This study focuses on conducting an ab initio evolutionary investigation to search for stable polymorphs of iron diborocarbides with the formula FeB 2 C 2 . We also examined other forms of C contents, including FeB 3 C and FeBC 3 . Our findings reveal that the lowest energetic structure of FeB 2 C 2 is a semimetallic monoclinic phase with a space group (s.g.) of C2/m and a metastable metallic phase of FeB 2 C 2 is an orthorhombic structure with s.g. of Pmmm. In addition, structural and relative properties of FeB 3 C and FeBC 3 are performed and discussed to compare with FeB 2 C 2 . All predicted structures are dynamically and elastically stable, verified without negative phonon frequency and Born criteria, respectively. We also analyzed the energetic stability through calculated cohesive and formation energies, which showed that C2/m- FeB 2 C 2 is stable at low pressure. Interestingly, the C2/m and Pmmm phases of FeB 2 C 2 are hard materials with Vickers hardness ( H v ) of 22.40 and 27.52 GPa, respectively. Additionally, we examined the electron–phonon coupling of both FeB 2 C 2 phases. Unexpectedly, we found that the semimetallic C2/m- FeB 2 C 2 phase is a superconductor with a significant superconducting temperature ( T c ) exceeding 6 K. These findings provide some novel results for the Fe–B–C system and pave the way for investigating other metal borocarbides and related ternary compounds.

With the advancement in computer performance, material science has witnessed tremendous growth in the field of exotic and complex material modeling and prediction. This has led to a deep understanding of material science and the ability to obtain novel and advanced applications. Many impactful discoveries based on material searching have been reported. For instance, the Na–Cl system predicted by an evolutionary algorithm can be formed with unusual bonding at high pressure.1 Similarly, sulfur hydride systems were predicted2 to hunt the room-temperature superconductor that was later proved by experiment3 with the superconducting temperature ( T c ) approaching 200 K. These works have motivated the researchers to competitively discover other promising high-temperature superconductors such as lanthanum superhydride,4,5 calcium hydride,6 Mg/Ca hexahydride,7 and yttrium superhydride.8 Additionally, some hard or superhard materials, such as FeB 4 ,9, WB 4 ,10,11 and CrB 4 ,12 were initially predicted and later confirmed through experiments.13–15 These examples demonstrate that this strategy is a consistent, effective, and inexpensive way for researchers to model advanced materials for specific demands.

In recent times, predicting the atomic crystal structures of compounds beyond binary materials, such as ternary and quaternary compounds, is a challenging task due to the vast possibility of chemical space and geometrical configurations. However, some investigations have revealed that a systematic strategy can be used to predict complex structures with a high degree of freedom. For instance, the evolutionary USPEX code (Universal Structure Predictor: Evolutionary Xtallography)16,17 has been validated and accurately predicted a post-perovskite of MgSiO 3 above 100 GPa. Similarly, using ab initio random structure searching (AIRSS),18 Zhang et al. reported a search for the high-pressure cubic phase of LaBeH 8 ,19 which was later successfully synthesized at around 80 GPa by Song et al.20 Furthermore, USPEX has been used to predict a chalcopyrite phase of AgGaTe 2 and its high-pressure phases,21 which correspond with available experiments.22,23 These examples have inspired many more predictions and investigations for specific purposes, such as searching for room-temperature superconductors24–28 and exploring hard/superhard materials.29–34 

Metal borocarbides are a class of ternary compounds that have demonstrated potential for utilization in numerous applications, with a particular focus on metal diborocarbide ( MB 2 C 2 ). Their crystal structures tend to process the graphite-like B–C layer with the insertion of metal elements like Li, Mg, Be, Ca, and Y.35–40 Among these compounds, LiB 2 C 2 has been the subject of significant investigation owing to its promising superconducting properties, with the calculated superconducting temperatures ( T c ) with different methods in a range of 42–92 K.37, CaB 2 C 2 was reported to have a high Curie-temperature ferromagnet (770 K) despite the absence of transition metal or rare earth ions.38 Yan et al. reported that BeB 2 C 2 is a potentially hard material with a Vickers hardness ( H v ) of 28 GPa.39 Alongside, its high-pressure phases exhibit the superhard structures ( H v 40 GPa).40 Moreover, rare earth metals diborocabide ( LnB 2 C 2 ; Ln = rare earth) were thoroughly theoretically investigated in their structural, mechanical, and thermodynamic properties by Bao et al.41 

The above-mentioned literary works have inspired us to explore transition metal diborocabides, which can be achieved by replacing carbon content instead of boron atoms within transition metal tetraborides. Therefore, this study uses an evolutionary algorithm to predict the modified recipe for a potentially hard and superconducting FeB 4 ,13,42 which is iron diborocarbide ( FeB 2 C 2 ). Additionally, we also searched for the polymorphs of FeB 3 C and FeBC 3 . The minimum energy structures were analyzed for their stability and mechanical and electronic properties. Our results show that FeB 2 C 2 possesses a unique structure without a graphite-like B–C layer, unlike the other metal diborocarbides.35–40 The estimated H v of predicted FeB 2 C 2 is 22 GPa, and its T c is calculated to be 6 K. This discovery might encourage further research on other transition metal diborocarbides and related ternary compounds.

This recent work employed the Density Functional Theory (DFT) to predict the stability of various structures using an unbiased search method that involved both an evolutionary algorithm (USPEX)16,17 and DFT optimization. The prediction began with using USPEX interfacing with VASP code.43,44 The study was conducted under ideal ground-state conditions, including zero temperature (T = 0 K) and zero pressure (P = 0 GPa). To initiate the predictions, we generated 40 randomized structures in the first generation, followed by creating 30 population structures by combining heredity and mutations of the lowest enthalpy structure from the previous generation, with 50% and 30%, respectively. The remaining 20% of the population was randomized. Spin polarization was included during all predictions because of an intense induced magnetic moment of d states in Fe atoms. The predictions continued until the lowest enthalpy structure survived continuously for 30 generations. We employed the CASTEP code45 to fully optimize the lowest energetic structure and accurately verify the candidate structures obtained from the searches. The thermodynamic stability of predicted structures was evaluated using the Generalized Gradient Approximation (GGA)46 and an ultrasoft pseudopotential.47 The electronic configurations of Fe, B, and C were Fe: 3d 6 4s 2 , B: 2s 2 2p 1 , and C: 2s 2 2p 2 . All calculations were verified by an energy convergence tolerance of 10 6  eV/atom. The energy cutoff ( E c u t ) was set to 500 eV, and the k-points mesh was fixed accurately at 0.03  Å 1 for all calculations. The corresponding k-points mesh resolutions are 4 × 13 × 4 for C2/m- FeB 2 C 2 , 11 × 12 × 7 for Pmmm- FeB 2 C 2 , 9 × 9 × 3 for Cm- FeB 3 C, and 14 × 9 × 4 for P 1 ¯ - FeBC 3 . We employed the BFGS algorithm with a force convergence tolerance of 0.001 eV/Å for full atomic and lattice relaxations. We used the finite displacement perturbation method to verify the dynamical stabilities of the predicted structures.

The cohesive energy ( E c o h ) of FeB 4 x C x , which is the energy required to separate atoms (so-called gas phase) in a crystalline material, as calculated by using Eq. (1),
E c o h = E F e B 4 x C x [ E F e + ( 4 x ) E B + x E C ] ,
(1)
where E F e B 4 x C x is the total energy of FeB 4 x C x , and E F e , E B , and E C are the total energies of isolated Fe, B, and C atoms, respectively. Moreover, to compute the formation energy ( E f o r m ), which indicates whether any materials are thermodynamic stable or not, it was necessary to replace the total energy of isolated Fe, B, and C atoms with the total energy of their bulk phases. These bulk phases comprise body-centered cubic Fe, α -rhombohedral B, and graphite C, respectively.

Furthermore, the elastic constants ( C i j ) were derived to evaluate the elastic stability in accordance with the Born criteria.48,49 The main two crystal structures consisting of monoclinic and orthorhombic structures have the conditions of stability given by

Monoclinic structure:

  1. C i j > 0 ; i=j=1,2,3,4,5, and 6,

  2. [ C 11 + C 22 + C 33 + 2 ( C 12 + C 13 + C 23 ) ] > 0 ,

  3. ( C 33 C 55 C 35 2 ) > 0 ,

  4. ( C 44 C 66 C 46 2 ) > 0 ,

  5. ( C 22 + C 33 2 C 23 ) > 0 ,

  6. [ C 22 ( C 33 C 55 C 35 2 ) + 2 C 23 C 25 C 35 C 23 2 C 55 C 25 2 C 33 ] > 0 ,

  7. 2 [ C 15 C 25 ( C 33 C 12 C 13 C 23 ) + C 15 C 35 ( C 22 C 13 C 12 C 23 ) + C 25 C 35 ( C 11 C 23 C 12 C 13 ) ] [ C 15 2 ( C 22 C 33 C 23 2 ) + C 25 2 ( C 11 C 33 -C 13 2 )+C 35 2 (C 11 C 22 C 12 2 ) ] + gC 55 > 0 ; g = C 11 C 22 C 33 C 11 C 23 2 C 22 C 13 2 C 33 C 12 2 + 2 C 12 C 13 C 23 .

Orthorhombic structure:

  1. C i j > 0 ; i=j=1,2,3,4,5, and 6,

  2. [ C 11 + C 22 + C 33 + 2 ( C 12 + C 13 + C 23 ) ] > 0 ,

  3. ( C 11 + C 22 2 C 12 ) > 0 ,

  4. ( C 11 + C 33 2 C 13 ) > 0 ,

  5. ( C 22 + C 33 2 C 23 ) > 0 .

Subsequently, the bulk moduli (B) and shear moduli (G) were computed utilizing the Voigt–Reuss–Hill (VRH) approximation.50 The Vickers hardness ( H v ) was approximated employing Chen’s model,51 as expressed by Eq. (2),
H v = 2.0 ( k 2 G ) 0.585 3.0 ; k = G / B .
(2)
Additionally, the electron–phonon coupling (EPC) of C2/m and Pmmm structures of FeB 2 C 2 were examined using the isotropic Eliashberg theory.52 To accomplish this, we employed the Quantum Espresso (QE) code53 with an E c u t of 80 Ry for the plane-wave basis set. In order to calculate the EPC matrix elements, we utilized 2 × 2 × 1 ( 3 × 3 × 3 ) q-meshes, sampling individual EPC matrices with 8 × 8 × 4 ( 12 × 12 × 8 ) k-point meshes within the first Brillouin zone (BZ) for the C2/m (Pmmm) structure. The critical superconducting temperature (T c ) was determined by solving the Allen–Dynes equation,54 as obeyed in Eq. (3),
T c = ω l o g 1.2 exp [ 1.04 ( 1 + λ ) λ μ ( 1 + 0.62 λ ) ] ,
(3)
where ω l o g , λ , and μ are logarithmically averaged phonon frequency, integrated electron–phonon coupling constant, and effective Coulomb pseudopotential, respectively. In this work, μ was set at 0.10.

Based on evolutionary structural searches, each FeB 4 x C x ’s lowest energetic structures are obtained from different formula units per cell (f.u./cell). Hence, they consist of a total of nine structures for FeB 3 C, FeB 2 C 2 , and FeBC 3 , as listed in Table I. It is found that the space groups of all three compounds decrease as the number of f.u./cell increases. The total energies of the direct outcome structures predicted by USPEX, which interfaces the VASP code with the PAW method, were compared with those obtained from re-optimization using the CASTEP code with ultrasoft (US) pseudopotential. The comparison revealed that the energy tendencies of the two different methods are identical. To ensure the lowest energetic structures of FeB 4 x C x , the more accurate norm-conserving (NC) pseudopotential (using E c u t = 1000  eV) was also performed, and it confirmed the original results obtained from the predictions. Therefore, each lowest energetic structure of FeB 3 C, FeB 2 C 2 , and FeBC 3 was chosen for further calculations. FeB 3 C and FeB 2 C 2 have monoclinic structures with space groups (s.g.) of Cm and C2/m, respectively, while FeBC 3 has a triclinic structure (s.g. P 1 ¯ ). These structures were obtained from the searches at 4, 4, and 2 f.u./cell for FeB 3 C, FeB 2 C 2 , and FeBC 3 , respectively. Figure 1 shows the ball-and-stick representation between atoms of these predicted materials. According to the main objective, our primary focus is the monoclinic structure of FeB 2 C 2 . It is worth noting that C2/m- FeB 2 C 2 does not have graphite-like B–C layers with the inclusion of Fe atoms [Fig. 1(a)], which sets it apart from other metal diborocarbides. Instead, it has a complex structure made up of Fe–B and Fe–C clusters, as shown in Fig. 1(a). However, upon examining the structures of FeB 2 C 2 , we discovered that an orthorhombic structure (s.g. Pmmm) obtained from searching 1 f.u./cell has a graphite-like B–C layered structure, as illustrated in Fig. 1(b). This structure has a higher energy compared to C2/m- FeB 2 C 2 by 62 meV/atom (GGA-US), which is fair to claim that Pmmm- FeB 2 C 2 should be a metastable structure. Turning to other recipes, FeB 3 C has a stacking structure that resembles FeB 2 and BC, as depicted in Fig. 1(c). This implies that it may have a higher tendency to form an alloy rather than the ternary Fe–B–C compound. Another compound, FeBC 3 , exhibits a hexagon C pallet with the decoration of FeB 2 , which corresponds to an assembly 2D material [shown in Fig. 1(d)]. Table II illustrates the fully optimized lattice parameters of these predicted compounds.

FIG. 1.

The crystal structures of (a) C2/m- FeB 2 C 2 , (b) Pmmm- FeB 2 C 2 , (c) FeB 3 C, and (d) FeBC 3 .

FIG. 1.

The crystal structures of (a) C2/m- FeB 2 C 2 , (b) Pmmm- FeB 2 C 2 , (c) FeB 3 C, and (d) FeBC 3 .

Close modal
TABLE I.

Total energies and space groups of the lowest energetic structures of FeB4−xCx (where x = 1, 2, 3) obtained from searching at different formula units per cell (f.u./cell). The total energies yielded by the USPEX interfacing VASP codes using the PAW method are listed in the first column of energy values. The total energies of structures from re-optimization using both ultrasoft (US) and norm-conserving (NC) pseudopotentials with the CASTEP code are in the second and third columns, respectively. Boldfaces denote the lowest energetic structure of each recipes obtained from different psedopotentials.

f.u./cell Space group Energy (eV/atom)
GGA-PAW GGA-US GGA-NC
FeB3 R3m  −7.4748  −250.4210  −730.0057 
  Pmc21  −7.5145  −250.4423  −730.0418 
  Cm  −7.5508  −250.5027  −730.0873 
FeB2C2  Pmmm  −7.9141  −265.9241  −745.4526 
  Cm  −7.9111  −265.9232  −745.4492 
  C2/m  −7.9722  −265.9873  −745.5440 
FeBC3  Pm  −8.2688  −281.2460  −760.6960 
  P 1 ¯   −8.3905  −281.4458  −760.9594 
  P1  −8.3140  −281.3879  −760.9250 
f.u./cell Space group Energy (eV/atom)
GGA-PAW GGA-US GGA-NC
FeB3 R3m  −7.4748  −250.4210  −730.0057 
  Pmc21  −7.5145  −250.4423  −730.0418 
  Cm  −7.5508  −250.5027  −730.0873 
FeB2C2  Pmmm  −7.9141  −265.9241  −745.4526 
  Cm  −7.9111  −265.9232  −745.4492 
  C2/m  −7.9722  −265.9873  −745.5440 
FeBC3  Pm  −8.2688  −281.2460  −760.6960 
  P 1 ¯   −8.3905  −281.4458  −760.9594 
  P1  −8.3140  −281.3879  −760.9250 
TABLE II.

Lattice constants, fractional atomic positions, bulk moduli (B), shear moduli (G), and Vickers hardness (Hv) of the predicted structures of FeB3C, FeB2C2, and FeBC3.

Lattice parameters Atomic positions Ecoh Eform Mechanical parameters (GPa)
(eV) (meV) B G Hv
C2/m-FeB2C2  a = 13.493 Å b = 2.570 Å c = 17.534 Å α = γ = 90 , β = 149.7   Fe 1 ( 0.288 0.500 0.590 ) 4 i Fe 2 ( 0.781 0.000 0.262 ) 4 i B 1 ( 0.130 0.500 0.266 ) 4 i B 2 ( 0.998 0.500 0.561 ) 4 i B 3 ( 0.199 0.500 0.638 ) 4 i B 4 ( 0.061 0.000 0.410 ) 4 i C 1 ( 0.602 0.500 0.769 ) 4 i C 2 ( 0.612 0.500 0.985 ) 4 i C 3 ( 0.528 0.000 0.905 ) 4 i C 4 ( 0.706 0.000 0.851 ) 4 i   −7.456  244.67  166.46  22.40 
Pmmm-FeB2C2  a = 3.039 Å b = 2.728 Å c = 4.622 Å α = β = γ = 90   Fe 1 ( 0.000 0.500 0.000 ) 1 e B ( 0.500 0.000 0.187 ) 2 s C ( 0.500 0.500 0.660 ) 2 t   −7.393  70  224.61  174.58  27.52 
Cm-FeB3 a = 5.508 Å b = 5.395 Å c = 11.220 Å α = γ = 90 β = 118.97   Fe 1 ( 0.776 0.751 0.201 ) 4 b Fe 2 ( 1.010 0.500 0.443 ) 2 a Fe 3 ( 1.018 0.000 0.443 ) 2 a B 1 ( 0.673 0.750 0.604 ) 4 b B 2 ( 0.682 0.773 0.359 ) 4 b B 3 ( 1.180 0.750 0.856 ) 4 b B 4 ( 0.924 0.500 0.605 ) 2 a B 5 ( 0.589 0.500 0.265 ) 2 a B 6 ( 0.587 0.500 0.027 ) 2 a B 7 ( 0.621 0.000 0.027 ) 2 a B 8 ( 0.434 0.000 0.108 ) 2 a B 9 ( 0.923 0.000 0.604 ) 2 a C 1 ( 1.264 0.500 0.935 ) 2 a C 2 ( 1.263 0.000 0.935 ) 2 a C 3 ( 1.513 0.749 0.695 ) 4 b   −7.045  −83  287.57  142.97  13.10 
P 1 ¯ -FeBC3  a = 2.467 Å b = 3.888 Å c = 8.769 Å α = 79.94 β = 82.89 γ = 89.80   Fe ( 0.934 0.698 1.087 ) 2 i . B ( 0.446 0.253 1.108 ) 2 i C 1 ( 0.160 0.195 0.680 ) 2 i C 2 ( 0.232 0.442 0.536 ) 2 i C 3 ( 0.375 0.945 0.250 ) 2 i   −7.841  232  190.26  104.09  11.95 
Lattice parameters Atomic positions Ecoh Eform Mechanical parameters (GPa)
(eV) (meV) B G Hv
C2/m-FeB2C2  a = 13.493 Å b = 2.570 Å c = 17.534 Å α = γ = 90 , β = 149.7   Fe 1 ( 0.288 0.500 0.590 ) 4 i Fe 2 ( 0.781 0.000 0.262 ) 4 i B 1 ( 0.130 0.500 0.266 ) 4 i B 2 ( 0.998 0.500 0.561 ) 4 i B 3 ( 0.199 0.500 0.638 ) 4 i B 4 ( 0.061 0.000 0.410 ) 4 i C 1 ( 0.602 0.500 0.769 ) 4 i C 2 ( 0.612 0.500 0.985 ) 4 i C 3 ( 0.528 0.000 0.905 ) 4 i C 4 ( 0.706 0.000 0.851 ) 4 i   −7.456  244.67  166.46  22.40 
Pmmm-FeB2C2  a = 3.039 Å b = 2.728 Å c = 4.622 Å α = β = γ = 90   Fe 1 ( 0.000 0.500 0.000 ) 1 e B ( 0.500 0.000 0.187 ) 2 s C ( 0.500 0.500 0.660 ) 2 t   −7.393  70  224.61  174.58  27.52 
Cm-FeB3 a = 5.508 Å b = 5.395 Å c = 11.220 Å α = γ = 90 β = 118.97   Fe 1 ( 0.776 0.751 0.201 ) 4 b Fe 2 ( 1.010 0.500 0.443 ) 2 a Fe 3 ( 1.018 0.000 0.443 ) 2 a B 1 ( 0.673 0.750 0.604 ) 4 b B 2 ( 0.682 0.773 0.359 ) 4 b B 3 ( 1.180 0.750 0.856 ) 4 b B 4 ( 0.924 0.500 0.605 ) 2 a B 5 ( 0.589 0.500 0.265 ) 2 a B 6 ( 0.587 0.500 0.027 ) 2 a B 7 ( 0.621 0.000 0.027 ) 2 a B 8 ( 0.434 0.000 0.108 ) 2 a B 9 ( 0.923 0.000 0.604 ) 2 a C 1 ( 1.264 0.500 0.935 ) 2 a C 2 ( 1.263 0.000 0.935 ) 2 a C 3 ( 1.513 0.749 0.695 ) 4 b   −7.045  −83  287.57  142.97  13.10 
P 1 ¯ -FeBC3  a = 2.467 Å b = 3.888 Å c = 8.769 Å α = 79.94 β = 82.89 γ = 89.80   Fe ( 0.934 0.698 1.087 ) 2 i . B ( 0.446 0.253 1.108 ) 2 i C 1 ( 0.160 0.195 0.680 ) 2 i C 2 ( 0.232 0.442 0.536 ) 2 i C 3 ( 0.375 0.945 0.250 ) 2 i   −7.841  232  190.26  104.09  11.95 

The structures predicted in this study were examined for dynamic stability through analysis of the phonon dispersions and the projected phonon density of states (PhDOS). The results indicate that all proposed structures are dynamically stable with no imaginary frequencies. The phonon frequency feature is considered to understand the effect of the number of C atoms on structural configurations. Results show that the heavier Fe atoms are mainly responsible for low frequencies up to 15 THz, while lighter B and C atoms dominate at higher frequencies. For instance, Fig. 2(a) presents a bulk Fe–B–C compound of C2/m- FeB 2 C 2 with the highest frequency of around 25 THz, and a high-frequency region of about 25–42 THz, originating from the oscillations of the B–B, B–C, and C–C pairs. Meanwhile, Pmmm- FeB 2 C 2 has an obvious separated region of Fe and graphite-like B–C layer frequencies, as displayed in Fig. 2(b). Figure 2(c) shows the coupling vibrations of B and C atoms at frequencies beyond 25 THz, which correspond to the vibration of the embedded BC in the Cm- FeB 3 C structure previously mentioned. Moreover, Fig. 2(d) reveals the phonon branches of the hexagon C pallet in FeBC 3 reaching 46 THz ( 160 meV), corresponding to the highest zone of frequency of graphite.55,56

FIG. 2.

Phonon dispersions and partial phonon density of states (PhDOS) of the predicted structures of (a) C2/m- FeB 2 C 2 , (b) Pmmm- FeB 2 C 2 , (c) FeB 3 C, and (d) FeBC 3 . The blue, red, and green solid lines in the PhDOS panels represent the contributions of Fe, B, and C atoms, respectively.

FIG. 2.

Phonon dispersions and partial phonon density of states (PhDOS) of the predicted structures of (a) C2/m- FeB 2 C 2 , (b) Pmmm- FeB 2 C 2 , (c) FeB 3 C, and (d) FeBC 3 . The blue, red, and green solid lines in the PhDOS panels represent the contributions of Fe, B, and C atoms, respectively.

Close modal

The cohesive ( E c o h ) and formation energies ( E f o r m ) were evaluated to guide the possible synthesis of some predicted compounds. It is found that the E c o h values of all presented structures are negative, which decreases with increasing C content. The E c o h values are 7.045, 7.456, 7.393, and 7.841 eV/atom for Cm- FeB 3 C, C2/m- FeB 2 C 2 , Pmmm- FeB 2 C 2 , and P 1 ¯ - FeBC 3 , respectively. On the other hand, only E f o r m of Cm- FeB 3 C is negative with the value of 83 meV/atom; meanwhile, C2/m- FeB 2 C 2 , Pmmm- FeB 2 C 2 , and P 1 ¯ - FeBC 3 have the E f o r m values of 7, 70, and 232 meV/atom, respectively. The opposite trend between E c o h and E f o r m values of these compounds arises from the larger E c o h of graphite ( 9.18 eV/atom) in comparison to α -B ( 6.68 eV/atom). This finding suggests that these compounds can be synthesized by considering the gaseous phase. However, only FeB 3 C is energetically stable in the solid phase under the ground-state condition. Moreover, the formation enthalpies ( H f o r m ) as a function of pressure based on the simple relation of H = E + PV were performed, as demonstrated in Fig. 3. The H f o r m of all calculated structures is reduced by increasing pressure. We found that C2/m- FeB 2 C 2 can be energetically stable at lower than 1 GPa. The H f o r m of C2/m- FeB 2 C 2 is approximately 11 meV/atom at 2 GPa. This finding suggests that C2/m- FeB 2 C 2 might be synthesizable in conditions that are not extreme.

FIG. 3.

Formation enthalpies ( H f o r m ) as a function of pressure of the predicted structures consisting of C2/m- FeB 2 C 2 , Pmmm- FeB 2 C 2 , FeB 3 C, and FeBC 3 .

FIG. 3.

Formation enthalpies ( H f o r m ) as a function of pressure of the predicted structures consisting of C2/m- FeB 2 C 2 , Pmmm- FeB 2 C 2 , FeB 3 C, and FeBC 3 .

Close modal

In order to access the electronic property of the predicted compounds, the electronic band structures and density of states are calculated, as illustrated in Figs. 4(a)4(d). It is observed that the d states of Fe mainly contribute around the Fermi level, along with the p-states of B and C. It is also found that FeB 2 C 2 [Figs. 4(a) and 4(b)] and FeB 3 C [Fig. 4(c)] are non-magnetic materials, whereas FeBC 3 is a ferromagnetic material with a magnetic moment of 3.05  μ B [Fig. 4(d)]. Intriguingly, the non-semiconducting state of both C2/m and Pmmm phases of FeB 2 C 2 deviates their electronic structures from other metal diborocarbides such as BeB 2 C 2 , CaB 2 C 2 , YB 2 C 2 , and LnB 2 C 2 (Ln = rare earth metals) that prefer to process the semiconductors. In particular, the considered C2/m- FeB 2 C 2 compound has a dangling band above the Fermi level with a few crossings around the middle points of L M and Γ Z paths, which fairly corresponds to a semimetal feature. To verify a possible bandgap opening, a hybrid HSE06 functional,57 denoted by the blue dashed line in Fig. 4(a), is performed, but it does not significantly change compared to the standard GGA-PBE band structure.

FIG. 4.

Electronic band structures and partial density of states (PDOS) of (a) C2/m- FeB 2 C 2 , (b) Pmmm- FeB 2 C 2 , (c) FeB 3 C, and (d) FeBC 3 . The blue dashed lines in band structure panels of (a) represent the HSE06 band structures around the Fermi level. Meanwhile, those lines of panel (d) indicate the band structures of spin-down with the spin-up of black solid lines.

FIG. 4.

Electronic band structures and partial density of states (PDOS) of (a) C2/m- FeB 2 C 2 , (b) Pmmm- FeB 2 C 2 , (c) FeB 3 C, and (d) FeBC 3 . The blue dashed lines in band structure panels of (a) represent the HSE06 band structures around the Fermi level. Meanwhile, those lines of panel (d) indicate the band structures of spin-down with the spin-up of black solid lines.

Close modal
To evaluate their elastic stability and other relative mechanical properties, including bulk modulus (B), shear modulus (G), and Vickers hardness ( H v ), the elastic constants ( C i j ) of FeB 3 C, FeB 2 C 2 , and FeBC 3 , are computed and displayed in the matrix form. The elastic stability based on Born criteria48,58 manifests that all the predicted structures of FeB 4 x C x are elastically stable. According to the mechanical parameters listed in Table II, we found that FeB 3 C has the highest B (287.57 GPa), caused by its high all C 11 , C 22 , and C 33 of 452, 601, and 466 GPa, respectively. This finding also reflects the packed atomic structure of FeB 3 C. However, due to the characteristic of stacking FeB 2 and BC layers, it results in a low G value of 142.97 GPa, leading to a low H v of 13.10 GPa. Meanwhile, FeBC 3 has the lowest G (104.09 GPa) and H v (11.95 GPa), which corresponds to its being an assembly 2D material. Therefore, it has a low H v of 11.9 GPa. Oppositely, Pmmm- FeB 2 C 2 , which has the highest G value of 174.58 GPa, is the hardest phase compared to others presented in this work, with the H v of 27.52 GPa. Moreover, the most energetic stable C2/m- FeB 2 C 2 has the B and G values of 244.67 and 166.46 GPa, respectively, which has a consequence in H v of 22.40 GPa. Hence, this phase of FeB 2 C 2 is a hard material ( H v 20 GPa). Compared with its relative materials, the H v of C2/m- FeB 2 C 2 is lower than those of Pnnm- FeB 4 (31.5 GPa) and Pmmn- OsB 4 (29.7 GPa), but it is comparably high regarding P6 3 /nmc- RuB 4 (18.7 GPa),59 
C 2 / m F e B 2 C 2 ; C i j = ( 456 129 166 0 127 0 129 721 88 0 37 0 166 88 360 0 14 0 0 0 0 163 0 72 127 37 14 0 214 0 0 0 0 72 0 178 )
P m m m F e B 2 C 2 ; C i j = ( 281 28 35 0 0 0 28 844 173 0 0 0 35 173 726 0 0 0 0 0 0 286 0 0 0 0 0 0 92 0 0 0 0 0 0 95 ) ,
F e B 3 C ; C i j = ( 452 134 271 0 49 0 134 601 132 0 2 0 271 132 466 0 83 0 0 0 0 189 0 33 49 2 83 0 152 0 0 0 0 33 0 155 ) ,
F e B C 3 ; C i j = ( 824 126 113 48 37 35 126 294 207 121 9 14 113 207 277 116 21 20 48 121 116 122 13 6 37 9 21 13 192 131 35 14 20 6 131 135 ) .

Electron Localized Function (ELF) analysis was conducted to study the chemical bonding characteristics of hard phases of FeB 2 C 2 . Figure 5(a) reveals a significant ELF between the B cages and C–C pairs, suggesting covalent bonding between these atoms. Moreover, the cleaved ELF at the (050) plane [Fig. 5(b)] shows a significant ELF pattern between Fe and the surrounding B and C atoms. Therefore, these ELF features support the moderately high hardness of C2/m- FeB 2 C 2 . Furthermore, the strong ELF, as shown in Fig. 5(c), demonstrates strong covalent bonds between B and C atoms in Pmmm- FeB 2 C 2 , which corresponds to its highest hardness.

FIG. 5.

Electron localized function (ELF) of C2/m- FeB 2 C 2 projected at planes of (020) and (050) as labeled by (a) and (b), respectively. The ELF of Pmmm- FeB 2 C 2 at the (200) plane is indicated by panel (c). The positions of Fe, B, and C atoms contributing significantly to ELF are shown on the presented planes.

FIG. 5.

Electron localized function (ELF) of C2/m- FeB 2 C 2 projected at planes of (020) and (050) as labeled by (a) and (b), respectively. The ELF of Pmmm- FeB 2 C 2 at the (200) plane is indicated by panel (c). The positions of Fe, B, and C atoms contributing significantly to ELF are shown on the presented planes.

Close modal

To extend the other important related properties in both phases of FeB 2 C 2 , the superconducting temperature ( T c ), as appearing in LiB 2 C 2 , is preliminary evaluated because of the evidence that the metallic state in the Pmmm phase and the dangling bands around the Fermi level in the C2/m phase. The isotropic Eliashberg equation is used to directly solve for the spectral function [ α 2 F( ω )] and the integrated electron–phonon coupling (EPC) parameter λ ( ω ) for both phases, and the results are presented in Figs. 6(a) and 6(b). For Pmmm- FeB 2 C 2 , it is found that the α 2 F( ω ) contributes to EPC up to 37 THz, resulting in the calculated λ reaching 0.35. This is mainly due to the contribution of Fe at low-frequency region (<9 THz), with a rapid increase at around 32 THz caused by the coupling of C–B vibration. The logarithmically averaged phonon frequency ( ω l o g ) is 587 K. Consequently, the T c of Pmmm- FeB 2 C 2 is calculated to be 1.01 K, which is very low compared to LiB 2 C 2 (42.3 K). However, the semimetal C2/m- FeB 2 C 2 surprisingly displays a higher λ (0.47) than its counterpart. The λ is not only dominantly enhanced by Fe but also linearly increased till the maximum frequency. As a result, the T c of C2/m- FeB 2 C 2 is calculated to be 6.1 K with an ω l o g of 514 K. It is unusual for a semimetal with a low charge carrier at the Fermi level to exhibit a superconducting state with a significant critical temperature, but this study has shown that the hard monoclinic C2/m structure of FeB 2 C 2 can also display superconducting properties. This opens up the possibility of further investigations into other metal borocarbides and related materials in order to design novel multifunctional materials.

FIG. 6.

The α 2 F ( ω ) (yellow shade), λ (dashed orange line), and the partial phonon density of states (PhDOS) of (a) Pmmm- FeB 2 C 2 and (b) C2/m- FeB 2 C 2 . The blue, red, and green solid lines represent the contributions of Fe, B, and C atoms in the PhDOS, respectively.

FIG. 6.

The α 2 F ( ω ) (yellow shade), λ (dashed orange line), and the partial phonon density of states (PhDOS) of (a) Pmmm- FeB 2 C 2 and (b) C2/m- FeB 2 C 2 . The blue, red, and green solid lines represent the contributions of Fe, B, and C atoms in the PhDOS, respectively.

Close modal

In summary, we thoroughly investigated the atomic crystal structures and the mechanical and electrical properties of FeB 4 x C x (where x = 1, 2, 3) using an ab initio evolutionary search. The results showed that FeB 3 C and FeB 2 C 2 have monoclinic crystal structures with space groups of Cm and C2/m, respectively. On the other hand, FeBC 3 has a triclinic structure with space group P 1 ¯ . Interestingly, C2/m- FeB 2 C 2 does not process a graphene-like BC layer like other metal diborocarbides, while another metastable Pmmm- FeB 2 C 2 does, and then it is chosen to study further. We also analyzed the atomic configurations of all predicted structures and found that FeB 3 C has a stacking structure of FeB 2 and BC, while FeBC 3 has a hexagonal C pallet with the decoration of FeB2. Our analysis of all predicted phases indicates that they are both dynamically and elastically stable, as confirmed by the absence of negative phonon frequency and Born criteria. In addition, the cohesive energy and formation energy are evaluated. It is found that only FeB 3 C is a stable phase when considering the formation energy, which respects the solid phase formation. However, C2/m- FeB 2 C 2 can be an energetically stable phase at low pressure with a formation energy of 11 meV/atom at 2 GPa. We also studied the electronic band structure and density of states, revealing that FeB 3 C is a non-magnetic metal, while FeBC 3 is a ferromagnetic material with a magnetic moment of 3.05  μ B . Furthermore, C2/m- FeB 2 C 2 is a semimetallic material, and Pmmm- FeB 2 C 2 is a metal. These C2/m- FeB 2 C 2 and Pmmm- FeB 2 C 2 have noteworthy Vickers hardnesses ( H v ) of 22.40 and 27.52 GPa, respectively, surpassing that of FeB 3 C (13.10 GPa) and FeBC 3 (11.95 GPa). At last, we also examined the electron–phonon coupling of both FeB 2 C 2 phases, resulting in the T c values of 6.1 and 1.1 K, for C2/m- FeB 2 C 2 and Pmmm- FeB 2 C 2 , respectively.

This work was supported by the Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation, Thailand Science Research and Innovation (TSRI) under Grant No. RGN63-243. T.B. acknowledges funding from Thailand Science Research and Innovation Fund Chulalongkorn University. The ThaiSC of Thailand and the SNIC of Sweden are acknowledged for providing computing facilities.

The authors have no conflicts to disclose.

Komsilp Kotmool: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Writing – original draft (equal). Udomsilp Pinsook: Formal analysis (equal); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Wei Luo: Funding acquisition (equal); Resources (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal). Rajeev Ahuja: Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal). Thiti Bovornratanaraks: Funding acquisition (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal).

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

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