## 1 Introduction  
The (Ln = La-Lu; M = Ti, Zr, Hf) systems with a pyrochlore structure can be used as potential oxygen ion-conducting electrolytes for solid oxide fuel cell (SOFC) applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The most promising electrolytes among pyrochlores are rare-element (RE)-based zirconates: the Li-doped gadolinium zirconate Gd\({}_{1.7}\)Li\({}_{0.3}\)Zr\({}_{2}\)O\({}_{6.7}\), with a conductivity of 1 \(\times\) 10\({}^{-3}\) S/cm at 450 \({}^{\circ}\)C (-8 \(\times\) 10\({}^{-3}\) S/cm at 700 \({}^{\circ}\)C) [1], and (Nd\({}_{2-x}\)Zr\({}_{x}\))Zr\({}_{2}\)O\({}_{7+x/2}\) (x = 0.4) (1.78 \(\times\) 10\({}^{-2}\) S/cm at 700 \({}^{\circ}\)C) [5]. It can be seen from previous results [4, 5] that the solid solutions in broad isomorphism ranges with a variable Ln/M ratio (Ln = La-Lu; M = Ti, Zr, Hf) in the titanate, zirconate, and haftrate systems can be used to prepare materials with high oxygen ion conductivity. The undoubtedly advantage of some RE zirconate solid solutions with the pyrochlore structure is pure oxygen ion conductivity persisting in a wide temperature range (600-900 \({}^{\circ}\)C) and a wide range of oxygen partial pressures, and absence of p-type conductivity at high oxygen partial pressure [4]. Undesirable electronic transport of materials used in PCFCs and PCECs leads to a decrease in their effectiveness [6; 7]. The \(Ln_{2}\)Hf\({}_{2}\)O\({}_{7}\) (\(Ln\) = Sm, Eu, Gd) compounds have the pyrochlore structure, and the highest oxygen ion conductivity is reached for Gd\({}_{2}\)Hf\({}_{2}\)O\({}_{7}\): \(\sim\)1.3 \(\times\) 10\({}^{-3}\) S/cm at 750 \({}^{\circ}\)C [8; 12]. The \(Ln_{2}\)Hf\({}_{2}\)O\({}_{7}\) (\(Ln\) = Dy, Y, Ho, Yb) heavy RE hafnates have a fluorite structure, and the Dy- and Ho-based compounds have the highest conductivity among them [8]. These results were essentially confirmed by [PERSON] et al. [9], who were able to separate the total conductivity onto the bulk (1.0 \(\times\) 10\({}^{-4}\) S/cm at 700 \({}^{\circ}\)C) and grain boundary (4.66 \(\times\) 10\({}^{-5}\) S/cm at 700 \({}^{\circ}\)C) components for the Ho\({}_{2}\)Hf\({}_{2}\)O\({}_{7}\) fluorite prepared by the conventional solid-state reaction technique. Recently, [PERSON] et al. [16] synthesized Eu\({}_{2}\)(Hf\({}_{2-x}\)Eu\({}_{x}\))O\({}_{7-x/2}\) (\(x\) = 0.1) stuffed pyrochlores, which showed the highest oxygen ion conductivity among the hafnate family: \(\sim\)7.5 \(\times\) 10\({}^{-4}\) S/cm at 700 \({}^{\circ}\)C. This is, however, lower than the conductivity of the RE zirconates and titanates [16; 17; 18; 19; 20]. The authors of [21] investigated the electronic structure of La\({}_{2}\)Hf\({}_{2}\)O\({}_{7}\) based on first-principle calculations with a particular focus on the anion Frenkel (anti-Frenkel) defect pair (V\({}_{\mathbf{O}}^{\bullet}\) + O\({}^{\prime}\),i) in terms of defect structures with correlated formation energies. Such anion Frenkel pair (V\({}_{\mathbf{O}}^{\bullet\bullet}\) + O\({}^{\prime}\),i) defects are the most stable ones in pure La\({}_{2}\)Hf\({}_{2}\)O\({}_{7}\). It is worth noting that, whereas the oxygen ion conductivity of the RE hafnates and related solid solutions has been the subject of intense research [8; 9; 10; 11; 12; 13; 14; 15; 16], there are few data on their proton conductivity, except for the Nd\({}_{2}\)(Hf\({}_{2-x}\)Nd\({}_{x}\))O\({}_{7-x/2}\) (\(x\) = 0; 0.1) system [22]. The proton conductivity of the \(Ln_{2}\)Hf\({}_{2}\)O\({}_{7}\) hafnates and related solid solutions is essentially unexplored. Proton conduction was first found comparatively recently in Nd\({}_{2}\)(Hf\({}_{2-x}\)Nd\({}_{x}\))O\({}_{7-x/2}\) (\(x\) = 0; 0.1) neodymium hafnates [22]. Proton conductivity was shown both in neodimium hafnate (1.25 \(\times\) 10\({}^{-6}\) S/cm at 700 \({}^{\circ}\)C) and Nd\({}_{2}\)(Hf\({}_{1.9}\)Nd\({}_{0.1}\))O\({}_{6.95}\) \"stuffed\" pyrochlore solid solution (\(\sim\)1 \(\times\) 10\({}^{-4}\) S/cm at 700 \({}^{\circ}\)C) [22].