Z. Phys. B 92, 3541 (1993) ZEITSCHRIFT FORPHYSIK B 9 Springer-Verlag 1993 Exciton-polaritons in halfspace K. Victor, V.M. Axt, A. Stahl Institut ffir Theoretische Physik B, RWTH Aachen, TH-Erweiterungsgeb/iudeSeffent-Melaten, D-52056 Aachen, Germany Received: 15 March 1993 Abstract. A rigorous method is presented describing the coupling between an exciton polariton in a halfspace semiconductor and the external driving field. The method is based on density matrix theory. It allows to consider realistic electron-hole interactions, spatial dis- persion and extrinsic surface potentials. Without invok- ing additional boundary conditions or an artificial subdi- vision of the semiconductor it is shown that the influence of the surface can be isolated from the bulk behaviour. This is accomplished by a symmetric continuation of the restricted configuration space to bulk geometry inspired by the image source method in electrostatics. As a demonstration the solution is worked out for a simplified polariton model. The results are compared with other theories and with experimental reflection spectra. PACS: 71.35; 71.36 1. Introduction The coupling through a surface between external light and exciton polaritons in a semiconductor is a compli- cated problem because the response of excitons to the driving electric field is nonlocal in a twofold sense. One type of nonlocality is caused by the finite exciton radius, the other one is due to the wavelike translational prop- agation of excitons [1]. The resulting anomalies in the optical properties of semiconducting samples have been studied extensively [2-5] ever since Pekar's pioneering work [6] on the problem of additional boundary condi- tions (ABC-problem). In a theoretical analysis of mea- sured spectra a frequently used approximation taking into account both types of nonlocality is the fit with an ABC plus an adjustable exciton-free surface layer. The method first proposed by Hopfield and Thomas [7] is convenient because of its simplicity, but it has the disadvantage of being rather sensitive to the choice of the physically ill defined dead-layer thickness parameter. It is understood, that a rigorous theory of the bound- ary value problem for exciton polaritons should be based only on electrodynamics and quantum theory without any additional assumptions [8,9]. Unfortunately the complexity of the problem has up to now prevented even for the simplest geometry of a semiconducting half space the construction of a solution that perfectly complies with the aforementioned requirement. This might be attributed to the fact that for a rigorous solution one must consider the boundary value problem in polarition configuration space. To overcome the complexity of the situation in existing theories for the half space problem a three layer geometry is assumed consisting of external space, a boundary zone and the bulk region of the semiconductor. The solution for the three zones then is constructed separately under certain simplifying assumptions and eventually connected by appropriately chosen interface conditions [5]. In contrast to the approach with an ad- justable dead layer a more sophisticated modelling of the surface region guarantees that the results do not depend on the position of the artificial interface separating the bulk material from the surface region [5]. In the present paper we shall describe for the first time a rigorous treatment of the halfspace problem without any artificial subdivision of the semiconductor. Our ap- proach will be based on density matrix theory for the semiconductor electrons. The technique applied will be a generalization of the technique of electrical imagies ex- tended to electron-hole configuration space. In Sect. 2 we shall present our model equations. The essence of our solution method is described in Sect. 3. There we show that by rigorous analytic manipulations the bulk part of the solution can be separated out, leav- ing us with an integral equation for the surface anomaly of the material response. In a certain sense the method is a rigorous counterpart to the above described three layer model. In Sect. 4 the new approach is applied to a simplified semiconductor model, and the results are compared with other theories and measured reflection spectra.