applied sciences Article  
Theoretical Study of the Multiferroic Properties of Ion-Doped CaBaCo\({}_{4}\)O\({}_{7}\)  
[PERSON]\({}^{\copyright}\), [PERSON] and [PERSON],*  
Footnote 1  
and lacks switchable permanent electric polarization [9], its application prospects remain promising, particularly in harnessing the substantial spin-induced polarization facilitated by a magnetic field slightly below \(T_{C}\)[7]. Multiferroics are materials that exhibit more than one primary ferroic property within the same phase, such as ferromagnetism, ferroelectricity, and ferroelasticity [13]. In magnetoelectric materials, an electric field can influence the magnetic properties and vice versa. A material is considered ferroelectric if it has a spontaneous electric polarization that can be switched by an applied electric field. This electric polarization typically results from a structural distortion that breaks inversion symmetry from a parent centrosymmetric phase, as seen in BaTiO\({}_{3}\). In lone-pair-active multiferroics, the ferroelectric displacement is driven by the A-site cation, while magnetism arises from a partially filled \(d\) shell on the B site, exemplified by BiFeO\({}_{3}\). Geometric ferroelectrics achieve a polar ferroelectric state through rotational distortion of the polyhedra, which drives the structural phase transition. In magnetically driven multiferroics, long-range non-centrosymmetric magnetic order induces macroscopic electric polarization. There are two categories of multiferroics: type I, where magnetism and ferroelectricity have separate origins, and type II, where ferroelectricity stems from the system's internal magnetism. Multiferroic materials have applications in actuators, switches, magnetic field sensors, new types of electronic memory devices, and nonvolatile memory. Additionally, the properties of ion-doped CBCO can be finely adjusted through doping with elements such as Zn, Y, Ni, Fe, and Al at the Co site [11; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. Neutron powder diffraction analyses of Zn-doped CBCO have revealed a reduction in the strength of magnetic interactions between the bitetrahedral cobalt \(c\)-axis chains, accompanied by a lowering of the ferrimagnetic transition temperature from 60 to 55 K [14]. Intriguingly, this doping-induced alteration also results in a discernible switching of electric polarization [14]. [PERSON] et al. [16] have documented a notable enhancement in the magnitude of magnetic field-induced electric polarization in Ni-doped CBCO. Additionally, [PERSON] et al. [17] observed a reduction in the ferrimagnetic transition temperature of CBCO from 60 to 55 K upon Zn substitution. [PERSON] et al. [18] delved into the effects of impurity substitution on the magnetoelectric properties of CBCO, revealing that increasing Fe doping results in a downward shift of the transition temperatures for both the ferrimagnetic- and ferroelectric-like phases. Notably, Y-doped CBCO at the Ca site exhibited magnetic properties and ferrimagnetism persisting up to a critical temperature (\(T_{C}\)) of 270 K [25]. Conversely, the replacement of Sr for Ba in CBCO disrupts its ferrimagnetic behavior due to the intense competition between long-range ferrimagnetic ordering and magnetic frustration [26]. The investigation of undoped CBCO properties through theoretical approaches remains relatively limited. [PERSON] et al. [9] conducted ab initio calculations coupled with the Landau theory to explore magneto-electric coupling in CBCO, while [PERSON] et al. [12] proposed a microscopic model that accounts for exchange interactions among Co ions and various anisotropies to elucidate the ferrimagnetic and multiferroic characteristics of CBCO. These studies contribute significantly to understanding the complex behavior of CBCO, shedding light on its intriguing physical phenomena.