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Are electrons running through electrical wires bound to any atoms? Or are they just free-flowing?

High school chemistry taught me that electrons exist as part of the "shell" of an atom. Whenever we spoke about electricity in physics class, I always got confused; how can these electrons just be flowing by themselves? Do they not need some sort of atom to exist within?
gpt
Electronic orbitals get a little more complicated when you are dealing with a solid rather than a single atom. High school chemistry only really taught you about isolated atoms. The lower electrons bound closer to the atom's nuclei are indeed fixed there. These do not contribute to electrical effects. It's the valence, or outer electrons, that come into play with electricity. 

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When you throw a whole bunch of atoms together, the atomic orbitals you learnt in chemistry no longer exist. Because of the pauli exclusion principle (the reason why in chemistry two electrons can't share the same orbital) two electrons cannot hold the same state within the solid. This leads to what are known as electronic bands, rather than orbitals. 

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Basically, when you throw two of the same atom together the valence electron states cannot overlap. This leads to two new states very close to the isolated atom energy level, but slightly off and not overlapping. Throw a few billion billion billion atoms together and you basically get a continuous bands of allowed energy levels. The bands are really made up of a lot of discrete possible states of energy and momentum that electrons can uniquely hold, but we treat them as a band. These bands really aren't localised to an atom, their wave function is more so for the composite solid.  

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Now just like an isolated atom has higher unfilled orbitals that an electron can be moved up into by addition of energy without being totally ionized from the atom, there are higher unfilled bands in solids that electrons can move up to given energy without being ionized. These higher unfilled bands have lots of empty states to any electrons that move up into them, so they facilitate electrons moving around. Which is what electricity is. So in electricity the electrons aren't bound to any specific atom, but they aren't ionized and totally free. They are just in a higher energy level, with a lot of very close by empty energy levels, which allow them to move between them freely. 

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In metals the valence band, the band with all the bound electrons that aren't free to move, overlaps the conduction band, the band of mostly empty states where electrons can freely move. This means in metals it takes basically zero energy to get electrons to hop up into the conduction band where they can move about. Simply applying a voltage is all it takes to get lots of electrons to move. 

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In insulators the story is a little different. The valance band, where electrons are at rest, and the conduction band above that have a high energy gap.  That means it takes a lot of energy to move an electron up. Very few electrons are free to conduct because of this, so they are poor conductors. The only way to make them really conduct is a very high voltage, which will give the electrons so much energy they will be pulled away. 

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Lastly there is semiconductors, which are the basis of electronics and computers. These are the same deal as insulator, but the energy gap is much smaller. This means at room temperatures, the thermal energy is enough to cause a useful amount of electrons to jump the gap into the conduction band. As well as the electrons that move up to the conduction band being able to conduct electricity, the empty space they left behind in the valance band allows other electrons that aren't moved up to fill their place. In essence, this empty place constantly being filled by electrons appears to move, so we actually treat it as a positive particle known as a *hole* that conducts electricity too. We can alter this energy gap with impurities, which is crucial to making electronics fucntion. Other things can cause the electrons to jump this gap, such as visible light photons giving them the energy. This is the basis behind solar-cells.