Non-zero current from a completely filled band

[nmbr28albert]

In band theory, one of the first things that is always stated is that a completely filled band, such as the valence band at low temperatures, will not contribute any current because the average velocity will equal zero. This is usually justified by showing a plot of a parabolic bandstructure around k = 0 and stating that for every electron of a certain positive velocity, there is another electron with an equal but opposite direction velocity. In general however, isn't the bandstructure symmetric only to a second approximation in the bandstructure Taylor expansion? For an asymmetric bandstructure, it seems that the average velocity will automatically be non-zero for a completely filled band and generate a current. What am I missing?

 

[Simon Bridge]

This is correct - and the symmetry may be diffreent in different angles in the crystal.
However - at absolute zero the approximation is very very good. The small differences from a quadratic profile occur above the top of the valence band.
Remember what "valance band" means?