${\Gamma }_1$ conduction electron $g$ factor and matrix elements in GaAs and ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ alloys

We report a new technique for determining $\left|g^{*}\right|$ of ${\Gamma }_1$ conduction electrons using optical-spin orientation techniques (polarization of luminescence, Hanle effect, and lifetime measurements). The variation of $\left|g^{*}\right|$ with $x$ in ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}(0\mathrm{}\le x\le 0.21)$ implies a negative $g^{*}$ in GaAs, in contrast to previous theoretical interpretations but in agreement with various recent measurements. The experimental values of $g^{*}$ and $m^{*}$ in GaAs are used to determine the momentum matrix elements connecting the ${\Gamma }_1$ state to the ${\Gamma }_{15}$ valence and conduction states. The results obtained for GaAs are employed to calculate matrix elements and $g$ factors in ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ alloys and in a number of zinc-blende crystals. Improved agreement between the calculated and measured values of $g^{*}$ is obtained in nearly all cases.