PHYSICAL REVIE%' 8 VOLUME 49, NUMBER 24 15 JUNE 1994-II NonthreshoM Auger recombination in quantum wells M. I. Dyakonov* Uni Uersi te de Montpellier II, Groupe d'Etudes des Semiconducteur, 34095 Montpellier Cedex 5, France V. Yu. Kachorovskii A F. Io+. e Physico Tech-nical Institute, 194021 St Pet. ersburg, Russia (Received 23 November 1993) We calculate the Auger recombination rate of carriers in a quantum well taking into account that due to the presence of the heteroboundaries the momentum-conservation law is violated. As a result, nonthreshold Auger recombination becomes possible, i. e. , low-energy carriers may recombine, the excess momentum being transferred to the well boundaries. This greatly enhances the Auger rate at low tem- peratures. We develop a method for calculating the overlap integrals in the limiting case which is of practical interest and obtain an explicit analytical formula for the nonthreshold Auger rate. The calcu- lated recombination rate weakly depends on temperature but increases drastically with decreasing well width and may become more important for narrow wells. I. INTRODUCTION Auger recombination is one of the most important recombination mechanisms for narrow-band-gap semi- conductors especially at high excitation levels, which are characteristic for semiconductor lasers. The Auger pro- cess is generally considered to be the predominant nonra- diative recombination mechanism for long-wavelength quantum-well lasers, and a number of papers was devoted to its theoretical investigation. ' In these works, the Auger process in a quantum well was considered quite analogously to the case of the bulk Auger recombination for which, as is well known, a threshold in the carrier en- ergy exists. Accordingly, the same threshold was ob- tained for the quantum-well case. The calculated recom- bination rate depends exponentially on temperature and becomes extremely small at low temperatures. The threshold character of the Auger recombination is a consequence of the energy and momentum- conservation laws. However, in a quantum well, the momentum-conservation law is violated, since one com- ponent of the momentum may be transferred to the well's heteroboundaries. As a result nonthreshold recombina- tion becomes possible, i.e. , low-energy carriers may recombine in a quantum well via the Auger process. Ob- viously this should greatly enhance the Auger rate at low temperatures. Thus, the heteroboundaries in a quantum well play the same role as phonons and defects in the bulk case, lifting the constraint imposed by the momen- turn conservation. This idea was put forward exphcitly by Zegrya and Kharchenko who calculated the velocity of surface nonthreshold Auger recombination due to the presence of a single heteroboundary. However, their results are not exact because they were based on the incorrect assurnp- tion that the main contribution to the Coulomb matrix element comes from the under-barrier region. It should be also mentioned that the dominant process is recom- bination with heavy holes, not with the light ones as it was assumed in Ref. 6, since the heavy-hole concentra- tion is substantially greater than that of the light holes. Earlier, the importance of Auger transitions from the ground bound state in a quantum well to the continuum of unbound states, particularly at small well widths, was pointed out in Ref. 7, and supported by numerical calcu- lations based on a simplified band-structure model (sim- ple parabolic bands} and some assumptions concerning the values of the overlap integrals. The purpose of the present work is to calculate the rate of nonthreshold Auger recombination in a quantum well. We will show that the recombination rate weakly de- pends on the temperature, but increases drastically with decreasing well width and may become most important for narrow wells. We will obtain an explicit analytical formula for the recombination rate in the practically in- teresting limiting case. II. PROBLEM FORMULATION. MAIN PARAMETERS AND ASSUMPTIONS We consider the CCCH Auger process (see Fig. 1; we use the conventional notations for different Auger pro- 4ii 0 1 3 Eo t 'E I Oh FIG. 1. Auger recombination in a quantum well. Electron (1) recombines with a heavy hole (2) exciting another electron (3) to the final state (4). 0163-1829/94/49(24)/17130(9)/$06. 00 49 17 130 1994 The American Physical Society