## 3 Numerical Results and Discussion
### Al Doping Dependence of the Polarization in CFO  
The compound CFO exhibits multiferroic behavior, wherein electric polarization \(P\) can be induced either through the application of an external magnetic field \(h\) or by chemical substitution at the Fe site with various dopant ions such as Al, Ga, Rh, Ni, Mn, or Co [3; 4; 5; 6; 7]. The dependence of the polarization component \(P_{[110]}\) on the Al concentration under zero magnetic field (\(h=0\)) and constant temperature (\(T=\) const) has been analyzed. As shown in Figure 3, the polarization \(P_{[110]}\) increases monotonically with doping concentration \(x\), remaining zero for \(x\leq 0.01\). This behavior indicates a threshold concentration necessary to induce spontaneous polarization in the absence of an external field. The calculated trend \(P(x)\) is in good agreement with experimental results reported by [PERSON] et al. [3], [PERSON] et al. [30], and [PERSON] et al. [8].  
Figure 2: Ion doping concentration dependence of the band gap energy \(E_{g}\) in CFO with (1) Al; (2) Ga; (3) In.  
To further substantiate the magnetoelectric coupling, the field dependence of the polarization \(P_{[110]}\) was investigated. In the ferroelectric incommensurate phase, a finite polarization along the \([110]\) direction emerges under the influence of a magnetic field applied parallel to the \(c\)-axis (\(h_{c}\)). This phenomenon has been examined through various experimental techniques, including bulk magnetoelectric measurements [3], synchrotron X-ray diffraction [15], neutron scattering [5], electron spin resonance [11], terahertz spectroscopy [10], and Moessbauer spectroscopy [33]. At zero doping (\(x=0\)), a polarization \(P_{[110]}\) is still induced when the magnetic field exceeds a threshold value, as depicted in Figure 4. Below approximately \(h_{c}=60\) kOe, \(P_{[110]}\) remains negligible, after which it increases with the field strength. However, at higher magnetic fields, polarization is gradually suppressed, indicating a magnetic-field-induced transition from an incommensurate polar phase to a nonpolar spin configuration. This field-induced suppression of polarization is characteristic of a spin reorientation transition, confirming the delicate balance between magnetic and electric orders in CFO. The calculated polarization response under magnetic field agrees well with experimental data by [PERSON] et al. [3] for Al-doped CFO. Comparable magnetoelectric responses under ion doping and magnetic fields have been observed for Rh- and Ga-substituted CFO, as reported by [PERSON] [34] and [PERSON] et al. [23], respectively.  
Figure 4: Magnetic field dependence of the polarization \(P_{[110]}\) of pure CFO for \(x=0\).  
Figure 3: Al doping concentration dependence of the polarization \(P_{[110]}\) in Al-doped CFO for \(h=0\).  
These findings underscore the tunability of multiferroic properties in CFO through external magnetic field and chemical perturbations, making it a promising platform for magnetoelectric device applications. Further studies on field orientation and temperature dependence could provide deeper insights into the phase boundaries and coupling mechanisms.