Spin and cyclotron energies of electrons in GaAs ∕ Ga 1 − x Al x As quantum wells

A five-level $\mathbf{P}∙\mathbf{p}$ model of the band structure for $\mathrm{GaAs}$-type semiconductors is used to describe the spin ${g}^{*}$ factor and the cyclotron mass ${m}_{c}^{*}$ of conduction electrons in $\mathrm{GaAs}∕{\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ quantum wells in an external magnetic field parallel to the growth direction [001]. It is demonstrated that the previous theory of the ${g}^{*}$ factor in heterostructures is inadequate. Our approach is based on an iteration procedure of solving 14 coupled differential $\mathbf{P}∙\mathbf{p}$ equations. The applicability of the iteration procedure is verified. The final eigenenergy problem for the conduction subbands is reduced to two differential equations for the spin-up and spin-down states of consecutive Landau levels. It is shown that the bulk inversion asymmetry of III-V compounds is of importance for the spin ${g}^{*}$ factor. Our theory with no adjustable parameters gives an excellent description of experimental data on the electron spin ${g}^{*}$ factor in $\mathrm{GaAs}∕{\mathrm{Ga}}_{0.67}{\mathrm{Al}}_{0.33}\mathrm{As}$ rectangular quantum wells for different well widths between 3 and $21\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. The same theory describes very well experimental cyclotron masses in $\mathrm{GaAs}∕{\mathrm{Ga}}_{0.74}{\mathrm{Al}}_{0.26}\mathrm{As}$ quantum wells for the well widths between 6 and $37\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$.