Batteries, fuel cells, and electrolyzers all involve:
Batteries typically use liquid electrolyte and variable-charge-state material that's insoluble in it.
A solid oxide electrolyte (hereafter SOE) is an electrolyte which is solid and conducts oxide ions, which I'd denote as [O]-2 if they weren't specially named. Oxide has high charge density, so it has strong interactions, so the activation energy for it shifting is high, so conducting it requires high temperatures.
Solid oxide fuel cells (SOFCs) using SOEs have been manufactured.
The main current SOE types are doped ceria (eg GDC) and doped zirconia. When an oxide crystal with +4 cations has some replaced with +3 cations, some of the oxygen positions in the crystal are vacant. At high temperatures, other oxygen atoms can move into those vacancies.
As the amount of dopant increases, conductivity increases, reaches a maximum, and decreases. Why does conductivity start decreasing before the crystal structure is destabilized? That's because dopants tend to form clusters at high concentrations, typically with 4 or 6 dopant atoms. Those clusters stabilize some oxygen vacancies, increasing the activation energy for oxygen moving into them.
The most important property of dopants besides their charge state is their ionic radius. Mismatch between dopant size and the crystal lattice causes strain, which often destabilizes the normal state of the crystal relative to the intermediate state of an oxygen moving into a vacancy, reducing activation energy.
Strain is relieved at crystal surfaces, so dopants have a tendency to move towards the surface, and that tendency is proportional to the strain they cause. As the crystal surfaces thus have a higher dopant concentration, dopant clusters first become a significant problem near the crystal surface.
By using 2 dopants with different ionic radius, generally using more of the one that causes less strain, it's possible to somewhat mitigate cluster formation and thus slightly increase peak conductivity. So, there are many papers on co-doped ceria.
People noticed that the conductivity of doped ceria can be greatly increased by mixing it with molten carbonate. There are many papers on such ceria-carbonate composite electrolytes. The enhanced oxide conductivity happens at the ceria-carbonate interface.
Often, composite materials are more expensive, but here, carbonate composites are easier to make. Maximizing density of ceramics is difficult, and normally, minimizing gaps between particles is important for high conductivity. With a ceria-carbonate composite, people can make low-density doped ceria, then simply soak it in molten carbonate to infiltrate it.
Ceria-carbonate composites also have lower electrical conductivity than doped ceria SOEs, which helps improve efficiency. Resistive loss from electron flow is wasted energy.
My view has been that the enhanced conductivity of these composites comes mainly from the cations solvating oxide on the surface. Some facts that agree with my view are:
Other molten salts would probably work too, such as chloride-carbonate eutectic or carbonate-cryolite eutectic. NaAlCl4 would probably work too if it didn't release AlCl3 gas at the relevant temperatures.
The molten carbonate salt itself also conducts protons, and it obviously can conduct carbonate ions. The conductivity of carbonate ions is even higher than the oxide conductivity of the ceria composites. That being the case, can you add CO2 to the side with oxygen and conduct carbonate? Yes; that design is called a molten carbonate fuel cell. They were made several decades ago, but they weren't economical because the electrodes corroded too quickly. There was a decent amount of research into electrode materials, and no solution was found.
Perhaps you see my concern about these ceria-carbonate composite electrolytes: why would electrode corrosion be any less of an issue for them than for molten carbonate fuel cells? If the electrode corrosion issue was solved, then people could just use molten carbonate electrolyte and not bother with these composites.
This brings me to a minor idea I proposed a few years back:
At high temperatures, BaCO3 and CaCO3 have fairly high solubility in eutectic LiNaK carbonate. If ceria was infiltrated by such a carbonate solution and then the temperature was lowered, BaCO3 or CaCO3 would precipitate. Ostwald ripening should cause crystals to mostly fill some pores in the ceria, with molten carbonate only remaining there at the ceria surface. This should mostly block the migration of electrode material (eg nickel ions) through the carbonate salt, slowing corrosion.
A commonly proposed application for SOFCs was combined heat and power generation. Gas turbines don't scale down to small sizes well, but SOFCs can be fairly small. The idea was that homes would have a SOFC in the furnace room connected to a natural gas line, and they'd generate electricity when heat was wanted, instead of just burning the natural gas. This was a major justification for SOFC research.
The currently-used systems to compare that to are:
SOFCs would need to be cheaper per watt than those systems to compete, and they aren't.
SOFCs operate at high temperature, so their waste heat is at a high temperature, so some of it could be converted to power with gas turbines, eg:
The net efficiency of such systems can be even higher than current combined-cycle gas turbines. It's been estimated at slightly over 70%. If SOFCs were cheaper, that might be practical.
With large-scale SOE production, high-temperature water electrolysis with SOEs would be cheaper than ambient-temperature water electrolysis. It's also more efficient, because waste heat drives more of the electrolysis reaction. But it's still too expensive. People aren't serious enough about water electrolysis to pay what it actually costs on a large scale.
SOEs can also do electrolysis of CO2 to CO, which is somewhat more valuable for chemical production than hydrogen, but that's also too expensive.
As I previously wrote, unlike water electrolysis, electrochemical reforming of methane seems competitive with current systems and worth pursuing. This involves a SOE transferring oxygen from steam or CO2 to methane. One side can produce pure H2 or CO, and the overall H2/CO ratio can be controlled which is valuable for eg methanol production.
H2 or CO can also be used for steel production. When people use coke for steelmaking, what actually does the reduction is generated CO; a blast furnace is just an inefficient way of making syngas, and coke is needed for its porosity and mechanical strength. It's much more efficient to use a purpose-built system to convert methane to syngas and use that to reduce iron ore. If cheap natural gas is available that's cheaper, so it's done now, but China prefers to use coal and make its own steel, and America doesn't make much new steel anymore. Iron electrolysis doesn't work well because of Fe+2/Fe+3 shuttling, so full electrification of steel production would involve water or CO2 electrolysis.