## 1 Introduction

In the past decades, the binary Fe-B alloy system has been a subject of numerous experimental and theoretical studies concerning hardness, melting point, wear resistance, corrosion resistance, and ferromagnetism [1–3]. According to the Fe-B equilibrium phase diagram, there are two stable iron borides at ambient temperature: single boride layer (Fe_{2}B) or double (FeB and Fe_{2}B) layers [4]. The metastable phase, Fe_{3}B, appears during formation of Fe_{2}B. The metastable o-Fe_{3}B phase has also been obtained in Fe-B glasses by quenching and annealing [5]. These compounds can be prepared using numerous equilibrium or non-equilibrium methods, such as ball milling, chemical vapor deposition (CVD), physical vapor deposition (PVD), magnetic sputtering and thermal chemical reactions. Fe_{2}B can also be prepared as a bulk single crystal [6]. FeB was prepared in a form of nanoparticles by chemical reduction method [7] in order to improve the cycle stability of PuNi_{3}-type hydrogen storage electrodes [8, 9]. The properties depending on process time and temperature, such as structure parameters, hardness, Young’s modulus and fracture toughness of iron boride layers have been investigated experimentally [10]. The electronic structure, stability and elastic constants of the three FexB compounds were calculated in the literature [11] using DFT. It was indicated that pressure affects the structure, mechanical and magnetic properties of iron borides, and spin-polarized calculations were important to obtain the correct ground state properties of FexB compounds. We tried to demonstrate that the structure properties and magnetic moment change strongly with pressure.

In this work, we performed the first principles calculations of anisotropic elastic properties for the three structures Fe_{x}B (x = 1, 2, 3) at 0 GPa pressure and at a critical pressure when a ferromagnetic material undergoes transition to a nonmagnetic state (NM). Anisotropy index A and directional dependences of bulk and Young’s moduli were investigated. From the anisotropic elastic properties, the easy and hard axes of magnetization of the three compounds were predicted, which revealed that for bcc Fe the highest density of atoms is in the 〈1 1 1〉 direction, and consequently 〈1 1 1〉 is the hard magnetization axis. In contrast, the atom density is the lowest in 〈1 0 0〉 direction, and consequently 〈1 0 0〉 is the easy magnetization axis. Certainly, since bcc iron is a cubic crystal, all six cube edge orientations 〈1 0 0〉, 〈0 1 0〉, 〈0 0 1〉, 〈

We hope that our study will provide a useful guidance for future works on the Fe-B compounds.

Finally, we concluded that spin-polarization and pressure are of significant importance in determining the anisotropic elastic properties of iron borides.

## 2 Structure aspects and calculation methods

FeB and Fe_{3}B belong to an orthorhombic space group Pnma (Fig. 1) [13–16]. Both structures contain four formula units per cell. In Fe_{3}B, the isotype of Fe_{3}C, iron atoms are distributed over two distinct lattice sites: the general Fe sites (Wyck-off position 8d) and the special Fe sites (Wyckoff position 4c). In contrast, Fe_{2}B (Fig. 1) belongs to the body-centered tetragonal Bravais lattice with I4/mcm space group where the unit cell contains four equivalent Fe atoms in the positions of point group mm and two equivalent B atoms in the positions of point group 42 [11]. The B atoms in Fe_{2}B are located between two layers of Fe atoms in a distorted, closely packed arrangement.

Total energy calculations were performed within the density functional theory (DFT) [17]. CASTEP code was used in this study. The last uses the plane wave in reciprocal space [18]. The ultrasoft Vanderbilt pseudopotentials were employed to represent the electrostatic interactions between valence electrons and ionic cores [19]. They were used with the following valence electronic configurations Fe: 3d^{6}4s^{2} and B: 2s^{2}2p^{1}. Generalized gradient approximation PBE-GGA was used for exchange-correlation energy calculations [20]. The kinetic energy cut-off value was selected as 500 eV, which was sufficient to obtain reliable results.

Total energies were evaluated in the first irreducible Brillouin zone with the following Monkhorst-pack grids [21]: (6 × 10 × 8) for FeB, (10 × 10 × 10) for Fe_{2}B and (10 × 12 × 8) for Fe_{3}B. It is known that the ground states of several Fe_{x}B compounds are ferromagnetic [22].

Structural and elastic properties were calculated for both FM and NM states in the three compounds. The convergence criteria of total energy and structure optimization were set to fine quality with the energy tolerance of 10^{−6} eV/atom. BFGS (Broydene-Fletchere-Goldarbe-Shanno) optimization method was used to obtain the equilibrium crystal structures of FexB with maximum atom displacement and force set to 0.002 Å and 0.001 eV/Å.

The cohesive energy (Ecoh) of a material, (which is a useful fundamental property), is a measure of the relative binding forces. The stability of our compounds was evaluated by calculating two energy parameters, cohesive energy Ecoh and formation energy E_{f} defined as:

where E_{coh} (Fe_{x}B) is the cohesive energy of Fe_{x}B per unit formula; E_{f} (Fe_{x}B) is its formation energy; Ecoh (Fe) is the cohesive energy of iron element per atom; E_{total} (Fe_{x}B, Cell) is the total calculated energy of Fe_{x}B per conventional unit cell; E_{iso}(Fe) is the total energy of an isolated Fe atom and finally n refers to the number of unit formula FexB in the conventional cell. The calculation method for E_{coh} (FexB) can also be used to evaluate the cohesive energy of pure elements B and Fe. Equation 1 and equation 2 require negative values of Ecoh (FexB) and Ef (Fe_{x}B) to refer to a thermodynamically stable structure. The crystal structures of Fe_{x}B studied in this paper were built based on experimental results.

## 3 Results and discussion

### 3.1 Structural properties and stability

The calculated lattice parameters, unit cell volume, bulk modulus, cohesive energy and the formation energy for FexB along with the available experimental and previous theoretical data for comparison have been discussed by Gueddouh et al. [11]. The results show that the calculated structure parameters are in good agreement with the experimental values. The calculated values of cohesive energies of FeB, Fe_{2}B and Fe_{3}B indicate that all of the Fe_{x}B compounds are stable. Furthermore, the cohesive energies decrease from FeB to Fe_{3}B, which is mainly caused by the increase in the volume concentration of Fe atoms.

### 3.2 Pressure effects

In order to induce a significant change in a structure, high pressure is usually needed to study the material. By increasing the pressure, a transfer from magnetic to nonmagnetic state occurs, which causes an extinction of the magnetic moment; the critical (transition) pressure was estimated as 77 GPa, 85 GPa and 55 GPa for FeB, Fe_{2}B and Fe_{3}B, respectively, as shown in Fig. 2. Usually, the critical pressure at which a ferromagnetic material undergoes transition to a NM state is defined as Pc = -ΔE/ΔV where ΔE is the difference between NM and spin-polarized (SP) equilibrium total energy by unit cell and ΔV is the respective difference between NM and SP equilibrium volumes. This definition of critical pressure (P_{c}) was first employed by Mohn et al. [23, 24] in their work on magneto-elastic anomalies in Fe— Ni Invar alloys [23] and NiFe_{3}N and PdFe_{3}N nitrides [24]. This definition is also used in the study on magnetic transition of intermetallic bilayers and substituted iron nitrides [25].

The calculated percentage change of volume at 0 GPa and at transition pressure for our three compounds [11] showed a volume compression of 15 %, 20 % and 19 % for FeB, Fe_{2}B and Fe_{3}B, respectively, at the applied pressure, which resulted in an increase in the bulk modulus of our compounds by 47.7 %, 62 % and 61.8 %.

The formation energy E_{f} was calculated to check the probability of thermodynamic existence of FexB under pressure. All formation energies are negative indicating that all the structures in the two pressure conditions are thermodynamically stable. The formation energies of FexB in magnetic state are less than those of Fe_{x}B in NM state by 4.7 %, 3.9 %, 7.1 %, implying that Fe_{x}B(FM) has better thermodynamic stability.

The calculated magnetic moments of our compounds at 0 GPa are in good agreement with theoretical and experimental values [26] and are 2.003 μB, 1.83 μB and 1.12 μB for Fe_{3}B, Fe_{2}B and FeB, respectively, which results from the difference between the density of spin-up and spin-down electrons and corresponds to the saturated magnetic moment, μ_{sat}, at T = 0, as shown Fig. 3. It appears that Fe_{3}B has the highest magnetic moment which leads to a shift in the Slater-Pauling curve [27]. These moments are smaller than the magnetic moments of pure bcc ferromagnetic iron which behaves as a weak ferromagnetic with a magnetic moment of 2.217 μB [28]. Indeed, when B atoms are inserted in Fe crystal, the volume concentration of metallic Fe—Fe bonds decreases and they are replaced by the newly formed covalent Fe—B and B—B.

The magnitude of the magnetic moment is strongly related to the volume. Thus, the values of equilibrium volume obtained in the magnetic state are larger than in NM state. A possible origin of this dependence is a magneto-volume effect [29]. Because the Pauli Exclusion Principle operates for parallel spins, the electron kinetic energy in the spin-polarized state is higher, and volume expansion relaxes the kinetic energy. Consequently, the magnetic (high-spin) state has a larger volume than the non-magnetic state [30]. However, the bulk modulus decreases from FeB to Fe_{3}B. Also due to the pressure effect, the bulk modulus has increased from 0 % to 47.7 % for FeB, from 0 % to 62 % for Fe2B and from 0 % to 61.8 % for Fe3B. In the NM state the bulk modulus B is in general larger than in the magnetic state. The low value of bulk modulus in the magnetic state points out to a larger compressibility. This means that the system is “softer” when it is magnetically ordered and “harder” when it is not. The spin-polarization calculations are important to obtain the correct ground state properties of Fe_{x}B ferromagnetic compounds. The calculated magnetic moment as a function of pressure is presented in Fig. 2. Total DOS at the Fermi level for Fe_{x}B increases under pressure by 21 %, 50 % and 32.5 % for FeB, Fe_{2}B and Fe_{3}B, respectively (Fig. 3). This enhanced N (E_{f}) is derived entirely from the Fe 3d states, with negligible contribution from the B 2p states. Following the above arguments we may predict that the appearance of superconductivity in Fe_{x}B under pressure is similar to the case of iron that undergoes a transition to superconducting phase above 30 GPa when it loses its magnetic moment [31]. It is easy to observe that the magnetic moment decreases with increasing pressure.

## 4 Elastic properties under pressure

It is well known that elastic properties reflect interatomic interactions and are related to some fundamental physical properties, such as thermal expansion, phonon spectra and equations of state [32]. The elastic constants of single crystalline Fe_{x}B compounds are presented in Table 1. Generally, the elastic constants C_{11}, C_{22} and C_{33} are very high, both at zero and critical pressure, which indicates high resistance to the axial compression in these directions. However, the magnitude orders in three axes are different. For orthorhombic FeB and Fe_{3}B, the order is C_{33} > C_{22} > C_{11}. For tetragonal Fe_{2}B the order is C_{11} > C_{33}. Since the tetragonal structure can be regarded as a special case of orthorhombic structure with an additional condition of a = b, the mechanical stability criteria can be represented in a uniform manner for orthorhombic structure [33]:

The calculated full set of elastic constants of Fe_{x}B (GPa) [11].

Species | Elastic constants | ||||||||
---|---|---|---|---|---|---|---|---|---|

C_{11} | C_{22} | C_{33} | C_{12} | C_{13} | C_{23} | C_{44} | C_{55} | C_{66} | |

FeB (0 GPa) | 389.82 | 438.36 | 557.07 | 286.85 | 183.12 | 239.76 | 218.8 | 132.25 | 212.01 |

373.7^{a} | 434.1^{a} | 503.4^{a} | 246^{a} | 184.4^{a} | 209.1^{a} | 207.4^{a} | 117.7^{a} | 193.7^{a} | |

FeB (77 GPa) | 718.5 | 752.6 | 993.9 | 583.2 | 350.1 | 492.4 | 296.4 | 243.3 | 354.4 |

Fe_{2}B (0 GPa) | 459.7 | 426.3 | 165.6 | 132.3 | 162.6 | 173.7 | |||

413^{b} | 389^{b} | 154^{b} | 132^{b} | 148^{b} | 157^{b} | ||||

Fe_{2}B (85 GPa) | 1010.4 | 839 | 541.3 | 488.4 | 298 | 288.4 | |||

Fe_{3}B (0 GPa) | 281.7 | 337.5 | 354 | 126.7 | 165.3 | 182.2 | 130.2 | 118.7 | 175.6 |

263.3^{a} | 302.7^{a} | 318.4^{a} | 133.5^{a} | 162.8^{a} | 178.6^{a} | 110.2^{a} | 101.2^{a} | 158.7^{a} | |

Fe3B (55 GPa) | 694 | 826 | 778.5 | 485.5 | 444.75 | 403.6 | 193.1 | 195.8 | 307.7 |

On the other hand, the mechanical stability leads to restrictions on the elastic coefficients under isotropic pressure as follows:

For tetragonal structure the elastic constants under pressure P are related to those under zero pressure, as follows [34]:

The stability criteria of material under pressure are similar to those under zero pressure, just replacing C_{ij} with C_{jj} (i = j = 1, 2, 3, 4, 5, 6) [35]. As for Fe_{2}B in the tetragonal structure, there are six independent elastic constants, C_{11}, C_{12}, C_{13}, C_{33}, C_{44}, C_{66}, because C_{22} = C_{11}, C_{23} = C_{13}, C_{44} = C_{55} as a result of the crystal symmetry. The single crystal elastic coefficients (C_{jj}) satisfy the stability criteria, which leads to the following restrictions on the elastic coefficients under isotropic pressure:

C_{ij} are the elements of elastic coefficient matrix. The arithmetic average of the Voigt and Reuss bounds is known as the Voigt-Reuss-Hill (VRH) average, which is regarded as the best estimate for the theoretical value of polycrystalline elastic modulus [11]:

The Young’s modulus and Poisson ratio can be computed from the formula [11]:

A larger B/G value (>1.75) for a solid indicates ductile behavior while a smaller B/G value (<1.75) usually means brittle material [36]. Similarly, Poisson ratio ν > 0.26 relates to ductile compounds usually [36]. At both pressures studied here 0 GPa and the critical pressure, B/G > 1.75 and ν > 0.26 is larger than 0.26 for FeB and Fe_{3}B (Table 3), which indicates that FeB and Fe_{3}B are ductile. The values of B/G and ν for Fe_{2}B are 1.52 and 0.23, respectively, at 0 GPa pressure which means that Fe_{2}B is brittle. In contrast, at critical pressure Fe_{2}B is ductile (B/G = 2.48, ν = 0.32).

Polycrystalline elastic properties and anisotropy factors of Fe-B system.

Species | By | BR | G_{V} | G_{R} | A_{1} | A_{2} | A_{3} | A_{u} | A_{G} | A_{B} |
---|---|---|---|---|---|---|---|---|---|---|

FeB (0 GPa) | 308.37 287.8 ^{a} | 302.80 284.6 ^{a} | 155.66 148.5 ^{a} | 125.47 130.9 ^{a} | 1.57 1.63 ^{a} | 1.04 0.91 ^{a} | 3.38 2.45 ^{a} | 1.22 0.68 ^{a} | 10.74 5.85 ^{a} | 0.91 0.62 ^{a} |

FeB (77 GPa) | 621.69 | 612.49 | 260.17 | 190.38 | 1.17 | 1.28 | 4.65 | 1.85 | 15.49 | 0.75 |

Fe2B (0 GPa) | 245.11 222.7 ^{a} | 244.07 221.8 ^{a} | 160.82 144.4 ^{a} | 160.21 143.6 ^{a} | 1.18 1.07 ^{a} | 1.18 1.07 ^{a} | 0.26 1.23 ^{a} | 0.02 0.03 ^{a} | 0.19 0.28 ^{a} | 0.21 0.20 ^{a} |

Fe2B (85 GPa) | 655.09 | 646.32 | 266.31 | 260.30 | 1.23 | 1.23 | 0.34 | 0.13 | 1.14 | 0.67 |

Fe3B (0 GPa) | 213.51 203.8 ^{a} | 207.55 199.5 ^{a} | 118.15 101.3 ^{a} | 107.50 90.6 ^{a} | 1.71 1.72 ^{a} | 1.45 1.53 ^{a} | 1.92 2.12 ^{a} | 0.52 0.61 ^{a} | 4.72 5.58 ^{a} | 1.41 1.07 ^{a} |

Fe3B (55 GPa) | 551.81 | 550.01 | 203.62 | 186.77 | 1.33 | 0.98 | 2.24 | 0.45 | 4.31 | 0.16 |

The calculated bulk, Young’s (E) and shear modulus (G) of Fe_{x}B (under 0 and critical pressure, in GPa), Poisson’s ratio (v) and B/G ratio along with other available values.

Species | B | E | G | v | B/G |
---|---|---|---|---|---|

FeB (0 GPa) | 305.6 | 365.7 | 140.6 | 0.30 | 2.174 |

286.6^{a} | 360.5^{a} | 139.7^{a} | 0.29^{a} | 2.05^{a} | |

FeB (77 GPa) | 617.1 | 567.5 | 225.3 | 0.34 | 2.76 |

Fe2B (0 GPa) | 244.6 | 395.2 | 160.5 | 0.23 | 1.52 |

222.3^{a} | 355^{a} | 144^{a} | 0.23^{a} | 1.54^{a} | |

Fe_{2}B (85 GPa) | 651 | 695.4 | 263 | 0.32 | 2.48 |

Fe3B (0 GPa) | 210 | 285.3 | 112 | 0.27 | 1.875 |

201.6^{a} | 248.5^{a} | 96^{a} | 0.3^{a} | 2.1^{a} | |

Fe_{3}B (55 GPa; | 551 | 524 | 195.2 | 0.34 | 2.82 |

## 5 Elastic anisotropy

It is known that elastic anisotropy correlates with anisotropic plastic deformation and behavior of microcracks in material. Hence, it is important to study elastic anisotropy in intermetallics structures in order to further understand these properties and improve their mechanical durability. Most of crystals exhibit elastic anisotropy to some extent, and several criteria have been developed to describe it. The elastic anisotropy of a crystal can be characterized by the universal anisotropic index A^{U} and by the indexes describing the behavior in shear and compression (AG and AB). The universal elastic anisotropy index A^{U} and indexes AG and AB for a crystal with any symmetry may be proposed as follows [37, 38]:

where B_{V} (G_{V}) and B_{R} (G_{R}) are the bulk modulus (shear modulus) in the Voigt and Reuss approximations respectively. A^{U} = 0 corresponds to the isotropy of the crystal. The deviation of A^{U} from zero defines the extent of single crystal anisotropy and accounts for both shear and bulk contribution, unlike all other existing anisotropy measures. Thus, A^{U} represents a universal measure to quantify a single crystal elastic anisotropy. A_{B} = A_{G} = 0 represents the elastic isotropic crystal, while A_{B} = A_{G} = 1 means the maximum elastic anisotropy [39]. From Table 2, it can be seen that the mechanical anisotropy of FeB is stronger than in other structures. In Fig. 5, we have outlined the projections of Young’s modulus in (0 0 1), (0 1 0) and (1 0 0) crystal planes. We can clearly show that the anisotropy of FeB is stronger than in Fe2B and Fe3B in the three planes. The results are also in good agreement with the calculated anisotropic indexes in Table 2.

The shear anisotropic factors provide a measure of the degree of anisotropy in atomic bonding in different crystallographic planes. The shear anisotropic factor for an orthorhombic crystal can be measured by three factors (Zener ratios) [40–42]:

- The shear anisotropic factor for the {1 0 0} shear planes between 〈0 1 1〉 and 〈0 1 0〈 directions is defined as:
- The anisotropic factor for the {0 1 0} shear planes between 〈1 0 1〉 and 〈0 0 1〉 directions is:and
${A}_{2}=\frac{4{C}_{55}}{{C}_{22}+{C}_{33}-2{C}_{23}}$ - The anisotropic factor for the {0 0 1} shear planes between 〈1 1 0〉 and 〈0 1 0〉 directions is:

The calculated values of anisotropic factors for iron borides are shown in Table 2. For an isotropic crystal, all three factors must be one, while any value smaller or greater than one is a measure of degree of elastic anisotropy possessed by the crystal. Our results thus indicate a very large shear anisotropy on the (1 0 0) and (0 0 1) planes of FeB and Fe3B due to the anomalously high C44 and C_{66} relatively to C55. Thus, for Fe_{2}B the large shear anisotropy is on the (0 0 1) plane due to high C44 compared to C66.

Taking also into account the strength characteristics of the studied compounds, which have low values of G/B ratio (0.46 for FeB, 0.66 for Fe_{2}B and 0.53 for Fe_{3}B), the ductility of the iron borides is a very important advantage and therefore they are intrinsically brittle.

The spin polarization and pressure increase the anisotropic factors A_{1} by 25 % and 22 % for FeB and Fe_{3}B, respectively, and A_{2} by 32 % for Fe_{3}B, but reduce the anisotropic factors A3 by 37 % and 17 % for FeB and Fe_{3}B, and A_{2} by 23 % for FeB, which means that the direction of easy axis of magnetization is 〈1 0 0〉 for FeB (C_{11} _{3}B the easy axis direction is 〈1 0 0〉 (C_{11}

The anisotropy is only dependent on crystal symmetry. The structure of the crystal has been changed under spin polarized moment which varied a, b and c. Therefore, the elastic anisotropy is different because of the variations of the elastic constants with magnetic moment.

The elastic anisotropy of a tetragonal crystal can be measured by two shear anisotropy factors (Zener ratios) [43]:

Fe_{2}B has very low anisotropy.

The spin polarization has reduced the anisotropic factors A_{1} and A_{3} by 4 % and 30 % for Fe_{2}B, which means that the direction of easy axis of magnetization is 〈0 0 1〉 (C_{33} _{ii} = C_{22}) >C_{33}).

The simplest way to illustrate the anisotropy of mechanical moduli is to plot them in the threedimensional space as a function of direction. Here, we have plotted the bulk modulus (B) and Young’s modulus (E) in different directions using spherical coordinates. For orthorhombic and tetragonal crystal class, the directional dependence of bulk modulus (B) or Young’s modulus (E) can be written as:

For an orthorhombic system [44]:

For a tetragonal system [44, 45]:

In the equations above, S_{ij} represents the compliance matrix and l_{1}, l_{2} and l_{3} are the direction cosines, which are given as l_{1} = sinθ cosφ, l_{2} = sinθ sinφ and l_{3} = cosφ in the spherical coordinates. The surface constructions of bulk and Young’s modulus of FeB, Fe_{2}B and Fe_{3}B compounds are shown in Fig. 4 and Fig. 6. The surface constructions of the bulk and Young’s moduli are similar to each other. The anisotropy of Young’s modulus shows strong directional dependence in three crystals planes, (0 0 1), (1 0 0) and (0 1 0), for FeB structure. The projections of the mechanical moduli are plotted in Fig. 5 and Fig. 7. The results indicate that for Fe_{2}B the contours of bulk modulus at (0 0 1) crystal plane is spherical, implying that the bulk modulus of this phase is nearly isotropic. In the same way, the Young’s modulus shows an isotropy at (0 0 1), (1 0 0) and (0 1 0) planes.

## 6 Conclusions

We have investigated the anisotropic elastic properties of Fe-B compounds with the help of first-principles calculations at two pressures: 0 GPa and at a critical pressure for each compound. The calculated elastic constants of all compounds clearly indicate that they are mechanically stable. Bulk modulus, shear modulus, Young’s modulus and Poisson ratio have also been calculated and discussed. Disappearance of ferromagnetic order decreases the volume of the unit cell and increases the bulk modulus and also makes the solid harder. The calculated values of B/G and ν indicate that Fe_{2}B is ductile while FeB and Fe_{3}B are brittle. The degree of the elastic anisotropy for the considered Fe-B compounds follows the order FeB > Fe_{3}B > Fe_{2}B. We have predicted the easy and hard axes of magnetization for three compounds.

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