Determination of Phonon Dispersions from X-Ray Transmission Scattering: The Example of Silicon

Abstract

A beam of monochromatic synchrotron x-ray incident on a silicon wafer creates a rich intensity pattern behind the wafer that reflects the cross section of scattering by thermally populated phonons. A least-squares fit of the patterns based on a lattice dynamics calculation yields the phonon dispersion relations over the entire reciprocal space. This simple and efficient method is suitable for phonon studies in essentially all materials, and complements the traditional neutron scattering technique. PACS numbers: 78.70.Ck, 63.20.Dj Phonons are the fundamental quanta of lattice vibration in a solid. They play a critical role in phenomena such as superconductivity and many types of phase transitions, and are the basis for the acoustic, thermal, elastic, and infrared properties of solids [1]. A fundamental description of phonons is the dispersion relation, which is determined traditionally through neutron scattering [2][3][4] or, more recently, through inelastic x-ray scattering [5]. However, these methods can be technically demanding, and alternative, complementary methods have been long sought. This study demonstrates a new approach based on x-ray intensity patterns produced by scattering from thermally populated phonons using silicon as a test case. Intensity patterns with a wide dynamic range were recorded in a matter of seconds at the third-generation synchrotron, the Advanced Photon Source. A least-squares analysis in terms of lattice dynamics yields dispersions for all six phonon branches in excellent agreement with neutron scattering results. The fast data acquisition rate, simplicity of the experiment, and a minimal requirement on sample volume make this method attractive for a wide range of applications in materials research.Although intensity distribution of x-ray scattering by thermally populated phonons has been long recognized as a possible measure of phonon properties [6][7][8], this method has remained impractical and attracted little interest. The situation has changed due to recent advances in synchrotron radiation instrumentation and computational power. Undulator beams at third-generation synchrotrons, such as the Advanced Photon Source, now yield a brightness about 8 orders of magnitude higher than a conventional laboratory source. In addition, the use of two-dimensional detectors such as the image plate used in this experiment allows parallel detection over a large scattering solid angle, effectively increasing the data collection rate by 6 orders of magnitude. The combined improvement makes it possible to carry out such measurements with a high degree of precision and efficiency. Intensity patterns recorded from both Si (111) and Si(100) in a transmission mode, displayed on a logarithmic scale, reveal details of intensity variation that are uniquely related to phonon modes over a wide range in the reciprocal space, thus enabling a determination of the phonon dispersions.Our experiment was performed at the undulator beam line of Sector 33 (University, Industry, and National Laboratory Collabor...

In-between Bragg reflections: thermal diffuse scattering and vibrational spectroscopy with x-rays

“…In the pioneering work of Holt [4] the thermal diffuse scattering from silicon was recorded for a set of high-symmetry orientations in order to fit the data by a Born-von Kármán model. A rather high photon energy (28 keV) and an image plate detector with rather long readout time was used.…”

“…Historically, phonon dispersion was first determined from thermal diffuse scattering (TDS) [3]; however, the potential of TDS remained largely unexploited and lattice dynamics studies were predominantly conducted using inelastic neutron (INS) and inelastic xray scattering (IXS). Recently, the advent of third generation synchrotron sources with their outstanding brilliance, in conjunction with the development of fast large area detectors has led to a revival of TDS studies [4,5].…”

DOE A9024 Final Report Functional and Nanoscale Materials Systems: Frontier Programs of Science at the Frederick Seitz Materials Research Laboratory

“…Detailed numerical analysis is underway, and a new algorithm is under development in collaboration with Zuo. In related work, Chiang and Zuo are developing methods based on x-ray and electron-beam thermal diffuse scattering to extract information about lattice dynamics and phase transitions in materials [43,44,45,46] . Fig.…”

Reflection thermal diffuse x-ray scattering for quantitative determination of phonon dispersion relations

“…[0 0 1] and [1 1 0]. Pioneering work by Chiang, Holt, and coworkers demonstrated the capabilities of TDS measurements by determining hω j (q) phonon dispersion curves for Si [10], Ga-stabilized Pu [11], TiSe 2 [12], and Cu [13] crystals. However, TDS experiments are presently carried out in transmission geometry and, as a result, samples are limited to bulk crystals.…”