# Why do unfilled energy bands in crystals result in electrical conductivity?

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.

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'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

• Maybe idea of a Fermi sphere would be of some help? – Žarko Tomičić Jan 19 '17 at 13:01
• Electric field puts electrons in new states if there are any states to populate. – Žarko Tomičić Jan 19 '17 at 13:02
• In any case you have to use quantum mechanics do describe conductivity. New states mean different kinetic energy for electrons. Means different momentum. Etc. So if you put electrons in new states Fermi sphere shifts in some definite direction. Difference betwen fixed Fermi sphere and one that is shifted gives conductivity. – Žarko Tomičić Jan 19 '17 at 13:10
• And, different k states correspond to different kinetic energies. – Žarko Tomičić Jan 19 '17 at 13:10
• Maybe the question is badly phrased. Why does a shift in the Fermi sphere result in conductivity? Is there a physical way to think about it? – nancy Jan 19 '17 at 14:02

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.
• I think you may be confusing real space and reciprocal ($k$-) space. The natural way of solving the Schrodinger equation is in reciprocal space with wavefunctions expanded as plane waves, which gives you a $k$-vector that encodes the frequency and polarisation of the wave-like electron. If you were to transform your k-space solution to real space, you would see an electron that is delocalised across the entire solid, but unfortunately this tends to be difficult to visualise. – Paraquat Jan 19 '17 at 14:12