Dataset and article "Second-order anisotropy due to magnetostriction for L10-FePt"

Solid State Sciences 160 (2025) 107782 Available online 4 December 2024 1293-2558/© 2024 The Authors. Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie Second-order anisotropy due to magnetostriction for L10-FePt D. Legut a,b,∗, P. Nieves c aIT4Innovations, VSB - Technical University of Ostrava, 17. listopadu 2172/15, CZ 708 00 Ostrava, Czech Republic bDepartment of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague 2, Czech Republic cDepartamento de Física, Universidad de Oviedo, C. Leopoldo Calvo Sotelo, 18, 33007, Oviedo, Spain ARTICLE INFO Keywords: Magnetic anisotropy Magnetostriction First-principles calculations ABSTRACT The effective magnetocrystalline anisotropy energy associated with magnetostriction is studied for tetragonal L10-FePt by means of first-principles calculations, which is expressed in terms of the intrinsic anisotropy for an undeformed crystal, the magnetostrictive coefficients, and the elastic tensor. A very small correction is found for the first anisotropy constant 𝛥𝐾1∕𝐾1= 0.07%, while a much more significant contribution is obtained for the second one 𝛥𝐾2∕𝐾2= 21.86%. General analysis of this effect for tetragonal crystals is provided, finding that 𝛥𝐾1will be always positive for any stable phase with this symmetry. The potential implications and applications of these results are discussed. 1. Introduction The magnetocrystalline anisotropy energy (MAE) plays a fundamental role in the design of modern magnetic materials in many technological applications since it allows to stabilize magnetization along some specific crystallographic directions [1]. Magnetoelasticity can also contribute to the overall anisotropy in addition to the intrinsic MAE (unstrained state). We can identify two types of magnetoelastic effects to anisotropy: (i) first-order anisotropy due to an external strain, and (ii) second-order anisotropy due to magnetostriction [2]. This second-order anisotropy naturally emerges from spontaneous magnetostriction due to magnetization process in an unconstrained sample (zero constant stress 𝜎= 0). Namely, as the magnetization is rotated, the unit cell is distorted due to magnetostriction modifying the MAE surface. The measurement or calculation of MAE that includes secondorder anisotropy due to magnetostriction is called MAE at constant stress 𝐸𝜎 𝐾, while intrinsic MAE of an unstrained unit cell (unchanged lattice parameters) is called MAE at constant strain 𝐸𝜖 𝐾[3], see Fig. 1. Typically, Density Functional Theory (DFT) calculations of MAE are performed by fixing spins along different directions on an unstrained unit cell giving MAE at constant strain 𝐸𝜖 𝐾(intrinsic MAE). However, we point out that many experimental measurements of MAE are done on free samples, where magnetostriction takes place, and consequently corresponds to MAE at constant stress 𝐸𝜎 𝐾. Therefore, for a consistent comparison of MAE between experiment and theory, it is necessary to carefully check whether the second-order anisotropy due to magnetostriction should be included or whether it is negligible. ∗Corresponding author at: IT4Innovations, VSB - Technical University of Ostrava, 17. listopadu 2172/15, CZ 708 00 Ostrava, Czech Republic. E-mail addresses: [email protected],[email protected] (D. Legut). In the case of cubic crystals, the MAE at constant strain and stress read [2,4] 𝐸𝜖 𝐾=𝐾1(𝛼2 𝑥𝛼2 𝑦+𝛼2 𝑦𝛼2 𝑧+𝛼2 𝑧𝛼2 𝑥), 𝐸𝜎 𝐾= (𝐾1+𝛥𝐾1)(𝛼2 𝑥𝛼2 𝑦+𝛼2 𝑦𝛼2 𝑧+𝛼2 𝑧𝛼2 𝑥),(1) where 𝛼𝑖are the direction cosine of magnetization, 𝐾1is the intrinsic MAE first constant, and 𝛥𝐾1is the second-order anisotropy due to magnetostriction giving by [2,4] 𝛥𝐾1=9 4[(𝐶11 −𝐶12)𝜆2 100 − 2𝐶44𝜆2 111],(2) where 𝐶𝑖𝑗 are the elastic constants, and 𝜆100(111) are the anisotropic magnetostrictive coefficients. Here, we see that 𝛥𝐾1could be positive or negative depending on the strength of factors (𝐶11 −𝐶12)𝜆2 100 and 2𝐶44𝜆2 111. For instance, in cubic crystals where |𝜆100|≈|𝜆111|like in bcc Fe and fcc Ni, the second-order magnetostrictive contribution to magnetic anisotropy is typically small (𝛥𝐾1∕𝐾1∼ 0.1 − 1%) [2,4]. However, note that at high temperatures close to 𝑇𝐶,𝛥𝐾1could be relevant also for the case of fcc Ni since it can change the sign of the effective 𝐾1[5]. Larger second-order magnetostrictive contribution to magnetic anisotropy is found in Rare Earth alloys, e.g. for TbFe2 𝛥𝐾1∕𝐾1≈ 20% [6], where |𝜆111|≫|𝜆100|. The analysis of second-order magnetostrictive contribution to magnetic anisotropy in symmetries lower than cubic is more complex [3], and it is still rather quantitatively unexplored in many magnetic materials despite on its possible significant effect on MAE. For example, this is the case of L10-FePt, a tetragonal crystal that exhibits a very large easy axis MAE ∼ 6.6MJ/m3[1] which plays a key role in magnetic recording https://doi.org/10.1016/j.solidstatesciences.2024.107782 Received 29 August 2024; Received in revised form 27 November 2024; Accepted 28 November 2024 Solid State Sciences 160 (2025) 107782 2 D. Legut and P. Nieves Fig. 1. Schematic showing the difference between (a)–(b) MAE at constant strain 𝐸𝜖 𝐾, and (c)–(d) MAE at constant stress 𝐸𝜎 𝐾for L10-FePt. In MAE at constant strain, the lattice parameters of the unit cell are not allowed to relax (null strain 𝜖= 0) when magnetization is rotated from (a) ground state easy axis along 𝑐-axis ∥[001], where there is no stress (𝜎= 0), to (b) hard plane along 𝑎-axis ∥[100] leading to an internal stress (𝜎≠0). In MAE at constant stress, the lattice parameters are allow to relax (null internal stress 𝜎= 0) when magnetization is rotated from (c) easy axis to (d) hard plane, leading to a deformation of the unit cell due to magnetostriction (𝜖≠0). applications [7,8]. In our recent work [9], we theoretically studied the origin of the intrinsic MAE and magnetostriction for this material. Here, we extend this analysis to study the second-order anisotropy due to the magnetostriction. The paper is organized as following; in next section we introduce main equations of the magnetoelastic theory for tetragonal crystals, and in Result and discussion section we use the ab initio obtained insights to determine the strength of the second order anisotropy due to the magnetostriction for L10-FePt. We finish with concluding remarks and general outlook in order to address the fact that the computational results should be performed as close to the conditions of the real experiments. 2. Methodology — theoretical The study of MAE at constant stress for tetragonal crystals is more difficult than for cubic ones, so that it is convenient to briefly introduce the main theoretical equations for such analysis in this section. For the total energy density (𝐸), energy per volume of a system, we include the elastic (𝐸𝑒𝑙 ), magnetoelastic (𝐸𝑚𝑒) and magnetocrystalline anisotropy at constant zero strain (𝐸𝜖 𝐾) terms [10–12] 𝐸(𝝐,𝜶) =𝐸𝑒𝑙(𝝐) +𝐸𝑚𝑒(𝝐,𝜶) +𝐸𝜖 𝐾(𝜶),(3) where 𝝐is the strain tensor and 𝜶is the normalized magnetization (|𝜶|= 1). Here, the elastic energy is considered up to second order in the strain, while the magnetoelastic energy contains only linear terms in the strain up to second order in the magnetization direction 𝛼. The MAE for the unstrained cell (at constant strain) includes the intrinsic first and second magnetocrystalline anisotropy constants 𝐾1and 𝐾2. For tetragonal single crystals with point groups 4 mm,422, 42𝑚and 4∕𝑚𝑚𝑚, these energy density terms read [10,12,13], 𝐸𝑒𝑙 =1 2𝐶11(𝜖2 𝑥𝑥 +𝜖2 𝑦𝑦) +𝐶12𝜖𝑥𝑥𝜖𝑦𝑦 +𝐶13(𝜖𝑥𝑥 +𝜖𝑦𝑦)𝜖𝑧𝑧 +1 2𝐶33𝜖2 𝑧𝑧 + 2𝐶44(𝜖2 𝑦𝑧 +𝜖2 𝑥𝑧) + 2𝐶66𝜖2 𝑥𝑦, (4) 𝐸𝑚𝑒 =𝑏11(𝜖𝑥𝑥 +𝜖𝑦𝑦) +𝑏12𝜖𝑧𝑧 +𝑏21 (𝛼2 𝑧−1 3)(𝜖𝑥𝑥 +𝜖𝑦𝑦) +𝑏22 (𝛼2 𝑧−1 3)𝜖𝑧𝑧 +1 2𝑏3(𝛼2 𝑥−𝛼2 𝑦)(𝜖𝑥𝑥 −𝜖𝑦𝑦) + 2𝑏′ 3𝛼𝑥𝛼𝑦𝜖𝑥𝑦 + 2𝑏4(𝛼𝑥𝛼𝑧𝜖𝑥𝑧 +𝛼𝑦𝛼𝑧𝜖𝑦𝑧), (5) 𝐸𝜖 𝐾=𝐾1(1 −𝛼2 𝑧) +𝐾2(1 −𝛼2 𝑧)2,(6) where 𝐶𝑖𝑗 are the elastic constants, 𝑏11 and 𝑏12 are the isotropic magnetoelastic constants (𝑏𝑖𝑠𝑜), and 𝑏21,𝑏22,𝑏3,𝑏′ 3and 𝑏4are the anisotropic magnetoelastic constants (𝑏𝑎𝑛𝑖) in Cullen et al. notation [10]. From the minimization of total energy Eq. (3) with respect to strain, one finds the following fractional change in length along a measuring length direction 𝜷[10,11] 𝑙−𝑙0 𝑙0||||| 𝜶 𝜷 =𝜆𝛼1,0(𝛽2 𝑥+𝛽2 𝑦) +𝜆𝛼2,0𝛽2 𝑧+𝜆𝛼1,2(𝛼2 𝑧−1 3)(𝛽2 𝑥+𝛽2 𝑦) +𝜆𝛼2,2(𝛼2 𝑧−1 3)𝛽2 𝑧+1 2𝜆𝛾 ,2(𝛼2 𝑥−𝛼2 𝑦)(𝛽2 𝑥−𝛽2 𝑦) + 2𝜆𝛿 ,2𝛼𝑥𝛼𝑦𝛽𝑥𝛽𝑦+ 2𝜆𝜖 ,2(𝛼𝑥𝛼𝑧𝛽𝑥𝛽𝑧+𝛼𝑦𝛼𝑧𝛽𝑦𝛽𝑧), (7) where 𝑙is the length at the saturated state, 𝑙0is the length at reference demagnetized state with randomly oriented atomic moments (hypothetically paramagnetic state), 𝜆𝛼1,0and 𝜆𝛼2,0are the isotropic magnetostrictive coefficients (𝜆𝑖𝑠𝑜), and 𝜆𝛼1,2,𝜆𝛼2,2,𝜆𝛾 ,2,𝜆𝛿 ,2and 𝜆𝜖 ,2are the anisotropic magnetostrictive coefficients (𝜆𝑎𝑛𝑖). The magnetostrictive coefficients are related to the elastic and magnetoelastic constants through 𝜆𝛼1,0=−𝑏11𝐶33 +𝑏12𝐶13 𝐶33(𝐶11 +𝐶12) − 2𝐶2 13 , 𝜆𝛼2,0=2𝑏11𝐶13 −𝑏12(𝐶11 +𝐶12) 𝐶33(𝐶11 +𝐶12) − 2𝐶2 13 , 𝜆𝛼1,2=−𝑏21𝐶33 +𝑏22𝐶13 𝐶33(𝐶11 +𝐶12) − 2𝐶2 13 , 𝜆𝛼2,2=2𝑏21𝐶13 −𝑏22(𝐶11 +𝐶12) 𝐶33(𝐶11 +𝐶12) − 2𝐶2 13 , 𝜆𝛾 ,2=−𝑏3 𝐶11 −𝐶12 , 𝜆𝛿 ,2=−𝑏′ 3 2𝐶66 , 𝜆𝜖 ,2=−𝑏4 2𝐶44 . (8) The reference demagnetized state with randomly oriented atomic moments (hypothetically paramagnetic state) is difficult to characterize both theoretically and experimentally. A more practical reference length (𝑙′ 0) corresponds to a demagnetized state with magnetic domains along all equivalent easy crystallographic directions (parallel and antiparallel to the lattice vector 𝒄since L10-FePt is an easy axis magnet). For this reference demagnetized state, the fractional change in length reads [13] 𝑙−𝑙′ 0 𝑙′ 0||||| 𝜶 𝜷 =𝜆𝛼1,2(𝛼2 𝑧− 1)(𝛽2 𝑥+𝛽2 𝑦) +𝜆𝛼2,2(𝛼2 𝑧− 1)𝛽2 𝑧 +1 2𝜆𝛾 ,2(𝛼2 𝑥−𝛼2 𝑦)(𝛽2 𝑥−𝛽2 𝑦) + 2𝜆𝛿 ,2𝛼𝑥𝛼𝑦𝛽𝑥𝛽𝑦 + 2𝜆𝜖 ,2(𝛼𝑥𝛼𝑧𝛽𝑥𝛽𝑧+𝛼𝑦𝛼𝑧𝛽𝑦𝛽𝑧). (9) In Mason notation the fractional change in length reads [3] 𝑙−𝑙′ 0 𝑙′ 0||||| 𝜶 𝜷 =1 2𝜆1[(𝛼𝑥𝛽𝑥−𝛼𝑦𝛽𝑦)2− (𝛼𝑥𝛽𝑦+𝛼𝑦𝛽𝑥)2 + (1 −𝛽2 𝑧)(1 −𝛼2 𝑧) − 2𝛼𝑧𝛽𝑧(𝛼𝑥𝛽𝑥+𝛼𝑦𝛽𝑦)] + 4𝜆2𝛼𝑧𝛽𝑧(𝛼𝑥𝛽𝑥+𝛼𝑦𝛽𝑦) + 4𝜆3𝛼𝑥𝛼𝑦𝛽𝑥𝛽𝑦 +𝜆4[𝛽2 𝑧(1 −𝛼2 𝑧) −𝛼𝑧𝛽𝑧(𝛼𝑥𝛽𝑥+𝛼𝑦𝛽𝑦)] +1 2𝜆5[(𝛼𝑥𝛽𝑦−𝛼𝑦𝛽𝑥)2− (𝛼𝑥𝛽𝑥+𝛼𝑦𝛽𝑦)2 + (1 −𝛽2 𝑧)(1 −𝛼2 𝑧)]. (10) Solid State Sciences 160 (2025) 107782 3 D. Legut and P. Nieves These magnetostrictive coefficients are related to those defined by Cullen [10] in Eq. (9) in the following way 𝜆1= −𝜆𝛼1,2+1 2𝜆𝛾 ,2 𝜆2=1 2𝜆𝜖 ,2−1 4𝜆𝛼2,2−1 4𝜆𝛼1,2+1 8𝜆𝛾 ,2 𝜆3=1 2𝜆𝛿 ,2−𝜆𝛼1,2 𝜆4= −𝜆𝛼2,2 𝜆5= −𝜆𝛼1,2−1 2𝜆𝛾 ,2. (11) The difference between MAE at constant stress (𝐸𝜎 𝐾) and constant strain (𝐸𝜖 𝐾) for tetragonal crystals was derived by Mason [3] 𝐸𝜎 𝐾−𝐸𝜖 𝐾=𝛥𝐾1(1 −𝛼2 𝑧) +𝛥𝐾2(1 −𝛼2 𝑧)2,(12) where 𝛥𝐾1=1 2𝐶44(4𝜆2−𝜆1−𝜆4)2, 𝛥𝐾2=𝐶11(𝜆2 1+𝜆2 5) +𝐶33𝜆2 4+ 2𝐶12𝜆1𝜆5+ 2𝐶13(𝜆1+𝜆5)𝜆4 −𝐶44(4𝜆2−𝜆1−𝜆4)2. (13) We see that the sign of 𝛥𝐾1is the same as 𝐶44, while the sign of 𝛥𝐾2 depends on the magnitude and sign of 𝐶𝑖𝑗 and 𝜆𝑖. Since 𝐶44 >0is required to stabilize a tetragonal crystal [14], then we conclude that 𝛥𝐾1is always positive for stable phase with this symmetry. 3. Results and discussion By means of first-principles calculations (for details see Appendix and Ref. [9]) we present in Table 1the elastic constants (𝐶𝑖𝑗 ), and anisotropic magneto-elastic and magnetostriction coefficients (𝑏𝑎𝑛𝑖 𝑖) and (𝜆𝑎𝑛𝑖 𝑖) in Cullen [10] and Mason [3] notation, respectively. We can compare our theoretical elastic constants 𝐶𝑖𝑗 reflected in the bulk modulus with experiment that could be derived for the tetragonal crystals from the 𝐶𝑖𝑗 as 𝐵=2 9(𝐶11 +𝐶12 + 2𝐶13 +𝐶33 2)[15]. The ab initio calculated values span the range of 𝐵≈ 196 − 222 GPa[16,17] as the experimental derived for 𝑇→0𝐾from polycrystalline is about 𝐵= 233 GPa[18]. Our calculated 𝐶𝑖𝑗 lead to the bulk modulus of 202 GPa which is within the range of other calculations, as well as slightly below the expected experimental value. The anisotropic magnetostrictive coefficients show a span within the range of (10−3–10−4) magnitude and one of them has even an opposite sign (𝜆4). We calculated intrinsic MAE constants 𝐾1and 𝐾2of the L10-FePt by fitting the total energy with different spin directions to the equation of MAE at constant strain 𝐸𝜖 𝐾for tetragonal crystal given by Eq. (6), see Ref. [9] for more details. The obtained results 𝐾1= 15.16 MJ/m3 and 𝐾2= 0.32 MJ/m3are in good agreement with previous DFT calculations [19–21]. This result is also close to the experimental value 𝐾1= 11 MJ/m3at T=4.2 K reported by Hai et al. [22]. We note that the inclusion of the Hubbard U on 3d(Fe) and 5d(Pt) states could improve the correspondence between calculated MAE and its experimental value [20]. Next, using the values of magnetostrictive coefficients 𝜆′𝑠and elastic tensor 𝐶𝑖𝑗 , we determine 𝛥𝐾1and 𝛥𝐾2via Eq. (13), see the Table 2, where also values of iron based Rare Earths cubic Laves phases are displayed from Ref. [6]. The obtained value for 𝛥𝐾1= 0.0102 MJ/m3is rather small in magnitude with respect to intrinsic 𝐾1, mainly due to the fact that 𝜆4and 𝜆1have different sign and therefore almost compensate the value of 4 ×𝜆2as well as the elastic constant 𝐶44 has a moderate value, i.e. more than 3× smaller than 𝐶11 or 𝐶33. For the more significant effect one would need a large difference between 𝜆2and 𝜆4+𝜆1and the latter two to have the same sign, unlike the situation here. In contrast to 𝛥𝐾1, the 𝛥𝐾2= 0.0699 MJ/m3is larger in magnitude itself, and especially with respect to the intrinsic 𝐾2, see Table 2. In general, the largest contribution to 𝛥𝐾2would arise from the fourth term in Eq. (13), however, several additional conditions would have to be fulfilled, at first e.g. all three 𝜆1,𝜆5, and 𝜆4will have to have Table 1 Calculated values of elastic constants (𝐶𝑖𝑗 ), anisotropic magnetoelastic constants (𝑏𝑎𝑛𝑖 𝑖), and anisotropic magnetostrictive coefficients (𝜆𝑎𝑛𝑖 𝑖) in Cullen (C) et al. and W. P. Mason (M) notations [3,10] for the ordered L10-FePt phase at zero temperature. 𝐶𝑖𝑗 𝐶𝑖𝑗 (GPa) 𝑏𝑎𝑛𝑖 𝑖 (C) 𝑏𝑎𝑛𝑖 𝑖 (MPa) 𝜆𝑎𝑛𝑖 𝑖 (C) 𝜆𝑎𝑛𝑖 𝑖 (×10−6) 𝜆𝑎𝑛𝑖 𝑖 (M) 𝜆𝑎𝑛𝑖 𝑖 (×10−6) 𝐶11 369.7 𝑏21 85.8 𝜆𝛼1,2−378.6 𝜆1350.3 𝐶12 79.5 𝑏22 −46.1 𝜆𝛼2,2542.1 𝜆259.0 𝐶13 155.5 𝑏316.4 𝜆𝛾 ,2−56.6 𝜆339.8 𝐶33 302.1 𝑏4−47.7 𝜆𝜀,2213.9 𝜆4−542.1 𝐶44 111.6 𝑏′ 370.5 𝜆𝛿 ,2−677.5 𝜆5406.9 𝐶66 52.0 Table 2 Calculated values (at 𝑇= 0𝐾) of anisotropy constants 𝐾1,𝐾2, their second order corrections due to the magnetostriction 𝛥𝐾1,𝛥𝐾2, and the ratios 𝛥𝐾1∕𝐾1,𝛥𝐾2∕𝐾2for L10-FePt. For comparison the experimental values [6] of cubic Laves phase iron-based Rare Earths alloys (at 𝑇= 300 𝐾) are assessed. Material 𝐾1 (MJ/m3) 𝐾2 (MJ/m3) 𝛥𝐾1 (MJ/m3) 𝛥𝐾2 (MJ/m3) 𝛥𝐾1∕𝐾1 (%) 𝛥𝐾2∕𝐾2 (%) FePt 15.16 0.32 0.0102 0.0699 0.067 21.857 TbFe2−6.30 −1.33 21.0 DyFe22.45 −0.35 −14.0 HoFe20.59 −0.008 −1.0 ErFe2−0.31 −0.02 6.0 TmFe2−0.04 −0.01 22.0 the same sign (not case for FePt), or at second 𝜆1≪0or 𝜆5≪0 and being highly anisotropic to each other |𝜆5|≠|𝜆1|. For current situation where 𝜆4is being negative, and both 𝜆5and 𝜆1are positive, the contribution from the fourth term of Eq. (13) results into negative one. However, the size of the first and second term (positive sign) is very large and positive. The 5th and 3rd terms basically cancels out as they have similar magnitudes and opposite signs, hence we ended up with overall positive 𝛥𝐾2for FePt. Taken into account that 𝐶11 ≈ 2𝐶13 the decisive coefficients are then 𝜆1and 𝜆5(𝜆4) which indeed are of the orders of ≈ 10−3, and therefore the 𝛥𝐾2is about 7 times larger than 𝛥𝐾1, see Table 2. 4. Summary and conclusions In this paper we have identified the second order anisotropy due to magnetostriction for 𝐾1and 𝐾2constants for L10-FePt alloy. The former one (𝐾1) is hardly affected as the 𝛥𝐾1is basically governed by 𝜆1and 𝜆4, these magnetostriction coefficients, however, have opposite signs. Interestingly enough, the theoretical analysis of 𝛥𝐾1reveals that it will be always positive for any stable tetragonal crystal (𝛥𝐾1∝𝐶44 >0), which facilitates the existence of an overall easy axis (𝐾1+𝛥𝐾1>0). The 𝛥𝐾2of L10-FePt is found to be greater than 𝛥𝐾1as well as the ratio 𝛥𝐾2∕𝐾2, affecting the anisotropy constant, i.e. increasing the effective 𝐾2by about 20%. However, since 𝐾1is much greater than 𝐾2, we conclude that the correction to MAE in unconstrained samples of L10-FePt arising from magnetostriction might be negligible at zero temperature. Although this correction can be neglected in this material, we point out that there are many other materials where these corrections could be comparable to intrinsic MAE, like in TbFe2. In fact, it might also change the type of MAE as in fcc Ni at high temperature [5]. Note also that MAE at constant strain is typically computed in DFT calculations while MAE at constant stress is typically measured in experiment, so that in general one needs to account for 𝛥𝐾 in the theoretical calculations for a fair comparison with experiment. The extension of the presented analysis to finite temperature, as well as to other L10phases could be interesting works for the future. Theoretical analysis of 𝛥𝐾1and 𝛥𝐾2could be exploited in the design of magnetic materials for applications where MAE plays a key role. For example, this could be particularly useful in computational search for novel Rare-Earth free permanent magnets based on high-throughput screening techniques as an additional material parameter which could influence overall MAE [23]. Solid State Sciences 160 (2025) 107782 4 D. Legut and P. Nieves CRediT authorship contribution statement D. Legut: Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. P. Nieves: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by projects e-INFRA CZ (ID:90254) and QM4ST (CZ.02.01.01/00/22_008/0004572) by The Ministry of Education, Youth and Sports of the Czech Republic and also by Czech Science Foundation of the Czech Republic by grant No. 22-35410K. P. N. acknowledges support by grant MU-23-BG22/00168 funded by The Ministry of Universities of Spain. Appendix Ab initio calculation The total energy was calculated by non-collinear version (spin– orbit coupling included) of Vienna Ab Initio Simulation Package code (VASP) [24–26] employing projected augmented wave pseudopotentials [27] using generalized gradient approximation parametrized by Perdew, Burke, and Ernzerhof [28]. The valence electronic configurations [Ar] 3𝑝63𝑑64𝑠2and [Xe] 5𝑑96𝑠1for Fe and Pt metals were used, respectively. The kinetic energy cut-off was set to 520 eV to ensure the total energy convergence of the order 10−9eV/f.u. together with Fermi–Dirac smearing of 0.1 eV, and 36 963 k-points in half of the first Brillouin zone was sufficient for convergence of the calculated MAE parameters 𝐾1and 𝐾2[9]. The anisotropic magnetoelastic constants 𝑏𝑎𝑛𝑖 were calculated with the interface between MAELAS (-mode 2, energy-strain method) [29] and VASP using the settings described above. To compute 𝐶𝑖𝑗 we make use of the energy-strain method as implemented in AELAS [30] interfaced with VASP. The anisotropic magnetostrictive coefficients 𝜆𝑎𝑛𝑖 are obtained by inserting calculated 𝑏𝑎𝑛𝑖 and 𝐶𝑖𝑗 in Eq. (8). Data availability Data will be made in open access repository. References [1] J.M.D. Coey, Magnetism and Magnetic Materials, Cambridge University Press, 2010. [2] R.C. O’Handley, Modern Magnetic Materials, Wiley, 2000. [3] W.P. Mason, Derivation of magnetostriction and anisotropic energies for hexagonal, tetragonal, and orthorhombic crystals, Phys. Rev. 96 (1954) 302– 310, http://dx.doi.org/10.1103/PhysRev.96.302, URL https://link.aps.org/doi/ 10.1103/PhysRev.96.302. [4] P.K. Baltzer, Effective magnetic anisotropy and magnetostriction of monocrystals, Phys. Rev. 108 (1957) 580–587, http://dx.doi.org/10.1103/PhysRev.108.580, URL https://link.aps.org/doi/10.1103/PhysRev.108.580. [5] E. du Tremolet de Lacheisserie, J. Rouchy, The magnetoelastic coupling in nickel, J. Magn. Magn. Mater. 28 (1) (1982) 77–87, http://dx.doi.org/10.1016/ 0304-8853(82)90031-2, URL https://www.sciencedirect.com/science/article/pii/ 0304885382900312. [6] A. Clark, Chapter 7 magnetostrictive rare earth-Fe2compounds, in: Handbook of Ferromagnetic Materials, vol. 1, Elsevier, 1980, pp. 531–589, http://dx.doi.org/ 10.1016/S1574-9304(05)80122-1, URL http://www.sciencedirect.com/science/ article/pii/S1574930405801221. [7] G. Varvaro, S. Laureti, D. Fiorani, L10 FePt-based thin films for future perpendicular magnetic recording media, J. Magn. Magn. Mater. 368 (2014) 415–420, http://dx.doi.org/10.1016/j.jmmm.2014.04.058, URL https://www. sciencedirect.com/science/article/pii/S0304885314004028. [8] D. Weller, G. Parker, O. Mosendz, A. Lyberatos, D. Mitin, N.Y. Safonova, M. Albrecht, Review Article: FePt heat assisted magnetic recording media, J. Vac. Sci. Technol. B 34 (6) (2016) 060801, http://dx.doi.org/10. 1116/1.4965980,arXiv:https://pubs.aip.org/avs/jvb/article-pdf/doi/10.1116/1. 4965980/19748252/060801_1_online.pdf. [9] T. Das, P. Nieves, D. Legut, Large magnetocrystalline anisotropic energy and its impact on magnetostriction of L10-FePt, J. Phys. D: Appl. Phys. 58 (3) (2024) 035004, http://dx.doi.org/10.1088/1361-6463/ad8001. [10] J.R. Cullen, A.E. Clark, K.B. Hathaway, Materials, Science and Technology, VCH Publishings, 1994, pp. 529–565. [11] P. Nieves, S. Arapan, S. Zhang, A. Kadzielawa, R. Zhang, D. Legut, MAELAS: Magneto-elastic properties calculation via computational high-throughput approach, Comput. Phys. Comm. 264 (2021) 107964, http://dx.doi.org/10. 1016/j.cpc.2021.107964, URL https://www.sciencedirect.com/science/article/ pii/S0010465521000801. [12] E.D.T. de Lacheisserie, Magnetostriction: Theory and Application of Magnetoelasticity, CRC Press, Boca Raton, FL, 1993. [13] P. Nieves, S. Arapan, S. Zhang, A. Kadzielawa, R. Zhang, D. Legut, MAELAS 2.0: A new version of a computer program for the calculation of magneto-elastic properties, Comput. Phys. Comm. 271 (2022) 108197, http: //dx.doi.org/10.1016/j.cpc.2021.108197, URL https://www.sciencedirect.com/ science/article/pii/S001046552100309X. [14] F. Mouhat, F.-X. Coudert, Necessary and sufficient elastic stability conditions in various crystal systems, Phys. Rev. B 90 (2014) 224104, http://dx.doi.org/10. 1103/PhysRevB.90.224104, URL https://link.aps.org/doi/10.1103/PhysRevB.90. 224104. [15] D. Legut, J. Pavlů, Electronic structure and elasticity of Z-phases in the Cr–Nb– V–N system, J. Phys.: Condens. Matter. 24 (19) (2012) 195502, http://dx.doi. org/10.1088/0953-8984/24/19/195502. [16] N. Zotov, A. Ludwig, First-principles calculations of the elastic constants of Fe–Pt alloys, Intermetallics 16 (1) (2008) 113–118, http://dx.doi.org/10.1016/ j.intermet.2007.08.006, URL https://www.sciencedirect.com/science/article/pii/ S0966979507001756. [17] M. Müller, P. Erhart, K. Albe, Thermodynamics of 𝐿10ordering in FePt nanoparticles studied by Monte Carlo simulations based on an analytic bond-order potential, Phys. Rev. B 76 (2007) 155412, http://dx.doi.org/10.1103/PhysRevB. 76.155412, URL https://link.aps.org/doi/10.1103/PhysRevB.76.155412. [18] N. Nakamura, N. Yoshimura, H. Ogi, M. Hirao, Elastic constants of polycrystalline L1-FePt at high temperatures, J. Appl. Phys. 114 (9) (2013) 093506, http: //dx.doi.org/10.1063/1.4819974,arXiv:https://pubs.aip.org/aip/jap/article-pdf/ doi/10.1063/1.4819974/13466789/093506_1_online.pdf. [19] J. Marciniak, W. Marciniak, M. Werwiński, DFT calculation of intrinsic properties of magnetically hard phase L10 FePt, J. Magn. Magn. Mater. 556 (2022) 169347, http://dx.doi.org/10.1016/j.jmmm.2022.169347, URL https:// www.sciencedirect.com/science/article/pii/S030488532200289X. [20] A.B. Shick, O.N. Mryasov, Coulomb correlations and magnetic anisotropy in ordered 𝐿10CoPt and FePt alloys, Phys. Rev. B 67 (2003) 172407, http://dx.doi.org/10.1103/PhysRevB.67.172407, URL https://link.aps.org/doi/ 10.1103/PhysRevB.67.172407. [21] P. Oppeneer, Magneto-optical spectroscopy in the valence-band energy regime: relationship to the magnetocrystalline anisotropy1contribution presented at the ‘International Workshop on Soft X-ray Magneto-Optics’, Institut für Angewandte Physik, Heinrich-Heine-Universität, Düsseldorf, 13 November 1997.1, J. Magn. Magn. Mater. 188 (3) (1998) 275–285, http://dx.doi.org/10.1016/ S0304-8853(98)00198-X, URL https://www.sciencedirect.com/science/article/ pii/S030488539800198X. [22] N. Hai, N. Dempsey, D. Givord, Magnetic properties of hard magnetic FePt prepared by cold deformation, IEEE Trans. Magn. 39 (5) (2003) 2914–2916, http://dx.doi.org/10.1109/TMAG.2003.815762. [23] P. Nieves, S. Arapan, J. Maudes-Raedo, R. Marticorena-Sánchez, N. Del Brío, A. Kovacs, C. Echevarria-Bonet, D. Salazar, J. Weischenberg, H. Zhang, O. Vekilova, R. Serrano-López, J. Barandiaran, K. Skokov, O. Gutfleisch, O. Eriksson, H. Herper, T. Schrefl, S. Cuesta-López, Database of novel magnetic materials for high-performance permanent magnet development, Comput. Mater. Sci. 168 (2019) 188–202, http://dx.doi.org/10.1016/j.commatsci.2019.06.007, URL https://www.sciencedirect.com/science/article/pii/S0927025619303489. [24] G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B (R) 47 (1993) 558, http://dx.doi.org/10.1103/PhysRevB.47.558. [25] G. Kresse, J. Furthmuller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1996) 15, http://dx.doi.org/10.1016/0927-0256(96)00008-0. Solid State Sciences 160 (2025) 107782 5 D. Legut and P. Nieves [26] G. Kresse, J. Furthmuller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169, http: //dx.doi.org/10.1103/PhysRevB.54.11169. [27] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59 (1999) 1758, http://dx.doi.org/10. 1103/PhysRevB.59.1758. [28] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. B 77 (1996) 3865, http://dx.doi.org/10.1103/PhysRevLett.77. 3865. [29] P. Nieves, S. Arapan, S. Zhang, A. Kadzielawa, R. Zhang, D. Legut, MAELAS 2.0: A new version of a computer program for the calculation of magneto-elastic properties, Comput. Phys. Comm. 271 (2022) 108197, http: //dx.doi.org/10.1016/j.cpc.2021.108197, URL https://www.sciencedirect.com/ science/article/pii/S001046552100309X. [30] S. Zhang, R. Zhang, AELAS: Automatic ELAStic property derivations via high-throughput first-principles computation, Comput. Phys. Comm. 220 (2017) 403–416, http://dx.doi.org/10.1016/j.cpc.2017.07.020, URL http://www. sciencedirect.com/science/article/pii/S0010465517302400.