Peierls transition

Peierls' theorem[2] states that a one-dimensional equally spaced chain with one electron per ion is unstable.

It can be proven using a simple model of the potential for an electron in a 1-D crystal with lattice spacing

(similar to the result of the Kronig–Penney model, which helps to explain the origin of band gaps in semiconductors).

In essence, the proof relies on the fact that doubling the period would introduce new band gaps located at multiples of

, the distortion due to the introduction of the new band gap will cause the electrons to be at a lower energy than they would be in the perfect crystal.

Of course, this effect will be noticeable only when the electrons are arranged close to their ground state – in other words, thermal excitation should be minimized.

The earliest written record of the Peierls transition was presented at the 1954 École de physique des Houches.

These lecture notes (shown below) contain Rudolf Peierls' handwritten equations and figures, and can be viewed [3] in the library of the Institut Laue–Langevin, in Grenoble, France.

Peierls’ discovery gained experimental backing during the effort to find new superconducting materials.

In 1964, Dr. William Little of the Stanford University Department of Physics theorized that a certain class of polymer chains may experience a high Tc superconducting transition.

With the introduction of new band gaps after the lattice becomes distorted, electrons must overcome this new energy barrier in order to become free to conduct.

The simple model of the Peierls distortion as a rearrangement of ions in a 1-D chain could describe why these materials became insulators rather than superconductors.

[6] Furthermore, the 1-D nature of the material causes a breakdown of the Fermi liquid theory for electron behavior.

Here are a few examples of both theoretical and experimental research efforts to illustrate the broad range of topics: