Microscopic theory of optical gain in small semiconductor quantum dots Y. Z. Hu, H. GieXen, and N. Peyghambarian Optical Sciences Center, University of Arizona, Tucson, Arizona 85721 S. W. Koch Fachbereich Physik und Zentrum fu ¨r Materialwissenschaften, Philipps-Universita ¨t Marburg, Renthof 5, 35112 Marburg, Germany Received 10 July 1995; revised manuscript received 3 November 1995 A microscopic theory is used to analyze optical gain in small semiconductor quantum dots. Based on a numerical matrix diagonalization method and subsequent solution of the optical Bloch equations, it is found that the quantum-dot gain is dominated by the stimulated transitions between biexciton and exciton states. The calculation shows that Coulomb interaction and valence-band mixing effects signiﬁcantly inﬂuence the spectral and dynamic gain properties in strongly conﬁned quantum dots. I. INTRODUCTION In almost all commercial semiconductor lasers the emitted light is generated by the stimulated recombination of electron-hole pairs in the high-density carrier plasma. Under standard laser operation conditions the threshold carrier den- sity is higher than the Mott density, so that bound electron- hole states are ionized because of the exchange interaction and screening of the attractive interband Coulomb potential. Therefore, it is usually a reasonable approximation to model the electron-hole plasma within the framework of the screened Hartree-Fock approximation. Such a treatment of the semiconductor gain medium allows us to explain many experimental ﬁndings and enables us to model laser and am- pliﬁer devices which are based on III-V semiconductor materials. 1 On the other hand, since the late 1970s and early 1980s it has been well known that optical gain in wide-gap bulk semiconductors has signiﬁcant excitonic and even biexci- tonic contributions, at least under low-temperature operation conditions. 2,3 Indications for the inﬂuence of such strong electron-hole correlations have also been discussed recently in connection with laser action in II-VI quantum-well structures. 4 In these II-VI materials, the exciton binding en- ergy is a few tens of meV, so that strong excitonic effects should be present even at laser threshold densities. A satisfactory theoretical understanding of electron-hole correlation effects in bulk and quantum-conﬁned semicon- ductor structures and their inﬂuence on the optical gain does not yet exist. This problem is not only interesting and chal- lenging because of its many-body aspects but is also of sig- niﬁcance for device development and optimization. As a step in the direction of understanding the inﬂuence of excitonic correlation effects on the gain in quantum-conﬁned semicon- ductors, in this paper we study very small semiconductor structures, i.e., quantum dots QD’s. The strongly conﬁned quantum dots are a model system for excitonic and biexci- tonic gain contributions in their purest form since the quan- tum conﬁnement leads to a complete absence of continuum states. There is substantial interest in QD’s as evidenced by the substantial number of theoretical and experimental studies conducted to understand the linear and nonlinear optical properties of such systems. 5 These investigations revealed many unique properties of QD’s compared with bulk semi- conductors or semiconductor quantum wells. As an impor- tant insight one recognized that with increasing quantum conﬁnement biexcitons play an increasingly important role in determining the optical nonlinearities. 6,7 In recent experimental studies on small CdSe quantum dots, optical gain with a bandwidth of approximately 0.5 eV has been observed. 8 The detailed experimental study shows that the gain in the effectively zero-dimensional QD system differs signiﬁcantly from that in bulk and quantum-well structures. For example, in QD’s spectrally very broad opti- cal gain can be realized from far below to well above the fundamental band gap, whereas in bulk and quantum-well structures plasma gain is possible only in the spectral region between the renormalized gap and the electron-hole qua- sichemical potential. It is our goal in this paper to present and evaluate an idealized but realistic model for excitonic lasing in QD’s. We demonstrate that the various biexciton-to-exciton transitions essentially determine the optical gain properties. To calculate the gain spectra, we ﬁrst compute the exciton and biexciton states using a numerical matrix diagonalization method, 7 in which the Coulomb interaction, valence-band mixing, 9 and surface polarization effect 10 are included. The optical transi- tion dipole moments are obtained from the computed exciton and biexciton wave functions. To study the optical gain dy- namics and compute pump-probe spectra in the gain regime we solve the spatially Fourier-transformed multilevel optical Bloch equations. The results explain the broad gain spectra observed experimentally and reveal some interesting qualita- tive differences between the electron-hole plasma lasing mechanism and the gain mechanism from a strongly corre- lated excitonic system in a QD. II. THEORETICAL MODEL The linear and nonlinear optical properties in the spectral region of the semiconductor band gap are determined by the electron-hole excitations. If one studies an intrinsic semicon- ductor system it is possible to describe the optical excitations PHYSICAL REVIEW B 15 FEBRUARY 1996-II VOLUME 53, NUMBER 8 53 0163-1829/96/538/48149/$06.00 4814 © 1996 The American Physical Society