Details about the delocalization of electron in metal

Question

Electrons in conduction band of metal is delocalized.

(1) How much is it delocalized? Will it be delocalized to the whole metal sample, or will it's wave function only cover a few/a dozen atoms nearby?

(2) Will extra (net negative charge on the metal sample) electrons be delocalized too? If so, then this seems to be a contradiction with the fact that the net charge will be on the surface of the metal. If not, considering there is no difference between every conducting electrons in the metal, then how can some of them be not delocalized and others delocalized?

Answer 1

1. Ignoring surface effects and and assuming no impurities or defects in the crystal structure, the energy eigenfunctions (i.e. Bloch waves) are delocalized across the entire crystal.
2. Excess charges deposited on a metallic sample will inhabit surface states.

If not, considering there is no difference between every conducting electrons in the metal, then how can some of them be not delocalized and others delocalized?

The presence of a surface - an interface between the crystal and the surrounding vacuum - introduces the possibility of non-bulk states which are localized to within a few atomic distances of the surface. These are the states which are inhabited by excess electrons under steady-state conditions, essentially for the reason which is given in Physics 102 - namely that an additional electron delocalized in the bulk would generate a net electric field which would push other electrons toward the surface.

It should also be noted that most metals (with exceptions like gold) will be coated with an oxide layer under ambient conditions, which somewhat complicates the discussion about surface states. The overall message remains the same, however - excess electrons will be localized to the surface rather than being delocalized throughout the bulk.