrons and magnetic moments can yield informaIt is often assumed that neutrons and x-rays interact
and in
interact
I
VI
tion the this
Foreword
on the magnetization densities of thin films and multilayers. I commend organizers for having organized a group of expert lecturers to present subject in a detailed but clear fa..shiou, as the importance of the subject
deserves.
S. K. Sinha
Advanced Photon Source
Argonne
National
Laboratory
December 1998
Contents
Part I.
1
Principles
Interaction of X-rays (and Neutrons) with FranVois de Bergevin
1.1
Introduction
1.2
Generalities and Definitions 1.2.1
Conventions
..................................
4
1.2.4
Scattering
........................................
and Flux
a
.............................
L,ength and Cross-Sections
1.2.5 The Use of Green
in
Functions
9
............................
11
Field
...............................................
16
Optical
an
Object
to the
..........................................
Theorem and Its Extensions
1.3.3 The Extinction
.................
Approximation Stronger
and the Born
Length
1.3.4 When the Interaction Becomes
X-Rays
Electromagnetic Propagation
13
1.3.1 Introduction
1.4
1.4.1
General Considerations
1.4.2
Classical
by
Description:
Free Electron
23 24
Scattering Description: by the Electrons of an Atom, Rayleigh Scattering 1.4.4 Quantum Description: a General Expression for Scattering and Absorption 1.4.5 Quantum Description: Elastic and Compton Scattering 1.4.6 Resonances: Absorption, Photoelectric Effect 1.4.7 Resonances: Dispersion and Anomalous Scattering 1.4.8 ’Resonances: Dispersion Relations X-Rays: Anisotropic Scattering
1.A
31
.....
34
..............
38
.........
41
........................
43
...............................
48
..........................................
48
.......................
51
...........................
52
........................
54
Magnetic Scattering Magnetic Scattering 1.5.4 Templeton Anisotropic Scattering 1.5.5 The Effect of an Anisotropy in the Appendix: the Born Approximation Anne
28
...........................
1.5.2 Non-resonant 1.5.3
27
Thomson
..........
1.5.1 Introduction
26 26
Scattering
.....................................
1.4.3 Classical
1.5
17
.......
.................................
Thomson
16
..................
....................................................
a
7
....
Scattering by
1.3.2 The
5
....................
Green Functions: the Case of the
Medium
3
..........................................
Equation Intensity, Current
1.2.6
3
4
1.2.3
From the
.........
................................................
1.2.2 Wave
1.3
Matter
Resonant
Senienac, Fran ois
de
Alain Gibaud and Guillaume
Index of Refraction
.....
54
.........................
56
Bergevin, Jean Daillant,
Vignaud
VIII
Contents
Statistical Aspects of Wave Senienac, Jean Daillani
2
Scattering
Rough
Surfaces
.
60
....................
60
......................
61
at
Anne 2.1
Introduction
2.2
Description of RandomlyRough Surfaces
...........................
2.2.1 Introduction ’2.2.2
2.2.3 2.2.4
....................
61
Height Probability Distributions Homogeneity and Ergodicity The Gaussian Probability Distribution
.........................
61
............................
63
and Various Correlation Functions
65
.....................
.......................
Complicated Geometries: Multilayers and Volume Inhomogeneities Description of a Surface Scattering Experiment,
2.2.5 More 2.3
Coherence Domains
................................
70
....................................
72
2.3.3
Coherence Domains To What Extent Is
a
Notions
...............
74
Statistical Formulation
of the Diffraction Problem Relevant? 2.4.2
68 69
2.3.2
2.4.1
67
...................................
Statistical Formulation of the Diffraction Problem
2.4
...........
.........................................
Scattering Geometry Scattering Cross-Section
2.3.1
......
(Specular) (Diffuse) Intensity
.....................
74
.......................
79
Coherent
on
and Incoherent
Statistical Formulation of the Scattered Intensity
2.5
Under the Born
...............................
80
.................
80
...................................
83
Approximation Scattering Cross-Section
2.5.1 The Differential
Flat Surfaces
2.5.2
Ideally
2.5.3
Self-Affine
Rough
Specular Reflectivity
3
Surfaces
..............................
from Smooth and
Rough
Surfaces
83
...
87
Alain Gibaud 3.1
.............
87 87
....................................
88
Concepts 3.1.2 Fresnel Reflectivity
3.1.1
Basic
Ideally Flat Surface
........................................
The Reflected Intensity from
an
............................
96
.................................
96
3.1.3 TheTransmission Coefficient
3.1.4 The Penetration 3.2
Depth
X-Ray Reflectivity in Stratified Media
.........................
99
99 :’’*--*******’ 103 3.2.2 The Refraction Matrix for X-Ray Radiation 104 3.2.3 Reflection from a Flat Homogeneous Material
3.2.1 The Matrix Method
..................
...............
..............
3.2.4 A
Single Layer
3.2.5 Two Layers 3.3 3.4
3.A
on a
on a
...........................
105
.............................
106
Substrate
Substrate
Dynamical to Kinematical Theory Influence of the Roughness on the Matrix Coefficients Appendix: The Treatment of Roughness in Specular Reflectivity Frangois de Bergevin, Jean Daillani, Alain Gibaud
.......................
108
...........
113
and Anne Sentenac
116
From
.........................................
Contents
Rayleigh Calculation Grating Treatment of Roughness in Specular Reflectivity
IX
3.A.1 Second-Order for
a
3.A.2 The
Sinusoidal
.................................
*
.......
118
...............................
118
.........................................
121
within the DWBA
..............................
3.A.3 Simple Derivation of the Debye-Waller and Croce-N6vot Factors 4
Diffuse
Scattering
116
Daillani, Anne Senienac Differential Scattering Cross-Section 4.1.1 Propagation Equation 4.1.2 Integral Equation
Jean 4.1
.................
122
..................................
122
......................................
123
for
X-Rays
4.1.3 Derivation of the Green Functions
Using
the
Reciprocity
4.1.4 Green Function in 4.1.5
a
Green Function for
Vacuum
a
..........................
125
............................
126
Theorem
Stratified Medium
Scattering Cross-Section Approximation .................................... 4.2.1 Expression of the Differential Scattering Cross-Section 4.2.2 Example: Scattering by a Single Rough Surface Distorted-Wave Born Approximation 4.3.1 Case of a Single Rough Surface 4.1.6 Differential
4.2
4.3
...................
.....................
First Born
4.3.2 General Case of
a
127 128
130
.....
130
............
131
..........................
132
..........................
133
Stratified Medium
.....................
134
.
.........
137
..........................................
140
4.3.3 Particular Case of
a
Film
......................
4.4
Polarisation Effects
4.5
Scattering by Density Inhomogeneities 4.5.1 Density Inhomogeneities in a Multilayer 4.5.2 Density Fluctuations at a Liquid Surface Further Approximations The Scattered Intensity ....................................... 4.7.1 Expression of the Scattered Intensity
4.6 4.7
.........................
141
..................
141
.................
142
.....................................
143
.....................
4.7.2 Wave-Vector Resolution Function Revisited
........................
4.1)
146 148
...........................
150
.................
152
..............................
153
4.A
Within the Born
144
........................................
Reflectivity Appendix: the Reciprocity Theorem 4.B Appendix: Verification of the Integral Equation in the Case of the Reflection by a Thin Film on a Substrate 4.C Appendix: Interface Roughness in a Multilayer 4.8
144
Approximation
Appendix: Quantum Mechanical Approach of Born and Distorted-Wave Born Approximations ..............................
155
........................................
155
Tilo Baumbach and Peir Mikulik 4.D.1
Formal
Theory
4.D.2 Formal Kinematical Treatment
by First-order
Born
Approximation
......................
157
X
Contents
4.D.3 Formal Tteatment
by 5
Neutron
a
Distorted Wave Born
Reflectometry
.
Approximation
................
158
...................................
163
Claude Fermon, Fir6d&te Ott and Alain Menelle 5.1 Introduction
................................................
5.2
..........
5.2.2 Neutron-Matter Interaction 5.3
.............................
Non-magnetic Systems Optical Indices 5.3.2 Critical Angle for Total External Reflection .5.3.3 Determination of Scattering Lengths and Optical Indices 5.3.4 Reflection on a Homogeneous Medium Neutron Reflectivity on Magnetic Systems Reflectivity
on
.........................
5.3.1 Neutron
5.4
163
165 Schr,5dinger Equation and Neutron-Matter Interactions 5.2.1 Schr6dinger Equation ................................... 165 165
168
................................
168
...............
169
....................................
170
....................
171
.....................
172
.....................
172
.....................
174
......................................
175
5.4.1 Interaction of the Neutron with
an
Infinite
5.4.2 Solution of the 5.4.3 5.4.4
General Solution
Continuity
5.4.5 Reflection 5.5
Homogeneous Layer Schr6dinger Equation
Conditions and Matrices on a
Magnetic Dioptre
......................
176
........................
179
Non-perfect Layers, Practical Problems and Experimental Limits
.....................................
5.5.1 Interface 5.5.2
Angular
Roughness Resolution
183
....................................
185
..........................
186
..........................................
186
..........................................
186
Analysis of Experimental 5.6 The Spectrometers 5.5.3
5.6.1 Introduction 5.6.2 Time of
Flight
Data
Reflectometers
5.6.3 Monochromatic Reflectometers 5.7 5.8
187
..........................
187 188
...............................
189
5.8.1 Absolute Measurement of
Bragg
...........................
..........................................
Polymer Examples Examples on Magnetic Systems 5.8.2
183
.....................................
Peaks of
a
Multilayers
............
189
.............................
190
Magnetic
Moment
5.8.3 Measurement of the In-Plane and Out-of-Plane Rotation of Moments. Measurement of the Moment Variation
Single Layer ....................................... Hysteresis Loops Conclusion on Neutron Reflectometry in
a
5.8.4 Selective 5.9
191
..............................
193
.........................
194
Contents
Part II.
6
XI
Applications
199 Physics at Crystal Surfaces Pimpinelli 199 Surface Thermodynamics 199 6.1.1 Surface Free Energy 200 6.1.2 Step Free Energy 6.1.3 Singularities of the Surface Tension ....................... 201
Statistical
.....................
Alberto 6.1
....................................
...................................
......................................
6.1.4 Surface Stiffness
.......................................
6.1.5 Surface Chemical Potential 6.2
Surface
.............................
a Crystal Morphology 6.2.1 Adatoms, Steps and Thermal Roughness of 6.2.2 The Roughening Transition 6.2.3 Smooth and Rough Surfaces 6.2.4 Diffraction from a Rough Surface 6.2.5 Capillary Waves Surface Growth and Kinetic Roughening 6.3.1 Equilibrium with the Saturated Vapour 6.3.2 Supersaturation and Vapour Deposition 6.3.3 Nucleation on a High Symmetry Substrate
of
..............................
Scaling
.............................
206
............................
206
.......................
208
Surface
210
..................
210
.................
210
................
211
..........................
...............................................
Experiments
on
210
.......................
6.3.6 Surface-Diffusion-Limited Growth Kinetics 7
204
204
6.3.4 Kink-Limited Growth Kinetics 6.3.5
202
......
a
.......................................
6.3
201
................
212
213 215 217
Solid Surfaces
Jean-Marc Gay and Laurent Lapena 7.1
....................................
217
...............................
217
..........................
223
Experimental Techniques Reflectivity Experiments 7.1.2 Roughness Investigations with Other Experimental Tools Examples of Investigations of Solid Surfaces/Interfaces 7.2.1 Co/Glass Self-Affine Gaussian Roughness 7.1.1
7.2
-
............
...............
8
223
7.2.2 Si Homoepitaxy on Misoriented Si Substrate. Structured Roughness
226
Conclusion
229
..................................
7.3
223
.................................................
X-ray Reflectivity by Rough Multilayers
..................
232
Tilo Baumbach and Petr Mikul%*k 8.1
Introduction
8.2
Rough Multilayers Multilayers 8.2.2 Multilayers with Rough Interfaces 8.2.3 Correlation Properties of Different Interfaces Setup of X-Ray Reflectivity Experiments
Description
................................................
of
8.2.1 Ideal Planar
8.3
.............................
................................
232 234 235
.......................
235
..............
235
.......................
237
XII
Contents
Experimental Setup Experimental Scans 8.4 Specular X-Ray Reflection 8.4.1 Roughness with a Gaussian Interface 8.3.1
....................................
8.3.2
...............................
Distribution Function 8.4.2
Stepped
Surfaces
238
....
239
....................................
241
;
..................................
241
......................................
248
8.4.3 Reflection
by "Virtual Interfaces" Between Porous Layers 8.5 Non-specular X-Ray Reflection 8.5.1 Interfaces with a Gaussian Roughness Profile 8.5.2 The Main Scattering Features of Non-specular Reflection by Rough Multilayers 8.5.3 Stepped Surfaces and Interfaces 8.5.4 Non-coplanar NSXR 8.6 Interface Roughness in Surface Sensitive Diffraction Methods 8.7 X-Ray Reflection from Multilayer Gratings
.................................
249
...............................
250
..............
251
..........
254
.........................
259
...................................
262
.....
264
.....................
266
.................................
267
............................................
272
8.7.1 Theoretical Treatments 8.7.2
Discussion
Reflectivity from Rough Multilayer Gratings Appendix: Reciprocal Space Constructions for Reflectivity 8.7.3
8.A
..............
8.A.1 Reflection from Planar Surfaces and Interfaces 8.A.2 Periodic
Multilayer Space Representation of DWBA
Reflectivity
of
Liquid
275
............
275
....................................
277
...............
278
,8.A.3 Reciprocal 9
273
.......
Surfaces and Interfaces
..............
281
.......................
281
Jean Daillant
Statistical
9.1
9.1.1
Description of Liquid Capillary Waves
9.1.2
Relation to Self-Affine Surfaces
Surfaces
.......................................
..........................
Bending Rigidity Experimental Measurement of the Reflectivity of Liquid Surfaces 9.2.1 Specific Experimental Difficulties 9.2.2 Reflectivity 9.2.3 Diffuse Scattering Some Examples 9.3.1 Simple Liquids Free Surface 9.3.2 Liquid Metals 9.3.3 Surfactant Monolayers Liquid-Liquid Interfaces 9.1.3
9.2
9.3
9.4
......................................
286
...........................................
287
........................
287
...........................................
288
.....................................
290
.............................................
294
.............................
295
.........................................
296
.................................
297
.....................................
300
.........................................
305
..............................................
305
Polymer Studies
10
282
285
G?inier Reiter 10.1
Introduction
10.2
Thin
Polymer
Films
.......................................
306
Contents
...................................
310
..................................
314
..........................................
315
10.3
Polymer,Bilayer Systems
10.4
Adsorbed Polymer Layers Polymer Brushes
10.5 10.6 10.7 10.8
Xiii
...................................
319
.....................................
320
.....................................
321
Interfaces
Polymer-Metal Spreading of Polymers Dewetting of Polymers
Main Notation Used in This Book
.............................
32.5
Index .......................................................... 327
List of authors
*
Dr. T. Baumbach
0
freie
Prof. A. Gibaud
Laboratoire de
Zerst6rungsPriifverfahren, EADQ Dres-
Fraunhofer Institut
Physique de FEW
Condens6, UPRESA 6087 Universit6 du Maine Facult6 des
den
Kriigerstraoe
sciences,
22
Dresden, Germany Present address: European Synchrotron Radiation Facility BP 220, F-3.8043, Grenoble Cedex France
72085 Le Mans Cedex
D-01326
*
France
Dr. L.
Lapena CNRS, Campus de Luminy
CRMC2
case
13288 Marseille Cedex
Dr. F. de
Bergevin Laboratoire de Cristallographie associ6 I’Universit6 Joseph Fourier
9,
40
9,
913, France
Dr. A. Menelle
CNRS Bitiment F
Laboratoire L6on Brillouin CEA
des martyrs, B.P. 166 38042 Grenoble Cedex 09, France
CNRS, CEA Saclay
25
avenue
91191 Gif sur Yvette
Cedex,
France
and
European Synchrotron Radiation Facility B.P. 220, 38043 Grenoble Cedex, France 0
Physique
dens6,- Orme CEA Saclay
Dr. P. Mikulik
Laboratory
de FEW Con-
91191 Gif sur Yvette
Cedex,
France
of Science
Masaryk Uni-
Kotliiski 2 611 37
des Merisiers
of Thin Films and Nanos-
tructures
Faculty versity
Dr. J. Daillant
Service de
0
Brno, Czech Republic
Dr. F. Ott
Laboratoire L6on Brillouin CEA Dr. C. Fermon
CNRS, CEA Saclay
Service de Physique de FEW Condens6, Orme des Merisiers
91191 Gif sur Yvette
CEA Saclay 91191 Gif sur Yvette Cedex, France
Prof. A.
Cedex,
Pimpinelli
LASMEA
Universit6 Blaise Pascal Dr. J.M.
Campus
Gay Luminy,
-
Cler-
mont 2
Les C6zeaux
CRMC2 CNRS, de
France
case
13288 Marseille Cedex
9,
913
France
63177 Aubi6re
Cedex,
France
XV1
List of authors
Dr. G. Reiter
Dr. G.
Institut de Chimie des Surfaces et Interfaces
CNRS, 15 rue Jean Starcky, B.P. 2488 68057 Mulhouse, France Dr. A. Sentenac:
-LOSCM/ENSPM Universit6 de St J6r6me 13397 Marseille Cedex
20,
France
Vignaud Bretagne 4 rue Jean Zay 56100 Lorient, France Universit6 de
sud
Acknowledgement s
school on reflectivity held in Luminy, France, 13th, 1997. The editors are particularly grateful to the Universit6 du Maine (Le Mans, France), to the Direction des Sciences de la Mati6re,of the Commissariat h I’Energie Atornique (C.E.A.), to the C.N.R.S. (D6partement Sciences Physiques et Math6matiques) and to the R6gion des Pays de la Loire for their help and sponsoring of this summer school. Many thanks are esp’ecially adressed to all of those who made this meeting possible: A. Radigois from the "d6l6gation C.N.R.S. Bretagne-Pays de la.Loire" who very kindly suggested the location of the school and who helped us through the admisnistrative tasks, J. Lemoine who made a wonderful job as the school secretary, and G. Ripault for his technical support at Luminy and a perfect
This book folllows
a summer
from June 9th to June
organisation of social events. We are also indebted to Dr. D. Bonhomme for helping us during the preparation of the manuscript, to Drs. N. Cowlam and T. Waigh for reading some chapters of this book, and to the staff of the C.N.R.S. center of Luminy for their hospitality.
Introduction
In his paper entitled "On
a
New Kind of
Ray, A Preliminary Communica-
relating the discovery of x-rays, which was submitted to the Wiirzburg Physico-Medical Society on December 28, 1895, R6ntgen stated the following about the refraction and reflection of the newly discovered rays: "The question as to the reflection of the X-ray may be regarded as settled, by the experiments mentioned in the preceding paragraph, in favor of the view that no noticeable regular reflection of the rays takes place from any of the substances examined. Other experiments, which I here omit, lead to tion"
the
same
conclusion.’"
This conclusion remained
unquestioned
out that if the refractive index of to be
ought
possible, according
reflection from
a
the
substance for x-rays
to the laws of
smooth surface of
optics,
tion that the reflection of x-rays
on
a
3
was
The demonstra-
obeying the laws and others who investigated the
surface
electromagnetism was pursued by Prins role of absorption on the sharpness of the limit
of
that it
to obtain total external
since the x-rays, on entering the submedium of smaller refractive index. This
it,
air, are going into a starting point for x-ray (and neutron) reflectivity.
stance from the was
a
2
Compton pointed is less than unity, it
until in 1922
was
indeed
of total reflection and showed
consistent with the Fresnel formulae. This work
was
continued
4
using nickel films evaporated on glass. Reflection on such thin films gives rise to fringes of equal inclination (the "Kiessig fringes" in the xthin film thicknesses, now the ray literature) which allow the measurement of most important application of x-ray and neutron reflectivity. It was, however, not until 1954 that Parratt5 suggested inverting the analysis and interpreting models of an inhomox-ray reflectivity as a function of angle of incidence via method The distribution. was then applied to several surface-density geneous by Kiessig
cases
of solid
or
liquid’
that "it is at first
interfaces. Whereas Parratt noticed in his 1954 paper that any experimental surface appears smooth
surprising
that, for good reflection, a mirror surface wavelength of the radiation involved..." appeared that effects of surface roughness were important, the most
to x-rays. One
frequently
hears
must be smooth to within about
it soon
A
more
complete
citation of
one
R6ntgen’s
paper is
given
in
an
appendix
to this
introduction.
Compton Phil. Mag. 45 1121 (1923) Prins, Zeit. f, Phys, 47 479 (1928); a very interesting developments is given in the famous book by R.W. James, of the diffraction of x-rays", Bell and sons, London, 1948. H. Kiessig Ann. der Physik 10 715 and 769 (1931). L.G. Parratt, Phys. Rev. 95 359 (1954). B.C. Lu and S.A. Rice, J. Chem. Phys. 68 5558 (1978).
A.H. J.A.
4 ,
6
J. Daillant and A. Gibaud: LNPm 58, pp. XIX - XXIII, 1999 © Springer-Verlag Berlin Heidelberg 1999
account of these
"the
early optical principles
XX
Introduction.
dramatic of them being the asymmetric surface reflection known
as
Yoneda
wings 7. These Yoneda wings were subsequently interpreted as diffuse scattering of the enhanced surface field for incidence or exit angle equal to the critical angle for total external reflection. The theoretical basis for the analysis of this surface diffuse scattering was established in particular through the pioneering work of Croce et al.’ In a context where coatings, thin films and nanostructured materials are playing an increasingly important role for applications, the number of studies using x-ray or neutron reflectivity dramatically increased during the 90’s, addressing vitually all kinds of interfaces: solid or liquid surfaces, buried solid-liquid or liquid-liquid interfaces, interfaces in thin films and multilayers9. Apart from the scientific and technological demand for more and more surface characterisation, at least two factors explain this blooming of x-ray and neutron reflectivity. First, the development of neutron reflectometers (Chap. 5) has been decisive, in particular for polymer physics owing to partial deuteration (Chap. 9), and an equally important contribution of neutron reflectivity can be expected for surface magnetism. Second, the use of 2nd and 3rd generation synchrotron sources has resulted in a sophistication of the techniq ue now such that not only the thicknesses but also the morphologies and correlations within and between rough interfaces can be accurately characterised for in-plane distances ranging from atomic or molecular distances to hundreds of microns. In parallel more and more accurate methods have been developed for data analysis. This book follows in June 1997. It i’s
school
a summer
organised
on
reflectivity held
into two parts, the first
in
Luminy (France)
being devoted to and the second the discussion to of one principles examples and applications. the school and the now Organising editing book, we had in mind that an number of are now non-specialists increasing using x-ray and neutron reflectometry and that the need for fulfilled. It is also true that
a
one
proper introduction to the field
if the
of
was
not
yet
principle reflectivity experiment is has the to measure extremely simple (one just intensity of a reflected beam), the technique is in fact really demanding. An important purpose of this book is therefore also to warn the beginners of experimental problems, often related to the experimental resolution, which are not necessarily apparent but may lead to serious misinterpretations. This is done in the second part of the book where specific aspects related to the nature of the samples are treated. An equally important purpose is also to share with the reader our enthusiasm for the many beautiful recent developments in reflectivity methods, and for the physics that can be can be done with it, and to give him/her the desire to do even more beautiful experiments. 7
8 9
even
a
Yoneda, Phys. Rev. 131 2010 (1963). N6vot, B. Pardo, C. R. Acad. Sc. Paris 274 803 and 855 (1972). For a recent review see for example S. Dietrich and A. Haase, Physics Reports 260 1 (1995) and the numerous examples cited in the different chapters of this Y.
P. Croce L.
book.
XXI
Introduction
As
,strongly suggested by
by considering
new
the short historical sketch
given above,
most
of x-rays (not only for interface studies) arise potentialitiesIO related to their nature of electromagnetic
of the revolutions in the
use
book therewaves, which was so controversial in the days of R6ntgen. The fore starts with a panorama of the interaction of x-rays with matter, giving both a thorough treatment of the basic principles, and an overview of more
topics like magnetic or anisotropic scattering, not only to give a following developments but also to stimulate reflection on new experiments. Then, a rigourous presentation of the statistical aspects of wave scattering at rough surfaces is given. This point, obviously important for understanding the nature of surface scattering experiments, as well as for their interpretation, is generally ignored in the x-ray literature (this chapter has been written mainly by a researcher in optics). The basic statistical properties of surfaces are introduced first. Then an ideal scattering experiment is described, and the limitations of such a description, in particular the fact that the experimental resolution is always finite, are discussed. The finiteness of advanced
firm basis to the
the resolution, leads to the introduction of ensemble averages for the calculation of the scattered intensities and to a natural distinction between coherent
(specular, equal to the average of the scattered field) and incoherent (diffuse, related to the mean-square deviation of the scattered field) scattering. These principles are immediately illustrated within the Born approximation in orcomplications resulting from the details of electromagnetic wave with matter. These more rigorous aspects of the scattering theory are treated in Chaps. 3 and 4 for specular and diffuse scattering. The matricial theory of the reflection of light in a smooth or rough stratified medium and its consequences are treated in Chap 3. This is used in Chap 4 for the treatment of diffuse scattering. The Croce approach to the distorted-wave Born approximation (DWBA) based on the use of Green functions is mainly used. This theory is currently the most popular for data analysis and is extensively used in the second part of the book’, which is devoted to applications, in particular in Chap 8. Howreviewed. The general ever, other methods used in optics are also shortly case of a stratified medium with interface roughness or density fluctuations is discussed using this DWBA, and different dynamical effects are discussed. Then, the theoretical aspects of a finite resolution function (the experimental aspects are treated in the second part of the book) are considered, as well as their implications for reflectivity experiments. The last chapter of this first part, principles, is devoted to neutron reflectometry whose specific aspects require a separated treatment. After an introduction to neutron-matter interactions, neutron reflectivity of non-magnetic materials is presented and the characteristics of the neutron spectrometers are given. Examples follow with der to avoid all the mathematical the interaction of
an
-
10
It is our opinion that fully exploiting reflectivity experiments would lead to
the
spectroscopic capabilities of interesting developments.
most
x-rays in
XXII
Introduction
particular emphasis put on the newly developed methods of investigation magnetic multilayers using polarised neutrons. The second part of the book is devoted to examples of the physics that can be done using x-ray eoad neuti7u, i -.reflectivity. The first three chapters are related to solid surfaces and multilayers, whereas the last two chapters deal with soft condensed matter. In both cases, a statistical description of the surfaces and of their properties is given first (Chap 6 and beginning of Chap 9) and examples follow. In (Ohap 7, the complete characterisation of the roughness of a single solid surface is considered. The experimental geometry, diffractometers, resolution functions are introduced first. Then, examples are given and and. the x-ray results are compared to the results obtained using complementary techniques like transmission electron microscopy and atomic force microscopy. More complicated cases of multilayers are discussed in Chap 8. The experimental setups’ are described and examples of reflectivity studies andnon-specular scattering measurements are discussed with the aim of reviewing all the important situations that can be encountered. Examples include rough multil4yers, stepped surfaces, interfaces in porous media, the role of roughness in diffraction experiments and multilayer gratings. Examples in soft condensed matter include liquid interfaces and polymers. This is a domain where the impact of reflectivity measurements has been very large because many of the very powerful complementary techniques which can be used with solid surfaces require high vacuum, and cannot be used for the characterisation of liquid interfaces. The specific aspects of liquid interface studies (mainly using x-rays) are discussed first. Experimental setups for the study of horizontal interfaces are described, and the implications of the specific features of liquid height-height correlation functions for reflectivity experiments are described. Examples include liquid-vapour interfaces, organic films at the air-water interfaces, liquid metal surfaces, and finally buried liquid-liquid interfaces. Finally, polymers at interfaces are considered in a last chapter. This is a domain where neutron reflectivity has made an invaluable contribution, in particular owing to the transparency of many materials to neutrons and to the possibility of contrast Variation. a
of
J. Daillant and A.
Saclay and May 1999
Le
Mans,
Gibaud,
Introduction
Appendix: R5ntgen’s report
on
the
importance from
of the
medium into
reflection of x-rays.
conditions here involved
general question
"With reference to the to the
XXIII
whether the
X-rays
on
can
the other
be refracted
it is most fortunate that this
another,
hand, and or
not
subject
on
passing be investigated in still another way than with the aid of prisms. Finely divided bodies in sufficiently thick. layers scatter the incident light and allow only a little of it to pass, owing to reflection and refraction; so that if powders are as transparent to X-rays as the same substances are in mass-equal amounts of material being presupposed-it follows at once that neither refraction nor regular reflection tak.es place to any sensible degree. Experiments were tried with finely powdered rock
one
may
salt, with finely electrolytic silver-powder, and with zinc-dust, such as is used investigations. In all these cases no difference was detected between the
in chemical
transparency of the powder and that of the substance in
mass, either
by observation
photographic plate... The question as to the reflection of the X-ray may be regarded as settled, by the experiments mentioned in the preceding paragraph, in favor of the view that no noticeable regular reflection of the rays takes place from any of the substances examined. Other experiments, with the fluorescent
which I here
with the
screen or
omit, lead
to the
same
conclusion.
as at first sight opposite. I exposed to the X-rays a photographic plate which and the glass side of which was turned was protected from the light by black. paper, towards the discharge-tube giving the X-rays. The sensitive film was covered, for the most part, with polished plates of platinum, lead, zinc, and aluminum, arranged in the form of a star. On the developing negative it was seen plainly that the dark-
One observation in this connection
it
seems
should, however, be mentioned,
to prove the
ening under
the
under the other
platinum,
particularly having exerted
the lead and
the aluminum
the
zinc,
was
stronger than
action at all. It appears,
plates, therefore, that these metals reflect the rays. Since, however, other explanations of second experiment, in order to be sure, I a stronger darkening are conceivable, in a metal the film and the sensitive plates a piece of thin aluminum-foil, between placed which is opaque to ultraviolet rays, but it is very transparent to the X-rays. Since the
same
result
substantially
the metals above named is
was
again obtained,
proved.
already mentioned that powders
are
If
we
as
no
the reflection of the X-rays from
compare this fact with the observation
transparent
as
coherent masses, and with
rough surfaces behave like polished bodies with reference to the passage of the X-rays, as shown as in the last experiment, we are led to the conclusion already stated that regular reflection does not take place, but that bodies behave toward X-rays as turbid media do towards light." the further fact that bodies with
X-rays (and Neutrons)
The Interaction of
1
with Matter
n,an ois
Bergevin
de
Cristallographie
Laboratoire de
assock
Joseph Fourier, CNRS,
l’Universit6
des martyrs, B.P. 166, 38042 Grenoble Cedex 09, France F, and European Synchrotron Radiation Facility, B.P-. 220, 38043 Grenoble Cedex,
B,itiment
25
avenue.
France
Introduction
1.1 The
propagation
generally presented according to an optical properties of a medium are described by a refractive the refractive index is sufficient to predict what will
of radiation is
formalism in which the
knowledge of happen at an interface,
index. A
that is to establish the Snell- Descartes’ laws and to
calculate the Fresnel coefficients for reflection and transmission. One of the objectives in this introduction will be to link the laws of propaof radiation and in particular the refractive index, to the fundamental
gation phenomena involved
in the interaction of radiation with matter. The main of the electromagnetic spectrum is process of interaction in the visible region for least an insulator). At higher energies molecules the polarisation of the (at
with x-rays, it is generally sufficient to take into account the interactions with the atoms and at the highest x-ray energies only the electrons need be considered in the interaction process. It is the nuclei of the materials which inas
teract with
neutrons, which also have
The conventions and
defined in Sect. 1.2. In the
a
second interaction with the electrons
magnetic moment. symbols which will be
for those atoms which carry
a
same
physical quantities introduced, together with
the
of Green functions. In Sect.
properties atomic scattering and the model and the
propagation will be scattering appropriate definitions
which characterise the
revised and the different of radiation will be
used in this book will be
section the basics of wave
of
a
1.3 the link between the
continuous medium
represented by
1.4 will be devoted to the a refractive index will be established. Section will include the inelastic That matter. with radiation of interaction x-ray will be described The the and elastic scattering and absorption. scattering, as
split into
a non
resonant and
a
resonant
part. Together with the questions
dispersion relations will be absorption the when the case scattering depends on the anisotropy 1.5, given. of the material will be briefly examined with reference to magnetic and to Templeton scattering. Neutron scattering will not be presented in detail in this chapter since it will appear in Chap. 5 of this book but we shall frequently
of
resonance
and
a
discussion of the
In Sect.
refer to it.
J. Daillant and A. Gibaud: LNPm 58, pp. 3 - 59, 1999 © Springer-Verlag Berlin Heidelberg 1999
Frangois
4
In the
de
present chapter, the bold italic font will be used
expressions
or
Bergevin
and the
emphasiZed
to
define
words
sentences will be in italic.
Generalities and Definitions
1.2
Conventions
1.2.1
Two conventions
be found in the literature to describe
can
a
propagating
wave, because complex quantities are not observed and the imaginary part has an arbitrary sign.. In optics and quantum mechanics a monochromatic
plane
wave
is
generally
written
as
A
e- i(wt-k.r)
oc
which is also the notation used in neutron
tallography. On the the plane wave as,
other
hand, A
x-ray
The
imaginary part
of all
scattering, even when doing cryscrystallographers are used to writing
+i(wt-k.r).
oc
complex quantities
(1.2)
are
the
opposite
of
one
another
in these two notations. Since the observed real quantities may be calculated from imaginary numbers, it is very important to keep consistently a unique convention. The
imaginary part f"
of the atomic
scattering factor
for
exam-
in x-ray crystallography is a positive number. This is correct provided that it is remembered that the complex scattering factor (f + fl + ifll) (f is the atomic form factor, also positive) is affected by a common minus
ple, used
sign, usually
left
implicit. In optics, the opposite convention is commonly quantity is the refractive index. Its imaginary part which is associated with absorption is always positive. The number of alternative choices is increased with another convention concerning the sign of the scattering wave-vector transfer q or scattering vector, which can be written as
used and the most useful
as
q
=
k,,,
-
kin
(1.3)
q
=
kin
-
k,,c
(1.4)
or
where kin and ksc the conventions
Only ter
one
5),
are
(1.3)
the incident and scattered and
(1.2)
exception will be made,
in which convention
describes the scattered be written in all
cases
used in
as
in
(1.1)
amplitude
(except
-_
vectors. In this
will be
book, adopted.
neutrons
(Chap-
will- be used. The structure factor which in the Born
with
f (q)
the
wave
crystallography, chapter devoted to
neutrons)
approximation will therefore
as,
I p(r) eiq.’dr
(1.5)
Interaction of
I
p(r)
where
X-rays (and Neutrons) with Matter
which will be discussed below. The real
scattering density,
is the
5
part of the refractive index is generally less than 1 with x-ray radiation and the refractive index is usually written as, n
I
--
-
6
where 6 and
i
-
0
are
(1.6)
positive.
imaginary part 0, equal to Ay/47r, is essentially positive (A is the and p is the absorption coefficient, see (1.84) and section 1.4.6). Note that because the opposite convention is used, the sign of the imaginary part of n is opposit ’e in visible optics. The waves will be assumed to be monochromatic in most instances, with the temporal dependence e’.". To satisfy the international standard of units, or SI units, the electromagnetic equations will be written in the rationalised
Indeed the
wavelength
MKSA system of units. The Coulombian force in
qq’147rEo r2
with -Opo
in
Propagation trons
or
which
be
propagation
a
yo
a
presented
in
propagation of a radiation whether neupartial differential equations
in
a common
form. We will discuss first the
Electromagnetic
radiation
0, 1, 2, 3)
defined
Ao
potential obeys AA,
--
(A,, A2, A3)
01c,
_o/-to , ’
potential
a
of
(1.7)
A,
-_
and A is the 3-vector
zA
potential. The charge
(9t2
02
1:
-
Xi=-C,Y,Z
For
case
represented by
in the Lorentz gauge and away from any
(92 A, =
be
by
where (P is the scalar electric 4-vector
can
potential
4-vector
A, (v
is in this system
series of second order
a vacuum.
the --
vacuum
47rlO-’.
-
Vacuum The
obeys
x-rays,
can
2,
Equation
Wave
1.2.2
c-.
=
5-X2i
I
EOILO ’
C2
Tf, the equivalent form of (1.8) is the Schr6dinger potential
neutron of wave function
equation without
any
h2
h,9Tf
2m
at
(using the convention of quantum mechanics for the sign of 2’, as discussed above). We shall consider essentially time independent problems and only then mono chromat ic’r adi ation which has frequency w127r. The time variable disappears from the equations, through 02
1
_W2
2 _C_ Ot2
.
ha
i-
C9 t
-
hw
h2 -
2m
of the relations
2
(electromagnetic field)
(1.10)
2
(Schr6dinger equation).
(1. 11)
ko
C2
use
ko
Frangois de Bergevin
ko is the the
wave
generic
vector in
field
a vacuum
or wave
function
and hw is the energy. In both cases, writing as A, yields the Helmholtz equation,
I/1 + _A
21 ko)
A
=
(1.12)
0.
The solutions to this
equation are plane waves with the wave vector ko. optics this equation is more usually expressed in terms of the electric and magnetic fields E and H, or the electric displacement and the magnetic induction D, B rather than the vector potential A,. E is related to the potential through In
OA
grad
E
-
c9t
-cgradAo
If the gauge is so that AO--O, E reduces to radiation is monochromatic, then E
--
0A -
c9t
-(OA/c9t).
If furthermore the
(1.14)
-iwA.
free field those conditions may usually be satisfied. Therefore, E and being proportional to each other, most of the discussion subsequent to
For A
a
equation (1.12) applies to E as well. Nevertheless, in the presence of electric charges, all the properties of the electromagnetic field cannot be described with the generic field written as a scalar. These particular vector or tensor properties will be addressed when necessary.
Propagation in
a
Medium Equation
(1. 12)
still
applies in
a
modified form
when the radiation propagates in a homogeneous medium rather than vacuum. All media are inhomogeneous, at least at the atomic scale, so for
even a
homogeneity will be taken as a provisional assumption whose justification will be discussed in Sect. 1.3. We also assume the isotropy of the medium, which is not the case for all materials. In the case of the electromagnetic radiation the medium is characterised by permeabilities e and p that replace -o and po in (1.8), although p can usually be kept unchanged. Though the static magnetic susceptibility can take different values in.various materials, we are concerned here with its value at the optical frequencies and above which is not significantly different from 1-to. In a medium equation (1.12) can be written as either, the moment the
( A + k 2)
A
--
(k
0
--
nko,
n
2
C[Z1,EO,,O
=
_
E/,EO)
(1-15)
or,
( A + ko2
-
U) A
The first form shows that the
=
wave
0
(U
-
vector has
2
ko
(I
-
n
2))
(1.16)
changed by a factor n, which is Schr6dinger equation
the refractive index. The second form is similar to the
Interaction of
I
X-rays (and Neutrons) with Matter
in the presence of a potential. Indeed in the case of the Schr6dinger equation, the material can be characterised by a potential V and the equation becomes, h 2m
which is
equivalent
to the
( A + k 2) 0
again
we
may define
1
2m
a
n2
h2
Yf
_
(1.17)
0
with
previous equation, U
and
+ V
(L i 8)
VI
refractive index =
I
2
-
Ulk 0
-
-
I
_
(1.19)
VlhW.
important to realise that describing the propagation in the medium by of the wave vector by a a Helmholtz equation, with just a simple change factor n or with the input of a potential U, is really just a convenience. In reality, each atom or molecule produces its own perturbation to -the radiation and the overall result is not just a simple addition of those perturbations. It happens in most cases that the Helmholtz equation can be retained in the form indicated above. How n or U depends on the atomic or molecular scattering has to be established. Before addressing this question we have to give some further definitions for the intensity, current, and flux of the radiation, and to introduce the formalism of scattering length, cross-section and Green functions which help to handle the scattering phenomena.. It is
1.2.3
Intensity,
Current and Flux
12
The square of the modulus of the field amplitude, i.e. JA , defines the intensity of the radiation, which is used to represent either the probability of of energy a quantum of radiation in a given volume or the density
finding transported by
the radiation.
vector direction to
correct in
vacuum
measure
JA 12
is also used when combined with the
the flux
density. These
but need to be revised in
a
definitions
are
wave-
trivially
material.
given surface is the amount of radiation, measured as unit time; an energy or a number of particles, which crosses this surface per that we shall current density this is a scalar quantity. The flux density or the electromagnetic energy flux also call the flow is a vector. For instance, in flux the an elementary surface do- is energy density (flow) is designed by S; to the flow by a relation connected is then S.do-. The density of energy u The
flux
across a
which expresses the energy conservation. The amount of energy which enters the energy inside that a given closed volume must be equal to the variation of volume:
aSX Ox
+
c9sy Oy
+
as, (9Z
Ou +
-
(9 t
0.
(1.20)
Frangois de Bergevin This
equation is no longer valid when the medium is absorbing. equation (1.20) can also be written in terms of the number of particles instead of the energy; for instance this is appropriate for the case of neutrons or for electromagnetic radiation if it is quantised. The same formalism stands for the flux, the density of current and the density of particles. The dimension of the density of flux is the one of the relevant quantity (energy, number of particles or other) divided by dimension L 2T. In the case of electromagnetic radiation, the quantities E, H, D and B, can be used instead of A as discussed above and the dielectric and magnetic Note that
-
permeabilities, and y, can density is then given by,
be used to characterise the medium. The energy
-
u
For
a
plane
wave
defined
(,-E.E*
--
by
pH.H*) /4.
+
H
and the energy
i along
the unit vector
(1.21) the
wave
vector,
(1.22)
E
x
density becomes, u
The energy flow is then S
=
equal
E
Note that these formulae
x
=
-
to the
H*/2
giving
(1.23)
Poynting
vector
cs-\I oyofty JE 12 i/2.
=
u
JE 12 /2.
and S
are
(1.24)
written in terms of
complex field
quantities
whose real part represents the physical field. The complex and the real formulations differ by a factor 1/2 in the expressions of second order in
the fields. The
change
in the
wave
vector
medium has been written above
so
that if y
--
po
(1.15), u
S
-
=
This shows that the
2
nc
(1.15)
k
=
n
_-
in
going
from
a
vacuum
into
in terms of the refractive index
nko
a
n
(1.25)
V______o P/ P 0
(1.26)
1
obtain
(,O/,to//,) JE 12 /2
-
(,-oyo/y) JEJ 2k/2
n260 JE 12 /2
--
nc,-o
JE 12 Q2.
(1.27) (1.28)
flux through a surface depends on both the amplitude E refractive index of the medium. A similar expression stands for neutrons (beware, in what follows as usual neutron physics i has the opposite sign). Here the probability density p
and in
n
we
length
also
on
the
and the current wave
a
density j
of
I Tf 12
j
particles
plane
-_
wave
Tfoe
,
P
(hil 2 m) (TfgradTf
--
ik.r
I TfO 12,
=
acterised
by
a
j
depends on potential V and
amplitude
both TfO and
n
and which from
+ V
to introduce
=
The above formulae
are
--
gradT/)
is the
along
(1.29)
.
k
rti:;
V
on
(1.30)
the medium which is char-
(1.31)
hw.
--
refractive index, which is
a
-
(1.19)
V
(1.32)
ru’.;
ko gives the length of the
j
*
(hk/m) I Tf012 .
--
h2 k2/2rn
optics, it is possible
TV
the unit vector
being
,
Here too, the current
As in
considered. The
are
9
function TV. P
For
X-rays (arid Neutrons) with Matter
interaction of
I
wave
vector k. Then
(nhko/m) I Tf012 j.
(1.33) it is
isotropic. When
valid when the medium is
anisotropic the flow of energy and the current are affected. In the electro-, magnetic case the direction of the flow does not always coincide with the direction of the
wave
vector-
Exercise 1.2.1. A beam a
impinging on amplitudes
transmitted beam. The
nel formulae
absorbing.
(see
section
As assumed
Check the conservation of the
Exercise 1.2.2. Let
jpoe(ik ,x-k z). 1.2.4
3.1.2).
a surface
us
consider
a wave
Calculate the current
Scattering Length
rise to
gives
of these beams
flux,
above,
are
reflected and
given by
the Fres-
the twa media
Tf, such
not
are ’
at least for the
function
a
as an
(s) polari’s’ation. evanescent
wave
density.
and Cross-Sections
scattering object (molecule, atom, electron), fully incident wave. The object reemits part of the incident radiation. We start with the assumption that its dimensions are small compared to the wavelength so that the scattered amplitude is the same in all the directions; for an extended object instead, direction-dependent phase shifts would appear between the scattered amplitudes coming from different regions in the sample. When examining the scattered amplitude at large distances r from the object, simple arguments yield the following expression of the scattered amplitude (see also the appendix 1.A to this chapter) Let
us
consider
an
isolated
immersed in the field of
an
A,3c
--
-Ailb
e-
ikr
r
(1-34)
Frangois de Bergevin
10
spherical symmetry (k and r are scalars), amplitude Ai,, and has locally the right wave-
Indeed this function which has the
proportional to the incident length 27r/k; the decay as a function is
of the inverse of
r guarantees the conserintensity decays as the inverse of the surface of a sphere of radius r. The remaining coefficient b has the dimension of a length; this coefficient characterises the scattering power of the sample and is the so-called scattering length. The notation b is rather used in the context of neutron scattering. Here we adopt it for x-rays as well. To be fully consistent with this notation we keep, as a mere convention, the minus sign in the definition of b. With this sign, the b value is positive for neutrons with most nuclei, and also for x-ray Thomson scattering. This length can have a complex value, since the wave can undergo a phase shift during the interaction process; we shall see that in our case if the sample is not absorbing then b is nearly real. A more rigorous justification of the expression (1.34) will be given in the next section. To justify that b has the dimension of a length, we have considered the
vation of the total flux since the related
scattered flux. The ratio of this flux to the incident
(flux density,
or
current)
has the dimension of
0-scat,tot This is the so-called total
47r
=
lb 12
a
one
per unit of surface
surface and is
general,
with
an
(1-35)
scattering cross-section
extended
the
object,
to
.
The scattered flux
the whole space is then equal to the one received by a surface which would be placed normal tothe incident beam. In
equal
equal
scattering depends fi, so that b, which
in
to crscat,tot
on
the direc-
depends fi, is written b(fi). Therefore it is useful to define a cross-section for this particular direction that is called the differential scattering cross-section tion of
observation, defined by
a
unit vector
also
on
(do-IdS?) (fi) which is for
a
equal
--
lb(fi) 12
to the measured flux in the solid
unit incident flux
(Fig. 1.1).
In this
case
(1.36)
angle
dQ directed towards
the definition
(1.35)
is
fi, replaced
by O"scat,tot
-
ff lb(fi) 12
(1-37)
dS?
integration is carried out over all the directions defined by fi. Any object (atom, molecule) also absorbs some part of the incident radiation without scattering it. Therefore one has to define the so-called crosssection of absorption, Cabs, equal to the ratio of the absorbed flux to the
where the
incident flux
notation
density.
(O"scat,tot)
We have used
in
to recall that it is
total cross-section
appellation,
(1.35) a
and
(1.37)
a
somewhat
scattering cross-section;
ut,,t, is also used to define the
clumsy
indeed the sum
of the
I
Interaction of
X-rays (and Neutrons) with Matter
11
dQ M
0
k
scattering length b(fl) and of the differential scattering -ik.r The incident plane wave is Ai,, and the scattered This last expression gives a well-defined flux in the wave Ai,#(fi)/OM)e-’kOM. The scattering length and the differential cone OM, whatever the distance OM. scattering cross-section in the direction Q are respectively b(d) and lb(fi) 12
Fig.
1.1. Definition of the
cross-section
(daldfl) (ft).
cross-section
concerning
inelastic
scattering);
all the interaction processes
it is the whole relative flux
O’scat,tot + Cabs-
O’tot
(absorption,
picked
up
by
the
elastic and
object,
(1.38)
The Use of Green Functions
1.2.5
scattering amplitude b in equation (1.34) has not been introduced very rigorously and it is possible to define it more formally. The field scattered by a point like object obeys the wave equation (1.15) everywhere except at the center of the object, which is both the source and a singular point. The simplest mathematical singularity is the Dirac J function. The Green function of equation (1.15), G(r) is a solution of the equation The
(, A + k2 )
G (r)
=
J (r).
(1.39)
Physically, G(r) represents the field emitted by the source normalised to unity. More, generally, any partial derivative equation which is homogeneous in A such
as
DA(x) (here plus
D a
--
0
(1.40)
represents a sum of differential operators with constant coefficients term, and x is a scalar or a vector variable), admits Green
constant
functions G which
satisfy
DG(x)
--
J(x).
Frangois de Bergevin
12
application of Green functions is the resolution of non homogeneous partial derivative equations. For example, if G(x) is a Green function and Ao (x) is any of the solutions of the homogeneous equation, the equation A
common
DA(x) admits the
following
can
be shown
--
Ao(x)
j G(x
+
by substitution
AX) and
(1.42)
solutions
A(x) This
f (x)
=
=
x) f (x’) dx’.
(1.42)
into
j 6(x
-
-
and
of the
use
x’)f (x’)dx’,
(1.43) equation,
(1.44)
by finally applying (1.41).
diverging wave (1.34) (or the converging wave is indeed, to a certain coefficient, a Green function solution of (1.39). Due to the spherical symmetry, it is worth using the spherical coordinates r, ft (r rfi; fi is defined by the polar angles 0, 0). The differential operators yield Let
having
us, now
the
check that the
opposite sign
k)
for
--
where d, and d2
perpendicular
grad
=
ZA
__
are
fi
a Or
Idi (0, 0),
+
02
2 0
Cqr2
fi.di
--
0
r
+ r
1
,_r
+
72
d2 (0)
0)
(1.45) (1.46)
1
differential operators relative to (0, 0) and d, is a vector moment, the above expressions are sufficient since
to ft. For the
only use functions having the spherical symmetry and therefore d, and d2 van-ish;’we shall also’use these expressions for less symmetrical functions, but in such a case we shall only consider the asymptotic behavior at large values of r where the exponent of 1/r is sufficient to make d, and d2 negligible. We then have exactly
we
ikr
(zA + k 2)
-
for
0
r
r
:A
0.
(1.47)
singularity with 6(r). It is possible to integrate a sphere of radius ro centered at the (1.47), of definition the Indeed from G, the integral of (A + k 2) G(r) must origin. be equal to 1 when performed over the whole volume including the origin. This calculation is proposed in the exercise 1.2.3 and yields -47r. The Green function of the, three dimensional Helmholtz equation is then At
r
=
0,
we
must compare the
the left hand side of
inside
G (r)
1 =
47rr
eikr
(1.48)
X-rays (arid Neutrons) with Matter
Interaction of
I
It is also useful to express the Green function in
specular reflectivity calculation yields
some
problems
can
Gld,(r) The Green function of Helmholtz
help harmonic, with with the
Tk
e
one
(1.49)
in two dimensions
equation
r-’-’ decay and
an
dimension. A similar
ikr
of Bessel functions. The a
dimension. Indeed in
one
be solved in
13
asymptotic additional phase
expressed large r is yet equal to (Ir/4). be
can
form at
shift
Exercise 1.2.3. Calculate in three dimensions
J’
+ k 2)
(1lr)e
’k’dr.
integral of the first term, A.... can be transformed into the integral of the gradient over the sphere of radius r ro ; the integral of the second, 2 k can be successively performed over spheres and then over r. Note that the independence of the result with respect to ro yields (1.47) and is sufficient Hints: The
-
....
to prove that this is
a
Green function.
Green Functions: the Case of the
1.2.6
While the Green functions of the Helmholtz
Electromagnetic
equation
are
Field
valid for scalar fields
electromagnetic field a is more complicated. potential is used) (if only which are the but also, represented by simplest sources are vibrating dipoles vectors and which cannot be described by a simple 6 function. The 4-vectors A, and j, represent respectively the potential, and the current-charge density, as follows as
for instance the neutron
wave
the field is
Not
Ao JO
function,
=
-
01c,
and j the electric current
it must fulfill the conservation
integrate
to
get
a
3-vector J. If
product
we
--
(il; j2) j3)
=:
as
1
NO
C
c9t
+
-
the current
consider
of the current
A
(1.50)
i
(1.51)
-
defined
(1.7),
describes the
p the
charge
charge
motion
relationship
J
is the
the
vector
previously density. Since j,
divj We shall have to
case
of the
(A,, A2, A3)
cp,
0 is the scalar electric poten tial
density
the
=
a
(1.52)
0.
density
over a
volume,
fj(r)dr conductor in which there is
intensity by
(1-53) a
the vector identified to
current, J a
portion
Frangois de Bergevin
14
a moving charge (an electron for instance), product charge by the velocity and if it is a vibrating dipole of amplitude d such as de iwt, ji (t) yields iwde"t. Such a vibrating dipole, if infinitesiinally small, is the sirnplestradiat-,,ng point source. It is characterised by the following current-charge density, which fulfills the conservation law:
of the conductor. If
J is the
consider
we
of the
Zc
io (r, t)
(J(t) (r))
’cJ(t).gradJ(r)
-_
(1.54)
J(t)J(r).
i(r,t) The
div
charge density jolc
(1-55)
has the form of the derivative in the direction
J, of
the scalar function V We write it into two different forms which both
are
useful. In the presence of the current-charge density j, , the potential A, (written help of the Lorentz gauge) verifies, instead of the four homogeneous
with
equations (1.8),
the
inhomogeneous ,,AA,
ones
I
&A,
C2
19t2
(1-56)
-
,OC2
as j, the dipole just described. We then keep as the useful soluA, those which have the same oscillating time dependence as j,. When
We take tions w
is
replaced by ck, (1.56)
transforms into the
inhomogeneous
Helmholtz
equations
(zA + k 2) Ao (r,t) (A + k 2 )
A
The solution of the second first
one
outgoing
can
be solved
by
So
J (t)
(r,t)
use
C2
,0
is
equation the
C2 k’
a
J(t).grad6(r)
(1.57)
6 (r).
(1.58) G(r) (1.48). proposed in (1.43).
Green function
of the method
The
The
solution is
Ao
(r,t)
A
(r,t)
E0 C2 k
J (t) 60 C2
where G_ is given by (1.48). Up to and (1.60) are the equivalent for the
J(t).gradG_(r)
(1-59) (1-60)
G
a
constant factor
I
J
I IcOC2, (1-59)
electromagnetic potential of
the Green
dipole, isolated in a vacuum, can be used to represent’what hapmicroscopic scale in a dielectric material in the range of a few atoms (for x-rays and any material the relevant scale lies inside a single atom). Once the average has been made over a larger volume, these microscopic currents disappear from the equations. They are implictly accounted for through the dielectric constant and the new fields D and H, otherwise equal to EoE and B/po. This is the point of view of chapter 4. This idealised
pens at
a
X-rays (and Neutrons) with Matter
Interaction of
I
15
particular expressions are due to both the dipolar character of the source. Other kinds of sources exist that we shall not describe here, as for example magnetic dipoles, or multipoles of higher order. One can also imagine the scalar field of multipole sources. For practical purpose we may need the electric field E. Following (1.13) field.
the scalar
function for
These
vector character of the field and to the electric
E
(r,t)
iW
J(t)G-(r)
-
nc2 0 C2
The second derivative
First,
since
we
I
(1.61)
T2 grad (J(t).gradG_(r))
+
gradJ.grad
can
be handled in two different ways. we look
often consider the radiated field far from the source,
an asymptotic value valid when kr >> 1. For this, the expression of the gradient (1.45) is used, but only the derivative according to r is kept, and in the derivative of G_ oc e-k’/r, only the derivative of e-ikr is calculated. All the other derivatives are of higher order in 1/kr. Thus we can write (fi is the unit vector along r)
for
grad (J (t).grad G_ (r))
-grad (J (t):hik
k
-
G_
(r))
k2 (J (t) fi) fi G.- (r),
(1.62)
-
kr-+c,o
and
The
the
recognize in this expression asymptotic form of E(r, t) is,
one can
(r, t) kr oo i.e. the scalar Green mal to
-
[J(t)
-
projection
(J(t)’16L)41
function multiplied by
of J
on
the vector
Zwe
(1.63)
47r,-o C2r
the component
r.
of the
current
nor-
fi and by iw1coc 2.
An other way to transform expression (1.61), now without any approximation, relies on the alternative form graddiv(JG-) (this equivalence is given by (1.54)) for the second derivative term. The following equation is also identically valid
grad div
=-
zA + curl
and since G_ is solution of Helmholtz be
replaced by
-k 2. As
E (r,
t)
=
a
result
we
curl curl
(1.64)
curl,
equation
away from the
origin,
A may
have
J (t)
ie- ikr
47r6owr
)
for
r
:A
0.
(1-65)
Frangois de Bergevin
16
1.3
Scattering by Propagation in a
Object
From the
an
to the
Medium
Introduction
1.3.1
progressive plane wave in a vacuum, at some fixed time, is replaced by an assembly of scattering objects, the field is modified by the scattered waves. We have said that, when the medium is sufficiently homogeneous, tbe result is still a plane wave. The wave in the medium has a different wavelenogth and the amplitude becomes e- inkox We shall derive this result ai.,id sho-sv what is meant by the term "sufficiently homogeneous". In addition, we shall link the optical index n to the scattering lengths of the ,obj ects. A way to approach that problem is to search directly for a partial derivative equation which is satisfied by the total field (incident field Ai" plus scattered As,). From the definition of the scattering length b (1.34) and the The
amplitude
is e- iko.r
.
of
If the
a
vacuum
.
Green function
(1.39, 1.48),
by Ai, independent
of the
equation (1.12) satisis verified if b is equation following
and from the Helmholtz
it may be shown that the
fied
scattering direction
[A
2
+ k0
-
47rp (r) b] (Ai,,
+
A,,)
--
(1.66)
0.
density of objects of scattering length b. This question is Appendix I.A to the present Chapter, about the Born apin If this equation p(r)b can be replaced by its space average proximation. refractive index n. obtain for the we pb, Here
p(r)
is the
discussed in the
n2 The
equation (1.66)
=
I
is exact for
-
a
(41r/k2) 0
(1.67)
pb.
scalar field with b
independent
of the
of neutrons and x-rays, because they small values of pb, the average can be made safely in most cases and the forward scattering is relevant (see next section). With these small
direction. It turns out that in the
give only values,
case
the vector character of Uhe
electromagnetic
radiation does not make
any difference and the same formula (1-67) applies to x-rays as well as to neutrons. Also its last term is small 9nough to make equivalent the writing
n’-- I A discussion
specific
to the x-ray
-
(27r/k2) 0
case
pb.
(1.68)
is found in the treatise of Landau and
[1], Although the method that we have just sketched is straightforward and gives the right answer, once the approxirnations are accepted as valid, we shall discuss the problem along another line. It will give a more intuitive insight to the building up of the total field from the scattered field. It will also give more handles to grasp and understand the role of the smallness of pb and of the Lifshitz
section 97.
and to discuss the
homogeneity, "small"
we
shall
see
with Matter
17
approximations. For pb to be considered as approximation must apply to a scattering least of the order of the wavelength and that
that the Born
volume whose dimensions
great simplifications
X-rays (and Neutrons)
Interaction of
I
are
at
then follow.
optical theorem (section 1.3.2), which exactly links an isolated object to its forward scattering length. of the optical theorem, valid under approxextension then We propose an imations which are discussed with some details. This extension yields the formula (1.68). The equivalence between those approximations and the Born approximation is commented (section 1.3.3). Finally (section 1.3.4) we shall discuss briefly the case of a strong interaction where these approximations are no longer valid. We start with the
the total cross-section of
1.3.2
The
Optical
Theorem and its Extensions
relation, known as the optical theorem, exists an object and its forward scattering amplitude. It is worth examining this relation because it looks like a partial solution to the problem of finding the optical index of a medium made of such objects. Indeed the total cross-section of the objects (atoms or molecules) which constitute a medium, approximately yields the attenuation of that medium, then the attenuation is linked to the imaginary part of the index. What we call attenuation includes the absorption and the loss of radiation due to scattering out of the direction of propagation. The relation between the molecular total cross-section and the attenuation is only approximate because the scattering cross-sections of all the molecules cannot.be simply summed, as can be the absorption. The optical theorem exactly relates the total cross-section (absorption and scattering) c-t,,t of an isolated object, with the imaginary part of the amplitude that this object scatters in the forward direction, i.e. Im [b(O)].
Optical
The
Theorem A
between the total cross-section of
at,,t
Here b may
depend
on
the
=
scattering direction, relies
(1.69)
2AIm[b(0)]. the
but
only
its value at
of the Green theorem
zero
applied
proof surrounding the object (see reference [21, section 9.14); we stress that no approximation iS’made. When the field, such as an electromagnetic field, has several components, b(O) represents the scattering having the same polarisation as the incident wave. We’cannot make use of the optical theorem as it is. First because it gives no access to the real part of the index, and second because of the approximation made when going from the cross-section of the objects to the
angle
to
a
is relevant. The
on
use
volume
attenuation of the medium. Its validity can hardly be discussed directly. We present a similar relation instead, which yields the complete refractive index of
a
medium,
with its real and
imaginary part. Again
it is related to the
Frangois de Bergevin
18
scattering on
in the forward direction. That calculation is not exact, but relies pb. We shall call this condition, "the weak interaction".
the smallness of
It is
equivalent
to
saying
that the index is close to 1. In
fact, for
x-rays and
neutrons, the difference to I is of the order of 10-5, or even less. This is not true for visible light, and in this case other formulations must be used. We write below the formulae for
interaction is
weak,
demonstration follows Jackson
Amplitude
scalar field. Under the condition that the
a
those will also be valid for the
electromagnetic
field. The
[2].
Scattered
by a Planar Assembly of Scattering Obpopulation of scattering objects homogeneously located in the surface of a plane P normal to the direction of propagation of the incident plane wave (Fig. 1.2). We shall consider the amplitude of the wave at a point M, far enough behind this plane but not at an infinite distance (like in Fresnel diffraction). The field is supposed to be a scalar. The
jects
Let
us
consider
a
P
plane wave coming from the left encounters a plane P containing an objects. The axis OM is normal to the wave planes. The value of scattering array of the field which is modified by the scattering objects, will be calculated at the point
Fig.
1.2. A
M
scattering objects and A(O) the 0, that is in the plane P. The objects located within the surface ds around a point X in the plane P will contribute an amplitude at the point M given by, Let p, be the surface amplitude of the
incident
density wave
of the
at
-
dAx where
b(XM)
expression r
is the
over
(M)
=
-
A (0) p, b (X M)
is called
-ikoXM
XM
ds,
(1-70)
scattering length in the direction J _M. To integr Iate this plane P, one first integrates around a ring of radius
the whole
and of thickness dr centered in 0. The
ring
e
-
b(20) (20 being
the
mean
value of
angle OMX).
The
b(Y-M)
around this
elementary amplitude
scattered
by
dA, (M)
integrated
This must be
XM2
+
r
2,
(1.71)
XM
XM dXM
whence
possible to integrate over XM instead by the entire plane P, appears as
it is
e-ikoXM
but since
over r
OM2
-
-A(O)p,b(20)2,7rrdr
=
19
is
elementary ring
this
X-rays (and Neutrons) with Matter
Interaction of
I
of
r
and the
--
r
(1.72)
dr,
amplitude A,,(M)
scattered
00
A,;,(M) If the
point
M is far
-A(O)p,27r
-_
at 0
A,c(M)
-
--
0.
b(20)
0 differs from
(27ri/ko) A(O)p, _
db (20) CIOe-ikoxm dXM 10,
as
(1.74)
-
00
JO’M
db
(20)
OMdcos20
by
(e- ikoXM )
which contains the factor
(Ilko)OM,
is very small if OM > A origin. The second term
extremum at the
a non
As for the first term, it
(1.76)
dXM.
singular b(20) presents (1.74) is thus negligible on this condition.
and if
is of the
(1.75)
XM2 dcos20
which allows to bound the second term
in
I
d
OM
d
integral
dXM
M
dXM
This
(1.73)
slowly with 0, it clearly appears that the second term the first one multiplied by A/OM. This comes from
varies
order
same
(20) e-’k11XMdXM.
behind the
[b (20) e-ikoXM]’ Om If
b M
plane P, the function to be integrated soon as 0; the integration easily gives almost constant and equal since b remains (1.78)), This can be rigorously proved by integrating by parts:
enough
oscillates very quickly as the result (see below Eq. to its value
10,
quickly
oscillates around
towards the infinite value of the upper bound
so
zero
that
when XM tends
one
can
make the
following approximation b
It is worth
larger
Fresnel
-
0.
XM-+00
(1.77)
that to average those oscillations, the upper bound value of the ring used in integrating over the plane P should be
noting
for the radius
much
(20) e-’kOxm
zone
r
than
a
characteristic
length.
This
radius which is -of the order of
length
(AOM)1/2.
is the so-called first
Frangois de Bergevin
20
Finally the forward
scattered
A,,, (M)
amplitude becomes
i.A(O)Ap,b(O)C k0om.
--
A(0)e-’1’-00M
The forward scattered field adds to the incident field
yields A If
total field
a
we now
in M and
A(M)
A (0) e -k’0M +
(M)
(1.78)
A,;, (M)
=
A (0)
(1
+
iAp, b (0))
e
-ikoOM
(1.79)
.
consider instead of
calculation remains valid
volume density p,
a plane a thin layer of thickness dx the above provided the surface density p, is related to the
by ps
The total field for such A
(M)
a
--
layer
_-
p, dx.
(1.80)
becomes
A(O) (I
+
iAp,b(O) dx)
e-
ikoOM
(1-81)
possible to deduce the optical theorem from this relation but this will be presented later. Note that the amplitude in M is outphased by 7r/2 relative to the one scattered by a volume element; that phase difference results from the summation of amplitudes in Fresnel diffraction. It is
The Propagation of a Wave in a Homogenous Population of Scattering Objects Let us now consider the plane P as an infinitesimal small
layer
of thickness dx made of
a
medium of index
n.
The
wave
vector in the
point 0 is located at the entrance of the layer, wave which has crossed the. thickness dx in the medium of index n amplitude at the point M given by medium is nko. If the
A
The
me-inhodx
e
-iko(OM-dx)
I_A2 p,b(O)/27r
Equations (1.82) and (1.81) plane in Fig. 1.3. As shown -
A(O) (I
-
i
(n
1) ko dx) e-’kOom.
plane
has
an
(1.82)
comparison of (1.82) with (1.81) shows that n-_
(n
, ’:z
a
1)
modifies the absolute
--
I
-
(27r/k 02) p, b (0).
(1-83)
schematically represented in the, complex figure, the imaginary part of b(O) or value of thefield amplitude in M, whereas the are
in this
real part modifies its
phase. Equation (1.83) links the scattering by elementary objects to the propagation in the medium which is considered to be continuous. It is an extension of the optical theorem. Indeed the imaginary part of n, i.e. 0, describes the attenuation of the radiation in the medium and 20ko is the absorption coefficient p:
JA(M) 12
-
dx-+O
JA(0) 12 (1
-
2#kf) dx)
=
JA(0) 12 (1
-
Cattenp,
dx),
(1.84)
Interaction of
I
X-rays (and Neutrons)
with Matter
21
Pmaginary
,A >
A
(0)
Real
complex plane. Up of the incident iAp,b(O)A(O) in the first calculation, field A(O) and of an infinitesimal field dA -i (n and dA 1) ko dxA0 in the second one. The component of the field dA 1 ) is turned by 7r/2 from the incident associated with the real part of b(O) or of (n field. This produces a phase shift of the total field, On the other hand, the imaginary I) decreases the amplitude of the total field part of b(o) or of (n
Fig.
1.3.
to the
e-"’00m,
factor
of the field in the
amplitude
of the
Representation
common
the total
field A(M)
is the
sum
=
=
-
-
-
where 0-atten is the attenuation cross-section of these objects in this medium. It is "almost" the optical theorem (1 .69). The "almost" means rep lacing atot,
the total cross-section of the isolated section in this
particular medium,
objects, by their attenuation
cross-
0atten.
approxielectromagnetic field. Indeed, only the forward scattering, which is usually independent of polarisation and conserving it, is retained. Beyond the above, approximations, and the scalar and one must take into account all the scattering direction’s behaviors. different vector fields display All the derivations above consider the field
(XM
mations made here
are
(1.82)
(1.82)
on
now
discuss the
the
on
and from
is valid if
(n and
scalar. Under the
approximations equality of the amplitudes calthe scattering (1.81). On the one
We must
argument relies
made. The
culated from the index hand
oo) they
Approximations
About the
which
-+
as a
be extended to the
can
the other hand
(1.81)
-
1) ko
dx < I
holds if
OM >> A. We
are
going
to show that in the
two conditions
can
case
of
simultaneously
hold
a
medium of finite thickness
under
some
restrictions
x
on n or
these b
(0).
An arbitrary thickness x of the material may be divided into thin layers of thickness *dx. Let 0 and M be the points taken at the entrance and at the exit of Let
0
layer j, such us
by (1.82)
into
(1-81)
as
dx
show that the as
far
as
I
=
OM
(see Fig. 1.4).
amplitude n
-
I
except in the global
at M
can
be deduced from the
one
at
I koOM < 1. Since OM does not appear phase factor, the expression of the total field
Frangois
22
Fig.
1.4. The
if
Bergevin
point M
is located at the border of the two layers (j, j+j) of the that the condition L, >> OM > A (see text) is satisfied. Note OM > A then NM > A for nearly all N. Then the amplitude at M only
material. We that
de
assume
from the layer j, and is given by (1.81) (with p, the surface density of the layer). Since L, > OM, (n I)ko dx is infinitesimally small, the approximation (1.82) does apply, and the material has an index given by (1.83) comes
-
holds whatever the value of OM
dx. Yet, for any point N even if OM points 0 and M, the condition NM > A has to be verified. Although there are some points N very close to M which do not verify NM > A, most of the points of the layer are at a distance from M --
located between the
larger
than A since the initial condition
was
OM > A.
In order for the
amplitude at M to be given by (1-81), it is also necessary for the back scattering coming from the layers j + I located behind the point M to be negligible. The different points of that layer scatter towards M with different phase shifts. It is possible to show that the ratio of the sum of scattered amplitudes in the backward direction to the one in the forward direction by layer j is of the order of A/OM. Therefore the condition OM > A is sufficient for equation (1.81) to be valid. Equations (1.81) and (1.82) are simultaneously satisfied if OM > A and In 11 kOOM < 1. The combination of these two inequalities yields -
A < OM <
This makes obvious that
L,
play
a
length L,
A 27r
In
-
11
-
A
27r
In
defined
V.
=
A
Ib(O)I
-
(1-85)
11
as
with
V,,
--
11p,
(1-86)
important role in the optical properties of the medium. It scattering process. With reference to the dynamical theory of x-ray diffraction we shall call this length the extinction length. We should remind ourselves that this length must be much larger than the wavelength, must
an
appears also in any
Interaction of X-rays
I
or
I n neutrons).
equivalently
that
-
I
I
with Matter
(and Neutrons)
must be very small
it is 1 0-5
(actually
for x-rays and Another condition must also be discussed. We have
23
or
replaced the
less
sum
to make the calculation of the scat-
or molecules by integrals amplitudes. This is allowed only if the intermolecular distances and than the range of more generally the dimensions of heterogeneity are smaller and in the transis OM that direction range integration. In the longitudinal the for integration, is the radius of verse direction the characteristic length
over
the atoms
tered
(OMA)1/2
The volume Vaver the first Fresnel zone, which is of the order of material the the (Vaver is dewhich is large enough to represent on average OM’A. As than less be must the fined as a volume larger than heterogeneity)
the
L2A of
inequality’OM =
Va2/A
scattering
can
function and
out of the
L,
must
must be very small
stand, Vaver
compared
to
I b(O) 12, where Va is the volume of the unit (namely the atom) 2 length b. If this condition Vaver < Le A is not fulfilled, the field
in the material wave
<
.
strongly around a value given by the ideal plane important fraction of the radiation may be scattered
fluctuate an
propagation
direction.
In condensed matter and for x-rays of energy 10 keV or thermal neutrons, 15, we have L,/A -- 10’. Va is of the order of a few A’. For x-rays and for Z condition L, >> A is The times ten For neutrons this ratio is about larger. thus well satisfied. If V,, is of the order of A’ 2
< L A is of the order of-
inequality Vaver last inequality can
,
the volume
10’ V,,
(10’0
L.2A
Va for
involved in the
neutrons).
The
materials.
reasonably homogeneous wave propagation according to the index given by (1-83) is with the continuous medium field equations consequently valid. be easily checked in
Provided these conditions ’are satisfied, the
1.3.3
The Extinction
The condition
(1.85) (L,
Length
>
and the Born
A)’shows
role in the evaluation of the
Approximation
that the extinction
strength
length plays
of the interaction of
a
a
major
radiation with
undergo ma phase shift of exactly one The substantial. becomes then results may terial; the scattering produced been which has material the of thickness the when different be qualitatively crossed is smaller or bigger than L,. For x-rays of energy 10 keV, the extincand it is one order I I < 10 tion length is of the order of a micron (I n for neutrons. of magnitude larger The approximation which has been made to relate n with b is connected first Born approximation. We have used a single scattering to produce the to the plane wave propagating in the medium. In addition the extinction length allows us to decide whether the Born approximation is valid for a given situation. When L, > A, the criterion is that the path travelled in the volume of the material giving rise to a coherent scattering must be less than L,. The kinematical theory of diffraction by crystals (equivalent to the Born
matter.-When the radiation has travelled a
measurable
a
distance L, it
begins
to
radian because of the crossed-
-
Frangois de Bergevin
24
approximation) is commonly used because the volume of the perfect crystal (coherently scattering) is often smaller than one micron cube. The property expressed in equation (1.85), which tells us that the extinction length is much larger
than the
wavelength,
also, presents beneficial effects for the physics of even if the kinematical
x-rays and neutrons. It is associated with the fact that
theory
is
longer valid, in perfect crystals, the dynamical theory remains optics, where this condition is not valid, the diffraction equations are most often not exactly solvable. In the domain of reflection in grazing incidence on a surface, the extinction length plays a major role. First it is related to the critical angle of total external reflection, discussed in chapter 3. Indeed the following relation stands no
calculable. In visible
I / I q,
I
--
L, sin 0, / 2
(1
+
n)
;z
L, sin 0, /4,
(1-87)
where I q, I is the scattering wave vector transfer corresponding to the critical specular reflection at the critical angle 0,. The left-hand side term represents (up to a factor I / 47r) a sort of wavelength perpendicular to the surface, and the right-hand side term (up to a factor 1 / 4) the extinction length projected on the perpendicular axis. The quasi equality of these two lengths is the sign that at the critical angle, the Born approximation is no longer valid. For less shallow angles, the perpendicular wavelength becomes smaller than the perpendicular extinction length and therefore the reflectivity becomes weak and calculable in this approximation. In the case of a rough surface, one must also compare the extinction length to the characteristic lengths of its waviness. If the waviness is
losses in
1.3.4
longer
or
shorter than the extinction
reflectivity and the scattering
are
different
When the Interaction Becomes
It. is -useful to know,
even
though
this does not
length,
the
(see appendix 3.A).
Stronger apply
to neutrons or.x-rays, the
kind of
propagation which arises when the interaction becomes stronger. In such: a. case, the representation by a -continuous medium can still be retained, but the -value of the index is
longer the one given above. In particular easily without making the supposition that the scattering length b(20) is independent of the 20 scattering direction. The formation of the index now implies that multiple scattering will be produced in all directions and not only in the forward direction. Also the scalar and vector fields do not have the same properties, since for the latter b depends on 20 because of polarisation (but we know how to take it into account provided no
the index cannot be calculated
that there is
no
scalar
other
anisotropy).
field, when b is independent of the angle, the calculation that leads to the amplitude (1.78) scattered by a plane is exact, even if OM is not much larger than A. Once the integration is made in the planes perpendicular to the propagation, it is then possible to work in one single dimension. Nevertheless the discussion which uses the decomposition of the For
a
Interaction of
I
X-rays (and Neutrons)
with Matter
25
layers (Fig. 1.4) must be revisited essentially because it is no longer possible to neglect the back scattering at M coming from the other layers which are located behind the point M. A calculation is proposed in exercise 1.3.3. As indicated in the introduction 1.3.1, it yields material in
n2 For vector fields
have found
an
the molecular
=
(the
1
-
2A 2p,b/27r
=
1
(47r/k2) p, b. 0
-
(1.88)
electromagnetic field), Clausius and Mossoti polarisability of the medium to similar expression due to Lorenz and Lorentz,
of the
case
which links the static
expression polarisability.
A
gives the refractive ind ex. Usi ng our notations, this is written Lorentz classical radius Of the electron, defined in section 1.4.2),
(n2
/(n2+ 2)
(47r/3) k2PVre. 0
(r,
is the
(1-89)
apply when the extinction length and the wavelength are of homogeneity must be verified at scales shorter than the wavelength. If it is not the case the propagation may be no longer possible; it is the phenomenon of localisation. These formulae the
order. The
same
Exercise 1.3. 1. A scalar
plane
wave, with the
interface
wave
ko, enters a ko. By dividing
-vector
0 with
the
angle planar making layers parallel to the interface, calculate the scattered amplitude at any point in the medium, as shown in (1.78). Find the direction of equiphase planes of the total amplitude and compare to Snell- Descartes’s law. For which values of 0, is the approximation improper ? Hint. One can show that the scattered amplitude at a point located at the back of an angled layer is given by expression (1.78) divided by sin 0. medium
through
a
the medium in
Exercise 1.3.2. In the
same
configuration
as
the
one
of the
previous exercise,
assuming b(20) constant, find with the same method the amplitude reflected by the interface. Compare with the exact Fresnel expression given in and
chapter 3, section
3.1. In the section 3.3 in the
imation is discussed
as
chapter the Born approxamplitude calculated negligible in the discussion of
same
in this exercise. Note that the
here is the bascliscattered one, considered as the approximations at the end of section 1.3.2. Hint. The
expressions
for the scattered
amplitudes
reflected and incident .
Notice. If the 0
vectors).
big enough to allow b(20) :A b(O), this Fresnel reflectivity expression is not exact.
angle
shows that the scalar Exercise 1.3.3. In
wave
symmetrical Only b may change angle between the
at two
points with respect to an infinitesimal layer are the same. from b(O) in one case to b(20) in the other (20 is the
a one
which receives the scalar
is
calculation
medium, the dx element located at x’, amplitude A, scatters the wave in the two opposite
dimensional
Frangois
26
de
Bergevin
directions
A7.7Gld(x Gld is the the
at
-
XI)
dimension Green
one
=
.-Apblddxe i1ko(x-x’)1
function,
a
q
constant coefficient and
pbld
power which in ger eral Is Imaginary. Find the relation between pbld and the refractive index in this medium
of
density
one
scattering
dimension.
Hints. One
can
negative from the
vacuum
by
and becomes
the
--
x
A
can
-e-""ox is incident
0 between the
--
wave
A(x)
--
vacuum
Aoe-’kox
at
comes
A’e-"’k0-T in the medium. The field
=
A (x) +
10"o A(x’)77Gld(X
the transmission at the interface A’
by
transmission One
positive
A(x)
x.
can
A’(x) -
interface at
an
be written into two ways: integral equation of the scattering
in the medium -
consider
and the medium at
x
coefficient,
which is
notice that the
at the
wave
(see
origin
--
-
x) dx’;
tA0 where
2/ (n + 1) (Chapter 3,
t is the Fresnel
section 3. 1,
Eqs. (3.18),
scattering is composed of two terms. The one in disappearance (so-called extinction) of the
of the
the extinction theorem
[3]).
X-Rays
1.4
General Considerations
1.4.1
electromagnetic radiation interacts principally with the electrons, and weakly with atomic nuclei (the ratio of the amplitudes is in the inverse of masses). The interaction is essentially between the electric field and the charge, but a much weaker interaction is also manifest between the electromagnetic field and the spin, or its associated magnetic moment. A photon which meets an atom can undergo one of the three following
The
very
events:
scattering, with no change in energy; scattering: part of the energy is transferred to the atom, the most frequently with the ejection of an electron (the so-called Compton effect); however it may happen that the lost energy brings the atom in an -
-
elastic
inelastic
excited state, without any ionisation (Raman effect); absorption: all the energy is transferred to the atom and the photon vanishes. Another photon can be emitted, but with a lower energy: this is -
the so-called
fluorescence.
These mechanisms James
[4]
found in
is
particularly
[5]).
described in many text books; the one of R. W. complete (except for the Raman effect which can be
are
Interaction of
I
X-rays (and Neutrons) with Matter
27
"To give an intuitive image, we shall begin with the classical mechanics theory which simply provides an exact result for the scattering by a free electron (Thomson scattering). When the electron is bound, this theory is still convenient enough. However the Compton scattering cannot be described by this classical theory. Also this theory does not describe correctly the motion of the electrons in the atom. Therefore we shall also review all the following processes in the frame of the quantum theory, i.e.: -
the elastic and inelastic
or an
electron bound to
atomic resonance-, the photo-electric -
-
the
dispersion
an
scattering (mainly Compton), -for
a
free electron
atom, when the radiation energy is well above the
absorption by an atom; brought to the elastic scattering by
correction
the atomic
resonance.
Finally we shall discuss the general properties of dispersion which are independent of a particular interaction or radiation. One can show that the real and imaginary parts of the scattering are linked by the Kramers-Kronig relations which are extremely general and probe the response of nearly every system to some kind of excitation. The origin of these properties lies in the thermodynamical irreversibility that can be introduced through the principle of causality. Classical
1.4.2
Thomson
Description: Scattering by
Free Electron
a
scattering by a free electron is simple and presents the main characters scattering by an atom. We shall start with this case. The electron undergoes an acceleration, which is due to the force exerted
The
of the
by
the incident electric field
Ei,,(t) Let
z
be the electron
Eo iwt
-
(1.90)
position and (-e) its charge, then Tni
--
iwt (-e) Eoe
The electron exhibits oscillations of small
amplitude, producing
a
localised
current
j (r,t)
e)
i
(1.92)
(r)
(_e)2 Ein (1) iwm The radiation of that
vibrating current,
J (r).
similar to
large
distances
(kr
>
E,,
a
dipole antenna,
(1.63)
discussed in section 1.2.6. From the formulae
(1.55),
and
has been
we
have at
1),’
hl: 00
[Ei,,
(Ei,,.r)
r
r2
I
(-e) 2e-ikr ,
-
47rEOMC2,r
’
(1-93)
Frangois de Bergevin
28
What is measured is the
projection of the field on some polarisation direcand iin is the unit vector which describes given by the unit vector the incident polarisation. These vectors are chosen so that ;&in is parallel or antiparallel to EO and’ s, normal to r (see Fig. 1.5) tion
Ein
-
(Ein-;40; in
and
rJ sc
In these conditions of polarisation the definition of the can
be
adapted
as
we
(1.94)
0.
scattering length (1.34)
follows
-Ein - in then
-_
e-ikr
b(; ,,c,i i,,),
(1.95)
have
b(
i,,c, iin)
::::::
re’ sc - iin
3
where r, is the Lorentz classical radius of the electron with charge 2.818 x 10-15 M).2 The charge of the e 2/47r6o rnC2 e and mass m., (r, -
-
electron appears twice, first in the movement and then for the emission of the radiation. Thus it appears as a square and b does not depend on its
opposite to the incident one because of (by convention, a positive value of b corresponds to such a sign reversal). If the ingoing polarisation is normal or parallel to the plane of scattering, the outgoing one has the same orientation. These polarisation modes are called (s)-(s) (or (o-)-(o-)) when perpendicular to the plane of scattering and (p)-(p) (or (7r)-(,7r)) when parallel. The polarisation The
sign.
scatteredfield
is however
its relation with the current
factor of the
scattering length
the latter. The process that
scattering.
1.4.3
case
and
cos
20
(Fig. 1.5)
in
have described is the so-called Thomson
.
Classical
by
is I in the former
we
Description: Thomson Scattering an Atom, Rayleigh Scattering
the Electrons of
scattering is exact, even for the bound frequency of the x-rays is large compared to the characteristic atomic frequencies. Nevertheless it is necessary to take into account both the number of electrons and their position in the electronic cloud when calculating the scattering from an atom. Every point of the electronic cloud is considered to scatter independently from the others and the scattered amplitudes add coherently. As in any interference calculation within the Born approximation (see the Appendix LA), justified whenever The
simple result
electrons of
an
of the Thomson
atom,
as
far
as
the
A system of units which is often used to describe microscopic 2 2 Gauss system. In this system we have r, e /mc =
phenomena
is the
I
Interaction of
X-rays (and Neutrons) with Matter
(n e in
29
2,0
(a) 6sc in
Wf:o
2,0
2
(b) Fig.
(a)
1.5.
Directions of incident and scattered
mode and
factor
is
(b)
respectively
scattering
the
the
(p)-(p) or(r)-(7r)
the total atomic
(a)
the
(s)-(s) or(a)-
amplitude polarisation
f (q)
(q),
--
scattering length b"t by
density p(r)
the Fourier transform of the electron
bat
for
I and cos20
weak,- one obtains
is
polarisations
mode, The associated
f
--
p
(r) eiq.rd(r)
(1.97)
the definition of q in. (1.25)). The quantity f (q) is called the atomic scattering factor or the atomic form factor. The integral of p(r) over
(see
all
r
values must be
equal
to the number of electrons in the atom:
f (0)
__
Z.
(1.98)
.
explanation to support the validity of this interference calculation. The justification comes from the alternative quantum calculation which gives the same result. The assumption that the frequency of the radiation is greater than the atomic frequency may not be valid especially for the inner electronic shells. The model can be improved by introducing the binding of the electron to the atom which is modelled by a restoring force of stiffness K -and a damping coefficient -/. The damping is the result of the radiation which is emitted by the electron, or of the energy transferred to other electrons. The equation of motion (1.91), still written for a single electron, now becomes
There is
no
safe
mi + One looks for
a
^/i
+
NZ
solution of the kind
(-MW
2
+
i’YW
+
-_
(-e)Eoe
(e"t)
2)Z
MW 0
iWt
which must satisfy =
(-e)E0eiWt’
(1-100)
Frangois de Bergevin
30
where
tc/m
(-e)i
-wo. The current
j (r, t)
is then
iw(-e)2 Ei,, (t) J (r) ,rn (W2 W2) 0
:--
(1.101)
-
As shown for the Thomson
scattering above,
this
yields the following
scat-
tering length 2
b
We shall
when
now
only
one
r,
W2
electron and
more
heavy
just
W20
i7W/M
e,,. ein
(1.102)
-
is modified for different
expression
are
considered
general. Actually it happens w with wo. For high energy
although that W,
expression.
Wo
energies this dis-
>>-Y/m
x-rays and not too
>>wo. Within these
or even w
(L 10 2) is just reduced to Thomson’s then b
_
one resonance
have to compare atoms we have w >wo
we
-
discuss how this
cussion could have been and
--
approximations
If on the other hand
W
< <W 0,
becomes, 2
b
-r,
--
(1-103)
_20 es,.ein-
This is the so-called
Rayleigh scattering, originally proposed to explain the scattering of visible light produced by gasses or small particles. Three important features of this kind of scattering should be noticed: the polarisation factor is the same as for the x-ray Thomson scattering; the scattered amplitude is proportional to the square of the frequency, and the cross-section is thus proportional to the fourth power of the fre-
-
quency;
the sign scattering. -
of the
scattering’ length
is
opposite
to the
one
of the Thomson
point explains the blue color of the sky (the highest frequency which from the first point may appear to be highly polarised. The change of sign noted in the third point is important, since it corresponds to a sign change of (n 1). We shall comment this further when we will dispose from a more quantitative theory. Again for x-rays, the scattering length (1.102), when summed over all the The second
in the visible
spectrum),
-
atomic electrons becomes similar to the and
imaginary
bat where
f
is the Thomson
the correction due to
correction
or
Originally, the
vicinity
it
one
of Thomson
(1.97)
but with real
corrections:
-
in
Y
+
scattering,
resonance.
anomalous was
re
optics
f’
if")6sc- &in,
+
whereas
f’
f",which
3 .
One must take into account
that the anornalous
dispersion
is
real, give. dispersion
are
This correction is the so-called
scattering
of resonances, the
and
(1.104)
as
in
dispersion was introduced. In opposite to the usual behavior for
I
Interaction of
X-rays (and Neutrons)
with Matter
31
the pure Thomson scattering the sum over all the electrons and their spatial distribution, but this discussion is difficult and uncertain in the classical
theory. We shall see that in the quantum theory f’ and f" only slightly depend on q and have an energy dependence that we shall discuss. Though crude, the classical model allows the calculation of the absorption as proposed in the exercise 1.4.1. In fact, one rather gets the total crosssection, including absorption and scattering. This result is very realistic, since it agrees with the prediction of the optical theorem discussed in section 1.3.2. To summarize, the classical model although simple describes most of the phenomena and provides exact values for a certain number of physical quantities. Nevertheless the values of the resonance frequencies and of the damping coefficients are not calculable within this framework and are left arbitrary. In addition, it does not give much indications about the q dependence of the scattering factor at resonance but more important it does not describe the scattering when an electron is ejected (Compton effect). Although it is possible to give a classical description of such an effect by considering the reaction on the scattering of a vibrating electron, only the quantum approach is correct. Therefore the only coherent and completely exact description is given by the quantum theory of the interaction between the radiation and atoms.
1.4.1. Calculate the total cross-section of an atom which exhibits resonance characterised by wo and -/. We assume that the power only taken by an atom from the radiation is the same as the one dissipated by the damping force -/-6 (do not forget that when complex numbers are used to describe the oscillation of real variables, the answer is twice the one obtained Exercise
one
with real
The initial power of the radiation is given in section 1.2.3. optical theorem (section 1.3.2) is satisfied.
numbers).
Check that the
Quantum Description:
1.4.4
a
General
Expression
for
Scattering
and
Absorption
description we shall assume that the radiation is quantised as photons. scattering and absorption probabilities are then the squared modulus of the probability amplitudes. The amplitudes are transformed into scattering lengths and the probabilities’into the scattering cross-section. The amplitudes themselves are derived from a perturbative calculation based on the
In this The
interaction Hamiltonian between the radiation and the electrons. which it is observed that the index of refraction varies in the energy. By extension one refers to "anomalous scattering". adjectives "anormale" and "anomale" are used. "Anormale" not follow the rule and "anomale"
the a
same
species. We
sense as
the
means
that it does
different from other individuals from
dispersion does not constitute only a usual behavior, the second expression seems to be more acknowledge B. Pardo for his comments.
law in itself but
appropriate.
means
same
In French the two
Since the normal behavior of the
Frangois de Bergevin
32
The
expression
contains the
potential
of the Hamiltonian of
following
of the
(1/2rn) (p
-
term
(we
electron in the radiation field
one
leave aside
other terms such
some
as
the
atom)
eA/C)2
P2 /2Tn + (C2 /2 MC2 ) A2
=
_
(e/mc)A.p.
(1.105)
The p and A operators are the momentum of the electron and the vector potential of the radiation. The first term of the right-hand side gives the kinetic energy of the electron and the two others the energy of interaction. In this expression, the spin has been neglected which is permitted when the energy of the radiation is weak
compared to the rest mass energy of the elecperturbation calculation made at the lowest order on the two interaction terms yields the scattered amplitude. This approximation is sufficient because the strength of the interaction, measured by the ratio of the coefficient e2/7,nC2 (or 47reor,) to the quantum size of the electron h/mc (or Ac defined further) is small. The perturbation terms are sketched in Fig. I.A. The smallest order of the perturbation is the first order for the 2 term in A and the second order for the term in A.p. These two terms give rise respectively to one and to two terms in the scattering length (with our convention for the sign of imaginaries, unusual in quantum mechanics): tron which is 511 keV. A
bat
7e <
8
1_; S*C e+ik- - iin -iki_r1i "’
SC
.p
e+ik_irJC
rn(Ec
C
<
S
-
Ei
>
CPin-P e -iki,,.rli hwin + i-V,/2)
>< -
1’ i*SC.p C-ik_ ’r1c >< CPin-Pe +ikin.rli m(E,, Ei + hw.c)
>
>
-
C
-
bTh
+
bdispl
+
I i > (respectively I s >) tering) electron states. These Here
bdisp2
scattering.
tron. In the last two
a sum
(bound
excitation and hwil In elastic and
hlFc
scattering
rows
or
wsc
-_
r
are
is the
is made
continuum
(hw,,)
(respectively
after scat-
identical for elastic
scattering
stands for the initial two states
and different for inelastic this electron
(1.106)
-
over
states). E,
position operator
all. the -
of the elec-
excited states
I
c
> of
Ei represents the energy of
is the energy of the incident (scattered) photon. wi,,. 1, is the width of the excited level I c >
its life time. The
polarisation vectors may be complex so they can represent elliptical polarisation states.’ The following discussion will show that the first term represents the Thomson scattering found in the classical theory. The two last terms, bdispi and bdisp2 define the dispersive part of the
scattering. In most instance in this book, only linear polarisations are considered and no complex conjugate is indicated. In the case of anisotropic scattering, section 1.5, the circular polarisation may be required.
I
intteraction of
X-..rapq (And Neutrons) with Matter
33
pho,ton
electron
(a) A2
(b) A.p
(c) A. p
(d) A. p
diagrams are the symbol of the amplitudes which are in formulae (1.107). A point represents a matrix element and a line the electron (1.1.06) element. For instance in or the photon in the initial or final state of the matrix and final states display initial the are elements represented, (b) where two matrix state intermediate one electron. In the the and only and electron one one photon, formulae as written in the text, the p4otonic states are not made explicit, but their contribution ( i, k) is present through the (-, e-ik.r) terms. For any of the four amplitudes, (a) bTh (b) bdispl (c) bdisp2, and (d) the absorption, the Hamiltonian
Fig.
These
1.6.
and
i
I
term is indicated
absorption cross-section is also derived from the interaction Hamiltonian, once again at the lowest order of perturbation, Ref. [6] section 44, The
Cabs
(hwi,,)
E
=
m
C
hwin Tc
(Ec
-
j< Cj’ in-P 6-ikinrli >. 1’
E,) 2 (Ec
-
E,
-
hW,n) 2
+
Ic2 /4
(1-107)
In this process the photon completely disappears. The A’ term in the Hamiltonian does not contribute and therefore only the A.p term is used. Every term of the sum corresponds to the excitation towards a I c > state. The numerator
electron,
suggests that the electric field transfers
and
changes
the
I
i > level into the
I
c
(1.106), one can say that the scattering bdispl + tion I i >-+I c >, then desexcitation I c > -4 1 in
bdisp2
created
,
since the
c
>
state, which is
>
some
bdisp2 s
virtual,
momentum to the
level. In
>.
is
a
similar way, in by excita-
is obtained
This order is reversed
destroyed
before
being
(Fig. 1.6).
scattering as well as the absorption, one generally uses dipolar approximation, that is to say one replaces the factors eik.r by one, supposing the wavelength much bigger than the atomic dimensions. This approximation which is excellent in the visible spectrum’, is still good for x-rays because the electronic levels which are excited are usually very much localised. Under certain conditions however, this approximation is not sufficient and the next term in the expansion of the exponential (ik.r, the To calculate the
the
quadrupolar term)
must be included.
When the energy of a photon is sensibly larger than all the excitation thresholds of the atom (w > w,), the first term in (1.106) which represents
Frangois
34
the Thomson
de
Bergevin
scattering
becomes
preponderant.
In the extremes of
light
atom
and very high energies, the scattering cross-section given by this first term is even greater than the absorption cross-section (1.107). We shall start the discussion of the Thomson
scattering b’l’h
to show that it
can
be
separated
(Compton) scattering. Then we shall describe which comes from (1.107). Finally we shall discuss
into the elastic and inelastic the the
absorption spectrum dispersive bdispl + bdisp2 scattering,
absorption.
Quantum Description: Elastic and Compton Scattering
1.4.5
For
in relation with
free electron and in the classical Thomson
scattering, the backward ignored. Compton performed a kinematical calculation which took into account the momentum and the energy carried by the radiation quantised as photons. For an electron initially at rest, the conservation of these two quantities implies that the photon releases an energy such that the wavelength after the scattering process A,,, becomes larger than the and satisfies the equation initial one Ai,, a
move
of the electron is
,
Asc
-_
Ai,,
+
A, (I
cos
-
20)
A,
--
27rh/mc
=
0.002426
(1.108)
nm
angle between the incident and scattered beams and A, is the Compton wavelength of the electron. In the present calculation, we are doing non relativistic approximations which are not valid if the photon energy becomes close to the rest energy of the electron. Neglected relativistic effects are the influence of the spin and a factor which diminishes the Compton scattering cross-section. When the electron is bound to an atom two Processes are possible: the radiation may be elastically scattered with the conservation of the electron state (the momentum being transferred to the atom which is assumed to have an infinite mass), or inelastically with the ejection of the electron, One must determine the respective probabilities of these two processes. We start first with the case of an atom which has only one electron. Let us evaluate the elastic, then the total scattering. The inelastic scattering will be obtained by subtraction. Keeping only bTh from (1.106) we have, where 20 is the
bat
-
re SC- iin fsi;
f’i
=<
sle iq.rli
where q is equal to ksc-kil (1.3). For elastic scattering, I i >-0(r) and p(r) be the wave function and the electron density then
Li
=
A
We have derived here
determined
by
-
1 0* (r) 0 (r)
more
the classical
,q.r dr
-f p(r)
ezq.rdr
(1.109) I
S
>. Let
(1.110)
rigorously, the form factor which we previously theory (for the atom having one electron). The
Interaction of
I
calculation is
completed by
plus scattering
of the
factor
1(si Since
35
scattering cross-section, by summing the square modulus final states of the electron,
the evaluation of the total
inelastic. This total is obtained
elastic
ones
X-rays (and Neutrons) with Matter
the
sum over
form
iq.r
all the
1,) 12
-iq.r
the final states is made
complete
a
e
over
set and
over
e-
iq.r
all the
ji)
(1.111)
.
possible states, these
the closure relation
satisfy
E is) (sl
Is) (SI
-
(1.112)
unitoperator.
13)
disappear from expression (1.111) which becomes equal to unity. The inelastic cross-section is obtained by subtraction and finally we have,
The final states
(do-IdQ),,ja ,
. &in)2 (re’ * -_ in )2 1fii 12
(do-1&2)i ,,j
(r, &*
(do-IdQ) elas+inel
re *
SC
SC
. in)2
SC
I
_
Ifii 12
This calculation prompts two remarks. We first observe that in the sum the terms which do not conserve the momentum seem to play no
(1.111),
part: they cancel the matrix element must be included in the
Next,
some
sum
iq.r
I
s
to enable the
use
I
i >. However these terms
of the closure relation
information about the conditions in which this To be correct
should be
given.
radiation,
with the
scattering
definition of the differential
obeyed. energies
<
Instead of
we
must
sum over
direction ft
sum
is
(1.112).
performed
all the final states of the
kept fixed (this
is
a
result of the
and with the energy conservation the exact expression is (E, and Ei being the
cross-section)
(1.111), states)
of the electron
I
E
I (S I
iq.r
1,) 12 6 (Es
-
-
Ej
I kc 1) dks,.
(1.116)
the two conditions used
by Comp-
-
he
I kin I
+ lie
Is) k_/Ik_I=fi Let
us
note that this
expression imposes
the energy conservation as shown by the 6 function, and the conservation of momentum in the matrix element. Performing the integral ton which
over
ksc
are
one
gets back the sum (1.111) k,;c/ I k,, 1= ft. One
with the condition
over
the final states of the
can see
that
being
electron,
in the 6 function
11 and q consequently has the same dependence. For consistent, this dependence must be neglected otherwise the closure relation (1.112) could not be used in (1.111). The approximation is very good but the small dependence of I k,,, I on the final state of the electron, which, through the momentum conservation, is also a dependence
ksc depends
on
<
the discussion to be
s
Frangois de Bergevin
36
on
the electron initial momentum,
distribution inside the atom
can
be used to
the momentum
measure
inside the solid. This
application
of
Compton develf)ped The inelastic scattering for which we have calculated the cross-section is frequently considered as the Compton scattering. This is not completely correct since the total scattering cross-section also includes the Raman seattering. In such a case the final state < s I of the electron is not a free plane wave but a bound excited state [5]. To be fully complete we must also consider another inelastic scattering process, the so-called resonant Raman scattering. This process does not appear in the above calculation, but rather in the- development of the second term in (1.106), bdispl) when one assumes < s I:A< i 1; it is obvious at energies close to an excitation edge. It is thus more associated with absorption and fluorescence than with Compton scattering. However far from resonances, the dominating inelastic process is usually the Compton scattering. The calculation that we have just carried out has to be changed for an atom having more than one electron. The electronic states are multi-electron states and each interaction operator is replaced by the sum of operators acting each on one electron. For an atom having two electrons,
scattering
will not be
f1further
I/ e
or
iq.r
v/2-) I TVI (r I ) TV (r 2Yfi (r ) Of2 (r 2
iq.rl e
Expression (1.117)
this book.
in
2
(1-117)
1
+,iq,r2.
is the Slater’s determinant which represents the antisympermutation of the electrons. The elastic
metric state with respect to the scattering factor becomes
sq.r].
f
I
With this
i >
given by (1.117)
---:
fl 1
+
the
orthogonality
between Tf, and Tf2,
fi 1
cross- section
as
(1 / v/2-) I TfI (r 1 ) T/2 (r 2)
Is)
(11V2) ITf,(r1)Tf2(r2) (1/v 2_) ITfi (r,)Tfx (r2) ( 1 / v/2-) I Tfx (r I ) Tfy (r2)
I s)
--
The last state corresponds to zero
*
i
(r) Tf, (r)
sum
the
zq.r dr.
(1.120)
amplitude
squares
in
s)
is)
’one must
TV
(1.111). Although it is not necessary to know write them for more complete view
all the final states
explicitly
where
f2 2
To obtain the total
a
using
sq.r2
yields f
we
and
+
amplitude,
but
once
-
over
them,
Tf, (r 2) Tf2 (r 1)
-
Tf,(r2)Tf2(r1))
x
0 1,2
(1.122)
-
Tf, (r2)Tf,, (ri))
x
0 1,2
(1.123)
-
Tf, (r 2 ) Tfy (r 1 ) )
x, y
a
0 1,
two electrons excitation and
again,
it must be included to
(1.124)
2.
gives
use
rise to
the closure
X-rays (and Neutrons) with Matter
Interaction of
I
relation. With this latter, the total cross-section is
proportional
37
to
(ij (e-iq.rj+e-$*q.r2 (e’ q.rj+ei.q.r,) ji)
(1.125)
Substituting ji) by (1.117) and using the definition (1.120),
this expression
becomes 2 +
f11f22
+
fllf22
-
jf12 12
1 f2 112
(1.126)
It is easy to extend this calculation to any number of electrons Z. The elastic
and inelastic
scattering cross-sections become,
.-6in)2
r6; *
(do-/dQ)e1as+ine1
SC
(z
(1.127)
Y
+
fj*j f, I
,
E I fjj 12)
-
1<jq61
1<j361
12
(dald O)elas
(r,; s*, -i in )2
(do-IdS?) inel
(r,; s* - iin )2
transform of wave
density (which
the electron
functions),
c.f.
(1.110).
Ifjj 12). I<j;41
is written
scattering factor
that the elastic
see
(1.129)
<j:5 z
1
can
(1.128)
Ifjj 12
(Z One
1: fjj 1<j
is the
When the terms
sum
the Fourier
as
of the densities of all the
I fjl 12
are
ignored,
the
case
of
the many electrons atom is naively deduced from the one having one electron: on one hand the elastic scattering lengths and on the other hand the inelastic cross
come
of
sections
in fact
a
are
added. The
They
are
called the
from matrix elements in which electrons
instance in the
case
of two
electrons,
(Tfl (rl)Tf2 (r2)1 They
are
jqj
....
subtracted because electrons
general evolution Fig. 1.7.
The in
I fjl 12 terms constitute exchange terms since they have been interchanged as for
all the electrons
modest correction.
To end this section
jTfi(r2)Tf2(ri)) are
fermions.
of the cross-sections is
we
now
(1.130)
.
presented
as
a
function of
easily the Compton scattering change of wavelength (1,.108) in the scattering vector transfer q
discuss how
is observed. It is suitable to rewrite the
following
way
as a
lki,, I A radiation of
-
function of the
I ksc,I
=
(Ac/41r) q’
wavelength Ac
of the electron at rest. For
a
has
+
an
(higher
order in
energy of 511
keV,
(1.131)
q). i.e. the
mass
radiation of energy 10 keV scattered at
energy
an
angle
Frangois de Bergevin
38
Z’12
C
wg
-
0
a
CD
W
U)
Z
2 R
jqj [arb. units]
Fig.
1.7. Schematic
sections
having
as a
jqj [arb. units]
representation
function of
jqj, (a)
for
of
elastic, inelastic (Compton) and total crossatom having one electron, (b) for an atom
an
2 any number of electrons in units of r,
one degree (typical of a grazing incidence surface experiment), the wavelength change is very weak since it depends on the square of q. Things are different in the range of medium and large values of scattering angles where this change is easily measurable, for instance by using an analyzer crystal. At energies greater than about 100 keV and medium angles of scattering this change is appreciable. The Compton cross-section varies in a similar way as shown in (1.115) (L 129) and in Fig. 1. 7. For radiation of 10 keV, it is negligible at small angles, but this is not true at wider angles. At higher energies, some tens of keV, the Compton scattering achieves its highest value already at medium angles. At those energies and for light elements it dominates the other processes. Indeed its proportion is larger for small Z scatterers. The table 1.1’ gives some values of the total scattering cross-sections (integrated over the whole angular space); the elastic cross-section is condensed in the forward direction
of say
in
a cone
1.4.6
which becomes
Resonances:
narrower
when the energy increases.
Absorption,
Photoelectric Effect
In the interaction process, part of the radiation disappears instead of being scattered. As shown in (1.107), the energy is transferred to an electron which is excited to an empty upper state I c >. Most frequently it is expelled from
the atom; this is the so-called
about
hlr,,
be radiative
experiment
the or
photoelectric absorption. After de-excites, according to various processes atom
not. The most obvious process in
a
diffraction
is the emission of fluorescence radiation. It
More cross-section values
crystallography [7],
can
or
a
delay
which
of
can
scattering
corresponds
to the
be found in the International Tables for x-ray
vol. III and IV
Interaction of X-rays
I
fall of
a
(and Neutrons)
with Matter
second electron of the atom into the level vacated
by the first
39
one.
Its energy is necessarily lower than the energy of excitation. The fluorescence yZeld, i.e. the fraction of excited atoms which are de-excited in this way,
depends
on
fluorescence
the elements and
yield
is 0.5. Let
on
us
levels;
the
note that for
for the K level of copper, the a given excited level I c >, the
cross-section varies with the energy and exhibits a Lorentzian behavior with a FWHM r,. In the x-ray domain, T, lies between about a bit less than 0.5 eV and
a
bit
than 5 eV.
more
10
(a
0_
_
C:
0
Cn 10
-10f
or
kb sor.
f
IVI 4-
-20
-20
-10
0,
E
-
10
20
30
EO[eV]
2 e V). The solid absorption edge, with a white line (r the imaginary and cross-section the both of absorption of real part of dotted the line and the next correction section), of (see dispersion part this correction in arbitrary units. The origin of energy, Eo, is taken at the edge. This schematic figure is not intended to show the real details of these curves in the -3 decay has vicinity and above the edge. On this short interval of energy, the E been neglected
Fig.
1.8. Schematics of
an
=
line shows the variations
important transitions for x-rays are those of the inner electrons which shells. The transitions may bring the excited elecK, L. belong tron towards the continuum of the free states; their spectral signature is then characterised by an absorption edge, located at the excitation energy, since also arise any level above the edge is equally accessible (Fig. 1.8). They can towards the first free bound levels. These states may have a large enough density to give rise to one (or several) well-defined absorption peaks superimposed to the edge, the so-called white line (s). The white line(s) spectrum is not an exact image of the density of free states of the atom, molecules or condensed system. The observed spectrum corresponds to a system which has lost a core electron and is deformed by the electric charge of the core The
to the
.
.
.
Frangois
40
de
Bergevin
hole. Peaks
can
lines
narrower
can
be
paradoxically and
then appear below the edge, and the white intense than the corresponding levels of the
more
ground state of the system. In condensed matter, the absorption above the edge exhibits oscillations, the so-called EXAFS (Extended x-ray Absorption Fine Structure), which are interpreted as arising from interference effects in the
function of the
ejected electron. These interference effects are due electron by the neighbouring atoms. ejected scattering In short, one can say that the absorption varies as E-3and as Z4 This does not take into account the discontinuities at the edges. The K edges produce a discontinuity of the absorption by a factor of about 5 to 10. Figure 1.9 shows for copper a discontinuity of a factor 7 at the K edge and of about the same amount for the three L edges together; one can see that the decay in between the edges is a bit slower than E -3 Table 1. 1 gives some values of the absorption and scattering cross-sections. wave
of the
to the
.
1 E+7
L ed 1 E+6 C
1 E+5
ps
N 1
(D M
-
1 E+4
111
"\
e dge
[\
2 E+3 0 .
ri
’
1 E+2
0 to
M Ca
10
1 E+2
1 E+3
1 E+4
1 E+5
1 E+6
Energy [eV] Fig. 1.9. Absorption cross-sections for the atom of copper in barns (10-22 MM 2), after Cromer-Libermann (above 10 keV) and Henke (below 10 keV). The slope of -3 the E power law is presented by a dotted line
practice one frequently needs the absorption coefficient ft rather than cross-section; this coefficient is defined by the fact that the transmission through a thickness i is given by e-At. It is also equal to 47r,3/A (,3 the imaginary part of the refractive index). For a homogeneous material made of a single element, p depends on the cross-section o- and on the atomic volume V through In
the
y
-_
0-/V.
(1-132)
interaction of
I
Cross-sections of
Table 1.1. cross-sections the
are
integrated
Compton scattering
elements
some
over
X-rays (and Neutrons) with Matter
cross-section
/
After [8];
see
function of energy (scattering In each case, are displayed
a
the elastic scattering cross-section
(i.e. 10-28 m2).
in barns
photoelectric absorption cross-section, in the evolution of the absorption are chosen energy.
as
angle 47r).
solid
a
41
The
/
the
irregularities
of an edge close to the for x-ray Crystallography
due to the presence
also the International Tables
[7] 100 keV
30 keV
10 keV
5 keV
Element
3.3/0.67/1.1 2.9/0.07/0.02 13.3/4.5/30 13/35.6/1090 19/117/6420 20.9/15.8/224 27.8/432/8420 33.2/60.8/1590
C (6) 2.7/3.2/39 2.1/5.8/371 Cu(29) 4.65/307/19500 8.2/153/22600 Ag(47) 6.5/820/132000 11.5/459/20600 Au(79) 8.22/2630/212000 15.3/1580/36100
absorption coefficient of a material, it is sometimes useful to introduce the mass absorption coefficient, given by p1p (p is the density) and commonly tabulated. This coefficient is characteristic of the element and independent of its density. If A is the molar mass and N Avogadro’s number To calculate the
p1p
(1.133)
No-IA.
--
absorption coefficient of a material composed of several elements i, each partial density pi, is simply given by
The
of them present with the
Pi
1.4.7
Dispersion
Resonances:
(1.134)
(P/A
and Anomalous
Scattering
scattering in which we had neglected equation (1.106). Actually we shall take dispersive part bdispl bdisp2 into account only bdispl, which represents the second line of this expression. For a term of the sum over c to be appreciable it is necessary for its denom-
We return
now
to the
case
the
of the elastic
in
+
inator to be small which
never occurs
in
son
bdisp2
reproduce here the formula (1.104), and dispersive contributions.
We
b,t
--
r,(f
+
f’
+
-
which
gives
the
separated
Thom-
(1.135)
if") &* .; jj,. SC
separation could appear artificial in the classical expression for b (1. 102), but arises perfectly naturally in the quantum mechanical one (1. 106). We have assumed that the polarisation contributes in bdispl that is f’ + if", through the same polarisation factor as in the Thomson term. This is not true in This
every case,
as
discussed in Sect. 1.5.
Each of the terms correction
f’
+
I
c
>
of the
sum
if" arises, corresponds
to
(1.106) an
from which the
excitation energy
dispersion
(or commonly
Frangois de Bergevin
42
a
resonance) E,
-
Ei. The associated correction is
re
(fc
+
ifc’ )
with The real and
x
imaginary parts
I
X
oc X
-
[hw
--
are
-
(E,
-
presented
(1.136)
+
T+X2 Ej)] / (-Vcl2)
1+ X2
2
in
Fig.
1.10.
1.2-
0 8.
0 4-
:3
.
CU
0 0.
f f
-0.4-
-10
-5
0
E
Fig.
1.10. Schematic
nance
at energy
Ec
-
-
5
10
EO/ (r/ 2)
representation of the dispersion correction for a single Eo. f’ and f" are given by b r,(f + f’+ if")
Ej
=
reso-
=
The formulae (1.106) and (1.107) show a correspondence between the dispersion correction in terms of the scattering length and the absorption cross-section. Exactly at the resonance energy, we check the optical theorem (Im is the imaginary part), Cabs
--
2A Im[b
(q
--
0)],
(1.137)
discussed in Sect. 1.3.2. 6 We obtain here
expression for the absorption cross-section, while the optical same expression for the total cross-section. The error comes from our calculation of the scattering length, made in the first order Born approximation. The next order is required to obtain an imaginary part which expresses the intensity loss due to scattering (see appendix LA). The calculation at that order is made intricate because of some difficulties of the quantum theory of ratheorem
an
yields the
(the renormalization of field theory). That error is negligible inasmuch absorption is the largest part of the cross-section, which is true up to moderate energies, but not at the highest. diation
as
the
1
Interaction of
The distribution of the
X-rays (and Neutrons) with Matter
energies E,
resonance
-
Ei, with the edges
-
as
43
main
absorption. Fig. 1.8 shows features, the comparison between the variations of the absorption cross-section close to an edge and the variations of the anomalous scattering. Let us now look how ’the real part of the scattering factor, f + f’, varies when the energy changes from x-rays to near infrared, that is to say from several tenths of keV to one eV. The highest energies are far above the edges of most elements and the Thomson scattering factor f is dominant. For lower energies, a negative contribution f’ appears at every edge and is more important below the edge than above because of the white lines. Low energy edges produce the most intense dispersion effects. Going to low energies, some edges for which f + f is negative are observed, and then a transition occurs towards 10 to 100 eV where f + f’ definitively changes its sign. In this range, the very intense absorption lines enormously reduce the propagation of light in matter, which makes it called vacuum ultraviolet radiation, because it propagates only in vacuum. When the sign of f + f’, which is also the sign of b, changes from positive to negative, the refractive index n goes from below to above the unit value (the link between scattering and the index is discussed has been
in section
1.4.8
previously
discussed about the
1.3).
Resonances:
Dispersion Relations
absorption cross-section is easily obtained directly by experiments as for example, the measurement of the transmission through a known thickness of a material. The imaginary part of the scattering length b is found at the same time. The real part of b however is more difficult to obtain accurately. Among the different methods, diffraction experiments but also reflectivity measurements have been used to extract the scattering length [9]. The drawback of such indirect methods can be overcome because it is possible to rebuild the real part of b if the imaginary part is known over the entire spectral range. Conversely the imaginary part can be deduced from the real one. For a single resonance, if the variation f"(E) as shown in (1.136) and Fig. 1.10 is known, then the energy, the amplitude and the width _V of the resonance are determined, and f(E) can be obtained. Note however that the sign of F is left ambiguous in this procedure and must be given. The question is to know whether such a reconstruction of f’(E) is still possible for the general case with several resonances. It will turn out to be possible, but not in the way it can be done for a single resonance whose shape is known. The key point is not the particular form of the function (1.136) but rather the well defined sign of F. Let us start from the classical model, namely the expression (1.102) for the scattering length,
The
2
b W
2
2
-
WO
_ .
i
7w ’rne ;,.ein-
(1-138)
44
Frangois
de
Bergevin
The
general case can be represented by summing many expressions of this corresponding to each different resonance wo with a different damping constant -y. Since this model is defined by two independent functions of WO, a distribution of the resonance densities and a distribution of the damping constants, one could expect the real and imaginary parts of the scattering length to also constitute two independent functions. However some constraints are imposed because the damping constants -/ are necessarily positive. Although kind
these constraints
seem
to be
weak,
it is remarkable that
to lead to
a
relation between the real and
that such
a
relation does not
come
from
they are sufficient imaginary parts of b. We shall see a particular scattering model; it is
general and concerns the response of any system to an excitation. For the proof, we return to the model with only one resonance (.1-138) but this could be easily extended to the general case. To prove the existence of a relation between the real and imaginary parts of b, it is necessary to make use of a mathematical trick, the analytical continuation of function b in the complex plane. The trick allows one to express some basic properties of complex functions. Indeed b is a complex function of the real variable w. If such a function can be represented by a series expansion which converges for any real value of the variable, then it can also be defined for complex values of this variable. Hence, the series still converges in a domain of the complex plane. Inside this domain, the function that we shall call here O(z) is analytic and follows Cauchy’s theorem. This theorem ensures that for any closed contour C inside the domain of analyticity and for any point z inside the contour, more
OW
I =
27ri
JC
dz’ z
z’ complex,
(z,
Z
C taken in the For this relation to be useful the
integral
positive sense).
must be taken
only
the
region known, plus a curve which continuously approaches infinity in the lower half-plane for instance a semi-circle with a radius approaching infinity (Fig. 1.11, drawn with variable w instead of z). We obtain the wanted relation provided that: (a) the function is analytic in all this half-plane, (b) it approaches zero when the modulus of the variable approaches infinity so that the integral taken over the semi-circle is zero. Under such conditions, relation (1.139) is expressed as an integral over z’ real. These conditions however impose z to be inside the. contour and therefore to have an imaginary part strictly negative though one would wish to have only real quantities. Nevertheless z can be on the real axis, but then the expression on the left-hand side is divided by two since z is at the border (a rigorous proof is available). Finally if P represents the principal part of the integral at the singularity x’ x, then where the function is
over
i.e. the real axis. Let C be the real axis
=
W
Ti
P
j
+00
-.0 -
dx’ X
X
(x, x’ real).
(1.140)
Interaction of
I
X-rays (and Neutrons) with
Matter
45
imaginary parts of this relation can be written separately. This if 0 (x) satisfies the above conditions (a) and (b), i. e. it is analytic in the lower half plane and tends to zero when jxj goes to infinity, some integral relations exist on the real domain of x between its imaginary and real
The real and
shows that
parts.
Imaainary
ci
Real
Fig. 1.11. Integration over a contour C defined by the real axis and a semi-circle having its radius approaching infinity. If a function does not have any pole inside the contour it satisfies relation (1,139) (w has the same role as the z variable). The poles of the scattering length b(w) (1.138) have been represented. They are outside the contour
The
written in
scattering length
(1.138),
which is
a
polynomial
fraction
of the variable w, is analytic over any domain which does not contain its poles, i.e. the zeros of its denominator. These zeros, indicated in Fig. 1. 11, are i7/2 w, P, depends on wo and -y). The scattering length b satisfies condition
(a)
because the
which b
by
are
w.
constant -y which is
damping
in the upper half
plane.
necessarily positive yields poles
satisfy
To
but it is better to divide
going addany pole
we are
by W2
to comment. Let
and does not
us
since
we
(b) one could divide b(w)lw (after replacing O(x)),
condition
The relations would then be valid in then
get
more
general
notice that the division of b
change
the domain of
by
relations w
2
as
does not
analyticity.
(1.140) is not yet completely convenient because the domain does not extend over the entire real axis but only over its,
A relation such
physical positive side (the oo
as
variable is the radiation
frequency). Integrating
from 0 to
is however sufficient since b verifies
b(-w) We should look if such
a
--
b*
(w).
(1.141)
symmetry of the scattering length is attached
to
A Fourier transform which transforms
particular model, or more general. expression from w to time space shows that this is simply the expression of a symmetry by time reversal. It is thus a general property which a
the above
Frangois
46
de
Bergevin
has however
a limitation: this symmetry does not hold for magnetic moments following expressions do not holdfior magnetic scattering. In that case, the equality (1.141) is written with a minus sign and different relations are obtained.’ With this equality ,tnd. a bit of algebra, one can rewrite the real and imaginary parts of (I. 1 0). Replacing 0 by b (w) lw’ and x, x’ by w, W’ yields so
the
Re [b (w) 1w
2
2]
2wp
_Emfb (W) /W 2] These relations
are
w’lm[b (wl) /U) /21dw’
P
7r
7r
W
0
JO
/2
_W
2
Re[b(W’)1W12] d.,.
’
W12
-
(1.143)
W2
the so-called Kramers and
Kronig
(1.142)
or
dispersion
relations for the
scattering length. model, the proof we have given
poles of b(W )/W 2 positive value of the damping constant but it can also be inferred from the principle of causality, which is of very general extent. To understand the equivalence of these two hypotheses, positive value of the damping constant and principle of causality, it is worth returning to the resolution of the differential equation (1.99), In this
assumes
all the
to be above the real axis. We have inferred this from the
which describes the movement of the electron in the incident field. We rewrite this
the
equation by noting z in
The
u
properties of
displacement
radiated field which is
proportional
mii +
A
u
instead of z to avoid any confusion
the present section; for simplifying, u will be a scalar. that we are going to discuss now are also the ones of the
with the variable
-lit
to
u.
2
+ mw,u
-
(-e)Eo e
iwt
(1.144)
systematic method to solve such a differential equation with a right-hand f (t) consists in using the Green function of the equation. This method
side
has been described in section 1.2.5. Let of
equation
is
us
recall that
a
solution of this kind
given by +00
U(t)
UO(t)
G (t
+
-
t) f (t’)
df
+00
G(t
-
t’)f (t’) dtl,
(1.145)
-00
function, G(t), is solution of the equation with J(t) instead f (t) right hand side; the solution uo (t) of the homogeneous equation the right hand side) becomes nearly zero after a certain amount of (without time due to damping. Writing the electron displacement u(t) as in (1.145)
where the Green of
7
in the
In practice the same dispersion relations can be written for magnetic and non magnetic scattering lengths provided that the magnetic part is affected by a factor i.
I
Interaction of
X-rays (and Neutrons) with Matter
47
to be
given. The displacement u of the excitation superposition given action f at any time t’; since the laws are invariant by time translation, the t’. G(t) is obtained through coefficient G only depends on the difference t its Fourier transform g (w). We replace the right-hand side of equation (L 144) by (t), whose Fourier transform is one. The derivatives in the left-hand side transform into powers of w, so we get allows the at
a
following physical interpretation
time t is the result of the linear
-
g
which
(W)=
21
1 -
MW
2
_W0_
(1.146)
i7w1rn)
yields G(t) +00
G (t)
27rm
eiwt W2
-
W20
-
i7W/M
(1.147)
dw.
integral, it is possible to integrate along a closed path in complex plane: if the function does not have any pole inside the path of integration, its integral over it is zero. The poles of g(w) are those of the scattering length that we have just discussed; the integral taken over the path of integration C (Fig. 1.11) is then zero. For t < 0 the integral over half the circle is also zero since the numerator is bound and the integral of 0 for t < 0. dw / I w12 goes to zero when I w I goes to infinity. Then G(t) It is important to mention that the proof depends on the position of the poles of g(w), and on the positive sign of the damping constant. Conversely if G(t)-- 0 for t < 0, it can be shown that g(w) does not have any pole below the real axis and the Kramers-Kronig relations can be applied to g(w) and u(W). The condition G(t) equal to zero at negative times constitutes the expression of a causality principle, according to which an excitation given at a certain Z.nstant cannot produce any effect before this instant. Making the dispersion relations to depend on this principle gives them a very general extent, beyond the cases where it is possible to clearly define some damping. In our world most of the phenomena are irreversible and time is therefore asymmetric. The positive character of the damping and the principle of causality as discussed here, both constitute two manifestations of the irreversibility. We have proved that one or the other of these two principles yields the dispersion relations (1.143). But a question still remains. It is generally admitted that microscopic laws in physics are mainly symmetric with respect to time reversal. On the opposite, the irreversibility manifests itself in macroscopic phenomena and in-statistical thermodynamics. We may be surprised that irreversibility is invoked in the scattering of radiation by an atom, which seems to be a rather elementary microscopic phenomenon. However one can also see that scattering involves some disorder. A plane wave travelling in vacuum, such as the incident wave constitutes a very unlikely state that can be considered as out of equilibrium. The ground state of a radiation is made of random waves in thermal equilibrium with neighbouring objects. Even considering a monochromatic radiation inside a perfectly reflecting box, it is only To calculate this
the
--
Frangois de Bergevin
48
possible
to find the initial
wavelength that has been scattered into multiple plane waves. One can see that the scattering of a plane wave by an atom is irreversible, a bit like the dilution of an alcohol droplet in a glass of water. The final state is the spherical wave moving away from the atom, superimposed to the incident wave which has a reduced amplitude. Therefore the unique incident plane wave has been changed into a superposition of plane waves travelling in all the directions. In addition, any of the plane components of the diverging wave can be associated to a particular movement of disordered
the atom since the momentum must be conserved. This is reminiscent of the dilution effect. If the
scattering were reversible, one could produce the reverse operation: starting from a spherical wave converging towards an atom and from a plane wave, one could see the plane wave coming out with an increased amplitude. This would be difficult to realise and may be impossible. For this, one should correlate the different plane components of the converging wave to some particular movements of the atom. The difficulty is similar to the one
hol
which would be faced in
droplet by imposing
initial conditions such
an
attempt
to invert the dilution of the alco-
to the molecules of the water-alcohol mixture
the mixture would demix into two
some
a phases the simplified problem not taking the absorption and the fluorescence in account. Multiple photons are then re-emitted for only one absorbed; in that case the radiation as
few instants. To what must be added the fact that
becomes still
more
after
have
we
disordered and this event is less reversible. As
a
matter
of fact it appears that absorption and resonant scattering contribute much more to the dispersion than pure elastic scattering. From the above arguments one can be convinced that even though the scattering looks like an
elementary phenomenon, it Zs actually something irreversible, which has obey the dispersion relations associated with irreversibility.
1.5
1.5.1
to
X-Rays: Anisotropic Scattering Introduction
we present briefly some other types of x-ray scattering, obessentially in crystalline materials. These are the magnetic scattering, which depends on the magnetic moment of the atom, and the Templeton anisotropic scattering, which depends on the neighbourhood of the atom in the crystal. A common feature to these two scattering effects is their anisotropy. The usual scattering amplitude which is described in the previous sections can be said isotropic because it depends on the incident and scattered polarisation directions through a unique factor, s,.Zi, independent of the orientation of the scattering object. The atomic scattering amplitudes which we discuss now can be said anisotropic because they depend on the
In this section
served
orientation of the characteristic
dent and scattered
polarisations.
axes
of the atom with
The characteristic
respect to the incimay represent the
axes
1.
Interaction of
X-rays (and Neutrons) with Matter
magnetic moment direction of the atom if it exists, or the directions crystal field which eventually perturbs the state of that atom. These
scattering
nisms. The first
effects
can
take their
origin
’49
of the
from two different mecha-
is the interaction between the
electromagnetic radiation spin of the electron. It produces some scattering, the so-called non resonant magnetic scattering. This one is essentially independent of the binding of the electron in the atom, as is the Thomson scattering. A second type of anisotropic scattering arises as a part of the anomalous or resonant scattering, presented earlier in section 1.4.7. The atomic states I i > and one
and the
I
>
c
in
(1.106),
that is the initial state and the
one
in which the electron is
anisotropic. If that anisotropy originates from a magnetic moment, the resulting scattering is called resonant magnetic scattering. If the anisotropy is some asphericity of the atom kept oriented by the peculiar symmetry of the material, it is the Templeton anisotropic scattering.
promoted
may be
X-ray magnetic scattering is It
can
be used with
some
a
useful
complement
elements whose
common
to neutron
scattering.
isotopes strongly
absorb
thermal neutrons. The very good resolution (in all respects, position, angle and wavelength) of x-ray beams is an advantage for some studies. Since xray
scattering depends
on
some
characters of the
magnetic
moment in
a
way different from neutrons, it may raise some ambiguities left by neutron scattering experiments. One of these ambiguities is the ratio of the orbital
spin moment of the atom, because they contribute to neutron scattering exactly in the same way and. cannot be discriminated from each other. xray amplitudes given by spin and orbital moment depend differently on the geometry of the experiment and they can be separated out. The resonant xray magnetic scattering is element dependent and eventually site dependent, which may give some useful information. It is also a spectroscopic method which probes the electronic state of the atom. The availability of small and brilliant x-ray beams compensates for the smallness of magnetic amplitudes in the study of thin films and multilayers. When the magnetic element has a very intense resonant magnetic scattering, even a single atomic layer can be probed. to
Applications of Templeton anisotropic scattering are not yet fully developed. It can give some information on the orbital state of the atom in a crystal. Quite recently, a strong interest has developed about the so-called orbital ordering, where an alternating orientation of atomic orbitals can be detected with the help of this type of scattering. In the present section we give a short description of the non resonant magnetic scattering, the resonant magnetic scattering and the Templeton anisotropic scattering. We also discuss the anisotropy of the optical index. The case of the magnetic neutron scattering is described in Chap. 5 of this
book.
Franigois de Bergevin
50
-e
Reradiation
Force
Driving
E
*
E
E
E-dipol.
H
/T\ it \-/
E
H-quadr. E
grad
H pt
E-dipol. H
H
/T\
Torque HxjL
H-dipol. H
-e
Hxp/m E-dipol.
Fig.
1.12. The electron
can
scatter the
electromagnetic
radiation
through
a
variety spin pair
of processes. In each of them, the incident field moves the electron itself or its through a driving force on the left. The back and forth motion is indicated by a
of thin
opposite
arrows.
In this
motion, the electron re-radiates through
a
mode
The first process is the well known Thomson scattering. the scattering by the spin, drawn as a double arrow. The describe 2 4 to Processes line is the fifth in a correction to Thomson scattering when the electron process
indicated
has
a
on
the
translation
right.
motion, indicated by the momentum p. When integrated gives rise to a scattering by the orbital
orbit of the electron in the atom, it
over
the
moment
Interaction of
I
1.5.2
can
51
Magnetic Scattering
Non Resonant
Similarly
X-rays (and Neutrons) with Matter
to the Thomson
the
scattering,
be found either in the classical
resonant
magnetic scattering
quantum theory. The quantum calcu-
or
[10].
non
spin of the electron is associated a a magnetic description, interacts with the which, radiation. The the of Fig. 1.12 shows schematically how magnetic component the interaction between the electromagnetic field and the electron, comprised of an electric charge and a magnetic moment can produce a magnetic dependent scattering. Having some interplay between spin and motion in space, and having some magnetic properties attached to an electrically charged particle lation
be found in reference
can
with
are
The
classical
in
moment
relativistic effects. That relativistic character introduces the scale factor
jhqj /27rme between the
magnetic
and Thomson
(1.148)
(A,/A) sin 0,
2
--
scattering amplitudes of an electron. In a typically of the order of 10-2 Since
that scale factor is
diffraction
experiment only unpaired electrons, which are at most one or two tenths of all electrons of a magnetised atom, contribute to the magnetic scattering, the magnetic amplitude is in favorable cases 10-3 10-4 of the Thomson amplitude. The intensity of magnetic Bragg peaks of antiferromagnets is then affected by a factor of the order of 10-7 The orbital moment contributes to the elastic .
_
.
spin moment and with the same order of magnitude, but with a different dependence on wave vectors and polarisations. We write below the scattering length of an electron of spin S and orbital moment L
scattering
as
bmag
well
ir,
-
The tensors
as
the
(A,/A)
[(ee*
sc
Ti S . &j.
Ts, TL simply help
polarisations. angle between
Their elements
iin
and
(8) (P)
(
are
sc
T:L
.’ jj) L] .
.
(1.149)
to write these bilinear functions of the
vectors. In their
expression below,
20 is the
sc: (P)
(8) TS
S+
isc X iin 2isc sin 2 0 -2iij sin 20 sc X iin
(1.150)
(P) TL
(8) (P)
(
(is, iin)
0 -
( s iin) c
+
+
sin
2
0
2isc
x
sin 20
(1.151)
iii sin 20
Remember that we use for i a sign opposite to the one used in quantum theory in the frame of which these equations are usually written. Magnetic Compton scattering is also present but results only from the spin. Since hq can be an important fraction of mc, the magnetic Compton amplitude can be significantly larger than the elastic one.
Frangois de Bergevin
52
Resonant
1.5.3
Magnetic Scattering
explained in sections 1.4.4 the resonant, or dispersive, part of the scattering is based on the virtual excitation oi. an electron from a core level to an empty state, which can be just above the Fermi level. In the subsequent discussion in section 1.4.7, we have assumed that the polarisation factor was This assumption in fact may the same as for Thomson scattering, be wrong. Let us write the numerator of a particular term in bdispl (1.106), while making the dipolar approximation (the exponentials are reduced to 1) As
<
SPS*C-PIC
Again we express this of writing this tensor reference frame
use a
><
CPin-Pli
bilinear function Of ’
(s)
with the
(x, y, z)
-
=
,*c, &in
attached to the
(X) J b3 (Y) b2 (z)
’&S*C -Tres -’ in with
-
iC3
-
iC2 bi
+
Tres
tensor
Instead
-
as a
we
(Z)
iC3 b2 + iC2
(1.153)
bi + ic,
a2 -
(1.152)
-
basis, may medium, generally a crystal.
(Y) b3
al
a
(p) polarisations
and
(X) Tres
>
ic,
a3
being nine real coefficients, this is the most general tensor expression (1.152). The actual structure of that tensor is determined by the symmetry of the scattering atom. The spherical symmetry is frequently a good approximation, though never completely exact in a With ai, bi, representing
the
crystal;
then
T,,, reduces
factor
sc *-ein
A
case
of
ci
and
we recover
the usual
of the symmetry is the presence of a magnetic moment. time inversion, which should not change the scattering
a
amplitude, exchanges the
magnetisation.
incident and
* 0 4C
reverses
-the
T,,,, that is the
the form
(1.154)
X-
magnetisation, at least in the simplest cases. arguments should be completed by an explicit discussion
the direction of
These symmetry
physical process. shortly explained in Fig.
of the
3SC Tres -’ in
i
and
of
antisymmetrical part magnetisation. That part has cc
just
scattering beams
This shows that the
is of odd order in the
where i is
[11]
-
lowering
We may observe that
ici’s,
to the unit matrix
0C
The mechanism is described in reference 1.13 and
&SC -’ in Vi I
F1
+ +
caption. -
1)
+ i
(" *SC
X
iin)
( isc -2) (’ in -i) (2F, o
-2 -
Vi 1 F, I
-
-
F1
F,
-
-
[12]
and
crystal field,
In the absence of any
1)
1)
,
(1.155)
where FI-1, Flo and F11 contain some transition probabilities. These transitions are described by two indices, the first one standing for the change in the orbital moment AL
(I
in the
dipolar term),
and the second
one
for
Interaction of X-rays
I
(and Neutrons)
with Matter
53
Energy down Fe rmi
Lui
2P3/2
magnetic scattering in the case of the Lrlr as platinum. Due to the magnetic moment the resonance occurs preferentially in the spin down (S -1/2) half valence band, on the right. Because of the strong ’Spin-orbit coupling in the core shell 2p, the 2P3/2 level is completely separated out from the 2p,/2 and contributes
Fig.
1.13. Mechanism of the resonant
resonance
of
a
third
row
transition element, such
=
alone to the is
resonance.
netisation)
Therefore the
(S
=
-1/2)
state involved in the
resonance
rather defined value of the z- component (:F the direction of magof the orbital moment in the initial state of the electron. For a given
coupled with
a
polarisation of the radiation, this makes the amplitude magnetisation direction, here the up direction
to
depend
on
the atom
isotropic anomalous scattering, discussed in section just discussed antisymmetrical part. The third one depends on the axis along which the magnetisation is lying, but not on its.sign; it, is responsible for the magnetic linear dichroism. One should not forget that the above expression is to be multiplied by a resonance function of the energy showed in (1.136) and Fig. 1.10. AL, The first
term is the
1.4.7. The second term is the
The spin orbit coupling is a key feature of this mechanism. In the example displayed in Fig. 1. 13, the spin orbit coupling interaction is very large in the core
state, but in
some cases
only in the excited state. In formula, a quadrupolar one neglected if it corresponds to a tran-
it may be present
term written in the above
dipolar Though smaller, it cannot be sition to a strongly magnetised atomic shell such as the 3d shell of transition element ’, or 4f shell of lanthanides. It shows a quadrilinear dependence on i SC iin, k,;,, iin addition to the also exists.
i
-
The order of
magnitude
of the resonant
magnetic scattering
may vary
on
wide range. The K resonances, accessible for example in the 3d transition elements, have amplitudes which are comparable to the non resonant. Indeed a
the K shell has
no
orbital moment
so
that the effect relies
on
the
spin
or-
54
Frangois
’
de
Bergevin
bit
coupling in the valence shell, which is much less efficient. Furthermore dipolar transition occurs to a weakly magnetised p valence shell, while the strongly magnetised d shell caD give only a quadrupolar transition. The latter drawback limits also t1le L.jjjjr resonances of lanthanides, but then the L shell is completely split by the spin orbit interaction. In that case the amplitude is typically ten times larger than the non resonant. The Ljl-,.rjj the
of transition elements
resonances
are
favorable in all respects and enhance
several orders of
amplitude by magnitude compared to non resonant. In the of 3d case elements, long wavelength (of the order of 1.5 nm) can only fit long periodicities, and give diffraction on multilayer, or be used in reflectivity experiments. The L resonances of 5d transition elements arise in the 0. 1 nm range but among those only the platinum group elements, and mainly the platinum itself, can take a magnetic moment. The Miv,v resonances of actinides offer the same favorable characters and are also quite effective, with amplitude enhancement by a factor of the order of one thousand. The wavelength, near 0.35 nm for the uranium allows for Bragg diffraction experthe the
iments.
Templeton Anisotropic Scattering
1.5.4
Even without any magnetic moment, the atom may show a low symmetry. Most often, this arises from the crystal field. A spontaneous orbital order
(that
arrangement resulting mainly from the electrostatic inneighbouring atoms) is also expected in some materials. As a consequence the symmetrical part of (1.153), aj, bi, differs from the unit matrix. Again a quadrupolar term may exist. The Templeton scattering produces some change in the intensity of Bragg peaks at the absorption edges and this can be used to get more structural information. A striking feature is the occurrence of otherwise forbidden reflections [13- 5]. is
an
orbital
teraction between orbitals of
When
reflection is forbidden because of
a
scalar
(that
axis
a screw
or
glide plane,
the
independent on axis, the atom rotates from one site to the next, so that the tensor amplitude in (1.152, 1.153) may not cancel. This breakdown of a crystallographic extinction rule should not be confused amplitude
of the
cancels
atom).
For
if it is
only example
a
with
is
the orientation
a screw
with the appearance of e.g. the 2 2 2 reflection in the diamond structure. In that case the structure factor is a scalar and the broken extinction rule is not a
general
rule for the space group; it
applies only
to
a
special position
in the
cell.
1.5.5
The Effect of
If the
is
tical
The
scattering properties.
an
Anisotropy in
anisotropic, non
so
the Index of Refraction
may be the index of refraction and the op-
resonant
magnetic scattering amplitude is
zero
in
the forward direction. From the discussion in section 1.3 it cannot contribute to the refractive index. We shall therefore discuss
only the
consequences of
Interaction of
I
X-rays (and Neutrons) with Matter k
55
Templeton scattering. We give only some brief inquestion which could deserve quite a long development. The propagation of neutrons in magnetised materials and the associated reflectivity is examined in the chapter 5 of this book. It is different from, and somewhat simpler than the propagation of the electromagnetic radiation in an anisotropic medium, especially when the interaction is strong. The thermal neutron has a non relativistic motion which allows for a complete separation of the space and spin variables. The direction of propagation is in particular independent of the spin state. The electromagnetic radiation instead is fully relativistic, which intermixes the propagation and polarisation properties. In that case, the direction of propagation depends on the polarisation. Several unusual effects are consequently observed. For example the direction of a light ray may differ from the normal to the wave planes, or a refracted ray may lie outside of the plane of incidence. resonant
magnetic
formation
and of
this
on
Starting from the Helmholtz equation (I 15) -
modifies the dielectric constant we
wrote the Helmholtz
be of fourth tensor we
order,
-_
n’)
of the medium
anisotropy
which becomes
a
tensor.
Though
for the 4-vector A and the tensor should
all the useful coefficients
are
contained in
in space, similar to (1.153). of crystal optics, described in several
a
third order
In the absence of
acting
have the
(c/,Eo
equation
the
case
magnetisation, textbooks, e.g. [3]. If
magnetised, the antisymmetrical part of the tensor is non zero and some phenomena occur, such as the Faraday rotation of the polarisation plane or,,the magnetooptic Kerr effect (that is polarisation dependent and polarisation rotating reflectivity). The basic theory can be found in [16]. The theory as exposed in the textbooks is drawn in the dipolar approximation, which is legitimate in the range of the visible or near visible optics. I am not aware of a complete description of anisotropic optics including the quadrupolar terms. It seems reasonable in practice to use instead of such a full theory, some perturbative corrections since the quadrupolar resonance terms are always small. The incidence of the quadrupolar term is clear in an effect well observed with visible light and discovered nearly two centuries ago: it is the optical activity, that is the rotation of the polarisation plane in substances the medium is
which lack
a
center of
symmetry. Indeed the combination of
a
effect. It is
dipolar
and
a
small
effect, produce quadrupolar terms is required since it corresponds to differences of the order of 10-4 between the indices of the two opposite circular polarisations. Yet it can be easily observed because the absorption of the visible light is still smaller and samples more than 10’ wavelengths thick can be probed. to
such
an
a
In the x-ray range, at energies of several keV and above, the refractive index differs from one by a small value and its anisotropic part is still smaller.
Only a limited ple terms. One dichroism. In
a
list of effects
are
observed and
they
are
interpreted magnetic
of the most studied of those effects is the
or ferrimagnet, where a net magnetisation is present, changes, according to the helicity of circularly polarised
ferro
the refractive index
in simcircular
Frangois de Bergevin
56
being parallel
antiparallel
to the
magnetisation. Indeed the optical (1.69) (1.83) yields the absorption atomic cross section or the dispersive part of the index of the medium, from the part of the scattering length written in (1.152, 1.155). For that we make ,,, equal to in and equal toi_ in. For a circular polarisation, the term (1.154) is real and x-rays
theorem
or
or
its extension
reads
+kin.i. The
for the
right handed helicity to + for the left complex resonance factor which we have left out from the formula. The difference in the real, S, component of the index, between both helicities gives rise to the Faraday rotation of the polarisation plane. Similarly the difference in the imaginary, , component gives rise to a difference in the absorption, called the magnetic circular dichroism. Once a circularly polarised radiation is available, it is relatively easy to measure that change of absorption, usually by switching the magnetisation parallel or antiparallel to the beam. Similarly to the resonant scattering, the dichroism shows a spectrum in the region of the absorption edge. At the L edges of the 3d elements, the resonances and their magnetic parts are quite large and the full optical theory, in the dipolar approximation, should be considered. Some reffectivity measurements have been done, e.g. sign
is switched from
(1.156)
handed. This is to be
-
multiplied by
the
[17]. LA
Appendix:
Anne
Sentenac, Fran ois Vignaud
Approximation
the Born de
Bergevin,
Daillant,
Jean
Alain
Gibaud,
and Guil-
laume
appendix we give the Born development for the field object. In absence of any object, the field (scalar for simplicity) is homogeneous Helmholtz equation,
In this
scattered
by
a
deterministic
( A + k2) Ain (r) 0
The
object introduces
(1.16-17).
In this
case
a
perturbation
V
on
the field is solution 2
(ZA + ko
-
--
solution of the
(LAI)
0.
the differential operator,
see
Eqs.
of,
V(r)) A(r)
--
0.
(LA2)
sum of an incident field Ain (r) homogeneous equation) and a scattered field A, which satisfies the out-going wave boundary condition. Following section 1.2.5, we transform Eq. (LA2) into an integral equation by
The total field A
can
Aine- iki,,.r
wave
(plane
be written
as
the
solution of the
=
(and Neutrons)
X-ray,
G_
outgoing
that satisfies
wave
(LA3)
-
47r
+J
power of the convolution
A(O)(r)
-ikor
I --
r
G
-
(r
AM
--
We obtain
r) V (r’) A (r) dr’.
-
write the solution of this
one can
A
with,
(r)
boundary condition.
Ai,, (r)
A
series in
57
the Green function
introducing
Formally,
with Matter
(LA4)
integral equation
in terms of
a
operator [G-V], +
AM SC
(2) + A SC +
(LA5)
Aijr), A(’) (r)
d 3r’G
SC
Jd r’J
A(2)
3
SC
d 3r"G_
(r
(r.
-
-
r’) V (r’) Ail (r%
-
r’)V(r)G- (r’
-
r")V(r")Ai1, (r").
When this series is convergent, one gets the exact value of the field. The main issue of such an expansion lies in its radius of convergence which is not easy
Physically,
to determine.
potential V(r)
the
combined with the
propagation
operator G_ represents the action of the particule (or polarisation density) at r/ on the incident wave, i. e. a scattering event. When the potential appears (in the first order term) the incident wave is singly scattered by the once,
object. When it appears twice (in the second order term), scattering events, etc. Eq. (LA5) can also be viewed as a perturbative development in which the scattering event [GV] is taken as a small parameter. The first Born approximation consists in stopping the development in Eq. (LA5) to the first order in V (thus assuming the of the
particules one
accounts for the double
predominance of single scattering). We now proceed by evaluating the scattered far-field and the scattering crosssection. We assume that the observation point r is far from all the points r’ constituting the object (with respect to an arbitrary origin situated inside the
object).
In this case,
one
has
Ir 8
In operator notation
1/1
_
which
X
yields A
with
+X+X2 +
=I
==
....
can
rl
make
Indeed,
,:z
r
an
-
(LA6)
ia.r’,
analogy with the Taylor expansion
the field
can
be written
as
A
=
Ai./i
the series
Ail +
[G-V]f
one
-
=
[G-V]Ain
f G-(r
-
+
[G-V][G-V]Ain
r’)V(r’)f (r)dr.
+... +
[G- V]n Ai. +...,
-
of
[G-VI
Frangois
58
so
de
Bergevin
that
G_
(r
-
e-ikor
r’)
Ir
47r
r’l
-
Using this far-field approximation given in section 1.2.4,
in
A,,,(r)
4r
Eq. (LA4),
-Ainb(i i)
eikoia.r’.
(LA7)
r
one
retrieves the
expression
(LA8)
r
Bearing in mind the Born development for the field, one can write the scattering length b in the form, b(fi) W)(ii) + b (2) (ji) + with, for example ...,
I
b
d 3,r/ V(r’) ei(kofi-ki.).rl
(LA9)
.
47r
The calculation of the differential
scattering cross-section, Eq. (1.36),
do-
dQ
lb(fi) 12
is then
straightforward. noting that the perturbative development of the energy (which is proportional to the square of the field) starts at second order in V. Hence, to be consistent in our calculation, we should always develop the field up to the second order to account for all the possible terms (of order two) in the energy. A striking illustration of this remark is that the first Born approximation does not satisfy energy conservation. This can be readily shown by injecting the perturbative development of b in the optical theorem which is a direct consequence of the energy conservation. The optical theorem relates the total cross-section to the imaginary part of the forward scattered amplitude. One has, see section 1.3.2, o-t,,t 2A_Tm[b(ki,1)]. If one disregards lossy media, the total cross-section is equal to the scattering cross-section, It is worth
=
O’S.c
-_
J I (iii) 12 b
dQ
=
2A_Tm[b(ki,1)].
(LA10)
expansion Eq. (LA5) being a formally exact representation of the field, optical theorem, written as a series, is verified at each order of the perturbative development. One gets, to the lowest order, The
the
II
b (1) (fi)
12 dQ
--
2A1m[bM(ki,,)
This last
equation shows clearly that if one
conserve
energy,
order in the
one
forward
+ b (2)
wants the Born
should calculate the scattered direction.
(ki,,)].
(LAII)
approximation to amplitude up to second
Interaction of
I
X-rays (and Neutrons)
with Matter
59
References 1.
Landau, E.M. Lifshitz, Course of theoretical physics vol. 8, Electrodynamics media, Pergamon Press, Oxford, 1960. L. Landau & E. Lifchitz. Physique th6orique. Tome VIII, Electrodynamique des milieux continus. Ed. Mir, Moscou (1969). J. D. Jackson. Classical Electrodynamics. 2nd Ed.. John Wiley & Sons (1975). M. Born & E. Wolf. Principles of Optics. 4th Ed.. Oxford, New-York (1980). R. W. James. The optical principles of the diffraction of X-rays. The cristalline state, vol 11, Sir Lawrence Bragg ed.. G. Bell & Sons ltd, London (1962). W. Schfilke. Inelastic scattering by electronic excitations. In Handbook of synchrotron radiation, vol. 3, G. Brown & D. E. Moncton ed.. Elsevier Science
L.
of continuous
2. 3.
4.
5.
Publisher 6.
7.
(1991).
Berestetskii, E.M. Lifshithz, L.P. Pitaevskii, Course of theoretical physics vol. 4, Quantum electrodynamics, Pergamon Press, Oxford, 1982. E. Lifchitz & L. Pitayevski (L. Landau & E. Lifchitz). Physique th6orique. Tome IV, Th6orie quantique relativiste, Premi6re partie. Ed. Mir, Moscou (1972). International tables for x-ray crystallography; vol. III Physical and chemical tables; vol. IV Revised and supplementary tables to vol. 11 and 111, The Kynoch
V.B.
Press, Birmingham, 8.
elements Z 9. F.
=
I to Z
Stanglmeier,
48, 626 10.
1968.
E. Storm & H. 1. Israel. Photon B.
=
cross
sections from I keV to 100 MeV for
A7, 565-681, (1970). G6bel, M. Schuster, Acta Cryst.
100. Nuclear Data Tables
Lengeler,
W.
Weber,
H.
A
(1992).
Blume, J. Appl. Phys, 57, 3615-3618, (1985). Templeton and L.K.Templeton, Acta Cryst. A 36, 237 (1980). J. P. Hannon, G. T. Trammel, M. Blume & Doon Gibbs, Phys. Rev. Lett. 61,
M.
11. D.H. 12.
15.
1245-1248, (1988). Dmitrienko, Acta Cryst. A 39, 29-35, (1983). D.H. Templeton and L.K. Templeton, Acta Cryst. A 42, A. Kirfel and A. Petcov, Acta Cryst. A 47, 180 (1991).
16.
A. V. Sokolov.
17.
C. C.
13. V. M. 14.
478
(1986).
Optical properties of metals. Blackie and Son Ldt, London
(1967). G.
Kao, C. T. Chen, E. D. Johnson, J. B. Hastings, H. J. Lin, G. H. Ho, Meigs, J.-M. Brot, S. L. Hulbert, Y. U. ldzerda & C. Vettier, Phys Rev. B
50, 9599-9602,
(1994).
Statistical
2 at
Aspects
of Wave
Scattering
Rough Surfaces
Anne Sentenael and Jean Daillant2 ’
2
LOSCM/ENSPM,
Universit6 de St Hr6me, 13397 Marseille Cedex 20, France, Physique de 1’Etat Condens6, Orme des Merisiers, CEA Saclay, 91191 Gif sur Yvette Cedex, France Service de
Introduction
2.1
The surface state of
objects
in any scattering experiment is, of necessity, of the most varied nature and lengthscales, ranging
rough. Irregularities from the atomic scale, where they are caused by the inner structure of the material, to the mesoscopic and macroscopic scale where they can be related are
to the defects in
processing in the case of solid bodies or to fluctuations in liquid surfaces (ocean waves, for example) The problem of wave scattering at rough surfaces has thus been a subject of study in many research areas, such as medical ultrasonic, radar imaging, optics or solid-state physics [1], [2], [3], [4]. The main differences stem from the nature of the wavefield and the wavelength of the incident radiation (which determines the scales of roughness that have to be accounted for in the models). When tackling the issue of modelling a scattering experiment, the first difficulty is to describe the geometrical aspect of the surface. In this chapter, we are interested solely by surface states that are not well controlled so that the precise defining equation of the surface, z z(x, y) is unknown or of little interest. One has (or needs) only information on certain statistical properties of the surface, such as the height repartition or height to height correlations. In this probabilistic approach, the shape of the rough surface is described by a random function of space coordinates (and possibly time as well). The wave scattering problem is then viewed as a statistical problem consisting in finding the statistical characteristics of the scattered field (such the
case
of
=
as
the
mean
the surface
value
or
field correlation
functions),
the statistical
properties of
being given.
In the first section of this contribution
we present the statistical techniques rough surfaces. The second section is devoted to the description of a surface scattering experiment from a conceptual point of view. In the third section, we investigate to what extent the knowledge of
used to characterise
the field statistics such
as
the
mean
field
for
or
field autocorrelation is relevant
interpreting the data of a scattering experiment which deals necessarily with deterministic rough samples. Finally, we derive in the fourth section a simple expression of the scattered field and scattered intensity from random rough surfaces under the Born approximation.
J. Daillant and A. Gibaud: LNPm 58, pp. 60 - 86, 1999 © Springer-Verlag Berlin Heidelberg 1999
Statistical
2
Description
2.2
2.2.1
Aspects
of Wave
Scattering
at
Rough
Surfaces
61
Randomly Rough Surfaces
of
Introduction
example of a liquid surface. The exact morphology rapidly fluctuating with time and is not accessible inasmuch as the detector will integrate over many different surface shapes. However statistical information can be obtained and it provides an useful insight on the physical processes. Indeed, these fluctuations obey Boltzmann statistics and are characterised by a small number of relevant parameters such as the density of the liquid or its surface tension (see Chap. 9). Let
us
first consider the
of the surface is
origin (such as metallic treatments (like similar technological undergone ) optical all the microscopic to it is Since and reproduce impossible cleaning). polishing factors affecting the surface state, these surfaces have complex and completely different defining equations z z(x, y). However, if the surface processing is well enough controlled, they will present some similarities, of statistical nature, that will distinguish them from surfaces that have received a totally We
now
consider
set of surfaces of artificial
a
that have
mirrors
--
different treatment.
examples,
In these two
we
are
faced with the issue of
describing
a
set of
real surfaces which present similar statistical properties and whose defining equations z (x, y) are unknown or of small interest (see Fig. 2.2. 1). It appears convenient [2] to approximate this set ’of surfaces by a statistical ensemble of surfaces that
are
realisations of
a
coordinates r1l (x, y), parameters of the physical processes size of the polishing abrasive in the
likely that
random continuous process of the plane properties depend on some relevant
whose statistical
-_
affecting case
the characteristic functions
the surface state
(like the grain origin). It is
of surfaces of artificial
z(rll)
of the surfaces
generated by
random process will be different from that of the real surfaces under but the statistical properties of both ensembles should be the same.
2.2.2
the
study,
Height Probability Distributions
Generally speaking, a random rough surface is completely described statistically by the assignement of the n-point (n -+ oo) height probability distribution p,, (r,11, zi r,, 11, z,,) where p, (r 111, zi r.11, z,,)dzi dz.,, is the probability the height coordinates for the surface points of plane rjjj,...r,jj of being at ...
...
...
dzi z,, + dz,,). However, in most cases, we restrict the description of the randomly rough surface to the assignement of the Inone and two-points distribution functions pi(r1j, z), and P2(rill, zi; r211, Z2)information. this need theories solely deed, most scattering From these probability functions, one can calculate the ensemble aver-
between
(zi
...
z,,)
and
(zi
+
...
age of any functional of the random variables
(zi
...
z,)
where zi
=
z(rjjj,,,,),
Anne Sentenac and Jean Daillant
62
7.0
-A_o
_=o
20-0
Fig.
Examples properties
of various
2. 1.
tistical
through
the
2-
rough surfaces
that present the
same
Gaussian sta-
integral, oo
(F)(rjjj r,,Il)
f
--
...
F (zi
...
z,,)p,, (r 111, z,
...
r,,Il,
zn)dzj
...
dz,,.
(2.1)
oo
The domain of integration covers all the possible values for (z, Zn). This quantity is equivalent to an average of F calculated over an ensemble of ...
surface realisations
Sp, N
(F)(rjjj r,,Il) ...
F(zpl..’.zPn),
(2.2)
surface realisation at
plane coordinates
lim
=
n
N-+oo
P=1
where
zjP
is the altitude of the
p-th
rjjj. With this
definition,
one
obtains in
particular the
mean
height
of the surface
through
(z) (r I I) The
mean
square
height
z
(rll)pl (r1j, z) dz.
of the surface is
(2.3)
given by
oo
(Z’ (rll) The
height-height
-
foo
Z2 (rll)pl (ril, z) dz.
correlation function
(2.4)
C,, is defined by
c’o
Cz,z(rjjj)r2jj)
ZIZ2)
f
ZI Z2P2 oo
(r 111, zi, r211, Z2) dzl dZ2,
(2.5)
where zj
--
Scattering
Statistical Aspects of Wave
2
z(rjll).
Rough
at
Surfaces
63
introduce the pair-correlation function
It is also usual to
g(rlll, r2ll) which averages the square of the difference points of the surface,
height
between two
C,O
g(rill, r2ll)
-
((ZI
-
Z2)2)
(Z1 -
g(rill, r2ll)
Note that
=
Homogeneity
2.2.3
2(z 2) (rll)
-
Z2
)2 P2 (rill, zi
r2ll,
,
Z2)dz,dZ2 .(2.6)
00
-
2C,, (rill,
r2ll)
Ergodicity
and
Randomly rough surfaces have frequently the property that the character of height fluctuations z does not change with the location on the surface. More precisely, if all the probability distribution functions pi are invariant under any arbitrary translation of the spatial origin, the random process is called homogeneous. As a consequence, the ensemble average of the functional F(z, z,) will depend only on the vector difference, rj 11 r, 11 between one of the n space argument rill and the (n 1) remaining others rjll, j 2 n.
the
-
...
--
-
(F) (r 111,
...,
r,,
11)
--
(F) (0 11
...
r,,Il
-
...
(2.7)
rill).
When the random process is isotropic (i.e has the same characteristics along reduce to the distance Jrjll rilll between any direction) the dependencies will only consider hoone of the argument and the others. Hereafter we -
space
mogeneous isotropic random processes and we propose for the various functions already introduced.
The can
mean
find
altitude
reference
a
o-,
a
depend
does not
plane surface such
of the surface is also
height
(z)(rll)
constant and
z)
as
--
on
a
the rll
0. The
position
mean
define the root
we
simplified
notation and
one
square deviation
mean
square
(rms)
as
00
a2
-
(Z2)
=
f
z
2pi (z) dz.
(2.8)
00
height is often used to give an indication of the "degree of roughlarger u the rougher the surface. Note that the arguments of the probability distribution are much simpler. Similarly, the height-height correlation function can be written as,
The
rms
ness",
the
C-,, (rill,
r2ll)
-
(_z(Oll)z(rll))
-
C,,(rll)
-j
Z1 Z2P2
where rll -- jr1l 1. We also introduce, with these point and two points characteristic functions,
Xi
(8)
-
(2.9)
(Zl; Z2, r1l) dzi dZ2,
simpler notations, the
foo pj(z)e"dz,
one
(2.10)
Anne Sentenac and Jean Daillant
64
00
X2
( S SI) 7’11) )
::::::
f.
P2
(Z) z’I) r1j) e "’+""’dzdz’.
One of the most
important attributes of a homogeneous random process is its P(qjj) that gives an indication on the strength of the surface fluctuations associated with a particular wavelength. Roughly speaking, the rough surface is regarded as a superposition of gratings with different periods and heights. The power spectrum is a tool that relates the height to the period. We introduce the Fourier transform of the random variable z, power
spectrum,
I
i(qjj) where trum
qjj
--
(q, qy)
is the
=
z(rll)e iqjj.rjj dril
7r2
in-plane
wave-vector transfer. We define the spec-
as
P(qjj)
--
fli(q11) 12)
The Wiener-Kintchine theorem
[5]
Fourier transform of the correlation
P(q1j) More
(2.12)
precisely,
one
1 -_
41r 2
I dr1le
--
(,i(qjj)i(-qjj)).
(2-13)
states that the power
spectrum is the
function,
iqll.ril
(z(0jj)z(rjj))
(2.14)
shows that
?(qjj)i(q’jj))
(i(-qjj)i(q’jj))
--
--
0 , ,(q11)6(q1I
-
q’11).
(2-15)
The Fourier components of a homogeneous random variable are independent random. variables, whose mean square dispersion is given by the Fourier transform of the correlation function. If the power spectrum decreases slowly with increasing q11, the roughness associated to small periods will remain important.
Thus, whatever the length scale, the
surface will present
irregularities.
In the real space, it implies that the correlation between the heights of two points on the surface will be small, whatever their separation. As a result, the correlation function will exhibit the derivative for
function sented in
(or
exemple). spectrum)
power
Fig.
a
singular
behavior about 0
(discontinuity
of
An illustration of the influence of the correlation on
the
roughness aspect of the surface is prespecial case of a Gaussian
2.2 and detailed in Sec. 2.2.4 in the
distribution of
heights.
have been interested solely in ensemble average, which neknowledge of the complete set of rough surfaces generated by the homogeneous random process (or the probability distributions). However, sometimes only a single realisation S,, (with dimension L, Ly along Ox and Oy) of the random process is available and one defines the spatial average of z,,) for this surface by, any functional F (zi, Until
now we
cessitates the
...’
FP(011’...’r,,11)
--
lim L ,,xL,
-+oo
LLY
JLxXLy drjjF[z(rjj)
...
z(rll
+
r,,11)]
Statistical Aspects of Wave
2
Scattering
at
Rough
Surfaces
65
happens frequently that each realisation of the ensemble carries the same homogeneous random process as every other realisation. The spatial averages calculated for any realisation are then all equal and coincide with the ensemble average. The homogeneous random process is then said to be an ergodic process. In this case, the following particular relations hold, It
statistical information about the
0’
C,,(rll) One
-
2
=
(Z2)
Jim L,,,L,-+oo
(z(0jj)z(rjj))
lim Lx,L,-+oo
L,LY 1
LxLy
z2 (rjj)drjj,
(2.17)
LxxLy
j
z
T X1 Lx
(r’ll)z(rll
+
rjj)dr’1. (2.18)
show that Eqs. (2.17) and (2-18) will be satisfied if the correla(r1j) dies out sufficiently rapidly with increasing r1l (see for
can
tion function Qz z
demonstration
rough
correlated
this
property implies that
one
realisation of the
can
so
realisation.
[5]). Indeed,
be divided up into subsurfaces of smaller area that are unthat an ensemble of surfaces can be constructed from a single
surface
Spatial averaging
amounts then to ensemble
averaging.
If the
ran-
dom process is homogeneous and ergodic, all the realisations will look similar while differing in detail. This is exactly what we expect in order to describe liquid surfaces varying with time or set of surfaces of artificial origin. The fact
spatial averaging is equivalent to ensemble averaging when the surface enough correlation lengths to recover all the information about the random process is of crucial importance in statistical wave scattering theory. that
contains
The Gaussian
2.2.4
Probability Distribution
and Various Correlation Functions
height probability distribution is taken to plays a central role because it has simple structure, and, because -of the central limit theorem, it is
In most
theories,
the
The Gaussian distribution
distribution that is encountered under If the
height
whose effects altitude will
a
be Gaussian.
especially probability
an a
great variety of different conditions.
number of local independent events cumulative, (like the passage of grain abrasive) the resulting obey nearly Gaussian statistics. This result is a manifestation of
z
of a surface is due to
a
large
are
the central limit theorem which states that if
a
random variable X is the
sum
independent random variables xi, it will have a Gaussian probability distribution in the limit of large N. Hereafter, we suppose that the average 0. The Gaussian height value of the Gaussian variate z(rll) is null, (z) of N
--
distribution function is written as,
Pi
W
=
o-v’2-7r
e xp
(
-
_02
)
(2.19)
Anne Sentenac and Jean Daillant
66
Gaussian variates have the remarkable property that the random process is entirely determined by the height probability distribution and the height-
height correlation function C,,.
All
higher
[5].
in terms of second order correlation is
given P2
in this
order correlations
The
two-points
expressible
by,
case
0’
(Z, Zf, r1j)
are
distribution function
exp
IF)
2,r,
2(Z2
+
Z/2)
-
2zz’C,,, (r1j)
-
2o-
4
-
2Q,2 ,(rjj)
I
-
(2.20) Other useful results
the Gaussian variates are,
on
XI
X2
(8, S,r1j)
-
(S)
-_
(e iSZ)
=
e_S20,2/2
(2.21)
e_0,2(S2+S,2 )12,ss’C_(1jj).
(e i(SZ_31Z1))
(2.22)
The correlation function
plays a fundamental role in the surface aspect. It the length scales over which.height changes along provides the surface. It gives in particular the distance beyond which two points of the surface can be considered independent. If the surface is truly random, C,,_,(rll) decays to zero with increasing r1j. The simplest and often used form for the correlation function is also Gaussian, an
indication
on
CZZ (r1l) The correlation larities
is the
length
(or bumps)
on
-
0.2 exp(-r 2g2). 11
(2.23)
typical distance between two different irreguBeyond this distance, the heights are not
the surface.
correlated. In certain
scattering experiments,
one can
retrieve the behaviour of the
cor-
relation function for q close to zero. We have thus access to the small scale properties of the surface. We have seen that the regularity of the correlation function at
mirrors the asymptotic behaviour of the power spectrum : high-frequency components of the surface decay to zero, the
zero
the faster the
smoother the correlation function about variations about cated
solely
zero
have the
for surfaces that
zero.
The Gaussian scheme whose 2
quadratic form o- (1 (rjj / ) 2) is thus present only one typical lateral length _
indiscale
161. For surfaces with structures down to the correlation function to be
affine
rough
surface for
arbitrary small scales, one expects singular at zero. An example is the self
more
which,
g(r1j) where A0 is
a
constant,
--
2h
(2.24)
Aor 11
or
C,, (r1j)
--
o’
2
(1
r,2,
h
_
2h
)
,
(2.25)
Statistical
2
with 0 < h < 1. The
Aspects
of Wave
Scattering
at
Rough
Surfaces
67
roughness exponent or Hurst exponent h is the key paheight fluctuations at the surface: small h values
rameter which describes the
produce very rough surfaces while if h is close to I the surface is more reguh lar. This exponent is associated to fractal surfaces with dimension D -- 3 as reported by Mandelbrodt [7]. The pair-correlation function given in Eq. -
(2.24) diverges are
represented
for r1l and the
oo.
Hence, all the lengthscales along the vertical axis
roughness
of the surface cannot be defined. We will
below that in that case, there is no specular reflection. However, very often, some physical processes limit the divergence of the correlation function, see
roughness saturates at some in-plane cut-off . Such by the following correlation function,
i.e. the
surfaces
are
well
described
C, (R)
-R 2h
-
,
liquid surfaces and surfaces
For
a2exp(-
close to the
functional forms described in Chaps. 6 and 9
(2.26)
2h roughening
are
transitions other
used.
0.4
0.2
0.0
VWW
-W _V"
-5.0
0.0
-0.2
z ’31,
";s
-0.4
0-6
-0.8
-1.0
-1.2 -15.0
_10.0
5.0
10.0
X
Fig.
2.2.
Various
rough
surfaces with Gaussian
correlation functions From bottom to top,
a2
exp(-L.2-), 2
2.2.5
C,,(R)
height distribution
C,,(R)
a2 4/( 2
but various
+ R2)2, C-,, (R)
0.2 exp(-
Complicated Geometries: Multilayers and Volume Inhomogeneities More
solely the statistical description of a rough surhomogeneous media. The mathematical notions that have been introduced can be generalised to more complicated problems such as stacks of rough surfaces in multilayer components. In this case, one must also consider the correlation function between the different interfaces, (zi (011)zj (r1j))
Up
to
face
now we
have considered
separating
two
Anne Sentenac and Jean Dw.Hant
68
where zi represents the height of the ith surface. A detailed description of the statistics of a rough multilayer is given in chap. 8, Sect. 8.2. One can also
describe in
(or
a
electronic
similar fashion the random fluctuations of the refractive index
density)
p. In this
case
p is
three-dimensional space coordinates Sect. 4.5.
Description
2.3
of
a
a
random continuous variable of the
(rij, z).
Surface
It will be introduced in
Chap.
4
Scattering Experiment,
Coherence Domains
We have seen how to characterise, with statistical tools, the rough surface geometry. The next issue is to relate these statistics to the intensity scattered by the sample in a scattering experiment. In this section, we introduce the main theoretical results that describe the interaction between and surfaces. Attention is drawn
waves
which takes
on
on
electromagnetic
the notion of "coherence domains"
particular importance in the modelling of scattering from foreword, we present briefly the basic mechanisms
random media. In this
that subtend this concept.
be shown
(bear
in mind the
Huygens-Fresnel principle or see Chap. by an electromagnetic incident field acts as a collection of radiating secondary point sources. The superposition of the radiation of those sources yields the total diffracted field. If the secondary sources are coherently illuminated, the total diffracted field is the sum of the complex amplitudes of each secondary diffracted beam. In other words, one has to account for the phase difference in this superposition. As a result, an interference pattern is created. The coherence domain is the surface region in which all the radiating secondary sources interfere. It depends trivially on the nature of the illuminating beam (which can be partially coherent), but more importantly, it depends on the angular resolution of the detector. To illustrate this assertion, we consider the Young’s holes experiment [8]. Light from a monochromatic point source (or a coherent beam) falls on two pinholes located in the sample plane (see Fig. 2.3). We study the transmitted radiation pattern on a screen parallel to the sample plane at a distance D. In this region, an interference pattern is formed. The periodicity A of the fringes, which is the signature of the coherence between the two secondary AD/d. sources, depends on the separation d between the two pinholes, A Suppose now that a detector is moved on the screen to record the diffrated intensity. As long as the detector width I is close to A, the modulation of the It
can
4 Sect.
4.1.6)
that
a
rough
surface illuminated
=
interference pattern will be detected. On the contrary, if I > 10A the intensity measured by the detector is the average of the fringe intensities. We obtain a constant equal to the sum of the intensities scattered by each secondary sources.
the
In this case, one may consider that, from the detector point of view, radiate in an incoherent way. We see with this simple experiment
sources
Statistical
2
Aspects of Wave Scattering
at
Rough
Surfaces
69
length is directly linked to the finite extent of the detector angular resolution).’ to a more accurate description of a surface scattering ex-
that the coherence
(equivalent We
to
now
finite
a
turn
periment.
Scattering Geometry
2.3.1
illuminating a rough a (perfectly sample along kin monochromatic) and detecting the flux of Poynting vector in an arbitrary small solid angle in the direction k c with a point-like detector located in the far-field region. We consider
an
ideal
scattering experiment consisting
in
beam directed
coherent
with
The interaction of the beam with the material results in
a
wavevector
transfer, q
=:
k,,
-
(2.27)
kin-
qz r)lane of incidence
sample plane
Fig.
Figure
2.3.
Scattering geometry
2.3 shows the
experiment.
The
plane
for
interpreting
scattering geometry
in the
surface
scattering
general
case
of
a
of incidence contains the incident wave-vector
the normal to the surface Oz. In
surface
kij and
it is usual to work
reflectivity experiment, 0--0. Yet the case 0 :A 0 is’ of diffrac: interest for surface ,tion experiments in grazing incidence geomspecial in the of incidence it is also useful to distinguish When plane working etry. the symmetric specular geometry for which Oin Os, and the off-specular The of which set for 0ij 0,,. equations following $ (2.28) gives the geometry components of the wave-vector transfer with the notations introduced in Fig.
in the
plane
a
of incidence and thus to have
--
2.3.
obviously linked to the degree of coherence fixed by, for example, the opening. However, for x-ray or neutron experiments the resolution actually generally limited by the detector slits opening.
It is also
incidence slit is
Anne Sentenac and Jean Daillant
70
i Scattering
2.3.2
q,
=
qy
=
qz
=
cos Oin) (cos 0,, cos (cos 0,, sin (sin Os, + sin Oi )
ko ko ko
(2.28)
m
Cross-Section
experimental setup presented in the previous section, one exactly measures scattering cross-section as described in Fig. 1.1 of isolated scattering object is the rough sample in this case). The Chap. 1, (the In the ideal
the differential
vectorial electric field E is written E
as
_-
the sum,
(2.29)
Ein + Esc
plus scattered field. We are interested by the flux of the Poynta surface dS located at the position R of the detector through ing for a unit incident flux. The precise calculations of the differential scattering cross-section are detailed in Sect. 4.1-6. In this paragraph, we simply introof the incident vector S
duce the main steps of the derivation. One assumes that the detector located at R is
(far-field approximation). We define see
placed far from the sample scattering direction by the vector ks,
the
Fig. 2.3, k c
-
koil
-
(2.30)
kOR/R.
It is shown in Sect. 4.1.6 that the scattered field
can
be viewed
as
the
sum
by the electric dipoles induced in the material by the incident field, (these radiating electric dipoles are the coherent secondary
of the wavelets radiated
sources
introduction). The strength of the induced dipole sample, is given by the total field times the permittivity 2 Let us recall that for x-rays, point, [k (r’) k2]E(r). 0 in the
presented
located at r’ in the contrast at this
-
2
(k (r/)
-
k2) 0
-
2[n2(r’)
k0
-
1]
=
-47rr,p,l (r/),
(2.31)
2 where p,1 is the local electron density and r, the classical electron radius. In the far-field region, the scattered field can be written as, see Eq. (4.19), (the 2
is only interested in materials with low atomic numbers for which the frequency is much larger than all atomic frequencies, the electrons can be considered as free electrons plunged into an electric field E. In this case, the movement of the electron is governed by m,dv/dt -eE, where m, v, -e, are the mass, the velocity and the charge of the electron. We find v (ze/m,W)E for a e"t time dependence of the electric field. Thus the current density is j -(ze 2pei/m,w)E where p,1 is the local electron density. Writing the -ep,iv 2 n C0-9E/(9t Maxwell’s equations in the form curIH j + rsoaElat aDlat
If
one
x-ray
=
=
=
=
or
I
as
_
a
on
whether the system is viewed
material of refractive index
(e2 /2m,co W2)p eI=
I
-
(A2/27r)r,p,1
the "classical electron radius", A in Ref.
[91
n),
one
;:Z
complete
=
=
=
(depending
as
a
set of electrons in
obtains I
-
and
a
vacuum
by identification that
10-6,
with
r,
=
n
=
(e2 /47reo rn_ C2)
rigourous demonstration is given
Statistical
2
far-field in
approximation Chap. 4). E,;, (R)
-
Wave
Aspects of
and its
domain
validity
exp(-ikoR)
Scattering
dr’(k 2(r’)
47rR
at
discussed in
are
more
71
details
(r /) eik,,.r’
2
k 0 )E-L
-
Surfaces
Rough
(2-32)
where
Ej-(r’)
E(r’) -: i.E(r’)ii
=
(2.33)
represents the component of the electric field that is orthogonal tion of
propagation given by fi. Expression
electric field E, (R)
k,c
--
koRIR
_-
E, (k,;,)
Poynting
plane
a
wave
[8]
with wavevector
E,,c (R).
-
,
The
(2.32)
approximated by kofi and amplitude, be
can
to the direc-
shows that the scattered
(2-34)
readily obtained,
vector is then
S
I =
2poe
I E , (R) 12 ia.
(2.35)
Poynting vector for a unit incident flux (or normalized by through a unit surface normal to the propagation direction) differential scattering cross-section in the direction given by k’,
The flux of the
the incident flux
yields the
do-
1
Note that
I [k (r’)
167r2lEi,112
dQ
duldfl
involves
do-
double
a
E-L (r).E*L (r +
the vertical
over
axis,
Es, (R)
--
2
-
that k 02)
ik,,
can
dr’
(2-36)
be cast in the
(k2 (r + r)
-
obtains
of a
(2.37) By integrating formally Eq. (2-32)
u.
surface
exp(-ikoR)
S
47rR
form,
k 02)
r)e ik.,,,.r’1
conjugate
one
ko]E_L (r’)e
integration,
dr
167r2lEi,112
where u* stands for the
-
I J dr’(k (r)
I _
dQ
2 2
2
i
integral,
(r’ll, k,,c , )
ik,,:Il.r ’1
dril,
(2-38)
with S
We
see
with
that
k,,,,
hand, the nated
k,,,)
[k 2(r/)
-
k 02]eik,c, "Ei
-
(r’) dz’.
Eq. (2.39) is a ID-Fourier transform, thus the variations directly linked to the thickness of the sample. On the
are
variations of Esc with
area
(
i.e. the
region
k,;,Il
for which
are
(2.39) of Eiother
related to the width of the illumi-
[k 2(r’)
-
k2]E 0
is
non-zero).
Anne Sentenac and Jean Daillant
72
Coherence Domains
2.3.3
to now, we have considered an ideal experiment with a point-like detector. reality, the detector has a finite size and one must integrate the differential scattering cross-section over the detector solid angle, zAodet- Since the
Up In
cross-section is defined
as
function of wavevectors, it is
a
more
convenient
integration over the solid angle -60det centered about the k,, into an integration in the (k, ky) plane. The measured inten-
to transform the
direction
sity (scattering cross-section convoluted given by, I
.1
x
where
R(kil)
is then
dkjjR(kjj)
-
167r2
function)
with the resolution
jEj,, I
I drjj I dr’ll S* (rjj L I
r’ll, k.,).9j (r1j, k,)e ikjj.r’jj
+
-
is the detector acceptance in the
(k, ky) plane.
(2.40)
The expression
of R in the wavevector space is not easily obtained. In an x-ray experiment, it depends on the parameters (height, width) of the collecting slits. The reader is referred to section 4.7 for
detector
shape.
detailed
expression introductory chapter it is
In this
a
Gaussian function centered about
R (k,,,,
k,,y)
=
C exp
of R
as
a
function of the
sufficient to take for R
a
k,,,11,
(k.,
-
ks,-,)2
(ky
-
k,,,y)
2./-A k 2
2Ak2X
2
(2.41)
Y
angular aperture of the detector. If one assumes that the integrant does not vary significantly along k, inside ’ Akx 3 zAky) the resulting intensity is given by,
zAk,,zAky
The variables
I
1
1
167r2
jEj,, I
-
govern the
drIldr’119*L (rjj
+
r
/11, ksc
),ik_jj.r’jj k(r 11)
(r1j,
(2.42) where
fZ(rjj) We
now
examine
fields radiated 3
by
-
27rCzAk,zAkye
j’Ak2 XX2 _.LAh2y2 2 2 Y
(2.43)
Eq. (2.38) that gives the scattered field as the sum of the all the induced dipoles in, the sample. We see that the
assumption is not straightforward. It is seen in Eq. (2.39) that the thicker sample, the faster the variations of 9 1 with k,. In an x-ray experiment, the sample under study is generally a thin film (a couple of microns) and we are interested by the structure along z of the material (multilayers). Hence, the size of the detector is chosen so that its angular resolution permits to resolve the interference pattern caused by the stack of layers. This amounts to saying that the k,-modulation of F*L (rjj + r’ll, k-.).Ej (r 11, k_.) is not averaged in the detector. This the
-
Statistical
2
Aspects of Wave Scattering
electric field radiated in the direction ksc
n +
r
Surfaces
73
the "effective"
by
whatever the distance between the
11
Rough
dipole placed at by another dipole placed points. The intensity, mea-
to the field radiated
point rll is added coherently at
at
ideal experiment (coherent source and point-like detector), is by double a integration of infinite extent which contains the incoherent given by term 191 (r1l,kc ,)l’) and the cross-product (namely the interference term) 91 (rll, k,c,)..E*1 (rll + r’ll, k,,,,) When the detector has a fi n ite size, the dou sured
an
-
-
ble
integration
is modified
by
the introduction of the resolution function
which is the Fourier transform of the detector. In is
our
example, fZ
roughly 11[zAkx
x
contribution of the main
is
a
radiated
by
detector
(the
two
points
that
belong
to this domain will add
while the fields
important),
term value is
cross
intensity is significant. This doScoh due to the detector. The fields
term to the total
be called the coherence domain
can
characteristic function of the
Gaussian whose support in the (x, y) plane This function limits the domain over which the
Ak.].
cross
angular
fZ
coherently in the coming from two
points outside this domain will add incoherently (the cross term contribution is damped to zero). The resulting intensity can be seen as the incoherent sum of intensities that are scattered from various regions of the sample whose sizes coincide with the coherent domain given by the detector. This can be readily understood by rewriting Eq. (2.42) in the form [10], I
C)c
E i=I,N L
I
(rill
drll
Sc.,j
+ rll +
Scoh
dr’ll
r’ll, ks,,).g (rill i
+ r1l,
k,,c, )eik_jj.r’jj ’&(r’
(2.44)
regions Scoh. Hence, integrating equivalent to summing the intensities (i. e. incoherent process) from various regions of the illuminated sample. This is the main result of this paragraph. The finite angular resolution of the detector introduces coherence lengths beyond which two radiating sources can be considered incoherent (even though the incident beam is perfectly coherent). Note that the plural is not fortuitous, indeed, the angular resolution of the detector can be different in the xOy and xOz plane, thus the coherent lengths vary along Ox, Oz and Oy. In a typical x-ray experiment (see Sect. 4.7.2, the sample is illuminated coherently over 5 mm’ but the angular resolution of the detector yields coherence domains of solely a couple of square microns. More precisely, it is shown in Sect. 4.7.2 that a detection slit with height 10 mrad 100 ym width 1 cm placed at I meter of the sample with 0, c limits the coherent length along Oz to 1 ym, the coherence length along Ox to 100 ym and that along Oy to 10 nm. Finally, in this introductory section, we have restricted our analysis solely to a detector of finite extent. In general, the incident source has also a finite angular resolution. However, coherence domains induced by the incident angular resolution is usually much bigger than that given by the detector angular resolution so that we do not where the
ri
is the center of the different coherent
intensity
over a
certain solid
angle
is
--
Anne Sentenac and Jean Daillant
74
consider it here.
(The
complete description
calculation scheme would be very
of the resolution function of the
similar).
experiment
is
A
more
given
in
Sect. 4.7.2.
2.4
Statistical Formulation of the Diffraction Problem
In this
section,
point out, through various numerical simulations, the perdescription of the surface and of the scattered power for modelling a scattering experiment in which the rough sample is necessarily deterministic. The main steps of our analysis are as follow: Within the coherence domain, the field radiated by the induced dipoles (or secondary sources) of the sample interfere. We call speckle the complicated intensity pattern stemming from these interferences. The angular resolution of the detector yields an incoherent averaging of the speckle structures, (the intensities are added over a certain angular domain). This angular integration can be performed with an ensemble average by invoking 1. the ergodicity property of the rough surface, (i.e. we assume that the sample is one particular realisation of an ergodic random process) and 2. the equivalence between finite angular resolution and limited coherence tinence of
a
we
statistical
domains. is
It appears finally that the diffused intensity measured by the detector adequatly modelled by the mean square of the electric field viewed as a
function of the random variable
Throughout this section, the numerical optical wavelength is about I pyn and incident beam is directed along the Oz axis.
in the
examples given the perfectly coherent are
To What Extent is
2.4.1
z.
domain. The
a
Statistical Formulation
of the Diffraction Problem Relevant? In Sect. 2.3 it has been shown how to calculate
formally the electromagnetic by the detector in a scattering experiment. To obtain the differential scattering cross-section, one needs to know the permittivity contrast at each point of the sample, and the electric field at those points, Eq. (2.37). If the geometry of the sample is perfectly well known (i.e. deterministic like gratings), various techniques (such as the integral boundary method [11], [12]) permit to obtain without any approximation the field inside the sample. It is thus possible to simulate with accuracy the experimental results. In the case of scattering by gratings (i.e periodic surfaces) the good agreement between experimental results and calculations confirms the validity of -
power measured
the numerical simulations We
study
faces s,,
(e.g.
beam. In this
[12].
the scattered
intensity from different rough deterministic surpresented in Fig. 2.1) illuminated by a perfectly coherent experiment, we suppose that the size of the coherence domains
those
2
Statistical
Aspects ofWave Scattering
at
Rough
Surfaces
75
0-40
o.oo-9o
o
-
-70
To
.
lo
so
... cl--)
Fig. 2.4. Simulations of the differential scattering cross-section for the surfaces presented in Fig. 2.1. The illuminated area covers 40 Mm which explains the large angular width of the speckle. The incident wavelength is I Mm, the refractive index is 1.5. Normal incidence. The calculations are performed with a rigorous integral n boundary method (no approximation in solving Eq. (2.37) other than the numerical ==
discretizations) [13]
induced minated
by
the finite resolution of the detector is close to that of the illu-
area
A. In other
words,
all the
fringes
of the interference pattern
by every illuminated by the detector. We observe in Fig. 2.4 that the angular distribution of the intensity scattered by each surface presents a chaotic behavior. This phenomenon can be explained by recalling that’the scattered field consists of many coherent wavelets, each arising from a different microscopic element of the rough surface, see Eq. (2.38). The random height position of these elements yields a random dephasing of the various’ coherent wavelets which results in a granular intensity pattern. This seemingly random angular intensity behavior, known as speckle effect, is obtained when the coherence domains include many correlation lengths of the surface, when the roughness is not negligible as compared to the wavelength (so that the random dephasing amplitude is important) and most importantly when the s2ze of the coherence domains is close to that of the illmninated area so that the speckle is not averaged in the detector. To retrieve the precise angular behavior of the intensity, one needs an accurate deterministic description of the surface [14]. In Fig. 2.4 the surfaces s,, present totally different intensity patterns even though they have the same statistical properties. However, some similarities can be found in the curves plotted in Fig. 2.4. For example, the typical angular width of the spikes is the same for all surfaces. Indeed, in our numerical experiment it is linked to the width L of the illuminated area (which is here equivalent to the coherence domain). The smallest angular period of the fringes formed by the (farthest-off) coherent point-source pair on
stemming
from the coherent
point of the surface
are
sum
resolved
of the fields radiated
Anne Sentenac and Jean Daillant
76
angular width A/L of the speckle spikes. Fig. 2.5, the larger the coherently illuminated angular speckle structures. In optics and radar imaging,
the surface determines the minimal
This is area
clearly
illustrated in
the thinner the
sufficiently coherent incident beams (lasers) combined with detectors with fine angular resolution permits to study this phenomenon [14]. In x-ray experiments, the speckle effect can also be visualised in certain configurations. I mrad), the apparent resolution of the detector At grazing angles (e. g. 0,, Sect. k00S0 Jq,, 4.7.2) may be better than 10-7 ko in-’. The size of (see the illuminated area being 5 mm, the speckle structures are resolved in the --
--
detector.
0.6
0.4
li,93
0.2
0.0 -so
-30
_10 ()
Fig.
2.5. Illustration of the
dependence
tures on the size of the illuminated
bution for case over
one
rough
10
_
area.
30
50
(Clegree.)
of the
angular
width of the
Simulation of the
surface illuminated in the first
30pm. The incident wavelength is I
specIde strucintensity angular distri-
case over
60 tim and in second
pm, the refractive index is
n
=
1.5,
normal incidence
now suppose that the illuminated area is increased enough so that typical angular width of the speckle structures will be much smaller than the angular resolution of the detector. The detector integrates the intensity over a certain solid angle and, as a result, the fine structures disappear. One notices then that the smooth intensity patterns obtained for all the different surfaces s,, are quite similar. This is not surprising. Indeed, we have seen in the previous paragraph that the finite angular resolution of the detector is equivalent to the introduction of a coherence domain Scoh (that is smaller than the illuminated area A). The measured intensity can be considered the incoherent sum of intensities stemming from the different subsurfaces of size Scoh that constitute the sample. We now suppose that the illuminated area is big enough to cover many "coherent" subsurfaces, A > 30Scoh. Moreover, we suppose that the coherence domain is large enough so that each subsurface presents the same statistical properties Lcoh > 30
We
the
Statistical
2
Aspects of Wave Scat ering
at
Rough
Surfaces
77
length and Lcoh the coherence length. If the set of by an ergodic stationary process, the ensemble of subsurfaces obtained from one particular realisation sj will define the same random process with the same ensemble averaging as that created from any other realisation sl,. Consequently, the scattered intensity from one "big" where
is the correlation
surfaces
f s,}
be described
can
surface sj can be seen as the ensemble average of the "subsurface" Sc,,h scattered intensity which should"be the same for all sk. This assertion is
supported by a comparison between two different numerical the’same scattering experiment [13] [15].
treatments of
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0 -90
-30
-60
Fig. 2.6.
Simulation of the differential
3mm
(roughly
one
scattering cross-section
realisation of
M. Saillard
In
Fig. 2.6 rough
ministic with
a
of
a
rough
deter-
a
random process are: Gaussian height distribution correlation function with Ipm. The incident
prof.
90
60
(Clegree)
random process. The illuminated area several thousand of optical wavelengths). The statistics of the
ministic surface which is covers
30
0 0
with, o0.2pm and Gaussian wavelength is lpm. Courtesy of =
[13]
we
have
surface
plotted
Sj
intensity obtained from a deterby a perfectly coherent Gaussian beam, The rough surface is one realisation of a
the diffuse
illuminated
detector of infinite resolution.
random process with Gaussian height distribution function and Gaussian correlation function with correlation length . The incident beam is chosen wide
enough
so
that the illuminated part of
Sj
is
representative
of the
ergodic
random process. In other words, Sj can be divided into many subsurfaces (with similar statistical properties) whose set describes accurately the ran-
dom process. The total length of the illuminated spot is 5000 . It is seen Fig. 2.6 that the scattered intensity exhibits a very thin speckle pattern.
in
Fig. 2.7 we have averaged degree, corresponding to the angular resolution of a detector. We compare in Fig. 2.7 the angular averaged pattern with the ensemble average of the scattered intensity from subsurfaces In
general
the diffuse
these fine structures
intensity
over an
are
not visible. In
angular
width of 5
Anne Sentenac and Jean Daillant
78
0.9
0.8
0.6
0.5
0.3
0.2
0.0 -84
;6
-56 0
Fig.
4
&Iegree )
Solid line:
Angular average over 5 degrees of the differential scattering ’big surface’ presented in Fig. 2.6, dotted line: ensemble average differential scattering cross-section of rough surfaces with the same statistics "big surface". Size of each realisation is 30pm, no angular averaging. Courtesy
2.7.
cross-section of the of the as
the
of Prof. Saillard
that
are
[13]
generated
with the
illuminated domain is induced
by
now
same
random process
as Sj but whose coherent (i.e. to the coherence domain detector). We obtain a perfect agree-
restricted to
the finite resolution of the
30
ment between the two
scattering patterns. In this example, we do not need precise value of the characteristic function z(rll) but solely the statistical properties of the random process that describes conveniently these particular surfaces. The integration of the intensity over the solid angle AQ will then be replaced by the calculation of the ensemble average of the intensity. This ensemble averaging appears also naturally in the case of surfaces varying with time (such as liquid surfaces like ocean) by recording the intensity during a sufficiently long amount of time. any
the
longer
Each subsurface (either spread spatially temporally) generates an electric field E. The
via the coherence domains latter
can
be viewed
as a
or
func-
tion of the random process z. The intensity measured by the detector is then mean (in the ensemble averaging sense) square of the field,
related to the
(JE 12
.
The purpose of most
ious moments of E. More mean
and
a
wave scattering theories is to evaluate the varprecisely, the random field can be divided into a
fluctuating part,
E
We
usually study separately the
--
(E)
+ JE.
different contributions to the
(2.45)
intensity.
Statistical
2
Notions
2.4.2
on
Aspects of Wave Scattering
Rough
Surfaces
79
(Specular) (Diffuse) Intensity
Coherent
and Incoherent In the
at
the scattered electric field E,, behaves like a plane wave with ks, and amplitude E(k,,) see Eq. (2.32). It can be written as the mean part and a fluctuating part,
far-field,
wavevector
of
sum
a
Esc
=
(E ,)
(2.46)
Es_
+
previous discussions have shown that the measured scattered intensity a rough sample (whose deterministic surface profile is assumed to be one realisation, of a given ergodic random process) can be evaluated with the ensemble average of the intensity JE,;,(k c) 12), The
from
(JE,, 12) The first term
intensity
on
the
right
--
I (E,,) 12
hand side of
(IJE" 12).
+
equation (2.47)
while the second term is known
as
(2.47) is called the coherent
the incoherent
intensity.
It is
sometimes useful to tell the coherent and incoherent processes in the scattered intensity. In the following, we show that the coherent part is a Dirac function
solely to the specular direction [4] if. the randomly rough statistically homogeneous in the (Oxy) plane. In most approximate theories, the random rough surface is of infinite extent and illuminated by a plane wave. Suppose we know the scattered far-field E,,c from a rough surface of defining equation z z(rll). We now address the is shifted horizontally by whole the surface when modified of how is issue E, not such shift will d. is clear that It a a vect’or modify the physical problem. the incident However wave amplitude acquires an additional phase factor and similarly each scattered plane wave Esc acquires, when reexp(ikin.d) the primary coordinates, the phase factor exp(-ik,d). Thus we turning to obtain, that contributes
surface is
=
z(r I, -d)
-i(k_-kj ,).d E Sz (-.11) C
ESC
(2,48)
suppose that the irregularities of the rough surface stem from spatially homogeneous process. In this case, the ensemble average invariant under any translation in the (xOy) plane.
We
now
random
(Ez(rll -d) SC
This
equality
is
only possible
(E"(ni)) Sc
a
is
(2.49)
if
(E,,)
--
AJ(k,cll
-
ki,11).
(2.50)
Hence, when the illuminated domain (or coherence domain) is infinite, the coherent intensity is a Dirac distribution in the Fresnel reflection (or transmission) direction. For this reason it is also called specular intensity. Note
Anne Sentenac and Jean Daillant
80
that unlike the coherent term, the incoherent intensity is a function in the k"cll plane and its co’ntribution in specular direction tends to zero as the detector
acceptance is decreased. In’real life, the incident beam is space-limited, the finite, thus the specular component becomes a function whose angular width is roughly given by A/Lcoh. coherence domain is
solely interested in the specularly intensity. configuration allows the determination of the zdependent electron density profile and is often used for studying stratified interfaces ( amphiphilic, or polymer adsorbed film). The modelisation of the coherent intensity requires the evaluation of the single integral Eq. (2-32) that gives the field amplitude while the incoherent intensity requires the evaluation of a double integral Eq. (2.37). It is thus much simpler to calculate only the coherent intensity and many elaborate theories have been devoted to this issue [4]. Chapter 3 of this book gives a thorough description of the main techniques developed for modelling the specular intensity from rough multilayers. However, it is important to bear in mind that the energy measured by the detector about the specular direction comes from both the coherent and incoherent processes inasmuch as the solid angle of collection is non zero. The incoherent part is not always negligible as compared to the coherent part especially when one moves away from the grazing angles. An estimation of both contributions is then needed to interpret the data. In many x-ray
reflected
2.5
experiments,
one
is
This
Statistical Formulation of the Scattered Under the Born
Intensity
Approximation
section, we illustrate the notions introduced previously with a widely used model that permits to evaluate the scattering crosssimple section of random rough surfaces within a probabilistic framework. We discuss the relationship between the scattered intensity and the statistics of the surfaces. The main principles of the Born development have been introduced in Chap. 1, Appendix LA, and a complementary approach of the Born approximation is given in Chap. 4 with some insights on the electromagnetic properties ofthe scattered field. In this last
and
2.5.1
The Differential
Scattering
Cross-Section
Eq. (2.32) that gives the scattered far field as the sum of the by the induced dipoles in the sample. The main difficulty of this integral is to’evaluate the exact field E inside the scattering object. In the x-ray domain, the permittivity contrast is very small 10-’) and one can assume that the incident field is not drastically perturbed by surrounding radiating dipoles. Hence, a popular assumption (known as the Born approximation ) is to approximate E by Ein With this approximation the integrant We start from
fields radiated
-
Statistical
2
Aspects
of Wave
Scattering
Rough
at
Surfaces
81
readily calculated. For an incident plane wave Eirle- iki,,.r , the differential scattering cross-section can be expressed as, is
do-
_1
M
167r2
I Eil _L 12 lEi,,12
I 1 dr’[k 2(r) dr
projection of the
where Ei,,j_ is the
-
k 02] [k 2 (r’)
-
k2]eiq.(r-r’)’ 0
incident electric field
on
the
to the direction of observation of the differential cross-section.
unit vectors in direction Ei,, and E,;,
)’ in
--
Ein/Ei,,
and
2
(’ c
(2.51)
plane normal Denoting the
E,,/E,,
-
re-
Ein ( iin . sc)2. In x-ray experiments, the incident spectively, we have I Ei,, I field impiges on the surface at grazing angle and one studies the scattered intensity in the vicinity of the specular component. In this configuration, the orthogonal component of the incident field with respect to the scattered direction is close to the total incident amplitude. Yet, we retain the projection in the differential scattering cross-section for completeness term and coherence with the results of Chap. 1. Bearing in mind the value of the permittivity contrast as a function of the electronic density, Eq. (2.31), equation (2.51) simplifies to, dor
2(; in. sc)2
dr’pel (r)pe (r’) e iq.(r-r’)
dr
e
(2.52)
with p.1 the electron density and r, the classical electron radius. ’ In the case of a rough interface separating two semi-infinite homogeneous media one gets,
do-
2
dQ
2
r. p. 1
-
(ein.e, c)
Integrating Eq. (2.53) term to
ensure-
2
100
over
2
Pei 7e,
dQ
q2
can
z(rli)
z
r
/11 dz’
dz -00
J dr11 J dr’l
(z, z’) (with the inclusion -oo) yields,
of
a
Ie
iq.(r-r’
small
(2.53)
absorption
the convergence at
do-
One
2
( in- isc )2
Z
J dr11 I dr’l
1e
iqll. (r11
-
r’11) eiq., [z (r11)
-
z
(r’11)].
(2.54)
general presentation of elastic scattering under the Born, approxscattering by an isolated object as presented in Qhap. I Sect. appendix I.A. The differential scattering cross-section can be cast in
make
a
imation from the 1.2.4 and
the form do"
dQ
be iq.rj
2
2
drpbe
iq.r
where p is the density of scattering troduced in Eq. (1.34). The complex
objects and b their scattered length as inexponential is the result of the phase shift between waves scattered in the ksc direction by scatterers separated by a vector r as shown in figure 2.8. For neutrons, b is the scattering length which takes into account the strong interaction between the neutrons and the nuclei ( we do not consider here magnetic materials); for x-rays, b r, (e2/47reo M’_ C2) 2.810-15m =
which is the classical radius of the electron.
=
=
Anne Sentenac and Jean Daillant
82
q
--------------
-
kin
k q
sc
r
Fig.
Phase shift between the
2.8.
rated
by
a
vector
r.
The
phase
waves
shift is
scattered
(k,,
-
by
ki,,).r
=
two
points
scatterers sepa-
q.r
equation concerns a-priori the scattering from any (deterministic or not) object. In this chapter, we are mostly interested by the scattering from surfaces whose surface profile z is unknown or of no interest. We have seen in the preceding sections that if z is described by a random homogeneous ergodic process, the intensity measured by the detector can be approximated by the ensemble average of the scattering cross-section. It amounts to replacing in Eq. (2.54) the integration over the surface by an ensemble average, f f (q) dr11 L., Ly (f , where L,,, Ly are the dimensions of the surface along Ox and Oy. One obtains, This
-_
P 2r2 el 6 LxL Y
do-
(6in sc)
2
-
q2
dQ
dr1le iqll.rll ( e
iq,
[z(r11)-z(011)]
(2.55)
z
Note that the
expression (2.55)
of the differential
scattering cross-section Hence, this integral
accounts for both the coherent and incoherent processes.
does not converge in the function sense, it contains a Dirac distribution if the surface is infinite. This property will be illustrated with various examples in the
following.
If the
probability density
of
z
is
Gaussian,
we
can
write the
differential cross-section as,
do-
2r e2LxL Pei
-
2
dQ
We
see
tering)
that,
qz
Y(-eir,-i ,,) 2
fdrp
iq11.r11 )e-
5q.2,([z(rll )_Z(011)]2) 2,
(2.56)
approximation (where we neglect multiple scatintensity is related to the Fourier transform of the ex-
under the Born
the scattered
pair-correlation function, g(r11) [z(rll) Z(011)]2). In the by studying the differential scattering crosssection for various pair-correlation functions. We start by the expression of the scattering differential cross-section in the case of a flat surface.
ponential of following we
the
illustrate this result
=
_
Statistical
2
ideally flat surfaces g(r1j) tering cross-section yields :
is
For
2
dQ
qZ
2
dS?
q,
is thus
intensity
expected,
for
a
1 drjj iqjj.rjj.
constant
a
2 2 L ,L Y 47r 2 r.p.
do-
direction. As
2
(; i.
is the Fourier transform of
The scattered
Surfaces
Rough
at
83
at the surface and the scat-
everywhere
2
2
-
integral
zero
r,p,,LL Y
do-
The
Scattering
Surfaces
Ideally Flat
2.5.2
of Wave
Aspects
(2.57)
that,
so
5
2
(2.58)
(; in .;&Sc) J(qll).
Dirac distribution in the Fresnel reflection
a
perfectly
surface, the reflectivity comes solely scattering is null (JE 2)
flat
a coherent process (Sect. 2.4.2), 0. Note that the reflectivity decreases
the incoherent
from
to the
more
a
complicated problem an homogeneous ergodic
described statistically by Self-Affine
2.5.3
-
power law with q,. We now turn of scattering from rough surfaces that are as
Rough
random process.
Surfaces
Surfaces Without Cut-Off We first consider self-affine
rough
surfaces with
2h pair- correlation function g given by Eq. (2.24), g(r1j) -- Aor 11 With this paircorrelation function, the roughness cannot be determined since there is no *
scattering
saturation. The
2
cross-section is in this case,
2
r. p., L,, L Y
do-
2
dQ
and
in
expressed
be
can
qZ
( &in . SC)2
polar coordinates
2
ro p2, L, Ly
do-
( &in isc)
-
2
dQ
2
-
qZ
g12 -1 A R2h
I drlle-
1
eiqjj.rjj
(2.59)
Jo (qjj r1j),
(2.60)
as, 2
"f AR 2h
drii e-
2
with q1I being the modulus of the in-plane scattering wave-vector, and J0 the zeroth order Bessel -function. The above integral has analytical solutions for 1 and has to be calculated
h
0.5 and h
h
1, the integration yields,
--
2
2
r. p., L, Ly
do-
2
dQ
and for h
-
dQ
is
purely Let
us
2
in other
4
SC)2e-qII/q.
cases.
For
(2.61)
0.5, dd-
The above
qz
(-&in .
numerically
r
(; in . &sc)2.
2
2
7rA
ePel LL Y 2
qz
(qjj
2
+
(A) 2
2
q4 Z
)3/2
(2.62)
expressions clearly show that for surfaces of this kind the scattering (no Dirac distribution, no specular component).
diffuse
recall that
6(qll)
=
=4,1 f e-iqll -’H dr1j. v
Anne Sentenac and Jean Daillant
84
Surfaces with Cut-Off
Rough
surfaces
when the correlation function C.,,,
(r1j)
are
said to present
tends to
a
when r1l
zero
cut-off
length increases, (for
2h
2 o example see Eq. (2.26), when C,,, (rj() exp J,-h ), the cut-off is ). In this general case an analytical calculation is not possible and the scattering cross-section becomes,
do-
r
2 6
2
Pel L , L Y
q 20,2
2
dQ
qz
gr’j’ SC)2
I driie
integrant in Eq. (2.63) does not tend to integration over an infinite surface does not exist The
do-IdO
q.2C_
(r1j) iqjj.rjj
(2-63)
0 when r1l is increased. The in the function sense. Indeed,
accounts for both the coherent and incoherent contributions to the
possible extract the specular (coherent) and (incoherent) compone4ts by writing the integrant in the form,
scattered power. It is
e
(or
The distributive part
ular
reflectivity
then cast in the
q ,C_(rjj) Dirac
while the
I +
=
(C q.2,,C,_(rjj) 1)
the diffuse
(2.64)
-
function)
.
characterises the coherent
regular part gives
the diffuse power.
or
spec-
Eq. (2.63)
is
form,
( )
do-
do-
-
dQ
dQ
dodQ
coh
)
(2-65) incoh
with
(M) do-
r
2
2
Pei LL Y
6
-
q.2Cr2
( &i. - sc)
drile iqll.ril
2
-
qz
c.h
2 2 4 7r 2 repelL
,Ly
q2
2
qz
U2
J(qll ) ( &,nj sc) 2,
(2-66)
and d t7 M
r
2 e
p
e2’l L,, LY
-
,
q 2U2
2
qz
incoh
-
&Sc)
2
drjj
e
q.2, C,,,, (r1j) -1
e
iqll.r1j.
(2-67) specular part is similar to that of a flat surface except that it is reduced 2’ The diffuse scattering part may by roughness Debye-Waller factor e-q be determined numerically if one knows the functional form of the correlation function. When qZ2C (r1j) is small, the exponential can be developed as I + qZ2C (r1j). In this case, the differential scattering cross-section appears to be proportional to the power spectrum of the surface P(qjj), The
2
the
.
Z
,
_
Zz
dodQ
)
-
incoh
r
2P 2jLj Y e- q 2 0,2 47r2p (qj I) ( iin jl sc) 2.
6
e
(2.68)
2
Statistical
Aspects of
Wave
Scattering
at
Rough
Surfaces
85
Eqs. (2.66, 2.68) that the Born assumption permits to evaluate scattering cross-section of rough surfaces in a relatively simple way. This technique can be applied without additional difficulties to more complicated structures such as multilayers or inhomogeneous films. Unfortunately, in many configurations, the Born assumption proves to be too restrictive and one can miss major features of the scattering process. More accurate models such as the Distorted-wave Born approximation have been developed and are presented in Chap. 4 of this book. Yet, the expressions of the coherent and incoherent scattering cross-se(Aions given here given by the first Born approximation provide useful insights on how the measured intensity relates to the shape (statistics) of the sample. The coherent reflectivity, Eq. (2.66), does not give direct information on the surface lateral fluctuations, except for the overall roughness o-, but it provides the electronic density of the plane substrate. Hence, reflectivity experiments are used in general to probe,, along the vertical axis, the electronic density of samples that is roughly homogeneous in the (xOy) plane but varies in a deterministic way along Oz (e.g. typically multilayers). Chapter 3 of this We
see
with
both the coherent and incoherent
book is devoted to this issue. On the other hand the incoherent
scattering height-height correlation function of the surface. Bearing in mind the physical meaning of the power spectrum, Sect. 2.2.3, we see that measuring the diffuse intensity at increasing q1I permits to, probe the surface state at decreasing lateral scales. Hence, scattering experiments can be a powerful tool to characterise the rough sample in the lateral (Oxy) plane. This property will be developed and detailed in Chap. 4. Eq. (2.68) is directly linked
to the
References. Spizzichino. The scattering of electromagnetic waves from rough surfaces. Pergamon Press, Oxford,UK, (1963). F. G. Bass and 1. M. Fuks. Wave -scattering from statistically rough -surfaces. Pergamon, New York, (1979). J. A. Ogilvy. Theory of wave scattering from random rough surfaces.’Adam Hilger, Bristol,UK, (1991). G. Voronovich. Wave scattering from rough -surfaces. Springer-Verlag, Berlin,
1. P. Beckmann and A.
2.
3.
4.
(1994). Optical coherence and quantum optics. Cambridge University Press, Cambridge USA, (1995)C. A. Gu6rin, M. Holschneider, and M. Saillard, Waves in Random Media, 7,
5. L. Mandel and E. Wolf.
6.
331-349,(1997). 7.
B.B.
Mandelbrodt,
"
The fractal geometry of
nature", Freeman, New-York
(1982). Principle of Optics. Pergamon Press, New York, (1980). Oxtoby, F. Novack, S.A. Rice, J. Chem. Phys. 76, 5278 (1982). 2740 (1998) S.K. Sinha, M. Tolan and A. Gibaud, Phys. Rev. B 57 M. Nieto-Vesperinas and J. C. Dainty. Scattering in Volume and Surfaces. Elsevier Science Publishers, B. V. North-Holland, (1990).
8.
M. Born and E. Wolf.
9.
D.W.
10. 11.
,
Anne Sentenac and Jean Daillant
86
12.
13. 14.
15.
R.
Petit, ed. Electromagnetic Theory of Gratings. Topics in Current Physics. Springer Verlag, Berlin, (1980). M.’Saillard and D. Maystre, J. Opt. Soc. Am. A, 7(6), 982-990, (1990). J. C. Dainty, ed. Laser speck* and related phenomena. Topics in Applied physics. Springer-Verlag, New York, (197151). M. Saillard and D.
Maystre, Journal of Optics 19, 173-176, (1988).
Specular Reflectivity and Rough Surfaces 3
from Smooth
Alain Gibaud Laboratoire de
Physique
Condens6, UPRESA 6087 9, France
de I’Etat
,
Universit6 du
Maine Facult6 des sciences, 72085 Le Mans Cedex
It is well known that
is reflected and transmitted with
direction of
an
ent
light propagation at
optical properties. The
to observe in the visible
used
(see
a
change
in the
interface between two media which have differ-
effects known
the introduction for
reflection and refraction
as
are
easy
difficult when x-ray radiation is historical presentation). The major reason
spectrum but a
more
for this is the fact that the refractive index of matter for x-ray radiation does not differ very much from unity, so that the direction of the refracted beam does not deviate much from the incident
The reflection of x-rays
one.
is however of great interest in surface science, since it allows the structure of the uppermost layers of a material to be probed. In this -chapter, we present
general optical rough surfaces and
the
3.1
The
from 3.1.1
It
was
is
(see
formalism used to calculate the
reflectivity
interfaces which is also valid for
x-rays.’
of smooth
or
Reflected’Intensity an Ideally Flat Surface
Ba.sic Concepts shown in
Chap.
Sect. 1.4.2 for
1 that the refractive index of matter for x-ray radiation more
details): n
The classical model of
pression
an
=
I
-
J
-
i,3.
bound electron
elastically
(3.1) yields the following
ex-
of J:
6
A2 ‘=
21r
reN7
(3.2)
where re is the classical electron radius (r, 2.810-5A), A is the wavelength and p, is the electron density of the material. This shows that the real part =
The basic concepts used to determine the reflection and transmission coefficients of an electromagnetic wave at an interface were first developed by A. Fresnel [1] in his mechano-elastic
theory
of
light.
J. Daillant and A. Gibaud: LNPm 58, pp. 87 - 120, 1999 © Springer-Verlag Berlin Heidelberg 1999
Alain Gibaud
88
of the refractive index
mainly depends on the electron density of the material 10-6 and 13 is ten on wavelength. Typical values for J are 10-5 times smaller. A similar equation holds for neutrons where r,p, has to be replaced by pb (see Chap. 5, Eq. (5.24)). A specific property of x-rays and neutrons is that since the refractive index is slightly less than 1, a beam impinging on a flat surface can be totally reflected. The condition to observe total external reflection is that the angle of incidence 0 (defined here as the angle between the incident ray and the surface) must be less than a critical angle 0,. This angle can be obtained 1, yielding in absence of by applying Snell- Descartes’ law with COS Otr absorption: and
the
_
-
cosO,
-_
n
I
-_
-
(3.3)
J.
10’, the critical angle for total external reflection clearly extremely small. At small angles, COO, can be approximated as
Since J is of the order of is I
-
OC/2
and
(3.3)
becomes
02
-
c
The total external reflection of
an
(or neutron)
x-ray
grazing angles of incidence reflectivity decreases very rapidly
(3.4)
2J.
only larger angles,
beam is therefore
observed at
below about 0 < 0.5’. At
the
as
mentioned above.
chapter, we will calculate the reflectivity as a function of the 47r sin O/A incident angle 0 or alternatively as a function of the modulus q of the wave-vector transfer q (see Eq. (2.28) and Fig. 2.3 with 0s, 0). This means that the following ratios, In this
=
=
R(O)
R(q)
1_(0)
(3.5)
-
10
=
I(q)
(3-6)
10
determined, where 1(0) or I(q) is the reflected intensity (Flux of Poynting’s vector through the detector area) for an angle of incidence 0 (or wavevector transfer q), wrid 10 is the intensity of the incident beam. The theory of x-ray reflectivity is valid under the assumption that it is possible to consider the electron density as continuous (see Chap. 1). Under this approximation, the reflection is treated like in optics, and the reflected amplitude is obtained by writing down the boundary conditions at the interface, i.e., the continuity of the electric and magnetic fields at the interface, leading to the will be
classical Fresnel relations.
3.1.2
Fresnel
Reflectivity
The reflection and transmission coefficients
conditions of continuity of the electric and
can
be derived
magnetic
by writing
the
fields at the interface.
Specular Reflectivity fi,oni Smooth
3
Rough
and
Surfaces
89
intensity, which is the square of the modulus of the reflection quantity measured in an experiment. Let us consider an coefficient, electromagnetic plane wave propagating in the xOz plane of incidence, with its electric field polarised normal to this plane along the Oy direction. The 0 as interface between air and the reflecting medium which is located at z better order to In be assumed to will be in 3.1 emphasize abrupt. shown figure that the same formalism applies for x-rays and visible optics we use in this section the angles defined from the surface normal as in optics, together with the grazing angles usually used in x-ray or neutron reflectivity. The reflected
is the
--
AZ
X
0 n=]
n=1-840
Fig. 3. 1. travelling
k
Reflection and refraction of in the xOz
plane
an
incident
wave
polarised along
y and
of incidence
expression for the electric field in a homogeneous medium is derived equations which when combined, lead to the propagation field known as Helmholtz’s equation (see Chap. 1, electric the of equation for details) Eqs. (1.12), (1.15) The
from Maxwell’s
kj2E
, AE + where
kj
-
0,
(3.7)
is the wave-vector in medium 1. The electric field which is solution of equation is given for the incident (in), reflected (r) and transmitted
Helmoltz’s
(tr) plane
waves
by,
Ej
=
Aje i(wt-kj.r),6Y
(3.8)
jk,j 27r/A jkt,J /n, and,&y is a unit vector 1ki,,j withj=in, r or tr, ko along the y axis (see figure 3.1). Note that the convention of signs used in crystallography is adopted here (see Part 1, Chap. 1, by F. de Bergevin for details). It is straightforward to show, that the components of the(in), (tr), =
and
(r)
wavevectors are,
=
=
--
Alain Gibaud
90
kin kr ktr The
=
--
=
tangential component
interface fields.
(z
=
0).
In
ko (sin ii 6, ko sin ij,&, + kon (sin i2 6x
that the medium is
Assuming completely absorbed,
beam to be
Aine i(wt-ko
sin
COS
COS
-
(3-9)
i2 6z
of the electric field must be continuous at the
the field is the
air,
ii,&,) il,&,,)
COS
-
the
sum
sufficiently thick following relation
ijx) + A, i(wt-kosinijx)
Equation (3.10) must be valid for hold,
of the incident and reflected
for the transmitted must be
fulfilled,
sin i,2x) At, i(wt-kon
--
any value of x,
so
that the
(3-10)
following
con-
dition must
sin il
This condition is result of
this,
--
n
Sin i2
simply the well-known Snell- Descartes’ second law. As a perpendicular component of the electric
the conservation of the
field leads to,
Ain + A, It will be assumed that the media
--
At,.
(3-12)
non-magnetic
are
so
that the
tangential
component of the magnetic field must also be continuous. According to the
Maxwell-Faraday equation,
17
the
x
tangential component Bt i.e.,
the unit vector 6,,
E-
OB -
_W
is the dot
(3.13)
-iwB
-
product
of the
magnetic
,
X
Bt
(3-14)
2W
Since the electric field is normal to the incident the y axis and the curl of the field
17
The
field with
tangential component
x
E
-
of the
plane,
it is
polarised along
gives,
OEy6z ax
_
magnetic
OEy6 ’. az field is then
(3-15) given by
Specular Reflectivity
3
Bt
from Smooth and
I
My
iW
C9Z
1 +
r
--
t
I
r
--
nt
-
by the
use
case
of
an
COS
i2
COS
Z1
reflected
COS
(3-17)
i2
and the transmitted
one
t
(3.18)
amplitude coefficient
COS
il
-
n COS
i2
COS
ii +
n COS
i2
in the
case
(3-19)
of the Snell- Descartes’ relation leads to,
Sin(i2 sin(ij
r
In the
nAt,
--
Combining these two equations, the of a (s) polarisation is found to be,
which
--
A,/Ain amplitude r following relations are obtained,
the reflected the
91
(3.16)
-
(Ain -A,) COS ii At,/Ain,
Surfaces
easy to show that the conservation of this
and from equation (3.10) it is quantity yields,
Writing
Rough
electric field
parallel
-
+
il) i2)
to the
(3.20) plane
of
incidence,
a
similar
calculation leads to,
r
tan(i2 tan(i2
(P)
-
+
il) il)
(3.21)
equations are known as the Fresnel equations [1]. It is easy to show r(’) -- r. Only (8) that at small grazing angles of incidence for x-rays r(P) the to plane of incidence) polarisation (electric field polarised perpendicular also be given for (p) will results will be considered in detail below but some
Those
polarisation)
-
grazing angle of incidence 0 that the incident beam makes with the reflecting surface is usually the experimental variable in a reflectivity measurement. It is therefore important to express the coefficient of reflection as from a function of this angle 0 and also of the refractive index n. Starting The
COS
il
COS
il +
-
n COS
i2
n COS
Z2
(3.22)
Alain Gibaud
92
using the fact that the 0 and il, and the Ot, angles as shown in Fig. 3.1, Eq. (3.22) becomes: and
sin 0
-
sin 0 +
Applying
the Snell- Descartes’
the
following
In the
case
n
sin
complementary
(3.23)
Otr’
sin 0
Otr
n cos
=
(3.24)
reflection,
v n_2
-
-
cos2 0
sin 0 +
V1n_2
(3.25)
angles (for
*
-
cos2 0
which cosO
=
for which the refractive index
2
1
--
-
2S
1
_
(in
02 /2) and for the absence of
02C.
1
(3.26)
general equation Eq. (3.25) becomes,
r
The is
n
are
_
of small incident
electromagnetic x-ray waves absorption) is given by,
The
0
coefficient of
(0)
sin Otr
i2
law, cos
produces
n
and
reflectivity given by,
if the
-
-
0+
VF02 VO-12
02 -
(3.27)
02C
which is the square of the modulus of the reflection
R
Finally,
0
(0)
(0)
absorption
the refractive index takes
=
r
r*
O
-
0 +
V/0 2 Oc2 V O2 C
2
coefficient,
(3.28)
of the x-ray beam by the material is accounted for, a complex value and the Fresnel reflectivity is then
written,
R
The q:
reflectivity
can
(0)
--
rr*
0
-
--
equally well
V02
0 +
be
given
-
02C
-
2i/3
_022i)3
(3.29)
C
in terms of the wave-vector transfer
from. Synooth and
Specular Reflectivity
3
Rough Surfaces
93
2
R (q)
q,
q2z-
-
2
32iir2o
2
32i7r2o
qc
>’2
(3.30)
=
q, +
Vq,
2
,
q,
\
2
compared
When the wave-vector transfer is very large following asymptotic behaviour is observed:
to qc
i.e. q >
3q, the
4
qc
R
(3.31)
16q4
101
5: 1071.
q,;
P 0 LU
107’
Fresnel
--I
Reflectivky Reflecfivky Refleowky
U_
LIJ 10, cl
4
6q /16%4 qc /I 4
z
LLJ
107’.
01075
3qc
< 0.00
0.
0.’10
0.65
%
Fig.
It
can
profile -
for
3.2. Calculated
a
be
seen
reflectivity
from the
of
a
0. O
5
0.25
(A ’) flat silicon wafer and
Fig. 3.2, that the reflectivity regimes:
asymptotic law
curve or
reflectivity
(see
Sect. 7.1.1
consists of three different
plateau of
more
total external reflection R
=
I when q < q,
details)
-
a
very
steep decrease when
q
-
a
11q’
power law when q >
3q,.
--
q,
noting that if the value of q, is measured experimentally, this immediately yields the value of the electron density in the material (see Part 1, Chap. I by F. de Bergevin) since,
It is worth
q,
=
3.75 10-2 \’F P e"
(3.32)
,
where p, is the electron density in the units Finally remembering that the reflectivity is observed under
tions, reference to the system of axes defined in Fresnel reflectivity R(q) can be written as:
Fig. 3.1,
specular
condi-
shows that the
Alain Gibaud
94
2
q,
R(q)
-
qz +
q2 2
qz
-
q2
_
32i7r2p
C
-
2
qC
-
(3-33)
q,,Jqy,
222N-5-21 .X2
equation (3.33), and the reflectivity of a flat surface is only specular direction. Equation (3.33) completely describes the reflectivity of a homogeneous material, showing in particular that the reflectivity differs from zero only for wave-vector transfers normal to the surface of the sample. 2 Figure 3.2 illustrates the calculated reflectivity curve for a silicon wafer in the power law regime and also in the case of a more complete dynamical calculation. The deviation from unity due to the absorption of the x-rays in the material can be seen to play a major role in determining the form of the curve in the region close to the critical edge at q q,. Equation (3.33) shows quite clearly that the calculation of a reflectivity curve requires only the electron density and the absorption of the material (for the wavelength used). Table 3.1 gives some useful data for calculating the reflectivity of various elements and compounds. A much wider data base of quantities relevant to reflectivity measurements can be found at the following web site, "http: / /www-cxro.lbl.gov/optical-const ants/". As a conclusion of this section we wish to stress some points concerning the validity of equation (3.33). It is important to realise that in a real experiment we never measure the theoretical reflectivity as given by Eq. (3-33) since the incident beam is not necessarily strictly monochromatic, is generally divergent, and the detector has a finite acceptance. For any instrument, the effects of the divergence of the x-ray source, of the slit settings or of the angular acceptance of the monochromator and analyser crystals used to collimate the incident and scattered beams (see Chap. 7 by J.M. Gay) must be taken into account. Those effects can be described using a 3-dimensional resolution since q
--
q, in
measurable in the
--
function which is
having
a
never
certain width
a
Dirac distribution but
(see Chaps.
4 and
7)
which
a
3-dimensional function
precisely depends on the reflectivity
setup characteristics detailed above. The value of the measured For this reason, the reffectivity of a flat surface is described. which is more normally used to describe the reflection by
"Specular", a term ordinary mirror. It seems that Compton [2] was the first to have foreseen the possibility of totally, reflecting x-rays in 1923 and that Forster [3] introduced equation (3.29). Prins [4] carried out some experiments to illustrate the predictions of this equation in 1928, using an iron mirror. He also used different anode targets to study the influence of the x-ray wavelength on the absorption. Kiessig also made similar experiments in 1931 [5] using a nickel mirror. An account of the historical development of the subject can be found in the pioneering work of L.G. Parrat [6] in 1954, and of Abeles [7]. The fundamental principles are discussed in the textbook by James
[8].
as
an
Specular Reflectivity
3
Rough
from Smooth and
Surfaces
95
examples of useful data used in reflectivity analysis. The table density p, the critical wave-vector q’_ the parameter J, the absorption coefficient 0, the structure of the material and its specific mass (6 and 1.54A). A useful formula for calculating the critical wave-vector ,8 are given at A Table 3.1. A few
contains the electron
,
=
transfer is
q,(A-1)
0.0375
Si
Si02 Ge
((e-//k11)) Vp--, P__
and
conversely
2
p,
711qr
Structure
P’,
q,
e-/A’
A‘
106
107
0.7083
0.03161
7.44
1.75
0.618
0.0294
6.5
1.425
0.0448 15.05
a=5.43A,
1.317 0.0431
13.9
4.99
2330
Z=8 2200
1.7 5
P
kg/m3 cubic diamond
cubic,diamond
a=5.658A, AsGa
=
cubic,diamond
a=5.66A,
5320
Z=8 5730
Z=8
67.5%SiO2 712%
Glass Crown
0.728
0.0328
8.1
1.36
B203 9%,Na2O,
2520
9.5%K20,2%BaO Float Glass
0.726
Nb
2.212
0.03201 0.056
7.7 24.5
1.3 15.1
-
cubic,bcc
a=3.03A, CU
2.271
24.1
0.0566
5.8
Au
4.391
46.5
0.0787
49.2
Z=2
cubic, fcc
a=3.61A,
8580
8930
Z=4
cubic, fcc
19280
a=4.078A 2.760
Ag
0.0624 29.25
0.0395
28
1.08
W03
1.723
0.0493 18.25
0.334
0.0217 3.61 0.123
H20
0.32
0.0212
-COOH
0.53
0.0273
CC14
0.46
0.0254
0.268
0.0194
-
CH30H
10500
11.8
Zr02
CH3 CH2
cubic, fcc a=4.09A
12
-
1000 -
1
t-0.3-77 .0.0233
be estimated through the convolution of equation (3.33) with the resolution function of the instrument. For measurements made in the incidence plane and under specular conditions, a first effect is that the convolution can
dependence of the reflectivity. This can generally be acconvolving R(q,) with a Gaussian function. Another, most important effect of the finite resolution is that beams outside the specular direction are accepted by the detector (in other words, the specular condition J(q,)J(qv) is replaced by a function having a finite width zAq, x zAqy). Then,
smears
out the q.,
counted for by
Alain Gibaud
96
if the surface to be
analysed is rough, the convolution with the resolution drastically changes the problem because part of the diffuse intenwhich arises from the roughness is contained in the resolution volume. It sity even happen for very rough surfaces that the diffuse intensity becomes may as intense as the specular reflectivity. When this is occuring, the only way to use equation (3.33) is to subtract the diffuse part from the reflected intensity to obtain the true specular reflectivity (see Sect. 4.7 for details). function
The Transmission Coefficient
3.1.3
As shown in
Eq. (3.18),
the
I+
the
amplitude of the transmission coefficient satisfies relation, straightforward to show by combining Eqs. (3.18) and (3.29), that the transmitted intensity must be given by, r
--
t. It is
T
(0)
tt*
=
20 --
0 +
V02
-
02
-
C
2i,8
2
(3-34) 2
tt*
T
2q , q, +
The transmitted
intensity
has
(3-35)
--
VqZ
2
maximum at
2
qC
2i7r2,8 \2
O=Oc
as shown in Fig. 3.3 which intensity as a function of the incident angle 0 (or qz) in the case of silicon, germanium and copper samples irradiated with the copper K, radiation. The transmitted intensity is nearly zero at very small angles in the regime of total reflection. It increases strongly at the critical angle and finally levels off towards a limit equal to unity at large angles of incidence. The maximum in the transmission coefficient, which is also a maximum in the field at the interface is the origin of the so-called Yoneda wings which are observed in transverse off-specular scans (see Sect.
gives
a
the actual variation of the transmitted
4.3-1). The Penetration
3.1.4
The
absorption
of
Depth
beam in
medium depends on the complex part of the penetration of the beam inside the material. The refractive index for x-rays, defined in equation (3. 1) is n 6 io. The I amplitude of the electric field polarised along the y direction ((s) polarisation) and propagating inside the medium of refractive index n is given by, a
a
refractive index and limits the
--
E
Since tion
n COS
can
Otr
=
COS
be written
0
--
Eoe i(wt-kon
(the
cos
Ot,x+kon sin
-
Otrz).
Snell- Desc artes’law) and sin Otr
-
(3.36) Ot,
,
this equa-
Specular Reflectivity from Sinooth and Rough Surfaces
3
97
4
Si 3
----
2
-
E
G.
I
-
-
T F_
0
,
0.000
% ( 3.3. Transmission coefficient in
Fig. and
germanium;
0.150
0.125
0.100
0.075
0.050
0.025
0.175
A ’)
intensity in different materials, silicon,
copper,
the maximum appears at the critical wave-vector transfer of the
material
E
The
absorption
is
--
EOe+i(wt-kocosOx)eikon0trz. the real part of
governed by
nOtr
-- ::
(I
-
The coefficients A and B
6
-
can
(3.37)
eikon0trz,
-2J- 2iP i#) V’_02 -
--
with
A + iB
be deduced from the above
equation
(3.38) and B is
given by, B
(0)
v "2_
rj(02
It follows that the electric field
E
Taking
-
2J
+
4p2
-
(02
-
26),
(3-39)
is,
Eoe i(wt-ho
cos
Ox+koAz)
-koB(O)z.
(3.40)
the modulus of this electric field shows that the variation of the in-
tensity 1(z) with depth into the material is given by,
1(z) The
absorption coefficient
oc
EE*
-_
is therefore
loe- 2koB(O)z
(3.41)
Alain Gibaud
98
y
(0)
=
-2koB
and the uated
penetration depth which by 1/e is given by, zj/
(0)
(0).
In
-41rB -_
-A
I
(0)
quantity depends particular, in the limit
zi/
(3.42)
1
-
p
(0)
A
is the distance for which the beam is atten-
=
Note that this B
(0)
on
0
-
47B
(0)
21?nk-,,l
the incident
(3.43)
angle 0 through the value 0, neglecting absorption,
-+
(01)
A __
(3.44)
*
47rO,
of
addition, the penetration depth is wavelength dependent since 3 depends the wavelength. Values of 3 are tabulated in the International Tables of Crystallography, vol. IV [9] or they can also be found at the web site which has already been referred to, "http: / /www-cxro. lbl. gov/optical-const ants/". In
on
100()0
TZ Uj
100D
z
A:’ 6A c =G.0311,1 A:’
_S1
Gla, qj0.0448 Ge 1 100
Cu C’,
-
A" !I-O.OW A
Li z LU 10 0.00
0.02
O.G4
0.06
0.08
GAO
0.12
0.14
0.16
% (k)
Fig.
3.4. Evolution of the
Ka line
fi ure
is
penetration depth in Si, Ge and Cu irradiated with the of a copper tube as a function of the wave vector transfer. Note that the 47r sin O/A presented as a function of q,
Figure
3.4 shows the variation of the
penetration depth as a function of angle in silicon, germanium and copper, for the case of CuKa radiation. The penetration depth remains small, that is below about 30A when 0 is smaller than the critical angle. This is this property which is exploited in surface diffraction, where only the first few atomic layers are analysed. The penetration depth increases steeply at the critical angle and finally slowly the incident
gr(jws when 0
>>
0,.
Specular Reflectivity from Smooth and Rough Surfaces
3
X-Ray Reflectivity in Stratified Media
3.2 The
99
simple
of
case
a
uniform substrate
exhibiting
a
constant electron den-
section. This situation is of
course not previous and media stratified multilayers are frethe most general one. For example, be considered as cannot interfaces encountered. generally Moreover, quently be interfaces Thick thick. and approximated by may steps, but are rough electron of constant slabs into them as many density as necessary dividing it is not possible in their to describe (continuous) density profile. Again, the reflectivity. calculate to coefficients Fresnel the this case to use directly the be boundary conditions for The calculation must performed by applying between the slabs interfaces the each of fields at and the electric magnetic result is The electron of constant usually presented as the product density. in account in the calculation taken reflections and are of matrices, multiple excellent descriptions of Several reflection. of the known as dynamical theory
sity,
was
considered in the
this kind of calculation
3.2.1
Let
us
can
be found in references
The Matrix Method
consider
a
plane
wave
polarised
plane of incidence ((s) polarisation) The in
axes are
Fig.
[10-14].
chosen
that the
so
and
wave
is
in the direction
perpendicular
to the
propagating into a stratified medium. travelling in the xOz plane as shown
3.5.
Air 0
-
IV
+
I 2
Zj-1
Fig. 3.5. Illustration of the plane of incidence for a stratified medium. The signs an’d + label the direction of propagation of the wave; Air is labelled medium 0 and the strata are identified by I<j
travel
The air is labelled electron densites
are
as
medium 0 and the strata
identified
by 1,< j _<
n
or
layers with different
downwards. In this notation the
Alain Gibaud
100
depth Zj+j marks the interface between the j and j+1 layers. The wave travelling through the material will be transmitted and reflected at each interface and the amplitudes of the upwards and downwards travelling waves will be defined as A+ and A- respectively. Tlbe electric field E- of the downwards travelling wave in the jth stratum for example, is given by the solution of the Helmoltz’s equation,
E-
The
following
=
A-e +i (wt
notation will be
kinx,j
=
kinzj
=
Note that the value of
-
h
x
adopted in
-
z),6Y
k
the
(3.45)
derivation,
kj cos Oj -
kilxj
kj
sin
k
Oj
j’
-
3
(3.46)
ki’inx,3’ j ,
is conserved at each interface since this condi-
tion is
imposed by the Snell-Descartes’ law of refraction. The upwards and downwards travelling waves are obviously superimposed at each interface so that at at a depth z from the surface the electric field in medium 3 is:
Ej (x, z) As
kinzj
takes
a
electric fields in
-
(Aj+
"’
InZ,3"+ A.
-
3
simplify
k,,j.
-ikinz,j-’
)e+4"
(3.47)
complex value, the magnitude of the upwards and downwards layer j will be denoted by,
U ( ki,,,,j, to
6
z)
-
A: eikinz,.?’z
(3.48)
,
the notation. In
addition, the quantity ki,,z,j will be replaced by continuity of the tangential component of the electric the conservation of k,,j at the depth Zj + 1 of the interface j, j +
The condition of
field and lead to:
U (kz,j,
Zj + 1)
+ U (- kz,j,
Zj + 1)
--
U (k-,,j + 1,
Zj + 1)
+ U (- k,,j + 1,
Zj + 1).
(3.49) It
was
shown in
(3.16),
that the
tangential component
of the
magnetic
is continuous when the first derivative of the electric field is conserved.
leads to the
equality below,
at the
j*j+1 interface,
kz,j [U(kz,j, 7,j+,)
kz,j+l [U(k-,,j+,, Zj+,) The combination of these two
that the
satisfy
magnitudes
field This
equations
-
U(-kj, Zj+,)] U(-kz,j+,, Z +,)]. -
can
be written in
of the electric field in media
J, j+1
at
a
matrix
(3-50) form,
depth Zj+j
so
must
3
Specular Refflectivity
U(k,,j,Zj+l) I [mj,j+l [ U(-k,j,Zj+j)j
fro-n-i Smooth and
Pjj+l mj,j+l
_ -
U(k,,j+l,Zj+l) LU(-kj+j,Zj+j)]
I [ j
pj,j+l
Rough Surfaces
101
(3.51)
with
Pj’j + i
k,,j
:--:
+ "
Mj’j+1
kj+1
1) 1.
k ’ 13 -
(3.52)
’, --’j+i’*,
2k,,3
The matrix which transforms the
magnitudes
of the electric field from the
medium j to the medium j+1’will be called the refraction matrix Rj,j+l. It is worth noting that Rj,j+l is not unimodular and has a determinant equal to
k,,j+llk,,j.
[
addition, the amplitude of the electric field within the medium depth as follows,
In
with
j varies
,j, z) U (- k,,j, z) U (k
I [
U (k ,j, O,jh ][ U(-k,,j,
e-ik,,jh
=
0
z
e
ik "
+
z
h)) I
+ h
(3.53)
.
The matrix which is involved here will be denoted the translation matrix T. The amplitude of the electric field at the surface (depth Z, =0) of the
layered material in Fig. as
by multiplying all the refraction and layer starting from the substrate (at z=Z,)
3.5 is obtained
the translation matrices in each
follows,
U(k.,,o, ZI) ZI)
U (- k.,, o)
]
1Z0,17i1R1,2
........
Z
,) I
(3.54)
product are 2x2 matrices so that their M, is also a 2x2 matrix. We thus
All the matrices involved in the above
product
[
RN, sU(k,,,, U (- k.,,,,
which is called the transfer matrix
have,
[
(- k,, zi)) I
U (k,, o,
o)
-
[
_
Z,
M1 1 M1 2 M21 M22
The reflection coefficient is defined
as
[ ][
M
Z1)) ] (k,,,, (- k,,,,Zs))
U (k,,,,
(- k,,,, Z,
(3.55)
U
U
Z,
the ratio of the reflected electric field
to the incident electric field at the surface of the material and is
r
U (k,, o, --
_
Zj)-
U(-k ,,o, Zj)
Zs) M21 U (k,,,, Z,) M, 1 U (k,, s)
+
M1 2 U
+
M22 U
k,, Z,) k,,,, Z,) ,,
*
given by,
(3.56)
102
Alain Gibaud
-
It is reasonable to strate if the x-rays
that
assume
no wave
penetrate only
a
will be reflected back from the sub-
few
Z’)
U (k,,,,
,
and therefore the reflection coefficient is
microns,
so
that
(3-57)
0,
=
simply
defined
as
M12
(3.58)
M22 The transmission coefficient is defined
as
the ratio of the transmitted electric
field to the incident electric field
t
and is
=
U(-k,,,, Z,)IU(-k,,o, Zi),
(3.59)
I/M22-
(3.60)
given by, I
-
This method for the derivation of the reflection and transmission coefficients
technique. It is a general method which is valid for any electromagnetic wave. However, it should be noted that for a plane of wave polarisation (p), the pj,j+l and mj,j+l coefficients must be modified in (3.52) by changing the wave-vector k,,,j in medium i by k-,,j/nj2. One obtains: is known
as
the matrix
kind of
(P)
P J+i
-
nJ23+1 k,,j + n3 kj+i k 2nj’3+1"z,3 nj+lkz,j nj2k.,,j+l 2nj2+1 k,,j
(3.61)
2
M(P) ’j+’
Aj
*
considering the passage from U(kz,j, Zj+,) possible to directly consider the passage from U(kz j+,, Zj+i), I matrix [12] is, The to corresponding Aj’+,.
Let to
-
us
remark
that, instead
of
it is also
[ Aj+] [pjj+jei(1 --,j+1- -,-,j)zj+1 Mj,j+le-i(k.,j+,+k.,j)Zj+l At 3
=
Mj’j + i e-
i(k,,j+l+k,,,j)Zj+l
Pj’j+16
-i(k ’j+j-h "j)Zj+:L
(3.62)
A+
,.j+l
Ajt+
I
In this case, it is no longer necessary to introduce the translation matrix. A third alternative consists in defining a matrix which links the electric field
and its first derivative at
a
depth Zj
to the
same
The matrix is unimodular and is defined for
an
quantities
at
(s) polarised
a
depth Zj+,.
wave as
[11,13],
from Smooth and
Specular Reflectivity
3
sin
Rough
Surfaces
6i+’
Jj+j k.,j+] -k,,j+l Sin Jj+1 cosJj+l Cos
Jj+j
with
--
k,,j+l (Zj
netic formalism to the
3.2.2
-
Zj+l).
case
As shown in the
previous
section
(3.63)
application of this general electromagreflectivity is discussed in the next section.
The
of x-ray
The Refraction Matrix for
matrix is defined
103
X-Ray Radiation
(equations (3.51)
(3.52)),
and
the refraction
as
l
pj,j+l Tnj,j+l mj,j+l Pj’j+1
I
with
Pj’j+1
Equation (3.46)
k,,j -
k,,j+1 2k,,,j
shows that
to the surface and that it is
k;,,j
of
is conserved and. is
ko
mj,j+l
-
k, j+j 2k ,j -
(3.64)
is the component of the wave-vector normal
equal to,
kj
k ,j k,,j
k,,j
+
equal
sin
to k
Fkjk? : kXkT.
cos
2
2
Oj
(3.65)
-
’j J
j 3
0. As
a
result of
this,
the
z
component
in medium j is
k,,j
=
- lk 2n 0
-
3
k 02 COS2
0,
where ko is the wave-vector in air. In the limit of small angles and the expression of the refractive index for x-rays, this becomes,
kz,j
-_
-ko
VF02 _: 2Jj
-
2iflj.
(3-66) substituting
(3.67)
that the coefficients pj,j+l and rnj,j+,, and as a consequence, the refraction matrix Rjj+1 are entirely determined by the incident angle and by the value of J and 3 in each layer. A similar
expression
can
be obtained for
k,,,j+l
so
Alain Gibaud
104
Reflection from
3.2.3
For
a
Flat
a
Homogeneous Material
homogeneous material, the transfer matrix between simply the refraction matrix, which means
medium I is
that the reflection coefficient
r
or
ro, 1
r
(medium 0) --
and
ROO.
so
becomes
U (k
M12 ,, 0, 0) U (- k,, o, 0)-M22-
-_
air
that M
k,,o k ,,o
rno,l P0, 1
-
k ,,i
+
k ,,,’
(3-68)
(neglecting absorption)
-
r
Vj2----’22 J koVj2 2Y
k0 0 + k0
-
-kOO
-
-
2, i0 ’
-
22i#
-
-’2, J
0 +
’2,Fi
--
26 24’# VOP--T-_ -
(3.69)
.
-
Equation (3.69) is of course identical to the one obtained by using the familiar expression for the Fresnel reflectivity (see Eq. (3.29)). Similarly, the transmission coefficient is simply given by,
to’l which is the
I I U(-k ,,,, 0) 2k ,,o U (- k,, o, 0)-M22 -P0, 1- k,,o + k,,, _
-
same
result
as
the
_
one
(3.70)
_
obtained earlier in
equation (3.34).
It
should be realised that these reflection and transmission coefficients have been derived for
with
an
wave-vector
a
treating
incident
kin. In
the distorted
wave
impinging on the surface of the material (for example in the next chapter when approximation), it is important to label
some cases
wave
Born
these coefficients to indicate which is the incident
wave
(Fig. 3.6).
The de-
tailed notation for the reflection and transmission coefficients will then be
rin, 0, The
when kin is concerned and r", and t’Ocj for the t", 0, 0, 31 explicit expressions for those coefficients are:
and
tin
0,1
isc
0,1
Let
also point out travelling with a
us
wave
E0 tin Oje
-
z,1z
vector
k,,,.
2ki,,,,o kinz,O + kinz,j.
(3.71)
2k,,z,o + k,, ,,j
(3-72)
-
k,,zo
that the field in medium wave
wave
vector
kinj
is El
1 associated
(kin,i, r)
=
with
a
plane-
U(ki,,,,, Z)e-k_in,xX
Specular Reflectivity
3
from,
S-niooth
and
Rough
Surfaces
105
X
k SC sc
kin K
k
Fig.
A
are
Single Layer
RO,17_1R1,2
and t" the calculation of tin 0’, O’l
angles for
on a
Substrate of
case
negative) deposited
on a
layer
a
of thickness
given
substrate is
PO’l rno,l
6-ik,,,h
MO’1 PO’l
0
and the reflection coefficient
Dividing
Oc 0..
Oin
The transfer matrix for the
k,,,,
i.0
in z,O
3.6. Definition of the
3.2.4
k.,’O
e
0
-
h
-_
Z1
as
P1,2 M1,2
+ik.,Ih
and
(3.73)
M1,2 P1,2
is,
ik ,jh + M1,2PO, le-ik.,,Ih
M12
MOJP1,26
M22
MO,1M1,2e
(3.74)
ik,,lh + P1,2PO,le -ikz,,.
numerator and denominator
by POJP1,2
tion coefficients rj_j,j = mj_jj/pj_j,j reflection coefficient of the electric field at the
and
introducing
the reflec-
for the two media i and i
r
(h
Z2
--
layer
2ikz,lh roj + rl,26-
1, the be,
-
is then found to
(3.75)
+ ro,lrl,2e- 2ikz,lh
noting that the denominator of this expression differs from unity by a term which corresponds to multiple reflections in the material, as shown by the product of the two reflection coefficients rojr12. It is also straightforward to determine the transmission coefficient since its value is given by 1/M22; this yields,
It is worth
t
tO,1 tl,2e-- ik ,jh +
In the
case
therefore,
when the
absorption
ro,lrl,2e-can
be
(3.76)
2ik ,,jh
neglected,
the reflected
intensity is
Alain Gibaud
106
2, + 2ro,11’1,2 1 + ro2, Ir?,2 + 2ro, 17’1,2 r2
R
=
O’l
+
r
1 2
cos cos
2k,,Ih
(3.77)
2k,,Ih’
The presence of the cosine terms in equation (3.77) indicates clearly that the reflectivity curve will exhibit oscillations in reciprocal space whose period will
be defined
by
the
equality
2k ,,Ih
Pz,,
q ,jh
=
(3.78)
2p7r,
or
2p7r q,z, I These oscillations waves
are
h
the result of the constructive interference between the
reflected at interfaces 1 and 2. The difference in
separates the two
waves
path length which
is
6 so
(3.79)
-
--
2h sin 01
--
(3.80)
pA,
that
qz, 1
27rp
(3.81)
=
h
3.7 which shows the
experimental reflectivitY of a copolymer deposited provides a good illustration of this type of interference phenomena. The experimental curve is presented in open circles and the calculated one as a solid line. The calculation is made by using the matrix technique in which we use equation (3.73) as starting point. The fact that the reflectivity is less than 1 below the critical angle is related to a surface effect. At very shallow angles, it frequently happens that the footprint of the beam is larger than the sample surface so that only part of the intensity is reflected (see Sect. 7.1.1 for details). A correction must then be applied to describe this part of the reflectivity curve. The roughness of the interfaces is
Figure
onto
a
silicon substrate
also included in the calculation
3.2.5
Two
Layers
on
a
as
discussed below.
Substrate
The calculation of the
also be made
in the
a
case
of two
reflectivity can layers deposited on
matrices of refraction and translation it is
coefficient as,
by
the matrix
substrate. After
possible
technique multiplying the five
to express the reflection
3
Specular Reflectivity from Smooth and Rough Surfaces
107
0
>
59A
P: -2
()
......
-
LU --I
-3
LL
rV
-4
LU -5
-
2 U)
-6
-
<
-71
_j
0.1
0.0
Fig.
3.7. Measured and calculated reflectivities of
deposited
PS-PBMA
r
on a
0.4
0.3
0.2
thin film of
a
a
diblock
copolymer
silicon wafer
’Se-2i(k ,2h2+k ,,1h.1) + rO,lrl,2r2 ’Se-2ik ,2h2 ,,2h2 + r2,sro,1C-2i(k ,jhj.+kz,2h2)
2ih;,,jhj + r2 ’rO,l + 7’1,2e+ 7’0,17’1,2e- 2ikz,lhl. + rl,2r2 ’SC-2ik
_
The above
expression clearly
(3,82)
shows that for two
layers
on a
substrate, multiple
reflections at each interface appear in the matrix calculation. The phase shifts depend on the path difference calculated in each medium and therefore on -
the thickness of each
of this kind
are
layer, and indirectly
on
the
angle
of incidence.
Examples
encountered in metallic thin films which tend to oxidise when
placed in air. The reflectivity curve of an oxidised, niobium thin film deposited on a sapphire substrate [15] is shown in Fig. 3.8. The upper layer obviously corresponds to the niobium oxide. The oxide layer grows as a function of time of exposure to air and reaches a maximum thickness of around 15A after
a
few hours. The
characteristic shape, *
0
short
wavelength
ences
within the
a
beating
reflectivity
curve
which includes the oscillations which
(thick)
presented in Fig. 3.8 displays following features, can
a
very
be identified with the interfer-
niobium
of the oscillations
layer. with a longer wavelength
in q, which
comes
from the presence of the oxide on the top of the niobium layer; this leads to two interfaces at nearly the same altitude from the surface of the
sapphire There
ing
are
substrate.
phenomenon and frequency.
similarities between this
of acoustic
waves
of similar
the characteristic beat-
Alain Gibaud
108
Oxide
13A
I _IA -2
-;5, (D
-3
Ir
1:1 Mil.,
-
-
4 Z
-4
-5 (n
0) 0
.
0.1
0.0
0.2
0.3
q7
0.4
0.5
(N)
3.8. ReflectivitY of Nb thin film on sapphire showing the beating of spafrequency between two comparable thicknesses which are the thickness of the
Fig. tial
niobium film and the thickness of the entire film
3.3
From
Dynamical
(niobium,
to Kinematical
and niobium
oxide)
Theory
dynamical theory described above is exact but does not clearly show the physics of scattering because numerical calculations are necessary. Sometimes, one can be more interested in an approximated analytical expression. Different approximations can be done [13,16-18], the simplest one being the Born approximation.’ We will start from the dynamical expression of the reflected amplitude calculated in the previous section (equation (3.77)) for a thin film of thickness h deposited on a substrate The full
r
and
degrade
it to obtain
tween the reflected -
2k_,, I h
-
h
_ro,l -
(3.83)
- ro,lrl,2
approximate expressions.
waves on
can
1
+ rl,2 6- 2ik ,Ih e- 2ik,,lh Here the
the substrate and the
be written
as
a
phase shift belayer denoted by 0
function of either k
or
q. The term
equation represents the effect of multiple reflections in the rc),j layer and a first step in the approximation consists in neglecting this term. This is illustrated in Fig. 3.9 which shows a comparison between the reflectivities calculated for a diblock copolymer film on a silicon wafer with the matrix method taking into account or not the multiple reflections at the r, ,2 0 0 in this
approach was first made by Rayleigh in 1912 in the context of the electromagnetic waves [16] but has since become known as the Born approximation since Born generalised it to different types of scattering processes.
This kind of
reflection’of
Specular Reflectivity
3
from Smooth and
Surfaces
Rough
109
(b)
(a)
2
oo,
0.02
.04
Q
M2
0.10
ma
0.00
0.14
0.02
0.04
O.OG
0)
0.08
0.10
0.12
0.14
QW)
Comparison between reflectivities calculated with the matrix technique (a) and (b)) and after neglecting the multiple reflections (squares in (full (a)) and in addition the refraction (triangles in (b)). Calculations are performed 0.022 A-’) 600 A thick on a silicon substrate for a diblock copolymer (g,
Fig.
3.9.
line in
=
interfaces’. It this
be
can
approximation
coefficient
r
for
a
7’
seen
is
that the two
quite good.
stratified medium
=
r0j +
7’1,2e
curves are
almost identical
Under this
approximation, composed of N layers is:
iq_ , 1 d,
+ 7’2,3 ei
(q ,, I d,
showing
+ q,,, 2 d2)
q ,k dk
+ In
...
+ rj,j+le
(3-84)
+
1-0
that
the reflection
Eq. (3.84) the ratio rj,j+lof the amplitudes of the reflected at interface j, j + I is
to the incident
waves
rj,j+l with the wave-vector transfer
q-,,j
--
in
q,,j
-
-
q,,j + medium
(47r/A) sin Oj
q,,j+l
(3.85)
q,,j+,’
j: =
qq 2,
2
-
(3-86)
qcj.
Finally,
R
rj j=O
A further
j+leiqzzj
with rj,j+l
=
q ,j
-
q,,j+l
q,,,j + q ,j+j
approximation consists in neglecting the refraction phase factor in Eq. (3.84):
tion in the material in the
and the
absorp-
110
Alain Gibaud
n
iq;,
d_
(3.87)
-0
e
rjj+l j=O
case the approximation is more drastic and this can be seen in figure showing that the region of the curve just after the critical angle is most affected, and in particular the positions of the interference fringes. A final approximation consists in assuming that the wavevector q, does not change significantly from one medium to the next so that the sum in the denominator of rj,j+l may be simplified:
In this
3.9 (b)
2
2
"j,j + i
Zj
--
-q2CJ
qc,j+l
__
(qz,j
with qc,j -tron. These
2
qz,j+l
-
qz,j+,)
+
2
47rre(pj+l
2-
2
4qz
qz
-
pj)
(3.88)
V/’1_6ir_rpj in
which r, stands for the classical radius of the elecapproximations lead to the following expression for the reflection
coefficient, n
r
(Pi + I
41r r,
--
Pj)
-
q2
d_
(3.89)
-=0
e
If the
at the upper surface
a
in the
origin of the z axis is chosen to be depth of Z, =0), then the sum over dn by the depth Zj+1 of the interface ,
r
--
47rr,
E
(medium
0 at
phase factor, can be replaced and the equation becomes,
(Pj+1
-
PA
q2
C
iq,,Zj+l
(3-90)
z
j=1
Finally, if we consider that the material is made of an infinite number of thin layers, the sum may then be transformed into an integral over z, and the reflection coefficient r has the form,
+00
47r r, r
-_
q2 z
I
(3-91)
dz.
dz
-00
A very
lq’ by z
useful, less drastic approximation is obtained by replacing (41rr,p, )2 (q,) in Eq. (3.91). Under this approximation the reflectivity can
RF
be written
as
[18], 2
+00
R
(qz)
--
r.r*
=
RF
(q,)
-I PS
I -
00
dz
e"7"dz
(3.92)
Specular Reflectivity from Smooth and Rough Surfaces
3
ill
expression for R(q,) is not rigorous but it has the advantage of being easily handled in analytical calculations. In addition, if the WienerKintchine theorem is applied to this result, we find
The above
R
(q,)
RF (qz) so
I _
p2
TF
[p’ (z)
p’ (z)]
(D
(3-93)
,
S
gives the autocorrelation function of the first density [19] or the Patterson function [20,21].
that the data inversion
tive of the electron
deriva-
iooA
,, W
0
NO
200
(K)
91
300 -
400
-SaoM
(A)
Reflectivity of a two layers system And its Fourier transby the Fresnel reflectivity of the substrate. In the calculation the two layers of different electron densities are 300’and 100A thick. The Fourier transform immediately gives the thickness of each layer without relying on any model. One can also note the an expected peak at z=400A in the
Fig.
3.10.’Calculated
form after division of the data
autocorrelation function
Figure 3.10,
illustrates the main features of this data inversion. It is based
[18]
sample consisting of two 100A on a substrate. The left hand side diagram gives the calculated reflectivity curve which shows a feature similar to the "beating" effect seen in Fig. 3.8, arising here because of the similar thicknesses of the two layers. The right hand side diagram gives the auto-correlation function, which has intense peaks at the interfaces where the derivative of the electron density is maximimum. In an ideally flat sample these peaks would be delta functions, but for a real case their width depends on factors such as the roughness and degree of interdiffusion at the interfaces. Equation (3.92) is a good starting point to introduce a last formulation for the reflected intensity. Starting from Eq. (3.92) on
a
calculation with
layers,
-
a
lower
one
model structure
a
of
300A
and
an
upper
for
one
a
of
2
+00
R
rr*
--
RF
(qz)
1 ps
I -00
dz
6iq , ’dz
(3.94)
Alain Gibaud
112
and
using
the
relation between the Fourier transform of
general
and the Fourier transform of its first
derivative,
-
j p(z) ei
RF(q-,)
function
2
+00
R(q-,)
a
have,
we
q,
’
dz
-00
-
if p(z)p(z’)e’q’(’-")dzdz.
RF (q,
(3-95)
summarise, it has been shown in this section that the kinematic theory is dynamical theory by three approximations (1) no multiple reflections at the interfaces, (2) that the effects of refraction can be neglected and, (3) -that the reflection coefficient at each interface is proportional to the difference of electron density. To
derived from the -
-
All the
expressions
discussed above have been derived under the assumpsamples. In such a case, the lateral position
tion of ideally flat interfaces in the
points at the interfaces is unimportant, since all of the points are depth from the surface. It is thus implicit that the intensity is localised along the specular direction. This means that the expression above can be considered as valid over the entire reciprocal space after multiplication by the delta functions Jq,, and Jqy which characterise the specular character of the reflected intensity. Therefore, the last equation of (3.95) for example of reflecting at the
may
as
same
well be written as,
R(q)
R_p(q ,)
-
if P(Z)P(Z,)e
iq ’
(’-")dzdz’Jq.,Jqy.
(3-96)
(z-")dzdz’Jq.,6qy.
(3-97)
Note that
R (q)
is the well-known Born
scattering.
It
can
if P(Z)P(Z,)e
2
qz
iq ,
approximation (or kinematical) expression for x-ray integration of the scattering cross-
be recovered from the
section
dor
dQ as 5
shown in
Chap. 4,
footnote
We may notice that if
iq.
Or-111)
(3-98)
8.’
applied
R(9)=qc (qx)J(qy)/16q4, 4
2jj drdr’p(r)p(r’)e to
a
flat surface this
expression
would lead to
Specular Reflectivity from
3
3.4
Influence of the on
Smooth and
Rough
Surfaces
113
Roughness
the Matrix Coefficients
shown in Chap. 2 that scattering from a rough surface/interface can separated into two contributions, coherent and incoherent scattering. In this chapter, we are only interested in the specular intensity, i.e. the coherent intensity given by the average value of the field. We give here a simple method to take roughness into account in the reflection by a rough multilayer, using the matrix method. We rely on a more complete and rigorous treatment of the case, of a single interface given in appendix LA to this chapter. In this appendix, it is shown that for roughnesses with in-plane characteristic lengths smaller than the extinction length ,z 11im for x-rays, introduced in Chap. 1,
It
was
be
rrough O’l
=
flat
rO,1
-
e
2k., ,ok,,Ia2I
exponential in Eq. (3-99) is known as the Croce-N6vot factor [22]. We now apply the method of section 3.A.3 to the matrix method. Starting from equation (3.62) The
Aj+ A’77 pj,j+le
i(k,,j+1-k,,j)Zj+1
mj,j+ I e-
Mj’j+1e-i(k ,j+1+k.,j)Zj+1
A+1
pj,j+le
A.? (3.100)
amplitudes of the electric field in two adjacent layers, we position of the interface Zj+1 between the j and j + I layers fluctuates vertically as a function of the lateral position because of the interface roughness. Following a method proposed by Tolan [23], we replace the quantity Zj+1 by Zj+1 + zj+l (x, y) in the above matrix and we take the average value of the matrix over the whole area coherently illuminated by the incident x-ray beam (in the spirit of Sect. 3.A.3, this amounts to averaging the phase-relationship between the fields above and below the interface). This leads to (as shown in Appendix LA, such expressions are only valid at which links the assume
that the
first order in
2
( zj
[AJ+ AJ- I ) ( [pj’j+1ei(k,,,j+1 _
mj,j+le
i(k ,j+i +k ,J)zi+’ e’ (k.,,j+1+k,,j)zj+1(x,y)
mj,j+le- i(k,,,j+l+k-,i)Zj+le-i(k,,j+,+k ,j)Zj+I(X,Y) i(k.,,j+l -k,,,j)Zj+, -i(k,,,j+, -k.,j)zj (x,y) pjlj+le-
[Aj++, A37+
1
(3.101)
Alain Gibaud
114
For Gaussian statistics, or at lowest order in uJ2, dependence of the different interface roughnesses:
[pj,j+,-
Aj+ A3-
e-
have, assuming
we
6-(k,,,j+3.-k ,,j)2aj2+.,/2
i(h:,,,j+,-kz,,i)Zj+-,
J+
Mj’j+1C
Mj,j+le-i(kz,j+,+kz,j)Zj+.Ie-(],.,j+l+kz,j)2cj2+1/2 ?+
pj,j+le The influence of the interface
coefficients rnj,j+l and pj,j+l C-(k,,,j+j+kz,j)2o, .7+ ./2 and es
the in-
[Aj++, A3+1
1/2
/2
(3.102)
roughness is apparent from this result.,The respectively reduced by the factors
are
1/2 It
was shown in the previous section, that the ratio mj,j+l/pj,j+l is the relevant quantity in the expression of the reflected intensity. This ratio which is the Fresnel coefficient of reflection at the altitude Zj+j is therefore reduced by the amount, ,
rough
rjJ+1 r at
-2k j+.jk ,,ja 3+
e- q
,j+jqz,jaj +112
(3-103)
3J+1
reduced
by the
In the
roughness.
in the presence of interface approximation holds (k,,j
k,,j+l
--
particular
(1/2)q,),
case
where the Born
the Fresnel coefficient is
amount rou
gh
rj,j +1 r 1 ff a t ,
-
-’:
-2k j+jk ,
ja 3+1
1
-
e
-q20rj2+1 /2
(3-104)
j’j+1
which is the
factor.
Debye-Waller
10, 10-, Z
t< EP
(D
10" la,
\
S%
10-4
rl (D
1 0.6
<
Roughness Roughness a, = cy, No
ir!. . ’*.i’. . .’,.-*. . .I. ".!.’-*. . . .,
44
SOOA
10-7
10" 0.0
0.1
0.3
0.2
q,
Fig. 3. 11. Influence of roughness deposited on a substrate
on
the
=
5A
Z 0.4
0.5
(M)
specular reflectivity
of
a
600A thin layer
Specular Reflectivity
3
from Smooth and
Rough’Surfaces
115
Fig. 3.11 how the introduction of the roughness at the inreflectivity curve. In particular, this figure shows that the reflectivity curve falls faster for rough interfaces and that the amplitude of the fringes is significantly reduced at high wave-vector transfers. We present in
terfaces modifies the
As
a
conlusion
reflectivity
can
be
we
chapter that the calculation of the by the matrix technique. This technique calculation of the specular r eflectivity for
have shown in this
properly
handled
widely used in the simple and exact. However the main drawback of this technique is that it is only valid in specular conditions, which is an important restriction. Incoherent scattering is discussed in the next chapter, using in particular the matrix formalism described above.
is the most the
reason
that it is
116
Alain Gibaud
3.A
Appendix: The Treatment Specular Reflectivity
Franqois
de
Bergevin,
The aim of this
which
can
Jean
Daillant,
of
Roughness in
Alain Gibaud and Anne Sentenac
appendix is to give an overview of the different methods roughness into account in specular reflectivity. We
be used to take
first present the second order Rayleigh calculation for a sinusoidal. grating in order to introduce the main ideas. Then, we discuss the Distorted-wave
approximation (DWBA) results (see Chap. 4 for a presentation of this approximation). Finally, we shortly discuss a simple method that allows one
Born
to retrieve the
laws,
Debye-Wall.er and Croce-N6vot factors which are respectively large and small in-plane correlation lengths. waves in all this appendix.
for
scalar
Second-Order
3.A.1
for
a
the
limiting
We consider
Rayleigh Calculation Grating
Sinusoidal
problem of the reflection by a -rough interface (here simplias a one grating of period A) separating two media and illuminated by a plane wave The Rayleigh method the in in both consists fields media as sets of plane waves expanding [24,25] and in writing the boundary conditions for the field and its first derivative. In order to write these boundary conditions, one has to calculate the values of the field and of its first derivative on the surface, as a series of terms like Let
us
consider the
fied
dimensional sinusoidal
a,, exp,
-
i
(kj,
,
x
+
k.,,,, z (x))
,
where q refers to both the medium (above or below the interface) and to the plane wave in the, expansion (in particular, the component of its wave vector
parallel
to the surface
a,, exp -i
(k,?,xx
+
describing
k?7,zz(x))
the
scattering order). One then expands:
a,, exp -i
(k,7, X) -
[I
-
ik,7,zz(’x)
2k’7zZ2 (X) +
...I.
of two exponentials (zo/2) exp(2i7rx/A) particular term in the Fourier expansion of the roughness), the expressions in the boundary conditions consist of sums of exponentials in x. For the boundary conditions to be satisfied for all x, it is necessary and sufficient that they are satisfied for each of these exponentials separately. We now have a series of equations, each corresponding to a scattering order: kinx) ki,,,, 27r/A,...
Since
(in
z(x)
the
can
general
be
expressed
case
as a sum
this would be
a
Specular Refiectivity from Smooth and Rough Surfaces
3
We define in medium 0
1:
or
k"
Vfk(o,17 (kinx 2
(0, 1) Z
and
117
-
similarly kb
cos
01
ko
--
cos
2r/A.
Oi,,
A series in zo appears in each equation, and the system will be solved perturbatively at each order. At zeroth order we get the Fresnel coefficients. At
(in amplitude),
first order in zo
get for the intensities in the 1 scattering
we
[26]:
orders
0
RF(Oi,,)
where for the
and
,k01VRF(0j,,)RF(0j),
intensity
01 respectively. in zo,
2ko, ,RF(Oin)Re (2k,,,
-10 Z 0
0
(3.A2)
’Z
the Fresnel reflection coefficients in
are
(in amplitude)
At second order
(2)
RF(01)
and
angles Oi,,
Z20 ko
get in the specular:
we
k+1
+
k+1
-
1,Z
0’Z
+
k-1
k-1.) 1,Z
-
0’Z
(3.A3)
now try"to find the change in reflectivity coefficient in the limiting cases large and small A values. Large A values For large A values, the diffracted orders in both media get close to the specular
We
of
and transmitted beams:
kO:L’
ko,z,
.z ’Z
k11
klz.
z ’Z
Then
R(Oi,,) Since
has
one
(z 2)
RF (0in) +
RF (0in) (1
0
2
2 Z2 0 ko
Z).
Z20 /2,
R(Oin) which is the first order
*
1(2)/10
;:
RF(0j,,)(1
expansion
of the
2
-
4ko
z
Z2))
Debye-Waller
factor in
(Z2).
Small A values
One has: 2
k1
0’Z
k1 1
kc
k0’Z1,
+ k
(3.A4)
111 z ’Z
k1 > ko-v/1--n2 is the critical wave vector. For small A values, k’, 1,Z 0’Z much is Since never < k,. klzl kl,z k, and therefore, using Eq. (3.A4), k01_1, ’Z smaller than k, it is the only term that survives in the sum in Eq (3.A3). where k,
=
-
Therefore,
R(Oin) which is the first order
--
RF(Oin)(l
expansion
UZe(ko,zki, ,) (z 2)),
of the Croce-N6vot factor
[22]
in
(z 2).
Alain Gibaud
118
3.A.2
The Treatment of
Roughness Specular Reflectivity within
in
the DWBA
The issue of the modification of the specular intensity due to surface scatbeen considered within the distorted-wave Born approximation in
tering has particular
in Refs.
treatment
given
[27,281.
in the
The results of Refs. [27,28] agree with the Rayleigh It is nevertheless interesting to note
previous section.
that:
Contrary
to what is sometimes
the
assumed,
specular intensity can be afappromixation includes both the reflected and transmitted fields. It is therefore possible that single scattering events transfer energy from one field to the other (in fact, energy would be conserved at this level of approximation for the sum of the reflected and transmitted fields, see Appendix I.A). In particular, the first order result of the DWBA Eq. (4.41) or [27] yields the Croce-N6vot facfected in the first-order DWBA. This is because the basis for this
tor at first order in
(z 2).
Exercise: Show this.
[28] shows, as did the Rayleigh calculation disDebye-Waller factor is obtained for large A values
The second order DWBA cussed
above,
that the
whereas the Croce-N6vot factor is obtained for small A values.
3.A.3
Simple Derivation
of the
Debye-Waller and, Croce-N6vot
Factors
The accuracy of approximated expressions for the reflectivity coefficient mainly relies on the quality of the approximations made on the local value of the electric field at the interface. Let
us
consider the two
limiting
with very small and very large in-plane characteristic If the characteristic length scale of the roughness is much nesses
extinction be written
length (we locally for
have
a
slowly varying
interface
the well-defined interface at
roughness characteristic
scale
(this
is the so-called
a
cases of roughlength scales. larger than the
height),
the field
can
scale smaller than the
tangent plane approxima-
tion): Ej (x, z)
--
(Aj+
e
ikj,,z +
The field will be reflected at different
Aj e -ikj, z) -
iwt-kjinjj.rjj
heights depending
on
(3.A5)
x, and the reflec-
tion coefficient is rough
-
(AO A0
where the average value is taken over the surface. Writing the boundary conditions, one obtains with the notations of Chap. 3 for a surface located at
z:
eik;"OZ + A-e -ikz,oz A +, 0
kz,oAo+e ik;,,oz0 kz,OAO
-
,
-
e-’
k,,,Iz A-e1
k ,Oz- _kz,,Al-e- k;,, -
1z
(3.A6)
Specular Reflectivity
3
from Smooth and
Rough
Surfaces
119
One obtains
r
rough
(A0+)x
=
ro, 1 (e
A0
2iko, z
roje-
2ko,,,(z2)
Debye-Waller factor as expected. We obtain this factor because roughness characteristic length is large enough for the incident and reflected fields to have a precise phase relationship. which is the the
length to be much smaller than the lym. Then, the electric field is length, not perturbed at the roughness scale (in other words, there are no short-scale correlations between the field and the roughness). There is only an overall perturbation of the electric field which can be written to a good approximation as the combination of upwards and downwards propagating plane waves, whose amplitude will however depend on the roughness: We
now
assume
the characteristic
which is
extinction
Ej (x, z) where the
=
Aj,eff Q
( Aj+
3,eff
A3- eff e- ikj,,zeiwt-kiinJJ.rII,
eikj,., z+
0, 1)
_-
the order of
on
are
unknown effective
interface. The reflection coefficient is defined
rrough
amplitudes
(3.A7) for the
rough
as:
A+ -
O,e
A-
O,eff
phase relationships between the field above and below the interface are only valid on average because the field "does not see" the local roughness (this is of course not a rigourous argument, the justification for Eq. (3.A8) are the calculations given in the two previous Now,
we
assume
that the
sections):
2ko,,A+0,,ff 0, Zko,zAo,eff
=
-
Eq. (3.A8) can be obtained method Eq. (3.62). Then
r
rough
(ko,, (ko,,
-
+
from
A+
O,eff
kl,,)ATi,eff(’ -i(ko,.+kz,1)z)
kl,,)Al,eff
Eq.
(3.A6),
or
flat,-2k;,,o
directly using the matrix
(Z2)
,
A
(3.A8)
(3.Ag)
I
O,eff
obtains the Croce-N6vot factor. Note that in this case, the transition layer method would give an equally good result. Note also, that the method one
could be 3 and 8.
applied
to the
averaging
of transfer
matrices,
as
it is done in
Chaps.
Alain Gibaud
120
References 1. A. Fresnel M6moires de IAcad6mie 2. 3.
4. 5. 6. 7. 8.
9.
11, 393 (1823). Compton, Phil. Mag, 45, 1121 (1923). R. Forster, Helv. Phys. Acta., 1, 18 (1927). H. Kiessig, Ann. Der Physik, 10, 715 (1931). J.A. Prins, Z. Phys., 47, 479 (1928). L.G. Parrat, Phys. Rev. 95, 359 (1954). F. Ab6l6s, Ann. de Physique 5, 596 (1950). R.W. James, The optical principles of the diffraction of x-rays, G. Bell and Sons, 9ondon, (1967). International tables for x-ray crystallography, The Kynoch Press, Birmingham, A.H.
1968. vol. IV. 10. R. son
11.
Petit, Ondes Electromagn6tiques
en
radio6lectricit6 et
en
optique, Ed. Mas-
(1989).
M. Born and E.
Wolf, Principles
of
Optics, Pergamon, London,
6th Edition
(1980). Vincent, Applied Optics, 23, 1794 (1984). Lelmer, Theory of reflection of electromagnetic and particle waves, Martinus Nijhoff Publishers (1987). 14. T. P. Russel, Mater. Science Rep. 5, 171 (1990). 15. A. Gibaud, D. McMorrow and P. P. Swadling, J. Phys. Condens. Matter 7, 12.
B. Vidal and P.
13.
J.
2645 16.
(1995). Rayleigh, Proc. Roy. Soc., 86, 207 (1912). Hamley and J. S. Pedersen, Appl. Cryst. 27,
J.W.S.
17. 1. W.
29
(1994).
Vignaud, Th6se de I’Universit6 du Maine (1997). 19. J. Als-Nielsen, Z. Phys. B 61, 411 (1985). 20. I.M. Tidswell, B.M. Ocko, P.S. Pershan, S.R. Wassermann, G.M. Whitesides, J.D. Axe, Phys. Rev. B 41, 1111 (1990). 21. G. Vignaud, A. Gibaud, G. Griibel, S. Joly, D. Ausserr6, J.F. Legrand, Y.’ Gallot, Physica B 248, 250 (1998). 22. L. N6vot and P. Croce, Revue de Physique appliqu6e, 15, 761 (1980). 23. M. Tolan, Rontgenstreuung an strukturierten Oberflachen Experiment &Theorie Ph.D. Thesis, Christian-Albrechts Universitat, Kiel, (1993) 24. Lord Rayleigh, Proc. R. Soc. London A 79, 339 (1907). 25. S.O. Rice, Com. Pure Appl. Math. 4, 351 (1951). 26. D.V. Roschchupkin, M. Brunel, F. de Bergevin, A.I. Erko, Nuclear Inst. Meth. 18.
G.
B 72, 471 27.
S.K.
(1992).
Sinha, E.B. Sirota, S. Garroff
and H.B.
Stanley, Phys. Rev. B, 38,
(1988). 28.
D.K.G. de
Boer, Phys. Rev.
B
49,
5817
(1994).
2297
4
Diffuse
Scattering
Jean Daillant’ and Anne Sentenac’
Condens6, Orme des Merisiers CEA Saclay, Cedex, France, LOSCM/ENSPM, Universit6 de St J&6me, 13397 Marseille Cedex 20, France Service de 91191 Gif
Physique
sur
de I’Etat
,
Yvette
Chap. 3 is sensitive to the average dena sample surface. Very often, one would also like to determine the statistical properties of surfaces or interfaces (i.e. the "lateral" structures in the (xOy) plane). We have seen in Chap. 2 that the scattered intensity depends on the roughness statistics of the sample (when
Specular reflectivity, as described sity profile along the normal (Oz)
the coherence domains
precisely,
under several
are
in
to
much smaller than the illuminated
simplifying assumptions,
area).
the differential
More
scattering
cross-section is related to the power spectrum of the surface. Many examples are given in the second part of this book where we shall see that, in particular because of the
grazing
incidence geometry, x-ray scattering experiments lengths of surface morphologies and
allow the determination of the lateral
of the correlations between buried interfaces
over
more
than five orders of
plane (see Sect. 4.7.2). Angtr6ms magnitudes In this chapter we present the theory of scattering by random media from an electromagnetic point of view. Starting from Maxwell equations we establish the volume integral equation giving the scattered field as the field radiated by the dipoles induced in the material (part of these results have been used without demonstration in Chap. 2). We then describe several perturbation techniques (Born approximation and Distorted-Wave Born Approximation, DWBA) that permit to obtain simple expressions for the differential scattering -cross-section. The latters are then applied to scattering problems of increasing complexity: scattering by a single rough surface, surface scattering in a thin film, scattering by rough inhomogeneous multilayers. Finally, special attention is paid to the resolution function and to the determination from
of absolute
(measured)
to tens of microns in
intensities. This is necessary if
one
wants to draw
experiment. chapter is"devoted to the discussion of the so-called distorted wave Born approximation (DWBA) which presently provides the most accurate analysis of x-ray and neutron data. It is a perturbation method in which the roughness is viewed as a random perturbation of a deterministic reference state. In the simplest version of the theory presented here [1,2], the unperturbated reference (ideal) state can be a plane (in the case of the study of a rough surface) or a perfect planar multilayer (in the case of the study of rough multilayers). In more sophisticated versions, the reference state of a rough surface can be a medium with graded index whose z-dependent di-
quantitative
information from
an
Most of the
J. Daillant and A. Gibaud: LNPm 58, pp. 121 - 162, 1999 © Springer-Verlag Berlin Heidelberg 1999
Jean Daillant and Anne Sentenac
122
electric constant varies continuously from the air value to the material value, following the average density profile [3,4]. The electromagnetic field is calculated exactly for these reference states, hence, we expect the theory to be accurate even close to the critical angle for total external reflection. On the other hand, the radiative contributions of the permittivity fluctuations (i.e. the perturbation) are restricted to single scattering events (first order approx-
imation)
within the reference Tnediuin. Second-order DWBA, which accounts doubly scattering processes, has been developed in Ref. [51 for specular reflectivitY.
for
Differential
4.1 In this
section,
we
Scattering
first establish the
field and show that its solution
tion
using Green development (see
Cross-Section for
propagation equation
be put in the form of
X-Rays for the electric
integral equaintegral equation is the basis for the Born the Born approximation and Sect. 4.3 for the can
an
functions. This
Sect. 4.2 for
(first order) DWBA).
The definition of the distorted-wave Born
tion then amounts to
the
choice of
an
unperturbated (ideal)
approxima-
state for which
the field in the
sample and the Green functions have to be evaluated exactly. difficulty of the technique give here a simple method, based on the reciprocity theorem, to cal-
The evaluation of the Green functions is the main and
we
culate them for various reference states:
approximation)
an
infinite
homogeneous
medium
(for
planar multilayer (for the DWBA). The developments made in this section are valid for complicated systems like multilayers with rough interfaces and possibly density inhomogeneities. However, for simplicity the reader can refer to the case of a single rough- interface separating two material media (0) and (1) depicted in Fig.4.1. the Born
4.1.1
Using
and
a
Propagation Equation Maxwell’s
equations: VxE
9B
(4.1)
=-
(9t M
’7xH=j+ one
obtains the
media
(0)
propagation equation -for the electric field no charges or currents:
and
(1) containing
17
17
x
(4.2)
x
E
-
n
2(r)WC22E
-
_,72 E 17
-
0,
x
17
-
x
n
in the
homogeneous
2(r) w2E C2
E
W
2
-
C2
(E _P) +
60
(4.3)
4
for
waves
having
a
e ‘ time dependence.
n
Scattering
Diffuse
123
index; the di-
is the refractive
n2; k nw/c is the wavevector; in the vacuum ko w1c 27/A (A is the wavelength). Note that all the possible complexity (roughness, inhomogeneities) of a sample is electric constant and the refractive index
In the a
case
related
by
c
-_
--
=
--
contained in
are
n’(r).
of neutrons,
one
has to solve
which exhibits
Schr6dinger equation
similar structure:
h2 2Tn
where
v2
21rh2
I: bi
+ M
(4.4)
90(r),
pi
bi is the scattering length of nuclei i whose
(number) density
sample is pi. In the following we shall work out a peTtubative solution for surface scattering. To do this, we decompose the index as
n2(r)
=
n
2 re f
(r)
+
the
in the
problem
Jn2(r),
of
(4.5)
precised. The reference simple enough for the electromagnetic field to be calculated exactly (vacuum, plane interface, planar multilayers). It represents the basis (zeroth order) of the perturbation development. Hence, it should be of as close as possible to the real medium in order to minimise the influence 2 2 Jn show that the perturbation. n ref yields a specular reflection, and we will is rewritten, yields the incoherent scattering. With Eq.(4.5), equation (4.3) where nref is the index of state is deterministic and
V
The
right-hand
x
V
x
a
reference situation to be
E(r)
side of Eq.
-
n
2 re f
(4.6)
(r) k2E(r) 0
can
=
Sn2(r) k2E(r). 0
be considered
as
a
(4.6)
fletitious dipole
coSn 2(r)E(r’) in the reference medium.’ Maxwell’s equations and Eq. (4.3) being linear, the electric field can be written as E source
SP(r’)
=
Eref + 6E where Eref is the field in the reference tion in the field radiated
4.1.2
by
the fictitious
’
source
perturba-
6P.
Integral Equation
The field radiated at the detector lated
dipole
case, and SE the
using
Note that
by
the fictitious
sources
6P
Green functions. We introduce the Green tensor
E(r’)
is the real unknown field at r’.
can
be calcu-
G(R, r’)
for the
Jean Daillant and Anne Sentenac
124
E(R)
kin
=
E
ref
8 E(R)
+
Yk
PA
ksc
Ein
-
=
e8 (r-R
ksc
,
2
P_ __S P P B=8S n E _
,
VOMM/M., (a)
(b)
Fig. 4. 1.
Illustration of the reciprocity theorem in the case of a single rough surrough surface (real state in (a)) is viewed as a perturbation (in gray) of the reference state (in (b)). The location of the planar interface in the reference case is arbitrary (it is here situated below the deepest incursion of the roughness, see footnote 9 on this subject). The total electric field E(R) existing in the real (rough) state is the sum of the specular field E,,f (R) coming from the reference medium and of the scattered field JE(R) radiated by the dipole density coJn’(r)E(r’), which is nonzero only within the grey region. In order JP(r’) to use the reciprocity theorem to calculate JE(R) we consider two distributions A unit dipole with moment ; placed at R (detector location) creof sources: E‘ d ,, (R, r’) in the reference state in (b), A ating a field distribution EA (r’) dipole density representing the perturbation brought by the roughness (gray recoJn’(r’)E(r) creating a field distribution ED(R) JE(R) in (a). gion), JP(r) The reciprocity theorem yields, JE(R).’ f dr’W(r’)E’ d, t(R, r’) face. The
=
-
=
-
=
=:
propagation equation in
17
that satisfies on
2
x
V
x
the reference
U(R, r’)
out-going
wave
(ideal)
2
-
2-
case as
nref (R)koG(]P,,
r’)
the solution of
k2
=
-"0 60
boundary conditions, (the
J(R-r’),
V
x
2
(4.7)
17 operator acts
R). 3
We have the identity 17 x V propagation equation reduces
=
to
graddiv a
-
A. In
a
vacuum, divE
set of three Helmholtz
0 and the
2 equations -AE-k 0 E
0. With this
defined in
sign convention, the outgoing Green tensor 9(r) reduces to -G-(r) Chap. 1 for the scalar field ob eying the Helmholz equation ZA E + ko2 E
0. 3
(in the sense of distributions) of Eq. (4.7) satisfies by construction boundary conditions in the system (e.g. the saltus conditions at each interface the reference state is a planar multilayer).
The solution the if
4
It is then
straightforward
(insert Eq. (4.8)
to show
E,,f (R) +
E(R)
Eref(R)
in
Scattering
125
Eq. (4.6)) that,
JE(R)
+ co
Eref (R) +
Diffuse
j
I
drJn
2
drg (R,
(r)g(R,r).E(r)
r). SP (r)
(4.8)
formally equivalent to Eq. (LA4) and (or DWBA) development. However, we first need an expression for the tensorial Green function. This can in particular be done using an elegant method due to P. Croce [6-11] based on the reciprocity
is the solution of
Eq. (4.6). Eq. (4.8)
is
will be the basis for the Born
theorem
[12].
Derivation of the Green Functions
4.1.3
Using
the
Reciprocity
Theorem
!9(R, r’) introduced in Eq. multilayer. For this, planar (4.8) we use the reciprocity theorem demonstrated for example in Appendix 4.A and Ref. [12-14]. The reciprocity theorem states that, in a given reference medium, two different distributions of dipole sources PA and PB creating the fields EA, and EB are linked by the relation,
In this
paragraph
we
determine the Green tensor
for two reference media ,
vacuum
and
a
fdrEA(r).PB(r) j drEB(r)-PA(r). =
In order to calculate the
perturbation
in the field at the detector
(4.9) SE(R),
we
following sources and field distributions ( see Fig. 4.1), EoJn2 (r’)E(r’) creating an The source with polarisation vector JP(r’) unknown field SE(R) at the detector in the real case of the rough interface, The unit dipole iS(R r’) located at the detector, creating a known field at point r’ in the roughness region (the field can be calculated E’ de t (R, r) exactly since the unit dipole radiates in the simple reference geometry). consider the
=
-
We
write, EA EB
and the
reciprocity
=
=
(RI r) PA E 3et d PB JE(R)
theorem
=
=
(R-r) coJn 2E(r’),
Eq. (4.9) yields,
j dr,EoJn2 (r’)E(r’).E’Lt (R, r’)
=
JE(R).’ .
(4.10)
Eq. (4.10) is equivalent to,
E(R)J
=
Eref (R)J +
I dr’On (r)E(r)-kddet(R,r’). 2
(4.11)
Jean Daillant and Anne Sentenac
126
The unit vector ’ being arbitrary, Eq. (4.11) is in fact a vector (and scalar) equation (choose & equal respectively to i, 5 and to calculate different field components). We retrieve formally Eq. (4.8), 4 E (R)
=
Comparing Eqs. (4.8)
E,,f (R) +
(4.11),
and
calculate the scattered field
can
j
Eo
dr’6n 2g (R,
not
the
r’).E(r’).
that the Green
function required to simply calculated as the field in r’ the reference case. In practice, we will
we see
therefore
be
due to a unit dipole in R (detector) in directly use this property in Eq. (4.10) to calculate the scattered field. This is particularly convenient in the far-field approximation within which the dipole field is easy to calculate. Note that Eq. (4.11) is an exact relation [2] from which approximations can be made. That Eq. (4.11) is exact is verified in Appendix 4.13 in the particular case of the reflection on a film. If the polarisation (or direction) ’ ,c of the scattered field in the detector is known, the unit dipole direction; is usually taken equal to; sc in order to directly obtain the scattered field amplitude,
6 E (R)
I dr’coJn
--
2
E (r’).
ledet (R, r’).
(4.12)
In the far-field
approximation, the polarisation vector -e,,, is necessarily per(meaningful in far-field) s ample- to-detector direction fi R/R. It appears convenient to introduce two main polarisation states. In polarisation (s), the field direction is normal to the scattering plane, (defined by the normal to the sample and the s ample-to- detector direction (Oz,fi) in polarisation (p) the field direction lies in the scattering plane. We now explicitly calculate the Greenfunction in a vacuum (which is the reference state in the Born approximation) and for a planar multilayer (which pendicular
to the
--
is the reference state for the
Green Function in
4.1.4
point r’
The electric field at an
homogeneous
I/jR
DWBA).
-
r’ 12 and
where ii
R we
-
I/jR
-
r’l / I R
assume
Note that the
Vacuum
created
infinite medium
by
dipole moment;
a
(vacuum)
can
be written
r’ J3 terms in the limit of
Ee det (R,
Hereafter
a
-
k2(fi 0
r) r’l
;)
e
-
X u
-
(neglecting r’j) [151,
the
ikoIR-r’l
4wcojR
-
(4.13)
r’ 1’
is the unit vector in the direction of observation.
that R >> r’
reciprocity
X
large IR
located at R in as
theorem
the Green tensor is
reciprocal
symmetry relations
on
so
that fi
R/R.
gives Eq. (4-11)
in the
sense
that
the Green tensor involve
Note that the
but does not tell
9(R, r’)
:=
U(r’, R).
transpositions,
see
dipole us
that
In fact the
e.g. Ref.
[14].
Diffuse
4
sample.
IR (see Fig. 4.1).5 tangent plane
r’l
-
we
(s)
choose ’
directions
’)
X
consider
a
; (8)
X
-
a
or
e
2
(R, r’)
Green Function for
now
R
-
k.,,.r’/ko
therefore be
developed
on
:he
the
-ikoR
(4.14)
e
47rEOR
scattering, for example along ’ (P), one simply has,
normal to the direction of
(’)’(P)
We
wave can
O(ia 0
(R r)
E det
4.1.5
fi.r’
-
wave,
(p) polarisation
and
R
dipole spherical
The
Eedet If
r’l
IR
127
is observed at the
position and that the field approximation, one can develop
is here located at the detector In the far-field
Scattering
k0
the
-ikoR
(4.15)
47rcOR
Stratified Medium
planar multilayer
reference state and
as
want the
we
point r’ by a unit dipole placed in R. expression We assume that the far-field conditions are satisfied, so that the direction k"c is meaningful, and we consider the two main states of polarisation, 3 and & -_’ i(P). The point r’ can be taken anywhere in the stratified medium, the general case of a stratified see Fig. 4.2. Using the same plane-wave limit in for or (p) polarisation for r’ lying be can (s) generalised Eq. medium, (4.15) of the electric field created at
in
layer j Edet (’)’
as,
(P)
6
(R,
k
2e-
ikoR
Pw
OT7rcoR El
The far-field, conditions
(or
(8), (P)
Fraunhofer
-k,,c,,,j,
diffraction)
Z/)- (s),(p)eikc
(4.16)
SC
are more
restricting of
Indeed,
needs
r
2//\
to
neglect
to be small
quadratic compared to
R.
c
=
-
6
than
only
e-’k,)JR-r’J
expansion Applying this approximation in r’ Eq. (4.8) yields a condition on the whole size of the scattering object (since in Chap. 2 has shown that the covers all the perturbated region). The discussion in Eq. (4.8) can actually be restricted to the supp ’ort of the integral appearing domain of coherence (induced by the incident beam and detector acceptance) of the scattering processes. In this case the far-field conditions can be written as 12. h/A < R. In a typical x-ray experiment, the sample-to- detector distance is IA. The total illuminated area is a few mm but R 1m, the wavelength is the coherence length is 1coh lpm, hence the far-field approximation is valid. When the coherence length is too important (very small detector acceptance) for the far-field conditions to be satisfied, we are in the frame of the Fresnel diffraction r’l [16] [17]. and one needs to retain the quadratic terms in the expansion of IR The electric field is the solution of the inhomogeneous differential equation 2 1 2 17 x V x E’ d ,, (r’) n ref(Z )koEdet (r) ’ &S(R-r’) that satisfies out-going wave the detector position lies in medium at unitdipole boundary conditions. The 0 as depicted in Fig. 4.1. In the homogeneous region 0, the electric field can be written as the sum of a particular solution and a homogeneous solution. The parR > r’.
one
term in the
the
e
-
=
Jean Daillant and Anne Sentenac
128
(SMP) 6ik.,11.r1l
PW
E
is the field in medium i for an incident plane wave with polarisation (s) or (p) which can be computed by using standard iterative procedures [18,16]. Using the notations of chapter 3, Eqs. (3.47), (3.48), one has, i
PW
(s)’(P)
(k,,,, z)
=
T-Tj(’) (P) (k
Zj) e-"kl,j’
-
+
U(’)’(P) (-k z,3) Z3-)e’k -
(4.17) where
r
--
(rij, z)
and where the calculated for
Differential
4.1.6
It is
with
now
z
possible
first choose the
Ede,t(R,r’)
z
coordinate with the
to
Scattering
give
field scattered in the i i of
is the
superscript "PW" an incident plane
vacuum as
in
an
--
has been used to
=
wave.
Cross-Section
exact
R/R
origin taken at z Zj, emphasize that EPW is
--
expression for the ; -component of the E k,,/ko direction, E,,, Eil’. We -
-
reference state.
Eq. (4.8),
one
Substituting the expression (4.14) obtains for the component along ; of the
given in Eq. (4.13) while the general homogeneous solutions simply up-going plane waves with wavector modulus ko. In media j with 3.0 0 0 s, the electric field is solution of the homogeneous vectorial Helmholtz equation and it can be written as a sum of up-going and down-going plane waves with wavevector modulus kj. In the substrate the general solutions are downgoing plane waves with wavector modulus k,. To obtain the amplitudes of these plane waves we write the boundary conditions at each interface. The far-field approximation permits to simplify greatly the problem. In this case, the expression of the particular solution at z Z, is given by Eq. (4.15). The dipole field close to the first interface can be approximated by an "incident" plane wave with wavevector k,,,. Hence, the amplitudes of the other plane waves (that are the general solutions of the homogeneous Helmholtz equations) are calculated easily with the transfer matrix technique presented in Chap. 3. The problem has been ticular solution is are
=
reduced to the calculation of the electric field in
by
plane
a
stratified medium illuminated
superscripts (s) or (p) is always unambiguous : it indicates the direction of the radiating unit dipole in a vacuum for a given position R of the detector. In other words, it indicates the polarisation state of the scattered plane wave with wavevector k,;,. Note that the directions of ; P) , and ks,_ will vary from layer to layer due to refraction whereas the directions given by k,,11 and ; 2) do not change. a
wave.
The
.
meaning
of
Diffuse
4
Scattering
129
scattered field in direction fi:
60f
E,, (R)J6
k20 e-
dr’Jn 2(r’) edet (R)
e-ikoR ,
47rR ikoR
dr’Jn 2(r)
I dr’(k (r’)
-
41rR
r’). E (r’)
ik-.r’
ko
-
(4.18) Writing Eq. (4.18)
for i
equal respectively e-ikoR
E,;, (R)
=
scattering.
E
-
(ii.E)fi
2,
one
obtains:
ko)E-L (r’)
r
47rR
of
and
ikoR
-I dr’(k where Ej-
i,
dr’Jn2 (r) E i (r /)e
47rR e-
to
(4.19)
is the component of the field normal to the direction case the incident field.
Note that the reference field is in this
scattering cross-section we proceed by expression. In the far-field approximation,
To calculate the differential
ing
the
Poynting
vector
1il
x
deriv-
E,
C
and the
Poynting’s
vector is:
JEF
S
fi.
2yoc
scattering cross-section is obtained by calculating the flux of Poynting’s vector (power radiated) per unit solid angle in direction k ,c across of Maxwell’s a sphere of radius R for a unit incident flux. Using the linearity equations, it can also be calculated for an incident field Eil, across a unit
The differential
surface. One gets,
This exact
ko
dQ
16 7r2 I Eil 12
expression
extended detector statistical
4
du
on
properties
j
dr’6 n 2 (r’) E _L (r’) e ik,,.r’
has been used in
Chap.
the measured scattered
of
a
I
2 .
(4.20)
2 to discuss the effect of
intensity
an
in relation with the
surface.
scattering from random media, we have seen in Chap. 2 that scattering can be separated into a coherent process and an incoherent process. The latter is the usual quantity of interest in a scattering If
one
considers the issue of
Jean Daillant and Anne Sentenac
130
experiment
and it is
do-
dS2
)
given by,
fl
4
ko. 2.
1 r)
incoh
I First Born
4.2
The first Born
2
dr’Jn 2(r’)E-L
(r/)eik_.r’) 2
dr’ Sn2 (r’) E _L
Approximation
approximation which neglects multiple reflections
can
only
be used far from the critical angle for total external reflection. Close to this point, the scattering cross-sect*ions are large and the contribution to the measured
intensity of at least multiple reflections cannot be neglected. The main advantage of presenting this approximation here is that it makes the structure of the scattered intensity very transparent. It has already been presented in Chap. 2 in a different context with the aim of illustrating how statistical information about surfaces or interfaces can be obtained in a scattering experiment.
4.2.1
Expression
In the Born
evaluated in
of the Differential
Scattering
Cross-Section
approximation, both the Green function and a vacuum, Eq. (4.11).
E%t (13, d
-
r’)
ko (ii
x
e)
the electric field
e- ikoR x u
-
47rcoR
(4.22)
e
Ejje- iki_r’.
E(r’)
(4.23)
-k,,c is the
wavevector orientated from the detector to the surface which
the
field of
dipole
EW
=
EinC
are
gives
Eq. (4.10). Then, substituting into (4.11), -iki_r’
+
(Ei,,
47rR
* sc); scjdr6n 2eiq.r,
(4.24)
with the wavevector transfer: q
For such
a
field
=
k,,c
-
(4.25)
kin-
dependence, the differential scattering cross-section (power angle, per unit incident flux) is [15]:
scattered per unit solid
k4
do-
dQ
02 ( iin . sc)2
167r
Note that for small wavevector
-
j
dr Jn 2e iq.r
transfers, (’6jn.’ ,c)
1.
12
(4.26)
Diffuse
4
Example: Scattering by
4.2.2
a
Scattering
131
Surface
Single Rough
apply Eq. (4.26) to the case of a single rough surface. complicated case of a rough multilayer is treated in Appendix 4.3. This example of the diffuse scattering by a rough surface within the Born approximation is the simplest one can imagine and is mainly treated here to show how height-height correlation functions arise as average surface quantitites in the scattering cross-section. The scheme of the calculations will always be the same within the Born or distorted-wave Born approximations, whatever the kind of surface or interface roughness considered. We start from proceed
To
The
we
will first
more
ko’
dodS?
The upper medium 1) is made slightly
grating
first
over z
(n
167r 2 qz2
dQ
Equation (4.28) k 04(n 2
1)2
2
167r qz2
dQ
vacuum)
or
_
dz
is medium
j drjj
_)2
e
iq.r
00
0, and
in order to make the
1)2
2
(r[l)
dril
2
(4.27)
the substrate
integrals
(medium
converge. Inte-
2 e
iq,, z (r1j)
iqjj.rjj
(4.28)
be written:
can
_
_
(air
I f
1)2 ( in -_; Sc )2
2
absorbing yields,
k40
do-
do-
z
(n 167r2
-
(; in sc) -
2
dr’ll e iq,, (z(rjj)-z(r’jj)) e iqjj.rjj -iqjj.ir1jj
drjj
(4.29) Making
the
change
do-
k 4A 0
dQ
167r 2 qz2
of variables
(n
2 _
R11
=
ril
-
1)2 (; in .-6sc )2
r’ll
and
integrating
over
R11:
dR11 (,iq.,(z(Rjj)-z(O))),iqjj.Rjj (4.30)
where A is the illuminated
area
and
we
have
simply
used the definition of
surface.’ Assuming Gaussian statictics of the height flucChap. 2), or in any case expanding the exponential to the order, we have:
the average over a tuations z(rjj) (see
lowest
(second)
(eiqz(z(Rjj)-z(O)))
-
--Lq.’,(z(Rjj)-z(O))’.
(4-31)
We then obtain: do-
k 40 A -
dQ
167r2q2
(n 2
2
-
1) (; in. &Sc)2
e
_,7.2
2
dR11 eq2 (z (R11)z
(0)) eiqll. R11
z
(4.32) 7
In
general,
this average over the surface will not be known and as discussed in Chap. 2.
ensemble average
we
will
use
an
Jean Daillant and Anne Sentenac
132
This
equation also includes specular (coherent) components because it has general solution of an electromagnetic field in vacuum. The diffuse intensity can be obtained by removing the specular
been constructed from the a
component:
8
k4 A
(dfl)c.h do-
where the
identity
"0"
4q2
has been used. The diffuse
dQ
_
choose
now as
interfaces
k4A 0
-
-
167r 2 qz2
incoh
present
an
(n
2 _
index
(qjj)
--
1)2 ( iin .,; sc)
(4.33)
(4-34)
is then:
2
(Z2)I dR11 (eq",:(z(Rjj)z(O)) 1)6iqjj.Rjj _
a
further order of
complexity.
We
the real one, but with smooth The Green function and the field in Eq. (4.11)
same
profiles).
(4.35)
Approximation
approximation with
reference state the
(step
. sc)2 S(qjj),
(incoherent) intensity
Distorted-Wave Born
We will
q 2 (Z) 2
e
j dR11 eiqjj.Rjj
xe-q.2
4.3
1)2
for Dirac S functions I
)
2
z
47r2
do-
(n
system
as
therefore those for smooth steep interfaces and the iterative methods discussed in Chap. 3 can be used to calculate the field and the Green function. are
This
approximation yields better results than the first Born approximation angle for total external reflection. It is currently the most popular approximation for the treatment of x-ray surface scattering data. A first change due to the new choice of reference state is that, because near
the critical
refraction is taken into account, the normal component of the wavevector now depends on the local index. Using Snell-Descartes laws:
k,,j
=
koVSin2 0
-
sin
2
Oc,’
(4.36)
Integrating (4.33)’ over the angular acceptance of the detector So SSdetector/R 2= dOdo (2/koq_,)dqjj, and normalising to the total incident flux through the area A (leading to a factor A sin 0, since contrary to the reflectivity coefficient daldfl is normAlised to a unit incident flux), one obtains for the
=
=
reflectivity coefficient R: 4
R
=
OA (n2 4
qz
)2
e
-q2(_)2
(- in -’ s ) c
2q,
2q.,
e
_q2(z)2 z
(- &i. I SC)2
which shows the well-known qz 4 decay. This expression also shows that within the Born approximation, the Brewster angle is 45 degrees.
Diffuse
4
where
OC,
vacuum
=
2(1
ni)
-
is the critical
and medium i with ni
imaginary
=
angle I
-
Ji
I -
v’2
Irn(k,,j)
ko
[(02
I- ko V[(02
-
i#j.
-
More
+
4#j2j1/2
2Si)2
+
40i’]1/2
-
precisely,
the real and
are:
2Ji)2
V2
-
133
for total external reflection between
parts of the wavevector in medium i
Re(k,,j)
Scattering
+
-
(02
(02
-
-
2Jj),
(4.37)
2Jj).
(4.38)
above, refraction also implies that the direction of the polarisation vector in (p) polarisation changes from layer to layer. To avoid the complications related to this point, unless otherwise specified, we will always limit ourselves to the case of scattering of a (s) polarised wave into (s) pocos 0, in every larisation in the rest of this section. Then, one has (; in -; sc) will be effects of given in secpolarisation layer. A more detailed discussion As mentioned
=
tion 4.4.
4.3.1
Case of
a
Single Rough
Surface
Considering only one rough interface between media (0) and (1) and placing the reference plane above the real rough interface (Fig. 4.1),’ we have, for 9
The choice of the reference erence
planes
is
important
medium, here. This
particular of the location of the refquestion did not arise in the discussion of
and in
approximation where the reference medium is the vacuum. Three different choices are a priori possible: place the reference plane above, below, or crossing the real rough interface (Fig. 4.2). In principle, all choices are equivalent for small roughnesses owing to the continuity of the field. For larger roughnesses, using the average plane of the rough surface might be the best choice. This has been done to calculate the specular reflectivity in Ref. [1] close to the critical angle. However, this approximation is not good for larger incident angles because Fresnel eigenstates are not a good approximation of’the real eigenstates of the system. In this regime, however, the first Born approximation is good far from Bragg peaks. In the present treatment, we have choosen to place the reference plane above the rough surface, hence the e. 1 and t’0 coefficients. This approximation is as good as that of the average plane close to the critical angle and of incidence. For a converges to the first Born approximation at larger angles multilayer, one might be worried by the phase factor corresponding to the small shift between the average plane, and the reference plane placed below the surface (this phase shift disappears in the cross-section for a single interface). On the other hand, when using the average plane, calculations become cumbersome. A reasonable solution is then to chose the average plane as reference plane, but to that use the analytical continuation of the field in one of the media, for example the first Born
n ,
above the interface.
C
Jean Daillant and Anne Sentenac
134
(s)
or
(p) polarisation: Ee
,c
det
k2e-ikOR 0
(R, r’)
4ri coR 2
k oe
-ikoR _
-
4rcoR E
eikscj.r’ &Sc,
tscl 0,
(4.39)
,
EjPW (kin z, 13 /) 6- ikjj.r’jj ; in
Ein =
Epw(-k,,,,l,z)e’’k 11-r1les, 1
-
z
in
Ein
toj
e- ikj.,j.r’ ’6i.
(4.40)
1
where t" and tsc
are the Fresnel transmission coefficients for polarisation (8) respectively the angle of incidence Oin and the scattering angle in the scattering plane Os,. Explicit expressions for those coefficients are given by equations (3.71) and (3.72). Putting Eqs. (4.39), (4.40) in Eq. (4.11) and following the same treatment of the integrals as in Sect. 4.2.2, we obtain a generalisation of Eq. (4.35):
for
(dQ)incoh do-
A
- L (n 167r2
X Cos
20
-
2 1
-
n
2)2 0
Itin,12 Itsc,12 0, 0,
e
.L(q2’j+q*2’1 (Z2) 2
-
iq,,112
1 dRjj 16 jq,,1j2(z(Rjj)z(O)) I Ciqjj.Rjj _
I
.
(4.41)
Eq. (4.41) differs from Eq. (4.35) by the additional transmission coefficients. This
expression is explicitly symmetrical in the source and detector posirequired by the reciprocity theorem. At the critical angle for total external reflection Oin 0, the transmission coefficients in Eq. (4.41) have a peak value of 2. The electric field is then at tions
as
--
its maximum value at the interface because the incident and scattered field 0. As the dipole source equivalent to roughness Co6n 2 E phase at z is proportional to E, there is a maximum in the scattered intensity. By using the reciprocity theorem, one can see that the Green function is also peaked near Osc 0,.’o Those peaks are the so-called Yoneda peaks [19]. They can be seen on Fig. 4.4. are
in
--
=
4.3.2
General Case of
In the
general
layer j
for
(s)
case
or
of
a
Stratified Medium
stratified medium
depicted in Fig. 4.2,
one
has
in
(p) polarisation: k 02e- ikoR
E’dlelt (R, E (r)
a
47rEoR "Zz
Efw(-k,,,,, ’i z’)e ik_ 11.r’jj-, s,:j 3
W Ein Ef (k-in z,31 3 .
Z/)e-ikj,,jj.r’jj-i ,nj.
Equivalently, the peak in the Green function can be dependence of the field emitted by a dipole placed
seen
to arise from the
below the interface.
(4.42) angular
4
Diffuse
Scattering
135
no
zi n
Z2
nj_j
zj (X,Y)
Z.
zi
nj j+1 n
N
ZN+1 Fig. 4.2. X-ray surface scattering in a stratified rough medium. Because of multiple reflections, there are waves propagating upwards (with an amplitude U(k,,j, z)) and downwards (with an amplitude U(-k;,,j, z)) in layer 3 where the total the field amplitude is EZ (there is an equivalent dependence of the Green function). in,3 Multiple reflections are considered within the DWBA but not within the first Born approximation. The perturbation method consists in evaluating the field scattered by the dipolar density equivalent to the index difference (nj 1 nj) between the real system where the rough interface profile is zj(rll) and the unperturbated system where the interface is located at Zj, and is placed here at the average interface plane. For interface J* the unperturbated and real index distributions differ in the hatched region -
-
The DWBA method consists then in in
Eq. (4.17) PW
Ej,
in each medium as, for
(’)
(k
,
31
Z)
U (’)
developing the EPW functions defined example in (s) polarisation, in layer j:
(k ,j) Zj )
U
ik; ,
e -j’
+
U()(-k ,, zj)e+ik ,jz ,
k ,j, Zj)jik ,jz,
(4.43)
magnitudes of the upwards and downwards are explicitly obtained in Chap. 3 of this book, Eq. propagating waves Vincent" representation of tranfer matrices [20]. and (3.48) using the "Vidal
where the U coefficients
are
the
which
The field is then written
(put Eqs. (4.42)
in
Eq. (4. 11) and
sum
over
all
Jean Daillant and Anne Sentenac
136
interfaces):
E(’)
--
E,, f + Ej "
J
(r1j)
z j+j
k2e-ikoR 0
dril eiqlIxII
47rEOR
j=O
dzE-O(n +j
-
3
0
1;,PW n2) i _j+I (ki,,,,,j+,, z) -’,’Pw(-ks,, 3+1
Z),
(4.44) where it has been assumed that the reference
interface, hence the EPPW 3+1 fields, zj+l of Eq. (4.41) is: dodS2
)
N
N
j=1
k=1
k4 ’0 DT7r2
incoh
plane is located above the negative). Then, the generalisation
is
U(’)(ki,,.,,j, Zj)U(’)(k , ,j, Zj)U(’)*(kj,, ,,j,, Zk)U(’)*(k,,,_,,k, Zk) Qj,k (kin zj
k , zj, kin z,k
k.,c z,k),
(4.45) with
(n
Qj,k (qz, qz)
3
n _ 1) (n 3
0
2
k
n
2
k
_1)* COS2
dril
j dr’l
1e
iqjj.(rjj-ir’jj)
(fzj(rll)
0
0
dz’e i (q;, z q,,,* z’)
dz
-
( fk(r’jj)
dze’.q,,
z
0
dz’e- iq.,"zl
(4.46) specular (coherent) contribution, obtained as an average over the Chap. 2, has been removed. Performing the integrations over z’ and making the change of variables r’ll RII as previously: r1l
where the field z
as
and
shown in
-
Qj,k(q ,, q’) z
=
A-
W3-
-
0 3
-
1) (nh2
_
n2_j* h
qz qz
dRii
e
iqjj.Rjj
COS
2
Oe- [qz2 (Z+ 2
2
(Z2)] k
(eq.,qz’*(zj(O)zj,(Rjj)) (4.47)
Because reflection at all interfaces is taken into
account, all the possible
com-
binations of the incident and scattered wavevectors appear in the formulae.
4
Diffuse
Scattering
137
100
.911
on a liquid surface as a function reflectivity curve. Note that in the case represcattering is peaked in the specular direction. The roughness spectrum (Fourier transform of the height-height correlation function) is directly obtained in a q., scan at constant q,, The possible extension of q., scans increases
Fig. 4.3. Intensity
scattered
of q., et q-.. I(q,, q , = sented here, the diffuse
0)
by
a
2nm thick film
is is the
with q-.. Out of this accessible range (black surface on the figure), either the detector or the source would move below the interface. Note the Yoneda peaks near the
extremety of q., scans, where 0j.
or
0,, is equal
to the very small critical
angle
for
total external reflection
Particular Case of
4.3.3
a
Film
particular case of a film has been considered in Ref. [2] where an explicit expression for the scattering cross-section as a function of the reflection and transmission coefficients has been given. In practice however, it is more convenient to write a program using Eq. (4.45) whatever the number of layers. In a rough thin film scattering occurs at both the film surface and at the film-substrate interface. The field at the film surface is proportional to 1 + r where r is the film reflection coefficient, and the field at the substrate-film The
interface is
depend
on
proportional to the film transmission coefficient 0ij et 0,,,, Eq. (3.75), (3.76): 2ik ,,jd roj + rl,2e2ik, +
r0,17’1,2e_
where rj,j+j ti,i+l i1i + 1, d the film Far from
an
reflection,
Then,
the Born
Green
function)
Therefore,
3.
r
and. t
d
(4.48)
+ rojrj,2e -2ik.,,ld
the reflection et transmission coefficients of interface
thickness,
incident
ternal
face.
are
10,1t1,2e_ ik.,,
; t
t. Both
or
and
exit
k,,, depends
angle
the reflection coefficients
approximation
is
on
0j,,
or
0,
close to the critical
valid,
are
angle
small and
and the
can
amplitude
for total be
ex-
neglected.
of the field
(or
is I at the upper interface and e- ik,,,ld at the lower interneglecting the polarisation factor, the scattering cross-section
138
Jean Daillant and Anne Sentenac:
will be
(compare
Eq. (4.C3)
also to
4
do-
ko 167r2q2
dQ
2
+(,17,2 Following
the usual
( dQ)incoh [(n,
appendix 4.C):
j drjj [(n
z,
2 1
n
_
ei(ksc, -k1nz)Z1 2) 0
nfle one
obtains-
I
(compare
A
2)2
-no
2
+(n2
e
-qf (Z:2L
dR11
(e q,’,(zi.(O)zj(R[j)) 1)6 jqjj.Rjj
dR11
(eq
2)2 e q2(,2) -
2
.
-ni
2
+2(n2
-
2) (n,2
nj
_
2
(Z2 (O)Z2(Rjj))
2)r- Lq 2(Z2)
-no
2
-
2
_
_
j dR11 (e qz2(zj(O)Z2(Rjj)) principle,
Eq. (4.CIO)):
to
167r q.
2
In
(4.49)
iqjj.rjj2
i(k,,: ,-kjz)(d+z2)j
2
-
procedure,
do-
in
_,
1)
lqz2(Z2) 2
f 2
z
_
e
jqjj.Rj1
(4.50)
cos(qd)
1) Cjqjj.RjjI
the correlation between the different interfaces
can
therefore be
determined because the contrast of the interference pattern directly depends on this correlation and the different contributions may be separated [21].
substrate, the substrate rougha film. Then, qz scans at conspectrum stant q, can be performed with the film. They consist of an oscillating and a non-oscillating component. The contrast of the interference pattern yields For
example, considering can
ness
single
a
film
on a
be first measured without
the cross-correlation between the film-substrate and film-vacuum interfaces at the
given
q, and. the
non-oscillating part yields
the
sum
of the film-substrate
and film-vacuum auto correlations. All the relevant correlation functions
therefore be determined. In
a
multilayer,
can
similar constructive interference ef-
interfaces, in Sect. 8.5. scattering" leading In addition to the Yoneda peaks, other dynamical effects can be observed in the case of a film (or better of a multilayer, see Sect. 8.5 for a thorough discussion). Because the reflection and transmission coefficients Eq. (4.48) depend on both the incident and scattering angle if the reflection coefficients are not too small, the scattered intensity can show oscillations with a characteristic period depending on the film thickness,11 even if both angles vary in a scan so that qz is kept constant (Fig. 4.4 left). This dynamical effect cannot be accounted for within the first Born approximation. More generally, similar dynamical effects will occur whenever the field is modified at an interface due
fects
can occur
between the beams scattered at different conformal
to what is called "resonant
Note that because of thee -2ik;,
associated to the cross-correlation
dfactor in
the reflection
coefficient,
the
periodicity
effect is different from that associated to the interface
dynamical term, allowing
to
distinguish
between them.
Diffuse
4
multiple reflections (see Fig.
to
right
4.4
for
are
not accounted for within the first Born
are
given
Scattering
139
example). Again such effects approximation. Many examples an
in Sect. 8.5.
10-1
10-1 10-1 5 2
10-7 5
10-6 0
8
.P
10-1
128 0 5
0
120-
0 0
.10-1
z,
-
I 0-
10-1
10-1
-
-
d-
0
o
10-5
o
10-1
1
%
10
1.0
0.5
0
-0.5
1.0
-6
-0.8
(X10’ m-)
Fig.4.4. Dynamical
0.4
0
0.8
(X 101)
effects.
Left:
"Rocking
(polystyrene-polymethylmetacrylate film is 18.9nm thick. Note in
-0.4
diblock
particular
curve"
copolymer)
the Yoneda
peak
its structure is related to interferences in the film
I(q,,) on a
for
at 0.75 X
(inset).
In
a
polymer
film
silicon substrate. The
10-8 m-1
particular,
’and that the oscil-
by arrows correspond to the dynamical effect discussed in the text. Right: "Rocking curve" I(q.,) for a CdTe multilayer (20 layers). The height-height
lations marked
correlation function is
(z(O)z(x))
=
a
2
exp
-
[Xg]2v
and the interfaces
are
assumed
0.25nm, fully correlated. The parameters used in the calculation are a v 300pm. The grey curve (divided by a factor of 2 for clarity) corre0.6, sponds to the first Born approximation and the black curve to the DWBA described in the text. Note the peaks in the DWBA intensity which are a dynamical effect to be =
and
=
=
occur
when the field is maximum at the different interfaces
Jean Daillant and Anne Sentenac
140
Polarisation Effects
4.4
According
to
Fig. 2.3,
the
polarisation
vectors
0
_’ ii(n
are:
sin
Oin
Cos
oin
in
0 -
sin
-
(P)
Cos
SC
-
SC
sin Os,
cos
(4.51)
0,,, sin Cos OSC
sin
0
-e(’) and scattering angles ’ is(c) in and almost are equal. (p) polarisation (s) reflection coefficients for the perpendicular a single interface:
At very small incident and and the EPW functions for
example, the ratio of the parallel polarisations is for
SC
-_
’r(P) where
00 and 01
are
with the surface. In
I +
angles between the incident and the general, at least one of the angles Oin
(p)
and
refracted beam Or
Os, is small,
reflection coefficients is also
generally not very important, but they Firstly, subtle effects can happen under conditions close to those responsible for the structure of the Yoneda peaks discussed above [4]. More importantly, polarisation effects must be taken into 10 account whenever the incident or the scattering angle are larger than degrees. This is the case when one tries to get information at small (atomic or molecular) lengthscales. The treatment of polarisatio’n effects will be different within the Born and distorted wave Born approximations. Within the Born approximation (see Sect. 4.2), the polarisation dependence is easily included in the differential scattering cross-section Eq. (4.26) small. can
Thus, polarisation effects
(s)
and
(4.52)
20o0i,
the,
and thus the difference between the
in
For
be noticeable in
some
are
instances.
du
k4
dS?
167r
2 .
2
02
dr
n 2 e iq.r
only dependence is through the scalar product ( in -’ 3sc). There is only simple, geometrical depolarisation corresponding to the projection of the incident polarisation on the final one. Generally, the polarisation of the scattered beam will not be known a priori, but we can calculate the relative scattering into (s) and (p) polarisations using Eq. (4.26). Then, since the a
dodQ
)
do- () _
tot
dQ
do- (P) +
dQ
*
(4.53)
4
Diffuse
Scattering
141
DWBA, we must always decompose the incident and scattered into (s) and (p) polarisations because the EPW functions depend on the polarisation. Moreover, for (p) polarisation the orientation of ; i,, ahd’ ., will differ from layer to layer because of refraction. Taking these two requirements into account, the scattering cross-section can be calculated using Eq. (4.53). A simple case" is that of the scattering of a (s) polarised incident wave into a (s) polarised wave since a unique cos’ 0 polarisation factor can be used in all the layers. This is the case which was considered in Sect. 4.3 for simplicity. Within the
field
Scattering by Density Inhomogeneities
4.5
Only surface scattering has been considered up to this point. However, the inhomogeneities leading to scattering can also be density fluctuations. This should always be born in mind when interpreting experiments. The scattering due to density inhomogeneities in a multilayer or at a liquid surface can be treated using a formalism similar to that used for surface scattering. The relevant correlation functions will be of the form (Jp(O, z’)Jp(rjj, z)). This problem was considered in the early paper of Bindell and Wainfan [22]. Again we limit the discussion to the scattering of a (s) polarised incident wave into a (s) polarised wave. dielectric index
4.5.1
Density Inhornogeneities in
a
Multilayer
are assumed to be perfectly smooth in this analysis. Within 13 DWBA, and assuming effective U functions within the layers, the differential scattering cross-section will be (cf Eq. (4.45)):
The interfaces the
4
do-
ko _
dS2
167r 2
N
N
J:EEj:ET U(’)(ki ,,,j,Zj)U()(k,;c ,,j,Zj) j=1
k=1
U(’)*(kinz,k)Zk)U(’)*(k,,c 12
13
Zk)j3Jj, k ( kinzj k,c ,j, ki,,z, 1,
k) (4-54)
kcz
example the case of horizontal scattering on a horizontal surface at a synchrotron source. Assuming effective U functions within the layers is only possible if the characteristic size of the inhomogeneities is much smaller than the extinction length, see appendix IA to Chap. 3. This might not be the case for multilayer gratings (see Sect. 8.7) or large copolymer domains [23]. This is for
Jean Daillant and Anne Sentenac
142
where
now
Bj,k (q,, q ,,)
-_
cos
2
drjj
j dr’jje
Z
zj+l-zj
JO
qll -(’11"10
I-Zk
dz’
dz 0
(6nj2’(rjj, z)Snk* (r’ll,
z
C
i(q,,z-q,,,*z’).
(4.55) Making
of variables r1l
change
the
Bj,k (qz, q,,)
A
--
cos
2o
I dR11 (
c
In the
case
U, (- k,
z,
of
a
semi-infinite
1, Zj)
=
Vc
are
-
r’ll
iqjj.Rjj
-4
JO
R11: Z
zj+’-Zj
1-Z
dz’
dz
(4.56)
0
Jn? (0, z) Jn k2* (1111, z’) j
e
i(q,,z q,,,*z’) .
t" and medium, only, U, (-kin, z, 1, Zj) Re Writing qz, 1 (qz, 1) +ilm (qz, 1), --
different from 0.
=
obtains:
one
B1, 1 (qz, 1,
qz, 1)
-_
A
cos
2o
I dR11
e
iqjj.Rjj
0
0
f foo
dzdz’
e
i7Ze(q ,,.)(z-z’)Clm(q.,,,)(z+z’)
Jn 2(0’
z)Jn
2*
(1111, Z’)
.
(4.57) i.e.,
the bulk fluctuations
are
integrated
over
the
penetration length
of the
beam.
Comparing Eq. (4.57) to Eq. (4-47), we note that contrary to bulk scattering, surface scattering is inversely proportional to the square of wavevector transfers. Therefore, surface scattering will generally be dominant at grazing angles whereas bulk scattering will ultimately dominate at large scattering angles (see Chap. 9, Sect. 9.3.1 for an example). Density Fluctuations
4.5.2
at
a
Liquid
Surface
interesting case is that of density fluctuations at a single liquid surface because an analytical calculation can be made. The liquid extends from -oo to 0 in z, and its vapor can be taken with negligible density. Eqs. (4.54),(4.57) above give: An
k
do-
dQ
167r
0
0
4
_"O
A(I 2
-
n
COS2 0 2) 2 Itinj 0, 12 0, 12 Itscl
f. f
dz’
dz
e
iq,,,Iz e- i q,,*,, z’
00
(Jp(O, z’)Jp(Rjj, z)) iqjj.Rjj e dR11 P2 (4.58)
4
where
we
have used
n
-
lation function for bulk
I
-
(A 2/21r)r,p. Inserting
liquid
-
-1/V((9V/OP)T do-
4 k 110
dfl
167r2
Further
4.6
A(I
the
Scattering
density-density
143
corre-
fluctuations
(p(r)p(r’)) where NT
Diffuse
=
P2 kBT KT6(r
is the isothermal
-
n
2)2
Itin12 ItscI2
-
r’),
(4.59)
compressibility yields, kBTr-T Cos
21m
2o.
(4.60)
Approximations
always intensity close to the critical angle for total external reflection [24]. Understanding scattering at grazing angles is highly desirable because bulk scattering is minimized under such conditions. This is critical because the signal scattered by surfaces or interfaces is generally very low. Differents approaches have been attempted to improve the DWBA. Only the first one has been extensively investigated. This approximation consists in taking into account the average interface profile in Eq. (4.11) [4]. The reference medium is now defined by the relative I/A f c(q) z) drjj in the case of a permittivity Cref W (6) W with (c) (z) random defined surface an by ergodic process. (In [3] the shape of the rough is an hyperbolic by tangent profile to simapproximated average permittivity main interest of this The Green the of reference the calculation tensor). plify
approximation enough representation
presented
here does not
The Distorted-wave Born
as
allow
of the scattered
an
accurate
--
new
--
reference medium is that the reference field Eref is that of the transition
directly the N6vot-Croce factor in the reflection co0 and perturbation 6n’(rll) is of null average (6n 2) we may expect to have minimised its value (and thus extended the validity domain of the perturbative development). This improvement has been shown to Yield much better results than the classical DWBA in the optical domain where the permittivity contrasts are important [25]. In the x-ray domain its interest is more questionable since it does not lead to simple expressions for the scattering cross-section. Indeed 6n 2(r1j) is no longer a step function and the integration along the z-axis cannot be done analytically. Another possibility would be to directly take into account multiple surface (roughness) scattering without using the effective medium approximation. It is then necessary to iterate the fundamental equation (4.11)[4], [26-28]. This has been done up to the second order in Ref. [5] for specular reflectivity, and the corrections might be important close to the critical angle for total layer
and thus. contains
efficient. Moreover the
=
external reflection.
Finally there exist many approximate methods that have been developed totally different contexts (optics, radar). In most methods, the field scattered by the rough surface is evaluated with a surface integral equation (given by the Huygens-Fresnel principle (or Kirchhoff integral) [29]). The integrant
in
144
Jean Daillant and Anne Sentenac
of the latter
contains
the field values and its normal derivatives at each
point approximation consists in replacing the field on the surface by the field that, would exist if the surface is locally assimilated to its tangent plane. This technique, when applied to the coherent field yields the famous Debye-Waller factor on the reflection coefficient. It is a single scattering approximation ( also called Physical Optics approximation). The perturbative theory (the small parameter is the rms height of the surface) has also been widely used. A possible starting point is writing the boundary conditions on the field and its derivative at the interface under the Rayleigh hypothesis. A brief survey of this method is given in Sect. 3.A.1. Note that the iteration of these methods permit to account for some multiple scattering effects, but the increasing complexity of the calculation limits their interest. It is now also possible to consider the resolution of the surface integral equation satisfied by the field without any approximation (and thus to account for all the multiple scattering). Preliminary results have been already presented in the Radar and optical domain. However in the x-ray domain those techniques have a major drawback: They only consider surface scattering (with a surface integral equation) and-the generalisation to both surface and volume scattering is not straightforward. The differential method [30] which consists in solving the inhomogeneous differential equation satisfied by the Fourier component of the field (in the k1l space) with a Runge-Kutta algorithm along the z-axis would be more adequate. It has already been used to calculate the diffraction by multilayer gratings and accounts for all multiple scattering (no approximation), but it remains difficult to use it for non-periodic (rough) surfaces because of the computing time and memory required. of the surface. The Kirchhoff
4.7 4.7.1
The Scattered
Expression
Intensity
of the Scattered
Intensity
specular reflection where the specular condition J(q,,,) implies that resolution effects amount to a simple convolution, the scattered intensity is proportional to the resolution volume for diffuse (incoherent) scattering. In order to achieve quantitative information from an experiment, it is necessary to measure (and calculate) absolute intensities, and therefore to have a detailed knowledge of the resolution function. The diffeFential scattering cross-section must be integrated over the detector solid angle Qd and the incident angle angular spread G AOin in the vertical and ’ Oy perpendicular to the incidence plane) (Fig. 4.5). Assuming a beam cross-section 1, x ly and a Gaussian angular distribution of the incident beam intensity: In contrast with
1C)e of width
incidence
so?
6 02
2.AO2
2AO2
i.
Y
zA0i,, in the plane of incidence and AOY normal to the plane of (Fig. 4.5), the scattered intensity ID is, using the definition of the
Diffuse
4
cross-section
scaftering
flux in direction
1D /-10
I
dMi. dS OY
-
X
unit solid
angle for
145
-
unit incident
a
ki,,):
-
IX
(power radiated per
Scattering
ly
27r, AOil,
X
’ Aoy
e
Be?
5,92
24,9? 2 19i.
2A02Y
do
Y
dS?
dQd.
(4.61) The normalisation factor
defined for
a
1/(I.,
x
1y)
is
required
because the cross-section is
unit.incident flux whereas the scattered intensity is normalized
to the total flux.
Since each experimental setup is different, it is impossible to give expression for the angular dependence of the resolution function. examples can be found in Chaps. 7, 8 and 9.
general
Different
Incident beam
Detector
AQ
a
d
OuridUe
Fig. 4.5.
Definition of the
angles
and solid
angles
used for
calculating
the scattered
intensity
The cross-section is defined
might the
be found
use
of
a
more
as a
convenient to
function of wave use wave
vectors;
and sometimes it
vectors instead of
wave-vector resolution function is very delicate
angles.
In
fact,
(see Ref.[31])
14
absolutely needs an analytical expresview, a numerical integration of Eq. (4.61) which very generally reduces to a multiplication with the detector solid angle is much more preferable. For this reason, we only give here a brief account of and should be avoided except if a computing point of
one
sion. From
how wave-vector resolution functions
The major into
a
problem
can
be dealt with.
is that the transformation of the
wave-vector resolution function leads to
separable
in q, and q,,
(hence
the
projection
in
a
angular
resolution function
function which is
Fig. 4.6).
generally
not
146
.
4.7.2
Jean Daillant and Anne Sentenac
Wave-Vector Resolution Function
Since a rigourous transformation of the angular resolution, function into a separable wave-vector transfer resolution function cannot be made generally, we use here a more simple approximation where only the resolution volume is conserved in the transformation, see Fig. 4.6. According to Fig. 4.6, close to the specular, a factor
60j" ’ Ao"
2 X
) o qZO
n
Osc’ Oin)] (and
must be introduced for resolution volume
ables, whereas there is no factor 2 if for scattering close to’the specular, an
diffuse
(in general zAO,
ly V2_7r, Aojn J
I
-TD / 10
conserva-
angular to wave-vector variexample Oin < 0,,. Considering approximation of Eq. (4.61) is: do-
2zAOi,,
I
1,
x
60in)- R(6q)
(4.62)
d6q R(6q)
-
-
>
intensity)
therefore
the transformation from
performing
tion when
k2oq,AO,, dQ
is the resolution function in the wave-vector
space.
Considering
for
1/10
k3 o =
the
simplicity
case
of
a
single rough interface,
(4.41), integrals
compute instead of Eq.
-
2)2
2
7r2 (7z Sin Oin
(n,.-
no
ItscJ12 Itin112 01 0,
e
--1(q2,j+q,,*211 )(z2) 2
I dJqll I dR11 [ejq ,jj2(z(Rjj)z(O)) 11 V2_7rko, AO,
integration since the
over
Jqz
integrant
resolution function is therefore:
tegration
over
6%,
yields
fZ,
-
R(R11) where
Aq_-NI -Log2
and
--
has been does not
7Z(Jqll)
=
C os
jqz, 112
_
where the
must
we
now
of the form:15
2
0
(4.63)
R(Sqjj)e i(qjj+6qjj).Rjj
replaced by
significantly e- 1/2
a
factor
vary
over
V2_7r, A qz Aqz. The
Jq,2 /Aq2-1/2 &qylAq 2Y. The in-
the Fourier transform of R:
27r,6_q.,zAqye_ 1/2(,dq2X2+,Aq2Y2) X
/AqyV/2_Log2
are
Y
(4.64)
the half-width-at-half-maximum of
According to Eq. (4.64), 11zAqx and 1/zAqy represent the coherence lengths along x and y, i.e. the lengths over which correla16 tions in the surface roughness can be observed, as discussed in Chap. 2. the resolution function R.
equation also shows that it is important to correct for the illuminated area, Chap. 7 and in particular Sect. 7. L In the case considered here, normalisation to the incident intensity instead of the incident flux in the differential scattering cross-section leads to the I/ sin Oi. factor in Eq. (4.63). For a typical experiment where the resolution is mainly determined by a slit of 0. Imm) x (hy H x V size (h,, lomm) placed in front of the detector, Im away
This see
16
=
=
Diffuse
4
Scattering
147
(b)
(a) Aq,
Aq,, T Z
Aq,
Aq,
AO
in
AO X
Fig. 4.6. Resolution surfaces in the plane of incidence (thick parallelograms) and their projections onto x and z in the wave-vector transfer space (grey rectangles) for scans in the plane of incidence. The parallelograms are obtained by convoluting the incident angular spread by the detector angular spread. With appropriate scaling (multiplication by ko), all points within the parallelogram correspond to possible final wave vectors, and the wave-vector transfer can be obtained through a -kin translation. Due to (x, z) coupling, the resolution area cannot be simply expressed as a function of Aq, and .6q__ In particular, because of the projection onto x Aqx x Aqz $ koq ,, AOi.AO, where koq, or z in the definition of Aqx and Aqz, is the Jacobian of the transformation. The approximation used here consists in using Aqx, Aqz, with an appropriate factor depending on the geometry for area (i.e. intensity) conservation. (a) Reflectivity geometry. The area of the parallelogram is 1/2ko q,, max (,60in lAOs,, 60 /,Min). (b) grazing incidence geometry. The area of the parallelogram is k 02q_ max(,!AOin/,AOsc1 AOsc/ZAOin)- In this last case, there is no coupling, and the transformation can be done via a straightforward Jacobian transformation. For scans limited to the plane of incidence as here, the slit openings normal to this plane only introduce an additional koA, in the resolution volume
intensity scattered by finally be written: The
a
interface close to the
single rough
specular
can
10mrad and a wavelength sample, for a typical scattering angle O , 10-2 rad in the horizontal O.Inm, the acceptance of the detector is ’AO’ 10-4 rad in the vertical. The coherence lengths along x, y and z are and AO-. lpm. This is 0.01pm, and A/AO,, 100mm, A/, AO, respectively A/O,,AO,
from the A
=
=
=
=
along of height
which allows the measurement
or
lengthscales.
incidence geometry correlations over such a large
due to the
x
range of
=
=
=
the enhancement in the coherence
density
grazing
148
Jean Daillant and Anne Sentenac
3
1/10
k0 2
87r q_, X
-
2
-(n 1 sin On,
I dR11 [e
In the limit of small qz
k0
1
10
81r 2 qz sin Oin
(n,
2)2 Itin, 0, 12 Itscl 0, 1
2 e
-
develop
one can
I
1 q ,, 1 12
)(Z2) Cos
2
116iq11.R11 ’&(R11).
the
v)
(4.65)
exponential in Eq. (4.65),
’in COS2 2)2 1 to, 1 12 1 tscl 0, 12
no
-
1(q2 2 -, +q,2
n0
lq,,, 12 (z(R11)z(0))
2
-
-
"
z(%)z(-q11)) (D R(qll). (4.66)
The scattered
roughness 4.8
intensity
is then
simply proportional function.
to
a
convolution
of
the
spectrum with the resolution
Reflectivity
Revisited
2 intensity decreases as qz for small q,, values, whereas it was shown in Chap. 3 that the specular (coherent) 4 intensity decreases as q,- One therefore expects that diffuse scattering will eventually dominate over the specular reflectivity. Of course the wave vector at which diffuse scattering becomes dominant will depend on the experimental resolution since the diffuse intensity is proportional to the resolution volume. In fact, for reasonable experimental conditions, the corresponding wave vectors are rather small, on the order of a few nm-1, and this leads to major difficulties in the treatment of reflectivity data. A "reflectivity curve" -I(q,,) is indeed never a pure specular reflectivity curve. Moreover, the diffuse intensity is often (but not always) peaked in the specular direction (Fig. 4.7) making the separation of the specular and diffuse components very difficult experimentally. This is a very difficult problem since the qz dependence of the diffuse intensity depends on the exact interface correlation function. A simple model can therefore no longer be used for the analysis of "reflectivity" curves. This is the situation found for the system of Fig. 4.8, an octadecyltrichlorosilane Langmuir film on water [32]. In this case, the surface spectrum can be calculated, and the specular and diffuse contributions to the reflectivity can be compared. The roughness spectrum (here thermally excited capillary waves) is obtained from thermodynamic considerations by Fourier decomposition of the free energy, see Chap. 9:
Equation (4.66)
above shows that the diffuse
.
(z(q11)z(-q11))
kBT
I -
L,
x
Ly zApg
2
4
+ -yq + Kq 11 11
(4.67)
L., x L. is the interfacial area, -y is the surface tension, and K is the bending rigidity modulus. The correlation function can be obtained by Fourier
Diffuse
4
Scattering
149
transformation.
(z(O)z(rjj)
--
kBT121r-y
[Ko(rjjVZ21p-g1 )
x
-
Ko(rjjV7--1K)],
(4.68)
where KO is the modified Bessel of second kind of order 0. Then, for a wave vector resolution Aq.,, the intensity measured in the 0,,, = Oi,, direction is smaller than the
7r- 1/2r
[2 I
reflectivity kBT q2
.
47r-y
z’
I
2
of
perfectly
a
Aq2
’
KI
x
flat interface
exp-
7
1
27r^/
In
by
(
a
factor:
e--1--
V/,-Y-/I,(
A/2-
Aqx
)I
4.69)
incomplete -V function, and -yE Euler’s constant. Note that _,2 ’ because diffuse scattering has been taken larger than e -q 2, in addition to specular reflectivity.
where F is the this factor is
,
into account
10-1
1
q7=1.5nrn-1 10-1
.11u=1t(n,) (0ff -1
z x
t=30nm
v=0.3
10-1 .S x
__P
-
-
.1
10-1
-
water
io-10
10-1 1
y=0.073 Nrn-1
11
0.03
L 1
L
-0.02
-0.01
0
0.01
0.02
0.03
% (nm
scattering from the water surface which is peaked in the specular capillary waves of longer wavelength cost less energy (only a calculated intensity is presented here because the large background due to bulk scattering prevents from a precise measurement, see below the chapter on liquid Fig. 4.7.
Diffuse
direction because
surfaces),
and
It
be
fuse
can
a
solid surface with
seen on
intensity
Fig.
a
4.8 that
flat power spectrum
even
for
relatively small wavevectors the difpossible to obtain physically
dominates. It would not have been
150
Jean Daillant and Anne Sentenac
101 10-1 10-2 10-1 10-4 ’I-10- 5
10-6 10-7 10-1 10-1 10-
V
10 r
0
2
4
q,
6
1
8
(nm-1)
Fig. 4.8. Reflectivity of an octadecyltrichlorosilane film on water. The broken line corresponds to specular intensity. It is dominated by diffuse intensity (grey line) for wavevectors larger than 2nm-’. The black line is the total (specular + diffuse) intensity. Inset: corresponding electron densities for the complete model Eq. (4.69) (thick line) and the simple box model with error function transition layers (thin
line)
reasonable parameters from the into account (see Fig. 4.8).
4.A
experiment without taking its contribution
Appendix: the Reciprocity Theorem
permittivity Cref (r) which is region of space. Let two different current distribution sources JA, JB (with same frequency w) be placed in this medium. We denote by the indices A, B the fields created by these sources, separately, in the medium. They satisfy Maxwell’s equations, We consider
a
medium described
assumed to be different from I in
V
X
V
X
EA(B) HA(B)
by
a
the relative
localised
=
=
-iwBA(B) JA(B) + iwDA(B)) (4.Al)
Diffuse
4
DA,(B)(r)
where
the Maxwell’s
--
COCref(r)EA(B)(r)
equations
’7.(EA
X
HB
-
and
X
HA)
HA(B)/Po.
--
151
Substituting
identity,
in the vectorial
EB
BA(B)
Scattering
HB.V
x
EA
EA.7
x
HB
HA-V
x
EB + EB.V
x
HA,
-
(4.A2) leads to V. (EA
X
HB
-
EB
X
HA)
:--
EA JB + iw (EB.DA
EB JA
+iw(HA.BB
The last two terms
on
V.(EA
X
Integrating Eq. (4.A4)
I and
d3,r,7. (EA the
using
now
X
HB
-
-
HA)
X
all space
over
HB
EB
-
FIB
(EB-JA
=
-
that
get,
we
EAJB)
EA.DB) HB.BA)(4.A3)
(4.A4)
-
gives
HA)
X
are zero so
-
I d3r (EB
-
JA
-
EA JB)
)
theorem
EB
x
HA)
-_
I
d3r (EB JA
-
EA JB)
-
(4.A5)
sources are limited to a finite volume, the surface of inteEq. (4.A5) is infinitely remote from them, and the electromagnetic be approximated by a plane wave with E and H orthogonal and
the current
gration field
HB
divergence
I d2 V(EA If
X
side
right-hand
the
-
in
can
transverse.
H
It follows
FW60-0-
fi
X
E.
that, EA
which
=
X
HB
-
EB
X
HA
-
0)
yields d 3rEA-JB,
d3rEB-JA
reciprocity theorem [12-14]. Eq. (4.A6) can (Iliw)J, one gets dipole density sources through P
which is the
(4.A6) also be written for
--
I d3rEB-PA jd3rEA-PB-
(4.A7)
152
Jean Daillant and Anne Sentenac
4.B
Appendix: Verification
of the
the Case of the Reflection
Integral Equation
by
Thin Film
a
on
in
a
Substrate It has been indicated in the niain text that the
integral equation Eq. (4.11) by applying the reciprocity theorem is an exact equation. In this appendix, we verify that this is indeed the case for a single film on a substrate. The reference situation an homogeneous medium of optical index no and we obtained
want to calulate the electric field in the
thickness d
the smooth film
r
are
2ik ,id roj + rl,26-
-_
ro,jrj,2e-2ik,,,1d
differs from the ideal
case
where there is
case
a
(1)
film
of
The reflection and transmission coefficients of
respectively, Eqs. (3.75), (3.76):
I +
The real
(2).
substrate
on a
10,14,2e_ ik
t
’
+
one
ro,jr1,2e-2ik ,1d
within the substrate where the
fractive index difference between the real and ideal 2
(n 1
the film where the difference is
_
n
2). 0
,jd
In
(n 22
is
case
_
n
2) 0
re-
and in
Eq. (4.11), we need the field in substrate, and is
the real case, which is the transmitted field in the PW
El
(Z)
to’l
=
1 + rojrl,26 2ikj_,jd
[e-illin,,11
in the film. We also need the Green function in
koe- ikoR 47r,EOR
Using Eq. (4-11),
the electric field
-
E
--
6
Eo +
ikoR
j drjj
47rR
can
-
10 koz ) f
e- k_’0Z
ko2z)
,
vacuum
I
(medium (0)),
ik_,0.r
be written:
jqjj.rjj
d
2
(klz
e
+ rl,2 e-ikj-,1(2d-z)
10, (e-ikin.,iz +
ikjn.,j(2d-z)
r12
+ roirl2e- 2i’kin.,Id
dz
d
2
+(k2z In
Eq. (4.Bl),
no2) to
=
kz2, j
ensure
-
we
kz2, 0,
e- kscz ’0zte- ikin.,2(z-d)
2
-
2,
have used that k z
Medium
(2)
The differential
-
Eo
e-ikR -
scattering
k?-k 2
=
is considered to be
the convergence of the
E
]1.
(4.131)
00
47rR
integration.
n k 02
k2
X
implies
k 02(n
slightly absorbing in
-
3
order
One obtains:
j drjj e’q11.rjj (2iko
cross-section is
-
thus:
sin
Oo r).
(4.132)
Diffuse
4
b2
do-
-0
And
we
Ir 12 Sin2
47r2
dQ
k 0 sin
2
00 1r 12Aj(qll).
153
(4.133)
find for the reflection coefficient:
R
as
2
Oo472Ajk- jj,kj ,,,
Scattering
LdS2
A sin 00
12
M
(4.134)
expected.
Roughness in Approximation
Interface
Appendix:
4.C
Within the Born
Multilayer
a
appendix we treat the case of a rough multilayer within the Born approximation in order to show some simple properties of the scattered intensity. In the case of the rough multilayer depicted in Fig. 4.2 Eq. (4.26) gives: In this
do-
ko Cos2
dz
drjj
167r 2
dQ
2
(n i
-
1)e iq.r
2
(4.Cl)
The upper medium (air or vacuum) is medium 0, and the substrate s) is slightly absorbing in order to make the integrals converge.
Let
Cos
167r2
dQ
which
8
k4 -0
do-
can
be written
dQ
167r 2 qz2
iq,,zi+i.
iqzj
_
iqjj.rjj
n?
(4.C2)
2
Cos
2o
1:
drjj In i+ 1 2
-
n?] 2
e
iq,,zieiqjj.rjj
(4.C3)
i=O
then define:
zi
where Zi is the can
height
=
Zi +
zi
(4.C4)
(r1j),
of the flat interface in the reference
case.
Equation
be written:
k4. 0
do-
dQ
e
iq,
N
ko
(4.C3)
1
as:
4
do-
us
1 drjj
2o
(medium
Mr 2 qz2 e
iq ,
N
COS2 0
X
N
E I: J drjj I dr’ll In i+ 2
i=O
1
-
n?] [n +, S
3
-
nfl 3
j=O
(Zj-Zj),iq , (zi(ril)-zj(r’ll)),iqll.(rll-rlll).
(4.C5)
Jean Daillant and Anne Sentenac
154
Making
of variables
change
the
0
-COS’
167r 2 qz2
dQ
zi(r1j),
(second) order,
we
over
r’ll:
N
2
-
S
,
nfl
-
3
3
j=O
,iq,, (zi(r1j)-zj (0)) ) z’qjj.rjj’
where A is the illuminated
fluctuations
r1l and integrating
-+
Y E j drjj [ni+l n?] [n +j
X
i=O
iq (Zi- Z5)
e
r’ll
-
N
k 4A 0
do-
r1l
area.
in any have: or
(4.C6)
Assuming Gaussian statictics of expanding the exponential to
case
(,iq,,(zj(rjj)-zj(O)))
-- 1q,2,:(zj(rjj )_Zj(o))2
-
the
height
the lowest
(4.C7)
We then obtain:
do-.
1,4 A Cos
-
dQ
161r2q., N
v)
N
11" E I i=O
n
nj2 [nj2+1 3
2
i+1
-
e’q- (Zi Zj) e 1q.2,(zj)"--Lq,2,(zj)’
nj2 ]
-
-
-
2
2
3
j=O
,q,2,(zj(rjj)zj(O))2,iqjj.rjj.
d rjj This
2
(4. C 8)
.
specular components because it has been congeneral solution of an electromagnetic field in a vacuum. intensity can be obtained by removing the specular component:
equation
also includes
structed from the
The diffuse
ko’A
do-
N
N
T Y 1: [n i+
dQ
2
n?] [n +,
i=O
-
3
,
aq _,
nfl 3
j=O
ei,j,(Zj-Zj)e- !q.2(,j)2- 1q.2 (,j)2 J(qll). 2
The diffuse
(dQ
intensity
is
then:
k4A 0
do-
__
incoh
16ir2q2 z
N
N 2
[ni+l i=O
Cos
For
has
(4.C9)
2
-
nj2] [nj2+1
-
nj2]
eiq (Zi-Zj)e- -g1q,2,(zj)2- 7q2,(zj)2
j=O
2o
j drjj (,q.2(zj(rjj)zj(O))
2 -
I
) eiqll.r1j.
(4.CIO)
single surface, we get Eq. (4.35). It is remarkable that equation (4.CIO) exactly the same structure as the reflectivity coefficient (Fig. 4.3),
a
4
do-
k0A
dQ
4qz
h
e
N-1 N-1
E 1: [ni+l 2
i=O
-
nj2] [nj2+1
-
nj2]
j=O
iq (Zi-Zj) .-
- 3q,2(zj)2- -g3q,2. (Zj)2 2
2
(4. C 11)
4
each term
simply being multiplied by I
47r 2
I
dr
a
Scattering
155
"transverse" coefficient:
(,q.(zi(rjj)zj(0))’
11
Diffuse
eiqlIx1j.
Appendix: Quantum MechanicalApproach Of Born and Distorted-Wave Born Approximations
4.D
T. Baumbach and P. Mikulfk
appendix we treat the formal quantum-mechanical approach to scattering by multilayers with random fluctuations. That can be interface roughthe ness, but also porosity or density fluctuations. In particular we develop kinematical in the section differential scattering cross approximation (first Born approximation) and in the distorted wave Born approximation in terms of the structure amplitudes of the individual layers and of their disturbances. This approach is written in a general way. In Chap. 8 it will be applied to the reflection and to diffraction under conditions of specular reflection under grazing incidence by rough multilayers and multilayered gratings. We would like to notice that we adopted here the phase-sign notation of this book, -ikr and Fourier transforms e +iqr, which is contrary to with plane waves that used in most publications using this formalism. In this
Formal
4.D.1
Theory
develop formally the incoherent approach for the scattering by multilayers with defects independently of the specific scattering method. We make use of the (scalar) quantum mechanical scattering theory and its approximations, in particular the first order Born approximation (kinematical theory) and the distorted wave Born approximation (semi-dynamical theory). Scattering of the incident wave JKO) by the potential V produces the total wave field JE), described by the integral equation [33]
Here
we
I E) where
00
transition
can
I Ko)
+
OJJE)
(4-1)1)
,
operator of the free particle. We define the I JE) and the transition rnatrix by the matrix
is the Green function
operator by
elements Tos The
--
=
tjKo)
=-
(Ks Itl Ko), characterising the scattering from I Ko)
differential scattering expressed by the matrix
cross
be
do-
section
a
into
an
into
I Ks).
elementary solid angle 6Q
elements of the transition matrix 1
=_
167r 2
JTos 12
dQ
.
(4.D2)
Jean Daillant and Anne Sentenac
156
Scattering by spatial
randomly
disturbed potential. including a random scattering potential, the differential cross section the statistical ensemble of all microscopic configurations a
fluctuation of the
averages
over
do-
(ITOS12
--
dQ
f
167r2
(4.D3)
We divide do- into coherent and incoherent contributions do-
1
(TOS) 12+
167r2
167r2
I COV(TOS’ TOS)12
dS2
=-
dO’coh
+
dUincoh
(4. D 4) by denoting the covariance
Cov(a, b) Defining the
potential)
=
(ab*)
(a) (b)*
-
(4.D5)
.
non-random part of the scattering potential by VA (unperturbed the random (perturbed) potential by VB’ the coherent part
and
of the differential
cross
section writes
dUcoh
1 -
167r 2
and the incoherent differential
dO’incoh
cross
A
IT
+
(T B) 12
(4.D6)
dQ
section
:::::::12 Cov(TB 167r
,
T
B)dQ
(4.D7)
If the random part VB causes only a small disturbance to the scattering B by VA, we can calculate T within the distorted wave Born approximation
(DWBA). is not
a
It is worth
noting that in contrast to the widely spread opinion it potential VB < VA which defines the validity of the DWBA, the scattering by VB which has to be weak.
small
but rather
Scattering by
randomly
a
disturbed
multilayer.
In
a
multilayer
represent each layer by the product of its volume polarizability the
X,,,, j
(r)
we
and
layer size function Qj(,r) N
X
1: Xj (r) j=1
N -
1: X. j H Qj (r)
(4.D8)
j=1
optical (or scattering) potential for X-rays can be expressed by the polarizability: V(r) -ko2X(,r). The contribution of the different layers to the section is distinguished by considering each layer as an indescattering cross scatterer pendent The
=
Vj (r)
(4.D9)
Diffuse
4
N
I
do-
E
--
j)
12
with Tj
(Ks I vj I E).
=
Separating
do-
vjB,
jE
3
j
(,rj, Tk)
Cov
(4.DIO)
dQ
,
N
N
TA j
+
1: (’rB) 12 j
+
j=1
j=1
where the r-:1 are
j=1
k=1
part of each layer, vj
the non-random and the random
N
i-6jr2
and TB
N
obtain
we
I =
N
+
j=1
vjA
157
Eq. (4.D4) writes
Then
+
Scattering
EE j=1
the contributions of the
are
those of the
N
Cov
(,rB,,FB k ) j 1;
(4.D11)
dQ
k=1
non-perturbed layers
to
scattering,
term is the coherent
layer disturbances. The first
part dUcoh; which consists of the contribution of the ideal multilayer and of the
averaged
transition elements of the
layer disturbances. The second, single layer
incoherent part do-incoh contains the covariance functions of all transition elements.
Formally the division of ’ into a sum of scatterers E, vj is arbitrary. The sticking point is to find a set of eigenstates, which is convenient to serve as basis for calculation, of the transition elements. Finally, we remind the reader, that until now no approximation has been made.
4.D.2
Formal Kinematical Treatment
by First Order
Born
Approximation
Within the kinematical treatment
(first
approximation)
order Born
we
ap-
scattering potential flK) P_- ’JK). The set of vacuum wave vectors JK) e-ikr provides an orthogonal basis for the calculation of the differential scattering cross section. The transition elements of the individual layers are proximate
by
the transition operator
the operator of the -
Tj
where q
--
ks
-
=
(KsjvjjKo)
ko. Defining the
-k02
I
structure
Sj (q) with the random one-dimensional
Fj(qz, r1j)
=
dr Xj
(r)
factor of
drIl Fj (q, r1j)
e
iqr
the
ez q1I r1l
(4. D 12)
layer
(4.DI3)
layer form factor dz Xj
(r)
iq, (z
-
Zj)
(4. D 14)
Jean Daillant and Anne Sentenac
158
the transition element becomes
-k20 eiq.,Zj S (q) The coherent
(7j)av,
and
scattering
cross
search for the
so we
incoherent differential
scattering
section
(4.D15)
(4.D9)
uses
the statistical averages The
layer form factor
mean
cross
Fj (q,,,rjj))av.
section contains the covariance
func-
tions
COV(SjSk) P
I
d’rjj
I drii’ciqll(rll-rll’)COV(Fj(q,,,rll),Fk(q,,,rll’))
Substituting (4.D13) scattering
differential
4.D.3
(4.D16)
and cross
1 1: (Sj) ’V
MT 2
iq ’ Zj
e
3.=I
12
(4.1)10),
into
section of
N
00
do-
(4.D16)
an
we
obtain the kinematical
arbitrary multilayer
N
N
E E & e’q;, (zi
+
j=1
-
ZO
dQ
.
(4. D 17)
k=1
Formal Treatment
by
a
The distorted
Distorted Wave Born Born
wave
Approximation
approximation takes all those effects of multiple are caused by the unperturbed potential V’.
into account which
scattering
right choice of VA, which decides enough transparent and sufficiently precise. We search for such a VA which enables to explain the essential multiple scattering effects. However, it should provide the simplest possible solutions EA used as orthonormal basis for the representation of scattering by the K disturbance (perturbed potential) VB. It is less the method
about the
itself, but
rather the
in order to be
success
Scattering by planar multilayers with sharp interfaces produces such simple solutions. It has been shown that rough multilayers as well as intentionally laterally patterned multilayers and gratings can be treated advantageously by starting with an ideal potential of a planar (laterally averaged) multilayer, splitting the polarizability in N
X
=
X
A
+ X
B
with
X
A
E XiAplanar
(4. D 18)
j=1
Coherent
scattering by the non-perturbed multilayer generates a wave field can be decomposed into a small number of plane waves within each plane homogeneous layer, both with constant complex amplitudes and A EK which ,
wave
vectors, A
EK ’i (r)
Ek,,,j
-
n=1
e- ik"jj’jTjj e- ik,_,,j(z-Zj)
S2A (Z) j
(4. D 19)
Diffuse
4
In
of
case
specular reflection
wave), for grazing tion 1=8. The
is,
it
transmitted and
(one
1=2
Scattering one
159
reflected
strongly asymmetric X-ray diffracnon-perturbed states for the estimation of
incidence diffraction and
A
EK (r)
are
used
as
TB (4.D6). Within the first order DWBA
TB,DWBA Again,
=
(E SA*IfrB JE 0A)
obtains
one
-k
-
-
0j
A
2
dr E S
A
(r )XB (r)E 0 (,r)
.
(4.D20)
it is recommendable to describe the contribution of the disturbance
within each
plane layer separately by B
73
-
2
A B E’ S (r) X3 EO’ (r)
dr
-k 0
(4.D21)
with Bplanar
j
(4.D14),
however
to the actual
now
scattering
expressions (4.D13) and corresponding
vector
km
=
Sj
-
(4.D23)
kn
Oj
layer
Fjm’(q’ 13’, r1j)
dz
=
Z
consists of I
Each
(4.D22)
Xj
with respect to the disturbance X)P and
q,mn inside the
-
similar to the
Sj"’, formally
We define Fmn and
Aplanar
XQA j
Xj
x
-
-
Zj)
(4.D24)
I terms
2
B
Tj
XP (r) e’q--, in (z 3
Em
-k 0
Sj
(4.D25)
n
E (Z) (Z)Smn Oj j
m=1 n=1
or
using
the matrix formalism B
Tj
-koEsj jEoj 2
-
rn
contains the where the column vector En Kj
(4.D26)
n
amplitudes of the
I
plane
waves
of
non-perturbed state in the jth layer and j is the structure factor matrix layer disturbance, respectively. Each term in (4.D25) represents the contribution of the disturbance to the scattering from one plane wave of the A A in another plane wave of the final state EK,. Each scattering initial state EKo of the according wave amplitudes process is characterised by the product
one
of the
E’ En Sj
03
and by the disturbance structure factors Smn. j average F the statistical ensemble and substitute these terms in (4.DIO).
In order to determine the coherent scattering
and S
over
cross
section
we
Jean Daillant and Anne Sentenac
160
The incoherent
section contains the covariance functions for each
cross
layer
pair Cov
B
B’
(7-j
4
E, m _E-n
ko
7k
Sj
S,
’P Qmn jk
E E’ Ok
(4.D27)
Ok
m,n,o,p
with OP Qmn jk
Cov
QOP (STn 5-k J 3
j drjj j drll’eiqjj(rjj-rjj’)
COV
(Fj (qm ’, r1j), Fk (ql
z,k
Z,3
,
(4.D28)
r1l
Each term represents the covariance of one scattering process in layer j and second scattering process in layer k. Adding up the contributions of all
a
scattering
processes and all
layers
obtain
we
finally 2
k 0’
do-
E Tj
167r2
+
1: E
EM
Sj
En (Smn) j Oj
(4.D29)
j=1 m,n=l
=1
N
I
E"
+
Sj
(E-Sj )* Qmn,, ik
* E, Ok (Epok)
dQ
j,k=l m,n,o,p=1
X-ray reflectivity, each eigenstate of the unperturbed potential consists
In
of a transmitted and reflected wave, thus _T
qll,. (8.48),
q22, corresponding
..,
to
(kscll
-
=
2. The four
kin1j; k,,c, ,
wave
ki,,,,)
in
vector transfers
(4.46),(4.47)
or
represented in the reciprocal space in Fig. 8.40. Further, The above expressions are written explicitly for diffuse scattering in Eqs. (8.46)(8.49) and for coherent reflectivity for deterministic (i.e. non-random) grating potential VB in (8.72). The covariance for grazing incidence diffraction is are
presented by (8.62).
Stmpler
DWBA
for multilayers.
The
expressions simplify enormously, approximate non-perturbed polarizability by its mean value in the multilayer, averaging vertically over the whole multilayer stack. We obtain a homogeneous "non-perturbed layer". The splitting of the potential in this way gives if
we
the
can
XA (1,)
-
(XML (r))
av
N
X, (r)
-
E XiBlaper (r)
with
XiBlay-(,r)
=
(X(,P)
_
(XML(,r)) ) oi.d(,r) av
j=1
(4.D30) Now the
non-perturbed wave only. In
transmitted
wave
field below the
consequence
sample exclusively the primary scattering prosurface consists of the
cesses
’Cov ( Tj!3 ;
TB) k
K4 tSt*S
’ ror*0
3k
(4.D31)
Diffuse
4
and the transmission function of the
sample
surface
are
Scattering
161
considered. Also the
effect of refraction is included.
References Sinha, E.B. Sirota,
S.K.
1.
Stanley, Phys.
S. Garoff and H.B.
Rev. B 38, 2297
(1988). B61orgey, J. Chem. Phys. 97, 5824 (1992). Karabekov, I.V. Kozhevnikov, B.M. Alaudinov, and V.E. Atyukov, Asadchikov, Physica B 198, 9 (1994). 4. S. Dietrich and A. Haase, Physics Reports 260, 1 (1995). 5. D.K.G. de Boer Phys. Rev. B 49 5817 (1994). 6. P. Croce, L. N6vot and B. Pardo, C.R. Acad. Sc. Paris 274 B, 803 (1972). 7. P. Croce, L. N6vot and B. Pardo, C.R. Acad. Sc. Paris 274 B, 855 (1972). 8. P. Croce and L. N6vot Revue Phys. Appl, 11, 113 (1976). 9. P. Croce J. Optics (Paris) 8, 127 (1977). 10. L. N6vot and P. Croce Revue Phys. Appl. 15, 761 (1980). 11. P. Croce J. Optics (Paris) 14, 213 (1983). 12. P. Lorrain and D.R. Corson "Electromagnetic Fields and Waves" W.H. Freeeman and Company (San Francisco) (1970) p.629. 13. L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media, Course of theoretical physics vol. 8, Pergamon Press, Oxford 1960, 69. 14. C.-T. Tai, Dyadic Green functions in electromagnetic theory, IEEE Press, New2.
J. Daillant and 0.
3.
I.A.
A. Yu.
’
York,
1994.
15.
J.D. Jackson "Classical Electro dynamics" 2
16.
M. Born and E.
Wolf, "Principles
of
d
optics"
Wiley (New-York) 1975. edition, Pergamon (London)
Edition 6
th
(1980) p.’51. Sinha, M. Tolan. A. Gibaud, Phys. Rev. B 57, 2740 (1998). Herpin, C. R. Acad. Sci. Paris 225 182 (1947). Y. Yoneda, Phys. Rev. 131, 2010 (1963). B. Vidal et P.. Vincent, Applied Optics 23 1794 (1984). I.M. Tidswell, T.A. Rabedeau, P.S. Pershan, S.D. Kosowsky, Phys. Rev. Lett.
17. S.K. 18. 19. 20. 21.
A.
66, 2108
(1991).
Waifan, J. Appl. Phys. 3 503 (1970) Cai, K. Huang, P.A. Montano, T.P. Russel,.J.M. Bai, and G.W. Zajac, J. Chem. Phys. 98 2376 (1993). 24. W. Weber and B. Lengeler, Phys. Rev. B 46, 7953 (1992). 25. A. Sentenac and J.J. Greffet, J. Opt. Soc. Am. A 15, 528 (1998). 26. G.C. Brown, V. Celli, M. Coopersmith and M. Haller Surface Science 129 507
22.
J.B. Bindell and N.
23.
Z.-h.
(1983) Brown, V. Celli, M. Haller and A. Marvin Surf. Sci., 136 381 (1984). Brown, V. Celli, M. Haller, A.A. Maradudin, and A. Marvin, Phys. Rev. B,
27. G.C. 28.
G.
31 4993
(1985).
in volume and surfaces, ElsePublishers, B.V. North-Holland (1990) 30. R. Petit, ed, Electromagnetic theory of gratings, Topics in current physics, Springer Verlag, Berlin (1980). 31. W.H. de Jeu, J.D. Schindler, E.A.L. Mol, J. Appl. Cryst. 29 511 (1996). 29. M.
Nieto-Vesperinas and J.C. Dainty, Scattering
vier Science
Jean Daillant and Anne Sentenac
162
32. L.
Bourdieu, J. Daillant, D. Chatenay, A. Braslau, and D. Colson, Phys. Rev. 72, 1502 (1994). A.S. Davydov, Quantum Mechanics, Pergamon Press, 1969.
Lett. 33.
1 149
(1991).
Neutron
5
Claude 1
Ferman’,
Reflectometry
Fr6d6ric Ott’ and Alain Menelle’
Physique de I’Etat Condens6, Orme des Merisiers, CEA Saclay, Yvette Cedex, France, 2Laboratoire Uon Brillouin CEA CNRS, CEA Saclay, 91191 Gif sur Yvette Service de
91191 Gif
sur
Cedex, France
Introduction
5.1
technique [1,2]. In the last years, problems like polymer solving extensively surface the of at the structure [5,6] for example. The liquids mixing [3,4] or small their is studies for absorption compared asset of neutrons polymer
Neutron
reflectometry is
a
relatively
new
used for
it has been
to x-rays and the
"labelling" by
large
contrast between
soft matter
’H and 2H which allows selective
deuteration.
80’s, a new field of application of neutron reflectometry has emerged. Following the discovery of giant magnet oresist ance in antiferromagnetically coupled multilayered -films [7] and new magnetic phenomena in ultra-thin films, there has been an interest in the precise measurement of the magnetic moment direction in each layer of a multilayer and at the interface between layers. Owing to the large magnetic coupling between the neutron and the magnetic moment, neutron reflectometry has proved to be a powerful tool for obtaining information about these magnetic configurations and for measuring magnetic depth profiles. In this chapter, we give an overview of the experimental and theoretical methods used for neutron reflectometry, focusing on specular reflectivity. The corresponding theory is partly derived from the previous work developed for of neutrons. x-rays, and we emphasize those aspects specific In the late
In
a
an
will review the neutron-matter interactions. We then non-magnetic scattering. In this case it is possible to introduce
first part,
describe the
optical
we
index and
give
a
treatment which is similar to x-ray
reflectometry
(Chap. 3). In a second part, the neutron spin is introduced. In this case, optical indices cannot be used any longer and it is necessary to completely solve the Schr6dinger equation. A detai,led matrix formalism is presented.
We then discuss the different aspects of data processing and the problems roughness. Two types of neutron reflectometers are
related to the surface
particular: fixed- wavelength two-axis reflectometers and timeof-flight spectrometers. The use of neutron reflectivity in the field of polymers films and of magnetic layers is then illustrated by several examples.
described in
J. Daillant and A. Gibaud: LNPm 58, pp. 163 - 195, 1999 © Springer-Verlag Berlin Heidelberg 1999
Claude Fermon et al.
164
Notation used in this
b, bj
bound
chapter
scattering length of
a
nucleus,
mean
scattering length
layer j b c*
bi bN V V
90, Sj 6
bound coherent
scattering length scattering length spin dependent scattering length real part of the scattering length imaginary part of the scattering length incoherent
energy of the neutron in the charge of the electron
d, dj
thickness of
9
Land6
Planck constant nuclear
k
wave
spin operator
vector
M, Mj magnetic
moment of
neutron
mass
M,
electron
mass
nj
refractive index of
an
electron and of
layer j conversion factor 2.696fm, p an effective scattering length scattering vector spin operator of the electron --
q S.
of
a
Pauli operator associated to the neutron
Vi
volume of the
V(T)
interaction Hamiltonian
gn
gn
A, A0
neutron
PB
Bohr magneton nuclear magneton
Pn
layer
magnetisation
0-
--
layer j
layer factor, (g =2)
I
P
and in
a
h
M
vacuum
to
spin
layer j
-1.9132, nuclear Land6 factor wavelength
of the neutron
density of the layer j (atoms per cm 3) absorption 0’j Oj, oj spherical angles of the magnetisation of the layer j Oin, Or incident and reflected angles of the neutron beam atomic
Pi
PB
Pn
=
=
O(x)
eh/(2m,) eh/(2mp)
is the
=
=
9.27
x
5.05
x
10-24 J.T-1 10-27 J.T-1
Heavyside function defined by:
O(X) O(x) O(X)
=
I
=
1/2
=
0
when
x
when
when
> x
x
<
0 --
0
0
of
a
5
Neutron
Reflectometry
165
"up" (resp. "down") the neutron polarisation parallel (resp. antiparallel) to the external applied magnetic field. "Down-up" designates a polarised "down" incident beam and polarised "up" We call
detected beam.
"Down-up" and "up-down"
5.2
are
called
spin-flip
processes.
Schr6dinger Equation and Neutron-Matter Interactions
Schr6dinger Equation
5.2.1
The neutron
can
be described
by
a wave
ko
of
wavelength A,
27r =
A
of
wave
vector:
(5.1)
1
and of energy
h2 ko2
so Its
function verifies the
wave
where
m
(5.2)
2m
Schr6dinger equation (1.17):
h2
d20
2m
-d_r2
+
[S
-
V(r)] 0
--
0,
is the neutron mass, 9 its energy Ind V the interaction potential. the base a spin 1/2 particle so that 0(r) can be expressed on
The neutron is of the two
spin
states:
0+ N R) When there is neutron
an
or
0- (0 H
-
(5.4)
magnetic field, an "up" (resp. "down") eigenstate 1+) (resp. I-)). In the following index will often be dropped.
internal
neutron in the
designates dependence (r) a
the space
5.2.2
external
+
of the
Neutron-Matter Interaction
The two main interactions
are
the strong interaction with the nuclei and
magnetic interaction with the existing magnetic moments (nuclear and electronic). There are a large number of second order interactions which are
the
described in
[8].
Claude Fermon et al.
166
Neutron Nucleus Interaction: Fermi Pseudo Potential The scatter-
ing of a neutron by a nucleus comes mainly from the strong interaction. The interaction potential is large but its extension is much smaller than the wavelength of the neutron. Hence this interaction can be considered as ponctual and
isotropic. Within the pseudopotential [9]:
Born
approximation,
it
can
be described
by
the
Fermi
VF(r)
(
b
=
27rh2 M
) S(r)
(5.5)
where b is the
scattering length and r is the position of the neutron. The scattering length b depends on the nucleus and on the nuclear nucleus. Formally it can be written
value of the
spin
of the
b N.B.: the
scattering length
_-
is
b, +
I 2
bNI-O’,
(5.6)
b’ + ib". a complex number: b scattering length. The second term corresponds to the strong interaction of the spin of the neutron (described by the operator 112o-) with that of the nucleus (operator I). The total spin J 112o- + -1 is a good quantum number for the neutron spin nucleus spin interaction 112o-.1. In he manifold f I 1/2}, the eigenvalues of the spinI 1+ 1/2) and -(I + 1) (for J dependent operator Lo- are I (for J 1/2). We name b+ and b- the two scattering lengths associated with these two eigenvalues, corresponding to the two states 1+ and I-) of the neutron spin. The nucleus spin-dependent scattering lengths can then be written [10]:
generally
--
The first term b, is called the coherent
=
-
=
t where I is the nuclear
b+
bo bo
b-
=
+ -
’bj
1 b,, (1+ 1)
-
(5.7)
2
spin quantum number. scattering cross section is given by (see Eq. (1.35):
We remind that the total
o-t,,t
in which the brackets
nuclear
designate
Neutron
the statistical average
(5-8) over
absorption
is
negligible
-‘
(47r/ko)
b".
energies
which
strongly
(5-9)
for thin films except for some elements: Gd, (n,^/) nuclear resonances at thermal
B and Cd. These elements present
neutron
the neutron and
Absorption The absorption of neutrons is described by the imagof the scattering length V. The absorption cross section is given
G’abs
The
41rflb 12),
spins.
inary part by:
Sm,
--
increase the
absorption.
Neutron
5
tribution of isotopes total
scattering
o-t,,t
nuclear
or
section
cross
47r(lbl’)
--
Incoherent
Scattering
Incoherent
47r
--
spin
scattering states in
comes
-
167
from the random dis-
material. In this case, the can be written:
a
(see equation 5.8)
((Ibl)’ + ((Ibl’)
Reflectometry
(Ibl)’))
:--
O’coh + O"incoh;
(5.10)
where 0"coh and O’incoh are called the coherent and incoherent scattering lengths. In the presence of isotope or spin disorder, the second term in Eq. (5.10) is not
zero.
If for
a spin (see 5.8) we have a spascattering lengths. In the case of an isotope material, the incoherent cross section is given by:
example
the nucleus carries
tial distribution b+ and b- of distribution b, in the
0-inc,isotope
47rEco, c,6 I b,
:--
-
bp 12
(5.11)
designates the fraction of isotope a in the material. Incoherent scattering appears as a q-independent background in the experiments and can be treated as an absorption plus a flat background. The incoherent scattering is particularly important for hydrogenated layers but it is small for deuterated layers. A more detailed discussion of incoherent scattering can be found in [11,12]. Tables of the different scattering lengths (coherent, incoherent, absorption) of the different elements can be found in [12]. where c,,
magnetic interaction is the dipolar iiytermagnetic field created by the unpaired This field contains two terms, the spin part
Interaction The main
Magnetic
with the
spin magnetic atoms.
action of the neutron electrons of the
and the orbital part:
B
where ji,
--
magneton,
110
47r
fie
17
e
v,, x R
(5.12)
111u13
IR 13
magnetic moment speed of the electron. magnetic moment is equal to:
-2PBO’
is the
of the
electron,
ILB is the Bohr
v, is the
The neutron
P
The
R
x
magnetic
::--:
interaction expresses
VM(r)
=
gn/-tnO’-
as
-M.13
(5.13)
:
:--
-9n/ino-.13.
(5.14)
reflectivity does not allow the separation of the orbital and spin contributions, it is only sensitive to the internal magnetic field.
Neutron
The Zeeman Interaction It is the interaction of the neutron
external
magnetic
spin with
an
field 130:
Vz(r)
=
-gnp,,c-.Bo.
(5.15)
Claude Fermon et al.
168
Reflectivity
5.3
Non-Magnetic Systems
on
For
non-magnetic systems we can introduce the notion of optical indices. It is approach similar to the x-ray formalism (Chaps. I and 2). It can be applied to neutron reflectometry on soft matter [13] and non-magnetic systems. We consider a neutron beam, reflected by a perfect surface with an incident angle 0. As in Chap. 3, the surface is defined by the interface between the air (n 1) and a material with an optical refractive index n. In a vacuum, the energy of the neutron is given by: an
--
e Let q
tering
_-
k,
wave
-
p12 k20 __
on
the
axis
z
(5-16)
2 MA2
2 rn
kil, be the scattering
vector
h2 -
wave
vector. The
(perpendicular 4 -7r
q,
sin
A
projection
of the scat-
surface)
given by:
to the
is
Oi..
(5.17)
ZAL
incident
wave
reflected
q
wave
k in
k,
air
(n=l)
Oin z
itted
Fig.
5.3.1
Neutron
5.1.
Optical
Reflectivity
’hwave
on a
ktr
perfect
=
0
medium
(n
surface
Indices
The neutron indices
are very different from the x-ray indices and we will determine their expression from the Schr6dinger equation. We suppose that the interaction potential V(r) in the medium is independent of the in-plane
coordinates
integration
x
and y. The
of the Fermi
mean potential pseudo-potential:
V
I -_
V
I V(r)d
3
V in the medium is
27rh r
-
?n
given by
2bp,
the
(5.18)
V
where p is the number of atoms per unit volume. In the absence of any magnetic field, the Schr6dinger
equation
can
be
written-
h2 d2 __ 2m dr2
+
[e
-
V] 0
=
0.
(5.19)
Neutron
5
equation (5.19) can be written in the form equation similar to the electromagnetic case:
of
The
d 20
a
Reflectometry
Helmholtz propagation
k2,0- 0,
dr2
169
(5.20)
with 2m
k2
We define the
optical index
- 2 [S
(5.21)
V].
-
as:
k n
The
optical
index
n can
cases
and thus
n can
V
2
Ti and
Mn).
(5.23)
-pb.
except for materials with
The
quantity
1
-
n
a
negative
is of the order of
10’
be written:
Critical
Angle
A
I
-
21r
(5.24)
pb.
for Total External Reflection
At the interface between two
media, the Snell’s cos
Oi,,
=
law
applies:
(5.25)
Ot,.
n cos
have shown that the index is smaller than one, for angles 0 < there is a total reflection of the incident wave like in the case of x-ray
Since
0,
A2
-
7r
one
n ;
5.3.2
1
-E
smaller than
scattering length (e.g.
(5.22)
2
ko
be written:
n
It is in most
--
we
reflection. The critical
angle 0,
is
the condition
given by 0,
cos
=
-
-
The
corresponding critical
wave
FrL,
0, i.e.:
a
Taylor expansion. Using (5.24)
A.
(5.27)
4Vqirpb.
(5.28)
vector is:
47rsinO, q,
=
(5.26)
n.
Since 0, is very small it is possible to use and (5.26) the expression of 0, is given by:
Oc
Otr
=
=
A
170
Claude Fermon et al.
5.3.3
Determination of
In the
case
of pure
material. In the can
Scattering Lengths
Optical Indices
and
materials, the knowledge of b and p fully characterises the of crystalline solids of the type A.,By for example, one
case
consider unit cells. The volume of the unit cell allows
to calculate
one
a
p of unit cells per unit volume. The average scattering length b", in the unit cell is simply given by b,,, = (xbA + ybB)I(x + y). The value pb,,,
density
be used to calculate the index of the material. The case of liquids and polymers is more complex since it is usually more difficult to define a "unit cell". Thus, the best method is to calibrate the index of each polymer or liquid that one wants to study. The Table 5.1 gives the scattering length, I n and critical wave vector q, (at 0.4 nm) for various optical index 6 elements and compounds.
can
--
Table 5.1.
-
Scattering length,
and critical wave-vector of
atomic
some
density, optical
common
index J
=
I
n
-
(at
found at "www. neutron. anl.gov"
Material
b,,
P
pb
j
qr-
28 (fm) ( 10 M-3 (iol3M-2) (10-6) (nm-’
(hydrogen) (deuterium) C (graphite) C (diamond)
-3.73
0
5.80
H D
6.67 6.64
11.3
75
19.1
0.19
6.64
17.6
117
29.8
0.24
0.10
Si
4.15
5.00
20.8
5.28
Ti
-3.44
5.66
-19.5
-5.0
Fe
9.45
8.50
80.3
20.45
Co
3.63
8.97
32.6
8.29
0.13
Ni
10.3
9.14
94.1
24.0
0.22
Cu
7.72
8.45
65.2
16.6
0.18
Ag
5.92
5.85
34.6
8.82
0.13
Au
7.63
5.90
45
11.5
0.15
H20
-1.68
3.35
-5.63
-1.43
D20
19.1
3.34
63.8
16.2
0.18
Si02
15.8
2.51
39.7
10.1
0.14
GaAs
13.9
2.21
30.7
7.82
0.12
24.3
2.34
56.9
14.5
0.17
42
10.7
0.14
23.2
0.61
14.2
3.6
0.084
106.5
0.61
65
16.5
0.18
A12 03
(sapphire)
pyrex
polystyrene polystyrene
(deuterated)
0.4
materials. More exhaustive data
-
0.20
-
nm),
can
be
Neutron
5
5.3.4
Reflection
on a
Homogeneous Medium
Reflectometry
171
,
by equatio n (5.19) the problem of the reflection of a neutron beam non-magnetic medium can be treated exactly in the same way as the reflection of x-rays. Since the potential V is only z dependent, the Schr6dinger equation (5.19) reduces to the I dimensional equation: As shown on
a
h2
d20Z
T2 __T Z2 2m M with
a wave
In the
[S"
+
medium,
general Oz
The transmitted
wave
solution is
-
vector
Ae’ ’-’
can
-
(5.29)
01
ei(kj_o+kinYY) b..
function of the form the
10-1
V
-
given by: Be-’kt-’-z.
+
(5-30)
be related to the incident
wave
vector
using
(5.21): k2 At the interface
we
2m
[9
_
have to
jj2
apply
-
V]
the
k?
=
-
in
(5.31)
47rpb.
continuity condition
on
0
and
VO.
In
that the paralway similar to the x-ray case, it is then possible to show lel components of the incident and reflected waves are continuous [9]. The a
continuity
of the
parallel components allows k2tr
k?
to write:
(5.32)
47rbp.
-
in z
z
us
Considering equations (5.20) and (5-32), the problems of neutron and x-ray reflectivity are formally the same. It is possible to use the same formalism as the one developed in Chap. 3 for x-ray reflectivity. In particular, it is possible to use the classical Fresnel formulae. The reflected and transmitted amplitudes are given by: sin Oin r
=
-.
SIn
-
Oin +
n
sin Ot.
n
sin
2 sin t
In terms of
scattering
wave
Fig.
5.2 shows
a
Oi +
n
sin
(5.34)
Otr’
vector, the reflected intensity is given by:
R
The
Oi
=
sin
(5-33)
Otr
typical
curve
ko,
-
ktrz
ko, + kt, ,
2
I.
calculated for
(5.35) a
perfect
surface.
Claude Fermon et al.
172
0 1 .
0.01-
0.001-
P4
0.0001
0.00001
i
i 0.2
0
0.6
0.4
Fig.
Reflected
5.2.
intensity
as
0.8
function of %, for
a
1
(mu-1)
q.,
a
silicon substrate
(at
A
0.4nm) 5.4
Neutron
Reflectivity
on
Magnetic Systems
If the system is magnetic or if there is an external magnetic field on the sample, we need to take into account the spin of the neutron. It is not posuse optical indices and it is always necessary to completely solve Schr6dinger equation [14-16]. In the case of homogeneous, infinite magnetic layers, the problem can be solved using a formalism very similar to the non-magnetic case developed in the previous part.
sible to the
5.4.1
Interaction of the Neutron with
order to
Infinite
Homogeneous Layer
magnetic layer of thickness d, the neutron interacts with the unpaired electrons. We perform a direct integration on the layer in obtain the potential V for the Schr6dinger equation.
We consider different
an
a
The Maunetic Interaction A first tron is sensitive to the internal
interaction
potential
approach magnetic field
is to in the
that the
neu-
magnetic layer.
The
assume
is then written: -gn/-tnc--
[PO (1
-
D)
M +
Bo],
(5.36)
magnetisation of the layer, D is the demagnetising factor magnetic field. In the case of an infinite magnetic M is thin film, (1 equal to the in-plane component of the magnetisation D) this result but the calculations are somedemonstrate to is It possible Mij. what lengthy. This is developed below for the interested reader but it can be skipped at the first reading. where M is the
and BO is the external -
Neutron Reflectometry
5
The
interaction
magnetic
-gnPno7.B
can
be written V
-9nPnU-
=
:
p, x R
X
173
JR1
3
1
e
v, x
R)
(5.37)
JR13
C
or
-gn/in
f
U-V
JR13R)
(
X
x
R
o- x
e
a x
R
(5.38)
P- Rr +TR-F’P"’
2rnc
with
(5.39)
P,
If
we
only the spin dependent part of
first consider V
(p,0 ) x r
x
=-Vx I
27r2
Integration
(1
f
q
2(q
(p,
Homogeneous Layer
on a
interaction,
we can
(1))
P’XV x
the
x
write:
(5.40)
q)) exp(zq.r)dq
We suppose
a
constant atomic den-
replace r by r + ro. where ro is the distance between the neutron and the center of the layer. r is the distance between the center and the volume dr in the layer. The spin dependent part of the interaction is: p. We
sity
29n/-tnPBO
f q2 f p(r)(q I
I -
27r2
(r)
-
x
x
q) exp (iq.ro) exp(iq.r)drdq.
(5.41)
V
where
p(r)
is the
density and - T(r)
in the volume dr. The two first
The third
f-L’I L
49n/-ZnPBP-f
z
=
- T is the
integrations
dr-,(q,,
calculation
same
Pa. M =
x
9
x
spin magnetisation
Dirac distributions:
q,) exp(iq,.ro;,) exp(iq_ r,,)dq,.
wave
on
+
L/2)
-
O(roz
-
the orbital part and
Mlip [O(ro,
+
L/2)
-
M 11 is given in PB per atom magnetisation and not necessarily
2.696f m.
ponent of the to the
give
(5.42)
2
27rh2 with p
and y
integration gives:
do the
can
value of the
2
87r9nPnPBPU-’ TJJ [O(rO_ We
mean over x
O(ro,
-
L/2)].
we
(5.43)
obtain
L/2)],
(5.44)
and represents the
the
in-plane commagnetisation perpendicular
vector.
equation (5.44), we can deduce two very important points: measure the in-plane magnetisation and the magnetic interaction is zero out of the layer. These two properties are essential, the first is the main limitation to the use of neutrons for the study of magnetic thin films, the second is the justification of solving the Schr6dinger equation in each layer, independently of the others. Thus the formalism developed for non-magnetic systems can be adapted to the magnetic case, however with some complications. Conclusion From
it is
only possible
to
Claude Fermon et al.
174
X
tolv
Fig. 5.3. Effective field B,ff,j in the layer, sum of the external field Bo and of the magnetisation of the layer Mj. Definition of the spherical coordinates Oj and 9j Solution of the
5.4.2
The interaction
potential
Vi We introduce
an
27rh2 -
M
Schr8dinger Equation for
layer j
a
pj bj
-27rh2 M
we
introduce the
Boy
+
Bo,
=
potential
V
can
-
g,,M,,
(5.45)
o-.Bo.
by:
(5.46)
MoM11.
0 and
O’to describe
the effective field:
Beff,j sin(O,,,) cos( o,) Beff,j sin(O,,) sin( o,)
(5.47)
then be written in the compact form:
21rh2 =
Pjbj
-
(5.48)
9nPno-.Beff,a-
Schr6dinger equation: 2m
a
o-.Mj,,
:
/-toMjy Beff,j COS (0j)
h2
is
Bo +
Box + 1-toMjx
Vi The
=
spherical angles
Beff,jx Beff,jy Beff,j, the interaction
pjp
effective field Beff defined
Beff If
given by
is
vectorial
equation
zAo
+
in the basis of the two
Schr6dinger equation (5.49) spinors components 0+ and 0-
to solve the
with
its two
It
27rh2
h2 2m gn/-tn
0-
(Beff,xG-x
M
+
pjbj)
Beff,yUy
(5.49)
V(r)V) -So
+
spin
states
a wave
can
J+)
and
I-).
We have
function
be written
expressed with explicitly as:
0-
Beff,zO’z)
( )=S(0+)’ (5-50)
5
where the Pauli
spin operators
(10)
o- are
We obtain the two
0+
and
=
0 -i
0-Y
=
Reflectometry
175
given by:
(i )
0 1
O-X
Neutron
0
(0 -1) 1
O-Z=
coupled equations involving the
0
spinor components
two
0-: h2 2rn +
27rh2 + M
(-9n ft,, Beff,x h2 2m
+
v2
v2
+
+ rn
-
-
bjPj
+
0+
g,, p,, Beff, z
ign Yn Beff,y) V)- (r)
27rh2
(-,qn/-InBeff,x,
bj pj
-
gn/-tnBeff,z
i,9n/-InBeff,y) ?P+(7)
Eo+ (r) 0- (r) 460’ H-
(5.51)
5.4.3
General Solution
We search solutions of the form
:
0+(r)
--
a+
exp(ik.r)
and
0-(r)
a- exp(ik.r) The possible values for k are given by the possibility of finding non zero solutions of the previous system (i.e. zero determinant condition). These conditions give four possible values for k :
2m
2
k.7. A
general
h2
9
-
41rpjbj
2mg,,/-In h2
jBjj.
(5.52)
solution of the form:
exp(ikjj,jrjj) (a exp(ik+,jZ +c
exp(ik;j z) Z,
+ b
exp(-ikZ,j z)
+ d exp(-ik; Z,j
z))
(5.53)
an external magnetic field because k1+1’0 0 k point there are two ways of solving the problem. The first way is solve the problem for each eigenstate 1+ and I-). The second way consists taking the general solution which is expressed as:
is not valid when there is At this
to
in
j+ z)) .,+ z) + b exp(-ik ZJ exp(ikl+,+J r1j) (a exp (ik ZJ + d exp(-ik;.,-. Z,j z)) exp(ik,-, 7 r1j) (c exp(ik;.,-. Z,j z) + f exp(-ikJ3 exp(ik,+,-.’i r1j) (eexp(ikJ3 ZJ z)) Z,j z) z) + h exp (- ik;j+ z)), exp (ik 11,3 F,+ ,j r1j). (g exp (ik;j+ ,3
,3
Z,3
Z,3
(5.54)
Claude Fermon et al.
176
with k +2
k
k+-2
k +2
k
+2
2
k++2
-
3
k37
=
k--2
=
.7
equation (5.4.2) quantisation axis.
-
1
,-
’2
1
i2 k7 :
2
k
k-2 ’0
k
k11 Ij2 k11 Ii
3
(5.55)
2
1110
k-2 1110
is the solution
The solution of of the
k3
+2 3
(5.54)
rotated
the
by
angles
Oj’+ (r) exp(i k Z,j ,+z) + bj+ exp(-i k j z)) exp(ikl+,+rll)(aj++ Ii ,j exp(ik,,+.-rll)(aj+-exp(ik ZjZ -,,j .-z)+b,+-exp(-ik 3.-,z)) ’i
+
3
,3
+
+
IF r1j) (a37j exp (ik IF,37 r1j) (a37 exp (ik IIJ 3
-
-
.3
k;,+ z)
exp (i
ZJ
k;,-. Z,j z)
exp (i
-
Z,aZ
a
3
3
b37
+ +
b37
+
+
k;j+ z))
exp (- i exp (- i
_, ZJ Z
k;3-.z ZJZ z))
cos(O,/2) cos(Oj/2)
e -’Wj e -’Wj
sin(O /2) sin (0, /2),
(5.56) and
0,7 (r)
j+.,z)) e"Pj sin(Oj/2) exp(ik+,,,+ r1j) (aj++ exp(i k Z,j,+ z) + bj++ exp(-i kZjZ 3-- z)) e’ sin(Oj/2) 3-- z) + b,+- exp(-i k ZjZ exp(ik,+,Ii7 r1j) (a,+- exp(i k Z,j exp(i k;,+z) + b.,-.+ exp(-i k;j z)) cos(Oj/2) exp(ik 11,3 F,+rll)(a3 i ,j 3
’Pj
+
,
3
,3
+
+
ZJ Z
ZJ
exp(i k;3 exp(ik IF --r1j)(a.7Z,j z) i
+
+
b37
+
exp (- i
k;3 ZjZ z)) z
cos
(Oj / 2). (5.57)
5.4.4
eight
The
constants
interface. This 8
x
and Matrices
Continuity Conditions
aj’
et
bj’
gives exactly
8 matrix but with 2
by the continuity of 0 and VO at the equations. The reflection matrix M is then a
are
8
4
non zero
fixed
x
4 blocks
(there
are no cross
terms between
"k,+,
components and the group with "k III 0 components in ,0 vector. It is possible to split the problem in two calculations using
the group with
I
their 4
x
wave
4 matrices. The
continuity
relations
can
be written:
at I+
at+
b3+
ajVj (rj)
3 +1 +
+
Oj --
aj
3+
It+
+
a,+, +
Vj+l(rj)
b37+1 I -
aj+,
Pj+1 a
b
3
aj+1
0-
j+1
5
(5.58)
Neutron
5
where the 8
8 matrix
x
Dj (rj)
is written
We
0
0
’DBj
(
write
we
ik+ +,r
’k+
0’
DAj
-
(we
Oj/2 )):
=
ik+ +r
+0
(0’) 6’ +r ik ik 11 k++e +I, Cos (0’) e Cos
e
(5.59)
of the two matrices ’DA and ’DB
explicit expression
here the
give
omit the index j and
177
:
DAj
Dj (rj)
Reflectometry
e
ik++z .
+r
+e ik,+, -k+ z
(0’) e ik +z icp e ikl+l+r e sin(fle- ikt+z
+z
z
+z j p ik++r 11 e in(fle ik +r k++e ikl+l eiW sin(fle ik
cos(O’)e-
e
-
Cos
k++ e ikl+l+r 6 i o sin (0’) e -ik +z
+z
z
(5-60) +r
ik -
e
e-"P
+re sin (0) -kz +e +r +z eik 11 Cos (O’)eik; ik
+r
ik -kZ +e 11
-e
Cos
-e
+z e ik;
(0’) eikz-
+r
ik
+z
sin(fle ik.
kz
e-4,
sin(O)e -ik.
+e’hIF,+re-i 0 sin(O)e -ik.- +,
ik,F, +r cos(O’)e- ik; + Iz +r ik +z cos(O’)e- ik,,kZ +e IF, e
+z
ikI-1-re- "P sin(O’)
ik,-- z
-e
j
ikI-1-re-ip Sin(OI)e-ik.
-kZ -e 11 -re-i ,,qin(0,)eik.--z kZ -e’ k11- e iP sin (0’) e eik,, -r cos(O’)e- ikZ -z ,%’k,, -r COS(O/ Wk. -Z ik
E) Bj
+z
=
kz-
-
e
ikil
r
-r
-
-
-kz--e’kl-l-r COS(OI)e-ik.
cos(o/) eik--z
-z
ik.
-z
-z
(5.61) ih+-r 6
11
Cos
(0’) e
k+e ikl+l-r z e
ik,+,11
r
COS
i k+-
(0’) e
eiw Sin(0
ik+-r 11 k+e z
iW e
,
e
’k+- r
ei
z
ik+-z
11
C0S
-k+-e z
ikt-z
e
Sin(o/)eik+--Z
The reflection matrix A4 is defined
’k+- z
-$
COS(OI)e-ik+-z z
i’k+ll-r e jW sin(fle- ikt-z
e’ik+jj-y -k+z
p sin(O’)e- ikz+-
z
by:
j=N
M
(01) e
ik+-r
JID, l(rj)’I)j+,(rj)
AIIA
0
0
MB
(5.62)
j=O
where N
N
MA
=
JJDAj ’E)Aj+l j=O
and
MB
JIDBi ’I)Bj+,. j=O
(5-63)
Claude Fermon et al.
178
We have the relation:
a++ 0 b++ 0 a.
a++ S b++
+
a.
b-+ 0
+
b-+ S
--
aO b0
a, b--
a+0
a+-
+_
b -
__
S
S
bo In the
case
of incident
S
"up" neutrons, equation (5.64) gives: I
t++ S
r++ 0
0
S+
0
ro0
For "down" neutrons
we
+
0
M
0
0
0
0
\0
0
have 0
0
0
0
0
0
0
0
M
0
t+ S +
"down"), efficients
-
0
VO +
(5.66)
t-
0
+ r+ 0
(5.65)
0
Q
Let
(5.64)
be the
reflectivity amplitudes for a neutron "up" (resp. "up" (resp. "down"). The corresponding transmission coWe deduce: given by t + + t
ro-
reflected are
,
r++ 0 r-+ 0
MA21MA33 MA11MA33 MA41MA33
MA11MA33
-
-
-
-
MA23MA31 MA13MA31 MA43MA31
(5-67)
MA13MA31
and
t++ S
MA33
MA13MA31 MAIIMA33 -MA31 -
t-+ S
MA11MA33
-’MA13MA3-1
(5.68)
Reflectometry
Neutron
5
179
We find similar relations for the 4 other coefficients. The reflected intensities are
given by: R++
cc
I ++12
(5.69)
R-+
oc
Jr- +12.
(5.70)
r
and
In the
case
of small external
Reflection
5.4.5
on a
magnetic field,
we
R-+
have R+-
Magnetic Dioptre
Let q,, be the (Oz) component of the consider the case of a reflection on a
scattering vector q magnetic substrate.
--
k,
-
kin We will
qO’2;
k,
kin
BO
Vacuum Im
Magnetic material
incident
Fig. 5.4. Neutron beam applied field Bo
M
ktr on a
magnetic
substrate of
M in
magnetisation
an
To
small
simplify the problem,
that q0z medium is given
qo ,
so
=
by (see 5.32):1 q,z
We will
that the applied magnetic field BO is component of the q vector in the magnetic
we assume
qo,- The
assume
’;Z
VqOz 2
-
(5.71)
b,,,).
167rp(b,,
magnetisation M (see Fig. 5.3). Let 0
that the external field BO and the to 0
90’
layer plane. This corresponds angle between BO and M. The expressions of the reflection deduced from the expressions of the A4 matrices are given by:
in the
--
the
r
++
COS
-
2
COS2
(qoz
2
0
(qo,
2
-
q+) (qo, Sz
+
qs+z) (qo,
+ qsz -) + +
sin2
sin qSz ) +
2
_ 2
0
(qo ,
(qoz
-
+
lie be
coefficients
q+) q- ) (qoz + q+
qSz )
(qoz
+
Sz
Sz
Sz
(5.72) we
remind that
to twice the
cc
qz
=
projection
2k,", the scattering of the incident
wave
wave
vector in the substrate is
vector
on
the
(Oz)
axis.
equal
Claude Fermon et al.
180
r
2qocos2
+_
COS2
(qO,
2
q+ ) (qO,
+
SZ
sin 0
(q+ -q-)
2
SZ
Sin2 q8Z ) +
+
SZ
(qO,
2
+
q8Z )
(qO,
+
q+
3Z
(5.73) The measured intensities
given by:
are
R++
Jr++ j2
--
R+-
et
--
jr+- j2
(5.74)
Case of a Non-Magnetic Substrate In this case, corresponding to a zero magnetisation (b, 0), the scattering vectors q,+ and q- are equal (eq. 5.71). The reflection coefficients simplify and can be written in the form of =
Z
SZ
classical Fresnel coefficients:
r++
qo,
-
qz
r+-
and
_
=
(5-75)
0.
qoz + q,Z
-
The reflected
is
intensity R++
given by: qo ,
-
2 qo,
q ,z
-
-
-
qoz + q,z
qo, +
.X,rq0Z q, j_2:__:::2 VqO qc 2 0z
C
a
2
(5.76)
wave vector q, is equal to -\1_167_rpb,, When q0z < qc qz imaginary number and one finds a reflected intensity equal to 1. 4 q0z is very large, one can show that the intensity decreases as llqo,.
where the critical is
2
-
-
,
pure
When
Case of
a
Aligned
expressions r
Magnetic Substrate in Magnetisation
qO,
++
-
q+
-
+
qo , + qSZ
expressions
cation introduced
critical
angles
for
r
,
Magnetic Field BO (,0 0) In this simple case, and can be written as: simplify
M
of the reflection coefficients
.
These
a
with the
qo,
-
=
q, , -
-
qo, +
and
qs-z
r
+-
the
(5-77)
still
correspond to Fresnel reflectivities. The only modifiby magnetism is a difference in the critical angle. The the reflectivity curves "up-up" and "down-down" are given the
by:
q C
The
spin-flip signal
(R:
and
V_16_Tp-(b,, b,). R::F)
is
N.B.: the coefficients r++ and r--
(5-78)
zero.
be deduced one from the other by a 0 rotation. The non spin-flip signals are plotted in solid lines on Fig. 5.5. We find classical shapes for the reflectivity curves, with a total reflectivity plateau followed by a sharp decrease. The main difference between the "up-up" and "down-down" curves is the extension of the total reflectivity plateau. 180’
can
Neutron
5
’it
Reflectometry
181
((MMLlp eli -
-
0.1
B) up-, B),Llp-t (M (M up-t up-up down-down d 0, B) do% (M e.. ej up-down/10 B) u (IM perp. (1\4 rp P. B) (M perp. up-up rp p. B)) U] p u]
0.01 B
CU 0. 001
0.0001
0.00001
q,-
0
0.2
qc"
0.8
0.6
0.4
qo
1
(nrn-1)
Reflectivity curves in the case of a magnetisation parallel and perpendicumagnetic field B0. When the magnetisation is parallel to the field, the non spin flip curves "up-up" and "down-down" are distinct (solid lines), the spin-flip signal is zero. When the magnetisation is perpendicular to the field Bo, the spin-flip curves superimpose (squares), a very large spin-flip signal appears (losanges) (the spin-flip signal has been divided by a factor 10 for clarity
Fig.
5.5.
lar to the
Case of
sation
a
(0
Magnetic =
(qo,
r
-
90’)
Field
In this
2
+
Magneti-
to the Substrate
the reflection coefficients become:
q,-j + (qo_, q,- ) (qo, (qo, + q+ ) (qo, + q- )
q ) (qo, SZ
Perpendicular
case
-
SZ
8Z
+
q ) 2
SZ
SZ
(
r
++
+,r--
)
(5.79)
r+-
R++
=
R--
-
Ir 12
-
are
I =
4
(5.80)
8Z
5Z
-
2
The reflected intensities
q(q+ q+ ) (qo,, +
q0, -
(qo
+
SZ
q8Z )
*
given by:
I r++ 12 +
1
4
I
jr-- 12 +
2
Re
(r++
x
r--)
(5.81)
notice that the up-up and down-down intensities are the sum of three terms. The first two correspond to the intensities of the non spin-flip signals
One
can
in the
case
a
magnetisation aligned
curve).
a
C
with the external
magnetic field
;
they
coefficient. These terms introduce two discontinuities
weighted by 1/4 the positions q+ et q-
are
at
of
C
in the
To these two terms,
an
reflectivity "
curve
interference
"
(see figure 5.5,
term adds
!Re 2
white square
(r++
x
r--)
Claude Fermon et al.
182
whose
analytical expression
is not
simple.
Its variations
are
plotted
on
Fig.
5.6. For qo,, = 0, this term is equal to 1/2 and the intensity is totally reflected. Its value decreases as soon as qo , increases and becomes negative around qc-. It becomes
positive again
around
this contribution does not curve
except that there is
q+, C
then decreases very quickly. However, of the non spin-flip
modify qualitatively the form no total reflectivity plateau.
0.5 04 .
03 .
0 2.
0 1.
0-
0 1.
-0.2-0.2
0.4
qo
Fig.
Contribution of the interference term
5.6.
amplitudes whose magnetisation and r--
The
in the is
0.8
0.6
(--C’)
Re(r++
x
r--)
between the r++
spin-flip intensities for a reflection on perpendicular to the applied magnetic field Bo
spin-flip intensity
non
is
a
substrate
given by: Z
R+_
1
12
qo ,
4
(qo.,
+
(q+
Sz
-
qSz
qs+z) (qo-,
+
(5.82)
qs-z)
The characteristic form of the
spin-flip signal (see figure 5.4) is given by the The variations of this term are plotted on the figure 5.7 term jq+ q- 12 (thick lines). Two successive regime changes appear at the points q. and qc+ They correspond to the points where q- and qs+z successively change from pure imaginary to real values. This signal is slightly modulated by the factor q0z which gives a linear increase. The factor 1 I(qoz + q+) (qo ’ + q- )12 gives a very fast decrease at large q,. Its variations are plotted on Fig. 5.7 (thin lines). In the- case where the magnetisation is not fully perpendicular to the applied magnetic field, the three terms in the R++ intensity are weighted by 4 2 4 sin and 2 COS 2 _0 sin _ factors, 0 being the angle between the field COS 2 2 2 and the magnetisation. In. the case of a magnetic layer deposited on a nonmagnetic substrate, the above considerations are not qualitatively modified. The main difference is that Kiessig fringes appear after the plateau of total -
Sz
Sz
.
-
8z
3z
,
reflection.
Sz
Neutron
5
Reflectometry
183
10
1
0.1
0.01
0.001
0,0001 0.2
0
qo
-
Sz
qSz
12;
(thin lines)
factor
Non Perfect
5.5
and
5.5.1
1/ 1 (qo,
1
spin-flip intensity: (bold lines) factor
q+ ) (qo_ Sz
+ qSz
)12
Layers, Practical Problems
Experimental
Interface
+
0.8
(nrdl)
Variations of two factors of the
Fig..5.7.
jq+
0.6
0.4
Limits
Roughness
Most of the studied systems show
imperfect
interfaces
depending
on
the
depo-
sition process of the layer. We will consider three roughness scales: interface roughness, atomic interdiffusion and homogeneity of the layer thickness. Let
represent the characteristic lateral length-scale for the roughness. A perfect
knowledge of the surface would correspond to the knowledge of z(x, y) for all in-plane lengthscales. The treatment of the roughness is very similar to that described in Chaps. 2 and 3. According to the resolution of a typical neutron reflectivity experiment, one can (somewhat arbitrarily) distinguish three typical types of roughness which have different origins. 9 Interdiffusion of the species between two successive layers. This happens during the deposition of a top layer which is miscible with the bottom material. This proce"ss is strongly temperature dependent. It corresponds to a typical lengthscale of < 0.5pm. * A roughness induced by rough edges on the substrate or by grains in the case of two successive layers. This roughness usually occurs during thin film growth. It is also the type of roughness which is difficult to take into account in models. It corresponds to Ipm < < 100yrn. Flatness of the sample. Depending on the deposition process, the atomic flux may have an angular dependence which can lead to an uneven thickness > 100prn. over the sample surface. It corresponds to These three
roughness
account for their effects very different effects
on
scales
on
the
can
be modelled in three different ways to reflectivity curves. They induce
the measured
experimental signals. One
limitation: if the lateral fluctuations
the
following
the
layers thicknesses the
following
treatments
are
are
has to
keep
not small
inadequate.
in mind
compared
to
Claude Fermon et al.
184
Thickness
Inhomogeneity of the Sample Thickness variation in a thin sample (usually between the middle and the sample edges) is a "large" lateral scale problem (a few mm). The experimental measured curve can be treated as the superposition of reflectivity curves calculated for the thicknesses spectrum weighted by the corresponding area. The resulting effect is a blurring of the coherent Oscillations for large q. N.B.: Since the Kiessig fringes period is inversely proportionnal to the wavelength of the incident beam, a thickness fluctuation (which reflects in the Kiessig oscillations period) can be taken into account as an incident wavelength spread JA. Figure, 5.8 (thin line, JA 10%) illustrates the effect of a wavelength spread; it also corresponds to what would be observed for a 10% sample thickness fluctuation. film
--
and Interdiffusion
Roughness
Specular reflectivity cannot distinguish roughness. The measurement of the coherent scattering length density pb probes a large planar scale compared to the size of the roughness: for a given z depth, one measures a mean value of pb averaged over a large surface. between these two type of
First solution: N6vot-Croce
factors
If
one
assumes
a
flat distribution of x,
the two types of interface can be treated by a single model where the step function is replaced by the following error function:
erf
(z-zj)
2
=
O’j
This
curves
shows
inverse of the
curve
an
inflexion
slope
at
Zj’.
OW In
(’ -zj)l’j
6_t2 dt.
(5.83)
point at zj. The value o-j is given by the The thickness is given by 2o-j. The effect of a
smooth interface surface described
by 5.83 is to multiply the reflectivity R of perfect flat interface by a Debye-Waller (or better N4vot-Croce, see Chap. 3, Appendix 3.A) factor [17]: a
2
exp(-2k,jk,j+jO-D In the
case
of
a
stack of
multilayers
each
having
(5-84) a
specific roughness, the Unfortunately, this preventing an easy
N6vot-Croce factor is applied to each transfer matrix. cannot be applied in the magnetic case, the formalism calculation of the
reflectivity R at each interface. However one can introduce apply this factor to each diagonal elements at each interface. In practice this works quite well except in the case of rather strong magnetic roughness like domains. The main effect of this factor is to decrease the reflectivity at high q. a
global
factor and then
Second solution: discretisation This
to model atomic
interdiffusion. The interface is
of discrete
technique is efficient replaced by a finite number
layers
5
describing the concentration index. Either tion profile can be used. For real systems an average pb usually works well.
Neutron
an error
Reflectometry
function
of thin solid
or a
films,
185
linear func-
one
layer with
Roughness In the case of the intermediate roughness, the previous methods are not completely satisfactory. Actually, this type of roughness not only decreases the speqular reflectivity but also creates a non specular diffuse background which can modify the results. In this case, the diffuse scattering should be measured and the specular reflectivity should be corrected accordingly. This treatment is quite complex and will not be detailed Intermediate
here.
Magnetic Roughness This problem is very complex. A. typical example where magnetic roughness appears is the case of a de ’magnetised sample. In this case each domain has an effective scattering length very different from its neighbour. This appears for neutrons as a giant roughness. There is no simple theoretical
way of
netic measurements
can
in
taking
this into account yet. Non
some cases
give
specular magmagnetic
the average size of the
domains.
5.5.2
Angular Resolution
expressions given above are valid for a perfect incident beam. experimental data, it is important to have a good knowledge of the beam divergence and homogeneity. The beam angular divergence and wavelength dispersion must be taken into account in the simulations. The divergence of the incident beam, 60, is usually determined by two slits if the beam is smaller than the effective width of the sample seen by the neutron beam, or by the first slit and the sample itself if the sample is small enough to be totally illuminated by the neutron beam. Usually, JO is fixed during the experiment. We have then to convolute the calculated reflectivity with a function which is the experimental shape of the beam divergence. However a of that function. In square function gives in most cases a good approximation the case of curved samples, 60 can be slightly adjusted during the treatment. JO has two effects: a decrease of the amplitude of the oscillations and a rounding of the discontinuity at the critical angle. Figure 5.8 gives an example of this effect. Wavelength dispersion is strongly dependent on the monochromator or on the time resolution in the case of time of flight spectrometers (see below). The effect of that dispersion is different from an angular divergence that if : the oscillations disappear at high angles (see Fig. 5.8). We remind similar is effect the (see a sample has a non,homogeneous thickness, very above). A wavelength dispersion can be used to model thickness variations over the sample surface.
The different
For the fit of
Claude Fermon et al.
186
perfect instrument 0 1 .
-50
=
0.04’
82,
=
10%
-
0 01 .
0
AON
0 001-
VYW
.
0 0001
V
.
0.00001 0.5
1
q,
,
Fig.
5.8. Effect of SO and SA.
strumental
SO,
and
a
SA for
a
1.5
(nm)
Comparison
between
measurement
on
a
a
perfect instrument, an in30 nm thick layer on a
single
substrate
5.5.3
Analysis
of
Experimental
Data
Reflectivity curves cannot be directly inverted. For a non magnetic system, it is even possible to build a family of profiles which give the same reflectivity curve. This is due to the fact that we measure only the intensity and loose the phase of the reflectivity [181. For magnetic systems, the problem of the signal phase is less critical. However, the main source of uncertainty on the result is in general due to the lack of intensity at high angles. The analysis of experimental data is done by adjusting the different parameters involved in the problem until a good fit is obtained. In the case of magnetic systems, we usually know rather well the composition of the different layers. We have then to adjust the roughness, the. thicknesses and the magnetic moments magnitude. It is in general very useful to have some external information like x-rays reflectometry and magnetic hysteresis curves.
5.6
5.6.1
The Spectrometers
Introduction
The spectrometers
can
be divided in two different groups: time of
flectometers like EROS at the Laboratoire L6on Brillouin SURF at
ISIS,
flight
re-
CRISP and
and monochromatic reflectometers like PADA at LLB and
ADAM at the Institut Laue necessary for
(LLB),
Langevin (ILL). Time reflectometry studies on liquids.
of
flight spectrometers
are
Neutron
5
5.6.2
Time of
Flight
Reflectometry
187
Reflectometers
The time of flight ’technique consists in sending a pulsed white beam on the sample. Since the speed of the neutron varies as the inverse of the wavelength, the latter is directly related to the time taken by the neutron to travel from the pulsed source to the detector (over the distance L) by : A
h t.
(5.85)
(p 8) 2527L(m
(5.86)
=
mL
This relation is also written
as
:
t
A (n m)
spallation source, the neutron beam is ,naturally" pulsed and the time of flight technique is used. On a reactor, pulsed neutrons are produced by a chopper. For a reflectivity measurement, the angle is fixed and the reflectivity curve is obtained by measuring the reflectivity signal for each wavelength of the available spectrum, each wavelength corresponding to a different scattering wave-vector magnitude. Sometimes, it is necessary to use several angles because the q, range is not large enough. An example of time of flight spectrometer is presented on Fig. 5.9. On
"
a
Neutron
Chopper
Collimator
Detector
sample
Incident beam
/
L
Moving
or
bi-
dimensional Reflected
detector
beam
0
Electronic
>h,,
Acquisition PC
Fig.
5.6.3
5.9.
Description
of the time of
flight
-----
reflectometer EROS at the LLB
[19]
Monochromatic Reflectometers
basically two axes spectrometers. The wavePADA) and the reflectivity curve is obtained by
Monochromatic reflectometers
length
q-------------------------------
is fixed
(0.4
nm
for
are
Claude Fermon et al.
188
changing
the incident
angle
0. In this case, the
sample is usually vertical. On
this type of reflectometers it is easy to put a polariser and an analyser in order to select the spin states of the incident and reflected neutrons.The flippers
be of Mezei type (2 orthogonal coils) [20]. They allow to flip the neuspin state from "up" to "down". An example of two-axis spectrometer is presented on Fig. 5.10. can
tron
white beam
polariser
flipper I
a
ple flipper 2 -7
P",
7
-analy-s-er 20
graphite monochromator
collimation slits
detector
M
Fig.
5.7
5.10. The two axis reflectometer PADA at the LLB
Polymer Examples
Reflectometry is widely used for soft matter studies due to the large penetradepth of the neutron beam and the large contrast between deuterated 6.67fm) and protonated (bH (bD -3.7fm) systems [21,22]. We shall illustrate here the use of selective deuteration’to study the polymerinterdiffusion. By spin coating, it is possible to deposit polymers layers on glass or silicon with a roughness below I mn. It is then possible to deposit a second layer on the first one. If one of the layers is deuterated, it is possible to study the interdiffusion as a function of time and annealing temperature. The diffusion will appear as a smearing of the interface between the two layers and thus a decrease of the Kiessig fringes amplitude. tion
=
: _-:
Interdiffusion Between Diblock
Copolymer Layers Diblock copolytogether (A B). These systems present a large variety of interesting properties. For example, if A and B are not miscible, they can form self-organised multilayers of a fixed thickness parallel to the surface where the solution is deposited. The observed structure is of the type (substrate; A B; B A; A B; B A ). We have studied diblock copolymers of the type (polystyrene polybuty1metacrylate: PS-PBMA). The initial system consisted in a layer of partially deuterated mers are
made of two chains A and B linked
-
-
-
-
-
...
-
Neutron Reflectometry
5
189
copolymer deposited on a trilayer of totally hydrogenated copolyreflectivity of this system is shown on Fig 5.11. The numerical fit shows a large index at the top of the system corresponding to the deuterated copolymer. The system has then been annealed for 12 hours at 400K and then remeasured (Fig.5.11). On this reflectivity curve, one can observe a 0.1 nm-’. This indicates the diffusion clear "Bragg" peak at the position q of the deuterated polymer to the inner layers. Since the diblock copolymers are ordered in multilayers, a periodic variation of the index appears (see insert on Fig. 5.11). Many other examples will be found in Chap. 10 which is entirely devoted to the discussion of polymer studies. PS-PBMA
mers.
The
-
--
niarledoyn-er
1
0.01
0.04-
o
suhtrate
0.1
0.1
O.W
Skstrat
>
0
0.01
0.01-
Thdaiess (rui
0.00 0
Mida-ess (m 0.001-
0.001-
o
0.0001
V
0.00M
o
-
o
oo
0.00001 0
q
q
(14
0.3
0.2
0.1
0
0.4
O’
(nnrl)
oo
o."I
0.00001
0.2
0.1
oo
(Poo 1
1
(nnrl)
Left: Reflectivity of a quadrilayer consisting in a partially deuterated copolymer layer deposited on a trilayer of totally hydrogenated polymer (measured on the EROS reflectometer at the Laboratoire L6on Brillouin.) Right: Reflectivity of the quadrilayer after annealing for I hour at 115’C
Fig. 5.11.
PS-PBMA
5.8
Examples
on
Magnetic Systems
give some examples in order to highlight the information that can be obtained by polarised neutron reflectometry. All the experiments shown here have been performed on the reflectometer PADA.
In this part,
shall
Absolute Measurement of
5.8.1 Neutron
atom).
we
reflectometry
can
be used to
sample. As an NiFe single layer. By fitting Fe with
an error
Magnetic
measure
independent example, Fig.
The obtained value is
surface of the
about
a
Moment
absolute moments
of the
(in
1-tB per
layer thickness and of the
5.12 shows
a curve
obtained
on a
the curves, we obtain the ratio between Ni and about of 2% and the absolute moment with a precision of
0.02/-tB. That
measurement took
only
15 minutes
on a
I
cm
2
sample.
Claude Fermon et al.
190
1000000
100000
10000
1000
100 0
0.2
0.4
0.6
q
Fig.
reflectivity R++ intensity and white 5.12.
5.8.2
curve
Bragg
on
a
25
squares R--
Peaks of
nm
0.8
1
1.2
1.4
(nm )
thick NiFe
layer. Black
squares
are
the
intensity
Multilayers
Periodic
Multilayer In the case of periodic multilayers, we can observe Bragg peaks corresponding to the period of the multilayer. In the case of antiferromagnetic coupling or variable angle coupling, it is possible to obtain directly a mean angle between the different magnetic layers. With polarised neutrons, it is possible to measure very rapidly a precise value of the average moments. If high"order Bragg peaks are observed, a good estimate of the chemical and magnetic interface can be obtained. In the literature, there is a large amount of results on magnetic multilayers [18,23,24]. 1000000
100000
10000
1000
100
10 0
0.02
0.04
0.06
0.08
q
Fig. spin
Example of a polarizing mirror. The two curves correspond respectively "up" (black squares) and "down" (white squares) state of incident neutron
5.13.
to the
Reflectometry
Neutron
5
191
Supermirrors [25] For technical purposes it is interesting to build systems exhibiting an articially large optical index. One can can build such a structure by stacking periodic multilayers with an almost continuous variation of the period. In such a system, if the periodicity range is well choosen, a large number of Bragg peaks follow the total reflectivity plateau. Since the periodicity of the multilayer is varying continuously, all these Bragg peaks add constructively. Using this technique it is possible to enhance the length of the total reflection plateau by a factor 3 to 4. Such mirrors are now widely used for neutron guides and for polarisation devices.’Figure 5.13 gives an example of a polarising mirror. Measurement of the In-Plane and Out-Of-Plane Rotation of Moments. Measurement of the Moment Variation in a
5.8.3
Single Layer perhaps the most important information given by polarised neutron reflectivity (PNR) for magnetic thin films. We shall give here two examples of determination of in-depth magnetic profiles.
This kind of measurement is
1000000
100000--
8 ML
Alloy (air/Au) d-down
Au
10000--
6.3
(111)
nm
5 ML
Alloy (Co/Au)
15
1000P-rip
101
-
100
-
__
-
-
-
-
-
-
-
-
R
-
0o
i
0
".
1.5
1
0.5
0
1000000
hcp (0001).
Co
%
2
Alloy (Co/Au)
flat
(degreE!s) Au
100000
(111)
28.4
Alloy (air/ float glass) Float
100001
nm
5 ML
glass
1000
100
10 i 0
1.5
1
0.5 0
2
(degrees)
Reflectivity curve of a Au/Co(3nm)/Au trilayer system. Empty squares: reflectivity; filled squares: down-down reftectivity; lines: best fits. Top left: fit assuming a uniformily aligned magnetisation through the layer. Bottom left: fit with a model allowing magnetisation rotation; this model provides a better fit than the model using a uniform magnetisation. Right: Thicknesses and moment
Fig.
5.14.
up-up
directions
giving
the best fit
parameters
Claude Fermon et al.
192
Out-Of-Plane Rotation of the Moments in
Au/Co/Au [26].
In the
of very thin cobalt layers, thinner than 2.5 mn, the magnetisation is perpendicular to the surface layer. We have studied a 3 nm thick Co layer case
sandwiched between two Au
layers. Magneto-optic Kerr effect measurements mainly in-plane but with a small out of order to understand the magnetic behaviour of such a
have shown that the moment is
plane contribution. In layer, we have fitted the one,
we
PNR
curves
with two different models: for the first
have considered that the whole
second one,
reflectivity
have allowed
we
curves
and the
a
layer
rotation in the
corresponding
uniformly aligned. In layer. The Fig. 5.14 gives
was
the the
model.
Rotation in Strained microstructure
throughout
Nickel Layers In single magnetic thin films, the magnetoelastic (ME) properties vary sample: a gradient of the ME coefficient B(z) can appear, be such that the
can
the
related to surface relaxation effects. It surface
anisotropy
constants
B(z) where
is the
can
be written in
a
form similar to
[27]: =
Bbulk +
(z
-
(5.87)
ZO)
depth
in the thin film and zo an adjustable parameter. When a applied on a magnetic thin film, the magnetisation tends to rotate either along or perpendicular to the applied strain. A ME coefficient gradient will then lead to a gradient of magnetisation rotation through the thin film. This has been measured on single nickel layers as illustrated on Fig. 5.15. The numerical fit shows that the average rotation under a 0.03% deformation is 30’ but there is a 15’ gradient of rotation between the surfaces z
mechanical strain is
and the bulk of the material
elongation strain
03"a%,
1000000
[28].
!-..wn--D.wn -
’%k
116
field
’P_Up
Up-Up
M
’I -U Up P -Down
100000
Down-Down o 75*
glass substrate
Oo
-o
10000
-----------
100’
T
-Z
Nickel
(40 run)
100 C)
0.2
0.4
0.6
0
Fig..5.15.
Left:
0.8
1
1.2
1.4
(&gr-)
Reflectivity curves on a strained nickel layer (thickness 40 nm) for spin. The deformation applied to the substrate is 0.03 %. Right: Diagram showing the magnetisation rotation gradient in the strained 40 nm Ni layer deduced from the neutron fit each state of neutron
Neutron Reflectometry
5
A
Hysteresis Loops
Selective
5.8.4
complete
hours to be
193
reflectivity performed. If we
set of
layers as a function too long compared
of field
or
curves
(R++, R--
and
R+-)
takes about 12
magnetisation of different total experiment would be far
want to follow the
temperature, the
to the time
usually allocated
on a
neutron reflectometer
week per year). So the idea is the following [29]: from the fits of (typically the reflectivity curves in the saturated state we know the different thicknesses one
and
magnetic
multilayer system. We
states of the
are
then able to calculate
magnetic perform complete reflectivity curves for each value of the magnetic field, but we can measure only the reflectivity for (n + 1) well chosen 0 values where n is the number of different magnetic layers. Comparison of the experimental values obtained and calculations using reflectivity curves layers. It is then not the
for different module and orientation of the
necessary to
the parameters obtained from the saturated state allows us to rebuild the magnetic evolution as a function of the applied external field.
(gB/atome)
moment 2
T-
-6
3
0
-3
field(mT)
Fig. uous
5.16.
Hysteresis loops measured and from the reflectivity
curve)
squares,
0.4’ black
for
a
single
cobalt
measurements at
layer : by MOKE (continonly one angle (0.3’ white
squares)
1.6 1.4
.2
1.2
0.8 0.6 0.4 0.2 0 10
5
H
Fig. 5.17. Magnetisati6n
/Co(3.8A) /Pt(32A)
of each
(kG)
magnetic layer
of
thin film. The white squares
a
/Pt(98A) /Co(7.6A) /Pt(33A)
correspondto
the thinnest
layer
Claude Fermon et al.
194
Figure 5.16 gives the example of an hysteresis loop obtained for a single magnetic layer. Figure 5.17 corresponds to a more complicated system: substrate
/Pt(98A) /Co(7.6A) /Pt(33A) /Co(3.8A) /Pt(32A).
Such a technique layer separately. The sum of the two magnetisations agrees well with the magnetisation given by conventional measurements and the saturation value of each layer corresponds to the values measured on other samples with just one layer. has
given
5.9
the moment of each
Conclusion
on
Neutron
Reflectometry
This
chapter has given an overview of the neutron reflectometry as a tool for investigation of surfaces. We have presented a matrix formalism which makes it possible to describe the specular reflectivity on non-magnetic and magnetic systems. Neutron reflectivity is especially suited for polymer and magnetic thin film systems. This has been illustrated with a few typical examples. We have not given here examples of non-specular and surface diffraction experiments. This kind of experiment has suffered until now from the lack of intensity on the neutron spectrometers. Moreover, the formalism necessary to analyse the experiments in the case of magnetic surface diffraction is still being developed. The neutron has a good energy for inelastic scattering on condensed matter but we have not spoken here on this aspect of reflectometry which is rather new. A beautiful example of inelastic scattering is the measurement of the Zeeman energy [30,31]. The problem of phase determination in neutron reflectometry is also an active field of research [32-34]. If not only the intensity but also the phase of the reflectivity could be measured a direct inversion of the reflectivity profile would be possible. the
References 1.
2.
3. 4.
G.P. Felcher, R.O. Hilleke, R.K. Crawford, J. Haumann, R. Kleb and G. Ostrowski, Rev. Sci. Instr, 58, 609 (1987). C. F. Majkrzak, J. W. Cable, J. Kwo, M. Hong, D. B. McWhan, Y. Yafet and J. Waszcak, Phys. Rev. Lett. 56, 2700 (1986). J. Penfold, R.K. Thomas, J. Phys. Condens. Matter 2, 1369-1412 (1990). T.P. Russel, Mat. Sci. Rep. 5, 171-271 (iggo); T.P. Russel, Physica B 221, 267-283
5.
6.
7.
8. 9. 10.
11.
(1996).
Lee, D. Langevin, B. Farnoux, Phys. Rev. Lett. 67, 2678-81 (1991). J. Penfold, E.M. Lee, R.K. Thomas, Molecular Physics 68, 33-47 (1989). M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friedrich, J. Chazelas, Phys. Rev. Lett. 61, 2472 ( 1988). V.F. Sears, Physics Report 141, 281 (1986). X.L. Zhou, S.H. Chen, Physics Reports 257, 223-348 (1995). H. Glittli and M. Goldman, Methods of experimental Physics, Vol 23C, Neutron Scatterinj (Academic Press, Orlando, 1987). S. Dietrich, A. Haase, Physics Reports 260, 1-138 (1995). L.T.
Neutron
5
Refiectometry
195
Sears, Methods of experimental Physics, Vol 23A, Neutron Scattering (Academic Press, Orlando, 1987); V.F. Sears, Neutron News 3, 26 (1992). J. Lekner, Theory of reflection of electromagnetic and particle waves (Martinus Nijhoff, Dordrecht, 1987). S.J. Blundell and J.A.C. Bland, Phys. Rev. B 46, 3391 (1992). C. Fermon, C. Miramond, F. Ott, G. Saux, J. of Neutron Research 4, 251
12. V.F.
13.
14. 15.
(1996). 16.
Z., Physica B 94, 233-243 (1994). Croce, Revue de Physique Appliqu6e 15, 761 (1980). C.F. Majkrzak, Physica B 221, 342-356 (1996). B. Farnoux, Neutron Scattering in the 90’1 Conf. Proc, IAEA in Jiilich, 14-18 january 1985, 205-209, Vienna, 1985, X.D. F. Mezei, Z. Phys. 255, 146 (1972). T.P. Russel, A. Menelle, W.A. Hamilton, G.S. Smith, S.K. Satija and C.F. Majkrzak, Macromolecules 24, 5721-5726 (1991). X. Zhao, W. Zhao, X. Zheng, M.H. Rafailovich, J. Sokolov, S.A. Schwarz, M.A.A. Pudensi, T.P. Russel., S.K.Kumar and L.J. Fetters, Phys. Rev. Lett. Pleshanov
17. L. N6vot and P. 18. 19.
20.
21.
22.
69, 776 23.
YY.
(1992).
Huang,
G.P. Felcher and S.S.P.
Parkin,
J.
Magn. Magn. Mater. 99,
31-38
(1991). Schreyer, J.F. Aukner, T. Zeidler, H. Zabel, C.F. Majkrzak, M. Schaefer Gruenberg, Euro. Phys. Lett., 595-600 (1995). P. B6ni, Physica B 234-236, 1038-1043 (1997). E. Train, C. Fermon, C. Chappert, A. Megy, P. Veillet and P. Beauvillain, J. Magn. Magn. Mater. 156, 86 (1996). 0. Song, C.A. Ballentine, R.C. O’Handley, Appl. Phys, Lett. 64, 2593 (1994). F. Ott, C. Fermon, Physica B 234-236, 522 (1997). C Fermon, S. Gray, G. Legoff, V. Mathet, S. Mathieu, F. Ott, M. Viret and P. Warin, Physica B 241-243, 1055 (1998). G.P. Felcher, S. Adenwalla, V.0. De Haan and A.A. Van Well, Nature 377,
24. A.
and P.
25. 26.
27. 28. 29.
30.
409 31.
G.P.
(1995). Felcher,
494-499 32. 33.
34.
S.
Adenwalla, V.0. de Haan, A.A.
van
Well, Physica
B 221,
(1996).
Majkrzak and N.F. Berk, Phys. Rev. B 52, 10827 (1995). C.F. Majkrzak and N.F. Berk, Phys. Rev. B 58, 15416 (1998). J. Kasper, H. Leeb and R. Lipperheide, Phys. Rev. Lett. 80, 2614-2617 (1998). C.F.
Statistical
6
Alberto
Physics
at
Crystal Surfaces
Pimpinelli
LASMEA, Universit6 Blaise Cedex, France
Pascal
-
Clermont 2, Les
C6zeaux,
63177 Aubi6re
1
Thermodynamics
Surface
6.1
Surface Free
6.1.1
Energy
According to thermodynamics, physical properties can be deduced from the knowledge of the free energy. In this lecture, the surface free energy is introduced in a simplified way (disregarding, in particular, elasticity). In order to create a surface, one has to break chemical bonds, and this costs energy. At finite
temperature, the free energy has
to be considered. It is
easy to give, a precise definition of the surface free energy: to break a crystal along a plane, it requires a work W. If L’ is the area of the crystal section,
the surface free energy for unit area or surface tension is ao W/(2L2), the factor of 2 coming from the two surfaces which are created this way. It is
straightforward
area
to
see
that the number of broken bonds per unit surface particular, in the broken-bond
varies with the surface orientation. In
approximation compact than open
surfaces
are
expected
to have
a
larger
surface tension
that, (Problem, temperature.) Prove
for
instance, crystal, Given this dependence on orientation, if the surface is not a plane the total surface free energy may be expected to be the integral of the energies of all surface elements. This is only true for an incompressible solid of large enough size (the interested reader will find a discussion of this statement in o-(111)
1.155o-(Ool)
o-(111)
[1].
ones:
>
for
o-(ool)
an
>
o-(11o)
fcc
at low
If x, y and z are Cartesian coordinates and z(x, y) is the height of the over the xy plane, the local surface tension is then a function o-(p, q)
surface of the
partial derivatives P
(9Z1,9X,
q
azlay
representing the local slopes of the surface profile. The total free large, incompressible solid body containing N atoms is then: F
--
Fo +
A.
on the book "Physics of crystal Villain, Cambridge University Press (1998). The should refer directly to it for delving deeper into the different
Pimpinelli
interested reader
a
I I o-(p, q)dS
Most of the material in this lecture is based
growth",
energy of
subjects quickly treated
& J.
here.
J. Daillant and A. Gibaud: LNPm 58, pp. 199 - 216, 1999 © Springer-Verlag Berlin Heidelberg 1999
Alberto
200
Pimpinelli
where the first term on the right-hand side is the bulk free energy, and the second term is the surface free energy. The integration is made over the surface of the solid and dS is the surface element. For
an
incompressible solid,
F(T, V)
one can
equivalently
use
the
(Helmholtz)
the Gibbs free energy (free enthalpy) G(T, P) PV. The latter is known from thermodynamics to be equal to yoN
energy
or
thermodynamic limit constant pressure we
shall
N
and in the absence of
-+ oo
adequately described
are more
surface).
a
--
free F +
(in
the
Processes at
if G is used. In the
future,
the free energy F because, in the absence of external it results directly from the interactions between the molecules of the
mainly
use
forces, solid. Sometimes,
we
will
drop the
where energy and free energy In formula
(6. 1),
are
word
"free", especially
at low
temperature
not very different.
the free energy is assumed to be independent of the only true for a large sample. (Problem. Why? Is the .
local curvature. This is
assumption appropriate
for
membranes?)
Another
simplification
comes
from
the fact that the surface itself may be difficult to define: its location may be rather imprecise. This point has been addressed by Gibbs [2]. Also, in formula
(6. 1) an
z (x, y) is analytic, which implies that fluctuations on ignored-or, better, averaged out (what is called coarse-
it is assumed that
atomic scale
graining
are
in the
jargon).
Free
6.1.2
Step
At
temperature,
zero
Energy
a
(001)
or
(111)
surface is flat. At
thermal fluqtuations will make atoms leave the surface
higher temperatures, plane and create holes
and mounds. In other words, steps are present on the surface at any finite temperature. In the same way as the surface free energy, a step free energy may be defined. This is how to proceed: a bulk solid is cut into two pieces of
cross
section L
x
L, but this time
a
step is also
cut. The work needed to
take apart the two pieces is proportional to the total surface exposed, which is now L2 + U, if b is the step heigth. The total free energy is thus W 0"-
b
-
L2
-
0" 0
(6.2)
+7L
The "excess" free energy due to the step, ^/, is called the step free energy or step line tension. If more steps are present, (6.2) shows that the total free energy is
proportional
to the number of
steps
per unit
length,
or
step density,
the steps are far enough from each other that their interactions can be neglected. Broken bond arguments show that the step free energy is a function of the step orientation. as
long
as
6
Singularities
6.1.3
Statistical
Physics
at
Crystal Surfaces
201
of the Surface Tension
Let us suppose that the plane z -- 0 is parallel to a high-symmetry orientation, for instance (001), and that the temperature is low. We will see the precise
temperature" when addressing the roughening transition experimental realisation of a flat high-symmetry a certain length. In other words, real surfaces are always slightly "misoriented" or "miscut" with respect to a given highsymmetry plane, so that they make a small angle 0 with that orientation (a typical lower bound for 0 is 0.1’). As said above, if step-step interactions are neglected-which is licit, if steps are rare-the projected surface free al cos 0 is simply equal to the step free energy Multiplied by the energy 0 step density I tan 0 1:
meaning
of "low
in Sect. 6.2. Of course, no surface is possible beyond
-
--
0(0)
-_
0(0)
+
-yJ tan 01
:--
0(0)
+
-yV/p2
+
q2
.
(6.3)
This shows that the surface free energy is non-analytic (as a function of the slopes p and # The line tension -/ is a function of the step orientation
local
y"
-_
dy/dx
--
-p1q.
As
a
matter of
fact,
it is
an
analytic function
at all
singular behaviour of the surface free energy at low temperature is typical of the high-symmetry, low-index orientations, which are therefore T. The
singular. These appear as flat facets in the macroscopic equilibrium crystal. The linear size of a facet is proportional to -/. These shape statements are proved using the so-called Wulff construction, a geometrical construction which relies the surface free energy of 4 crystal to its equilibrium shape [3]. Using the projected free energy 0 is useful to derive the analytic equivalent of Wulff’s contruction from a variational principle: the equilibrium shape of a crystal minimises the surface free energy for a given volume [4,3]. called
of the
6.1.4
Surface Stiffness
equation (6.3) may hold at all temperatures. Steps can 0. At finite temperature, deriving (6.2), only at T they must fluctuate-in what is called step meandering. It may well be that a step becomes so "delocalised" that the concept of step looses its meaning altogether, above a given temperature. Indeed, we will see in Sect. 6.2 that such a temperature exists, and the corresponding transition is the roughening transition; in the rough, high-temperature phase, the step line tension ^/ actually vanishes. What is the form of the surface free energy at high temperature? Our experience with critical phenomena suggests that it may be useful to assume for the free energy an analytic expansion a la Landau in the disordered, high temperature phase [4]. To do that, it is convenient to transform the surface integral in (6.1) into an integral over Cartesian coordinates x and y. For a closed surface, this requires that the surface be cut into a few pieces and different projection planes be used for the various pieces. Locally, It seems doubtful that
be
straight
as
assumed in
--
Alberto
202
i. e. for
Pimpinelli
given piece, it is convenient 0 of equation (6.3),
energy
0(p, q) so
that
(6.1)
can
be rewritten
F
Take
to introduce the
a
reference the
now as
--
=
a(p, q)
as
follows:
+
surface free
q2
I 0(p, q)dxdy
F0 +
plane
11 + p2
projected
0.
z
(6.4)
Expanding 0
and q, and noting that the linear terms in p and q well as the mixed second derivatives 0’12 = 0’21 =
to
can
quadratic order in p vanish, as
be made to
02 alOpOq, (6. 1)
and
(6.3)
become F
where o-1
(within
--
an
_-
Fo +
Oulap,
0.11
1 2
11 dxdy [(0.
=
we
0.11) P2 +(0- + 0’22) q2]
a20.1OP2, etc. Letting constant):
0"11
-
0’22
-
0’//,
one
obtains
immaterial additive
-_I& (P2+ q 2)
(p, q) where
+
introduced the
(6.5)
2
surface stiffness &
=
0-
+ 0-"
.
obtaining (6.4) we started from an analytic surface free energy. In other words, we treated the solid as a continuum, completely disregarding the discrete structure of the crystalline lattice. Does it matter? The answer is no, above the roughening temperature, as we argue in Sect. 6.2. For
6.1.5
Surface Chemical Potential
The chemical
potential
is the Gibbs free energy per
particle (here,
per
atom),
and it is defined from y (T,
(ON) OG
P)
OF
y(T, V)
ON
The chemical
(6.6a) T,P
)T,V
(6.6b)
potential determines, at equilibrium, the shape of a crystal. Indeed, it is also useful to describe the time evolution of a surface not too far from equilibrium. At equilibrium, the chemical potential of an infinite system must be constant everywhere (if more than one kind of particle are present, the chemical potential of each species must be constant): Y
-
Yo
Statistical
6
Physics
Crystal Surfaces
at
203
finite size system such as a crystal of fixed volume V? Let us locally modify the crystal surface so as to change isothermally and at constant volume the number of particles by a quantity How is this relation. modified for
a
appropriate thermodynamic potential to use is the Helmoltz yJN. Introducing the atomic Then, p is found from (JF)T,V
JN. The most free energy. volume v --
=
VIN,
let
z(x, y)dxdy
N V
If
z(x, y)
then
by Jz,
varies
.
(JF)T,V
y(T, V)JN
--
=
V
if Jz(x, y)dxdy.
(6.7)
Varying (6.4) yields: JF
10 =
V
if
Jz (x,
y) dxdy
Sp I I ( LO Op
+
integral
The term within brackets in the second
provided 0
c9p
C9X We
Sp) Op
=
Jz,
can
--
if JZ(X’ Y) I
Finally, equating Mullins
the
following we
volume
integrated by parts,
so
(
a
Ox
00
dxdy.
OP
that the first term vanishes. The
(00)
-
OX
preceding equation
to
( )] 00
’9
aq
ay
Op
(6.7)
same
one
way, and
we
dxdy.’
gets (Herring 1953,
1963): Y
In the
(6.8)
c9z/Ox,
=
be handled in the
Ito V
JZ
-
find: JF
dxdy.
0 (Jz)) dxdy I f (L Op
dxdy
(6.8)
be
)
OX
local variations
only consider
second term in brackets in
Jq
’9
dxdy
( 0 JZ)
Oq
can
is at least twice differentiable. Since p
00
00
+
v
potential potential
--
at the
P0
-
[c9x (2p)
V
will often choose
our
+
C9 Y
(6-9)
c9 q
units in such
a
way that the molecular
potential po is the chemical bulk solid, according to (6.1). At equilibrium, the chemical surface must also be equal to yo, and from (6.9) the surface
VIN
of the
=
is 1. The reference chemical
must be flat.
If
(6.4) holds,
the chemical
potential
(6.9), given by 61-t
is P +
(92Z
a2Z
=
-V&( TX-2
Jp,
+
9y2
where
Jp is, according
to
(6-10)
Alberto
204
Pimpinelli
where & is the surface stiffness. In’the &
=
o-.
Locally,
surface
convex
any
Z(X’ Y) where R, and R2
Inserting (6. 11)
are
into
the two
(6. 10)
particular,
for
a
sphere
of
case
(x. y) I
(x2
-
2
an
may be
R,
+
isotropic surface tension, approximated as follows
Y2
(6.11)
R2
principal radii
of curvature of the surface.
find the Gibbs-Thomson relation
we
61t In
ZO
z
=
V&
(Ri 1
1
+
(6.12)
R2
of radius R 2vo-
R A similar
expression yields the
excess
chemical
potential
in the
vicinity
curved step on the surface, as a function of the step stiffness = d27/dO2’ where ^/ is the step line tension. Expliciting the atomic area
letting
Note the factor 2 in
6.2 6.2.1
-Y(O) a
2 ,
R be the local radius of curvature of the step, the relation reads
Jy
(6.13)-a
Morphology
of
Adatoms, Steps
a
=
a
of
2
a
+
and ,
(6.14)
*
R
consequence of the 2 terms in
(6.9).
Crystal Surface
and Thermal
Roughness
of
a
Surface
a high-symmetry surface-e.g. (100) or (I I 1)-at equiperfectly flat, i.e. it should contain no step. At low temperature, there are a few free atoms, or adatoms, and vacancies. Their density depends on the energy one has to pay to create an adatom-and, as a consequence, a vacancy-from a terrace. In the approximation where bond energies are additive (broken bond approximation), and only nearest-neighbour interactions are considered, this energy is equal to 4E on an fcc-(001) surface and to 60 on a (111) surface, if E is the bond energy. At higher temperature, clusters of atoms start to appear. Such clusters are closed terraces bounded by steps, at equilibrium with a two-dimensional gas of adatoms. Indeed, adatoms are continuously absorbed at and released from steps. Atoms emission from steps requires an energy Wad) much smaller than the energy cost for extracting an atom from a terrace-indeed, on an fcc surface the former is just half of the latter. Then, we expect the equilibrium adatom density neq to be given by a Gibbs formula
At
zero
temperature,
librium should be
neq
--
exp,
(-#Wad)
1
(6.15)
where
1/#
=
Physics
Statistical
6
k.8T and kB is the Boltzmann
at
Crystal Surfaces
205
constant.
The step density (total step length per unit surface) increases with temperature. This increase is not easily seen directly by microscopy. Indeed, most of
microscopy techniques work best
at low
temperature, where
matter
transport
equilibrium. On the easily other hand, the atom scattering or x-ray diffraction signal does exhibit a dramatic change when temperature is increased and the surface roughens. Above some temperature, the lineshape, which is lorentzian at low temperature, undergoes a qualitative change. One can for instance measure the "specular" reflection, i.e. that whose reflection angle almost equals the incidence angle (speculum is the Latin for mirror). The specular peak (as well as the Bragg peaks) is narrow for a smooth surface while a rough surface scatters radiation attain thermal
is hard and the surface does not
in all directions.
The
[5-7]
interpretation
patterns from
of diffraction
a
hot surface is difficult
roughness to broaden clear, from a quantitative analysis, that the greatly increased by heating.
because the effect of atomic vibrations adds to
the reflected beam. It is however total step
The
is
length
reason
is
essentially the following.
At
temperature, creating a W, does not change much,
zero
an energy (per length) the step entropy increases, so that the free energy decreases. A simple estimate is possible, if we consider on the (001) face of a cubic crystal a zig-zag
step
W1. Even if
unit
costs
step, whose average direction makes
angle
an
of ’45’ with the bond directions.
configurations of a possible approximate each left to from walk random step of the random right, going square-lattice walker being parallel to a lattice bond, and backward steps being forbidden. If the width of the system in the direction of the walk is uniformly equal to the bond length multiplied by Lv _2, all configurations have the same energy An
if
calculation is
2LW,. Since there
are
2 2L
we
configurations,
consider the
the entropy is 2L In 2 and the free
energy per bond is
W, The free energy per unit length Since a, the lattice parameter, is
-y1a
(6.16)
kBTln2.
-
is called the line tension chosen
generally
lecture notes, the term line tension is often
as
employed
the
length
of steps.
unit in this
for 7 itself.
When -y is positive, one has to provide mechanical work to introduce astep into the surface. If the total step free energy L7 vanishes, steps can that appear spontaneously-they cost nothing. Thus, equation (6.16) tells us the surface
undergoes
a
transition at
a
temperature TR approximately given
by TR
This transition is called the
W, -
kB In 2
roughening transition.
(6.17)
Alberto
206
6.2.2
The
As
above,
seen
Pimpinelli
Roughening the
Transition
roughening
transition temperature may be defined as the steps vanishes. According to the
at which the line tension of
temperature
experiment [8]
the line tension does vanish at
some
temperature, and its
behaviour agrees with (6.16). Near TR, the experimental curve bends away from the straight line predicted by formula (6.16): one can wonder whether it is
instrumental effect
an
section, it is
fundamental
or a
As will be
one.
fundamental effect. But before
a
seen
discussing that,
we
in the next would like
to make three remarks.
1. The
*
entation,
roughening
transition temperature
i.e. it is different for
a
(111)
and for
a
depends
(1,1,19)
on
the surface ori-
surface. This
point
will be addressed in Sect. 6.2.6. 2. For
*
given
a
orientation,
the step line tension vanishes at the
3. Formula
ative for T >
so
(Problem: prove this.) (6.16) suggests that the step
step orientation. *
surface
temperature for all step orientations,
same
TR. Actually, it is
not so, if
that Tp is
independent
of the
line tension may become negstep is defined in a model-
a
independent way. (Problem: could you think of the appropriate definition?) Indeed, the concept of a step looses its meaning above TR.
Smooth and
6.2.3
The
roughening
Rough
Surfaces
transition is much
more complicated than suggested by the Indeed, thermally excited steps are not isolated objects, they are closed loops. In this Section, we shall try to give an idea of what a rough surface really is. The reader will find more details about the roughening transition in the monographies by Balibar & Castaing (1985) [9], Van Beijeren & Nolden (1987) [10], Lapujoulade (1994) [11], Nozi6res (1991)
discussion of Sect. 6.1.2.
[12]
and Weeks As
seen
(1980) [13].
in the
previous section,
TR. It is therefore appropriate way. Consider
the concept of step is not useful above
to characterize
roughness
in
an
alternative
infinite surface of average orientation perpendicular to the z axis. Let (x, y, z) be a point of the surface. The "height" z will be assumed to be a one-valued function of x and y, so that "overhangs" are excluded. Let r1l
--
an
(x, y)
height-height
be
a
point of the two-dimensional (x, y)
g(R11)
-=
( [z(rll)
-
The interest of this function is that it
tatively
space. We define the
correlation function
z(r11
has,
+
if
R11 )]2
gravity
(6-18) is
neglected,
different behaviours above and below TR:
lim
JR111-+oo
g
(I R11 1)
finite for T <
TR,
(6.19a)
two
quali-
Statistical
6
lim
JR111-+oo
g
(I R11
g(JR11 I)
The finiteness of
Crystal Surfaces
at
207
(6.19b)
for T > TR
oo
at low
Physics
temperature, in agreement with
(6.19a),
low-temperature expansion [3]. It is more difficult to and only a plausibility argument will be given. It is indeed prove (6.19b), reasonable to assume that, at sufficiently high tempqrature, say kBT > W1, the discreteness of the crystal lattice becomes negligible, so that the surface height z(x, y) may be regarded as a differentiable function of x and y, as it is for a liquid. Thus, we just forget that we have a crystal, and we write the can
be
proved by
a
surface energy as if it were a continuous medium. The surface free energy F ,,,,f is then simply proportional to the surface area, and the proportionality
(cf. 6.5). Introducing
surface stiffness
coefficient & is the
the local
slopes
p
and q,
F,u If
dxdyVF17+i-p
6
(thermal fluctuations)
or, for small undulations
FE;u,f
-
Co.nst +
=
dxdy
2
+q2
of the
surface,
(P 2
2)
+ q
(6.20a)
However, in our world subject to gravity, thermal fluctuations of the surface acquire an additional energy from gravity. The effect of gravity is irrelevant below TR (Problem. Why?). It is not so above the transition temperature. The energy of a column of matter of cross section dxdy, whose height is between z, and z, is
pgdxdy
f
Z
(d(
=
pgdxdy(Z2.
_
Z2I )/2
’.
where p is the specific mass and g the gravity acceleration. The term containing z, is constant and will be omitted. The energy excess associated with surface
shape fluctuations
and
resulting
from both
is
F
2JJ dxdy 1
(6.20b) q2)] Y
by using
g(R11)
the
-
2
7.
(19Z2 -
C9 X
(9Z +
ay
and surface tension
2)
(6.20b)
readily obtained by Fourier transforming 2 kBTI[pg+ &(q + equipartition theorem (I Z2q 1)
(6.18)
The correlation function and
pgz
gravity
is
=
2kBT
jila dq,dqy n 0
cos(q R11) + &(q2 + q2) Y
X
-
P9
(6.21)
X
approximation of g(R) may be obtained if the lower limit of integration is replaced by an appropriate cutoff, below which the numerator is almost zero. This allows us to replace the cosine by its average value 0, and one obtains An
g(R11)
_-
27r
kBT &
In
(
pg +
2
&/a &/ JR11 12
pg +
(6.22)
Alberto
208
If
gravity
goes
Pimpinelli
be
can
which is
neglected,
possible
JR11 I
for
< A
=
V1o_-1pg, (6.22)
as:
g(Rll)=Const.+(47rkBT/ ,-)I-nlR,llI This proves function
(6.19b).
A surface is called
g(R11) diverges
Previously,
as
(6.23),
in
defined the
we
(6.23)
rough if the height-height correlation no divergence.
and smooth if there is
roughening
transition in terms of the vanish-
ing of the step free energy 7. We should worry about the equivalence of the two definitions. Fortunately, they are equivalent. (Problem. Prove this equivalence.
proof of (6.23) relies on the use of (6.20b). If the step positive, equation (6.20b) cannot be true). Indeed, if -/ > 0,
Hint: the
line tension is
the formation of
terrace of size L has
a free energy cost which diverges free forbidden means diverging energy configurations-i.e., whose statistical weight vanishes-and the correlation function g(R11) is finite for a
with L. A
JR11 I
Therefore, the roughening transition can be defined (at least on high-symmetry surface) either by (6.19a,b) or by the vanishing of ^/. The role of gravity, which kills the roughening transition, becomes negc Ipg. The ligible at lengthscales shorter than the capillary length A order of magnitude of & is typically the energy of a chemical bond, i.e. IeV per atom. The resulting value of A is a few centimetres, i.e. much larger than the distance over which equilibrium can be reached at a crystal surface. For this reason, gravity is usually neglected in surface science. -* oo.
a
--
Diffraction from
6.2.4
Another
a
atom
scattering
do-
A
dS2
q’
where A
h(X, Y) mal
cross
peculiarity of the roughening transia scattering experiment. The x-rays section has the form (see Chap. 2)
dXdY
_2
depends
on
describes the
averaging
over
Surface
consequence of the
important
tion is the form of diffraction or
Rough
peaks
e
-
i’l [h(X,Y)-h(O)]
the electronic
shape
in
density
e
_i(g"X+qYY)
(6.24)
of the surface atoms and
of the surface. The
angular brackets
z
--
denote ther-
surface fluctuations. In order to compute this average, it assumption that fluctuations have a Gaussian prob-
is customary to make the
ability distribution. With this assumption. it
is
straightforward
to show that
(Chap. 2)
r(R11) where
we
wrote
equation (6.25)
preceding
-=
Rjj we
(e_ iq ,[h(Rjj)-h(O)] ) =
(X, Y).
recognize
In the
the
section. The function
q2 -
e
2
exponent
height-height
r(R)
([h(Rjj)-h(O)]2)
takes the
at the
right-hand
(6.25) side of
correlation function of the name
of
pair correlation
Statistical
6
Physics
Crystal Surfaces
at
209
appreciate the relation between the height-height and the pair functions, the latter being the quantity directly measured in a scattering experiment, it is instructive to consider a few examples: i) g(R11) C. The pair correlation function is also a constant,
function.
To
correlation
=
r(R11) and the scattered
ii) g(R11)
intensity
CIR11 I.
=
The
’pair
Lorentzian in this
a
cross
cg2’ -_
us
e-.
z
iii)
g
(R11)
an
exponential
JR111
(6.26) intensity 1, which
2
+
4q211
the components of the momentum transfer wavevector
parallel to the average surface. pair correlation function has I R11 1.
respectively orthogonal -_
delta function.
to find the scattered
C2q4 are
a
case:
2Cq
where q, and q1I
is
correlation function is
section allows
scattering
is
,
(6.24)
obtained from
r(R11) The
C’
=
C In
and
The
a
power-law
behaviour
1’(RII) as
well
as
the scattered
=
IRIII-Cq,2
e- Cq -2InIR111
(6.27)
intensity 2
2+Cq.
q
The
case
of
a
vicinal
(stepped) crystal
surface will be treated here
[14].
At
peaks expected. g(R11) a constant, Beyond TR, the diffraction peaks are expected to acquire a power-law shape. However, a vicinal surface has two non equivalent directions, parallel and orthogonal to the steps. When computing the scattered intensity we must integrate only along a vector orthogonal to the steps. Then one finds and delta-function
is
low temperature
I+Cq 2,
q
At the
peaks
position behave
perature, C
of
as
--
I
largest sensitivity, -
q 11
+C7r2
/a2,
are
qz
=
7r/a,
where C is
the
intensity
actually
a
of the diffraction
function of the tem-
QT). According to (6.23), at TR the height-height correlation case (iii) above, and C(TR) is equal to the universal
function has the form of
value 2a 2/7r2
.
Thus,
the scattered
intensity
at the transition
temperature TR
reads q
and the
roughening temperature
can
d T
-_
d
1+2
be -found
(In 1)
(In q11)
as
the
temperature where
Alberto
210
Pimpinelli
Capillary
6.2.5
In the
of
case
an
Waves
isotropic surface, such as the surface of a liquid, (6.20b) surface, the so-called capillary waves. in Sect. 6.2.5, equation (6.22) implies that the reflected
describes thermal excitations of the
As
have
we
seen
of e.g. x-rays from the surface allows a direct measurement of the surface tension o- (which coincides with & for a liquid).
intensity
Surface Growth and Kinetic
6.3
Equilibrium
6.3.1
At any temperature, at pressure
crystal
a
is in
Psat. Assuming the
::::::
Vapour
equilibrium
with its saturated vapour
ideal gas with density Psat, rate of atoms onto the surface [4,1]:
vapour to be
theory dictates the impingement
kinetic
Rimp
with the Saturated
Roughening
an
Psat1’/2__7rMkBT-
that all the atoms are adsorbed-equivalently, that the sticking unity, which is not necessarily true with molecular speciesdetailed balance gives the equilibrium adatom density neq once the saturated
Assuming
coefficient is
vapour pressure is known. If Tev is the average lifetime of
evaporation,
then
Rimp
neq/Tev,
Revap
--
an
adatom before
and
neq
Psat
Tev
v ’27rmkBT
(6.28)
right-hand side of this equation is called the evaporation rate of the crystal. The saturated vapour pressure for a given solid is obtained from the equation of state of an ideal gas; the detailed derivation is in [1]. The result The
is
Psat
Inserting (6.29)
into
(k BT)
-
(6.28) neq
we
obtain the
M
’rev
( 27rh2 )2 exp(-PWcoh) M
2
47r2h3
evaporation
(6.29)
-
rate in the form
(6.30)
exp(-13Wcoh)
time-’]. Its exponential has dimension [length the to exponential, so that it is compared temperature dependence coefficient T -’. Numerical values of I/To usually written as a T-independent s-1 ’k-2 at for most elements, room temperature. are of order 1014 The
quantity
-2
in front of the
is weak
6.3.2
In the
Supersaturation equilibrium state, equal: 1-tsolid
must be
-
and Vapour
the chemical Yvapour
=
Deposition
potential
of the solid and of its vapour we increase the
Peq- What happens if
Statistical
6
Physics
at
Crystal Surfaces
potential in the vapour? The vapour atoms go potential is lower, i.e. to the solid. The solid grows! The chemical
/-1vapour
-
-
211
where the chemical difference
Yeq
supersaturation. The chemical potential is not easily controlled experimentalist. It is easier to change the pressure. For a vapour
is called the
by
the
treated
as an
ideal gas, the pressure P
Therefore,
for
0Ap
---::
can
Psat
rate i of
growth
as
exp (OzAp)
1,
<
P
flzA The
be written
a
-
Psat
(6.31)
Psat
rough crystal surface is then simply determined by impinging from the vapour and those re-
the balance between the atoms
evaporating, =
where
we
used
Rimp
(6.28).
-
From
Rev
(P
:--
(6.29)
we
-
Psat)/V/’2__7rrnkBT,
find
P M
Ps at From
(6.33)
we see
that the
when the latter is not too
(6-32)
472 V
(6.33)
exp(-Mcoh)
growth rate is proportional to the super-saturationsmall,
see
Ref.
[15]
-and that it is
thermally
ac-
barrier must be overcome, which is given by the cohesion energy. In particular, one sees that the growth rate does not depend on the surface tivated:
a
by [15]. However, (6.33) crystal. Indeed, equation (6.33) is valid for a surface which is rough at equilibrium. Impinging atoms find plenty of favorable sites (kinks) for being incorporated. When the starting surface is Parallel to a high-symmetry direction below the equilibrium roughening orientation. It is not
gives
an
always so, as growth
upper limit to the
it has been shown rate of
a
temperature, steps and kinks have to be created before the incorporated. This is the phenomenon of nucleation.
6.3.3
Nucleation
on a
High Symmetry
atoms
can
be
Substrate
a close packed, step-free high-symmetry surface like (I 11) or (00 1), below roughening temperature, atoms impinging from a vapour-as well as from an atomic or molecular beam-do not find energetically favourable sites to be incorporated into. Indeed, they first have to condense into aggregates, which will then grow by capturing other atoms. If the supersaturation ZAP is small, i.e. if the system is not far from equilibrium, atom condensation will be ruled by the free energy gain in forming a two-dimensional aggregate of size R,
On ’its
Alberto
212
, AG&,, step
on
=
Pimpinelli
-irzApR’,
the surface
which is
,_AGc,,,,t
=
opposed by the free energy cost of forming a 21r7R. Adding the competing contributions one
finds
AG(R) This function of R has
a
--
27r-yR
maximum at R
-
=
2
7rzApR R,
=
-ylAy.
(6.34) It is the
phenomenon
of nucleation: aggregates of size smaller than the critical radius Rc can decrease their free energy by shrinking, while those whose radius is larger than
R, will grow further. The nucleation rate J,,uc is expected to be dominated by the probability of finding an aggregate at the nucleation barrier AG* --
Z G(R,)
-
7r-y’/zAp,
i.e.
Jnuc
-
exp
(-7r’ ) Ay
(6-35)
-
Small variations in the supersaturation are enormously amplified by the exponential dependence. The growth rate is now determined by the nucleation rate (6.35), and by the spreading velocity of the supercritical aggregates. Qualitatively, one expects to observe random nucleation of two-dimensional seeds, lateral spreading of the aggregates, and coalescence. The process will then. start again on the freshly created layer. The growth rate will thus be an oscillating function of time, the period being equal to the formation time of a whole layer. The oscillation is a consequence of the discrete (layered) structure of a crystal, which is kept by its surface as long as it is below its roughening temperature. However, if growth is continued, after a system-dependent time span the oscillations die out, and a quasi-stationary distribution of surface steps sets in. Indeed, the oscillations demand that each layer is started and completed in succession, with a high degree of temporal correlation. Randomness in deposition and nucleation destroy such correlations, and a disordered state, that we would be tempted to call rough, appears. In fact, it is found that after transients have died out, any growing surface is rough, with power-law correlations, as well in space as in time. This is kinetic roughening.
6.3.4
Kink-Limited Growth Kinetics
Kinetic
roughening
is the outcome of the
competition of two different mechadeposition and nucleation, which, following the current jargon, we ,Vill call noise, and matter transport processes. The effect of noise can be easily seen by picturing growth without matter transport at the growing interface. Atoms just stick where they hit, which happens at a rate F per unit surface and time. The number of deposited atoms in time t is thus nisms: randomness in
N
Ft. The statistical fluctuation of this number is W
One would thus expect the
rms
fluctuations of the surface
-\/_N
height (or
VT. surface
Statistical Physics at Crystal Surfaces
6
213
width), W(t) to be
proportional
JN,
to
V([Z(J)
=
that
so
W(t) This is
example of the
an
(6.36)
Z(o)]2)
-
_
t1/2
(6.37)
power laws mentioned at the end of the
previous
sec-
tion. Of course, the random deposition model without smoothening processes is an extreme example, though it may be appropriate to very low tempera-
experimental situations, possible. Indeed, the Gibbs-Thomson equation (6.12) states that an excess of chemical potential is stored where an excess of particle has accumulated: the surface will thus tend to relax by transferring these particles towards places with lower chemical potential. In a reference frame moving at the average growth velocity of the interface, F, the surface height z(t) evolves according to the laws of linear thermodynamics: ture, where surface mobilities
one
seeks to obtain
as
flat
a
very slow. In real
are
surface
-
and
v
of atoms
02 Z
02 Z
(19X2
V’3
expression (6.10)
where the linearized
used,
as
+
(6.38)
y2
for the chemical
potential
has been
kinetic coefficient related to the rate of capture and emission kinks, from and to the vapour.
is
a
by randomizing noise and the smoothing term (6.38) may be put together obtain a Langevin equation for describing growth of a fluctuating interface a vapour phase:
The to
in
where
q(x, t)
is
19t
_X2
+
-02Z. (9 Y2
0,
--
(77 (x, t),q (x’, t’))
)
properties
FS (x
--
(6.39)
+ Tj
-
x) 6 (t
t’)
-
(6.40).
.
Scaling equation (6.40)
Since the
tion
[1]. Instead,
by
factor
a
a2Z
random function with the
a
(TI)
6.3.5
(9Z
A,
z
by
is linear, it is readily solved by Fourier tranformaperfo "rm a scaling analysis [16,1]. Rescale x (x, y) factor Ac, and the time by a factor A’. Equation (6.39)
will
we a
=
becomes
Aa-z where the
(6(ax)
=
Dividing
19Z
at
rescaling of
11&Aa-2
(a2Z
5_X2
+
(92Z. OY 2
)
+
the noise follows from the
A-1-z/2 properties
(6.41) of &functions
J(x)la). both sides
by A",
0Z =
at
we
V&Az-2
get
02Z
(5-X2
(92Z +
V
)
+
Az/2-1-a 77
(6.42)
Alberto
214
Pimpinelli
2 and a 0. What is the meaning of equation coincides with (6.39) if z To little with scaling relations. see we exponents? this, need playing a Consider the surface width (6.36). Under rescaling z’ A’z, t’ A’t, w
This
=
=
these
=
behaves
AO V([Z(Azj)
W(tl) On the other
(6.37).
--
as
hand,
we
expect that
w(t)
We introduce another exponent,
w(t) In critical
be
Z(o)]2)
(6.43)
power-law function
a
of t
as
in
such that
3,
-
-
to
(6.44)
.
phenomena, power-law behaviour
at the critical
point
is
a
con-
sequence of the absence of a characteristic lengthscale in the problem, the correlation length being infinite at T,. We conclude that also in the case of
growth
of an interface no typical lengthscale exists, except for the lattice parameter a, which is immaterial at long wavelengths, and the size L of the L in equation (6.43), we find system. Letting A -_
w(t) We
see
LV _(_[z(L2t)- z(O)]2)
that for times t of the order Of tcrossover
reaches the
-
LV([z(l)
-
Z(o)]2)
(6.44), which implies Uisaturation L’,8, equation (6.46) requires L’ L’P, or
To be coherent with
see now a
the surface width
L’
w
(6.46)
.
W(tcrossover)
-
t c3rossover
a/,8.
Z
*
L’,
time-independent (saturation) value Wsaturation --
We
(6.45)
.
(6.47)
how the exponents a,
is the
and z can be intrerpreted: roughness exponent, which characterizes the increase
saturation value
(the
value at t >
tcrossover)
of the
of the surface width with the
system size L: Wsaturation *
P
is the
growth exponent, which characterizes the increase of the
surface width with the time t
(at
z
is the
tcrossover):
t <
w
crossover
(t)
dynamic exponent,
-
0
-3
which characterizes the increase of the
time with the system size L:
tcrossover The
L’ ;
-
-
L’
equality (6.47) relates the three exponents. Random deposition, as seen beginning of Sect. 6.3.4, gives # 1/2. In this extreme model, a and
at the
--
Statistical
6
are
defined,
not
due to the strict
Physics
locality of
Crystal Surfaces
at
deposition
the
215
process: the
system’s behaviour is completely insensitive to the system size. The resulting interface is completely-and maximally-uncorrelated. The growth model of equation (6.39), known as the Edwards-Wilkinson model[17], has z = 2 and
oz
#
-_
-_
is, for a two-dimensional surface. 0 means in reality logarithmithat a 0 equation (6.23) of equilibrium roughening.
0 in three dimensions-that
Indeed, the direct solution shows cally diverging correlations, like Not all models behave like that, as
=
seen
--
in next section.
Surface-Diffusion-Limited Growth Kinetics
6.3.6
Another situation of interest is when the matter transport process charged to smoothen the growing surface is conservative: in this case, the evolution
equation (6.38)
is
longer valid,
no
and it has to be
i
=
replaced with
(6.48)
-divi,
latter, we invoke again now that the exrequire equation (6.10). However, relaxed is atom local to due excess a chemical through diffusion cess potential to the grais the current other In interface. the proportional j words, along chemical local the of dient potential:
where
j
is the surface diffusion current. To find the
the Gibbs-Thomson
we
j where D is
a
(collective)
--
-DWy
surface diffusion coefficient. From
inserting
this into
X2
(6.48) yields, 0 Z-
at where
we
known we
as
defined the
V
((92/aX2
Mullins model
find
Aa-z
Dividing
=
both sides
az =
at
at
+
+
instead of
7,72(V2Z)
get
a2/ay2).
(6.39): (6.49)
+ 71
equation (6.49) is rescaling as before,
The model of
[18]. Performing
we
we
ay2
-D&Aa-4,72(V2Z)
by A‘,
az
-R
(6.10)
2Z
2Z
j and
,
the
+
same
A-I-z/2,
get
D&A’ -4,72(,72Z) +
Az/2-1-a,
(6.50)
1. The relation (6.47) 4 and a equation coincides with (6.49) if z in this case faster increases width surface the w now 1/4. Hence, yields kinetics kink-limited for t1/4 than at or (w In t). equilibrium ), (W
This
_
=
=
-
Alberto
216
Different
Pimpinelli
physical
situations will thus
give
different exponents.
Again,
this resembles very much the case of critical phenomena. Indeed, universality classes appear, where different models such as (6.39) or (6.49) find their place, characterised
by different sets of vallues of a, 13 and z. Symmetry arguments, physical considerations, dictate the form of the equation which rules the evolution of the interface, and thus the universality class where it belongs to. A lot more of details on this fascinating subject will be found in as
well
Refs.
as
[16]
and
[1].
References 1.
2.
Pimpinelli, J.. Villain Physics of Cristal Growth Al6a-Saclay series n’ 4, Cambridge University Press, Cambridge (UK), 1998. B. Caroli, C. Caroli, B. Roulet in Solids far from equilibrium, C. Godreche ed. A.
(Cambridge University Press, 1991).
3.
4. 5.
6.
7. 8.
9. 10.
11. 12.
13.
14. 15.
16.
17. 18.
Instabilities of planar solidification fronts. Villain, A. Pimpinelli. Physique de la croissance cristalline, Coll. A16aSaclay, Eyrolles (Paris), 1995. L. Landau, E. Lifshitz Statistical Physics, Pergamon Press, London, 1959. G. Blatter Surface Sci. 145, 419 (1984). A.C. Levi Surface Sci. 137, 385 (1984). G. Armand, J.R. Manson Phys. Rev. B 37, 4363 (1988). F. Gallet, S. Balibar, E. Rolley J. de Physique 48, 369 (1987). S. Balibar, B. Castaing Surface Sci. Reports 5 (1985), 87. H. van Beijeren, 1. Nolden in Structure and Dynamics of Surfaces 11, W. Schommers and P. von Blanckenhagen eds., Topics in Current Physics 43 (Springer, Berlin, 1987). J. Lapujoulade Surf. Sci. Rep. 20, 191 (1994). P. Nozi6res in Solids far from equilibrium, C. Godr6che ed. (Cambridge University Press, 1991). Shape and growth of crystals. J.b. Weeks in Ordering in Strongly Fluctuating Condensed Matter Systems T. Riste ed. (Plenum, New York, 1980) p. 293. The roughening transition. J. Villain, D. GremPel, J. Lapujolade J. Phys. F 15, 806 (1985). W.K. Burton, N. Cabrera, F. Frank Phil. Trans. Roy. Soc. 243, 299 (1951). A.-L. BarabAsi, H.E. Stanley Fractal Concepts in Surface Growth Cambridge University Press, Cambridge (UK), 1995. Edwards S.F., Wilkinson, D.R. Proc. Roy. Soc. A 381, 17 (1982). ’, Mullins, W.W. J. Appl. Phys. 30, 77 (1959). J.
Experiments
7
Jean-Marc CRMC2
Gay and
on
Laurent
CNRS, Campus
de
Solid Surfaces
Lapena
Luminy,
case
913, 13288 Marseille Cedex 9, France
reflectivity studies has largely increased in the last years so technique is nowadays well developed using various x-ray sources sealed tubes and rotating anodes to last generation synchrotrons from ranging with setups adapted to liquid or solid surfaces [1,2]. This chapter is focussed on experimental solid surface studies. Let’s just mention that solid surfaces allow more flexibility than liquid ones since they can be oriented in any direction’without deviating the incident beam. In addition, some other questions about resolution with long range correlations on liquid surfaces are generally avoided with rough solid surfaces. The number of
that this
7.1
Experimental Techniques
7.1.1
Reflectivity Experiments
Measurement Setups and Procedures Setups for x-ray reflectivity experiments are now rather common. Figure 7.1 schematically shows a typical experimental system for standard reflectivity measurements which can use the more or less divergent beam emitted by a conventional or a synchrotron source. Slits and a monochromator produce a collimated monochromatic beam which impinges onto the sample surface under the incidence angle Oin Various monochromators are available depending on the desired resolution, and intensity. The sample is mounted on a goniometer with precise motors (the angular displacement accuracy is at least 1 mdeg) which control the sample surface’(plane (x, y)) and the detector positions. Slits or an analyser crystal are set at the detector side to reduce the background and the divergence of the outgoing beam. The detector position is given by the 0,,, and 0 angles, polar angles in the plane of incidence (x, z) and out of this plane in the y-direction respectively (see Sect. 2.3.1). The components of the -
wave-vector transfer
are
then qx
qy q,
k (cos 0,,, cos 0
-
Oil sin
cos
ko cos 0 ko (sin Oi,, + sin Os,,)
Oi,,)
(7.1)
Oi,1 and 0, that makes Specular reflection is characterised by Os, (0, 0, 2k sin Oin). The angle Oin + Osc is often also denoted 20, whereas q the incident angle Oi,, defined by the orientation of the surface is named w. =
=
J. Daillant and A. Gibaud: LNPm 58, pp. 217 - 231, 1999 © Springer-Verlag Berlin Heidelberg 1999
Jean-Marc
218
Gay and Laurent Lapena
two-circle
Fig. 7.1. Standard setup of a triple-axis specular diffuse scattering. From [3]
goniometer
diffractometer for
specular reflectivity and
off
The rotations detector
simply describe the position of the surface and the incoming beam. For small scattering angles, the
and 20
w
relatively
to the
wave-vector transfer is q
(q ,
=
2koO(w
-
Different combinations of w and 20
(q,,q.,) w
shown in
as
20
=
scan
Fig.
or
*
are
k,O,
used for
q,
,zz
2kOO)
mapping
the
reciprocal
space
7.2:
specular
Oin In keeps the condition 0,;c the specularly reflected beam and radially in the normal z-direction, ie q
scan, that
this geometry, the detector maps the reciprocal space
(0, 0, q,2koO). 0 + ::AOo/20 w
0), qy
-
-
measures
-_
longitudinal diffuse scan, with offset ’ AOO. The radially mapped, but in a direction with an angle ,A00 from the surface normal. This type of scan is useful for measuring the diffuse scattering contribution close to the specular peak. Subtracted from the measured specular reflectivity, it allows to get the true specular =
reciprocal
scan or
space is still
reflection. *
0, w
=
in the *
rocking scan at fixed 20. The rotation is limited to the range W scattering angles, the reciprocal space is measured transverse q,-direction of the accessible area at different q, levels.
w scan or
20
-
--
20. For small
scan
or
detector
both varied with
0
scan
at fixed
limitation
w.
In this 20 >
geometry,
q., and q,
are
addition, sample area is illuminated during the scan. 0. In this geometry, the reciprocal space is measured 0 scan at fixed w in the qy-direction normal to the incidence plane. This type of measurement is generally less used than the others because most reflectometers have no -motion out of the incidence plane even though it offers a full qy range accessibility [4]. It is usually preferred to let slits widely opened in the y-direction leading to an effective integration over qy. -
no
=
as soon as
w.
In
a
constant
7
Experiments
on
Solid Surfaces
219
0.25
0.20
015
2
r
o
o
specular reflectivity
longitudinal diffuse scan
oooc’oo oo
/
detector
A
00=o.1,
ooooo rocking scan
scan
0 9*
/20=1.5"
0.10
0.05
rinaccessible q-
0.00 -2.0
-1.5
inaccessible q-area
area
-1.0
-0.5
0.0
1.0
0.5
1.5
2.0
qX(j 0-3A-1) Scans in
Fig. 7.2. The line
scans are
reciprocal
space with
of the inaccessible
representation
areas.
shown for the CuKal radiation
Resolution The choice of slit widths or monochromator and analyser crystals is essential for setting the resolution of the measurements for given configurations [3,5-7]. The calculations of different resolution functions can be
Considering the case of in-plane scattering scattering angles, one gets from Eqs. (7.1):
found in the literature.
with small
I
Jq, Jq,
kO(Oj2nzAOj2n + 02 ko(ZAO + //\02 in
SC
in
A02
0)
(7.2)
SC
SC
in
monochromatic radiation, and incoming and outgoing beams with anguOsc -- 0 and acceptance zAO,,c respectively. Assuming Oin
for
a
lar
divergence AOin
and ZAOin
;z
zA0,;,
;: -,
zAO,
one can
estimate the resolutions
Jq, q_,
q ,z_AO
as
(7.3)
2kOAO.
experimental results reported in the following section from meaon a triple axis spectrometer using flat Ge(111) monochromator and analyser and the CuK, radiation, the typical values for resolution are 2.10-3A-1 and Jq,, 2.10-’q,. Jq, The resolution functions determine the maximum length scales which can be coherently proben by the experimental measurements. For the above men107A. 10’A and Xinax ,: 104 / q, tionned resolution, they are Zmax
For the
surements
,
220
Jean-Marc
Gay
and Laurent
Lapena
Data
Analysis Before any comparison with theoretical simulations of reflectivity, the data may have to be corrected for various geometrical factors depending on the measurement, configuration [6,7]. Alternatively, the simulations may include the corrections and the data are considered as they are. The specular reflectivity is defined (see Eqs. (3.5), (3.6)) as the ratio of the reflected intensity at the scattering angle 20 to the intensity of the direct through beam. For very grazing incidence angles Oin, the sample surface (length I along the x-direction) is almost parallel to the beam so that a fraction of the incoming beam (width bin) does not illuminate the surface and cannot be consequently reflected. This geometrical effect is responsible for the bump experimentally observed at low angles. A plateau of total external reflection below the critical angle, as theoretically expected, is obtained upon renormalisation of the experimental data. Assuming a uniform rectangular flux distribution of the primary beam, the correction factor is simply expressed as a function of Oj0,I arcsin(bin/1) by: =
in
fsp ec, in fspec,in On the detector
side, slits
=
Sill Oin / sin
=
I
or
and
Os,
is
in
for for
Oin Oin
<
’
Oj’n 019n
(7.4)
in
the
beam width b,,,. A correction
angles Oin
Ojon
analyser crystal size may limit the measured factor, independent of the incidence and exit
given by:
fspec,sc fspec,sc
bin/bs’
if
I
if bsc > bin
bc
<
bin
(7-5)
The actual flected
specular reflectivity is therefore deduced from the measured re0 using the formula: intensity Ispec(Oin) at the incident angle Oin -
R(O)
--
R(Oin)
fspeclin (Oin)fspec,sc
Isp (Oin ) ec
10
where I0 is the
intensity of the incident beam. On the other hand, the meaintensity is proportional to (i) the incident beam intensity, (ii) the illuminated area of the sample, and (iii) the resolution volume (see Sect. 4.7), whereas the calculation of the diffuse intensity is usually based on the expression of the scattering cross-section for a unit incident flux. The illuminated area which covers the full length I of the sample at low angles, decreases as soon as Oil, > Ofn. For the data normalisation, one has to consider a correction factor proportional to the actual illuminated area: sured diffuse
in
fdiff,in (Oin) hiff, in (Oin) This is the normalisation
for Oin <
I
sin
prefactor
O n /sin Oin in
for
the slit widths
or
Oin. Oj’n
the size of the
on
(4.61).
the exit side
analyser crystal.
(7-6)
in
mentioned in formula
Similar corrections have to be taken into account on
Oin !
depending
A correction factor is
7
required for
Oin
!
when the width of the illuminated
OiOn) in
bs,l sin Orc).
is .
larger
than the
fdiff,sc (Oin) Osc) experimental Idiff (Oin function of the
)
-
-:::::
diffuse
Osc)
:::::
area
(I
Solid Surfaces
on
for Oin <
"seen"
by actually can be expressed by:
area
The correction factor
fdiff,sc (Oin) Osc)
The
Experiments
for
1 b_
Ifdiff,i.(9i.)
0,,,
<
arcsin
for Os, < arcsin
intensity finally
Oin, in
or
the detector
221
bi/Oin (width
b-
Udiff, in (0 in).
(7.7)
b_
lfdiff,i. (Oi-)
appears with the form:
10 hiff, in (Oin ) hiff, sc (Oin
scattering cross-section integrated
)
Osc)
I
over
the
dodS2
dQsc,
angular
resolution
function of the detector.
Specular Reflectivity Once the true coherent specular reflectivity has been experimentally determined (see above the description and interest of the longitudinal diffuse scan), it is adjusted against a simulation with various parameters describing the investigated surface. Very often the surface of an homogeneous material is in fact made of a thin surface layer of different density, resulting from various reasons: oxidation, mechanical treatment, inhomogeneous deposition, etc. As a consequence, the formalism for stratified media is the most commonly used even for single solid surfaces. Basically the thickness of the layer, its density and the surface and interfacial roughnesses When diffuse are expected to come out from the specular reflectivity study. data set is looked all the fit of simultaneous a data are available, scattering the of full to order in a (see Sect. configuration sample description get for,
thereafter). Exact theoretical
stratified
media)
are
descriptions available for
of x-ray reflectivity from solid surfaces (of sharp surfaces and interfaces either from
technique (see Chap. 3) or from the equivalent recursive approach initially developped by Parratt [8]. In reality, roughness of surfaces and interfaces can significantly alter the specularly reflected intensity. A rough interface can be seen as made of locally flat areas at different heights (see Chap. 2). A classical approximation considers a Gaussian height-distribution probability. The N6vot-Croce factors [9] which depict the root-mean-square roughness of each interface can be easily introduced in the formalism derived for smooth interfaces (see Appendix 3.A). The analysis of the experimental data with this kind of model of can be rather fast since the free parameters which describe each layer j the considered system are restricted to the r.m.s roughnesses o-j, the layer thicknesses dj and densities Jj. The final result of the analysis can then be shown as the density profile J(z) of the investigated system. At the surface and at each interface, the profile looks like an error-function deduced from the gaussian probability of the height distribution. Each interface is treated the matrix
Jean-Marc Gay and Laurent
222
independently
from the
others,
Lapena
that is
expressed with the condition O-j
<
dj.
This
approach does not hold when the r.m.s. roughness o-j is on the same magnitude that the layer thickness dj. For large roughnesses, the density profile does not show clearly identified plateaus corresponding to the different layers. The analysis is a little more complicated since one has first to guess the profile J(z). With this initial guess, the investigated system is seen as a series of very thin layers p with sharp interfaces and density Jp vaying from one to the other following the profile J(z). This parametrisation of the system relies on a rather large number of density parameters which makes the calculation long, but still simply tractable with the exact Parratt formalism. A slightly different and faster procedure has been proposed by the group of Press [2]. This parametrisation is based on an effective density model which allows to consider the profile has made of individual layers order of
"
"
even
when the condition o-j <
dj
is not fulfilled.
The above formalisms are based on an optical approach of light (not specially x-ray) scattering by stratified media. A different approach is also available for describing x-ray scattering. The so-called Born or kinematical approximation valid for the weak scattering regime does not hold when refraction effects are important and cannot be neglected, i.e. for angles close to the critical angle of total reflection. This drawback is usually overcome within the Distorded Wave Born Approximation (see Chap. 4). This theoretical framework is used for modelling both specular and off-specular diffuse scattering.
Off-Specular Diffuse Scattering Coherent specular and off-specular difscattering are complementary for providing a complete set of parameters describing stratified media. In the simple case of a single layer on a semiinfinite substrate, besides the layer density, thickness and r.m.s. roughness parameters deduced from specular reflectivity, one can have access to a more detailed representation of the morphology of the sample through the lateral surface and interface height-height correlation functions, and the correlation function between the surface and the buried interface (see Sect. 2.2, and Sects. 8.2, 8.4, 8.5 for multilayers). Many isotropic solid surfaces are self-affine so that the height-height correlation function Q, can be simply expressed with three parameters (Eq. (2.26)): the r.m.s. roughness o-, the correlation length which shows the scale on which the surface is rough and the Hurst parameter h related to the fractal dimension of the surface which describes how jagged or smooth it is fuse
[4,101.
7
Experiments
Roughness Investigations with
7.1.2
other
on
Solid Surfaces
Experimental
223
Tools
Microscopy Near Field Microscopies have been now rather investigating the roughness of solid surfaces. Scanning Tunneling Microscopes (STM) and Atomic Force Microscopes (AFM) are particularly well suited for imaging surface morphology [11-13]. Combining a large number of line scans along the surface, they yield a detailed map of its roughness at various scales to some hundreds of ym’. A statistical treatement of the images provides the power-spectral-density (PSD) of the surface from which can be determined the parameters o-, , and h ’of the height-height correlation function mentionned in the previous section. Like x-ray reflectivity, near field microscopies are non-destructive techniques which can be used in rather fast measurements in laboratories. They are nevertheless local probes as compared to x-ray scattering. Studies with a satisfactory precision often require a very large amount of recorded data, that can be finally very timeand computer memory-consuming. Images of real space are always very appealing even if they only show the surface, ignoring the underlying interfaces. Near Field
common
tools for
Microscopy Transmission Electron Microscopy (TEM) can be investigating the morphology of surface and interfaces of stratified solid materials [14]. This technique requires a delicate destructive sample preparation in order to get a thin slab cut normal to the surface. Different recipes are used depending on the investigated material which must be kept undammaged during the preparation process. Real space images of cross section TEM can clearly show the thickness of the layers and the morphology of the interfaces. R.m.s. roughness can be estimated from grey contrast profiles through the interfaces. More quantitative analysis of the TEM images would require calibrations against known standards. Information is always local. Electron used for
Examples of Investigations
7.2
of Solid 7.2.1
Surfaces/ Interfaces
Co/Glass
-
Self-Affine Gaussian
Roughness
experimental study of a Co film, 150A thick, deposited at room temperature on a glass substrate (see Sect. 6.2 for a description of the morphology of crystal surfaces). We report in this section the
X-Ray Reflectivity The x-ray scattering measurements were performed on a triple-axis diffractometer with flat Ge(111) monochromator and analyser crystals using the CuKaj radiation emitted by a rotating anode operated at 1OkW
the
[15].
specularly
The detector
was a
reflected intensity
Fig. 7.3a shows scattering angle. The
standard Nal scintillator. as
a
function of the
Jean-Marc
224
Gay
and Laurent
Lapena
Spacular B&R&&Jvfty
L-Oud. DW. So.
40.-0.25
lop
f
I
IQ,
1
2
1
3
to,
-
5
a
[d.9.1
2e
Rooklngmcm
4
?.a
20-1.80 dog.
a
tdO9,1
Rockingocon
20-2,00 dog,
A MG w
015
[dog.)
Riockfnpcan
i’s
(dog.]
a
20-2.20 deg.
Rockingsaun
26-2.+D dog.
icr,
lar,
9
-e
do
L-dog-1
Detector Soon
i
dec.
a
2
3
20
7.3. Co
tal data: diffuse
(150 A)
symbols,
scan
with
4
[deg.]
film
on
1&_
IV
’s.
.6v.
40
glass :
offset AOo
’C"
too
z
and best fit an
1d’vJ
-
W-1.11
10r,
1
Fig.
I G,
CXI
measured with the CuKai radiation
(experimenline). (a) Specular reflectivity. (b) Longitudinal 0.25’. (c) to (f) Rocking scans at different scatwith fixed incident angle W. (h) &profile deduced
solid
tering angles. (g) Detector scan from the adjustment -of the experimental data
Experiments
7
on
Solid Surfaces
225
using the effective density model with two layers on top of a glass substrate (fixed density Jo). The best fit parameters 7.3h. are given in Table 7. 1. The corresponding J-profile is represented in Fig. the medium of 6 is proportional to the electron density (see Chap. 1). The effective of the the that use density model. surface layer is rather thin, justifies The density air. in the since was It is certainly made of cobalt oxide, sample
simulation has been calculated
layer is in quite good agreement with that of bulk cobalt. The r.m.s. roughness of the glass/Co interface is small as expected for the surface of a bare glass substrate. of the cobalt
Table 7.1.
Co/glass. Summary
of the different parameters deduced from the x-ray Sj and dj are the density and thickness of
AFM and TEM studies.
reflectivity, 0 for the substrate). aj, j, hj are the r.m.s., height-height layer J’respectively (3’ correlation length and Hurst parameter describing the interfacial roughness between layer Jand layer J*+1. Vertical correlation between successive interfaces is considered with the aij coupling coefficient. In the adjustment of the x-ray reflectivity study, fixed parameter all the above parameters are free except Jo. =
X-Rays Substrate
(Glass)
10660
[A] [A]
o-o
0
layer
ai
6
[A] [A] [A]
layer
0’2
2
6.8
0.5 0.5 0.3
13.5
h2
0.1 0.05
45.8 9.2
1200 0.6
[A]
not
accessible
700
1200
8.9
[A] [A] [A]
di + d2
not
accessible
0.05
171.4
I
a12
d2
TEM
0.2
0.2
10662
surface
500
24.4
hi Oxide
0.3
0.43
1091 di
5.1 1060
0.5
ho a0i
Co
AFM
7.48(*)
217.2
0.5 0.5 0.6
700 1430 0.3 1
9.1
0.8
13.2
2.0
210/5000
60
0.1
215
Off-specular scattering has been measured in different modes shown in Fig. 7.3b-g. The longitudinal diffuse scan (Fig. 7.3b) has been used for extracting the true specular reflectivity reported above. Its oscillatory behaviour indicates correlated rough interfaces. A detector scan is also shown along with rocking curves at various scattering angles. All show the specular peak together with the diffuse scattering contribution. Yoneda wings (see Sect. 4.3.1) are observed on both sides of the rocking curves. They appear for incident and exit angles equal to the critical angle for total external reflection, for
Jean-Marc Gay and Laurent
226
Lapena
which refraction effect is clear. The
off-specular diffuse scattering curves are against a DWBA model with a self-affine roughness with gaussian probability. The best fit parameters can be found in Table 7.1. The adjustment performed with several diffuse scattering curves yields parameters for a complete description of the surface and interface morphology. fitted
Study of the Surface Roughness AFM images have been recorded sample areas of 3.5 x 3.5ym 2. Figure 7.4a shows a typical AFM image and Fig. 7.4b the height profile measured along an arbitrary line of the surface. A computational statistical analysis of the AFM images of the proben areas gives the r.m.s. roughness and the PSD. For isotropic self-affine rough surfaces described with the three parameters (o-, and h), the PSD is expected to be almost constant for low q (ie large scale in direct space) and to decrease like q- 2(1+h) for frequencies larger than a cutoff frequency associated to the correlation length. Figure 7.4c shows the experimental PSD with three regimes and two correlation lengths 6 and *. The investigated sample surface is 2 presumably more complicated than the simple self-affine model description. It is however worthwhile noting that the shortest correlation length and the Hurst parameter h agree pretty well with those deduced from the x-ray study AFM on
(see
Table
7.1).
Study Cross section Transmission Electron Microscopy measurements performed on the same Co/glass sample. Figure 7.5a shows a TEM which can be seen the substate material, the intermediate layer on image surface transition layer. The image is quantitatively analysed and the (Co) the by plotting average grey level profile normal to the surface (Fig. 7.5b), which is representative of the density contrast through the sample. A smooth decrease of the profile at the surface prevents from distinguishing a surface oxide layer on top of the Co layer. After normalisation of the grey levels of the substrate and the Co layer, the profile can be compared to that deduced from the x-ray specular reflectivity work. The shape of the TEM and x-ray profiles are in rather good agreement indicating the same total thickness and similar r.m.s. roughnesses. TEM
have been
7.2.2
Si
Hornoepitaxy on Misoriented Roughness
Si Substrate.
Structured
are very often well suited for statistical descriptions isotropic growth-induced roughening of deposited films, as that presented in the above section. A quite different class of roughnesses is constituted with (quasi-) periodic undulations that make laterally structured rough surfaces/ interfaces. For instance, such cases can result from growth on misori-
Self affine surface models
of
Experiments
7
Solid Surfaces
on
227
1215
E M W
;T
10-
00
2000
1000
-3.5
3000
-3.0
-2.0
-2.5
LogIO[
X(nm)
-1,5
%(nm") ]
Fig. 7.4. Co (150A) film on glass. (a) AFM image of the surface (3.5 x 3.5p m2), (b) height profile along an arbitrary line of the surface, (c) PSD from the AFM study
0.6
A
OA
d
50
100
150
200
250
z(A)
Fig. 7.5.
Co
(150A)
film
on
glass. (a)
Cross section TEM
image (335 x 335A2) and line) along with the
(b) normalised grey level profile normal to the surface (solid delta-profile from the x-ray reflectivity study (dashed line)
Jean-Marc
228
Gay
and Laurent
Lapena
A
A
0.4
1000 500 0
0.9
Fig.
7.6. AFM
oriented
image
Si(111)
0.8
.4
0.4
of the surface of
substrate. The
image
a
0
Si
film,
is 1.0
X
500
nm
thick, deposited
ented surfaces with the influence of on
miscut-generated steps specially designed surface gratings. The work
reported thereafter deals with
on
mis-
1.OMra2
a
or
from
growth
sample obtained by Si homoepi-
Si Czochralski grown wafer. The substrate surface is misoriented 10’ around the[I-10] axis toward the [-1-12] axis with respect to the (111)
taxy
on
a
by plane. After cleaning, the initial substrate surface array of triatomic
steps with
terraces about 5.5
struction. A total thickness of 500 at 700’ C with
a
rate of
nm
of Si is
nrn
is
composed
of
wide and the 7
deposited in
a
a
x
regular 7
recon-
MBE chamber
0.15nm-s-1 [16].
Si(500nm)/Si sample surface is shown in Fig. 7.6. A long period (about ’250 nm) undulation with an amplitude varying from 1 to An AFM
image
of the
clearly observed. X-ray specular and off-specular diffuse- scattering investigations have been performed in order to provide a full description of the surface morphology. 10
nm
is
The measured
specular reflectivity can be easily modelled with a density surface transition layer 6.6 nrn thick on top of bulk Si. This thickness is in quite good agreement with the mean amplitude of the surface undulations seen in the AFM images. Off-specular diffuse scattering has been measured with rocking scans at different azimuthal orientations of the surface, ie at different angles a between the grazing incoming beam and the grooves of the surface undulations of period do (see Fig. 7.7). The apparent period of the surface grating given by the projection of do on the (x, z) scattering plane changes with a and gives rise to satellites in the rocking 5 curves, the position of which allows the precise determination of do: 215 nm. A careful examination of the satellite intensities shows that they are not symmetric. A simulation of the measured rocking curves is proposed (Fig. 7.8) in the simple frame of the kinematical approximation for the calculation profile
made of
a
of the structure factor of each satellite. It is based upon an asymmetric surface profile shown in Fig. 7.8, the shape of which is adjusted to reproduce Ahe experimental data. This profile of period 215 nm and amplitude 6.6 nrn shows extended facets which make 0.3 deg. with the average surface. This determination is an angle of 2.6
ExperLments
7
on
Solid Surfaces
229
Zt-x Y
ZI
26=1. deg. 107
-
-2
+O.Sd g.
103
-ro
-2
101
_X..
+2
-1
I
a
-4.5d.9. -7.Sd.g.
10-1 -0.5
to)
Fig. 7.7.
Surface of
a
Si
film,
0.5
0.0
500
-
nm
91(deg.)
thick, deposited
on
misorientea Si(111) sub-
Azimuthal orientation of the surface grooves with respect to the incoming and reflected wave vectors. (b) X-ray off specular diffuse scattering measured with strate.
(a)
rocking
scans
recorded at 20
=
1.5deg.
for various azimuthal
angles
a
(CuKal
wavelength)
quite consistent with surface facets with
7.3
an
cross
section TEM
angle
of 2.8
deg.
images
of this
sample which
reveal
with the average surface.
Conclusion
X-ray specular and off-specular diffuse scattering have now become rather common tools for investigating microscopic surface and interface morphology. Measurements can be performed on setups coupled with classical x-ray generators as well as synchroton light sources. The feasibility of the experiments and the associated information which can be deduced from the experimental data heavily depend on the quality of the investigated samples. Macroscopic faceting of a surface can completely obscure reflectivity. One has also to keep in mind that reflectivity decreases dramatically with r.m.s. roughness which consequently limits the measurable q-range. The extensive study of two types of roughness (self-affine fractal and structured grating surfaces) have been reported in this chapter as an illustration of experimental x-ray studies. When possible, a comparison is proposed with investigations with AFM or TEM
Jean-Marc Gay and Laurent
230
Lapena
surface
profile
10deg. lateral
position
103-
W
0.=+5,5deq.
OL=-4.5dg. 4-1
C),
a-7.5deg.
100 -0.5
0.0
0.5
1-01(deg.) Fig.
7.8. Surface of
a
Si
film,
500
nm
thick, deposited
on
(see also Sects. 8.4 and 8.5). Simulation (solid line) (dashed line) with the surface profile shown in the inset
strate
which
give
consistent parameters.
non-destructive tool for
looking
X-ray reflectivity
at buried interfaces.
misoriented of the
Si(111)
sub-
experimental data
is nevertheless the
only
7
Experiments
on
Solid Surfaces
231
References 1.
on Surface X-ray and Neutron Scattering, Ed., North Holland, Physica B 221 (1996). M. Tolan, in X-Ray Scattering from Soft-Matter Thin Films, Springer Verlag,
Proceedings of
the 4th Intern. Conf.
G.P. Felcher and H. You
2.
1998 in press.
3. L. 4. T.
Briigemann, R. Bloch, W. Salditt, C. Brandt, T.H.
51, 5617
Press and M.
Metzger,
Tolan,
Acta
Cryst. A 48, 688 (1992). Peisl, Phys. Rev. B
U. Klemradt and J.
(1995).
Jeu, J.D. Shindler and E.A.L. Mol, J. Appl. Cryst. 29, 511 (1996). 6. M.F. Toney and D.G. Wiesler, Acta Cryst. A 49, 624 (1993). 7. A. Gibaud, G. Vignaud and S.K. Sinha, Acta Cryst. A 49, 642 (1993). 8. L.G. Parratt, Phy8. Rev. B 95, 359 (1954). 9. L. N6vot et P. Croce, Revue de Physique Appliqu6e 15, 761 (1980). 10. S.K. Sinha, E.B. Sirota, S. Garoff and H.B. Stanley, Phys. Rev. B 38, 2297 5.
W.H. de
(198 ). Bonnell, J. Mater. Res. 5, 2244 (1990). Rohrer, Surface Science 299-300, 956 (1994). 13. C.F. Quate, Surface Science 299-300, 980 (1994). 14. P. Schwander, C. Kisielowski, M. Seibt, F.H. Baumann, Y. Kim, and A. Ourmazd, Phys. Rev. Lett. 71, 4150 (1993). 15. J.M. Gay, P. Stocker and F. R6thor6, J. Appl. Phys. 73, 8169 (1993). 16. M. Ladev6ze, 1. Berbezier and F. Arnaud dAvitaya, Surface Science 352-354, 11. M.W. Mitchell and D.A. 12. H.
797
(1996).
X-ray Reflectivity by Rough Multilayers
8
Tilo Baumbach’ and Petr ,
Mikulik’
Fraunhofer Institut Zerst6rungsfreie Priifverfahren, EADQ Dresden, Kriigerstra)3e 22, D-01326 Dresden, Germany, Present address: European Synchrotron Radiation Facility BP 220, F-38043, Grenoble Cedex France, Laboratory of Thin Films and Nanostructures, Faculty of Science Masaryk University, KotlAfskA 2, 611 37 Brno, Czech Republic
2
Introduction
8.1 One
tendency
in
present material research is the increasing ability
ture solids in one, two and three dimensions at on
various material systems artificial
a
to struc-
sub-micrometer scale. Based
rnesoscopic layered superstructures
such
quantum wires and dots have
multilayers, superlattices, layered gratings, successfully. This has opened new perspectives for manifold technological applications (e.g. for anticorrosion coating and hard coating, micro and optoelectronic devices, neutron and x-ray optical elements, magnetooptical recording). The perfection of mesoscopic layered super-structures is characterised by as
been fabricated
1. the
perfection
thickness
...
2. the interface
sion 3.
of the super structure
(grating shape, periodicity, layer
), quality (roughness, graduated
hetero- transition, interdiffu-
...
crystalline properties (strain, defects, mosaicity
Roughness is of crucial importance for the physical behaviour of interfaces. Roughness reduces the specular reflectivity of mirrors and wave guides for xray and neutron optics. Moreover it creates unintentional diffuse scattering. In magnetic layers it changes the interface magnetisation. Roughness promotes corrosion and influences the hardness of materials. It disturbs the electronic band structure in semiconductor devices. Interface
roughness supports the generation of crystalline defects in layered structures. In multilayers already the roughness of the substrate or the buffer layer influences the quality of all subsequent layers. Depending on the growth process the roughness profile can be partially replicated from interface to interface. Interface roughness is a random deviation of the layer shape from an ideally smooth plane. We consider here roughness with correlation properties of mesoscopic (sub-micrometer) scale. Irradiating a macroscopic area of the sample, surface sensitive x-ray scattering allows the investigation of the statistical behaviour of the roughness profile. Interface roughness in multilayers can be studied by all surface sensitive xray scattering methods (x-ray reflection (XRR), grazing incidence diffraction
J. Daillant and A. Gibaud: LNPm 58, pp. 232 - 280, 1999 © Springer-Verlag Berlin Heidelberg 1999
X-ray Reflectivity by Rough Multilayers
8
(GID), strongly asymmetric similar to the
principles
case
x-ray diffraction
of
(SAXRD)) employing physical
simple surfaces. They
1. the reduction of the information
depth
at
233
are
based
grazing angles
on
of incidence and
exit,
by the individual interfaces incidence,
2. reflection of x-rays
small
angles
of
3. interference of the 4. diffuse
scattering
waves
reflected
of x-rays
of
a
multilayer (ML)
at
different
interfaces, by interface disturbances.
by
Specular x-ray reflection (SXR) as the most frequently used method studdepth profile of the electron density. It detects the density gradient at the interface between two layers, where from we conclude on the r.m.s. roughness. Grazing incidence diffraction and strongly asymmetric x-ray diffraction detect interface roughness via the strain and the depth profile of the Fourier components of the electron density. The measurement of diffuse x-ray scattering (DXS) gives a clear evidence of interface roughness, distinguishing between roughness and graduated interfaces due to transition layers, interdiffusion or graduated hetero-transitions. Up to now DXS has frequently been ies the
observed in the XRR mode
[1-10].
First measurements of DXS in the diffrac-
reported recently [11,?]. DXS by multilayers enables one to characterise the lateral correlation properties of interfaces similar to DXS by surfaces. Moreover it allows to detect vertical roughness replication from interface to interface. DXS at grazing incidence occurs under condition of simultaneous intense specular reflection. This gives rise to strong effects of multiple scattering [5,7,8,13,14,10,12]. That is why semi-dynamical methods such as the distorted wave Born approximation (DWBA) are more appropriate to explain the DXS features than kinematical treatments. The paper intends to give an introduction into theoretical and experimental aspects of x-ray reflection by solid multilayers with rough interfaces, illustrated by various examples. We start in section 8.2 with a short presentation of rough multilayers and of the notations used in this chapter. tion mode have been
In section 8.3 we will introduce in the experimental set-up and usual experimental scans and in the following sections we apply the results of the chapters 3 and 4 on multilayered samples with different types of interface correlation properties. There we discuss typical features of the reflection curves and reciprocal space maps by various experimental examples. Afterwards, we mention the investigation of roughness by surface sensitive diffraction methods and at the end we study the reflectivity by intentionally laterally structured multilayers (gratings). Throughout the chapter the reciprocal space representation of the optical potential and the scattering processes allows us to outline the scattering principles in a geometrical way. The basic principles of it are surnmarised in the appendix.
Tilo Baumbach and Petr Mikulik
234
Description
8.2
The
of
scattering potential
of
Rough Multilayers sample
be
represented by the polarizability by the dielectric function c(r). In classical optics it is common to use n(r) or c(r), x-ray optics uses also I S(r) n(r). In order to pronounce similarities in the procedures and expressions for all x-ray scattering methods, thus reflection and diffraction, we preferred to use in this chapter the polarizability X(r). We recall the
X(r), by =
a
the refractive index
can
n(r)
as
X(r)
--
well
as
-
relation between X and S
Furthermore,
we
will make
use
of the
V(r) where ko
=
27r/A
is the
-2S(r)
-
(8.1)
optical potential defined by 2
-k 0 X
vacuum wave
(8.2)
vector and A is the
wavelength
of the
scattered radiation. We represent
X(r)
of the
multilayer by
the
polarizability of the
individual
layers (see Fig. 8.1) N
1: Xj H
XH
(8-3)
j=1
In order to
distinguish between the interface properties of the layers and properties, each layer is presented by the product of the volume polarizability x,, j (r) and the layer siZe (shape) function Qj (r) their volume
Xj
(r)
X. j
=
(r) Qj (r)
.
(8.4)
X-ray reflection methods measure the scattered intensity in the region near the origin of reciprocal space (000) There, only the mean polarizability plays a role and we can replace x,,,, j (r) by the zero order Fourier component Xo j (r) which is not sensitive to crystalline properties. -
’7
X XZ" Zi Zj,l
Fig.
Z,
V Z
8. 1. The schematic
"ideal"
set-up of
planar multilayer (left). Its optical potential is characterised by the polarizability depth profile (right) an
-,,. y
235
Multilayers
Ideal Planar
8.2.1
Rough Multilayers
"ideal" multdayer with sharp and a laterally extended id interfaces. Then x0j (,r) will be constant within each "ideal" layer. The layer size (shape) function of a smooth layer with sharp interfaces is the difference of two Heaviside functions corresponding to the upper and lower, interfaces, Let
first deal with
us
smooth
0i.d 13
(,r)
-
H(z
-
Zj)
-
H(z
(8-5)
Zj+,)
-
Sharp interfaces do not allow any overlapping of neighbouring layers, thus DO (r) 0 for all other layers k 1 predicts f2id j. k (r) 3 --
Multilayers with Rough Interfaces
8.2.2
Similar to the smooth multilayer we express the layer contributions
polarizability by
the
sum
of
the individual
N
X (r)
1: X0 j Qj Or)
-
(8.6)
-
j=1
vertically layered structures with a random defect to be laterally statistically homogeneous. We concentrate on defects, which vary the layer shape and interface sharpness S?j (r) (interdiffusion and roughness) in contrast to those influencing the layer volWe will further consider
structure, which
we assume
properties X,, j (porosity, inclusions), Interdiffusion and graduated heterotransition between
ume
neighbouring layhave all, values Then can interfaces. vertically (r) graduated f2j produce between 1 and 0. The layer is defined within the region Qj (r) :A 0. We allow an intermixing of neighbouring layers only, in order to keep the layer sequence. We define here by interface roughness the random profile of locally sharp interfaces. The vertical shift of the, actual interface position with respect to its mean position is characterised by the displacement function Zj (q) Zj of each interface, Fig. 8.2(a), modifying the actual layer zj (r1j) size function, ers
--
-
Qj (r)
--
and the actual
H
(z
=
+ zj
were
Zj+j
(,rll)]) is
-
of
-
tj (r1j)
H
-_
(z tj
-
[Zj+l
+ zj+l
+ zj+l
(rjj)
-
(rll)])
zj
(r1j),
(8.7) where the
Zj.
Properties
properties
ter 2. There
lateral
tj
Correlation
Correlation
[Zj
layer thickness
"ideal" thickness is
8.2.3
-
of Different Interfaces
single rough interfaces have been studied in chapprobability density of heights pl(z) and
introduced the
height-height
correlation function
C_,,,(rjj,rjj’)
=
(z(,rjj)z(r)j’))
for
Tilo Baumbach and Petr Milculik
236
(a)
(b) Zj
Vj(T10
C,
’11,
; An)
Fig.
8.2. Notation of the interface
the correlation function of
(a)
one
Zk-
displacements
and schematical
and of two interfaces
representation of
(b)
interface, i.e. for a substrate. In this section, we will treat the correlaproperties between different interfaces of a multilayer. We introduce the two-dimensional probability density of two interfaces, Fig. 8.2(b), one
tion
P2(Zj)Zk1) -P(Zj(’Pjj))Zk(’Pjj1))
(8-8)
-
and
height-height correlation
Cjk(’rll
function
-
V111)
-
(Zj(’rjj)Zk(’rjj1))
(8.9)
Usually the perfection of interfaces in multilayers is essentially influenced by the quality of the substrate or buffer surface. The surface defects can be replicated in growth direction. Different replication behaviours have been observed, depending on the material system, layer setup and the growth conditions. The following replication model has been proposed in [15]: 1. during the growth of the jth layer, the roughness profile zj+l (,rll) of the lower interface is partially replicated and 2. other defects, an intrinsic roughness Aj (,rll), are induced by imperfections of the growth process Zj
where 0 denotes
(ril)
a
I dril’
=
/-Aj Orli)
+
=
zAj (r1j)
+ zj+l
convolution
zj+l (ril’) aj
(,rjj)
product.
0 aj
Here
a
(rjj
-
r1l’)
(r1j)
(8.10)
non-random
replication
func-
tion aj (,rll) has been introduced, determining the "degree of memory" of the interface at the top for the roughnes profile at the bottom interface. If the
replication function is zero, the upper interface of a layer "forgets" the interprofile at the layer bottom and its profile is entirely determined by the intrinsic roughness (no replication). Identical profile replication is achieved for zero intrinsic roughness and full replication (aj (r1j) equals the delta funcface
tion).
Other
cases
are
discussed in detail in
within the discussion of the In later sections
we
will
experimental use
[15]
and will win
our
interest
results.
the Fourier transformation of the interface
correlation functions
Cjk (qll)
dRjj Cjk (RIJ) ei" R11
-
( j (qjj) k* (qjj))
(8.11)
X-zay Reflectivity by Rough Multilayers
8
237
with
ij (q) In the
following
zj+,(,rll)
Zj (qll)
=
+
(8.12)
ij+l (qll) &j (qll)
neglect any statistical influence of the interface profile roughness Aj(,rll). Also the intrinsic roughness of shall be statistically independent. Then we find the recur-
we
the intrinsic
on
different interfaces
sion formula for the Fourier transform of the correlation function
Cjk (qll) where
kj (qjj)
trinsic
roughness
=
Cj+l,k+l(qll) aj (qll) ak (qll)
+
Jjk Kj (qll)
(8.13)
is the Fourier transform of the correlation function of the in-
,,(j(rii
’ril)
-
=
(zAj(,r10,Aj(,r11,))
(8.14)
replication function a (rij) and the same in(replicated substrate roughness ZN (’r1j), for instance) roughness we get the explicit expressions for the Fourier transforms of the in-plane correlation function
If we
assume
for all
layers
the
same
A (r1j)
trinsic
Cjj(qll)
__
ONN(qll) [a(qjj)]
2(N _j)
+
K(qll)
(CNN (Q11)
is the correlation function of the
correlation
function
Ok>j(qll)
=
[d(qll )]2
substrate)
1
and of the
Okk(qjj) [&(qjj)](1’_j)
(8.15) inter-plane
(8.16)
physical meaning of the particular terms in (8.15) is obvious. The first on the right hand side represents the influence of the substrate surface modified by the replication function) the second term is due to the intrinsic roughness of the layers beneath the layer j. Knowing Ojj(qll) we can calculate the mean square roughness O-J2 of the jth interface: The
term
O_j2 8.3
=
(Zj2(r,,))
dqjj Ojj(qll)
(8,.17)
Setup of X-Ray Reflectivity Experiments
experimental setup to investigate the fine strucintensity pattern in vicinity of the origin of reciprocal and exit with respect space (000) under conditions of small angles of incidence to the sample surface.
In this section
we
outline the
ture of the reflected
Tilo Baumbach and Petr Miki-ilik
238
Experimental Setup
8.3.1
A conventional x-ray reflectometer is drawn in Fig. 8.3. The x-ray source (a conventional x-ray tube or a synchrotron) emits a more or less divergent and
polychromatic beam. The -nonochronuytvr i1a crystal or a multilayer mirror) and entrance slits produce a sufficiently monochromatic and parallel beam, hitting the sample surface under the incident angle Oil,. Its angular divergence is characterised by the spatial angle AQin. The sample is mounted on a goniometer, which allows one to change the incident angle Oin by the rotation w. The x-rays are reflected (scattered) by the sample. The coherently reflected beam leaves the sample in specular direction (under the exit (final) angle 0,,c 0,,c in the plane of incidence). Due to roughness there occurs diffuse scattering into the upper half space of the sample. A detector rotates around the sample and measures the Pux of photons (in units of counts per second) through the detector window, which defines the spatial angle interval AQdt around a certain spatial angle S?.,c (sufficiently defined by Os, in the coplanar case). If we suppose a perfectly monochromatic and parallel incident beam of intensity -10 then the idealised flux through the detector window is related with the differential scattering cross section by =
J
Taking
the
account,
i
=
divergence
we
JAd?j.
Actually,
in
lo
I
0
do-
-_
+AQdet/2
lo
and the
(dQ
intensity profile
do-
dQ
(8.18)
.
of the incident beam into
obtain d2
d,/-A Qj 1, lo (A Qjj
of
+Afldet/2
)Jf2,_,-cAd2det/2
large sample,
do-(S2jj +zAQj, Q) dQ
dQ
.
(8.19)
the detector slits select another
angular sample area. That can be overcome replacing the detector slits by an analyser (also a perfect crystal or a multilayer mirror) in front of the detector similar a triple-crystal diffractometer (TCD). The monochromator is the "first crystal", the sample the "second case
interval for each
a
point
on
the illuminated
Monochromator
Analyser
A- ,‘D&tectoir
ar
Sample
Fig. 8.3. Schematic setup of an x-ray reflectometer (source, monochromator, sample, slits and detector) and of a triple-crystal-like diffracton-leter (source, monochromator, sample, analyser and detector)
X-ray Reflectivity by Rough Multilayers
8
crystal"
and the
analyser
the "third
crystal".
The flux measured
by
239
the TCD
is
L where
d A Qin 10 (ZA f2in) i.
D(zAS2)
8.3.2
is the
f
dS2
do-(0i, +, Af2in, 0) dS2
V (S2
-
Qs, )
(8.20)
,
reflectivity of the analyser.
Experimental
Scans
Mapping the measured flux for different angles of incidence and exit we can or by plot the measured scattering pattern in angular space, J(f2in, three reciprocal space coordinates and one angular coordinate of the sample, kin, Vin). Restricting ourselves on coplanar reflection (ks, kin e.g. j (k,;, and the surface normal are in the same plane), the angular representation J(Oin, 0,;,) and the reciprocal space representation J(q) with the scattering vector q kin are equivalent. k 3, The principal rotations of a (coplanar) TCD are: -
=
-
arrangement in the coplanar scattering measures the scattering angle (20 Oi,,+O,,,), plane around the sample: the variation of 20 changes 0,,, (A20 AO,c). The rotation w of the samp le around the same axis: Oin 20-w, a w, O , variation of w changes simultaneously Os, and Oin (ZAW Oin -AOsc)
1. The rotation 20 of the detector
20
--
2.
-
AWA20
*
Q.
Fig.
Q11
QZ
experimental scans in the reciprocal space. Right figure enlargement around its origin, where x-ray reflection takes place. The
8.4. Illustration of the
shows the 20-scan
-
(detector-scan)
represents rocking with qjj
=
0
follows the Ewald circle of the incident
scan, which is transversal for XRR. For 2W
(speculax scan)
wave.
=
The
20 it is
a
W-scan
q-,-scan
Tilo Baumbach and Petr Miktdik
240
Different
experimental
Fig. They are:
detector
be
scans can
8.4 the most usual
In
scans are
performed by coupling both rotations. reciprocal space.
illustrated in real and
20-scan. The incident wave vector kij opens out the Ewald keep the angle of incidence fixed (w const) and rotate the detector arrangement, we move in reciprocal space along the Ewald sphere c. w-scan or constant q-scan. The w-scan rotates the Ewald sphere around the origin of reciprocal space. Fixing the scattering angle 20, we fix the modulus of the scattering vector. Then the w-scan represents a constant q-scan since we move in reciprocal space on a circle of radius q q around the origin. zAO/A20-scan or radial scans. Rotating the sample and the detector arrangement in a ratio zAw/zA20 1/2, we drive the TCD in reciprocal space in radial direction from the origin of reciprocal space. 0/20-scan on the q,, axis or specular scan. This special radial scan with 0., and performs a q,-scan at w120 1/2 keeps the condition Ojj 0. This experimental mode is also called specular scan, since the q., detector selects always the specularly reflected beam. q.,-scan and q,,-scan. These scans go parallel to the q., and q,, axes at fixed q,, and q., position, respectively. scan or
sphere
c.
If
we
=
--
--
-
=
Sometimes it is useful to
reciprocal space map, i.e., to meaintensity by combining different scans, e.g.
measure a
the map of the scattered measuring a series of w-scans sure
(rocking-scans)
in the interval from W--O to
varying position-sensitive detetor (PSD), one would Using for different PSD-spectra omega positions. The angular region investigated by a reflection experiment is limited by the horizon of the sample. The limiting cases for grazing incidence (0ij 0) and grazing exit (0, 0) are illustrated in Fig. 8.5. The situation in is reciprocal space represented by the two limiting half spheres co and ’E". X-ray reflection experiments are usually realised at very small scattering angles. In Fig. 8.4(right) we show the introduced experimental scans in the x-ray reflection mode and their restrictions due to the sample horizon. Especially the w-scans are narrowed down. In the accessible region of reflection, w--20 for
20.
a
detect
-_
--
Grazing
incidence
Grazing
exit
OK.
K
2n& 4./)L
Fig. 8.5.
Situation of
and real space
grazing
incidence
(left)
and
grazing
exit
(right)
in
reciprocal
X-ray Reflectivity by Rough Multilayers
8
i.e.
near
the
transversal
of the
origin
reciprocal
space,
241
they perform approximately
a
(qll-scan).
scan
Specular X-Ray Reflection
8.4
In this section
we
discuss
experimental examples
theoretical and
some
of
structures with the aim to show
specular x-ray reflection by features created by different
layered surface. roughness point properties. The coherent scattering intensity is concentrated along the specular rod. That means, the appropriate experimental scan is the specular or 0/20-scan. coherent
typical
with
Roughness
8.4.1
a
Gaussian Interface Distribution Function
Single Surface The predominant number of samples have been successfully characterised assuming a Gaussian probability density of the interface roughness profile (see (2.19)) Pi
In this case,
(e.g.
a
(Z)
o-- ,/_27r
e---’ /2
or
2
(8.21)
Chap. 3, Eq. (3.103), we obtain for a single amplitude ratio of dynamic reflection [16,17]
shown in
as
substrate)
the
r,,h dyn with the
=
amplitude
flat
rdyn
::--:
e
-2k ,,Ok.,
,or
ratio of the flat substrate
2
dyn
see
(8.22)
being
reflection coefficient of the substrate surface rflat
surface
the
dynamical
Fresnel
(k;,,o-k_’,j)1(k,,o+k_’,j),
Eq. (3.68). The ratio of kinematical
reflection
rcoh kin
’::::: ’
coefficients is
flat
rkin
(Eq. (3.104))
2
-
0-
2k. ’00,2
with the kinematical Fresnel reflection coefficient of the surface
(8.23) rflat kin
=
214q 2
qC
Z;
Eq. (3.9 1). Both the kinematical and the
multiplied
with
a
dynamical Fresnel reflection coefficients are containing the r.m.s. roughness 0- in the
diminution factor
exponent. The kinematical diminution factor decreases with the
square of
proportional to the angle of incidence. Its scattering form resembles the static Debye-Waller factor. The dynamical diminution factor contains the product of the scattering vector in vacuum q,,o and that in the medium q,,,. The angular dependence of the diminution factors in the dynamical and the kinematical theory differs substantially for small angles near the critical angle of total external reflection 0, see Fig. 8.6. Neglecting absorption, the scattering vector q,,, becomes purely imaginary below 0,. Consequently there is no influence of roughness on the reflectivity in this the
vector q, which is
Tilo Baumbach and Petr Mikulik
242
101L
10,
10-1
1Oo 10-, 10
10 0 0.0
-2
0
0.1
0.2
0.3
OA
10‘
10, 10-4
10-4
107,
e
10"0!0* ’07 ’0:4’0.6 angle
Fig. 8.6.
0.8
1.0
of incidence
The coherent
1.2
1.4
1.6
1
7; .0
0.2
0.4
(deg]
0.6
angle
1.0
1.2
of incidence
0.8
[deg]
1.4
1.6
reflectivity
of a rough Si surface. In the left panel the refleccompared with that for the roughness a =I nm, calculated by the "dynamical" theory (8.22) (full) and the kinematical theory (8.23) (dotted). The kinematical reflectivity diverges at grazing incidence. The "dynamical" curve coincides nearly with that of the flat surface below the critical angle 0,-. In the subfigure, the dashed line represents the coherent reflectivity of a rough surface calculated with dynamical Fresnel reflection coefficient and kinematical diminution factor. Thus the reflectivity decreases also,below 0,. In the right figure, influence of different roughness, calculated by dynamical formulae, is demonstrated. Close to 0,_ (see subfigure), no essential change is observed
tivity of
a
flat surface
(dashed)
is
angular range within the dynamical diminution factors coincide. A
description.
At
detailed discussion of both formulae
large
incident
(8.22)
and
angles
(8.23)
both
is
given scattering specular direction has been studied by means of second order DWBA, showing its dependence on the lateral correlation length A. Concluding therefrom, the specularly reflected intensity can be described by the "dynamical" equation (8.22) for short A below I pm. For larger A the kinematical formula (8.23) becomes more appropriate. with r flat kin Surface roughness of numberless samples of amorphous, polycrystalline and mono-crystalline material systems has been studied by SXR. In Fig. 8.7 we plotted one experimental example, the reflectivity of a rough GaAsin
more
[13].
There the contribution of the incoherent
to the
substrate.
Multilayer Conventional SXR-simulation and fit programs are today based a multilayer model with independent r.m.s. roughness profiles of each interface supposing a Gaussian probability density. This leads to effective on
Fresnel reflection and transmission coefficients
rj,j+l
-
r
flat
3,3+1
e
and
tj,j+l
(Eq. 3.103): --
tjflat+,,(k,,j-k,,j+l )2 7
.1/2
3,3
(8.24) for each interface. The influence
according
on
the transmission function is
to the small difference in the vertical
scattering
vector
rather
small
components
X-ray Reflectivity by Rough Multilayers
8
243
10, 10
0 Z
r \\
0
O.S.
I Z=CY
Z=
0.0
\20 -40
(D
1074 10-,
-20
020 20
’0 (A) z
40 40J
(points) and (line) reflectivity curves 12 A [18]. of a GaAs substrate, a Measured
F ig. 8.7.
calculated
=
0.2
0.0
0.4
0.8
0.6
1.0
1.4
1.2
1.6
In the inset the the surface is
angle of incidence [deg]
mean
coverage of
plotted
layers. However, the interface reflection is exponentially diminished by roughness, creating a strong change in the interference pattern. The effect of interface roughness versus surface roughness is shown in Fig. 8.8. The surface roughness mainly decreases the specular intensity of the whole curve progressively with q, where the interface roughness gives rise to a progressive dampening of the interference fringes (thickness oscillations). However, locally the variation in the Fresnel coefficients can cause more pronounced oscillations, too. In Fig. 8.9 we plotted the experimental and simulated curves of a magnetic rare earth/transition metal multilayer (Cr/TbFe2/W on sapphire A1203), grown by laser ablation deposition. It shows a quite complicated non-regular interference pattern. A good agreement with the simulation was realised by considering a thin oxide film at the sample surface. of the
1 100.1
10-2 0.01
10-3 10
A,‘.’,A
-4
0.00,
10-5 10-1
14\ 0
0.5
1
1.5
angle
2
2.5
3
of incidence
0.0001 3.5
4
4.5
[deg]
5
0
0.25
0*5
angle
1.25 1 0.75 of incidence [deg)
1.5
specular reflectivity of a single layer (20 nm tungsten) r.m.s. roughness and diminution factors. (a) (sapphire) Dynamical diminution factor. F om the upper to the lower curve: without roughboth surface and interness, interface roughness 0.5nm, surface roughness 0.5nm, face roughnesses 0. 5 nm. Surface roughness yields a faster decay of the refiectivity, while interface roughness attenuates the peaks. (b) Different diminution factors. Surface roughness 1.2 nm and interface roughness 0.3 nm calculated for the kinematical "slow" roughness (lower curve), dynamical "rapid" roughness (middle curve),
Fig.
8.8. Calculation- of the
on a
substrate
and without
for different
roughness (upper curve)
Tilo Baumbach and Petr Mikulik
244
.
..........
100000
2
10000 t
A12 03
1000
100 0
0.5
1
1.5
2
angle of incidence [deg]
8.9. Measurement (points) and the fit (full curve) of the specular reflectivity of Cr/TbFe2 /W multilayer [19]. We determined the thicknesses (34.6 nm W, 4.8 nm TbFe2, 5 0.5 nm Cr, 3 nm oxidised Cr) and the roughnesses (0. 2 nm above sapphire, 2.0 nm W, 0.9 nm TbFe2, 2.2-nm Cr)
Fig.
a
Periodic Multilayer The main feature of the specular scans of a periodic multilayer are the multilayer Bragg peaks, giving evidence for the vertical
periodicity,
see
Fig.
8.10 and Sect. 8.A.2.
10000
10
(a)
1000
b)
100
10
0.1
0
!A ;14
0.1
0.01
0.01
0.001
0.001 0.0001
0.0001 0
0.25
of incidence
angle
Fig. 8.10. Specular for
a
reflection
[GaAs (13 nm)
strate, flat interfaces
curve)
0.75
0.5
1
0
[deg]
by
an
0.25
angle "ideal"
0.5 of incidence
0.75
[deg]
periodic multilayer-calculated
curves
(7 nm)] superlattice with 10 periods on a GaAs sub(no roughness). (a) Comparison of the dynamical theory (full AlAs
with the kinematical
theory. The kinematical multilayer Bragg peaks
cor-
positions (000) RLP. The curve diverges at low incident angles. The dynamical calculation shows the plateau of total external reflection below the critical angle. Due to refraction the multilayer Bragg peaks are shifted to larger angles. The first multilayer Bragg peak broadening is caused by multiple reflection (extinction effect). (b) Comparison of the dynamical theory with the semi-dynamical approximation (single-reflection approximation [18]). The satellite positions of all Bragg peaks coincide, also the shape and intensities except for the intense Bragg peaks respond
to the
of the satellites of the
X-ray Reflectivity by Rough Multilayers
8
245
intensity ratio of the Bragg peaks depends on the layer set-up within multilayer period. The difference in the electron density determines the Fresnel coefficients, and the thickness ratio of the layers characterises the phase relations of the reflected waves of different interfaces. The laterally averaged gradual interface profile caused by interdiffusion or interface roughness leads to a damping mainly of the multilayer Bragg-peaks progressively with q, whereas the roughness of the sample surface reduces the intensity of the whole curve. This is demonstrated in Fig. 8.11. The
the
roughness roughnesses surface roughness no
10-1 10 t5 A
interface
----------
------
2
10-3 10"4 10-5
1
1
1
1
1
1
0
0.2
0.4
0.6
0.8
1
angle
of incidence
1 V 3 H9
1.2
1.6
1.4
[deg]
Fig. 8.11. Simulation of coherent reflectivity of a [GaAs (7nm) / AlAs (15 nm)]10x periodic multilayer with no roughness (full curve) or I nm roughness of surface
(dashed
lower
curve)
or
of all interfaces
(dotted)
Fig. 8.12 we plotted the measured SXR curves of an epitaxial CdTe superlattice on a CdZnTe substrate. Due to the low contrast of the electron density of both layer materials the first order Bragg peak appears only as a very weak hump on the slope of the surface. The other Bragg peaks have a shape similar to a resonance line. From the best fit we obtain the mean compositional profile. In
CdMnTe
decreasing roughness in multilayers The influence or decreasing during the growth from the substrate increasing roughness towards the surface can be described. by use of the roughness replication
Increasing
and
of
model introduced in Sect. 8.2. We start the ness
layer growth from
a
substrate with
a
Gaussian surface
rough-
profile, CNN (’rjj
For the non-random
-
replication
2
(8.25)
AN
ON
’rjj
function in
(8.10)
we
choose for all
layers
a
Gaussian function
a(ril
-
r1l’)
1,11 _11 112
1 =
-
7r L2
C
2L2
(8.26)
Tilo Baumbach and Petr Mikulik
246
10-
0.75
Ca) a)
0.5
10
>’
0
10-
PL re Pure
0.25
> C.)
LC(djTe
Pure
-2
3
T CdTe
4
0
8
12
16
position [nm]
z
10-4 10-5
10-6 0
0.25
0.5
1.5
1.75
2
[deg]
8.12. Measured and calculated
CdMnTe is
1.25
of incidence
angle Fig.
1
0.75
specular reflectivity of a [CdTe (14.2 nm) CdZnTe [20]. In the subfigure, the roughness effective MnTe concentration depth proffle
(2.5 nm)] 20 x superlattice
represented by
an
on
The factor L determines the loss of memory from interface to interface. This choice arises from the aim to explain the different limiting cases of roughness
by one class of functions. It is not supported by any physHowever, the model allowed to describe measured curves of SXR and NSXR showing good agreement [15,8].
replication ical
models
reason.
-
We
the intrinsic correlation function
assume
(8.13) ’
K(’rjj _,r111) Now
we
1,11 -11 ) (’ Ao.)2 e ( I
of all interfaces
2
(8.27)
-
AA
-
continue like in Sect. 8.2. The Fourier transform of the
correlation
function
is under these
N-1
Z2 1
Cjj (q)
-
(ONAN 2
2
in-plane
assumptions +
2
(Ao-zAA )2 2
(gAk )2 e
4
(8.28)
k=j
where
The
we
have denoted
A’
j
VA2
inter-plane correlation
is then
IV
Ci j, k (q) We obtain the
mean
?
O’j
square
=
+ 4L2 (N
j)
(8.29)
simply given by
Cjj (q)
roughness
dq Cjj (q)
-
2
ON
(qL)2 -
2
e
of the
AN2 W.:
i-k
(8-30)
jth interface N-1
+
(AozAA )2
E k=j
I
A k’2
(8-31)
X-ray Reflectivity by Rough Multilayers
8
VIA
0.1
(a) 0.1
2:1
0,01
0.01
.5
0.00,
0 .001
10-4
10-4
10-5
10 1
0.5
0
angle
Fig.
247
2
1.5
of incidence
-5
0
3
2.5
0.25
0.5
2
1 1.25 1.5 1.75 of incidence [deg],
0.75
angle
[deg)
(points) and simulation (full curve) of the specular reflecperiodic Nb/Si multilayer’ of 10 periods [19]. (a) Sample A, fitted by of constant roughness, (b) sample 13, fitted by the model of increasing
8.13. Measurement
tivity
of
a
the model
roughness
Let
us see
what does it
for
give
some
limiting
cases
of the model:
interface roughness is achieved with maximum replication and 0. Consequently o-j 0 and AoON, and all no intrinsic roughness: L substrate of the the ZN (X) surface, Zj (X) interfaces reproduce profile maximum obtained is surface the free towards by Increasing roughness o- > 0). From 0 and replication and a non-zero intrinsic roughness (L
1. Identical
=
--
=
2.
(8.28)
and
(8.31)
we
find 2
O’j
3.
-
-
2
2
ON +
(N
-
(8.32)
j)
describing the roughening during the growth. 0) leads Partial replication and no intrinsic roughness (L > 0 and zAosurface free the towards (smoothing of the to decreasing r.m.s. roughness described by multilayer during growth), --
?
17j
0’ -
2
N
-
1+4 (_L_) AN
2(N
(8.33) -
j)
replication occurs for diverging L, where a(,rll roughness profile of each interface is independent.
4. No
-
r1l’)
goes to
zero.
The
experimental example of two periodic Si1Nb rnultilayers, studies. The multilayer is grown by magnetosputtering for superconductivity thick with a Si02 layer and an Al buffer layer. deposited on a Si substrate buffer The roughness of the layer depends on its thickness and influences the
We compare here the
quality
of the interfaces. Two
investigated
and the results
samples
are
of different Al thickness have been
shown in
Fig.
8.13. The
multilayer period-
icity generates the multilayer Bragg peaks or reflection satellites, which are dampened by interface roughness. The roughness of the substrate and the
Tilo Baumbach and Petr Mikulik
248
buffer
layers has less influence on the reflection pattern. Sample A can be by a roughness model of constant r.m.s. roughness for all interfaces. The peak widths of the first intense Bragg peak is broadened by extinction due to dynamical multiple scattering. For all higher order Bragg peaks we observe a narrower (kinematical) peak width. The satellite reflections of sample B are also rapidly damped, indicating a large interface roughness. Besides the widths of the peaks increases with q.,. That can not be explained by model 1. The satellite intensities and shape can be successfully reproduced by supposing increasing roughness according to (8.32). Due to their increased roughness, the upper layers near the surface contribute with decreasing effective Fresnel coefficients to ’the reflected wave. Within the Bragg position the contributions of all interfaces are still in phase, however, slightly away from the Bragg condition the contribution of interfaces near the substrate and those near the sample surface do not cancel completely, giving rise to the peak broadening. fitted
Stepped Surfaces
8.4.2
The surface
morphology of monocrystalline samples can also be described by a discrete surface probability distribution following .LIj- L_j L__J .............X Z. the concept of terraces or small separated I islands. In the simplest case, the two-level 8.14. Multilayer with random Fig. consists of surface randomly placed islands two-level islands of uniform height d, so that the displacement z (r1j) has two possible values z, and Z2 d + z, with the corresponding 1 probabilities pi and P2 pi, see Fig. 8.14 [21]. The surface probability ..
I..dn.................n
F
.............
..
........
........................
...........
--
--
distribution function
p(z)
-
for this
P(Z) Since
(z (,Pll)-)
=
0, then Zi
-
:---
case
writes
P16(Zl)
-P2 d and
+
Z2
P26(Z2) =
(8-34)
pi d. The
mean
square
roughness
is 0’
2
=
p1Z21
and the characteristic function
X(q ,)
-
+ P2
(2.10)
e-iq
dP2
Z22
-
2
(8-35)
d)
(8.36)
PlP2 d
is
(PI
+ P2
e
iq,,
Putting this in the formulae for the reflected amplitude ratio of rough faces, we get the amplitude ratio of kinematical specular reflection rc9h kin A surface
forming
an
=
e-iq ,dp2
region perturbed in
upper and
a
(pi ro, 1
+ P2 ri 2e
this way acts
iq d)
as a
sur-
(8.37)
thin, homogeneous layer
lower interface with the Fresnel reflection coefficients
8
1
0o
..
...
.
X-ray Reflectivity by Rough Multilayers
...
10, V
:
...
111.11,11.1
........ ......
-two level surface -------
0 .............
10
gaussian surface
10
fiat surface 2:
t
(D
10-,
=
............
10
W
pl=0.4
.............
W
m
angle of
2.0
1.5
incidence
d
50 o
A
d
10
=
30
A
-,
10
10-, 1.0
u
surface level l two tw 0 I.,
10-,
pl=0.5 0.5
....
...........
-6
0.0
I
fl at surface flat Sul s
-2
10-3
10
....
249
2.5
3.0
ir r..
0.0
0.5
angle
[deg]
t
........
1.0
1.5
of incidence
2.0
2.5
3.0
[deg]
Fig. 8.15. Coherent reflectivity of a two level surface calculated within the kinemat5 nm (left) theory for two values of the probability pi’and the step height d 0.5 and for two values of d and a symmetrical probability distribution pi P2
ical
=
=
=
(right)
piroj and P2r,,2. They height d (Fig. 8.15).
give
rise to interference
fringes
which represent the
example of a thin surface layer of porous silicon fits approximately this simple model, if its thickness is smaller than the vertical correlation lengths of the crystallites (Fig. 8.16 (a)) [22]. Since the surface "layer" density is quite different from that of the substrate, we can observe two critical angles 01 and 02. The second one, 02, corresponds to silicon, the first one, 01, to the averaged surface region. Above 01 the wave can penetrate into the perturbed The
region, however total external reflection occurs at the "interface" with the non-perturbed region. That is why very intense fringes appear in this region between 01 and 02, which drop rapidly above 02. The whole curve is similar to that of a homogeneous layer of much less density or to that of a surface grating. In the fitted curves a small Gaussian deviation of the actual displacement around the z, and Z2 has been supposed, which leads to roughness diminution factors of the Fresnel reflection coefficients similar to
surface
(8.24). 8.4.3
Reflection by "Virtual Interfaces" Between Porous Layers
layers are fabricated by electrochemical etching in, a monocrystalline silicon wafer. By a variation of the anode voltage, multilayers of modulated porosity can be produced. Following our division of the layer polarisability we can distinguish between the porous layer volume and the size of the layer of equal porosity. The interface between two layers of,different porosities is not a microscopic laterally continuous and sharp interface between two media of different density, but an interface of two degrees of porosity. According to the coherent approach (used also in Sect. 3.4) we take for the coherent reflection an effective averaged refractive index into account. Porous silicon
Tilo Baumbach and Petr Mikulik
250
(a)
(b)
-2-
-2-
3
-
-3
I
4[
0 0,
01
-4-
.
0.4
0.2
0.0
angle
of incidence
0.6
-5 0.0
0.2
angle
[deg]
0.4 of incidence
0.6
[deg]
(full) and fitted (dashed) reflectivity curves of a thin porous layer (a) and of a porous silicon double layer (b) on silicon substrate [22]. Positions 011 02 are the critical angles of the porous layer and the substrate, respectively Fig.
8.16. Measured
silicon surface
Layers of statistically homogeneous porosity
are
assumed. We treat the slow
of the transition between two
layers of different porosity by a "roughness" function results Same are obtained by introducing a probability transition from of to layer layer. An experimental examporosity graduated for double is in a layer sample [22]. The thickness of the Fig. 8.16(b) ple given surface layer is much smaller than that of the buried layer. The fast oscillating fringes represent the total thickness. The fringe amplitude is modulated by a period, which corresponds approximately to the thickness of the surface layer. It has been found from the simulation that the interface between the two layers of different porosity is much sharper than the interface with the substrate (which is the end front of the etching process). The occurrence of the modulation of thickness oscillations in Fig. 8.16(b) is a direct proof for the validity of the coherent scattering approach. Between the two porous layers there is nowhere a real roughly smooth lateral interface between two media. Nevertheless the x-rays are specularly reflected at this "microscopically non-existent interface" showing all features of the continuum theory of dynamical reflection by multilayers. Gaussian
8.5
Non-Specular X-Ray Reflection
scattering approach (2) within the explicit expressions for the incoherent scattering cross section for x-ray reflection by rough multilayers. We discuss the main features of the scattering patterns illustrated by experimental examples. The representation of the scattering in reciprocal space allows a simple interpretation of the findings by the various scattering processes. We will treat samples with interfaces having a Gaussian roughness profile, diffuse scattering from terraced interfaces and finally non-coplanar diffuse scattering. In this section
DWBA
we
(Chap. 4)
use
the incoherent
and derive
some
8
Interfaces with
8.5.1
X-ray Reflectivity by Rough Multilayers
Gaussian
a
251
Profile
Roughness
having a Gaussian roughness profile. We start with the scattering from a single surface. Then we continue with a multilayer showing the effects of different roughness replication as well as dynamical scattering effects on reciprocal space maps. We will deal with interfaces
Surface
Single
distribution. The case
(see [1,23-25],
for
instance) Z2 + Z/2
1
(Z, Z
P2
Firstly we will deal with surfaces of a gaussian probability pair probability distribution function is in the stationary
27r
0
4
-
exp
C2 ZZ (,rl
2o-2[l
r,
2zz’C
_
-
ZZ
-LC2z(,rll ,4
-
’ril) r111)] (8.38)
with the two-dimensional characteristic function X,, -,,
i(qz-qY))
(q, q’)
e_0,2(q2+q’2)/2 eqq’C_(,r11-r11’)
_
(8.39)
One correlation function, which has been successfully applied to interpret experimental findings, follows from similarities between the description
the
roughness properties and the Brownian motion, position by time. Supposing a behaviour like [1,24]
of interfaces with fractal
replace
we
the lateral
[Z(,rll) leads
together
_
Z(,Pll/)] 2)
=
A
I
Jr11
r11
12h
0 < h <
if
(8.40)
1,
with
[Z(’rll) Z(,rlll)]2)
2 0.2
_
-
2C,,(,r11
-
(8.41)
rll/)
correlation function, which only depends on the distance 1,P11 rl,’J. The so-called Hurst factor h describes the jagged shape of the interface, h. For h -- 1 determining the fractal dimension D of the interface, D = 3
to
a
-
-
corresponds
the fractal dimension is 2 and an
interface
(without
a
fractal
structure).
to the
topological dimension of diverges for large
This function
Thus it is. suitable to introduce a cut-off radius . Below the correlation function s4all approximately behave like (8.40), but above
distance
Jr11
-
r1l’J.
it should converge to
C..- (ril The cut-off radius Let
us
now
,
zero.
A function with such
r1l’)-Wrll)
-
a
behaviour is
Z(’r11T-0-2e_(1rJJ_’rJJ111 )2h
(8.42)
.
represents the lateral correlation length of the interface.
determine the incoherent
cross
section for
a
surface with such
incoherent scattering
properties. Using Eq. (4.41) we find for the of a single rough surface within the full DWBA dCincoh
=
dQ
k40 1672
01 Itinj 0, 12 0, 12 Itscl
cross
section
(8.43)
Tilo Baumbach and Petr Mikuhk
252
with the covariance function
Q,
--
A
2
In,
e
212
(q." +q.* i
2
no
-
1
f d(,rll
X
(4.D28)
-
(8.44)
X
12
r1l’) eiqll (rjj -,rll’)
A
16
1 q1z 12Czz (ril -11,)
where A is the area of integration, that means the illuminated surface of the sample. The result can be interpreted as follows: the incident wave transmits through the surface considered by the Fresnel transmission coefficients. This "distorted wave" is diffusely scattered by the surface disturbance. Thus the non-specularly reflected intensity depends on the r.m.s. roughness and is proportional to the Fourier transform of
le jqjz1’C--(r11-,11’) Taking the correlation function (8.42), we have C,, (,rjj r1l’) < 0.2 For small roughness or small q,, fulfilling (o-q,, )2 < 1, we can approximate (8.42) by the first two terms of its Taylor series and obtain finally -
.
4
dO’incoh i.e.
an
--
dQ
koAInj2
_
n0212
167r 2
expression, which
is
Itin 01 12 Itsc 01 12
proportional
-
e
1.0,2
,2F
,
(q,2, I+q
.2
(8.45)
,,i
to the Fourier transform of the
cor-
relation function. The
according
kinematical
mission coefficients
medium q,,l
equal by the scattering
expressions
to I and
are
substituting
found the
by setting the transscattering vectors in the
vectors in vacuum, q,,.
Multilayer with no vertical roughness replication In case of independent roughness profiles of all different interfaces we have the replication function a,(rll) 0 (L --+ oo in (8.26)). There is no inter-plane correlation, that is why only the in-plane correlation functions have to be considered. We can proceed for each interface like in the case of a single surface described above. However, now we take four scattering processes (coresponding to downwards and upwards propagating incident and scattered waves), see (4.D27), into account instead of one in (8.43). Consequently, we consider 4 x 4 covariance functions for each interface. The incoherent scattering cross section adds up the contribution of all single interfaces --
(dQ )
4
do-
ko incoh
167r 2
N
EEEET,
(8.46)
j=0
Uj(kin,,j,Zj)Uj(k,,,,j,Zj)Uj(k’ ,,j,Zj)Uj(k* Qjj ( kin zj k,c j, kin zj kjc zj) in
Sc
j,
Zj)
X-ray Reflectivity by Rough Multilayers
8
253
100
jo
Yoneciawing
.
Yoneda
wing
Fig.
shifted 0
2
1.5
1
0.5
angle of
incidence
2.5
layer
[deg]
fit
and down
at 2e
w-scan
0.1
Measurement
8.17.
(points)
W
2
2.630 single
=
(11. 1
(full line, x) of an
nm)
on
Si
substrate
with A
Qjj(q,,q") z
Ixoj+l q, (ql )
xoj
-
12
e
2
.7
[q2+ (q ’, )*2] .
(8.47)
X
*
z
x
I d(rij
-
r1l’) eiqII(,II-,II’) eq;;(qzl)* Cjj(rII-TII’),
A
polarisabilities XOj+j-XOj-_n3 +j-n 3 instead of the optical indices. Assuming the same in-plane correlation functions for all interfaces the jj of different interfaces differ only by the scattering vectors and the differences of polarisability. Figure 8.17 shows a measurement and fit of an w-scan from a single layer sample. where
we
have used the
Multilayer with partial vertical roughness replication In case of parroughness replication also the covariance functions of scattering at different interfaces have to be included. We get (cf. (4.45)-(4.46) and
tial vertical
(4.D28)-(4.D29)) do-
k4
M
16/1
incoh
N
N
1: 1: 1: j=0
-
’
-
’
-
(8.48)
’
k=O
Uj(ki,, ,,j,Zj)Uj(ksc,,j,Zj)Uk(k z,k)Zk)Uk(k* z,,,,Zk)’ SC
in
k,;c,,j, kin z,k
Qjj, (kil zj with the covariance function
A
Qj k (q,, q’)
(Xoj+l
-
k c,,k)
(see Fig. 8.18)
Xoj)(Xo k+j
z
-
Xok)*-e-2![u q
2
+,72 (q’
x
z
x
I d(,rll
_
V111) eiqll (III -III’)
A
Here o-j and o-k are the termined by (8.31), Cjk
(8.49)
roughnesses of the corresponding interfaces detheir inter-plane correlation functions. Restricting
r.m.s. are
(eq.(qlz)* Cjk(rll-Tll’) 1)
Tilo Baumbach and Petr Mikulik
254
Fig.
8.18. lRustration
variance
tering other
function
considering
C?jl, (q., q’) z
of
process qz at the interface
scattering
process
qz’
the
one
co-
scat-
j and
an-
at the interface
k
ourselves lar to
roughness (o-q,)2
small
on
(8.45) using
<
1,
we can
make
approximations simi-
the Fourier transform of the correlation functions
obtained in section 8.4, Eqs. (8.28) and (8.30). The treatment of the corresponding expressions of the
Cjk (Q11)
simpler D WBA for multilayers (p. 160), is straightforward. It neglects the influence of specular interface reflection on the diffuse scattering. Only the primary scattering processes
8.5.2
are
taken into account.
The Main
Scattering
Features of
Non-Specular
Reflection
by Rough, Multilayers Let
give
overview of the main features in the
non-specular reflected physical origin. The diffuse x-ray scattering (DXS) pattern is characterised by the transmitted / reflected wave amplitudes Uj (k,) of the incident and final wave fields in the layers and by the 16 covariances of the scattering processes, Ojj, (q, q.,) for each pair of interfaces j, k. We want to study the features of the DXS pattern under the aspect whether they are particularities of scattering by the roughness profiles, caused by the correlation properties, or of the excited non-perturbed wave amplitudes. With other words, we want to distinguish between effects of the random disturbance potential and of the non-perturbed potential. The latter effects do not depend on the statistical roughness properties, we call them dynamical scattering effects. us
an
intensities and discuss their
Resonant diffuse
scattering First we investigate the influence of the interroughness correlation. One essential characteristics caused by the interplane correlation is the so-called resonant diffuse scattering (RDS). We simplify the discussion of this phenomena by introducing a simpler model of vertical roughness correlation [26], where the inter-plane correlation function Cjk depends on the in-plane correlation function C11, I max(j, k) of the lower interface, by face
=
Cjk(,rll In this
-
rlll)
=
C11(,rjj
-
rjj’) e-lZj-Zkl/A_L
(8-50)
phenomenological model the vertical correlation of the roughness proby a vertical correlation length A I The model does not explain
files is limited
.
X-ray ReflectivitY by Rough Multilayers
8
0,20
255
0,20
5
....... .........
Q
0,15
0,10
0,05
0,15
............
..........
g
..
.................
6,0,10
I ....................
....................
0,00 0,004
.
J
............
-0,002
..
................
0,002
0,000
0,05
...................
................
....................
-0,002
T
.........
0,000
....................
0,002
0,004
0,20
L"’.f
IW
0,15
.
a. [1 /A]
Q. [1 /A] 0,20
...................
...........................
0,00. -0,004
0,004
...........
...........
0 15
..........
-
..............
............
IN
0,10
0,05
...........................
0,10
....................................
0,05
....................
....................
0,00 0,004
...................
-0,002
0,000
0,002
0,001
-0,004
0,004
I.
....................
...................
.............
-0,002
0,000
0,002
0,004
Q. [1/A]
Q. [1/A]
Fig. 8.19. Reciprocal space maps of the diffusely scattered intensity calculated for the DWBA method and the a [GaAs (7nm) / AlAs (15nm)].10x multilayer using simpler replication model (8.50) [18]. All the interfaces have the same r.m.s. roughvertical correlation lengths ness 1 nm, the correlation lengths 50 nm and different 0. Upper right panel: full replication, Aj-. Upper left panel: no replication, Aj_ 100 nm. Bottom right panel: full replication, A oo. Bottom left panel: Aj_ The full lines represent the arcs of the A oo, calculated by the simpler DWBA. 0. The RDS disappear, 0 and 0., Ewald spheres for the limiting cases of Oi. left not if the roughness profiles are panel). Bragg-like resonance replicated (upper lines are visible in all maps calculated by the full DWBA. They axe not reproduced by the simpler DWBA (bottom right panel) =
=
=
=
smoothening and roughening studied in section 8.4, since it neglects the interdependence of the r.m.s. roughness and the lateral correlation length (8.30). However, it makes the calculation and the discussion simpler. In Fig. 8.19 we see some calculated reciprocal space maps of the diffusely scattered intensity for a GaAs/AlAs superlattice assuming this vertical repliInm, cation model. All the interfaces have the same r.m.s. roughness 050 nm. It shows the cases of no repliand the lateral correlation length A the effects of
=
=
Tilo Baumbach and Petr Mikulik
256
01 ME
R
T
’
13
c 4
......
E
F: 300000
o
5
_o.oi
-0.015
Qx [1 /A]
-0.010
-0.005
0.000
0.005
o.010
o.ols
Qx/Q;:
Fig. 8.20.
Measured reciprocal space maps (top). periodic multilayer [Si (3.Onm) / Nb (5.8nm)] 10x starting from a rough Si substrate of o, 0.46nm and with interface roughness decreasing towards the free surface [19]. Right map: periodic multilayer with the setup corresponding to that of Fig. 8.19 with interface roughness increasing towards the free surface [8]. Left schema: the reciprocal space representation of diffuse scattering by a multilayer with interface roughness replication. The essential features are 1. the multilayer truncation rod through the RLP (000) with the multilayer satellite peaks and 2. horizontal sheets crossing the TR in the satellite positions Left map:
=
...........
.......... .
...
......
......
Q11
cation, partial replication and full replication. In the first case all interfaces independently, the diffuse intensities of all individual interfaces superpose. The other two cases give rise to scattering with partial coherence, the resonant diffuse scattering. It occurs due to the vertical replication of the roughness profiles of different interfaces. The partial phase coherence of the waves diffusely scattered from different interfaces leads to a concentration of the scattered intensity in narrow sheets. These sheets of resonant diffuse scattering intersect the specular rod in the multilayer Bragg peaks. Neglecting refraction the sheets would be horizontally oriented with the centre fulfilling the one-dimensional Bragg conditions scatter
q,
=
ko (sin Oil + sin 0,;c)
27r
rn
(8-51)
=
DML
schernatised in
treatment. Due
to the
curved
Fig. 8.20, which is the case in a kinematical angle dependent refraction of x-rays the sheets are
forming
X-ray Reflectivity by Rough Multilayers
8
following
"RDS-bananas"
(q,)ML
the modified
( /sin
A 72 2 _./ \ Oin + (xo)ML
ko
-
.
Bragg law
(XO)ML)
-
+
257
-sin2 Osc 0,,, + (Xo ML Vsin 2
27r
m
-
DML
)
(8.52)
is the mean polarisability of the multilayer (XO)ML T_jl j=j Xoj/DML period and (q,)ML q1 (0i- 0sc) (XO)ML ) the mean scattering vector in the
where
:--
-
medium. The tional to
length Aj
I
of the RDS-bananas in q, direction is inversely proporeffective correlation length Aeff depending on the correlation
length
some
of the interfaces. If all interfaces have the
same
Aeff would equal Aj. The widths of the RDS-bananas in
correlation
length,
q,, direction repre-
degree of replication. In the simple model it depends inversely on large A 1 on the total thickness of the multilayer. The sheets 0, turning into a broad verdisappear if there is no vertical replication, Aj_ tical maximum similar to that for a single surface. The RDS-bananas have no dynamical nature; their existence is not related with any kind of multiple scattering. They are also produced by the kinematical theory and by the simpler DWBA. RDS has been experimentally observed at amorphous, polycrystalline as well as epitaxial multilayers as it is shown in Fig. 8.20. The RDS sheets are clearly visible, bent is due to the refraction. Their existence and narrow vertical width gives evidence for full roughness replication in both samples.
sent the
A,
and for
=
Dynamical scattering effects One typical dynamical feature is known by rough’surfaces. The so-called Yoneda wings arise if the in0,. The wings are cident or the exit angle equals the critical angle, Oin/sc at the inner transmitted the of wave enhancement the amplitude by generated sample surface, Figs. 4.4, 8.17 In the case of a single layer structure interference fringes can also be created due to the wave guide behaviour of the two interfaces in the layer structure. In general, this behaviour can produce dynamical fringes in w-scans as well as in 20-scans. In case of periodic multilayers we call them Bragg-like resonance lines, since the amplitudes of the reflected waves exhibit a maximum if the incident or exit wave fulfills the refraction-corrected Bragg-law from NSXR
=
’.
ko
where ’Min, Tnsc resonances
are
are
Vsin 2Oin/sc + (X0)
integers.
7rrnin/sc -
DML
(8.53)
It is easy to proove that the zero order Bragg-like wings. The resonance lines have a
identical with the Yoneda
particular maximum, the so-called Bragg-like peak (BL), where the incident and exit waves are simultaneously in Bragg condition and the Bragg-like
258
Tilo Baumbach and Petr Mikulik
0.150 9
.j
-
7,’’
:-,-:. ... .
4MH, . .
-1"’
sitions
P"
18 -27
0.125
N,
45,1
-36
Fig. 8.21.
.........
.
-ai 54
63
’72
170
the
W
w
the
43
-0.001
0.000
numbers
orders
and
deof to
Min
The dotted lines denote the
positions of the Yoneda wings. The
52:
0.001
intersect. That is
are
Bragg-like
lines, corresponding
0.002
Q. [1 /A]
resonances
RDS-bananas
Bragg-like peaks according
full lines
0.100 0.002
the The
m ,
(8.54).
16
25
The schema of the pothe Bragg-like peaks
(points) and (grey areas).
",80
note
6i.
-26.,,.
A
of
to
resonance
Min/sc
=
4
and 5
at the
positions +
k 02
QJI’Minm-
-
(;m ,727,r/D)")2 + ko (X0) + Vk02 _L
7.2
-
(m,,c27r/D )2
+
ko2 (X0)
(8-54) The existence of the Yoneda
wings, dynamic fringes and Bragg-like peaks is of completely dynamical origin. They occur independent of the actual interface correlation function. However, their form and intensity is influenced by the interface correlation.
Fig. 8.22.
Bragg-like peaks on the RDS-sheets and interpretation by Umweganregung. On the left side, both the incident and final nonperturbed state fulfill the Bragg condition (8.53). Simultaneously all four diffuse scattering processes are in the situation of resonant diffuse scattering (8.52). On the right side, the situation of RDS (8.52) is fulfiilled for the primary scattering process. The incident wave is out of Bragg condition, consequently also the final state is out of Bragg condition. Additionally all three secondary diffuse scattering Generation of
the concept of
processes
are
out of
resonance
8
X-ray Reflectivity by Rough Multilayers
259
vertically replicated roughness we see with (8.52)-(8.54) that Bragg-like peaks of an even number Tnin+,rnsc are situated on RDS-sheets, Fig. 8.21. These Bragg-like peaks are very pronounced with respect to the others. That can be interpreted by the concept of Umweganregung (excitation of a reflection by another reflection), well known from x-ray diffraction and outlined in Fig. 8.22. In our experimental map of Fig. 8.20 the Yoneda wings and the Bragg-like resonances are well resolved. Along the RDS-sheets we observe intense Bragg-like peaks. All the features are reproduced by the calculation using the full DWBA treatment for multilayers. In
case
of
all
.Not In
always is
general
it
w-scans
Already one replication.
possible and
necessary to
measure a
at different q., and offset-scans
offset-scan
or
20-scan is sufficient to
or
full well resolved map.
20-scans
give
are
employed.
evidence for vertical
Stepped Surfaces and Interfaces
8.5.3
nearly uniform height discussed in section 8.4.2 is the simplest case for a discrete stepped n-level surface. An infinite number of levels exist at a terraced surface, see Fig. 8.23, which is mostly the case of multilayers grown on slightly miscut substrates [27-29]. The miscut angle a equals the mean ratio of the step height (h) and the terrace widths (L): a (h) / (L). The lateral correlation properties of such a stepped surface are determined by the conditional probability p(Ax, z) giving the probability of displacement z for two surface points with the distance Ax. The twoThe model of islands of
=
dimensional characteristic function Xzz, of such a stair-like surface can be described based on the approach of stationary random processes [21]. Using
calculate the covariance function lQ and with (8.46) the differential scattering cross section for the diffuse scattering by the stair-like surface. In Ref. [27] the gamma-distribution of order M has been supposed
(8.47)
-
one can
[00 1orientation of the
terraces
X n -surface
normal
(X
Y
r
X
z
Fig.
8.23.
(left)
U
mean
space
mean
replication surface
X3 a step-like surface. (right) Illustration of the stair-like superlattice and of the corresponding fine structure in the
Model of
interface pattern in the
reciprocal
-
direction
X2
260
Tilo Baumbach and Petr Mikulik
Cn
-0,05
-0,05
0,05
0,00
0,00
Q. [nm-1]
Fig. 8.24. sizes, (b)
w-scans
of
for different
0,05
Q. [nm
30 miscut GaAs surface. (a) calculation for different dispersion of the terrace size [27]
a
terrace
for the distribution of the terrace widths L p (L)
with the
dispersion
M
I’ (M)
length
was
between the terraces h
persion
Ch.
For such
.characteristic
M ML
CPT L
M_1
(8.55)
of the distribution
0"
The terrace
( (L) )
I --
a
2 L
described
was
L )2
_
(8.56)
-
M
by
a
similar distribution. The step height normally distributed with the dis-
assumed to be
model the correlation function and the two-dimensional
function have been calculated
[271
and
implemented
in the
pressions of the DWBA. The terrace size and its statistical distribution
ex-
can
be
determined
by transversal scans in reciprocal space or by w-scans. In Fig. 8.24 the DXS intensity has been calculated for a terraced surface of GaAs with a slight miscut of 0.3’. Between the Yoneda wings there occur maxima, which are equidistant in reciprocal space and their distance is inversely proportional to the mean terrace size. The positions of these maxima correspond to the grating satellites of a mean surface grating with the lateral grating period DG. The DXS peaks are broadened with increasing dispersion of the terrace lengths and of the step height. Growing an epitaxial layer on a miscut substrate, the staircase profile can be replicated from the substrate/layer interface to the sample surface. In a superlattice on off-oriented substrates, the staircase profile can be replicated from interface to interface [28,29]. The direction of the replication may be inclined with respect to the growth direction (see Fig. 8.23). For simplicity we suppose first laterally uniform terrace lengths and perfect interface replication, giving the recursion formulae for the layer size functions
Qj (r)
ee:
Qj
-
2
(r
+
DSL i2 +
D11 6)
,
(8-57)
8
X-ray Reflectivity by Rough Multilayers
261
0.4
0.35
0.3
0.25
0.2
0.15
0.1
Fig.
-
8.25.
Calculated map for
(7 nm GaAs / perlattice
-
15
grown
GaAs substrate. 0.05 0
-0.004 -0.002
Qx
where
0.002
0.004
[A-’]
AlAs) 10 x
a
su-
0.50
miscut
Averaged
terrace
on a
distance is
(L) =500 nm.
face steps
are
and inter-
fully replicated
at
400.
is the lateral shift of the stair-like
D11
nm
pattern during the growth bilayer superlattice period).
superlattice period (here two- dimensionally periodic morphological superstructure creates a two-dimensional fine structure, similar to later discussed multilayer surface gratings. In this case the whole reflected intensity would be concentrated along so-called grating truncation rods perpendicular to the sample surface, representing the lateral periodicity. Each truncation rod would contain the multilayer Bragg peaks due to the multilayer periodicity. An inclined replication direction of the interface profile creates inclined branches of multilayer Bragg peaks. All that is shown schematically in Fig. 8.23. In reality there will be a rather partial interface replication, characterised by an effective replication length A I In the Gaussian roughness model (discussed in Sect. 8.5. 1) the vertical replication in the periodic multilayer caused horizontal bananas of resonant diffuse scattering, crossing the multilayer Bragg-peaks in the specof
we
one
Such
assume
a
a
.
ular
scan.
In the
present
case
of the lateral correlation of the interface steps
similar horizontal sheets appear. However, they are, in addition, horizontally structured by lateral DXS maxima, which indicate the laterally and vertically
interfaces, see Fig. 8.25. dimensionally structured pattern
correlated stair-like In result
a
two
is obtained with
tering periodicity
longitudinal
of resonant diffuse scat-
DXS-satellites due to the
superlattice
and transversal DXS satellites 27r
)av
(8-58)
which represent the more or less periodic lateral morphological order of the interfaces. Both together form longitudinal stripes perpendicular to the mean
sample surface, which remind to the grating truncation rods of multilayer surface gratings (see Sect. 8.7). Considering the q,-dependence of the diffuse intensity one observes, that the envelope of the intensity follows with its
Tilo Baumbach and Petr Mikulik
262
ZI
-2
-1 (a-2e/2
[deg]
Fig. 8.26.
Measured
of
a
and its fit
by the theory using
a
(dashed) [27].
steps
w-scan
The left-hand
GalnAs/ GaAs/ GaAsP /GaAs multilayer (dots) single type of steps (full), and two sets of the figure shows a possible microscopic structure of
terraces
maximum the direction of the terrace orientation. However, the simultaneous existence of large terraces formed by step bunching and atomic scale micro-
modify the DXS pattern (see Fig. 8.26) [27-29]. investigation of step-like interface morphology by interface diffraction methods is briefly discussed in Sect. 8.6.
terraces
can
The
8.5.4
Non-Coplanar
sensitive
NSXR
and
simple to realise with conintensity distribution is resolved in the q_,/q,-plane which contains the surface normal. The region in the q,/qplane accessible by coplanar reflection geometry is restricted by the Ewald spheres for the limiting cases of grazing incidence and grazing exit, which represent the horizon of the sample surface. Especially for small XRR in
coplanar geometry
is
most
common
ventional diffractometers and reflectometers. The
values of q, the measurable lateral momentum transfer decreases and consequently the information is cut about roughness with small lateral dimensions of
nanoscopic By use of
scale.
non-coplanar scattering geometry this limitation has been 1, [30,3 10]. The equipment requires monochromatic beam collimated an in two directions, which can be provided by synchrotron radiation sources. First experiments used the setup of a small angle scattering instrument with a well collimated beam and a two dimensional position sensitive detector. Other setups are based on surface diffraction instruments, working usually in a strongly non-coplanar (grazing incidence diffraction) geometry, see Fig. 8.27. The detection of the diffusely scattered intensity up to a parallel momentum transfer of I A‘ enables to study the correlation properties up to a few k a
overcome
The
diffusely Fitting
scale.
law, the
Hurst
intensity is usually drawn in a double logarithmic asymptotic intensity decay with increasing Q11 by a power factor introduced in section 8.5.1 can be determined with good
scattered the
8
X-ray Reflectivity by Rough Multilayers
263
lind
afl
Fig. 8.27. Schema of non-coplanar reflectivity setup [30]
x-
ray
precision,
wherefrom
one
can
conclude
on
the
validity of different growth
models.
Fig. 8.28(a)
shows measured 0,,,-scans of
an
amorphous W/Si superlattice
for different q1j. They cross the RDS-sheets indicated by roman numbers. For increasing qjj the width of the RDS-sheets increases and finally the resonant
scattering disappears, indicating a reduction of the vertical replication length L for the higher frequencies of the roughness profile. In Fig. 8.28(b), the decrease of the intensity of the first RDS-sheet is plotted. The measurements prove the validity of a logarithmic scaling behavior as predicted by the Edward-Wilkinson equation [32]. diffuse
10’
V.
-
III.
10,
roughness dominated. scattering
IV.
10-1
B
W
amorphous scattering 107,
103
102
0
1
2
3
4
af [deg]
0.01
0.1
1
Q,, [A-,]
non-specular x-ray reflectivity.of an amorphous W/Si superlattice. (a) 0,,-scans for different qjj [10]. Intersections with the RDS-sheets are indicated by roman numbers. (b) Intensity profile of the first RDS-sheet [30]
Fig.
8.28. Measurement of
Tilo Bauidbach and Petr Mikuhk
264
Interface
8.6
in Surface Sensitive
Roughness
Diffraction Methods
epitaxial multilayers surface and interface roughness can also be by surface sensitive x-ray diffraction methods such as grazing incidence diffraction (G ’ID) and strongly asymmetric x-ray diffraction (SAXRD). Beside reflection at the interfaces there occurs diffraction by the layer lattices. The principles of diffraction by rough multilayers are similar to those In
of
case
studied
described in can
its
more
detail for
x
ray reflection. All used theoretical treatments
be extended.
The polarisability of each layer reciprocal lattice vectors
can
be
X1 yer(,,)
developed
in
a
Fourier series after
gj (r) Cig’
3
(8.59)
9
Measuring
the
intensity pattern
of
a
Bragg-reflection
with the
reciprocal
lat-
conventional diffraction geometry (so-called two-beam case) the Fourier components with the indices h, -h and 0 are of importance.
tice vector h in
only Crystal
a
truncation rods
through each reciprocal lattice point characterise amplitude of a crystalline layer. All truncation rods of a periodic multilayer contain the fine structure of equidistant superlattice satellites similar to the schema in Fig. 8.20. The non-perturbed wave field of diffraction by a planar epitaxial multilayer under conditions of grazing incidence consists in each plane layer of 8 plane waves for each polarisation the structure
4
Ej’PI( )
[Tn
=
-ikOjjrjj,-ihO’_j(z-Zj) 0.
j
+
n
Rj
ikhjjrjje-’k)’1 j(’--Zi) hz
e
I
Q,
I
N
n=1
(8-60) rough interface shape function Qj"(z) H(z [Zj + zj]) H(z superlattices with rough interfaces the layer disturbance includes Zj). the variation of the Fourier components of the polarisability and the lattice displacement zAu(r) due to the lattice deformation created by the interface roughness profile. In layer j, with the
=
-
-
-
For
BIay,r
-igr
AXg,j (r)
ZAxi
(8-61)
with
g=O,-h,h ,
Axg,j (r)
1Xg’j ( eigAU(T)
-
1
+zAXgj eigAu(r)
I
’guo(z)
S2j’l i W
-
Similar to x-ray reflection by the rough interfaces the disturbances give rise to scattering. The number of possible diffuse scattering processes between
diffuse two
non-perturbed
states at
one
interface,
see
(4.D23),
increases up to 64.
y Rough Multilayers
X-ray Reflectivity
8
265
Fortunately a certain number of them is almost negligible. If the roughness profile is replicated, the diffusely scattered intensity is concentrated in horizontal sheets of resonant diffuse scattering crossing the crystal truncation rods in theposition of the diffraction satellites. Their origin arises now from partially coherent diffraction and reflection by the interface disturbances. For weak strain the covariance functions are formally quite similar to (8.49) found for x-ray reflection
_AzAxgj(, Axg/,k)* mnop jk )* Jqml (Jqop Z,k1 Z’3
1 d(rjj
_
(,11 -,11’) P11 1) eiqll
[X,,,Zk (Jqm ’, (6qOP Z’3
however
now
with the reduced
X.,
z,k
scattering
ing process in the layers, which depend in the layers by Sqz,j = q,,j g,,j.
(8.62)
X
OP
(JqmZ’3’)XZk ((6q z,k corresponding scatterreciprocal lattice vectors
vectors of the
the local
on
-
Fig. 8.29 the scattering geometry in reciprocal space and the corresponding experimental results of strongly asymmetric diffraction by a GaAs/ AlAs superlattice are shown. The measured sheets of resonant diffuse scattering (RDS) of the diffraction mode are clearly visible. It is an advantage of the AXRD measurements, that the RDS sheets are not limited by the samIn
.
.
.
.
.
.
.
.
.
.
.
3.50
3.45-
*hk K,
hkl 622
3.40
3.35
1.590
1.600
1.595
Q
-
1.605
(A-,)
scattering by rough interfaces in the strongly asymmetric reciprocal space. Right: reciprocal 1.47 A. The of diffraction of a superlattice for A GaAs/AlAs (113) space map coherent crystal truncation rod (CTR) is crossed by horizontal RDS sheets, indicating correlated roughness. The sheets are laterally not limited by the experimental
Fig. 8.29.
Diffuse x-ray
diffraction mode. Left: schematic situation in
=
geometry
Tilo Baumbach and Petr Mikuhk
266
0.24
0.22
ple horizon, in transfer The
can
contrast to
be detected in
application
coplanar XRR. So the full range coplanar scattering geometry.
of momentum
a
of x-ray diffraction methods is limited on epitaxial struchand, x-ray reflection experiments are less successful
tures. On the other
for many semiconductor systems due to the missing contrast in the electron density modulation. Thus the choice of suitable Bragg-reflections allows inthe contrast between the
layers in the diffraction mode. non-coplanar surface sensitive diffraction method, was successfully applied for the measurement of RDS by rough multilayers in Ref. [12]. Beside Gaussian roughness correlation behaviour, the step-like interface morphology was also investigated by various diffraction methods. In Fig. 8.30 we show the measured 200-reciprocal space map of a GaInAs/GaAs/GaAsP/ GaAs-superlattice on a 2’ off-oriented GaAs substrate, measured by grazing incidence diffraction. This reflection is highly sensitive for the morphological ordering, since the scattering contrast of the corresponding Fourier components of the susceptibility. is much larger than that in the above discussed reflection mode. Similarly to Figs. 8.23 and 8.25, the diffuse scattering is concentrated in stripes, resonant diffuse scattering along so-called grating truncation rods, which are perpendicular to the averaged surface. The grating rods are therefore inclined with respect to the crystallographic orientation, which is simultaneously the orientation of the terraces. Each grating rod contains multilayer Bragg-peaks. The Bragg-peaks of the same vertical order but of different grating rods form branches which are inclined with respect to the sample surface according to the inclination of the morphological interface replication via the surface normal. The envelope maximum of the diffuse scattering follows the 001-direction, which is the orientation of the terraces.
creasing
GID,
8.7
a
X-Ray Reflection from Multilayer Gratings
In this-section
we
discuss the calculation of the x-ray reflection from multi8.31. Gratings are etched into planar multilayers
layer gratings (MLGs), Fig.
X-ray Reflectivity by Rough Multilayers
8
267
-9
W’PP_ 17
Fig.
by
that their lateral structure is formed
with
a
fan
a
multi-
consisting waves
wires distributed
equidistantly study mainly on the about micrometers, which are
the surface. We focus the present
d
period along short-period gratings with
A sketch of
of four diffracted-reflected
Z
so
8.31.
layer grating
X
of most interest in
periodicity semiconductor physics. with the
at
The part etched out (dips between wires) can be several hundreds nanodeep. Thus these structures can be considered as a special case of
meters
roughness or as an artificial lateral one-dimensional cryscrystals periodic in all three directions. Thus the reflectivity from gratings can be treated by approximate as well as rigorous methods [33-35,?,37,38,?,39], thus making possible to treat and compare the adequateness of various approximations. In this section, we formulate the approximate perturbative treatment by the kinematical theory and by DWBA and compare them to the exact dynamical calculation. We determine region of validity of DWBA and we show that the correct choice of the eigenstates can lead to good results even when the perturbed potential is present in the most volume of the sample, contrary to the small roughness of interfaces. huge
deterministic
tals contrary to the
Theoretical Treatments
8.7.1
MLG possesses the translation symmetry so that it is fully sufficient to determine its d
period (- 2 we
j.
susceptibility X(r)
<
x
<
fl 2
only.
in
S-2 i’
=
0
1
d
one
Therefore tj
first describe it for any of the layer The period consists of two parts
dj6 (I -.Vj)d d3q _ id Notation of the variables de8.32. Fig. grating, scribing a laterally structured layer is the air). We denote their susceptid d Tbilities Xj, Xj and their widths dq (1 Tj) d with 0 < _Pj < 1. 3 3 3
(wires)
of
an
j (for
named jj and
etched
one
the
case
of the parts
=
-
-
We introduce the
Qj" (r)
-
of the material
qj in the period.
occupied by the material jj and it is 8.32. Then the susceptibility of one period is
inside the volume
equals unity elsewhere, see Fig. It
shape function
’
X 3 (r)
=
Xqfl "(x, z) 3 3
+
X)
(I
-
fl3 ’(x, z))
.
zero,
(8-63)
Tilo Baumbach and Petr Mikulik
268
By
f2al h (q,) j ’
function of
denote the two-dimensional Fourier transform of the
we
one
shape
period.
Because of the presence of two
types of minterfaces, horizontal and verti-
cal ones, different theories treat the
respective reflectivities using different the single-scattering approaches those related’to the lateral diffraction case. We treat separately the perturbative (single-scattering) and rigorous dynamical (multiple-scattering) theories.
approximations. Further,
we mean
by
Perturbative treatments MLG is periodic along the axis ic with the lateral periodicity d, Fig. 8.31. Then the scattering potential V(r) of the sample, as defined in Sect. 8.2, can be given as a convolution of the scattering potential of one period V’(r) (defined on the interval A2 :5 x : 2d) with a periodic arrangement of J-functions -
V(r)
V’(r)
--
0
E J(x
-
nd)
(8.64)
n
Its Fourier transform is
V (q)
dr
product
of two terms
A
V(,r) eiqr
=
d
(A
denotes the
’ ’ (qx, q.,)
sample area)
Jq.,,h Jqy,O h=
2,-’
(8-65)
m
(summation) term expresses the reciprocal lattice of the grating, grating truncatton rods (GTRs) in Q, direction positioned equidis2’ where m is an integer hn tantly along the axis qx at points q, d m,
The second
which
are
=
(see Fig. 8.33). one
The first term
period, behaving like
0
is the Fourier transform of the
envelop
an
-
function for the
potential in
fields associated
wave
to. the GTRs. In a multilayer grating, the potential of one period V1 (r) and similarly for their potentials of individual layers W(r), 3
W(h, q,). 3
Fourier transforms nent
to the
proportional
Fourier
is the
sum
of the
two-dimensional
The latter separates into the zeroth compothe discrete
laterally averaged susceptibility and to components proportional to the susceptibility contrast
f/’jl (h, q,)
--
K2 d f dz xoj (z) -K
2
(Xja
f2al X3 3
iq,, z ’
h
(q,
for
h
0
for
h
0
(8-66)
Now we will consider the scattering from the sample we characterised generally above. We first determine the directions Kh of scattered waves. We use the principles for the Ewald construction, discussed in Appendix 8.A, which state that the wave vector end-points lay at the intersection of the sample reciprocal lattice and the Ewald sphere of the incident wave. Thus the incident
wave
associated to each
is scattered into the fan of reflected and transmitted
GTR,
see
Figs.
8.31 and 8.35.
waves
8
X-ray Reflectivity by Rough Multilayers
269
amplitudes. Scattering potential of amplitude of all GTRs comes from the coherent scattering only (even though into non-specular directions). It is expressed similarly to the coherent specular reflection amplitude of rough MLs calculated by DWBA. Using the formalism from Appendix 4.1), the h TOhl2iKh,,A. The scattering amplitude at the sample surface is R (Kh) matrix element 70h EhJV(r)jEo) can be decomposed into the sum over the individual layer contributions r . The sample reflectivity along GTR h 3 Further,
we
calculate the reflection
MLG is- deterministic and thus the reflection
-
-
h finally IR 12 Khz lKz The reflection amplitude then depends on the approximation used in the evaluation of the scattering matrix element. We discuss briefly the calculation by the kinematical theory and by the first-order DWBA applying the approach of Sects. 4.D.2 and 4.D.3, respectively.
is
-
theory is equivalent to the first Born approximation [36,37] thus calculating the scattering process as the singlee-’KO’ into scattering transition of the incident vacuum plane wave JEo) e- iKhr ,see Fig. 8.35(a). The scatthe diffracted vacuum plane wave jEh) tering matrix element for one period and one layer is proportional to the Fourier transform of the layer potential in one period (with the scattering vector Qh Kh Ko) Kinematical calculation Kinematical
=
=
=
-
iKh
Th
(e-2
j
r
IW j (r)le
W
-’
0’)
-
Vj (h, Qhz)
(8-67)
-
According to (8.66), we can see that the Fourier transform for h--O is by the profile of laterally averaged susceptibility. Thus the Specular reflectivity profile coincides with a kinematical reflection from laterally averaged planar multilayer and the specular reflectivity curve exhibits the
determined
same
feature’s
ory and the
V
(a)
as
those calculated in the framework of the kinematical the-
stationary phase method (SPM) [19,39]. (SPM helps
VQ
(b)
Qz
Fig.
8.33. Schematical
odic
grating etched
drawing.of
into
a
periodic
(C)
to avoid
VQ.
reciprocal space maxima of a laterally perimultilayer. The "Bragg" sheets are parallel to
the
the q, axis in the kinematical treatment (a), whereas they are curved and shifted upwards in the DWBA (b) and dynamical (c) calculations due to refraction. In ad-
dition, the subfigure (c) illustrates the multiple scattering simultaneously excited GTRs w1lich is taken dynamical theory
fields of the
interaction among wave into account within the
Tilo Baumbach and Petr MikWik
270
the Fraunhofer
approximation
which is not suitable for
laterally
extended
samples.) Considering the intensity of the non-specular scattering matrix contribution is
Tjh
-
k 0 (X a j
-al h
b
2
-
-
X )
truncation rods
(Qhz)
(h:AO),
the
(8.68)
By calculating the kinematical scattering integral by the stationary phase method we generalise the kinematical Fresnel reflection coefficient for lateral diffraction case k20
rh,kin
j’j+1
For
specular reflection
reflection coefficient for
(3.91),
as we
Xj 3 2Khz Qhz
(8-69)
it perfectly coincides with the kinematical Fresnel O,kin k20 (Xo,j_Xo,j+l ) IQ 2, cf. planar multilayers rj,j+, -
-
Z
said above.
As all the kinematical theories, also in the present case the effects of absorption and refraction are not comprised. Thus the kinematical intensity is much larger than unity below the critical angle and it diverges for the specular scan at the origin of the reciprocal space. Further, the kinematical period of oscillations of a MLG converges slowly to that calculated by a theory including the refraction. Let us figure out the positions of maxima of a periodic multilayer grating using a reciprocal space schema, Fig. 8.33. They lay on the intersections of the grating truncation rods (reciprocal lattice of the grating represents the lateral periodicity) and. the sheets passing through the ML maxima on the specular truncation rod (which represents the vertical periodicity)
Calculation
by
DWBA
We follow the basis of the DWBA
detector
source
as treated for the roughness and we split the MLG potential V(,P) into two parts, see Fig. 8.34. We choose the ideal (unperturbed) potential VA (,r)
+
IV, 1( r 1)
VA(,r)
Fig.
VB
8.34.
planar laterally averaged multilayer and thus calculating the see (4.D19), according to (3.47). For the simplicity of the further treatment we restrict ourselves to the rectangular gratings only [401. From (8.65) and (8.66) it follows that the ideal potential V:A is constant 3 in each etched layer, VA (r) W(O, O)Idtj, whilst the perturbed potenas
that of
eigenstates
a
J_EA K ),
j
tial
VjB
VB(,r)
_-
3
VA(r) is V(,r) V.1 (h, 0) eihx Idtj. h 4_ 0 3 =
Consequently intervene into the
_
the
sum
scattering element specular term
the
h=O
A JEo) (E 0 I V:A 3
of
non-zero
of the
+
Fourier components,
perturbed potential does
A A (EO I vB j JEO
not
(8.70)
X-ray Reflectivity by Rough Multilayers
8
271
amplitude from the whole MLG then equals the (dylaterally averaged multilayer. From this it clearly follows that this DWBA considers multiple scattering between the horizontal interfaces of averaged layers by using the dynamical Fresnel reflection coefficients, but neglects the influence of multiple scattering by the The
specular
namically)
reflection
calculated reflection from the
vertical side walls. The
amplitude
of the
wave
scattered into
h
,rj
laterally
The contribution of each -K
2
Tk,,jSI’Tk 3
-
O’j
A
B
A
=
non-specular GTR h :A
a
Eh IVj A ) structured
Rk 0,3 + + Tkh,j S12 3 -
0 is
(8.71)
-
layer consists
Rkh,jS?’Tk 3
-
0,3
of four terms
+
Rk,,j S?2 3
Rko,j) (8.72)
where the
amplitudes Tk,, Rkj
(4.013)
structure factor
four fuse
scattering scattering,
wave
is
are
equal
Sj"
vectors
Uj (: k_,j)
to
Sj (qj") 22
11
qj
qj
=
are
(xj’
(3.48)
in
and the
(-q’-,3’). X) ) D’-’. q. ’j
-
defined
as
in the
case
layer The
of dif-
(4.D23) and Fig. 8.40. We draw them in the reciprocal Fig. 8.35(b) while demonstrating there the single-scattering
se
space schema in
character of the diffraction from the incident to the diffracted Because the
eigenstates
wave
fields.
potential are calculated using the specular reflectivity from a planar mul-
of the ideal
dynamical matrix formalism for tilayer, thus the effects of absorption and refraction are taken into account. Then the maxima of a periodic multilayer grating, Fig. 8.33(b), lay on the intersection of the truncation rods and the refraction- curved sheets passing through the maxima on the specular truncation rod. usual
tZ
(a)
U
h
(b)
I
U I
Q.
q
W
Q.
Fig. 8.35. Single-scattering approaches, i.e. kinematical (a) and DWBA (b), calas a single-scattering process from GTR 0 to a GTR h, while the multiple-scattering approaches (c) take the contributions from all the culate the diffracted field
GTRs into account
by the dynamical theory The dynamia multilayer grating by rigorously solving
MU ’Itiple
scattering
cal
treats the reflection from
theory
treatment
Tilo Baumbach and Petr Mik-ulik
272
the
wave
equation
under the condition of
X(r)
"
XO(Z)
+
a
one-dimensional
E. h(Z) e-
periodicity
ihx
(8.73)
h
There
are
miscellaneous
approaches
literature, reviewed
found in the
e.g. in
light [33], while gratings [38] using inteapplied they from XRR formulae. multilayered gratings is studied gral Rayleigh-Maystre method. modal matrix in Dynamical theory takes into deeply [19,39] using account the multiple scattering among the wave fields (each consisting of pair
[19,39].
Their formulation
in XRR
have been
of
a
comes
from the
only
wave),
transmitted and reflected
of visible
optics
for surface
which
are
associated to all truncation
as shown in Fig. 8.35(c). rods, including that for planar’multilayformalism similar to matrix convenient a Using lateral Fresnel the the diffraction case, for coefficients generalisation of ers,
the real
compare
(3.68) hg
rj,j+l
khZ ,i kh
Z,j
Here,
well
(3.70),
and
-
-
as
as
evanescent GTRs
has been found
k9
Zj+1
and
+ k9
&
[19,39] 2k h Z’j
-
-
3,3+1
-
kh
.
Z’J +1
Zj+1
+ kg Zj+1
(8-74)
the indices h and g relate the transmission and reflection processes to wave fields of two GTRs h and g. Wave
simultaneous diffraction between
point to a spherical Ewald sphere, but a dynamical theory of x-ray diffraction. dispersion surface In the dynamical theory the energy is conserved. Therefore a strong wave field corresponding to a certain GTR can influence significantly the intensity profile of another GTR. This may be the case, for instance, in the angular region where the wave field of the first GTR changes from evanescent to real (near the intersection of the Ewald sphere with the GTR +1, see Fig. 8.33(c)). There the specular intensity can be enhanced with respect to the specular intensity of an averaged planar multilayer. vectors to
kh of scattered
waves
do not
like in
so-called
Discussion
8.7.2
For the
following discussion
(period
around
one
we
period rectangular gratings wavelength at about one Angstrom. kinematical theory does not involve the
will consider short
micrometer)
and the
already mentioned that the refraction, which is of crucial importance in XRR, we will further devote our discussion to the comparison of DWBA to the dynamical theory. We choose the ratio _V of the wire width with respect to the period one half. Then we Since
can
we
find truncation rods of three types:
0). Here, the DWBA and dynamical theory Specular truncation rod (h the known angular region of the enhanced for the same except profiles, give --
interaction with GTR +1
as
discussed earlier.
Weak, kinematically forbidden Fourier coefficients
are
(h is even). The associated single-scat-tering theories, including
truncation rods
zero, and therefore
X-ray Reflectivity by Rough Multilayers
8
0.040
10-3
/I
0.035
GTR -1
dynamical aynamicai
273
DWBA calculation caicuiation
0.030
10-
4
F_
,
0
0.025 0.020
5
10
0.015 0.010
10-
6
0.005 0 0.2
10-7 0.1
0
0.2
angle
0.4
0.3
of incidence
0.225
0.5
0.25
0.275
0.3
0.325
0.35
angle of incidence [deg]
(deg]
a GaAs surface grating (thickness and for 5 of 0.8 wavelength 1. 54A. In the former period (b) (a) Mm pm nm) case, DWBA gives the same results as the dynamical theory. In the latter case the multiple scattering starts to be important and DWBA of the first order gives only approximative result
Fig.
8.36. Calculation of the odd-order GTRs for
for
300
the kinematical
GTRs their
are
profile
and DWBA, predict zero intensity for them. Thus these by multiple scattering in the etched layers and consequently be calculated by the dynamical theory or by higher-order
one
excited can
DWBA.
Strong truncation -rods (h is odd). Here, both DWBA and dynamical thecoincide, see Fig. 8.36(a). The good coincidence depends on the force of the dynamical interaction between diffracted wave fields. There are more GTRs excited in the Ewald sphere of the incident wave for large periods or small wavelengths, thus the dynamical effects will be enhanced and DWBA starts to be only approximative, see Fig. 8.36(b). We found possible to formulate a condition separating the two cases using a two-beam approximation of the dynamical theory [19]. ory
Reflectivity
8.7.3
from
The influence of interface
Rough Multilayer Gratings
roughness
on
grating, reflectivity
can
be studied
within all three theoretical treatments discussed earlier. Within the matrix
dynamical theory [19,39], the generalised Fresnel coefficients roughness were found formally similar to those f6r rough planar multilayers (8.24)
approach
(8.74) hg
of the
corrected for
r,,j+,
hg,flat
rj,j+l
-
e
2,k
"
9, j+l’j+l and
hg
tj3,3+1 j
_
-
thg,flat e(k h, -kg,j+,)’a’ + 1/2 jj+1
(8.75)
Roughness in gratings decreases the scattered intensity for the incidence angles even below the critical angle. Furthermore, there is different sensitivity to the surface and interface roughnesses for weak and strong GTRs, respectively. Finally we can notice that the kinematical reflection coefficients (8.69) QL crj2+1/2 similarly to are attenuated by e(3.104). h
Tilo Baumbach and Petr Mikulik
274
.
Fig.
In
8.37
we
show XRR results of
a
periodic W/Si multilayer grating.
Structural parameters (lateral periodicity and wires width, layer thicknesses and interface roughnesses) of the sample were obtained by fitting the measured GTR
profiles employing
the
dynamical theory for rough gratings. Finally, the calculation (by DWBA) of the diffuse scattering from MLGs, such as simulation of the map in Fig. 8.37(c), is even more tricky procedure which requires the preliminary calculation of the eigenstates either using the DWBA for perfect MLG or the dynamical theory. I
.
:
I
(a)
10-1
measurement
......
.......
fit 10
GTR 0 Jar scan) e cular (specular (sp Uscan)
IO_
measurement measuremeqt fit T113 -1 T TF3
0
0.05 0.1
0.15 0.2 0.25 0.3 0.35 0.4
Q,
10-
.......
5
A
(YV
I
10-7
JS
1
......
4
10-6
-5
10-6
(b) M
-2
10-
3
10-4 10
10
10-3
-2
10-
10-1
8
0.01
J
TR
x
+
1
0.1
[1/A]
0.15
0.2
0.25
0.3
Q, [I /A]
(C) Fig.
0,
ment
0
6 S
GTR +2
p GTR +1
0
1 e5
GTR -1
(a),
8.37.
and
fit
and +2 from
(b)
of
Measure
GTRs
Si (6.23nm)llox multilayer grating (lateral period 780nm, wire width to period ratio 0.7) [41]. The measured reciprocal space map
(c)
shows
the
coherent
intensity scattered into and the diffuse
0-0.004
of -0.002
0
Qx [1/A]
0.002
0.004
resonant
in
in-
sheets
diffuse
which indicates lated
GTRs
(incoherent)
tensity concentrated C0
0
-1,
[W (1.5 nm)
a
scattering, vertically corre-
roughness
Acknowledgments supported by the Deutsche Forschungsgerneinschaft (grant BA1642/1-1),bythe LiseMeitner Fellowship ofFWF, Austria (project M428PHY) and by the grant VS 96102 of the Ministry of Education of the Czech Republic. The work
was
8
X-ray Refiectivity by Rough Multilayers
275
Appendix: Reciprocal Space Constructions Reflectivity
8.A
for
previous chapters in the book, the recZProcal space representadrawing the experimental scattering geometry: experimental scans and inaccessible regions for coplanar reflectivity (Figs. 7.2, 8.4 and 8.5). In addition, throughout this chapter we use the reciprocal space to describe graphically the scattering events of x-ray reflection. Since this approach may not be common to the reader who is not accustomed to that representation, we give here some schematic interpretations of the reflection by multilayers in reciprocal space, which help in finding the intuition for an easy understanding of the scattering features in a simple geometrical way. We start by the In
of the
some
tion
was
used for
interpretation
of fundamental laws of reflection and refraction at interfaces.
We relate the reflection
curves
of thin films and
particular reciprocal space features and discuss considered within the treatment by a DWBA.
periodic multilayers multiple scattering
to their as
it is
The idea to represent x-ray scattering by reciprocal space constructions has been introduced by P.P. Ewald in the early stage of the dynamical theory of x-ray diffraction. The goal is to relate the directions of the scattered waves and the symmetry of the sample represented by the Fourier transform of the
crystal lattice and/or
space) 1.
2.
the
shape function of the scatterers. Ewald (reciprocal physical principles:
construction visualizes two basic
Energy conservation. x-ray reflection is an elastic scattering process, conserving the wave vector length. Then the end-points of all scattered waves can lay only on the Ewald sphere of the radius of the wave vector length, Fig. 8.38(a). Momentum conservation except of a reciprocal lattice vector if the diffraction condition is fulfilled. This reflects the symmetry properties of the
sample. In this book
we use
Ewald construction for the illustration of the reflection the wave vectors in the vacuum and in
by layers and multilayers, including the medium.
8.A.1
Reflection from Planar Surfaces and Interfaces
reciprocal space, Fig. 8.38, propagation in the vacuum and in the media is determined by the different length of the wave vectors. In case of a homogeneous half space of a slightly absorbing medium with a flat surface is or of planar layers with smooth interfaces their reciprocal space structure normal to and the rod truncation so-called defined by a origin passing through the surface (i.e., it usually coincides with the axis q,,). We call the truncation rod through the origin of the reciprocal space here specular rod, since it defines Let
by
us
use
discuss the reflection and refraction laws in of Ewald construction. The
wave
Tilo Baumbach and Petr Mikulik
276
(a)
(b)
(C)
(d)
-KX,, i /2
Fig.
Graphical representation of the laws of reflection and refraction by an by means of the Ewald construction. (a) The law of reflection, (b)-(d) Snell’s law: (b) above, (c) at and (d) below the critical angle. Below the critical angle the lateral component of k is larger than the radius of the Ewald sphere of the medium j thus it has purely imaginary k-. component (neglecting absorption) 8.38.
interface
and the
wave
is called evanescent
the conditions for
specular reflection. It intersects the vacuum Ewald sphere wave ko in two points, which pin down the wave vectors of the reflected wave h, and of the transmitted wave in the vacuum, see subfigure (a). Therefrom we obtain the the law of reflection-the reflected Cvac of the incident
wave
makes the
angle with the surface as the incident one. The Ewald specular rod represents the symmetry of the sample and scattering process, which permits a momentum transfer only along the same
construction with the of the
q,, direction
(along
the surface
normal).
8
Inside
a
layer j
of
a
X-ray Reflectivity by Rough Mulfilayers
multilayer (or
in
a
substrate)
the
wave
277
vectors
are
determined 1.
by the dispersion relation kj
-_
within the medium cj, 2. by the continuity of the lateral
njko giving the wave
vector
These two conditions lead to the Snell’s law
radius of the Ewald
components
(also
sphere
at the interface.
refraction
law)
for the
subfigures (b)-(d). The tie points Tj and Rj of the transmitted and reflected wave in the layer j, respectively, are located at the intersections of the specular rod and the "inner" Ewald sphere Ej. For x-rays is n < I (X < 0), thus three distinct cases may happen in each layer. Case (b) marks the refraction law above the critical angle: two waves, reflected k,j and transmitted ktj, propage in the layer. The case (c) visualises the situation at the critical angle for total external reflection in the layer. There is one tie point T,j only and the wave in the layer propagates parallel to the interface, kj k1j. Case (d) interprets the generation of the evanescent wave in the layer, propagating parallel to the interface and exponentially damped perpendicularly to it. According to the Fresnel formulae, see (3-68) and (3.70), the reflected and transmitted wave amplitudes depend exclusively on the complex wave vectors of the media bordering the interface. transmitted
wave as
outlined in
=
8.A.2
Periodic Multilayer
Reciprocal lattice of a periodic multilayer, Fig. 8.39(a), is a set of points positioned equidistantly along the q, axis, subfigure (c). Thus the "superperiodicity" in real space causes a periodic fine structure along the specular rod, and we find so-called multilayer Bragg peaks on the specular reflectivity curve, see Fig. 8.11 for instance. Following from Fig. 8.38 the refraction in the layers causes a shift of the actual multilayer multilayer Bragg peaks with respect to the position of the reciprocal lattice points. This is shown by the comparison between the kinematical the dynamical reflection curve of a smooth multilayer in Fig. 8. 10. The position of the kinematical Bragg-peaks coincide exactly with the reciprocal lattice points. The finite total multilayer thickness gives rise to additional side maxima, so-called Ifiessig fringes between the multilayer Bragg peaks (not shown in the figure). There are p-2 maxima in between two Bragg peaks for a flat multilayer with p periods. Reciprocal lattice of a laterally periodic multilayer grating etched into a planar periodic multilayer is shown in Fig. 8.33. The lateral periodicity gives rise to a grating rod pattern. The grating rods are equidistantly positioned along the direction of patterning With the specular rod in the center.
Tilo Baumbach and Petr Mikulik
278
ML
I
M
-
M,M
-
-
LED,,
X0
27c/D,,
000
(a)
*z
QZ
(b)
Fig.8.39. Schematic set-up of a periodic multilayer: (a) polarizability profile, (c) in reciprocal space
Reciprocal Space Representation
8.A.3
Q11
(c)
in real space,
(b)
the
of DWBA
The formulae for the calculation of the first order DWBA have been derived in
Chap. 4. Here, we show the graphical representation of the corresponding scattering events. Each of the two eigenstates of the unperturbed potential V’ consists of a transmitted and reflected wave T U(+k-,), R U(-k,). =
The four
ing
to
wave
(k,,,Il
ki,,11, k ,,,
-
--
qll,.., q" defined by (4.D23) and correspondki,,,,) in (4.41),(4.46) or (8.48), are represented
vector transfers
in the
reciprocal space by the four intervening scattering processes. They are schematically drawn in Fig. 8.40. We call the first (transmission-transmission) term the Primary scattering process %P, since it is directly excited by the incident wave and it corresponds to the measured scattering vector in vacuum q k, ki, The other three terms are secondary scattering processes. They are of purely dynamical nature, called Umweganvegung (detour or nondirect excitation), which occurs exclusively due to multiple scattering (direct --
or
-
non-direct
T
Fig.
S"T,
excitations)
-
R,,S"T, T,,S"R.
representation of the four x-ray reflection processes in real reciprocal space (right) of the first order DWBA. The fun circles denote the dynamical reflection and transmission in the ideal multilayer, open circles indicate the diffuse scattering due to the interface roughness. The process with the indices 11 is the primary scattering process, described also by the kinematical approximation. The other three are processes of Umweganregung 8.40.
space
(left)
Schematic
R,,S"R,
and in the
X-ray Refiectivity by Rough Multilayers
8
279
perturbed potential VB into the layer disturbances represent the scattering in terms of structure factors Sj, an advantage usually reserved for the kinematical theory. The Eq. (4.DI3), contribution of one scattering process in a single layer to the amplitude reflected by the whole sample depends on the structure factor of the layer disturbance and on the amplitudes of the participating waves. Reciprocal space representation of the scattering processes in the Born approximation, DWBA and dynamical theory for reflection by gratings is shown in Fig. 8.35. The division of the
V:B allowed
to
References 1.
Sirota, S. Garoff, and H.B. Stanley,
S.K. Sinha, E.B.
,
Phys. Rev.
B 38, 2297
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2.
D.G. Stearns,
3.
J.B.
4.
,
J.
,
J.
,
5.
6. 7.
,
,
,
15896
(1993).
Hol and T. Baumbach, Phys. Rev. B 49, 10668 (1994). Schlomka, M. Tolan, L. Schwalowsky, O.H. Seeck, J. Stettner, and W. Press, Phys. Rev. B 51, 2311 (1995). T. Salditt, D. Lott, T.H. Metzger, J. Peisl, G. Vignaud, P. Hoghoj, J.0. Sch irpf, P. Hinze, and R.’Lauer, Phys. Rev. B 54, 5860 (1996). J. Phys. D 28, A220 (1995). V. Hol , G.T. Baumbach, and M. Bessi6re, S.A. Stepanov, E.A. Kondrashkina, M. Schmidbauer, R. K6hler, and J.-U.
8.
V.
9.
J.-P.
,
,
10.
,
11. 12.
,
Phys. Rev. B 54, 8150 (1996). Boer, Phys. Rev. B 49, 5817 (1994). G.T. Baumbach, V. Hol , U. Pietsch, and M. Gailhanou,
Pfeiffer,
,
13. D.K.G. de 14.
,
,
Physica, B 198,
249
(1994).
19.
Stearns, and M. Krumrey, J. Appl. Phys. 74, 107 (1993). N6vot, Revue Phys. Appl. 11, 113 (1976). Revue Phys. Appl. 15, 761 (1980). L. N6vot and P. Croce, V. Hol , U. Pietsch, and G.T. Baumbach. High-Resolution X-Ray Scattering from Thin Films and Multilayers. Springer-Verlag, Berlin, 1999. P. Mikuhk. PhD thesis, Universit6 Joseph Fourier (Grenoble) and Masaryk
20.
University (Brno), 1997. J. Eymery, J.M. Hartmann, and G.T. Baumbach,
15. 16.
17. 18.
E.
Spiller,
D.
,
P. Croce and L,
,
J.
Cryst. Growth 184,
109
(1998). 21.
22.
23. 24.
Lent, and P.I. Cohen, Surf. Sci. 161, 39 (1985). D. Buttard, Dolino, D. Bellet, T. Baumbach, and F. Rieutord, submitted to Appl.,Phys. Lett., (1998). Int. J. Mod. Phys. 9, 599 (1995). J. Krim and G. Palasantzas, G. Palasantzas and J. Krim, Phys. Rev. B 48, 2873 (1993). P.R. Pukite, C.S. G.
,
,
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25.
G.
Palasantzas, Phys. Rev. B 48, 14472 (1993). Ming, A. Krol, Y.L. Soo, Y.H. Kao, J.S. Park, and K.L. Wang, Rev. B 47, 16373 (1993). V. Hol , C. Giannini, L. Tapfer, T. Marschner, and W. Stolz, Phys. ,
26. Z.H.
27.
,
55,
55
,
Phys.
Rev. B
(1997).
Hol , A.A. Darhuber, J. Stangl, G. Bauer, J.F. Niitzel, and G. Cimento19D, 419 k1997). V. Hol , A.A. Darhuber, J. Stangl, G. Bauer, J.F. Niitzel, and G. Semicond. Sci. & Technol. 13, 590 (1998). T. Salditt, T.H. Metzger, and J. Peisl, Phys. Rev. Lett. 73, 2228 T. Salditt, T.H. Metzger, Ch. Brandt, U. Klemradt, and J. Peisl,
28. V.
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Abstreiter,
,
30. 31.
,
,
B 51, 5617 32.
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(1995).
S.F. Edwards and D.R.
Wilkinson,
,
Proc. R. Soc. London, Ser. A 381, 17
(1982). 33.
34. 35.
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,
(1991). Andr6, J. Opt, Soc. Am. A 10, 2324 (1993). Bac, P. Troussel, A. Sammar, P. Guerin, F.-R. Ladan, J.-M.Andr6, D. SchirX-ray Sci. Technol. 5, 161 (1995). mann, and R. Barchewitz, M. Tolan, W. Press, F. Brinkop, and J.P. Kotthaus, Phys. Rev. B 51, 2239
36. A. Sammar and J.-M. 37.
,
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,
38.
,
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P. Mikuli& and T.
and E.
Pindk,
Reflectivity
9
of
Liquid
Surfaces and
Interfaces
Jean Daillant Service de
Gif
sur
Physique de I’Etat Condens6, Orme Cedex, France
des
Merisiers, CEA Saclay,
91191
Yvette
reflectivity experiments liquid surfaces and interfaces. This is a field where reflectivity techniques are widely used, in particular because the range of available techniques is relatively less important than for solid surfaces (no high-vacuum techniques, no scanning tunneling microscopy, atomic force microscopy is difficult). Reflectivity experiments on liquid surfaces present specific features both experimentally and conceptually. Experimentally, the liquid surface is always horizontal, and therefore requires adapted experimental setups. Morevover, the subtraction of the high background scattering in the bulk liquid phases imposes severe constraints on the experiments. Conceptually, the distinctive property of liquid surfaces is the low q divermakes the separation gence of the height-height correlation functions which of specular reflectivity and diffuse scattering impossible. On the other hand, analytical expressions for the height-height correlation functions are available, at least in the capillary regime when the physics is governed by surface tension (see chapter 6 for the surface energy of solid surfaces). This allows a thorough analysis of x-ray surface scattering methods through exact model calculations which can be interesting even for readers having no particular interest in liquid surfaces. After a short introduction to liquid interfaces, we shall comment on the specific features of liquid surface reflectivity studies and then give some examples.
The aim of this
chapter
is to make
a
presentation
of
on
Statistical
9.1
Description
of
Liquid
Surfaces
microscopic structure of the interfacial region reflectivity or surface scattering experiment. A first approach to the structure of liquid surfaces was initiated by van der Waals and consists in describing the liquid-vapour interfacial region as a region from smooth transition from the liquid density to that of the gas [1]. A complete description of van der Waals and related theories is given in Ref. [2]. The principle of such density functional theories is to minimise a free energy (or grand potential) functional taking into account both the local free energy of the fluid at a given density and temperature, and the effect
In this as
it
section,
can
we
consider the
be determined
by
a
J. Daillant and A. Gibaud: LNPm 58, pp. 281 - 304, 1999 © Springer-Verlag Berlin Heidelberg 1999
Jean Daillant
282
of
density gradients (as
theory).
the
a
square
term in the most
gradient
The minimisation of this functional
simple version of yields the liquid-gas inter-
face
density profile and the surface tension which is the surface excess of the grand potential (or of the free energy if the Gibbs dividing surface is chosen [3]), i.e. the total free energy minus that of bulk liquid and gaseous phases* extended up to an arbitrary dividing surface [2]. The alternative capillary wave model of Buff, Lovett, and Stillinger [4] assumes a step-like profile for the liquid-vapour interface. Then all the Aructural information about the interface is contained in its profile z(rll), or, since only a statistical description is meaningful (see chapter 2), in the height correlations which are assumed to result from the propagation of capillary waves (i.e. surface deformation modes). There is now a good experimental evidence using different techniques that this model gives an accurate description of the liquid surface structure for in-plane lengthscales larger than one micron, and also describes the mean surface roughness better than the van der Waals theory. We will therefore limit the discussion below to a description of this model.
9.1.1
Capillary
Fig. 9. 1. Capillary The,corresponding
Let are
z(rll)
Waves
waves.
average
the surface tension and z(x, y) the interface height. density profile is given on the left
- is
be the interface
characterised
by
a
height
spectrum
z(r1j)
in r1l
(Fig. 9.1).
(z(qjj)z*(qjj))
E z(qll )e q
-
-
The work necessary to deform the interface
Ap gzdz
dxdy
W A
0
The interface fluctuations
for the wavevector qjj.
+ 7
z(rll)
is:
Reflectivity of Liquid Surfaces and Interfaces
9
p is
density and -/ the surface tension. Developping the transforming, one obtains:
283
square root and
Fourier
W
=
-yA
+
z(qll)z(q’Il)
2
q111
qjj
fA d2re’ ((qjj+qjj).rjj) I
pg
-
-yq1I.q 11 I
I
(9.3) where the modes qjj are multiples of The terms :A -qjj vanish and
27r/L,
with L the surface dimension.
q’11
W
=
-1
7A
I
E z(qll)z(-qll) qjj
The The
zAp g
+ q
2
^/
I
(9.4)
length V1_-__y1zApg, on the order of Imm is the so-called capillary length. equipartition of energy (Gaussian Hamiltonian) among the degrees of
freedom of the system in thermal
equilibrium gives:’ kBT
1
(z(qjj)z(-qjj))
=
A
Apg
2
(9.5)
1
+ ^/q 11
represented in Fig. 9.2 left. This spectrum has been well characterised for down to wavelengths in the micrommany liquid surfaces by light scattering eter range [6] and is valid is the limit of small in-plane momentum qjj. It describes thermally excited capillary waves, limited by gravity for lengthscales larger than the capillary length, and by surface tension for smaller lengthscales. The resulting surface structure is isotropic in plane. Then, the all the modes: rms roughness of the interface is obtained by summing over 1
2) where the summation
molecular
length.
we
may
=
+
27r/L
=
In
(9-6)
-Iqll to qmax
2
(71’49).
2
(-YIApg)
I + qmax
+ q Mi.
2j,1(-y1Apg)
assume
73mN/m,
kBT
__
(Z 2) For -yHo
Apg
ql,>O
from qmi,,
runs
kBT
1:
=
21r/a
where
is
a
qm
kBT _
In
47-y
one
obtains
[I
< 1 and
+
(z 2)
(9.7)
q2
max
=
(71‘49)]
0.4nm
.
(9.8)
Also
interesting
are
limits:
’
For
a
a
In the continuous limit:
z 2) Generally,
A
rigorous calculation
of the statistical average
see
for
example Ref. [5].
the
Jean
284
Daillant
to-,,
2.0
10
10
10
1.5 -30
+
Z 1.0
0.5
10
to
10
-401
0
to,
to,
to, q
Fig.
9.2. left:
to,
to,
10
10
to
-12
10-10
to-,
(m-’)
r
Amplitude
of the
73mN/m (continuous line)
capillary
wave
to-,
10
-4
to-,
(m)
spectrum for surface tension
-y
7.3mN/m (broken line). right: The heightheight correlation function (z(O)z(rjj)) for water (continuous line) and the surface of a liquid having the same density but a 10 times smaller surface tension (long dashed line). The dotted line is obtained by attenuating the spectrum for water at a
molecular cut-off q
and y
=
27r/10-lorn-1
=
A
oo
0
g
(z2) (z 2)
_
_171g,
-
InA.
These
logarithmic divergences do not imply that there is no interface. The exist, but is not localised in space [7], in agreement with the fact that the divergence is due to small wavevector modes q -+ 0". An open question is that of the corrections to the surface energy at very small lengthscales. This is an important question since, as shown by Eq. (9.5) the (deformation) surface energy determines the surface structure. Taking into account the coupling between capillary modes [8] would lead to a renormalisation of the surface tension equivalent to a larger effective surface tension at very small lengthscales. The opposite trend is displayed when we take into account the effect of long-range dispersion forces which lead to a interface does
[9].
smaller effective surface tension The
height-height
correlation function
forming
the spectrum
Eq. (9.5):
g(r1j)
--
Z(o)]2)
=
(z(O)z(rjj))
=
([z(rll)
_
2(Z2
can
be obtained
k.BT _
KO
7r’Y
by Fourier
(ril Izpgl^l)
trans-
(9.9)
and
(figure 9.2 right). Ko is Ko(x).,,o -_ Log2 7, -
kBT
27r-/
Ko
(ril
(9.10)
the modified second kind Bessel -function of order 0. -
Logx and lim Ko(x).,-,,,
=
0.
Reflectivity of Liquid Surfaces and Interfaces
9
285
Chap. 2, there is a specular reflection on a liquid surface only gravity limits the logarithmic divergence. This logarithmic divergence with distance of the roughness, only limited by gravity or the finite size of the surface is a distinctive property of liquid surfaces. Substituting Eq. (9.9) into Eq. (4.41), it is possible to find the following approximation for the scattering cross-section which is valid in the limit q11 >> As indicated in because
lZp_gl-l [10-13]: do-
A
-
dQ
k 04(1
"1
-
and q
area
=
12,
q11 qmax
(kB T/21r7) I q,, 1 12.
Relation to Self-Affine Surfaces
Many solid surfaces
are
Such surfaces
are
known
exponent (see chapter
length.
by correlation functions of the form:
well described
(z(O)z(r11))
tion
(
kBT
’
2 Ito’ll 1 tsc, 0, 12 Tq-
167r2
where A is the illuminated
9.1.2
n2)2
-
.
as
2).
=
0.2e-(ril/ )21’
;0
self-affine surfaces and h is the a
A self-affine fractal differs from
the
and
roughness,
is the
(9.12)
h < 1.
<
roughness or Hurst roughness correla-
self-similar fractal in that all
a
directions of space are not equivalent for the self-affine case. In contrast to self-similar fractals, self-affine fractals do not have a unique fractal dimension As a whole, they are surfaces, d 2, but they can alternatively be de-
[14].
scribed by of the
a
roughness
an
-
spectrum of such surfaces A
(z (q )2) which is
3
local fractal dimension D
=
0.5
[15] 0’
2705 (1
exact relation for h
h 2. A
+
(see
fairly good approximation
is
2 2
(9.13)
aq211 2)(1+h) also section
6.2.4).
The smaller
h,
rougher the surface appears. The correlation function of a liquid surface is obtained in the limit h -+ 0. As compared to a solid surface described by the correlation function Eq. (9.12) the range of the correlations of a liquid the
surface is much
longer (logarithmic divergence). The correlation length
is
on
the order of Imm.
2
The local fractal dimension constant 1. If
Globally
h(ax, y)
N(1)
by mapping the object on a lattice of limi_’o [1n(N(1)11n(111)]. occupied, then D 0 thus the dimension is 2. Locally however,
be defined
are
1im,,_+oh(x) a hh (x, y) and N (1) OC
h < I and oc
can
lattice sites
=
=
-(3-h)
; D
=
3
-
h.
Jean Dafflant
286
Bending Rigidity
9.1.3
If
a
film is present at the interface, it will reduce the surface tension but also bending. The simplest treatmentof these effects which will be most im-
resist
portant for very low Surface tensions, is due to Helfrich [161, and is described below. Very low surface tensions can be achieved for example in microemulsions composed of brine, oil (e.g. alkanes) and surfactants where interfacial tensions
as
low
as a
few thousandths of
Very
low surface"tensions
films
(vesicles,
lamellar
a
mN/m
can
sometimes be obtained.
also obtained for self-assembled
are
phases)
in water
brine. The molecular
or
amphiphilic area
in such
systems results of a balance between attractive hydrophobic interactions and repulsive interactions (hydrophilic, steric, electrostatic) between headgroups.
equilibrium OFIOA 0, and surface tension is therefore not a relevant for the describing parameter system. In all these systems fluctuations play an important role. This for example is the case of lamellar phases which can be diluted down to very low concentrations where the separation between lamellae (> 100nm) is larger than the range of electrostatic forces. Such structures are stabilised by the so-called Helfrich entropic interaction (undulation forces): the fluctuations of a lamella are limited by the neighbouring lamellae, resulting in a repulsive effective interaction. For such systems having a very low surface tension, the fluctuations are no longer limited by the surface tension but by the bending stiffness. The curvature is defined by two parameters independent of the surface parametrisation, the mean curvature C and the Gaussian curvature G. If R, and R2 are the principle radii of curvature, then: At
--
C
I =
2
(
G=
I
Ri
I
+
R2
)
(9.14)
,
-
(9.15)
.*
RIR2
The deformation free energy can now be develo ed as a function of the curvature and the Gaussian curvature to the second order:
F
/,c
is the
--
bending rigidity
Eq. (9.16)
can
be
_-
CO is the spontaneous
integral
dA
(Fo + AC + 2nC2
TG)
+
modulus and -k Gaussian
(9.16)
.
bending rigidity
modulus.
alternatively written: F
the
I
mean
of G is
a
j
dA
(FO’+ 2n(C
_
CO)
2
curvature. If the surface constant
+
-I G)
topology
(Gauss theorem).
(9.17)
.
does not
change,
Methods similar to those
9
previously
Reflectivity
Liquid Surfaces and Interfaces
287
used lead to:
kBT
(z(qjj)z(-qjj)) Note that
of
a
A
Apg
comprehensive understanding
+
-/q211
4
+ Kq 11
bending rigidity in terms lacking. This is a central systems are often composed
of the
of molecular order and chain conformations is still
problem of films
9.2
9.2.1
in soft condensed matter
(monolayers
physics where
and the role of fluctuations is dominant.
bilayers)
Experimental Measurement Liquid Surfaces
of the
Reflectivity
of
Specific Experimental Difficulties
Considering tube,
reflectivity of liquid surfaces, we first necessarily be horizontal. If the source is a probe the scattering vectors, whereas using
the measurement of the
note that the
sealed
or
liquid
it
can
surface must
be moved to
synchrotron radiation sources alternative solutions must angular spread of a rotating anode source can be used to change the incidence on a fixed point by displacing a monochromator on a circle containing the source and the target point (Fig. 9.3). a
rotating
anode
or
be considered. The
09
0
Monochromator
0
Sample nple Source
Fig. 9.3. Experimental setup for changing the angle of incidence at a fixed point using a divergent x-ray source, e.g. a rotating anode. The monochromator is moved on a circle containing the source and target point
Using synchrotron radiation, two different solutions have been found. For grazing incidence a mirror is generally used. For higher incidences, a crystal deflector can be used (Fig. 9.4) to deflect the beam [17], for example a very thin silicon crystal. By rotating the crystal around the incident beam, the diffracted beam describes a cone of opening angle 40B if OB is the Bragg angle. The whole diffractometer must then be used to keep a fixed point of impact. This crystal can be bent to fit the divergence requirements of the incident beam.
experimental problem posed by reflectivity measurements in the bulk. If on liquid surfaces is that of the background due to scattering efficient method consists one is interested in specular reflectivity, the most
The other crucial
Jean Daillant
288
in
scanning in background.
q., for each q, value in order to determine and subtract the
D
;r’, ac
IInCid( Incident nCid(
bearr beam
Diffracted iff te d bear beams
Fig.
9.4. Beam Deflector.
diffracted beam describes
9.2.2
7c
Crystal
-20 20
B
By rotating the crystal around the incident beam, of opening angle 40B if OB is the Bragg angle
the
a cone
Reflectivity
interpretation of reflectivity curves can be tricky because for liquid surintensity is generally peaked in the specular direction as shown in Fig. 9.5 [18]. This is of course a consequence of the long-range correlations. This point was already discussed in Chap. 4 where it was shown in particular that in a reflectivity experiment diffuse scattering will always eventually dominate the true specular (coherent) component for large wavevector transfers. A reflectivity experiment should therefore never only consist in the The
faces the diffuse
10-1
q.,=1.5nm-’ 10-1
z(O)z(x)>=or 11=
1.4nm
0-
e-’ ]21 v=0.3
WN0
X
I
10-1
10-10
10-11
2
C=30nm
water
Y=0.073 Nm_1
1
-0.03
1
1
-0.02
1
-0.01
0
qx
Fig.
9.5. Diffuse
scattering
0.01
1
0.02
11
1
0.03
(nm-’)
from the bare water surface and
a
solid surface
Reflectivity
9
measurement of the
of
and Interfaces
Liquid Surfaces
289
specular intensity, since the measured signal is always dependence of the height-height correlation function
sensitive to the exact
(see
Sect. 4.8 for
details).
Fig. 9.6. In that case, we calculated that, using the intensity measured in the specular direction for a spectrum Eq. (9.18), the in resolution :Aq., plane of incidence is smaller than the refiectivity of an interface smooth by a factor equivalent This is illustrated in
the
7r
2
1
kBTq L
1
1/2_V
2
-
2
47r^/
2
zAq X7
I
X
I
exp-
27r-/
Log
"’ / N/2-
zA q.,
)I
I
0.577 is Euler’s constant. incomplete _V function and -1E 2(z2 ) because diffuse scattering has been taken ,O-q than larger into account in addition to specular reflectivity. One can see on Fig. 9.6 that the diffuse intensity dominates the specular intensity for q,, > 2nM-1. Note also that a data analysis not taking into account diffuse scattering leads to
where F is the
=
This factor is
an erroneous
.
estimation of the structural parameters.
101
5
10-1
4
10-2
3
10-3
2
water
Si
10-4
C
10-1
2
-1
1
0 z
10-
chain
1 2
3
4
(nm)
6
10-7 10-1 10-9 10-
10
0
4
2
q,
6
(nm-1)
Fig. 9.6. Interference pattern resulting from the reflection of an x-ray beam on an octadecyltrichlorosilane monolayer at the air/water interface and its corresponding electron density model (inset, black curve). The broken gurve represents the specular reflection, the long-dashed curve the diffuse intensity, and the thick line the total intensity. The grey curve in the inset is obtained when the data are analysed using a "box model" with error function transition layers, not taking into account diffuse scattering
Jean Daillant
290
Fig.
"Rocking curve" geometry. q,, is approximately kept by "rocking" the incident beam and the detector
9.7.
varied
9.2.3
The
the
Diffuse
case
constant
as
q,
is
Scattering
of diffuse
scattering
is
even
kind of measurements that
more
difficult. One would expect that
successfully carried out on solid surfaces could be also applied to liquid surfaces. In particular, "rocking curves" (equivalent to q__ scans at fixed q,,, Fig. 9.7) would yield a very good resolution 3 along q,: same
/A q,,
27r =
A
sin
are
0i"zA0i" + sin O,C, Osc)
(9.20)
In fact (Fig. 9.8, left) such measurements lead to "flat" spectra for q, 2nm-’. This is because when the incidence angle becomes larger than the
critical
angle for total external reflection, bulk scattering dramatically inlarge q,, (Fig. 9.8, right). This example shows that it is in practice necessary to fix the incidence angle below the grazing angle for total external reflection 0,. It is then possible to measure the scattered intensity either in the plane of incidence (projected on q,) or in the horizontal sample plane. In the first case, q, and q,, are varied together and it is possible to measure the normal structure of, for example, a film, and verify that surface scattering is indeed measured. However, one has to decouple the structural effects from creases
at
the fluctuation
Measuring whenever
one
spectrum.
diffuse
scattering
in the
plane
of incidence should be considered
is interested in the determination of the normal structure of
thin films using synchrotron radiation. This has two main advantages over reflectivity (see Fig. 9.11): The reduced background. The much lighter experimental setup (only a mirror is required instead of
-
-
a
beam
deflector).
The resolution function is in in Refs.
[19,20].
particular discussed in section
4.7.2 of this book and
9
Reflectivity of Liquid Surfaces and Interfaces
10-1
101
5
10-1
2
10-2
107’
10-15
10- --’
11 .
2
10-5
10
10-1 0
10
0.5
0
-0.5
-1.0
Fig. 9.8.
left: Diffuse
2.5nm-’ (dark
i.
-7
1.0
10
101
101
IV qx
scattering (rocking curves)
grey
circles)
-
..
’10 10J
% (x Im-’)
q,
291
and q_
=
101
le
lop
(.-’)
from the bare water surface for
3nm-’ (light
grey
circles)
as a
function
peaks. Calculated surface 2.5nm-’ (black line). A constant background has been added to signal at q, calculate the dashed line, giving a better agreement with the experimental data. COOH) film (CH2)18 Right: Diffuse scattering by an arachidic acid (CH3 at the air/water interface. The fixed angles of incidence are respectively 2mrad the reflected beam is at q.,
of
=
0. Note the Yoneda
=
-
-
(black symbols),
6mrad
2.4mrad. Note that for
(dark-grey symbols) a
grazing angle
and 10mrad
of incidence
(light-grey symbols). O’_
equal
to
sensitivity revealed on the other curves by the constructive 10’m-1 is lost because of bulk scattering
=
10mrad, the surface interference for q.,
"Z
only interested in the roughness spectrum, a second kind of sample plane) which directly yields a signal propor(in tional to the roughness spectrum should be preferred (Fig. 9.10). A last important point which is not specific to liquid surfaces is that the diffuse intensity is proportional to the resolution volume (Fig. 9.12). It is therefore necessary to precisely determine the resolution function as a function of slit openings and of the footprint of the beam on the surface to precisely determine the magnitude of this intensity. When
scan
we are
the horizontal
Jean Daillant
292
Z
q
Nal(TI)
Od X
VacuA "
0
SC
Path
7
Vacuum
Fath F
C*(, 11
S.1C
Langmuir trough Fig. 9.9. Schematics of experiments (Troika beamline, ESRF). (q, q,,) plane of incidence geometry. in-plane q, geometry. C*(111): diamond monochromator, SiC: mirror, Nal(TI): scintillation detector. Typical distances are: Sample-to-Sd distance 700 mm, Sr--Sd distance 500
Sj, S,_ and Sd mm
x
0.250
are wi x
mm.
hi: 0.4
Typical horizontal x
mm
0.2 mm, w,
x
x
vertical
hr-: 2
mm
openings
x
of the slits
2 mm, Wd
x
hd: 10
mm
Y
d
X
Vacuum Path I
P.S.D.
Schematics of experiments (Troika beamline, ESRF). In-plane q, geC*(1-11): diamond monochromator, SiC: mirror, PSD: position sensitive gas-filled (xenon) detector. The experimental curve represents the scattered intensity (horizontal axis) as a function of the vertical position on the PSD. Typical
Fig.9.10.
ometry.
distances izontal mm, w,
are:
x x
Sample-tO-Sd distance 700 MM, Sc-Sd distance 500 mm. Typical horopenings of the slits Si, S,_ and Sd are wi x hj: 0.3 mm x 0.2
vertical
h,-: 0.3
mm
X
100 MM, Wd
x
hd: 0.5
MM
X
100
HIM
9
Reflectivity of Liquid Surfaces and Interfaces
293
10-3
10-4 10-5 10-6 10-7 10-8 10-9
10-101 0
0.25
0.75
0.50
1.00
1.25
10" m-1) q. (x Fig. 9.11. Laboratory (empty circles) and synchrotron (filled circles reflectivity experiments (top). Diffuse scattering experiment in the plane of incidence (filled squares, bottom) for the same arachidic acid monolayer on a CdC12 subphase 100 0 0
10-2
(b
10-4 <’I 0-1
10-8
10-
10
10-121 106
107
10a
109
10
10
qY (M-1) Fig. 9.12. Diffuse scattering from the bare water surface. The dependence of the intensity above 10-7h results from the convolution of the reflected beam with the experimental resolution. Even the slope of the intensity scattered by the rough water surface is modified by the integration over the resolution volume (-3 instead of -2 for the scattering cross-section)
Jean Daillant
294
Some
9.3
Examples
The first reflectivity experiments on liquid surfaces were carried out on liquid metals [21,22] in the 70’s and the roughness of the free surface of water was demonstrated to be consistent with the
capillary wave model in the mid 80’s reflectivity however, only density profiles averaged over [23-25]. the surface can be measured. From the beginning of the 90’s, the analysis of diffuse scattering from the surface, which gives access to the roughness spectrum has been undertaken [2 6, 11]. The spectrum could be measured up to wave-vectors of the order of 107rn-1 The method has been more recently extended down to molecular length-scales giving access to new phenomena [27]. After a short Presentation of the method we will discuss the results pertaining to the bare water surface andgive some examples for liquid metal surfaces, surfactant monolayers, and liquid-liquid interfaces. References [2830] contain recent review articles on the scattering by liquid surfaces. studies
In
.
101
-Filied Rued 8 ac -----Background
Vground
A
t-00
100
(a)
(b)
RIO-’
qz(A71)1) q.(K qz(A
-
I
.2
0.75-
ICF1
10 ,0.10 0.10 ,0, + 15 +0.15 0.15 +0, .0.20 *0.20 0.20 .0.25 *0-25 0.25 0.30 ,
o
A 0.50-
&10
0.25-
1-
10-7 ----------
W -j W ir
IC 4
0.002
(c)
-------
G003
-
OL004 1
,F5
......
0-
(b)
-----
NO 0
Reso[uUon
XIC 4
106
"s
10F,
001 0
0
1’
0.02
0.03
U04
0.05
.0
0
107,
INCIDENT ANGLE (RADIANS)
10-1,
qY
Fig.
Left: First
reflectivity experiment on the bare water free surface by Hasylab in 1985 [23]. (a) Measured x-ray reflectivity (circles), calculated Fresnel reflectivity (continuous line); the dotted line takes the capillary wave roughness into account, (b) xpanded version around the critical angle for total external reflection. (c) same as (a) but measured with use of a rotating anode generator. With kind permission of A. Braslau and P.S. Pershan. Right: Diffuse scattering by the ethanol -free surface (Sanyal et al. [11], NSLS beamline X22B). q,, scans at constant q-,; The background is subtracted in Fig. (b). With kind permission of M.K. Sanyal 9.13.
Braslau et al. at
Reflectivity
9
9.3.1
Simple Liquids
Liquid Surfaces
of
and Interfaces
295
Free Surface
The bare water free surface
was
studied for the first time in 1985
et al. at Hasylab, Hamburg (Fig. 9.13, left) [23]. fully interpreted within the frame of the capillary
Their results wave
by
Braslau
were success-
model and
they found
a rms roughness (z 2) proved in Refs. [24] and [25]. In Ref. [26] diffuse scattering from the surface was also measured, and it was demonstrated that the capillary wave model could be applied down to distances as small as 50nrn. This model was applied -1 in Ref. [11] for’the ethanol surface up to wavevectors on the order of 107 M =
0.32nrn. These results have been confirmed and im-
(Fig. 9.13, right). In both cases, the limit was fixed by the source flux and background subtraction. This is no longer the case for the experiments of Ref. [27] carried out at
European Synchrotron Radiation Facility. The intensity plane of incidence is presented in Fig. 9.14a.
the
’
-
scattered in the
1 OMM MM
1 "
E
1MM M 0.1
X x
c?E ,
2
<1 10
8
c.4
Cr
(b) 107
108 qxy (m-
101
10
10
Scattering by the bare water surface in the plane of incidence (a) and plane of the surface (b). For measurements in the plane of incidence q,, is the projection of the wave-vector transfer on the horizontal. "Imm" (circles) and 4(10mm" (triangles) is the opening of slit 53 in Fig.9.9. Squares represent the background which was not subtracted. In (b) the background was subtracted following the procedure indicated in text. The full line is calculated by using the capillary wave spectrum and the acoustic wave scattering cross section Eq. (9.21) has been included to calculate the dashed line curve. Note that the great dynamic range of the y-scale has been compressed by a multiplication of the measured scattered 2 intensity by q. or qy Fig.
9.14.
in the
Jean Daillant
296
The data extend to q > 108m-’ and are well described by the capillary spectrum. The data taken in the interface plane (qy) extend to even
wave
larger wave-vector ( magnitude than that
Wom’,
note that this is
more
than two orders of
of the best
previous measurements [26]) but can only be described with the spectrum of Eq. (9.5) up to q 5 x 10’m-’. The excess scattering at q > 10’m-’ indicates either a smaller effective surface
large wavelengths [9] or another source of scattering. This scattering has no measurable dependence on q (the apparent q-1 dependence on Fig. 9.14 is due to the resolution function), and can be attributed to density fluctuations (acoustic waves) within the penetration depth of the beam. The corresponding scattering cross-section [171 can be calculated from the density-density correlation fUnCtion4 at or near the interface which can itself be determined by using the linear response theory [6]: tension at such
00
A(1-n2)2jtinjj2jtscjj2kBTKT 0, 0,
167r2
21m (q,,, 1)
do-
dQ
where
(9.21)
is the refractive index of water, lin and t,,, are the transmission coefair/water interface for the incident and scattering beams, IOT is
n
ficients of the
compressibility of water (4-58 x 10-10rn2 N-1), and lm(q,,,) imaginary part of the normal component of the wave-vector transfer in the liquid (inverse of the penetration length). Including this contribution gives a better agreement (Fig. 9.14b), without discarding the possibility of the isothermal
is the
other corrections.
extensively studied [31]. In liquid phase by a crystal phase has been
Alcohol and alkane surfaces have also been
particular, partial wetting
of the
discovered.
9.3.2
Liquid
The first
Metals
reflectivity investigation
Lu and Rice in 1978. A
more
of the mercury surface was carried out by experiment is that of Magnussen et al.
recent
The’ data extend to 25nm-1 and the structuration of the unambiguously demonstrated. Except for the first layer which is shifted towards the vapor, the distance between layers is smaller than the distance between atoms in the liquid, but larger than the distance between atoms in the solid. The decay length of the order is on the order of 0.35nm and the peaks are smeared out by capillary waves.
,(Fig. 9.15) [321. interface is
4
see
Sect. 4.5 for details.
Reflectivity of Liquid Surfaces and Interfaces
9
297
10, 10-1 10-2 10-1 IG-1
25
-
’ .00
-
10-,
1.65
o
1.85
2.21
2.41
-
(a)
.
0.75
% 0.50
0.25
10-7
0
10-8
-10
-5
0
20
15
1
5
10-1 1.2
(b).
\(b)
1.0
0.8 c’
0.6 0.4 0.2
0. 1.0
0.5
0
1.5
2.5
2.0.
Reflectivity of the mercury surface according to Ref. [32] (a). In reflectivity curve has been divided by the Fresnel reflectivity of a smooth surface of a liquid that would have the bulk mercury density up to its surface in order to enhance the structures in the refl6ctivity profile. Different reflections are shown in (c). Note that the data extend up to 25nm-1. Right: Model profile of the density at the mercury free surface obtained from the previous reflectivity curves.
Fig.
(b)
9.15. Left:
the
With kind
9.3.3
permission
of P.S. Pershan
Surfactant
Monolayers
Reflectivity Studies Surfactant monolayers have been the subject x-ray and neutron reflectivity studies. We shall only discuss one of illustrate the
manner
in which information
ity experiment. A very comprehensive study
air/water
of
can
be extracted from
temperature. The data
density changes’at
long-chain alcohols (Clo -+ C16) at the by Rieu et al.. [33] (Fig. 9.16). From
(Fig. 9.16).
were a
sufficiently
Diffuse
The
Scattering
When
a
surfactant
measure as
a
the den-
function of
well resolved to evidence thickness
two-dimensional
bending rigidity determination is necessarily very rough now be performed by grazing incidence
sition
reflectiv-
interface has been carried out
their very precise measurements, the authors were able to sity of both the aliphatic chains and of the headgroups and
a
of many them to
liquid-solid (rotator phase)
modulus
was
reflectivity experiment, diffuse scattering. in
a
monolayer is present
its first effect is to reduce the surface tension: -y
=
tran-
also evaluated. Such
-yH,o
-
but
a
can
interface, IT, where H is
at the
Jean Daillant
298
qy
K,
18
16.
q.
E12 C
C’. 00
14
C11
-
0,
AIR
12
.0
.......
...
10
0
40
30
20
50
60
70
80
Encl
T"Perature
CH2 LAYER
28 27.5 27
E
i’ _Vf CU
0.84
CH
0.86’1
C
C
t *26.5
OH
_IL A
’
26 25.5
WATER
25
C
0.88
-
,
>1
0.9 0.92 0.94
24.5 0
10
20
30
40
50
60
70
80
E*C3
Fig. 9.16. Left: Schematics of the alcohol monolayer at the air/water interface. Top, right: Film thickness as a function of temperature. The arrows indicate the phase transition. Bottom, right: Volume per CH2 as a function of temperature. Note the density jump at the 2-d liquid to solid (rotator phase) transition. With kind permission of B. Berge and J.P. Rieu
the surface pressure,
(CH2)18
-
COOH)
as
illustrated in
Fig. 9.17
for
an
arachidic acid
(CH3
-
film.
order corrections to the spectrum, i.e. effects of the bending stiffof the film are also apparent. Results for a L,, di-palmitoylphosphatidyl-
Higher ness
choline
(DPPC)
film
on
pure water
are
presented
in
Fig.
9.18. Whereas at
small qy values the scattered intensity scales with the surface tension as expected, this is no longer true at large qy due to the effect of bending stiffness.
Fig. 9.18 have been analysed using the spectrum Eq. (9.18) including the additional term Kq’ in the denominator. For the more compressed film of Fig. 9.18 it is found that x (5 2)kBT, smaller than generally expected in condensed DPPC films [34]. The observed wave-vector range is not large enough to allow the precise determination on the exponent 4. Smaller exponen ts are however found with the very rigid films[27] formed by fatty The data of
--
acids (here behenic acid CH3 (CH2)20 COOH) on divalent cation subphases (5 x 10-’mol/I CdC12) at high pH (8.9) and low temperature (5’C). 3.3 Uncompressed, such films exhibit a qpower law which has been attributed to the coupling between in-plane (phonons) and out-of-plane elasticity [27]. Finally, in systems with more than one interface, it is possible to measure -
-
the correlation between the interfaces soap films constants
[35] can
and also for free be measured
[36].
(see
standing ,
Sect.
4.3.3).
This is the
case
for
smectic films for which the elastic
all’
Reflectivity
9
Surfaces and Interfaces
Liquid
299
10
E
tr
0.1 10
E
X
a,?, 0.1
2
5
le
2
q.
5
108
2
5
(m-’)
scattered by an arachidic acid film (black curves) and (grey curves). The surface tensions are (top to bottom) 33 mN/m (diamonds), 43mN/m (triangles), 53mN/m (squares), 69mN/m (circles) and 73mN/m. (b): The same data normalised by 7/7water in order to illustrate the scaling I oc y in the
Fig. 9.17. (a): Intensity
water
range 3 x106,rn
-
1<
q,
spectra. The fringes in the
8 x106,rn-1 where
:,
capillary
waves
dominate the fluctuation
due to the normal film structure since q;, is not constant
axe
(x,z) configuration
2
108
2
5
109
5
2
10",
qY (.-,)
Fig. 9.18. Intensity a
plane by a bare water surface (grey compressed at 20mNlrn (grey circles) and
scattered in the horizontal
DPPC film at VC
triangles) 40mN/m (black circles). and
Lines
are
the best fits
as
indicated in text. Note that the
intensity scales with the surface tension at low q, but that this is no longer true at large q. due to the effect of bending stiffness (the black curve passes below the grey curves). Inset: corresponding molecular area surface pressure isotherm of
scattered
-
the DPPC film
Jean Daillant
300
Liquid-liquid
9.4
Only
a
few x-ray
Interfaces
reflectivity experiments
(neutron reflectivity experiments possibility of
have been
will not be discussed
attempted
here).
up to now In Ref. [37], the
using the high energy bremstrahlung of a tungdemonstrated. Two geomeries are possible with the incident coming either through the top or the side (Fig. 9.19). The experiment
sten tube
beam
measurements
was
done at
fixed
angle with an energy sensitive detector. The main difficulty through the 7cm wide cell (Fig. 9.20, left). Quite surprisingly, this classical setup allowed the measurement of a very nice reflectivity curve at the water-cyclohexane interface (Fig. 9.20, right). was
a
is related to the transmission
type (a)
Upper phas
M\ ME\,
Lower phase,
it
"(b)
Upper $F upper. Upperphase
I N
Fig.
9.19. Possible
Lower "a Laws Lowerr
geometries for liquid-liquid surface scattering (a) coming from coming from the top of the upper liquid.
the side of the upper liquid. (b) beam With kind permission of S.J. Roser
100[ ’o-1
-
-
w-’
I.,
I.,
..0
wavelengffi (A)
Fig.
9.20. Left:
Absorption through
air and
’
O.M5
I
I 0.25
liquid. The curves represent the transwavelength. Right: Reflectivity of the water/cyclohexane interface and model fit using a Fresnel reflectivity profile with additional roughness. With kind permission of S.J. Roser mitted
intensity
as
a
function of the
9
[38]
McClain et al.
Reflectivity of Liquid Surfaces and Interfaces
studied
a
301
surfactant film in contact with the mi-
decane, water, triethylene glycol monooctyl ether (C8E3) ternary mixture at 17keV using synchrotron radiation (NSLS, beamline X2013). They measured both reflectivity (Fig. 9.22, left) and diffuse scattering (Fig. 9.22, right). The rms roughness extracted from the reflec8.5nm. In addition, the diffuse scattering tivity experiment was (z 2)1/2 O.IlrnN/m and K < 0.5kBT. Note however that the experiment yielded 7 curves of Fig. 9.22 suffer from background subtraction problems at large q, values which represent in fact the main difficulty of such experiments. croemulsion middle
phase in
a
=
=
B
Tamp--
c.-Ikd c6i
Fig.
9.21. The
in the
angle
experimental
X20B, NSLS. a angle of reflection. The decane-water-C8E,3 miequilibrium with decane and water. With kind per-
cell of McClain et al. at beamline
of incidence. 8 is the
croemulsion middle
is in
phase
mission of B.R. McClain
lo, Ra
Data
lo, lo,
W
’N:
13
1,
le
10
,
o.
%
0.005
0.010
0.015
0.025
’0.020
0.030
16, .5
-1.0
-0.50
0.5
1.0
I.S
Specular reflectivity from the water-micro emulsion interface. The reflectivity of the corresponding flat interface, and the solid line is calculated with a r.m.s. roughness of 8.5nm. Raw data (circles) and background (squares) are displayed in the inset. Right: Diffuse scattering from the
Fig.
9.22. left:
dashed line is the Fresnel
water-micro emulsion interface lines
are
calculated with
of B.R. McClain
a
(q,,
scans
at fixed indicated q-.
surface tension -y
=
O.HmN/m.
value).
The theoretical
With kind
permission
Jean Daillant
302
Another illustration of
Fig.
9.23
[39].
liquid-liquid interface measurement is given in Langmuir trough made of glass (lower teflon (upper part containing oil) to avoid leak-
a
A dedicated two-barrier
part containing
water)
and
age, equipped with very thin (50pm) teflon windows for the x-ray beam was used. The optimal cell width resulting of a balance between absorption and the requirement of a flat meniscus is 7cm. A high energy has to be used 0.068nm, for which the transmission through the 7cm wide (18keV, A film of hexadecane is 0.1). Detector scans in the plane of incidence are represented Fig. 9.23. The amphiphile used is the phospholipid di-palmitgylphospshatidyl-choline (DPPC) which forms very stable films at the water/oil interface and can therefore be compressed to high pressures (i.e. low surface tensions), thus giving rise to a large diffuse scattering signal. The fluctuations of this amphiphilic film (-y MmN/m) at the oil water interface could be measured up to wave-vectors 10’m-’. The background is very large but the subtraction procedure is sufficiently efficient and reliable to allow the measurement of very small signal to background ratios. The structural parameters used to analyse the data were similar to those of compressed DPPC films at the water/air interface, and the film fluctuations could be analysed using Eq. (9.5) with WmN/m. =
=
10-1
10-1
Nk
1--’
:; LLq
Nal(TI)
11
*
Sd
.
SC
10
bar ier
11-1
\
10-1
-9
Pt lo-
10
10-11 wler
1.
10-121 101
101
101 q.
1. 1
101
101
(m-1)
Fig. 9.23. (a) Schematics of the experiment (ESRF, BM32). (b) Diffuse x-ray scattering at the hexadecane-water interface. Detector O c scans in the vertical plane of incidence. Grey symbols and curve: bare hexadecane-water interface. Black symbols and curve: compressed L-a-dipalmytoilphosphatidylcholine film (-Y 1OMNIM). These curves are divided by a factor of 1000. Empty circles: signal, empty squares: background mainly due to bulk hexadecane scattering, filled cirles: signal minus background =
Liquid Surfaces and Interfaces
303
der Waals Verhandel. Konink. Akad. Weten. Amsterdam 1, 8
(1893).
9
Reflectivity
of
References 1.
J. D.
2.
J.S. Rowlinson and B.
van
Widom,
"Molecular
Theory of Capillaxity" Oxford,
Press,1982. On the equilibrium of heterogeneous substances in The scientific papers of J. Willard Gibbs reprinted by Dover Publications, New-York, 1961. 4. F.P. Buff, R.A. Lovett and F.H. Stillinger Jr. Phys. Rev. Lett. 15 621 (1965). 5. R.F. Kayser Phys. Rev. A 33, 1948 (1986) 6. See for example R. Loudon, "Ripples on liquid interfaces" in "Surface excitations" edited by V.M. Agranovich and R. Loudon, Modern Problems in Condensed Matter Science, vo.l. 9, North-Holland, Amsterdam, 1984. 7. R. Evans, Molecular Physics 42 1169 (1981). 8. J. Meunier, J. Physique 48 1819 (1987). 9. M. Niap6rkowski, S. Dietrich, Phys. Rev. E 47 1836 (1993). 10. S.K. Sinha, E.B. Sirota, S. Garoff, and H.B. Stanley, Phys. Rev. B 38, 2297 Clarendon
3.
J.W. Gibbs,
(1988). 11.
M.K.
Sanyal, S.K. Sinha, K.G. Huang, B.M. Ocko, Phys, Rev. Lett.
66 628
(1991). Fukuto, R.K. Heilmann, P.S. Pershan, J.A. Griffiths, S.M. Yu, and D.A. Tirrell, Phys. Rev. Lett. 81, 3455 (1998). 13. H. Tostmann, E. DiMasi, P.S. Pershan, B.M. Ocko, O.G. Shpyrko, M. Deutsch, Phys. Rev. B 59, 783 (1999). 14. "Dynamics of fractal surfaces" edited by F. Family and T. Vicsek, World Scientific, Singapour, 1991. 15. G. Palasantzas, Phys. Rev. B 48 14472 (1993). 16. W. Helfrich, Z. Naturforschung 28 c 693 (1973). 17. J. Daillant, K. Quinn, C. Gourier, F. Rieutord, J. Chem. Soc. Faraday Trans., 12. M.
92 505. 18.
L. Bourdieu, J. Daillant, D. Lett. 72, 1502
Chatenay,
A.
Braslau,
and D.
Colson, Phys. Rev.
(1994).
Daillant, 0. B61orgey J. Chem. Phys. 97 5824 (1992). Jeu, J.D. Schindler, E.A.L. Mol, J. Appl. Cryst. 29 511 (1996). 21. B.C. Lu, S.A. Rice, J. Chem. Phys. 68 5558 (1978). 22. L. Bosio, M. Oumezine, J. Chem. Phys. 80 959 (1984). 23. A. Braslau, M. Deutsch, P.S. Pershan, A.H. Weiss, J. Als-Nielsen, J. Bohr, Phys. Rev. Lett. 54 114 (1985). 24. A. Braslau, P.S. Pershan, G. Swislow, B.M. Ocko and J. Als-Nielsen, Phys. 19.
J.
20. W.H. de
25.
Rev. A, 38, 2457 (1988). J. Daillant, L. Bosio, B. Harzallah, J.J. Benattar, J.
Phys. France 111
149
(1991). Schwartz, M.L. Schlossman, E.H. Kawamoto, G.J. Kellog, and P.S. Pershan, Phys. Rev. A 41 5687 (1990). 27. C. Gourier, J. Daillant, A. Braslau, M. Alba, K. Quinn, D. Luzet, C. Blot, D. Chatenay, G. Griibel, J.F. Legrand, G. Vignaud, Phys. Rev. Lett. 78 3157. 28. S. Dietrich, A. Haase, Physics Reports, 260 1 (1995). 29. R.K. Thomas, J. Penfold Current opinion in colloid and interface science 1 23 26. D.K.
(1996). 30.
S.K. Sinha Current opinion in solid state and material science 1 645
(1996).
Jean Daillant
304
Many references concerning this work can be found in M. Deutsch, B.M. Ocko, Wu, E.B. Sirota, S.K. Sinha, in "Short and long chains at interfaces" edited by J. Daillant, P. Guenoun, C. Marques, P. Muller, J. Tran Thanh Van, Editions Fronti6res, Gif-sur-Yvette 1995, p.155. 32. O.M. Magnussen, B.M. Ocko, M.J. Regan, K. Penanen, P.S. Pershan, and M. Deutsch, Phys. Rev. Lett. 74 4444 (1995). 33. J.P. Rieu, J.F. Legrand, A. Renault, B. Berge, B.M. Ocko, X.Z. Wu, M. Deutsch, J. Phys. 11 France 5 607 (1995). 34. E. Sackmann in "Handbook of biological physics",, vol. IA edited by R. Lipowsky and E. Sackmann, Noth-Holland, Amsterdam, 1995. 35. J. Daillant, 0. B61orgey J. Chem. Phy8. 97 5837 (1992). 36. E.A.L. Mol, J.D. Schindler, A.N. Shaldginov, W.H. de Jeu. Phys. Rev. E 54 536 31.
X.Z.
(1996). 37.
S.J.
38.
B.R.
Roser, S. Felici, A. Eaglesham, Langmuir 10 3853 (1994). McClain, D.D. Lee, B.L. Carvalho, S.G.J. Mochrie, S.H. Chen, J.D. Litster, Phys. Rev. Lett. 72 246 (1994). 39. C. Fradin, D. Luzet, A. Braslau, M. Alba, F. Muller, J. Daillant, J.M. Petit, F. Rieutord, Langmuir 14, 7329 (1998).
10
polymer Studies
Giinter Reiter Institut de Chimie des Surfaces et Interfaces CNRS, 15 rue Jean Starcky, B.P. 2488, 68057 Mulhouse, France
10.1
Introduction
In this chapter I would like to present some examples for the great success of neutron and x-ray reflectometry in polymer science. These techniques are unique 'for the determination of interfacial density profiles, even of buried interfaces. The vertical resolution of these techniques is at least comparable with SFM (scanning force microscopy) but at the same time they take averages over large enough areas to give a representative and characteristic information of the system. Other techniques may be more direct (like SFM or NRA (nuclear reaction analysis)) but they have severe disadvantages concerning either vertical resolution or lateral sampling. While SFM may provide detailed information on small sample sizes this may not be representative for the whole sample. NRA and other ion beam techniques are certainly more direct as they work in direct space (and not in Fourier space as neutron and x-ray reflectivity). However, their vertical resolution is in many cases insufficient to detect all important features of polymeric interfaces. Comparing the ,vertical depth resolution of neutron and x-ray reflectometry (of the order of Angstroms) to the typical size of a polymer (some hundreds of Angstroms) clearly shows the possibility to measure changes at a submolecular level. Here I will give several examples where neutron and x-ray reflectivity have been successfully used to investigate interfacial problems in polymer science. It should be noted that samples generally need to be quite large (several cm2) and homogeneous over this area. In some cases this may present a major difficulty concerning sample preparation. For stratified systems the interfaces need to be extremely parallel. Otherwise averaging of the large areas illuminated by the incident beam will lead to smearing effects. As a consequence, the high vertical resolution of this technique would be lost. I have selected mostly examples regarding problems from polymer sciences which are demonstrating the many possibilities x-ray and neutron reflectometry offer. They will show how powerful, versatile and unique these techniques are. Due to space limits, I have not been able to give an exhaustive survey and focused on some rather unique, successful and convincing setups. It was also not my intention to give a review on polymer physics and thus most of the explanations and interpretation are short and limited.
Daillantand andA. A.Gibaud: Gibaud:LNPm LNPm58, 58,pp. pp. 305 305--323, 323, 1999 1999 J.J.Daillant © Springer-Verlag Springer-VerlagBerlin BerlinHeidelberg Heidelberg 1999 1999 0
G nter Reiter
306
Note that k used in this
chapter is deflned
q,/2 for specularreflectivity. neutron
k
=
k,
=
(27r/A)sinO
--
by the
community.
Thin
10.2
as
This convention is mostly used
Polymer Films thin
films.
there
simpler polymer neutron reflectometry are nonetheless used. This comes from the possibility to distinguish between film thickness and film density, as well as the density profile, even for films as thin as some nanometers. One can thus determine density and thickness changes separately if e.g. the film is measured at different temperatures. I will start with work
on
polymer
Although
(and faster) techniques available to measure the films (like e.g. ellipsometry) x-ray and especially
are
thickness of thin
typical reflectivity curve for a thin polymer polystyrene film deposited by spincoating onto a glass slide and measured by x-ray reflectometry. The thickness can be determined quite precisely (better than 0.1 nm) from the well pronounced interference fringes. Assuming an error-function density profile for the substrate/film and the film/air interfaces one obtains roughness values of 0.6 and 0.3 mn, respectively. The specific density of the film (which shows up mostly at the critical wavevector for total reflection, see Sects. 3.1.2 and 5.3.2) corresponds well with the density of polystyrene in bulk samples (1.05 g/cm 3). In
Fig.
film. It
10.1
one
represents
a
can
see
49.3
nm
a
thick
10
0.5
1.0
k
[nm-11
1.5
curve for a polymer film: polystyrene spinglass. Experimental data points are shown with error-bars. The dotted curve is the best fit yielding a thickness of 49.3 nm, and 0.3 and 0.6 nm for the roughness of the polymer-air and the polymer-substrate interface, respectively. The density of the film corresponds to the bulk density of polystyrene
Fig.
10.1. A
typical
x-ray
reflectivity
coated from toluene onto float
10
polymer Studies
104
307
1
1
103-
PS 660k
I
1o2 101 100 > *;_0 CD
cc
10"’
14.9nm -
10-2. 1 Onm
10-310 10
-4
5
as
10
4.7nm ’+. 11 F1111
prepared
after
6
iLoLur at 80’ ,L.
Wavevector k
Fig.
10.2.
0.4
0.3
0.2
0.1
[nm-1]
660k) X-ray reflectivity curves for 3 different polystyrene films (M. annealing for one hour at 80’C, i.e. below the bulk glass transition
before and after
temperature. The thicknesses
The
high
indicated in the
are
figure. (adapted
vertical resolution of x-ray reflectometry is
from Ref.
[1])
extremely useful and
necessary for the investigation of changes in polymer films after (or during) thermal treatments. In Fig. 10.2 reflectivity curves for three polystyrene films
(differing
in initial
thickness)
are
shown. The
curves were
temperature but after the sample was annealed for time the curve is compared with the result just after can
clearly
see
that the
curves
have
attributed to relaxations of the
changed
after
one
measured at room
hour at 80’ C. Each
sample preparation. One heating. These changes are
polymers.
high sensitivity of x-ray reflectometry has been used to measure the thickness of an ultrathin PS-film at different temperatures (see Fig. 10.3). The
expansion extremely thin films rather showed rechanged by more than 10%. X-ray and neutron, reflectometry can be used not only to investigate featureless thin films but these techniques are also able to provide information on the internal structure of the films. Using specular reflection one may obtain results of e.g. on the multilayer structure induced by surface directed orientation self assembling block copolymers (see Figs. 10.4-10.6). These examples show Contrary
to usual thermal
versible contraction. The film thickness
polystyrene-poly(methyl methacrylate) may either be air
polymer layer). In Fig. 10.4 one
(i.e.
a
can see
dent wavevector for
a
block
copolymers (P(S-b-MMA) debounding medium deposited Si02-layer (confined
onto silicon wafers. The second
posited by spincoating
FREE
surface)
or
a
intensity (R) as a function of the inciP(S-b-MMA) film. Several features can
the reflected
556
nm
thick
First, the intensity drops quite abruptly at about 0.01A-1. This passing the critical angle of total reflection for the copolymer film, i.e. the x-ray beam is penetrating the film. The slow increase of R below 0.01A-1 is due to an increase of the fraction of the incident beam hitting the
be noted. is due to
G nter Reiter
308
U)
(D C
20
30,
40
50
60
70
80
Temperature (’C)
Fig.
10.3. Thickness of
a thin polystyrene film as a function of annealing temannealing (to remove non-equilibrium conformations due to spincoating) the sample was heated incrementally to 80’ C (full circles), was cooled to room temperature, and re-heated to 80’ C (open circles) (adapted from ref. [2])
perature. After
a
first
10B 0 _E
’a;
5.0
7
106
1
1
.
I
I
.
.
.
.
.
.
.
.I
I
I
I
I
2.0 -
;’
1.0
R
0A
1()4
r
io, 0.00
0.05
0.10
k,’O (,C)
F.
0.20
,
,
,
I
.
,
,
k
Fig.
.
0.30
0.25
,,,
0.35
0.40
V)
X-ray reflectivity curve of a P(S-b-MMA) diblock copolymer that has a multilayer structure after annealing at 170’ C. The right curve expansion of the curve on the left (adaptedfrom ref. [3])
10.4.
self-assembled into shows
an
sample. Around 0.015A-’ the intensity drops again as now the substrate is not totally reflecting anymore. Beyond 0.015A-’ the reflection curve consists of essentially two features: The interference
fringes (separated by
the total film thickness of 556 The
multilayer Bragg period of 44.5 nm. Thus this film consists of
which
are
proportional
to
nm
reflections
(see Chap. 8) corresponding
to
a
lamellar
exactly 12.5 layers. Fig. 10.5 shows a similar P(S-b-MMA) (150nm thick) film, now investigated by neutron reflectometry below and above the order-disorder transition of the system. Again, the clearly visible multilayer reflection peak around 0.02 A-’ reflects the layering of the lamellae parallel to the substrate surface. A good fit is obtained by using the model shown in the inset. Above the order-
polymer Studies
10
309
IDO 10-1 -2
1()-2
R
R
10-3 I
10-5
-
’0-5
0.’
1 0-1
I
00
I
L-
0.02
...
10-06.00
0.08
0.02
0.06
0.04
ka(A-1)
0,08
ko(A-1)
reflectivity curve of a P(S-b-MMA) diblock copolymer film. weight of the copolymer is about 30k. The sample was annealed at 176’ C (left panel) and at 1400 C (right panel), respectively. The insets show the scattering length density profiles that yielded the best fits, indicated by the solid lines (adaptedfrom ref. [3])
Fig.
10.5.
Neutron
The molecular
disorder transition less
ordered can
(ODT)
the lamellae further away from the interfaces are this sample below the ODT (i.e. in the
developed. However, annealing be
region)
seen.
obtained
leads to
Due to the
increase of the
an
large
contrast
Bragg peaks.
(see, Sect. 5.3.2)
higher orders scattering length
Even
of the
by deuteration of the PMMA block, details of the interfaces between particular, the interfacial width between
the lamellae could be resolved. In PS and PMMA
found to be 5 0.2nm.
was
100
10-1
-
6.73 6.93
10-6
7.05
10-1
10-10
lo-12
-
7.43 -
7.
7.74 -
lo-141 0.000
Fig.
10.6. Neutron
7.88 I
I
0.010
reflectivity
0.020
equilibrium
(adapted
lamellar
from ref.
-
A-
0.040
0.050
0,060
0.070
k,.,o (A-’) curves
fined between two surfaces where the
distance is indicated
0,030
of a P(S-b-MMA) diblock copolymer conseparation distance has been changed. This
by the t/Lo values (where t is the film thickness period of the block copolymer) on the right side
[3])
and Lo is the of each
curve
G nter Reiter
310
Polymer films
can
also be confined between two solid walls. The second
wall may be produced by evaporating Si02 onto the polymer film. The only technique which is able at present to measure the density distribution in such confined thin films with the necessary precision is neutron reflectometry. Fig.
example for P(S-b-MMA) films of different thicknesses. For increasing film thicknesses (inO’icated at the right side of the figure by t/Lo) where LO is the thickness of an unperturbed lamella) the shift of the third order reflection peak corresponds to an increase of the period. The occurrence of a double peak indicates two distinct lamellar thicknesses. Thin films may also prepared from solutions of polymer mixtures. In Fig. 10.7 one can see the results of neutron reflectometry from thin PS/PB films containing 44% and 12% of deuterated PS. In this case the interference fringes are due to segregation of PS to the substrate interface creating a sharp PS-PB interface. The interface is the sharper the lower the amount of PS is. 10.6 shows
an
U
-2 -3
P 2 40
-4
-6
........
......................
0.01
0
0.02
0.03
Neutron wavevector, k
Fig.
10.7. Neutron
tion of
a
refiectivity
volume fraction of d-PS
respectively (adapted
10.3
curves
mixture of deuterated was
from films
polystyrene
and
0.44 for the upper
from ref.
0.04
(V)
spincoated from a toluene solupolybutadiene (d-PS/PB). The
curve
and 0.12 for the lower curve,
[4])
Polymer Bilayer Systems
X-ray and neutron reflectometry have been extensively used to study polymerpolymer interfaces and interdiffusion. For this purpose double layer samples have been prepared. Usually, a first film is spincoated onto a silicon wafer. Using a highly selective solvent, the second film may be spincoated directly onto the first one. Alternatively, one may prepare the second film on a different substrate and then float off this film onto
floating film
now can
be
picked
up
by
a
clean water surface. This
the substrate coated with the first film.
polymer Studies
10
311
technique double layer samples up to some 10 x 10cm’ have the technique appears to be rather crude the width (or the roughness) of the polymer-polymer interface can be as low as Inm. In most cases neutron reflectometry is more favorable due to the much larger scattering length density (see Sects. 1.2, 1.3.1, 3.1.2, 5.3.2) contrast as compared to x-rays. Fig. 10.8 gives a calculated example for an incompatible
Based
on
this
prepared. Although
been
system of PS and-brominated PS (=PBrS). Due to the many electrons of the Br-atoms this system has already a quite measurable contrast for x-rays. But deuteration allows to improve this contrast for neutrons by almost an order of
magnitude. Consequently, the reflectivity curves (via their fringe spacing, Chap. 3) mainly reflect the thickness of the overall system (in the case of
see
x-rays)
or
the thickness of the deuterated
layer (in
the
of
case
neutrons).
40neutrons
iX
0- 30-
PS/PBrS
10
R
Siq
PBrS(H)
PS (D)
V
101
X-rys
1
20..........
10,
01
.................... ...............
neu
luu
Z[n.)
Fig.
10.8. Calculated
files shown
on
the left
reflectivity panel
0
0.2
ns
Q4 k
curves
(right panel)
Inm’)
for the refractive index pro-
polymers neuonly possible reflection technique. Deuteration of the molecules of one film does not strongly modify the system (unless the polymers are extremely long, where deuteration may lead to incompatibility). A system studied by several groups is polystyrene interdiffusing into polystyrene. The most important question in this context is how do polymers diffuse across an interface. The reptation model by de Gennes predicts significantly different interfacial profiles with respect to the interdiffusion of simple (small) molecules. As the polymer chains are supposed to cross the interface first via their ends, the interface should stay rather sharp at its center for the characteristic "reptation time". Only few molecules or parts of the molecules are able to cross initially. With progressing time more and more chain segments will be able to diffuse across the interface and the density profile eventually can be described by an error-function, typical for Fickian If
tron
one
is interested in the interdiffusion,. between identical
reflectometry
diffusion.
is the
312
G nter Reiter
R
1
X Inm
Fig.
"I
10.9. Measured neutron
reflectivity curves (full lines) for a bilayer system of protonated polystyrene on glass. Results are for a) the unannealed, b) 2 min at 120’ C, and c) 3900 min at 120’ C annealed sample. The broken lines represent the best fits using the refractive index profiles shown in Fig. 10. 10 for the polymer-polymer interface (adapted from ref. [5]). deuterated and
3(
15-
ic
IL-113
Fig.
10.10.
Refractive index
fit the data of
(adapted
Fig.
from ref.
profiles for the polymer-polymer interface used to reptation alone
10.9. The inset shows the contribution due to
[5]).
Figures 10.9-10.11 show typical results for the early stages of interdiffusion, where reptation is "visible" [5]. The system consists of a deuterated and a protonated film of about the same thickness. In curve a) (just after preparation) the fringe spacing (Chap. 3) is mainly determined by the deuterated
polymer
10
Studies
313
-2
10 R
error
function
fit
M.,
10
2 .10-3 modified fit 10
0.20
0.15
k
Fig.
10.11.
the modified
Fig.
Comparison of the best fits from profile (full line) as described in
an
0.25
[nm-11
error-function
the text for
a
(broken line)
section of
curve
and c
of
10.9
top. The protonated layer is only visible through the distortions of the interference fringes. As the interface between the two polymer layers gets smeared out (a consequence of interdiffusion) the fringe spacing gets smaller
layer
(the
on
number of fringes
doubles).
The
air/polymer
and the
polymer/substrate
annealing procedure and eventually Thus, the fringe spacing corresponds double the to the overall thickness of layer system. These reflectivity curves the following refractive index profile used have been fitted to a model which the polymer/polymer interface: for error-functions n(z) of two superposed the whole
interfaces stay sharp during will dominate the reflection of neutrons.
n
(z)
=
with erfc (z)
no +
=
I
An f (1 -
-
p) (2
erf (z), no
-
erfc (zlc,)) +
being
p(2
-
the refractive index of the first
,An the maximum difference in refractive index between the two and ut -
are
the widths of the
error
(10.1)
erfc (zlut) 1
layer, layers, cr,
functions and p described the relative
weights. (The indices c and t represent core and tail contribution). Such a profile is based on the possibility of restricted local movements between the entanglement points (responsible for the core part) and the pure reptation contribution which shows up in the tails of the profile (Figure 10.10 shows the profiles used to fit the curves of Fig. 10.9. The inset shows the "pure" reptation part). In, order to show how powerful neutron reflectometry is, we enlarged a section of curve (c) of Fig. 10.9. The broken and the full lines represent the best fit using a simple error-function profile and the profile given above, respectively. Although the modified profile is deviating only slightly from an error-function profile (see inset of Fig. 10.11) it gives a significantly better
G nter Reiter
314
[5] interdiffusion has been followed as a function of time and (in particular the profiles) have been successfully compared to the
fit. In reference
the results
reptation theory. At this
point,
I want to add
characteristic features of
a
word of caution. In order to be able to resolve
interfacial
profile the reflectivity curve has to appropriate k-range. In Fig. 10.12 simulated reflectivity curves for step (broken lines) and linear (full lines) interfacial profiles of different widths are compared. The system and the k-range chosen are the same as for the example of interdiffusion shown in Figs. 10.9-10.11. One can clearly see that fine details of a profile can only be determined if the k-range is sufficiently large, e.g. a linear profile of 4 nm width and a step profile of 2 nm width can hardly be distinguished if the reflectivity curve is limited to an
be measured in the
k
=
0.6nm-1.
10.4
Adsorbed Polymer Layers
problems mentioned above where the interfaces have been sharp, adsorbed polymer layers are usually diffuse. Nonetheless, neutron reflectometry can be rather sensitive and, especially for liquid systems, is a unique technique for determining characteristic features of such layers. Maximum contrast or an improved contrast can be achieved by contrast matching the environment of the adsorbed polymer layer to the bounding medium (air or the substrate). In many cases, one has also the possibility to invert the contrast (deuterated polymer layer in protonated environment and vice versa). Results of two complementary curves have to be described by a single model. This helps to reduce ambiguities in analysing the data. Due to the monotonic and featureless decay of the reflectivity such curves need significant theoretical support (predicting a model for the interfacial profiles) to enable a satisfying analysis. In Fig. 10. 13 a result from one of the first successful experiments in this context is shown. The full lines represent the model fits based on a powerlaw decay over a distance of 60nm (for further details, see ref. [6]). X-ray reflectometry has also been used to measure self-assembled surface micelles of end-functionalised AB diblock copolymers. For this purpose a polymer solution has been spread onto a water surface on a Langmuir trough. The entire film balance system was placed in a Plexiglas container with kapton windows for the x-ray beams. Figure 10.14 shows a schematic drawing of the setup and Fig. 10.15 gives some typical results (including least-squares fits based on the model shown in right part of Fig. 10.15) for polystyrene- alkylated polyvinylpyridine terminated by iodine (PSP4VP-C81-). The different curves are for different compression (i.e. different areas per molecule) of the surface layer. In contrast to the
rather
polymer Studies
10
315
...........
V
k
Fig. 50
[nm"’]
reflectivity curves for a double layer system, representing nm protonated polystyrene on glass. The interfacial profile on polymers is assumed to be a step function (dotted line) or a linear
10.12. Calculated
nm
deuterated
between the two
50
profile (full line). The three 10
V
nm/20
10.5
nm
sets compare widths of 2
for the step and the linear
nm/4
nm, 5
nm/10
nm, and
profile, respectively
Polymer Brushes
polymer physics which has been extensively investigated by refiectometry is the formation and the properties of polyif many mer brushes, either at a solid or a liquid interface. A brush is formed molecules The interface. arpolymers are anchored with one endgroup at an of number the other. with each Increasing riving first are not yet interacting molecules these between leads to overlap grafted molecules (per unit area) and eventually to a stretching of the chains if more than one chain per crosssection of an unperturbed polymer chain is grafted onto the substrate. Fig. 10.16 gives a schematic drawing of this process. The degree of stretching depends (for a given number of grafted molecules) on the quality of the solvent. In Fig. 10.17 one can see how this behavior was observed by neutron reflectometry. Increasing the temperature of a polystyrene
A
major
area
in
x-ray and neutron
G nter Reiter
316
r- -
M
-----
-
1*.
le-
1.0
% 41 0.02
0.01
3
A,
0.02 .
(A-)
Fig.
10.13. Experimental (crosses) and calculated (full lines) values of ’ AR (R(q) RFresnd) for polydimethylsiloxane (Mw=4200k) adsorbed at an air/deuterated toluene interface. q is the momentum transfer normal to the surface with total reflection stopping at q=O (adapted from ref. [6]) =
-
Fig.
10.14. Schematic
adsorbed at from ref.
a
drawing
for the system used to
liquid/air interface, allowing
block copolymers monolayer (adapted
measure
to compress the
[7])
brush in deuterated
cyclohexane increases the interaction between polymer swelling of the brush. It should be noted that in this example the neutron beam passed through the substrate (silicon single crystal, entered at one edge at an angle of about 90’) and then is reflected at the interfaces of the brush. The solvent on top of the brush is sufficiently thick (of the order of mm), deuterated and the solvent-glass interface is not necessarily parallel to avoid contributions from reflection from this interface. and solvent and thus leads to
Brush formation
can
also be followed
as
a
function of time allowed for the
grafting. For this purpose one can, for example, start from a thin film containing a mixture of deuterated non-functionalised and protonated functionalised molecules. As the brush forms the deuterated and the protonated molecules are progressively separated. This leads to a distinct two layer system which be
clearly seen in the reflectivity curves shown in Fig. 10.18. One starts containing homogeneously distributed deuterated and protonated molecules. As the protonated molecules are grafted one obtains a layer of mainly deuterated molecules with a thickness which is less than the initial can
with
a
film
10
polymer
Studies
317
to,
L-,23,:0,,A .7-3 2..4 I"
__
0 A
to,
jj.
.
.3",
:3.6
23 0 32.4 .356
A-109 A " A_’I A A 79
"’,
AA A-69
A," 2 :355:67A 7 A-48 A
’5 5 ’.. " f, . 55 .
.
’ -36.5 ’r’A 365A A-42
1-40.7 _4..7 ..7
to-,
A-37 A. A. -31
10-
0.0
0.1
0.2
%
0,3
[V] Distonce from top Joyer
Fig.
10.15.
Measured x-ray
reflectivity
curves
(symbols)
(A)
for PS-P4VP-C81- ad-
the water surface for different surface pressures. The lines represent best fits based on the two-layer model sho*wn on the right. The dashed line represents
sorbed the
on
from the pure water surface
(adapted
Drawing showing the transition from grafted polymers during grafting (adapted
from ref.
specular refiectivity
Fig.
10.16.
stretched
film thickness
on
from ref.
isolated to
[7])
overlapping and
[8])
a mainly protonated layer. Thus the fringe spacing protonated layer is not contributing significantly to the
top of
increases because the
reflected intensity. At the same time the critical wavevector for total reflection shifts to higher values due to an increase of the volume fraction of deuterated molecules in the top
possible to use a single sample to measure not affecting (e.g. degrading) the polymer. different samples are always slightly different due to their film thickness). Using a single sample further-
layer.
It is
the kinetics because neutrons This is
advantageous preparation etc. (e.g.
as
in
are
allows to compare refiectivity curves directly. All differences between the curves are necessarily due to the grafting process and not to differences of the samples. Thus, one can. directly see, without having to use a model more
G nter Reiter
318
0.0
-2.0
*C "C
0
31.5
&
53A
+
d_TJ d_TQI d-Toluenr, a ene
j
1.0 0.50
-
1.5
2A
21.4
3.11
---
---
,
I
1.50 0,40 1.25
j’
-4.0
D.20
-6.0
........
..
.
(T-B)/T
.P
!L
0.10
25
-8.0 0.0
0.;1
0 tQ3
0.0
q, (
Fig.
250.0
500,0
750.0
Dpth z (A)
Ot
1000.0
A:I)
10.17. Neutron
reflectivity from a polystyrene brush grafted onto a silicon cyclohexane for different temperatures. The lines using the brush profiles shown on the right side (adapted from
wafer and dissolved in deuterated
give
the best fits
Ref.
[9])
for the
fit, if significant changes quantify these changes.
have occured. The fit is
101
..........
only
necessary to
...
POWR
102
-
P$Zk-N
75MN1140T
b)
440rnjn/140*C
19OOnVnII40*C
d)
i0l
.........
t5
.
............
d 10-1
...
..
..................
C
10.2
El
..
10-3
...
..
..
...
.......
....
.
.....
...........
.......
......
b
....
V.
,A-rich
0.10
0.16
wavevector k
Fig.
10.18. Neutron
reflectivity
alised and non-functionalised
from
a
thin film
0.20
containing
(deuterated) polystyrene
0.26
nm-l] a
mixture of function-
molecules. In the
course
of
annealing the functionalised molecules graft onto the substrate. This leads to a separation of deuterated and protonated molecules as indicated by the schematic on
the left
10
10.6
Polymer-Metal
polymer
Studies
319
Interfaces
investigation of polymers is the low polymers. Usually the contrast is of the order of 10-6 for most polymers. The refractive index for x-rays depends mostly on the electron density. Thus it is quite obvious that the contrast can be improved by more than an order of magnitude if one deals with polymer-metal interfaces. Here I want to give an example where the increased contrast has-been used to investigate polymer interdiffusion between (protonated) polystyrene molecules of different lengths. As the molecular weight does not affect the refractive index the interface between the two layers of polymers was marked by evaporating a thin layer (about 5 nm) of gold. The surface tension of the polymer is much lower than the one of gold. Thus, gold only partially wets the polymer. Consequently, one obtains tiny droplets or particles of gold on top of the polymer film. The individual particles cannot be resolved by the is of the order of x-ray beam as its lateral coherence length (Sects. 2.3, 4.7.2) some microns. The reflectivity curve from such a thin layer of gold on top of a polymer film clearly represents two fringe periods (see Fig. 10.19). The well pronounced interference pattern originating from the gold layer allows one to obtain an average density (or equivalently a mean coverage, in the present example it is about 50%) and a density profile of the gold layer indicating the shape of the particles. A
major disadvantage
of x-rays for the
difference in’the refractive index between various
Z
Vi
V
3
4
771’= 77777M 71-77)777
0.2
0.1
0.6
0.0
1.0
1.2
14
kinim"I
klml)
10.19. Left panel: x-ray reflectivity curves from various steps of preparation four-layer gold-polystyrene (Mw=198k)-gold-polystyrene (Mw=660k) system as indicated in the inset. Right panel: Comparison of curves from the system shown in the inset after different annealing steps (T 120’C): curves a): 1=675sec, from ref. [10]) 2=658 000sec 1=245 curves (adapted 400sec, 2=6830sec, b):
Fig. of
a
=
G nter Reiter
320
Putting a second polymer layer on top of this system does NOT much change the reflectivity curve. It would be impossible to measure changes e.g. of the thicknesses of the polymer layers. Thus, a second gold layer has been evaporated onto this tri-layer system. Now the two gold layers mainly determine the reflectivity of the system and thus the distance between these two layers is clearly visible from the interference fringes below about 0.6 nm-’. This four-layer system has been used to investigate the "fast" interdiffusion of the shorter polymer into the layer of longer molecules. The imbalance in the fluxes led to a "swelling" of the top layer. This is clearly visible in a change of the spacing of the interference fringes (see Fig. 10.19). However, such a swelling only occured after an induction period which could be related to the reptation time of the shorter molecules.
Spreading
10.7 In most
cases
have rather
of
of x-ray and
large samples
Polymers especially
ble to collimate the x-ray beam. The measure much smaller areas. Such
a
reflectometry, it is necessary to is, however, also possiremaining intensity is still sufficient to
neutron
of at least several cm’. It
collimated system
(see Fig. 10.20)
has been used to
Fig. 10.20. Schematic view of (a) a microscopic droplet and (b) experiment studied by x-ray reflectometry (adapted from ref. [11])
measure
a
final
capillary rise
stages of spreading of polymer droplets. As the beam could be moved across the droplet this setup also allowed to detect the shape of the droplet and to compare it with theoretical predictions. In particular, the existence of a "pancake", i.e. a homogeneous, quite dense and flat film of molecular thickness
(-- 0.8nm)
could be detected. It has to be
nated
depends
area
on
the
angle
this has to be taken into account the
noted, however,
of incidence. For
by
a
that the illumi-
correct
interpretation
film thickness which may vary with between such a model and the measured data is a
angle. The agreement satisfying (see Fig. 10.21).
very
A similar setup allows also to measure a thin film climbing up a solid substrate in a capillary rise experiment. In both cases, the well collimated beam
10
polymer Studies
321
0A
0.6
"s
5
0
15
10 6
Fig.
10.21.
is 10nm, the
and the
2.0
is
30
(.,d)
Reflectivity curve recorded at the edge of a droplet. The mean thickness slope is 2nm/mm, and the beam size is 20 x 10OMm (adapted from ref.
possibility
variations with
a
to translate the
sample provide
lateral resolution of the order of
a means
mm.
to
height advantage
measure
The main
technique (in comparison with ellipsometry, which has a better lateral resolution, of the order of 50pm) is the sensitivity for density-variations in addition to the high vertical resolution which can be better than 0.1nM. For the case of spreading of polymers one thus can distinguish if the final stage is a dense and homogeneous film or an incomplete layer covering only partially of this
the substrate.
10.8
Dewetting
of
Polymers
as the opposite process to spreading. One homogeneous film and ends up with many droplets. In some cases these droplet may sit on top of a remaining thin layer (e.g. a monolayer of adsorbed or grafted polymers). Under certain conditions, x-ray reflectometry is extremely well suited for a) demonstrating that the film has
Dewetting
may be understood
starts from
a
smooth and
become unstable and holes have been created and
(--droplets)
has been reached but
a
b)
thin and compact
that the final stage on the
layer remains
substrate. In the first case,
one
takes
advantage
of the fact that the lateral coherence
of the incident beam allows to average over areas the fact that the film contains holes is reflected in a
length (Sects. 2.3, 4.7.2) of several
pM2 Thus, .
density profile schematically shown in Fig. 10.22. The decrease of the average density of the film is proportional to the fraction of the film which is now replaced by holes. The material removed from the holes is deposited in rims around the holes. These rims lead to tails of the density profile which have an effect similar to roughness. It should be noted however, that the film between the holes remains unperturbed as can be seen from the fringe spacing representing this thickness. Some typical examples for unstable films of endfunctionalised polystyrene are shown in Fig. 10.23. As dewetting proceeds
Gfinter Reiter
322
Z
rim
am
Wr or wwum
air
polymer film .........
or vamum
p-lyme""m
....
X
P.
0
Pe substrate
hole t. wb.trat.
air
or
air or --M -.m
p
cuum
ly..
I
4m
Z
Fig. 10.22. Schematic representation dewetting was deduced
Z
of how the
density profile
used in the
case
of
this
spacing disappears and, because a grafted layer is formed, a new larger spacing evolves giving the thickness of this layer. At the final stage where only droplets remain on a monolayer one only detects this monolayer. The droplets are not significantly contributing. Firstly, because they are occupying only a minor fraction of the surface (typically 10%, but this depends sensitively on the initial film thickness) and secondly, because they scatter the x-rays in directions off the specular direction. (Off-specular scattering can be used to
droplets)
detect the size and the distribution of the
’0
. =
4
k
Fig.
10.23.
30
40
-
nm
X-ray reflectivity
13k, 18k, 19.5k. Left
A"
0.,
0.5
Inm-’]
1:0 k
1:5
2’, 0
Inm")
from thin films of w-barium sulfonato
polystyrene of weight varies from bottom to top: Mw=2.8k, panel: BEFORE annealing, right panel: AFTER annealing for inset shows the model used for the fits, shown by the solid lines
thickness. The molecular
200h at 175’ C. The
I
4
(adaptedfrom ref. [12])
10
conclusion,
In
x-ray and neutron
reflectometry
polymer Studies are
323
extremely versatile,
powerful and in certain cases, unique techniques for the investigation of thin films, or interfacial problems in general, in polymer science. References 1.
G.
Reiter, Macromolecules, 27, 3046 (1994). Orts, J.H. van Zanten, W.-L. Wu, S.K. Satija, Phys. Rev. Lett., 71,
2. W.J.
867
(1993).
3.
4.
5. 6.
7. 8. 9.
Russell, Physica B, 221, 267 (1996). see also: T.P. Russell, Materials Sci. Rep. 5, 171 (1990). M. Geoghegan, R.A.L. Jones, A.S. Clough, J. Penfold, J. Polym. Sci.: B: Polym. Phys., 33, 1307 (1995). G. Reiter, U. Steiner, J. Phys. 11, 1, 659 (1991). X. Sun et al., Europhys. Lett., 6, 207 (1988). Z. Li et al., Langmuir, 11, 4785 (1995). R.A.L. Jones et al., Macromolecules, 25, 2352 (1992). A. Karim, S.K. Satija, J.F. Douglas, J.F. Ankner, L.J. Fetters, Phys. Rev. Lett., T.P.
73 3407 10.
G.
(1994).
Reiter, S. Hiittenbach, M. Foster, M. Stamm, Macromolecules, 24,
(1191).
Daillant, J.J. Benattar, L. Uger, Phys. Rev. A, 41, 1963 (1990). J. Daillant, Benattar, L. Bosio, L. Uger, Europhys. Lett., 6, 431 (1988). G. Henn, D.G. Buclinall, M. Stamm, P. Vanhoorne, R. J6r6me, Macromolecules,
11. J.
J.J. 12.
1179
29, 4305
(1996).
Main Notation Used in This Book
Direction normal to the surface
z
-T Y
Directions in the
plane
Used to describe
a
interface
of the surface
component parallel
to the
plane
Plane of incidence
XOz
Label of
layer. Numbering
(upper medium) Average location
layers goes from 0 layer. s is the substrate 1, j interface
of
to N the last
of the
j
Zi zj (X, Y)
Fluctuations of the interface location around
k
Wave vector
-
ki, kr, ktr, k., Incident, reflected, transmitted ki,,
z
j
component of the incident
and scattered wavectors
wavevector in the
jth layer
unambiguous
k ,j
when
q
Wave-vector transfer
q
Modulus of the wave-vector transfer
Components of the wave-vector Scattering direction
q,,,qll,q, U
Reflection and transmission coefficients in
r, t
R,
Zj
Intensity
T
Reflection coefficient in
rj-l,j
medium
amplitude
reflection and transmission coefficients
j
-
amplitude
1 to medium
when-
passing
from
j
amplitude
when
passing from
tj-l,j
Transmission coefficient in
E
I to medium j medium j Electric field
6in) 6sc
Polarisation vectors of the incident and scattered fields
B
Magnetic field Current density Electric polarisation Vector potential Poynting’s vector Amplitude of the upwards and downwards propagating electric fields in layer j Aj ekjn ;,jz
-
i P
A
S
Aj’ U ( ki,,
j,
z)
M
T ransfer matrix
Pn
n-point probability distribution 2 r.m.s. roughness. a (Z2)
9
=
Notation
Czz (X 1
X2) Y1
i
Y2) Height-height
326
correlation function
WX1, Yl)Z(X2) Y2)) 2C,, (x 1, X2) Y1 Y2)
Also denoted
g(r)
20-2
G
Green function
-
Green tensor
e’ (wt-k.r)
are
waves
i
(electromagnetic case)
used except in
Chap.
5 devoted to neutron reflectiv-
ity (see Sect. 1.2.1 for details related
to the conventions used in this
and Sect. 5.1 for the notation used in
Chap. 5).
Table 1.
Typical length scales for
x-ray
reflectivity experiments Value
Definition
A
I
Wavelength A Scattering length
book,
b
r,
=
2.81810-1,5
m
for I electron Extinction
length
A
Le
A:,/JA length opening Detector slit opening normal to the plane of incidence (y) h, Detector slit opening h in the plane of incidence L Sample-to-detector distance Transverse coherence length A/’AO’ normal to the plane of incidence (y) with A9, h,IL the fixed by detector) (when Transverse coherence length (0,A 0) with in the plane of incidence h ,IL projected on the surface (x) (when fixed by the detector) Longitudinal
I Mm
27rln-11
coherence
Mm
O.1mm
Incidence slit
10
0.1
mm
-
Imm
IM
10
nm
=
Illuminated
(length
x
100 Pm
for 0
10 mrad
(0.1 mm/0) X (I 10 mm)
area
width)
Absorption length
=
-
M
=
\/47ro
0.1
for
mm
-
)3 =10-7
I
mm
_
10-8
Index
Absorption, -
26
neutron, 166
Compton effect, 26 wavelentgth, 34 Conformal interfaces, 138 Correlated roughness, see replicated roughness Correlation function, 62, 63 Gaussian, 66 self-affine surface with cutoff, 67 self-affine surface, fractal surface, 66 Correlation length lateral, 251 vertical, 254 Covariance, 156 Covariance function, 158, 252, 253, 265 Critical angle, 88 neutron reflectometry, 1169, 170 Croce-N6vot factors, 118
-
photoelectric, 38 Absorption edge, 39 Amphiphilic molecules, 297 Angle of incidence, 88 Anomalous scattering, 30 Atomic force microscopy, 223, 226, Atomic form factor, 29 Atomic scattering factor, 29
-
-
228
Beam deflector
-
-
-
-
studying liquid surfaces, 287 Bending rigidity, 286, 298 Born approximation, 108, 157, 222 -
-
for
distorted wave, see Distorted Born approximation
wave
first, 57, 80, 130 scattering cross-section, 130 planar interface, 83 scattering cross-section, 80 self-affine surface, 83 single rough interface, 81 single rough surface, 131 Born development, 56 Boundary conditions, 100 wave function, 171, 176 Bragg-like peaks, 257 Brewster angle, 132 Broadening peak, 248 Broken bond approximation, 204
-
--
-
-
-
-
-
Cross-section
absorption, 10 total, 10 Current density,
-
-
7
-
-
-
-
Debye-Waller factor, 118, 184 Density fluctuations liquid surface, 142 Density inhomogeneities, 67, 141 in a multilayer, 141 Density profile, 222 Deuteration, 167, 188 Diffuse scattering, 250
-
-
-
Capillary
waves,
210,
282
Central limit theorem, 65 Characteristic function, 63
two-dimensional, 251 potential, 202 Classical electron radius, 329 Coherence domains, 74 detector angular resolution, 72 X-ray experiment, 73 Young’s holes experiment, 68 Coherent and incoherent scattering, 79,
-
Chemical
-
(DWBA), 121, 132,458, 222, 251, 270,
-
-
-
-
-
82, 84, 132, 136
resonant, 265
Dipolar approximation, 33 Dispersion correction, 30 Dispersion relations, 46 Distorted wave Born Approximation single rough surface, 133 Distorted wave Born approximation
-
-
278
film, 137 dipole source, second order, 118, 122 simpler, 160, 254 case
of
a
fictitious
123
Index
328
stratified media, 134 Dynamic exponent, 214 Dynamical scattering effects, 254, scattering in a thin film, 138 Dynamical theory, 108
-
-
257
-
Gibbs-Thomson relation, 204, 213, 215 Grating truncation rods, 268
Gratings, 266 Grazing incidence diffraction, 264 Green function, 11, 123-128 determination using the reciprocity theorem, 125
Edwards-Wilkinson
model, 215 density model, 222 Elastic scattering, 26 Electric dipole field, 126 Energy conservation, 58 Energy density, 7 Equilibrium adatom density, 204 Ergodicity, 63, 74 Effective
-
-
in vacuum, 126
-
multilayer,
-
Evanescent wave, 277 Evaporation rate of a Ewald
construction, Ewald sphere, 275 Experimental setup,
crystal,
Green tensor
-
see
also Green function
Growth exponent, 214 Growth rate of
a
crystal,
211
Height distribution, 222 Height-height correlation function, 206,
210
275
222,
236
intrinsic, 246 liquid interfaces, 283 logarithmic divergence, 281, 285 of two interfaces, 236 Helmholtz equation, 6, 89, 169
238
-
-
length, 22 theorem, 26
Facets,
127
125
Extinction -
two-points probability distribution, 66
-
-
201
Hurst exponent,
222, 251,
285
Far-field
approximation, 71 Faraday rotation, 56 Flow, 7 Fluorescence, 26, 38 yield, 39 Flux, 7, 238 density, 7 Fractal dimension, 222, 251
Incoherent Inelastic
Integral equation (for the Intensity, 7 Interdiffusion, 184 polymers, 188 Interface roughness,
-
-
equations, 91 reflectivity, 92
-
attenuated, formulas, 171
approximation, approximation
Kinematical
-
attenuated, for
gratings,
Fresnel
zone
Kinetic
-
241, 273 272
radius, 19,
23
Gaussian curvature, 286 Gaussian
roughness,
251
Gaussian variates -
-
see
theory, 108, 157, roughening, 212 and scaling, 213 Kramers Kronig relations, 46
242
Fresnel coefficients -
characteristic functions, 66 height probability distribution,
field,
183
Kinematical
coefficients
--
electric
-
Fresnel
-
166
123
-
-
scattering (neutron), scattering, 26
65
Langevin. equation, 213 Langmuir films, 297 Law of reflection, 276 Layer form factor, 157 Layer size function, 234 Layer structure factor, 157 Liquid metals, 296
Born
269
Index
Liquid surfaces and. interfaces capillary waves, 282 height-height correlation function,
-
-
broadening,
reflectivity,
288
roughness spectrum, 283 Liquid-liquid interfaces, 300
-
Lorentz electron radius
Bragg-like, 257 depth, 98 Perturbed potential, 156, Polarisability, 234
-
,
-
-
-
193
-
moment, 189
-
profile, 191, 192 Magnetic circular dichroism, 56 Magnetic interaction (neutron), Magnetic scattering (x-rays)
-
-
-
-
non
resonant,
-
167
in surface
copolymer, 188, 307-309 films, 309 contrast matching, 314 dewetting of polymers, 321-323 interdiflusion, 188, 311 confined
internal structure, 307
polymer bilayer, 310 polymer brushes, 315-318 polymer mixtures, 310 polystyrene film, 188, 306 reptation model, 311 spreading of polymers, 320-321 Porous samples, 249 Power spectral density, see Spectral density
-
-
-
-
Power spectrum, 64, 66
-
Poynting vector, 8, 71 Probability distribution height, 61 Projected surface free energy, 201, 202 Propagation equation (electric field),
-
-
factors, 184,
adsorbed polymer layers, 314 block
-
176
Mean curvature, 286
N6vot-Croce
140
-
method, 99, 221 polarised neutron reflectometry,
--
scattering,
film
-
resonant, 49
Microemulsions, 301 Mullins model, 215 Multilayer DWBA, 134 Green function, 127 periodic, 244, 277 surface scattrering Born approximation,
-
-
49
Matrix -
270
Polarisation. effects
28
Polymer
Magnetic hysteresis loop, -
248
Peaks
Penetration
283 -
Peak
329
153
-
221
Neutron
absorption, 166 magnetic interaction, 167 neutron-matter interaction, optical index, 168, 170 Non-coplanar reflectivity, 262 Non-specular reflection, 250 Nucleated crystal growth, 212
12?
-
Quadrupolar terms,
-
Raman Raman -
effect, 26 scattering,
deposition, 213 Rayleigh scattering, 30 Rayleigh theory, 116 Reciprocal lattice, 275 Reciprocal space, 218 Reciprocal space construction
Optical index neutron reflectometry, 168, 170 Optical potential, see scattering potential Optical theorem, 17, 58
-
see
Ewald construction 275
Reciprocity
probability distribution, 236, Parratt formalism, 221 Patterson function, III
36
resonant, 36
Random
-
Pair
33
165
251
-
theorem
determination of Green functions, 125
Reflection
Index
330
non-specular, 250 specular, 241 Reflection coefficient, 101 Reflectivity, 88 effects of surface scattering, -
-
-
-
-
-
-
-
-
148
liquid surfaces, 288 magnetic systems, 172 non-coplanar, 262 off-specular, see Scattering, off-specular on a magnetic system, 179 specular, 94, 220, 221
57
scattering and the roughening transition, 209 Scattering by a rough surface, Surface scattering Scattering Cross-Section, 112 Scattering cross-section Born approximation, 81
Reflectometer -
-
see
-
monochromatic, 187 time of flight, 187
coherent, 156 differential, 10, 70, 122, 155 first Born approximation, 130 incoherent, 156 total, 10 Scattering length, 10 neutron, 166, 170 Scattering potential, 156 Schr6dinger equation, 123 Self affine roughness, 223 relation to liquid surfaces, 285 Self-assembled amphiphilic films, Setup experimental, 238 Simple liquids surface, 295 Singular surface, 201 Snell’s law, 277 Solid surfaces, 221 Speckle, 74, 75 Spectral density, 223 Spectrometers neutron reflectometry, 186 Specular reflection, 241 Step density, 201, 205
-
-
Refraction
-
and
polarisation, 133 Refractive index, 70 Replicated roughness, 253 -
Scattering by density inhomogeneities, 141 d ffu..se, see Scattering, off-specular off-spec-ular, 220, 222 single scattering, multiple scattering,
-
-
Resolution
-
-
angular,
185
Resolution function, 144 -
wave
vector, 146
Resonant diffuse
scattering, 254, 257,
265
Resonant
scattering, 138 Root mean square height (RMS height), 62,
63
Rough and smooth surfaces, 208 Roughening temperature, 205, 206 Roughening transition, 201, 205, 206, 208
Roughness, 114, 221 Gaussian, 241 intrinsic, 236 replicated, 236, 247, 254 Roughness exponent, 214 Roughness spectrum and scattered intensity, 148 liquid interfaces, 283 -
-
-
-
-
-
-
-
-
-
-
Satellite
-
intensity, 228 Scanning tunneling microscopy (STM),
-
223
Scans -
experimental, 239 field, 70 exact integral equation, 71,
free energy, 200 line tension, 200, 205
stiffness, 204 Stepped surfaces, 248, 259, 266 Stratified media, 67, 221 DWBA, 134 Stratified media see’ also Multilayer Supermirror (neutron), 191 Supersaturation, 211
-
-
Scattered -
286
128
127
Index
Surface
-
of
a
rough surface,
207
Thin film
misoriented, 226 undulations, 228 Surface diffusion, 215
-
surface
scattering, 137 scattering, 28 Transfer matrices, 101, 176
-
-
Thomson
Surface energy, 282 Surface free energy, 199
Transition operator, 155
scattering, 205, 208 approximate expression of the intensity as a function of the roughness
Surface
Transmission
coefficient,
102
Transmission electron microscopy, 223, 226
spectrum, 148
approximation, 80, 131 DWBA, 133 effects on reflectivity, 148 geometry, 69 in a thin film, 137 polarisation. effects, 140 scattered intensity, 144 Surface stiffness, 202 Surface tension, 199, 282, 297 Surface width, 213 Surfactant monolayers, 297 Susceptibility, see polarizability
-
Born
-
Umweganregung, 259, 278 Unperturbed potential, 156,
270
-
-
-
-
-
Van der Waals
theory for liquid interfaces, Vapour pressure, 210
-
281
-
Templeton anisotropic scattering, 49 Terraced surfaces, see stepped surfaces Thermal fluctuations
331
Water -
surface, 295 lines, 39
White
Wiener-Kintchine theorem, 64 Yoneda peak, 134, 225, 253, 257 Yoneda wings, see Yoneda peak Zeeman
interaction,
167
