A mixed conductor represented by Formula 1:
A.sub.4±xTi.sub.5−yG.sub.zO.sub.12−δ  Formula 1 wherein, in Formula 1, A is a monovalent cation, G is at least one of a monovalent cation, a divalent cation, a trivalent cation, a tetravalent cation, a pentavalent cation, or a hexavalent cation, with the proviso that G is not Ti or Cr, wherein 0<x<2, 0.3<y<5, 0<z<5, and 0<δ≤3.

A mixed conductor represented by Formula 1:

A.sub.xTi.sub.5yG.sub.zO.sub.12Formula 1 wherein, in Formula 1, A is a monovalent cation, G is at least one of a monovalent cation, a divalent cation, a trivalent cation, a tetravalent cation, a pentavalent cation, or a hexavalent cation, with the proviso that G is not Ti or Cr, wherein 0<x<2, 0.3<y<5, 0<z<5, and 0<3.