IsGISAXS - Version 2.4

IsGISAXS :
a tool for Grazing Incidence Small Angle X-ray Scattering analysis for nanostructures
Version 2.4



Rémi Lazzari


CEA-Grenoble
Département de Recherches Fondamentales sur la Matière Condensée
Service des Matériaux et Microstructures
Interfaces et Rayonnement Synchrotron
17 Rue des Martyrs
38054 Grenoble, France
Tel : (33)-4-38-78-57-61
Fax : (33)-4-38-78-51-38
lazzari@drfmc.ceng.cea.fr

European Synchrotron Radiation Facility
BP 220, 38043 Grenoble, France
Collaborating Research Group
Beamline BM32-IF









Contents

1  Formalism for the scattered intensity in GISAXS
    1.1  The scattering geometry
    1.2  The scattered intensity : analysis in terms of form factor and interference function
    1.3  The form factor
    1.4  Reflection-refraction effects and the Distorted Wave Born Approximation
        1.4.1  Islands sitting on a substrate or on over-layer
        1.4.2  Buried particles in a substrate
        1.4.3  Particles encapsulated in a layer
    1.5  The interference function
    1.6  The X-ray reflectivity for a particle layer
        1.6.1  Islands on a substrate
        1.6.2  Buried particles
2  How to use the IsGISAXS software ?
    2.1  The IsGISAXS software environment and outputs
    2.2  Description of the input file *.inp and the underlying parameters
        2.2.1  The file contents
        2.2.2  Some useful remarks
    2.3  The morphology file *.mor
    2.4  The treatment of experimental data *.dat and the fitting file *.fit
    2.5  The outputs files *.out,*.pro,*.ima,*.ki2,*.cor,*.dwba
3  IsGISAXS : Typical examples
    3.1  The form factor
        3.1.1  Distorted Wave Born Approximation and the refraction effect
        3.1.2  The island facetting
        3.1.3  The island sizes distribution
    3.2  The interference function
        3.2.1  Non regular lattice and the pair correlation function
        3.2.2  Regular lattice
4  Future developments-improvements








Introduction

The techniques of elastic scattering of radiation, in particular X-rays or neutrons, are widely used in the field of condensed matter as they provide valuable tools to probe the order properties of matter. The diffraction range is reached when the incident radiation wavelength is closed to the interatomic distances in a crystal and when the scattering angles are wide. By a careful analysis of the diffracted intensities, one can access to the intimate atomic structure i.e. the positions of nuclei and the spread of the electronic cloud around atoms. Usually, the range of scattering involves techniques which allow to get statistical information at scales that are greater than the interatomic distances. With X-rays or neutrons with a wavelength of a few angstroms, this domain is restricted in between the first Bragg peak which overlaps with the direct beam and the diffraction peaks (see Fig. 1). Usually, this domain of small momentum transfer is reached in the small angle range. The scattering at small angles with X-rays [8,6] or neutrons [19,9] is a widespread technique in particular for getting structural information in the range 1-100 nm. The typical probed sizes imply that the atomic nature of the wave-matter interaction can be ignored. The scattering comes essentially from strong variations of the mean electronic density for X-rays or the mean scattering lengths for neutrons as an homogeneous medium does not scatter. However, one has to keep in mind that scattering and diffraction are two intimately linked phenomena as they are answerable to the same wave-matter interaction.

figures/scatt.gif
Figure 0.1: The scattering and diffraction ranges versus the wavevector transfer.

Up to fifteen years ago, the scattering techniques were limited to three dimensional samples as the strong penetration depth of the radiations and the small signal to noise ratio hampered the surface sensitivity. Quite recently, thanks to the increasingly use of synchrotron radiation, these techniques [1,4] were extended to surface geometry using the phenomenon of total reflection of X-rays in the grazing incidence range. The diffraction which is now routinely used in surface science allows to get accurate information on reconstruction of surfaces, surface relaxation and on the atomic position [32,5] thanks to the precise measurements of crystal truncation rods. In a parallel way, the field of semiconductors and thin films growth brought a need of knowledge about layer morphology and sizes of quantum dots, supported islands or buried particles which has pushed the development of Grazing Incidence Small Angle X-Ray Scattering. If the use of reflectivity is widespread in this field, this technique provides only information about the dependence of the electronic density perpendicular to the surface. On the contrary with off-specular measurements, valuable information can be extracted from the scattering curves like the roughness of a surface [33,11,10,3], the lateral correlations, sizes and shapes of semiconductors dots [22,23,16,34,38], of metallic islands [18,31,29] or of the self organized dots superlattices [17,35,13] or wires [12]. Such information are of prime interest in understanding the link between morphology and physical or chemical properties like light emission in quantum dots or catalytic properties of clusters, or more fundamentally in understanding the phenomenon of layer growth. Even though, the near-field microscopies can give some answers to these questions for surfaces, X-rays present several advantages: (i) they give an averaged statistical information over the whole sample surface (ii) they can be applied in various environments ranging from ultra-high vacuum to gas atmospheres and in situ and in quasi real time when kinetics phenomena are involved (iii) using the variable probed depth as function of the incidence angle, X-rays offer the opportunity to characterize from surface roughness to buried particles. By combining the advantages of synchrotron radiation and two-dimensional detectors with an in situ sample preparation, the full potentiality of such a method can be obtained. Thus, quite recently, some experiments [29] used the GISAXS technique to characterize in situ and in ultra-high vacuum the growth process of metal/oxide interfaces and of self organized cobalt clusters on the herringbone reconstruction Au(111) surface. The quality of the data with a very low background opens the door to a real quantitative analysis, even in the very thin film range. Up to now, the GISAXS data analysis was often performed in a crude way forgetting either the reflection-refraction effects in the simple Born approximation or the interplay between interference function and form factor or the particle size distributions. For instance, most of the time for particles, the interparticle spacing is directly extracted from the "Bragg" law, method which could induce up to 30% of error. Moreover, concerning particle size, the coupling between the interference function and form factor greatly increases the complexity of the analysis and prevents the use of classical Guinier or Porod approaches. Thus only direct modelling of the data is appropriate. In fact, the needed theoretical background is simply derived from classical small angle scattering [8,6]. A peculiar attention has to be payed for the refraction of the beam at the surface. The theory in the Distorted Wave Born Approximation was recently derived for rough surfaces [33,11,10,3,4], buried particles [28] and supported islands [27]. The use of such a theory is mandatory, as it will be shown herein, for describing correctly the influence of the substrate in the scattering phenomenon. However a complete program for easily analyzing data and simulating off-specular scattering is still lacking in this rapidly growing field. The aim of this instructions is to give the main theoretical elements enclosed in the program IsGISAXS  1. In a first theoretical part, the scattering cross section is reminded and decomposed in terms of form factor of the island and interference function. In order to have tractable expressions, two approximations are described for the size-position coupling: the Decoupling Approximation and the Local Monodisperse Approximation. The implemented interference functions cover the problem of uncorrelated islands characterized by their pair-correlation function, that of paracrystal with loss of long range order and that of regular or defective lattice. In a second part, the capabilities of the software are illustrated with various types of examples which originally motivated the elaboration of such a program [29].








Chapter 1
Formalism for the scattered intensity in GISAXS

The aim of this chapter is to give the necessary theoretical background for deriving the cross section in the case of scattering by a plane of nanoparticles and excited by an electromagnetic wave at grazing incidence.

1.1  The scattering geometry

In a grazing incidence experiment (see Fig. 1.1), a monochromatic beam of wavevector ki in the X-ray range (wavelength l - wave number k0=[(2p)/(l)]) is sent on a surface with an incident angle with respect to the surface ai in the range of a few tenth of degrees. Possibly, the in-plane direction of the incident beam 2qi is different from zero. The reference cartesian frame with its origin on the surface is defined by its z-axis pointing upwards, its x-axis perpendicular to the detector plane and its y-axis along it. The light is scattered along kf by any type of roughness on the surface in the direction (2qf,af). Because of energy conservation, the scattering wave vector q is defined by:
q =  2p
l
 æ
ç
ç
ç
è
cos(af)cos(2qf)-cos(ai)cos(2qi)
cos(af)sin(2qf)-cos(ai)sin(2qi)
sin(af)+sin(ai)
 ö
÷
÷
÷
ø 		.
(1.1)

figures/geometry.gif
Figure 1.1: Sketch of the grazing incidence geometry: an incident wave of wavevector ki is scattered in the direction kf.

The scattering intensity is recorded on a plane ensuring that the angles are in the few degrees range and thus enabling the study of lateral sizes of a few nanometers. The detector can be punctual (0D), linear (1D) or even bidimensional (2D). The direct beam is often suppressed by a beam stop to avoid the detector saturation as several orders of magnitude in intensity separate the diffuse scattering from the reflected beam.

1.2  The scattered intensity : analysis in terms of form factor and interference function

The goal of this section is to compute the scattering cross section defined by:
 dS
dW
 (q) =  N
N I0 DW
 ,
(1.2)
with N the number of photons scattered per second into the solid angle DW around (2qf,af), I0 the flux of incident photons and N the total number of scatterers i.e. particles.
On a perfectly flat surface, all the intensity is concentrated in the reflected beam. In fact, the off-specular scattering appears when any type of surface roughness or scattering density is present on the surface. In the present case, the roughness is restricted to small islands on a surface or to a plane of clusters embedded in a host matrix. Each particle is characterized by its position on the substrate Ri|| and its shape function Si(r) equal to one inside the object and zero outside. The scattering density (electronic density) is given by:
r(r) = r0
å
i 
 Si(r) Äd(r-Ri||),
(1.3)
where Ä is the notation for the convolution product and r0 is the mean electronic density. This writing implicitly implies that the exact distribution of electrons around the nuclei is of no special interest as the scattering angles are in the small angle regime.

In the framework of the kinematic approximation, the scattered cross section is proportional to the modulus square of the Fourier transform of the electronic density. The polarization effect for X-rays leads to a term in:
P = ì
ï
ï
í
ï
ï
î
1
s-polarization
cos2y
p-polarization
 1
2
 (1+cos2y)
unpolarized
(1.4)
with cosy = ki.kf/k02. However, it can be dropped out safely as the scattering angles are small. Thus, by normalizing to the mean electronic density and by the classical Thomson scattering cross section re2, the scattering cross section ds/dW(q) can be written as:
 ds
dW
 (q)
=
 1
re2r02
  dS
dW
 (q) =  1
N
 ê
ê
å
i 
 Fi(q)  exp(-iq·Ri||) ê
ê 		2
 
=
 1
N
å
i 
å
j 
 Fi(q) Fj,*(q)exp[-iq·(Ri||-Rj||)].
(1.5)
In the simple Born approximation (BA), Fi is the Fourier transform of the shape function:
Fi = ó
õ

Si 
 exp( -iq ·r)  d3r.
(1.6)
If the reflection-refraction effects have to be accounted for, Fi has to be computed in the Distorted Wave Born Approximation (DWBA) and has a more complex expression (see Sect. 1.4). By isolating the self term, one has:
N  ds
dW
 (q) =
å
i 
 |Fi(q)|2 +
å
i,j ¹ i 
 Fi(q) Fj *(q)exp[-iq·(Ri||-Ri||)].
(1.7)
If only the statistical quantities [40] are known for the considered system (i.e. the size distribution and the position disorder), one has can write the previous expression Eq. (1.7) with a continuous integral:
 ds
dW
 (q) =
å
a 
 pa |Fa(q)|2 +  rS
S
å
a, b 
 papb   Fa(q) F*b(q)
(1.8)
ó
õ 		ó
õ 		d2Ria d2Rjb   Gab(Ria||,Rjb||) exp[-iq·(Ria||-Rjb||) ].
rS is the number of particles per unit of surface. S is the surface sampled coherently by the beam. The particles have been sorted out in classes a of sizes and shapes of occurrence probability pa. The probability per unit of surface to find a particle of class a in Ri|| knowing that there is a particle of class b in Rj|| is called rS2 papb   Gab(Ria||,Rjb||). Gab(Ria||,Rjb||) is known as the partial pair correlation function. The condition i ¹ j of Eq. (1.7) is of course implicitly included in this function as a hard core type effect. In practice, the previous equation is unusable as it implies the knowledge of all the Ga,b. The equation Eq. (1.7) have been directly implemented in the program IsGISAXS for simulating GISAXS results from known morphologies, like from transmission electron microscopy picture. To go further on, when the morphology is not exactly known, or when in particular data fitting is involved, some hypothesis need to be made.

Decoupling Appproximation
A current hypothesis called Decoupling Approximation (DA) is to suppose that the nature of the scatterers and their position are not correlated in such a way that the partial pair correlation functions depend only on the relative positions of the scatterers and not on the class type:
Gab(Ria||,Rjb||) @ g(Rij||).
(1.9)
This genuine random substitutional mixture leads to:
 ds
dW
 (q)
@
< |Fa(q)|2 > a
(1.10)
+
| < F(q) > a |2 rS ó
õ 		d2Rij   g(Rij||)exp[-iq·Rij|| ],
where < ¼ > a is the mean value over the size-shape distribution. Finally, the cross section appears as the sum of two terms, a coherent one and a diffuse one:
 ds
dW
 (q) @ Id(q) + | < F(q) > a|2 ×S(q)
(1.11)
Id(q) = < |F(q)|2 > a - | < F(q) > a|2
(1.12)
S(q) = 1 + rS ó
õ 		d2Rij   g(Rij||) exp[-iq·Rij|| ].
(1.13)
Id(q) is the diffuse part of the scattering which is linked to the disorder in the scatterers nature (size, shape). S(q) is the total interference function; it describes the statistical distribution of the objects on the surface and thus their lateral correlations. It is the Fourier transform of the island position autocorrelation function:
z(r) =  1
N
å
i,j 
 d(r-ri)Äd(r-rj) = d(r) +
å
i ¹ j 
 d(r-ri + rj).
(1.14)
S(q) will be detailed in the following.

The Local Monodisperse Approximation
To partially account for the coupling between the position and the nature of the particles, the Local Monodisperse Approximation (LMA) is often used in the literature. It consists in replacing the scattering power of each particle by its mean value on the size distribution:
 ds
dW
 (q) @ < |F(q)|2 > a ×S(q).
(1.15)
This expression is asymptotically equal for large q to the Decoupling Approximation as:
< | F(q)|2 > a q® ¥
@
 
 | < F(q) > a |2.
(1.16)
This expression is obtained from Eq. (1.7) by supposing that all the surrounding particles for each origin particle are, approximatively, of the same size in such a way that the size-shape of the islands vary slowly across the sample. In some way, in the LMA, the intensity originates from an incoherent sums of the scattering intensities from monodisperse subsystems weighted by the size-shape probabilities. Such an approximation is expected to reproduce better the data than Eq. (1.11) as it includes in some way the coupling between the position and the nature of the scatterers.
To conclude, this size-position coupling is, by far, the most delicate point in the quantitative analysis of experimental data as it varies from one situation to an other leading to strong variations of the scattered intensities in particular close to the specular beam (see Chap. 3).

If the incident beam has a finite divergence (distribution on (2qi,ai)) and wavelength resolution i.e. a finite coherence length, for each scattering directions (2qf,af), one has to perform an incoherent sum of the intensity scattered by each plane wave with a weight p(l) p(2qi) p(ai):
d
^
s
 
dW
 (q) = ó
õ

l 
 ó
õ

qi 
 ó
õ

ai 
d
~
s
 
dW
 (q,l,2qi,ai) p(l) p(2qi) p(ai)  dldqi dai .
(1.17)

1.3  The form factor

The form factor Eq. (1.6) is only the Fourier transform of the shape of the particle. In some particular cases with special symmetries, the 3D-integral can be reduced to 1D-integral or even expressed analytically. The shape depicted in Fig. 1.2-1.3-1.4 are supported in the IsGISAXS program.

figures/form.gif
Figure 1.2: Supported geometries for island shapes in IsGISAXS (Left: side view - Right: top view).

figures/form1.gif
Figure 1.3: Supported geometries for island shapes in IsGISAXS (Left: side view - Right: top view).

figures/form2.gif
Figure 1.4: Supported geometries for island shapes in IsGISAXS .

In the cartesian frame attached to each island with its origin at the center of the bottom of the island, its x-axis aligned along one side of the island,and its z-axis pointing upwards, the mathematical expressions for the form factor are the following:

with sinc(x)=sin(x)/x the cardinal sine, J1(x) the Bessel function of first order.
Except for the facetted sphere or simple shapes for which the form factor can be expressed analytically, the 1D-integration is performed, in IsGISAXS , by Gauss-Konrod algorithm [25] with an autoadaptative sampling of the integration range in order to reach the user desired accuracy. When the frame linked to the island is not aligned with the x- axis of the impinging beam, the rotation matrix has to be applied to the scattering vector in order to apply the previous formulae:
R
 
 (z) q = æ
ç
ç
ç
è
cos(z)
-sin(z)
0
sin(z)
cos(z)
0
0
0
1
ö
÷
÷
÷
ø 		æ
ç
ç
ç
è
qx
qy
qz
ö
÷
÷
÷
ø 		.
(1.32)
The phase factor in qz of Eqs. (1.18-1.29) seems to be useless but it finds all its importance in the averaging process over the size distribution which implies the use of a common origin of the frame. Indeed, to perform the averages of Eqs. (1.11-1.12), one has to define the distribution probabilities of each parameter which characterizes the island : lateral size R, height H, orientation z and to compute the integrals:
< |F(q)|2 > = ó
õ

z 
 ó
õ

R 
 ó
õ

H 
 p(z) p(R) p(H) |F(z,R,H,q)|2  dzdR dH
| < F(q) > |2 = ê
ê 		ó
õ

z 
 ó
õ

R 
 ó
õ

H 
 p(z) p(R) p(H) F(z,R,H,q)  dzdR dH ê
ê 		2
 
 .
(1.33)

1.4  Reflection-refraction effects and the Distorted Wave Born Approximation

Because of the presence of the substrate and of the closeness of ai from the critical angle of total external reflection ac, the Born Approximation has to be modified in order to account for reflection-refraction effects at the surface of the substrate. The appropriate theory called Distorted Wave Born Approximation is nothing else than the application of first order perturbation [21,33,28] induced by the island roughness at the substrate surface or contrast variation to the correct unperturbed wave i.e. the trio of incident-reflected-refracted waves. The following geometries Fig. 1.5 are encompassed in the IsGISAXS program: a) islands supported on a substrate b) islands on a continuous layer on a substrate c) buried particles in a substrate d) particles buried in an over-layer. Notice that the particles are always gathered in one plane which defines the origin for the form factors.

figures/layer_geome.gif
Figure 1.5: The particle layer geometry developed in IsGISAXS . The geometry D is planned for future development.

1.4.1  Islands sitting on a substrate or on over-layer

Historically, the IsGISAXS program was developed to handle mainly this geometry. A physical picture of the full calculation [27] for the scattering cross section in the DWBA for an island is depicted in Fig. 1.6. ki and kf are respectively the incident and outgoing wavevectors.

figures/dwba.gif
Figure 1.6: The four terms involved in the scattering by a supported island. The fist term corresponds to the simple Born approximation.

The four terms involved in the scattering process are associated to different scattering events which involve or not a reflection of either the incident beam or the final beam collected on the detector. These waves interfere coherently giving rise to the following effective form factor where comes into play the classical form factor but computed with specific momentum transfers:
F(q||,kiz,kfz) = F(q||,kfz-kiz) + RF(ai)F(q||,kfz+kiz) + RF(af)F(q||,-kfz-kiz) + RF(ai)RF(af) F(q||,-kfz+kiz).
(1.34)
Thus in the DWBA, the form factor does not simply depend on the wavevector transfer q but on (q||,kiz,kfz). Each term is weighted by the corresponding reflection coefficient, either in incidence RF(ai) or in reflection RF(af) which are defined by the Fresnel formulae:
RF =
kz-
~
k
 
z 
kz+
~
k
 
z 
     with    
~
k
 
z 
 = -
Ö
 
ns2 k02 - |k|||2
 
 .
(1.35)
ns = 1 - ds - ibs is the complex refractive index of the substrate. Possibly, the Fresnel reflectivity of the substrate Eq. (1.35) can be reduced, in a classical way [1], by an uncorrelated roughness of mean standard deviation s = Ö{ < h2 > }:
RS = RF exp æ
è 		-2 s2 kz
~
k
 
z 
 ö
ø 		.
(1.36)
Moreover, if the substrate is covered with a continuous layer of thickness D, the reflectivity RS [1] becomes:
RS =  R01 + R12 exp( i qz D)
1 + R01 R12exp( i qz D)
,
(1.37)
with R01,R12 the reflectivity of the vacuum-layer and layer-substrate interfaces. Thus, the form factor is modulated by the Kiessig fringes of this over-layer.
To conclude, if the absorption has to be accounted for inside the island, the correct scattering cross section of one island is:
 ds
dW
 (q) = ê
ê 		 k02
4pre r0
 (1-ni2) F(q) ê
ê 		2

 
 ,
(1.38)
with ni=1-di + ibi the index of refraction of the island. If bi=0, as the scattering is connected to the refraction index via di=2pr0 re/k02, the prefactor is equal to one to first order in di.
An effective and approximative way called Layer Born Approximation (LBA) to account for this phenomenon is to simulate only the strong dependance of the scattered intensity as function of kzf by multiplying the Born term by a transmission coefficient in an effective layer:
F(q) @ F(q||,kfz-kiz)×T(kzf)
T(kzf) =  2kzf
kzf +
~
k
 
 f
z 
 .
(1.39)
In this last case the index of refraction is that of the effective layer, where the islands are supposed to be buried.
Because of the sharp variation of the reflection coefficient around the critical angle for total external reflection ac=Ö{2d}, the DWBA is expected to be important only when the incident ai or exit af angles are closed to ac. More precisely, the influence of the reflection-refraction effect is depicted in Fig. 1.7.

figures/dwbacompa.gif
Figure 1.7: The interference fringes for the form factor of a cylinder of height H=5 nm (l = 1 Å-d = 5 10-6,b = 2 10-8) as function of the exit angle af normalized by the angle of total external reflection ac within the various approximations: BA, DWBA(ai=ac/2,ac,2ac), layer BA(ai=ac).

Because of a complex interference between the four terms Fig. 1.7-1.8-1.9, neither the Born Approximation nor the effective layer model are able to catch the exact position of the form factor minima and the overall curve intensity, in particular when ai is closed to ac (maximum of intensity). This interference phenomenon blurs the sharp minima of the cardinal sine function which is found in the simple Born Approximation as the phase factor Fig. 1.9 is shifted from one term to the other by ±kzi. However, if ai > ac and af > ac, the Born Approximation and the effective layer model can give a good approximation as shown in Fig. 1.7. Indeed, in this case the dominant term of the DWBA form factor F(q) is the first one.

figures/dwbaalphac.gif
Figure 1.8: The modulus square of the four terms Fig. 1.6 for a cylinder as function of af for various ai: Term 1(red)-Term 2(green)-Term 3(orange)-Term 4(blue) (same parameters as in Fig. 1.7).

figures/dwbaphase.gif
Figure 1.9: The phase of the four terms Fig. 1.6 for a cylinder as function of af for various ai: Term 1(red)-Term 2(green)-Term 3(orange)-Term 4(blue) (same parameters as in Fig. 1.7).

1.4.2  Buried particles in a substrate

The scattering cross section for a buried particle was derived by Rauscher [28]:
 ds
dW
 (q) = ê
ê 		 k02
4pre r0
 (ns2-ni2) F(q) ê
ê 		2

 
 .
(1.40)
F(q)=T(kzi) T(-kzf)F(q||,[(qz)\tilde]) exp(i[(qz)\tilde]d ) is the form factor computed with momentum transfer inside the substrate [(qz)\tilde] multiplied by the transmission function in incidence and emergence. The transmission functions are given by Eq. 1.39 and can be modified as for the reflection coefficient by an uncorrelated roughness. The function T(kzf) gives rise to the Yoneda peak behavior i.e no intensity is present below the critical angle in emergence. This term is often explained on the basis of the reciprocity theorem for light propagation [5]. Moreover, comes into play the phase factor linked to propagation and attenuation of the plane wave during its path to the buried particle layer at depth d.

1.4.3  Particles encapsulated in a layer

Still to be developed !!

1.5  The interference function

In the evaluation of the interference function Eq. (1.13), three main useful cases can be distinguished:

The pair correlation function

When the islands do not present a long range-order as for instance in a classical nucleation-growth-coalescence process, the only relevant statistical quantity in the interference function is the total pair correlation function, as underlined in the DA and LMA (see Sect. 1.2). Let assume that dP(r||) is the number of particles at r|| knowing that a particle is at the origin. As for a completely random distribution, this value tends towards the surface per particle times the elementary surface area around r||, one defines the pair correlation g(r||) function as the departure from this mean value:
dP(r||) = rS g(r||)d2r||.
(1.41)
with rS the particle density per surface unit. In fact, the autocorrelation function of the island-island position can be written in terms of the pair correlation function:
z(r||) = d(r||) +rS g(r||).
(1.42)
The Dirac function represents the particle at the origin. Using Eq. (1.14), one sees that:
g(r||) =  1
S
 <
å
i ¹ j 
 d(r-ri + rj ) > ,
(1.43)
where < ¼ > is a configuration average. As the long-range order is absent, g(r||) ® 1 when r|| ® ¥. Thus writing Eq. (1.42) in the following way:
z(r||) = d(r||) +rS + rS (g(r||)-1)
(1.44)
lets appear the oscillating part of g(r||). By Fourier transform of Eq. (1.44), one obtains the interference function Eq. (1.13):
S(q||) = 1 + rS d(q||) + rS ó
õ 		(g(r||)-1) exp( - i r||·q|| )  d2r|| .
(1.45)
The first term leads to the specular reflectivity which will be described in Sect. 1.6; it will be ignored in the following as it is often difficult to measure it as the same time as the diffuse scattering because of the intensity differences. Moreover, for an homogeneous and isotropic sample, the pair correlation function and the interference function depend only on the modulus r|| and q||, respectively. Thus in two dimensions:
S(q||) = 1 + 2prS ó
õ 		¥

0 
 (g(r||)-1) J0( r|| q|| ) r||  dr||.
(1.46)
An inverse Fourier transform leads to the following result:
g(r||) = 1 +  1
2prS
 ó
õ 		¥

0 
 (S(q||)-1) J0( r|| q|| ) q|| dq||.
(1.47)
By building, the limit g(r||)® 1 when r|| ® ¥ insures the convergence of the previous equation. The size of the coherently irradiated area Si was not accounted for in the previous derivation; it should lead to a convolution product of the right hand side of Eq. (1.45) with the Fourier transform of the autocorrelation function of Si and to a broadening to the specular reflectivity rS d(q||). To conclude, one has to take care to the fact that, after eliminating the "Dirac peak" at the origin, S(q||=0) is not zero but equal to the relative fluctuations on the number of particles in the irradiated surface [7,14].

Some arbitrary pair correlation functions have been implemented in the program. Although their shape catch the main features for this type of function, they do not represent a physical situation in all the parameter space as they are not self-convolution products of a function in 2 dimensions. Moreover, the island density for these functions is not specially linked to the first neighbor distance D as sone would expect.

By convention, the hard core radius R0 is set to the mean value for the parallel rotational radius of the island. For instance, for a cylinder R0= < R > , for a parallelepiped R0=Ö2 < R > . In the case of the hard core function pf0,pf1,pf4,HC, the hard core diameter is given by the parameter sigma and thus its value can be different from 2R0.
No special normalization can be applied to these function except for pf3,pf4 where the w parameter can be found through the conservation of the number of particles:
rS ó
õ 		¥

0 
 2pr (g(r)-1)  dr = 1.
(1.55)

The example of the hard-core pair correlation function and the interference function is given in Fig. 1.10. The main parameter is the effective surface coverage h = rS ps2/4. Obviously, the pair correlation function is equal to zero for r < s. Its shape is close of the Debye behavior (pf0 or pf4) for the small coverage whereas for higher coverage the function is more and more structured with peaks in the interference function at values close to multiples of 2p/s. Notice that in this case, the maximum in g(r) is found at s, independently of the particle density.

figures/HC.gif
Figure 1.10: The hard-core pair correlation function a) and the interference function b) as function of the surface coverage h = rS ps2/4, with s the hard core diameter. The functions are plotted against reduced parameters r/s and qs/2p.

The regular bidimensional lattice

In the case of a regular lattice, a pattern is linked to each node of the lattice defined by its base vectors a,b. In the interference function Eq. (1.13), the intensity is concentrated in Bragg rods perpendicular to the substrate surface. For infinite crystal, these rods are Dirac peaks at the nodes of the reciprocal lattice given by the vectors a*,b*:
a* = 2p  bÙn
a ·[ bÙn ]
 ,     b* = 2p  nÙa
b ·[ nÙa ]
 ,
(1.56)
with n is the normal to the surface.
The orientation x of the first lattice vector a with respect to the x- axis implies to rotate the momentum transfer as:
R
 
 (x) q = æ
ç
ç
ç
è
cos(x)
-sin(x)
0
sin(x)
cos(x)
0
0
0
1
ö
÷
÷
÷
ø 		æ
ç
ç
ç
è
qx
qy
qz
ö
÷
÷
÷
ø
(1.57)
before its decomposition on the basis vectors a,b. Moreover, in the case of a regular lattice made of different variants rotated from one to the other, an incoherent sum of intensities has to be applied with the weights of each variant.

Defective crystal: finite size effect
In reality, the crystal is always defective and the first defect that is encountered is the finite size : Na,Nb cells in both direction a,b. Thus, by decomposing q on the a*,b* basis (q = aa* + bb*), the interference function is equal to:
SL(q)
=
 1
NL
 ê
ê
å
i 
 exp(-i q·Ri||) ê
ê 		2
 
=
 1
Na Nb
 ê
ê 		Na-1
å
n=0 
 Nb-1
å
m=0 
 exp[-i (aa* + bb*) ·(na + mb)] ê
ê 		2
 
=
 1
Na Nb
 ê
ê 		Na-1
å
n=0 
 exp(-i 2 pan) ê
ê 		2
 
 ê
ê 		Nb-1
å
m=0 
 exp(-i 2pbm) ê
ê 		2
 
=
 1
Na
 ê
ê 		 sin(paNa)
sin(pa)
 ê
ê 		2

 
  1
Nb
 ê
ê 		 sin(pbNb)
sin(pb)
 ê
ê 		2

 
 .
(1.58)
In the case of various types of domain sizes, one has to make an incoherent sum of the diffracted intensities from the various domains as in the case of variants.

Defective crystal: correlation length
To account for various kinds of defects in diffraction, it is usual to asses that the correlation between two unit cells decreases with their distance in such a way that:
SL(q) =  1
Na Nb
å
i 
å
j 
 exp[-iq·(Ri||-Rj||) ] C(Ri||- Rj|| ).
(1.59)
with : C( Ri||- Rj|| ) ® 0 when | Ri||- Rj|| | ® ¥. However when the correlation function depends only on the distance between nodes, the summation in Eq. (1.59) has no analytical simple expression and is untractable numerically. One other approximate and tractable way is to write:
C( Ri||- Rj|| ) = C( (ni-nj) a + (mi-mj)b )
= exp æ
è 		-  2 pa|ni-nj|
La
 ö
ø 		exp æ
è 		-  2 pb|mi-mj|
Lb
 ö
ø 		.
(1.60)
With these expressions knowing that q = aa* + bb*, one finds for a size limited lattice:
SL(q) =  1
NaNb
 Sa(q) Sb(q)
(1.61)
with:
Sa(q) = Na-1
å
ni,nj=0 
 exp( -i 2pa(ni-nj) ) exp æ
è 		-  2 p|ni-nj|a
La
 ö
ø
Sb(q) = Nb-1
å
mi,mj=0 
 exp( -i 2pb(mi-mj) ) exp æ
è 		-  2 p|mi-mj|a
Lb
 ö
ø 		.
(1.62)
The summation can be carried out and leads to:
 Sa(q)
Na
 = Real é
ë 		1 -  2
1-exp[ 2 p(a/La +i a)]
 +  2
Na
 ×  1 - (exp[ 2 p(-a/La + i a)] )Na
2 - exp[ 2 p(a/La - ia) ] - exp[2 p(-a/La + i a) ]
 ù
û 		.
(1.63)
and the same expressions for Sb(q). For a Gaussian dependence in C( Ri||- Rj|| ), the sum is untractable analytically but converges rapidly on a numerical point of view.

Defective crystal: reciprocal space approach
The other way is to work directly in the reciprocal space by convoluting the nodes of the reciprocal lattice with special shapes (Gaussian, Lorentzian, ...) as it is done, in a similar way, in the case of the analysis of powder diffraction data:
SL(q) =
å
n 
å
m 
 S(q-n a* - m b*),
(1.64)
with S is the desired shape for the diffraction rod. As a matter of convenience in IsGISAXS , the peak shape has been decomposed in the following way:
S(q) = Sa*(a) Sb*(b)     with     q = aa* + bb* .
(1.65)
Whatever the peak shape is, it implies the use of a reciprocal length which accounts for the size of the coherent domain in real space. For a Lorentzian or a Gaussian shape:
Sa*(a) =  sa*
p
  1
a2 + sa*2
Sa*(a) =  1
sa*
Ö
2p
 exp æ
è 		-  a2
2sa*2
 ö
ø 		.
(1.66)

The structure factor and the Decoupling Approximation
In the case of a regular lattice, a pattern which is made of Ni islands is attached to each node of the lattice. By using the hypothesis of a full decorrelation between the position of the unit cell and its contents , the form factor in Eqs. (1.11-1.13,1.34) has to be replaced by the structure factor of the unit cell which describes its content:
FS(q) =
å
k 
 Fk(q) exp(-i q||·r||k ),
(1.67)
where r||k = xk a + yk b is the position of the kth island in the unit cell. Moreover, the mean values in Eqs. (1.11-1.12) should account for the intra-cell position and for the scatterer type disorders. With a lack of intra-cell correlation for these parameters, one is led to:
< FS(q) > = < Fk(q) > N <
å
k 
 exp(-i q||·r||k ) > P.
(1.68)
with the indexes N,P stand for nature and position. It is possible to write r||k = r||k,0 + dr||k as the sum of its mean value and a deviation from it; after an expansion of the exponential term with < dr||k > =0, the classical Debye-Waller term appears:
< FS(q) > = < Fk(q) > N
å
k 
 exp(-i q||·r||k ) exp æ
è 		-  Wk
2
 ö
ø 		,
(1.69)
with Wk = < ( q||·dr||k )2 > P = q||[`([`(Bk)])] q||.
[`([`(Bk)])] is the symmetric tensor of standard deviations of the kth island position. In an analog way,
< | FS(q) |2 > =
å
k 
 < |Fk(q)|2 > N + | < Fk(q) > N |2
å
k ¹ l 
 exp(-i q||·(r||k - r||l ) ) é
ë 		exp æ
è 		-  Wk
2
 ö
ø 		+ exp æ
è 		-  Wl
2
 ö
ø 		- 1 ù
û 		.
(1.70)
By gathering Eqs. (1.69-1.70) in Eqs. (1.11-1.12), one ends up to first order in the disorder Debye-Waller factors with:
 ds
dW
 (q) @ Id(q) + Ic(q)
(1.71)
Id(q) = < |Fk(q)|2 > N - | < Fk(q) > N |2
å
k 
exp æ
è 		-  Wk
2
 ö
ø
Nk
(1.72)
Ic(q) = | < Fk(q) > N |2 ×SC(q) ×SL(q)
(1.73)
SC(q) =  1
Ni
 ê
ê
å
k 
 exp(-iq||·r||k )exp æ
è 		-  Wk
2
 ö
ø 		ê
ê 		2

 
 .
(1.74)
It appears that the decrease of intensity in the coherent part Ic(q) of the intensity induced by the Debye-Waller factor is found in the diffuse scattering Id(q) spread uniformly over all the reciprocal space in the total decoupling approximation.

The structure factor and the Local Monodisperse Approximation
In this approximation, the diffuse scattering is forgotten and each node is decorated with a structure factor accounting for the Debye-Waller term but with each island replaced by its mean value over size and shape distributions:
 ds
dW
 (q) @ < |Fk |2 > N SC(q) SL(q).
(1.75)

The bidimensional paracrystal

The paracrystal theory is fully described in the book of Hosemann [14]. The type of disorder (called of first kind) described previously in the interference function of a regular lattice SL(q) does not affect the long range order but only the intensity in the Bragg peaks. On the contrary, in the model of paracrystal, the long-range order is destroyed gradually in a probabilistic way. This model allows to make the link between the regular lattice and fully disordered structures.
To understand it, the example of the one-dimensional disordered lattice [7] is instructive. To build the autocorrelation function g(x) for the island positions, the distance between two successive points An-1,An is chosen to be independent of the previous and next one and to obey a statistical distribution p(x) with:
ó
õ 		¥

-¥ 
 p(x) dx = 1
ó
õ 		¥

-¥ 
 x p(x) dx = D.
(1.76)
Thus, after having placed the first island at the origin A0 and second island A1 at a mean distance D from the first one (see Fig. 1.11), the probability of placing the third one A2 at a distance x from the first one is given by the occurrence of a distance y between the first and the second and a distance x-y between the second and the third.

figures/1DDL2.gif
Figure 1.11: The schematic view of the one dimensional paracrystal.

By integrating over all the y possibilities, one is led to:
p2(x) = ó
õ 		+¥

-¥ 
 p(y)p(x-y) dy
(1.77)
which is the self convolution product p(x) Äp(x). By generalizing,
g(x) = d(x) + p(x) + p(x) Äp(x) + p(x) Äp(x) Äp(x) + ¼
(1.78)
The interference function is then given by the Fourier transform of Eq. (1.78):
S(q) = 1 + P(q) + P(q) ·P(q) + P(q) ·P(q) ·P(q) ¼
(1.79)
By writing the Fourier transform of p(x) as P(q) = fexp(i u), one finds:
S(q) = 1 + 2 ¥
å
n=0 
 fn cos( n u) =  1 - f2
1 + f2 - 2 fcos(u)
 .
(1.80)
For a gaussian probability distribution which is the small disorder limit in case of any type of distribution,
p(x) =  1
w
Ö
2 p
 exp é
ë 		-  (x-D)2
2 w2
 ù
û
P(q) = exp( pq2 w2 ) exp(- i q D ).
(1.81)
The results for this gaussian disorder are depicted in Fig. 1.12 in direct and reciprocal space. The broadening of the peaks with increasing the ratio w/D reflects the transition from an ordered lattice to a disordered lattice.

figures/1DDL.gif
Figure 1.12: The pair correlation function a) and the interference function b) in the case of the 1D disordered lattice for various disorder parameter w/D.

In two dimensions, the paracrystal is constructed on a pseudo regular underlying lattice with basis vector a,b (see Fig. 1.13). In IsGISAXS , only the case of perfect paracrystal i.e. with parallelogram cell is implemented.

figures/2DDL.gif
Figure 1.13: The schematic view of a paracrystal in two dimensions. Each circle represent the area where the probability of finding one island is maximum.

In an analogous way as for 1D, the probability of finding a particle at a position around the basis vector a,b are defined by pa(r),pb(r), respectively with:
ó
õ 		pa,b(r)  d2r = 1,   ó
õ 		r pa(r)  d2r = a,   ó
õ 		r pb(r)  d2r = b.
(1.82)
If the Fourier transform of probability distributions are defined by Pa,b(q||), assuming that all the directions behave independently, the interference function appears as:
SL¥(q||) =
Õ
k=a,b 
 Real æ
è 		 1+Pk(q||)
1-Pk(q||)
 ö
ø 		.
(1.83)
As a matter of convenience [37], the Fourier transforms of the probability distributions Pa,b are decomposed in the program IsGISAXS along the a = q||·a and b = q||·b as:
Pa(q||) = Pa,a(a) Pa,b(b)
Pb(q||) = Pb,a(a) Pb,b(b).
(1.84)
The available function for Pa,a,Pa,b,Pb,a,Pb,b are chosen as gaussian, lorentzian,.... However, with this formalism, a divergence appears close to the origin of the reciprocal space when one approaches along a direction perpendicular to one basis vector. This divergence is cured by the convolution brought by the finite sizes of the coherent domains [14,15,20,24]. For a practical point of view, the finite size of the paracrystal is incorporated not by the shape function of the crystal but by explicitly accounting for the number of cells Nk in the k direction:
SL(q||) =
Õ
k=a,b 
 Real æ
è 		2  1-Pk(q||)Nk
1-Pk(q||)
 -1 ö
ø 		.
(1.85)
Notice that Eq. 1.85 contains the "Dirac peak" at q=0 evoked in Eq. 1.45 (term in Pk(q||)Nk) in the form of the Fourier transform of the paracrystal shape.

One convenient way of getting a "physical" isotropic interference function is to average the paracrystal interference function Eq. 1.85 over all the azimuthal directions x.

figures/2DDLSqgr.gif
Figure 1.14: The interference function a) and the pair correlation function b) for a paracrystal of hexagonal symmetry averaged over all the azimuths z. The gaussian disorder parameter s/D is indicated on the figure. The axis are normalized by the lattice parameter D and the expected peaks position 1,Ö3,2,Ö7,3,2Ö3,Ö{13},4 are marked with a circle.

Fig. 1.14 presents the S(q||) function and the corresponding g(r||) for an hexagonal paracrystal of parameter D with various gaussian isotropic disorder s/D. The peak at q||=0 is, as explained above, the result of the Fourier transform of the crystal shape function whose oscillatory parts appear clearly at small disorder. The back Fourier transform Eq. 1.47 for computing g(r||) leads to unphysical results below r||/D=1. Besides this problem, by reducing s/D, the interdistances in the hexagonal lattice 1,Ö3,2,Ö7,3,2Ö3,Ö{13},4 (in reduced units) shows up more and more clearly in the g(r||) function. It is very instructive to notice that, depending on the disorder value, the first peak in the interference function is shifted from the expected value q||=2p/D. This shift is simply linked to the module of the reciprocal basis vector which is 2/Ö3 @ 1.15. Thus, as the close packing in 2D leads to a deformed local hexagonal symmetry, taking rS = (qm/2p)2 as the particle density (qm position of the interference peak) induces an overestimation of 15%. This error often encountered in the literature is induced by an hidden square lattice hypothesis.

1.6  The X-ray reflectivity for a particle layer

1.6.1  Islands on a substrate

The Dirac term in the interference function Eq. (1.45) gives rise to the reflectivity signal. In the DWBA, the reflectivity amplitude is given by:
RT = RS + i  k0
2 sin(ai)
 (1-ni2) rS < F(q||=0,kiz,kfz) > ,
(1.86)
with RS the reflectivity of the bare substrate. In the absence of absorption in the islands bi=0,
RT = RS + i  2 pr0 re
k0sin(ai)
 rS < F(q||=0,kiz,kfz) > .
(1.87)
The reflection coefficient in intensity is given by: |RT|2.
Notice that a "good" criterion of the DWBA validity should be that the bare substrate reflectivity is greater than the introduced correction.

1.6.2  Buried particles

The equation Eq. 1.86 is still valid except that the contrast of index of refraction 1-ni2 has to be replaced by nS2-ni2 and the form factor by that of Eq. 1.40 taken at q||=0.







Chapter 2
How to use the IsGISAXS software ?

The program IsGISAXS runs under Windows 9x,2000,XP,NT operating system in a windows mode. The development language is Fortran 90 under Compaq Visual Fortran.

2.1  The IsGISAXS software environment and outputs

The philosophy of the program IsGISAXS is to use special input files for the simulation of GISAXS images or cuts or for the fit of experimental data. These files are named *.inp, *.fit and *.mor will be described in the following sections. On starting, the program reads automatically some information files *.inf located in the *.exe directory. The file gisaxs.inf contains: The other file graph.inf contains a set of parameters for the graphical outputs that are set-up with windows based dialog boxes. The only think to do is to modify possibly the path for the grfont.dat file (furnished with the distribution) for the PGPLOT outputs.

The menu bar is made of the following items and sub items:

The short keys for the menus are accessible by Alt-Key. The simulation, pre-fit and fit can be launched by the F5,F6,F7 keys, respectively.
In general, one loads the name of the main files *.inp,*.fit,*.dat,*.mor, launches a simulation or a fit and then visualizes the results of the calculation. The *.inp,*.fit,*.dat file are read at each calculation allowing the user to modify them between two runs.

2.2  Description of the input file *.inp and the underlying parameters

2.2.1  The file contents

The main input file named *.inp is reproduced in Fig. 2.2. Be careful : the format is rather strict for a correct reading !

###########################################
#
#GISAXS SIMULATIONS : INPUT PARAMETERS
#
###########################################
# Base filename
results
# Framework,Diffuse,DWBA_terms
DWBA LMA F
# Beam Wavelength :Lambda(nm),Wl_distribution,Sigma_Wl/Wl,Wl_min(nm),Wl_max(nm),nWl, xWl
0.1 none 0.2 0.08 0.12 10 -2
# Beam Alpha_i :Alpha_i(nm),Ai_distribution,Sigma_Ai/Ai,Ai_min(nm),Ai_max(nm),nAi, xAi
0.2 none 0.05 0.1 0.3 10 -2
# Beam 2Theta_i :2Theta_i(nm),Ti_distribution,Sigma_Ti/Ti,Ti_min(nm),Ti_max(nm),nTi, xTi
0. none 0.05 -0.1 0.1 10 -2
# Substrate :n-delta_S, n-beta_S, Layer thickness(nm), n-delta_L, n-beta_L, RMS(nm)
6.11642E-06 3.46012E-080. 6.11642E-06 3.46012E-08 0
# Island : n-delta_I, n-beta_I, Depth(nm)
1.23632E-05 5.34090E-07 0
# Output q(nm-1) :Two theta min-max(deg),Alphaf min-max(deg),n(1),n(2)
0     1.5 0     1.550 50
# Number of different island types
1
# Island type,Probability
cylinder 1
# Fixed geometrical parameters :Base angle(deg),Height ratio,FS-radii/R
54.7356 1. 0.8     0.8
# Orientation of island :Zeta(deg),Z_distribution,SigmaZ(deg),Zmin(deg),Zmax(deg),nZ,xZ
0. none 20. -45. 45. 10 -2
# H_uncoupled
T
# Size of island : Radius(nm),R_distribution,SigmaR/R,Rmin(nm),Rmax(nm),nR, xR
6. gaussian 0.5 0.5 10 20 -2
# Aspect ratio : Height/R,H_distribution,SigmaH/H,Hmin/R,Hmax/R,nH, xH
0.3 none 0.2 0.25 1. 10 -2
# Island distribution types
1DDL
# Pair correlation function : Density(nm-2),Peak position D(nm),w(nm or au),D1(nm)Sigma(nm)
0.016 20. 6. 25. 20
# Lattice parameters : L(1)(nm),L(2)(nm),Angle(deg)
7.7 17. 90.
Xi(deg),Xi_distribution,SigmaXi(deg),Ximin(deg),Ximax(deg),nXi, xXi
88. none 120. -120. 120. 3 -2
Correlation lengths (nm),Rod shape
400     400 lorentz
Domain sizes DL(nm),DL_distribution,SigmaDL/DL,DLmin(nm),DLmax(nm),nDL, xDl
4000     4000none 0.2   0.2 200   200400   40010   10-2   -2
Disorder factors w/L
0.05     0.05     0.05     0.05
g     g     g     g
Particles per pattern
2
Positions xp/L,Debye-Waller factors
0.     0. 0.    0.     0.
0.4     0.5 0.     0.     0.
Figure 2.2: A typical input file *.inp

This file is read each time the Run button is activated allowing the modification of parameters between each run. All the parameters are necessary for a correct reading although not all of them are used for the calculation.

As introduced in the theoretical chapter Chap. 1, the following options are available :

2.2.2  Some useful remarks

The grid for the image calculation is set up in such a way that the scale in x-y is proportional to the sine of the scattering angle as on a 2D-detector. Thus the number of points may be different than the required number of points n(1),n(2).

The distribution probabilities p(x) for the wavelength l, the incidence angle ai, the island radius R, aspect ratio H/R , orientation angle z and for the lattice dimensions DL and orientation angle x are defined by:

The chosen points where the distribution probabilities are sampled are defined by an arithmetic progression:
xi = x1 + i ×  xmax-xmin
N-1
 .
(2.6)
As the number of classes N is finite, the normalization of the probability distribution is numerically ensured by the condition:
A N
å
i=1 
 p(xi) = 1 ® p(xi) Þ p(xi)/A.
(2.7)
The boundaries xmin,xmax are defined in two ways depending on the sign of xX value (with X=Wl,Ai,Ti,Z,R,H,Xi,DL). If xX is negative, the value of xmin,xmax in the input file *.inp are used. If positive, these limits are reset accordingly to the central value x0 and to the full width at half maximum of the distribution which depends on s parameter:

The position probability distribution for the 2D paracrystal with loss of long-range order are defined through:
g(gaussian)
:
pg(x) =  1
s
Ö
2p
 exp æ
è 		-  x2
2 s2
 ö
ø
e(exponential)
:
pe(x) =  1
2 s
 exp æ
è 		-  |x|
s
 ö
ø
d(gate)
:
pd(x) =  1
2 s
     for     |x| < s
t(triangle)
:
pt(x) = -  |x|
s2
 + sign(x) s    for     |x| < s
(2.8)
v(pseudo-Voigt)
:
pv(x) = hpg(x) + (1-h)pe(x)     with     |h| < 1.
(2.9)

The reflectivity coefficient for the island layer is obtained as a by-product when a simulation with n1=1,2 Theta max=0 is launched. A new column appears then in the *.out file.

2.3  The morphology file *.mor

The morphology file *.mor is only used for simulation and contains all the parameters described in the *.inp file. It allows the mixture of any types of island shape with any types of morphological parameters. For each class of islands, the user gives the associated probability of occurrence. After reading, the positions of the islands are recalculated from the center of the box containing all the particles.

##############################################
#
# GISAXS SIMULATIONS : MORPHOLOGY PARAMETERS
#
##############################################
# Total number of island
2
# Probability-Island type-Positions(nm)-Orientation(deg)-Radius(nm)-Height/R-Base angle(deg)-Height ratio-FS-radii/R
0.5 pyramid 0     0 0 2 1 54.7356 0 0     0
0.5 cylinder 5     5 0 2 1 54.7356 0 0     0
Figure 2.3: A typical morphology file *.mor

The reading of such a file is activated by putting Number of different island type=0.
The GISAXS intensity is calculated in the following way:
 ds
dW
 (q) @ Id(q) + | < F(q) > |2 ×S(q)
Id(q) =
å
m 
 < ( Fi(q) - < F(q) > ) (Fi+m(q) - < F(q) > ) > i exp(-i q|| ·Rm,|| ).
(2.10)
The summation in the diffuse scattering Id(q) is calculated up to a critical Rc radius of separation between the islands. This radius is asked to the user before the calculation. The interference function is calculated either in a direct way (morif option in the choice of interference function):
S(q) =  1
N
 ê
ê 		N
å
i=1 
 exp(-i q|| ·Ri,|| ) ê
ê 		2
 
 .
(2.11)
or using the other available possibilities. If calculated from experimental positions, the interference function is filtered by multiplying the box enclosing all the islands by a Hann function.

2.4  The treatment of experimental data *.dat and the fitting file *.fit

The experimental data need to be processed before being used by IsGISAXS . The main treatment is to find the position of the intersection of the detection plane and the x-axis. In principle, it should be in between the transmitted and the reflected beam. The a = 0 line correspond to a zero intensity as shown by the DWBA formula whereas the 2q = 0 line is just in the middle of the two symmetric interference peaks. Once localized and as the sample-detector distance is known, this origin allows to extract some line cuts I=f(sin(a),sin(2q)) which are the main inputs for the fits.

An example of *.fit file is sketched in Fig. 2.4.

###########################################
#
# GISAXS FIT : INPUT PARAMETERS
#
###########################################
# Cut :Number of cuts,# points,Fit type,Error Bar,Epsilon,# cycles
2 50 0 1 0.1 5
# Scale and shift factors :ALASLSSt(deg)LStSa(deg)LSa
6e5T0.F0.T 0T
# Beam Wavelength :Lambda (nm),Sig_Wl
F F
# Beam Alpha_i :Alpha_i,Sig_Ai,2Theta_i,Sig_Ti
F F FF
# Substrate :n_delta_S,n_beta_S,Thickness(nm)n_delta_L,n_beta_L, Roughness(nm)
F F F F F F
# Island : n-delta_I, n-beta_I, Depth
F F F
# Fixed geometrical parameters :Base angle,Height ratio,FS-radii/R
F F F     F
# Orientation of island :Zeta(deg),SigmaZ(deg)
F F
# Size of island :Radius(nm),SigmaR/R
T F
# Aspect ratio : Height/R,SigmaH/H
T F
# Pair correlation function :Density(nm-2),Peak position D,w,D1
F F FF
# Potential g(r):Sigma(nm),Epsilon
F F
# Lattice parameters :L(1)(nm),L(2)(nm),Angle(deg)
F F F
Xi(deg),sigmaXi(deg)
F F
Correlation lengths (nm)
F     F
Domain sizes DL(nm),SigmaDL/DL
F     F F     F
Disorder factors w/L
F     F F     F
Positions xp/L,Debye-Waller factors
F     F F     F     F
F     F F     F     F
Figure 2.4: A typical input file *.fit for data fitting

This file has more or less the same structure as the *.inp file except that it gives the constraints for the fitted parameters. T ~ .true. means that the corresponding parameter will be fitted whereas a kept fixed parameter is marked by F ~ .false.. Be careful to fit only the necessary parameters as fitting parameters that do not influence the final result will lead to a singular matrix error.

It is possible to fit simultaneously a variable number of line cuts by the minimizing the c2 with the Levenberg-Marquardt algorithm [26]. c2 is defined by:
c2
=
n
å
k=1 
    1
wk2
  c2k
c2k
=
 1
d
   N
å
ik=1 
    (Icalck,ik-Iobsk,ik)2
(sk,ikobs)2
 ,
(2.12)
where:

The following items are accessible in the *.fit file The c2 minimization is always applied to the function f(A×[(I)/(Imax)] + S). The outputs (file or graphics) concern only this function f.
In an analogous way to the c2 definition, to characterize the goodness of a fit, the RB factor for cut number k is defined by :
RB(k) =
N
å
ik=1 
  |Icalck,ik-Iobsk,ik |
N
å
ik=1 
  |Icalck,ik |
 .
(2.13)
Once again, the RB is defined on f(I).

The various experimental data are put in a separated file *.dat in the format sin(2q),sin(a),I,sI. sI is optional and if absent it is calculated from the intensity following the code Error Bar. Below the four lines of comments, one puts:

Fig. 2.5 depicts the format for the data file.

#################################
# Parallel cut at alphaf=1deg
# Image test
# DeltaOmega(deg),Weight
0. 1.
0 0.0174524 69290.1
0.000468164 0.0174524 68691.2
0.000936329 0.0174524 66955.3
0.00140449 0.0174524 64264.3
...............................
0.0688201 0.0174524 11.6726
0.0692883 0.0174524 11.4518
0.0697565 0.0174524 11.2475
##################################
# Perpendicular cut at 2theta=1deg
# Image test
# DeltaOmega(deg),Weight
0. 1.
0.01745240.00046816470.1678
0.01745240.000936329284.976
0.01745240.00140449658.752
...............................
0.01745240.068820133.54
0.01745240.069288332.1555
0.01745240.069756530.7835
###################################
Figure 2.5: A typical data file *.dat

2.5  The outputs files *.out,*.pro,*.ima,*.ki2,*.cor,*.dwba

The outputs are written in ASCII files with the Base filename plus a special extension: The name of the output file (*.out) as well as the c2 file (*.ki2) is automatically incremented with a counter to avoid scratching the results of work.







Chapter 3
IsGISAXS : Typical examples

In the following examples, all the images are displayed in logarithmic scale with the horizontal axis proportional to sin(2q) and the vertical one to sin(af). The chosen wavelength is l = 1 Å and the angles of incidence is fixed at ai=ac=0.2°,qi=0° if not specified. The index of refraction of the substrate are fixed to d = 5. 10-6,b = 2. 10-8.

3.1  The form factor

3.1.1  Distorted Wave Born Approximation and the refraction effect

As shown in Sect. 1.4 and in particular in Fig. 1.8, the treatment in DWBA is a prerequisite for a correct description of the scattering phenomenon. The figure Fig. 3.1 shows that the maximum of intensity is obtained at the critical angle when the four DWBA terms interfere coherently.

figures/cylinder.gif
Figure 3.1: Calculated form factor for a cylinder (R=5 nm; H/R=1) for 0 < 2 q < 2°; 0 < af < 2° in the Distorted Wave Born Approximation for two angles of incidence a) ai=ac and b) ai=2ac. Same color scale.

3.1.2  The island facetting

An anisotropic shape is able to generate an anisotropy of scattering as function of the angle z between island edge and incident beam. Some rods of scattering by facets appear clearly in Fig. 3.2 for a pyramidal island. For z = 0°, the direction of scattering is exactly at the complementary angle of the facets angle (35.27°=90°-54.73°).

figures/facette.gif
Figure 3.2: Calculated form factor for a truncated pyramid (R=5 nm; H/R=1; a = 54.73° angle between (111) and (100) planes in cubic system) at ai=ac for 0 < 2 q < 3°; 0 < af < 3° in the Distorted Wave Born Approximation for two angles between the direct beam and the island edge a) z = 0° b) z = 45°.

3.1.3  The island sizes distribution

For a monodisperse size distribution, sharp interferences fringes (called Kiessig fringes in reflectivity) appear at roughly q|| ~ 2p/R or q^ ~ 2p/H. These fringes are associated to the zero of the sine cardinal or Bessel function for simple shapes like parallelepiped or cylinder. The figure Fig. 3.3 illustrates the smoothing of these fringes upon increasing the width of the size distribution. In this case a Guinier analysis at small scattering vector is tractable [8,7] as interference function does not perturb the form factor. The change in scattering underlines a transfer of intensity from the lobes to the minima which smoothes out all the curve.

figures/cylinder_size.gif
Figure 3.3: Cut along q|| at a = ac for the mean form factor < |F|2 > of a cylinder (Born Approximation). The size distribution is gaussian (R0=5 nm) with various broadening s/R).

The size and height distribution for the pyramidal shape sheds light on the rod of scattering by facets as underlined in Fig. 3.4 by smoothing out all the sharp minima which appeared in Fig. 3.2.

figures/pyramid_size.gif
Figure 3.4: Mean form factor < |F|2 > for a gaussian distribution of pyramidal shape island (R0=5 nm-s/R=0.2). Same parameters as Fig. 3.2.

3.2  The interference function

3.2.1  Non regular lattice and the pair correlation function

The main influence of the interference function is to shift the maximum of the intensity of the form factor located at qy = 0. The figures Fig. 3.5-3.6 illustrate this in the case of the simple Debye hard core model and of the isotropic paracrystal interference function. Notice that, because of the slope of the mean form factor, the maximum of the intensity is not located at the maximum of the interference function which means that this position can not give with certainty the mean interparticle distance. As usual in scattering at small angles, this interplay between the interference function and the form factor often prevents to perform the classical Guinier analysis [8]; it implies a full treatment of all the scattering curve to extract reliable parameters. Moreover, the DA model generates a strong diffuse scattering Id(q) around the specular beam contrary the LMA model (see Fig. 3.6).

figures/interference.gif
Figure 3.5: a) GISAXS pattern computed between -1.2° < 2 q < 1.2°;0 < af < 1.5° using the LMA model (see Sect. 1.2). The DWBA form factor is that of a cylinder with the morphological parameters: R=5 nm-H/R=1-s/R=0.2 (gaussian). The interference function is given by the Debye model with a hard core radius of R0=7.5 nm and a density rS=0.001 nm-2. b) Line cut at af=0.2° of left image showing the intensity, the form factor and the interference function.

figures/DALMA.gif
Figure 3.6: GISAXS patterns computed between -0.5° < 2 q < 0.5°;0 < af < 1° using the a) DA b) LMA models (see Sect. 1.2). The DWBA form factor is that of a cylinder with the morphological parameters: R0=5 nm-H0/R0=1-s/R=0.5 (gaussian). The interference function is given by the isotropic paracrystal model model with a preferential distance D=20 nm and a gaussian disorder s/D=0.25. c) Line cut at af=0.2° of both images showing the intensity, the mean form factor, the incoherent signal and the interference function.

3.2.2  Regular lattice

For a regular lattice, the interference function is built on diffraction rods perpendicular to the surface whose widths are inversely proportional to the correlation length L. In this case, the GISAXS figure can be interpreted in terms of the Ewald construction as shown in Fig. 3.7.

figures/ewald.gif
Figure 3.7: The Ewald construction in the case of bidimensional regular lattice.

figures/lattice.gif
Figure 3.8: The GISAXS image in the case of a regular square lattice of size a=b=10 nm with a correlation length of L = 200 nm seen between 0 < 2 q < 1.5°; 0 < af < 1.5° for two orientations with respect to the incident beam a) x = 0° b) x = 1.5°. The form factor is that of a cylinder (R=2 nm;H/R=1).

The directions of scattering are given by the intersection of the Ewald sphere and of the rods of scattering. In the small angle range and for typical island interspacing, only the first and second order of diffraction can be seen. For an angle between ki|| and the lattice x = 0, these rods are tangential to the Ewald sphere leading to rods on the GISAXS image (Fig. 3.8-a). By rotating slightly the lattice, the first and second rods drills the Ewald sphere at af ¹ 0 leading to a concentration of intensity in a out of plane spot (see Fig. 3.8-b).







Chapter 4
Future developments-improvements

The planned developments for the hard core program IsGISAXS are the following:

Of course, a constant improvement of the working environments is under progress.







License agreement

The program IsGISAXS was developed by Rémi Lazzari in collaboration with Gilles Renaud, Christine Revenant-Brizard and Fréderic Leroy from CEA-Grenoble (Départememt de Recherche Fondamentale sur la Matière Condensée/Service de Physique des Matériaux et Microstructures/Interface et Rayonnememt Synchrotron). It is freely available for non-commercial use, and is provided as-is without any warranty. Any publication resulting from this software should acknowledge its use. The author disclaims any problems which could result from the use of the program.









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List of Figures

    1  The scattering and diffraction ranges versus the wavevector transfer.
    1.1  Sketch of the grazing incidence geometry: an incident wave of wavevector \@mathbf ki is scattered in the direction \@mathbf kf.
    1.2  Supported geometries for island shapes in IsGISAXS (Left: side view - Right: top view).
    1.3  Supported geometries for island shapes in IsGISAXS (Left: side view - Right: top view).
    1.4  Supported geometries for island shapes in IsGISAXS .
    1.5  The particle layer geometry developed in IsGISAXS . The geometry D is planned for future development.
    1.6  The four terms involved in the scattering by a supported island. The fist term corresponds to the simple Born approximation.
    1.7  The interference fringes for the form factor of a cylinder of height H=5 nm (l = 1  A-d = 5 10-6,b = 2 10-8) as function of the exit angle af normalized by the angle of total external reflection ac within the various approximations: BA, DWBA(ai=ac/2,ac,2ac), layer BA(ai=ac).
    1.8  The modulus square of the four terms Fig. 1.6 for a cylinder as function of af for various ai: Term 1(red)-Term 2(green)-Term 3(orange)-Term 4(blue) (same parameters as in Fig. 1.7).
    1.9  The phase of the four terms Fig. 1.6 for a cylinder as function of af for various ai: Term 1(red)-Term 2(green)-Term 3(orange)-Term 4(blue) (same parameters as in Fig. 1.7).
    1.10  The hard-core pair correlation function a) and the interference function b) as function of the surface coverage h = rS ps2/4, with s the hard core diameter. The functions are plotted against reduced parameters r/s and qs/2p.
    1.11  The schematic view of the one dimensional paracrystal.
    1.12  The pair correlation function a) and the interference function b) in the case of the 1D disordered lattice for various disorder parameter w/D.
    1.13  The schematic view of a paracrystal in two dimensions. Each circle represent the area where the probability of finding one island is maximum.
    1.14  The interference function a) and the pair correlation function b) for a paracrystal of hexagonal symmetry averaged over all the azimuths z. The gaussian disorder parameter s/D is indicated on the figure. The axis are normalized by the lattice parameter D and the expected peaks position 1,Ö3,2,Ö7,3,2Ö3,Ö{13},4 are marked with a circle.
    2.1  The "Display Image Parameters" and "Graph Parameters" boxes.
    2.2  1214.5 plus3 minus7 plus3 plus3.5 minus3 plus2.5 minus plus4 minus6 plus2.5 minus plus2.5 minus plus4 minus6 plus2.5 minus A typical input file *.inp
    2.3  A typical morphology file *.mor
    2.4  A typical input file *.fit for data fitting
    2.5  A typical data file *.dat
    3.1  Calculated form factor for a cylinder (R=5 nm; H/R=1) for 0 < 2 q < 2°; 0 < af < 2° in the Distorted Wave Born Approximation for two angles of incidence a) ai=ac and b) ai=2ac. Same color scale.
    3.2  Calculated form factor for a truncated pyramid (R=5 nm; H/R=1; a = 54.73° angle between (111) and (100) planes in cubic system) at ai=ac for 0 < 2 q < 3°; 0 < af < 3° in the Distorted Wave Born Approximation for two angles between the direct beam and the island edge a) z = 0° b) z = 45°.
    3.3  Cut along q|| at a = ac for the mean form factor < |F|2 > of a cylinder (Born Approximation). The size distribution is gaussian (R0=5 nm) with various broadening s/R).
    3.4  Mean form factor < |F|2 > for a gaussian distribution of pyramidal shape island (R0=5 nm-s/R=0.2). Same parameters as Fig. 3.2.
    3.5  a) GISAXS pattern computed between -1.2° < 2 q < 1.2°;0 < af < 1.5° using the LMA model (see Sect. 1.2). The DWBA form factor is that of a cylinder with the morphological parameters: R=5 nm-H/R=1-s/R=0.2 (gaussian). The interference function is given by the Debye model with a hard core radius of R0=7.5 nm and a density rS=0.001 nm-2. b) Line cut at af=0.2° of left image showing the intensity, the form factor and the interference function.
    3.6  GISAXS patterns computed between -0.5° < 2 q < 0.5°;0 < af < 1° using the a) DA b) LMA models (see Sect. 1.2). The DWBA form factor is that of a cylinder with the morphological parameters: R0=5 nm-H0/R0=1-s/R=0.5 (gaussian). The interference function is given by the isotropic paracrystal model model with a preferential distance D=20 nm and a gaussian disorder s/D=0.25. c) Line cut at af=0.2° of both images showing the intensity, the mean form factor, the incoherent signal and the interference function.
    3.7  The Ewald construction in the case of bidimensional regular lattice.
    3.8  The GISAXS image in the case of a regular square lattice of size a=b=10 nm with a correlation length of L = 200 nm seen between 0 < 2 q < 1.5°; 0 < af < 1.5° for two orientations with respect to the incident beam a) x = 0° b) x = 1.5°. The form factor is that of a cylinder (R=2 nm;H/R=1).

Footnotes:

1The program with instructions is available on simple request to the author or at http:\\www .esrf.fr.

2As no dependence of the index of refraction is accounted for, the wavelength distribution should not be two large or should not cross an absorption threshold.

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