Quantum‐chemical perspective of nanoscale Raman spectroscopy with the three‐dimensional phonon confinement model

Abstract

Raman spectroscopy of crystalline/molecular systems is well backed with quantum chemical calculations and group theory, making it a unique characterization tool. For the "intermediate" case of nanoscale systems, however, the use of Raman spectroscopy is limited by the lack of such theoretical bases. Here, we suggest to couple a scaled quantum-mechanical (SQM) calculation with the phonon confinement model (PCM) to construct a universal and physically consistent basis for nanoscale Raman spectroscopy. Unlike the commonly used one-dimensional dispersion PCM, we take into account the confinement along all the three dimensions of the k-space. We apply it to diamond nanoparticles of sub-50nm size, a system with pronounced anisotropy of dispersion for which the use of three-dimensional dispersion is a requisite. The model excellently reproduces sizesensitive spectral features, including the peak position, bandwidth and asymmetry of the sp 3 C-C Raman band. This fundamental approach can be easily generalized to other nanocrystalline solids to hopefully contribute the future development of quantitative nanoscale Raman spectroscopy. Keywords: phonon confinement, nanoparticlesThe continuously growing interest to nanoscale matter drives a strong demand for fast, noninvasive and statistically reliable characterization methods. This makes Raman spectroscopy an indispensable tool for nanoscience. The difficult challenge of Raman spectroscopy of nanoscale materials is to establish a consistent way to link the Raman pattern to the characteristic size of the crystallites. The straightforward approach is to directly calculate the normal modes of nanoparticles as big molecules [1,2] . This requires assumptions of the shape, size and symmetry. The calculated spectra are the set of narrow lines. Since the number of normal modes for nanoparticles is in the order of thousands, such calculations do not seem to be routinely availbale for real systems. The other common approach is the elastic sphere model (ESM), in which the nanoparticle is approximated by a homogeneous isotropic freestanding elastic sphere. The vibrations of such system can be easily derived from first principles calculations [3] . For the nanoparticles of other shapes, as well as systems with the size smaller than ~5nm, the applicability of ESM is limited. For more details on ESM the reader is referred to the review [4] . The third general approach starts from the Raman spectrum of a bulk crystal, for which the selection rules allow only the phonons close to the Brillouin zone (BZ) center. Nano-size crystallites lack the translational symmetry and the quasimomentum preservation does not hold. As a result, BZ points away from the center can also contribute to the Raman spectra. To what extent the selection rule breaks depends on the size of the crystallite. This effect is named the "phonon confinement" after the work of Richter [5] , who first suggested the physical model for describing it. The approach by Richter's is to multiply the phonon wavefunction of the crystal φ...

Localized phonons in Raman spectra of nanoparticles and disordered media

“…Spreading of allowed q‐vectors upon different confinement (a) for 32, 8, and 2 lattice constants; corresponding calculated Raman spectra of nanodiamond (b) and phonon dispersion of diamond (c), where the areas with the dominant contribution for the three confinement scales are marked. In (c), solid lines correspond to calculations, whereas markers are the experimental data…”

Phonon confinement and size effect in Raman spectra of ZnO nanoparticles

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Raman Spectra of Nonpolar Crystalline Nanoparticles: Elasticity Theory-like Approach for Optical Phonons

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