Quasiparticle energy of semicore $d$ electrons in $\mathrm{ZnS}$: Combined $\mathrm{LDA}+U$ and $GW$ approach

We present a first-principles study of quasiparticle energies in $\mathrm{ZnS}$, with particular emphasis on the semicore $d$ electrons that are located too shallow by $\sim 2.8\mathrm{eV}$ compared to experiment in the local density approximation (LDA). Although the many-body correction in the $GW$ approximation pulls down the $d$ band, the correction $(-0.7\mathrm{eV})$ is too small to reproduce measured values. The $\mathrm{LDA}+U$ method also shifts the $d$ band down compared to LDA. With a reasonable choice of $U$, $d$-state energy in agreement with experiment may be achieved. Subsequent quasiparticle calculation within the $GW$ approximation performed to the $\mathrm{LDA}+U$ mean-field solution, however, pushes the $d$ band back close to the $GW$ result. These results show that the standard $GW$ method is insensitive to the reference mean-field Hamiltonian for this class of materials and suggest that going beyond $GW$ may be needed for an accurate description of the $d$ electron level in this system.

Figure 1

LDA band gap as a function of energy cutoff.

Figure 2

The position of $d$-states (at $\Gamma$) as a function of the $U$ parameter. The $\mathrm{LDA}+U$ results (open circles) and the $\mathrm{LDA}+U+GW$ results (closed circles). The dashed line is a guide for the eye.

Figure 3

Band structure obtained by (left) LDA and (right) $\mathrm{LDA}+U$.

Figure 4

(Color online) Energy levels at $\Gamma$. The upper (lower) $d$ levels are doubly (triply) degenerate. The dotted lines indicate measured values for the band gap and $d$-state energy, respectively. The energy is measured from the top of the valence band (also indicated by a dotted line.)