Anisotropic elastic properties of Fe_xB (x = 1, 2, 3) under pressure. First-principles study

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 (Fe₂B) or double (FeB and Fe₂B) layers [4]. The metastable phase, Fe₃B, appears during formation of Fe₂B. The metastable o-Fe₃B 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. Fe₂B 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 PuNi₃-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_xB (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〉, 〈1¯$\bar 1$ 0 0〉, 〈0 1¯$\bar 1$ 0〉 and 〈0 0 1¯$\bar 1$〉 are in fact equivalent easy axes [12].

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.

FeB and Fe₃B belong to an orthorhombic space group Pnma (Fig. 1) [13–16]. Both structures contain four formula units per cell. In Fe₃B, the isotype of Fe₃C, 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, Fe₂B (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 Fe₂B are located between two layers of Fe atoms in a distorted, closely packed arrangement.

Fig. 1

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: 3d⁶4s² and B: 2s²2p¹. 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 Fe₂B and (10 × 12 × 8) for Fe₃B. It is known that the ground states of several Fe_xB 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:

EcohFexB=EtotalFexB,Cell−xnEisoFe−nEisoBn$${E_{coh}}\left( {F{e_x}B} \right) = \frac{{{E_{total}}\left( {F{e_x}B, Cell} \right) - xn{E_{iso}}\left( {Fe} \right) - n{E_{iso}}\left( B \right)}}{n}$$(1)

EfFexB=EcohFexB−xEcohFe−EcohB$${E_f}\left( {F{e_x}B} \right) = {E_{coh}}\left( {F{e_x}B} \right) - x{E_{coh}}\left( {Fe} \right) - {E_{coh}}\left( B \right)$$(2)

where E_coh (Fe_xB) is the cohesive energy of Fe_xB per unit formula; E_f (Fe_xB) is its formation energy; Ecoh (Fe) is the cohesive energy of iron element per atom; E_total (Fe_xB, Cell) is the total calculated energy of Fe_xB 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_xB) to refer to a thermodynamically stable structure. The crystal structures of Fe_xB studied in this paper were built based on experimental results.

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_xB compounds are presented in Table 1. Generally, the elastic constants C₁₁, C₂₂ and C₃₃ 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 Fe₃B, the order is C₃₃ > C₂₂ > C₁₁. For tetragonal Fe₂B the order is C₁₁ > C₃₃. 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]:

Cii>0(i=1;2;3;4;5;6)$${C_{ii}}\:{\rm{ > 0 \:(}}i{\rm{ = 1; 2; 3; 4; 5;6)}}$$(3)

C11+C22+C33+2C12+C13+C23>0,C11+C22−2C12>0;C11+C33−2C13>0$${C_{{\rm{11}}}}{\rm{ + }}{C_{{\rm{22}}}}{\rm{ + }}{C_{{\rm{33}}}}{\rm{ + 2}}\left( {{C_{{\rm{12}}}}{\rm{ + }}{C_{{\rm{13}}}}{\rm{ + }}{C_{{\rm{23}}}}} \right){\rm{ > 0,}}\left( {{C_{{\rm{11}}}}{\rm{ + }}{C_{{\rm{22}}}} - 2{C_{{\rm{12}}}}} \right){\rm{ > 0; }}\left( {{C_{{\rm{11}}}}{\rm{ + }}{C_{{\rm{33}}}} - 2{C_{{\rm{13}}}}} \right){\rm{ > 0}}$$(4)

C22+C33−2C23>0$$\left( {{C_{{\rm{22}}}}{\rm{ + }}{C_{{\rm{33}}}} - 2{C_{{\rm{23}}}}} \right)\;{\rm{ > }}\;{\rm{0}}$$(5)

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

C~ii=Cii−P>0,(i=1,2,3,4,5,6)$${\tilde C_{ii}}\;{\rm{ = }}\;{C_{ii}} - P\;{\rm{ > }}\;{\rm{0, \:(}}i\;{\rm{ = }}\;{\rm{1,2,3,4,5,6)}}$$(6)

C11+C22−2C12−4P>0$$\left( {{C_{11}} + {C_{22}} - 2{C_{12}} - 4P} \right) > 0$$(7)

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

C~ij=Ciji=1,2,3;j=4,5,6$${\tilde C_{ij}} = {C_{ij}}\left( {i = 1,2,3;j = 4,5,6} \right)$$(8)

C~ii=Cii−Pi=1,2,3,4,5,6$${\tilde C_{ii}} = {C_{ii}} - P\left( {i = 1,2,3,4,5,6} \right)$$(9)

C~12=C12−P$${\tilde C_{12}} = {C_{12}} - P$$(10)

C~13=C13+P,C~23=C23+P,C~45=C45,C~46=C46$$\begin{array}{l} {{\tilde C}_{13}} = {C_{13}} + P,\\ {{\tilde C}_{23}} = {C_{23}} + P,\\ {{\tilde C}_{45}} = {C_{45}},\\ {{\tilde C}_{46}} = {C_{46}} \end{array}$$(11)

C~56=C56$${\tilde C_{56}} = {C_{56}}$$(12)

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 Fe₂B in the tetragonal structure, there are six independent elastic constants, C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, C₆₆, because C₂₂ = C₁₁, C₂₃ = C₁₃, C₄₄ = C₅₅ 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~ii>0,i=1,2,3,…6,C~11+C~33−2C~13>0$${\tilde C_{ii}} > 0,\left( {i = 1,2,3, \ldots 6} \right),{\tilde C_{11}} + {\tilde C_{33}} - 2{\tilde C_{13}} > 0$$(13)

2C~11+C~33+2C~12+4C~13>0$$2{\tilde C_{11}} + {\tilde C_{33}} + 2{\tilde C_{12}} + 4{\tilde C_{13}} > 0$$(14)

(C~11−C~12)>0$$({\tilde C_{11}} - {\tilde C_{12}}) > 0$$(15)

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]:

GH=(GR+GV)/2$${G_H} = ({G_R} + {G_V})/2$$(16)

BH=(BRBV)/2$${B_H} = ({B_R}{B_V})/2$$(17)

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

E=9BG/(3B+G)$$E = 9BG/(3B + G)$$(18)

ν=(3B−2G)/(6B+2G)$$\nu = (3B - 2G)/(6B + 2G)$$(19)

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 Fe₃B (Table 3), which indicates that FeB and Fe₃B are ductile. The values of B/G and ν for Fe₂B are 1.52 and 0.23, respectively, at 0 GPa pressure which means that Fe₂B is brittle. In contrast, at critical pressure Fe₂B is ductile (B/G = 2.48, ν = 0.32).

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]:

AU=5GVGR+BVBR−6⩽0$${A^U} = 5\frac{{{G_V}}}{{{G_R}}} + \frac{{{B_V}}}{{{B_R}}} - 6 \leqslant 0$$(20)

AG=GV−GRGV+GR×100$${A_G} = \frac{{{G_V} - {G_R}}}{{{G_V} + {G_R}}} \times 100$$(21)

AB=BV−BRBV+BR×100$${A_B} = \frac{{{B_V} - {B_R}}}{{{B_V} + {B_R}}} \times 100$$(22)

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.

Fig. 4

Fig. 5

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:

A3=4C66C11+C22−2C12$${A_3} = \frac{{4{C_{66}}}}{{{C_{11}} + {C_{22}} - 2{C_{12}}}}$$(24)

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 C₆₆ relatively to C55. Thus, for Fe₂B 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 Fe₂B and 0.53 for Fe₃B), 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 A₁ by 25 % and 22 % for FeB and Fe₃B, respectively, and A₂ by 32 % for Fe₃B, but reduce the anisotropic factors A3 by 37 % and 17 % for FeB and Fe₃B, and A₂ by 23 % for FeB, which means that the direction of easy axis of magnetization is 〈1 0 0〉 for FeB (C₁₁ <C₂₂ <C₃₃) and the hard axes directions are 〈1 0 0〉 and 〈0 1 0〉 (Fig. 6a and Fig. 6d). For Fe₃B the easy axis direction is 〈1 0 0〉 (C₁₁ <C₂₂ <C₃₃) and the hard axes directions are 〈0 1 0〉, 〈0 0 1〉 (Fig. 6c and Fig. 6f).

Fig. 6

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]:

A1=2C66C11−C12=A2$${A_1} = \frac{{2{C_{66}}}}{{{C_{11}} - {C_{12}}}} = {A_2}$$(25)

A3=C44C11+C33−2C13$${A_3} = \frac{{{C_{44}}}}{{{C_{11}} + {C_{33}} - 2{C_{13}}}}$$(26)

(A↑u≅0)$$({A^{ \uparrow u}} \cong 0)$$

Fe₂B has very low anisotropy.

The spin polarization has reduced the anisotropic factors A₁ and A₃ by 4 % and 30 % for Fe₂B, which means that the direction of easy axis of magnetization is 〈0 0 1〉 (C₃₃ <C₁₁) and the directions of hard axes of magnetizations are 〈1 0 0〉, 〈0 1 0〉 ((C_ii = C₂₂) >C₃₃).

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]:

1B=S11+S12+S13l12+S12+S22+S23l22+S13+S23+S33l32$$\frac{1}{B} = \left( {{S_{11}} + {S_{12}} + {S_{13}}} \right)l_1^2 + \left( {{S_{12}} + {S_{22}} + {S_{23}}} \right)l_2^2 + \left( {{S_{13}} + {S_{23}} + {S_{33}}} \right)l_3^2$$(27)

1E=S11+S22+S33l14+2S12+S66l12l22+2S23+S44l23l32+2S13+S55l12l32$$\frac{1}{E} = \left( {{S_{11}} + {S_{22}} + {S_{33}}} \right)l_1^4 + \left( {2{S_{12}} + {S_{66}}} \right)l_1^2l_2^2 + \left( {2{S_{23}} + {S_{44}}} \right)l_2^3l_3^2 + \left( {2{S_{13}} + {S_{55}}} \right)l_1^2l_3^2$$(28)

For a tetragonal system [44, 45]:

1E=S11l14+l14+2S13+S44l12l32+l22l32+S33l34+2S12+S66l12l22$$\frac{1}{E} = {S_{11}}\left( {l_1^4 + l_1^4} \right) + \left( {2{S_{13}} + {S_{44}}} \right)\left( {l_1^2l_3^2 + l_2^2l_3^2} \right) + {S_{33}}l_3^4 + \left( {2{S_{12}} + {S_{66}}} \right)l_1^2l_2^2$$(29)

1B=S11+S12+S13l12l22−2S13−S33l32$$\frac{1}{B} = \left( {{S_{11}} + {S_{12}} + {S_{13}}} \right)\left( {l_1^2l_2^2} \right) - \left( {2{S_{13}} - {S_{33}}} \right)l_3^2$$(30)

In the equations above, S_ij represents the compliance matrix and l₁, l₂ and l₃ are the direction cosines, which are given as l₁ = sinθ cosφ, l₂ = sinθ sinφ and l₃ = cosφ in the spherical coordinates. The surface constructions of bulk and Young’s modulus of FeB, Fe₂B and Fe₃B 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 Fe₂B 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.

Fig. 7

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 Fe₂B is ductile while FeB and Fe₃B are brittle. The degree of the elastic anisotropy for the considered Fe-B compounds follows the order FeB > Fe₃B > Fe₂B. We have predicted the easy and hard axes of magnetization for three compounds.