Concept of a home made icy transistor

Some time ago I found out that ice (solid water) is a semi-conductor, like f.e. silicon and germanium. So this means that theoretically a diode or transistor can be made out of it, even at home with common products.

What follows is a back of the envelope reasoning, not a proof of concept, as I haven't tried it, but it might work.

First some rudimentary background about semi-conductors and how to make a simple component, like a diode or a transistor. A semi-conductor is basically an insulator that conducts electricity only in some circumstances, for instance when a semi-conductor is doped with another material to add electrons (negative, n-type) or to create 'holes' (positive, p-type), which can conduct electricity.
If a n-type semiconductor is joined by a p-type, you get a diode. If another p-type or n-type is added to create a npn or pnp-sandwich, you get a transistor.

So if frozen water (pure water (H2O), distilled) forms a semi-conductor lattice, it needs to be doped with something else to create the holes and electrons. More research is necessary to find suitable candidates, but for instance kitchen salt consists of sodium (Na) and chlorine (Cl) atoms which, when diluted in (pure) water, become positively and negatively charged ions.
By adding an electrical charge on two sides of the solution (pure water with f.e. a little bit of kitchen salt), most of the positively charged ions would be transfered to one side and the negatively charged ions to the other side. Freezing the solution at that moment would create an ice cube with a p-type and an n-type side, thus a diode.
Cutting a second diode ice cube in half and joining it with the other one, creating a npn or pnp sandwich results in a transistor.

This is just an idea, a lot of things need to be solved to make it work. For instance, what materials should be used to make n-type and p-type semi-conductors out of frozen water? How much of the material is needed? Will this work on a macroscopic scale (regular sized ice-cubes), or should the ice-cube be very tiny?
Where should the electrodes be positioned to separate the ions and what charge is necesarry?
Should the ions be evenly spread, and how can this be achieved?
And if it works, is it a practical diode? It would need subzero temperatures to remain solid, but it could melt by the heat dissipation of the electrical current flowing in the icy diode.