Semiconductors, Vol. 38, No. 1, 2004, pp. 1–22. Translated from Fizika i Tekhnika Poluprovodnikov, Vol. 38, No. 1, 2004, pp. 3–25. Original Russian Text Copyright © 2004 by Asryan, Suris. REVIEW Theory of Threshold Characteristics of Semiconductor Quantum Dot Lasers L. V. Asryan1, 2^ and R. A. Suris1^^ 1Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia 2Department of Electrical and Computer Engineering, State University of New York, Stony Brook, NY 11794-2350, USA ^e-mail: [email protected] ^^e-mail: [email protected] Submitted March 6, 2003; accepted for publication April 30, 2003 Abstract—A comprehensive theory of threshold characteristics of quantum dot (QD) lasers, which provides a basis for optimization of their design, is reviewed. The dependences of the gain, transparency current, threshold current, characteristic temperature, and multimode generation threshold on the parameters of the QD ensemble (surface density and size dispersion of QDs), cavity (stripe length and thickness of the waveguide region), het- erocontacts (band offsets), and temperature are considered in detail. The limiting characteristics of the laser (optimum structure parameters, minimum threshold current density, and characteristic temperature of the opti- mized structure) are discussed at length. The results of the analysis may serve as direct recommendations for the development of QD lasers that significantly outperform the semiconductor lasers currently in use. © 2004 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION gain) and to achieve lasing (gain equal to loss). Conse- quently, the transparency current (or inversion current, Heterostructures and devices based on them consti- i.e., the injection current at which the population inver- tute one of the most important objects of the modern sion is zero) and the threshold current (injection current physics of semiconductors and semiconductor elec- at which the gain is equal to the loss and lasing begins) tronics [1, 2]. Presently, advances in the field of micro- decrease and their temperature dependences become and optoelectronics are largely made possible by the weaker. The decrease in the threshold current and use of low-dimensional heterostructures. In quantum dots (QDs), heterostructures with spatial confinement increase in its temperature stability reflect one of the of carriers in three dimensions, we have the limiting main areas of development and improvement of injec- case of quantum confinement, and the energy spectrum tion lasers (see, e.g., [3–8] for the evolution of the is discrete. The extraordinary interest in QDs, from threshold current densities of QW lasers). Owing to the both the fundamental and practical point of view, is pri- continuous nature of the carrier spectrum within the marily due to the dramatic difference between the car- allowed subbands, the use of QWs [9–12] or QWRs rier spectra in them and in heterostructures based on [12] as active medium for optical transitions can only bulk crystals, quantum wells (QWs),1 and quantum quantitatively improve the parameters of devices based wires (QWRs),2 in which the spectrum is continuous on them compared with devices with a bulk active within the bands or subbands of allowed states. region [9, 13]. It can be seen from Fig. 1 that a funda- mental change in the density of states and gain spectra As they are similar to transitions between exactly is only achieved in a zero-dimensional active region. discrete levels of a single atom, transitions between lev- Consequently, a fundamental decrease in the threshold els in a QD composed of several thousands or tens of current and weakening of its temperature dependence thousands of atoms seem to be ideal for lasing. can only be achieved by using QDs. The use of QDs as With decreasing dimensionality of the active region active medium in injection lasers is the most topical of an injection laser, the density of states and gain spec- application of nanotechnology to the development of tra become narrower (Fig. 1), which leads to a decrease devices that present great commercial interest. Thus, in the number of states to be filled to make the active semiconductor (diode) QD lasers are the most promis- region transparent (zero population inversion and zero ing generation of injection lasers with fundamentally improved operating characteristics [14, 15]. Among the 1 A QW is an ultrathin layer in which carriers are spatially confined advantages of QD lasers over the presently used QW in a single (transverse) direction and move freely in the other two lasers are their narrower gain spectra, much lower directions (in the well plane). 2 In QWRs, carriers are spatially confined in two (transverse) threshold currents, and ultrahigh temperature stability, directions and move freely in the third (longitudinal) direction as well as the wider possibilities for controlling their (along the wire). lasing wavelength. 1063-7826/04/3801-0001 $26.00 © 2004 MAIK “Nauka/Interperiodica” 2 ASRYAN, SURIS Lasing from QDs (first with optical pumping [19] 3D Density of states and then with current injection [20]) was made possible Bulk by the fabrication of dot arrays that satisfy rather strin- gent requirements for the uniformity of dot size and Cavity loss shape. The commercial prospects of QD lasers are stim- ulating research in this field. By now, much has been achieved in the development of QD lasers [21]. A num- Gain ber of research groups have reported successful fabrica- tion of QD lasers [19–43]. The record-breakingly low, for all kinds of injection lasers, threshold current den- 2 0 Eg Energy sity jth = 19 A/cm at room temperature in the CW mode has been demonstrated [28]. 2D Density of states Experimental advances in the design of QD lasers have made it practically relevant to develop a compre- hensive theory of their operating characteristics, which could give practical recommendations for realizing Cavity Gain QW their potential advantages over the lasers currently in loss use. Such a theory must include analysis of the basic processes (generation–recombination, capture into QDs and thermal escape therefrom, diffusion in the waveguide region), take into account inhomogeneous ε + ε 0 Eg + n p Energy line broadening, and make it possible to determine the limiting parameters of the lasers and to optimize their design. This paper reviews a detailed theory of thresh- Cavity old characteristics of interband (bipolar) semiconduc- loss QWR tor QD lasers based on original studies by the authors [44–57]. The development of such a theory assumed that the following basic problems, which determine the 1D Density of states structure of the paper, are to be solved. It is necessary —to determine how the inhomogeneous line broad- Gain ening caused by dispersion of QD parameters affects the threshold characteristics (Sections 2–6); ε + ε 0 Eg + n p Energy —to reveal the influence of parasitic recombination (recombination outside QDs) on the threshold charac- teristics (Sections 2–6); QD —to determine how charge neutrality violation in QDs affects the threshold characteristics (Sections 3 Cavity 0D δ-Density of states and 4); loss and Gain —to study the temperature dependence of the threshold current jth and calculate the characteristic temperature T0 of the laser (Section 4); —to determine the influence of optical transitions 0 E + ε + ε Energy from excited states in QDs on the threshold character- g n p istics (Section 5); Fig. 1. Modification of the density of states and the shape of —to study the effect of spatial hole burning (SHB) and the gain spectrum with decreasing dimensionality of the active region. the phenomenon of multimode generation (Section 6); —to find ways of optimizing the QD laser structures in order to minimize jth and raise T0 and the threshold of Achieving each of these advantages was the aim of multimode generation; to calculate the best possible research in the field of semiconductor lasers from the characteristics of the laser (Sections 2–4, 6). very beginning. For example, lasing in various spectral ranges in a continuous-wave (CW) mode at high tem- perature was one of the reasons for using heterostruc- 2. INHOMOGENEOUS LINE BROADENING ture lasers instead of homojunction devices [16]. It AND THE THRESHOLD CURRENT DENSITY should be noted that the low threshold currents of het- The advantages of QD lasers over the QW lasers erostructure lasers compared with homojunction currently in use are due to the delta-function-like den- devices were demonstrated at an early stage [17, 18]. sity of states in QDs. In the ideal case of identical QDs, SEMICONDUCTORS Vol. 38 No. 1 2004 THEORY OF THRESHOLD CHARACTERISTICS 3 the gain spectrum would also be a delta function. In real (a) structures, there occurs inhomogeneous line broaden- ing caused by the inevitable scatter of parameters (pri- marily, size) of QDs.3 In structures with QDs fabricated by electron-beam lithography [29], this scatter is due to QD the “noise” of the lithographic process. Such fluctua- tions are also characteristic of QD ensembles formed via self-assembling in molecular-beam epitaxy [58] QD and metal-organic chemical vapor deposition. Inhomo- geneous line broadening is a key factor limiting the characteristics of a QD laser. The dispersion of QD Cladding layer Cladding layer parameters and deviation of the gain spectrum from the QD ideal case adversely affect these characteristics by low- a ering the maximum gain, raising the threshold current, Optical confinement and making its temperature dependence more pro- layer 4 nounced. The advantages of QD structures can only be b realized in the case of QDs that are sufficiently uniform in size and shape. Theoretical studies of QD lasers have (b) been reported previously [59–61].5 However, the ques- tion of how the threshold characteristics of a laser depend on QD size fluctuations, i.e., on the perfection ∆E ε of the laser structure, remained unanswered.