Dielectric engineering of heterostructures made from two-dimensional van der Waals semiconductors is a unique and powerful tool to tailor the electric and optical band gaps solely via the dielectric environment and the crystal thickness modulation. Here, we utilize high quality MoTe2 monolayer and bilayer crystals as a candidate for near-infrared photonic applications. The crystals are exfoliated on various technologically relevant carrier substrates: silicon/silicon dioxide, poly(methyl methacrylate), hexagonal boron nitride, silicon carbide, and silicon nitride. These substrates provide a large range of high frequency dielectric constants from 2.1 to 7.0 for MoTe2-containing heterostructures. We assess the relationship between the environmental dielectric function and Coulomb screening by combining detailed spectroscopic measurements, utilizing low-temperature and high-spatially resolved photoluminescence and contrast reflectivity, with microscopic many-body modeling, to explore the potential of this less-recognized material platform for applications in optoelectronics at photon wavelengths above 1 μm. We observe a redshift of the optical gap emission energy from the monolayer to bilayer regime on the order of 30 meV. Furthermore, the thickness controlled shift is slightly larger than the one induced by the local dielectric environment, which ranges on the order of 20 meV for the MoTe2 monolayers and on the order of 8 meV for the MoTe2 bilayers. We also show that the local dielectric screening barely affects the trion binding energy, which is captured by our microscopic model, accounting for the screened Coulomb potential for the heterostructures.
Following the initial studies on graphene,1 two-dimensional (2D) atomically thin materials have attracted enormous interest driven by their unique physical properties.2–5 Among the countless 2D materials obtained to date, the 2D transition metal dichalcogenide (TMDC) semiconductors, with their most recognizable members MoS2, MoSe2, WS2, and WSe2, are intensively studied, showing new exciting physical phenomena promising future generation of low-energy consumption, nanoscale, and efficient optoelectronic devices.4,6,7 New device functionalities arise from the low dimensionality of TMDC crystals, with most of them based on the enhanced Coulomb interaction due to weak dielectric screening. Therefore, Coulomb-bound complexes, such as neutral excitons and charged excitons (trions), are the fundamental excitations and dominate the absorption and emission spectra even at room temperature.
However, potential TMDC-based optoelectronics in the technologically highly relevant near-infrared spectral range () generally suffers from the lack of material solutions with the one particular exception of MoTe2. The direct optical bandgap () of the most commonly studied 2D TMDCs WS2 (), MoS2 (), WSe2 (), MoSe2 (), ReS2 (), and ReSe2 () is settled far above the , resulting in absorption and emission threshold . Breaking this important energy scale can provide opportunities to render 2D TMDCs compatible with today's established optoelectronic material platforms (Si, SiN, and 4H-SiC). It opens the route to small-footprint and on-chip integrated optoelectronics without lattice-matching constraints and with newly gained functionalities from hybridization, e.g., Si/TMDC nanolasers,8 III-V/TMDC room-temperature polaritonic emitters,9 and hybrid Si/TMDC photodiodes.10,11
In this respect, the less studied member of the TMDC family, molybdenum ditelluride MoTe2, finds its application, although many of its properties are still largely unknown. Bulk MoTe2 can be synthesized in several polymorphs,12 with the semiconducting hexagonal one (2H) being the most suited to photonic applications. The 2H-MoTe2 (considered here and from now on referred to as MoTe2) is composed of Mo atoms arranged in hexagonal sheets, sandwiched between two atomic planes with the hexagonal ordered Te atoms, forming trigonal prismatic coordination within a Te–Mo–Te layer sequence called a monolayer (ML). The bulk crystal consists of stacked MLs weakly bound by van der Waals forces, but with strong ionic bonds in plane. By thinning the crystal down from bulk to bilayer (BL) or ML, the MoTe2 optical bandgap evolves from an indirect one at () and (Ref. 13) to a direct one at () and .14–17 At room temperature, the optical gaps for ML and BL MoTe2 are, respectively, ()18,19 and ()20 (see the supplementary material, S-IV), making this material the only TMDC with device applicability in the near-infrared spectral range. Indeed, in 2017, a ML MoTe2-based nanolaser integrated with the Si nanobeam was presented,8 and a gain mechanism mediated by trions has been suggested for the interpretation of the observed room-temperature lasing21 at under a very low pumping power (a few W/cm2). In 2018, a near-infrared photodiode operating at was shown, revealing record quantum efficiency (9.8% at ) among all TMDCs.22 This finding was predeceased by presenting MoTe2-based photodetectors.23 With MoTe2 being the candidate for near-infrared optoelectronics, the possibility of band structure engineering to further tailor its properties is highly desirable. So far, this issue has been addressed by modulating the 2D crystal thickness,14,19,24 electrostatic gating,25 and strain engineering.26 The potential of extensive alloying, heterogeneous stacking, and dielectric screening is yet unknown.
In this work, we examine the applicability of Coulomb engineering27 to ML and BL MoTe2. This term refers to the possibility to tailor the Coulomb interaction that directly determines the optical properties via the excitonic and trionic binding energies by tailoring the local dielectric environment. This approach has been studied theoretically27–35 and experimentally29,34,36–47 concerning MoS2,27–30,32,33,37,43,47 MoSe2,28,29,31,36,39,81 WS2,28,34,40,44–47 and WSe2.28,31,35,38,39,41–44,46 Despite this history of publications, studies on the application of Coulomb engineering to ML and BL MoTe2 were omitted so far.
Here, we present the results of a combined experimental and theoretical study on the role of dielectric environment on the fundamental neutral exciton and trion properties in ML and BL MoTe2. To tune the effective dielectric constant of the TMDC environment, we use a wide variety of materials having application capabilities that could potentially arise from hybrid-type TMDC/organic/inorganic heterostructures. These include poly(methyl methacrylate) (PMMA), hexagonal Boron Nitride (hBN), group-IV semiconductors (4H-SiC), and wide-bandgap semiconductors and isolators (SiN, SiO2), which are all transparent for the MoTe2 emission. We exfoliated the ML and BL MoTe2 from chemical vapor transport (CVT)-made bulk crystals and combined them with the dielectric materials using the dry stamp technique, resulting in a mean high-frequency dielectric constant (κ) spanning a range between 1.55 and 5.74. The impact of the mean dielectric surrounding is examined by tracking the exciton and trion fundamental transitions in MoTe2 layers using highly spatially and spectrally resolved photoluminescence and reflectivity contrast. The theoretical approach utilizes semiconductor Bloch equations (SBEs) with additional contributions for three-particle complexes, providing access to the exciton and trion binding energies on a microscopic level. We show that, similar to other TMDCs, the excitonic bandgap in ML and BL MoTe2 is barely affected by the dielectric surrounding due to the lowering of the quasi-particle bandgap accompanied by the simultaneous decreasing of the Coulomb binding energy between electron and hole. We also show that the trionic bandgap is weakly susceptible to the existence of local dielectric screening that only barely affects the trion binding energy, which is captured by our theoretical model.
II. EXPERIMENT AND THEORY
A. Stacks fabrication
The ML and BL MoTe2 crystals were prepared by mechanical exfoliation from the CVT-grown bulk crystals in the ambient atmosphere at room temperature. Subsequently, the 2D layers were stacked using the deterministic all-dry stamping method on different commercially available substrates (Si/SiO2, 4H-SiC, and Si3N4). Thin hBN crystals were exfoliated from bulk structures, similarly as MoTe2 ones. The typical exfoliated hBN has a thickness of a few tens of nanometers. To improve the contact between transferred layers, immediately after the deposition of each subsequent layer, the structure was annealed for at on a hot plate in air. After the transfer of the topmost layer, each stack was annealed for at in air and immediately transferred to an optical cryostat and kept in the vacuum for the measurement session.
For data presentation purposes, we choose the high-frequency dielectric constant that characterizes the environment for exciton complexes confined to ML and BL MoTe2 crystals. It can be justified based on the impact of frequency-dependent environmental screening of excitons in 2D materials.33,48–51 At low temperatures, the neutral exciton is not sensitive to a certain frequency, sampling the dielectric function over a range of hundreds of meV, beginning at zero frequency. The dielectric function of materials in the stack can be split into a high-frequency contribution from electronic band-to-band transitions and low-energy resonances from phonons, especially considering dynamical screening due to optical phonons in polar substrate materials like hBN.50 Since the bandgap energy of substrate materials is on the order of electronvolts, the high-frequency part has a weak frequency dependence for several hundreds of meV above the phonon resonances. Therefore, one can use the value of the dielectric function at a frequency that lies above the phonon resonance and clearly below the substrate bandgap. Notably, the interaction with plasmons or phonons in the substrate has little impact on the energy of the neutral exciton ground state.33,50 However, higher exciton states and the quasi-particle gap are affected more strongly. Since this work focuses on the absolute exciton energies, it is justified to model substrate screening with a high-frequency dielectric constant. We assume that the energy of three-particle complexes reacts similarly to the dynamical screening effect.
The high-frequency dielectric constants for the materials are shown in Table I and spread between 2.1 and 7.0, whereas the mean dielectric constants for the stack κ are presented in Table II, spanning the range of 1.55–5.74.
|Stack (bottom/top) .||a .||a .||κb .||Stack symbol .|
|Stack (bottom/top) .||a .||a .||κb .||Stack symbol .|
, where b, t for bottom and top materials with parameters as in Table I, respectively.
κ is calculated according to Eq. (3).
B. Optical experiments
For the high spatially (single μm scale) and spectrally resolved photoluminescence (μPL) in the near-infrared, the structures were kept in a helium-flow cryostat with temperature control in the range of . A continuous-wave laser line at was employed for the non-resonant excitation of the MoTe2-containing structures through a long working distance (WD), high-resolution, infinity-corrected microscope objective (×50 magnification, 0.65 numerical aperture, ). The diffraction-limited excitation spot was in diameter. The same objective collected photon emission and directed it to a 0.5-m-focal-length monochromator equipped with the thermoelectrically cooled 2D InGaAs-based camera. The stacks were inspected by an optical microscope at room temperature. The excitation laser power, , measured before the objective was kept below , which provides the weak excitation regime with the electron–hole pair density below (for more details, see the supplementary material, S-I). The high spatially resolved contrast reflectivity (CR) experiment was performed in the same μPL setup. The broadband halogen lamp source was used to probe the reflectivity spectrum on the flake (μPL) and outside the flake () with the resultant curve presented as .
To calculate the trion binding energy , which we define as the energetic separation between the neutral and charged excitons, we apply the microscopic theory introduced in Ref. 29. In this approach, the linear optical response of MoTe2 is obtained from the semiconductor Bloch equations (SBEs) for the microscopic interband polarizations ,
Here, denotes the carrier population in band λ. The SBEs are augmented by equations of motion for trion amplitudes contained in , which amounts to solving a generalized three-particle Schrödinger equation.29 The generalized SBEs are solved in frequency space by matrix inversion using a Monkhorst-Pack mesh with 84 grid points along Γ–M to achieve sufficient convergence. The material properties enter via band structures , Coulomb matrix elements and dipole matrix elements that are projected in the polarization direction of the electric field . Density functional theory (DFT) calculations for freestanding ML MoTe2 are carried out using Quantum Espresso.59,60 We apply the generalized gradient approximation (GGA) by Perdew, Burke, and Ernzerhof (PBE)61,62 and use an optimized norm-conserving Vanderbilt pseudopotential63 at a plane wave cutoff of 80 Ry. Uniform meshes (including the Γ-point) with k-points are combined with a Fermi–Dirac smearing of 5 mRy. The calculations are performed based on the MoTe2 lattice structure from the Materials Project.64 The band structure for the two lowest (highest) conduction (valence) bands is obtained by a lattice Hamiltonian, for which we utilize a localized basis of Wannier functions that we obtain with the help of the Wannier90 package.65 The corresponding dipole transition matrix elements are calculated using a Peierls approximation as discussed in Ref. 66. For the Coulomb interaction, we use Coulomb matrix elements in a 2D layer,
where the contribution of the Wannier orbital α to the band λ is given by . The matrix elements are assumed to be independent of the orbital character and modeled by a macroscopic dielectric function , which is obtained by solving Poisson's equation for the respective dielectric structure. A narrow gap between the TMDC and the encapsulating layers has been taken into account, and a layer substrate distance of 0.3 nm has been found to be an appropriate value.29
III. RESULTS AND DISCUSSION
Figures 2(a)–2(h) show the low-temperature () μPL and CR spectra recorded for ML and BL MoTe2 embedded in different dielectric environments defined by the bottom and the top material surrounding the 2D MoTe2 crystal. Each μPL spectrum exhibits two prominent high- and low-energy emission bands for both ML and BL MoTe2. We attribute the high-energy band to emission from the neutral exciton XA, which arises from Coulomb-bound electron–hole pairs in the vicinity of K/K′ valleys of the Brillouin zone in the ML/BL MoTe2 band structure sketched in Fig. 1(c). In the literature, the XA transition is often referred to as the A-exciton, underlying the difference from the B-exciton that involves an electron at the minimum of the conduction band and a hole at the maximum of the spin–orbit split valence band at K/K′ points of the Brillouin zone. The XA transition energy defines the optical bandgap of the material [see Fig. 1(c)]. For the recorded μPL spectra, the XA emission is accompanied by a second spectral peak labeled T and observed at its low-energy side. This feature is attributed to a radiative recombination of trions, which are Coulomb-correlated three-particle complexes consisting of a neutral exciton and an additional electron or hole. The presence of trions is a common feature in TMDC emission spectra and often results from unintentional residual doping of the MoTe2 crystals. We believe residual doping in the investigated stacks is generated mainly by the unintentional intrinsic dopants, defects or uncontrolled charge transfer across the MoTe2/surrounding material interfaces upon the stack fabrication procedure. A dominant photo-doping effect was excluded due to the weak excitation regime (see also the supplementary material, S-I, S-II, and S-V). We estimated the residual charge concentration in the range of (see the supplementary material, S-V). The small magnitude of residual doping is manifested in the CR response where only the neutral exciton-related features are present in the recorded spectra.67,68
The presence of trions is essential for the present work, as the strength of the Coulomb interaction in the presence of different dielectric environments is directly reflected in the trion binding energy .
To extract the and energies, the μPL bands are fitted by pseudo-Voigt line shape profiles that account for the inhomogeneities contributing to the natural transition linewidth, which are shown by the shaded areas in Figs. 2(a)–2(h). The extracted transition linewidths and transition energies are plotted as a function of the high-frequency mean dielectric constant,
in Figs. 3 and 4, respectively. In Eq. (3) is defined via the high-frequency dielectric constants of the different bottom () and top materials () surrounding the 2D MoTe2 crystal as listed in Table II. It is important to note that κ represents the mean dielectric constant of the sub- and superstate environment. The screening of Coulomb interaction within exciton and trion states in the MoTe2 bilayer is also affected by a second MoTe2 layer, which the estimated κ value from Eq. (3) might not accurately reflect. However, for comparing datasets for ML and BL MoTe2, we use the same κ values for both types of stacks.
We start our discussion with the transition linewidth that reflects inhomogeneities due to fluctuations of the local dielectric environment,44 charge environment, or strain field.26,71 As such, it is a good measure for the uncertainty of and . Note that the impact of the mentioned local inhomogeneities is more pronounced for the high-quality crystals, where the exciton and trion linewidth is narrow. Such an example is plotted in Fig. 2(h). Two PL spectra (color shade and gray) are presented for hBN-encapsulated stacks with ML and BL MoTe2. The indicated energy difference () within the given stack family is comparable to the inhomogeneous spectral broadening, which we took as the uncertain measure for the energy position of exciton and trion.
Figures 3(a) and 3(d) show the broadening of exciton () and trion () emission bands, respectively, with the results of statistical analysis presented in Figs. 3(b) and 3(c) and 3(e) and 3(f). The analysis confirms that the stacks having a 2D MoTe2 crystal exposed to ambient for a longer time before encapsulation or enclosing in an optical cryostat exhibit significantly larger mean broadening and of the XA and T transition, than the stacks with hBN on top or the hBN-encapsulated 2D crystals (see the supplementary material, S-III). In this case, decreases from to , while changes from to . It is worth noting that the 2D MoTe2 crystal covered by hBN can be of similarly high quality in the optical emission lines than its fully hBN-encapsulated version. The minimal is within the studied stacks and is obtained for SiO2/ML MoTe2/hBN material combination, which is comparable to those for hBN/BL MoTe2/hBN () (see the supplementary material, Table S1). These values match the calculated and measured coherent neutral exciton linewidth of for molybdenum-based 2D TMDC at low temperature.72,73 Moreover, the minimal for ML and BL MoTe2 is comparable to the lowest reported to-date exciton linewidth for hBN-encapsulated ML MoTe2 crystal ( in Ref. 69) with all the stacks having below , confirming a consistently high sample quality.
The trion linewidth is much less discussed across the literature. For example, in Ref. 14, reaches for the hBN-encapsulated ML MoTe2 structure, which is comparable to for the investigated stacks with 2D MoTe2 crystals exposed to air. However, for the majority of hBN-encapsulated stacks, we find and report the record minimal value for SiO2/ML MoTe2/hBN, which is comparable to for hBN-encapsulated ML MoTe2.
Now, we shift to the analysis of the exciton and trion transition energies for ML and BL MoTe2 as a function of the dielectric environment. Figure 4(a) summarizes the results for the neutral exciton transition energy. At first glance, one can find a clear distinction between the optical gap energy for ML (gray points) and BL MoTe2 (red points). The transition from ML to BL crystal lowers the excitonic bandgap by , translated to a roughly redshift in the emission wavelength. This spectral shift agrees with previously obtained .24,74 Importantly, increasing the 2D crystal thickness to three and more layers transforms MoTe2 to an indirect bandgap material, preventing the efficient light emission process.24 In contrast to the more frequently studied TMDCs (WS2, WSe2, MoS2, and MoSe2), the crystal thickness of MoTe2 is a limited tuning knob of the excitonic bandgap. For example, transition from ML to BL MoS2 causes a redshift of the excitonic bandgap  of ,74 for MoSe2,74 and for WSe2.74 On the contrary to the optical gap variation with the crystal thickness modulation, Fig. 4(a) presents a weak trend for the in the case of ML MoTe2 and no distinctive trend within the scattering range of the data points for the BL MoTe2. Figure 4(a) also includes available literature data (open symbols). Since the is determined by the difference between the quasi-particle energy and the exciton binding energy : [see Fig. 1(c)], the lack of systematic trend in Fig. 4(a) can be attributed to similar magnitude shrinkage of the and with κ.
For ML MoTe2, the shrinkage has been experimentally verified, showing reduction of and from ,14 and (Ref. 25) for to (Ref. 70) and (Ref. 70) for , respectively. Theory predicts that for ML MoS2, MoSe2, WS2, and WSe2, a slight imbalance between changes in and with κ results in a small redshift of the neutral exciton transition energy, being one of the sources of the optical gap tunability. For the previously studied molybdenum-based TMDCs, the imbalance generates .37,39 For ML MoTe2, the optical gap tunability amounts to , while for the BL MoTe2, it can be estimated to be as high as , according to our results.
The environmental control over the neutral exciton transition energy through the screening of the Coulomb interactions should also play a role as a tuning knob for other Coulomb-bound complexes. We particularly focus on the evolution of the trion and its binding energy , which was previously thoroughly theoretically and experimentally addressed only for the molybdenum-based TMDCs MoS2 and MoSe2.29 The trion binding energy is evaluated from the datasets presented in Fig. 4, using the following equation:
The equation does not account for the correction factors to the trion binding energy related to charge carrier/exciton/trion localization induced by strain, structural defects, or charge distribution within a substrate and to the phase space filling, free carrier screening, and bandgap renormalization, triggered by the presence of residual carrier density in the crystal. The functional form of the correction factor induced by residual carriers has been partially evaluated only for the case of absorption-type experiments75–77 and cannot be directly transferred to the emission-type experiment. We can only speculate that its magnitude can be as large as the estimated Fermi energy (see the supplementary material, S-V). The correction factor related to the localization is reflected in the emission band broadening parameter, depicted as error bars in Fig. 5.
From Fig. 5, one can see that both trends for ML and BL MoTe2 are hindered within the experimental uncertainities. Therefore, the tunability for both ML and BL MoTe2 can be within a few meV.
To find the expected trion binding energy tunability range, we employed macroscopic models to describe the screened Coulomb potentials (see Sec. II C). Within the theoretical framework, the macroscopic dielectric function for the respective dielectric structure is obtained by solving Poisson's equation. Herein, the dielectric function of the TMDC material is described by an effective dielectric constant, for which two different approaches are used that capture (1) the strong anisotropy between in- and out-of-plane direction of the dielectric tensor and (2) the reduced dimensionality of the atomically thin layer. For the first model (model 1), we follow Ref. 78 and calculate the dielectric constant as a geometric mean of the parallel and perpendicular directions with values given in Ref. 28. For the second model (model 2), only the in-plane component is considered. In addition, while for ML MoTe2, the dielectric structure is given by the encapsulated single layer, we model a BL by including an additional MoTe2 layer in the dielectric environment between the active layer and the substrate. We assume that enhanced screening is the dominant effect of the second layer, while electronic coupling between the layers can be neglected as a result of the large interlayer distance.79 For the additional MoTe2 layer, we use the same dielectric constant and layer width as for the active layer. The layer width is taken from Ref. 28. Both the results from model 1 and model 2 are presented in Fig. 5 together with the experimental data points obtained in this study (filled symbols) and in the literature (open symbols). First, we find that the theoretical results give a reasonable estimate for the trion binding energy for ML and BL MoTe2, where the results using model 1 (model 2) serve as an upper (lower) bound for the experimentally observed values. In our macroscopic description of screening, local-field effects are neglected, which admix components of the dielectric tensor according to the orbital composition of electronic states. Including only the large in-plane component in model 2 will, therefore, tend to overestimate the screening of the TMDC. On the other hand, weighting all components equally in model 1 underestimates screening, which suggests that the in-plane component is the dominant one. Second, the theory clearly distinguishes the trion binding energy between ML and BL MoTe2. Finally, the models allow extracting for κ ranging from 1.5 to 5.8, which is for ML MoTe2 and for the BL. These values are consistent with the experimental observations, confirming that it is challenging to observe the tunability of trion binding energies in the experimental data. We attribute the weaker κ-dependence of for the BL to the enhanced internal screening, which leads to a saturation of screening efficiency.
Our study reveals the following central features of the two-dimensional semiconductor MoTe2. The encapsulation with hexagonal boron nitride yields excellent optical properties of MoTe2 mono- and bilayers, for which we obtain photoluminescence linewidths close to the radiation limit of . This is a strong indicator that MoTe2 bilayers are direct bandgap semiconductors, in contrast to trilayers, which do not show substantial PL. These excellent optical properties are retained by direct mono- and bilayer transfer on semiconducting substrates, including the technologically relevant SiO2, Si3N4, and 4H-SiC, which are transparent for MoTe2 emission. We observe a significant spectral redshift of the optical gap emission energy from the monolayer to the bilayer regime on the order of . By using various selected carrier substrates, we find that the thickness-controlled shift is only slightly higher than the one induced by the local dielectric environment. For monolayer MoTe2, the redshift amounts to in comparison to for the bilayer. Having the opportunity to study the impact of dielectric screening on the Coulomb-bound complexes, we show that the trion binding energy in mono- and bilayer MoTe2 is only barely affected by the local dielectric environment, which is fully captured by our microscopic model accounting for the screened Coulomb potential of the heterostructure.
Finally, we conclude that the combination of layer- and dielectric engineering of MoTe2 renders insufficient to shift the optical gap of MoTe2 all the way to the technologically relevant wavelength of 1300 nm. We therefore suggest to use extensive alloying,26 applying strain,26 or combining MoTe2 monolayers or bilayers with other 2D materials in van der Waals heterostructures80 to reach the telecommunication O-band wavelength. An alternative approach could involve the resonant hybridization to a microcavity resonance, where the normal-mode coupling can renormalize the system eigenenergies substantially,9 which may allow to approach the 1300 nm emission regime.
See the supplementary material for the following: S-I—the excitation power dependence of the neutral and charged exciton energy and the charged exciton binding energy for mono- and b-layer MoTe2; S-II—the neutral and charged exciton energy and the charged exciton binding energy dependence on the thickness of hexagonal boron nitride for stacks with mono- and bilayer MoTe2 exfoliated on Si/SiO2; S-III—the neutral and charged exciton emission linewidth dependence on an environment dielectric constant for mono- and bilayer MoTe2; S-IV—the temperature dependence of mono- and bilayer MoTe2; S-V—the estimation of the Fermi energy in mono- and bilayer MoTe2.
These studies were largely carried out from the OPUS 18 research Project No. 2019/35/B/ST5/04308 financed by the Polish National Science Center (NCN) and the project SCHN1376 11.1 funded by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG). The project was also partially funded by the QuanterERA II European Union's Horizon 2020 research and innovation programme under the EQUAISE project, Grant Agreement No. 101017733. M.S., E.Z., and D.B. acknowledge support from the Polish National Agency for Academic Exchange (NAWA) (No. PPI/APM/2018/1/00031/U/001). M.F. acknowledges support by the Alexander von Humboldt foundation. C.G. and C.S. gratefully acknowledge funding from the priority program SPP2244 of the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) via the projects Gi1121-4/1 and SCHN1376-14/1. C.G. further acknowledges support from the DFG graduate school 2247 (QM3). M.F. and A.S. acknowledge support for computational time at the HLRN (Berlin/Göttingen). We thank Professor Joanna Jadczak for a fruitful discussion.
Conflict of Interest
The authors have no conflicts to disclose.
Joanna Jadwiga Kutrowska-Girzycka: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal). Christopher Gies: Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Sefaattin Tongay: Resources (equal). Christian Schneider: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Marcin Syperek: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Emilia Zięba-Ostój: Data curation (equal); Formal analysis (equal); Investigation (equal). Dąbrowka Biegańska: Data curation (equal); Formal analysis (equal); Investigation (equal). Matthias Florian: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Writing – review & editing (equal). Alexander Steinhoff: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). Ernest Rogowicz: Data curation (equal); Formal analysis (equal); Investigation (equal). Pawel Mrowinski: Data curation (equal); Formal analysis (equal); Investigation (equal). Kenji Watanabe: Resources (equal). Takashi Taniguchi: Resources (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.