X-ray reflectivity (sometimes known as X-ray specular reflectivity, X-ray reflectometry, or XRR) is a surface-sensitive analytical technique used in
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
,
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, and
materials science
Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries.
The intellectual origins of materials sci ...
to characterize
surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surface or surfaces may also refer to:
Mathematics
*Surface (mathematics), a generalization of a plane which needs not be flat
* Sur ...
,
thin film
A thin film is a layer of materials ranging from fractions of a nanometer ( monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many ...
s and
multilayers.
[J. Als-Nielsen, D. McMorrow, ''Elements of Modern X-Ray Physics'', Wiley, New York, (2001).] It is a form of
reflectometry
Reflectometry is a general term for the use of the reflection of waves or pulses at surfaces and interfaces to detect or characterize objects, sometimes to detect anomalies as in fault detection and medical diagnosis.
There are many different ...
based on the use of
X-ray
An X-ray (also known in many languages as Röntgen radiation) is a form of high-energy electromagnetic radiation with a wavelength shorter than those of ultraviolet rays and longer than those of gamma rays. Roughly, X-rays have a wavelength ran ...
s and is related to the techniques of
neutron reflectometry and
ellipsometry
Ellipsometry is an optical technique for investigating the dielectric properties (complex refractive index or dielectric function) of thin films. Ellipsometry measures the change of polarization upon reflection or transmission and compares it ...
.
The basic principle of X-ray reflectivity is to reflect a beam of X-rays from a flat surface and to then measure the intensity of X-rays reflected in the specular direction (reflected angle equal to incident angle). If the interface is not perfectly sharp and smooth then the reflected intensity will deviate from that predicted by the law of
Fresnel reflectivity. The deviations can then be analyzed to obtain the density profile of the interface normal to the surface.
History
The earliest measurements of X-ray reflectometry were published by Heinz Kiessig in 1931, focusing mainly on the total reflection region of thin nickel films on glass. First calculations of XRR curves were performed by
Lyman G. Parratt in 1954. Parratt's work explored the surface of copper-coated glass, but since that time the technique has been extended to a wide range of both solid and liquid interfaces.
Approximation
When an interface is not perfectly sharp, but has an average electron density profile given by
, then the X-ray reflectivity can be approximated by the so called Master formula:
:
Here
is the reflectivity,
,
is the X-ray wavelength (e.g. copper's
K-alpha peak at 0.154056 nm),
is the density deep within the material and
is the angle of incidence.
The Fresnel reflectivity,
, in the limit of small angles where polarization can be neglected, is given by:
:
Here
is the wavevector
inside the material,
and the critical angle
, with
the Thomson scattering length.
Below the critical angle
(derived from
Snell's law
Snell's law (also known as the Snell–Descartes law, the ibn-Sahl law, and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing th ...
), 100% of incident radiation is reflected through
total external reflection,
. For
,
. Typically one can then use this formula to compare parameterized models of the average density profile in the z-direction with the measured X-ray reflectivity and then vary the parameters until the theoretical profile matches the measurement.
Oscillations
For films with multiple layers, X-ray reflectivity may show oscillations with Q (angle/wavelength), analogous to the
Fabry-Pérot effect, here called
Kiessig fringes. The period of these oscillations can be used to infer layer thicknesses, interlayer roughnesses, electron densities and their
contrasts, and complex
refractive indices
In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
(which depend on
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of its atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (''n''p) or the number of pro ...
and
atomic form factor
In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident ...
), for example using the
Abeles matrix formalism
The transfer-matrix method is a method used in optics and acoustics to analyze the propagation of electromagnetic or acoustic waves through a stratified medium; a stack of thin films. This is, for example, relevant for the design of anti-refle ...
or the recursive Parratt-formalism as follows:
:
where X
j is the ratio of reflected and transmitted amplitudes between layers j and j+1, d
j is the thickness of layer j, and r
j,j+1 is the
Fresnel coefficient for layers j and j+1
:
where k
j,z is the z component of the
wavenumber
In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of ...
. For specular reflection where the incident and reflected angles are equal, Q used previously is two times k
z because
. With conditions R
N+1 = 0 and T
1 = 1 for an N-interface system (i.e. nothing coming back from inside the semi-infinite substrate and unit amplitude incident wave), all X
j can be calculated successively. Roughness can also be accounted for by adding the factor
:
where
is a standard deviation (aka roughness).
Thin film thickness and critical angle can also be approximated with a linear fit of squared incident angle of the peaks
in rad
2 vs unitless squared peak number
as follows:
:
.
Curve fitting
X-ray reflectivity measurements are analyzed by fitting to the measured data a simulated curve calculated using the recursive Parratt's formalism combined with the rough interface formula. The fitting parameters are typically layer thicknesses, densities (from which the index of refraction
and eventually the wavevector z component
is calculated) and interfacial roughnesses. Measurements are typically normalized so that the maximum reflectivity is 1, but normalization factor can be included in fitting, as well. Additional fitting parameters may be background radiation level and limited sample size due to which beam footprint at low angles may exceed the sample size, thus reducing reflectivity.
Several fitting algorithms have been attempted for X-ray reflectivity, some of which find a local optimum instead of the global optimum. The
Levenberg-Marquardt method finds a local optimum. Due to the curve having many interference fringes, it finds incorrect layer thicknesses unless the initial guess is extraordinarily good. The derivative-free
simplex method
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
also finds a local optimum. In order to find global optimum, global optimization algorithms such as simulated annealing are required. Unfortunately, simulated annealing may be hard to parallelize on modern multicore computers. Given enough time,
simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. ...
can be shown to find the global optimum with a probability approaching 1, but such convergence proof does not mean the required time is reasonably low. In 1998, it was found that
genetic algorithm
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to g ...
s are robust and fast fitting methods for X-ray reflectivity. Thus, genetic algorithms have been adopted by the software of practically all X-ray diffractometer manufacturers and also by open source fitting software.
Fitting a curve requires a function usually called fitness function, cost function, fitting error function or figure of merit (FOM). It measures the difference between measured curve and simulated curve, and therefore, lower values are better. When fitting, the measurement and the best simulation are typically represented in logarithmic space.
From mathematical standpoint, the
fitting error function takes into account the effects of Poisson-distributed photon counting noise in a mathematically correct way:
:
.
However, this
function may give too much weight to the high-intensity regions. If high-intensity regions are important (such as when finding mass density from critical angle), this may not be a problem, but the fit may not visually agree with the measurement at low-intensity high-angle ranges.
Another popular fitting error function is the 2-norm in logarithmic space function. It is defined in the following way:
:
.
Needless to say, in the equation data points with zero measured photon counts need to be removed. This 2-norm in logarithmic space can be generalized to p-norm in logarithmic space. The drawback of this 2-norm in logarithmic space is that it may give too much weight to regions where relative photon counting noise is high.
Neural network analysis of XRR
The application of neural networks (NNs) in X-ray reflectivity (XRR) has gained attention for its ability to offer high analysis speed, noise tolerance and its ability to find global optima. Neural networks offer a fast and robust alternative to fit programs by learning from large synthetic datasets that are easy to calculate in the forward direction and providing quick predictions of material properties, such as layer thickness, roughness, and density. The first application of neural networks in XRR was demonstrated in the analysis of thin film growth, and a wide range of publications has explored the possibilities offered by neural networks, including free form fitting, fast feedback loops for autonomous labs and online expeirmnet control.
One of the main challenges in XRR is the non-uniqueness of the inverse problem, where multiple Scattering Length Density (SLD) profiles can produce the same reflectivity curve. Recent advances in neural networks have focused on addressing this by designing architectures that explore all possible solutions, providing a broader view of potential material profiles. This development is critical in ensuring that solutions are not confined to a single, potentially incorrect branch of the solution space.
Open source software
An up to date overview over current analysis software can be found in the following link. Diffractometer manufacturers typically provide commercial software to be used for X-ray reflectivity measurements. However, several open source software packages are also available: Refnx and Refl1D for X-ray and neutron relectometry, and GenX are commonly used open source X-ray reflectivity curve fitting software. They are implemented in the
Python programming language
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation.
Python is dynamically type-checked and garbage-collected. It supports multiple prog ...
and runs therefore on both Windows and Linux. Reflex is a standalone software dedicated to the simulation and analysis of X-rays and neutron reflectivity from multilayers. Micronova XRR runs under
Java
Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...
and is therefore available on any operating system on which Java is available.
Documented neural network analysis packages such as MLreflect have also become available as an alternative approach to XRR data analysis recently.
References
{{X-ray science
X-ray scattering