Self-consistent solution of Hedin's equations: Semiconductors and insulators

The band gaps of a few selected semiconductors/insulators are obtained from the self-consistent solution of the Hedin's equations. Two different schemes to include the vertex corrections are studied: (i) the vertex function of the first order (in the screened interaction $W$) is applied in both polarizability $P$ and self-energy $\mathrm{\Sigma }$, and (ii) the vertex function obtained from the Bethe-Salpeter equation is used in $P$ whereas the vertex of the first order is used in $\mathrm{\Sigma }$. Both schemes show considerable improvement in the accuracy of the calculated band gaps as compared to the self-consistent $GW$ approach ($\mathrm{sc}GW$) and to the self-consistent quasiparticle $GW$ approach ($\mathrm{QS}GW$). To further distinguish between the performances of two vertex-corrected schemes one has to properly take into account the effect of the electron-phonon interaction on the calculated band gaps which appears to be of the same magnitude as the difference between schemes (i) and (ii).

Figure 1

Relative deviation of the calculated (in GGA) band gaps of selected semiconductors/insulators from the experimental band gaps (excluding the electron-phonon interaction). Experimental data has been taken from Refs. [17, 18, 19]. The horizontal line represents the experimental data corrected by the effects of electron-phonon and spin-orbit interactions (zero deviation). The corrections were taken from Refs. [20, 21, 22, 23]. The GGA results have been cited from Ref. [24].

Figure 2

First-order approximation for the three-point vertex function. Direct lines represent Green's function G and the wavy line represents the screened interaction W. The dot is for the trivial part of the vertex function.

Figure 3

Ladder sequence for the three-point vertex function with the kernel $\mathrm{\Theta }$ (see Fig. 4) as the rung of the ladder.

Figure 4

The GW approximation for the irreducible four-point kernel $\mathrm{\Theta }$. Direct lines represent the Green's function and wavy lines represent the screened interaction W.

Figure 5

Dependence of the density of states of MgO (from the effective Hartree-Fock problem) on the quality of the representation of the band states. Line “FULL” corresponds to the properly normalized band states (in this case they are represented with the sets of plane waves and $\varphi$'s as in $GW$ part; see Table 2). The rest of the lines have been obtained with the sets of plane waves and $\varphi$'s of different completeness. Each line is marked with the corresponding set of $\varphi$'s (first entry, Mg/O), and the number of plane waves $N_{\mathbf{G}}$ in the interstitial region (second entry). Two vertical lines show the range of the band states ($\sim 20$ bands) used in the vertex part as a basis set. Formula (D4) from Ref. [10] has been used to evaluate the spectral function. Smearing parameter was set to 0.005 Ry to avoid extremely sharp peaks. The $k$ mesh $3\times 3\times 3$ corresponding to the vertex part was used.

Figure 6

The same as Fig. 5 but for the low-energy part of the spectrum.

Figure 7

Convergence of the vertex function with respect to the iteration number when one solves the Bethe-Salpeter equation. Shown are the ratios of the largest element of the correction to the vertex at a given iteration and at the first iteration.

Figure 8

Calculated electron energy loss spectrum of LiF at $q=0.5\mathrm{GX}$. Symbols represent the EELS data reproduced from the Ref. [31]