# Tuning electronic properties of transition-metal dichalcogenides via defect charge

## Abstract

Defect engineering is a promising route for controlling the electronic properties of monolayer transition-metal dichalcogenide (TMD) materials. Here, we demonstrate that the electronic structure of MoS2 depends sensitively on the defect charge, both its sign and magnitude. In particular, we study shallow bound states induced by charged defects using large-scale tight-binding simulations with screened defect potentials and observe qualitative changes in the orbital character of the lowest lying impurity states as function of the impurity charge. To gain further insights, we analyze the competition of impurity states originating from different valleys of the TMD band structure using effective mass theory and find that impurity state binding energies are controlled by the effective mass of the corresponding valley, but with significant deviations from hydrogenic behaviour due to unconventional screening of the defect potential.

## Introduction

Since the discovery of graphene, there has been significant interest in the development of ultrathin devices based on two-dimensional (2D) materials. In contrast to graphene, which is a semimetal when undoped, monolayer transition-metal dichalcogenides (TMDs) with the chemical formula MX2 (M = Mo, W; X = S, Se, Te) are semiconductors with a direct band gap1,2. Monolayer TMDs have been used as channel materials in field-effect transistors3,4 and microprocessors5, as well as absorbers in solar cells6 and as sensors7,8, with promising results.

Defects play a critical role in the performance of devices under realistic conditions9,10,11. Analogously to conventional bulk semiconductors, impurities with shallow donor or acceptor states can be used to control the carrier concentration in TMDs via defect engineering12,13. Adsorbed atoms and molecules are a particularly promising class of impurities in TMDs as they tend to only weakly perturb the atomic structure of the TMD substrate, thereby limiting any degradation of carrier mobility that may result from impurity scattering or trapping14,15, and experimental fabrication of adsorbate-engineered samples is straightforward16.

A detailed theoretical understanding of the properties of charged adsorbates on TMDs is important to enable the rational design of new devices. On the one hand, many groups have used ab initio density-functional theory (DFT) to study the interaction of adsorbed atoms and molecules with TMDs. Such calculations yield important material-specific insights about adsorption geometries, adsorbate binding energies and charge transfer17,18,19,20,21. However, ab initio calculations are limited in terms of the size of the systems that can be considered (typically containing up to several hundred or a few thousand atoms), which are much too small to describe properties of shallow defect states that can extend up 100 Ångstrom (Å) or more, as has been observed recently for Coulomb impurities in graphene using scanning tunnelling spectroscopy (STS)22.

On the other hand, continuum electronic structure methods, such as Dirac theory for graphene or effective mass theory for bulk semiconductors, can describe the behaviour of extended impurity states, but require parameters from experiments or ab initio calculations, such as Fermi velocities, effective masses23,24,25,26,27 and rather importantly, the defect potential that is typically screened by electrons of the host material.

In this paper, we study properties of shallow impurity states induced by charged adatoms on monolayer MoS2. Using large-scale tight-binding models and screened defect potentials calculated from ab initio dielectric functions, we reveal a surprising diversity of bound defect states resulting from the unconventional screening present in reduced-dimensional materials and the interplay between multiple valleys in the TMD band structure. We present results for impurity wavefunctions and binding energies as function of the impurity charge and also compute the local density of states (LDOS) in the vicinity of the adatom, which can be measured in STS experiments. For both donor and acceptor impurities, we find that impurity wavefunctions have similar nodal structure to 2D hydrogenic states, but with radii that lie on the nanoscale. We find that that the orbital character of the most strongly bound impurity state switches as a function of the impurity charge strength Z due to the different effective masses associated with different valleys in the monolayer TMD band structure. We compare our results to the 2D hydrogen atom and also to effective mass theory calculations and discuss the limitations of these continuum models. Whilst an approach based on the effective mass model is able to describe some of the general behaviour with reasonable accuracy, we find significant discrepancies from our tight-binding model which arise from short-range features of the defect potential. Our calculations demonstrate the potential of adsorbate engineering for ultrathin devices based on TMDs and the importance of first-principles based description of their properties.

## Methods

To describe the electronic structure of the MoS2 monolayer, we employ the three-band tight-binding (TB) model by Liu et al.28. This model uses a basis of transition-metal $$4{d}_{{z}^{2}}$$, 4dxy and $$4{d}_{{x}^{2}-{y}^{2}}$$ orbitals which give the dominant contribution to the states near the conduction and valence band extrema and includes hoppings up to third-nearest neighbours as well as spin-orbit interactions. The various parameters were determined by fits to DFT band structures.

The charged adatom is described as a point charge Q = Ze (with e being the proton charge) located a distance d above the plane of the transition-metal atoms. The charge gives rise to a screened potential in the TMD sheet. Within linear response theory, the screened potential is given by

$$V(\rho ;\,Z,d)=Z{e}^{2}{\int }_{0}^{\infty }{\rm{d}}q\,{\varepsilon }_{{\rm{2D}}}^{-1}(q){J}_{0}(q\rho ){e}^{-qd},$$
(1)

where ρ denotes the in-plane distance from the adatom and $${\varepsilon }_{{\rm{2D}}}^{-1}(q)$$ is the inverse 2D dielectric function of a single TMD monolayer. The 2D dielectric function can be obtained from the inverse dielectric matrix $${\varepsilon }_{{\bf{GG}}^{\prime} }^{-1}({\bf{q}})$$ of an infinite system of stacked TMD sheets (simulated in an electronic structure calculation that employs periodic boundary conditions) via29

$${\varepsilon }_{{\rm{2D}}}^{-1}({\bf{q}})=\frac{q}{2\pi {e}^{2}{L}_{z}}\sum _{{{\bf{G}}}_{z}{{\bf{G}}}_{z}^{^{\prime} }}\,{\varepsilon }_{{{\bf{G}}}_{z}{{\bf{G}}}_{z}^{^{\prime} }}^{-1}({\bf{q}}){v}_{{\rm{trunc}}}(|{\bf{q}}+{{\bf{G}}}_{z}^{^{\prime} }|).$$
(2)

Here, Gz and $${{\bf{G}}{\boldsymbol{^{\prime} }}}_{z}$$ denote reciprocal lattice vectors along the out-of-plane (z) direction, vtrunc is a slab-truncated Coulomb interaction30 and Lz denotes the distance between the stacked sheets. The inverse dielectric matrix is computed for a MoS2 monolayer using the random-phase approximation31 (RPA) with Kohn-Sham wave functions and energies from ab initio DFT (see Supplementary Materials for details). Calculations were carried out using the Quantum Espresso32 and BerkeleyGW software packages33. For small wave vectors, which are relevant for describing shallow impurity impurity bound states, we find that the right hand side of Eq. (2) depends only on the magnitude of the wave vector.Figure 1 shows the screened (calculated from Eq. 1) and unscreened potentials of a charged adatom with Z = 1 and d = 2 Å above the Mo-layer in the MoS2 sheet. While there are clear differences at short distances, the two potentials both converge to the unscreened case at long distances from the adatom which is characteristic of screening in 2D semiconductors. This short-range discrepancy corresponds to significant differences between the Fourier transforms of these potentials at large wavevectors, shown in the inset of Fig. 1.

To study shallow bound states of the screened adatom potential, we construct a 51 × 51 TMD supercell containing 7803 atoms and diagonalize the resulting TB Hamiltonian with the adatom potential as an on-site term22,23. Note that the adatom is placed above a transition-metal site as this is the preferred adsorption geometry for many adatom species, such as alkali metals17,18,19.

To analyze the results of our atomistic tight-binding simulations, we have also carried out calculations using effective mass theory. In this approach, which has been used routinely to study shallow bound states of charged impurities in bulk semiconductors25,34,35, the impurity states are expressed as $${{\rm{\Psi }}}_{n\nu }({\bf{r}})=\int {\rm{d}}{\bf{k}}\,{\varphi }_{n\nu }({\bf{k}}){\psi }_{n{\bf{k}}}({\bf{r}})$$. Here, ψnk denotes an unperturbed Bloch state with band index n and crystal momentum k of the host material and ϕnv(k) is an envelope function determined by36

$${\varepsilon }_{n{\bf{k}}}{\varphi }_{n\nu }({\bf{k}})+\int {\rm{d}}{\bf{k}}^{\prime} \,\langle {\psi }_{n{\bf{k}}}|V|{\psi }_{n{\bf{k}}^{\prime} }\rangle {\varphi }_{n\nu }({\bf{k}}^{\prime} )={E}_{n\nu }{\varphi }_{n\nu }({\bf{k}}),$$
(3)

where εnk describes the band structure of the host material and V(r) denotes the screened impurity potential. In bulk semiconductors, V can be accurately approximated36 by Ze2ε−1(q = 0)/r and the resulting equation for the impurity state envelope function reduces to the Schrödinger equation of a hydrogen atom with a reduced Bohr radius $${\tilde{a}}_{0}=({m}^{\ast }/{m}_{0})Z{a}_{0}{\varepsilon }^{-1}(q=0)$$ (with m* and m0 denoting the effective and bare mass of the electron, respectively, and a0 is the Bohr radius). In this approximation, the impurity state envelope functions take the form of the 2D hydrogenic states37 give by

$${\varphi }_{nl}^{({\rm{2DH}})}(\rho ,\theta )=\frac{{e}^{il\theta }}{{N}_{nl}(Z,{m}^{\ast })}{(\rho {\lambda }_{n})}^{|l|}{e}^{\rho {\lambda }_{n}\mathrm{/2}}{L}_{n-l-1}^{\mathrm{2|}l|}(\rho {\lambda }_{n}),$$
(4)

where Nnl is a normalization constant, $${L}_{j}^{k}$$ are the generalized Laguerre polynomials, and $${\lambda }_{n}=\frac{2}{2n+1}\frac{Z{m}^{\ast }{e}^{2}}{4\pi {\varepsilon }_{0}{\hslash }^{2}}$$. We compare these solutions to the wavefunctions extracted from our TB model to identify similarities in nodal structure.

The screened impurity potential in a 2D semiconductor, such as a TMD monolayer, however, cannot be accurately approximated by a bare Coulomb interaction divided by a constant dielectric function (see Fig. 1). A well-known model for the screening of a point charge embedded in a thin dielectric film was derived by Keldysh38 and is given by

$${\varepsilon }_{{\rm{Keldysh}}}(q)=1+{\rho }_{0}q$$
(5)

where ρ0 is the screening length. We calculate the screened potential VKelysh(ρ) using the Keldysh model by substituting $${\varepsilon }_{{\rm{Keldysh}}}^{-1}(q)$$ for the inverse dielectric function in Eq. 1. The value of ρ0 = 45 Å is obtained by fitting to the RPA-screened potential of Fig. 1. The Keldysh model has been frequently used to study excitons in TMDs29,39,40 and we also use it here for comparison to our tight-binding results.

To simplify the integration over k-points in Eq. (3), Bassani et al.36 divided the first Brillouin zone into subzones Ωi centered on critial points ki, typically associated with band extrema. The impurity states Ψnv(r) are then constructed as linear combinations of subzone states

$${{\rm{\Psi }}}_{n\nu i}({\bf{r}})\approx {\varphi }_{n\nu i}({\bf{r}}){\psi }_{n{{\bf{k}}}_{i}}({\bf{r}}).$$
(6)

To determine the subzone envelope functions ϕnvi(r), we minimize the expectation value of the Keldysh Hamiltonian $$\hat{H}=\frac{-{\hslash }^{2}}{2{m}_{i}^{\ast }}({\partial }_{x}^{2}+{\partial }_{y}^{2})+{V}_{{\rm{Keldysh}}}(r)$$ (where $${m}_{i}^{\ast }$$ denotes the effective mass associated with the relevant conduction or valence band at ki) using the following ansatz for the most strongly bound impurity state

$${\varphi }_{1s,i}(\rho ;\alpha )=\frac{(2\alpha ){e}^{\alpha d}}{\sqrt{2\pi (2\alpha d+1)}}{e}^{-\alpha \sqrt{{\rho }^{2}+{d}^{2}}},$$
(7)

where α is a variational parameter, which we use to define the impurity radius aimp = α−1. Once the subzone states are obtained, the full impurity states are found by including interactions between different subzones. As the coupling is usually weak, it can be treated using perturbation theory36.

## Results and Discussion

### Acceptor States

Figure 2(a–e) show the wavefunctions (specifically, their squared magnitudes sampled at the Γ-point of the first Brillouin zone) of the five most strongly bound impurity states for an adatom with Z = −0.3, situated d = 2 Å above the Mo-site, as calculated from our tight-binding model with an RPA-screened impurity potential. To label the impurity states, we compare them to the 2D hydrogenic states37. While the two most strongly bound impurity states (Fig. 2(a,b)) have 1s character, the states in Fig. 2(c,e and d) resemble the 2p and 2s states of the 2D hydrogen atom, respectively. We also present the corresponding 2D hydrogenic states in Fig. 2(f–j) for a nuclear charge Q = −0.3ζ, where ζ ≈ 0.26 is the ratio of the screened and unscreened potentials at r = 0 in Fig. 1. Surprisingly, the more strongly bound 1s states of Fig. 2(a) is significantly more delocalized with an impurity radius of aimp = 12.6 Å than the less strongly bound 1s state in Fig. 2(b), which has a radius of aimp = 5.19 Å. We determine aimp by fitting the impurity state to an exponential decay as in Eq. 7, and extracting the inverse decay scale $$\alpha ={a}_{{\rm{imp}}}^{-1}$$. The 2p impurity states exhibit an angular modulation caused by the trigonal warping of the valence states near the band edge41. Note that the modulation is different for the two 2p states and we therefore label the second state distinctly as 2p′. In contrast to the 2D hydrogen atom, the 2s, 2p and 2p′ are not degenerate, as indicated by their binding energies given in the top right corner of Fig. 2(a–e), because the impurity potential is screened and no longer follows a simple 1/r behaviour.

To further analyze the impurity states, we projected their wavefunctions onto unperturbed states of the MoS2 monolayer (see Supplementary Materials for details) and find that the most strongly bound 1s state and also the 2p and 2s states are composed of valence states from the K and K′ points of the MoS2 bandstructure, see Fig. 3(b). In contrast, the second 1s state originates from the valence band near the Γ-point of the unperturbed band structure. We label the states in Fig. 2(a–e) by their origin in the Brillouin zone (BZ), in addition to their 2D hydrogenic orbital character. We have subsequently labelled Fig. 2(f–j) by the effective mass of the valence band maxima (VBM) from which the corresponding TB states originate.

Figure 3(a) shows the dependence of the impurity state binding energies Eb = E − EVBM (energy E with reference to the primary valence band maximum EVBM) on the adatom charge Z for negatively charged adatoms. We have fitted the 1s binding energies to a power law of the form −B + AZη, see Table 1, where B = 0 for 1s (K/K′) and B = 0.071 eV for 1s (Γ), and find that the 1s (K/K′) and 1s (Γ) states have exponents of η = 1.30 and η = 1.25, respectively. These are significantly smaller than the exponent for a 2D hydrogen atom where the binding energy is given by $$E(Z)=-\,4\frac{{m}^{\ast }}{{m}_{0}}{Z}^{2}$$ Ry. Interestingly, the different Z-dependences of the 1s (K/K′) and 1s (Γ) binding energies result in a crossover at Z = −0.32, where the order of the two states switches. As the character of 1s (K/K′) is dominated by Mo 4dxy and $$4{d}_{{x}^{2}-{y}^{2}}$$ orbitals, while Mo $$4{d}_{Z}2$$ orbitals make up the 1s (Γ) state1, our calculations suggest the possibility of controlling the orbital character of low-lying electronic states via defect engineering with potentially interesting consequences for optical properties.

To further analyze the results of the tight-binding calculations, the bound impurity states were studied with effective mass theory. Specifically, we determined the impurity states associated with the subzones near Γ, K and K′ using Eq. (7). For the acceptor states, each subzone acts as an independent 2D hydrogen-like system as the different spin states of the degenerate valence band maxima at K and K′ prohibit interactions between the subzones. The resulting binding energies agree reasonably well with the tight-binding results, see dashed lines in Fig. 3(a) and Table 1. We see that the discrepancy between these two models increases with Z, as the RPA-screened potential in Fig. 1 is deeper than the screened potential in the Keldysh model, resulting in more strongly bound states. In particular, effective mass theory also predicts a crossover of 1s (K/K′) and 1s (Γ) near Z = −0.45. The binding energy of 1s (Γ) increases more quickly with Z because the effective mass near Γ is about 5.5 times larger than the effective mass near K or K′. This also explains the differences in impurity radii, see Fig. 2(a,b).

### Donor States

Next, we study the shallow impurity states induced by positively charged adatoms. Figure 4(a–h) show the wavefunctions of the eight most strongly bound impurity states for an adatom with Z = 0.3 and d = 2 Å. The states are labelled based on their similarity to the eigenstates of the 2D hydrogen atom. In contrast to the acceptor case, we find a pair of states corresponding to each solution of the 2D hydrogen atom, with different binding energies, indicated at the top right corner of each subfigure in white. The states of each pair are distinguished by a “+” or “−” subscript.

Figure 4(i) shows the binding energies Eb = ECBM − E of the most strongly bound states (with energy E) with respect to the conduction band minimum (with energy ECBM) as function of the impurity charge Z. At low values of Z, the 1s(K/K′) and 1s+(K/K′) states are almost degenerate, but their binding energy difference increases with increasing Z. A third impurity state originating from the local conduction band minimum at the 6 Q points of the Brillouin zone crosses the two 1s (K/K′) states near Z = 0.6 and becomes the most strongly bound state for higher values of Z. The crossover is again caused by the larger effective mass at Q point compared to the K and K′ points. We have fitted the binding energies of these states to a power law of the form B + AZη, see Table 2, where B = 0 for states from K/K′ and B = 0.267 eV for states from the Q-points. As for the acceptor impurity states, the exponents of the donor states are significantly smaller than the 2D hydrogen value η = 2.

Again, we compare the tight-binding results to effective mass theory. We first determine the subzone envelope functions, Eq. (7), for the regions near the critical points at K and K′. In contrast to the valence bands, there is no spin-orbit splitting of the conduction band states at K and K′. As a consequence, the conduction band states at K and K′ with equal spin are degenerate and this gives rise to the observed pairs of impurity states with same symmetry in Fig. 4. The subzone impurity states can couple and the resulting binding energy splitting is given by36

$${{\rm{\Delta }}}_{KK^{\prime} }\approx 2|{\varphi }_{1s,{\bf{K}}}^{\ast }(r=0){\varphi }_{1s,{\bf{K}}^{\prime} }(r=0)V({\bf{q}}={\bf{K}}-{\bf{K}}^{\prime} )|.$$
(8)

We evaluate the splitting with the Keldysh approximation for V, using the Fourier transform of the screened Coulomb potential in the Keldysh model. We find that the splitting is several orders of magnitude smaller than the splitting found in the tight-binding model. This discrepancy is caused by the inaccurate behaviour of the Keldysh model at large wave vectors, which is shown in the inset of Fig. 1, where the vertical black line indicates |K − K′|. We note, however, that for such large wavevectors (corresponding to positions in the immediate vicinity of the impurity) the use of the 2D dielectric function can cause inaccuracies, as Eq. 2 assumes that the distance from the impurity is significantly larger than the width of the MoS2 sheet29. We show the binding energies, found from effective mass theory using the Keldysh screening model for the splitting (see Fig. 4(i) as blue dashed an green dot-dashed lines). The fitting parameters of the binding energies to a power law are compared to the tight-binding results in Table 2.

The 1s impurity state wavefunctions from effective mass theory are given by

$${{\rm{\Psi }}}_{1{s}_{\pm }(K/K^{\prime} )}({\bf{r}})=\frac{1}{\sqrt{2}}({\varphi }_{1s,{\bf{K}}}(r){\psi }_{{\bf{K}}}({\bf{r}})\pm {\varphi }_{1s,{\bf{K}}^{\prime} }(r){\psi }_{{\bf{K}}}({\bf{r}})),$$
(9)

where $${\psi }_{K/K^{\prime} }({\bf{r}})$$ denote the Bloch states of the unperturbed MoS2 band structure at K and K′. Notably, the states with an s-character (Fig. 4(a,b,d and h)) exhibit an intensity modulation with a period of three unit cells along the directions connecting nearest neighbours. Projecting the impurity states onto unperturbed Bloch states reveals that all states originate from both the K and K′ points of the Brillouin zone, where the minimum of the conduction band occurs, see Fig. 3(b). The corresponding probability densities contain a term with a cos((K − K′) ·r) factor which gives rise to the oscillatory pattern in Fig. 4(a,e,d,h). In contrast to the impurity states with s-character which derive from unperturbed states directly at K and K′, the states with p-character mostly derive from conduction band states in the vicinity of the band edges. As a consequence, the coupling between K and K′ is weaker for the p-states and the spatial modulation is not observed. We find that this modulation does not occur when the defect is not placed on the transition-metal site.

### Local density of states

Scanning tunnelling spectroscopy (STS) provides spatially-resolved information about the electronic structure of surfaces and has been used to study the properties of shallow impurity states induced by charged adatoms experimentally. The dI/dV curves obtained in STS are often assumed to be proportional to the local density of states (LDOS) of the sample. We have calculated the LDOS for values of Z and d that represent lithium (Li) and carbon (C) atoms adsorbed on a MoS2. For Li, Chang et al. found an impurity charge of ZLi = 0.63 from a Bader charge analysis42 of the DFT charge density18. Using a similar procedure, Ataca et al. determined ZC = −0.58 for a C atom adsorbed to MoS2 above the Mo site19,43. We modelled adsorbed atoms sitting above the Mo site at a height of dLi = 3.1 Å and dC = 1.58 Å17,18,19,43. Screening by a SiO2 substrate is included via a substrate dielectric function of 3.7.

Figure 5(a,b) show the tight-binding LDOS for a C adatom on MoS2 in the vicinity of valence band maximum and the conduction band minimum, respectively. A 6 × 6 k-point mesh and a Gaussian broadening of 0.01 eV were used. Near the VBM, several peaks originating from bound acceptor states can be observed in the band gap. The peak from 1s (Γ) disappears more quickly as a function of distance from the adatom than the 1s (K/K′) peak. This is a consequence of the stronger localization of this state, see Fig. 2. At a distance of ~66 Å from the adatom, the LDOS of the perturbed system has converged to the LDOS of the pristine TMD. In the vicinity of the CBM, no impurity states are present. However, the screened potential created by the adatom leads to a shift of the unperturbed LDOS.

Figure 5(c,d) show the tight-binding LDOS for a Li adatom on MoS2 in the vicinity of valence band maximum and the conduction band minimum, respectively. The peaks near the CBM in the vicinity of the adatom originate from bound donor states and can be observed up to a distance of ~25 Å from the adatom. Note that the splitting of the two impurity states from the K and K′ points is too small to be resolved. No impurity state peaks are found in the vicinity of the VBM, but again the impurity potential causes a shift of the TMD LDOS.

## Conclusions

In summary, we have studied the electronic properties of charged defects in transition-metal dichalcogenides. Using tight-binding simulations with screened impurity potentials on unit cells containing up to 8,000 atoms, we have calculated the binding energies and wave functions of shallow impurity bound states. Our key finding is that the orbital character of the lowest lying impurity states depends sensitively on the magnitude of the defect charge. For acceptor states, i.e., negatively charged defects, a crossover of impurity states with different orbital characters occurs at a critical defect charge of Q = −0.32 e (with e being the proton charge). For defect charges above this value, the lowest impurity state from the Γ valley of the TMD band structure, which is dominated by contributions from Mo $$4{{\rm{d}}}_{{{\rm{z}}}^{2}}$$ orbitals, is more strongly bound than the degenerate impurity states from the K and K′ valleys which are dominated by Mo 4dxy and Mo $$4{{\rm{d}}}_{{x}^{2}-{y}^{2}}$$ orbitals. For donor states, i.e., positively charged defects, a crossover between hybridized impurity states from the K and K′ valleys and impurity states from the Q valleys occurs at a critical impurity charge of +0.6 e. To understand the competition between different impurity states, we analyze their properties using effective mass theory. We find that the impurity binding energies can be described by power laws of the defect charge, but with significant deviations from hydrogenic behaviour due to screening. Importantly, the prefactor of the power law is determined by the effective mass and the significant differences of the effective masses in the different valleys of the TMD band structure give rise to the observed crossovers. Our calculations thus establish the defect charge as an important control parameter for tuning the electronic structure of TMDs via defect engineering.

## Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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## Acknowledgements

This work was supported through a studentship in the Centre for Doctoral Training on Theory and Simulation of Materials at Imperial College London funded by the EPSRC (EP/L015579/1). We acknowledge the Thomas Young Centre under grant number TYC-101. This work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), and the Imperial College London High-Performance Computing Facility.

## Author information

J.L. and A.A.M. proposed the work, M.A. performed the calculations and all authors contributed to analyzing the results. All authors reviewed and contributed to the writing of the manuscript.

Correspondence to Johannes Lischner.

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