Z. Phys. B 98, 39-47 (1995) ZEITSCHRIFT FORPHYSIK B 9 Springer-Verlag 1995 Polaritons in anisotropic semiconductors F. Bassani, G. Czajkowski*, A. Tredicucci Scuola Normale Superiore, Piazza dei Cavalieri 7, 1-56126 Pisa, Italy Received: 8 August 1994 / Revised version: 18 November 1994 Abstract. We show how to compute the optical properties (reflection and absorption) of anisotropic semiconductors in the exciton energy region, taking into account polariton and electron-hole coherence effects. The method is applied to a GaAs/Gal_xAlxAs superlattice, and the modifications in the optical properties with respect to GaAs are related to the anisotropy. PACS: 78.66.-w; 71.35.+z; 71.36.+c 1. Introduction Optical properties of bulk semiconducting crystals in the en- ergetic region of excitonic resonances were studied for sev- eral years (see, for example, [1-8]). Most of these studies dealt with semiconductors with isotropic valence- and con- duction bands. Recently more attention is paid to anisotropic semiconductors. The anisotropy appears, for instance, in semi- conductor slabs where the degeneracy of the valence band is removed and anisotropic heavy-hole and light-hole excitons appear. Other examples of great interest are semiconductor superlattices where the superlattice exciton can be treated like an exciton in an effective anisotropic medium (see, for exam- ple, [9-10]). In what follows we discuss the optical properties of anisotropic semiconductors, applying Stahl's density matrix approach [7, 11-14]. The advantages of this method concern- ing the computation of optical functions of finite crystals are described extensively elsewhere [14,15]. Here we show that the method is appropriate also for bulk crystals, where it al- lows the computation of the susceptibility as a function of the crystal anisotropy, implying also the coherence of the electric field with both the electron and the hole motion, and leading to corrections on the usual theory. This aspect was disregarded in former papers applying Stahl's approach, due to a special choice of the dipole density M. * Present address: Institute of Mathematics and Physics, Academy of Technology and Agriculture, Kaliskiego 7, PL-85790 Bydgoszcz, Poland We solve the basic constitutive equation, which corre- sponds to an anisotropic SchrSdinger equation, using an appro- priate Green's function, which depends on an anisotropy pa- rameter defined in terms of excitonic effective masses. Though in principle this is equivalent to the "fractional dimensional- ity approach" adopted by Mathieu et al. [16-20], we find it to be advantageous for introducing the interaction with the elec- tromagnetic field. In our approach we go beyond the "Fermi golden rule" and can discuss the polaritonic aspect, giving a simple, combined treatment of the excitonic resonances and the continuum. We calculate the dispersion rule for polaritons in anisotro- pic semiconductors, then the absorption and the reflectivity as functions of the anisotropy parameter. Applications to the case of a GaAs/Gal _xAl~As superlattice are given for various choices of the relevant parameters. The paper is organized as follows. In Sect. 2 we briefly recall the basic equation of Stahl's density matrix approach for bulk semiconductors and derive expressions for the sus- ceptibility of anisotropic semiconductors in terms of an ap- propriate Green's function as well as of a sum of oscillator contributions. In Sect. 3 we calculate the optical properties as functions of the anisotropy parameter, and apply them to the case of GaAs/Gal_xAlzAs superlattices in Sect. 4. We present our conclusions in Sect. 5. 2. Excitons in anisotropic bulk crystals In Stahl's approach the optical properties of semiconductors near the fundamental gap are described by a set of constitutive equations for the coherent amplitudes Yau (1,2) of the electron and the hole of coordinates 1 and 2 respectively: OtY;~u + (i/h) H~,**Y),~. = (i/h) [Mx~." E- F~.Y;~.] , (1) where A is the index of an occupied valence band and > la- bels an empty conduction band, E denotes the electric field, M~, is a coupling coefficient defined as the interband transi- tion dipole moment, and F~u is a phenomenological damping coefficient. The two-band Hamiltonian H;~I, with gap Eg~f, is H,x~, Eg~,~,-(h2/2m~,) V2-(h2/2m~) 2 = Ve+Veh (1,2) , (2)