First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe₈₀Si₁₀Nb₆B₂Cu₂

Figure 1 shows the structure model of the Fe-based nanocrystalline alloy Fe₈₀Si₁₀Nb₆B₂Cu₂, with its lattice constant a=5.7328, b=c=14.332, and the crystal face angle α = β = γ = 90°. It is a 1K107 material with different element ratios. This paper establishes a 2 × 5 × 5 Fe supercrystal cells, a total of 100 Fe atoms, and then randomly replaced Fe atoms with 10 Si atoms, 6 Nb atoms, 2 B atoms and 2 Cu atoms, respectively, thereby establishing the Fe₈₀Si₁₀Nb₆B₂Cu₂ model.

Fig. 1

The calculation in this paper is carried out by the Cambridge sequential total energy package (CASTEP) [14] software package based on the plane wave geese potential method. For ensuring the reliability and stability of the calculation, and to obtain a crystal model close to the actual structure, we first use DFT to process the interaction between ions and electrons, and use generalised gradient approximation (GGA) PBE [13] basis set which deals with the exchange correlation energy between atoms and then optimise the geometric structure of the Fe-based nano-alloy to obtain a stable crystal structure. Finally, the density of states, energy band structure, elastic constants, magnetic properties and population analysis are calculated and analysed for the stable crystal structure.

In this paper, the cut-off energy of the plane wave is set to 330 eV, select 3 × 3 × 1 for k-point sampling, and set the spin polarisation conditions to calculate and analyse the spin-down and spin-up electrons in the Fe₈₀Si₁₀Nb₆B₂Cu₂ crystal. In the iterative process, the single-atom energy convergence accuracy is 2 × 10⁻⁵ eV/atom, the maximum stress is 0.10 GPa and the convergence standard for the maximum displacement between atoms is 0.002 Å. In this calculation, the valence electron of each atom is Fe-3d⁶4s², Si-3s²3p², Nb-4d⁴5s¹, B-2s²2p¹ and Cu-3d¹⁰4s¹.

Figure 2 shows the spin-up and spin-down band structures of this material. One of the sources of electronic magnetism is the magnetic moment produced by the electron spin. The spin is closely related to the material's magnetic properties, so the spin-up and spin-down band structures are calculated. The Fe-based nanocrystalline alloy Fe₈₀Si₁₀Nb₆B₂Cu₂ has a total of 786 electrons near the Fermi level (energy is zero), of which there are 565 spin-up electrons and 221 spin-down electrons. After optimising the structure of Fe₈₀Si₁₀Nb₆B₂Cu₂, the energy band structure is calculated. This paper mainly intercepts the energy band range of −10~110 eV. It can be seen in Figure 2(a) and (b) the spin-up and spin-down energy band structure diagram of Fe₈₀Si₁₀Nb₆B₂Cu₂. Figure 2 shows that the Fermi energy level passes through the energy band in both the spin-up and spin-down directions, and the energy gap is 0, showing metallicity.

Fig. 2

In addition, it can be seen from Figure 2(a) and (b) that the spin up and down band structure dispersion of Fe₈₀Si₁₀Nb₆B₂Cu₂ is significantly different, indicating that the alloy has spin splitting. Therefore, this shows that Fe₈₀Si₁₀Nb₆B₂Cu₂ has strong magnetic properties. Near the Brillouin zone Q, the energy band is very gentle, so the effective mass m of carriers will be relatively large, resulting in a relatively large Seebeck coefficient S of the alloy system.

Figure 3 shows the total density of states of Fe₈₀Si₁₀Nb₆B₂Cu₂ near the Fermi level and the partial density of states of each atom. This paper mainly intercepts the density of states diagram with energy from −15 eV to 25 eV. It can be seen from the figure that the density of states of Fe₈₀Si₁₀Nb₆B₂Cu₂ is mainly composed of Fe-4s, Si-3p, Nb-4d, B-2P state and Cu-3d state. In the low-energy region with energies ranging from −15 eV to 0 eV, that is, at the left side of the Fermi level, the density of states is mainly composed of Fe-4s state, Si-3s state, Nb-4d state, B-2s state and Cu-3d state. On the right side of the Fermi level, the density of states is mainly composed of Fe-3p, Si-3p, B-2P and Nb-4p state. And there are two obvious peaks near the Fermi level, and the distance between the two peaks is called the pseudo-energy gap. The wider the pseudo-energy gap, the stronger the ability to form bonds between atoms and the stronger the covalent nature of the substance. Therefore, this pseudo-energy gap is mainly provided by the localised Fe-3d electron orbital of the transition metal, in which the density of each partial wave state crosses the Fermi level, showing obvious metallic properties.

Fig. 3

When the energy is −13.41 eV, the Si atom has a peak, which is mainly composed of electrons in the Si-3s orbital. Here, there is also a tiny peak of Fe-3d orbital, which overlaps with the peak of Si-3s orbital. It can be explained that there is a charge transfer between Si-3S and Fe-3d orbital, and then a chemical bond is formed. The typical population numbers and magnetic moments of bonds between several adjacent atoms are shown in Table 1.

Table 1 shows that the magnetic moment of the chemical bond formed between adjacent atoms is very small, close to 0, indicating that the alloy has a full-shell structure due to the transfer of electrons.

Figure 4 is the electron spin state density diagram of Fe₈₀Si₁₀Nb₆B₂Cu₂. According to quantum mechanics, spin is a basic characteristic of electrons. After the magnetised ferromagnetic material, the electrons will generate two kinds of carriers with spin-up and spin-down [14]. It can be seen from the figure that the spin up and down density of states of Fe₈₀Si₁₀Nb₆B₂Cu₂ near the Fermi level is not symmetrical, and the difference is very large. In both the spin up and down state densities, the peak of the state density appeared near −1.77 eV, resulting in the phenomenon of spin splitting, indicating that the alloy is magnetic.

Fig. 4

There is a large peak near the Fermi level; Figure 5 shows that this is mainly caused by the hybridisation of the 4s of Fe atoms and the 4d electron orbitals of Nb atoms, and Fe atoms in particular play most of the role. It can be seen from the figure that the upper and lower spins of Fe atoms are obviously asymmetric, and the upper and lower spins of other elements are slightly different, but they are basically symmetric, indicating that the magnetic moment of the alloy is mainly contributed by iron atoms.

Fig. 5

This paper calculates the population, electronic charge and magnetic moment of all atoms in Fe₈₀Si₁₀Nb₆B₂Cu₂, as shown in Table 2. Table 2 shows that the distribution of electrons in different orbits and the magnitude of the magnetic moment of each iron atom are different. This may be because the distance between the iron atom and the surrounding atoms is different, which leads to mutual coupling with each atom and lattice distortion. Table 2 shows that the total magnetic moment of Fe₈₀Si₁₀Nb₆B₂Cu₂ is 84.15 μ. Among them, the magnetic moment of B atoms is −0.16 μ, the magnetic moment of Cu atoms is −0.17 μ, the magnetic moment of Si atoms is −1.16 μ, the magnetic moment of Nb atoms is −3.11 μ and the magnetic moment of Fe atoms is 88.75 μ. Therefore, it can be concluded that Fe atoms play a decisive role in the magnetic moment of the entire alloy, which is also consistent with the conclusion drawn from the previous analysis of the spin state density.

For the optimised Fe₈₀Si₁₀Nb₆B₂Cu₂, first-principles DFT is still used to calculate its elastic constant C_ij. Since the crystal structure of the Fe-based nanocrystalline alloy belongs to the cubic crystal system, its mechanical stability is only related to the values of elastic constants C₁₁, C₁₂and C₄₄. The elastic constants of Fe₈₀Si₁₀Nb₆B₂Cu_2, which were calculated, are shown in Table 3.

The elastic constant determines the mechanical stability of the material. Judging by the mechanical stability of cubic crystals [15]: C11>0, C44>0, (C11−C12)>0, (C11+2C12)>0 {C_{11}} > 0,\;{C_{44}} > 0,\;({C_{11}} - {C_{12}}) > 0,\;({C_{11}} + 2{C_{12}}) > 0 it can be seen that Fe₈₀Si₁₀Nb₆B₂Cu₂ meets the mechanical stability. According to the theory of elasticity [16], for the cubic crystal structure, the elastic constants satisfy the following equations: (1) B=C11+2C123 B = {{{C_{11}} + 2{C_{12}}} \over 3} (2) G=3C44+C11−C125 G = {{3{C_{44}} + {C_{11}} - {C_{12}}} \over 5} (3) E=9GB3B+G E = {{9GB} \over {3B + G}} (4) μ=−1+E2G \mu = - 1 + {E \over {2G}} where G is the shear modulus, B is the body modulus, μ is Poisson's ratio and E is Young's modulus.

Generally speaking, if the material's B, G and E are larger, the hardness of the material is stronger. According to calculations, the body modulus B is 127.3713 GPa, G is 20.1133 GPa and E is 57.3226 GPa, indicating that the iron-based nanocrystalline alloy Fe₈₀Si₁₀Nb₆B₂Cu₂ has a relatively strong resistance to deformation and high hardness. Poisson's ratio μ, also known as the transverse deformation coefficient, can be used to reflect the ductility and brittleness of materials, just like Cauchy pressure. According to the criterion of mechanical theory, when μ < 1/3, the material is brittle. The calculated μ is 0.425, indicating that the alloy has better flexibility. In order to further study the mechanical properties of Fe₈₀Si₁₀Nb₆B₂Cu₂, the ductility and brittleness of the material can be judged according to the ratio of the bulk modulus B and the shear modulus G. When the ratio B/G is greater than 1.75, the material exhibits toughness. When the ratio B/G is less than 1.75, the material exhibits brittleness. The calculated ratio B/G is 6.3327, indicating that Fe₈₀Si₁₀Nb₆B₂Cu₂ has very strong toughness, which is also consistent with the conclusion drawn by Poisson's ratio μ.