Charge ordering

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, charges are localized on different sites leading to a disproportionation and an 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. This long range order phenomena was first discovered in magnetite (Fe 3 O 4 ) by Verwey in 1939. He observed an increase of the electrical resistivity by two orders of magnitude at T CO =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 Paul Attfield with the results published in Nature. Periodic lattice distortions associated with charge order were later mapped in the manganite lattice to reveal striped domains containing topological disorder.

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, charges are localized on different sites leading to a disproportionation and an 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. This long range order phenomena was first discovered in magnetite (Fe 3 O 4 ) by Verwey in 1939. He observed an increase of the electrical resistivity by two orders of magnitude at T CO =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 Paul Attfield with the results published in Nature. Periodic lattice distortions associated with charge order were later mapped in the manganite lattice to reveal striped domains containing topological disorder. 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} where t describes the transfer integral or the kinetic energy of the electron and c^{\dagger}_{i,\sigma} and c_{i+1,\sigma} are the creation and annihilation operator, respectively, for an electron with the spin \sigma = \uparrow , \downarrow at the i th or i+1 th site. n_{i,\downarrow, \uparrow} denotes the density operator. For non-dimerized systems, \delta_d can be set to zero Normally, the on-site Coulomb repulsion U stays unchanged only t and V can vary with pressure. ==Examples== The transition temperature T CO for the TMTTF subfamily was registered over two order of magnitudes for the centrosymmetric anions X = Br, PF 6 , AsF 6 , SbF 6 and the non-centrosymmetric anions X= BF 4 and ReO 4 . In the middle of the eighties, a new "structureless transition" was discovered by Coulon et al. conducting transport and thermopower measurements. They observed a suddenly rise of the resistivity and the thermopower at T CO 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 13 C-NMR and dielectric measurements. Different measurements under pressure reveal a decrease of the transition temperature T CO 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. At the turn of the 20th century, first NMR measurements on the θ-(BEDT-TTF) 2 RbZn(SCN) 4 compound uncovered the known metal to insulator transition at T CO = 195 K as an charge order transition. ===Alkali metal oxides=== The alkali metal oxides rubidium sesquioxide (Rb 4 O 6 ) and caesium sesquioxide (Cs 4 O 6 ) display charge ordering. The 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. 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 spectrum depends on its symmetry. In the case of BEDT-TTF, the most sensitive ones are the Raman active ν 3 , ν 2 and the infrared out of phase mode ν 27 . Their frequency is linearly associated to the charge per molecule giving the opportunity to determine the degree of disproportionation. 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. ==References==

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