## 2 Materials and Methods  
La\({}_{2}\)Hf\({}_{2}\)O\({}_{7}\) and La\({}_{2}\)(Hf\({}_{1.9}\)La\({}_{0.1}\))O\({}_{6.95}\) were synthesized by reacting appropriate oxide mixtures (La\({}_{2}\)O\({}_{3}\) + HfO\({}_{2}\)) after mechanical activation in an Aronov ball mill [33; 34]. The parameters of the Aronov ball mill were as follows: an oscillation amplitude of 0.5 cm, a frequency of 50 Hz, a vial volume of 120 cm\({}^{3}\), and a ball/powder weight ratio of 15. The La\({}_{2}\)O\({}_{3}\) powder resource was annealed at 1000 \({}^{\circ}\)C for 2 h to remove absorber gases and then placed in a desiccator after cooling to 850 \({}^{\circ}\)C. The calcined mixtures were milled and pressed at 650 MPa and then sintered at 1600 \({}^{\circ}\)C for 10 h in air. The relative densities of the obtained ceramic samples were determined by measuring their weight and dimensions; they were found to be 92% above the theoretical levels for both La\({}_{2}\)Hf\({}_{2}\)O\({}_{7}\) and La\({}_{2}\)(Hf\({}_{1.9}\)La\({}_{0.1}\))O\({}_{6.95}\) compounds. In the case of La\({}_{2}\)(Hf\({}_{2-x}\)La\({}_{x}\))O\({}_{7-x/2}\) (\(x\) = 0.1) \"stuffed\" Infante pyrochlore, the ceramic's relative density was determined by hydrostatic weighing in toluene (94.8%). The phase composition features and structural properties of the powder samples were analyzed by the X-ray diffraction (XRD) technique using a DRON-3M diffractometer (Bragg-reflection geometry, Cu \(K_{\alpha}\) radiation) in the range 20 = 10\({}^{\circ}\)\(-\)75\({}^{\circ}\) at ambient condition. Scanning electron microscopy (SEM--using a JEOL JSM-6390 LA device) was employed to analyze the morphology of the ceramic samples as well as to estimate their chemical compositions. Energy dispersive X-ray microanalysis (EDX) was used to obtain X-ray concentration maps of elements. The La\({}_{2}\)(Hf\({}_{2-x}\)La\({}_{x}\))O\({}_{7-x/2}\) (\(x\) = 0, 0.1) ceramics were subject to thermogravimetric analysis in air using a NETZSCH STA 449C system (50-1000 \({}^{\circ}\)C, heating rate of 10 K/min, Al\({}_{2}\)O\({}_{3}\) plate). To verify whether the La\({}_{2}\)(Hf\({}_{1.9}\)La\({}_{0.1}\))O\({}_{6.95}\) ceramics contained structurally bound water, which might be responsible for proton conduction, we performed TG scans from 50 to 1000 \({}^{\circ}\)C in an oxygen atmosphere at a heating rate of 10 \({}^{\circ}\)C/min. To assess the reproducibility of the observed effects, the sample was reheated after cooling in the oxygen atmosphere. The disk-shaped ceramic samples (diameter \(\sim\)9 mm and thickness \(\sim\)2 mm) were fabricated for electrical measurements. Electrode contacts to the sample were made by firing ChemPur C3605 paste with colloidal platinum at 950 \({}^{\circ}\)C for 0.5 h. The conductivity of La\({}_{2}\)Hf\({}_{2}\)O\({}_{7}\) and La\({}_{2}\)(Hf\({}_{1.9}\)La\({}_{0.1}\))O\({}_{6.95}\) was studied by impedance spectroscopy analysis (EIS) via a two-probe AC method in both dry and wet air atmospheres. A combination of P-5X potentiostat/galvanostat and a frequency response analyzer module (Electrochemical Instruments Ltd., Chernoglovka, Russia) was purposefully used to provide electrical conductivity measurements. The impedance spectra between 100 and 900 \({}^{\circ}\)C were recorded over a frequency range of 10\({}^{-1}\)-5 \(\times\) 10\({}^{6}\) Hz at a signal amplitude of 150 mV. The dry atmosphere was created by passing air through a KOH, while the wet atmosphere was regulated by passing air through a water saturator held at 20 \({}^{\circ}\)C, which ensured a constant humidity of \(\sim\)0.023 atm (2.3% H\({}_{2}\)O). Air flow rate was 130 mL/min. To obtain stable water vapor pressure before measurement, the sample was kept at each temperature for 40 min. The impedance data fitting was performed using Zview software (Scribner Associates, Southern Pines, NC, USA). In addition, the total conductivity of La\({}_{2}\)(Hf\({}_{1.9}\)La\({}_{0.1}\))O\({}_{6.95}\) the was evaluated by the four-probe DC method.