Peierls transition

A Peierls transition or Peierls distortion is a distortion of the periodic lattice of a one-dimensional crystal. Atomic positions oscillate, so that the perfect order of the 1-D crystal is broken.

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

It was asserted in the 1930s by Rudolf Peierls. It can be proven using a simple model of the potential for an electron in a 1-D crystal with lattice spacing ${\displaystyle a}$. The periodicity of the crystal creates energy band gaps in the ${\displaystyle \epsilon -k}$ diagram at the edge of the Brillouin zone ${\displaystyle ka=\pm \pi }$ (similar to the result of the Kronig–Penney model, which helps to explain the origin of band gaps in semiconductors). If the ions each contribute one electron, then the band will be half-filled, up to values of ${\displaystyle ka=\pm \pi /2}$ in the ground state.

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