Abstract
Twodimensional (2D) materials combine many fascinating properties that make them more interesting than their threedimensional counterparts for a variety of applications. For example, 2D materials exhibit stronger electronphonon and electronhole interactions, and their energy gaps and effective carrier masses can be easily tuned. Surprisingly, published band gaps of several 2D materials obtained with the GW approach, the stateoftheart in electronicstructure calculations, are quite scattered. The details of these calculations, such as the underlying geometry, the starting point, the inclusion of spinorbit coupling, and the treatment of the Coulomb potential can critically determine how accurate the results are. Taking monolayer MoS_{2} as a representative material, we employ the linearized augmented planewave + local orbital method to systematically investigate how all these aspects affect the quality of G_{0}W_{0} calculations, and also provide a summary of literature data. We conclude that the best overall agreement with experiments and coupledcluster calculations is found for G_{0}W_{0} results with HSE06 as a starting point including spinorbit coupling, a truncated Coulomb potential, and an analytical treatment of the singularity at q = 0.
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Introduction
The isolation of graphene in 2004 can be regarded as a milestone in materials science that initiated the research field of atomically thin 2D materials^{1}. Compared to their 3D counterparts, 2D materials have a higher surfacetovolume ratio, making them ideal candidates for catalysts and sensors^{2,3}. Due to the confinement of electrons, holes, phonons, and photons in the 2D plane, the electronic, thermal, and optical properties of 2D materials present unusual features not found in their 3D counterparts^{4,5,6}. For instance, their electronic structure – especially band gaps – can be easily adjusted by acting on the vertical quantum confinement through, e.g., the number of atomic layers, or external perturbations, such as an external electric field, and strain^{7,8}. The sensitivity to strain, i.e., to structural details, implies that 2D materials also exhibit strong electronphonon coupling^{8}. In addition, exciton binding energies are significantly larger than in 3D materials, and they can be tuned by the dielectric environment, e.g., by encapsulation or deposition on substrates^{9,10,11}. All these characteristics make them outstanding components in novel applications for electronics and optoelectronics^{12,13,14,15,16,17}.
For a deep understanding of 2D materials, an accurate description of their bandstructure is a must. Manybody perturbation theory within the GW approach has become the stateoftheart for ab initio electronicstructure calculations of materials. In this sense, many studies have employed GW to investigate the electronic properties of 2D materials^{18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65}. Surprisingly, as illustrated in Fig. 1 for monolayer MoS_{2}, they show a wide dispersion in the fundamental band gap. The same has been found for a number of 2D materials that have been extensively studied in the last years. Results for MoS_{2}, MoSe_{2}, MoTe_{2}, WS_{2}, WSe_{2}, BN, and phosphorene are summarized in the Supplementary Information. In the extreme cases of MoS_{2}, WS_{2}, WSe_{2}, and BN, the calculated band gaps are scattered between 2.31–2.97, 2.43–3.19, 1.70–2.89, and 6.00–7.74 eV, respectively; in the worst case, the deviation (ratio between largest and smallest values) is as much as 61%. Moreover, for some materials, such as for MoS_{2}, MoTe_{2}, WS_{2}, WSe_{2}, and BN, not even the gap character is uniquely obtained—being direct or indirect, depending on the details of the calculation.
Many factors contribute to this confusing and unsatisfactory situation:

In various works, different geometries have been adopted. In this context, it must be said that the properties of 2D materials are highly sensitive to structural parameters^{18,19,66}. Small changes in the lattice constant a already have a large impact on the energy gap, as seen in Fig. 1 and Supplementary Tables 4–10. Moreover, often the lattice parameter alone is not sufficient to unambiguously determine the structure of a 2D material. For instance, phosphorene is characterized by four structural parameters; transition metal dichalcogenides require, besides a, the distance between two chalcogens (for MoS_{2}, d_{SS} as depicted in Fig. 2). These “other” structural parameters have a notable effect on the electronic properties as well^{66,67}. Unfortunately, in several studies, only the lattice parameter is reported, which prevents not only a fair comparison between published results but also reproducibility.

A second reason can be attributed to the various ways of performing G_{0}W_{0} calculations. First, there is the wellknown starting problem^{68,69,70,71,72,73,74,75,76}. Then, especially for 2D materials, G_{0}W_{0} energy gaps converge very slowly with respect to the vacuum thickness and the number of kpoints. Even slabs with a vacuum layer of 60 Å together with a 33 × 33 × 1 kgrid have been shown to be insufficient to obtain fully converged results^{28}. However, by truncating the Coulomb potential, convergence can be achieved with a reasonable amount of vacuum^{49}. The number of kpoints can be drastically reduced by an analytic treatment of the q = 0 singularity of the dielectric screening or by using nonuniform kgrids^{24,38}.

Last but not least, also spinorbit coupling (SOC) plays an important role in many cases. Besides decreasing the size of the fundamental gap mainly through a splitting of the valence band^{77}, in some 2D materials and for certain geometries and methods, disregarding or including this effect may change its character from indirect to direct or vice versa^{66}.
In this manuscript, we address all these issues and provide a benchmark data set of density functional theory (DFT) and G_{0}W_{0} calculations, taking monolayer MoS_{2} as a representative 2D material. Due to its unique properties, it can be considered the most important 2D material after graphene. MoS_{2} exhibits high electron mobility^{14,78}; moderate SOC that can be exploited in spin and valleytronics^{79,80,81,82,83,84,85}; a direct fundamental band gap with intermediately strong exciton binding, which is suitable for (opto)electronic devices operating at room temperature^{14,29,31,38,78,86}. For these reasons, there are many experimental and theoretical works in the literature that investigate MoS_{2}, allowing for a better comparison with our results.
We employ the linearized augmented planewave + local orbital (in short LAPW+LO) method as implemented in the exciting code. LAPW+LO is known to achieve ultimate precision for solving the KohnSham (KS) equations of DFT^{87} and highlevel GW results^{88}. Besides the local and semilocal DFT functionals LDA^{89} and PBE^{90,91} respectively, we include HSE06^{92,93,94} both for geometry optimization and as a starting point for G_{0}W_{0}. So far, HSE06 has not often been used for such calculations of MoS_{2}^{20,22,29,33}, and to the best of our knowledge, neither a Coulomb truncation nor an adequate treatment of the singularity at q = 0 was applied. For brevity, hereafter, we will refer to HSE06 as HSE. In our G_{0}W_{0} calculations, we truncate the Coulomb potential^{28,49,95}, and apply a special analytical treatment for the q = 0 singularity^{24}. Moreover, we investigate the role of SOC at all levels. We carefully evaluate the impact of all these elements and conclude what leads to the most reliable electronic structure of this important material. Besides a detailed analysis of energy gaps, we address effective masses and spinorbit splittings.
Results
Groundstate geometries
The geometry of MoS_{2}, depicted in Fig. 2, is determined by the inplane lattice parameter a and the distance between sulfur atoms, d_{SS}. In Table 1, we list these structural parameters as obtained with LDA, PBE, and HSE, and include the MoS bond length d_{MoS} and the angle θ between Mo and S atoms as well. As expected, LDA underestimates the lattice spacing, PBE slightly overestimates it, and HSE shows the best performance with respect to experiment. All three exchangecorrelation (xc) functionals underestimate the SS bond length, PBE being closest to its measured counterpart. Comparison with computed literature data reveals good agreement.
Electronic structure
Table 2 summarizes the energy gaps obtained with different functionals for the different geometries. We consider here the direct gap at the K point (E_{g}(KK)) as well as the indirect gaps between Γ and K (E_{g}(ΓK)) and between K and T (E_{g}(KT)). For the definition of the T point, see Fig. 2. For each geometry and methodology, the bold font highlights the fundamental gap. For the calculations that include SOC, Fig. 3 displays the energy gaps given in Table 2 with respect to the lattice parameter. In the DFT calculations, regardless of the calculation method, E_{g}(KT) (squares) shows a weak dependence on the geometry. The fundamental gap obtained with LDA, PBE, and HSE, is always direct at K. In G_{0}W_{0}, the fundamental gap is E_{g}(KT) for the structures with smaller lattice parameter and E_{g}(KK) for larger lattice constants.
As to be expected and also observed in ref. ^{38}, for a fixed geometry, the energy gaps obtained by LDA and PBE are quite similar, with the largest difference being 0.02 eV. The two functionals also agree on the location of the valenceband maximum (VBM) and the conductionband minimum (CBm). For both, the band gap is direct if SOC is included, and indirect otherwise for the PBE and HSE geometries. In contrast, HSE gives a direct gap, regardless of whether SOC is considered.
Also G_{0}W_{0}@LDA and G_{0}W_{0}@PBE are very close to each other, the largest difference being 0.04 eV. This can be attributed to the similarity between LDA and PBE when the same geometry is adopted. However, the similarity between G_{0}W_{0}@LDA and G_{0}W_{0}@PBE results is material dependent, as observed in other works^{96,97,98,99}.
For a given geometry, the locations of the VBM and the CBm are independent of the starting point, with the only exception being the HSE geometry when SOC is included. When comparing the three geometries, we encounter three different scenarios. First, for the LDA geometry, the fundamental gap changes from a direct KS gap at K to an indirect QP gap (between K and T), independent of the starting point. Second, for the PBE and HSE geometries, when SOC is disregarded, the indirect gap ΓK obtained withII D LDA and PBE becomes direct and located at K upon applying G_{0}W_{0}. Third, for the HSE geometry, and SOC being included we observe an indirect band gap for G_{0}W_{0}@LDA while it is direct for PBE and HSE as starting points. This can be understood in terms of the small differences between the KK and KT gaps, Δ_{KT}, which are 0.01 eV, 0.05 eV, and 0.15 eV for G_{0}W_{0}@LDA, G_{0}W_{0}@PBE and G_{0}W_{0}@HSE, respectively. Including SOC, splits the conduction band state at T (K) by ~0.07 eV (~3 meV), decreasing Δ_{KT} by ~0.03 eV. This is enough to make Δ_{KT} negative for G_{0}W_{0}@LDA, but not for G_{0}W_{0}@PBE, and G_{0}W_{0}@HSE. A more detailed discussion about Δ_{KT} can be found in Section “Discussion of energy gaps: comparison with experiment”.
Figure 4 shows the band structures obtained for the HSE geometry including SOC. As expected, the differences between G_{0}W_{0}@HSE and HSE bands are less pronounced than those between G_{0}W_{0}@LDA (G_{0}W_{0}@PBE) and LDA (PBE) bands. For all three starting points, the G_{0}W_{0} corrections are not uniform over all kpoints, i.e., a simple scissors approximation is, strictly speaking, not applicable. We will explore this in a more quantitative fashion further below. The SOC splitting in the valence band is zero at the Γ point and increases toward the K point. The SOC effect on the conduction bands is much less pronounced. These observations are in agreement with other theoretical^{25,29,100,101,102,103} and experimental works^{86,104}.
The impact of the selfenergy correction on the energy gaps, \(\Delta {{{{\rm{E}}}}}_{{{{\rm{g}}}}}={{{{{\rm{E}}}}}_{{{{\rm{g}}}}}}^{{G}_{0}{W}_{0}}{{{{\rm{E}}}}}_{{{{\rm{g}}}}}^{{{{\rm{DFT}}}}}\), is shown in Fig. 5 for the case when SOC is included. Clearly, ΔE_{g} is more significant for (semi)local DFT starting points than for HSE. Interestingly, for a given starting point, ΔE_{g} hardly depends on the geometry. For LDA and PBE as the starting points, the ranges of ΔE_{g}(KK), ΔE_{g}(ΓK), and ΔE_{g}(KT) are 0.84–0.90, 1.0–1.1, and 0.61–0.65 eV, respectively. The dependence is even weaker for G_{0}W_{0}@HSE with values of 0.64–0.66, 0.74–0.78, and 0.39–0.40 eV, respectively. Very similar results are observed when SOC is disregarded.
Spinorbit splittings and effective masses
In Table 3, we report for the HSE structure the spinorbit splitting Δ_{val} (Δ_{cond}) at the K point for the highest occupied (lowest unoccupied) band. Effective electron (hole) masses m_{e} (m_{h}) calculated at the K point along different directions are shown as well. We observe that neither the spinorbit splittings nor the effective masses are very sensitive to the geometry (see Supplementary Table 11 for more details).
Δ_{val} exhibits a very narrow spread among all the methods employed here (DFT and G_{0}W_{0}), i.e., a range of 143–149 meV. These values are in excellent agreement with the experimental counterparts of Δ_{val} = 130–160 meV^{86,102,104,105,106,107,108}. The value for the conduction band, Δ_{cond}, is 3 meV for LDA, PBE, G_{0}W_{0}@LDA, and G_{0}W_{0}@PBE; it is only slightly higher for HSE and G_{0}W_{0}@HSE, namely 4 meV. Again, there is excellent agreement with the measured value of Δ_{cond} = 4.3 ± 0.1 meV^{109}.
The effective hole mass, m_{h}, exhibits minor variations, not larger than 0.04m_{0}, when going from KΓ to KM. This is in line with other calculations^{27,110}. The measured value for freestanding MoS_{2} is m_{h} = (0.43 ± 0.02)m_{0}^{111} and, apart from LDA and PBE, all theoretical results show excellent agreement. The electron mass, m_{e}, is more isotropic than m_{h}. For LDA, it differs by at most 0.02m_{0} between KΓ and KM. Our values are in line with other calculated results^{18,21,27,34,101,110,112}. The measured counterpart of (0.67 ± 0.08)m_{0}^{113} is significantly larger than the calculated value reported here and in other theoretical works^{18,21,27,34,101,110,112}. As discussed in ref. ^{34}, the difference could originate from the heavy doping of the measured sample, which may introduce metallic screening.
Discussion of energy gaps: comparison with experiment
The experimental band gap for freestanding MoS_{2}, determined by photocurrent spectroscopy, is 2.5 eV^{86}. In order to compare with experiment, it is important to account for the zeropoint renormalization energy of 75 meV^{114}. This means that the theoretical value computed without this correction should be 2.575 ≅ 2.6 eV to match its measured counterpart. For our discussion here, we consider the HSE and PBE geometries, which are closer to experiment than that obtained by LDA. For the following assessment, we refer to the values in Table 2. For the structure optimized with HSE, G_{0}W_{0} performed on LDA, PBE, and HSE as starting points gives E_{g}(KK) of 2.52, 2.52, and 2.76 eV, respectively, i.e., G_{0}W_{0}@LDA and G_{0}W_{0}@PBE underestimate the measured value by about 0.08 eV, whereas G_{0}W_{0}@HSE overestimates it by 0.16 eV. However, even though G_{0}W_{0}@LDA agrees better with experiment than G_{0}W_{0}@HSE, it erroneously predicts an indirect band gap E_{g}(KT) which is 0.02 eV smaller than E_{g}(KK). G_{0}W_{0}@PBE shows the best agreement with experiment and also predicts the gap to be direct. Interestingly, considering the PBE geometry, as done in several works^{19,21,22,23,24,28,31,32,33,35,40,41,42,43,112}, G_{0}W_{0}@HSE, giving a direct band gap of E_{g}(KK) = 2.68 eV, agrees best with experiment, the deviation being 0.08 eV only. With LDA and PBE as starting points, the calculated G_{0}W_{0} band gap is direct, but 0.16 and 0.15 eV, respectively, below the experimental value.
Other relevant aspects of the bandstructure concern relative energy differences, in particular Δ_{KT} (introduced above) as well as the maximum energy at the Γ point wrt the VBM at the K point, Δ_{ΓK} = E_{g}(KK) − E_{g}(ΓK) (see Fig. 4). Experimentally, Δ_{KT} is expected to be ≳ 60 meV^{101,113} and Δ_{ΓK} ≈ 140 meV^{101,115}. Taking our calculations with SOC at the HSE geometry, (see Table 4), the G_{0}W_{0}@HSE value of 0.12 eV reproduces Δ_{KT} best. On the other hand, HSE satisfies Δ_{ΓK} best with a value of 0.12 eV (0.02 eV smaller than in experiment), while the values obtained with the other methods differ from experiment by 0.09 eV (G_{0}W_{0}@HSE), −0.09 eV (LDA and PBE), 0.10 eV (G_{0}W_{0}@PBE), and 0.11 eV eV (G_{0}W_{0}@LDA), respectively. At the PBE geometry, G_{0}W_{0}@HSE and G_{0}W_{0}@PBE give the same value for Δ_{ΓK}. With an overestimation of 0.07 eV, it is closer to experiment than the value at the HSE geometry. Again, HSE is the only starting point for which G_{0}W_{0} predicts Δ_{KT} in agreement with experiment.
In summary, considering the band gap as well as the energy differences Δ_{KT} and Δ_{ΓK}, we conclude that at the PBE geometry, G_{0}W_{0}@HSE including SOC shows the best overall agreement with experimental data. Also for the HSE structure, HSE is the best starting point, with results that are overall only slightly worse. Overall, G_{0}W_{0}@HSE at the HSE geometry can be considered more appropriate, since only one xc functional is needed for providing decent results for both, the geometry and the electronic properties and thus the most consistent picture. Also for other materials, HSE has been found to be a superior G_{0}W_{0} starting point^{76,116,117,118} compared to LDA and PBE. For such materials with intermediate band gaps,^{71,72,119,120,121,122,123,124}, this functional better justifies the perturbative selfenergy correction^{68,73,116}. Figure 5 confirms this for MoS_{2}.
Discussion of energy gaps: comparison with theoretical works
By employing coupledcluster calculations including singles and doubles^{125,126}, Pulkin et al. obtained energy gaps of E_{g}(ΓK) = 2.93 eV and E_{g}(KK) = 3.00 eV with an error bar of ± 0.05 eV^{112} for the PBE geometry of ref. ^{103}; SOC was not included. For our PBE geometry and also omitting SOC, the G_{0}W_{0}@HSE results are the ones closest to these values, with E_{g}(ΓK) differing by 0.04 eV and E_{g}(KK) by 0.24 eV.
For a fair comparison with other published G_{0}W_{0} values with LDA and PBE as starting points, we restrict ourselves here to results obtained by using a Coulomb truncation in combination with either a special treatment of the q = 0 singularity or a nonuniform kgrid sampling, since these methods ensure well converged gaps. In ref. ^{24}, disregarding SOC and adopting the PBE geometry (a = 3.184 Å, d_{SS} = 3.127 Å), a direct band gap of 2.54 eV was reported for G_{0}W_{0}@PBE which is very close to ours (E_{g}(KK) = 2.52 eV), i.e., differing by only 0.02 eV. Including SOC and the thus slightly changed PBE geometry (a = 3.18 Å, d_{SS} = 3.13 Å), a G_{0}W_{0}@LDA value of 2.48 eV was obtained in ref. ^{23}; at basically the same geometry (differences in the order of 10^{−3} Å), our results of E_{g}(KK) = 2.44 eV is only 0.04 eV smaller.
For a lattice parameter of 3.15 Å, Rasmussen et al. calculated a G_{0}W_{0}@PBE band gap of 2.64 eV without SOC^{24}. For the same lattice constant, but including SOC, Qiu et al. reported a G_{0}W_{0}@LDA band gap of^{38} of E_{g}(KK) = 2.59 eV with the plasmonpole model and E_{g}(KK) = 2.45 eV with the contour deformation method. In our case, the structure optimized with HSE (a = 3.160 Å) is closest to a = 3.15 Å. For this structure, without including SOC, we compute a G_{0}W_{0}@PBE band gap of E_{g}(KK) = 2.60 eV, which agrees quite well with the one by ref. ^{24}, differing by less than 0.04 eV. When we include SOC, we obtain E_{g}(KK) = 2.52 eV with G_{0}W_{0}@LDA, although at this geometry, we obtain an indirect gap that is 18 meV smaller than E_{g}(KK). This is in good agreement with ref. ^{38}, with a difference of 0.07 meV only.
For the experimental geometry and neglecting SOC, ref. ^{28} reported values of E_{g}(KT) = 2.58 eV and E_{g}(KK) = 2.77 eV for G_{0}W_{0}@LDA. In our case, at the HSE geometry, we obtain E_{g}(KT) = 2.61 eV and E_{g}(KK) = 2.60 eV. As the HSE geometry is close to experiment, we may attribute the discrepancies mainly to the different underlying KS states. Indeed, at the LDA level, the energy gaps in ref. ^{28} are E_{g}(KK) = 1.77 eV and E_{g}(ΓK) = 1.83 eV^{28}, while ours are E_{g}(KK) = 1.73 eV and E_{g}(ΓK) = 1.71 eV. The values for ΔE_{g}, however, compare fairly well (ΔE_{g}(KK) = 1.00 eV, ΔE_{g}(KT) = 0.6–0.7 eV in ref. ^{28}, compared to ΔE_{g}(KK) = 0.87 eV, ΔE_{g}(KT) = 0.63 eV in the present work).
When it comes to G_{0}W_{0}@HSE, there are only a few results for MoS_{2} in the literature, neither obtained with Coulomb truncation nor by any special treatment of the q = 0 singularity. For MoS_{2}, these two aspects lead to opposite effects, competing with each other when converging band gaps with respect to the vacuum size and the number of kpoints^{24}: Neglecting them, band gaps increase when the vacuum layer is enlarged, whereas denser kgrids make them decrease. Hence, due to fortunate error cancellation, an insufficient vacuum length combined with a coarse kgrid may lead to G_{0}W_{0} band gaps that agree well with those obtained in a highly converged situation^{24,29}. In ref. ^{33}, using the PBE geometry and taking SOC into account, a KK gap of 2.66 eV was reported. With 15 Å of vacuum, a 6 × 6 × 1 kgrid, and adopting the PBE geometry, in ref. ^{22}, band gaps of 2.05 and 2.82 eV at the HSE and G_{0}W_{0} levels, respectively, have been obtained. The HSE band gap agrees quite well with ours (2.04 eV), whereas our G_{0}W_{0}@HSE06 gap is 0.14 eV smaller. The band gap of 2.72 eV reported in ref. ^{29} is based on the experimental lattice parameter of 3.16 Å, a 12 × 12 × 1 kpoints grid, and a vacuum layer of 17 Å, and includes SOC effects. The authors state to have chosen these settings to take advantage of error cancellation in the band gap^{29}, and, surprisingly, our band gap of 2.76 eV obtained with G_{0}W_{0}@HSE at the HSE geometry (a = 3.160 Å) agrees quite well.
Closing remarks
We have employed the LAPW+LO method to provide a set of benchmark G_{0}W_{0} calculations of the electronic structure of twodimensional MoS_{2}. We have addressed the impact of geometry, SOC, and DFT starting point on the energy gaps, spinorbit splittings, and effective masses. We find that the selfenergy corrections to the band gaps hardly depend on the adopted geometry. As could be expected, employing LDA and PBE as starting points does not make a significant difference when the same structure is used. The best agreement with experimental results is achieved by G_{0}W_{0}@HSE at either the HSE or PBE geometry, considering SOC. The spinsplittings obtained with all methods agree well with experimental results. This also holds true for the effective hole mass, using either HSE or G_{0}W_{0} on top of any of the considered starting points (LDA, PBE, and HSE). In line with other theoretical works, we highlight the importance of a Coulomb truncation and an adequate treatment of the Coulomb singularity around q = 0 as being fundamental for highquality calculations. Our findings are expected to be valid for other twodimensional materials as well.
Methods
Linearized augmented planewave + local orbital methods
The fullpotential allelectron code exciting^{127} implements the family of LAPW+LO methods. In this framework, the unit cell is divided into nonoverlapping muffintin (MT) spheres, centered at the atomic positions, and the interstitial region in between the spheres. exciting treats all electrons in a calculation by distinguishing between core and valence/semicore states. For core electrons, assumed to be confined inside the respective MT sphere, the KS potential is employed to solve the Dirac equation which captures relativistic effects including SOC. The KS wavefunctions \(\left\vert {\Psi }_{n{{{\bf{k}}}}}\right\rangle\) for valence and semicore states, characterized by band index n and wavevector k, are expanded in terms of augmented planewaves, \(\left\vert {\phi }_{{{{\bf{G}}}}+{{{\bf{k}}}}}\right\rangle\), and local orbitals, \(\vert {\phi }_{\gamma }\rangle\), as
\(\left\vert {\phi }_{{{{\bf{G}}}}+{{{\bf{k}}}}}\right\rangle\) are constructed by augmenting each planewave with reciprocal lattice vector G, living in the interstitial region, by a linear combination of atomiclike functions inside the MT spheres. In contrast, the LOs \(\vert {\phi }_{\gamma }\rangle\) are nonzero only inside a specific MT sphere. They are used for reducing the linearization error^{127,128}, for the description of semicore states, as well as for improving the basis set for unoccupied states^{87,88}. The quality of the basis set can be systematically improved by increasing the number of augmented planewaves (controlled in exciting by the dimensionless parameter rgkmax) and by introducing more LOs^{87,127}. With all these features, exciting can be regarded as a reference code not only for solving the KS equation^{129}, where it is capable of reaching microHartree precision^{87}, but also for G_{0}W_{0} calculations^{88}.
G _{0} W _{0} approximation
In the G_{0}W_{0} approximation, one takes a set of KS eigenvalues {ε_{nk}} and eigenfunctions {Ψ_{nk}} as a reference, and evaluates firstorder quasiparticle (QP) corrections to the KS eigenvalues in firstorder perturbation theory as
where Z_{nk} is the renormalization factor, V_{xc} is the xc potential, and Σ is the selfenergy. The latter is given as the convolution
with G being the singleparticle Green function and W the screened Coulomb potential.
In this nonselfconsistent method, the quality of the QP eigenvalues may depend critically on the starting point. In many cases, LDA and PBE have been proven to be good starting points for G_{0}W_{0}, leading to QP energies that agree well with experiments^{73,130,131}. However, e.g., for materials containing d electrons, such as Mo, hybrid functionals, like HSE, usually provide an improved reference for QP corrections compared to semilocal functionals^{71,72,119,120,121,122,123,124}. Here, we evaluate the quality of each of these three as a starting point.
Coulomb truncation
In calculations with periodic boundary conditions, for treating 2D systems, a sufficient amount of vacuum is required to avoid spurious interaction between the replica along the outofplane direction. Local and semilocal density functionals have a (nonphysical) asymptotic decay much faster than 1/r, facilitating convergence of unoccupied states –and thus KS gaps– with respect to the vacuum size. In G_{0}W_{0}, the 1/r decay of the selfenergy complicates this task. Specifically for MoS_{2}, even a vacuum layer with 60 Å thickness turned out not being sufficient to obtain a fully converged band gap^{28,49}. Truncating the Coulomb potential along the outofplane direction z, however, leads to wellconverged G_{0}W_{0} band gaps with a considerably smaller vacuum size^{28,49}.
Here, we truncate the Coulomb potential with a step function along z. Setting the cutoff length to L/2, where L is the size of the supercell along z (Fig. 2), the truncated Coulomb potential can be written in a planewave basis as^{95}:
where Q = q + G, and q is a vector in the first Brillouin zone (BZ).
Treatment of the q = 0 singularity
On the downside, truncating the Coulomb interaction slows down the convergence in terms of kpoints because of the nonsmooth behavior of the dielectric function around the singularity at q = 0^{23,24,41}. To bypass this problem, we follow an analytical treatment of W^{c}, the correlation part of W, close to the singularity as proposed in ref. ^{24}. Without this treatment, the correlation part of the selfenergy Σ^{c}(ω) can be written as^{132}
where \({\tilde{\epsilon }}_{n{{{\bf{k}}}}}={\epsilon }_{n{{{\bf{k}}}}}+{{{\rm{i}}}}\,\eta \,{{{\rm{sign}}}}({E}_{{{{\rm{F}}}}}{\epsilon }_{n{{{\bf{k}}}}})\) and E_{F} the Fermi energy. \({M}_{nm}^{i}({{{\bf{k}}}},{{{\bf{q}}}})\) are the expansion coefficients of the mixedproduct basis, an auxiliary basis to represent products of KS wavefunctions. Like LAPWs and LOs, they have distinct characteristics in the MT spheres and interstitial region^{132,133}. To treat the q = 0 case separately, the corresponding term in Eq. (5) is replaced by
where Ω_{0} is a small region around q = 0. Analytical expressions for \({W}_{ij}^{{{{\rm{c}}}}}({{{\bf{q}}}},{\omega }^{{\prime} })\) in the limit q → 0^{24} are then employed to calculate the integral in Eq. (6).
Spinorbit coupling
In this study, SOC is treated via the secondvariational (SV) scheme^{134}. In LDA and PBE calculations, the conventional SV approach is utilized^{127,135,136,137}: First, the scalarrelativistic problem, i.e., omitting SOC, is solved. A subset of the resulting eigenvectors is then used as basis set for addressing the full problem. The number of eigenvectors is a convergence parameter. In this work, the SOC term is evaluated with the zeroorder regular approximation (ZORA)^{138,139}.
A groundstate calculation with HSE is performed via a nested loop^{140}: In the outer loop, the nonlocal exchange is computed for a subset of KS wavefunctions, using a mixedproduct basis. The inner loop solves the generalized KS equations selfconsistently, where only the local part of the effective potential is updated in each step. Within this inner loop, the SV scheme is applied selfconsistently to incorporate SOC. The corresponding term, evaluated within the ZORA, is based on PBE, which is justified by the minimal contribution due to the gradient of the nonlocal potential^{141,142,143}.
G_{0}W_{0} calculations with SOC are performed in two steps on top of groundstate calculations that include SOC. First, the QP energies are computed as explained in Section “G_{0}W_{0} approximation”, using the scalarrelativistic KS eigenvalues and eigenvectors. In the second step, the obtained QP energies are used, together with the SV KS eigenvectors, to evaluate SOC through the diagonalization of the SV Hamiltonian. This approach is sufficient in the case of MoS_{2} since SOC does not cause band inversion^{133,143}.
Computational details
In our calculations, we employ the allelectron fullpotential code exciting^{127}. The only exception is for obtaining the HSE equilibrium geometry, where FHIaims^{144,145} is used, since so far exciting lacks geometry relaxation with hybrid functionals. Even though exciting and FHIaims implement very different basis sets to expand the KS wavefunctions, the two codes have been shown to be among those with the best mutual agreement^{129}. Moreover, a comparison of energy gaps and geometry relaxations for MoS_{2} confirms this finding (see Supplementary Information  Section I).
In all calculations, the unitcell size L along the outofplane direction (Fig. 2) is set to 30 bohr. Different flavors of xc functionals are applied, namely LDA, PBE, and HSE. In the latter, we use the typical parameters^{94}, i.e., a mixing factor of α = 0.25 and a screening range of ω = 0.11 bohr^{−1}. To determine the respective equilibrium geometries, we relax the atomic positions until the total force on each ion is smaller than 10 μHa bohr^{−1}. For these geometries, the electronic structure is calculated with all three functionals, with and without SOC, giving rise to a set of 18 calculations. These calculations are followed by G_{0}W_{0} calculations, taking the respective DFT solutions as starting points.
The dimensionless parameter rgkmax that controls the exciting basisset size is set to 8. In the LDA and PBE calculations, we use a 30 × 30 × 1 kgrid. In HSE and G_{0}W_{0} calculations, we employ 400 empty states and an 18 × 18 × 1 kgrid. In G_{0}W_{0}, the correlation part of the selfenergy is evaluated with 32 frequency points along the imaginary axis, and then analytic continuation to the real axis is carried out by means of Pade’s approximant. For the bare Coulomb potential, we use a 2D cutoff^{95} combined with a special treatment of the q = 0 singularity as described in Section “Treatment of the q = 0 singularity”. Furthermore, we carefully determine the minimal set of LOs, sufficient to converge at least the lowest 400 KS states. This is achieved with 2 and 6 LOs for sulfur s and p states, respectively, as well as 3, 6, and 10 LOs for molybdenum s, p, and d states, respectively. Overall, we estimate a numerical precision of 50–100 meV in the energy gaps obtained with our G_{0}W_{0} calculations. To determine effective masses, we use parabolic fits within a range of 0.05 Å^{−1} around the VBM and the CBm at the K point of the BZ. Depending on the respective case, K can host either global or local extrema.
Data availability
All input and output files are available at NOMAD^{146,147} under https://doi.org/10.17172/NOMAD/2023.09.161.
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Acknowledgements
This work received funding from the German Research Foundation, projects 182087777 (CRC HIOS) and 424709454 (SPP 2196, Perovskite Semiconductors). I.G.O. thanks the DAAD (Deutscher Akademischer Austauschdienst) for financial support. Partial funding is appreciated from the European Union’s Horizon 2020 research and innovation program under the grant agreement No. 951786 (NOMAD CoE). Computing time on the supercomputers Lise and Emmy at NHR@ZIB and NHR@Göttingen as part of the NHR infrastructure is gratefully acknowledged.
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R.R.P. carried out the G_{0}W_{0}(@LDA and @HSE) calculations with SOC; collected, analyzed, and interpreted all the results; and wrote the first version of the manuscript. C.V. carried out the majority of calculations involving HSE with exciting; tested the Coulomb truncation and singularity treatment for G_{0}W_{0}@HSE. S.L. carried out G_{0}W_{0} (@LDA and @PBE) calculations with SOC. I.G.O. carried out the FHIaims calculations. B.A. implemented the Coulomb truncation in exciting. C.D. initiated and guided the overall work. All authors contributed to regular discussions, the next steps to be taken, and to the writing.
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Rodrigues Pela, R., Vona, C., Lubeck, S. et al. Critical assessment of G_{0}W_{0} calculations for 2D materials: the example of monolayer MoS_{2}. npj Comput Mater 10, 77 (2024). https://doi.org/10.1038/s41524024012532
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DOI: https://doi.org/10.1038/s41524024012532
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