Efficient electronic structure calculations for extended systems of coupled quantum dots using a linear combination of quantum dot orbitals method

We present a novel ``linear combination of atomic orbitals''-type of approximation, enabling accurate electronic structure calculations for systems of up to 20 or more electronically coupled quantum dots. Using realistic single quantum dot wave functions as a basis to expand the eigenstates of the heterostructure, our method shows excellent agreement with full 8-band $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ calculations, exemplarily chosen for our benchmarking comparison, with orders of magnitude reduction in computational time. We show that, to correctly predict the electronic properties of such stacks of coupled quantum dots, it is necessary to consider the strain distribution in the whole heterostructure. Edge effects determine the electronic structure for stacks of $\ensuremath{\lesssim}10$ quantum dots, after which a homogeneous confinement region develops in the center. The overarching goal of our investigations is to design a stack of vertically coupled quantum dots with an intraband staircase potential suitable as an active material for a quantum-dot-based quantum cascade laser. Following a parameter study in the ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}\text{/}\mathrm{GaAs}$ material system, varying quantum dot size, material composition, and interdot coupling strength, we show that an intraband staircase potential of identical transitions can, in principle, be realized. A species library we generated for over 800 unique quantum dots provides easy access to the basis functions required for different realizations of heterostructures. In the associated paper [Mittelst\"adt et al., Phys. Rev. B 103, 115301 (2021)], we investigate room temperature lasing of a terahertz quantum cascade laser based on a two-quantum-dot unit cell superlattice.