Symmetry ofk∙pHamiltonian in pyramidalInAs∕GaAsquantum dots: Application to the

Abstract

A method for the calculation of the electronic structure of pyramidal self-assembled InAs/GaAs quantum dots is presented. The method is based on exploiting the C 4 symmetry of the 8-band k · p Hamiltonian with the strain taken into account via the continuum mechanical model. The operators representing symmetry group elements were represented in the plane wave basis and the group projectors were used to find the symmetry adapted basis in which the corresponding Hamiltonian matrix is block diagonal with four blocks of approximately equal size. The quantum number of total quasi-angular momentum is introduced and the states are classified according to its value. Selection rules for interaction with electromagnetic field in the dipole approximation are derived. The method was applied to calculate electron and hole quasibound states in a periodic array of vertically stacked pyramidal self-assembled InAs/GaAs quantum dots for different values of the distance between the dots and external axial magnetic field. As the distance between the dots in an array is varied, an interesting effect of simultaneous change of ground hole state symmetry, type and the sign of miniband effective mass is predicted. This effect is explained in terms of the change of biaxial strain. It is also found that the magnetic field splitting of Kramer's double degenerate states is most prominent for the first and second excited state in the conduction band and that the magnetic field can both separate otherwise overlapping minibands and concatenate otherwise nonoverlapping minibands.

Semianalytical model for simulation of electronic properties of narrow-gap strained semiconductor quantum nanostructures

“…Another important result is that the degeneracies of CB and VB first excited states ("CB 1P" and "HH 1P") are lifted by the coupling to remote bands. The same result was obtained for QD with C 4v geometry [16]. It is not related to atomistic, strain or piezoelectric effects [10] but simply to the fact that the symmetry of the system in the 8-band description is represented by the z F quantum number instead of the irreducible representations of the C ∞v symmetry group.…”

Optically pumped intersublevel MidInfrared lasers based on InAs-GaAs quantum dots

“…With the envelope functions of the final and initial state given by (6) the matrix element is equal to (7) where is the volume of the embedding box and (8) are the Fourier transforms of the perturbation Hamiltonian matrix elements. They can all be expressed analytically in terms of the components of the vectors , and , the material parameters of InAs and GaAs and the Fourier transform of the quantum-dot characteristic function (see [19] or [22] for its definition). The same recipe as in [19] for the order of differential and multiplication operators was used to ensure the hermiticity of the perturbation Hamiltonian matrix.…”

“…They can all be expressed analytically in terms of the components of the vectors , and , the material parameters of InAs and GaAs and the Fourier transform of the quantum-dot characteristic function (see [19] or [22] for its definition). The same recipe as in [19] for the order of differential and multiplication operators was used to ensure the hermiticity of the perturbation Hamiltonian matrix.…”

Quantum Dots as Sources and Detectors οf Mid- and Far-Infrared Radiation: Theoretical Models

“…One then obtains the eigenvalue problem of the Hamiltonian, where the eigenvectors consist of N envelope functions, where N is the number of bands in the expansion. While possibly limited in the description of some subtle effects, the k·p method can inherently incorporate the effects of band mixing, strain, piezoelectricity, as well as the influence of external fields, keeping a lower computational cost (which can be further reduced by exploiting the symmetry of the Hamiltonian [23][24][25][26]) compared to atomistic methods. This is the reason why k·p is mostly used when modeling of optoelectronic devices is concerned.…”