Why are there electron and hole currents in semiconductor, but only electron current in metal? Why are there electron and hole current in semiconductor, but only electron current in metal ?
I know that in metal there is an overlap between valence and conduction band,
while for semiconductor, there is no overlap, but that does not help me to understand my point of the question.
I have read the thread :
Electron holes in metals
and
Why doesn’t a conductor have hole current?
but the explanations were not so much convincing to me.
I search for an explanation connected with the band theory.
 A: In case of semiconductors, the valence band is filled (majority). Electrons in the valence bands are localised around the parent atom. This means to have conduction, we need to promote an electron from the valence band to the conduction band that is vacant. So that electron contributes to the charge transport. At the same time, we leave behind a hole in the valence band. This hole can be filled by a neighbouring valence electron causing a second means of charge transport. 
However, in a metal, the electrons are already in the conduction band (that is partially filled). Here the electrons are no longer localised to any atom. So the concept of hole no longer makes sense. 
A: As Pieter noted, in some metals, arguably, all the carriers are holes. This can be seen by measuring a material's Hall coefficient, which tells you the effective carrier charge, and for a number of metals, a Hall measurement tells you that the charge carriers are positively charged.
The reason for this is somewhat complicated. In semiconductors, the chemical potential is usually close to a band extrema, and we typically approximate the band structure as being parabolic near the extrema. That gives us an effective mass (by taking the second derivative of the band structure at the extrema) that can be positive or negative. In metals, the Fermi level (chemical potential) can be far from the band edge, where the band structure can have a quite complicated shape, and the concept of an effective mass doesn't really work. The details of the Fermi surface determines the sign of the Hall coefficient.
However, you could also argue that the concept of electrons vs holes makes no sense in a metal. If an effective mass doesn't exist, this simple view of charge carriers doesn't exist because it relies on looking at the sign of the effective mass. Since an effective mass doesn't exist in metals (in general), how can you even talk about electrons and holes?
For example, here's the Fermi surface of Pb. The Fermi surface is not a sphere, so you have no hope extract a meaningful effective mass.

(Taken from https://iopscience.iop.org/article/10.1088/0031-8949/91/5/053009 )
Ashcroft and Mermin cover this, but I don't have my copy handy to tell you the pages.
