The splitting in zero magnetic field between the up- and down-spin electrons in a two-dimensional electron gas is obtained for a series of three different ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}/{\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$ modulation-doped heterostructures with high electron densities [$n_s\sim \mathrm{}(1.5-1.8)\mathrm{}\times {10}^{12}$ ${\mathrm{cm}}^{-2\mathrm{}}$]. We have observed a characteristic beating modulation in the amplitude of the Shubnikov-de Haas oscillations in this system and up to six nodes have been measured in the Shubnikov-de Haas data for magnetic fields in the range $0.15 \mathrm{T}<B<\mathrm{}1.0\mathrm{} \mathrm{T}$. Analysis of these data indicates that one subband is primarily occupied and the two beating frequencies arise from a spin splitting of the lowest subband. A spin splitting of 1.5-2.5 meV as $B\to 0$ is deduced from the data. For magnetic fields applied at an angle $\theta$ to the interface, the beat positions scale as $cos\theta$ for small angles but increase steeply after a critical angle.

• Received 19 April 1988

DOI:https://doi.org/10.1103/PhysRevB.39.1411