Interrelation of structural and electronic properties in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ quantum dots using an eight-band $\mathbf{k}\bullet \mathbf{p}$ model

We present an eight-band $\mathbf{k}\bullet \mathbf{p}$-model for the calculation of the electronic structure of wurtzite semiconductor quantum dots (QDs) and its application to indium gallium nitride $\left({\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}\right)$ QDs formed by composition fluctuations in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ layers. The eight-band $\mathbf{k}\bullet \mathbf{p}$-model accounts for strain effects, piezoelectricity and pyroelectricity, and spin-orbit and crystal-field splitting. Exciton binding energies are calculated using the self-consistent Hartree method. Using this model, we studied the electronic properties of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ QDs and their dependence on structural properties, i.e., their chemical composition, height, and lateral diameter. We found a dominant influence of the built-in piezoelectric and pyroelectric fields, causing a spatial separation of the bound electron and hole states and a redshift of the exciton transition energies. The single-particle energies as well as the exciton energies depend heavily on the composition and geometry of the QDs.