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We present a very efficient scheme to calculate the eigenvalue problem of the time-independent Schrödinger equation. The eigenvalue problem can be solved via an initial-value procedure of the time-dependent Schrödinger equation. First, the time evolution of the wave function is calculated by the finite-difference time-domain method. Then the eigenenergies of the electron system can be obtained through a fast Fourier transformation along the time axis of the wave function after some point. The computing effort for this scheme is roughly proportional to the total grid points involved in the structure and it is suitable for large scale quantum systems. We have applied this approach to the three-dimensional GaN quantum dot system involving one million grid points. It takes only 7 h to calculate the confined energies and the wave functions on a standard 2-GHz Pentium 4 computer. The proposed approach can be implemented in a parallel computer system to study more complex systems.

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We present modelling results investigating the carrier dynamics of an SOA composed of an inhomogeneous array of quantum dots designed to produce broad gain amplification when optically pumped. We use a set of rate equations that describe the QDs inhomogeneity and include an energy dependent occupation factor within each inhomogeneously broadened level and numerically solve them with the propagation equation to investigate the amplification of optical signals in the waveguide. By treating the carrier filling according to the quasi-equilibrium distribution, we are able to investigate the effect of band-filling (BFE) on the gain and refractive index. The linewidth enhancement factor (α) is computed and analysed with respect to optical signal intensity as well as electrical current density.

A model portraying the carrier dynamics for an inhomogeneous array of quantum dots (QDs) interacting with a number of photon modes is presented. The model treats an ensemble of QDs with one confined level coupled to a wetting layer or quantum well level and explicitly considers only the electrons. The model is derived by numerically solving a set of rate equations that includes the inhomogeneity of the dot size, multimode photon modes and temperature dependence. Explicitly the inhomogeneous size distribution is included within a inhomogeneous broadening parameter and the temperature dependence within the homogeneous broadening parameter as well as carrier thermal escape. This is similar to the well-known Sugawara model 1 but in the Sugawara model the carriers are assumed to occupy the inhomogeneous quantum dots equally at all temperatures. Experimental and theoretical work in ref.(2) and (3) believes this is true only for a low temperature regime. Above the temperature where a global minima exists, Fermi-Dirac statistics have been used. This results in different gain and lasing behaviour for higher temperatures from those calculated using the Sugawara model.