Electronic band dispersion determination in azimuthally disordered transition-metal dichalcogenide monolayers

Generally, the lack of long-range order in materials prevents from experimentally addressing their electronic band dispersion by angle-resolved 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 Γ-K and Γ-M. 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. The complex nature of polycrystalline materials mean that characteristics such as their electronic band structure can be more challenging to interpret than their single crystal counterparts. Here, the authors present a framework based on angle-resolved photoemission spectroscopy to reveal the band dispersions for azimuthally disordered transition metal dichalcogenide polycrystalline monolayers.

T ransition-metal dichalcogenide (TMDC) monolayers have gained attention as semiconductors for next-generation nanoscale electronic devices owing to their unique electronic properties that originate from their low-dimensionality and high-symmetry 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, angle-resolved 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 two-dimensional (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 1D-DOS) 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 angle-dependent DOS 21 , should be challenged. According to the quasi 1D-DOS 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 1D-DOS 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 1D-DOS and Van Hove singularities.
In this contribution, we evidence high-symmetry 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 high-symmetry directions [Γ-K and Γ-M 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 high-symmetry directions and high-symmetry 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 high-symmetry directions Γ-K-M (path 1) and Γ-M-Γ (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 high-symmetry directions.
Calculation of photoelectron intensity. The measured photoelectron intensity I E; k r ð Þ 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 E; k r ð Þ. Given a sufficiently large N with random orientation, we can assume that an angular (φ) interval of Δφ ¼ 2π=N represents one flake. For each interval Δφ, the corresponding absolute energy change ΔE j j due to dispersion along φ can be calculated based on its derivative and is given by: Þjis the partial derivative of the energy with respect to φ direction. Using the above relation, we can count the N E; k r ; φ ð Þthat contribute to a certain energy interval δE (e.g., the experimental energy resolution): Due to the azimuthal disorder, I E; k r ð Þ is obtained by summing over all φ directions, leading to: From Eq. (3) it becomes clear that I E; k r ð Þis inversely proportional to ∇ φ Eðk r ; φÞ . This implies that the intensity measured in ARPES at a certain (E, k r ) is high for ∇ φ Eðk r ; φÞ ! 0, because this corresponds to a large number of flakes contributing to the same E (within the range of E ± δE=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 high-symmetry 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 two-dimensional 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 high-symmetry directions for TMDCs, energy maps of the valence band (VB) and 1/ ∇ φ E VB ðk r ; φÞ 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 2H-phase classified by its 3-fold dihedral symmetry group (D 3h ). Figure 2e-h show the corresponding 1= ∇ φ E VB ðk r ; φÞ 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 high-symmetry directions (Γ-K and Γ-M). Thus, the band structure along these inevitably has local maxima and minima, where ∇ φ Eðk r ; φÞ tends towards zero, returning high values of 1= ∇ φ E VB ðk r ; φÞ as clearly seen in Fig. 2e-h. High values of 1= ∇ φ E VB ðk r ; φÞ are mostly found in the colored areas that form "lines" along Γ-K and Γ-M. Therefore, only these two directions contribute notably to I E; k r ð Þwhere N E; k r ð Þis counted along the φ direction using Eq. (3). Consequently, while information on non-high-symmetry directions is missing (or appears with small intensity in ARPES), the high values of 1= ∇ φ E VB ðk r ; φÞ allow measuring the electronic band dispersion in the two high-symmetry 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 E; k r ð Þ with our measured spectra, we first calculate I E; k r ð Þfor selected k r values (see Fig. 3a with color-coded 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 E; k r ð Þ 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= ∇ φ E VB ðk r ; φÞ 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 above-mentioned arguments about azimuthal integration and the observation of the band structure along the (single-crystal) high-symmetry 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 φ-angular 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 high-symmetry lines, Fig. 4d is constructed by performing rotational symmetry of the two line-cuts along the two highsymmetry directions (Γ-K and Γ-M). 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  Energy map . d Experimental angle-resolved photoemission spectroscopy (ARPES) spectrum (hν = 21 eV) of azimuthally disordered WSe 2 on highly oriented pyrolytic graphite as a function of k r together with high intensity points from c. The calculated electronic bands are rigidly shifted in energy to match the measured spectra at Γ 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 high-symmetry directions (Γ-K-M and Γ-M-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 well-resolved electronic band dispersion along two high-symmetry 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 high-symmetry directions (Γ-K and Γ-M) 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.  ky (Å -1 )

Methods
Angle-resolved 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 Universitaẗ 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 free-standing TMDC monolayer using the Vienna ab initio simulation package (VASP) with the Perdew-Burke-Ernzerhof (PBE) functional including spin-orbit 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 free-standing 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 ultra-high 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.