On the Raman shift in nanosized crystals

An analytical form of the Raman shift dependence on size of nanocrystals is presented. Based on the hard confinement model, it works in those cases where the average phonon curve shows a quadratic dependence on the phonon quasi-momentum in the range of interest. PACS : 78.67.Bf, 78.30.-j, 78.30.Am, 36.20.Ng Raman spectroscopy is an important tool in gathering information about molecular and crystal vibrational properties. Raman spectra can show significant changes when sizes of investigated crystals go down to the nanometer scale. Strain, stress and non-stoichiometry can be invoked to explain these changes [1, 2]. However, many published works have shown that in most cases the main cause of spectra changes is to be ascribed to the phonon confinement (PC) effect [3 − 9]. Actually, the confinement of the q0 = 0 phonon wave function in nanometric-sized crystals makes accessible to Raman investigation a significant portion of the Brillouin Zone (BZ), whose extension increases as the crystal size decreases [3 − 5]. Besides some specific cases [5, 10, 11], a direct relation, even when approximated, which explicitly shows the connection between Raman shift and crystal size, depending on suitable model parameters, is still not available. In this letter we present a detailed numerical analysis of the hard confinement model (HC) which yields an unexpectedly simple analytical form capable of closely reproducing the model predictions. It uses the quadratic approximation of the crystal phonon curve ω(q) which, in several practical cases, holds down to a few nanometers of crystal sizes. Formally, the PC effect can be explained by considering the modulation of the phonon wave function in the infinite crystal with a suitable ”weighting” function [4]. On expansion by Fourier integrals, we can calculate the first order Raman spectrum of a nano-sized crystal as