Numerical computation of pyramidal quantum dots with band non-parabolicity

• Polynomial matrix is formed from quantum dots’ band non-parabolicity by finite difference method.
• Eigen-energy scanning method is derived for solving an arbitrary polynomial matrix.
• Eigen-energy levels from isolated and vertically aligned array quantum dots are obtained.
• Coupling effect is shown for variable distances between quantum dots and different size.

This paper presents an effective and feasible eigen-energy scanning method to solve polynomial matrix eigenvalues introduced by 3D quantum dots problem with band non-parabolicity. The pyramid-shaped quantum dot is placed in a computational box with uniform mesh in Cartesian coordinates. Its corresponding Schrödinger equation is discretized by the finite difference method. The interface conditions are incorporated into the discretization scheme without explicitly enforcing them. By comparing the eigenvalues from isolated quantum dots and a vertically aligned regular array of them, we investigate the coupling effect for variable distances between the quantum dots and different size.