[Physics] Why do unfilled energy bands in crystals result in electrical conductivity

Question

Edit: This question is similar and possibly presents the question in a more approachable way, and this answer has given me a more real-space way of considering the movement of electrons.

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I'm looking for an intuitive way to think about how electrons in a conduction band (or in an unfilled valence band) make a material a conductor, whereas filled bands result in an insulator.

What I think I understand:

A metal is a lattice of atoms with a 'sea' of delocalised electrons. If we consider these electrons as particles, an electric current can be thought of as the drift of these electrons in a particular direction when a potential difference is applied.

Energy bands arise due to the mixing of atoms' orbitals into molecular orbitals. The periodicity of the lattice results in band gaps forming. That is, certain electron energies aren't allowed.

The highest occupied electron energy band is the valence band. If this band is full (all $k$ states are full), the material is an insulator as an applied electric field has no effect on the states of the electrons (bottom row on figure) – there's nowhere for electrons to go as all states are taken. However, if the valence band is partially filled, only a small amount of energy is required to shift some electrons into higher energy states (top row on figure). Thus the material is a conductor.

What I'm confused about:

What's the link between electrons being able to populate new $k$ states, and the macroscopic property of conductivity, or a flowing current? After some reading of similar questions, I've come to the rough idea that if an electron can easily access empty $k$ states, it can "hop" around the crystal lattice relatively easily as it has empty spots to hop into. So different $k$ states correspond to different spatial locations? This is obviously treating the electron as a particle rather than a wave.

Is it more "right" in this situation to consider electrons as waves, thus filled bands result in standing waves in the electronic wavefunctions, and travelling waves for conductors? (I'm very unclear on this take of things.)

Pic http://users-phys.au.dk/philip/pictures/solid_metalquantum/blochconduction.gif

Answer 1

The first point to note is that in a periodic solid, you are essentially representing a solid with an infinite number of atoms, so you have a continuum of states, i.e. bands. The second point is that in a solid, the bands will be filled up to the top of the conduction band, each containing two electrons. In an electrical insulator, the electrons can be considered to be confined to their states, unless you give them enough energy (e.g. via photons or phonons) to cross the band gap from the valence band to the conduction band. In a metal, however, there is no band gap, so electrons can occupy conduction states without giving them a kick; instead, they occupy states up to the Fermi energy or Fermi level.

In a metal, the electrons can no longer considered to be localised. The $k$ in the $k$-states refers to a wave vector, a solution of the Schrödinger equation. In an insulator, these wave vectors are localised, akin to a Gaussian function, but in a metal, they are more plane wave-like and are delocalised. The fact that these states are wave-like, and therefore delocalised across the notionally infinite solid is what allows the flow of current.