1. IntroductionRecently, boron–carbon compounds have been widely applied in refractory and shielding material, semiconductor, thermoelectric materials, abrasive powder, and neutron detector due to their high hardness, high melting point, high elastic modulus, low density, and chemical inertness.[1] In recent decades, researchers have investigated various boron–carbon systems using experimental methods.[2–4] It was found that boron–carbon system is a very complicated material that can be identified as rhombohedral phase in the range of 8–20 at.%C. B4C is the third hardest material in nature and the most widely accepted boron–carbon binary system with space group
. A diamond-like BC3 was synthesized in diamond anvil cell at 2033± 241 K and 50 GPa.[5] By using a laser-heated diamond anvil cell and large-volume multi-anvil apparatus, a cubic BC5 was successfully synthesized.[6] Theoretically, Saal et al.[7] calculated the enthalpy of formation and lattice parameters for boron–carbon systems through the ab initio calculations. Their results show that for the carbon-rich 13.33% C composition, the B11C-CBC structure is the most stable structure. Using the ab initio pseudopotential local orbital approach, Tomanek et al.[8] predicted a layered BC3 with graphitic structure. The monolayer of BC3 is a semiconductor, and the metallic energy band of the corresponding bulk structure stems from interactions between the neighboring layers. By performing the particle swarm optimization (PSO) simulations, Luo et al.[9] predicted seven stable two-dimension boron–carbon structures (BC5, BC3, BC2, BC, B2C, B3C, and B5C). Li et al.[10] performed the first-principles calculations and obtained a hexagonal BC with two atom layers. Through the PSO algorithm and first-principles method, Xu et al.[11] predicted two novel structures of BC7 with space groups Amm2 and P-4m2. They claimed that P-4m2 BC7 is a super-hard material. Based on the ab initio density functional theory, Hu et al.[12] studied seven possible stacking configurations in hexagonal BC5. They proposed that BC5-I with AA stacking sequence is the most stable structure among the seven possible candidates. Although many experimental and theoretical studies have been carried out so far, the bulk boron–carbon binary compounds with an atomic ratio of B/C = 1 have not yet been reported in experiments. Thus, this prompts us to search for a potential BC structure with atomic ratio B/C = 1, and then study its electronic structure and thermal properties, which would further enrich the family of bulk boron–carbon binary compounds. The first principles method can predict new materials and their various properties,[13–18] so it is selected for the purpose.
3. Results and discussion3.1. Structure propertiesIn order to obtain the preferable structures of BC, we build the initial seed structures using the permutation method, then the local structural search is performed by the USPEX.[20] Through further optimizing and symmetrical analysis of the structures obtained from the USPEX, we obtain the final structure with the local minimum energy. This structure belongs to the orthorhombic crystal system with the space group Imm2. The atomic coordinates and lattice lattices of Imm2 BC that are calculated from the GGA and local-density approximation (LDA) are presented in Table 1 for comparing their differences.
Table 1.
Table 1.
 Table 1. Calculated lattice parameters a, b, and c with different exchange correlation functionals for Imm2 BC. .
|
Lattice constant/Å |
Deviation/% |
| LDA |
GGA |
| a |
2.6696 |
2.6954 |
0.96 |
| b |
13.6300 |
13.8546 |
1.62 |
| c |
4.4239 |
4.4624 |
0.86 |
|
Atomic coordinate |
Deviation/% |
| B1(4d) |
0.0000 |
0.0000 |
0.00 |
|
0.8913 |
0.8923 |
0.11 |
|
0.4050 |
0.4009 |
1.02 |
| B2(4d) |
0.0000 |
0.0000 |
0.00 |
|
0.7430 |
0.7427 |
0.04 |
|
0.6644 |
0.6702 |
0.87 |
| C1(4d) |
0.0000 |
0.0000 |
0.00 |
|
0.5500 |
0.5497 |
0.05 |
|
0.9292 |
0.9211 |
0.88 |
| C2(4d) |
0.0000 |
0.0000 |
0.00 |
|
0.2208 |
0.2200 |
0.36 |
|
0.3268 |
0.3331 |
1.89 |
| Table 1. Calculated lattice parameters a, b, and c with different exchange correlation functionals for Imm2 BC. . |
From Table 1, it is found that the lattice parameters show good agreement between the GGA and LDA values and their maximum deviation is smaller than 2%. Thus, the other properties of Imm2 BC, e.g., electronic structure and heat transport properties, are calculated by the GGA functional in the next.
The super-cell of Imm2 BC is shown in Fig. 1. It can be seen that the predicted Imm2 BC has an orthorhombic structure and possesses a cavity of 4.5 Å×6.1 Å (i.e., 27.5 Å2). Thus, Imm2 BC would be a potential molecular sieve material. Along the [010] direction, the C1 atom has three nearest neighbors, i.e., two B7 and one C15. The bond length of C1–B7 is about 1.572 Å. The bond length for C1–C15 is 1.376 Å, which is equal to that of 1, 3-butadiene and indicates that the C1–C15 bond is a typical C=C double bond. Compared to the C1 atom, the C3 atom has a different coordination number (4). The bond lengths of C3–B6, B7–C3, and B3–C3 are 1.591 Å, 1.585 Å, and 1.563 Å, respectively. Their maximum difference is only 0.028 Å, which implies that the C1 atom locates in a tetrahedron of B and has a sp3 hybridization bonding with four B atoms. Each B atom has three nearest neighbor C along the [001] and [010] directions. The calculated bond lengths are 1.572 Å, 1.584 Å, 1.563 Å, and 1.591 Å for the B7–C1, B7–C3, C6–B6, and B6–C3 atom pairs, respectively. The angles of C3–B7–C1, C3–B7–C1, C6–B6–C6, and C6–B6–C3 are 120.9°, 118.0°, 119.1°, and 120.3°, respectively. This suggests that the B atoms occur sp2 hybridization and have covalent bonding with the around C atoms. Next, the electronic localization function is calculated to well illustrate this problem.
The B atom usually exhibits a sp2 bonding with the others in its compounds,[25] and C usually shows two bond types, e.g., sp2 in graphite and sp3 in diamond. In this work, the obtained Imm2 BC per formula unit has seven valence electrons, which could form a stable orthorhombic structure. This is very different from the case of bulk BN that has eight electrons per formula unit. Therefore, the bonding analysis is helpful for understanding the electronic characteristic of the orthorhombic BC. The electronic localization function (ELF) is considered an important tool for studying the lone pair electrons and bonding electrons, and can be written as[26]
where
χBE(
r) =
D(
r)/
Dh(
r),
Dh(
r) = 3/5(6
π2)
2/3ρ(
r)
5/3, and
ρ is the electron density.
The volume ELF distribution of Imm2 BC is shown in Fig. 1(a). It is observed that the localized electron density appears between the C6–B6, B6–C3, C3–B3, C3–B7, B7–C1, C1–C15, and C15–B9 atom pairs, which suggests that the covalent bonding occurs between these atoms. This conclusion is in good agreement with the analysis of bond length and bond angle above.
We also investigate the different charge density to analyze the electronic properties of Imm2 BC as shown in Fig. 1(b). We find that the pz electrons transfer from the B7, C1, C15, B9, B3, and B6 atoms to the bonding region of the B–C (and C–C) bonds, suggesting that these atoms occur sp2 hybridization, while the C6 and C3 atoms exhibit obvious sp3 hybridization. From the above analysis on the ELF and different charge density, we can know that two different orbital hybridizations of C appear in Imm2 BC, and they are the important factor for the structure stability.
In order to understand the electronic properties of Imm2 BC, we calculate its total and partial density of states (DOS) at zero pressure as shown in Fig. 2. It is found that both the C1 and C4 (or both the C2 and C3) atoms are in the same atomic environment, leading to the same electronic states. This also applies to the B1 and B4 atoms as well as B2 and B3 atoms. As shown in Fig. 2, the valence band (VB) is dominated by the C1 and C2 s states mixing a few B1 and B2 s states in the energy range from −16.0 eV to −11.0 eV, suggesting that there exists covalent bonding between the C and B atoms. The s, px, and py states of the B1 and C1 atoms resonate in the range from −16 eV to 10 eV, indicating that they occur sp2 hybridization in this region. The energy states at the top of VB consist of the C1 and C4 pz states which can be assigned to C̿C double bonds. The C2 px and py states are almost identical with the pz state, illustrating that the C2 atom occurs sp3 hybridization. The B2 pz state is almost the same as the px state, and its py state shows the unoccupied case. This implies that the B2 atom occurs sp2 hybridization. Moreover, there are some energy states around the Fermi level, which indicates that Imm2 BC may be a half-metal.
The projected energy band is believed to be a useful tool to analyze the energy states, thus we plot the atomic projected energy band in Fig. 3 to clarify whether Imm2 BC is a half-metal. From Fig. 3, we see that the C and B px states cross the Fermi level, then form the unoccupied states in the W–X direction. The C and B pz states cross the Fermi level, resulting in that the occupied energy bands appear in the W–X direction.
Although the B and C px states overlap with their pz states as shown in Fig. 3, the electron transition between the px and pz states is symmetry forbidden. Therefore, Imm2 BC is a semiconductor and the calculated band gap is 1.3 eV. Because the B and C atoms form the covalent bonds and their pz states have the same symmetry that fulfills the transition rule, the electron transition would occur between the C and B pz bonding states and antibonding states at the S (or W) point in the Brillouin zone.
Figure 4(a) presents the simulated x-ray diffraction (XRD) spectra (using copper radiation). It is clear that Imm2 BC has three peaks at about 13°, 21°, and 26°. It can be seen from Fig. 4(b) that there are three Raman peaks at 20.6 THz, 30.2 THz, and 34.5 THz, respectively. The Raman peaks (33.1 cm−1 and 34.5 cm−1) are too close to be distinguished. The simulated XRD and Raman spectra can be used as the reference for experiments.
3.2. Elastic propertiesIn order to confirm the mechanical stability of Imm2 BC, we calculate its elastic constant with a 6× 6 symmetric matrix as follows:
For an orthorhombic crystal, the generalized elastic stability criteria
[27] are given by
One can see that the calculated
c11,
c22,
c33,
c44,
c55, and
c66 are greater than zero, which meets the elastic stability criteria
cii > 0. The other criteria are
M1 = 584 GPa,
M2 = 1336 GPa,
M3 = 666 GPa, and
M4 = 243 GPa, and they are also greater than zero, which satisfies the elastic stability criteria given by Eqs. (
4)–(
7). Therefore,
Imm2 BC is mechanically stable under the ambient condition.
For Imm2 BC, there are nine independent elastic constants as shown in Eq. (2). From the calculated elastic constants cij, some mechanical parameters of Imm2 BC, e.g., bulk modulus (B) and shear modulus (G), can be calculated through Voigt–Reuss–Hill (VRH) approximation.[28] The VRH BH is defined as the average BH = (BV +BR)/2 between the Voigt BV and Reuss BR, where
The VRH
GH = (
GV+
GR)/2 is the average between the Voigt upper
GV and Reuss
GR, where
The calculated
BH and
GH for
Imm2 BC are 133 GPa and 57 GPa, respectively. The
BH/
GH ratio is usually used to distinguish the brittleness or ductility of materials. The
BH/
GH ratio has a value larger than 1.75, suggesting that this material is ductile. Conversely, it is brittle. The obtained
BH/
GH ratio of
Imm2 BC is 2.33 (larger than 1.75), which indicates that
Imm2 BC is brittle.
BH,
GH, and
BH/
GH are suitable for describing polycrystalline systems, while Young’s modulus (
E) can be used to estimate the elastic anisotropy of single crystal materials.
[29] As shown in Fig.
5, the calculated
E of
Imm2 BC along the [100] direction is 502 GPa at zero pressure, considerably larger than the values (134 GPa and 90) along the [010] and [001] directions, which indicates that
Imm2 BC shows a strong elastic anisotropy.
3.3. Phonon propertiesThe calculated phonon dispersion and phonon partial densities of states are presented in Fig. 6, in which no imaginary phonon modes are found, confirming the dynamical stability of Imm2 BC. The B atoms are mainly responsible for the frequency vibrations in the range of 9–21 THz, and the C atoms contribute to the low frequency parts in the range of 2–9 THz. The maximum frequency peak at 42.3 THz is predominantly from the C=C stretching vibration. Moreover, the acoustic-optical frequency gap is not found in the phonon dispersion, and the significant acoustic-optical coupling in Imm2 BC is observed in the range of 5–15 THz. Thus, the lattice thermal conductivity of Imm2 BC contains some contribution from the optical modes. The dynamical stability of Imm2 BC is also examined by first-principles molecular dynamics simulations with a 1 fs time step. Imm2 BC is structurally complete when the temperature is below 850 K, and the break of the B–C bonds is observed when the temperature is higher than 870 K. Thus Imm2 BC is structurally stable between 300 K and 850 K.
3.4. Lattice thermal conductivityThe lattice thermal conductivity (LTC) is calculated by the following equation:[22]
where
V0 is the volume of the unit cell,
Cλ is the mode dependent heat capacity,
νλ is the phonon group velocity, and
τλ is the phonon lifetime. Based on the second and third order interatomic force constants obtained from the super-cell method, the LTC of
Imm2 BC along the
x,
y, and
z directions (i.e., [100], [010], and [001] directions) are calculated as shown in Fig.
7. To understand features of the intrinsic LTC of
Imm2 BC, the other contributions (e.g., boundary scattering) are not included in the present work.
From Fig. 7, it is found that at 300 K, the kb and kc of Imm2 BC have an almost equal value (i.e., 31 W·m−1·K−1 and 34 W·m−1·K−1), suggesting that Imm2 BC has almost the same atomic distribution along the [010] and [001] directions. Along the [100] direction, the ka at room temperature is 186 W·m−1·K−1, which is about 5 times higher than that along the [010] and [001] directions; this indicates that the LTC of Imm2 BC is anisotropic. To illustrate the LTC anisotropy in Imm2 BC, we calculate the average outer product of group velocity ν⊗ν along the [100], [010], and [001] directions. The ν⊗ν in the [100] direction is 25169 THz2·Å2, which is about 5 times higher than that (i.e., 4360 THz2·Å2 and 5296 THz2·Å2) in the [010] and [001] directions, suggesting that the LTC anisotropy in Imm2 BC is caused by the different group velocity along the crystal direction.
In order to understand the relationship between the thermal conductivity and phonon lifetime of Imm2 BC, we calculate the phonon lifetime τ by the first-principles super-cell method, and the results are shown in Fig. 8. It is found that the integral τ at 300 K is about 100 ps, and then decreases to 44 ps when the temperature increases to 600 K. When the temperature increases to 1000 K, the integral τ decreases to 25 ps, which is about 3 times smaller than that at 300 K. Through Eq. (12), it is well known that the k is proportional to the phonon lifetime τ. The average k of 84 W·m−1·K−1 at 300 K is about 3 times that at 1000 K, thus the decrease in phonon lifetime with the temperature is the major cause of the decrease of LTC. Moreover, it is found that the phonon lifetime in the region of 0–7.5 THz mainly comes from the contribution of three acoustic modes, and the other region is dominated by the optical modes. This indicates that the LTC might be from the contribution of both the acoustic and optical modes.
In order to verify the preliminary inspection above, we calculate the LTC of Imm2 BC with respect to the phonon branch as shown in Fig. 9. It is found that the maximum LTC contribution in the [100] direction comes from the contribution of the first phonon mode (23%), while the largest LTC along the [010] direction is derived from the second phonon mode (30%). These two maximum LTCs originate from the contribution of acoustic modes in common. The largest LTC in [001] direction stems from the fifth phonon mode, i.e., the optical mode, and the corresponding contribution is up to 22%. This is obviously different from the case in both the [100] and [010] directions. The calculated contribution of optical phonons to LTC along the [100], [010], and [001] directions are 49%, 59%, and 61%, respectively, suggesting that the acoustic-optical coupling is important for phonon transport in Imm2 BC.
