Importance of the spin-orbit split-off band on the tunneling properties of holes through ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ and ${\mathrm{I}\mathrm{n}\mathrm{P}/\mathrm{I}\mathrm{n}}_y{\mathrm{Ga}}_{1-y}\mathrm{As}$ heterostructures

The influence of the spin-orbit split-off band on the tunneling of holes across heterostructures is studied starting with the $6\times 6$ Luttinger-Kohn Hamiltonian. The latter is diagonalized into $3\times 3$ blocks (upper and lower Hamiltonians) using a unitary transformation. We consider ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ and ${\mathrm{I}\mathrm{n}\mathrm{P}/\mathrm{I}\mathrm{n}}_y{\mathrm{Ga}}_{1-y}\mathrm{As}$ material systems, and study the tunneling of holes through a one-dimensional δ scatterer and across abrupt potential steps. In each case, we show that the presence of the spin-orbit split-off band has a profound influence on the transmission coefficients of holes, even for holes with energy much lower than the threshold for free propagation in the spin-orbit split-off band. For the potential steps, we show that the results can be quite different with upper and lower Hamiltonians. Furthermore, we stress the importance of the spin-orbit split-off band by comparing the results with those obtained with the $4\times 4$ Luttinger-Kohn Hamiltonian which neglects the importance of the spin-orbit split-off band.