Magnetocrystalline anisotropy of Fe5PB2 and its alloys with Co and 5d elements: a combined first-principles and experimental study
Mirosław Werwiński Alexander Edström Ján Rusz Daniel Hedlund Klas Gunnarsson Peter Svedlindh Johan Cedervall Martin Sahlberg





[Corresponding author: ]werwinski@ifmpan.poznan.pl
Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, 60-179 Poznań, Poland


Materials Theory, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland


Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden




Department of Engineering Sciences, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden



Department of Chemistry – The Ångström Laboratory, Uppsala University, Box 538, SE-751 21 Uppsala, Sweden


























The Fe_5PB_2 compound  offers tunable magnetic properties via the possibility of various combinations of substitutions on the Fe and P-sites. 

Here, we present a combined computational and experimental study of the magnetic properties of .

Computationally, we are able to explore the full concentration range, 
while the real samples were only obtained for 0 ≤ x ≤ 0.7.

The calculated magnetic moments, Curie temperatures, and magnetocrystalline anisotropy energies (MAEs) are found to decrease with increasing  Co concentration. 

Co substitution allows for tuning the Curie temperature in a wide range of values, from about six hundred to zero kelvins. 

As the MAE depends on the electronic structure in the vicinity of Fermi energy, the geometry of the Fermi surface of  and the k-resolved contributions to the MAE are discussed. 

Low temperature measurements of an effective anisotropy constant for a series of  samples determined the highest value of 0.94 MJ m^-3 for the terminal  composition, which then decreases with increasing Co concentration, thus confirming the computational result that Co alloying of  is not a good strategy to increase the MAE of the system.

However, the relativistic version of the fixed spin moment method reveals that a reduction in the magnetic moment of , by about 25%, produces a fourfold increase of the MAE.

Furthermore, calculations for (Fe_0.95X_0.05)_5PB_2 (X = 5d element) indicate that 5% doping of  with W or Re should double the MAE. 

These are results of high interest for, e.g., permanent magnet applications, where a large MAE is crucial.










71.20.Be, 
75.30.Gw, 
75.50.Bb, 
75.50.Cc, 
75.50.Ww  


Magnetocrystalline anisotropy of Fe_5PB_2 and its alloys with Co and 5d elements:
 a combined first-principles and experimental study
    Martin Sahlberg
    December 30, 2023
=====================================================================================================================================









§ INTRODUCTION

 




Many sectors of modern technology depend on magnetic materials,

which are used in such ubiquitous applications as electric motors, power generators, transformers, and recording media.

Hence, magnetic materials are crucial, not only for the digital technology revolution observed in past decades, but also for the green energy revolution expected within the years to come.

The fundamentally and technologically most important intrinsic parameters of magnetic materials include the Curie temperature (T_C), saturation magnetization (M_s), and magnetocrystalline anisotropy energy (MAE).

These parameters are important in a wide variety of applications, including hard and soft magnetic materials for energy conversion, spintronics, and information storage.

Thus, the ability to predict these basic magnetic parameters from first principles is of utmost importance, and accurate modern electronic structure calculations provide an indispensable tool for exploring new materials with desired properties. 

In parallel, experimental synthesis and characterization retains its fundamental importance and a close interplay between computational and experimental work is of ever increasing value in modern materials discovery.

One example of an area in which the search for new magnetic materials, with specific combinations of properties, has been intense in recent years is that of permanent magnets. In this field it is typically desirable to have large M_s, T_C and MAE. This combination is obtained in the commonly used rare-earth transition metal compounds, such as NdFe_14B_2. However, the so called Rare-Earth Crisis <cit.> 
triggered immense international research initiatives in search for new substitute permanent magnet materials with reduced amounts of, or no, rare-earth elements <cit.>. The main challenge in this context is obtaining a sufficiently large MAE in transition metal compounds, where a uniaxial (e.g. tetragonal or hexagonal) crystal structure is a crucial prerequisite. Other areas of applications depend upon other combinations of properties. For example, for magnetocaloric solid state cooling, it is desirable to be able to tune the ordering temperature such that it coincides with the operating temperature (often room temperature)<cit.>. 

Various works have shown how strain engineering or alloying can be used to carefully tune the properties of magnetic materials to obtain desired functionality. For example, it was shown that a careful control of strain and alloy concentration allows for a large MAE in bct FeCo alloys <cit.>. The potential route to FeCo-based permanent magnets offered by that work inspired subsequent studies aiming to stabilize tetragonality in FeCo by B or C-impurities <cit.>. Also the tetragonal (Fe_1-xCo_x)_2B compound has been carefully studied due to its tunable MAE as function of x <cit.>  which, furthermore, has an intriguing temperature dependence <cit.>. It was 
also shown, in both calculations and experiments, that small amounts of 5d substitutions on the Fe/Co site allowed a large increase in the MAE of this material <cit.>.  

The tetragonal family of compounds with compositions (Fe_1-xCo_x)_5P_1-ySi_yB_2 has also been the subject of numerous recent studies <cit.>. Additionally, other chemical substitutions, including Mn on the Fe/Co site <cit.>, have been considered. Due to the broad range of chemical compositions available, this material offers wide tunability of its magnetic properties. Furthermore, the tetragonal crystal structure could potentially allow for a large MAE and, thus, make the compounds interesting within the context of permanent magnet applications. The materials also exhibit other interesting aspects, such as the temperature dependent spin-reorientation transition in Fe_5SiB_2 <cit.>.






The aim of the work is to investigate the effect of the Co and 5d dopants on the tunable magnetic properties of the technologically promising semi-hard  Fe_5PB_2 compound.





 crystallizes in the Cr_5B_3-type structure with a body-centered tetragonal (bct) unit cell, space group I4/mcm <cit.> (see Fig. <ref>).

The unit cell of  consists of 4 formula units (32 atoms).

Fe atoms occupy two inequivalent sites Fe_1 (16l) and Fe_2 (4c).

Fe_1 atoms are distributed on the 16-fold position, Fe_2 and P on the 4-fold, and B on the 8-fold position.





One of the motivations to investigate the  system are our previous results for isostructural  system (with Si in place of P), 
for which we have predicted the highest MAE = 1.16 MJ m^-3 for Co concentration x = 0.3. <cit.>

Next, the (Fe_0.8Co_0.2)_5SiB_2 sample (with Co concentration x = 0.2) was synthesized by McGuire and Parker <cit.> and
their magnetic measurements showed an increase of the anisotropy field after Co substitution, which supports our prediction.

All the previous experimental studies conducted on the  system are limited to the Fe-rich compositions, 

while Co_5SiB_2 is not known to form. <cit.>

For melt-spun samples Fe_5(Si_0.75Ge_0.25)B_2
Lejeune et al. were determined a relatively high anisotropy constant K_1 of about 0.5 MJ m^-3 at room temperature, which is about double the value for . <cit.>

Recently, we also presented a combined experimental and theoretical study of the Fe_5Si_1-xP_xB_2 system, which showed the highest  anisotropy constant for the terminal  composition. <cit.>






 has high T_C of about 655±2 K, magnetic moment of 1.72 μ_B/Fe atom (8.60 μ_B/f.u.), and anisotropy constant K_1 of 0.50 MJ m^-3 measured at 2 K for single crystal. <cit.>

The value of an effective anisotropy constant K_eff of  obtained in our previous work is however significantly higher and equal to ∼0.9 MJ m^-3 at 10 K. <cit.> 

An important parameter, in context of permanent magnets, is magnetic hardness, defined as:

    κ = √(|K|/μ_0 M_S^2),


where K is the magnetic anisotropy constant and M_S is the saturation magnetization.

An empirical rule κ > 1 specifies whether the material have a chance to resist self-demagnetization. <cit.>

From the experimental values of K_eff∼ 0.65 MJ m^-3 and M_S=0.87 MA/m <cit.>, we determined for  κ = 0.69 (at 300 K).

It implies, that without a further engineering of the anisotropy constant,
  will stay in a category of semi-hard magnets. <cit.>


In this work we consider alloying of  with Co and 5d elements. 

In our recent study of  alloys we observed a reduction in magnetization and Curie temperature with an increase of Co concentration. <cit.>

McGuire and Parker also found that 20% Co alloying in  leads to decrease in magnetization, Curie temperature and anisotropy field. <cit.>



Previously we showed also that increase of the MAE of 3d alloys can be achieved through doping with 5d elements. <cit.>

In this work we follow this idea and calculate the resultant MAEs of -based alloys with 5% substitutions of each 5d element in place of Fe.









§ COMPUTATIONAL AND EXPERIMENTAL DETAILS




 §.§ Computational Details




The electronic band structure calculations for  and (Fe_0.95X_0.05)_5PB_2 (X = 5d element) systems
were carried out with use of the 
full-potential local-orbital electronic structure code FPLO14.0-49 <cit.> using a fixed atomic-like basis set.

The FPLO was an optimal choice for the accurate calculations of MAE
due to
the full potential and fully relativistic character of the code.

To model the Co and 5d alloying we used the supercell method.

The generalized gradient approximation (GGA) was used in the Perdew-Burke-Ernzerhof form (PBE). <cit.>

A 16 × 16 × 16 k-mesh was found to lead to well converged results of the MAE.

For k-point integration, the tetrahedron method was used. <cit.>

The energy and charge density convergence criteria of ∼ 10^-7 eV and 10^-6, respectively, were applied simultaneously.



The lattice parameters and Wyckoff positions were optimized for  and  within a spin-polarized scalar-relativistic approach.

The crystallographic parameters for compositions with intermediate Co concentrations were taken from calculations of full lattice relaxation carried out previously in virtual crystal approximation. <cit.>

For the (Fe_0.95X_0.05)_5PB_2 supercells we used the same crystallographic parameters as for the .



The MAE was evaluated as a difference between the fully relativistic total energies calculated for quantization axes [100] and [001].

In the adopted sign convention the positive sign of MAE corresponds to an easy magnetization axis along the [001] direction.



The Fermi surface (FS) of  was calculated on a 28^3 𝐤-mesh in a boundary of the first Brillouin zone.



Using the fully relativistic fixed spin moment (FSM) scheme <cit.> we study the MAE as a function of total magnetic moment (m) for .






A supercell method was used to model the chemical disorder. <cit.>

To build a supercell, multiplication of the basal unit cell and replacement of an appropriate amount of atoms of one type by atoms of the other type were made.

The Fe atoms were replaced by Co or 5d atoms forming the  and (Fe_0.95X_0.05)_5PB_2 compositions (X = 5d element).

For  the considered intermediate compositions were: x = 0.2, 0.4, 0.6, and 0.8.

The MAE calculations based on the supercell method <cit.> are uncommon, as they are time-consuming even for relatively simple alloys.

The reason for that is significant increase in the number of inequivalent atomic positions generated for the supercell model.

Additionally, accurate results require averaging over several different large supercells. <cit.>

It limits the size of supercells which we can use for MAE calculations.

Hence, we study only the supercells including symmetry operations and consisting of up to 16 inequivalent atoms.

The considered crystal structures are presented in Fig. <ref>.

Three configurations were considered for Fe_4Co_1PB_2 and  Fe_1Co_4PB_2 and one for Fe_3Co_2PB_2 and Fe_2Co_3PB_2 compositions.

For the considered supercell models, the energy convergence with a number of 𝐤-points was carefully tested.



The supercell method was also employed to calculate the MAE 
of (Fe_0.95X_0.05)_5PB_2 compositions
with various 5d elements X.

To construct the models, 
one of twenty Fe atoms  in the basal  supercell was replaced by the dopant. 

It led to the crystal structures containing 10 inequivalent atomic positions.

For calculations of the systems with 5d dopants a relatively dense 20 × 20 × 10 k-mesh was used, in order to get the well converged results of MAE.




To compute the Curie temperatures within the mean-field theory (T_C^MFT) for the whole series of  compositions the FPLO5.00 version of the code was used. <cit.>

The T_C^MFT is proportional to the total energy difference between the ferromagnetic and paramagnetic configurations <cit.> 

according to:



    k_B T_C^MFT = 2/3E_DLM - E_FM/c,


where E_DLM and E_FM are total energies for the paramagnetic and ferromagnetic configurations, 
k_B is Boltzmann constant, and
c is total concentration of magnetic atoms. 

In case of  containing five Fe atoms (considered as magnetic ones) within a formula unit consisting of eight atoms, the concentration parameter c is equal 5/8.

To model the paramagnetic state the disordered local moment (DLM) method was used, <cit.>

in which the thermal disorder among the magnetic moments is modeled by using the coherent potential approximation (CPA). <cit.>

The FPLO5 is the latest public version of the code allowing for the CPA calculations and does not have implemented the GGA.

Thus, the local density approximation (PW92) <cit.> form of the exchange-correlation potential had to be chosen.

For the calculations within FPLO5, a scalar-relativistic mode and a 12 × 12 × 12 k-mesh were used.

In the FPLO5 the magnetically ordered state (resulting in E_FM) was artificially modeled within the CPA, to avoid numerical discrepancies between the ordered (in principle non-CPA) and DLM (CPA) models.

In calculations using the FPLO5 code, the minimum basis have been optimized for the terminal compositions  and , 
subsequently the resultant compression parameters were used for intermediate compositions modeled with CPA.



The VESTA code was used for visualization of crystal structure. <cit.> 







 §.§ Experimental Details




The samples in the series  (x from 0.0 to 0.7) were synthesized by mixing stoichiometric amounts of the master alloys  and .


The master alloys were prepared, in accordance with previous studies <cit.>, 
from pure elements of 
iron (Leico Industries, purity 99.995%, surface oxides reduced in H_2-gas), 
cobalt (Johnson Matthey, purity 99.999%), 
phosphorus (Cerac, purity 99.999%), 
and 
boron (Wacher-Chemie, purity 99.995%). 

This was done by forming first the TM_2B (TM = Fe, Co), using a conventional arc furnace, and subsequently dropping the phosphorus in a melt of the metal boride in an induction furnace using the drop synthesis method. <cit.> 

All samples were subsequently crushed, pressed into pellets, and heat treated in evacuated silica ampules at 1273 K for 14 days after which they were quenched in cold water.

At x higher than 0.7 the correct crystalline phase could not be produced, 
all attempts resulted in a decomposition to other crystalline phases.





To study the phase content and to perform crystal structure analysis of all samples, a powder X-ray diffraction (XRD) was used. 

The measurements were done using a Bruker D8 diffractometer equipped with a LynxEye position sensitive detector (4° opening) using CuKα_1 radiation (λ = 1.540598 Å) at 298 K in a 2θ range of 20°–90°. 

The crystal structures were evaluated with the software FullProf <cit.> using refinements according to the Rietveld method. <cit.> 

The unit cell parameters were precisely studied using the least square refinements of the peak positions, employing the software UnitCell. <cit.>




The synthesized samples were magnetically studied using a Quantum Design PPMS 6000.

Samples were immobilized in gelatin capsules with varnish.

The magnetization at 3 K was measured between applied magnetic fields of 0 and 7.2 MA m^-1.

The magnetization in SI units was calculated from magnetic moment using the sample weight and the crystallographic volume obtained from the XRD measurements at 298 K. 

When approaching magnetic saturation the magnetization process is described by the law of approach to saturation (LAS). <cit.>

LAS has been formulated in several ways <cit.>,

but it takes a general form 

    M/M_S = ∑_j a_j H^j,

where j is usually an integer, a_j are coefficients, M and M_S are magnetization and saturation magnetization, and H is the applied magnetic field.

The LAS was used to determine an effective anisotropy constant | K_eff| in the same implementation as we used before. <cit.>

The interval 93%–98% of the magnetic saturation was used.

The applied formula was 


    M/M_S = 1 + aH + b/H + c/H^2.


The experimental data was fit with four models in which a and b coefficients can be zero or non-zero

and since 1/H^2 term is used to extract | K_eff| this part is always considered as non-zero.

| K_eff| is given here by 


    | K_eff| = √(15c/4)μ_0 M_S.


The difference in results between all four models are relatively small (max. 0.20 MJ m^-3), 
thus in the experimental section we present only the | K_eff| for the simplest model with the coefficients a = b = 0.









§ RESULTS AND DISCUSSION



The results of first-principles calculations of technologically important magnetic parameters for the considered systems are shown.

For  the M_S, T_C, and MAE are presented.

For (Fe_0.95X_0.05)_5PB_2 (X = 5d element) the results are limited to MAE and partial magnetic moments.

For the main phase –  – a detailed analysis of  electronic structure, magnetic moments, Fermi surface, and MAE is given.

The theoretical efforts  are complemented by experimental synthesis and measurements of the considered  compositions.






 §.§ Crystal Structure and Electronic Structure of Fe_5PB_2 and Co_5PB_2













The optimized crystallographic parameters of  and  are compared in Table <ref> with the results of measurements.

For  the agreement between the GGA and experiment is good and for the  the GGA underestimates a and overestimates c. 

The disagreement may originate from both theory and experiment.

The lattice parameters of  were last refined by Rundqvist back in 1962. <cit.> 

Unfortunately, we did not manage to synthesize the  sample.

According to comprehensive study of Haas et al. the PBE remains the best GGA functional for most of the solids containing 3d transition elements. <cit.> 

However, it has a tendency to overestimate the lattice constants. <cit.>
The presented PBE results for  go against this trend.

PBE underestimates also a volume of .

The observed underestimation of lattice parameters/volumes is similar to the results obtained from GGA for bcc Fe <cit.> and fcc Co <cit.>, 
for which the use of GGA leads to about 0.5 - 1.0% underestimation of the lattice parameters (which is equivalent to about 1.5 - 3.0% underestimation in volume). 

In case of  and  the calculated (with spin polarization) PBE volumes are 2.2 and 1.8% underestimated, respectively.

It is surprising, however, that when the c/a ratio for  is in agreement with experiment (both values are equal to 1.887), the corresponding result for  from (spin polarized) GGA is significantly different (1.995 against the experimental value 1.882).

The non-magnetic GGA calculations leads for  to c/a equal to 1.960 – also significantly different from the measured value.

We can only give a very general explanation for this discrepancy as coming from the insufficient treatment of corrections in PBE functional. 

 




The spin projected partial and total densities of states (DOS) for Fe_5PB_2 and Co_5PB_2 are presented in Fig. <ref>.

The valence bands of these two metallic systems start around -9 eV.

In a range from -9 to -3 eV the main contributions to a valence band come from the P 3p and B 2p orbitals, while from -5 eV up to above E_F the dominant role play the 3d orbitals.

The observed spin splitting (proportional to the magnetic moment) is bigger for Fe_5PB_2 than for Co_5PB_2,

which is related to a higher filling of the valence band for Co_5PB_2 than for Fe_5PB_2.

The majority spin channels of the two compounds are similar and nearly completely occupied.

The additional electrons in the Co_5PB_2 fill mainly the minority spin channel, reducing the magnetic moment.

The weak spin polarization of the P 3p and B 2p orbitals is induced by the 3d orbitals.

The spin polarization on the Fermi level is defined as P=|D_u-D_d/D_u+D_d|, where D_u is the density of states at the Fermi level of the majority spin channel, and D_d for the minority spin channel.

The calculated spin polarization on the Fermi level (a total value including Fe, Co, P, and B contributions) is about 0.46 for Fe_5PB_2 and 0.60 for Co_5PB_2.




 §.§ Magnetic Moments of 

















The calculated Co concentration dependence of the total magnetic moment (a sum of spin and orbital contributions) for the  system is presented in Fig. <ref>
together with the experimental results at low temperature (3 K). <cit.>

Whereas the results presented here are based on the supercell approach <cit.>,
in our previous work one can find the corresponding m(x) plots based on the virtual crystal approximation (VCA) and coherent potential approximation (CPA).

The calculated and experimental m(x) curves presented in Fig. <ref> stay in good qualitative agreement, showing a linear decrease of magnetic moment with Co concentration.

Nevertheless, they differ by about 0.5 – 1.0 μ_B/f.u., 
where the lower values come from measurements.

The reasons for this discrepancy should be sought on both experimental and theoretical sides.

Looking at the experiment, it is worth noting that the samples produced in this work are slightly non-stoichiometric and with a small amount of impurities. <cit.>

Our measurements at 3 K for a powder sample of  showed a total magnetic moment equal to 8.29 μ_B/f.u.
in comparison to 8.6 μ_B/f.u. obtained by Lamichane et al. for a  single crystal at 2 K. <cit.>

It leads us to the conclusion that the magnetic moments we have measured may be slightly underestimated.

The calculated total magnetic moment of Fe_5PB_2 (8.85 μ_B/f.u.) using GGA is closer to the result obtained for the single crystal than for the powder sample. 

The discrepancy between the result of GGA calculations and single crystal measurements can then be attributed to the insufficiency of the GGA in description of correlations, however the calculations still provide an acceptable level of agreement with experiment.






The calculated spin, orbital, and total magnetic moments (m_s, m_l, m) for Fe_5PB_2 and Co_5PB_2 are collected in Table <ref>.

For Fe_5PB_2 the calculated magnetic moments on Fe_1 and Fe_2 sites are equal to 1.81 (1.62) and 2.16 (2.16) μ_B, respectively,

where in parentheses are given estimations from the magnetic hyperfine fields. <cit.>

The induced spin magnetic moments on P and B are relatively small and oriented antiparallel to the dominant 3d moments on Fe/Co.

The total magnetic moments of Fe_5PB_2 and Co_5PB_2 are almost entirely of spin character, where the 3d orbital magnetic moments (m_l's) are nearly quenched.

The m_l's of Fe_1 and Fe_2 of the Fe_5PB_2 (calculated for the [001] quantization axis) are equal to 0.033 μ_B and 0.052 μ_B, respectively.

These values surround the m_l value calculated for bcc Fe (0.043 μ_B) and are reduced in comparison to the experimental value for the bcc Fe (0.086 μ_B). <cit.>

The underestimation of the orbital magnetic moment in transition metals is recognized as a general weakness of the LDA and GGA.

Finally, almost no orbital contributions are observed for P and B atoms (m_l∼ 10^-3 μ_B).



The calculated m of Co_5PB_2 is equal to 2.20 μ_B/f.u. (0.44 μ_B/Co atom).

For comparison, the experimental magnetic moment of hcp Co is equal to 1.67 μ_B/atom. <cit.>

The calculated m_l's of Co_1 and Co_2 of the Co_5PB_2 are equal to 0.011 μ_B and 0.013 μ_B, respectively, and are one order of magnitude smaller than the m_l measured for hcp Co (0.13 μ_B). <cit.>

Although the theoretical values of magnetic moments for Co_5PB_2  have been presented above,
the magnetic ground state of this system has not been unambiguously resolved, 
which will be discussed in the next section.









 §.§ Curie Temperature of 








The Curie temperatures (T_C^MFT(LDA)) calculated for the whole concentration range of the  system within the mean-field theory and with the LDA functional are presented in Fig. <ref>.



The observed overall decrease of the calculated T_C^MFT with increase of Co concentration is consistent with experimental observations. <cit.> 

However, in the whole range in which it is possible to compare the MFT-LDA results with the experiment (0.0 ≤ x ≤ 0.7), theoretical values are smaller.

For example, the calculated T_C^MFT(LDA) of  is equal 547 K, whereas the corresponding experimental value is 622 K for the powder sample <cit.>, or 655±2 K for the single crystal. <cit.>



This difference is due to the limitations of the MFT approach and insufficiency of the LDA in description of correlations.

By calculating Heisenberg exchange interactions, one could extract accurate critical temperatures using the random phase approximation (RPA) or Monte Carlo simulations. <cit.>

The insufficiency of the LDA  manifests in underestimated values of the calculated magnetic moments of ; 7.30 μ_B/f.u. versus 8.6 μ_B/f.u. from experiment for a single crystal. <cit.>

As it has been shown in the previous subsection, a much better description of magnetic moments of  in relation to the experimental result can be obtained by using the GGA functional instead LDA.

Thus, we suggest that the negative effect on T_C^MFT coming from the limitations of the LDA can be partially corrected by using the correction parameter based on the magnetic moments obtained from GGA.

In Heisenberg model,  T_C^MFT is proportional to squared effective moment (m_eff^2).

Defining b = m^GGA/m^LDA the corrected Curie temperature is b^2 T_C^MFT(LDA), 

where in case of  b is about 1.2.

Figure <ref> shows that for the region of intermediate Co concentrations the b^2 T_C^MFT(LDA) curve is in a better agreement with experiment than the uncorrected MFT-LDA results.




Unfortunately, we were unable to get experimental results of T_C for Co concentrations x > 0.7.

Because of that, we can not unambiguously resolve the issue of the magnetic ground state of terminal composition .

Linear extrapolation of experimental magnetic moments for  system suggests non-zero moment for , see Fig. <ref>.

On the contrary, linear extrapolation of the measured Curie temperature suggests a transition from ordered to disordered magnetic state at about x = 0.9, 
and therefore a non-magnetic ground state of , see Fig. <ref>.

Furthermore, experimental results reported by McGuire and Parker suggested absence of magnetic ordering for . <cit.>

From theoretical point of view, both uncorrected and corrected approaches show the non-zero values of T_C for Co-rich region (T_C^MFT(LDA) = 37 K for ).

Taking into account (1) the problems with synthesis of the  phase, (2) preliminary character of the measurements reported by McGuire and Parker, (3) issues mentioned in previous subsection regarding optimization of the structural model of , and (4) limitations of LDA/GGA in description of correlations of Co-rich phases, 
we conclude that 
based on existing data
the magnetic ground state of 
can not be definitively determined.










 §.§ Fermi Surface of 




Figure <ref> presents the calculated Fermi surface (FS) of  in a boundary of the first Brillouin zone.

The FS of  reflects the tetragonal symmetry of the crystal.

The FS consists of nine sheets and is relatively complex.

The states at the Fermi level (E_F) have a Fe 3d character, 
as can be read from the DOS plots in Fig. <ref>.

The observed FS sheets can be divided into two groups.

The first group consists of four nested sheets of hole-type, see panels (a)–(d) of Fig. <ref>, and

the second group includes the remaining five sheets of electron-type nested in a multiwalled way around the high symmetry point Z, see panels (e)-(i) of Fig. <ref>.

While the sheets (c)–(f) form rather tubular shapes, allowing for open orbits along the symmetry axis, the remaining sheets, (a)–(b) and (g)–(i), take the form of pockets enabling only for closed FS orbits. <cit.>

Because the band structure was calculated with spin-orbit coupling, 
the FS sheets cannot be unambiguously attributed to a particular spin channel.







 §.§ Magnetocrystalline Anisotropy of 



The results of investigating the MAE of  carried out in this work are: 
the band structure in vicinity of the Fermi level,
one- and two-dimensional 𝐤-resolved MAE plots,
and the cross-section of FS.

Our inquiry is complemented by considerations of MAE engineering, as for example reduction of total magnetic moment.



The calculated MAE of the  is 0.52 MJ m^-3.

It indicates a uniaxial magnetocrystalline anisotropy with an easy axis along the tetragonal axis.

This result stays in a good agreement with the experimental value of anisotropy constant measured at 2 K (0.50 MJ m^-3) and with the previous theoretical findings (0.46 MJ m^-3). <cit.>

Previously reported results for  show that K_1 first increases with temperature starting from 2 K up to about 100 K and then decreases to zero at T_C. <cit.>

The well known origin of the magnetocrystalline anisotropy is the spin-orbit coupling, which is taken into account in the fully relativistic full potential calculations.

In comparison with scalar relativistic approach, 
the fully relativistic one results in additional splitting of the electronic bands.

Since the spin-orbit coupling constant of 3d-metals is of the order of 0.05 eV, the spin-orbit splitting also does not exceed this value.



The spin-orbit splitting leads to slightly different band structures for different quantization axes (e.g. for the orthogonal [001] and [100] axes).




Figure <ref> presents the band structures calculated for  in the proximity of E_F,

together with the MAE contributions per 𝐤-point obtained with the magnetic force theorem <cit.> from the formula:


    MAE = E(θ = 90^∘) -  E(θ = 0^∘) =
    
    = ∑_occ'ϵ_i(θ = 90^∘) -  ∑_occ”ϵ_i(θ = 0^∘),



where θ is an angle between the magnetization direction and the c axis, E(θ) is a total energy for a specific direction; and ϵ_i is the band energy of the ith state.

The spin-orbit splitting is most easily observed for the energy window of a tenth eV around E_F.

The 𝐤-point resolved MAE takes positive and negative values, depending on the spin and orbital character of the bands near the Fermi energy. Generally, negative MAE-contributions coincide with occupied bands for a [100] spin quantization axis (solid red line) being pushed below corresponding bands for a [001] spin quantization axis (dashed blue line), and vice versa for positive contributions. For example, at the Z-point, there is a negative MAE contribution and at approximately -0.3 eV one can observe a solid red line below the dashed blue line. A more detailed analysis of the MAE contributions is in principle straight forward but somewhat complicated due to the complex band structure. Nevertheless, one can clearly observe the characteristic jumps where the bands cross E_F, confirming the usual behavior that the MAE is determined by the electronic structure around the Fermi energy. Thus, controlling the MAE around E_F also allows for control of the MAE, as is practically 
possible, for example, via alloying. 





The same form of presentation of the 𝐤-resolved MAE, as we have shown in Fig. <ref>, dominates in literature.

However, it is possible to plot the MAE(𝐤) data within a three dimensional Brillouin zone, 
similar like the FS.

Recently, the 3D MAE(𝐤) maps were presented for  and FeNi. <cit.>



In Fig. <ref> (a) we show a cross-section of the MAE(𝐤) (single plane going trough the Γ-point).

The selected profile is perpendicular to the easy axis [001], crosses the high symmetry point Γ, and is limited by the Brillouin zone boundaries.

The MAE(𝐤) cross-section is a relatively complicated map of symmetric regions consisting of positive and negative contributions. 

The MAE contributions observed in Fig. <ref> along the orthogonal axes [100] and [010] are not equal,
because the [100] direction is distinguished as quantization axis resulting in breaking of the four-fold symmetry. 




As the E_F is an upper integration boundary of total MAE, the FS sheets coincide with sharp changes in the 𝐤-resolved MAE contributions.

It can be seen in Fig. <ref> (b), where the MAE(𝐤) 2D plot is overlapped by the corresponding section of the FS.

As many of 𝐤-resolved MAE contributions is in order of 10^-3 eV per 𝐤-point,

the total MAE value of about 10^-4 eV/f.u.  (83 μeV/f.u. or 0.52 MJ m^-3) indicates a fine compensation of many bigger components.

Unfortunately, this extra fine compensation and the complexity of the MAE(𝐤) makes the ways to increase the MAE of the material difficult to predict.






 §.§ Fully Relativistic Fixed Spin Moment Calculations for 




The MAE value for  (0.52 MJ m^-3) is calculated with the equilibrium value of the magnetic moment (8.85 μ_B/f.u.). 

In the fixed spin moment (FSM) method <cit.> the value of spin magnetic moment is considered as a parameter.

The fully relativistic implementation of FSM method allows to calculate the MAE as a function of spin magnetic moment.

Previously, we presented the MAE results as a function of FSM and Co concentration for the  alloys. <cit.> 



Figure <ref> presents the evolution of the MAE with the total magnetic moment m for the ,

together with the previous results for . <cit.>

The two MAE(m) plots are similar in shape.

Going down from an equilibrium m the corresponding MAE first increases, then it reaches maximum, to decrease finally to zero at m equals zero. 

For  the maximum MAE(m) is 1.94 MJ m^-3 

for a fixed total magnetic moment of 6.7 μ_B/f.u.,

which means that the optimal magnetic moment has to be reduced by about 25% with respect to the equilibrium value (8.85 μ_B/f.u.).




Thus, the question arises, how to stabilize this reduction.

A simple solution would be alloying the magnetic Fe by a non-magnetic element,
which often results in a linear decrease of magnetization.

However, alloying with a new element can severely affect the band structure,

which would change also the expected value of the MAE.

The smallest impact on the electronic structure should have 
substitutions chemically most similar to Fe and for this purpose we suggest Ru and Os of the Fe group.

Another strategy could be alloying of Fe (Z_Fe = 26) with two elements at the same time, e.g. Cr (Z_Cr = 24) and Ni (Z_Ni = 28), keeping a constant number of the valence electrons, which should affect the band structure the least.

The above considerations, however, take into account only the band structure and neglect further issues like the crystal structure and size of the atoms, for example.






 §.§ Magnetocrystalline Anisotropy of 




The effect of Fe/Co alloying on the MAE is not obvious in advance,
whereby the first-principles calculations are of great value in predicting the results, as has been shown previously for the  <cit.> and  <cit.> alloys.




Figure <ref> presents the MAE(x) dependence for the  system as calculated with use of the supercell method.

The MAE calculations based on the supercell method proved to be one of the most accurate method for evaluation the MAE. <cit.>

However, our calculations were limited by computational challenges of the supercell method.

Thus, in practice we were able to consider only a relatively small number of configurations, see Sec. <ref>.

The scattering of individual data points for x = 0.2 and x = 0.8 is in a similar range as observed by Däne et al. <cit.>
or Steiner et al. <cit.> and shows that an averaging for several configurations is needed for accurate results. 

In Fig. <ref> the regions of positive and negative MAE (of perpendicular and in-plane anisotropy) are separated at Co concentration x≃0.5.

The calculated MAE is equal 0.52 MJ m^-3 for  and -0.51 MJ m^-3 for .

Whereas, the anisotropy value close to zero, observed for x≃0.5, indicates a good soft magnetic material.

Figure <ref> presents also the low temperature measurements of the effective anisotropy constant | K_eff| carried out at 3 K for several  compositions within the boundaries of 0.0 ≤ x ≤ 0.7.

The value of | K_eff| is the highest (0.94 MJ m^-3) for  and the the lowest for a Co concentration x∼0.6.

| K_eff| measured for  is significantly larger than the K_1 = 0.5 MJ m^-3 measured at 2 K for the single crystal. <cit.>  

The decrease of | K_eff| with x is in agreement with the previous measurements for (Fe_0.8Co_0.2)_5PB_2 suggesting that 20% Co substitution reduces the anisotropy field. <cit.>

Previously we also showed the corresponding | K_eff| results for the Fe_5Si_1-xP_xB_2 system. <cit.>

The presented values of | K_eff| for  were ∼0.9 MJ m^-3 at 10 K and ∼0.65 MJ m^-3 at 300 K. <cit.>

Notice that LAS is unable to determine the sign of | K_eff| and thus the negative values of MAE predicted for x ≳ 0.6 cannot be confirmed by this method. 

Other methods, such as magnetometry measurements in different directions for single crystals or torque magnetometry would be preferable. 

Here, single crystals were not available, and up to 10 wt% of impurities were present in the samples. 

Therefore, given the limitation in the model and the starting material the results presented from these should be seen as semi-quantitative. 

Taking into account the limitations of the LAS and the supercell method, the differences between theoretical and measured MAE(x) results are acceptable.

We conclude, that Co alloying of  is not a good strategy to increase the MAE of this system.






A typical magnetization (M) versus applied field (H) curve measured at 3 K is shown in Fig. <ref>.

The inset of Fig. <ref> presents a plot of M/M_S versus 1/H^2 as used to determine the | K_eff| within the LAS method.

More details on the implementation of the LAS method can be found in Sec. <ref>.




 §.§ Doping Fe_5PB_2 with 5d Elements


One of the methods of tailoring the MAE is doping with 5d elements. <cit.>

Previously, we have confirmed that the 5d elements can significantly affect the MAE due to a large spin-orbit coupling. <cit.>

From the Fe_5Si_1-xP_xB_2 and  systems, the highest MAE is found in the  phase. <cit.>

Thus, it is considered as the parental compound for a further MAE engineering.

The MAE of (Fe_0.95X_0.05)_5PB_2 compounds (X = 5d elements) is calculated using the supercell method.





The results are shown in Fig. <ref>, with the 5d element marked on the x axis and dashed line indicating the MAE of undoped Fe_5PB_2.

The 5d doping has sometimes beneficial and sometimes adverse effect on MAE. <cit.>

Significant increase of MAE is observed for W or Re doping, similar like in the case of (Fe_1-xCo_x)_2B alloys investigated experimentally in our previous work <cit.>.

The MAE grows from 0.52 MJ m^-3 for Fe_5PB_2 to about 1.1 MJ m^-3 for the compositions with W or Re, with 5% Fe substitution.

Previously we have shown, that the increase in MAE observed for W and Re dopants is mainly due to the strong spin-orbit coupling of the 5d atoms, however other variations in electronic structure also affect the MAE. <cit.>





Although in our calculations the 5d elements are initially considered as non-magnetic, the dopants undergo spin polarization in a ferromagnetic medium and contribute to the total magnetic moment of the system.

The calculated spin and orbital magnetic moments on 5d impurity show clear trend along the increasing atomic number of 5d element, see Fig. <ref>.

The spin magnetic moment of 5d impurities are antiparallel to the Fe moments in the early 5d series, while they are parallel in the late 5d series.

Corresponding trends for 5d atoms in magnetic 3d hosts have been found previously computationally <cit.> and experimentally. <cit.>






§ SUMMARY AND CONCLUSIONS




Our considerations began with a detailed theoretical analysis of the  compound.

The Fe 3d orbitals are dominant in the valence band and responsible for the formation of large magnetic moments.

For the 
the fully relativistic band structure in the vicinity of Fermi level
was considered
to better understand the origin of the high value of magnetocrystalline anisotropy energy (MAE).

The calculated Fermi surface requires experimental confirmation.



The results of fully relativistic fixed spin moment calculations suggested that reduction of the magnetic moment of  should induce about fourfold increase of the MAE.

For practical realization of magnetic moment reduction it is suggested to alloy Fe with a non-magnetic element Ru or Os from the Fe group,
or to partially replace Fe with two elements at once, Cr and Ni, for example, keeping constant number of valence electrons.





Three critical parameters for technological applications: saturation magnetization (M_S), Curie temperature (T_C), and MAE
were calculated
for the whole concentration range between  and .

The calculated M_S and T_C decreased with Co concentration and for the terminal composition  a weakly ordered magnetic ground state was predicted.

The calculated  M(x) and T_C(x) were in decent agreement with the measurements, although the ferromagnetic ground state of  is questionable.

The Co doping in  system gives the possibility of tuning the T_C in a range from about six hundred kelvins to almost down to zero.

The calculated MAE was positive for , negative for , and 
went through zero around 50% Co concentration.



This picture of MAE(x) behavior was in overall agreement with the experimental study of the effective anisotropy constant | K_eff| for the  alloys.

The measurements showed the highest | K_eff| value for stoichiometric  which decreased with Co doping.

We concluded then that Co alloying is not a good strategy to increase the MAE of  alloy.

The measured | K_eff| of about 0.94 MJ m^-3 at 3 K was, however, the highest value obtained so far for , giving a hope for potential application of its other alloys.

It was also calculated how the 5% doping of Fe with 5d elements affects the MAE of the .

It was shown that  doping with W or Re results in significant increase of the magnetocrystalline anisotropy energy.







MW and JR acknowledge the financial support from the Foundation of Polish Science grant HOMING. 

The HOMING programme is co-financed by the European Union under the European Regional Development Fund.

JC and MS acknowledge the financial support from the Swedish Research Council.

DH, PS, KG acknowledge Swedish Foundation for Strategic Research for financial support. 

Part of the computations were performed on resources provided by the Poznań Supercomputing and Networking Center (PSNC).

We thank Bartosz Wasilewski for help with language editing and Dr Jakub Kaczkowski for reading the manuscript and helpful discussion. 



 


        

