## 4 Numerical Results and Discussion
### Dependence of the Magnetization in CBCO on the Dopant Concentration at the Co Site  
The primary objective of this study is to explore the influence of doping on the multiferroic characteristics of the CBCO compound. It is important to note that the exchange interaction constant \(J\) is intricately linked to the spatial arrangement of spins, relying on lattice parameters, lattice symmetry, and the number of nearest neighbors, while also being susceptible to various strains. Specifically, \(J\) is inversely proportional to the distance separating adjacent spins within the lattice. Doping introduces a disparity in ionic radii between the dopant and the host Co ions, thereby inducing strain. In scenarios where the doping ion possesses a larger ionic radius, tensile strain ensues, leading to an expansion of lattice parameters and a consequent reduction in the exchange interaction constant under doping, denoted as \(J_{d}\), \(J_{d}<J\). Conversely, when a compressive strain occurs due to a smaller dopant ionic radius, lattice parameters contract, increasing \(J_{d}\), \(J_{d}>J\).  
The influence of the doping concentration is quantified through the factors \((1-x)\) and \(x\) in the first two terms of Equation (1), respectively. Here, \(x\) represents the doping concentration, wherein the exchange interaction constant \(J\) progressively diminishes from its maximal value in the absence of doping (\(x=0\)) to the exchange interaction constant \(J_{d}\) in the scenario where all host ions are replaced by dopants (\(x=1\)).  
Initially, our investigation focuses on the substitution of Fe into the Co site of CBCO. The Fe atoms preferentially occupy the sites at the triangular layer as shown by [PERSON] et al. [30]. Given that the ionic radius of the Fe dopant exceeds that of the host Co ion, inducing tensile strain, we anticipate the relation \(J_{d}<J\). This strain-induced modulation is reflected in the observed enhancement of magnetization, \(M\), with increasing Fe dopant concentration \(x\), as depicted in Figure 5, curve 1. This trend is consistent with observations reported by [PERSON] et al. [18] and [PERSON] et al. [11].  
We calculated the Fe doping dependence of the specific heat \(C_{p}\). The results are shown in Figure 6. It can be seen that \(C_{p}\) increases with increasing \(x\). There is also a peak at \(T_{C}\), which is shifted to higher \(T\) values because \(T_{C}\) is enhanced for larger \(x\) values. The results are in agreement with the experimental data of [PERSON] et al. [18].  
Continuing our investigation, we turn our attention to Zn as a doping ion within CBCO. The Zn atoms prefer occupying the kagome layer as shown by [PERSON] et al. [30]. Given the larger ionic radius of the Zn\({}^{2+}\) ion (0.88 A) compared to that of Co\({}^{2+}\) one (0.79 A), i.e., a tensile strain arises, necessitating the application of the relationship \(J_{d}>J\). Consequently, this strain-induced effect manifests as a reduction in magnetization, \(M\), as depicted in Figure 5, curve 2, aligning closely with the experimental findings by [PERSON] et al. [17], [PERSON] et al. [14], and [PERSON] et al. [30]. Notably, for higher Zn doping concentrations \(x\), such as \(x=3\), the magnetization, \(M\), reaches a critical point of vanishing, indicative of a transition from ferrimagnetic CBCO to antiferromagnetic Zn-doped CBCO. Further  
Figure 5: Dependence of the magnetization on the dopant concentration \(x\) of CBCO for different doping ions: (1) Fe, (2) Zn, (3) Mn, (4) Ni.  
more, similar reductions in the magnetization, \(M\), are observed upon ion doping with Ni (\(r\) = 0.83 A) or Mn (\(r\) = 0.81 A), both possessing larger radii than Co ions, thereby inducing tensile strain. This corroborates with the findings depicted in Figure 5, curves 3 and 4, as reported by [PERSON] et al. [31], [PERSON] et al. [16], and [PERSON] et al. [32], indicating a weakening of ferrimagnetic interactions upon Mn doping, and a decrease in the magnetization, \(M\), upon Ni doping.