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Efficient quantum dot $\mathbf{k}\cdot\mathbf{p}$ in wurtzite systems including spatially varying elastic and dielectric constants and smooth alloy profile

We present Fourier-space based methods to calculate the electronic structure of wurtzite quantum dot systems with continuous alloy profiles. We incorporate spatially varying elastic and dielectric constants in strain and piezoelectric potential calculations. A method to incorporate smooth alloy profiles in all aspects of the calculations is presented. We demonstrate our methodology for the case of a 1D InGaN quantum dot array and show the importance of including these spatially varying parameters in the modeling of devices. We demonstrate that convergence of the lowest bound state energies is to good approximation determined by the largest wave vector used in constructing the states. We also present a novel approach of coupling strain into the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian, greatly reducing the computational cost of generating the Hamiltonian.