Metallic behaviour explained by Nearly free electron model

Question

I am having trouble understanding what causes metallic behaviour in context of Nearly free electron model. Is it because

1) In 2d material with fermi level equation is $k_F=\frac{√2πz}{a}$. If for a certain valency the fermi level (circle) doesn't touch the Brillouin zone there is no band gap, thus electrons can move freely between bands.

2) Is it because in 2D material (for example) we can have two different reciprocal lattice length, thus two different band gaps depending on the direction. And if the second band starts at B below the top of the first band at A. Two electrons per primitive cell would fill exactly one band. But some states in the 2nd band at B lie lower in energy than some 1st band states at A. The fermi energy will straddle both first and second bands. We get partially full first and second bands.

Or is it a combination of this

Answer 1

A combination of that. In more than one dimension, there are two mechanisms for metallic electronic structures.

1. Even in the presence of a common gap at every point of the Brillouin zone, a system with n odd number of valence electrons will have partly filled bands and metallic behavior. Such a mechanism is the only one possible in $1D$ systems.
2. Systems with an even number of valence electrons may have metallic character if there is more than one band cut by the Fermi energy in such a way that more than one band is partly filled (as in the example in your figures), although in general, such phenomenon may happen everywhere in the Brillouin zone, not necessarily at the zone boundary.