Controlling the electronic properties of van der Waals heterostructures by applying electrostatic design

Van der Waals heterostructures based on the heteroassembly of 2D materials represent a recently developed class of materials with promising properties especially for optoelectronic applications. The alignment of electronic energy bands between consecutive layers of these heterostructures crucially determines their functionality. In the present paper, relying on dispersion-corrected density-functional theory calculations, we present electrostatic design as a promising tool for manipulating this band alignment. The latter is achieved by inserting a layer of aligned polar molecules between consecutive transition-metal dichalcogenide (TMD) sheets. As a consequence, collective electrostatic effects induce a shift of as much as 0.3 eV in the band edges of successive TMD layers. Building on that, the proposed approach can be used to design electronically more complex systems, like quantum cascades or quantum wells, or to change the type of band lineup between type II and type I.


Introduction
Research on two-dimensional (2D) materials has been growing rapidly since the successful exfoliation of graphene [1]. Nowadays the full range of electronic properties can be realized in various classes of 2D materials. Amongst them, semiconducting transitionmetal dichalcogenides (TMDs) have proven to be especially interesting, on the one hand, for gaining fundamental physical insights and, on the other hand, for optoelectronic applications. This is associated with the emergence of spin and valley physics [2][3][4] and strong light-matter interactions [5][6][7] observed in these systems. The latter result in large absorption coefficients and, for properly designed band-offsets, either in significant electron-hole pair creation or exciton recombination, two properties highly relevant for optoelectronics. Of particular interest in this context is the combination of semiconducting organic molecules with 2D materials to change their characteristics either via doping [8,9] or by the exploitation of dielectric effects [10].
In their monolayer form, the TMDs in the focus of the present paper [MoX 2 and WX 2 (X = S, Se)] have a direct band gap ranging between 2.0 to 2.5 eV [11]. Additionally, the exciton binding energies [11][12][13][14] of these layered semiconductors are an order of magnitude larger than in conventional bulk semiconductors such as Si or GaAs resulting in an increased stability of the excitons against thermal dissociation. As a consequence, the strongly bound excitons of MoX 2 and WX 2 have been shown to be stable at room temperature [15][16][17]. This makes TMD monolayers promising candidates for optoelectronics. Even more interesting are vdW heterostructures [15,[18][19][20], which consist of different TMD layers stacked on top of each other along the direction perpend icular to the 2D plane. Individual layers in such vdW heterostructures are held together by strong in-plane covalent bonds, while comparably weak vdW interactions are present between the layers. Because of this weak interlayer bonding, 2D crystals with different lattice constants and crystal structures can be easily combined to form vdW heterostructures, which are chemically stable, flexible [21,22] and have atomically sharp interfaces. Consequently, it is easy to mix and match 2D materials with different electronic properties such as semiconductors (e.g. TMDs), semimetals (e.g. graphene) or insulators (e.g. hexagonal boron nitride). In the resulting structures, the variation of the charac- Electrostatic design applied to van der Waals (vdW) heterostructures. (a) vdW heterostructure consisting of two semiconducting TMDs which show symmetric type II band alignment (i.e. the valence-and conductionband offsets are of equal magnitude and have the same sign). When the band offset of this system is too small, an intralayer exciton created in MX 2 cannot be separated. (b) When electrostatic design is applied to this heterostructure by inserting a polar layer, the bands of the upper TMD layer can be shifted and, thus, the band offsets (valence band offset Δ VB and conduction band offset Δ CB ) can be changed. This promotes exciton dissociation. (c) van der Waals heterostructure with type II band alignment and very different band offsets (Δ VB is much larger than Δ CB ). (d) By applying electrostatic design to such a heterostructure, the states of the upper layer can be shifted such that the band alignment changes from type II to type I (i.e. the signs of the valence-and conduction-band offsets differ). teristics of individual layers can be exploited to create new types of devices.
A crucial property that determines the functionality of van der Waals heterostructures is the alignment of the valence band maxima (referred hereafter as VBM) and conduction band minima (CBM) of consecutive layers. For example, to obtain TMD-based heterobilayer systems that promote charge carrier separation, the band extrema of the two TMD layers should have type II band lineup with a sufficiently large band offset to overcome the exciton binding energy; otherwise the system would fail to localize charge carriers onto the two layers as is shown in figure 1(a). Consequently, in order to realize a functional vdW heterostructure with specific properties, one possibility would be to search the available pool of 2D materials [23] for a combination of layers, where the sizes of the band gaps and the band lineup suite the intended purpose. This can be tedious and sometimes requires the application of materials that are difficult to obtain/handle. Therefore, in this paper, we suggest an alternative approach for tuning the band alignment of vdW heterostructures, which is based on electrostatic design. This concept has been introduced by Kretz et al [24] for controlling the energetics of the electronic states of self-assembled organic monolayers (SAMs). Later it has been applied to modify the electronic structure of graphene [25]. The approach relies on collective electrostatic effects, which arise from a regular and periodic arrangement of dipolar groups in an organic monolayer resulting in a step in the energy landscape [26][27][28][29][30][31]. In the present paper, we show that this design principle holds high promise for vdW heterostructures. In particular we discuss, how the intercalation of a monolayer consisting of oriented polar organic molecules triggers a step in the electrostatic energy, which leads to a shift between the electronic states of the TMD layers above, respectively, below the molecules. The shift can then be employed to tune the electronic structure, which can, for example, be exploited for increasing the band-offsets in neighboring layers to a degree that exciton-dissociation becomes energetically favorable (see figure 1(b) in comparison to figure 1(a)). Alternatively, one can use electrostatic design to switch between type II band lineup favoring exciton dissociation (figure 1(c)) and type I band lineup promoting exciton recombination ( figure 1(d)).
In the following, we use quantum mechanical modeling in the framework of dispersion corrected density-functional theory (DFT) to discuss the enormous potential of electrostatic design. First, we describe prototypical stacks consisting solely of MoS 2 layers. Subsequently, we present the possibility to electrostatically design more complex systems such as quantum cascades and quantum wells. Finally, we show how for a system consisting of WSe 2 and MoSe 2 the above-mentioned qualitative change in the band alignment from type II to type I can be achieved.

Results and discussion
2.1. Choice and setup of the model system The first system whose electronic properties are investigated contains only MoS 2 as 2D semiconducting layers. As dipolar molecules intercalated between the MoS 2 sheets we chose titanyl-phthalocyanine (TiOPc; see inset in figure 2(c)) as a prototypical example. This choice is motivated by the observation of Fukagawa et al [32] that on weakly interacting substrates (explicitly shown for highly-oriented pyrolytic graphite, HOPG) up to monolayer coverage all TiOPc molecules align with their dipole moments pointing in the same direction (which is perpendicular to the HOPG surface). As a result, the work function of the sample continuously increases with coverage until a closed monolayer is formed. For that case, a work function modification of 0.29 eV is obtained in the experiment [32]. The observed arrangement of the molecules is a consequence of maximized van der Waals bonding when the Ti = O groups point away from the surface. Indeed, when fully relaxing the geometry of an MoS 2 /TiOPc interface in the configuration shown in the supporting information (stacks.iop.org/ TDM/5/035019/mmedia) the slightly bent molecular backbone is aligned essentially parallel to the substrate at a distance of 3.35 Å between the average planes of the topmost S layer and the C atoms. This results in a van der Waals attraction of 3.61 eV between the MoS 2 and the TiOPc. A configuration with the Ti = O group pointing towards the substrate would not only be at variance with the experimental observations on HOPG [32], we also calculate it to be less stable by 0.8 eV per molecule on MoS 2 , which we attribute to a larger average distance between the substrate and the atoms in the molecular backbone. The experiments on HOPG also imply that the increased van der Waals attraction in the Ti = O up configuration is sufficient for overcoming intermolecular dipole-dipole repulsion between the molecules.
For the TiOPc intercalated van der Waals stack sketched in the inset of figure 2(c), no such driving force exists, as there are MoS 2 layers above and below the TiOPcs. Thus, the thermodynamically most stable configuration would be molecules aligned in an antiparallel fashion. In the experiment, this problem can, however, be overcome by realizing a kinetically trapped configuration where first a TiOPc layer is deposited onto a MoS 2 surface and, subsequently, the (modified) TMD layer is covered with a second MoS 2 flake [33,34]. For stabilizing the orientation of the molecules in the intercalated layer, an option would also be to substitute them with cross-linkable side groups.

Electronic structure of the MoS 2 |TiOPc|MoS 2 system
To characterize the electronic structure of the investigated systems, it is particularly useful to discuss densities of states projected onto either individual TMD layers or onto the intercalated molecules (PDOSs); the corresponding band-structures can be found in the supporting information. The PDOS for a geometry relaxed bilayer system is shown in figure 2(a). Inserting the TiOPc layer slightly distorts especially the upper TMD layer, but, more, importantly also significantly increases the interlayer distance. To highlight the consequences of these changes in the geometry of the TMD layers, figure 2(b) shows the PDOS of the system obtained when first relaxing the MoS 2 |TiOPc|MoS 2 geometry (in a twostep process, for details see supporting information) and subsequently calculating the electronic properties of the stack with the TiOPc layer removed (figure 2(b)). In spite of the loss of mirror symmetry due to the buckling of the top layer, the densities of states projected onto the two MoS 2 monolayers still overlap. The differences between figures 2(a) and (b), especially in the region of the valence-band edge arise from quantum-mechanical interactions in the close-packed bilayer [35,36]. They result in a plateau in the PDOS below the valence-band edge (figure 2(a)), which is fundamentally different from the steep increase of the DOS in the two separated monolayers (figure 2(b)), respectively, in an isolated monolayer (see supporting information). The reason for that is that the occupied states around the Γ point are dominated by p z orbitals localized on S atoms and d z 2 orbitals on Mo [37] (see also species-projected DOS contained in the SI), whose 'out of plane' nature makes them particularly sensitive to inter-layer interactions. In the densely packed MoS 2 bilayer, this leads to a destabilization of the occupied electronic states around the Γ-point, resulting in an energy window, where only the highest occupied band is present (see supporting information). Its roughly parabolic shape then causes the small but constant DOS close to the VBM (see figure 2(a)) that is expected for a 2D electron gas. When the inter-layer spacing increases, as in figure 2(b), the coupling diminishes and a DOS essentially corresponding to that of an isolated monolayer is recovered.
The situation changes fundamentally in the MoS 2 |TiOPc|MoS 2 heterostructure (see figure 2(c)). Here, one observes a shift of the band edges of the two MoS 2 layers, which is caused by the step in electrostatic energy due to the aligned TiOPc dipoles (see figure 3). The shift amounts to 0.28 eV, which is very close to the above-mentioned measured (calculated) work function change for TiOPc on HOPG amounting to 0.29 (0.30) eV [32]. This is consistent with the band offset being caused exclusively by the electrostatic shift induced by the TiOPc layer. This assessment is further supported by a vanishingly small bond-dipole (amounting to −0.01 eV) resulting from charge rearrangements upon interface formation. The particularly small value of the latter is primarily caused by a largely symmetric transfer of electrons from the TiOPc film to both MoS 2 layers. For a system with the top MoS 2 layer lifted by 0.5 Å, this symmetry is bro-ken and a considerable net bond dipole of −0.09 eV is observed (see supporting information). This results in a concomitant reduction of the shift between the states in the two MoS2 layers to 0.2 eV. In passing, we note that in all our simulations we assumed a defectfree MoS 2 layer, which is not necessarily the case in all experiments. Defects changing the position of the Fermi level and potentially also providing localized interaction sites can, for example, trigger much more significant charge rearrangements between strongly S-deficient MoS 2 layers and TiOPc molecules, as recently discussed by Park et al [38]. The above considerations, however, show that as long as these charge rearrangements are largely symmetric between the molecular layer and the two MoS 2 sheets, this would not significantly change the electrostatic shifts.
Another interesting observation in figure 3 is that there are no significant local potential variations induced by the TiOPc layer, which extend into the MoS 2 sheets. This is associated with screening within the MoS 2 layers, as can be inferred from a comparison between the electrostatic energy shown in figure 3 with that of a hypothetical free-standing TiOPc layer in the absence of the 2D semiconductors (see supporting information).
To more clearly visualize the spatial distribution of the states of the MoS 2 |TiOPc|MoS 2 system, a contour map of the local density of states (LDOS) as a function of the position along the stacking direction is contained in figure 4. It shows how the electronic states in the system are distributed as a function of energy and position in stacking direction. The data not only confirm the type II band alignment of the MoS 2 layers, but also clearly show that the largest contribution to the DOS at the band edges stems from regions close to the transition metal (molybdenum for this system) [37,[39][40][41], together with contributions from the PDOS and LDOS plots show that the occupied states of the TiOPc layer lie in the gap of both MoS 2 layers (similar to what has been found for ZnPc on MoS 2 ) [42]. This observation is very likely not a methodological artefact, as it prevails when reducing the selfinteraction error in the DFT calculations by employing hybrid functionals [43,44] (see supporting information). For TiOPc intercalation, these intra-gap states are π-states corresponding to the molecular frontier orbitals localized nearly exclusively on the C atoms (for the HOMO) and on the C and N atoms (for the LUMO). This can be concluded from a comparison between calculated molecular orbital shapes and the local densities of states in the periodic stack as well as from atom-projected DOSs of the intra-gap features (see supporting information). Electronic states of the intercalated molecules in the TMD gaps could lead to trap states especially for electrons and holes, which would be detrimental for applications in optoelectronic devices geared at splitting excitons into free electrons and holes (like photodetectors or solar cells). In practice this problem can be, however, overcome by choosing dipolar molecules with larger ionization energies and gaps, provided that the molecules are designed such that they still align with dipoles parallel to the stacking direction.

More complex quantum-structures: cascades and wells
The idea of using collective electrostatic effects to shift the electronic states in vdW heterostructures can be extended to more complex systems containing more than two TMD layers in combination with intercalated molecular layers with identical or opposing dipole orientations. In this way, for example, quantum cascades and quantum wells can be created: Having parallel dipole arrangements in successive polar layers (as shown schematically in figure 5(a)) results in a cascade of electronic states [24]. This is illustrated by the LDOSs of a correspondingly designed 3-layer stack in figure 5(b). The cascade creates a situation interesting for spatially separating electrons and holes.
Upon reversing the dipole orientation in one of the polar films, the situation shown in figures 5(c) and (d) is achieved. For the dipole orientations considered here, this would lead to an accumulation of holes in the central TMD layer. In practice, such a structure could possibly be realized by stacking a TMD layer (containing adsorbed polar molecules) in a 'face down' fashion onto a two-layer stack. In passing we note that due to the 'robust' conceptual nature of the electrostatic design approach, a slip or rotation of consecutive layers should not adversely affect the electronic structure of the stack.
Notably, for the examples shown in figure 5 we employed molecular systems with identical dipole moments for both intercalated layers; as an alternative, locally varying step sizes can be realized using molecular layers with different dipole densities. Moreover, one is not restricted to only three layers, which implies that conceptually there is no limit to the complexity of electronic structure of the systems that can be realized employing electrostatic design.

Electrostatically designed heterobilayers
So far only systems containing chemically identical TMD layers have been described. When extending the discussion to heterogeneous stacks, in the simulations we are restricted to the combination of materials with very similar lattice constants, due to the employed periodic boundary conditions. Otherwise, considerable strain would have to be applied to the layers in order to study the heterostructure in reasonably small unit cells. This would blur the picture, because even small amounts of strain can significantly change the electronic structure of TMDs [45]. Interestingly, for WSe 2 and MoSe 2 the calculated lattice constants match within 0.08%. Thus a simple unit cell containing only one TiOPc molecule can be constructed. For this construction the lattice constant of the lower layer (WSe 2 in this case) has been used. Due to the very low lattice mismatch the effects of strain on the MoSe 2 layer are negligible. WSe 2 |MoSe 2 bilayer systems are interesting also from an application point of view, as they have been used in solar cells [46] and interlayer exciton optoelectronics [47,48]. The WSe 2 |MoSe 2 bilayer displays type II band alignment [36,[48][49][50][51][52][53] with band offsets of 0.30 eV for the valence [50] and 0.45 eV for the conduction-band [54] edges (where the valence-band offset is taken from angular resolved photoelectron spectroscopy experiment and the conduction-band offset is derived from GW calculations). As these values are in the range of the shifts induced electrostatically by the TiOPc layers discussed above, we tested, whether electrostatic design can be used to qualitatively change the system from type II to type I band alignment. The PDOSs of the relevant systems are shown in figure 6. For the bonded WSe 2 |MoSe 2 bilayer (figure 6(a)), type II band alignment is indeed obtained, but with smaller band offsets than reported in experiment [50] and in previous theoretical studies [11,54]. The latter is very likely a consequence of differing interlayer distances. They, on the one hand, depend on the stacking motif of consecutive TMD layers. On the other hand, the interlayer distance in the simulations is influenced by the choice of the van der Waals correction scheme.
In the experiments it is impacted by organic residues trapped between successive TMD layers [55]. The situation displayed in figure 6(a) is obtained for C7 (2H) stacking, which is the lowest energy configuration in our calculations (in agreement with the calculations of Zhang et al) [56]. It is also the stacking motif of MoSe 2 and WSe 2 in bulk crystals [57,58]. For AA stacking with coplanar layers and a slightly increased interlayer distance the valence band offset is larger (approximately 0.2 eV; see supporting information for details). Consistently, when reducing the interlayer electronic coupling by positioning the layers at the distance they will eventually adopt in the TiOPcintercalated situation, the energetic offsets of the band edges increase even more (to 0.2 eV for the valence band edges and to 0.3 eV for the conduction band edges, see figure 6(b)).
Again, the situation changes qualitatively when including the TiOPc molecules in the simulations: the electrostatically induced shift of the electronic states between the WSe 2 and MoSe 2 due to the presence of the molecular layer amounts to 0.25 eV. Consequently, the conduction band edge of MoSe 2 still lies below that of WSe 2 , while for the valence band edges that of WSe 2 is lower. In this way, a situation with type I band alignment is realized ( figure 6(c)). Conversely, when reversing the stacking order while maintaining the TiOPc orientation ( figure 6(d)), particularly large type II offsets are achieved.
Finally, we note that the concept of electrostatic design of vdW heterostructures is not restricted to stacks consisting of semiconducting TMD layers, but also prevails for example for stacks of graphene sheets. Corresponding data are presented in the supporting information.

Dielectric screening and band-gap renormalization
It has been demonstrated that band-gaps and exciton binding energies of semiconducting TMDs depend strongly on the surrounding dielectric environment [59][60][61][62]. To quantify these effects, one has to go beyond the single particle picture using methods like GW [63]. Such calculations, for instance, reveal that, when comparing a freestanding MoS 2 monolayer in vacuum to a MoS 2 layer sandwiched between two dielectrics with a dielectric constant of 5, the gap is reduced by approximately 24% [60]. Within the bandgap center alignment scheme [64], this affects the positions of the conduction and valence band edges equivalently, as one assumes a quantitatively similar effect of the selfenergy correction on the VBM and the CBM (albeit with different signs) [65,66]. This assumption has been shown to hold for molybdenum and tungsten based dichalcogenides when using GW calculations [64]. On the basis of these findings we conclude that in simple heterostructures, like MoS 2 |TiOPc|MoS 2 , interlayer screening should have no significant effect on the TiOPc-induced changes of the band alignment, since in such systems both MoS 2 layers feel a similar dielectric environment. Only the energy of the transport gap decreases and the exciton binding energy is reduced compared to an isolated monolayer. However, it still remains one order of magnitude larger than in conventional 3D semiconductors-at least for MoSe 2 on bilayer graphene and HOPG (the systems investigated in [59]). These considerations show that for stacks like MoS 2 |TiOPc|MoS 2 , screening and excitonic effects are certainly relevant for their detailed optical properties, but should not significantly impact the modifications of the electronic structure arising from the electrostatic design idea presented in the present paper. Similar considerations apply to heterostructures consisting of different transition metal dichalcogenide layers provided that their dielectric constants are not dramatically different [67].
In cases, where the two TMD layers feel different dielectric environments [61,62], e.g. because the vdW stack is deposited onto a (dielectric) substrate, or the TMD layers themselves have significantly different dielectric constants, stacks with type I and type II band alignment are affected differently: for heterostructures with type II band alignment non-symmetric di electric environments will result in the increase of one of the band offsets and a decrease of the other (see figures 7(a)-(c)). In the type I case, both band offsets are either increased or decreased, depending on which layer faces the environment with a higher dielectric constant (see figures 7(d)-(f)).
For the quantum cascade and the quantum well structures, the situation is somewhat more involved, since there (in the absence of a dielectric surrounding), the band gap renormalization will be larger for the central layer compared to the outer ones.  nitude of that effect: assuming that the outer MoS 2 layers are facing vacuum on one side, and the rest of the heterostructure on the other side and that all the MoS 2 layers have the same effective dielectric constant (here we take κ E equal to 10, to obtain an upper boundary), the difference in band-gap renormalization between the central and the outer MoS 2 layers amounts to 0.2 eV. In the framework of the bandgap center alignment scheme, this results in changes of the level alignment by 0.1 eV (for more details see supporting information).

Conclusions
In conclusion, our calculations reveal that applying electrostatic design to van der Waals heterostructures is a viable way to tune the electronic structure of such systems. Especially the band alignment of successive TMD layers can be changed via the insertion of ordered polar molecular layers. The suggested approach also allows the realization of more complex heterostructures and a change of the type of bandalignment. From a practical point of view, it will be necessary in the future to explore strategies for stabilizing the intercalated layers either through crosslinking between molecules or through direct bonding the dipolar molecules to the TMD layers. Nevertheless, the versatility of electronic properties that can be realized by combining the presented approach with the huge pool of available layered materials to be stacked into vdW heterostructure is enormous and appears to be limited only by the imagination of the materials' designer.

DFT calculations
All calculations are performed using density functional theory (DFT) as implemented in the FHIaims package [68]. The exchange and correlation part is treated using the PBE functional [69]. A k-point grid of 8 × 8 × 1 is employed. Regarding the basis set, 'tight' settings with full 'tier2' for all included atomic species have been used. Van der Waals interactions are considered based on the pairwise correction scheme proposed by Tkatchenko and Scheffler [70]. The occupation of the Kohn-Sham eigenstates was modelled by applying a Gaussian broadening function [71] with a width of 0.01 eV. Details on the convergence settings of the self-consistent field procedure and of the geometry optimization can be found in the supporting information. Spin-orbit coupling (SOC) is not considered in our calculations, as it is not important for the effects discussed in this paper and calculations including SOC would be very memory intensive and, thus, computationally expensive. Ovito [72] and Mayavi [73] were used for graphical visualization.

Geometric details of the heterostructures
The lateral arrangement of the TiOPc molecules in the above considerations is again inspired by the situation on HOPG: there, for a CuPc monolayer Wang et al [74] found a square unit cell containing one molecule with a lattice constant of 13.8 Å. This unit cell cannot be directly adopted in our calculations, as a square unit cell is incompatible with the hexagonal symmetry of the MoS 2 substrate. As even minor (unidirectional) strain would significantly modify the properties of the TMDs (e.g. triggering a direct to indirect gap transition [45]), we fully relaxed the substrate and then picked the smallest rectangular supercell, which was large enough to accommodate a single TiOPc molecule. The only expected impact on the properties of the resulting minor distortion of the TiOPc layer is a minute decrease of the dipole density, which has no relevant consequences for the presented considerations. The supporting information contains a detailed description of the structure of the resulting unit cell, the strategy employed to set-up and optimize the structures of the quantum cascades, quantum wells, and heterobilayers, and a compilation of the resulting structures.

Acknowledgments
We thank K Bolotin for stimulating discussions and O T Hofmann for providing continuous support when performing the simulations. We are grateful for financial support by the Austrian Science Fund (FWF): P28051-N36 and by the European Union Seventh Framework Programme Initial Training Network THINFACE under grant agreement 607232. Electronic structure calculations have been performed using the Vienna Scientific Cluster.

Associated content
The Supporting Information is available free of charge and contains: computational details, and details on the system set-up and optimization; densities of states of isolated monolayers and for WSe 2 |MoSe 2 heterobilayers for different stacking motifs; band structures of the van der Waals heterostructures and the individual subsystems and of the MoS 2 homobilayer; DOSs for the hybrid-functional calculation and for electrostatically designed graphene stacks; speciesand orbital angular momentum projected DOSs; charge rearrangements and bond dipole of the MoS 2 |TiOPc|MoS 2 heterostructure with the top MoS 2 layer being (not) lifted; comparison of the electrostatic energy generated by the TiOPc layer in the absence, respectively, presence of the MoS 2 sheets; species-projected and angular momentum resolved DOS of the TiOPc layer as well as the LDOS of the intra-gap states and the corresponding molecular orbitals; species-projected DOS of the MoS 2 bilayer; depictions of the relaxed geometries; details regarding the quantitative effect of dielectric screening in the quantum cascades and wells.

Notes
The authors declare no competing financial interest

Author contributions
CW performed all calculations presented in the paper, evaluated and interpreted the data, and wrote a first draft of the manuscript. SSH supported the simulations, contributed to the general setting-up of the project, and revised the manuscript. EZ conceived the project idea, steered the work, and significantly contributed to the interpretation of the data and the writing of the manuscript.