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A theoretical model of electron transport in quantum cascade lasers subjected to a magnetic field is developed. The Landau level electronic structure was calculated from the envelope-function Schr\"odinger equation within the effective-mass approximation. The electron transport in a magnetic field was modeled using the self-consistent rate-equation description for the full period of the cascade and its interaction with adjacent periods. The scattering processes included in the model are electron\char21{}longitudinal-optical-phonon, electron\char21{}longitudinal-acoustic-phonon, and electron-electron scattering. All these processes show oscillatory behavior with magnetic field, and their interplay determines the electron transport and the output characteristics of quantum cascade lasers in magnetic field. The model was applied to investigate the influence of magnetic field on the performance of a $\mathrm{GaAs}∕\mathrm{AlGaAs}$ quantum cascade laser emitting at $\ensuremath{\lambda}\ensuremath{\approx}11.4\phantom{\rule{0.3em}{0ex}}\mathrm{\ensuremath{\mu}}\mathrm{m}$ [P. Kruck et al., Appl. Phys. Lett. 76, 3340 (2000)]. The calculated results show good overall agreement with the available experimental data.

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 quasiangular 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.