Structural, Electronic, and Magnetic Properties of Neutral Borometallic Molecular Wheel Clusters

1. Introduction

Boron, a versatile element with applications spanning many fields, has a rich history that traces its origins to the isolation of simple boranes by Stock et al. [1]. Over the past two decades, a noteworthy chapter in this narrative has unfolded with the synthesis and characterization of various low-dimensional boron nanomaterials, including nanoclusters, nanowires, nanotubes, nanobelts, nanoribbons, and monolayer crystalline sheets [2,3,4,5]. Diverging from the traditional bulk boron crystals found in $\alpha$-, $\beta$-, and $\gamma$-rhombohedral and $\alpha$-tetragonal forms, these newly crafted boron-based nanomaterials have distinctive bonding patterns [6]. This departure gives rise to captivating physical and chemical properties, making them a focal point of fascination within materials science. Of particular note is the capability of boron-based nanomaterials, such as clusters, to serve as superatoms or as fundamental building blocks for crafting nanostructures imbued with novel functionalities and properties. These systems are characterized by having diverse structures, including planar, quasi-planar, bowl, cage, tubular drum-like forms, multiple ring tubes, and fullerenes [7,8,9,10]. Electron deficiency of the boron atom enables its involvement in localized and delocalized bonding patterns across various geometric configurations. The neutral B${‌}_n$ clusters with $n<20$ prefer a planar or quasi-planar structure [11]. On the other hand B${‌}_n^{-}$ clusters retain planarity (or quasi-planarity) for $n\le 40$ [12]. The perfectly planar B${‌}_8$ and B${‌}_9$ molecular wheels, with a hepta- or octacoordinated central boron atom, respectively, laid the foundation of borometallic clusters [13]. In the borometallic cluster, the central B atom is replaced with a metal atom, creating a similar doubly aromatic cluster [14,15]. Experiments in cluster beams have demonstrated that transition metal (TM) atoms can be incorporated into the center of the planar boron clusters [16,17,18].

More recently, Chang Xu et al. [19] conducted a theoretical investigation on TM-centered double-ring clusters, M@B${‌}_{2n}$ (where M = Ti, Cr, Fe, Ni, Zn and n = 6, 7, 8). They explored the factors contributing to the stability of these clusters in their study. Miao Yan et al. [20] introduced the concept of fluxional bonds in various structures, including planar B${‌}_{19}^{-}$, tubular Ta@B${‌}_{20}^{-}$, and cage-like B${‌}_{39}^{-}$. These bonds form and break continuously on these systems’ extremely flat potential energy surfaces. Ren et al. [21] used the CALYPSO method combined with density functional theory (DFT) calculations on niobium-doped boron clusters. Their research revealed that the global minima for the neutral clusters correspond to half-sandwich structures at $n=10-17$ and tubular-type structures at $n=18-20$. Chen et al. [22] study offered a comprehensive overview of the geometric configurations of both pure (B${‌}_n$) and metal-doped boron clusters, which exhibit structures such as planar, nanotube, bilayer, fullerene-like, and core-shell forms across a wide range of cluster sizes (n). They also explored the potential for developing boron-based nanomaterials with specific functionalities derived from metal-boron clusters. Liu et al. [23] identified ten potential clusters made up of rings formed by 12 to 15 atoms of boron-carbon doped with heavier alkaline-earth (Ae) metals (Ae = Ca, Sr, Ba). Their chemical bonding analyses reveal the important role of Ae d orbitals in facilitating covalent interactions between the central Ae atom and the surrounding boron-carbon rings.

Linear magnetic chains are becoming crucial in quantum information involving entanglement and correlation [24,25]. Carbon and boron-nitride-based nanotubes are used as support surfaces to stabilize unstable isolated atomic chains [26]. Photoelectron spectroscopy and theoretical studies identified metal centered tubular ionic structures MnB${‌}_{16}^{-}$, CoB${‌}_{16}^{-}$, RhB${‌}_{18}^{-}$, and TaB${‌}_{20}^{-}$ with a magnetic moment of 2, 2, 0, and 0 ${\mu }_{\mathrm{B}}$, respectively [27,28,29,30]. Neutral boron-based tubular structures, including FeB${‌}_{14}$, FeB${‌}_{16}$, UB${‌}_{20}$, NpB${‌}_{20}$, and PuB${‌}_{20}$, were examined theoretically and found to have a quenched magnetic moment of $2{\mu }_{\mathrm{B}}$ for the iron-based structures and relatively high magnetic moment of 3.69, 4.92, and 6.07 ${\mu }_B$, respectively, for actinide centered tubular structures [31,32]. These research works showed that the size and stability of the tubular structures greatly depend on the encapsulated metal atoms through the covalent bonds forming between the metal ions and the boron frame (MB${‌}_n$) [33]. Metal-centered tubular boron clusters have been suggested as the building blocks for one-dimensional metallo-nanotubes [33]. However, the magnetic moments of TM-centered tubular boron clusters remain unexplored, with limited discussion on their magnetic behaviors.

In this work, by using DFT calculations, we study the structure and properties of borometallic molecular wheel structures derived from planar boron B${‌}_8$, B${‌}_9$, and B${‌}_{10}$ wheel clusters. For all these clusters, we analyze their magnetic behavior using spin-polarized calculations and analyze charge density differences and the contribution of metal and boron atoms to the density of states via the projected density of states (PDOS) analysis.

2. Computational Details

A DFT-based study is carried out using the Quantum ESPRESSO package [34]. The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [35] is used in the calculations combined with projector-augmented wave (PAW) pseudopotentials. Boron-based rings centered with TM atoms (TMB${‌}_n$) are modeled by varying the value of n from 8 to 10 for single rings and subsequently increasing it to 16, 18, and 20 by taking the same ring twice and sandwiching the metal atom between them (TMB${‌}_{2n}$). The TMB${‌}_n$ and TMB${‌}_{2n}$ clusters are shown in Figure 1. A vacuum distance of 16 Å is introduced to avoid interactions between cluster replicas. A plane-wave basis set with an energy cutoff of 70 Ry is used. The convergence threshold for the self-consistent energy calculations is ${10}^{-12}$, and the atomic positions are optimized until the forces on atoms are smaller than ${10}^{-4}$ Ry/Bohr. The convergence threshold during phonon self-consistent calculations is ${10}^{-14}$ eV. All the computations are carried out at the $\Gamma$-point of the Brillouin zone. Assessing the stability of clusters often relies on the binding energy, $E_b$, as an important parameter. The calculation of $E_b$ for the TMB${‌}_n$ and TMB${‌}_{2n}$ clusters is done using the equation:

$$E_b=E\left({\mathrm{TMB}}_m\right)-E\left({\mathrm{B}}_m\right)-E\left(\mathrm{TM}\right),$$

where $E\left({\mathrm{TMB}}_m\right)$ is the total energy of the optimized boron ring with the TM atom in the center, $E\left({\mathrm{B}}_m\right)$ is the total energy of the boron cluster in the ring ($m=n$) or drum ($m=2n$) forms, and $E\left(\mathrm{TM}\right)$ is the energy of the isolated TM atom. Spin-polarized calculations are carried out to study the magnetic properties of the clusters.

3. Results and discussion

We became interested in the high symmetry found in aromatic boron rings, which are represented as B${‌}_n$ where n ranges from 8 to 10. This led us to conduct thorough investigations involving replacing the central B atom with TM counterparts, specifically TM = Ti, Cr, Mn, Fe, Co, Nb, and Mo. Our exploration focused on two main configurations: the "ring configuration" (TMB${‌}_n$), featuring a TM atom at the center of a ring composed of 8, 9, or 10 boron atoms, and the "drum configuration" (TMB${‌}_{2n}$), involving two rings of the same size stacked on top of each other, with the metal atom situated between the centers of two rings comprising 8, 9, or 10 boron atoms. Our system is visually represented in Figure 1, with boron atoms highlighted in dark red and forming a ring, and the central TM atom shown in blue.

3.1. Structural Properties

In line with the structural arrangement shown in Figure 1, the bond lengths between adjacent boron atoms ($d_{\mathrm{B}-\mathrm{B}}$) and those connecting boron with central metal atoms ($d_{\mathrm{B}-\mathrm{TM}}$) are summarized for all the studied clusters in Table 1. It is worth noting that all the designed configurations are planar except for TiB${‌}_8$, TiB${‌}_9$, and FeB${‌}_8$, where the central atom deviates slightly out of plane by 0.8387 Å 0.6168 Å and 0.8394 Å respectively, giving rise to a distinctive bowl-like structure, which aligns well with previous research. The distances between metal and boron atoms ($d_{\mathrm{B}-\mathrm{TM}}$) for $n=8$, 9, and 16 are small compared to their covalent bonds, indicating minimal steric repulsion between them. The bond length $d_{\mathrm{B}-\mathrm{B}}$ is influenced by the number of B atoms present in the boron ring and the size of the central metal atom, especially for $n=8$ and 9 (refer to Table 1). For TM = Cr, Mn, Fe, or Co, the drum configuration with the maximum number of boron atoms is only possible with B${‌}_{18}$ because, beyond that limit, an elliptical geometry prevails over a circular shape. In Table 1, we summarize the point group (PG) symmetry for each cluster, which was determined using the automatic flow for material discovery (AFLOW) software. Point group $D_{4h}$ is assigned to TMB${‌}_8$, except for the non-planar configuration. This outcome is in close agreement with the existing literature. For TMB${‌}_{10}$ clusters, the PG symmetry is $D_{2h}$ due to its nearly circular geometry or $C_{2h}$ due to its elliptical geometry. Drum configurations mostly belong to the $C_1$ PG symmetry.

In Table 1, we have summarized the values of $E_b$ for all the clusters. The $E_b$ values consistently increase from TMB${‌}_8$ to TMB${‌}_9$ (or, in some cases, TMB${‌}_8$ to TMB${‌}_{10}$) and then decrease gradually until reaching TMB${‌}_{20}$ across all seven systems for each TM. Overall, the $E_b$ values are relatively high for Co-doped configurations but notably lower for Mn-doped structures. It is important to note that all structures have negative $E_b$ values, indicating the stability of the clusters since energy is required to separate the clusters into TM and B${‌}_n$.

3.2. Electronic Properties

In Table 1, we have summarized the HOMO-LUMO gaps (HOMO refers to "Highest Occupied Molecular Orbital" and LUMO refers to "Lowest Unoccupied Molecular Orbital") of the studied clusters. Although DFT-GGA underestimates energy gap values, most clusters still exhibit nonzero HOMO-LUMO gaps. The highest energy gaps for the ring configurations are found in CoB${‌}_9$, NbB${‌}_{10}$, and FeB${‌}_9$, with values of 1.055 eV, 0.907 eV, and 0.785 eV, respectively. Among the drum configurations, the HOMO-LUMO gaps for MoB${‌}_{20}$, CrB${‌}_{16}$, and MnB${‌}_{16}$ (1.088 eV, 0.822 eV, and 0.804 eV, respectively) are larger compared to other drum structures. Structures with large HOMO-LUMO gaps are associated with high kinetic stability and low chemical reactivity, making them promising candidates for future applications.

The PDOS spectra for all the clusters can be found in Appendix A. These spectra are valuable for studying the contribution of different atomic orbitals to the density of states in the studied clusters, particularly near the Fermi level. They also aid in visualizing the contribution of atomic orbitals to magnetization. In general, the p-orbitals of boron and the d-orbitals of transition metals make the most significant contributions to the density of states near the Fermi level. However, in specific cases such as TiB${‌}_9$, CoB${‌}_8$, NbB${‌}_{10}$, and NbB${‌}_{20}$, the contribution at the Fermi level comes solely from boron s-orbitals. A combination of s and d orbitals contributes to the density of states at the Fermi level for CrB${‌}_8$, MnB${‌}_9$, FeB${‌}_8$, FeB${‌}_{16}$, MoB${‌}_9$, and MoB${‌}_{10}$.

3.3. Vibrational Properties

The vibrational properties of the structures were analyzed using phonon computations to assess their stability. These calculations provide valuable insights into the dynamic behavior and structural integrity of the systems under study. According to the literature [36], a cluster is considered stable if all its real vibrational frequencies are positive, and the minimum vibrational frequency, denoted as $f_{min}$, is significantly large. Table 2 presents the values of $f_{min}$ for each cluster. For the ring configuration, TMB${‌}_8$ and TMB${‌}_9$ with TM = Ti, Cr, Mn, Fe, or Co have the highest values of $f_{min}$, whereas TMB${‌}_9$ and TMB${‌}_{10}$ clusters have the highest $f_{min}$ for TM = Nb and Mo. Interestingly, the FeB${‌}_{10}$ cluster in our calculations is at the stability border, as its $f_{min}$ value is small but negative. For the drum configurations, there is less regularity, and large values of $f_{min}$ are obtained for CrB${‌}_{16}$, while the lowest value is for MnB${‌}_{20}$. Our results for $f_{min}$ are in close agreement with a study by Pu et al. [18], where the values of $f_{min}$ for FeB${‌}_9$ and CoB${‌}_9$ were 110.5 cm${‌}^{-1}$ and 96.7 cm${‌}^{-1}$, respectively. The full list of phonon frequencies for each cluster is provided in Appendix B.

3.4. Dynamic Properties

To further verify the stability of the ring and drum structures, we performed quantum molecular dynamics (MD) simulations at two different temperatures with the choice of 1 fs for the time step. Each run was 3.5 ps long. At a temperature of 300 K, the ring and drum structures maintain their shape during the MD run, and no significant deformations occur. At a temperature of 600 K, both structure types deform but retain their wheel-like structure. This is shown in Figure 2 on the example of CoB${‌}_9$ and MnB${‌}_{18}$. It is worth noting to mention that MnB${‌}_{18}$ has one of the smallest $E_b$ in absolute value.

3.5. Magnetic Properties

The magnetic moment values, denoted as m, for each cluster, are listed in Table 1. Generally, the value of m decreases as the number of boron atoms in the ring increases from 8 to 10. In contrast, an opposite trend is observed for drum structures, with m tending to increase as the number of boron atoms in the TMB${‌}_n$ structure increases (from 16 to 20). Interestingly, the m values for CoB${‌}_n$ and MoB${‌}_n$ drum structures are independent of n, remaining equal to 1${\mu }_B$ and 0${\mu }_B$, respectively.

The contribution from boron and TM atoms to magnetization is illustrated in Figure 3. In general, the contribution from TM atoms to total m is large. However, in most cases, the boron atoms become polarized, resulting in a non-zero contribution to m. Specifically, for TMB${‌}_8$ clusters, the boron contribution (represented by the yellow line) to m is consistently greater than the TM contribution (represented by the blue line). Conversely, in the case of TMB${‌}_{10}$ structures, metals demonstrate dominance. For drum configurations, both metal and boron atoms consistently exhibit positive (or zero) contributions to m, except for the case of Mn, where boron atoms have slightly negative local magnetization. The case of NbB${‌}_n$ clusters is particularly interesting, as the magnetization is induced by TM atoms but is entirely localized on the boron atoms.

The result presented in Figure 3 can be spatially visualized by drawing the difference between the spin-up charge density (${\rho }^{\uparrow }$) and the spin-down charge density (${\rho }^{\downarrow }$):

$$\mathit{spin\; polarization\; density}={\rho }^{\uparrow }\left(\overrightarrow{r}\right)-{\rho }^{\downarrow }\left(\overrightarrow{r}\right).$$

This is done in Figure 4 where the positive and negative values of the spin polarization density are represented with yellow and cyan, respectively. This figure clearly shows that the boron atoms in all clusters get polarized, and this polarization significantly influences the total value of m. This is quantitatively demonstrated by our Löwdin population analysis provided in Appendix C. This analysis evidenced, in most cases, a charge transfer from/to the TM atom, which gives rise to polarization effects on the boron atoms that, as a result, contribute to the total magnetic moment. The results suggest that the polarization of boron atoms is important in modulating the electron distribution, which may affect the cluster’s reactivity and interactions in different environments.

4. Summary and Conclusions

The successful integration of TM atoms into the core of planar boron clusters B${‌}_n$ creates a fascinating class of borometallic molecules. In our study, we present the results of first-principles calculations aimed at examining the electronic and magnetic properties of boron rings with a TM atom in the center, TMB${‌}_n$ (with TM being Ti, Cr, Mn, Fe, Co, Nb, or Mo, and n being 8, 9, or 10), as well as boron double-rings sandwiching the metal atom, TMB${‌}_{2n}$. In these molecular structures, each B atom in the circumference provides two electrons to the B–B peripheral covalent bonds and one electron to the delocalized B–TM bonds. Consequently, a TM atom with the appropriate number of valence electrons can be accommodated in the center of the boron wheel to form the TMB${‌}_n$ clusters, resulting in a structurally and dynamically stable borometallic cluster compound. Additionally, most of the studied clusters possess relatively high HOMO-LUMO gaps, which may suggest low chemical reactivity. Our investigation also indicates that the overall magnetic moment of the clusters is a combination of the magnetic moment of the central TM atom and the induced magnetic moment of the peripheral boron atoms. These molecular-based magnets provide a platform for studying magnetism in the zero-dimensional limit. Furthermore, the TMB${‌}_{2n}$ clusters can function as building blocks for one-dimensional tubular forms.