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A charge density wave Is a periodic modulation of a density of electronic charges, which forms mainly in a low-dimensional systems as a result of electron-phonon coupling. The key role for CDW formation plays the topology of the Fermi surface of the crystal, which may cause the Peierls transition.Under equilibrium conditions CDW formation can be described phenomenologically by effective mean field models [ref], but on a microscopic level a description is far from trivial, because the electronic and lattice contributions can hardly be disentangled due to their intrinsic coupling [ref].

Most CDW's in metallic crystals form due to the wave-like nature of electrons - a manifestation of quantum mechanical wave-particle duality - causing the electronic charge density to become spatially modulated, i.e., to form periodic "bumps" in charge. This standing wave affects each electronic wave function, and is created by combining electron states, or wavefunctions, of opposite momenta. The effect is somewhat analogous to the standing wave in a guitar string, which can be viewed as the combination of two interfering, traveling waves moving in opposite directions (see interference (wave propagation)).

The electrons within a CDW create a standing wave pattern and can sometimes collectively carry an electric current. The electrons in such a CDW, like those in a superconductor (see superconductivity), can flow through a linear chain compound ‘en masse’, in a highly correlated fashion. Unlike a superconductor, however, the electric CDW current often flows in a jerky fashion, much like water dripping from a faucet due to its electrostatic properties. In a CDW, the combined effects of pinning (due to impurities) and electrostatic interactions (due to the net electric charges of any CDW kinks) likely play critical roles in the CDW current's jerky behavior, as discussed in sections 4 & 5 below.The CDW in electronic charge is accompanied by a periodic distortion - essentially a superlattice - of the atomic lattice in a quasi-1-D or quasi-2-D layered metallic crystal. The metallic crystals look like thin shiny ribbons (e.g., quasi-1-D NbSe3 crystals) or shiny flat sheets (e.g., quasi-2-D, 1T-TaS2 crystals). The CDW's existence was first predicted in the 1930s by Rudolf Peierls. He argued that a 1-D metal would be unstable to the formation of energy gaps at the Fermi wavevectors ±kF, which reduce the energies of the filled electronic states at ±kF as compared to their original Fermi energy EF. The temperature below which such gaps form is known as the Peierls transition temperature, TP.

Fermi surface nesting
Fermi surface nesting means that there are two parallel pieces of a Fermi surface, such that a single q-vector can connect many points. A one dimensional metal automatically has perfect nesting because the Fermi surface consists of two points at k=±kF. In higher dimensions, good nesting will not occur for a free-electron-like single-band metal (which will have an almost circular or spherical Fermi surface), but systems with stronger interactions (or quasi 1D bands) can have Fermi surface nesting (bottom image below). As a disclaimer, not all charge density waves are driven by nesting, and not all Fermi surface topologies with a good nesting condition result in a CDW instability.

Lindhard response function
The formation of a CDW is driven by an instability of the electronic system to a spatially periodic perturbation. Particularly in quasi one-dimensional systems parallel parts of the Fermi surface (FS) are nested by an ordering vector qCDW, and lead to a divergence of the electronic Lindhard susceptibility (see Lindhard theory). Electron-phonon (e-ph) coupling imprints this ordering tendency on the lattice and freezes a soft phonon mode into a periodic lattice distortion, leading to a complex many-body problem. While this is a widely considered explanation for CDW formation1,17, momentum-dependent e-ph coupling may modify this picture3,18,19. This coupled charge and lattice periodicity 1/qCDW creates an energy gap 2D in the electronic structure due to Bragg scattering at the modulated charge density. Thereby the periodic lattice distortion and the CDW are stabilized by the strength of e-ph coupling and the diverging susceptibility, that is, the FS topology.

General references

 * Grüner, George. Density Waves in Solids. Addison-Wesley, 1994. ISBN 0-201-62654-3
 * Review of experiments as of 2013 by Pierre Monceau. Electronic crystals: an experimental overview.