Multiscale Modeling of a Quantum Dot Heterostructure
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Multiscale Modeling of a Quantum Dot Heterostructure P. Sengupta, S. Lee, S. Steiger, H. Ryu, and G. Klimeck Dept. Of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA ABSTRACT A multiscale approach was adopted for the calculation of confined states in self-assembled semiconductor quantum dots (QDs). While results close to experimental data have been obtained with a combination of atomistic strain and tight-binding (TB) electronic structure description for the confined quantum states in the QD, the TB calculation requires substantial computational resources. To alleviate this problem an integrated approach was adopted to compute the energy states from a continuum 8-band k.p Hamiltonian under the influence of an atomistic strain field. Such multiscale simulations yield a roughly six-fold faster simulation. Atomic-resolution strain is added to the k.p Hamiltonian through interpolation onto a coarser continuum grid. Sufficient numerical accuracy is obtained by the multiscale approach. Optical transition wavelengths are within 7% of the corresponding TB results with a proper splitting of p-type sub-bands. The systematically lower emission wavelengths in k.p are attributable to an underestimation of the coupling between the conduction and valence bands. INTRODUCTION Self-assembled semiconductor quantum dots (SAQDs) are being employed as active metamaterials in high-speed semiconductor lasers, which achieve high-speed data transmission while utilizing minimal power. SAQDs are also one of the simplest means of exploring the physics of carriers in a three-dimensional confined regime. In this contribution the electronic properties of such dots are explored through atomistic, continuum and a combination of both approaches. Traditionally quantum dots have been modelled within an effective mass approach using the multi-band k.p formulation. 1 While the results obtained through these methods work well, the details of the atomic arrangement are disregarded. 2 To preserve the microscopic character of the dot and its interfaces, the atomistic tight-binding approach can be adopted for energy calculations. Local strain, which considerably modifies the energy spectrum, is accounted for through the atomistic valence force field (VFF) method.3 Results from combinations of these methods are presented and analysed here regarding their symmetry and their deviation to experiment. After a brief summary of the methods and a description of the studied QD, some numerical aspects are highlighted. The shortcomings of some of these methods are demonstrated and it is shown that their origin lies in the assertion upon which the theory is constructed. THEORY In this section the strain and electronic structure models are briefly reviewed, combinations of which are applied to the structure described in the following section.
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The atomistic TB Hamiltonian is constructed by choosing a relevant set of orbitals localized on an atom. The wave function of the system is expanded in the basis of these localized orbitals
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