## 2 Model and Method  
The compound CFO crystallizes in the R3m space group, characterized by a layered structure composed of alternating non-magnetic Cu\({}^{+}\)-O\({}^{2-}\)-Cu\({}^{+}\) planes and magnetic Fe\({}^{3+}\) layers aligned along the crystallographic \(c\)-axis. In this arrangement, the Fe\({}^{3+}\) ions occupy a two-dimensional triangular lattice, giving rise to geometrically frustrated antiferromagnetic interactions. In addition to its magnetic behavior, CFO also exhibits semiconducting characteristics, which can be effectively modeled using the s-d exchange interaction:  
\[H=H_{sp}+H_{el}+H_{sp-el}. \tag{1}\]  
\(H_{sp}\) is the Heisenberg Hamiltonian:  
\[H_{sp} = -\sum_{ij}(1-x)J_{1 ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{ij }xJ_{dij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{il}J_{2 il}\mathbf{S}_{i}\cdot \mathbf{S}_{l} \tag{2}\] \[- D\sum_{i}(S_{i}^{z})^{2}-g\mu_{B}\mathbf{h}\sum_{i}\mathbf{S}_{ i}.\]  
In this model, \(\mathbf{S}_{i}\) and its \(z\) component \(S_{i}^{z}\) represent the spin operators associated with the localized magnetic moments at lattice site \(i\). The exchange interaction between nearest-neighbor spins is characterized by the parameter \(J_{ij}\), which includes both intra-layer (\(J_{1}<0\)) and inter-layer (\(J_{2}<0\)) antiferromagnetic couplings. Next-nearest-neighbor interactions within the hexagonal planes are neglected due to their relatively weak influence. The parameter \(J_{d}\) accounts for the exchange interaction introduced by dopant ions, and \(x\) denotesthe concentration of dopants at the Fe sites. The single-ion anisotropy is described by a positive parameter \(D>0\), favoring spin alignment along the easy axis, while \(h\) represents an external magnetic field applied to the system. The magnetic properties of the system originate from the Fe\({}^{3+}\) ions, which possess a spin quantum number \(S=5/2\). Incorporating dopant effects through \(J_{d}\) and \(x\) enables a detailed analysis of how chemical substitution influences the magnetic phase stability and spin dynamics in the compound CFO. The term \(H_{el}\) denotes the Hamiltonian for the conduction electrons expressed in the Wannier representation and is given as follows:  
\[H_{el}=\sum_{ij\sigma}t_{ij}c_{i\sigma}^{+}c_{j\sigma}, \tag{3}\]  
where \(t_{ij}\) represents the hopping integral, quantifying the probability amplitude for an electron to transfer from site \(i\) to site \(j\). The operators \(c_{i\sigma}\) and \(c_{i\sigma}^{+}\) correspond to the fermionic annihilation and creation operators, respectively, for an electron with spin \(\sigma\) at lattice site \(i\). This Hamiltonian captures the kinetic energy contribution of the conduction electrons within the crystal lattice. Including the hopping terms allows for the analysis of electron mobility and its coupling with the localized magnetic moments in the system. The Hamiltonian \(H_{sp-el}\) describes, within the framework of the single-band approximation, the exchange interaction between conduction electrons and localized magnetic moments, and is expressed as follows:  
\[H_{sp-el}=-\sum_{i}I_{i}\mathbf{S}_{i}\mathbf{s}_{i}, \tag{4}\]  
where \(I\) denotes the strength of the short-range exchange interaction between the localized spin operator \(\mathbf{S}_{i}\) at site \(i\) and the conduction electron spin \(\mathbf{s}_{i}\) at the same site. Due to its localized nature, this interaction is limited to on-site coupling, implying that a conduction electron interacts only with the magnetic ion at the same lattice position. The components of the conduction electron spin \(\mathbf{s}_{i}\) can be expressed in terms of the fermionic creation and annihilation operators as follows: \(s_{i}^{+}=c_{i+}^{+}c_{i-},s_{i}^{z}=(c_{i+}^{+}c_{i+}-c_{i-}^{+}c_{i-})/2\). This Hamiltonian plays a central role in the s-d model, containing the spin-dependent scattering processes that influence both the magnetic and electronic transport properties of the material.