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Diamond nanomembranes to change electronics?

No, diamond is a 3D structure (same crystal structure as silicon) which is an insulator but happens to be the best heat conductor. Using diamond films as substrates for hot devices is an idea that has been around for decades. You can also dope diamond, like silicon, to make high temperature electronics.

Mostly these ideas have not worked, a combination of incompatible temps in processing and cost of large scale films. Membranes might separate the processing concerns, leaving only cost?

A micron thick membrane is several thousand layers of graphene, and graphene is directional. It is difficult to make thermal or electrical connections upwards through the layers.
 
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