## 2 The Model  
The Hamiltonian describing the multiferroic properties of CBCO is as follows:  
\[H=H_{m}+H_{f}+H_{mf}. \tag{1}\][PERSON] et al. [12] proposed a Heisenberg model for the magnetic properties of pure CBCO with 12 nearest-neighbor interactions and associated anisotropies. The magnetic state of CBCO can be described as a triangular array of ferrimagnetically aligned, bitetrahedral \(c\)-axis chains with net moment along \(b\) (see Figure 1). Competing interactions within each chain produce a non-collinear spin state. Each magnetic unit cell of CBCO contains 16 Co ions on two kagome layers and two triangular layers. Moreover, the cobalt atoms can be either in a fluctuating Co\({}^{2+}\)/Co\({}^{3+}\) valence state or Co\({}^{2+}\) and Co\({}^{3+}\) ions. We extend this model here in order to study the magnetic properties of ion-doped CBCO:  
\[H_{m} = -\frac{1}{2}\sum_{i,j}(1-x)I_{i,j}^{Co-Co}\mathbf{S}_{i}^{Co} \cdot\mathbf{S}_{j}^{Co}-\frac{1}{2}\sum_{i,k}xI_{i,k}^{Co-DI}\mathbf{S}_{i}^{ Co}\cdot\mathbf{S}_{k}^{DI} \tag{2}\] \[+ D^{bag}\sum_{i,kag}S_{ic}^{2}+D^{tri}\sum_{i,tri}S_{ic}^{2}-C^{ bag}\sum_{i,kag}(\mathbf{o}_{i}\cdot\mathbf{S}_{i})^{2}\] \[- C^{tri}\sum_{i,tri}(\mathbf{n}_{i}\cdot\mathbf{S}_{i})^{2}-g \mu_{B}\mathbf{h}\cdot\sum_{i}\mathbf{S}_{i}^{Co},\]  
where \(\mathbf{S}_{i}\) is the Heisenberg spin operator of the Co spin at the site \(i\); \(J\) is the exchange interaction constant between the Co-Co or Co-doping ions (DI),; and \(x\) is the ion doping concentration. Six of these exchange interaction constants (\(J_{1}\)-\(J_{6}\)) couple the kagome and triangular layers as shown in Figure 1; the other six exchange interaction constants (\(J\)-\(J_{12}\)) couple the spins within a kagome layer. \(D\) and \(C\) are easy-plane and easy-axis anisotropy constants, whereas \(\mathbf{o}_{i}\) and \(\mathbf{n}_{i}\) are the unit vectors for the kagome and triangular layers, respectively [12]. The hexagonal anisotropy can be neglected, it is very small. \(\mathbf{h}\) is an external magnetic field. Below the ferrimagnetic transition, CBCO is reported to develop a very large spin-induced polarization, \(P\), which classifies it as a type I multiferroic. This polarization must be generated by the dependence of the exchange interactions \(J_{ij}\) on an electric field, called magnetostriction. The electric polarization, \(P\), produced between two magnetic moments is given by the following:  
\[\mathbf{P}=a\sum_{ij}\mathbf{e}_{ij}\times(\mathbf{S}_{i}\times\mathbf{S}_{j}). \tag{3}\]  
\(\mathbf{e}_{ij}\) is the unit vector connecting the sites \(i\) and \(j\). The proportional constant \(a\) is determined by the spin exchange and spin-orbit interactions [27; 28]. An orthorhombic distortion above the phase transition temperature (\(T_{C}\)) partially relieves the geometric frustration on the kagome and triangular layers and allows ferrimag  
Figure 1: Structure of the elementary cell of CBCO with the 12 nearest-neighbor spin interactions, 6 of which (\(J_{1}\) through \(J_{6}\)) couple the kagome and triangular layers, the other 6 (\(J_{7}\) through \(J_{12}\)) couple the spins within a kagome layer. netism and ferroelectricity to coexist below \(T_{C}\) in CBCO. Although occupying a triangular lattice, bitetrahedral \(c\)-axis chains are ferrimagnetically ordered in the \(ab\) plane. Competing interactions within each chain produce non-collinear spin states. Sets of bonds coupling those chains are responsible for the large spin-induced polarization of CBCO. The magnetoelectric coupling is taken to be linear [9] due to the coincidence between the magnetic and ferroelectric phase transition temperatures:  
\[H_{mf}=-g\sum_{ikl}P_{i}M_{k}M_{l}. \tag{4}\]  
\(g\) is the magnetoelectric coupling constant.