Brillouin zone momentum conservation

G. Herzberg, Molecular Spectra and Molecular Structure, Vol. II - Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold, New York, 1945 In the crystalline state, it is more convenient to speak about multi-phonon processes since the modes from the whole dispersion range of the first Brillouin zone are allowed to contribute according to the conservation of energy and momentum of the phonons involved in the process... 

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

Different models have been used to derive the particle size from Raman spectra As an example, we shah briefly explain the phonon confinement model (PCM). The scattering of one photon by n phonons is governed by the momentum conservation. Only vibrations from the center of the Brillouin zone (BZC) should therefore be active in one phonon process (first-order Raman spectrum) and this is actually the case in large and flawless crystals, where... 

Ax 1 cm, the uncertainty of the momentum, Ap, is small. Thus, the momentum is a good quantum number and its conservation has to be obeyed in electronic transitions. In silicon, which has an indirect band gap, the recombination of electrons and holes requires a creation or annihilation of phonons and is therefore predominantly of non-radiative nature.If, however, the size of an Si nanocrystal approches 1 nm. Ax 1 nm and Ap spans a significant part of the Brillouin zone. The momentum conservation is relaxed and electronic transitions (absorption of a photon with the formation of an electron-hole pair or their radiative recombination) become efficient. 

With regard to Raman scattering or infrared absorption, one of the major tasks consists of assigning the experimentally observed lines to the different irreducible representations (species) predicted by a previous group theory analysis (see Sec. V.F) based on the supposedly known space group of the molecule or crystal. The species are labeled by the irreducible representations of the point group at the center of the first Brillouin zone because of the momentum conservation rule evoked earlier. 

The phonon confinement model [23] attributes the redshift of the asymmetric Raman line to relaxation of the -vector selection rule for the excitation of the Raman active phonons due to their locahzation. The relaxation of the momentum conservation rule arises from the finite crystalline size and the diameter distribution of the nanosolid in the films. When the size is decreased, the rule of momentum conservation will be relaxed and the Raman active modes will not be limited at the center of the Brillouin zone [21]. The large surface-to-volume ratio of a nanodot strongly affects the optical properties mainly due to introducing surface polarization states [28]. 

Because of the momentum conservation rule during the light scattering process, first-order Raman scattering is caused by phonons at the Brillouin zone center. The Raman scattering efficiency S with polarization detection is given by ... 

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