Electronic band structure of rhenium dichalcogenides

The band structures of bulk transition metal dichalcogenides ReS2 and ReSe2 are presented, showing the complicated nature of the interband transitions in these materials, with several close-lying band gaps. Three-dimensional plots of constant energy surfaces in the Brillouin zone at energies near the band extrema are used to show that the valence band maximum and conduction band minimum may not be located at special high symmetry points. We find that both materials are indirect gap materials and that one must be careful to consider the whole Brillouin zone volume in addressing this question.


INTRODUCTION
The field of two-dimensional few-layer and monolayer materials has generated intense and sustained interest since the isolation of graphene in 2004 [1], though bulk (three-dimensional) van der Waals layered materials and, in particular, the transition metal dichalcogenides (TMDs), were known and studied much earlier [2]. In the last few years, many more two-dimensional materials have been identified or proposed and there are currently around 1000 candidates for two-dimensional metals, semiconductors, superconductors and charge density wave materials. Besides graphene, the TMDs are still the most actively studied members of this family and many prototype devices (e.g, field effect transistors, sensors, and photo-detectors) based on MoS 2 have been demonstrated successfully, as well as device paradigms being proposed (e.g, spin-and valleytronics) based on the band structure of mono-or few-layer MoS 2 [3][4].
Although MoS 2 is arguably the archetypal semiconducting TMD, there are many TMDs that offer contrasting properties to MoS 2 and therefore add significantly to the diversity of TMD devices and heterostructures that can be explored [2]. The rhenium chalcogenides ReS 2 and ReSe 2 are prime examples of this [5] since they differ markedly from MoS 2 in crystal structure and symmetry [6], electronic band structure [7], and lattice dynamics [8]. The crystal structure of these compounds is shown in Figure 1, based on early crystal structure determinations via X-ray diffraction [6] and optimisation of the structures via first-principles calculations as described in the Methods section. In the ReX 2 structure (X=S, Se) Re atoms group into diamond-shaped clusters of 4 atoms and each cluster is linked to the next one by Re-Re bonds to form chains running along the crystallographic axis direction we define as a [9]. The resulting structure has only inversion symmetry and is highly anisotropic in all physical properties in-plane, in contrast to the hexagonal TMDs.
The stacking of layers is such that the c axis is not perpendicular to the layer plane but a and b do lie in the layer plane, so that the reciprocal space c* axis is normal to the real space layers and thus also normal to the principal surface of bulk crystals; a* and b* lie out of the layer plane. Thus, angle-resolved photoemission experiments (ARPES), which preserve information about electron momentum normal to the sample surface, will probe the valence band dispersion in an approximately planar section through the Brillouin zone that does not contain any of the reciprocal lattice vectors (this is well illustrated in Ref. [10], Figure 2).
Recently, we have presented experimental data on the electronic valence band structures of ReS 2 [11] and ReSe 2 [12] as determined by nanoscale ARPES, and closely related work has been reported also by other groups, on both bulk ReS 2 [13] and also few-layer ReS 2 [10]. This work addresses controversies over (i) the nature of the bandgaps in the ReX 2 family and (ii) the degree to which the ReX 2 bulk materials can be considered as noninteracting stacked layers. It has become clear that the details of a particular first-principles calculation influence the conclusions about the existence of a direct gap and the locations of the conduction and valence band extrema; these materials present a very flat band structure with a large number of bands in a small energy range (due to the large 12-atom unit cell) and, as we shall show, the band extrema are not necessarily located at any high symmetry points of the Brillouin zone (BZ). Calculations typically focus on high symmetry paths in the BZ and may miss the true band edges; here, we take a different approach, calculating the band energies over the whole volume of the BZ and tracing out constant energy surfaces rather than dispersions. In particular, we present data for the conduction band of bulk ReSe 2 , which has not previously been discussed in detail.

RESULTS AND DISCUSSION
No direct comparison of the band structures of ReSe 2 and ReS 2 over the whole Brillouin zone (BZ) has yet been presented and so we provide such a comparison here. This is useful as a guide to the electronic properties of these materials but also gives insight into the difficulties in making first-principles calculations of band structure in this system; our results show that it is necessary to consider the whole BZ and not to focus only on high-symmetry paths.
To make this comparison, we use consistently two types of pseudopotential; although more sophisticated calculations of band structure can be performed, and the exact positions of the band extrema depend on the choice of pseudopotential, the above comments will apply at any level of approximation.

Rhenium sulphide
We consider first ReS 2 , being the more well-studied of the two materials, and the material with the wider bandgap. It should be noted that here we take the bulk unit cell to contain just one monolayer and 4 formula units; this contradicts the conclusions of some early XRD studies of ReS 2 [6] but is consistent with more recent XRD [21] and photon-energy dependent ARPES studies [10,13,20]; the latter technique probes the valence band dispersion in the direction normal to the sample surface (which is here the layer plane) and is thus sensitive to the lattice periodicity in that direction. In both cases, the smaller (red) surface encloses the band extremum and the larger (yellow) surface shows how the bands develop at energies further from the band extremum. To obtain these figures, the band energies were calculated for a grid of points covering the whole BZ. This process is not usually required in more high-symmetry materials and can be computationally expensive; each of our datasets required 24 hours with 64 processors, which is acceptable, but there is a cubic scaling with k -space resolution which makes the use of a very fine grid impractical. Therefore, we do not attempt to extract band dispersions from the same dataset, and we are only able to locate the band extrema to the resolution of our k -point grid. However, we propose that presenting the band structure in this way is useful, since this process searches for the band extrema without prejudging where they are located. As Fig. 2 makes clear, this calculation implies that both the VBM and CBM are not located at Γ or at any particular high-symmetry point on the BZ boundary.
Most reported calculations of the band structure so far have presented the dispersion only along a special set of directions. In some cases, these have been chosen connecting theΓ, K andM points of a hypothetical two-dimensional BZ; however,K andM are not special points in the three-dimensional BZ, but are projections of the latter onto the layer plane [7, 5 22-25] (and so cannot be represented on Figure 2). In other cases, paths have been chosen in the 3D BZ connecting three-dimensional special points but not exploring the whole BZ [26][27]. The 2D projection onto the layer plane is very useful when considering how the band structure evolves from bulk to monolayer materials and is also necessary when modelling ARPES experiments, in which a 2D section through the 3D BZ is measured.
However, by discarding the c* component of the wavevector, it is impossible to determine whether the band extrema occur at the same crystal momentum values.
The results of Figure 2 suggest that the lowest-energy inter-band transition is indirect in the bulk material, since the VBM and CBM do not coincide. This question has been actively discussed recently on the basis of optical, ARPES and transport measurements, and is not yet resolved. Briefly, for bulk ReS 2 , electron energy loss spectroscopy (EELS) results gave a room-temperature direct gap of 1.42 eV [24] whilst optical absorption-edge measurements suggested indirect gaps of 1.35 and 1.38 eV (for polarisations parallel and perpendicular respectively to the Re chains) [28] and electroluminescence data suggested an indirect gap of 1.41 eV [29]. Recently, a combined photoacoustic and modulated reflectance spectroscopy study concluded that both ReS 2 and ReSe 2 have indirect bandgaps of 1.37 and 1.18 eV respectively and higher-lying direct gaps of 1.55 and 1.31 eV [30].
Only one ARPES study has reported observation of the CB states via rubidium doping, and found at 10-20 K a direct gap located at the BZ boundary along the c* direction though with a substantially lower magnitude, of around 1.2 eV [13], assumed to be due to bandgap renormalisation at the high doping levels required. This special point in the BZ, labelled Z [13,20] (as in Fig. 2)  We note that our calculations are not in very good agreement with two other DFT calculations for bulk or many-layer ReS 2 which report a (T=0 K) direct gap of 1.35 eV at theΓ point [7,22], though we do find, in agreement with [13] that, along the c* direction, the VB has a local maximum at the Z point.This is most clearly seen in Fig. 2(b) and was reported earlier, where the direct gap at Z was found to be 1.525 eV [20]. It is also generally found that the VB at the true Γ point shows a local minimum, giving an M -shaped dispersion, as shown by the bifurcation of the constant energy surface in Figure 2 It is clear from Fig. 2(a) and (c) that the CB minimum in this level of approximation does not appear at Z but is indeed located near the plane containing Z and is displaced towards the 3D special point 5. This is in qualitative agreement with the calculations of [10] (their Figure 4d) and is potentially compatible with the ARPES results [13].
Rhenium selenide Fig. 3 shows the equivalent constant energy surfaces for ReSe 2 and, as might be expected from the chemical similarity between S and Se, the overall structure is similar to that of ReS 2 . Focusing first on the results obtained via the LDA, Fig. 3 (a) and (b), (i) we find the conduction band minimum again lies within the volume of the BZ (rather than at the surface or at Γ) and is closer to the plane containing Z than to that containing Γ, and (ii) the valence band maxima are offset either side of the Γ point. We have discussed ARPES data for ReSe 2 in detail elsewhere and reported already the form of one constant energy surface (yellow) for its VB [12] but we show this again in Fig. 3 (b) for comparison with the CB structure. We have also expanded Fig. 3 (b) to include a second constant energy surface (red) closer to the calculated VBM. Just as for ReS 2 , we find that the three-dimensional Γ point is a local minimum in energy with a bifurcation of the constant energy surfaces around it. This is borne out by the experimental ARPES data [12] which shows that the VBM is displaced in the layer plane away from the projection of Γ and agrees also with other DFT calculations (Fig. S1 of the Supplementary material of ref. [23]). Turning to our results obtained in the GGA, we find that the same remarks apply, with one exception; in the GGA, two VB maxima appear, one in the same location as for the LDA results and one very close to Z. This does not, however, affect the conclusion that ReSe 2 is indirect, The recent photoacoustic spectroscopy study gives an indirect gap of 1.18 eV and direct gap of 1.31 eV [30]. However, both computational studies only report band dispersions along specific paths in the BZ and it would be interesting in future to apply these more sophisticated models to the whole BZ.
The findings of the present calculations are summarised in Figure 4 This accounts for the left hand halves of Fig. 4(a-d); in the right hand halves, we then follow the same procedure, from Γ to the VB maximum, on to the BZ boundary (VBM') and then across the BZ surface to Z and finally back to Γ. This shows clearly that the VBM and CBM do not coincide so that, considering first ReS 2 , Fig. 2 (a) and (b), the material is formally an indirect semiconductor, but that there is a gap at Z which is formed between a local VB maximum and a very flat CB minimum and is therefore a slightly larger direct gap. In the region of the VBM, it appears that there is also a close-lying local minimum in the CB, providing another nearly direct gap. It is hardly surprising, therefore, that measurements of optical reflection or absorption have difficulty in identifying the nature of the band gap.
Similar conclusions follow from the analogous data for ReSe 2 shown in Fig. 4 (c) and (d). demanding since direct and indirect transitions lie close in energy and within the range of DFT results arising from different choices of pseudopotential. In the case of ReSe 2 , computational and experimental results are in better agreement though, it should be noted, there is less experimental data available; the current consensus is that bulk ReSe 2 is an indirect semiconductor.