Many particle approach to resonance Raman scattering in crystals: Strong electron–phonon interaction and multi-phonon processes A.M. Yaremko a , V.V. Koroteev a , V.O. Yukhymchuk a , V.M. Dzhagan a , H. Ratajczak b,⇑ , A.J. Barnes c , B. Silvi d a Institute of Semiconductor Physics of the National Academy of Sciences of Ukraine, 45 Prospect Nauky, Kyiv 03028, Ukraine b Faculty of Chemistry, University of Wrocław, ul. F. Joliot-Curie 14, 50-383 Wrocław, Poland c Institute of Materials Research, University of Salford, Salford M5 4WT, UK d Laboratory of Theoretical Chemistry, University Pierre and Marie Curie, Place Jussieu, 75252 Paris Cedex 05, France article info Article history: Received 2 March 2011 In final form 20 July 2011 Available online 29 July 2011 Keywords: Raman scattering Electron–phonon interaction Multi-phonon excitations abstract Raman scattering (RS) of light by crystals was studied theoretically taking into account the electron–elec- tron and electron–phonon interactions. The partial diagonalization of the Hamiltonian using unitary transformation was fulfilled. It allowed the structure of the many phonon repetition of bands to be described as a function of the electron–phonon interaction constant. It is shown that the spectral rela- tions obtained for the scattering intensity can describe both the RS and the resonance RS (RRS) processes. Numerical modelling calculations for different parameters were carried out and comparisons with the experimental data for CdS and ZnO crystals were made. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Raman scattering (RS) and absorption of light are among the best known methods used for the study of optical properties of solids. Absorption by a crystal is related to the one photon process of interband transitions. In semiconductors, the interband light absorption results in a structure of bands which are connected with Wannier excitons [1]. If this process is also accompanied by a strong electron–phonon interaction, the structure of bands can become more complicated due to many-phonon replicas (repetitions) arising from such interaction which was first studied by Toyozawa [2]. The Raman scattering process is more complicated. It is a two photon process, in which one photon, x k,k , is absorbed by the system and another photon, x k 0 ;k 0 , is emitted. This problem was first theoretically considered by Loudon [3]. The next step was made by Ganguly and Birman [4], who calculated the Raman tensor including excitons as intermediate states for deformation potential and Fröhlich electron–phonon interaction. Then Martin [5] repeated the calculations of Ganguly using the Green function (GF) formalism and studied only the energy region below the band gap. Zeyher et al. [6] calculated the first order Raman tensor con- sidering only one valence band and the corresponding continuum of excitonic states. Trallero and Sotolongo [7] treated the hot- exciton problem and studied the region above the band gap. In the following works of Cantarero et al. [8] and of Trallero-Giner et al. [9], the problem of RS with participation of exciton states as intermediate ones was considered in detail. The Wannier–Mott exciton model (effective mass approximation) was used and all discrete and continuous states were taken into consideration. The use of this approach was developed in later work on RS in quantum dots [10]. The approach used in these works is the following: the unperturbed Hamiltonian is a sum of electronic, H E , lattice, H L , and radiation operators, H R , i.e. H 0 = H E + H L + H R , while the pertur- bation part is a sum of electron-lattice, electron-radiation and lattice-radiation terms (the last one is negligible in resonance conditions [10]), H int = H EL + H ER + H LR . The process with partic- ipation of phonons is then considered by using the matrix element of third order according to the following consistent perturbations (H ER ? H EL ? H ER ). A similar approach was developed in works (see [11] and references therein). The approach considered in the present work is somewhat different. Taking into account that the perturbation potential describing interaction of an electromagnetic field with the crystal (neglecting term H LR in resonance conditions) is proportional to the first power of the electromagnetic field, H ER  Ap, the process of RS should be described by the second order of perturbation the- ory on the electromagnetic (EM) field. Therefore to consider the RS by any system we have to study the response of the system to the EM field in the second order perturbation theory, but the detailed description of scattering (in particular the effects of phonons, impurities, etc.) should depend on the Hamiltonian which in the present work describes a many-particle system of all the electrons and phonons of the crystal interacting with each other. 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.07.023 ⇑ Corresponding author. E-mail address: henryk.ratajczak@gmail.com (H. Ratajczak). Chemical Physics 388 (2011) 57–68 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys