Effect of inversion asymmetry on the conduction subbands in ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}-\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ heterostructures

Spin splitting of conduction subbands in ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}-\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ heterostructures due to both bulk and structure inversion asymmetry is considered theoretically using a five-level $\mathbf{k}\mathbf{\cdot }\mathbf{p}$ model. A formula for the splitting related to the structure inversion asymmetry is derived and it is explicitly demonstrated that this splitting is not proportional to the average electric field in the system. The theory is shown to describe well existing Raman data on anisotropic spin splitting in ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}-\mathrm{G}\mathrm{a}}_{1-\mathrm{x}}{\mathrm{Al}}_{\mathrm{x}}\mathrm{As}$ heterostructures for various directions of the Fermi wave vector. It is shown that the splitting is dominated by the bulk inversion asymmetry at low two-dimensional electron densities and by the structure inversion asymmetry at high densities. Various simplifications of the presented complete theory are discussed.