Energy-Gap Anomaly in the Semiconductor Sequence PbS, PbSe, and PbTe

The $f$-sum rule has been examined at the conduction- and valence-band edges of PbS, PbSe, and PbTe. Experimental values of electron and hole effective masses, and of energy gaps were used. For both two-band and six-band models, it was found that PbTe does not follow the same $f$-sum rule as do PbS and PbSe. However, it was also found that, for the III-V and II-VI semiconductor sequences (InSb, InAs, InP), (GaSb, GaAs), and [CdSe, CdS (hexagonal)], the members of each sequence do follow the same $f$-sum rule. It is concluded that the well-known anomaly in the values of the energy gap $E_G$ of the $\mathrm{Pb}X$ sequence [$E_G\left(\mathrm{PbS}\right)>\mathrm{}{E}_G\left(\mathrm{PbTe}\right)>\mathrm{}{E}_G\left(\mathrm{PbSe}\right)$] is due to an irregular value of $E_G$ for PbTe. It appears likely that this anomalous value of the energy gap of PbTe reflects, through the $L_6^{-}$ conduction-band edge, an irregular value of the $5s$ electron energy of the tellurium atom. It is proposed that the $L_6^{-}$ conduction-band edge states in PbTe differ from the equivalent states in PbS and PbSe. This difference results in PbTe having an $f$-sum matrix element different from that for PbS and PbSe; this is the reason that PbTe does not follow the same $f$-sum rule as do PbS and PbSe. The $f$-sum-rule plots of the experimental data for these semiconductors have been used to calculate matrix element values.