A comparison between different numerical methods used to solve Poisson’s and Schroedinger’s equations in semiconductor heterostructures

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Abstract
A comparison between different numerical methods which are used to solve Poisson’s and Schroedinger’s equations in semiconductor heterostructures is presented. Considering Schroedinger’s equation, both the Rayleigh–Ritz method and the finite difference method are examined. The accuracy and the computational speed are investigated as a function of both the mesh size and the number of Rayleigh–Ritz functions and the numerical results are compared with analytical solutions for special cases. To solve Poisson’s equation, direct and iterative methods are implemented and the advantages and limitations of each method are discussed. The previous methods are used to solve Poisson’s and Schroedinger’s equations self-consistently in typical heterostructures to obtain the wave functions, the carrier distribution, and the subband energies.