**A possible ground state of a metal in which the conduction-electron charge density is sinusoidally modulated in space.** The periodicity of this extra modulation is unrelated to the lattice periodicity. Instead, it is determined by the dimension of the conduction-electron Fermi surface in momentum space.

#### Description

The conduction-electron charge density ρ_{0}() would ordinarily exhibit a dependence on position having the same spatial periodicity as that of the positive-ion lattice. A metal with a charge-density wave (CDW) has an additional charge modulation described by Eq. (1).

The fractional amplitude of the charge-density wave is *f* and typically has a value of approximately 0.1. The wave vector of the charge-density wave is determined by the conduction-electron Fermi surface. In a simple metal, having a spherical surface of radius *p*_{F} in momentum space, the magnitude of is approximately the value in Eq. (2),

where *h* is Planck's constant. Although the wavelength of a charge-density wave is comparable to the spacing between lattice planes, their ratio is not a rational number. The charge-density wave is then said to be incommensurate. In such a case, the total energy of the metal is independent of the phase ϕ in Eq. (1).* See also: ***Fermi surface**

#### Origin

In a quasi-one-dimensional metal, for which conduction electrons are mobile in one direction only, a charge-density wave can be caused by a Peierls instability. This mechanism involves interaction between the electrons and a periodic lattice distortion having a wave vector *Q* parallel to the conduction axis. The linear-chain metal niobium triselenide (NbSe_{3}) is prototypical, and exhibits nonlinear conduction phenomena arising from electrically induced dynamic variations of the phase ϕ in Eq. (1).

For isotropic metals, and quasi-two-dimensional metals, Coulomb interactions between electrons are the cause of a charge-density wave instability. The exchange energy, an effect of the Pauli exclusion principle, and the correlation energy, an effect of electron-electron scattering, both act to stabilize a charge-density wave. However, the electrostatic energy attributable to the charge modulation in Eq. (1) would suppress a charge-density wave were it not for a compensating charge response of the positive-ion lattice.* See also: ***Exchange interaction**; **Exclusion principle**

#### Lattice distortion

Suppose that () is the displacement of a positive ion from its lattice site at . Then a wavelike displacement given by Eq. (3)

will generate a positive-ion charge density that almost cancels the electronic charge modulation of Eq. (1). A typical value of the displacement amplitude *A* is about 1% of the lattice constant. Ion-ion repulsive interactions must be small in order to permit such a distortion. Consequently, charge-density waves are more likely to occur in metals having small elastic moduli.

#### Detection

The unambiguous signature of a charge-density wave is the observation of two satellites, on opposite sides of each Bragg reflection, in a single-crystal diffraction experiment, employing either x-rays, neutrons, or electrons. The satellites are caused by the periodic lattice displacement, Eq. (3). Charge-density waves were first seen by electron diffraction in layered metals like tantalum disulfide (TaS_{2}) and tantalum diselenide (TaSe_{2}), which have three charge-density waves. At reduced temperature, transitions from incommensurate to commensurate 's are observed. The length of a charge-density wave in the latter case is then an integral multiple of some lattice periodicity. The charge-density wave in the elemental metal potassium has been observed by neutron diffraction. The wavelength of the charge-density wave in this case is 1.5% larger than the spacing between close-packed atomic planes.* See also: ***Electron diffraction**; **Neutron diffraction**; **X-ray diffraction**

#### Fermi surface effects

If simple metals like sodium and potassium did not have a charge-density-wave structure, their conduction-electron Fermi surface would be almost a perfect sphere. Many electronic conduction phenomena would then be isotropic, and the low-temperature magnetoresistance would be essentially zero. The presence of a charge-density wave leads to a dramatic contradiction of such expectations. Conduction-electron dynamics will be modified by the presence of a new potential having the periodicity of Eq. (1). The Fermi surface will distort and become multiply connected. In high magnetic fields some electrons will travel in open orbits rather than in closed, cyclotron orbits. The low-temperature magnetoresistance will then exhibit sharp resonances as the magnetic field is rotated relative to the crystal axes. Such phenomena have been observed in sodium and potassium and have been explained with charge-density-wave theory.* See also: ***Magnetoresistance**

#### -domains

Generally, the direction will not be the same throughout an entire sample of, for example, a cubic crystal having a single charge-density wave. There will be a domain structure analogous to magnetic domains in a ferromagnet. The direction will (of course) prefer some specific axis described by direction cosines α, β, γ. In a cubic crystal there would be 24 equivalent axes and, therefore, 24 -domain types. As a consequence, some physical properties of a sample will depend markedly on the orientation distribution of its -domains. For example, the low-temperature resistivity of a potassium wire might be several times larger if its -domains are oriented parallel to the wire, than the value obtained if the domains were oriented perpendicular. Since -domain distribution can be altered by stress-induced domain regrowth, some physical properties will vary significantly from experiment to experiment, even on the same sample. Such behavior is observed in alkali metals. Techniques for the control of -domain orientation have not yet been developed.* See also: ***Domain (electricity and magnetism)**; ** Electrical resistivity**

#### Phasons

The energy decrease caused by the existence of an incommensurate charge-density wave is independent of the phase ϕ, Eq. (1). It follows that there are new low-frequency excitations which can be described by a slowly varying phase modulation of the charge-density wave, as represented approximately in Eq. (4).

Quantized excitations of this type are called phasons. They exist only for small , ; and their frequency ω() approaches zero linearly with *q*.

A number of physical phenomena caused by phasons have been observed. Phasons give rise to a low-temperature anomaly in the heat capacity. The charge-density waves in lanthanum digerminide (LaGe_{2}) were first suspected from its anomalous heat capacity. The phason spectral density of potassium has been directly observed in point-contact spectroscopy. Phasons reduce the intensity of charge-density-wave diffraction satellites with increasing temperature and contribute a diffuse-scattering cloud which surrounds each satellite. The phason velocity in tantalum disulfide has been determined from measurements of the phason diffuse scattering. Conduction electrons are strongly scattered by phasons, and such processes modify the temperature dependence of the resistivity at low temperature. * See also: ***Band theory of solids**; **Crystal structure**; **Spin-density wave**