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Home > Nanotechnology Columns > UAlbany College of Nanoscale Science and Engineering > Goldilocks, X-rays, and Nanotechnology

Richard Matyi
Professor of Nanoscience
UAlbany - College of Nanoscale Science & Engineering

Abstract:
X-rays - electromagnetic radiation with wavelengths that are typically of the order of 0.05 nm to 1 nm - are finding growing importance in nanoscale measurement technology and metrology. Their sub-nanometer wavelengths and their typical weak interactions with solids make X-ray probes a nearly ideal way of studying the structural characteristics of thin layer and nanoscaled structures that underlie much of modern nanotechnology. The fact that the probing wavelength is commensurate with the sizes of nanostructured objects results in interactions (in particular, scattering processes) that occur over practically measureable length and angular scales.

September 30th, 2008

Goldilocks, X-rays, and Nanotechnology

X-rays - electromagnetic radiation with wavelengths that are typically of the order of 0.05 nm to 1 nm - are finding growing importance in nanoscale measurement technology and metrology. Their sub-nanometer wavelengths and their typical weak interactions with solids make X-ray probes a nearly ideal way of studying the structural characteristics of thin layer and nanoscaled structures that underlie much of modern nanotechnology. The fact that the probing wavelength is commensurate with the sizes of nanostructured objects results in interactions (in particular, scattering processes) that occur over practically measureable length and angular scales.

But the second characteristic - the non-perturbing, weak interactions between X-rays and solids - is equally important. Mathematically, this property means that one can treat X-ray scattering using a wide array of mathematical tools and describe in full quantitative detail all phases of the scattering process. A notable counter-example is found in the interactions of electron beams with solids, where the strong coupling between energetic electrons and a target's atoms can make a full description of the scattering process a difficult task. Fortunately, the X-ray scattering cross-section for most atoms is not vanishingly small, so meaningful information in the form of reasonably intense scattering can be gleaned from many X-ray experiments in a tolerable time frame (minutes down to seconds) even in a conventional laboratory environment. It is this favorable combination that leads to the "Goldilocks principle": the interactions of X-rays with solids are not too strong, not too weak, but just right.

For years scientists have labored to work around any complications caused by the weak X-ray cross-section in order to best utilize the wealth of analytical approaches that X-ray scattering provides. X-rays are scattered primarily by the electrons in a solid, and quantitatively the amplitude of the single electron scattering is proportional to a value known as the classical electron radius, a fundamental physical parameter with a magnitude of about 2.818x10^-15 meter. This means that the scattering from a single electron is incredibly weak; fortunately, a material such as pure crystalline silicon has about 5x10^22 atoms in a cubic centimeter, and each Si atom is surrounded by 14 electrons. In a crystalline solid all of these electrons can be persuaded to scatter synchronously, so that the amplitudes from all of theses individual electrons will sum together - but this only happens in certain angular directions with respect to the incident beam direction. As a result of this cooperative scattering (known as X-ray diffraction), the angular positions and the intensities of diffracted X-ray beams are determined by the types of atoms in crystalline solid and their relative positions. X-ray diffraction has thus become an indispensible tool in the study of the structure of solids in fields ranging from biology and chemistry to materials science and physics.

Nanotechnology has, of course, benefited from conventional X-ray diffraction methods just like these more traditional fields. It turns out that, in many cases, the atomic structure of a given nanomaterial is relatively well known; what is needed is information on form or morphology. For instance, the performance of many structured nanomaterials requires the synthesis of crystallites with sizes in the range of nanometers. X-ray scattering has long been used for crystal size measurements, because the angular range over which a single crystal diffracts in inversely proportional to crystal size - the smaller the crystal, the greater the angular breadth.

But what can we do if a nanomaterial is not crystalline? In this case we can deal with the weak scattering of X-rays by performing our measurements in the regime where the intensity is bound to be high: in the direct vicinity of the primary transmitted X-ray beam. If an X-ray beam can be transmitted through a nanostructured solid, then heterogeneities in that solid can scatter X-rays away from the main beam direction. Small-angle X-ray scattering (SAXS) has a direct analogue to the physical processes the appearance of the "haze" that surrounds a streetlight on a foggy night - the smaller the water droplets in the fog, the further away the haze will extend, and the "thicker" the fog, the brighter the haze will be. SAXS is a powerful method for characterizing the structure of nanomaterials and yields information such as the average particle size, particle shape and size distribution, surface area, and concentration - all critical parameters for materials used in nanotechnology.

Even weakly-interacting laboratory X-rays will be absorbed by most solids after a few tens of microns of penetration, however, thus preventing the transmission of X-rays through many samples of interest. Fortunately, in many cases we can utilize the "index of refraction", which is the measure of how much a beam of electromagnetic radiation is "bent" when it enters a medium, to our advantage. The index of refraction n is unity for light in vacuum, while n is about 1.5 for materials such as glass. One consequence for this is that light inside a glass prism can experience total internal reflection at relatively large angles of incidence.

In contrast, the weak interaction of X-rays with matter leads to an X-ray refractive index that differs from unity by a very small amount (a few parts in 100,000) and is slightly less than unity. X-rays impinging on a solid surface will thus experience total external reflection if they impinge on a solid at relatively small angles of incidence. For instance, with a material such as silicon and typical laboratory X-rays, the "critical angle" for total external reflection is usually around a quarter of a degree. Above this angle, X-rays will begin to penetrate the solid and will be absorbed, thus reducing the reflected intensity - but the transmitted X-rays can be reflected from materials interfaces within the solid and interfere with those reflected from the top surface. In the case of thin film samples, this can lead to a complex reflectivity pattern of angle-dependent intensity maxima and minima, with the angular positions and intensities being highly sensitive to the thickness and the chemical structure of the thin film system. Specular X-ray reflectometry (XRR) has thus emerged as a popular method for analyzing thin film systems, and like other X-ray methods, the relatively weak interactions allow us to mathematically model the resultant reflected intensity from films with any physical structure - single crystal, polycrystalline, or amorphous - with high accuracy and confidence.

Often in nanotechnology we encounter complex materials structures that are stratified in depth but also contain lateral inhomogeneities. Interrogation of a sample like this with grazing incidence X-rays would be expected to generate multiple signals: a specular reflection signature that carries the information of the thin film structure, and an off-specular signal formed by the spatial heterogeneities. X-ray scattering theory is sufficiently robust to handle the interactions of these mixed contributions, making this analytical method - grazing incidence small angle X-ray scattering, or GISAXS - an extremely powerful tool for characterizing complex nanostructures.

Thus X-ray scattering offers numerous avenues that can lead to fully quantitative measurement options for nanomaterials systems. At their essence, these techniques all rely on Goldilocks principle - making X-ray methods "just right" for metrology at the nanoscale.

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