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 wavefunctions as basis to expand the eigenstates of the heterostructure, our method shows excellent agreement with full 8-band $\boldsymbol{k}\cdot\boldsymbol{p}$ calculations, exemplarily chosen for our benchmarking comparison, with an orders of magnitude reduction in computational time. We show that, in order 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 $\lesssim$ 10 QDs, 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 intra-band staircase potential suitable as active material for a quantum-dot-based quantum cascade laser. Following a parameter study in the In$_{x}$Ga$_{1-x}$As/GaAs material system, varying quantum dot size, material composition and inter-dot coupling strength, we show that an intra-band 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 manuscript entitled "Room Temperature Lasing of Terahertz Quantum Cascade Lasers Based on a Quantum Dot Superlattice", we investigate room temperature lasing of a terahertz quantum cascade laser based on a two-quantum-dot unit cell superlattice.