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{{About|ground state spin structures|the excitation mode of magnetic materials|spin wave}}

'''Spin-density wave''' (SDW) and [[charge-density wave]] (CDW) are names for two similar low-energy ordered states of solids.  Both these states occur at low temperature in [[anisotropic]], low-dimensional materials or in metals that have high densities of states at the Fermi level <math>N(E_F)</math>.  Other low-temperature [[ground state]]s that occur in such materials are [[superconductivity]], [[ferromagnetism]] and [[antiferromagnetism]].  The transition to the ordered states is driven by the condensation energy which is approximately <math>N(E_F) \Delta^2</math> where <math>\Delta</math> is the magnitude of the [[energy gap]] opened by the transition.

Fundamentally SDWs and CDWs involve the development of a [[superstructure (condensed matter)|superstructure]] in the form of a periodic modulation in the density of the electronic [[Spin (physics)|spins]] and charges with a characteristic spatial frequency <math>q</math> that does not transform according to the symmetry group that describes the ionic positions.
The new periodicity associated with CDWs can easily be observed using [[scanning tunneling microscopy]] or [[electron diffraction]] while the more elusive SDWs are typically observed via [[neutron diffraction]] or [[Electric susceptibility|susceptibility]] measurements.  If the new periodicity is a rational fraction or multiple of the [[lattice constant]], the density wave is said to be [[Superstructure (condensed matter)|commensurate]]; otherwise the density wave is termed [[Superstructure (condensed matter)|incommensurate]].

[[Image:Crnest.png|thumb|A sketch in k-space of a (001) section of the Fermi surface of Cr.  The band structure of Cr yields an electron pocket (green) centered at Gamma and a hole pocket (blue) centered at H.  The surrounding black square indicates the boundary of the first [[Brillouin zone]].]]

Some solids with a high <math>N(E_F)</math> form density waves while others choose a superconducting or magnetic ground state at low temperatures, because of the existence of [[nesting vectors]] in the materials' [[Fermi surface]]s.  The concept of a nesting vector is illustrated in the Figure for the famous case of [[chromium]], which transitions from a paramagnetic to SDW state at a [[Néel temperature]] of 311 K.  Cr is a [[body-centered cubic]] metal whose Fermi surface features many parallel boundaries between electron pockets centered at <math>\Gamma</math> and hole pockets at H. These large parallel regions can be spanned by the nesting wavevector <math>q</math> shown in red.  The real-space periodicity of the resulting spin-density wave is given by <math>2\pi/q</math>.  The formation of an SDW with a corresponding spatial frequency causes the opening of an energy gap that lowers the system's energy.  The existence of the SDW in Cr was first posited in 1960 by [[Albert Overhauser]] of [[Purdue]]. The theory of CDWs was first put forth by [[Rudolf Peierls]] of [[Oxford University]], who was trying to explain superconductivity.

Many low-dimensional solids have anisotropic Fermi surfaces that have prominent nesting vectors.  Well-known examples include layered materials like NbSe<sub>3</sub>,<ref name="Grüner pp. 1129–1181">{{cite journal | last=Grüner | first=G. | title=The dynamics of charge-density waves | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=60 | issue=4 | date=1 September 1988 | issn=0034-6861 | doi=10.1103/revmodphys.60.1129 | pages=1129–1181 | bibcode=1988RvMP...60.1129G}}</ref> TaSe<sub>2</sub><ref name="Mutka Zuppiroli Molinié Bourgoin pp. 5030–5037">{{cite journal | last1=Mutka | first1=H. | last2=Zuppiroli | first2=L. | last3=Molinié | first3=P. | last4=Bourgoin | first4=J. C. | title=Charge-density waves and localization in electron-irradiated 1T−TaS<sub>2</sub> | journal=Physical Review B | publisher=American Physical Society (APS) | volume=23 | issue=10 | date=15 May 1981 | issn=0163-1829 | doi=10.1103/physrevb.23.5030 | pages=5030–5037| bibcode=1981PhRvB..23.5030M }}</ref> and K<sub>0.3</sub>MoO<sub>3</sub> (a [[Chevrel phase]])<ref name="Pouget Hennion Escribe-Filippini Sato pp. 8421–8430">{{cite journal | last1=Pouget | first1=J. P. | last2=Hennion | first2=B. | last3=Escribe-Filippini | first3=C. | last4=Sato | first4=M. | title=Neutron-scattering investigations of the Kohn anomaly and of the phase and amplitude charge-density-wave excitations of the blue bronze K<sub>0.3</sub>MoO<sub>3</sub> | journal=Physical Review B | publisher=American Physical Society (APS) | volume=43 | issue=10 | date=1 March 1991 | issn=0163-1829 | doi=10.1103/physrevb.43.8421 | pages=8421–8430| pmid=9996473 | bibcode=1991PhRvB..43.8421P }}</ref> and quasi-1D organic conductors like TMTSF or TTF-TCNQ.<ref name="Patton Sham pp. 631–634">{{cite journal | last1=Patton | first1=Bruce R. | last2=Sham | first2=L. J. | title=Conductivity, Superconductivity, and the Peierls Instability | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=31 | issue=10 | date=3 September 1973 | issn=0031-9007 | doi=10.1103/physrevlett.31.631 | pages=631–634| bibcode=1973PhRvL..31..631P }}</ref>  CDWs are also common at the surface of solids where they are more commonly called [[surface reconstruction]]s or even dimerization.  Surfaces so often support CDWs because they can be described by two-dimensional Fermi surfaces like those of layered materials. Chains of Au and In on semiconducting substrates have been shown to exhibit CDWs.<ref>{{cite journal|last=Snijders|first=P. C.|author2=Weitering, H. H. |title=Electronic instabilities in self-assembled atom wires|journal=Rev. Mod. Phys.|year=2010|volume=82|issue=1|pages=307–329|doi=10.1103/RevModPhys.82.307|bibcode = 2010RvMP...82..307S |url=https://zenodo.org/record/1233973}}</ref> More recently, monatomic chains of Co on a metallic substrate were experimentally shown to exhibit a CDW instability and was attributed to ferromagnetic correlations.<ref>{{cite journal|last=Zaki|first=Nader|title=Experimental observation of spin-exchange-induced dimerization of an atomic one-dimensional system|journal=Phys. Rev. B|year=2013|volume=87|issue=16|page=161406(R)|doi=10.1103/PhysRevB.87.161406|bibcode = 2013PhRvB..87p1406Z |display-authors=etal|arxiv=1208.0612|s2cid=118474115}}</ref>

The most intriguing properties of density waves are their dynamics.  Under an appropriate electric field or magnetic field, a density wave will "slide" in the direction indicated by the field due to the electrostatic or magnetostatic force.  Typically the sliding will not begin until a "depinning" threshold field is exceeded where the wave can escape from a potential well caused by a defect.  The [[hysteresis|hysteretic]] motion of density waves is therefore not unlike that of [[dislocations]] or [[magnetic domain]]s.  The current-voltage curve of a CDW solid therefore shows a very high electrical resistance up to the depinning voltage, above which it shows a nearly [[Ohm's Law|ohmic]] behavior. Under the depinning voltage (which depends on the purity of the material), the crystal is an [[Insulator (electrical)|insulator]].

==See also==

* [[Peierls transition]]
* [[Superstructure (condensed matter)]]

==References==
<references />

==General References==
#A pedagogical article about the topic: [https://doi.org/10.1038/scientificamerican0494-50 "Charge and Spin Density Waves,"] Stuart Brown and George Gruner, ''Scientific American'' 270, 50 (1994).
#Authoritative work on Cr: {{cite journal | last=Fawcett | first=Eric | title=Spin-density-wave antiferromagnetism in chromium | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=60 | issue=1 | date=1988-01-01 | issn=0034-6861 | doi=10.1103/revmodphys.60.209 | pages=209–283 | bibcode=1988RvMP...60..209F}}
#About Fermi surfaces and nesting: ''Electronic Structure and the Properties of Solids,'' Walter A. Harrison, {{ISBN|0-486-66021-4}}.
#Observation of CDW by [[ARPES]]: {{cite journal | last1=Borisenko | first1=S. V. | last2=Kordyuk | first2=A. A. | last3=Yaresko | first3=A. N. | last4=Zabolotnyy | first4=V. B. | last5=Inosov | first5=D. S. | last6=Schuster | first6=R. | last7=Büchner | first7=B. | last8=Weber | first8=R. | last9=Follath | first9=R. | last10=Patthey | first10=L. | last11=Berger | first11=H. |display-authors=5| title=Pseudogap and Charge Density Waves in Two Dimensions | journal=Physical Review Letters | volume=100 | issue=19 | date=2008-05-13 | issn=0031-9007 | doi=10.1103/physrevlett.100.196402 | page=196402| pmid=18518466 | s2cid=5532038 | url=https://www.dora.lib4ri.ch/psi/islandora/object/psi%3A19112 | arxiv=0704.1544 | bibcode=2008PhRvL.100s6402B }}
#[http://www.techfak.uni-kiel.de/matwis/amat/semi_en/kap_a/advanced/ta_4_1.html Peierls instability.]
#An extensive review of experiments as of 2013 by Pierre Monceau. {{cite journal | last=Monceau | first=Pierre | title=Electronic crystals: an experimental overview | journal=Advances in Physics | publisher=Informa UK Limited | volume=61 | issue=4 | year=2012 | issn=0001-8732 | doi=10.1080/00018732.2012.719674 | pages=325–581|arxiv=1307.0929 | bibcode=2012AdPhy..61..325M| s2cid=119271518 }}
{{Condensed matter physics topics}}

[[Category:Condensed matter physics]]
[[Category:Electric and magnetic fields in matter]]