Magnetic Anisotropy Tailoring by 5d Doping in (Fe,Co)₅SiB₂ Alloys

1. Introduction

Magnetic materials are crucial for technological development in many sectors, such as electric motors, transformers, power generators or recording media. High performance permanent magnets with large saturation magnetization, high Curie temperature and large uniaxial magnetic anisotropy are necessary, these requirements being currently fulfilled by the rare earth permanent magnets such as Sm-Co and Nd-Fe-B [1]. The volatile prices and restricted accessibility to the rare earth resources raised a strategic issue, directing the research efforts in the last two decades to the search of new magnetic materials with close performance to Sm-Co and Nd-Fe-B magnets, but with lower rare earth content. Also, the search for new permanent magnets without rare earth elements arouse interest in the magnetic materials research [2,3,4]. Due to their reduced costs and small supply risks, their use in less-demanding applications can be efficient. This research direction is motivated by the large gap in the price-performance diagram between the hard ferrite/Alnico magnets and the high performant RE which could be filled in by new rare-earth free permanent magnets. In this last category, the Fe compounds with anisotropic crystal structure are important candidates for functional rare-earth free permanent magnets due to their lower price, standardized preparation procedure and good mechanical properties. Such compound is Fe₅SiB₂ discovered in the 1960’s by Aronsson et al. [5], having a Cr₅B₃ structure type (I4/mcm space group). According to the recent research, a saturation magnetization of 9.2 µ_B/f.u. has been theoretically predicted for this compound [6], whilst the magnetic measurements at 10 K show a value of 1.1 MA/m, with large values of Fe spin magnetic moments on both Fe crystal sites (2.31 µ_B for Fe on 4c site and 2.10 µ_B for Fe on 16l site) determined by neutron diffraction at 16 K [7]. A magnetic anisotropy constant K₁ of -0.28 MJ/m³ [6] and experimentally determined Curie temperature over 800 K [7,8] have been also reported for this compound. The in-plane anisotropy and the spin reorientation at ~ 170 K [7] makes this compound inefficient for permanent magnets (PM) development, but further research found that the Co for Fe substitution in (Fe,Co)₅SiB₂ is able to turn the sign of MAE from negative to positive, enabling an axial easy magnetization direction for the corresponding alloys, as well as the decrease of the spin reorientation transition temperature down to the lowest temperature [9]. Previous studies showed that doping with 5d elements (M) is effective by building stronger uniaxial anisotropy in Fe alloys, due to the stronger spin-orbit coupling of Fe-3d and M-5d states. Earlier research on Fe₅PB₂ doped with W of Thakur et al. [10] showed that the anisotropy constant K₁ increases from 0.152 MJ/m³ in Fe₅PB₂ to 1.135 MJ/m3 in Fe₄WPB_2, this enhancement being attributed to the strong spin-orbit coupling of Fe-3d states and W-5d states. Still, 5d doping could decrease the magnetization and the Curie temperature in Fe alloys, the doping amount in this case having to be optimized to maintain them at values compatible with functional applications. Our previous theoretical investigations showed for Fe₄WSiB₂ alloy an increased anisotropy constant of 0.218 meV/f.u. but lower total magnetic moment (7.97 µ_B/f.u.) and Curie temperature (747 K) compared with the undoped Fe₅SiB₂ compound [11]

Also, other 5d elements could be effective for the magnetic anisotropy tailoring. For instance, 5d doping in Fe₂B alloy showed larger efficiency for Re doping (K₁ = 1.5 MJ/m³, magnetization µ₀M_s = 1.22 T and estimated Curie temperature of 595 K for Fe_1.52Re_0.48B), whilst the W doping effects on anisotropy are much weaker [12].

In this study, we present the magnetic anisotropy behaviour by 5d doping in the Fe_5-x-yCo_xM_ySiB₂ alloys (M = W and Re) in the range 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1. Two-dimensional 2D maps of the anisotropy constant are built, allowing to select the composition range for compounds with large uniaxial anisotropies. Further investigations on the magnetization and Curie temperature behaviour vs. Co and M content (M = W and Re) enables a detailed view on the intrinsic magnetic properties of these alloys. Fully relativistic theoretical calculations done in the framework of density functional theory (DFT) using the exchange-correlation potential under general gradient approximation with the Perdew et al. (PBE) parametrization [13] were employed for the current investigations.

2. Results and Discussions

2.1. Crystal Structure

The crystal structure of the Fe_5-x-yCo_xM_ySiB₂ alloys (Figure 1) consists of a body-centered tetragonal unit cell (space group I4/mcm, no. 140), with Fe atoms on two different sites (4c and 16l), the B atoms on 8h sites, whilst Si atoms occupy the 4a crystallographic sites. The lattice parameters determined by X-ray diffraction are a = 5.5541 Å and c = 10.3429 Å [7].

The electronic band structure calculations for the Fe_5-x-yCo_xM_ySiB₂ alloys (for 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1) in tetragonal structure (space group I4/mcm) using the experimental lattice parameters [2,14] have been performed. The experimental free parameters of 8h and 16l crystallographic sites determined by Cedervall et al. for Fe₅SiB₂ have been considered [7]. The lattice constants of the W and Re-doped alloys have been evaluated using the lattice constants of Re₅SiB₂, W₅SiB₂ and Co₅SiB₂ [7,14] and considering their linear dependence by doping. The occupation of Co atoms on 16l crystal sites has been considered, according to our previous investigations [8]. Similar preferential occupation has been obtained for Re atoms, whilst a random occupation of the 4c and 16l sites has been derived for W atoms by total energy calculations.

Figure 1. The body-centered tetragonal unit cell (space group I4/mcm, no. 140) of Fe₅SiB₂ alloy, with Fe atoms on two different sites (4c and 16l), the B atoms on 8h sites and Si atoms on 4a crystallographic sites. Graphic program XCrySDen [15] was used.

Figure 1. The body-centered tetragonal unit cell (space group I4/mcm, no. 140) of Fe₅SiB₂ alloy, with Fe atoms on two different sites (4c and 16l), the B atoms on 8h sites and Si atoms on 4a crystallographic sites. Graphic program XCrySDen [15] was used.

2.2. Magnetic Moments

The band structure calculations for Fe₅SiB₂ alloy have been performed using different exchange and correlation parametrizations (local density approximation LDA, as well as GGA) within atomic spheres approximation as well as the full potential approach. The results of the calculations are listed in Table 1 together with the results of experimental measurements [7] and with other theoretical results [6].

According to our calculations, the Fe₅SiB₂ alloy is ferromagnetic with a total magnetic moment between 8.46 and 9.41 µ_B/f.u., depending significantly on the exchange-correlation potential and on the potential symmetry (spherical in ASA and no shape approximation in full potential approach). As can be seen in Table 1, the VWN-ASA approach gives an underestimated total magnetic moment compared with the experimental data. Better agreement is obtained by the use of the GGA approach, with total magnetic moment of 9.34 µ_B/f.u. (GGA-ASA) and 9.41 µ_B/f.u. (GGA-FP). Also, the SPR-KKR shows similar values of individual magnetic moments with FPLO calculations, with only minor deviation determined most probably by the lattice constants used for calculations. The Fe magnetic moments show higher values for Fe 4c, but the ratio between Fe 4c/16l moments depend on the calculation approach.

Based on these results, the GGA-ASA calculation mode has been chosen to deal with the combined Co and M (M = Re, W) doping. Moreover, the GGA-ASA calculated value of µ₀M_s (1.36T) is close to the value determined experimentally by McGuire et al. (1.31 T) [9].

The total magnetic moments calculated for the Fe_5-x-yCo_xM_ySiB₂ alloys for 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1 are used to build a two-dimensional map of magnetization vs. Co and M content (x,y), shown in Figure 2. According to the contour maps for several selected values of total magnetic moment, the Re and W doping have similar effect in combination with Co doping on the magnetization of the Fe_5-x-yCo_xM_ySiB₂ alloys. The values of magnetization decrease by both combined doping (Co and Re; Co and W, respectively) from 9.34 µ_B/f.u. to values lower than 5 µ_B/f.u. The decrease of magnetization was evidenced already for doping with Co [6, 8]. Still, the use of combined doping (Co + Re; Co + W) is able to enhance this tendency, as shown in Figure 2. If the desired values of magnetization are set to values over 6 µB/f.u., the corresponding alloys have Co content x ≤ 1.5 and M content y ≤ 0.6.

2.3. Magnetic Anisotropy

The earlier reports show that the anisotropy constant of (Fe,Co)₅SiB₂ alloys is changing the sign for a certain range of Co amount [6,8]. This effect is accounted in the present study, as an easy-axis anisotropy is obtained by Co doping, followed by expected anisotropy enhancement by further doping with 5d elements. The validity of this mechanism is discussed in the following, using the representation of the anisotropy constants K₁ calculated for Fe_5-x-yCo_xM_ySiB₂ alloys. The K₁ values obtained for both M = Re, and W in a 5x3 grid for 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1 (Δx = Δy = 0.5) have been used to build the K₁(x,y) 2D maps. The K₁ isolines between -0.3 meV/f.u and 0.25 meV/f.u. have been evidenced in Fig.3. As can be seen, much different K₁ 2D maps have been obtained by Re/W doping in Fe_5-x-yCo_xM_ySiB₂ alloys. In the case of Re, values of K₁ over 0.25 meV/f.u. can be obtained for a combination of Co doping with approx. x ≥ 1.8 and Re doping with y ≥ 0.7. The disadvantage of the Re doping is visible by overlapping the M_s 2D map (Figure 2) and K₁ 2D map (Figure 3). The (x,y) areas with large K₁ values almost overlap with the areas with low magnetization values, as M_s < 5 µ_B/f.u. for x ≥1.5. This behaviour of magnocrystalline anisotropy is opposite to that evidenced in Fe₂B alloy, where Re doping promoted large axial anisotropy [12].

On the other hand, W doping can increase the anisotropy of the Fe₅SiB₂ alloy, values of K₁ ≥ 0.2 meV/f.u. being obtained for x = 0 and y > 0.8. Also, values of K₁ ≥ 0.25 meV/f.u. are seen for x = 0 and y ~ 1. By overlapping the 2D maps of M_s and K₁ in the case of M = W, one can find magnetization values between 8 - 8.5 µ_B/f.u. It is interesting to note that in the case of Re doping, the area (x = 0, y > 0.9) corresponds to largest value of in-plane magnetocrystalline anisotropy, whilst W doping alone is able to switch the anisotropy to easy-axis, without additional Co doping.

2.4. Curie Temperatures and Exchange-Coupling Parameters

The Curie temperatures (T_c) have been determined by mean field approach (MFA) using the procedure described in Section 3. The grid data for T_c have been used to build isolines with constant values of T_c, which are represented in Figure 4. According to our calculations, Fe₅SiB₂ alloy is ferromagnetic with a Curie temperature of 1162 K calculated by mean field approach. As the mean field method is known to overestimate by ~20% the Curie temperature values, the estimated T_c would be comparable with the experimental value of 850 K [8].

From the T_c representation in Figure 4, it is obvious that the combined Co + M doping is going to decrease the Curie temperatures for both M = Re and W. This decrease is slightly steeper in the case of Co + Re. For Re-doped alloys, by overlapping the areas with K₁ ≥ 0.2 meV/f.u. of the K₁ 2D map to the Curie temperatures map, the corresponding alloys have T_c values under 400 K, which makes the Re-doping rather unsuited for PM applications.

On the other hand, the expected T_c values of the W-doped alloys with anisotropy constant K₁ ≥ 0.2 meV/f.u. (obtained for x = 0 and y > 0.8) are between 850 – 900 K, compatible with PM applications.

Due to the promising intrinsic properties Fe_1-yW_ySiB₂ alloys in doping range y > 0.8, we investigated into more details the magnetic properties of such W-doped alloy. We used a complementary approach by considering particular microscopic models making use of magnetic interaction based on the classical Heisenberg Hamiltonian described in Section 3. Using this approach, the Fe-Fe exchange-coupling parameters are obtained and represented as a function of distance (in units of lattice constant a). The Fe-Fe exchange interaction for Fe₅SiB₂ and Fe_4.1W_0.9SiB₂ alloys are presented in Figure 5. As can be seen in Figure 5, the dominant exchange interactions are positive, favouring the ferromagnetic ordering. Negative exchange interactions are also seen, mostly between Fe 16l spins at distances of between 0.6a and 0.9a. The largest exchange interactions are between Fe 4c - Fe 16l nearest neighbours. These exchange interactions are affected by the W doping and a decrease between the nearest neighbour’s exchange-coupling parameters in Fe_4.1W_0.9SiB₂ compared with Fe₅SiB₂ can be seen. Also, a decrease of the exchange coupling parameters values by W doping is observed for the interactions between the Fe 4c - Fe 4c spins. The exchange interactions between nearest Fe 4c - Fe 4c spins are much smaller, as their interatomic distance are considerably larger than those of the other pairs. The Curie temperature is mostly determined by the Fe 16l - Fe 16l and Fe 4c - Fe 16l exchange interactions.

The spin and orbital magnetic moments, magnetization µ₀M_s, anisotropy constant K₁, Curie temperature and hardness parameter for Fe_4.1W_0.9SiB₂ alloy are listed in Table 2. In particular, the spin and orbital moments on Fe sites (2.43 µ_B and 0.07 µ_B for Fe 4c; 1.95 µ_B and 0.06 µ_B for Fe 16l) and are higher compared with the corresponding values in Fe₅SiB₂ (see Table 1) but the weight of the magnetic atoms decreases with about 18 %, resulting in a magnetization decrease (8.37 µ_B/f.u.) by W doping, which correspond to a µ₀M_s value of 1.17 T.

The results from Table 2 show that Fe_4.1W_0.9SiB₂ alloy is a semihard permanent magnet, with µ₀M_s > 1T and magnetic anisotropy energy of 0.46 MJ/m³, close to the values in Sr-ferrite [4]. The magnetic hardness parameter captures the potential of the present alloy with adequate magnetization M_s (1.17 T) to be developed into a compact permanent. The value obtained according to formula $\kappa =\sqrt{K_1/{\mu }_0{M}_s^2}$ is 0.65, corresponding to a semi-hard magnetic material. The Curie temperature is expected to be overestimated by MFA, but a value over 680 K is still expected for the Fe_4.1W_0.9SiB₂ alloy. The high value of Curie temperature allows the use of this alloys for relative high temperature applications.

3. Materials and Methods

Spin-polarized fully relativistic band structure calculations based on the Korringa-Kohn-Rostoker (SPRKKR) method has been used for theoretical calculations [16]. This calculation method is based on the KKR Green’s function formalism that makes use of the multiple scattering theory. Within this approach, the so-called t-scattering matrices describing the scattering properties of each scattering centre (atom) are determined. In the second step, the multiple scattering matrices by all atoms in the lattice are calculated accounting that that the incident wave at each atom is the sum of the outgoing waves from all other atoms. The central role in the KKR Green’s function method is played by the crystal Green’s function, which is efficiently related to the Green’s function of free space via the Dyson equation. The SPR-KKR calculation method supplies all the information about the electronic structure, similar to other band structure methods which are using the energy eigenvalues and eigenfunctions to represent the electronic structure. In addition, the SPR-KKR band structure method has an important advantage by dealing with chemical disorder, as the coherent potential approximation (CPA) [17] was efficiently integrated by defining a Green’s function of average crystal medium, which is determined self-consistently through the condition that the concentration average of the various atom types should not produce any additional scattering in this medium [ 17]. In this way, the SPR-KKR is a more efficient and reliable alternative to more tedious supercell techniques, requiring enlarged unit cells and atomic configuration averaging in the supercells in order to describe the alloys properties. More details about the SPR-KKR method are extensively described elsewhere [16].

The local spin density approximation (LSDA) [18], as well as the generalized gradient approximation with the parametrization of Perdew et al. (GGA-PBE) [13] have been used to account on the exchange and correlation effects. The k-space integration was performed using the special points method [19]. The study of the magnetic anisotropy has been performed by magnetic torque calculations. The magnetic torque acting on the magnetic moment $\overrightarrow{{\mathbf{m}}_{\mathbf{і}}}$ of the atomic site i, oriented along the magnetization direction $\overrightarrow{\mathit{M}}$ generates a component ${\mathit{T}}_{\widehat{\mathit{u}}}\left(\mathit{\theta },\mathit{\phi }\right)=-\partial \mathit{E}(\overrightarrow{\mathit{M}(}\mathit{\theta },\mathit{\phi }))/\partial \mathit{\theta }$ with respect to axis $\widehat{\mathit{u}}$, where θ and ϕ are the polar angles [20,21]. The magnetic anisotropy energy, defined as the energy difference between the in-plane and out-of-plane magnetization directions and the magnetic torque are related by a special geometry. By setting the polar angles to θ = π/4 and ϕ = 0, the calculated magnetic torque is ${\mathit{T}}_{\widehat{\mathit{u}}}\left(\mathit{\pi }/\mathrm{4,0}\right)={\mathit{E}}_{\left[100\right]}-{\mathit{E}}_{\left[001\right]}$ [21].

The magnetic behaviour of the Fe_5-x-yCo_xM_ySiB₂ alloys is investigated by a complementary approach based on the classical Heisenberg Hamiltonian described by the expression:

$${\mathit{H}}_{\mathit{e}\mathit{x}}=-\sum _{\mathit{i}.\mathit{j}}{\mathit{J}}_{\mathit{i}\mathit{j}}{\widehat{\mathit{e}}}_{\mathit{i}}·{\widehat{\mathit{e}}}_{\mathit{j}}$$

(1)

where the summation is performed on all lattice sites i and j and${\widehat{\mathit{e}}}_{\mathit{i}},{\widehat{\mathit{e}}}_{\mathit{j}}$ are the unit vectors of magnetic moments on sites i and j, respectively. The total energy difference $∆{\mathit{E}}_{\mathit{i}\mathit{j}}$ due to infinitesimal change in angle between magnetic moments of (i, j) spin pair has been calculated in scalar-relativistic mode and expressed to lowest order with respect of the orientation angle between ${\widehat{\mathit{e}}}_{\mathit{i}}$ and ${\widehat{\mathit{e}}}_{\mathit{j}}$. Using the expression derived by Liechtenstein [20] based on the magnetic force theorem, a one-to-one mapping between the exchange-coupling energy ΔE_ij and the Heisenberg Hamiltonian is obtained, allowing to determine the J_ij exchange coupling parameters. The J_ij exchange coupling parameters have been calculated for all magnetic atoms as a function of distance up to 15 Å around each lattice site. The mean field approach has been used to derive the Curie temperatures, using the calculated J_ij exchange coupling parameters, according to the expression [20,22]:

$${\mathit{T}}_{\mathit{c}}^{\mathit{r}\mathit{o}\mathit{u}\mathit{g}\mathit{h}-\mathit{M}\mathit{F}\mathit{A}}=\frac{2}{3{\mathit{k}}_{\mathit{B}}}\sum _{\mathit{i}}{\mathit{J}}_{0\mathit{i}}$$

(2)

where ${\mathit{J}}_{0\mathit{i}}$ is the exchange-coupling parameters i^th site sum over all coordination shells.

4. Conclusions

The present research aims to find the M 5d doping elements and their amount able to switch to the easy-axis anisotropy and to enhance it by enabling a strong spin-orbit coupling between the Fe-3d and M-5d states. Theoretical studies on the Fe_5-x-yCo_yW_xSiB₂ alloys show the decrease of total magnetic moment and Curie temperatures of the alloys by increasing W and Re content x. Also, the theoretical calculated anisotropy constant turns from planar for Fe₅SiB₂ alloy to axial for Fe_5-x-yCo_yW_xSiB₂ in different zones of the 2D map, showing rather opposite impact of Re/W doping on the magnetocrystalline anisotropy behaviour in these alloys. The superposition of the 2D maps for magnetization, MAE and Curie temperatures allowed us to conclude that W doping is favourable for development of intrinsic properties compatible with PM applications, even without Co doping. In the case of Re + Co doping, such concentrations range was not found, due to the reversed effects of combined doping on the magnetization and Curie temperature on one side and magnetocristalline anisotropy, on the other side. Despite of its strong spin-orbit coupling with Fe-3d states, the Re doping determined large anisotropy constants for the alloys having low Curie temperatures and magnetizations, incompatible with PM applications. On the other hand, the W-doped Fe₅SiB₂ alloys can be optimized in order to allow its use as semi-hard magnet in nanocomposite magnetic materials.