Numerical simulation of three dimensional pyramid quantum dot

We present a simple and efficient numerical method for the simulation of the three-dimensional pyramid quantum dot heterostructure. The pyramid-shaped quantum dot is placed in a computational box with uniform mesh in Cartesian coordinates. The corresponding Schrödinger equation is discretized using the finite volume method and the interface conditions are incorporated into the discretization scheme without explicitly enforcing them. The resulting matrix eigenvalue problem is then solved using a Jacobi–Davidson based method. Both linear and non-linear eigenvalue problems are simulated. The scheme is 2nd order accurate and converges extremely fast. The superior performance is a combined effect of the uniform spacing of the grids and the nice structure of the resulting matrices. We have successfully simulated a variety of test problems, including a quintic polynomial eigenvalue problem with more than 32 million variables.