Structure, stability, mechanical and electronic properties of Fe–P binary compounds by first-principles calculations

Abstract

The equilibrium crystal structures, stability, elastic properties, hardness and electronic structures of Fe–P binary compounds (Fe₃P, Fe₂P, o-FeP-1, o-FeP-2, FeP₂, m-FeP₄-1, o-FeP₄, m-FeP₄-2) are investigated systematically by first principles calculations. The calculated formation enthalpy is used to estimate the stability of the Fe–P binary compounds. Fe₃P has the largest formation enthalpy at −44.950 kJ mol⁻¹ and o-FeP-2 has the smallest at −78.590 kJ mol⁻¹. The elastic constants are calculated by the stress–strain method and the Voigt–Reuss–Hill approximation is used to estimate the elastic moduli. The mechanical anisotropy of Fe_xP_y compounds are studied using the anisotropic indexes and by plotting the 3D surface contour of Young’s modulus. The electronic structures and chemical bonding characteristics of the Fe–P binary compounds are interpreted by the band structures and density of states. Finally, the sound velocity and Debye temperatures of the Fe–P binary compounds are discussed.

---

1. Introduction

The iron–steel industry has been developing fast since the last century, which is representative of modern industry. A common non-metal element such as phosphorus would form enrichment regions during the casting process or some heat treatment conditions, (for example, grain-boundary segregation). Phosphorus would combine with iron to form Fe–P binary compounds including Fe₃P, Fe₂P, FeP, FeP₂, and FeP₄ according to the Fe–P equilibrium phase diagram.¹ Moreover, phosphorus is usually considered hazardous to the properties of steel. But phosphorus existing in steel is inevitable. In order to control the effect of phosphorus in iron and steel, first of all, we should know the structure, stability and properties of the iron phosphides.

As a traditional industry, most research about the properties of steel has depended on experimental discovery. Therefore, it is difficult to explore the performance of these compounds at the electron-atomic level. The development of first-principles calculations based on density functional theory^2,3 has made it possible to study the fundamental properties at the electron-atomic level of materials under difficult experimental conditions. Nevertheless, there are only a few reports on iron phosphides. Li et al.⁴ calculated the elastic constants and formation enthalpy of Fe₂P by the first principles pseudo-potential plane wave method. They found that both of the hexagonal and orthorhombic Fe₂P intermetallic compounds were ductile. The hexagonal structure was more stable than the orthorhombic structure for Fe₂P. Chen et al.⁵ investigated the magnetic properties of Fe₂P. They deduced the magnetic properties of the compound even under considerably high pressures above 5.0 GPa. Tobola et al.⁶ investigated the magnetism and band structure of Fe₂P by neutron diffraction experiments and the KKR-CPA calculation method. However, most physical and chemical properties of Fe–P binary compounds are rarely reported systematically, so far. In this paper, the stability, mechanical properties and electronic structures of all the Fe–P binary compounds are investigated, and discussed for the first time, with a first-principles calculation method.

2. Calculation methods and models

The first-principles calculations of Fe–P compounds have been performed by density functional theory (DFT) which was implemented in Cambridge sequential energy package (CASTEP) code.⁷ A generalized gradient approximation (GGA) approach in the form of Perdew–Burke–Ernzerhof (PBE) was used to calculate the exchange and correlation functional.⁸ The interactions between the ionic cores and the valence electrons were indicated by ultra-soft pseudo potentials. For Fe and P, the valence electrons considered were 3p⁶4s²3d⁶ and 3s²3p³, respectively. A plane wave expansion method was applied for the optimization of the crystal structure. A special k-point sampling in the first irreducible Brillouin zone was confirmed by the Monkhorst–Pack scheme, and the k point mesh was selected as 3 × 3 × 9, 9 × 9 × 12, 6 × 6 × 12, 6 × 12 × 6, 9 × 6 × 15, 6 × 3 × 3, 6 × 3 × 6 and 9 × 3 × 6 for Fe₃P, Fe₂P, o-FeP-1, o-FeP-2 (CoAs-structure and MnP-structure, simplified as o-FeP-1 and o-FeP-2), FeP₂, m-FeP₄-1, o-FeP₄ and m-FeP₄-2, respectively. A kinetic energy cut-off value of 400.0 eV was used for the plane wave expansion in reciprocal space. The selected k point was three times as much as the default values, and the cut-off energy was tested. The result is shown in Fig. 1, the total energy will stay constant when the cut-off energy is larger than 380 eV for these compounds. So the selected values are suitable for the chosen system. The Broyden–Fletcher–Goldfarb–Shannon (BFGS) method was applied to relax the whole structure based on total energy minimization. The total energy changes during the optimization processes were finally converged to 1 × 10⁻⁶ eV and the forces per atom were reduced to 0.05 eV Å⁻¹.

Fig. 1 Cut-off energy testing.

By using the stress–strain method according to the generalized Hooke’s law, the elastic constants of the Fe_xP_y compounds were calculated. Several different strain modes were imposed on the crystal structure, and then the Cauchy stress tensor for each strain mode was estimated. Finally, the related elastic constants were identified as the coefficients in the strain–stress relations as shown in eqn (1):⁹

here, c_ij is the elastic constant, and τ_i and σ_j are the shear stress and normal stress, respectively. The total number of independent elastic constants is determined by the symmetry of the crystal. For high symmetry point groups, the required different strain patterns for the c_ij calculations can be greatly reduced.⁹

The cohesive energy and formation enthalpy were calculated to estimate the chemical stability of the Fe_xP_y compounds. These two energy parameters are defined in eqn (2) and (3):¹⁰

where E_coh(Fe_xP_y) and Δ_rH(Fe_xP_y) are the cohesive energy and formation enthalpy of Fe_xP_y per atom, respectively. E_tot(Fe_xP_y) is the total energy of the Fe_xP_y phase, E_iso is the total energy of a single Fe or P atom and E_bin refers to the cohesive energy of the Fe or P crystal.¹⁰

3. Results and discussion

3.1 Stability

Fig. 2 shows the Fe–P equilibrium phase diagram¹ and the five Fe_xP_y binary phases are Fe₃P, Fe₂P, FeP, FeP₂, and FeP₄. With the increase of phosphorus content, the melting points of the Fe–P compounds increased. FeP₄ has the highest melting point, at 1700 °C, among the Fe–P compounds. The melting point of Fe₃P is the lowest which may be ascribed to the weak covalent bonding between the Fe and P atoms. In fact, one example of this is Fe₃P as a common phase in the grain-boundary segregation in some types of steel materials.

Fig. 2 The Fe–P equilibrium phase diagram.

Fig. 3 shows the crystal structures of the Fe–P compounds. The Fe_xP_y binary compounds contain four different types of crystal system, including tetragonal (Fe₃P), hexagonal (Fe₂P), orthorhombic (o-FeP-1, o-FeP-2, FeP₂, o-FeP₄), and monoclinic (m-FeP₄-1, m-FeP₄-2). The calculated lattice parameters of Fe, P and the Fe–P compounds have been optimized, and are listed in Table 1. By comparing these results, it can be seen that the calculated lattice parameters of the o-FeP-2 compound are in good agreement with other calculated results and experimental values.^11–20 The average deviation of our results to the experimental data for lattice parameters is less than 5.6%, which can be attributed to the approximation method in the work and thermodynamic effects on the crystal structures.

Fig. 3 The crystal structure of the Fe–P compounds. The blue balls represent Fe atoms and the purple balls refer to P atoms.

Table 1 The space group, calculated lattice parameters, V_cell, cohesive energy (kJ mol⁻¹), formation enthalpy Δ_rH (kJ mol⁻¹) and the calculated spin-polarized formation enthalpy Δ_rH* (kJ mol⁻¹) data for the Fe–P binary compounds

Substances	Space group	Composition at % P	a (Å)	b (Å)	c (Å)	V cell (Å^3)	E coh	ΔrH	ΔrH*
a Cal. in ref. 11. b Exp. in ref. 12. c Exp. in ref. 13. d Cal. in ref. 4. e Exp. in ref. 14. f Exp. in ref. 15. g Exp. in ref. 16. h Exp. in ref. 17. i Exp. in ref. 18. j Exp. in ref. 19. k Cal. in ref. 20.
Fe	Im3̄m	0	3.402, 3.430^a	3.402, 3.430^a	3.402, 3.430^a	39.369, 40.350^a	−898.415	0	0
Fe3P	I4̄	25.000	8.598, 9.107^b	8.598, 9.107^b	4.277, 4.460^b	316.109, 369.900^b	−848.235	−67.068	−44.950
Fe2P	P6̄2m	33.333	5.531, 5.690^c, 5.704^d	5.531, 5.690^c, 5.704^d	3.399, 3.458^c, 3.431^d	90.062, 96.960^c	−834.725	−81.957	−61.510
o-FeP-1	Pna21	50.000	5.055, 5.193^e	5.680, 5.792^e	2.931, 3.099^e	84.146, 93.210^e	−791.300	−95.236	−78.580
o-FeP-2	Pnma	50.000	5.053, 5.191^f	2.932, 3.099^f	5.680, 5.792^f	84.152, 93.180^f	−791.300	−95.255	−78.590
FeP2	Pnnm	66.700	4.883, 4.973^g	5.536, 5.657^g	2.647, 2.723^g	71.560, 76.600^g	−720.855	−80.674	−69.200
m-FeP4-1	C12/c1	80.000	4.968, 5.054^h	10.261, 10.407^h	10.939, 11.069^h	557.632, 582.120^h	−651.375	−56.443	−49.553
o-FeP4	C2221	80.000	4.923, 5.005^i	10.077, 10.213^i	5.434, 5.530^i	269.551, 282.670^i	−649.445	−55.275	−48.395
m-FeP4-2	P121/c1	80.000	4.540, 4.619^j	13.444, 13.670^j	6.914, 7.002^j	413.839, 433.270^j	−650.410	−56.260	−49.370
P	Cmca	100.000	3.289, 3.241^k	10.832, 10.192^k	4.390, 4.239^k	156.408, 140.000^k	−526.890	0	0

---

The cohesive energy and formation enthalpy are used to evaluate the stability of the Fe–P binary compounds. As defined in eqn (2) and (3), the lower the values for these two thermodynamic parameters, the more stable the compound. The cohesive energy mainly reflects the stability of the combination of two atoms, while the formation enthalpy mainly reflects the stability of the formation of compounds. As shown in Table 1, the calculated cohesive energy and formation enthalpy values of the studied Fe–P binary compounds are negative, which shows that the Fe–P compounds are thermodynamically stable. The formation enthalpy values without spin-polarization of the Fe–P compounds are −67.068, −81.957, −78.580, −78.590, −69.200, −49.553, −48.395 and −49.370 kJ mol⁻¹, and the calculated spin-polarized formation enthalpy values of the Fe–P compounds are −44.950, −61.510, −78.580, −78.590, −69.200, −49.553, −48.395 and −49.370 kJ mol⁻¹ for the Fe₃P, Fe₂P, o-FeP-1, o-FeP-2, FeP₂, m-FeP₄-1, o-FeP₄, and m-FeP₄-2 phases, respectively. Fig. 4 shows the formation enthalpy (ΔH) and the calculated spin-polarized formation enthalpy (ΔH*) curves as a function of the P atom content for the Fe–P binary compounds. Obviously, with the increase of phosphorus content, the formation enthalpy of the Fe_xP_y compounds decreases at first and then increases. Which may be due to the proportion of anti-bonding states decreasing at first and then increasing. For the ΔH, among all the Fe–P binary compounds, o-FeP₄ has the largest formation enthalpy value at −55.275 kJ mol⁻¹, and o-FeP-2 has the smallest formation enthalpy value at −95.255 kJ mol⁻¹. The stability sequence of the eight Fe–P phases forms the following order: o-FeP-2 > o-FeP-1 > Fe₂P > FeP₂ > Fe₃P > m-FeP₄-1 > m-FeP₄-2 > o-FeP₄. For the ΔH*, Fe₃P has the largest formation enthalpy value at −44.950 kJ mol⁻¹, and o-FeP-2 has the smallest formation enthalpy value at −78.590 kJ mol⁻¹. The stability sequence of these Fe–P phases forms the following order: o-FeP-2 > o-FeP-1 > FeP₂ > Fe₂P > m-FeP₄-1 > m-FeP₄-2 > o-FeP₄ > Fe₃P. The spin-polarization increased the formation enthalpy of the Fe_xP_y compounds, but the effect on the formation enthalpy was weakened with an increased phosphorus content. The stability sequence of the compounds changes with and without the spin-polarization. From the results of the stability sequence it can be seen that the stability of Fe₃P and Fe₂P is reduced with the spin-polarization, which may be due to the magnetic characters of Fe₃P and Fe₂P. According to the density of states, we know that Fe₃P and Fe₂P have magnetic characters. In general, o-FeP-2 is the most stable compound among the Fe–P compounds.

Fig. 4 The formation enthalpy (ΔH) and the spin-polarized formation enthalpy (ΔH*) of the Fe–P binary compounds.

3.2 Elastic constants and polycrystalline moduli

The calculated elastic constants of the α-Fe, phosphorus and Fe–P compounds are listed in Table 2. The elastic stability conditions in various crystal systems can be expressed by the following equations (eqn (4–6)):^21,22

(1) Orthorhombic (for o-FeP-1, o-FeP-2, FeP₂ and o-FeP₄):

c₁₁ > 0, c₁₁c₂₂ > c₁₂², c₁₁c₂₂c₃₃ + 2c₁₂c₁₃c₂₃ − c₁₁c₂₃² − c₂₂c₁₃² − c₃₃c₁₂² > 0, c₄₄ > 0, c₅₅ > 0, c₆₆ > 0. (4)

(2) Tetragonal (for Fe₃P) and hexagonal (for Fe₂P):

c₁₁ > |c₁₂|, 2c₁₃² < c₃₃(c₁₁ + c₁₂), c₄₄ > 0, 2c₁₆² < c₆₆(c₁₁ − c₁₂). (5)

(3) Monoclinic (for m-FeP₄-1 and m-FeP₄-2):

c₁₁ + c₂₂ + c₃₃ + 2(c₁₂ + c₁₃ + c₂₃) > 0, c₃₃c₅₅ − c₃₅² > 0, c₄₄c₆₆ − c₄₆² > 0, c₂₂ + c₃₃ − 2c₂₃ > 0, c₂₂(c₃₃c₅₅ − c₃₅²) + 2c₂₃c₂₅c₃₅ − c₂₃²c₅₅ − c₂₅²c₃₃ > 0, 2[c₁₅c₂₅(c₃₃c₁₂ − c₁₃c₂₃) + c₁₅c₃₅(c₂₂c₁₃ − c₁₂c₂₃) + c₂₅c₃₅(c₁₁c₂₃ − c₁₂c₁₃)] − [c₁₅²(c₂₂c₃₃ − c₂₃²) + c₂₅²(c₁₁c₃₃ − c₁₃²) + c₃₅²(c₁₁c₂₂ − c₁₂²)] + c₅₅g > 0 (g = c₁₁c₂₂c₃₃ − c₁₁c₂₃² − c₂₂c₁₃² − c₃₃c₁₂² + 2c₁₂c₁₃c₂₃) > 0, c_ii > 0(i = 1–6). (6)

Table 2 Single crystalline elastic constants (c_ij, in GPa) of the Fe–P binary compounds

Substances	Fe	Fe3P	Fe2P	o-FeP-1	o-FeP-2	FeP2	m-FeP4-1	o-FeP4	m-FeP4-2	P
a Cal. in ref. 7. b Cal. in ref. 23. c Cal. in ref. 4.
c 11	266.4^a, 279.2^b	418.2	460.3	502.5	506.4	570.0	335.9	367.5	279.9	188.4
c 22		418.2	460.3	476.4	280.1	685.5	325.2	310.8	285.6	42.9
c 33		513.5	474.7	277.4	529.9	382.4	344.2	374.3	285.5	39.3
c 44	96.3^a, 93.0^b	98.1	183.9	176.5	171.3	105.9	91.6	76.7	147.5	19.9
c 55		98.1	183.9	177.1	170.0	228.4	109.3	121.1	136.8	55.0
c 66		136.6	88.9	165.6	176.6	227.1	188.3	189.2	160.0	31.9
c 12	146.5^a, 148.8^b	348.4	282.4	136.9	147.6	239.0	99.5	104.0	90.4	−1.2
c 13		230.6	195.7, 172.0^c	144.5	166.7	180.5	41.1	44.0	73.2	36.0
c 15							−7.3		16.6
c 23		230.6	195.7	183.8	210.0	97.2	44.1	20.9	75.7	2.5
c 25							18.5		21.8
c 35							−15.9		−26.5
c 46							17.7		26.1

---

As shown in Table 2, the calculated elastic constants of each Fe–P compound satisfy the above criteria, which indicates that all of the Fe–P compounds are mechanically stable. The calculated c₁₁ and c₂₂ values of FeP₂ are larger than the other elastic constants which indicates that FeP₂ is highly incompressible under uniaxial stress along the crystallographic a (ε₁₁) and b (ε₂₂) axes. o-FeP-2 has the largest c₃₃ value, which shows that o-FeP-2 is very incompressible under uniaxial stress along the crystallographic c (ε₃₃) axis. The largest elastic constant value is c₂₂ for FeP₂ with 685.5 GPa. For the two FeP polymorphs, the c₂₂ value of o-FeP-1 is close to the c₃₃ value of o-FeP-2 and the c₃₃ value of o-FeP-1 is close to the c₂₂ value of o-FeP-2, which indicates that in terms of the incompressible performance the b (ε₂₂) axis of o-FeP-1 is similar to the c (ε₃₃) axis of o-FeP-2, and the c (ε₃₃) axis of o-FeP-1 is similar to the b (ε₂₂) axis of o-FeP-1. c₄₄, c₅₅ and c₆₆ represent the shear modulus on the (100), (010) and (001) crystal planes. From Table 2, one can see that Fe₂P has the largest c₄₄ value and o-FeP₄ has the smallest c₄₄ value for all the Fe–P binary compounds. For the three FeP₄ polymorphs, m-FeP₄-2 has the largest c₄₄ value at 147.5 GPa.

The bulk modulus (B), shear modulus (G), Young’s modulus (E) and Poisson’s ratio (σ) of polycrystalline crystals are estimated with independent single crystal elastic constants according to the Viogt–Reuss–Hill (VGH) approximation.²⁴ The Voigt method is based on the assumption of uniform strain throughout a polycrystal, which is given by:

9B_V = (c₁₁ + c₂₂ + c₃₃) + 2(c₁₂ + c₁₃ + c₂₃) (7)

15G_V = (c₁₁ + c₂₂ + c₃₃) − (c₁₂ + c₁₃ + c₂₃) + 3(c₄₄ + c₅₅ + c₆₆) (8)

The Reuss method assumes a uniform stress and gives B and G as functions of the elastic compliance constants s_ij, which are given by the inverse matrix of c_ij.

1/B_R = (S₁₁ + S₂₂ + S₃₃) + 2(S₁₂ + S₁₃ + S₂₃) (9)

1/G_R = 4[(S₁₁ + S₂₂ + S₃₃) − (S₁₂ + S₁₃ + S₂₃)]/15 + (s₄₄ + s₅₅ + s₆₆)/5 (10)

The Voigt–Ruess–Hill (VRH) approximation is considered as a good estimation method for the elastic modulus of polycrystalline materials.

G_H = (G_R + G_V)/2 (11)

B_H = (B_R + B_V)/2 (12)

The Young’s modulus (E) and Poisson’s ratio (σ) can be calculated as follows:^25,26

E = 9BG/(3B + G) (13)

σ = (3B − 2G)/(6B + 2G) (14)

The calculated bulk modulus (B), shear modulus (G), Young’s modulus (E) and Poisson’s ratio (σ) of these Fe–P binary compounds are shown in Table 3. The bulk modulus reveals the compressibility of the solid under hydrostatic pressure and the values are 315.8, 304.1, 235.5, 250.7, 284.2, 152.5, 153.6 and 147.7 GPa for Fe₃P, Fe₂P, o-FeP-1, o-FeP-2, FeP₂, m-FeP₄-1, o-FeP₄ and m-FeP₄-2, respectively. The bulk modulus values of Fe₃P and Fe₂P are higher than those of other carbides, such as Fe₃C (255 GPa),²⁸ TiC (242 GPa)²⁹ and Cr₃C (287.5 GPa),³⁰ but lower than those of h-WC (393.0 GPa)³¹ and diamond (436.8 GPa).³² Fe₃P has the largest value of bulk modulus among the Fe–P binary compounds, which may be owed to the fact that it has the strongest ionic bond between the Fe and P atoms. Because the ionic bonds have no direction. The calculated values of the shear modulus are 84.0, 132.2, 148.7, 148.0, 176.8, 127.0, 129.4 and 125.4 GPa for Fe₃P, Fe₂P, o-FeP-1, o-FeP-2, FeP₂, m-FeP₄-1, o-FeP₄ and m-FeP₄-2, respectively. The largest value of shear modulus belongs to FeP₂, and the largest value of Young’s modulus is attributed to FeP₂. In addition, Fe₃P with the lowest P atom content has the largest bulk modulus at 315.8 GPa and the smallest shear modulus at 84.0 GPa. This result can be explained by the strong ionic bonding and weak covalent bonding of Fe₃P, because the ionic bonds have no direction, but the covalent bonds have directionality.

Fig. 5 presents the bulk modulus, shear modulus and Young’s modulus curves as a function of the P atom content for the Fe–P systems. With an increase in phosphorus content, the three moduli of the Fe–P compounds first increase and then decrease. The ratio of B/G (here B_H and G_H are used) can be used to indicate the ductile or brittle nature of the compounds, a high value is associated with ductility and a low value is associated with brittleness, the critical value is about 1.75. Fig. 6 shows the B/G and Poisson’s ratio curves as a function of the P atom content for the Fe–P systems. When comparing the B/G values, it’s clearly implied that Fe₃P and Fe₂P are considered to be ductile compounds since the values of B/G are larger than 1.75, m-FeP₄-2 has the smallest value at 1.18, indicating that it’s the most brittle. The results are in good agreement with the analysis of the density of states. Meanwhile, a Poisson’s ratio of larger or smaller than 0.25 can also be used to indicate the ductile or brittle nature of the compounds. From Fig. 6, we know that Fe₃P and Fe₂P can be classified as ductile, which may be owed to their strong metallic bonding. While the three FeP₄ compounds, two FeP compounds and FeP₂ should be classified as brittle, since their values of B/G are smaller than 1.75 and their values of the Poisson’s ratio are smaller than 0.25. This may be owed to their strong covalent bonding.

Fig. 5 Polycrystalline elastic moduli of the Fe–P binary compounds.

Fig. 6 The B/G and σ values of the Fe–P binary compounds.

The hardness (H_v) is very important in the applications of Fe–P binary compounds. In the present paper, the hardness of the two FeP compounds was estimated by a relative semi-empirical equation (eqn (15)).^33,44 The equation is defined as follows:

where H_v denotes the hardness, A is a proportional coefficient and α is a constant. A = 14 and α = −1.191. n_i is the number of i atoms in the cell, Z_i is the valence electron number of the i atoms, v is the cell volume, x_A is the electronegativity of atom A and d is the bond length in angstroms. The calculated hardness of the FeP compounds is shown in Table 3.

Table 3 Polycrystalline elastic properties of Fe–P binary compounds, including the bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (σ) and Vickers hardness (H_v)

Substances	Fe	Fe3P	Fe2P	o-FeP-1	o-FeP-2	FeP2	m-FeP4-1	o-FeP4	m-FeP4-2	P
a Cal. in ref. 7. b Cal. in ref. 4. c Cal. in ref. 27.
B V (GPa)	186.5^a	315.8	304.8	243.0	262.8	296.8	152.7	154.5	147.7	38.4
B R (GPa)	186.5^a	315.7	303.5	228.1	238.7	271.7	152.3	152.7	147.6	21.7
B H (GPa)	186.5^a	315.8	304.1	235.5	250.7	284.2	152.5	153.6	147.7	30.0
G V (GPa)	81.7^a	102.6	139.5	156.6	156.4	187.0	132.6	136.3	129.6	36.9
G R (GPa)	77.5^a	65.4	125.0	140.9	139.6	166.5	121.7	122.5	121.2	26.2
G H (GPa)	79.6^a	84.0	132.2	148.7	148.0	176.8	127.0	129.4	125.4	31.5
E H (GPa)	209.1^a	231.5	346.4	368.5	371.0	439.3	298.2	303.1	293.2	70.0
B H/GH	2.34^a	3.76	2.30, 2.18^b	1.58	1.69	1.61	1.20	1.19	1.18	0.95
σ	0.31^a, 0.29^c	0.38	0.31, 0.30^b	0.24	0.25	0.24	0.17	0.17	0.17	0.11
H V (GPa)	35.82			33.39	33.39					19.23

---

3.3 Anisotropy of elastic properties

The mechanical anisotropy is important in the applications of Fe–P materials. The fracturing cracks of materials form not only in the substrate, but also in the inclusion. The formation and propagation of micro cracks is often related to the elastic anisotropy. Knowing the anisotropy of Fe_xP_y compounds would be helpful in designing and enhancing some special device. To describe the degrees of anisotropy of Fe–P binary compounds, a numbers of indexes, including the shear anisotropic factors (A₁, A₂ and A₃), the percentages of anisotropy under compression and shear stress (A_B and A_G) and the universal elastic anisotropy (A^U), were calculated using the following equations.^34–36where B_V, B_R, G_V and G_R are the bulk and shear moduli calculated with the Voigt and Reuss methods, respectively. The calculated results are shown in Table 4. For A₁, A₂ and A₃, a value of unity implies isotropy and a non-unity value implies anisotropy for a crystal. For isotropic structures, the values of A_B, A_G and A^U are zero. Meanwhile the large discrepancies from zero indicate highly mechanical anisotropic properties.

Table 4 Anisotropic factors of the Fe–P binary compounds

Substances	A 1	A 2	A 3	A B (%)	A G (%)	A ^U
Fe	2.71	2.71	2.71	0	11.45	1.29
Fe3P	0.83	0.83	3.91	0.03	22.43	2.89
Fe2P	1.35	1.35	1.00	0.21	5.48	0.58
o-FeP-1	1.44	1.83	0.94	3.16	5.28	0.62
o-FeP-2	0.97	1.74	1.44	4.80	5.68	0.70
FeP2	0.72	1.05	1.17	4.42	5.80	0.71
m-FeP4-1	0.61	0.75	1.63	0.13	4.29	0.45
o-FeP4	0.47	1.18	1.61	0.56	5.33	0.58
m-FeP4-2	1.41	1.30	1.66	0.03	3.35	0.35
P	0.51	2.85	0.55	27.79	16.96	2.81

---

From Table 4, it can be seen that Fe is isotropic, Fe₂P and m-FeP₄-2 have strong isotropy, and Fe₃P has the strongest anisotropy especially in the (001) plane. Fe₃P has the largest value of A_G at 22.43% among all the Fe–P binary compounds, implying that the anisotropy in the shear modulus for Fe₃P is the strongest. However, the A₁, A₂, A₃, A_B and A_G values can’t fully describe the elastic anisotropy. The A₁, A₂ and A₃ values describe the anisotropy of the shear modulus in the different crystal planes, and the index A^U is considered as an appropriate parameter to describe the degrees of elastic anisotropy of the compound. From Table 4, Fe₃P has the strongest anisotropy of the Fe–P binary compounds, since the value of A^U is 2.89, and m-FeP₄-2 has the strongest isotropy among the Fe_xP_y compounds, since the value of A^U is 0.35.

The most straightforward way to illustrate the elastic anisotropy is to plot the Young’s modulus and shear modulus in three dimensions (3D) as a function of the crystallographic direction. The directional dependence of Young’s modulus and the shear modulus is given by eqn (22–25):^{10,34,37,38,42,43}

Hexagonal crystal (for Fe₂P)

Orthorhombic crystal (for o-FeP-1, o-FeP-2, FeP₂ and o-FeP₄)

Monoclinic crystal (for m-FeP₄-1 and m-FeP₄-2)

Tetragonal (for Fe₃P)

where the s_ij values are the elastic compliance constants, and l₁, l₂ and l₃ are the directional cosines. The surface contours of the Young’s modulus and shear modulus of the Fe–P binary compounds are illustrated in Fig. 7 and Fig. 9. For an isotropic system, the graph would be a sphere. Obviously, Fe₃P, Fe₂P, FeP₂, m-FeP₄-1 and o-FeP₄ show a strong anisotropic character in the Young’s modulus, and Fe₃P, the two FeP compounds, FeP₂ and o-FeP₄ show a strong anisotropic character in the shear modulus. The surface contours of the two FeP compounds and m-FeP₄-2 in Fig. 7 are similar to a cylinder, which means that the Young’s modulus of these compounds have weaker anisotropy than the other compounds. Projections of the Young’s modulus on the (100), (001) and (110) planes show more details about the anisotropic properties of the Young’s modulus as shown in Fig. 8. Obviously, the Young’s modulus has a strong directional dependence on these planes. From Fig. 8, we can see that FeP₂ shows the maximum Young’s modulus along the [010] and [110] directions, and that Fe₃P shows the minimum Young’s modulus along the [010] direction on the (100) and (001) planes. For Fe₂P, the planar contours on the (001) plane is similar to an ellipse, which means that the Young’s modulus of Fe₂P on the (001) plane has weaker anisotropy than the other compounds.

Fig. 7 Contour plots of the Young’s modulus of the Fe–P compounds in 3-D space.

Fig. 8 Planar projections of the Young’s modulus of the Fe–P compounds on the (100), (001) and (110) crystallographic planes.

Fig. 9 Contour plots of the shear modulus (GPa) of the Fe–P compounds in 3-D space.

3.4 Electronic structures

The electronic structures and chemical bonding characteristics of the Fe_xP_y compounds are indicated by the electron density distribution map, total density of states (TDOS) and partial density of states (PDOS).

The calculated total electron density distribution maps of the Fe–P compounds are shown in Fig. 10. The electron density is mainly concentrated on the iron atoms. For o-FeP-1, o-FeP-2, Fe₂P and Fe₃P, the electron density values are larger than zero even in the interstitial regions, which indicates the metallic nature of these compounds. The elongated contours along the Fe–P bond axes show the covalent interactions. m-FeP₄-1 has the strongest covalent bonding between the P atoms.

Fig. 10 Total electron density distribution through (−0.563, 0.318, 0.763), (0.561, −0.764, −0.318), (−0.021, −0.020, 0.999), (−0.039, −0.355, −0.934), (1, 0, 0), (0, 0, 1) and (−0.012, −0.012, 0.999) slices intersecting both the Fe and P atoms for the (a) o-FeP-1, (b) o-FeP-2, (c) FeP₂, (d) o-FeP₄, (e) m-FeP₄-1, (f) Fe₂P and (g) Fe₃P compounds, respectively.

The TDOS and PDOS of the Fe_xP_y compounds are shown in Fig. 11. The nature of the magnetic characters can be understood from the spin-polarized total density of states. Comparing the up with the down densities, it can be seen that the up and down states are not symmetric for Fe₃P and Fe₂P, which indicates that they have magnetic characters. Actually, the low and high valence bands are almost symmetric, and it is near to the Fermi level that the up and down states are dissymmetric. While for other compounds, the up and down states are symmetric, so they have no magnetic character. No energy gap near to the Fermi level can be seen for Fe₃P, Fe₂P, o-FeP-1 and o-FeP-2, which indicates metallicity and electronic conductivity for these binary compounds. Based on the analysis of the band structure, the energy gaps near to the Fermi level for FeP₂, o-FeP₄, m-FeP₄-1 and m-FeP₄-2 are 0.573, 0.903, 0.924 and 1.16 eV, respectively. So these four compounds are considered to be semiconductors, and the electroconductibility of m-FeP₄-2 is the worst. The other compounds are conductors. We can see that the TDOS values at the Fermi level increases as the Fe atom content increases. Fe₃P has the strongest metallicity, which is consistent with the values of the B/G and Poisson’s ratio (σ). The Fe-d bands of m-FeP₄-1 have a peak which shows that the electrons of the d bands have relatively local state density. From Fig. 11, it can be seen that the ground state properties of the Fe–P binary compounds are determined by the 3d bands of Fe. At the low energy part, the band from −9 eV to −5 eV is mainly due to the 3p bands of P. The Fe-d bands are overlapped with the P-p bands in the energy range from −4 eV to 3 eV for the three FeP₄ compounds, which indicates covalent interactions because of the strong hybridization between the Fe-d bands and the P-p bands. It’s similar to the energy range from −2 eV to 3 eV of FeP₂. The Fe-d bands are hybridized weakly with the P-p bands in the energy range from −5 eV to −2 eV for o-FeP-1 and o-FeP-2. We can see that the FeP₄ compounds have the strongest covalent interactions among all the Fe_xP_y compounds from Fig. 11, which lends to the higher melting points. This result is consistent with the Fe–P equilibrium phase diagram. The valence electrons of the Fe and P atoms considered are 3p⁶3d⁶4s² and 3s²3p³. For Fe₃P, the position of the Fe-d bands does not coincide with the P-p bands, which shows an interaction between the electrons of the P and Fe atoms. The Fe 3d orbitals lose an electron and form the partially filled state, the P 3p orbitals get three electrons and form the entirely filled state. In addition, the partially filled and the entirely filled states are stable states. The electron transfer path implies ionic interaction in the Fe₃P compound. In other words, with the increase of phosphorus content, the covalent interactions of the Fe–P binary compounds strengthen, and the ionic interactions and the metallicity weaken.

Fig. 11 The total density of states (TDOS) and partial density of states (PDOS) for the Fe_xP_y compounds, dashed lines represent the Fermi level.

According to the above discussion, Fe₃P and Fe₂P have magnetic characters. m-FeP₄-2 is considered to be a semiconductor. The bonding behaviors of the Fe–P binary compounds are combinations of metallic, covalent and ionic bonds. For Fe₃P and Fe₂P, the chemical bonding is dominated by the Fe–P ionic bonds. The chemical bonding of o-FeP-1, o-FeP-2, FeP₂, m-FeP₄-1, o-FeP₄ and m-FeP₄-2 is dominated by the Fe–P covalent bonds but also possesses ionic and metallic bond character, which may lead to the high melting points.

3.5 Debye temperature

It is certain that the sound velocity and Debye temperature can be used to evaluate the chemical bonding characteristics and thermal properties of compounds. The sound velocity and Debye temperature at low-temperature are calculated with the previously obtained bulk modulus (B) and shear modulus (G) values. The Debye temperature can be calculated by the following equation:^39,9where, Θ_D represents the Debye temperature; h and k_B are the Planck and Boltzmann constants, respectively; n is the total number of atoms per formula; N_A is the Avogadro constant; M is the molecular weight per formula; ρ is the theoretical density, and the v_m is the average sound velocity defined as:where, v_s is the transverse sound velocity and v_l is the longitudinal sound velocity; B and G are the isothermal bulk modulus and shear modulus values.^22,40,41.

The values of the Debye temperature and sound velocities of the Fe–P binary compounds are listed in Table 5. The Debye temperature reflects the strength of the chemical bonding in a crystal structure. From Table 5, we can see that the Debye temperature increases as the P atom content increases in the Fe–P binary compounds except for FeP₂. Moreover, Fe₃P has the lowest Debye temperature and the highest B/G ratio, which indicates the strongest metallic character. This is consistent with the previous calculations on the density of states. Besides, the average sound velocities of these Fe_xP_y compounds are relatively large at about 5500 m s⁻¹ except for Fe₃P. A reasonable explanation is that these compounds have high bulk and shear modulus values, and a low density, and that the v_l and v_s values are correlated to the bulk modulus, shear modulus and density.

Table 5 The theoretical density (ρ, g cm⁻³), longitudinal sound velocity (v_l, m s⁻¹), transverse sound velocity (v_s, m s⁻¹), average sound velocity (v_m, m s⁻¹) and Debye temperature (Θ_D, K)

Species	ρ	v l	v s	v m	Θ D
Fe3P	8.343	7160.8	3173.1	3581.1	497.1
Fe2P	5.665	9208.4	4830.8	5402.9	668.6
o-FeP-1	6.853	7955.9	4658.2	5165.0	702.1
o-FeP-2	6.853	8085.6	4647.2	5161.3	701.6
FeP2	5.467	9752.1	5686.8	6307.8	822.4
m-FeP4-1	4.282	8669.5	5446.0	5996.0	742.2
o-FeP4	4.430	8580.2	5404.6	5948.6	744.7
m-FeP4-2	4.327	8530.9	5383.4	5924.1	735.9

---

4. Conclusions

In general, we have investigated the stability, mechanical properties and electronic structures of all the Fe–P binary compounds with first-principles calculations. The cohesive energy and formation enthalpy indicate that they are thermodynamically stable. The elastic constants of the Fe_xP_y compounds satisfy the mechanical stability criteria. Fe₃P has the largest bulk modulus at 315.8 GPa and the smallest shear modulus at 84.0 GPa. FeP₂ exhibits the largest shear and Young’s moduli at 176.8 and 439.3 GPa, respectively. The hardness of the two FeP compounds is 33.39 GPa. Fe₃P and Fe₂P are considered to be ductile compounds, which may be owed to their strong metallic bonding. The 3D surface contour maps of the Young’s modulus and shear modulus were plotted to verify the mechanical anisotropy of the Fe–P binary compounds, Fe₃P has the strongest anisotropy among the Fe_xP_y compounds. The bonding behaviors of the Fe–P binary compounds are combinations of metallic, covalent and ionic bonds. Fe₃P and FeP₂ have the smallest and largest Debye temperatures at 497.1 K and 822.4 K, respectively.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 51171074 and 51261013).

References

1. H. Ohtani, N. Hanaya, M. Hasebe, S. Teraoka and M. Abe, CALPHAD: Comput. Coupling Phase Diagrams Thermochem., 2006, 30, 147–158.
2. W. Zhou, L.-J. Liu, B.-L. Li, P. Wu and Q.-G. Song, Comput. Mater. Sci., 2009, 46, 921–931.
3. P. Hohenberg and W. Kohn, Phys. Rev. B: Condens. Matter Mater. Phys., 1964, 136, 864.
4. J.-Y. Li, H.-M. Chen, Q.-Z. Gao, Q.-L. Xie and J.-L. Huang, J. Guangxi Univ., Nat. Sci. Ed., 2009, 34, 565–570.
5. L. Chen, N. Kurita, T. Fujiwara, M. Abliz, M. Hedo, I. Oguro, A. Matsuo, K. Kindo, Uwatoko and H. Fujii, J. Magn. Magn. Mater., 2007, 310, 219–221.
6. J. Tobola, M. Bacmann, D. Fruchart, S. Kaprzyk, A. Koumina, S. Niziol, J. L. Soubeyroux, P. Wolfers and R. Zach, J. Magn. Magn. Mater., 1996, 157–158, 708–710.
7. H.-L. Liu, W.-L. Wang, L. Hu and B.-B. Wei, Intermetallics, 2014, 46, 211–221.
8. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
9. J. Feng, B. Xiao, J. Chen, Y. Du, J. Yu and R. Zhou, Mater. Des., 2011, 32, 3231–3239.
10. X.-Y. Chong, Y.-H. Jiang, R. Zhou and J. Feng, J. Alloys Compd., 2014, 610, 684–694.
11. J. Haglund, A. Fernandez Guillermet, G. Grimvall and M. Korling, Phys. Rev. B, 1993, 48, 11685–11691.
12. S. Rundqvist, Acta Chem. Scand., 1962, 16, 1–19.
13. A. Koumina, M. Bacmann, D. Fruchart, J. L. Soubeyroux, P. Wolfers, J. Tobola, S. Kaprzyk, S. Niziol, M. Mesnaoui and R. Zach, Annales de Chimie Science des Materiaux, 1998, 23, 177–180.
14. K. Selte and A. Kjekshus, Acta Chem. Scand., 1972, 26, 1276–1277.
15. S. Rundqvist and P. C. Nawapong, Acta Chem. Scand., 1965, 19, 1006–1008.
16. E. Dahl, Acta Chem. Scand., 1969, 23, 2677–2684.
17. M. Evain, R. Brec, S. Fiechter and H. Tributsch, J. Solid State Chem., 1987, 71, 40–46.
18. M. Sugitani, N. Kinomura, M. Koizumi and S. Kume, J. Solid State Chem., 1978, 26, 195–201.
19. W. Jeitschko and D. J. Braun, Acta Crystallogr., Sect. B, 1978, 34, 3196–3201.
20. R. Ahuja, Phys. Status Solidi B, 2003, 235, 282–287.
21. F. Mouhat, F. X. Coudert, B. Phys Rev, 2014, 90, 224104.
22. X.-Y. Chong, Y.-H. Jiang, R. Zhou and J. Feng, Comput. Mater. Sci., 2014, 87, 19–25.
23. S.-L. Shang, A. Saengdeejing, Z.-G. Mei, D. E. Kim, H. Zhang and S. Ganeshan, et al. , Comput. Mater. Sci., 2010, 48, 813–826.
24. W. Zhou, L. Liu, B. Li, P. Wu and Q. Song, Comput. Mater. Sci., 2009, 46, 921.
25. R. Hill, Proc. Phys. Soc., London, Sect. A, 1952, 65, 349–354.
26. B. Xiao, J. Feng, C. T. Zhou, Y. H. Jiang and R. Zhou, J. Appl. Phys., 2011, 109, 023507–023509.
27. T. Frechen and G. Dietz, J. Phys. F: Met. Phys., 1984, 14, 1811–1826.
28. C. T. Zhou, B. Xiao, J. Feng, J. D. Xing, X. J. Xie, Y. H. Chen and R. Zhou, Comput. Mater. Sci., 2009, 45, 986.
29. J. J. Gilman and B. W. Roberts, J. Appl. Phys., 1961, 32, 1405.
30. Y. F. Li, Y. M. Gao, B. Xiao, T. Min, Y. Yang, S. Q. Ma and D. W. Yi, J. Alloys Compd., 2011, 509, 5242–5249.
31. Y. F. Li, Y. M. Gao, B. Xiao, T. Min, Z. J. Fan, S. Q. Ma and L. L. Xu, J. Alloys Compd., 2010, 502, 28–37.
32. Y. C. Liang, W. L. Guo and Z. Fang, Acta Phys. Sin., 2007, 56, 4847.
33. F. M. Gao, J. L. He, E. D. Wu, S. M. Liu, D. L. Yu, D. C. Li, S. Y. Zhang and Y. J. Tian, Phys. Rev. Lett., 2003, 91, 015502.
34. K. B. Panda and K. S. Ravi Chandran, Acta Mater., 2006, 54, 1641–1657.
35. X. Q. Chen, H. Y. Niu, D. Z. Li and Y. Y. Li, Intermetallics, 2011, 19, 1275–1281.
36. N. H. Miao, B. S. Sa, J. Zhou and Z. M. Sun, Comput. Mater. Sci., 2011, 50, 1559–1566.
37. A. L. Ding, C. M. Li, J. Wang, J. Ao, F. Li and Z. Q. Chen, Chin. Phys. B., 2014, 23, 096201.
38. J. Feng, B. Xiao, R. Zhou and W. Pan, Acta Mater., 2013, 61, 7364–7383.
39. J. Feng, B. Xiao, R. Zhou, W. Pan and D. R. Clarke, Acta Mater., 2012, 60, 3380–3392.
40. Y. Z. Liu, Y. H. Jiang, R. Zhou and J. Feng, J. Alloys Compd., 2014, 582, 500.
41. J. Feng, B. Xiao, C. L. Wan, Z. X. Qu, Z. C. Huang, J. C. Chen, R. Zhou and W. Pan, Acta Mater., 2011, 59, 1742–1760.
42. X. Y. Cheng, X. Q. Chen, D. Z. Li and Y. Y. Li, Acta Crystallogr., 2014, 70, 85–103.
43. Z. Q. Chen, J. Wang and C. M. Li, J. Alloys Compd., 2013, 575, 137–144.
44. J. C. Phillips, Rev. Mod. Phys., 1970, 42, 320.

---