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| '''Charge ordering''' (CO) is a (first- or second-order) [[phase transition]] occurring mostly in strongly correlated materials such as [[transition metal oxides]] or [[organic conductors]]. Due to the strong interaction between electrons, [[charge (physics)|charges]] are localized on different sites leading to a disproportionation and an [[Order (crystal lattice)|ordered]] [[superlattice]]. It appears in different patterns ranging from vertical to horizontal stripes to a checkerboard–like pattern, and it is not limited to the two-dimensional case. The charge order transition is accompanied by [[symmetry breaking]] and may lead to [[ferroelectricity]]. It is often found in close proximity to [[superconductivity]] and [[colossal magnetoresistance]].
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| [[File:ChargeOrderPattern.jpg|thumb| Charge order patterns]]
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| This long range order phenomena was first discovered in magnetite (Fe<sub>3</sub>O<sub>4</sub>) by Verwey in 1939.<ref>{{cite journal|last=Verwey|first=E.J.W.|title=Electronic conduction of magnetite (Fe<sub>3</sub>O<sub>4</sub>) and its transition point at low temperatures|journal=Nature|year=1939|volume=144|pages=327–328|doi=10.1038/144327b0|issue=3642 | bibcode=1939Natur.144..327V}}</ref><ref>{{cite journal|last=Verwey|first=E.J.W.|coauthors=Haayman, P.W.|title=Electronic conductivity and transition point of magnetite (Fe<sub>3</sub>O<sub>4</sub>)|journal=Physica |year=1941|volume=8|pages=979–987|doi=10.1016/S0031-8914(41)80005-6|issue=9|bibcode = 1941Phy.....8..979V }}</ref>
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| He observed an increase of the [[electrical resistivity]] by two orders of magnitude at T<sub>CO</sub>=120K, suggesting a phase transition which is now well known as the Verwey transition. He was the first to propose the idea of an ordering process in this context. The charge ordered structure of magnetite was solved in 2011 by a group led by Prof. J.P. Attfield with the results published in Nature.
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| == Theoretical description ==
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| The extended one-dimensional [[Hubbard model]] delivers a good description of the charge order transition with the on-site and nearest neighbor Coulomb repulsion U and V. It emerged that V is a crucial parameter and important for developing the charge order state. Further model calculations try to take the temperature and an interchain interaction into account.<ref>{{cite journal|last=Yoshioka|first=H.|coauthors=Tsuchuuzu, M; Seo, H.|journal=Journal of the Physical Society of Japan|year=2007|volume=76|pages=103701|doi=10.1143/JPSJ.76.103701|title=Charge-Ordered State versus Dimer-Mott Insulator at Finite Temperatures|issue=10|bibcode = 2007JPSJ...76j3701Y |arxiv = 0708.0910 }}</ref>
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| The extended Hubbard model for a single chain including inter-site and on-site interaction V and U as well as the parameter <math> \delta_d </math> for a small dimerization which can be typically found in the (TMTTF)<sub>2</sub>X compounds is presented as follows:
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| <math>H = -t \sum_{i} \sum_{\sigma} \left ( \left [ 1+ \left(-1 \right)^i \delta_d \right ]c^{\dagger}_{i,\sigma}c_{i+1,\sigma}+ h.c \right)+ U \sum_i n_{i,\uparrow}n_{i,\downarrow} + V \sum_{i}n_i, n_{i+1} </math>
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| <!-- Deleted image removed: [[File:Extend1DHubbardModel.jpg|thumb|Illustration of the 1D Hubbard model for a slightly dimerized chain.]] -->
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| where t describes the transfer integral or the kinetic energy of the electron and <math> c^{\dagger}_{i,\sigma}</math> and <math> c_{i+1,\sigma} </math> are the creation respectively the annihilation operator for an electron with the spin <math> \sigma = \uparrow , \downarrow </math> at the <math> i </math>th or <math>i+1</math>th site. <math> n_{i,\downarrow, \uparrow} </math> denotes the density operator. For non-dimerized systems, <math> \delta_d </math> can be set to zero Normally, the on-site Coulomb repulsion U stays unchanged only t and V can vary with pressure.
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| ==Examples== | |
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| === Organic conductors ===
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| Organic conductors consist of [[Electron donor|donor]] and [[Electron acceptor|acceptor]] molecules building separated planar sheets or columns. The energy difference in the [[ionization]] energy acceptor and the [[electron affinity]] of the donor leads to a charge transfer and consequently to free carriers whose number is normally fixed. The carriers are delocalized throughout the crystal due to the overlap of the molecular orbitals being also reasonable for the high anisotropic conductivity. That is why it will be distinct between different dimensional organic conductors. They possess a huge variety of ground states, for instance, charge ordering, spin-Peierls, [[spin-density wave]], [[antiferromagnetism|antiferromagnetic]] state, [[superconductivity]], [[charge-density wave]] to name only some of them.<ref>{{cite book|last=Ishiguro|first=T.|title=Organic Superconductors|year=1998|publisher=Springer-Verlag|location=Berlin Heidelberg New York|isbn=3-540-63025-2}}</ref><ref>{{cite book|last=Toyota|first=N.|title=Low-dimensional Molecular Metals|year=2007|publisher=Springer-Verlag|location=Berlin Heidelberg|isbn=978-3-540-49574-1}}</ref>
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| ==== Quasi One-dimensional Organic Conductors ==== | |
| The model system of one-dimensional conductors is the [[Bechgaard salt|Bechgaard]]-Fabre salts family, (TMTTF)<sub>2</sub>X and (TMTSF)<sub>2</sub>X, where in the latter one [[sulfur]] is substituted by [[selenium]] leading to a more metallic behavior over a wide temperature range and exhibiting no charge order. While the TMTTF compounds depending on the counterions X show the conductivity of a semiconductor at room temperature and are expected to be more one-dimensional than (TMTSF)<sub>2</sub>X.<ref>{{cite journal|last=Jeromé|first=D.|title=THE PHYSICS OF ORGANIC SUPERCONDUCTORS|journal=Science|year=1991|volume=252|pages=1509–1514|doi=10.1126/science.252.5012.1509|pmid=17834876|issue=5012|bibcode = 1991Sci...252.1509J }}</ref>
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| The transition temperature T<sub>CO</sub> for the TMTTF subfamily was registered over two order of magnitudes for the centrosymmetric anions X = Br, PF<sub>6</sub>, AsF<sub>6</sub>, SbF<sub>6</sub> and the non-centrosymmetric anions X= BF<sub>4</sub> and ReO<sub>4</sub>.<ref>{{cite journal|last=Nad|first=F.|coauthors=Monceau, P.|title=Dielectric response of the charge ordered state in quasi-one-dimensional organic conductors|journal=Journal of Physical society of Japan|year=2006|volume=75|pages=051005|doi=10.1143/JPSJ.75.051005|issue=5|bibcode = 2006JPSJ...75e1005N }}</ref>
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| In the middle of the eighties, a new "structureless transition" was discovered by Coulon et al.<ref>{{cite journal|last=Coulon|first=C.|coauthors=Parkin, S.S.P.; Laversanne, R.|title=STRUCTURELESS TRANSITION AND STRONG LOCALIZATION EFFECTS IN BIS-TETRAMETHYLTETRAHTHIAFULVALENIUM SALTS [(TMTTF)<sub>2</sub>X]|journal=Physical Review B|year=1985|volume=31|pages=3583–3587|doi=10.1103/PhysRevB.31.3583|issue=6|bibcode = 1985PhRvB..31.3583C }}</ref> conducting transport and thermopower measurements. They observed a suddenly rise of the resistivity and the thermopower at T<sub>CO</sub> while x-ray measurements showed no evidence for a change in the crystal symmetry or a formation of a superstructure. The transition was later confirmed by <sup>13</sup>C-NMR<ref>{{cite journal|last=Chow|first=D.S.|coauthors=et al.|title=Charge Ordering in the TMTTF Family of Molecular Conductors|journal=Physical Review Letters|year=2000|volume=85|pages=1698–1701|doi=10.1103/PhysRevLett.85.1698|pmid=10970592|issue=8|arxiv = cond-mat/0004106 |bibcode = 2000PhRvL..85.1698C }}</ref> and dielectric measurements.
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| Different measurements under pressure reveal a decrease of the transition temperature T<sub>CO</sub> by increasing the pressure. According to the phase diagram of that family, an increasing pressure applied to the TMTTF compounds can be understood as a shift from the semiconducting state (at room temperature) to a higher dimensional and metallic state as you can find for TMTSF compounds without a charge order state.
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| {| class="wikitable sortable"
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| !Anion X !! T<sub>CO</sub> (K)
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| | (TMTTF)<sub>2</sub>Br || 28
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| |-
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| | (TMTTF)<sub>2</sub>PF<sub>6</sub> || 70
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| | (TMTTF)<sub>2</sub>AsF<sub>6</sub> || 100.6
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| | (TMTTF)<sub>2</sub>SbF<sub>6</sub> || 154
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| | (TMTTF)<sub>2</sub>BF<sub>4</sub> || 83
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| |-
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| | (TMTTF)<sub>2</sub>ReO<sub>4</sub> || 227.5
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| |-
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| | (DI-DCNQI)<sub>2</sub>Ag|| 220
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| |-
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| | TTM-TTPI<sub>3</sub> || 120
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| |}
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| ==== Quasi Two-dimensional Organic Conductors ====
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| A dimensional crossover can be induced not only by applying pressure, but also be substituting the donor molecules by other ones. From a historical point of view, the main aim was to synthesize an [[organic superconductor]] with a high T<sub>C</sub>. The key to reach that aim was to increase the orbital overlap in two dimension. With the BEDT-TTF and its huge π-electron system, a new family of quasi two-dimensional organic conductors were created exhibiting also a great variety of the phase diagram and crystal structure arrangements.
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| <br /> | |
| At the turn of the 20th century, first [[NMR]] measurements on the θ-(BEDT-TTF)<sub>2</sub>RbZn(SCN)<sub>4</sub> compound uncovered the known metal to insulator transition at T<sub>CO</sub>= 195 K as an charge order transition.<ref>{{cite journal|last=Miyagawa|first=K.|coauthors=Kawamoto, A.; Kanoda, K.|title=Charge ordering in a quasi-two-dimensional organic conductor|journal=Physical Review B|year=2000|volume=62|pages=R7679|doi=10.1103/PhysRevB.62.R7679|issue=12|bibcode = 2000PhRvB..62.7679M }}</ref>
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| {| class="wikitable sortable"
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| ! Compound !! T<sub>CO</sub> (K)
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| | α-(BEDT-TTF)<sub>2</sub>I<sub>3</sub> || 135
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| |-
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| | θ-(BEDT-TTF)<sub>2</sub>TlCo(SCN)<sub>4</sub> || 240
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| |-
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| | θ-(BEDT-TTF)<sub>2</sub>TlZn(SCN)<sub>4</sub> || 165
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| |-
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| |θ-(BEDT-TTF)<sub>2</sub>RbZn(SCN)<sub>4</sub> || 195
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| |-
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| |θ-(BEDT-TTF)<sub>2</sub>RbCo(SCN)<sub>4</sub> || 190
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| |}
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| === Transition metal oxides ===
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| The most prominent [[transition metal oxide]] revealing a CO transition is the [[magnetite]] Fe<sub>3</sub>O<sub>4</sub> being a mixed-valence oxide where the iron atoms have a statistical distribution of Fe<sup>3+</sup> and Fe<sup>2+</sup> above the transition temperature. Below 122 K, the combination of 2+ and 3+ species arrange themselves in a regular pattern, whereas above that transition temperature (also referred to as the Verwey temperature in this case) the thermal energy is large enough to destroy the order.<ref>{{cite journal | doi=10.1126/science.276.5314.911 | title=Materials Science: Charge Ordering in Manganates | year=1997 | last1=Rao | first1=C. N. R. | journal=Science | volume=276 | issue=5314 | pages=911–912}}</ref>
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| {| class="wikitable sortable"
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| |-
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| ! Compound<ref>{{cite journal|last=Attfield|first=J.P.|title=Charge ordering in transition metal oxides|journal=Solid State Sciences|year=2006|volume=8|pages=861–867|doi=10.1016/j.solidstatesciences.2005.02.011|issue=8|bibcode = 2006SSSci...8..861A }}</ref>!! T<sub>CO</sub> (K)
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| |-
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| | Y<sub>0.5</sub>NiO<sub>3</sub> || 582
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| |-
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|
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| | YBaCo<sub>2</sub>O<sub>5</sub> || 220
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| |-
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| | CaFeO<sub>3</sub> || 290
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| |-
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| | TbBaFe<sub>2</sub>O<sub>5</sub> || 282
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| |-
| |
| | Fe<sub>3</sub>O<sub>4</sub> || 122
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| |-
| |
| | Li<sub>0.5</sub>MnO<sub>2</sub> || 290
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| |-
| |
| | LaSrMn<sub>3</sub>O<sub>7</sub> || 210
| |
| |-
| |
| | Na<sub>0.25</sub>Mn<sub>3</sub>O<sub>6</sub> || 176
| |
| |-
| |
| | YBaMn<sub>2</sub>O<sub>6</sub> || 498
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| |-
| |
| | TbBaMn<sub>2</sub>O<sub>6</sub> || 473
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| |-
| |
| | PrCaMn<sub>2</sub>O<sub>6</sub> || 230
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| |-
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| | α'-NaV<sub>2</sub>O<sub>5</sub> || 34
| |
| |}
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| === Detection of charge order ===
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| * [[NMR]] spectroscopy is a powerful tool to measure the charge disproportionation. To apply this method to a certain system, it has to be doped with a nuclei, for instance <sup>13</sup>C as it is the case for TMTTF compounds, being active for NMR. The local probe nuclei is very sensitive to the charge on the molecule observable in the [[Knight shift]] K and the [[chemical shift]] D. The Knight shift K is proportional to the spin [[spin susceptibility]] χ<sub>Sp</sub> on the molecule. The charge order or charge disproportionation appear as a splitting or broadening of the certain feature in the spectrum.
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| * The [[X-ray crystallography|X-ray diffraction]] technique allows to determine the atomic position, but the extinction effect hinders to receive a high resolution spectrum. In the case of the organic conductors, the charge per molecule is measured by the change of the bond length of the C=C double bonds in the TTF molecule. A further problem arising by irradiating the organic conductors with x-rays is the destruction of the CO state.<ref>{{cite journal|last=Coulon|first=C.|coauthors=Lalet, G.; Pouget, J.-P.; Foury-Leylekian, P.;Moradpour, A.|title=Anisotropic conductivity and charge ordering in (TMTTF)(<sub>2</sub>)X salts probed by ESR|journal=Physical Review B|year=2007|volume=76|pages=085126|doi=10.1103/PhysRevB.76.085126|issue=8|bibcode = 2007PhRvB..76h5126C }}</ref>
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| * In the organic molecules like TMTTF, TMTSF or BEDT-TFF, there are charge-sensitive modes changing their frequency depending on the local charge. Especially the C=C double bonds are quite sensitive to the charge. If a vibrational mode is infrared active or only visible in the [[Raman spectroscopy|Raman]] spectrum depends on its symmetry. In the case of BEDT-TTF, the most sensitive ones are the [[Raman spectroscopy|Raman]] active ν<sub>3</sub>, ν<sub>2</sub> and the infrared out of phase mode ν<sub>27</sub>.<ref>{{cite journal|last=Dressel|first=M.|coauthors=Drichko, N.|title=Optical Properties of Two-Dimensional Organic Conductors: Signatures of Charge Ordering and Correlation Effects|journal=Chemical Reviews|year=2004|volume=104|pages=5689–5715|doi=10.1021/cr030642f|issue=11}}</ref> Their frequency is linearly associated to the charge per molecule giving the opportunity to determine the degree of disproportionation.
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| * The charge order transition is also a metal to insulator transition being observable in transport measurements as a sharp rise in the resistivity. Transport measurements are therefore a good tool to get first evidences of a possible charge order transition.
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| ==References==
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| {{reflist}}
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| [[Category:Electric and magnetic fields in matter]]
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| [[Category:Phase transitions]]
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