Abstract
Manybody perturbation theory methods, such as the G_{0}W_{0} approximation, are able to accurately predict quasiparticle (QP) properties of several classes of materials. However, the calculation of the QP band structure of twodimensional (2D) semiconductors is known to require a very dense BZ sampling, due to the sharp qdependence of the dielectric matrix in the longwavelength limit (q → 0). In this work, we show how the convergence of the QP corrections of 2D semiconductors with respect to the BZ sampling can be drastically improved, by combining a Monte Carlo integration with an interpolation scheme able to represent the screened potential between the calculated grid points. The method has been validated by computing the band gap of three different prototype monolayer materials: a transition metal dichalcogenide (MoS_{2}), a wide band gap insulator (hBN) and an anisotropic semiconductor (phosphorene). The proposed scheme shows that the convergence of the gap for these three materials up to 50meV is achieved by using kpoint grids comparable to those needed by DFT calculations, while keeping the grid uniform.
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Introduction
The GW approximation^{1,2,3,4} is a wellestablished method for first principle calculations of electronic excitations of materials^{5,6,7}. It provides access to quasiparticle energy bands as measured in ARPES experiments^{8}, satellites^{9,10}, lifetimes^{11}, and spectral functions^{12,13}. Since its development, the GW approximation has been applied to a large class of systems ranging from bulk crystals to nanostructures and molecules^{5,6}. During the last decades, since the isolation and characterization of graphene^{14}, large attention has been devoted to the study of 2D materials, due to their remarkable electronic and optical properties^{15,16}. Since then, the GW approximation has been extensively applied to predict quasiparticle properties of these materials^{17,18,19,20,21,22,23,24,25,26,27}.
Often, 2D systems are treated using plane waves within the supercell approach, in which an amount of vacuum is added in the non periodic direction in order to remove spurious interactions among replicas. In principle, accurate GW calculations require the inclusion of a very large vacuum extension due to the longrange nature of the Coulomb interaction. This difficulty has been mitigated using truncated Coulomb potentials^{28,29} that allows one to obtain converged results considering manageable interlayer distances (e.g., in the range of 10–20 Å). Furthermore, characteristic properties of 2D screening, such as the dielectric function approaching unity in the longwavelength limit (see below), are correctly reproduced in the supercell approach only if the potential is appropriately truncated^{30,31}. However, once the Coulomb potential is truncated, the resulting sharp behaviour of the screened potential can make the integration over the Brillouin zone (BZ) rather inefficient^{22,31}. Thus very large kpoint grids are needed to obtain converged results, making the computation of quasiparticle (QP) properties within the GW method for 2D systems computationally expensive^{32}.
More in details, in a planewave basis set description, the screening properties are described by the matrix elements of the Fourier transform of the inverse dielectric function \({\epsilon }_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{1}({{{\bf{q}}}})\), where G is a reciprocal lattice vector and q a reciprocal vector of the first BZ. In 2D systems, as already pointed out in the literature^{30,33,34,35,36,37}, the head [\({{{\bf{G}}}}={{{{\bf{G}}}}}^{{\prime} }=(0,0,0)\)] of the dielectric matrix sharply approaches unity in the long wavelength limit (Fig. 1, left panel), and it is clear that with coarse meshes it is not possible to correctly reproduce such limit with a regular discretization procedure. In addition, the first matrix elements associated with lattice vectors along the confined direction (G_{⊥}) show a dispersion in the longwavelength limit, differently from the matrix elements with lattice vectors oriented in the periodic directions (G_{∥}) which are approximately constant with respect to ∣q∣ (see Fig. 1). This trend originates from the fact that \(\min  {{{{\bf{G}}}}}_{\perp }\) is significantly smaller than \(\min  {{{{\bf{G}}}}}_{\parallel }\), due to the amount of vacuum added in the perpendicular direction. Furthermore, we note that in 2D systems the longwavelength limit of the wing matrix of the dielectric matrix, \({\epsilon }_{{{{\bf{G}}}}{{{\bf{0}}}}}^{1}\), goes to zero as ∣q∣ → 0 (Fig. 1 right panel) leading to possible numerical instabilities when these terms are multiplied by the diverging Coulomb potential. All these features of the dielectric matrix contribute to slow the convergence of the QP properties with respect to the number of sampling points in the BZ, usually discretized on a uniform grid.
In the last years, different strategies have been proposed to accelerate the convergence of GW results for 2D systems with respect to the number of kpoint sampling. Rasmussen et al.^{35} proposed an analytic model for the longwavelength limit of the head of the inverse dielectric function \({\epsilon }_{00}^{1}\). This model has been used to integrate the screened potential in a small region around the Γ point, thus reducing the size of the kpoint mesh needed to converge the quasiparticle gap. However, denser meshes with respect to DFT are still required (e.g., for the band gap of MoS_{2} converged results within 50meV were reported^{35} using 18 × 18 × 1 grids) as the analytic model is applied only to the \({{{\bf{G}}}}={{{{\bf{G}}}}}^{{\prime} }=0\) matrix element.
Da Jornada et al.^{36} proposed instead a fully numerical approach, where a nonuniform qsampling is used to increase the sampling close to the Γ point. This approach has been applied not only to the \({{{\bf{G}}}}={{{{\bf{G}}}}}^{{\prime} }=0\) element of the dielectric matrix, but to a submatrix (G_{⊥}, \({{{{\bf{G}}}}}_{\perp }^{{\prime} }\)) such that ∣G_{⊥}∣,\( {{{{\bf{G}}}}}_{\perp }^{{\prime} } < \min  {{{{\bf{G}}}}}_{\parallel }\). In Xia et al.^{38}, the two previous strategies are combined by performing a nonuniform subsampling of the Brillouin zone around Γ followed by a nonlinear fitting procedure to model the qdependence of the selfenergy terms (both exchange and correlation) instead of modelling the behaviour of the dielectric matrix or screened potential elements. Xing et al.^{39} proposed a staggered mesh method for accelerating the BZ sampling convergence of the correlation energies evaluated with diagrammatic perturbation theory. While this work may be extended also for the calculation of QP corrections, to the best of our knowledge, it has not yet been applied in this context.
The methods of refs. ^{36,38} showed that meshes of size similar to those needed to converge the DFT ground state calculations were enough to obtain reliable results as demonstrated for the gap of MoSe_{2} bilayer and MoS_{2} monolayer. However, both methods rely on a nonuniform sampling, which add a convergence parameter to be managed (the number of subsampling points). Moreover, the region around Γ in which the additional sampling is performed (and consequently the nonlinear fitting in Xia et al.) depends on the size of the uniform grid. This may cause inconsistency problems when comparing results from different grids, e.g., in a convergence set of calculations, as increasing the grid the region around Γ becomes smaller and smaller.
Motivated by these works, we show that the convergence of QP properties of 2D semiconductors with respect to the number of kpoints in the BZ sampling can be accelerated, at the same level of previous methods found in the literature^{36,38}, by combining the Monte Carlo integration techniques with an interpolation scheme of the screened potential. Unlike the methods described above, the proposed method allows one to accelerate the convergence of the QP properties overcoming the need to rely on a nonuniform sampling. In addition, the same integration procedure (see below) is applied to the full BZ, thereby avoiding the need to treat the Γ region differently from the remaining part of the BZ. The proposed method has been implemented in the Yambo package^{40,41}.
The work is organized as follows: In “Methods”, we present the main ideas of the proposed method and its implementation. In “Results”, we show the performance of the method for three prototype 2D semiconductors: a transition metal dichalcogenide (MoS_{2}), a wide band gap insulator (hBN), and an anisotropic semiconductor (phosphorene). In section “Computational details”, we provide the computational details, and in “Discussion”, we draw the conclusions.
Results
The Wav method
Within manybody perturbation theory, quasiparticle energies are usually calculated either by solving numerically the QP equation:
where {nk} are the KS wavefunctions and \({v}_{xc}^{{{{\rm{KS}}}}}\) is the exchangecorrelation potential, or by linearizing the equation at the first order:
where the renormalization factor Z_{nk} is defined as:
To obtain the QP correction of a singleparticle state \(\left\vert n{{{\bf{k}}}}\right\rangle\) in the GW approximation, we need to evaluate the diagonal matrix element of the selfenergy,
where the screened interaction W is obtained from the expression W(ω) = v + vχ(ω)v = ϵ^{−1}(ω)v, with the reducible polarizability χ(ω) treated at the RPA level. The self energy can be split into the exchange (x) and correlation (c) parts as
Notably, both terms of the selfenergy involve an integration over the momentum transfer q. If we discretize the BZ with a 2D uniform kgrid (centred at Γ), following the Monkhorst–Pack scheme^{42}, the momentum transfer q is discretized with the same uniform grid, and the q integrals can be evaluated as finite sums. Thus, the x selfenergy is written as:
where Ω is the volume of the unit cell in real space, v labels the occupied bands, the ρ_{nm} matrix elements are defined as \({\rho }_{nm}({{{\bf{k}}}},{{{\bf{q}}}},{{{\bf{G}}}})=\left\langle n{{{\bf{k}}}}\right\vert {e}^{i({{{\bf{q}}}}+{{{\bf{G}}}})\cdot {{{\bf{r}}}}}\left\vert m{{{\bf{k}}}}{{{\bf{q}}}}\right\rangle\), and N_{q} is the number of points of the q grid. In order to eliminate periodic image interactions for a 2D system, we take the Coulomb potential in Eq. (6) as a truncated potential in a slab geometry. Its Fourier transform reads^{28,29}:
where L is the length of the cell in the nonperiodic z direction. As the qgrid is 2D, we have q_{z} = 0.
Nevertheless, Eq. (6) cannot be directly applied due to the divergence of the Coulomb interaction at G = q = 0. There are several approaches to treat such divergence^{40,43,44,45,46}. Among these, we select the vaverage (vav) method (called random integration method and described in ref. ^{40}). In this method, it is assumed that the matrix elements ρ_{nm}(k, q, G) are smooth with respect to q, and Eq. (6) is discretized in the following way
where \(\bar{v}\) is the average of the Coulomb interaction within a region of the BZ centred around q of the Monkhorst–Pack grid:
D_{Γ} is the small area of the Monkhorst–Pack grid centred around Γ (red area in Fig. 2). The integrals in Eq. (9) are evaluated via a 2D Monte Carlo technique.
In addition, Eq. (8), as compared with Eq. (6), leads to a faster convergence of the the exchange selfenergy with respect to N_{q}, since Eq. (8) takes into account the variation of the bare potential within the region of the BZ centred around each q point and it has been extensively applied also to three dimensional semiconductors^{47}. For practical purposes, it is sufficient to evaluate the averages up to a threshold \( {{{\bf{G}}}}{ }^{2}/2\, < \,{E}_{{{{\rm{cut}}}}}^{{{{\rm{vav}}}}}\), since the Coulomb interaction becomes a smooth function of q at large ∣G∣.
We now consider the correlation part of the selfenergy, that is the most problematic term for 2D semiconductors. Within the plasmonpole approximation (PPA) (we here adopt the Godby–Needs formulation^{48}) the correlation part of the screened Coulomb potential, W^{c}(ω) = W(ω) − v, is written as:
where the limit η → 0^{+} is implicitly assumed. Then, Σ^{c} can be expressed as:
where the matrix elements \({g}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{n{{{\bf{k}}}}}\) are defined as:
In Eq. (11) \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}({{{\bf{q}}}})\) is the static component of the screened Coulomb interaction. In particular, \({g}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{n{{{\bf{k}}}}}\) are smooth functions of q for ω far from \({{{\Omega }}}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\), where the PPA is justified.
For small G vectors, the correlation part of the screened potential, \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}({{{\bf{q}}}})\) shows a sharp q dependence due to the behaviour of both the bare interaction v_{G}(q) and the inverse dielectric function \({\epsilon }_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{1}({{{\bf{q}}}})\), as shown in Fig. 1. For this reason, following a similar procedure already applied to \({{{\Sigma }}}_{n{{{\bf{k}}}}}^{{{{\rm{x}}}}}\), we discretize Eq. (11) as
where
is the average of the correlation part of the screened potential in the miniBZ around q of the Monkhorst–Pack grid. The evaluation of the correlation part of the selfenergy via Eq. (14) is referred to in the following as the Wav method. The integrals in Eq. (14) are calculated using a 2D Monte Carlo integration method, where \({\overline{{W}^{{{{\rm{c}}}}}}}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}({{{\bf{q}}}}+{{{{\bf{q}}}}}^{{\prime} })\) is evaluated considering typically ≈ 10^{6}\({{{{\bf{q}}}}}^{{\prime} }\) points in the region around Γ (red area of Fig. 2) making use of an interpolation scheme, that is discussed in details in the next section. The number of random points used to evaluate the integrals guarantees a statistical error that does not interfere with the accuracy of the calculation. In practice, the Monte Carlo average is performed for a limited number of matrix elements of W such that \( {{{\bf{G}}}}{ }^{2}/2 \,< \,{E}_{cut}^{{{{\rm{Wav}}}}}\), i.e., for the matrix elements presenting a sharp behaviour as a function of q, while for the remaining G vectors the screening is evaluated on the q grid determined by the kpoint sampling.
Importantly, the Waverage correction performed for \( {{{\bf{G}}}}{ }^{2}/2 \,< \,{E}_{cut}^{{{{\rm{Wav}}}}}\) is applied to every q point of the BZ, at variance with other proposed methods where corrections are applied to the q = 0 term only^{35,36,38}.
We note the Wav method, here derived along with the PPA, may be easily generalized to fullfrequency treatments of the selfenergy, by averaging the dynamical screened interaction, and not only its static part, with a generalized version of Eq. (14). However, this is out of the scope of this work.
Interpolation of the static screening
In this Section, we describe a procedure to interpolate the correlation part of the static screened potential \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}({{{\bf{q}}}})\) as a function of q, for the computation of the average \({\overline{{W}^{{{{\rm{c}}}}}}}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) according to Eq. (14). The head of the screened potential, \({W}_{00}^{{{{\rm{c}}}}}\), can be exactly written as:
where f(q) is an auxiliary function defined in the Supplementary Methods. The expression of Eq. (15) suggests that it is possible to use f(q) for the interpolation of \({W}_{00}^{{{{\rm{c}}}}}({{{\bf{q}}}})\). In fact, while \({W}_{00}^{{{{\rm{c}}}}}({{{\bf{q}}}})\) shows a sharp behavour as a function of q, the function f(q) is smoother, as it resembles the irreducible response function \({\chi }_{00}^{0}({{{\bf{q}}}})\) plus some corrections due to localfield contributions. In fact, \(f({{{\bf{q}}}})={\chi }_{00}^{0}({{{\bf{q}}}})\) if local field effects are neglected. Guided by this argument, we propose to represent the matrix elements of the static screening as
where the Coulomb interaction v_{G} is given by Eq. (7) and \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) is an auxiliary function. We note that Eq. (16) is the simplest generalization of Eq. (15) for the case G ≠ 0, \({{{{\bf{G}}}}}^{{\prime} }\,\ne\, 0\). We remind that in our notation q is a point of the MonkhorstPack grid, while \({{{{\bf{q}}}}}^{{\prime} }\) belongs to D_{Γ} (red region in Fig. 2). By inverting Eq. (16) on the qpoints of the mesh we have:
In order to compute \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}({{{\bf{q}}}}+{{{{\bf{q}}}}}^{{\prime} })\) without requiring a dense mesh of q points, the function is numerically determined by interpolating between the given q point and its four nearest neighbours in reciprocal lattice coordinates v. A sketch of the interpolation scheme is shown in Fig. 2.
The auxiliary function is parametrized as:
where
and,
v and \({{{{\bf{v}}}}}^{{\prime} }\) being q and \({{{{\bf{q}}}}}^{{\prime} }\) in reciprocal lattice coordinates. The polynomial dependence of \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}({{{\bf{v}}}}+{{{{\bf{v}}}}}^{{\prime} })\) with respect to \({{{{\bf{v}}}}}^{{\prime} }\) is inspired by the Taylor expansion of \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) around v. In Eq. (18), there are six coefficients that must be determined. As there are only four nearest neighbours, thus four conditions to apply, we set for simplicity \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{12}({{{\bf{v}}}})={f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{21}({{{\bf{v}}}})=0\). This choice corresponds to adopt a bilinear interpolation.
We note that f_{00}(q → 0) is the most relevant element in the integration of the selfenergy^{35}. For semiconductors, it is possible to exploit the known behaviour \(\mathop{\lim }\nolimits_{{{{\bf{q}}}}\to 0}{f}_{00}({{{\bf{q}}}})\propto  {{{\bf{q}}}}{ }^{2}\) (see Supplementary Methods and ref. ^{49}) to impose a specific and more accurate functional form to the head (G = 0 and \({{{{\bf{G}}}}}^{{\prime} }=0\)) at q = 0. Following the model for the inverse dielectric function adopted by IsmailBeigi^{28} we consider for \({f}_{00}({{{{\bf{v}}}}}^{{\prime} })\) the expression:
where \({\bar{\bar{f}}}_{\lim }\) is a 2 × 2 tensor which describes the anisotropy of \({\chi }_{00}^{0}\) and of \({W}_{00}^{{{{\rm{c}}}}}\), and \({{{{\bf{q}}}}}^{{\prime} }={{{{\bf{q}}}}}^{{\prime} }({{{{\bf{v}}}}}^{{\prime} })\). We note that in Eq. (21) the \({\bar{\bar{f}}}_{\lim }\) tensor is represented in cartesian coordinates. However, we stress that the representation basis is arbitrary, as the tensorial scalar product does not depend on the coordinate choice. This choice, differently from the reciprocal lattice unit representation, makes the \({\bar{\bar{f}}}_{\lim }\) proportional to the identity matrix in the case of isotropic systems. We can partially account for the anisotropy of the auxiliary function by keeping the diagonal form (\({f}_{\lim }^{xy}={f}_{\lim }^{yx}=0\)) but relaxing the proportionality to the identity matrix (\({f}_{\lim }^{xx}\,\ne \,{f}_{\lim }^{yy}\)).
By substituting Eq. (21) into Eq. (17) and taking the ∣q∣ → 0 limit along the x and y directions, respectively (the periodic directions), we have
Otherwise, we may neglect the anisotropy of the auxiliary function adding the following approximation: \({f}_{\lim }^{xx}\approx {f}_{\lim }^{yy}\equiv {f}_{\lim }\), where
In Eq. (23), the limit is performed along the inplane 110 cartesian direction, in order to partially average between the x and y directions. The α and β coefficients in Eq. (21) are obtained by interpolation, using the nearest neighbours of the q = 0 point. We note there are only two independent nearestneighbour conditions to be applied, due to the symmetry property f_{00}(q) = f_{00}(−q) [which can be derived from the symmetry property \({\chi }_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{0}({{{\bf{q}}}})={\chi }_{{{{{\bf{G}}}}}^{{\prime} }{{{\bf{G}}}}}^{0}({{{\bf{q}}}})\)].
Accuracy of the interpolation
We now present the results obtained with the interpolation scheme derived in “Methods” for three prototype monolayer materials starting from the transition metal dicalcogenide MoS_{2}. Electronic properties of MoS_{2} have been extensively studied in the literature, including several calculations using the G_{0}W_{0} approach^{31,34,50,51}. It is a direct gap material with hexagonal structure having the gap localized at the \(K/{K}^{{\prime} }\) point in the BZ. The valence band at \(K/{K}^{{\prime} }\) is split due to spinorbit coupling^{52}, but since we are interested in the convergence behaviour with respect to the qpoint sampling, and for sake of simplicity, we have not included spinorbit effects in the present calculations. In addition, MoS_{2} has been used to test two other convergenceaccelerator schemes^{35,38}, which allows for a direct comparison with our approach.
In Fig. 3, we show some matrix elements of the correlation part of the screened potential \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}({{{\bf{q}}}})\) and the auxiliary function \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) [see Eq. (17)] as a function of the momentum transfer q. The interpolation functions (orange lines) are computed starting from the data on a coarse grid (6 × 6 × 1), and compared with the same quantities computed with a denser grid (60 × 60 × 1), taken here as a benchmark. The matrix element of W contributing the most to the GW correction is the \({{{\bf{G}}}}={{{{\bf{G}}}}}^{{\prime} }=0\) term (Fig. 3 left panels), being \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\) at least two order of magnitude larger than the other elements. As shown in Fig. 3, for the \({{{\bf{G}}}}={{{{\bf{G}}}}}^{{\prime} }=0\) element there is a very good agreement between the results obtained interpolating the coarser grid (orange line) and the values calculated using the denser grid. For all the matrix elements considered, the auxiliary function \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) is smoother than \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\) which supports the choice of interpolating \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) instead of \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\). Fig. 3
shows no clear trend between the interpolation accuracy of \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) and of \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\), in particular in the region q ≈ 0. Since the different Gcomponents of the bare Coulomb potential, Eq. (7), have different limits and slopes for ∣q∣ → 0, the error associated with the interpolation of \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) can be both enhanced or quenched when propagated to \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\). Despite this, we find a very good agreement between the interpolated values of \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\) and the results obtained with the denser grid for all the considered matrix elements.
The grey shaded areas represent the integrals of \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\) as obtained by applying the trapezoidal rule to the coarser grid together with the regularization of the Coulomb potential at \({{{\bf{G}}}}={{{{\bf{G}}}}}^{{\prime} }={{{\bf{q}}}}=0\), given by Eq. (9). For the sake of comparison, the same integrals, now obtained by using the interpolation, are highlighted in orange. The trapezoidal rule, due to the regularization of the bare Coulomb potential, misses completely the integral contribution at \({{{\bf{G}}}}={{{{\bf{G}}}}}^{{\prime} }=0\) because of the vanishing value of \([{\epsilon }_{00}^{1}({{{\bf{q}}}}=0)1]\), while \({\bar{v}}_{0}({{{\bf{q}}}}=0)\) remains finite.
Therefore, averaging the whole W^{c}, as we do in Eq. (14), instead of averaging v, Eq. (9), and multiplying by \([{\epsilon }_{00}^{1}({{{\bf{q}}}})1]\), is mandatory to have a contribution different from zero in this region.
We also note that the trapezoidal rule misses the integral contributions of \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}\) for G = 0 or \({{{{\bf{G}}}}}^{{\prime} }=0\) (wings) in the longwavelength limit (q → 0), since \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}({{{\bf{q}}}})\to 0\) as q → 0. Finally, when \({{{\bf{G}}}},{{{{\bf{G}}}}}^{{\prime} }\,\ne \,0\), the trapezoidal rule overestimate the integral in the region q ≈ 0. The orange areas, obtained with the interpolation functions, give instead a good description of the areas under the dense grid data. This justifies the accuracy of the interpolation method with fairly coarse grids, as detailed in the following.
Convergence of the fundamental gap
In Fig. 4, we show the results for the QP band gap as a function of the qpoint sampling for MoS_{2}, hBN, and phosphorene. As for the case of MoS_{2}, also hBN^{53,54,55,56,57,58,59} and phosphorene monolayer^{23,60,61} have been extensively studied using GW approach. Moreover, due to its high anisotropy, the phosphorene monolayer is an ideal system to test the two proposed treatments of the \({f}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}\) anisotropy, given in Eqs. (23) and (22).
In Fig. 4, the convergence of the fundamental gap for the three materials as a function of the qsampling is shown using the proposed accelerated method (Wav) and the vav method. In the latter case, only the q = G = 0 term of the Coulomb interaction has been averaged, in order to regularize the Coulomb divergence. We verified that the use of the vav method to treat the q ≠ 0 an G ≠ 0 terms of the Coulomb interaction does not significantly affect the results of the fundamental gap for the considered systems. The vav method shows a very slow convergence with respect to N_{k}, as expected, and the gaps in the limit of an infinitely dense grids have been obtained by using an 1/N_{k} extrapolation. For all the three cases we note that the gap is overestimated when unconverged grids are used, mainly due to the lack of the long wavelength contributions of the correlation parts of the screened potential, as explained in the previous section (see Fig. 3). Using the vav method, to obtain a gap value within less than ±50 meV with respect to the extrapolated value, kgrids of 54 × 54 × 1, 36 × 36 × 1, and 36 × 50 × 1 are required for MoS_{2}, hBN, and phosphorene, respectively.
With the proposed Wav method, the convergence of the gaps is greatly accelerated, and we obtain converged results already using 6 × 6 × 1, 6 × 6 × 1 and 8 × 12 × 1 grids for MoS_{2}, hBN, and phosphorene, respectively, comparable with those required to obtain converged DFT results. These grids are respectively 80, 40 and 20 times smaller than the ones required to have similar accuracy without acceleration. As an example, the timetosolution (on a single node) for the calculation of the converged GW energy gap in the case of MoS_{2}, reduces from about 11.5 h to less than a minute, thanks to the reduction of the meshsize (from 54 × 54 to 6 × 6) obtained through the application of the Wav method. Converged results using similar size of kgrids were also obtained with alternative accelerator schemes^{36,38}. However, within the present method, differently from the other proposed strategies^{36,38}, no additional subsampling points are required to be computed in the region q ≈ 0.
The orange dots in Fig. 4 are obtained with a parametrization of the head of the auxiliary function given by Eq. (23), which accounts for the anisotropy of the system by simply interpolating along the direction (110). Nevertheless, for phosphorene, that is highly anisotropic, we have also taken explicitly into account the anisotropy of W^{c}, using a parametrization of the auxiliary function given by Eq. (22) (red dots). Although the longwavelength limits of \({W}_{00}^{{{{\rm{c}}}}}\) are different, the average of the correlation part of the potential, see Eq. (14), is very similar in the two schemes and the resulting quasiparticle corrections do not show substantial differences. Despite the present results for phosphorene show that the explicit anisotropic treatment does not affect the value of the computed band gap, this does not exclude the fact that it can be potentially relevant for other systems and deserves further investigation.
Next, we turn the attention on the role of the number of matrix elements of \({W}_{{{{\bf{G}}}}{{{{\bf{G}}}}}^{{\prime} }}^{{{{\rm{c}}}}}({{{\bf{q}}}})\) averaged through Eq. (14), identified by the parameter \({E}_{cut}^{av}\). In Fig. 5, we plot the convergence of the band gap of MoS_{2} with respect to \({E}_{cut}^{av}\), or, alternatively, with respect to the number of G shells for which the averaging procedure is employed. In the plot, points at \({E}_{\rm{cut}}^{\rm{av}}=0\) refer to gaps obtained with the vav method, i.e., blue triangles shown in the top panel of Fig. 4. The W averaging of the first element gives the largest contribution to the convergence acceleration, closing the gap to 1.28 and 0.60 eV for the 6 × 6 × 1 and 12 × 12 × 1 grids, respectively. The W averaging of the first G_{⊥} matrix elements is also important to obtain converged results with coarser grids. By looking at Fig. 3, it is evident that the standard integration technique (black shaded area) misses the q = 0 contribution of the wing elements, as \({W}_{0{{{{\bf{G}}}}}_{\perp }}^{{{{\rm{c}}}}}({{{\bf{q}}}}\to 0)=0\). The Wav method (orange shaded area) provides instead a finite contribution, improving the convergence trend. In particular, the coarser the grid, the more important is the averaging of \({W}_{{{{{\bf{G}}}}}_{\perp }{{{{\bf{G}}}}}_{\perp }^{{\prime} }}^{{{{\rm{c}}}}}\), as shown by the comparison of the 6 × 6 × 1 with the 12 × 12 × 1 grids. Still, in both cases, the convergence is reached for a small number of G shells, which translates into a nearly negligible added computational cost required to perform the averaging of the screened potential. According to our results, \({E}_{\rm{cut}}^{\rm{av}}\approx 1\)–2 Ry is a reasonable choice for all the systems considered.
Comparison with the literature
Finally, in Table 1, we show the G_{0}W_{0} converged gaps of MoS_{2}, hBN, and phosphorene and compare them with the data present in the literature. We emphasize that the results presented in Table 1 have been obtained with an increased number of bands and cutoff energy of the dielectric function with respect to the data shown in the previous figures, as explained in detail in section “Computational details”. The MoS_{2} gaps found in the literature ranges from 2.41 to 2.78 eV. Our value, 2.62 eV, lies within this range. Instead, the fundamental gaps of hBN found in the literature differ considerably from each other, with discrepancies most likely due to the different approximations employed in the calculations. Coming to the case of phosphorene, the gap computed within this work is generally in agreement with the results found in the literature.^{35,36,62,63} Only the value found in ref. ^{64} deviates significantly from the others. Notably, in this work the fundamental gap of the periodic structure is obtained by extrapolating the thermodynamic limit from finite size systems of increasing size, at variance with the other works, where a periodic structure is considered. Such extrapolation procedure may be the cause of the observed discrepancy.
Discussion
Accurate results for the calculation of quasiparticle energies in the GW approximation for 2D semiconductors can be obtained only by using very large kpoint grids, making calculations computationally very demanding. We have provided here a technique based on a stochastic averaging and interpolation of the screened potential to accelerate the convergence of the self energy with respect to the qpoint sampling. We have tested the proposed scheme for the calculation of the QP gap of three prototypical monolayer semiconductors: MoS_{2}, hBN, and phosphorene. We find that grids such as 6 × 6 × 1, 6 × 6 × 1 and 8 × 12 × 1 are enough to obtain converged results for the fundamental gap up to 50 meV for MoS_{2}, hBN, and phosphorene, respectively. These grids are 80, 40 and 20 times smaller than those required to achieve a similar accuracy when averaging only the bare coulomb potential (vav method). Taking the k and q grids to be identical, G_{0}W_{0} typically scales^{65} as \({N}_{k}^{2}\). When this is the case, with the proposed method the computational cost of a G_{0}W_{0} calculation is reduced by at least two orders of magnitude, without loss of accuracy. The proposed Wav method is able to describe the anisotropy of the screened potential at different levels of approximations, and differently from other methods recently proposed does not rely on any subsampling of the BZ.
The possibility to extend the present methodology to metals and systems with different dimensionalities (1D or 3D) is envisaged and will be explored in a future research.
Methods
Computational details
DFT calculations were performed using a plane wave basis set as implemented in the Quantum ESPRESSO package^{66}, and using the PerdewBurkeErnzerhof (PBE) exchangecorrelation functional^{67}. We have considered supercells with an interlayer distance L = 10 Å for MoS_{2} and L = 15 Å for hBN and phosphorene, which are enough to obtain converged results with respect to the cell size, in agreement with ref. ^{35}. The kinetic energy cutoff for the wavefunctions was set to 60 Ry and we adopted normconserving pseudopotentials to model the electronion interaction.
G_{0}W_{0} calculations were performed with the Yambo package^{40,41}. In the calculations reported in Figs. 1, 3–5, we used a cutoff of 5 Ry for the size of the dielectric matrix, including up to 400 states in the sumoverstate of the response function. The same number of states has been employed in the calculation of the correlation part of the self energy. To accelerate the convergence with respect to the number of empty states we have used the algorithm described in refs. ^{41,68}. Despite the cutoff used to represent the dielectric matrix is not enough to provide highly converged QP properties, it is sufficient to provide accurate convergence trends with respect to the qsampling. In the calculations reported in Table 1, we included up to 600 states for MoS_{2} and phosphorene, while 1200 for hBN, both in the response function and in the Green’s function, with a size of the dielectric matrix of 25 Ry. We employed kpoint grids of 9 × 9 × 1, 12 × 12 × 1, and 10 × 14 × 1 for MoS_{2}, hBN and phosphorene, respectively.
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Code availability
The code developed in this work is available since the 5.1 version of the Yambo package.
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Acknowledgements
We acknowledge stimulating discussions with Dario A. Leon, Miki Bonacci, Simone Vacondio and Matteo Zanfrognini. This work was partially supported by SUPER (Supercomputing Unified Platform  EmiliaRomagna) from EmiliaRomagna PORFESR 20142020 regional funds. We also thank MaX  MAterials design at the eXascale  a European Centre of Excellence, funded by the European Union programme H2020INFRAEDI20181 (Grant No. 824143), HORIZONEUROHPCJU2021COE1, Grant No. 101093324. We also acknowledge financial support from ICSC  Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by European Union  NextGenerationEU  PNRR, Missione 4 Componente 2 Investimento 1.4. D.V. also thanks the Italian national programme PRIN2017 2017BZPKSZ “Excitonic insulator in twodimensional longrange interacting systems”. Computational time on the Galileo machine at CINECA was provided by the Italian ISCRA programme.
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A.G. implemented the method and performed the calculations. A.F., P.D.’A., and D.V. conceived the original idea of the work. All authors contributed to the method development, analysis of the results, wrote the manuscript, and critically discussed the paper.
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Guandalini, A., D’Amico, P., Ferretti, A. et al. Efficient GW calculations in two dimensional materials through a stochastic integration of the screened potential. npj Comput Mater 9, 44 (2023). https://doi.org/10.1038/s41524023009897
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DOI: https://doi.org/10.1038/s41524023009897
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