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## Main Question or Discussion Point

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?