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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 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

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  • $\begingroup$ Maybe idea of a Fermi sphere would be of some help? $\endgroup$ – Žarko Tomičić Jan 19 '17 at 13:01
  • $\begingroup$ Electric field puts electrons in new states if there are any states to populate. $\endgroup$ – Žarko Tomičić Jan 19 '17 at 13:02
  • $\begingroup$ 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. $\endgroup$ – Žarko Tomičić Jan 19 '17 at 13:10
  • $\begingroup$ And, different k states correspond to different kinetic energies. $\endgroup$ – Žarko Tomičić Jan 19 '17 at 13:10
  • $\begingroup$ 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? $\endgroup$ – nancy Jan 19 '17 at 14:02
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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.

Fermi energy

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.

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    $\begingroup$ But from what I've read and understood, the wavefunction of electrons in (periodic) insulators is still plane wave-like and delocalised across the entire solid, it's just that these states are stationary so there's no transfer of energy. $\endgroup$ – nancy Jan 19 '17 at 13:37
  • $\begingroup$ You're right, you can mathematically expand the wavefunction of an electron in an insulator in terms of plane waves, but they can still be localised --- there will be an energy penalty involved if you try to move it to a delocalised state in the conduction band. By the same token, you can expand a delocalised wavefunction in terms of localised functions such as Gaussians. In a material without a band gap, there is no such penalty. $\endgroup$ – Paraquat Jan 19 '17 at 13:50
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    $\begingroup$ Thanks for the clarification! But I don't understand the difference between electrons in the valence band and the conduction band. Is saying the wavefunction is delocalised correspond to a classical particle that can "move" through the materal? I feel I'm missing something obvious or I'm thinking about this in completely the wrong way. $\endgroup$ – nancy Jan 19 '17 at 14:01
  • $\begingroup$ 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. $\endgroup$ – Paraquat Jan 19 '17 at 14:12
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    $\begingroup$ If I may rephrase my question, how can one intuitively picture what an electrical current is, in terms of the delocalised electron wavefunction? Or alternatively, what's the distinction between an electron in the valence band and one in the conduction band? (Why does an electron in the conduction band mean the material is now a conductor?) $\endgroup$ – nancy Jan 19 '17 at 14:22

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