# Numerical simulation of the electronic structure of quantum dots

## Heinrich Voss

In semiconductor nanostructures free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero.

The governing equation characterizing the relevant energy states *λ* and
corresponding wave functions *ψ* is the Schrödinger equation

*- ∇ ( (h*

^{2}) / (2m(λ)) ∇ ψ ) + V ψ = λ ψ,
which depends nonlinearly on the eigenparameter if we assume a non-parabolic electron
effective mass *m*.

Taking advantage of variational properties of the eigenproblem (which are inherited by finite element discretizations) it can be solved efficiently by iterative projection methods of Arnoldi or Jacobi-Davidson type combined with safeguarded iteration.