## 4 Conclusions  
This study systematically explored the magnetic properties of Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) (\(0.1\leq y\leq 0.175\), \(0.35\leq x\leq 0.45\)) solid solutions. X-ray diffraction measurements revealed three distinct structural phases within the studied compositional ranges, highlighting the role of Mn in stabilizing the polar rhombohedral phase at lower concentrations and facilitating transitions to antipolar or nonpolar orthorhombic phases at higher concentrations. Scanning electron microscopy analyses further supported the absence of impurity phases. Investigation of magnetic properties revealed the correlation between crystal structure and magnetization behavior. Samples containing the rhombohedral phase exhibited metamagnetic transitions with threshold fields decreasing as the Mn content increased or temperature decreased. Orthorhombic samples were weakly ferromagnetic below their Neel points and demonstrated no metamagnetic features. These findings not only contribute to the fundamental understanding of magnetic phase transitions in BiFeO\({}_{3}\)-based multiferroic materials but also suggest potential avenues for optimizing their magnetoelectric properties through controlled chemical substitutions.  
The following supporting information can be downloaded at [[https://www.mdpi.com/article/10.3390/app14219991/s1](https://www.mdpi.com/article/10.3390/app14219991/s1)]([https://www.mdpi.com/article/10.3390/app14219991/s1](https://www.mdpi.com/article/10.3390/app14219991/s1)): Figure S1: Examples of observed and calculated X-ray diffraction patterns for Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) samples. (**a**) \(y=0.15\), \(x=0.35\); (**b**) \(y=0.1\), \(x=0.45\); and (**c**) \(y=0.175\), \(x=0.35\). Figure S2: SEM images of Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{0.55}\)Mn\({}_{0.45}\)O\({}_{3}\) ceramics (\(y=0.10\), \(0.15\), and \(0.175\), from top to bottom), captured in secondary electron mode (left column) and backscattered electron mode (right column). Figure S3: Magnetic hysteresis loops measured for the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) samples at different temperatures: (**a**) \(y=0.1\), \(x=0.35\); (**b**) \(y=0.1\), \(x=0.4\); (**c**) \(y=0.1\), \(x=0.45\); (**d**) \(y=0.15\), \(x=0.35\); (**e**) \(y=0.15\), \(x=0.4\); (**f**) \(y=0.15\), \(x=0.45\); (**g**) \(y=0.175\), \(x=0.35\); (**h**) \(y=0.175\), \(x=0.4\); (**i**) \(y=0.175\), \(x=0.45\).  
Conceptualization: V.A.K.; investigation: V.A.K. and M.D.; formal analysis: V.A.K. and M.D.; visualization: V.A.K.; writing--original draft: V.A.K.; writing--review and editing: V.A.K.; methodology: J.A.P.; validation: J.A.P.; resources: J.A.P.; funding acquisition: J.A.P. All authors have read and agreed to the published version of the manuscript.  
This work was supported by FCT--Fundacao para a Ciencia e Tecnologia, I.P. through the projects UIDB/04564/2020 and UIDP/04564/2020, with DOI identifiers 10.54499/UIDB/04564/2020 and 10.54499/UIDP/04564/2020, respectively. Access to the TAIL-UC facility funded under QREN-Mais Centro project ICT_2009_02_012_1890 is gratefully acknowledged.  
Not applicable.  
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.  
The authors would like to thank [PERSON] for his assistance with sample preparation.  
The authors declare no conflicts of interest.