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 natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, propertie ...
,
physics
Physics is the natural science that studies matter, its 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 which re ...
, and
materials science 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
* Su ...
,
thin film
A thin film is a layer of material 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 based on the use of
X-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&n ...
s and is related to the techniques of
neutron reflectometry
Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films, similar to the often complementary techniques of X-ray reflectivity and ellipsometry. The technique provides valuable information over a wide va ...
and
ellipsometry.
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 technique appears to have first been applied to X-rays by
Lyman G. Parratt in 1954. Parratt's initial 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:
:
Here
is the reflectivity,
,
is the X-ray wavelength (typically copper's
K-alpha peak at 0.154056 nm),
is the density deep within the material and
is the angle of incidence. Below the critical angle
(derived from
Snell's law
Snell's law (also known as Snell–Descartes law and 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 through ...
), 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 (which depend on
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of ever ...
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 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 (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
. 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. I ...
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 ge ...
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.
Open source software
Diffractometer manufacturers typically provide commercial software to be used for X-ray reflectivity measurements. However, several open source software packages are also available: GenX is a commonly used open source X-ray reflectivity curve fitting software. It is 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-typed and garbage-collected. It supports multiple programming par ...
and runs therefore on both Windows and Linux. Motofit runs in the
IGOR Pro
IGOR Pro is a scientific data analysis software, numerical computing environment and programming language that runs on Windows or Mac operating systems. It is developed by WaveMetrics Inc., and was originally aimed at time series analysis, but ha ...
environment, and thus cannot be used in open-source operating systems such as Linux. Micronova XRR runs under
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
and is therefore available on any operating system on which Java is available.
Reflex
is a standalone software dedicated to the simulation and analysis of X-rays and neutron reflectivity from multilayers. REFLEX is a user-friendly freeware program working under Windows, Mac and Linux platforms.
References
{{Reflist
X-ray scattering