Proceedings of COMCOL Conference 2007 Boston Modeling of the Quantum Well and Cascade Semiconductor Lasers using 8-Band Schrödinger and Poisson Equation System Mikhail V. Kisin Electrical Engineering, SUNY Stony Brook, NY 11794-2350 mvk@ece.sunysb.edu Abstract: Eight-band quantum-mechanical model of the electron and hole energy spectrum in nanoscale semiconductor heterostructures with phenomenological boundary conditions for the charge carrier multicomponent effective wave function at the heterostructure interfaces has been formulated using symmetry considerations. COMSOL programs based on this model were developed and used for design and simulation of mid-infrared quantum well diode lasers and intersubband/interband cascade lasers. Keywords: semiconductor nanostructures, mid- infrared optoelectronics, quantum well lasers, cascade lasers. 1. Introduction Simulating of the modern optoelectronic devices is extremely complicated task. Carrier transport, light generation and absorption, optical mode propagation, and heat dissipation processes are all strongly interconnected and usually described on the different scales – from nanoscale modeling of the optically active regions to full-size device simulation for the heat management task. The core of the modeling problem is the adequate quantum-mechanical description of the charge carrier energy spectrum in the optically active regions and subsequent calculation of the device optical characteristics. Electron energy spectrum of narrow-bandgap materials used in mid-infrared optoelectronics can be correctly described only in multi-band quantum-mechanical model taking into account strong band mixing and nonparabolicity effects. In A 3 B 5 semiconductors, the direct optical gap is formed by eight nearest Bloch states of scalar {S} and vector {V} symmetries, so that the effective Schrödinger equation takes the form of a system of 8 differential equations with complex matrix structure. Formulation of the boundary conditions for such a multi-band phenomenological model still presents an open question [1]. In this work, both the effective Hamiltonian of the model and the boundary conditions for corresponding multicomponent wave function are formulated phenomenologically starting from symmetry considerations. The equations of the 8-band quantum-mechanical model are solved self- consistently with the Poisson equation using COMSOL. The developed software was applied to modeling and simulation of heavily strained Sb-based quantum well mid-infrared lasers [2] which recently demonstrated record high-power characteristics at room temperature [3]. The program was also applied to modeling and design of tunable cascade lasers, including both type-I quantum cascade (QC) and type-II interband cascade (IC) devices [4]. The software utilizes an extensive database of A 3 B 5 material parameters with biquadratic interpolation algorithm for ternary and quaternary compounds. 2. Governing Equations In 8-band model, the effective 8-component wave function of an arbitrary mesoscopic state is represented by a column of smooth envelopes ψ n which consists of a scalar and vector parts, that is of one even and three odd bi-component spinors: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Ψ ψ 0 ψ ; . (1) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = 3 2 1 ψ ψ ψ ψ The effective Hamiltonian of an isotropic 8-band model can be written in the matrix form ( ) p σ J p J J σ p P ⋅ + ⋅ − − ⋅ ∆ + ⋅ + − + = ) ˆ ˆ ( ) ˆ ( 2 1 ˆ 3 1 ˆ ˆ ˆ ˆ ˆ ) ( ˆ 2 3 3 2 3 0 2 0 γ γ γ e e p e p E H g (2) Here, p = −i∇ is the momentum operator, and we use the unit system with = = 1. Symbols with hats represent 8x8 matrices. The matrices ê 0 and ê 3 are diagonal unit matrices with non-zero elements only in scalar or vector subspaces correspondingly. These matrices reveal the intraband nature of related operators while non- diagonal matrix operator describes the interband ψ P ˆ 0 -ψ mixing and is represented by the 489