Boron carbide (B$4$C) has been well studied both theoretically and experimentally in its bulk form due to its exceptional hardness and use as a high-temperature thermoelectric. However, the properties of its two-dimensional nanosheets are not well established. In this paper, using van der Waals-corrected density-functional theory simulations, we show that bulk B$4$C can be cleaved along different directions to form B$4$C nanosheets with low formation energies. We find that there is minimal dependence of formation energies on cleavage planes and surface terminations, even though the bulk is not van der Waals layered. This anomalous stability of B$4$C nanosheets is found to be a result of surface reconstructions that are unique to B-rich systems. While the density of states of the bulk B$4$C indicate that it is a semiconductor, the B$4$C nanosheets are found to be predominantly metallic. We attribute this metallic behavior to a redistribution of charges on the surface bonds of the films. The Seebeck coefficients of the B$4$C films remain comparable to those of the bulk and are nearly constant as a function of temperature. Our results provide guidance for experimental synthesis efforts and future application of B$4$C nanosheets in nanoelectronic and thermoelectric applications.

## I. INTRODUCTION

Boron-rich compounds have garnered significant scientific attention over the last several decades.^{1–3} Boron, unlike its neighbor on the periodic table, carbon, has decidedly unusual covalent bonding patterns. Even in the simplest multi-boron compound, diborane(B$2$H$6$), the “banana” bond between the boron orbitals hybridized with those of hydrogen, leave only two electrons per bond to be shared between three atoms.^{4} These unusual bonding patterns often result in functional physical and thermoelectric properties in boron-rich compounds, for instance, in boron carbide (B$4$C).^{5–7}

The interesting properties of B$4$C have led to extensive studies of this material. Bulk B$4$C is an excellent high-temperature thermoelectric with functional Seebeck coefficients between 200 and 400 $\mu $V/K that remain almost invariant from 300 up to 1500 K.^{8,9} It has large electrical conductivity ($103$–$104$ $\Omega \u22121m\u22121$) at reasonable carrier concentrations.^{10} Finally, it is also one of the hardest known materials with a hardness of 9.3 on the Mohs scale,^{11} making it harder than steel (8 Mohs) and softer than diamond (10 Mohs).^{12} It is also semiconducting, with an experimentally measured bandgap of 1.6–2 eV.^{13–16}

Figures 1(a) and 1(b) show the rhombohedral primitive cell of B$4$C with a C–C–C chain along the body diagonal. The crystal structure is characterized by 12-atom boron icosahedra at the vertices of the primitive cell. The icosahedra are more clearly seen in the orthorhombic conventional cell shown in Fig. 1(c). Due to the lack of electrons needed to form pairwise B–B bonds within the icosahedra, the B atoms are bonded in a unique bonding scheme called triple center bonding.^{17,18}

While most of the work in generating two-dimensional (2D) boron carbide nanostructures has employed bottom–up approaches like chemical vapor deposition (CVD) on a variety of substrates,^{19,20} in a recent article,^{21} using both experiments and first-principles simulations, we demonstrated that 2D B$4$C nanosheets can also be synthesized using top–down synthesis techniques. We successfully synthesized ultrathin nanosheets of B$4$C by sonication of bulk B$4$C powders in organic solvents. Atomic-force miscroscopy, transmission electron microscopy, Raman spectroscopy, and density functional theory (DFT) simulations revealed that the nanosheet surfaces were oriented along several different crystallographic planes with similar energetic favorability.

Our success in the formation of B$4$C nanosheets and without any preferential cleavage plane^{21} is surprising since B$4$C has no weak vdW bonds that can be broken to form low-dimensional forms of materials. In order to gain insight and determine the origin of this seemingly anomalous behavior, we examined the structures of the near-surface atoms of the nanosheets and found a rearrangement of surface atoms into smaller cage structures. In this article, using first-principles simulations, we elucidate the mechanism for this anomalous stability of the B$4$C nanosheets. Furthermore, employing DFT and excited state theory $G0W0$ simulations, we predict their electronic and thermoelectric properties. Our work can be used to guide the future application of B$4$C nanosheets in nanoelectronics, thermoelectric devices, as reinforcement materials, and as lightweight armor.

## II. METHODS

B$4$C nanosheets were generated by cleaving the conventional cell of B$4$C along four planes, (300), (001), (101), and (012), see Fig. 1(c). All the possible unique surface atomic terminations, including those that break the B-icosahedra, were considered. This resulted in 4, 7, 8, and 9 uniquely terminated nanosheets with surfaces normal to (300), (001), (101), and (012) planes, respectively (total of 28 unique terminations). The formation energy for each plane and termination was computed as the difference between the van der Walls-corrected DFT computed energies of the bulk and 2D nanosheets, normalized to the number of atoms in each unique termination. Details of the method involved in the generation of nanosheet slabs and DFT simulation parameters can be found in our previous article.^{21} In the same article, we reported that DFT computed formation energies along all the cleavage planes and surface terminations were similar, indicating no preferential direction of B$4$C cleavage during B$4$C nanosheet formation via exfoliation. The minimum formation energies of B$4$C nanosheets cleaved along (300), (100), (101), and (012) planes were found to be 0.113, 0.056, 0.116, and 0.105 eV/atom, respectively, which are all below the thermodynamic stability threshold energy of 0.2 eV/atom for a free-standing single-layer or few-layer nanosheets.^{22}

All the density functional theory (DFT) simulations performed for this article are based on the projector-augmented wave method as implemented in the plane-wave code VASP.^{23–25} DFT simulations were performed using the vdW-DF-optB88 exchange–correlation functional,^{26} with a cutoff energy for the plane wave basis at 600 eV, a $k$-point density of 60 Å$\u22121$ in the $x$ and $y$ directions and only one $k$-point in the $z$ direction for all slab simulations. Structural relaxations have been performed with energy convergence within $10\u22126$ eV in each ionic step and until the forces are converged to 0.005 eV/atom. The charge densities on individual atoms were determined using the Bader charge method^{27–30} and the charges on the atoms were converged to within 0.5 electrons.

The quasiparticle (QP) energies were obtained by using many body perturbations theory within the G$0$W$0$ approximation for the self-energy operator.^{31,32} The basis set size for the response functions and screened Coulomb potential, $W$, was chosen to include all plane waves up to an energy cutoff of 250 eV. The number of unoccupied bands included in the GW calculation was set to 160 such that the QP gap is converged to within 0.1 eV. For the frequency integral in the calculation of self-energy, we have used 80 frequency points. To obtain the QP band structure, we have used the interpolation formalism implemented in WANNIER90 package.^{33}

## III. RESULTS AND DISCUSSIONS

### A. Bond lengths and charge densities of surface atoms

The cleavage of covalent materials along different directions leads to the formation of surfaces that have a variety of dangling bonds and surface-atom densities and thus varying surface stabilities. The low formation energies of the B$4$C nanosheets reported in our previous article indicate that there are perhaps little-to-no high-energy dangling bonds on their surfaces. The lack of dependence of formation energies of B$4$C on the cleavage plane indicates uniformity of surface bonding-type (covalent and ionic instead of weak vdW) between different plane terminations. To test this hypothesis, we examined the bonding environment and atomic charge densities of nanosheet surfaces in comparison to their bulk.

Figures 2(a) and 2(b) show the B–B bonds, B–C bonds, and the C–C bond of the near-surface atoms of one of the terminations along the (300) plane. The color bars denote the lengths of the bonds. The bond length maps of all the other 27 B$4$C nanosheet slabs are shown in Figs. S1–S4 in the supplementary material.

Figure 2(c) shows the charge density isosurface of the same structure normalized to a saturation level of 0.13 $e$/Å$3$ such that the charge densities on atoms are clearly defined.^{34} The boron(carbon) atoms are labeled as B$(n)$(C$(n)$) where ($n=1,2,$…), so that individual atoms can be clearly referenced in the discussion. All the three plots are constructed along the same projection.

We note that most boron atoms at the surface, for example, B$(3)$–B$(7)$ in Fig. 2(a), rearrange into smaller cage structures with atoms fewer than 12 in the icosahedra of bulk B$4$C. The B–B bond for these boron atoms are partly larger ($\u223c2.0$ Å) and partly smaller ($\u223c1.5$ Å) compared to those in the icosahedra of bulk-like atoms (for example, $\u223c1.7$ Å bond length of B$(8)$–B$(15)$ that are located far from the surface).

However, examination of the charge density at these sites in Fig. 2(c) reveals that there is a significant charge transfer between the B atoms, suggesting an existence of an ionic bond between B$(3)$ and B$(5)$ and strong covalent bonds between the other B atoms in the cage. We find that some of the surface B atoms, like B$(1)$ and B$(2)$ in Fig. 2(a), have smaller bond-lengths than the B atoms in the bulk and thus are strongly bound. This strong bonding in the surface B-atoms due to the formation of cage structures provides an inherent stability to B$4$C nanosheet surfaces.

We find that C–C bond lengths at the surface of the nanosheets to be similar to those far from the surface, i.e., similar to the C–C bond lengths in bulk B$4$C. For instance in Fig. 2(b), the C$(4)$-C$(5)$-C$(6)$ bond lengths are comparable to C$(7)$-C$(8)$-C$(9)$ bond lengths. However, the surface C chains are bent in contrast to the straight C chains in bulk B$4$C. The bending of the C chains can be explained by inspecting the charge density distribution, for instance, between B$(2)$, C$(4)$, and C$(5)$ in Fig. 2(c). We see there that there is charge accumulation between B$(2)$, C$(4)$, and C$(5)$ atoms, indicating a strong bonding of C chains with solitary boron atoms at the surface—this holds the surface boron atoms to the nanosheet structure. In comparison, the bend in the surface C$(1)$–C$(2)$–C$(3)$ compared to the bulk C$(7)$–C$(8)$–C$(9)$ can simply be explained by the asymmetry of B atoms on the top and bottom of the chains.

Notably, the overall charge distribution on the C chains is unaltered due to the creation of the surface of the nanosheet that explains the nearly constant C chain bond lengths at and far from the surface of the nanosheets. Additionally, for atoms near the C chains, we see that the electron density is larger on the C atoms than the B atoms, implying that the bonds between the C-chains and the icosahedra are ionic in nature. Hence, the inter-icosahedra bonds are just as strong, perhaps stronger, than the intra-icosahedra bonds. This unique bonding scheme leads to an extremely stable structure both in bulk B$4$C and at atoms far away from the surface of nanosheets. The bond length maps of all other nanosheets show a similar trend.

Thus, using the bond length and charge density maps, we show that the B$4$C nanosheets remain highly stable despite the breakage of the strong bonds of bulk B$4$C. We find three primary reasons for their anomalous stability and their cleavage plane and termination independent formation energies. First, we find that the B$4$C nanosheets maintain strong covalent and ionic bonds both at the surfaces and also far from the surfaces of the nanosheets. Second, the surface B atoms of any broken B-icosahedra restructure into stable B-cages, with fewer than the 12 B atoms of the original icosahedra. Finally, the isolated surface B-atoms also bind strongly to C-chains, imparting additional stability to the B$4$C nanosheets. Altogether, we discern that the anomalous stability of the B$4$C nanosheets is a result of surface reconstructions.

### B. Electronic structure of B_{4}C nanosheets

To find potential application for B$4$C nanosheets in nanoelectronic applications, we calculated the electronic properties of B$4$C nanosheets.

Figure 3 shows the element-projected electronic density of states (DOS) for the B$4$C nanosheets computed from DFT. The DOS belonging to B-atoms and C-atoms are shown by blue and red shaded sub-figures, respectively. The figure shows the DOS of the lowest formation energy termination for each cleavage direction, i.e., (001), (300), (101), and (012). The DOS of all the other nanosheets studied in this work can be found in Figs. S5–S8 in the supplementary material. We find that there are states at the Fermi level for 16 of the 28 B$4$C nanosheets, indicating that they are metallic in nature. Figure S14 in the supplementary material shows the value of bandgaps and formation energies of all the 28 nanosheets. Since the B$4$C planes have low formation energies,^{21} both metallic and semiconducting phases of B$4$C nanosheets are likely to form. The previously measured range of bandgaps of B$4$C nanosheets are in good agreement with our study. However, a direct comparison cannot be made between bandgaps obtained in this work with those in the literature, as complete information of the cleavage plane and termination is unavailable for the existing studies.^{35,36}

Figure 4 compares the DFT-computed and many-body perturbation theory $G0W0$ computed band structure and the density of states of bulk B$4$C. The DFT-computed bandgap is found to be 1.53 eV, similar to the theoretically computed bandgaps reported in the literature.^{15} The $G0W0$ bandgap is found to be 2.3 eV, comparable to the experimentally measured values of 2.09 eV reported by Werheit.^{37} $G0W0$ simulations are often an order of magnitude more expensive than standard DFT simulations, but they are found to be remarkably successful in predicting bandgaps of semiconducting and insulating materials. We emphasize that this is the first study to report the $G0W0$ computed band structure and DOS of bulk B$4$C. While $G0W0$ provides better agreement of bandgaps with experimental measurements, we need not perform $G0W0$ simulations for the B$4$C nanosheets as they are metallic and because $G0W0$ are computationally not tractable for the study of large number of atoms such as those in the B$4$C nanosheets (46–176 atoms/simulation cell).

To understand the role of surface atoms in the semiconducting-to-metallic transition in B$4$C nanosheets, we examined the DOS of the surface atoms. We defined the surface-atoms as the atoms within the top and bottom 15% of the total slab thickness and the remaining atoms are considered bulk-like. The DOS of the bulk-like and surface-like atoms are projected separately in Fig. 3.

We can see that the DOS of the bulk-like atoms in the nanosheets (i.e., atoms far from the surface) remain similar to that of bulk B$4$C states for all lowest energy terminations of the four cleavage planes. This result is expected as the bonding of the atoms in the far-from surface of the nanosheets closely resembles the bonding in bulk B$4$C. The Fermi level, i.e., the valence band maxima energies, are slightly different for different planes as the nanosheets deviate nominally from the B–C ratio of 4:1 (see Table 1 in the supplementary material) for various plane terminations.

The states attributed to the surface of the nanosheets have a drastically different nature than that of the bulk-like atoms. Figure 3 shows that for most terminations, there are now states at the Fermi level. These additional states are contributed by both boron and carbon atoms. In the process of rearrangement of atoms close to the cleaved surface, the band structure of the material changes leading to new hybridized states near the Fermi level. Thus, 2D B$4$C shows metallic character on its surface. The emergence of similar surface reconstruction induced metallicity has been observed in other materials as well.^{38,39}

### C. Seebeck coefficients of B_{4}C nanosheets

B$4$C is an excellent thermoelectric material in the bulk form. Thermoelectric materials can directly convert waste heat into electrical energy based on the Seebeck effect. B$4$C has a high Seebeck coefficient, which remains invariant with temperature fluctuations.^{9,10} Here, we computed the Seebeck coefficient of the B$4$C nanosheets to understand the impact of reduction in the dimensionality of B$4$C .

The Seebeck coefficient, $S$, is given by the following equation:

where $\Delta V$ and $\Delta T$ are the voltage and temperature differences across the material, respectively.

We estimated the Seebeck coefficients of the B$4$C nanosheets using the BoltzTrap2 code,^{40} which solves the semiclassical Boltzmann transport equation (BTE) under constant relaxation time approximation. Figure 5 compares the Seebeck coefficients of bulk B$4$C with four lowest energy B$4$C nanosheets. The Seebeck coefficient for all other terminations can be found in Figs. S10–S13 in the supplementary material. Notably, the Seebeck coefficient for all the nanosheets remain almost invariant across the temperature range of 300–600 K. Thus, just like the bulk B$4$C, the B$4$C nanosheets are also temperature-insensitive Seebeck materials and hence could have a large range of operational temperatures in thermoelectric devices.

In addition, the Seebeck coefficients for minimum energy terminations along (001) and (012) are similar in magnitude to that of bulk B$4$C as opposed to $S$ values for terminations along (300) and (101) that are about 75% smaller than that of the bulk B$4$C. We examine the semiclassical BTE to understand this deviation. The semiclassical BTE computes physical parameters of a macroscopic system by expressing them in terms of specific moments $L(\alpha )$.^{41} For example, the electrical current, $j$, is given by

where $e$ is the electronic charge, $T$ is the temperature of the system, and $L(0)/(1)$ are the aforementioned moments. In the situation where $j=0$, we can represent the Seebeck coefficient using Eq. (1) as

The $\alpha th$ moment, $L(\alpha )$, is calculated by summing over all the bands, $n$, of integrals over the entire reciprocal $k$-space, given by Eq. (4),

where $\mu $, $e$, and $\tau n$ are the chemical potential, electronic charge, and the relaxation time in the $n$th band, respectively. The $g0$ is the Fermi distribution function and $\epsilon n$ is the energy relation for the $n$th band.

The derivative of the energy in the reciprocal $k$ space [highlighted in bold in (4)] changes the value of the moment $L(1)$ and in turn of $S$ given by Eq. (3). Thus, materials with drastically different features in the energy band structure near the Fermi level, will have drastically different Seebeck coefficients.

Figure S9 in the supplementary material shows the band structures for four minimum energy terminations. A comparison of those band structures and the band structure of the bulk, as shown in Fig. 4, clearly illustrates that the band structures for the bulk, (001), and (012) are very similar to each other and different from those of (300) and (101) terminated nanosheets. Thus, we conclude that the differences in the Seebeck coefficients for nanosheets are a result of the differences in the curvature of bands near the Fermi level in their respective band structures.

## IV. CONCLUSIONS

In summary, in this work, we have established that 2D B$4$C nanosheets are energetically stable despite having a non-van der Waals bonded bulk counterpart due to a unique reconstruction of B-bonds at the nanosheet surfaces. Additionally, we show that they exhibit a semiconductor-metal transition due to reduction in dimensionality. We find that they have theoretical Seebeck coefficients of upto 1600 $\mu $V/K, while also maintaining the encouraging thermoelectric properties of its bulk counterpart. The work provides first-principles atomic-scale insight into stability, electronic properties, and thermoelectric properties of 2D B$4$C nanosheets, guiding future application of B$4$C in applications such as mechanical strengthening, nanoelectronics, and thermoelectrics.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the surface bond length maps, element and orbital projected surface and bulk density of states, table of carbon percentages, and temperature-dependent Seebeck coefficient plots for all the 28 uniquely terminated 2D nanosheets simulated in this work. It also contains the band structures of the four minimum energy terminations for cleavage along the planes (001), (300), (101), and (012). Additionally, the electronic bandgaps and the formation energies for all 28 B$4$C nanosheets are shown.

## ACKNOWLEDGMENTS

The authors acknowledge the San Diego Supercomputer Center under the NSF-XSEDE Award No. DMR150006 and the Research Computing at Arizona State University for providing HPC resources. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. A.G. and A.K.S. acknowledge support by NSF DMR under Grant No. DMR-1906030. The authors thank start-up funds from Arizona State University, USA. The authors thank Q. H. Wang for fruitful discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**A. Gupta:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **T. Biswas:** Formal analysis (supporting); Writing – review & editing (supporting). **A. K. Singh:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request