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Is it possible to create a thin $\ce{BaTiO3}$ crystal layer on a copper plate by putting the plate in a beaker, pouring in heated hydrofluoric acid that has been saturated with $\ce{BaTiO3}$ powder and letting it cool?

An alternative method could be CVD, though it's more expensive.

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I don't think your first mentioned method is possible due to a few reasons :

  1. The hydrofluoric acid we're talking about is highly reactive and so, corrosive. It can etch and in fact, dissolve several metals including Cu and the fact that it is heated aggravates the situation leading to degradation of surface of copper rather than deposition.
  2. As you've mentioned it as saturated, there may be some part undissolved suspended in the solution or settled below. By simply just immersing in the solution, there is a possibility that the particles settle out or be just suspended rather than adhering to the surface of the plate.
  3. Even if the particles are deposited on the surface somehow, chances for achieving precise uniform crystalline film is very slim.

On the other hand, the second method though not pocket friendly it gets the job done. It even offers greater control over the film thickness, composition and crystallinity.

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