Quantum-mechanical spin states and Zeeman-level diagrams of the positively charged exciton

We consider the spin interaction in the Hamiltonian for the positively charged exciton $X^{+},$ determining the spin states and Zeeman-level diagrams for $X^{+}$ in which the heavy- and light-hole bands are degenerate and nondegenerate. The former case results in an $X^{+}$ with quintuplet and septet states in addition to the singlet and triplet states that are also observed for the negatively charged exciton $X^{-}.$ When the heavy- and light-hole bands are split $X^{+}$ can comprise two heavy holes, two light holes, or a heavy and a light hole. The heavy-hole $X^{+}$ Zeeman-level diagram is found to be completely analogous to that of $X^{-},$ while the light-hole $X^{+}$ has more optical transitions and a different Zeeman splitting in photoluminescence. $X^{+}$ consisting of a heavy and a light hole has no coupled hole states, and is truly a heavy- (or light-) hole exciton plus a light (heavy) hole.