X-ray diffraction and theoretical study of the transition 2H-3R polytypes in Nb_1+xSe₂ (0 < x < 0.1)

Crystal growth and chemical analysis

Single crystals of 2H and 3R-Nb_1+xSe₂ were obtained by chemical vapor transport (CVT) method during our attempts to prepare ternary HgNbSe₂ and SnNbSe₂. A stoichiometric amount of pure elements with an excess of Se were mixed and sealed in an evacuated quartz tube (length ∼18 cm) using iodine (<5 mg/cm³) as a transport agent to favour crystallization. The mixtures (charge ∼1g) were placed into a zone tube furnace. The temperature was first increased slowly at 500 °C and held then for 48 h in order to avoid thermal runaway caused by the exothermic reaction between Nb and Se. After that the temperature is turned up between 760−1,050 °C for 15 days. Finally the furnace was allowed to cool slowly to room temperature. Crystals of appreciable sizes were grown in the cold end of the tube; 2H+3R crystals are obtained at ∼760 °C while the 3R crystals are obtained at ∼1,050 °C. The analysis results by X-ray energy dispersive spectroscopy (XEDS) with the TEM (EDX Oxford Instruments) on several different crystallites of each polytype sample gave an approximate atomic ratio of 1:2 for Nb and Se [for a selected 3R-crystal: Nb (at%) = 29,9 and Se (at%) = 70,1].

Single crystal structure determination

Single crystals data were recorded on a Kappa Apex II CCD X-ray diffractometer (Bruker, 2006) with graphite-monochromated MoKα (λ = 0.71071 Å) radiation. The reflection intensities were integrated with the SAINT (Bruker, 2006); SADABS was used for empirical absorption correction (Sheldrick, 2002, and Petříček, Dušék & Palatinus, 2006) was performed for the structure refinement.

The crystal structure of 3R-Nb_1.071Se₂ was obverse/reverse twinned while the 3R-Nb_1.085Se₂ was twinned by inversion; with a refined twin domain fraction of 0.611(5): 0.389(5) and 0.85(8): 0.15(8) respectively. The CIF files containing details of the structure refinements are available in the File S1.

In the final cycles of refinement, all the atoms in the different structures were refined anisotropically.

Details concerning the structure refinement and final results are presented in Table 1 while atomic coordinates, anisotropic displacement parameters and selected bond distances are listed in Tables S1–S3.

Theoretical calculations

The DFT calculations were performed using the plane-wave pseudopotential method implemented in the Cambridge Sequential Total Energy Package (CASTEP) code (Clark et al., 2005) of the Material Studio program from Accelrys (Materials Studio CASTEP Manual Accelrys 2010, 2010). The GGA-PBE functional (Hammer, Hansen & Nørskov, 1999) was used to model the exchange and correlation interactions. The Broyden–Fletcher–Goldfrab–Shanno (BFGS) method was used to carry out the geometrical optimization (Shanno, 1985). The plane-wave cut-off energy was adopted to be 500 eV and the Monkhorst–Pack scheme (Perdew, Burke & Ernzerhof, 1996), k-point grid sampling was set to 7 × 1 × 1 in the Brillouin zone. As far as self-consistent condition setting is concerned, the total energy was less than 3.10⁻² eV atom, and the maximum displacement and maximum stress allowed was 10⁻³ Å  and 5.10⁻² GPa respectively. The valence electronic configurations considered for atomic pseudopotential calculation are Nb: 4s² 4p⁶ 4d⁴ 5s¹ and Se: 4s² 4p⁴.

To investigate the relative stability of the Nb-SI in the 2H and 3R polytype with an excess of niobium (x = 0.1), 5 × 1 × 1 and 5 × 2 × 1 supercells respectively are adopted, corresponding to the ordered configurations Nb₁₀Se₂₀ (x = 0) and Nb₁₁Se₂₀ (x = 0.1) to simulate both pure and SI systems.

The formation enthalpy ΔE at T = 0 K is expressed as the difference in total energy of the supercell calculated by DFT and the chemical potentials E of Nb (cubic Fm-3m) and Se (trigonal P3₁21) calculated from their respective bulks in the same computation conditions. ${\displaystyle \Delta E=\frac{1}{\left(x+y\right)}\left[E_{DFT}-x{E}_{Nb}-y{E}_{Se}\right]}$ where x and y, respectively represent the Nb and Se atoms number in the supercell structure model.

The calculated free energy ΔG at typical synthesis temperatures of 760 and 1,050 °C only include configurational entropy due to intercalation, according to (Ivanova et al., 2019).

X-ray experimental crystal structure

The polytype 2H-Nb_1.031Se₂ is almost stoichiometric (Fig. 1A). Additional metal Nb2 atoms with a refined occupancy about 7.3% are located into the octahedral sites within the van der Waals VDW gap with the stacking sequence AcAδBcB (A,B refer to the selenide layers; c to the completely filled niobium layers and δ to the partially filled niobium layers).

Polyhedrons around the Nb atoms are not distorted with six equivalent Nb-Se distances. The Nb1- 6Se = 2.597(2) Å  around Nb1 (in a trigonal prismatic coordination) are somewhat larger than the Nb2-6Se = 2.477(2) Å, observed around Nb2.

A characteristic feature of this 2H- structure is the short Nb1-Nb2 = 3.1425(6) Å  distance, suggesting the formation of strong metal–metal repulsions between the adjacent layers, but can also be analyzed in terms of electrostatic repulsion (Fig. 2A).

The relaxation probably occurs following a defect model by introducing vacancy (about 4%) in the Nb1 filled layers. The Nb1 atoms are then displaced forward the created vacancy to minimize the formation of Nb1-Nb2 pairs interactions (Tronc & Moret, 1981; Amzallag et al., 2007).

The composition obtained with the structure refinement is: 2H-(Nb_0.96(2) Δ_0.04) Nb_0.073(13)Se₂

(Δ denotes Nb vacancy in the fully layers) (ie) 2H-Nb_1.031Se₂.

The Nb1-Se distances in the 3R-Nb_1.071Se₂ polytype are very comparable [2.594(9) –2.601(8) Å] and close to the values [2.59(8)–2.62(8) Å] reported by Brown & Beerntsen (1965). The refined occupancy (sof) of the additional Nb2 atoms (7.1%) is almost equal to that observed in the previous 2H structure with the stacking sequence AbAδBcBαCaC β (A, B, C refer to the selenide layers; a, b, c to the completely filled niobium layers and α, β, δ to the partially filled niobium layers).

The octahedrons Nb2Se₆ are slightly distorted with three short [2.288(10) Å] and three long [2.698(14) Å] distances. The Nb2 atoms are then shifted from the center of the octahedrons in order to minimize the Nb1-Nb2 [3.425(17) Å] interactions.

In order to avoid the direct stacking Nb1—Nb2—Nb1, a relaxation occurs following the transition 2H-3R model and consequently the Nb1-Nb2 distance increases to 3.425(17) Å. Indeed, in the 2H structure the relaxation by introducing high concentration of vacancy in the filled Nb layers may destabilize the structure by breaking the Nb-Nb bonding across the face sharing polyhedrons.

In the 3R-Nb_1.085Se₂ (Fig. 1B), the refinement reveals a high occupancy, about 11.8%, of additional Nb2 atoms. By comparing with the previous 3R structure, the Nb2Se₆ octahedrons are less distorted Nb2-Se = 2.362(11)-2.621(14) Åand the Nb1-Nb2 distance (3.36(2) Å) becomes shorter (Fig. 2B). This can be explained by a shrink of the Nb-Nb layers with high content of extra Nb2 atoms.

In this case, both models exhibit relaxation: 2H-3R transition followed by a defect model introducing about 3% of vacancy in the Nb filled layers. A comparable defect was observed in the 3R-Nb_1.06S₂ (Powell & Jacobson, 1981).

The composition obtained with the structure refinement is: 3R-(Nb_0.97(2) Δ_0.03) Nb_0.118(13)Se2 (Δ denotes Nb vacancy in the fully layers) (ie) 3R-Nb_1.085Se₂.

In Nb_1+xSe₂, as in the layered transition metal dichalcogenides, vacancies in the sublattice of the metal (Nb) and chalcogen (Se) are expected.

It is quite clear that the increase in the additional Nb induces changes in the pressure of saturated Se vapor above the surface of the growing crystal, and then affects the concentration of selenium vacancies. However, a free refinement of the occupancy of Se atoms in the three polytypes leads to a fully occupied sites, possibly due to the excess of Se used as starting material. Therefore, the generation of vacancies in the sublattice of Nb should be correlated to the location of Nb interlayer.

By the presence of extra Nb atoms, a significant residual electron densities are observed in the crystal structure refinements stacking in a way that violates the ideal 2H (ABAB..) and 3R (ABCABC..), about 7.3% for the former and 7.1–11.8% for the latter. The remaining 92.7% and 92.9–88.2% are unfaulted (see CIFs in Electronic Supplemental Files). Certainly, a stacking fault is a common feature in this kind of layered structures, with strong diffuse scattering observed along the c^∗ axis in electron diffraction patterns (see Fig. S1).

An indication about the relative stability of 2H and 3R-Nb_1+xSe₂ structures, can be obtained by comparing the variation of c/a and (c/n)/a ratios with x, where a and c are the unit cell parameters and n the number of layers in the stacking sequence; n = 2 and 3 for 2H and 3R respectively.

As shown in Fig. 3, this ratio change is relatively small with the increase of x for both polytypes with a discontinuity at the 2H to 3R transition around x = 0.07. It is assumed that both polytypes may co-exist in the range 0<  x < 0.07, and above this limit only the form 3R predominates. Comparable results were observed in the study of the non-stoichiometric 2H and 3R-Nb_1+xS₂ (0.07 <  x < 0.18) by powder diffraction (Fisher & Sienko, 1980). The mechanism involving the 2H-3R transition is still unclear, and may be attributed to the phase limit of the non-stoichiometric Nb_1+xSe₂. The difference in energy between polytypes is small, thus the possibility to obtain a mixture of polytypes by changing the conditions of synthesis cannot be ruled out.

The present results are in agreement with earlier studies by Huisman, Kadijk & Jellinek (1970) and Selte, Bjerkelund & Kjekshus (1966). The mixture 2H and 3R polytype were prepared at temperature of 760 °C in the range 0.07 < x < 0.118, yet the 3R phase at x = 0.07 required higher temperature 1,050−1100 °C. Further experimental and theoretical works are mandatory to determine the x range of 2H-3R transition accurately.

Theoretical results

To corroborate the single X-ray conclusions, the stability of four structures: the pure 2H- NbSe₂ and 3R- NbSe₂, 2H-Nb_1.1Se₂ and 3R-Nb_1.1Se₂ with 10% of extra Nb-SI in the pure 2H and 3R, respectively, have been investigated. The optimized structural parameters of the supercell structures are summarized in Table 2.

The calculated c lattice parameters of 2H and 3R-NbSe₂ are slightly overestimated compared to the experimental values by 5.28% and 4.38% respectively. This discrepancy is due to the failure of GGA (local and semi-local approximations for the exchange–correlation) in describing the van der Waals interactions.

Structural geometrical optimization revealed that additional Nb2 atoms incorporated in 2H and 3R-Nb_1+xSe₂ with x = 0.1, obviously increase the c parameter by 7.61% and 9.73% respectively while the a parameter remains almost constant.

The Nb1-Se distance slightly increases in pure 2H and 3R-NbSe₂ (2.587–2.600 Å  and 2.597–2.600 Å) compared to 2H and 3R-Nb_1.1Se₂ (2.569 –2.687 Å  and 2.592–2.697 Å) while the Nb2-Se distances range from 2.564–2.759 Å  and 2.610–2.797 Å  respectively.

The calculated formation energies at T = 0 K (Table 2) indicate that the four structures polytypes can be thermodynamically stable. However, due to the strong Nb-Nb interactions the Nb (x = 0.1) SI system is found to be the least favorable. The close free energy ΔG values at high temperature suggest that facile phase transition between the two polytypes may occur, as established between Nb₂Se₃ and Nb_1.33Se₂ polytypes Ivanova et al. (2019). These results can explain the experimental stability of both 2H and 3R between 0 <  x < 0.07, and confirm the hypothesis of a phase transition from 2H to 3R beyond x = 0.1.

The band structures and density of states (PDOS) of the foregoing four cases have been presented to study the impact of the Nb-SI on the electronic structure for both 2H and 3R-NbSe_2.. Figures 4A and 4B show the band structure of the pure 2H and 3R respectively. The presence of several bands across the Fermi level E_F reveals the metallic nature of the pure polytypes.

Partial density of states (PDOS) (see Figs. S2A and S2C), indicates that the major contribution in DOS at E_F comes from the hybridization between the Nb-4d and Se-4p orbitals which is responsible for the covalent Nb-Se bonds.

For both 2H and 3R Nb-SI, the number of electronic bands around E_F apparently increases compared to the pure 2H and 3R (Figs. 4C and 4D), and consequently the gap around 2 eV in the Conduction Band disappears. Moreover, the PDOS at E_F decreases and upshifts (see Figs. S2B and S2D) compared to the pure polytypes which implies that a large degree of electrons is transferred by Nb extra atoms into these pure polytypes.

It is worth noticing, that the PDOS of Nb-4d is rather broad and almost located at the same energy with Se-4p, suggesting strong interactions. These phenomenon have been observed in some doped and intercalated 2H-NbSe₂ (Chen et al., 2014; Hongping et al., 2014a; Hongping et al., 2014b; Kouarta, Zanat & Belkhir, 2019; Pervin et al., 2020; Xiao-Chen et al., 2020).