The design of two-dimensional superatomic materials, which form their atomic structures through covalently bonded clusters with variable chemical compositions, will enable the development of new materials with promised electronic properties that are beneficial for modern nanoelectronics. This paper presents ab initio calculations of the atomic and electronic structures of both bulk and 2D Re₆Se₈Cl₂. The calculations were carried out using density functional theory, incorporating noncollinear spin density and the pseudopotential method. The results include data on the atomic structure, band gap value, formation energy of the Re₆Se₈Cl₂ 2D layer, and the redistribution of atomic charges within the structures. The differences in effective masses for electrons and holes in the two-dimensional and bulk Re₆Se₈Cl₂ materials are demonstrated, along with an explanation of how these differences impact their transport properties. The findings are expected to be of great significance for the design, synthesis, and implementation of new two-dimensional superatomic materials with controlled properties in modern nanoelectronics.

superatomic 2D materials, atomic and electronic structure, ab initio calculations, band gap, atomic charges, effective masses

The superatomic compound Re₆Se₈Cl₂ is a two-dimensional structural analog of the Chevrel phase materials MₓMo₆E₈ (where M = metal, E = S, Se, Te) [1–5]. These materials are referred to as "superatomic crystals" because they are composed of molecular clusters, and some of them are characterized by a layered structure in which [Re₆Se₈] clusters are covalently bonded into layers capped by terminal chlorine atoms [6–8]. In [8], it was demonstrated that strong in-plane intercluster bonding and weak interlayer interactions allow for mechanical exfoliation of Re₆Se₈Cl₂ layers. The authors of [8] also showed that bulk Re₆Se₈Cl₂ is an indirect bandgap semiconductor with an electronic bandgap of 1.58±0.03 eV, an optical bandgap of 1.48±0.01 eV, and a large exciton binding energy of around 100 meV. Strong coupling of electrons with intercluster optical phonons [9, 10] leads to the emergence of superconductivity in this material. It was shown that polaron formation protects excitons from scattering by lattice phonons, resulting in quasi-ballistic electron energy transport over several micrometers [10]. Furthermore, an exceptionally long exciton free path of approximately 1 mm was discovered, which opens up the possibility of creating ballistic exciton transistors. In a recent study by Shih et al. [11], it was found that the narrow exciton bandwidth, which can be explained by temperature-dependent renormalization due to optical phonons, plays a crucial role in the stability of acoustic polarons in Re₆Se₈Cl₂. Within this mechanism, the polaron binding energy decreases with decreasing temperature.

Transition this material into a two-dimensional state and altering its chemical composition may yield even more intriguing properties, with the potential to design and develop advanced and promising nanoelectronic materials based on it. Therefore, the aim of our research was to theoretically investigate and detail understand the differences in atomic and electronic properties between the bulk and two-dimensional states of Re₆Se₈Cl₂.

The calculations of atomic structures and electronic properties were carried out using the VASP package [12–14]. The generalized gradient approximation in the form of GGA–PBE [15] was employed in the PAW pseudopotentials [16, 17]. Spin-orbit coupling and spin polarization were taken into account in the calculations [18]. Van der Waals interactions were included using the Grimme DFT-D3 semi-empirical method [19] to account for interlayer interactions. For testing the properties of the bulk unit cell of Re₆Se₈Cl₂, a k-point mesh of 9×9×7 was used. For the monolayer, a k-point mesh of 9×9×1 was applied using the Monkhorst–Pack scheme [20]. The optimization of the atomic structure was continued until the forces converged to a precision of 10^-4 eV/nm.

The bulk structure of Re₆Se₈Cl₂ is characterized by a triclinic space group P-1, with experimental lattice parameters of a = 0.65784(7) nm, b = 0.66194(8) nm, c = 0.88010(9) nm, and angles α = 76.708(9)°, β = 70.204(9)°, and γ = 86.368(9)° (Fig. 1). The volume of the unit cell is V = 0.35088(6) nm³ [8]. In our study, a full structural relaxation of the unit cell was performed, maintaining its symmetry while allowing the atomic coordinates to change. As a result, our theoretical calculations yield the following values for the lattice parameters: a = 0.65599 nm, b = 0.66031 nm, and c = 0.87116 nm, with angles α = 76.478°, β = 69.883°, and γ = 86.532°, and a unit cell volume of V = 0.34442 nm³. Figure 1 shows the bulk unit cell with the specified atomic positions and lattice parameters. The interlayer distance in the Re₆Se₈Cl₂ unit cell is d = 0.25930 nm. It is evident that our theoretical data agree well with the experimental data, with errors in the lattice constants of only ε_a = 0.28%, ε_b = 0.25%, and ε_c = 1.02%. For the angles, the errors are ε_α = 0.30%, ε_β = 0.46%, and ε_γ = 0.19%, respectively.

To build the most stable 2D layer from the bulk structure of Re₆Se₈Cl₂, we used the methodology proposed by Vahdat [21]. This methodology is based on a machine learning approach and allows to determine of the most favorable two-dimensional structure from the three-dimensional bulk material. The resulting 2D layer for Re₆Se₈Cl₂ is shown in Fig. 2. Next, the atomic relaxation of the layer was performed, also preserving symmetry and allowing for atomic displacements. For this 2D layer, the equilibrium lattice parameters were determined to be a = 0.65807 nm and b = 0.66075 nm, with angles α = 89.6500°, β = 89.7870°, and γ = 86.0720°. It is observed that during the relaxation of the 2D layer, to relieve surface stresses, the lattice parameter a changes more significantly compared to parameter b, resulting in an increase in the cross-sectional area in the XY plane. To eliminate interactions between Re₆Se₈Cl₂ layers during their translation along the Z-axis, we set the lattice parameter c to 2.86201 nm. According to our calculations, the formation of a 2D layer from bulk Re₆Se₈Cl₂ by mechanical exfoliation requires an energy expenditure of approximately 0.65 eV per formula unit. This energy value was determined using the following expression:

E _form = E_2D – E_bulk,

where E_2D is the total energy of the 2D layer of Re₆Se₈Cl₂ and E_bulk is the total energy of the bulk material [22]. Thus, it is evident that the obtained value for the energy required to form a two-dimensional layer of Re₆Se₈Cl₂ is significantly higher than that for the monolayer ReSeCl, which is 0.22 eV/fu [23], and the monolayer ReSe₂, which is 0.23 eV/fu [24].

Figure 1.

The bulk structure of Re₆Se₈Cl₂

Figure 2.

The 2D layer of Re₆Se₈Cl₂

The electronic structure analysis reveals that for the bulk material the bandgap is 1.11 eV and the Re₆Se₈Cl₂ is an indirect bandgap semiconductor. The "top" of the valence band is localized at the k-point Γ (0, 0, 0), while the "bottom" of the conduction band is located at the k-point T (0, 0.444, 0.5) in the Brillouin zone. For the 2D structure of Re₆Se₈Cl₂, the bandgap increases to 1.34 eV. Thus, the bandgap of the 2D layer is larger compared to the bulk material. The 2D structure remains an indirect bandgap semiconductor, but the localization of the "top" of the valence band and the "bottom" of the conduction band changes. Specifically, the "top" of the valence band is now located at the k-point (0.111, –0.111, 0), while the "bottom" of the conduction band is at the k-point (0, –0.444, 0). The obtained bandgap values for both the bulk and 2D Re₆Se₈Cl₂ are consistent with other reported data [25]. For an accurate assessment of the bandgap, we employed the local potential approximation (LMBJ-potential) [26, 27], which is optimal for two-dimensional materials.

Next, the analysis of the charge distribution for the Re₆Se₈Cl₂ structures was conducted. The charge on the atoms was calculated using the Bader method [28]. It is observed that the charges on the rhenium atoms remain almost unchanged when transitioning from bulk to 2D state. This is likely due to the presence of unpaired electrons on Re atoms, which are shielded by Se atoms through the formation of stable Re–Se bonds [8, 29]. The charge on Se atoms increases by approximately 0.007e, while the charge on Cl atoms decreases by 0.023e (Table 1). This charge transfer influences on the bandgap increase in the two-dimensional structure of Re₆Se₈Cl₂. According to Bader's analysis, during forming bulk and 2D materials of Re₆Se₈Cl₂ from the chemical elements Re, Se, and Cl, there is a charge transfer from Re atoms to Se and Cl atoms.

Next, to understand the processes of energy and information transfer in these materials, we calculated the effective mass for electrons and holes in the layer planes formed by the Re₆Se₈Cl₂ clusters, for both the bulk and 2D materials. We computed the effective masses along the high-symmetry directions G → X and G → Y. For the bulk material, the calculated values are as follows: the effective mass of the electron is 1.571m₀ and for the hole is 0.701m₀ along the G → X direction, where m₀ is the free electron mass. Along the G → Y direction, the effective masses of the electron and hole are 9.545m₀ and 1.105m₀, respectively. However, for the 2D material, the effective masses for electrons and holes differ and are as follows: the effective mass of the electron is 1.294m₀ and for the hole is 1.643m₀ along the G → X direction, while along the G → Y direction, the effective masses of the electron and hole are 5.644m₀ and 2.216m₀, respectively. Thus, it is evident that the 2D material Re₆Se₈Cl₂ exhibits enhanced electron transport properties along the G → X direction compared to the bulk material. However, for holes, the transport properties are reduced in both the G → X and G → Y directions. These results are consistent with those in [10], where the authors first predicted enhanced transport properties for single crystals of Re₆Se₈Cl₂. We are the first to show that the 2D materials of Re₆Se₈Cl₂ have better electron transport properties compared to the bulk material. Table 2 presents the calculated values for the effective masses of electrons and holes for both the bulk and 2D Re₆Se₈Cl₂ materials.

In this work, quantum mechanical calculations of the atomic and electronic structure for both bulk and two-dimensional layers of Re₆Se₈Cl₂ have been performed. The calculations show that during the relaxation of the 2D layer, to relieve surface stresses, the lattice parameter a changes more significantly (increasing by 0.31%) compared to parameter b, leading to an increase in the surface area of the layer. It is demonstrated the band-gap of the Re₆Se₈Cl₂ layer increases comparably to the bulk material, and the 2D structure remains an indirect bandgap semiconductor. Additionally, the charge transfer from Re atoms to Se and Cl atoms occurs during the formation of both the bulk and 2D Re₆Se₈Cl₂ materials from the chemical elements Re, Se, and Cl. Analysis of the effective masses for electrons and holes indicates that the 2D material Re₆Se₈Cl₂ exhibits enhanced electron transport properties along the G → X direction compared to the bulk material. However, for holes, the transport properties are reduced in both the G → X and G → Y directions.

Computations were performed using methods and techniques developed under the State assignment for research work implementation from the Computing Centre Far Eastern Branch of the Russian Academy of Sciences. A.S. Fedorov, who carried out calculations of the carriers effective masses in the materials under study, thanks the state assignment of the L.V. Kirensky Institute of Physics for support. The authors would like to thank them for providing access to the HPC cluster at the Joint Supercom-puter Center of the Russian Academy of Sciences (JSCC RAS).