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The typical classification of electronic materials is metal-semiconductor-insulator. Is there any actual difference between a semiconductor and an insulator, besides the size of the bandgap?

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Basically they are the same from a physics point of view, if you only look at crystalline materials. A semiconductor is defined to be insulating at $ 0 K $, while conducting at room temperature, although I don't recall, what level of conductivity is required to count as semiconductor.

Technically, insulators are a more general group of materials, since they could also be amorphous, while semiconductor materials are in the best case monocrystalline with very few defects or at least polycrystalline with a certain grain size in order to show proper semiconducting behavior due to the bandstructure. If there are to many defects, grain boundaries, impurities, the real bandstructure can differ from the ideal one by a lot.

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