Due to their excellent soft magnetic property and low cost, Fe-based metallic glasses have captured much attention experimentally and theoretically.^[1–6] Metallic glasses are generally produced by rapid cooling methods, where the viscosity increases exponentially until glass transition occurs.^[7] Egami et al. put forward the concept of atomic level stress, and described the glass transition in terms of topological instability.^[8] However, the structure details of metallic glasses remain ambiguous owing to the lack of three-dimensional (3D) atomic structural model. Molecular dynamic (MD) simulation^[9] especially ab initio molecular dynamic (AIMD) simulation^[10] is regarded as an effective way to investigate the structural evolution during rapid cooling on an extremely short timescale.

During the past decades, a lot of work has been done to improve the saturation flux density and the underlying mechanism has been investigated. Earlier study suggested that moment variation in transition metal–metalloid (T-M) alloy is simply determined by the different local symmetry of the M atom or the electronic valence.^[11,12] However, these rough approximations could not explain the strong magnetovolume effect observed in Fe-based alloys. Rahman et al.^[13] studied the influences of impurities in magnetism and electronic structures of bcc Fe. They found that the average magnetic moment generally follows a Slater–Pauling curve, and attributed the discrepancy to the weak ferromagnetic nature of bcc Fe. Recently, Kiss et al.^[14] investigated the anomalous magnetic behaviors of Fe–Zr metallic glasses in detail and elaborated this anomaly in terms of critical Fe atomic volume. Based on AIMD simulation of Fe₈₅Si₂B₈P₄Cu₁ (NANOMET) amorphous alloy,^[15] it is revealed that magnetic moment is both electron charge dependent and atomic volume dependent. Our results found that the magnetic moment of Fe of typical Fe-based amorphous alloy Fe₈₀P₉B₁₁ is strongly affected by the charge quantity.

The calculation was performed by using the Vienna ab initio simulation package (VASP).^[16] Projected augmented wave (PAW) parameterized by Perdew, Burke, and Ernzerhof was adopted to describe interactions between particles.^[17,18] Only Γ -point was used to sample the Brillouin zone. The simulation was performed in a canonical ensemble (NVT) with a Nosé thermostat^[19] for temperature control. The initial structure contained 100 atoms that were totally randomly distributed. The melt was kept at 2000 K for 10 ps in time-steps of 5 fs to reach thermal equilibrium. Afterwards, the system was gradually cooled down to 1600 K, 1200 K, 800 K, 600 K, 400 K in 1000 MD steps. The cooling rates were 8×10¹³ K/s and 4×10¹³ K/s when the temperature ranged from 2000 K to 800 K and 800 K to 400 K, respectively. The external pressure was turned to essential zero by adjusting the volume of the supercell. Finally, an additional 5000 MD steps were performed to collect atomic structure information. For accurate electron density of states (DOS) calculation, a 2×2×2 k-points mesh were adopted.

The mean square displacements (MSDs) are calculated at each temperature. Using the Einstein relationship: D = lim_t→∞〈r²(t)〉/6t, the values of self-diffusion constant D for Fe, P, and B are obtained and shown in Fig. 1. By comparison, the diffusion rates can be ordered as B > P > Fe in magnitude, which is in accord with the results reported previously.^[20] It is concluded that glass transition occurs at about 800 K, since the diffusion rate at 800 K is only 4%–6% of that at 2000 K. Figure 2 shows the total pair distribution functions (PDFs) at various temperatures. The positions of first peak of the total PDF are almost the same for various temperatures, while the intensity of first peak slightly increases. Based on topological fluctuation theory,^[21] the increase of the intensity of g(r) peak is due to the reduced topological fluctuation. However, the second peak begins to split at about 800 K and the splitting becomes more obvious with the decrease of temperature, which has been reported previously.^[22,23] It seems that the left sub peak becomes more prominent with temperature decreasing from 800 K to 400 K, while the right sub peak varies slightly. The relative position of peak at 400 K is approximately 1:1.70:1.96, which follows the universal scaling behaviors of R_i/R₁ pointed out by Liu et al., where R_i is the i-th peak position of PDF.^[24] Previous work^[25,26] suggested that the unevenness of the connecting style of atomic clusters results in the splitting of the second peak. Thus the liquid–glass transition may be a process of the increase of local topological order combined with the formation of cluster connection.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Temperature dependence of self-diffusion constant of Fe, P, and B in Fe80P9B11 alloy.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) Plots of total PDF (g(r)) versus position (r) at various temperatures.

A more detailed study of structure evolution of Fe–P–B amorphous alloy can be done through partial PDFs and the results are shown in Fig. 3. Considering the fact that Fe is the dominating component, only the Fe–Fe, Fe–P, and Fe–B pairs are presented. It can be seen that the Fe–Fe partial PDFs show the same tendency as total PDFs upon cooling. The position of the first peak for Fe–P partial PDF at 400 K is 2.32 Å, which is shorter than the sum of Goldschmidt atomic radii of Fe and P. The first peak of Fe–B partial PDF is located at 2.12 Å, comparable to the sum of Goldschmidt atomic radii of Fe and B. This actually indicates the existence of Fe–P and Fe–B bonds, and the Fe–P bond is stronger than Fe–B bond.^[27] It is worth noting that Fe–P pair and Fe–B pair show different behaviors below the glass transition temperature. The left sub peak of the second peak of Fe–B partial PDFs becomes more pronounced, and its position shifts to small value. When a B atom is embedded in Fe-based amorphous matrix, the B atom will suffer compressive stress due to the local topological instability.^[8] In addition, the external stress which is equal to the average atomic level stress is zero. Thus, tensile stress is predominant in the second nearest atomic shell and another B atom tends to occupy this site, which may reduce the atomic level stress and hence a denser structure is formed. Dai et al.^[28] have observed that a positive sign and a negative sign for volumetric strain ε alternate from short range order to medium range order; this phenomenon verifies our findings. In this case, the behavior of Fe–B atomic pair can be attributed to the oscillation of stress (strain). Lashgari et al.^[29] have reviewed that B can promote the glassy structure to a very great degree. That B can promote the glass formation ability (GFA) of Fe-based amorphous alloy may lie in the unique behavior of Fe–B atomic pair apart from what is so-called “solute–solute avoidance”.^[27] Because of the relatively large atomic radius of P, this effect is much slighter than that of B.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) Variations of partial PDF g(r) with position for (a) Fe–Fe, (b) Fe–P, and (c) Fe–B atomic pairs at various temperatures.

The total magnetic moment of our simulated supercell is 165 µ_B. Since it is impossible to contain a domain wall as illustrated by Wang et al.,^[30] the saturation flux density can thus be calculated from B_s = M/V = 1.67 T where B_s, M, and V denote saturation magnetization, calculated magnetic moment, and supercell volume, respectively. Figures 4(a) and 4(b) illustrate total electron DOS and the partial electron DOS of Fe atom in Fe₈₀P₉B₁₁ amorphous alloy, respectively. It is obvious that the electron DOS near the Fermi level is dominated by Fe-d states and that the d band splits into two main peaks.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) (a) Spin-polarized total electron DOS of Fe80P9B11 amorphous alloy, and (b) partial electron DOS of Fe atom.

The Fermi level is located right above the majority spin band and falls near the minimum of spin-down band, indicating the existence of ferromagnetism in this alloy. Moreover, both spin bands are partially filled and in this case, exchange splitting energy is less than the energy difference between the Fermi level E_F and the top of the d band. Therefore, Fe₈₀P₉B₁₁ alloy is a weak ferromagnet. Figure 5 shows partial DOSs of Fe, P, and B atoms. The s-states of P and B lie far below Fermi level and are fully filled. Besides, there is hybridization between Fe 3d states and 2p states of P and B, which is consistent with the results from partial PDFs. This can explain the small negative magnetic moments of metalloid atoms.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) Partial DOSs of Fe, P, and B atoms in Fe80P9B11 amorphous alloy.

The average magnetic moments per atom for Fe, P, and B are listed in Table 1. The Fe has an average magnetic moment of 2.1 µ_B, while P and B each obtain a magnetic moment of about −0.1 µ_B. As is well known, the magnetic moment depends on the interplay between a local packing or topologically dependent electronic band structure and the number of valence electrons.^[31] The existence of covalent coupling and the larger electronegativity of metalloid atoms result in the charge transfer from Fe atoms to metalloid atoms. The relationship between Fe magnetic moment and charge is illustrated in Fig. 6. One can see that magnetic moment follows nearly linear dependence on charge. The larger the magnetic moment, the less the electric charge quantity that Fe will obtain. Atom-averaged moment in Bohr magnetons is the difference between the atom-averaged number of spin-up and that of spin-down electrons, that is, µ_av = N^↑ − N^↓.^[9] For systems with strong ferromagnetism whose spin-up d-band is full, and if the contribution from sp-bands is ignored, N^↑ will be equal to a constant of 5. In this case, the increase of charge indicates the augment of N^↓, namely the reduction of magnetic moment of Fe. The deviation from linear correlation is due to the fact that Fe₈₀P₉B₁₁ amorphous alloy is just a weak ferromagnet so that the magnetic moment is sensitive to the Fe–Fe atomic distance. This can also explain that Co₉₁Zr₉ manifests an almost pressure-independent magnetic behavior, while the magnetic property of Fe₉₁Zr₉ shows an extreme sensitivity to volume change^[14] in this simple method. According to the relevant work,^[15] Fe atom pairs with longer distance are supposed to have stronger ferromagnetic exchange, resulting in higher magnetic moment.

Table 1

Table 1

Table 1 Average magnetic moments mi i = s, p, and d of s, p, and d states and the total magnetic moments mtotal of Fe, P, and B atoms in Fe80P9B11 amorphous alloy. . Atom ms/µB mp/µB md/µB Mtotal/µB Fe –0.001 –0.024 2.125 2.100 P 0 –0.105 0.001 –0.104 B –0.020 –0.104 0.002 –0.121

Table 1 Average magnetic moments mi i = s, p, and d of s, p, and d states and the total magnetic moments mtotal of Fe, P, and B atoms in Fe80P9B11 amorphous alloy. .

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. (color online) Fe magnetic moment versus charge quantity.

The high saturation flux density is preferred for Fe-based metallic glasses. The concentration of Fe atoms dominates the saturation flux density, however, the high concentration of Fe atoms leads to the poor GFA. Therefore, it is worthwhile to investigate the micro-alloying effects in Fe-based amorphous alloys. As discussed above, an alloying element with larger electronegativity may benefit the magnetic property since it will make Fe atoms more positively charged. In contrast, the hybrid Fe–B or Fe–P bonds will replace the polarizable d–d bonds and deteriorate the magnetic property. Therefore there is an optimal concentration of metalloid atoms in Fe-based amorphous alloy to obtain high saturation flux density. To further testify this point, more work needs to be done, and this work provides a theoretic viewpoint for developing Fe-based amorphous alloy.

We perform AIMD simulation to investigate the structure evolution of Fe₈₀P₉B₁₁ amorphous alloy during rapid cooling and evaluate its magnetic property. It is found the intensity of first g(r) peak increases monotonically with the decrease of temperature. The second peak of PDF starts to split around 800 K. The left sub peak of the second PDF peak of Fe–B becomes more pronounced, and the position shifts to small value below the glass transition temperature, which may be the major reason why B promotes the glass formation ability significantly.

The DOS calculation indicates that the Fe₈₀P₉B₁₁ is weak ferromagnet. The calculated average magnetic moment of Fe is about 2.10 µ_B, while P and B obtain the magnetic moments of −0.10 µ_B and −0.12 µ_B, respectively. Our results reveal that in this system, Fe magnetic moment varies almost linearly with electric charge. Alloying elements with large electronegativity may be an effective method of improving the magnetic properties of Fe-based amorphous alloy.