Ab initio study of magnetocrystalline anisotropy, magnetostriction, and Fermi surface of L10 FeNi (tetrataenite)

TL;DR

This study uses relativistic first-principles calculations to analyze the magnetic and electronic properties of L1$_0$ FeNi (tetrataenite), highlighting its potential for rare-earth free permanent magnets.

Contribution

It provides detailed relativistic calculations of magnetocrystalline anisotropy, Fermi surface, and magnetostriction for L1$_0$ FeNi, offering new insights into its magnetic characteristics.

Findings

Significant discrepancies between calculated and experimental MAE values.

Detailed analysis of Fermi surface and k-resolved MAE.

Calculated magnetic moments and magnetostrictive coefficients.

Abstract

The ordered L1$_0$ FeNi phase (tetrataenite) is recently considered as a promising candidate for the rare-earth free permanent magnets applications. In this work we calculate several characteristics of the L1$_0$ FeNi, where most of the results come form the fully relativistic full potential FPLO method with the generalized gradient approximation (GGA). A special attention deserves the summary of the magnetocrystalline anisotropy energies (MAE's), the full potential calculations of the anisotropy constant $K_3$, and the combined analysis of the Fermi surface and three-dimensional $\mathbf{k}$-resolved MAE. Other calculated parameters presented in this article are the magnetic moments $m_{s}$ and $m_{l}$, magnetostrictive coefficient $\lambda_{001}$, bulk modulus B$_0$, and lattice parameters. The MAE's summary shows rather big discrepancies between the experimental MAE's from literature and also between the calculated MAE's.

Tables

Table 1. Table 1: The lattice parameters ( a 𝑎 a and c 𝑐 c ), spin ( M S subscript 𝑀 𝑆 M_{S} ) and orbital ( M L subscript 𝑀 𝐿 M_{L} ) magnetic moments, magnetocrystalline anisotropy energies (MAE), magnetostrictive coefficients ( λ 001 subscript 𝜆 001 \lambda_{001} ), and bulk moduli (B 0 ) of the L1 0 FeNi phase.

quantity	$a$	$c$	MS(Fe)	ML(Fe)	MS(Ni)	ML(Ni)	MAE or $K_1$	MAE or $K_1$	${\lambda }_{001}$	B0
unit	Å	Å	${\mu }_B$	${\mu }_B$	${\mu }_B$	${\mu }_B$	$\mu$eV formula-1	MJ m-3	10-6	GPa
experiment	3.57 Frisk et al. (2017)	3.57 Frisk et al. (2017)	2.54±0.16 Cable and Wollan (1973)	$\sim$0.05 Kotsugi et al. (2013)	0.73±0.04 Cable and Wollan (1973)	0.10 Kotsugi et al. (2013)	—	0.58–1.3 Mizuguchi et al. (2011); Lewis et al. (2014)	$\sim$9 Bozorth and Walker (1953)	—
FLAPW-GGA Wu and Freeman (1999)	3.58	3.58	2.71	0.052	0.69	0.038	32	0.22	9.7	—
VASP-GGA Lewis et al. (2014)	—	—	—	—	—	—	110	0.78	—	—
VASP-GGA Miura et al. (2013)	3.556	3.584	2.65	—	0.61	—	78	0.56	—	—
WIEN2k-GGA Miura et al. (2013)	—	—	—	—	—	—	69	0.48	—	—
WIEN2k-GGA Edström et al. (2014)	3.56	3.58	2.69	—	0.67	—	69	0.48	—	—
WIEN2k-GGA (this work)	—	—	2.69	0.052	0.66	0.036	67	0.47	—	—
ASA-SPR-KKR-GGA Edström et al. (2014)	—	—	2.73	—	0.62	—	110	0.77	—	—
FP-SPR-KKR-GGA (this work)	—	—	2.69	0.053	0.60	0.036	32	0.23	—	—
FPLO14-GGA (this work)	3.56	3.58	2.72	0.054	0.53	0.039	48	0.34	9.4	194

Equations

$$\frac{\Delta l}{l_0}=\frac{3}{2}{\lambda }_{001}\left[\sum _{i=1}^3{\alpha }_i^2{\beta }_i^2-\frac{1}{3}\right]+3{\lambda }_{111}\sum _{i\ne j}^3{\alpha }_i{\alpha }_j{\beta }_i{\beta }_j.$$

$$\frac{\Delta l}{l_0}=\frac{3}{2}{\lambda }_{001}\left[{\alpha }_z^2-\frac{1}{3}\right]$$

$${\lambda }_{001}=\frac{1}{3}\frac{l_0(\theta =0^{\circ })-l_0(\theta =90^{\circ })}{l_0(\theta =0^{\circ })+l_0(\theta =90^{\circ })},$$

$$E(\theta =0^{\circ })=a{l}^2+bl+c;$$

$$E(\theta =90^{\circ })=a{l}^2+bl+c+\mathrm{MAE},$$

$${\lambda }_{001}=-\frac{2}{3}\frac{\frac{\mathrm{d}(\mathrm{MAE})}{\mathrm{d}l}}{b}.$$

$$\mathrm{MAE}=E(\theta =90^{\circ })-E(\theta =0^{\circ }).$$

$$\mathrm{MAE}=K_1{\sin }^2\theta +K_2{\sin }^4\theta +K_3{\sin }^4\theta \cos 4\varphi ,$$

$$\mathrm{MAE}-K_1-K_2=K_3\cos 4\varphi .$$

$$2K_3=E_{100}-E_{110}.$$

$$\begin{gathered} \mathrm{MAE}=E(\theta =90^{\circ })-E(\theta =0^{\circ })= \\ =\sum _{oc{c}^{\prime }}{ϵ}_i(\theta =90^{\circ })-\sum _{oc{c}^{\prime \prime }}{ϵ}_i(\theta =0^{\circ })+O(\delta {\rho }^n), \end{gathered}$$

Full text

I Introduction

The electric power generators, motors, and transformers are just a few examples where the magnetic materials find an application in the modern technology. The hard magnetic materials used the most are the alnicos, hexaferrites, and Nd-Fe-B alloys. The economical event from 2011 called The Rare-Earth Crisis Bourzac (2011) destabilized i.a. the prices of neodymium motivating efforts to find new rare-earth free permanent magnets. The ongoing search for hard magnetic materials free from rare-earth elements is summarized in several review articles. Kuz’min et al. (2014); McCallum et al. (2014); Niarchos et al. (2015); Hirosawa (2015); Skomski and Coey (2016); Li et al. (2016) Some of the promising candidates studied recently are e.g. Fe/Co nanowires Bran et al. (2015); Palmero et al. (2016), Fe-Co alloys doped with B and C Reichel et al. (2014); Giannopoulos et al. (2015); Salikhov et al. (2017); Kuswik et al. (2017), (Fe/Co)2B Edström et al. (2015); Däne et al. (2015); Belashchenko et al. (2015); Wallisch et al. (2015), (Fe/Co)5XB2 Cedervall et al. (2016); Werwiński et al. (2016); Lamichhane et al. (2016), MnBi Antropov et al. (2014), and the L10 phases such as FePt, CoNi, MnAl, and FeNi. Ayaz Khan et al. (2016); Gutfleisch et al. (2005); Edström et al. (2014) The last candidate on the list is a subject of this work.

Existence of an ordered L10-type FeNi phase (tetrataenite) was confirmed in the sixties by Néel et al. Néel et al. (1964) in the study of a single crystal ordered by neutron bombardment and once again in the seventies by Paterson et al. Petersen et al. (1977) in the study of taenite lamellae from the iron meteorite. The L10 is the strukturbericht designation of the CuAu I-type ordered tetragonal phase. The L10 unit cell confined by solid lines in Fig. 1 consists of two formula units. Two faces of its are occupied by one type of atoms. The third face and corners are occupied by the second type of atoms. A detailed study of crystallographic aspects of L10 magnetic materials can be found in a paper of Laughlin et al. Laughlin et al. (2005)

In this work we investigate theoretically the magnetocrystalline anisotropy constants $K_{2}$ and $K_{3}$, Fermi surface, and bulk modulus of the tetrataenite. This efforts are followed by the theoretical reinvestigation of the magnetostrictive coefficient $\lambda_{001}$ and magnetocrystalline anisotropy energy (MAE) treated with a full potential fully relativistic method based on the full four component representation of the Bloch states. Eschrig et al. (2004) Also the experimental Wu and Freeman (1999); Mizuguchi et al. (2011); Kotsugi et al. (2013); Lewis et al. (2014) and theoretical Wu and Freeman (1999); Miura et al. (2013); Edström et al. (2014) results from literature of the magnetocrystalline anisotropy of the L10 FeNi are considered. One of the most important parameters from the perspective of permanent magnets application is a magnetocrystalline anisotropy constant $K_{1}$. The measured values of $K_{1}$ of the L10 FeNi are relatively high Lewis et al. (2014); Poirier et al. (2015) and equal up to 1.0–1.3 MJ m*-3*. Skomski and Coey Skomski and Coey (2016) suggest that even such high values of $K_{1}$ are insufficient to raise FeNi off the category of semi-hard magnets. However it has been shown for the L10 FeNi films, that their intrinsic magnetic properties can be altered e.g. by engineering larger strains. Sakamaki and Amemiya (2013); Frisk et al. (2017) Furthermore, the composition and microstructure of the L10 FeNi may be tailored as well to improve the FeNi potential for rare-earth-free permanent magnet application. Lewis et al. (2014) Skomski points out the beneficial self-organized microstructure of the L10 FeNi being reflected in a relatively high coercivity of about 120 mT. Skomski (2016) Some other characteristics indicating the L10 FeNi as a good candidate for hard magnets are magnetization approaching that of the Nd-Fe-B and relatively high Curie point near 550° C Wasilewski (1988), however preceded by the critical temperature of the ordered state of about 320° C. Néel et al. (1964) Considering from the application point of view, a serious weakness of the FeNi remains achieving and retaining the L10 atomic order. Frisk et al. (2016, 2017)

II Computational details

The determination from the first principles of the magnetocrystalline anisotropy constants and the magnetostrictive coefficient $\lambda_{001}$ requires the fully relativistic electronic band structure calculations. The calculations are carried out by using the full-potential local-orbital minimum-basis scheme FPLO-14.0-49 Eschrig et al. (2004); Koepernik and Eschrig (1999) with the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form. Perdew et al. (1996) The calculations are performed up to a 803 k-mesh with tetrahedron method for integration, an energy convergence criterion $10^{-8}$ Ha, and a charge density convergence criterion $10^{-6}$. The two-dimensional maps of MAE($\mathbf{k}$) are constructed on the 1000 × 1000 $\mathbf{k}$-mesh and the three-dimensional plot of MAE($\mathbf{k}$) is based on 250 × 250 × 170 $\mathbf{k}$-mesh within a selected one-eighth part of the full Brillouin zone. An initial spin splitting is applied assuming the ferromagnetic structure. The bct representation of the L10 unit cell is used (as described e.g. by Edström et al. Edström et al. (2014)) with a space group $P$4/$mmm$ and atomic coordinates Fe (0, 0, 0) and Ni (0.5, 0.5, 0.5). Both the volume and $c$/$a$ ratio are optimized. The calculated structural parameters are in good agreement with the corresponding experimental and theoretical values, see Table 1. For the visualization of crystal structure the VESTA code Momma and Izumi (2008) is used. The magnetostriction is calculated with the same scheme as used by Wu et al. Wu and Freeman (1999); Wu (2001); Zhang and Wu (2011); Wu (2007) In the latter references one can find the detailed description of the method but also the magnetostrictive coefficients calculated for several materials that stay in a good agreement with the experiment. In order to calculate the magnetostrictive coefficient of the L10 FeNi we use the approach developed for cubic geometry as the tetragonal distortion in this system is very small (c/a = 1.0036). To determine the magnetostrictive coefficient for a cubic material one has to calculate the strain dependences of the total energy ($E$) and the magnetocrystalline anisotropy energy (MAE), see Fig. 3. The dependence of the fractional change in length can be written based on the direction cosines of the magnetization ($\alpha$) and of the strain measurement ($\beta$):

[TABLE]

If the measurement is made along the [001] direction the equation simplifies to:

[TABLE]

and for a single domain system it takes a form:

[TABLE]

where $\theta$ is the angle between the magnetization direction and the $c$ axis. If the ab initio calculated total energies are fitted in a quadratic form:

[TABLE]

the equation for magnetostrictive coefficient can be written as:

[TABLE]

In this work the MAE’s are evaluated based on the total energies calculated self-consistently for two perpendicular quantization axes:

[TABLE]

In addition to the FPLO calculations, the Korringa-Kohn-Rostoker (KKR) approach as implemented in the Munich SPR-KKR package (non-public full potential version 7.6.0) is used to calculate the magnetocrystalline anisotropy constant $K_{3}$. Ebert et al. ; Ebert et al. (2011) The advantage of using full potential method for MAE calculations has been discussed before. Edström et al. (2015); Ke et al. (2013) The ab initio calculations of $K_{3}$ are today still a numerically demanding task. But what distinguishes the SPR-KKR among the other ab initio codes is a numerical accuracy on the level of about 0.1 $\mu$eV in calculating total energy, which is the same order of magnitude as expected for the $K_{3}$ value of the L10 FeNi. Another argument in favor of the SPR-KKR is that it has been successfully applied before to calculate $K_{3}$ for the magnetic shape memory Fe-Pd alloys. Kauffmann-Weiss et al. (2012) In order to get a converged values of $K_{3}$ for the L10 FeNi, the FP-SPR-KKR parameters of 10*-10* Ry energy convergence criterion and up to 225 × 225 × 158 $\mathbf{k}$-points (about 8 million) are necessary. For Brillouin zone integration the special point method with a regular k-point grid is used. Khan et al. have shown for L10 FePt phase Ayaz Khan et al. (2016) that the KKR calculations of total energies are also quite sensitive to the angular momentum expansion $l_{max}$ cutoff used for the multipole expansion of the Green function. Khan et al. concluded that the angular momentum $l_{max}$ = 3 cuttoff yields to a qualitatively correct value of the MAE, however even for $l_{max}$ = 7 a full convergence is still difficult to reach. Taking this conclusion into account we choose for our calculations a maximum angular momentum value $l_{max}$ = 4 (NL = 5 in the KKR configuration file). Our decision is further motivated by results of a convergence test of $K_{1}$ with respect to the angular momentum performed up to $l_{max}$ = 6. The test indicated that problems with convergence occur above the $l_{max}$ = 4 leading to divergence of the $K_{1}$ value. This behavior may come from numerical problems in evaluating the Madelung potential and near-field corrections, pointed out by Khan et al. Ayaz Khan et al. (2016) The energy integrals are evaluated by contour integration on a circular energy path in complex plane (GRID = 5), using 40 points of the E-mesh. The calculations within the FP-SPR-KKR are carried out with the PBE exchange-correlation potential and with the same crystallographic parameters of FeNi as used for FPLO MAE calculations. The magnetocrystalline anisotropy energy in tetragonal crystal can be described by the following equation Buschow and Boer (2004):

[TABLE]

where Ki are the anisotropy constants, $\theta$ is the angle between the magnetization direction and the $c$ axis, and $\phi$ is the angle between the magnetization and the $a$ axis within the basal plane of a tetragonal lattice. For $\theta$ = 90*∘* the Eq. 7 takes the following form:

[TABLE]

The $K_{3}$ is then evaluated based on the total energies $E_{100}$ and $E_{110}$ calculated self-consistently for $\phi$ equal to 0*∘* and 45*∘* (directions [100] and [110] in the bct unit cell) from equation:

[TABLE]

The computational parameters used to obtain K1 and K2 with FP-SPR-KKR are the same as presented above for the K3 calculations, with an exception that lower number of $\mathbf{k}$-points, about 1.5 million, is used in the whole Brillouin zone.

Furthermore the full-potential augmented plane-wave method FP-LAPW as implemented in the WIEN2k code Peter Blaha et al. (2014) is used to calculate a reference value of MAE. Muffin-tin radii RMT are 2.29 a0 for Fe and 2.29 a0 for Ni atoms, where a0 is Bohr radius. The PBE exchange-correlation potential is used. Plane wave cut-off parameter $RK_{max}$ is set to 10, which leads to above 245 basis functions. Relativistic effects are included with the second variational treatment of spin-orbit coupling. The total energy convergence criterion is set to $10^{-8}$ Ry. The 55 × 55 × 39 k-points (about 120 thousand) are used in the whole Brillouin zone.

Although, the reproducibility of the results in density functional theory calculations of solids has been well established on the level of scalar relativistic estimation of the lattice parameters Lejaeghere et al. (2016), the accurate calculations of the magnetocrystalline anisotropy energy MAE and its derivatives, like magnetostrictive coefficient, remain a challenge. Ayaz Khan et al. (2016) It is even harder to calculate the values of magnetocrystalline anisotropy constant $K_{3}$ which happens to be two orders of magnitude smaller than MAE. Kauffmann-Weiss et al. (2012) In this work we managed to calculate the $K_{3}$ of the L10 FeNi with the full potential thanks to the very accurate convergence tests.

III Density Functional Theory Calculations

The L10 FeNi phase has been studied ab inito several times before. Wu and Freeman (1999); Ravindran et al. (2001); Miura et al. (2013); Edström et al. (2014) In this work we reinvestigate the magnetocrystalline anisotropy energy MAE and the magnetostrictive coefficient $\lambda_{001}$ of the L10 FeNi phase and calculate its magnetocrystalline anisotropy constant $K_{2}$, $K_{3}$, Fermi surface, and bulk modulus. Our results together with the literature data are summarized in Tab. 1.

III.1 Bulk Modulus

We start from the calculations of the equilibrium volume (22.686 Å3), followed by the calculations of equilibrium $c$/$a$ ratio (1.0036), see Fig. 2. The optimization leads to a tetragonal structure with the lattice parameters $a=3.56$ Å and $c=3.58$ Å. The energy-volume data, see Fig. 2 (a), allows to calculate the bulk modulus B0 for the L10 FeNi by fitting the third-order Birch-Murnaghan equation of state. Birch (1947) It leads to a B0 equal to 194 GPa at 0 K. The previous calculations with coherent potential approximation CPA, of the Fe0.5Ni0.5 random alloy have given the B0 value of about 220 GPa. Abrikosov et al. (1995) For comparison, the experimental values of B0 for the Fe0.5Ni0.5 alloy measured at room temperature vary between 165 GPa and 177 GPa Bonnenberg et al. (1986) and the experimental B0 values for Fe and Ni are about 170 GPa and 180 GPa, respectively.

III.2 Magnetostrictive Coefficient $\lambda_{001}$

The total and magnetocrystalline anisotropy energies versus the length of the lattice parameter $c$ are presented in Fig. 3 as necessary to evaluate the magnetostrictive coefficient $\lambda_{001}$ from Eq. 5. The total energy and MAE are calculated based on the optimized crystal structure in a constant volume mode adopted for the distortion. The MAE calculated for the equilibrium L10 FeNi structure equals 48 $\mu$eV formula*-1* (0.34 MJ m*-3*). Initial tests performed with volume relaxation for 503 k-points have shown that the constant volume mode underestimates the $\lambda_{001}$ of the L10 FeNi by about 10%. For the price of this inaccuracy, in constant volume mode we can perform calculations up to 803 k-points. The resultant $\lambda_{001}$ obtained for 603, 703, and 803 k-points are equal to 11.3 $\times$ 10*-6*, 8.7 $\times$ 10*-6*, and 9.4 $\times$ 10*-6*, respectively. The variation of the estimated $\lambda_{001}$ with number of k-points comes from the numerical inaccuracy in the evaluation of a very small quantity as MAE. Nevertheless, the calculated here $\lambda_{001}$ = 9.4 $\times$ 10*-6* stays in a good agreement with the previous theoretical result by Wu and Freeman Wu and Freeman (1999), $\lambda_{001}$ = 9.7 $\times$ 10*-6*, and with the experimental value by Bozorth Bozorth and Walker (1953) $\lambda_{100}\sim 9\times 10^{-6}$. $\lambda_{001}\sim 10\times 10^{-6}$ for the L10 FeNi is rather small value. It is of the same order of magnitude as for elements Fe and Ni and three orders of magnitude smaller than for the magnetostrictive material Terfenol-D.

III.3 Magnetic Moments

The spin magnetic moments calculated with the FPLO for the L10 FeNi are equal 2.72 $\mu_{\mathrm{B}}$ for the Fe atom and 0.53 $\mu_{\mathrm{B}}$ for the Ni atom, see Tab. 1. The calculated spin magnetic moments on Fe and Ni stay in relatively good agreement with the values measured on Fe equal to 2.54±0.16 $\mu_{\mathrm{B}}$ and on Ni equal to 0.73±0.04 $\mu_{\mathrm{B}}$ for the L10 FeNi phase. Cable and Wollan (1973) The calculated orbital magnetic moments are equal to 0.054 $\mu_{\mathrm{B}}$ for the Fe atom and 0.039 $\mu_{\mathrm{B}}$ for the Ni atom. They are also close to the experimental values on Fe equal to $\sim$0.05 $\mu_{\mathrm{B}}$ and on Ni equal to 0.10 $\mu_{\mathrm{B}}$ as measured for the L10 FeNi phase. Kotsugi et al. (2013) The calculated and measured orbital magnetic moments on Fe in the L10 FeNi are reduced in comparison to the experimental value of 0.086 $\mu_{\mathrm{B}}$ for the bcc iron. Chen et al. (1995) The calculated orbital magnetic moments on Ni (0.039 $\mu_{\mathrm{B}}$) in the L10 FeNi are also reduced in respect to the experimental value of 0.055 $\mu_{\mathrm{B}}$ for the fcc nickel. Mook (1966) The calculated total magnetic moment is equal to 3.34 $\mu_{\mathrm{B}}$ formula*-1* (1.67 $\mu_{\mathrm{B}}$ atom*-1*) which is not particularly high value for 3$d$-based magnetic materials. From the perspective of hard magnetic materials this reduction of magnetic moment in comparison to e.g. pure bcc Fe is beneficial for the magnetic hardness but adversely affects the energy product.

III.4 Magnetocrystalline Anisotropy Energy

Getting consistent MAE results from different first principles codes is still a challenge. The difficulties come from such factors as a complex shape of the valence band structure or a demand of very high numerical accuracy. Ayaz Khan et al. (2016) The differences between the results from various codes may come from application of different approximations like the atomic sphere approximation, the lack of crystal structure optimization, or the insufficient number of k-points. Edström et al. (2015); Kuz’min et al. (2015); Ayaz Khan et al. (2016) Some recent papers discuss however the reproducibility of the MAE between WIEN2k and KKR methods Ayaz Khan et al. (2016) and between FPLO and WIEN2k. Edström et al. (2015)

The calculated in GGA MAE’s of the L10 FeNi taken from literature are equal to 0.22 Wu and Freeman (1999), 0.48 Edström et al. (2014), 0.56 Miura et al. (2013), and 0.78 Lewis et al. (2014) MJ m*-3*. The experimental determination of the L10 FeNi magnetocrystalline anisotropy constant $K_{1}$ is ambiguous as well, with a spread in the $K_{1}$ values from 0.58 Mizuguchi et al. (2011), trough 0.67 Néel et al. (1964), up to 1.3 Lewis et al. (2014) MJ m*-3*. The MAE’s of the L10 FeNi calculated in GGA in this work are equal to 0.23, 0.34, and 0.47 MJ m*-3* from FP-SPR-KKR, FPLO, and WIEN2k, respectively.

Based on the above theoretical results we expect that the accurate GGA value of the MAE for the L10 FeNi is below 0.5 MJ m*-3*. Such value could be an argument for removing the L10 FeNi from a list of candidates for rare-earth free permanent magnets. Some authors suggest however that bare GGA is insufficient for describing the L10 FeNi and a consideration of orbital polarization corrections is necessary. Ravindran et al. (2001); Miura et al. (2013) It has been shown that inclusion of the orbital polarization causes a significant increase of the MAE of the L10 FeNi (from $\sim$0.55 to $\sim$1.23 MJ m*-3*) Ravindran et al. (2001) and (from 0.48 to 0.84 MJ m*-3*) Miura et al. (2013). Unfortunately, even the MAE of about 1 MJ m*-3* may be insufficient to raise the L10 FeNi from the category of semi-hard magnets. Skomski and Coey (2016) To reach that goal further efforts on increasing the magnetocrystalline anisotropy have to be made.

III.5 Magnetocrystalline Anisotropy Constant $K_{3}$

The magnetocrystalline anisotropy constant $K_{3}$ of the L10 FeNi is particularly interesting from a perspective of spintronic applications of the material, e.g. for the magnetic tunnel junction. Miura et al. (2013) In a tetragonal crystal the $K_{3}$ can be defined by Eq. 7. Unfortunately we cannot calculate the $K_{3}$ with the FPLO code, which was applied to obtain previous results. To get the $K_{3}$ we use then the FP-SPR-KKR package which produces the value of $K_{3}$ equal to 110 J m*-3*. It is four orders of magnitude smaller from the value of MAE or $K_{1}$ for the L10 FeNi. For comparison, the values of $K_{3}$ calculated in ASA-SPR-KKR for magnetic shape memory alloys Fe-Pd Kauffmann-Weiss et al. (2012) have similar order of magnitude ($\sim 10^{3}$ J m*-3*) as the one presented above. The details of calculation method and the motivations for using FP-SPR-KKR package for evaluating $K_{3}$ are presented in the introductory section.

As the $K_{3}$ is $\mathbf{k}$-mesh sensitive the convergence test is made, see Fig. 4. It can be noticed that large number of $\mathbf{k}$-points is necessary to get a satisfactory convergence of $K_{3}$.

To complete the FP-SPR-KKR analysis also the K1 and K2 magnetocrystalline anisotropy constants are calculated. In Fig. 5 we present the energy dependence as a function of the polar angle $\theta$ between the magnetization direction and the $c$ axis. Parameters from a fit to $K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta$ are $K_{1}$ = 0.23 MJ m*-3* and $K_{2}$ = -2.0 kJ m*-3*, see Eq. 8. The lowest energy in Fig. 5 corresponds to [001] quantization axis and the highest energy to [100] axis.

III.6 Fermi Surface

Figure 6 presents the Fermi surface of the L10 FeNi. The tetragonal crystal structure of the L10 FeNi is reflected in a uniaxial anisotropy and a four-fold symmetry of Fermi surface sheets. The seven sheets are divided into two groups of closed hole pockets and open sheets. The nested hole pockets (a), (b), and (c) are centered at the high symmetry point M. One another hole pocket is centered at point $\Gamma$, see panel (e). When the hole pockets permit for only closed orbits the mainly open sheets (d), (e), and (f) allow for both the opened and closed orbits.

We have calculated the Fermi surface of the L10 FeNi both as the basic characteristic of the material and as the basis for a k-resolved MAE analysis which will be presented in the next paragraph. Basic knowledge on the Fermi surface of the crystal lets us think on manipulating of its properties using the emerging method called the Fermi surface engineering. The method makes relations between a shape of the Fermi surface and external factors like doping or strain. The Fermi surface engineering technique has been successfully applied e.g. to transparent conductors Katayama-Yoshida et al. (2003) and superconductors. Hsu et al. (2016)

III.7 MAE Analysis in Reciprocal Space

Most often the anisotropic magnetic properties of the materials are analyzed in the real space. The spin and orbital moments and magnetocrystalline anisotropy constants are presented as functions of the atomic position or angle. Miura et al. (2013); Antropov et al. (2014); Thersleff et al. (2017) The magnetocrystalline anisotropy energy (MAE) is one of the most important intrinsic properties of the hard magnetic crystals which can be analyzed in reciprocal space as a $\mathbf{k}$-resolved quantity. The MAE can be determined with the magnetic force theorem Liechtenstein et al. (1987); Wang et al. (1996); Wu and Freeman (1999) from a formula:

[TABLE]

where $\epsilon_{i}$ is the band energy of the $i$th state and $\theta$ is the angle between the magnetization direction and the $c$ axis. The modern computer technique allows to visualize the MAE($\mathbf{k}$) structure in a three-dimensional (3D) Brillouin zone. The 3D maps of magnetocrystalline anisotropy have been recently calculated for (Fe/Co)2B alloys Belashchenko et al. (2015) – another candidates for the rare-earth free permanent magnets. However usually the MAE($\mathbf{k}$) dependences are presented along one-dimensional $\mathbf{k}$-path. Wu and Freeman (1999); Wu (2007); Edström et al. (2015) Although various authors attempt to interpret the MAE($\mathbf{k}$) without a 3D-resolution, in our opinion the $\mathbf{k}$-path or single $\mathbf{k}$-point analyses do not cover properly the complexity of magnetocrystalline anisotropy. We think that in order to get a full picture of it the entire Brillouin zone should be considered. Edström et al. (2015) For a hard magnetic material the calculated 3D-MAE($\mathbf{k}$) plot is a unique characteristic as for example the Fermi surface is for a metal. Interesting is that the connections between the 3D-MAE($\mathbf{k}$) and the Fermi surface go beyond the above comparison and a fact is a close relation between these two. Both, the 3D-MAE($\mathbf{k}$) and Fermi surface, are reciprocal space electronic structure characteristic. Furthermore, as the Fermi level is an upper integration boundary of the MAE, the sheets of the Fermi surface indicate the borders between the distinct regions of the MAE. Because of that the Fermi surface is a link between the calculated 3D-MAE($\mathbf{k}$) and experiment. The 3D-MAE($\mathbf{k}$) analysis is relevant for hard magnetic materials also because it can determine directions to improve the MAE. In relation to the Fermi surface engineering an improvement technique based on the MAE($\mathbf{k}$) analysis could be called magnetocrystalline anisotropy engineering.

In Fig. 7 we present the calculated 3D-MAE($\mathbf{k}$) plots of the L10 FeNi. We can see that the whole Brillouin zone is filled by the positive (red) and negative (blue) contributions. The magnitude of the $\mathbf{k}$-resolved contributions is in order of $10^{-3}$ eV per $\mathbf{k}$-point, where the value of the total MAE is three orders of magnitude smaller (48 $\mu$eV formula*-1*). A comparison between the MAE contributions and total value indicates a fine compensation of the relatively large negative and positive components. From Fig. 7 it is easy to notice that the overall shape of the MAE($\mathbf{k}$) is dictated by the Fermi surface. The Fermi surface sheets divide the Brillouin zone into several mainly positive or negative regions. The two most prominent positive contributions come from (1) the vicinity of the R-Z line along the [100] quantization axis and (2) from the spherical region around the M point. Actually, around the M point we observe a sequence of negative, positive, and again negative regions filling the volumes between the corresponding nested hole pockets of the Fermi surface. A similar alternating ordering of the $\mathbf{k}$-resolved contributions to anisotropy constant $K$ has been presented for the (Fe/Co)2B alloys. Belashchenko et al. (2015)

In Fig. 8 the cross-sections of the MAE(k) for planes $\Gamma$-X-M and R-Z-A are shown together with corresponding cross-sections of the Fermi surface. At these plots it is even easier to notice how the Fermi surface splits the regions with differing MAE(k) components. The MAE cross-sections also confirm that the first hole pocket around the M point contains mainly negative $\mathbf{k}$–resolved contributions and the regions between the next two hole pockets are respectively positive and negative. Although the MAE($\mathbf{k}$) structure reflects the uniaxial anisotropy of the crystal, it is easy to notice how the four-fold symmetry is broken. A reason for this is that the [100] direction has been distinguished as a quantization axis.

The above-described MAE($\mathbf{k}$) analysis reveals details of the structure of magnetocrystalline anisotropy of the L10 FeNi which are the distribution of positive and negative components, the magnitude of these shares, or the detailed relationship between the MAE($\mathbf{k}$) and the Fermi surface. Unfortunately, an extra fine compensation of the large MAE($\mathbf{k}$) contributions makes it difficult to predict the roads to increase the MAE of the L10 FeNi. Because of the compensation effect, even large changes of the MAE($\mathbf{k}$) structure will finish as hard to predict small changes to the MAE. Based on the combined 3D-MAE and Fermi surface analysis we are able however to point out the sheets of Fermi surface which should be extended or reduced to enlarge the regions of positive MAE contributions. Unfortunately, this simple line of reasoning is difficult to realize in practice. Taking into account the innovative nature of the combined 3D-MAE and Fermi surface analysis, we do not exclude that this approach will be more fruitful for another hard magnetic material.

IV Summary and conclusions

The magnetocrystalline anisotropy energy MAE of the L10 FeNi calculated in this work within GGA goes below 0.5 MJ m*-3*. It has been shown in literature that the more reliable model including orbital polarization corrections gives about twice this value, whereas the experimental values of the anisotropy constant $K_{1}$ from literature oscillate around 1.0 MJ m*-3*. Regarding application of the L10 FeNi as rare-earth free permanent magnets, the above numbers of MAE/$K_{1}$ are rather discouraging. Other known limitations of tetrataenite are practical difficulties with synthesis of the ordered phase and relatively low critical temperature of the ordered state. However the L10 FeNi has still a potential to improve its MAE by modifications, like e.g. a tetragonal strain or an interstitial or substitutional alloying. In favor of the L10 FeNi speak also the high saturation magnetization and Curie temperature. The L10 FeNi with well defined anisotropic parameters may also find applications in modern electronics. From point of view of computations it can be also used as a reference standard for advanced ab initio calculations due to relatively simple crystal structure.

In the reciprocal space analysis of the hard magnetic materials it is common to present the MAE($\mathbf{k}$) contributions together with the bandstructure along the $\mathbf{k}$-path. In this work we have shown the three-dimensional $\mathbf{k}$-resolved analysis of the MAE (3D-MAE) which gives a complete picture of the MAE contributions in the Brillouin zone. Unfortunately, in case of the L10 FeNi we cannot use the $\mathbf{k}$-resolved analysis to give specific suggestions. In return we show the close interconnection between the 3D-MAE and the Fermi surface. We expect that analysis of the effect of strain or alloying on the $\mathbf{k}$-resolved MAE may allow to improve the properties of hard magnetic materials.

The calculated spin and orbital magnetic moments and magnetostrictive coefficient $\lambda_{001}$ stay in an acceptable agreement with the previous theoretical results and measurements. The calculated bulk modulus $B_{0}$, magnetocrystalline anisotropy constants $K_{2}$ and $K_{3}$, and Fermi surface require experimental confirmation.

The conducted full relativistic calculations can be also used to formulate several general conclusions regarding the computational parameters. The fact that that spin-orbit coupling is the origin of the orbital magnetic moment, magnetocrystalline anisotropy, and magnetostriction is well known. In order to accurately calculate the magnetic parameters it is also crucial to consider the full potential. The significant discrepancies between the calculated MAE’s even from various full potential GGA implementations show how sensitive the calculations are. To improve the reliability of the results we suggest to choose the computational parameters very carefully and use a second code for cross-checking. Though it is well known that the orbital magnetic moment on Fe is underestimated in GGA, it is not obvious that it leads to underestimation of the MAE. For better description of the magnetic moments in considered system the orbital polarization corrections and dynamical mean field theory can be used. In this work we have also shown that it is technically possible to calculate the anisotropy constant $K_{3}$ with full potential. The calculated value of $K_{3}$ is four orders of magnitude smaller than the MAE and a huge number of $\mathbf{k}$-points is indispensable to get a consistent value.

V Acknowledgements

We thank A. Edström and B. Wasilewski for discussions. We also 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.