Conductor band overlap definition question

[td21]

i search all over the web but i cannot find the definition. Although I find all the literature saying that metal have valence band and conduction band overlapped, (and I cannot find a metal example with valence band and conduction band not being overlapped), I wonder if there is a conductor(including metal) example with valence band and conduction band being separated with an infinitesimal width such that valence electron can jump easily thermally.
I know this is semiconductor properties but with gap being very small then does this make the situation being conducting at temperature bigger than but not equal to 0K?
I appreciate the helps provided by physicsforums and thank you so much.

 

[rigetFrog]

I've seen the phrase "overlapping conduction and valence band", but I think it's a historical appendage that's not useful anymore. See if you can find the full k-dependent electronic structure of various semiconductors and metals and compare them. The solution to your question should become apparent.

The only very small gap I know of is doped graphene.

 

[analogdesign]

The definition of a metal as a solid with "overlapping conduction and valence band" is still useful (and used).

There is a gradual progression from conduction to insulation and small bandgap materials will seem like conductors (perhaps poor conductors) and common temperatures. One example is PbS, or Galena, which was the first semiconductor used commercially (before it was known it was a semiconductor, or even what a semiconductor was!). It can be used as a conductor in many situations.

Whether a material is a conductor or insulator or semiconductor can depend on your point of view. Most people think of diamond as an insulator but it can be used as a semiconductor if it is biased appropriately.