Raman Spectroscopy of nanomaterials - institut de chimie et des ...

ARTICLE IN PRESS+ MODEL16 G. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56a consequence of the long range Coulomb interaction. In spherical crystals of Blendetype semiconductors [223,226,228]:vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi‘ þ 1‘ n2 TO þ 3 Nn 2 LO3n SO ¼ uMt ‘ þ 1þ 3 ð14ÞN‘ 3 MIn Eq. (14), ‘ ¼ 1; 2; 3; .; 3 N is the high frequency dielectric constant of the semiconductorand 3 M is the frequency-independent dielectric constant of the surrounding medium.The radial dependence of the surface modes is in the r ‘ 1 form and only the ‘ ¼ 1 surfacemode (the so-called Fröhlich mode), with its constant amplitude, is thus expected to makea significant contribution. Surface optical modes were similarly described for semiconductorsof the Wurtzite type [225] and metals [229]. Surface/interface modes for the specificgeometries of nanocylinders and nanowires [221], spherically capped QDs/Quantum Wells(QWs) [230] and Multiple QWs (MQWs) [231,232] have also been addressed.(iv) The confinement function has limited physical meaning: most criticisms formulated againstthe PCM concerned the arbitrariness of the confinement function. As a matter of fact, thePCM only is a phenomenological tool and even its best advocates acknowledge it cannotbe expected to fully account for the lineshape when it relies on propagating phonons(‘‘bulk’’ dispersion curves) to describe confined modes. Alternative and more ‘‘physical’’descriptions of the optical modes confined in nanocrystals were mostly proposed by semiconductorspecialists. First, it was predicted [233] and confirmed experimentally for PbS[234] that the TO and LO modes are coupled in nanocrystals. Besides, the dipoles generatedin polar semiconductors by optical vibrations generate electromagnetic fields and thusmay interact with electrons (polarons) or electronehole pairs (excitons). This so-calledFröhlich interaction is weak in single crystals but leads to strong resonances of the LOand SO modes in confined semiconductors: quantum dots, wires and superlattices[195,219,223,233,235e239]). Pusep et al. [240] tried to adapt the PCM to the electroneLO coupling in doped semiconductors but an accurate description would require a continuousapproach like the one proposed by Roca et al. [233], later improved by Vasilevskiyet al. [241]. Note that in conducting/coloured nanomaterials (carbon, conducting polymers,etc.) and superlattices, an additional Fano coupling is possible between the continuum ofelectronic states and the discrete energy levels of the phonons [211,242e245].3.3.2. The Elastic Sphere Model (ESM)The use of bulk dispersion curves is questionable when there is a lot of reduction in particlesize [9,237]. Meyer et al. [246] used Molecular Dynamics Simulations to calculate the nanoparticleVDOS and demonstrated the importance of grain boundary-related contributions. Analternative to considering the vibrations as ensuing from a disturbed infinite crystal (what thePCM does) is to adopt a first principle description of vibrations in a free sphere. This problemwas theoretically discussed in 1882 by Lamb [247]. A brief description will be given but detailedinformation can be found elsewhere (with a broad spectrum of notations) [248e250,252,253,255,256].Imagine a sphere of radius R with vector ~uðM; tÞ as the displacement field induced by propagatingwaves at a given point (M) and for a given time (t). Assuming the sphere isPlease cite this article in press as: G. Gouadec, Ph. Colomban, Prog. Cryst. Growth Charact. Mater. (2007),doi:10.1016/j.pcrysgrow.2007.01.001