An effective approach for describing the electronic structure of InGaAs/GaAs quantum dots (QDs) is presented. We model QDs based on a single sub-band approach with an energy dependent electron effective mass. The model assumes that the total effect of inter-band interactions, strain and piezoelectricity can be taken into account by an effective potential. Using this approximation, we define a strength parameter of the effective potential to reproduce capacitance-gate-voltage (CV) experimental data for InAs/GaAs QDs (see Fig.4a). In the present work, we expand the model to describe InxGa1-xAs QDs with significant Ga fractions. We find that our model accurately describes CV and photoluminescence (PL) data for QDs, assuming 22% Ga fractions, and also reproduces the experimental data for Coulomb shifts of exciton complexes (X,X,XX) (Fig. 4b). We compared our results with those from atomistic pseudopotential and 8-band kp-Hamiltonian approaches. The strength of the electron and heavy hole confinements is found to be weaker in the kp-model than in the atomistic pseudopotential approach.
Fig. 4 (a) Calculated energies and effective masses of the electrons occupying s-, p-, and d-orbitals. (b) Coulomb shifts of the transition energies for positively (X+) and negatively (X–) charged trions and biexcitons (XX) as functions of the exciton recombination energy (solid lines – simulations, dashed lines –root-mean-square fits of experimental data). (c) Cross sections of the single electron/hole probability distributions in different orbital states in the InAs/GaAs quantum dot.
Spherical shaped Si quantum dots embedded into SiO2 substrate are considered under single sub-band effective mass approach. In this model the electron and heavy hole sub-bands are taken into account (Fig.5). The energy dependence electron effective mass is especially applied for case of QD with small size. Calculations of low-lying single electron and hole energy levels are performed. For QD having small sizes (diameter d < 6nm) there is a strong confinement regime when the number confinement levels is restricted by several levels (Fig. 6-7).
Fig.5 Band structure for the Si/SiO2 QDs modeling.
We used first order of perturbation theory to calculated neutral exciton recombination energy taking into account the Coulomb force between electron and heavy hole. For the Si/SiO2 QDs the PL experimental data are available for comparison with the theoretical consideration. The PL exciton data are reproduced well by our model calculations (Fig.8). We also compare the results with those obtained within the more sophisticate models. The comparison shows that inter-band interaction has character of second order term and cannot be defined by the PL data.
Fig. 6 a) Spectrum of electron low-lying levels (energy measured from the Si conduction band edge). D= 3.1 nm b) Transition energies with ∆n=1, ∆l=1.
For weak confinement regime (QD diameter d>10 nm), when the number of confinement levels is limited by several hundred, we considered the statistical properties of the confinement. Evidence of chaotic properties of the electron spectrum is demonstrated and reason of this effect is discussed.
Fig. 7 Wave functions of several electron levels.
Fig. 8 Neutral exciton recombination energy of Si/SiO2 spherical QDs. Experimental data are shown by various symbols. Calculated values are depicted as triangles.
Nanosized quantum objects manifest atom like electron structure due to size confinement. The electronic structure of these objects is restricted to a few electron and hole levels. We model InGaAs/GaAs and Si/SiO2 quantum objects, such as quantum dots (QD) and quantum wire (QW) based on single band effective mass approximation. We shown that spectrum of electron energy levels demonstrate chaotic properties (the preliminary hypotheses is the Brody-parameterization) (see Fig. 9 a)). The strength of the effect depends on QD geometry and ratio of effective masses for different direction in QD.
Fig. 9a) Distribution function for energies of spherical Si/SiO2 QD. The statistic is for 117 states. Diameter of QD is 10 nm. The effective mass ratio s=m*t/m*r=0.2 .
We considered the Si/SO2 QD super lattice (see Fig. 9b) in 3D. The periodical potential is schematically presented in Fig. 9c). Calculated low-lying electron energy levels are also shown in Fig. 9c). We found that the effect of interactions between QDs can be neglect up to very small distances between QDs (see Fig. 10). This effect is limited by values of 2%-3%. In super lattice QDs can be considered as set of independent single QDs.
Fig. 9b) QD superlattice 9c) Few electron low-lying levels (energy measured from the Si conduction band edge). The potential model for QD lattice is presented schematically. Diameter D = 3.1 nm.
Fig. 9d) Relative changing of neutral exciton recombination energy Eex of Si/SiO2 spherical QD superlattice along distance between QDs surface DL. Eexs means energy of single QD. D is diameter of QDs.
We found that single electron/hole level of single QD is transformed to the miniband structure in the QD superlattice.