Gain optimization in intersubband quantum well lasers by inverse spectral theory S. Tomic ´ a, * , V. Milanovic ´ b , M. Tadic ´ b , Z. Ikonic ´ b a VINC ˇ A Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Minor Yugoslavia b Faculty of Electrical Engineering, University of Belgrade, Bulevar revolucije 73, P.O. Box 3554, 11120 Belgrade, Minor Yugoslavia Received 26 August 1999; accepted 17 September 1999 by R.T. Phillips Abstract A systematic procedure is described for the maximization of gain in optically pumped intersubband lasers, via tailoring the quantum well potential. The procedure relies on the inverse spectral theory, which enables shifting bound states in a quantum well. By varying the free parameters appearing in this procedure one can design a convenient optimized structure. An design example is presented for an Al x Ga 1-x As alloy based well, and the band nonparabolicity effects are accounted for.  1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Nanostructures; A. Quantum wells; D. Optical properties 1. Introduction Since the first proposal of unipolar semiconductor laser, based on intersubband transitions in quantum wells (QW), by Kazarinov and Suris in 1971 [1], there has been consid- erable research effort in this area. This was particularly boosted by the development of molecular beam epitaxy (MBE), which made possible the realization of such struc- tures. Various types of structures for this purpose have been proposed and realized, the most important of which is the quantum cascade laser (QCL), demonstrated by Faist et al. [2]. It is current pumped and covers the mid-infrared range 4–11 mm [3,4]. Recently the QWL comprising superlattices were demonstrated, that lase in the 3–5 mm range [5], and also antimonide based QW lasing down to 1.9 mm [6], because of the large band offset in this system. In the range of longer wavelengths the optically pumped intersubband lasers have some distinct advantages, notably the high selectivity of excitation of relevant states, and free- dom from losses due to electron migration towards electrical contacts. In the field of optically pumped lasers the multiple quantum well structures have been considered for far- infrared applications [7], and also the asymmetric coupled QWs for the mid-infrared range [8]. Although both the intra- conduction-band and intra-valence-band transitions may be used, the former is more convenient because of higher scattering rates among the valence subbands [9]. Among the most important issues in the design of inter- subband lasers is the maximization of gain. This may be accomplished via careful tailoring of the QW profile. In Ref. [10] the asymmetric step QW has been optimized so to obtain the largest values of dipole matrix elements relevant for pump absorption and for lasing. Later investi- gations have indicated that it might be convenient to take the gain/loss ratio as the optimization criterion [11,12]. In order to obtain large population inversion it is essential to have fast relaxation of the lower laser level, via optical phonons, which depends on both the level spacing and the wave functions. Further, the spacing between the upper laser state and the ground state has to be matched to the pump laser wavelength, and this transition has to be allowed, just as is the lasing transition, hence the structure must be asym- metric. In this paper we propose a systematic procedure for optimizing the QW shape in optically pumped laser which will provide maximal gain, taking into account that both the dipole matrix elements and the relaxation times depend on the wave functions of relevant states. The procedure relies on the inverse spectral theory (IST) [13], which enables one to shift a bound state of some potential as appropriate and Solid State Communications 113 (2000) 221–226 0038-1098/00/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S0038-1098(99)00434-2 PERGAMON www.elsevier.com/locate/ssc * Corresponding author. E-mail address: stomic@rt270.vin.bg.ac.yu (S. Tomic ´)