# Thermodynamics and Magnetism of SmFe\({}_{12}\) Compound Doped with Co and Ni: An Ab Initio Study
## 1 Introduction  
Three main material parameters determine the intrinsic properties of hard magnetic materials: (i) the saturation magnetization, \(M_{s^{z}}\) (ii) the Curie temperature, \(T_{c}\); (iii) the anisotropy field, \(H_{a}\)[1]. These three parameters all need to be of a significant value for an economically compelling permanent magnet, i.e., \(\mu_{0}M_{s}\sim\geq\) 1.25 T, \(T_{c}\sim\geq\) 550 K, and \(\mu_{0}H_{a}\sim\geq\) 3.75 T (\(\mu_{0}\) is the vacuum permeability). By mixing transition-metal (\(T\)\(M\)) with rare-earth-metal (\(RE\)) atoms in various intermetallic compounds [1, 2], one can produce a material with these important magnetic properties. The advancement of Nd-Fe-B sintered magnets in the 1980's by [PERSON] et al. [3, 4, 5, 6] led to the introduction of iron-containing compounds that were of interest due to their considerable magnetic moments and anisotropies, which are necessary for industrially functional permanent magnetic materials [7, 8, 9, 10, 11, 12]. Currently, the Nd\({}_{2}\)Fe\({}_{14}\)B\({}_{1}\)-based sintered magnets (or so-called Neomax magnets) are considered the technological standard due to their robust performance as a result of their considerable saturation magnetization (\(\mu_{0}M_{s}=\) 1.61 T, anisotropy field, \(\mu_{0}H_{a}\) = 7.6 T, sufficient coercivity, \(\mu_{0}H_{c}\) \(\geq\) 0.8 T) at room temperature [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and enormous maximum energy production (\(\left\lvert{BH}\right\rvert_{max}\sim\)515 kJ/m\({}^{3}\)) [14]. A deficiency of the Nd\({}_{2}\)Fe\({}_{14}\)B\({}_{1}\) magnets is their low Curie temperature, \(T_{c}=\) 588 K, and the notable deterioration of the anisotropy field with temperature [12]. Additionally, as was mentioned by [PERSON] et al. [15], the corrosion resistance of the Nd\({}_{2}\)Fe\({}_{14}\)B\({}_{1}\) magnets is limited, which may lead to their faster aging. Nevertheless, the permanent magnet market is currently largely immobilized by \(RE\)-based alloys, namely Nd-Fe-B magnets and Sm-Co magnets for high-temperature operation [16, 17, 18]. However, all \(RE\) elements are considered critical materials, and their use in wind generators, electric vehicles, and other applications that require powerful magnets is often minimized. Because some \(RE\) elements are not naturally abundant, numerous studieshave been committed to the creation of high-performance permanent magnets with a diminished content of the critical \(RE\) materials. The Nd content of the Nd\({}_{2}\)Fe\({}_{14}\)B\({}_{1}\) compound is 11.8 at.%. Aside from the Nd\({}_{2}\)Fe\({}_{14}\)B\({}_{1}\) compound, the \(RE\)(Fe,Co)\({}_{11}M_{1}\) compound, which crystalizes with the ThMn\({}_{12}\)-type structure, has been recognized as a possible contender for high-performance permanent magnets where the content of 3\(d\) metals produces very large \(\mu_{0}M_{S}\) and \(T_{c}\), and a considerable \(a\)/\(c\) ratio results in a significant magnetocrystalline anisotropy energy (MAE) [16; 17; 18; 19; 20]. \(RE\)(Fe,Co)\({}_{11}M_{1}\) alloys have a decreased \(RE\) content (7.7 at.%) as compared with Nd\({}_{2}\)Fe\({}_{14}\)B\({}_{1}\) magnets. However, the SmFe\({}_{12}\) compound is thermodynamically unstable as a bulk material. In order to stabilize the \(RE\)Fe\({}_{12}\) phase in bulk form, the replacement of Fe with a stabilizer \(M\), where \(M\) = Ti [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 39; 30; 32; 37; 38; 39; 40; 41; 44; 49; 82; 85; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105; 106], Al [32; 49; 105; 106], Si [27; 30; 32; 35; 49; 107; 108; 109; 110; 111; 112; 113; 114; 115], Ga [40; 82; 85], Cr [27; 29; 32; 37; 49; 92], Mn [37; 92], Mo [27; 29; 32; 37; 39; 49; 72; 106; 116; 117; 118; 119; 120], Nb [121], W [27; 29; 32; 49], Re [49], H [19; 56; 57; 58; 59; 60; 61; 62; 67; 77], N [11; 43; 59; 61; 65; 66; 68; 72; 77; 92; 93; 101; 122; 123; 124], and C [59], has been studied.