On Improving Accuracy of Finite-Element Solutions of the Effective-Mass Schrödinger Equation for Interdiffused Quantum Wells and Quantum Wires

We use the Galerkin approach and the finite-element method to numerically solve the effective-mass Schrödinger equation.The accuracy of the solution is explored as it varies with the range of the numerical domain.The model potentials are those of interdiffused semiconductor quantum wells and axially symmetric quantum wires.Also,the model of a linear harmonic oscillator is considered for comparison reasons.It is demonstrated that the absolute error of the electron ground state energy level exhibits a minimum at a certain domain range,which is thus considered to be optimal.This range is found to depend on the number of mesh nodes N approximately as α₀ log_e^α₁(α₂ N),where the values of the constants α₀,α₁,and α₂ are determined by fitting the numerical data.And the optimal range is found to be a weak function of the diffusion length.Moreover,it was demonstrated that a domain range adaptation to the optimal value leads to substantial improvement of accuracy of the solution of the Schrödinger equation.