A fast and robust iterative method for obtaining self-consistent solutions to the coupled system of Schrödinger’s and Poisson’s equations is presented. Using quantum mechanical perturbation theory, a simple expression describing the dependence of the quantum electron density on the electrostatic potential is derived. This expression is then used to implement an iteration scheme, based on a predictor-corrector type approach, for the solution of the coupled system of differential equations. We find that this iteration approach simplifies the software implementation of the nonlinear problem, and provides excellent convergence speed and stability. We demonstrate the approach by presenting an example for the calculation of the two-dimensional bound electron states within the cross section of a GaAs-AlGaAs based quantum wire. For this example, the convergence is six times faster by applying our predictor-corrector approach compared to a corresponding underrelaxation algorithm.

A quasi-three-dimensional (3D) simulation of a quantum waveguide coupler has been performed, computing the self-consistent transverse potential along the electron waveguides and then solving the transport problem with a modified recursive Green's-function method. Results have been obtained for the tunneling conductance between the two waveguides as a function of coupling length and gate biases. A clear structure of conductance peaks is observed, strongly dependent on both the drain and the source biases. Such dependence has been investigated in greater detail for an idealized model, allowing a fast numerical simulation. A ridgelike conductance pattern has been obtained, which can be interpreted as a characteristic signature to be looked for when searching for the evidence of 113-to-1D tunneling in experimental data.

We present a new algorithm for the numerical simulation of electrons in a quantum wire as described by a two-dimensional eigenvalue problem for Schrödinger’s equation coupled with Poisson’s equation. Initially, the algorithm employs an underrelaxed fixed point iteration to generate an approximation which is reasonably close to the solution. Subsequently, this approximate solution is employed as an initial guess for a Jacobian-free implementation of an approximate Newton method. In this manner the nonlinearity in the model is dealt with effectively. We demonstrate the effectiveness of our approach in a set of numerical experiments which study the electron states on the cross section of a quantum wire structure based on III-V semiconductors at 4.2 and 77 K.

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