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Semiconductors are called that due to their elemental makeup, not because current is or is not flowing through them. Basically, semiconductors have a "good" number of electrons in their outer ring (valence electrons), making them (as my chemistry professor said...) "fat and happy" as they are.

Silicon is an example. It has 4 valence electrons, so it is happy. Elements in the third row of the Periodic Table want to have 0 (actually 8 in row 2), 4, or 8 valence electrons, and will try to steal or give away electrons to get there. This "stealing" or "giving away" causes elements to group together into molecules, but it also can cause an electrical charge build up if you can get enough atoms of one element that want electrons or want to give away electrons in a specific area.

This is why semiconductors are so important for modern circuits. Since they're "fat and happy", they don't influence whatever the designer is trying to build; they act as an unbiased base to build on.

To create electrical circuits, silicon is used as the foundation, and specific locations of the silicon disk have elements with (usually) 3 or 5 valence electrons placed onto it (called "doping"). This creates electrical potentials across the silicon; when put together in specific ways, this forms diodes, transistors, micro capacitors, exc.

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