PHYSICAL REVIEW B VOLUME 51, NUMBER 16 15 APRIL 1995-II Optical nonlinearities in multiple quantum wells: Generalized Elliott formula Domenico Campi and Claudio Coriasso CSELT — Centro Studi e Laboratori Telecomunicazioni, Via G. Reiss Rorno/i 274, 10148 Torino, Italy (Received 22 July 1994; revised manuscript received 5 December 1994) Excitonic nonlinearities due to plasma effects in quantum-well structures under quasistationary excita- tion can be reproduced by appropriate many-body treatments, which usually require numerical calcula- tions that may become computationally intensive. An alternative approach is based on analytical ap- proximations; however, this has not been examined carefully so far. In this paper we present the analyti- cal calculation of the optical properties of quasi-two-dimensional, type-I semiconductor quantum wells, at varying plasma densities and accounting for one conduction band and two valence subbands. This has been developed based on the two-dimensional version of the Elliott formula and on some analytical ap- proximations already known in part. The obtained analytical results are scalable to a considerable range of constituent materials and of quantum-well thicknesses. These results are compared with the numeri- cal solutions achieved within a more complete many-body approach, based on the Bethe-Salpeter equa- tion, and with experimental results obtained in a pump-probe experiment. The comparison provides general guidelines on the accuracy and on the limitations of the analytical approach applicable to the case of quasi-two-dimensional excitons when multisubband and finite-size effects are included. I. INTRODUCTION Nonlinear optical properties in quasi-two-dimensional electron systems have been studied extensively during the past years, both theoretically and experimentally. ' Almost all experiments in which (light-} intensity- dependent changes of optical properties are measured, or used to obtain optical bistability, are carried out in the photon-energy range of the excitonic absorption peak. These changes are due to the photogeneration of a carrier plasma by the incident radiation. Within this context, the inclusion of many-body effects is thus essential to any calculation of optical properties of quantum wells (QW's}, since these dictate in large measure the form of the ab- sorption edge and its evolution with the incident pump power. There have been several investigations concern- ing many-body theory in semiconductors, in which plas- rna effects give rise to the large room-temperature non- linearities in QW's. Actually, compared to the wealth of literature concerning the basic aspects, few simplified ex- pressions in practical structures have been reported so far. Even though some earlier works on nonlinearities in QW's included a few approximate analytical or phenome- nological evaluations of many-body effects, it is fair to say that, to our knowledge, there has not been any sys- tematic analytical approach to nonlinear optical proper- ties in QW's, although sparse elements can already be found, scattered in the existing literature. It has been suggested ' that waveguides based on quantum-well materials should display interesting dynamical behavior, that could be analyzed, for example, through equations governing the evolutions of two coun- terpropagating fields, coupled with one carrier density equation. ' However, one crucial aspect in solving dynamical equations describing optical nonlinearities is the proper treatment of terms describing the field-matter interaction. In principle, this would require a complete description of many-body effects at each computational step and at each wavelength. Most of these calculations are done numerically using algorithms that are not par- ticularly simple and require recursive or iterative ap- proaches that become computationally intensive. In practice, one might use a phenomenological expression, with parameters derived either from theory or experi- ment: even this is not completely satisfactory, since it would involve interpolations that might become particu- larly dangerous in treating nonlinear problems. In this respect, a simple formulation of the optical properties, accounting analytically for plasma effects under quasista- tionary excitation, might be useful. This paper proposes a systematic formulation of the nonlinear optical properties of QW structures, providing a generalized form of the two-dimensional Eliott formula, and enlightening strengths and weaknesses in the ap- proach. The relevant equations are derived as physically interpretable analytical approximations of more complete many-body treatments that can be found in the litera- ture. This model is tested against numerical calculations in a more complete many-body picture, and against ex- perirnental results. Both numerical and experimental re- sults have been obtained as part of this work. Wherever its implementation is regarded as convenient, this method eliminates the use of computationally intensive numerical solutions which previous techniques employed, since even differentiation and integration and other potentially time-consuming operators are avoided in the final expres- sions. The outline of the paper is as follows: in Sec. II our generalized expression of the Elliott formula will be pro- vided, and the various terms entering the formula will be described and explained, leaving algebraic details for the appendixes. In Sec. III we will discuss the analytical re- sults, obtained with our formula, and we will compare these results with experimental spectra obtained in a 0163-1829/95/51(16)/10719(10)/$06. 00 51 10 719 1995 The American Physical Society