From Two- to Three-Dimensional van der Waals Layered Structures of Boron Crystals: An Ab Initio Study.

A remarkable recent advancement has been the successful synthesis of two-dimensional boron monolayers on metal substrates. However, although up to 16 possible bulk allotropes of boron have been reported, none of them possess van der Waals (vdW) layered structures. In this work, starting from the experimentally synthesized monolayer boron sheet (β12 borophene), we explored the possibility for forming vdW layered bulk boron. We found that two β12 borophene sheets cannot form a stable vdW bilayer structure, as covalent-like B-B bonds are formed between them because of the peculiar bonding. Interestingly, when the covalently bonded bilayer borophene sheets are stacked on top of each other, three-dimensional (3D) layered structures are constructed via vdW interlayer interactions, rather than covalent. The 3D vdW layered structures were found to be dynamically stable. The interlayer binding energy is about 20 meV/Å2, which is close to the weakly bound graphene layers in graphite (∼16 meV/Å2). Furthermore, the density functional theory predicted electronic band structure testifies that these vdW bulk boron crystals can behave as good conductors. The insights obtained from this work suggest an opportunity to discover new vdW layered structures of bulk boron, which is expected to be crucial to numerous applications ranging from microelectronic devices to energy storage devices.

Introduction

Following the successful synthesis of graphene, it has created a plethora of research activities in exploring the growth, structure, and properties of a diverse array of two-dimensional (2D) materials. 2D materials exhibit extraordinary physical and chemical properties that are being exploited in nanoelectronics, optoelectronics, energy storage, and thermoelectric and thermal management.[1−6] As the nearest neighbor of carbon in the periodic table, boron is expected to possess free-standing monolayer allotropes. However, a graphene-like honeycomb sheet is unstable for boron because of its trivalent outer shell four orbitals, hindering the formation of a closed-shell electronic structure.[7,8] Following the initial theoretical works,[9−13] monolayer boron polymorphs (i.e., borophene) are the subject of interest owing to its extraordinary physical and chemical properties and potential applications, and various ways to improve its stability are proposed.[14−30] Theoretical studies reveal that borophene possesses a number of unique characteristics, including highly anisotropic electronic structure,[31] mechanical compliance,[32] superconducting ability at relatively high temperature,[33,34] ultrahigh thermal conductance,[35−37] and optical transparency.[38]

Because of the multicenter bonding nature, where the bonds involve multiple atoms sharing a certain amount of electrons, boron exhibits up to 16 bulk allotropes,[8,39] including rhombohedral α-phase, β-phase, tetragonal T-phase, and γ-phase. The multicenter characteristics of boron–boron bonding also lead to the formation of configurationally varied 2D borophene sheets, with different densities of atomic holes being indispensable for stabilizing borophene structures.[11,40] For example, different possible borophene phases have been investigated, including hexagonal borophene, β12, α, δ6, χ3 sheets, and so forth.[40] Recently, several boron monolayer sheets and boron nanoribbons have been experimentally synthesized on metal substrates.[41−46] Among the recently synthesized borophene allotropes, β12 borophene possesses ultrahigh thermal conductance[35] and high property tenability.[47] In particular, Dirac fermion states are predicted theoretically[15] and observed experimentally.[43] Thus, intense research activity has led to a series of studies in exploring physical and chemical properties of this allotrope.[48−50]

These theoretical and experimental studies have motivated increasing interest in seeking new layered structures of boron. Naturally, a rising question is whether there is bulk boron allotrope that can possess layered structure. Moreover, is the interlayer interaction likely van der Waals (vdW) type or not? This question is not trivial, as bulk boron structures are not layered, although up to 16 phases were reported.[39,48] In this work, we report the first vdW layered bulk boron structure constructed with covalently bonded bilayer β12 borophene sheets. Its thermodynamic stability is confirmed by phonon dispersion analysis, and a metallic band structure is revealed through first-principles calculation.

Computational Methods

We performed the calculations of structural and electronic properties based on density functional theory (DFT)[51] with the Vienna Ab initio Simulation Package (VASP).[52] The projector augmented wave method[53,54] and the generalized gradient approximation (GGA) with Perdew, Burke, and Ernzerhof (PBE)[55] functional for the exchange–correlation functional were used. The cutoff energy of plane wave is set to 400 eV. The DFT calculation including Grimme’s D3 dispersion correction scheme[56] was employed to account for the vdW interaction because the long-range weak interaction is important for the construction of layered structures. A vacuum layer of 30 Å was introduced to eliminate the interactions between periodic images. A Monkhorst–Pack k-point[57,58] grid of 9 × 15 × 1 was adopted for a 2D structure and 9 × 15 × 6 was used for a 3D structure. The criteria for energy and atom force convergence were set to 10–5 eV per unit cell and 0.01 eV/Å, respectively, during the optimization.

The interatomic force constants (IFCs) and phonon dispersion relation were obtained using Quantum ESPRESSO package[59] within the frameworks of density functional perturbation theory (DFPT). The GGA–PBE functional was adopted together with ultrasoft pseudopotential. The Brillouin zone for 2D structures was sampled with a 6 × 10 × 1 grid of k-point and a 3 × 5 × 1 grid of q-point, and the 6 × 10 × 4 grid of k-point and the 3 × 5 × 2 grid of q-point were used for 3D structures. Cutoff energies of 40 Ry for plane wave and 400 Ry for electronic density were adopted.

Results and Discussion

We begin with the well-known single-layer borophene (β12) sheet. The β12 sheet has a planar structure that contains honeycombs plus additional boron atoms, which are present in one column along the zigzag direction but absent in the next column. The relaxed structure is shown in Figure , where the calculated lattice constants are a1 = 5.076 Å and a2 = 2.928 Å, which are in good agreement with previously reported calculation and experimental results.[22,23,42] Similar to graphene, no buckling appeared in the monolayer β12 borophene.

Figure 1

(a) Top view and side view of the optimized geometry structure and (b) phonon dispersion relationship of AA-stacking bilayer β12 sheets. (c) Top view and side view of the optimized geometry structure and (d) phonon dispersion relationship of AB-stacking bilayer β12 sheets. In (a,c), the boron atoms in top (bottom) layers are denoted in different colors. The optimized interlayer distance is 3.25 (a) and 3.33 Å (c).

To construct the possible bulk layered structure, it is natural to start from the monolayer sheet and stack the multilayer structures layer-by-layer through vdW interaction. Considering its similarity with graphene, we only consider two stacking patterns (AA- and AB-stacking) of β12 with high symmetry in this work. Figure a shows the top view and side view of the AA-stacking vdW bilayer β12 borophene sheets. We then shift the top layer with respect to the bottom one along the x-axis with a transverse displacement of 0.5 lattice constant to obtain the AB-stacking vdW bilayer β12 borophene, as shown in Figure c. This method is usually adopted to search for the stable stacking patterns in vdW layered structures.[60,61] For each structure, all atoms were allowed to relax. The optimized interlayer spacing is 3.25 and 3.33 Å for AA- and AB-stacking vdW bilayer β12 borophene sheets, respectively, confirming the vdW-type interaction. However, although monolayer β12 borophene is thermodynamically stable,[35] both AA- and AB-stacking vdW bilayer β12 borophene sheets are unstable, which is demonstrated by the obvious imaginary frequencies in phonon dispersion, as shown in Figure b,d. This reveals the low possibility to form a stable vdW layered structure based on monolayer β12 borophene.

To explore the possible stable structure of bilayer β12 borophene, we further reduce the interlayer spacing of AA- and AB-stacking vdW bilayer β12 borophene from 5 to 1.5 Å and using the optimized in-plane lattice constant of separated monolayer β12 as the initial lattice constants. When the interlayer spacing is about 2 Å, stable AA- and AB-stacking bilayer β12 borophene structures are formed, as shown in Figure a,b. The optimized lattice constants are a1 = 4.984 Å and a2 = 2.933 Å for AA-stacking bilayer β12 borophene and a1 = 4.939 Å and a2 = 2.928 Å for AB-stacking bilayer β12 borophene. The formation of these bilayer structures thus shorten the B–B bonds across rows and retain those within rows, compared to those in the free-standing monolayer sheet. Importantly, unlike the thermodynamic instability observed in the vdW bilayer, there is no sign of imaginary frequencies in the phonon dispersion as shown in Figure e,f, accordingly confirming the thermodynamic stability of these novel bilayer borophene structures. The highest optical phonon mode at Γ point for AA-stacking bilayer β12 and AB-stacking bilayer β12 corresponds to a B–B stretching mode with a frequency of 1126 and 1120 cm–1, respectively. As a comparison, that frequency is 1147 cm–1 for monolayer β12 borophene.[35]

Figure 2

Top view and side view of the optimized geometry structure of covalently bonded bilayer β12: (a) AA-stacking 2-β12-AA and (b) AB-stacking 2-β12-AB. The interlayer binding energy (meV/Å2) of bilayer borophene 2-β12-AA (c) and 2-β12-AB (d). Note that the black squares represent the data obtained from DFT-D3 calculations, whereas the red solid lines are the fitting curves based on the Buckingham potential. Phonon dispersion and density of states of (e) 2-β12-AA sheet and (f) 2-β12-AB sheet.

The binding energy Eb is shown in Figure c,d as a function of interlayer spacing, which is defined as[62]Here, Emonolayer and Ebilayer(d) are the total energy of free-standing monolayer and the corresponding double layer with interlayer spacing d, respectively, and S is the area of the supercell in AA- and AB-stacking configurations. According to this definition, a larger absolute value of Eb stands for a more stable stacking configuration. The resulting interlayer spacing and binding energy are 1.90 and 91.99 meV/Å2 for AA-stacking and 1.82 and 115.93 meV/Å2 for AB-stacking bilayer β12 sheets (Table S1), respectively, indicating the presence of a strong interaction between the adjacent β12 borophene sheets. On the basis of the DFT results, the distance–energy curve can be described by the empirical Buckingham potential[63]

The fitting parameters A, B, C for Buckingham potential are shown in Table S2.

Compared with the binding energy of well-studied vdW-attracted layered materials, for example, graphite (−16.84 meV/Å2), black phosphorene (−21.85 meV/Å2), and MoS2 (−26.99 meV/Å2),[64] the binding energy in bilayer β12 borophene is remarkably higher, indicating that the interlayer interaction in β12 borophene stack is primarily covalent attraction, rather than a likely vdW-type interaction. Moreover, the absolute value of the binding energy of AB-stacking bilayer β12 borophene is 23.94 meV/Å2 higher than that of the AA-stacking counterpart. Thus, the AB-stacking β12 borophene is the most stable energetically. To distinct from the usually discussed vdW layered structures, the covalently bonded bilayer β12 borophene sheets are denoted as 2-β12-AA or 2-β12-AB sheets.

To further understand the covalently bonded bilayer structures, we calculated the electronic localization functions (ELFs). The results are shown in Figure a,b. The values of ELF reflect the degree of charge localization, with 1 representing the localization and 0 means a free electronic state. It is clear that a significant amount of electrons are distributed between B–B bonds along the z-axis, indicating that covalent-like B–B bonds are formed between two β12 sheets.

As the 2-β12-AB sheet is more stable energetically than its AA stacking counterpart, in the following, we start from it to explore the possible layered bulk boron. Figure a shows the top view and side view of the layered bulk formed by the 2-β12-AB sheets via AA-stacking. Only the AA-stacking is found to be stable, while obvious negative frequency modes appear in phonon dispersion of AB-stacking counterparts (not shown here). The lattice constants of the bulk structure obtained from the DFT calculation are 4.95 Å in the x direction and 2.93 Å in the y direction. Figure c shows the phonon dispersion relation of this structure. We find that the cutoff frequency is 1117 cm–1, which is much lower than that of graphene (1600 cm–1). Significantly, unlike the bulk structure formed by monolayer β12 sheets via vdW interaction, there is no negative frequency observed in the phonon dispersion, demonstrating thermodynamic stability. Moreover, as shown in Figure b, the equilibrium interlayer gap is around 3 Å, and interlayer binding energy is about 20 meV/Å2, close to those of other vdW bulk materials as shown in Table . Therefore, although the monolayer borophene sheet cannot form a stable layered bulk via vdW interaction, the covalently bonded bilayer β12 sheet is a promising candidate to realize this purpose.

Figure 3

(a) Side view of the optimized geometry structure of the bulk boron formed by 2-β12-AB sheets. (b) Interlayer binding energy (meV/Å2) as a function of interlayer spacing. The black squares represent the data obtained from DFT-D3 calculations, whereas the red solid line is the fitting curve based on the Buckingham potential. (c) Phonon dispersion and density of states of bulk boron formed by 2-β12-AB sheets.

Table 1

Comparison of the Binding Energy of Covalently Bonded Bilayer Borophene with Other 2D Materialsa

	2-β12-AA	2-β12-AB	graphene	h-BN
Eb (meV/Å2)	–91.99	–115.93	–16.84	–16.19

The data of other 2D materials are adopted from ref (64).

On the other side, the 2-β12-AA sheet can also form a stable bulk layered structure via vdW interaction (AA-stacking), as shown in Figure . In this structure, two planes of rectangular lattice form a plane structure. The lattice constants are 4.998 Å in the x direction (the ridge direction), 2.937 Å in the y direction (the across-ridge direction), and 7.468 Å in the z direction. Figure b shows Eb as a function of interlayer distance. The distance–energy points can be fitted to the empirical Buckingham potential. By using DFPT, we obtained the IFC and hence the phonon dispersion relation. Moreover, the optical phonon modes along the x direction are generally more flat than that along the y direction. This phenomenon reveals that the bond strength is highly anisotropic. Compared to carbon, boron lacks one electron. As a result, the bulk structure of boron is complex, exhibiting up to 16 reported bulk allotropes, including clusters and cages. Although several boride compounds such as MgB2 exhibit graphene-like boron layers,[48] these layers are strongly bound with metal atoms, unlike the weakly vdW-bonded sheets in graphite. Therefore, no naturally layered bulk boron structures are reported. Our results demonstrate that covalently bonded bilayer borophene is a possible building unit to form a stable layered bulk boron allotrope.

Figure 4

(a) Side view of the optimized geometry structure of the bulk boron formed by 2-β12-AA sheets. (b) Interlayer binding energy (meV/Å2) as a function of interlayer spacing. The black squares represent the data obtained from DFT-D3 calculations, whereas the red solid line is the fitting curve based on the Buckingham potential. (c) Phonon dispersion and density of states of bulk boron formed by 2-β12-AA sheets.

Finally, we explore the electronic band structures of 2-β12-AA sheet, 2-β12-AB sheet, and the corresponding layered bulk structures, as shown in Figure . Compared with the case of monolayer borophene, the Fermi level crosses more bands because of the band splitting. Therefore, the robust metallic feature is maintained. Moreover, because of the anisotropic atomic structure, layered borophene is expected to possess anisotropic electronic properties, and the electrical conductivity is confined along the uncorrugated armchair direction. More importantly, the vdW layered bulk structures also possess metallic behavior. The unique metallic behavior may have remarkable impact for fundamental research and technological application of borophene itself, and it also holds significance in the context of vdW heterostructure design. The fruitful development and applications of 2D materials have inspired great interest in exploring their heterostructures, in which different 2D layers are stacked in a certain sequence via vdW interaction.[65] Such vdW heterostructures have recently been fabricated and display potential in several distinct applications because many intriguing electronic, optical, and thermal properties that are absent in individual 2D units.[66−68] Graphene, MoS2, h-BN, and phosphorene were reported to be a building unit for vdW heterostructures. However, no metallic unit was reported. Thus, the stable vdW boron crystal and the preserved metallic characteristic reported in the current work opens new route to expand the family of 2D vdW heterostructures.

Figure 5

Band structures of (a) 2-β12-AA, (b) 2-β12-AB, (c) bulk boron formed by 2-β12-AA sheets, and (d) bulk boron formed by 2-β12-AB sheets. The Fermi levels are set to zero.

Summary

In this work, first-principles calculations are employed to understand the interface interaction, structural stability, and electronic band structure of layered borophene sheets. We found that a strong interlayer interaction occurs in the bilayer borophene sheets, forming a covalently bonded bilayer structure. Compared to AA-stacking, the AB-stacking is more stable in energy. We also explore the possible layered bulk boron crystals and find that metallic nature is remarkably robust against the weak vdW interlayer interaction. The vdW layered structures of bulk boron are dynamically stable and promising for applications ranging from nanoscale electronics devices to energy storage.