Abstract
Generally, the lack of longrange order in materials prevents from experimentally addressing their electronic band dispersion by angleresolved photoelectron spectroscopy (ARPES), limiting such assessment to single crystalline samples. Here we demonstrate that the ARPES spectra of azimuthally disordered transition metal dichalcogenide (TMDC) monolayers with 2 H phase are dominated by their band dispersion along the two high symmetry directions ΓK and ΓM. We exemplify this by analyzing the ARPES spectra of four prototypical TMDCs within a mathematical framework, which allows to consistently explain the reported observations. A robust base for investigating TMDC monolayers significantly beyond single crystal samples is thus established.
Introduction
Transitionmetal dichalcogenide (TMDC) monolayers have gained attention as semiconductors for nextgeneration nanoscale electronic devices owing to their unique electronic properties that originate from their lowdimensionality and highsymmetry structure^{1,2,3}. Recently, heterostructures of monolayer TMDC combinations have been employed in novel devices with superior optoelectronic performance^{4,5,6,7,8,9,10}. Furthermore, they can be used to explore physical phenomena like topological edge states^{11} or the spin Hall effect^{12,13,14}. The key for their fundamental understanding, and notably also for rational device design, is understanding the electronic structure of TMDC monolayers and their heterostructures. To obtain detailed information on semiconductor band structures, angleresolved photoemission spectroscopy (ARPES) is a very effective method and it has been widely applied to single crystalline materials for decades^{15}. However, sufficiently large single crystalline samples are required for conventional ARPES measurements (ca. mm^{2} range), which is still challenging for TMDC monolayers and thus limits obtaining the desired information on the electronic band structure for many interesting systems. Specifically, TMDC monolayers fabricated by various methods generally comprise flakes (typical lengths ranging from a few 10 nm to several 10 µm) with azimuthal (φ) disorder, i.e., samples are twodimensional (2D) powders^{16,17,18,19,20}. Accordingly, the electronic band dispersion of each azimuthally rotated flakes contributes to the ARPES spectra, generally prohibiting band dispersion observation due to the angular averaging. Notably, Zhou et al. reported the observation of dispersing bands in ARPES of highly oriented pyrolytic graphite (HOPG; in terms of surface structure also a 2D powder) and they suggested Van Hove singularities as reason for the defined electronic band dispersion in the “angle dependent” density of states (quasi 1DDOS)^{21}. Using the same proposition, others have observed and explained ARPES spectra exhibiting energy dispersion for azimuthally disordered MoS_{2} and WSe_{2} samples^{22,22}.
These reported ARPES spectra are the result of a summation over a large number of single crystal flakes with random azimuthal orientation. However, the existence of Van Hove singularities along certain directions over extended energy and momentum values, as invoked earlier to explain 2D powder spectra using the concept of an angledependent DOS^{21}, should be challenged. According to the quasi 1DDOS approach proposed by Zhou et al.^{21}, Van Hove singularities should occur along the highsymmetry directions in disordered samples. Since Van Hove singularities do only occur at specific points of the Brillouin zone of a single crystal (see Supplementary Note 1), this approach might not be appropriate to explain the ARPES features of disordered TMDC samples, calling for a revision of the quasi 1DDOS concept. Here, we propose an alternative explanation for the observed dispersing bands in ARPES of 2D powders, without the need of using the concept of quasi 1DDOS and Van Hove singularities.
In this contribution, we evidence highsymmetry induced sharp dispersion for TMDC monolayer samples with azimuthally disordered flakes by ARPES measurements and we consolidate our observations with the help of density functional theory (DFT) calculations and angular averaging considerations. With highresolution ARPES spectra of single crystalline and azimuthally disordered WSe_{2} monolayers we reveal the impact of the azimuthal disorder and rationalize that the two highsymmetry directions [ΓK and ΓM of the Brillouin zone (BZ)] dominate the ARPES spectra of azimuthally disordered samples. This is a consequence of the angular integration of the single crystal TMDC monolayer band structure. The same is evidenced for three further prototypical semiconducting TMDCs (MoS_{2}, MoSe_{2}, and WS_{2}), allowing the experimental determination of band dispersion in the two most important BZ directions for azimuthally disordered TMDC monolayers.
Results
Sharp band dispersion in azimuthally disordered TMDCs
Figure 1a shows the BZ of a single crystalline TMDC with D_{3h} symmetry, with its highsymmetry directions and highsymmetry points. In typical ARPES measurements the detection area is on the order of 1 mm^{2}, while a TMDC monolayer sample typically comprises many single crystalline flakes with typical lengths of several 100 nm up to a few 100 μm with random azimuthal orientation (see Supplementary Note 2). This leads to a spectral summation over the individual flakes’ contributions along φ, as schematically shown in Fig. 1b. As a consequence, the angularly superimposed BZs of individual flakes lead to a spherical symmetry around Γ. As shown in Fig. 1c, remarkably, sharp electronic band dispersion is observed in an ARPES spectrum as a function of radial momentum (k_{r}), corresponding to “path 1 + path 2” of Fig. 1b. Notably, the measured band dispersion compares well with the calculated band structure along the highsymmetry directions ΓKM (path 1) and ΓMΓ (path 2) shown in Fig. 1d, strongly suggesting that these directions dominantly contribute to the ARPES spectrum. As explained in the following, the measured dispersion is due to the fact that the angular summation returns an effective spectrum that contains only k_{r} as parameter and is dominated by contributions from the highsymmetry directions.
Calculation of photoelectron intensity
The measured photoelectron intensity I\(\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\) as a function of energy E and radial momentum k_{r}, shown in Fig. 1c, is proportional to the number of flakes N that emit photoelectrons with this E and k_{r}. For an azimuthally disordered sample, it is necessary to obtain the number of flakes that contribute at a certain E to \(I\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\). Given a sufficiently large N with random orientation, we can assume that an angular (φ) interval of \(\Delta \varphi = 2\pi /N\) represents one flake. For each interval Δφ, the corresponding absolute energy change \(\left {\Delta E} \right\) due to dispersion along φ can be calculated based on its derivative and is given by:
\(\nabla _{\mathrm{\varphi }}E\left( {{\mathbf{k}}_{\mathbf{r}},\varphi } \right)\) is the partial derivative of the energy with respect to φ direction. Using the above relation, we can count the N \(\left( {E,{\mathbf{k}}_{\mathbf{r}},\varphi } \right)\) that contribute to a certain energy interval δE (e.g., the experimental energy resolution):
Due to the azimuthal disorder, \(I\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\) is obtained by summing over all \(\varphi\) directions, leading to:
From Eq. (3) it becomes clear that \(I\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\)is inversely proportional to \(\left {\nabla _\varphi E({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\). This implies that the intensity measured in ARPES at a certain (E, k_{r}) is high for \(\left {\nabla _\varphi E({\mathbf{k}}_{\mathbf{r}},\varphi )} \right \to 0\), because this corresponds to a large number of flakes contributing to the same E (within the range of \(E \pm \delta E/2\)) at the corresponding (k_{r}, φ). Note that the derivation of Eq. (3) is exact only if the number of flakes approaches infinity and the energy and angular resolution approach zero. Since in any experiment δE, Δφ, and N are finite, (lower) intensity between the highsymmetry directions may still be observed in ARPES spectra. Importantly, even though our formula reflects the one used by Zhou et al.^{21}, we do not rely on a discussion of the DOS, which remains twodimensional for the 2D powders.
Symmetry of TMDC band structure and photoelectron intensity
To better understand why measured spectra of 2D powders are dominated by the highsymmetry directions for TMDCs, energy maps of the valence band (VB) and 1/\(\left {\nabla _{\mathrm{\varphi }}E_{{\mathrm{VB}}}({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\)  maps are calculated for the four different TMDCs. Figure 2a–d show the calculated valence band (VB) energy maps for single crystal BZs of monolayer TMDCs as a function of momentum (k_{r}, φ), and they agree well with previous calculations^{23}. The energy maps of all four TMDCs are very similar, since they have the same 2Hphase classified by its 3fold dihedral symmetry group (D_{3h}).
Figure 2e–h show the corresponding \(1/\left {\nabla _{\mathrm{\varphi }}E_{{\mathrm{VB}}}({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\) maps of the four monolayer TMDCs. Looking along the φ direction, the VB energy map in a single crystalline BZ with D_{3h} symmetry must be symmetric with respect to the two highsymmetry directions (ΓK and ΓM). Thus, the band structure along these inevitably has local maxima and minima, where \(\left {\nabla _{\mathrm{\varphi }}E({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\) tends towards zero, returning high values of \(1/\left {\nabla _{\mathrm{\varphi }}E_{{\mathrm{VB}}}({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\) as clearly seen in Fig. 2e–h. High values of \(1/\left {\nabla _{\mathrm{\varphi }}E_{{\mathrm{VB}}}({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\) are mostly found in the colored areas that form “lines” along ΓK and ΓM. Therefore, only these two directions contribute notably to \(I\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\) where N\(\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\) is counted along the φ direction using Eq. (3). Consequently, while information on nonhighsymmetry directions is missing (or appears with small intensity in ARPES), the high values of \(1/\left {\nabla _{\mathrm{\varphi }}E_{{\mathrm{VB}}}({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\) allow measuring the electronic band dispersion in the two highsymmetry directions of azimuthally disordered TMDC monolayer flake samples, which is a key information to understand the behavior of electrons in a solid.
Interpretation of photoelectron intensity
To directly compare the calculated \(I\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\) with our measured spectra, we first calculate \(I\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\) for selected k_{r} values (see Fig. 3a with colorcoded k_{r} radii), using Eq. (3) based on the valence energy map of a single crystal WSe_{2} monolayer (see Fig. 3b). In Fig. 3c, which depicts the calculated \(I\left( {E,{\mathbf{k}}_{\mathbf{r}}} \right)\) as a function of energy for the (φintegrated) circles with corresponding k_{r}, high intensity points are observed. In Fig. 3d, the experimentally measured electronic band dispersion of WSe_{2} is shown. The calculated high intensity points at the corresponding k_{r} values are superimposed on the spectrum. The comparison yields very good agreement between calculated and experimental results, evidencing the relation between \(1/\left {\nabla _{\mathrm{\varphi }}E_{VB}({\mathbf{k}}_{\mathbf{r}},\varphi )} \right\) and the ARPES signal for dominating band dispersion directions for 2D powders of WSe_{2}. It should be mentioned at this point that matrix element effects can modify the spectral intensity distribution. Notably, the matrix element effects in photoemission can be influenced by several experimental factors such as the excitation energy, light incidence angle, orbital distribution, and their binding energy. However, this does not affect the abovementioned arguments about azimuthal integration and the observation of the band structure along the (singlecrystal) highsymmetry directions in disordered samples.
Comparison of single crystal and azimuthally disordered WSe_{2}
To consolidate the single crystal versus 2D powder picture developed above, Fig. 4 shows measured ARPES spectra of (a) single crystalline and (b) azimuthally disordered WSe_{2} monolayer samples as a function of momentum and sliced at different constant binding energies (E_{B}). The E_{B} values were chosen such that the spectra correspond to clearly different, but matching regions of the single crystal and φangular averaged spectra. Note that the E_{B} values differ for the two samples since different substrates were used (WSe_{2} is physisorbed on both substrates and the band structure is virtually the same except for a rigid shift), i.e., the electronic levels are differently aligned with respect to Fermi energy^{22}. Next to the experimental data, simulated spectra of a single crystalline and an azimuthally disordered WSe_{2} monolayer are plotted in Fig. 4c, d, respectively. In 2D powders, since almost all of the photoemission signal stems from the two highsymmetry lines, Fig. 4d is constructed by performing rotational symmetry of the two linecuts along the two highsymmetry directions (ΓK and ΓM).
Excellent agreement between theory and experiment is observed for the single crystal case, as seen from Fig. 4a, c. For the azimuthally disordered monolayer (Fig. 4b), the ARPES spectra sliced at different E_{B} exhibit no appreciable dependence on the azimuthal parameter φ. Overall, the simulated energy maps in Fig. 4d have the same circular shape as the ARPES spectra sliced at the corresponding E_{B} values in Fig. 4b. Illustrative is a comparison of the two bottommost plots in Fig. 4a, b, where a clear difference in the intensity distribution is observed, which underscores the impact of an integration along the φ direction. Likewise, the comparison of Fig. 4b, d clearly demonstrates that the measured ARPES spectra dominantly consist of the two highsymmetry electronic band dispersion lines, which is in excellent agreement with the predictions above.
Electronic band dispersion of different TMDCs
Finally, we demonstrate that the symmetry induced sharp electronic band dispersion can as well be observed experimentally for other TMDC monolayer 2D powders, as shown in Fig. 5 for (a) WSe_{2}, (b) WS_{2}, (c) MoSe_{2}, and (d) MoS_{2} on HOPG. The measured spectra are overlaid with the calculated electronic band structure of the two highsymmetry directions (ΓKM and ΓM Γ); the agreement is very satisfactory. In addition, also the electronic structure of the HOPG substrate can be discerned in the spectra of Fig. 5, which is known to be nearly linear around the Fermi energy at the K point (green circles, 1.70 Å^{−1})^{21}. To facilitate a clear discrimination between spectral contributions from the HOPG substrate and the TMDC monolayers, the spectra of bare HOPG, MoS_{2} on sapphire (without HOPG contributions), and MoS_{2} on HOPG are displayed in the Supplementary Note 3. Consequently, it should be possible to determine the electronic band structure of various TMDCs independently of the substrate.
Discussion
We evidence that wellresolved electronic band dispersion along two highsymmetry directions of the BZ can be determined in ARPES for azimuthally disordered TMDC monolayer samples. By comparing data of single crystalline and 2D powder monolayers, we derive, with the help of calculated band structures, that two highsymmetry directions (ΓK and ΓM) dominate the intensity in ARPES spectra. In addition, we demonstrate that the materials’ dihedral group symmetry enables ARPES measurements in azimuthally disordered monolayer TMDCs. The insight provided here constitutes a solid base for investigating the electronic band dispersion of 2D powders featuring appropriate φ direction energy dispersion and lifts the restrictions of finding a sufficiently large single crystal.
Methods
Angleresolved photoemission
The spectra of single crystalline and azimuthally disordered WSe_{2} monolayers were measured at the beamline PM4 (BESSY II, Germany), the beamline BL7U (UVSOR, Japan) and the Humboldt Universität zu Berlin using Scienta DA30 analyzer, employing 100 eV, 21 eV, and 21.22 eV (He I) photon energy, respectively^{24}. The resolution determination and energy calibration of the instruments were done by measuring the Fermi edge of a clean Au sample. The total energy resolution was 110 meV, 100 meV, and 105 meV for the beamline PM4, beamline BL7U, and Scienta DA30, respectively.
Density functional theory calculations
Density functional theory (DFT) calculations were performed for a freestanding TMDC monolayer using the Vienna ab initio simulation package (VASP) with the PerdewBurkeErnzerhof (PBE) functional including spinorbit coupling implemented in the model^{25,26,27,28}. The electronic iteration convergence condition was 1 × 10^{−6} eV, and a 11 × 11 × 3 Γ centered Kpoint mesh was used with an energy cutoff of 500 eV. The lattice constants of MoS_{2}, MoSe_{2}, WS_{2}, and WSe_{2} monolayers were 3.18 Å, 3.32 Å, 3.19 Å, and 3.32 Å, respectively, corresponding to the relaxed lattices after structure optimization. The use of a freestanding TMDC and PBE functional might lead to small discrepancies when comparing calculations and experiments.
Sample preparation
Monolayer TMDCs were fabricated by chemical vapor deposition on sapphire substrates and were transferred to the HOPG substrate using poly (methyl methacrylate) (PMMA)^{29,30}. The samples were annealed overnight at 350 °C in an ultrahigh vacuum chamber (10^{−9} mbar) to remove contaminants and PMMA residue before ARPES measurements.
Data availability
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by the DFG (SFB951 and AM 419/11) and the Alexander von HumboldtStiftung. We thank the HZB and the IMS for allocating synchrotron radiation beam time (Bessy ΙΙ, PM4 and UVSOR, BL7U). We are grateful to Prof. Torsten Fritz for insightful discussions.
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S.P., T.S., and P.A. wrote the manuscript and derived equations with the help of M.M. and S.K. under the supervision of N.K.; S.P., T.S., P.A., X.X., R.O., P.B., T.Y. and A.O. measured and analyzed ARPES spectra of the TMDCs. A.A., A.H. and L.J.L. prepared 2D TMDCs samples. All authors commented on the manuscript.
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Park, S., Schultz, T., Han, A. et al. Electronic band dispersion determination in azimuthally disordered transitionmetal dichalcogenide monolayers. Commun Phys 2, 68 (2019). https://doi.org/10.1038/s4200501901660
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