Does quantum mechanics predict that there is any probability of finding excess charge inside a conductor?

I've read an explanation about the distribution of excess charge placed on conductors. The explanation says that at electrostatic equilibrium, all of the excess charge migrates to the surface of the conductor. The justification for this is that if any excess charged was found locally inside the conductor, then a Gaussian surface around the charge would imply an electric field flux and perpetual movement of charge.

However, I also know that quantum mechanics predicts that electrons are delocalized around a nucleus with a probability function for being at any arbitrary distance away from the nucleus, even for the lowest energy level. I've also read that electrons can tunnel through energy barriers.

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    $\begingroup$ I'm having great difficulty finding a connection between your second paragraph and your third, and even more difficulty with the presence of 'doped semiconductors' in the title. $\endgroup$ – Jon Custer Nov 1 '18 at 12:44
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    $\begingroup$ Thanks for pointing that out. This is rambling a little, but I was thinking about the possibility of charge domains existing in a conductor, and it occurred to me that in a pure metal solid, some models of this might have no dipoles. But then it occurred to me that in a doped semiconductor, there are different elements with different electronegativities and there could be a dipole. But then I decided to limit my question to a pure conductor. $\endgroup$ – lamplamp Nov 1 '18 at 15:42