Adequacy of approximations in $\mathit{GW}$ theory

Following the usual procedure of the $GW$ approximation (GWA) within the first-principles framework, we calculate the self-energy from eigenfunctions and eigenvalues generated by the local-density approximation. We analyze several possible sources of error in the theory and its implementation, using a recently developed all-electron approach based on the full-potential linear muffin-tin orbital (LMTO) method. First we present some analysis of convergence in some quasiparticle energies with respect to the number of bands, and also their dependence on different basis sets within the LMTO method. We next present a new analysis of core contributions. Then we apply the GWA to a variety of materials systems to test its range of validity. For simple $sp$ semiconductors, GWA always underestimates band gaps. Better agreement with experiment is obtained when the renormalization $\left(Z\right)$ factor is not included, and we propose a justification for it. We close with some analysis of difficulties in the usual GWA procedure.

Figure 1

(Color online) LDA energy bands in Si, computed by different methods. APW bands from Ref. 19 are denoted by “$+$” and can be regarded as near exact. Dotted lines denote bands calculated by the same authors using the LAPW method, without local orbitals. Solid lines denote bands computed by the present generalized LMTO method, including local and floating orbitals as described in the text (180 energy bands were included in the basis). On the right is a table of the RMS deviation relative to the APW bands for several energy windows, computed along the $L\text{−}\Gamma$ line. The first column denotes the range of bands used to compute the RMS deviation; the horizontal dashed lines denote the approximate energy window for each range. The second and third columns denote how the present method and the LAPW method, respectively, deviate from the APW bands (in eV), as described in the text.

Figure 2

(Color online) ${\Gamma }_{25v}\to {{X}}_{1c}$ gap in Si in GWA as a function of the number of unoccupied states $N^{\prime }$. Filled (yellow) squares: LAPW GWA taken from Ku and Eguiluz (Ref. 26). The authors presented data only for $N^{\prime }<31$. Also, their data was given for the minimum gap, so we shifted their results by $+0.14\mathrm{eV}$ to estimate the ${\Gamma }_{25v}\to {{X}}_{1c}$ gap. Some checks show that the shift should be $\sim 0.14\mathrm{eV}$, approximately independent of $N^{\prime }$. Filled (green) pentagons: LMTO results varying $N^{\prime }$ both $G$ and $W$. LMTO results were shifted by $-0.02\mathrm{eV}$ to correct for incomplete $k$ convergence (Ref. 25). The $\mathrm{Si}2p$ was included in the valence using a local orbital (the LAPW calculation of Ku and Eguiluz did not). The total dimension of the LMTO basis is 180. Results from the same basis are shown as filled (green) pentagons in Fig. 4, and also filled (green) pentagons No. (13) in Fig. 5. Open triangles: PPGW from Ref. 9, which included the $\mathrm{Si}2p$ levels as part of the valence, and in which $W$ was computed using the plasmon-pole approximation. Open circles: LMTO results varying $N^{\prime }$ in $G$ but not $W$.

Figure 3

(Color online) ${\Gamma }_{25v}\to {{X}}_{1c}$ gap in Si as a function $1∕n_k^3$, where $n_k^3$ is the number of $k$ points in the full Brillouin zone. The dependence on $1∕n_k^3$ shown for ${\Gamma }_{25v}\to X_{1c}$ is essentially the same for all of the unoccupied QPEs we examined. Gaps for $n_k=6$, $n_k=5$, and $n_k=4$ exceed the $n_k=8$ case by 0.02, 0.04, and $0.10\mathrm{eV}$, respectively. We can also estimate what the $n_k\to \infty$ gap would be by extrapolating the approximately linear dependence on $1∕n_k^3$ to zero (dashed line). The $n_k=8$ case apparently overestimates the converged result by $0.01\mathrm{eV}$.

Figure 4

(Color online) The left panel shows ${\Gamma }_{25v}\to {{X}}_{1c}$ gap in Si as a function of the number of unoccupied states $N^{\prime }$ for a smaller basis (filled squares) and a larger basis (pentagons). The latter are redrawn from Fig. 2. The top horizontal scale shows an approximate relation between energy and $N^{\prime }$ in the large basis (interpolated from levels at $\Gamma$). The right panel contains the same data but reverses the top and bottom horizontal scales. Had $N^{\prime }$ in the upper horizontal axis been drawn for the small basis (squares), the scale would be a little different: the last data point corresponds to $N^{\prime }=82$ instead of 120. In the right panel the small-basis QPE’s (filled squares) were shifted by $-0.01\mathrm{eV}$ (Ref. 29) to clarify how large- and small-basis data diverge as the energy increases. The inset compares convergence in ${{X}}_{1c}$ as a function of $1∕N^{\prime }$ to a PPGW calculation that includes $2p$ states in the valence (Ref. 9) (open triangles) and a PPGW calculation that does not (Ref. 30) (open diamonds). LMTO data were shifted by $-0.02\mathrm{eV}$ to correct for incomplete $k$ convergence (Ref. 25). Some of the differencess between PP and LMTO data (triangles and hexagons) may be related to incomplete $k$ convergence; see Sec. 3.

Figure 5

(Color online) QPE in the GWA at ${\Gamma }_{15c}$ (top), $L_{1c}$ (middle), and $X_{1c}$ (bottom) in Si relative to the valence-band maximum, using different basis sets in the present FP-LMTO. Abscissa is the total number of basis functions $N$. Yellow diamonds show a minimal basis (see text). All results depicted by blue (filled) circles contain no local orbitals. Those depicted by filled (empty) squares or pentagons include the $\mathrm{Si}2p$ $\left(\mathrm{Si}3p\right)$ as local orbitals. See the text for further description.

Figure 6

(Color online) Energy bands in Ge for $\mathbf{k}=2\pi ∕a\left[00k\right]$ for small $k$ within the GWA, using $Z$=1. Spin-orbit coupling was omitted. Three approximations are compared: LDA (dashed blue line), $GW$ in the diagonal-$\Sigma$-only approximation, Eq. (6) (black line with circles), and $GW$ with $\Sigma$ computed according to Eq. (21) (solid green line). In all three cases the three states of $p$ character (${\Gamma }_{{25}^{\prime }}$ symmetry) form the valence-band maximum; this was taken to be the energy zero. The LDA predicts the conduction band, the ${\Gamma }_{2^{\prime }}$ state of $s$ character, to be slightly negative, causing the energy bands to be wrongly ordered at $\Gamma$. For $k>0$, the ${\Gamma }_{{25}^{\prime }}$ state of $p_z$ symmetry couples to the ${\Gamma }_{2^{\prime }}$ state, and the two repel each other. Both kinds of $GW$ put ${\Gamma }_{2^{\prime }c}$ at approximately the correct position, $1\mathrm{eV}$. However, the diagonal-only $GW$ must follow the topology of the LDA: the eigenvectors are unchanged from the LDA. Therefore the band starting out at ${\Gamma }_{2^{\prime }c}$ sweeps downward, while the $p_z$ band starting out at ${\Gamma }_{{25}^{\prime }v}$ sweeps upward, and the two bands cross near $k=0.02$. When the off-diagonal parts of $\Sigma$ are included, these two bands repel each other as they should.