## 2 Material and Methods  
In order to precisely control the atomic ratio of the designed compound, we prepared the La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) powders by the chemical co-precipitation method with polymer precursors. In this study, Lanthanum acetate (0.4429 g, 0.001307 mol) and Barium acetate (0.1113 g, 0.000436 mol) were dissolved in 20 mL glycerin to form the La + Ba precursor solution. The ferric acetylacetonate (6 g, 0.016989 mol) was dissolved in a mixture solution of 70 mL Aceton and 50 mL ethanol to form an Fe precursor in a 250 mL three neck flask. The La + Ba precursor was poured into the flask to mix with the iron precursor. Then 45 mL of ammonia was added into the flask to co-precipitate the La + Ba and Fe ions. The precipitate paste was separated from the solution by a centrifuge and then calcined at 450 \({}^{\circ}\)C and 800 \({}^{\circ}\)C, respectively. After a series of steps, Lathan-doped M-type barium hexagonal ferrite (La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\)) powders being composed of 0.75 lanthanum and 0.25 barium was prepared. In this designed compound, 75% Ba was replaced by 75% La from BaFe\({}_{12}\)O\({}_{19}\). Powders of 0.06 g were poured into a \(\Phi\)6.24 mm round mold and held at a pressure of 1000 msi for 30 s to form a pellet, which was sintered in air at 1400 \({}^{\circ}\)C for 3 h into ceramics. However, the ceramic samples made in this way had a large number of oxygen vacancies that were formed during the sintering process; meanwhile, the Fe ions may not have been able to be fully oxidized into Fe\({}^{3+}\), instead becoming Fe\({}^{2+}\). Either oxygen vacancies or Fe\({}^{3+}\)/Fe\({}^{2+}\) ions may cause current leakage during the _P-E_ measurements, and thus would generate unsaturated or banana-shaped _P-E_ loops. Therefore, it was necessary to perform oxygen annealing for the ceramic samples after sintering in a sealed furnace. The specific operation process was to anneal the ceramic samples in a pure oxygen atmosphere at 800 \({}^{\circ}\)C for 6 h, then turn it over to be upside down and perform the same oxygen heat treatment at 800 \({}^{\circ}\)C for 6 h. The specimen was finally annealed at 750 \({}^{\circ}\)C in a pure oxygen atmosphere for 3 h once more. Both sides of the ceramic samples were coated with a silver paste, which was solidified at 820 \({}^{\circ}\)C for 30 min to form silver electrodes.  
A structural analysis of the La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) powder and bulk ceramic samples was performed using X-ray diffraction (XRD). The morphology of the ceramic surfaces was checked by scanning electron microscopy (SEM). The _P-E_ hysteresis loops were measured on a Sawyer-Tower-circuit-based ferroelectric measurement system. Temperature-dependent dielectric spectra were measured by a Microtest PRECISION 6630 LCR meter. Magnetic properties were measured by a Vibrating Sample Magnetometer (VSM). Lambda 750 S UV-vis-near infrared spectroscopy was used to measure the optical absorption spectrum in the wavelength range of 200-2500 nm. The polarization performance, magneto-dielectric (MD) response, and magnetoelectric coupling (ME) were recorded using a Microtest PRECISION 6300 LCR meter by applying a variable DC magnetic field over a frequency range of 10 Hz to 10 MHz.