Finite Element and Finite Difference Algorithms for Calculations of Electron Spectra in the System of Isolated Donor-Quantum Dot

Numerical methods to calculate electronic states of isolated donor coupled to electrically defined quantum dot are developed. The donor-dot system is described with the use for stationary Schrodinger and Laplace (Poisson) equations. Energy spectra of an electron bound to the donor and quantum dot are calculated using the finite difference and finite element methods. It is shown that when developing the finite difference method, the singularity at the donor location should be taken into account using cusp condition. Asymptotic boundary conditions to calculate confining electric fields are proposed. Main sources of errors related to numerical calculations are analyzed. Dependences of critical parameters on the size of a disc-shaped controlling gate and external magnetic field are determined for donor qubit.