Detecting the sign reversal of the $g$-factor of two-dimensional electrons in narrow GaAs/Al$_x$Ga$_{1-x}$As quantum wells

By means of polarization-resolved photoluminescence spectroscopy, the sign reversal of the $z$ component of the $g$-factor of two-dimensional electrons in a GaAs/Al$_{0.33}$Ga$_{0.67}$As quantum well in a strong perpendicular magnetic field in the vicinity of the filling factor $\nu=3$ is observed as the well width is reduced. The energy splitting between the spectral lines with opposite circular polarizations corresponding to the recombination of electrons from the lowest Landau level is analyzed. Owing to the exchange interaction, this splitting has either a maximum or a minimum as a function of the magnetic field in the vicinity of the electron filling factor $\nu=3$, the type of extremum depending on the sign of the $z$ component of the electron $g$-factor. The suggested method makes it possible to reliably recognize the sign reversal of the $g$-factor upon the variation of its magnitude by less than $\Delta g_e \approx 0.06$. It is found that the sign reversal occurs for a quantum-well width of $65\pm 3$ Å.