Polytypism of MoS2
ETH Zürich, Swiss Federal Institute of Technology, Department of Chemistry and Applied Biosciences, Institute of Inorganic Chemistry,
Vladimir-Prelog-Weg 8093 ,1 Zürich, Switzerland
The polytypism of molybdenite, MoS2, has been reported as early as the 20s when Dickinson and Pauling  investigated the naturally occurring crystal structure and mentioned the different symmetry exhibited by the as-synthesized compound. In 1957 Bell and Herfert  clarified that the synthesized MoS2 “showed a threefold symmetry, instead of the six-fold symmetry characteristic of the naturally occurring molybdenum disulfide”. They suggested a CdCl2-type structure, which has the space group R-3m (IT No 166). Actually, as Jellinek pointed out , the X-ray diffraction (XRD) data were better compatible with the rhombohedral space group R3m (IT No 160). The rhombohedral modification was then reported by Traill  from a natural occurrence at the Con mine, Yellowknife, Canada. Later, Takéuchi and Nowacki  reported that a rhombohedral modification of molybdenite was found in Binnatal, Switzerland. As Bell and Herfert suggested , other modifications, differing only in the stacking sequence, might exists depending upon the mode of preparation. The polytypism of MoS2 was theoretically studied by Wickman and Smith , who derived 112 theoretically possible polytypes with less than seven layers. They applied systematically a simple concept, introduced a few of years earlier [5, 7], that by using the two stacking operations, screw and translation, it is possible to derive different polytypes, which are described by using Ramsdell’s notation . At the same time, Frondel and Wickman  described the 108 specimens of naturally-occurring MoS2. Up to then, there was no mention of the 1T-MoS2 modification. It was first reported, to our best knowledge, in 1992 by Wypych and Schöllhorn  as “a new metastable metallic MoS2 polytype with a distorted layer structure and octahedral coordination of the metal atoms”. Unfortunately, they did not suggest any space group even though they reported the XRD pattern and the DTA curve showing the transition from 1T-MoS2 to a disordered 2H-MoS2. But they suggested that the “relation between the unit cells of 1T-MoS2 and 2H-MoS2is a1T ≅ a2H √3,c1T = ½ c2H”. In a later paper, Wypych et alt.  investigated the 1T-MoS2 via scanning tunneling microscopy (STM) and suggested that the external factor can affect the degree of distortion within the [MoS6] octahedra. Actually, it was already known  that one of the chemical methodologies to exfoliate MoS2 was by lithium inter- calation. The resulting product upon exfoliation and flocculation was, as later proposed by Dungey at al. , the 1T-MoS2, with P3 symmetry. The restacked MoS2 aroused a lot of confusion about its exact symmetry representation whether it would be represented in the trigonal or orthorhombic systems . The growing number of publications and monographs amply demonstrates the variety of structural and chemical properties exhibited by MoS2 , from bulk to monolayer, suitable for a variety of applications. The references cited in the present work are clearly far from being exhaustive and complete. Although a recent renewed and growing interest in MoS2 , a systematic computational study of the possible polytypes is, to our best knowledge, still missing. The correspondence between the polytypes suggested by Wickman  and the naturally-occurring ones reported by Frondel  is not clearly defined. In the present work, we studied the polytypism of MoS2 via first-principles calculations, by combining the crystal structure prediction method, AIRSS, and total energy calculations, CASTEP, as described in the section below. The methodology has been successfully applied to other chemical systems, e.g. [16–18]. The crystal structure prediction (CSP) of solid state compounds from first-principles implies that no other information about
the compounds except the knowledge of the chemistry of their constituent atoms is used as only input. That means to establish a priori how the atoms or molecules arrange to form the crystalline phase. Therefore, the problem reduces to search for crystalline structures of compounds either with variable composition,when the atomic ratio is allowed to vary, or with fixed atomic ratio, in the case of polymorphism. The polytypism of MoS2 falls in the latter situation, being the polytypism a special case of polymorphism. The flourishing of methodologies to tackle the problem is not new and the debate about which one is most likely to succeed in achieving its goal dates back to several decades . Particularly for compounds of industrial interest, the polymorphism has a paramount importance for the different physicochemical properties exhibited by different polymorphs. Along the decades, several approaches and methodologies have been developed and tuned to solve a particular class of compounds, e.g. molecular crystals, ionic crystals, alloys. After a decade of theoretical assessment in the 90’s, the subsequent decade in 2000’s was witnessed to a prolific implementation of algorithms and techniques aimed at laying the machinery of the CSP problem. Since then and up to date an increasing need for validation criteria are going to be established in order to define the rule by which select the putative crystal structures. The exhaustive list of methodologies would be too long, but we can safely group the different methodologies in (i) global optimization methods, which include simulated annealing, genetic and evolutionary algorithms; (ii) metadynamics; (iii) data mining [20– 22].
We investigated the phase space of Mo-S system, at the point of atomic ratio Mo: S equal to 1:2, via first-principles crystal structure prediction approach. In particular, we combined the random searching method, AIRSS  and the first principles total energy calculations, based on density functional theory as implemented in CASTEP . This is a general approach to search for possible polymorphs at any composition in the phase space. In the particular case of the polytypism of MoS2, the atomic ratio 1:2 was accounted as the unique constraint in the AIRSS calculations. In the present work, we used two formula units in the AIRSS calculations. The entire ensemble of structures modeled via AIRSS was then consistently optimized by using CASTEP , as implemented in Materials Studio 6.0 [25–31]. The periodic lattice structures were optimized by relaxing the atomic positions and the lattice parameters, without any symmetry constraints. In CASTEP we used norm-conserving pseudopotentials and reciprocal space representation forall atoms. The valence shells of the atoms, Mo and S, contain 4d5 5s1 and 3s2 3p4 electrons, respectively. The Brillouin zone was sampled by using a fine mesh commensurate to the specific lattice dimensions, but with the actual spacing below 0.03Å−1. The energy threshold, the maximum atomic displacement, the maximum atomic force and the lattice stress were set to 0.001 meV/atom, 0.001 Å, 0.001 eV/ Å and 0.002 GPa, respectively. We used the Perdew-Burke-Ernzerhof96 (PBE) and thegeneralized gradient form (GGA) of the exchange-correlation fun ctional. Being the dispersion term one of the important long-range contributions to the total energy, noncovalent forces and in particular van der Waals interactions, were introduced by using damped atom-pairwise dispersion corrections of the form C6 R-6, proposed by Grimme [32, 33], as implemented in CASTEP . We used the Reflex powder diffraction module, in Materials Studio 6.0, to simulate the XRD patterns of the optimized structures. In the simulations, we used the X-ray radiationfor αCu source with λ1= 1.5405 Å. We ascertained the lattice stability via phonon calculations in a quasi-harmonic approximation. In particular, we used the finite displacement method, as implemented in CASTEP, to calculate the vibrational frequencies and eigenvectors at a selected set of q-vectors. For finite displacement calculations, we chose an appropriate cutoff radius, to be used to construct the supercell, dependent on the actual unit cell size, but at least double of the cell size used in the geometry optimization. We adjusted the density of q-vectors along the high symmetry directions in order to have a high quality dispersion curves, by using the actual spacing of the Monkhorst-Pack grid commensurate to the grid used to sample the Brillouin zone. The results of the phonon calculations were used to compute the thermodynamic properties, in particular the entropy, S and the lattice heat capacity, Cv, as a function of temperature. From the thermodynamic relations ti is possible to evaluate the thermal contribution to the enthalpy and the entropy of a compound. At p = 0, Cv= Cp.
The lattice contribution to the heat capacity is:
The infrared spectra of the semiconductor MoS2 structures were calculated in terms of the dynamical matrix and the Born effective charges within the linear response formalism, as implemented in CASTEP. The infrared intensities were obtained from the atomic polar tensors A of all atoms, being , where qi and μi are the Cartesian coordinates and the dipole moment of i-th atom. moments and expressed in terms of the A matrix and the eigenvectors of the mass-weighted Hessian, F, being
Thus Ii= Σi,j(Fi,j Aj,k )2 . The charts display the infrared intensities as a function of the wave numbers, cm−1, and the Lorentzian broadening is applied to the eigenvalues obtained from the CASTEP phonon calculations.
The symbolic description of polytypes was introduced by Ramsdell  for SiC. The chemical formula is preceded by a number and a capital letter, which represent the number of layers in the stacking sequence and the crystal system of the lattice. The notation was then extended by Gard  and modified by Kato et al. as reported in the Report of the I.M.A.I.U.Cr. Joint Committee on Nomenclature[36, 37]. In the following, we use the modified Gard notation. The stacking sequence is described by using the capital letter for sulfur atoms and the low case letter for molybdenum. In our calculations, out of the 280 structures, modeled by combining AIRSS and CASTEP, themost recurring are represented in the trigonal system, 1T, and 3T; orthorhombic system, 2O ; monoclinic, 1M; rhombohedral, 3R and 6R; and tetragonal, 2Q. The polytypes exhibit either a metallic or semiconductor band structures. The band gap, although underestimated, is in the range 0.928-1.007 eV. In Table. 1 we report the calculated enthalpy of formation and the structured data of the different polytypes. The reference states in our calculations are the body-centered cubic structure of molybdenum and the orthorhombic structure of sulfur, α-S, space group Fddd (IT No 70). The standard enthalpy of formation of MoS2 reported in the Lange’s Handbook of Chemistry (Table 6.3 section 6)  is -235.1 kJ/mol, but which polytypes it refers to is not specified.
In our calculations, the semiconductor polytypes are the lowest energy structures, in agreement with the literature data. Interestingly, the crystal structure prediction calculations suggested that 3R-polytype crystal structure, space group R3m (IT No 160), is the lowest energy structure. Actually, this is the rhombohedral lattice representation of the trigonal 3T-polytype with the hexagonal lattice. In fact, the 3R– and the 3T-polytype structures share the same stacking sequence, AbA BcB CaC, but the 3T-structure exhibits a small instability along Γ(0,0,0) →A(0,0,0.5) path. This is the antiparallel mode along the crystallographic c-direction of the sulfur atoms laying on the AbA and the BcB layers. The detailed crystal structure of the 3R-MoS2 was reported by Takéuchi et alt.  with lattice constants: a = 3.166 Å and c = 18.41 Å. The two three-layer structures, 3R, and 3T are iso-energetic, up to room temperature, with the 2H-polytype structure, space group P63/ mmc (IT No 194), which was optimized starting with the reference structure . The 3R-polytype converts into the 2H-polytype by sliding the second layer respect to the first in order to make the second-layer Mo atoms to overlap with the first-layer S atoms. This leads to the stacking AbA BaB, present in the 2H-polytype. In Figure. 1 and Figure. 2, the comparison of the 2H and the 3Rpolytypes shows that a sliding of one layer to the other followed by a rotation along the principal axes brings the 2H structure in the 3Rone. The almost zero energy cost of this sliding motion explains the good lubricant properties of MoS2. The two polytype, 2H and 3R, differ in the XRD pattern. Additional diffraction peaks are present in the XRD pattern of the 3R-polytype, because of the reduced symmetry, as shown in Figure 5. They differ in the IR spectra, as shown in Figure 6. The anti-parallel mode at 407 cm−1 in the 3R-polytype is absent in the 2H-polytype. A 3R-structure was also reported by Traill  as the new rhombohedral polytype of MoS2 and then confirmed by Graeser . In addition, we found a 2O-polytype represented in the space group Fmm2 (IT No 42) and almost iso-energetic to the 2H-polytype.
The orthonormal lattice angles make the atoms on the second layer to slide laterally, for which the stacking sequence becomes AbA CdC instead of AbA BaB in the 2H-polytype. This 2O-structure exhibits vibrational lattice stability, being the vibrational modes all real, but a translational instability due to the sliding motion of one layer to the other. The first three translational modes are imaginary with the wave number of -48.2 cm−1, -41.7 cm−1, -7.3 cm−1.
The lowest energy structure of the metallic polytypes is the 1T-structure represented in the space group P3m1 (IT No 156) with AbA stacking sequence.
This is the optimized structure of the 1T-MoS2 represented by Dungey et al.  in the space group P3 (IT No 143). In addition, we found a monoclinic, 1M, space group P21/m (IT No 11) and an orthorhombic, 2O, space group Amm2 (IT No 38), polytypes with metallic band structures and lattice stability, but higher in energy than the trigonal 1T by 25.25 kJ/mol and 75.94 kJ/mol, respectively. Of the metallic polytypes, the 1T, space group P-3m1 (IT No 164) and the 6R, space group R-3m (IT No 166) exhibit translational lattice instability.
The unstable 1T structure differs from the 1T stable for the inversion operation, for which the lattice parameters a, b of P-3m1 are half of those of P3m1. The 6R, rhombohedral structure with 6-layer stacking sequence, AbC AcB CaB CbA BcA BaC shows large lattice instabilities non only along the Γ(0,0,0) →A(0,0,0.5) path, which is along the stacking direction, but also along A(0,0,0.5)→ H(0.333,0.667,0.5) path, on the plane perpendicular to the stacking direction. Interestingly, Wickman et al.  already reported, in their theoretical
Table 1. The calculated structure data of the different polytypes of MoS2 modelled by combining AIRSS and CASTEP calculations. The modified Gard notation is used for the stacking sequence. The stacking sequence is described by the capital letter, for sulfur atoms, and the low case letter for molybdenum. The structures are identified whether stable or unstable according to the corresponding phonon dispersion results. The unstable structures exhibited the translational instability, but real modes of vibration, except the 2Q-structure, which has two imaginary modes. For comparison, the lattice parameters of the 2H, 3R and 1Tstructures, reported in the literature, are shown in square brackets. For the 2H and the 3R structures the lattice parameters are the mean values of those reported in the Inorganic Crystal Structure Database (ICSD). The reference of the 1T structure is the P3structure .
Figure 1. A topview of the 2H– and 3R-polytypes. The translation of 1/3 along the a-direction followed by a rotation of 60 degrees along the normal to the plane of the AbA layer in 2H-structure changes the stacking sequence from AbA BaB in the 2H-polytype to AbA BcB CaC in the 3R-polytype.
Figure 2. A 3D-view of the 2H– and 3R-polytpes. (a) 2H-MoS2. The average bond length of Mo-S is 2.433 Å, the Mo-S coordination is a trigonal prism, with null distortion index; the volume of the polyhedra is 14.077 Å3. (b) 3R-MoS2. The average bond length of Mo-S is 2.433 Å, the Mo-S coordination is a slightly distorted trigonal prism, with distortion index of 0.0002 in bond lengths; the volume of the polyhedra is 14.084 Å3.
work, the 1T, space group P3m1, and the 6R, space group R-3m. The tetragonal 2Q, space group I4/mmm (IT No 139) has one imaginary vibrational mode, the antiparallel motion of the sulfur atoms on the adjacent slabs with the wave number of -155.03 cm−1. The results suggest that the transition from the stable, low-energy 1T-P3m1 to the high energy 2O-Amm2 polytype can occur via the tetragonal structure, 2Q-I4/mmm. These three polytypes differ in the coordination of the Mo atoms: octahedral in the 1T and cubic in the 2Q structures. In fact, the dual of the octahedron is a cube and both have the Oh symmetry point group.
The octahedral coordination in 1T can become cubic as found in the 2Q-structure. The reorientation of one of the two slabs of sulfur atoms by a rotation of 45 degrees along the normal to the AbC layer, makes them stack above each other. The resulting polyhedra are a face-sharing cube, as found in the 2Q-structure. But this induces translational instability because of the relatively short S-S distance, 2.968 Å, compared to3.278 Å in 1T-structure. The instability is released by sliding laterally the sulfur slab by 1/2 of the S-S distance, as shown in Figure 3.
The resultant structure has still orthonormal lattice angles and is represented in the orthorhombic system. This is the 2O-polytype, space group Amm2.
Figure 3. The polyhedra of coordination in (a) 1T-P3m1, (b) 2Q-I4/mmm, (c) 2O-Amm2 polytypes. The sulfur atoms (in yellow) depicted with black dots are representative of those involved in the reorientation.
Figure 4. A 3D-view of the 1T-, 1M-, 2O-polytypes. (a) 1T-MoS2. The average bond length of Mo-S is 2.457 Å, with three shorter Mo-S bond lengths of 2.400 Å and three longer, of 2.513 Å; the Mo-S coordination is a distorted octahedra with distortion index of 0.024 in bond lengths; the volume of the polyhedra is 19.055 Å3. (b) 1M-MoS2. The average bond length of Mo-S is 2.423 Å, the Mo-S coordination is an edge-sharing distorted squared-based pyramid with distortion index of 0.006 in bond lengths; the volume of the polyhedra is 10.251 Å3. (c) 2O-MoS2. The average bond length of Mo-S is 2.456 Å, the coordination of Mo-S is a distorted, face-sharing trigonal prism, with distortion index of 0.029 in bond lengths; the volume of the polyhedra is 14.491 Å3.
The different stacking sequence, which manifests in the different electronic properties – semiconductor versus metallic band structure –is caused by the different coordination of molybdenum atoms to sulfur atoms. In fact, the optimized structures, which also exhibit structural stability, with coordination different from the trigonal prism show a metallic band structure. They are the 1T and 1M, AbC stacking sequence and space group P3m1 and P21/m, respectively; and the 2O with stacking sequence AbC CdC, space group Amm2. In Figure. 4 we show a 3D-view of the three different polytypes.
The calculated IR spectra of the 2H-, 3R-, 3T- and 2O-polytypes are shown in Figure 6. The fundamental aspect which differentiates the 2H-semiconductor polytype from the 1T-metal polytype is the stacking of the ideal slabs of the sulfur atoms.
In the 2H-structure they stack directly above each other, making a trigonal prismatic hole for the Mo atoms. In the 1T structure, the slabs stagger and form octahedral holes for Mo atoms. In Figure.5 we report the calculated XRD patterns of selected polytypes.
Figure 5. The calculated XRD patterns of the semiconductors polytypes, 2H and 3R and the metallic polytypes 1T, 2O, 1M. The number in brackets is the IT No of the space group.
Figure 6. The calculated IR spectra of the semiconductor polytypes. The number in the brackets is the IT No of the space group.
Figure 7. The slabs of the sulfur atoms (in yellow) stack above each other in the 2H-structure and form a trigonal prism. If one slab (the second one from the bottom) is staggered, the sulfur atoms form an octahedron, as in the 1T-structure.
The AA stacking sequence of the slabs of the sulfur atoms in the 2H-structure gives rise to the energy gap, while the staggered sequence in the 1T-structure enables the lateral Mo-Mo interactions which cross the Fermi level and thus close the energy gap. Figure. 7 shows the trigonal prism and the octahedron formed by the overlaid and staggered sulfur slabs, respectively. A study of the trigonal-prismatic versus the octahedral coordination in transition metal dichalcogenides was reported by Hoffmann and co-workers  by using the rigid band model.
In Figure. 8 we compare the band structures of the 2H-polytype, semiconductor, and 1T-polytype, metallic, The two structures differ by 61.27 kJ/mol at T = 0 K, without including the zero-point energy. At T = 298 K the difference of the free energy of formation, calculated by including the zero-point energy and the thermal contribution via phonon calculations is 75.42 kJ/mol.
Figure 8. Band structure (left) with partial density of states, PDOS, (right) of the 2H– and the 1T-polytypes at high-symmetric points in the irreducible Brillouin zone. The PDOS panels show the contribution to the electronic structure according to the angular momentum of thestates. The states with the p(S) and d(Mo) characters dominate the uppermost region of the valence band.
We investigated the polytypism of MoS2 via first-principles crystal structure prediction approach and mapped the phase space at constant composition by keeping the atomic ratio Mo:S constant at 1:2. In agreement with the study reported by Trigunayat et al. on the symmetry-allowed polymorphs of MX2-type structures , we found that the structures of MoS2
are represented in the five allowed space groups.
In particular, we found the iso-energetic semiconductor polytypes, 2H, 3R and 3T, space groups P63/mmc, R3m, P3m1, respectively; and the metallic polytypes, 1T, and 6R, space groups P3m1, P-3m1, and R-3m, respectively. The three iso-energetic structures, 2H-, 3R-, 3T-polytypes, exhibit different stacking sequences but the same coordination of molybdenum atoms to sulfur atoms, edge-sharing trigonal prism, and similar band structure, the mean band gap is 0.93 eV. In addition, we found one monoclinic, 1M, two orthorhombic, 2O and one tetragonal, 2Q, polytypes, which show a different Mo-S coordination and band structures than the 2H and the 1T. We found that our 1M-polytype could satisfy the relation of the lattice parameters suggested by Wypych et al.  for 1T– and 2H-polytypes. Closely related to the semiconductor structures, in terms of enthalpy of formation and band gap, the orthorhombic structure, 2O-polytype, space group Fmm2, exhibits a translational instability, showing that the minimum energy stacking sequence is principally governed by the relative interactions of the sulfur atoms. In fact, in the 2H-polytype, the stacking sequence, AbA BaB, maximize the π-interactions of the sulfur atoms on the adjacent (MoS2) slabs. In the 1T-polytype, the sulfur slabs are staggered one to each other and the resultant stacking sequence AbC not only creates an octahedral hole for the Mo atoms but makes possible the lateral Mo-Mo orbital overlapping.
As a consequence, the band gap closes and the 1T-polytype, space group P3m1, shows a metallic band structure. The energy cost of the structural transition from 2H– to 1T-polytype is 61.27 kJ/mol at T = 0 K and 75.42 kJ/mol at T = 298.15 K, when accounting in the respective zero-point energies and the thermal contributions via phonon calculations. The study of polytypism of layered dichalcogenides, and in particular of MoS2, has been of relevant interest along decades due to a unique combination of valuable structural, electronic and optical properties exhibited by the different polytypes. The first-principles investigation of polytypism can help to rationalize the structure restacking of MoS2 [14, 43] and provide an atomistic insight to experimental results .
The author thanks, Professor Reinhard Nesper for his valuable comments and Professor Adem Tekin for carefully reading the manuscript. The computer facilities at ETH, Zürich are highly appreciated. The present work is part of the research done by
the author during the period 2010-2015 at ETH Zürich, Department of Chemistry and Applied Biosciences, Institute of Inorganic Chemistry.
5. Takéuchi Y, Nowacki W. Detailed crystal structure of rhombohedral MoS2 and systematic deduction of possible polytypes of molybdenite. Schweizerische mineralogische und petrographische Mitteilungen. 1964, 44: 105.
10. Wypych F, Schöllhorn R. 1T-MoS2, a new metallic modification of molybdenum disulfide. J Chem Soc Chem Commun. 1992, 19: 1386-1388.
16. Mali G, Patel M U M, Mazaj M, Dominko R. Stable crystalline forms of Na polysulfides: Experiment versus ab initio computational prediction. Chem Eur J. 2016, 22(10): 3355-3360.
17. See KA, Leskes M, Griffin JM, Britto S, Matthews PD et al. Ab initio structure search and in situ 7Li NMR studies of discharge products in the Li-S battery system. J Amer Chem Soc. 2014, 136(46): 16368-16377.
21.Oganov AR. Modern methods of crystal structure prediction. Wiley-VCH. 2010: 274.
26. Caputo R. Exploring the structure-composition phase space of lithium borocarbide, LixBC for x ≤ 1. RSC Advances. 2013,26(3): 10230-10241.
27. Caputo R, Kupczak A, Sikora W, Tekin A. Ab-initio crystal structure prediction by combining symmetry analysis representations and total energy calculations. An insight into the structure of Mg(BH4)2. Phys Chem. 2013, 15(5): 1471-1480.
28. Zeng G, Caputo R, Carriazo D, Luo L, Niederberger M. Tailoring two polymorphs of LiFePO4 by efficient microwave-assisted synthesis: A combined experimental and theoretical study. Chem Mater. 2013, 25(17): 3399.
33. Grimme S, Antony J, Ehrlich S, Krieg H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J Chem Phys. 2010, 132: 154104.
37. Guinier A, Bokij GB, Boll-Dornberger K, Cowley JM, Durovic S et al. Nomenclature of polytype structures. Report of the International Union of Crystallography ad-hoc committee on the nomenclature of disordered, modulated and polytype structures. Acta Cryst A. 1984, 40: 399-404.