Spin splitting in a p-type quantum well with built-in electric field and microscopic inversion asymmetry

The strain dependence of the spin splitting of hole subbands in modulation-doped asymmetric lattice-matched ${\mathrm{In}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As/${\mathrm{In}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{As}}_{\mathrm{y}}$ ${\mathrm{P}}_{1\mathrm{-}\mathrm{y}}$ quantum wells on lattice-mismatched ${\mathrm{In}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{As}}_{\mathrm{y}}$ ${\mathrm{P}}_{1\mathrm{-}\mathrm{y}}$ substrates is investigated theoretically using a 6×6 Luttinger-Kohn Hamiltonian. The influence of the built-in electric field, the microscopic inversion asymmetry of the zinc-blende lattice, and the strain are taken into account and analyzed for different widths of the quantum wells. The spin splitting is dominated by the effects of the electric field for compressive strain and small tensile strain. For large tensile strain the microscopic inversion asymmetry is the most important origin of spin splitting. A local maximum of spin splitting is located at small tensile strain. For large compressive strain the spin splitting is strongly suppressed whereas for large tensile strain the spin splitting increases with the absolute value of strain. However, the spin splitting vanishes completely in some directions for tensile strain.