Novel hard and unusual superconducting monoclinic phase of FeB₂C₂: An ab initio evolutionary study Open Access

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 superconductors^24–28 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.

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. $( C 22 + C 33 −2 C 23 )>0$⁠.

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.

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

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.

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.

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.

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.

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.