IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29, NO. 9, SEPTEMBER 1993 2433 zyx Calculating the Optical Properties of Multidimensional Heterostructures: Application to the Modeling of Quatemary Quantum Well Lasers zyx D. Gershoni, C. H. Henry and G. A. Baraff Abstract-A method for calculating the electronic states and optical properties of multidimensionalsemiconductor quantum structures is described. The method is applicable to hetero- structures with confinement in any number of dimensions: e.g. bulk, quantum wells, quantum wires and quantum dots. It is applied here to model bulk and multiquantum well (MQW) InGaAsP active layer quaternary lasers. The band parameters of the quaternary system required for the modeling are inter- polated from the available literature. We compare bulk versus MQW performance, the effects of compressive and tensile strain, room temperature versus high temperature operation and 1.3 versus 1.55 pm wavelength operation. Our model shows that: compressive strain improves MQW laser performance. MQW lasers have higher amplification per carrier and higher differential gain than bulk lasers, however, MQW performance is far from ideal because of occupation of non-lasing mini- bands. This results in higher carrier densities at threshold than in bulk lasers, and may nullify the advantage of MQW lasers over bulk devices for high temperature operation. I. INTRODUCTION HIS PAPER is basically concerned with calculating T the optical absorption and gain spectra of some quan- tum well laser structures. To do zyxwvutsrq so, we first need to be able to calculate their electronic states and optical prop- erties. The paper starts by describing a method for cal- culating electronic band structure and optical properties of multidimensional heterostructures realized in semicon- ductor compounds. The method used here is based on the k zyxwvutsrqp - p method [ 13 uses envelope functions [2] and has been described in partial detail elsewhere [3]. Lattice mis- match strain, which is commonly used in heterostructures grown by modem epitaxial techniques, is incorporated into this calculation using deformation potentials zyxwvut [4]. Our method provides a common approach for calculat- ing the optical properties of heterostructures with confine- Manuscript received October 15, 1992 D. Gershoni is with Physics Department Technion, Haifa, 32000 Israel. C. H. Henry and G A. Baraff are with AT&T Bell Laboratories, Mur- ray, Hill, N.J. 07974. IEEE Log Number 9211356. ment in any number of dimensions, e.g., bulk semicon- ductors, one dimensional quantum heterostructure systems (quantum wells, multiquantum wells and superlattices), two dimensional quantum systems (quantum wires) and three dimensional systems (quantum dots). For all these systems, we provide essentially the same means of cal- culating the spectra of both types of optical transitions, namely, across the fundamental band-gap (interband tran- sitions) and also within the conduction or valence band (intersubband transitions). We apply this method to the InGaAsP /InP quaternary heterostructure system which is currently used for optical communications. Relevant earlier calculations of gain in AlGaAs /GaAs and InGaAs/GaAs active layer lasers, taking into account the valence band structure were made by Corzine et al. zyxw [5]. Strain effects on the valence band structure and consequently on performance of InGaAsP /InP quantum well lasers were considered by Loehr and Singh [6] and also in [5]. Our approach is more accurate in the sense that it takes into account the con- duction band structure as well. It permits us also to model any shape potential structure in the active layer in one or more dimensions. We calculate miniband dispersion curves and optical matrix elements and use them to obtain gain spectra, curves of peak gain versus carrier density and threshold dependence on temperature. We compare MQW and bulk lasers, room temperature and high tem- perature operation, 1.3 and 1.55 pm wavelength opera- tion lasers, and compressively strained, tensilely strained and unstrained MQW lasers. We also compare ideal and actual MQW laser performance. In this way, we try to draw conclusions about the advantages and disadvantages of quaternary MQW lasers. The manuscript is organized as follows: In Section I1 we describe the theory and define the parameters needed. In Section 111, we describe the software code that we have developed for the actual calculation of the multidimen- sional heterostructure band structure and for optical tran- sitions between the different bands. Section IV details the calculation of optical spectra from the calculated elec- 0018-9197/93$03.00 zyxwvuts 0 1993 IEEE