Band-Gap Problem in Semiconductors Revisited: Effects of Core States and Many-Body Self-Consistency

A novel picture of the quasiparticle (QP) gap in prototype semiconductors Si and Ge emerges from an analysis based on all-electron, self-consistent, GW calculations. The deep-core electrons are shown to play a key role via the exchange diagram—if this effect is neglected, Si becomes a semimetal. Contrary to current lore, the Ge $3d$ semicore states (e.g., their polarization) have no impact on the GW gap. Self-consistency improves the calculated gaps—a first clear-cut success story for the Baym-Kadanoff method in the study of real-materials spectroscopy; it also has a significant impact on the QP lifetimes. Our results embody a new paradigm for ab initio QP theory.

Figure 1

(a) The PM used to solve the DEq on the $\tau$ axis is defined by two integers: “$p$” is the “order” of the underlying nonuniform mesh, whose width doubles in each step; “$u$” is the number of uniform intervals into which the nonuniform intervals are partitioned. (b) Typical $\tau$ dependence of $G$ and of the particle-hole bubble (GG); their exponential localization (and discontinuity) at $\tau =0$ and $\beta \hslash$ is efficiently accounted for by our PM [19]. (c) Spectral function $A_{\mathbf{k},j}(\omega )$ for Si along $\Gamma L$; to aid visualization, small numerical broadening has been introduced. (d) $A_{\mathbf{k},j}(\omega )$ for the hole state at the midpoint of the occupied band; the solid/dashed line denotes the self-consistent/first-iteration solution of Eq. (2).

Figure 2

Solid lines: Real and imaginary parts of the density-response function of Ge, obtained as outlined in Ref. [22]. The dashes correspond to the elimination of transitions from the $3d$ states. The $3d$ onset is highlighted in the inset.