Screening in two dimensions: $GW$ calculations for surfaces and thin films using the repeated-slab approach

In the context of photoelectron spectroscopy, the $GW$ approach has developed into the method of choice for computing excitation spectra of weakly correlated bulk systems and their surfaces. To employ the established computational schemes that have been developed for three-dimensional crystals, two-dimensional systems are typically treated in the repeated-slab approach. In this work we critically examine this approach and identify three important aspects for which the treatment of long-range screening in two dimensions differs from the bulk: (1) anisotropy of the macroscopic screening, (2) $\mathbf{k}$-point sampling parallel to the surface, (3) periodic repetition and slab-slab interaction. For prototypical semiconductor (silicon) and ionic (NaCl) thin films we quantify the individual contributions of points (1) to (3) and develop robust and efficient correction schemes derived from the classic theory of dielectric screening.

Figure 1

(Color online) Repeated-slab approach for surfaces and thin films schematically, with $s$ being the slab thickness and $v$ the slab separation.

Figure 2

Dielectric tensor for repeated-slab systems according to effective-medium theory as a function of the $s/c$ ratio ($s$ slab thickness, $c$ total size of the simulation box): component ${\epsilon }_{\parallel }$ parallel to the surface and ${\epsilon }_z$ perpendicular to it.

Figure 3

Convergence of the fundamental gap of a H-saturated four-layer Si(100) slab with respect to the number of $\mathbf{k}$ points $N_z$ perpendicular to the surface for (a) the isotropic and (b) the anisotropic approach. The quantitative behavior depends on the sampling in the parallel direction $\left(N_{\parallel }\times N_{\parallel }\right)$. Note the different scales of the two graphs.

Figure 4

Band gap for a two-layer NaCl(100) slab and a hydrogen-saturated four-layer Si(100) slab with a $N_{\parallel }\times N_{\parallel }\times 1$ $\mathbf{k}$-point sampling as a function of $1/N_{\parallel }$. The vacuum region is 20 bohr wide for both systems. The fit function is explained in the text.

Figure 5

(Color online) The nonlocality of two-point functions, such as $\Sigma$, is defined within the interaction cell. In the space-time method, its size is linked to the $\mathbf{k}$-point discretization grid.

Figure 6

(Color online) (a) Computation of the screened potential from the image charges. (b) Periodic repetition of the image charges to simulate the effect of the ${\mathbf{k}}_{\parallel }$ discretization. $a$ denotes the lateral extent of the interaction cell.

Figure 7

(Color online) Additional potential $\Delta V^{\text{im}}$ introduced by the ${\mathbf{k}}_{\parallel }$ discretization of the screened interaction (see text). The dielectric model corresponds to a two-layer NaCl(100) slab in a $c=30\text{bohr}$ unit cell with a $3\times 3$ ${\mathbf{k}}_{\parallel }$ sampling. The (blue) shaded areas indicate the position of the slabs; the horizontal dashed line marks the average potential. The isolated-slab case is shown for comparison.

Figure 8

(Color online) Dependence of the band gap of H-saturated Si(100) slabs with four, six, and eight layers (two layers $\approx 5\text{bohr}$) on the total cell height $c$. Solid lines: $\text{DFT}+\text{GW}$ results. Dashed lines: $\text{DFT}+\text{GW}$ results with finite-vacuum corrections derived from the dielectric model. The DFT gaps (not shown) are independent of the vacuum thickness.

Figure 9

(Color online) Image potential for an isolated slab and a repeated-slab system in comparison. The blue-shaded regions indicate the position of the slabs. The model parameters correspond to a two-layer NaCl slab.

Figure 10

(Color online) Left: ${\mathbf{k}}_{\parallel }$ convergence for the gap of a two-layer NaCl slab for different total cell heights $c$. Right: DFT-LDA gap (bottom) and extrapolated quasiparticle correction ${\Delta }_{\text{qp}}$ as well as its value corrected for the finite vacuum (top) as a function of $c$.