Optical properties of excitons in quantum dots: diffraction of an electromagnetic plane wave by a spherical quantum dot L. Silvestri a,b, * , F. Bassani a,b , G. Czajkowski c a Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy b Istituto Nazionale di Fisica della Materia, Rome, Italy c University of Technology and Agriculture, Kaliskiego 7, PL-85796 Bydgoszcz, Poland Received 4 October 1999; accepted 22 June 2000 Abstract We show how to compute the optical spectra resulting from the scattering of a plane electromagnetic wave on a quantum dot (QD) in the region of the excitonic resonances. The method uses the microscopic calculation of quantum dot eigenfunctions and the macroscopic Stahl's density matrix approach to compute the electromagnetic ®eld within and outside the QD, taking into account the coherence between the carriers and the electromagnetic wave. Numerical results are obtained for the case of GaAs/ Ga 12x Al x As QD. Oscillator strengths and lineshapes are shown to have a typical dependence on the QD radius, and an increase in the scattering intensities near resonance is found. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Nanostructures; A. Semiconductors; D. Optical properties 1. Introduction Zero-dimensional structures which are usually called quantum dots (QD) have been extensively investigated. Particular interest has been recently focused on their linear and nonlinear optical properties [1±5]. The optical properties of QDs can be interpreted in terms of con®ned excitons. The con®nement of quasi-particles in a dot increases the binding energy and the oscillator strength per unit volume of excitons and thus leads to enhancement of excitonic effects. Much effort has been devoted to the calculation of excitonic binding energies and oscillator strengths, but this is only a preliminary step for the under- standing of the optical behaviour of a medium containing a density of quantum dots in the anomalous dispersive region. For this purpose we must consider the scattering of an inci- dent linearly polarized wave, which becomes a spherical one within the dot and interacts with all spatial components of the excitonic dipole. As a consequence, the scattered waves will be strongly affected by the excitonic properties of the dot. This aspect was not dealt with in previous works on QDs which were mainly devoted to the study of electronic states [2±17,52]. Variations of polarization of an incident plane wave due to scattering on a sphere were extensively discussed in the past (see the classical works of Mie and Debye [18,19], and, for example, recent contributions of Borghese et al. [20], Fucile et al. [21], Ruppin [22], Lacoste et al. [23] for new developments). Our purpose is to discuss in detail the effect of excitonic resonances on the character of the scattered wave, using a spherical model for the QD shape. We compute an average (isotropic) susceptibility of the QD in order to determine the amplitude and phase of the scattered wave. The main motivation of the present paper is to study the optical properties of a material composed of a matrix containing QDs, taking into account the excitonic nature of the QD and the correlation between the radiation ®eld and the carriers. We also aim at a calculation of the optical properties including the size distribution of the QD. Recent experimental work [24,63] requires more detailed calcula- tion of the optical response function than now available. In what follows we discuss the optical properties inside the QD, using a microscopic model for the calculation of the eigenstates and taking into account the coupling with the electromagnetic ®eld with the Stahl's density matrix Journal of Physics and Chemistry of Solids 61 (2000) 2043±2053 0022-3697/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0022-3697(00)00206-7 www.elsevier.nl/locate/jpcs * Corresponding author. Tel.: 139-050-509111; fax: 139-050- 563-513. E-mail address: silve@cibs.sns.it (L. Silvestri).