$s-d$ coupling in zinc-blende semiconductors

Most zinc blende semiconductors have a single anion-like s state near the bottom of the valence band, found in density-of-states (DOS) calculations, and seen in photoemission. Here, we discuss the case where two s-like peaks appear, due to strong $s-d$ coupling. Indeed, away from the $\mathbf{k}=0$ Brillouin zone center, cation d states and anion s states can couple in zinc blende symmetry. Depending on the energy difference $\Delta E_{\mathrm{sd}}{=E}_s^{\mathrm{anion}}-E_d^{\mathrm{cation}},$ this interaction can lead to either a single or two s-like peaks in the DOS and photoemission. We find four types of behaviors. (i) In GaP, GaAs, InP, and InAs, $\Delta E_{\mathrm{sd}}$ is large, giving rise to a single cation d peak well below the single anion s peak. (ii) Similarly, in CdS, CdSe, ZnS, ZnSe, and ZnTe, we see also a single s peak, but now the cation d is above the anion s. In both (i) and (ii) the $s-d$ coupling is very weak. (iii) In GaN and InN, the local density approximation (LDA) predicts two s-like peaks bracketing below and above the cation d-like state. Correcting the too low binding energies of LDA by LDA+SIC (self-interaction correction) still leaves the two s-like peaks. The occurrence of two s-like peaks represents the fingerprint of strong $s-d$ coupling. (iv) In CdTe, LDA predicts a single s-like peak just as in case (ii) above. However, LDA+SIC correction shifts down the cation d state closer to the anion s band, enhancing the $s-d$ coupling, and leading to the appearance of two s-like peaks. Case (iv) is a remarkable situation where LDA errors cause not only quantitative energetic errors, but actually leads to a qualitative effect of a DOS peak that exists in LDA+SIC but is missing in LDA. We predict that the double-$s$ peak should be observed in photoemission for GaN, InN, and CdTe.