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 ^[1][2] , 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.

Charge order patterns

This long range order phenomena was first discovered in magnetite (Fe₃O₄) by Verwey in 1939.^[3][4] 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.^[5] Periodic lattice distortions associated with charge order were later mapped in the manganite lattice to reveal striped domains containing topological disorder.^[6]

Theoretical description

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.^[7] The extended Hubbard model for a single chain including inter-site and on-site interaction V and U as well as the parameter ${\displaystyle \delta _{d}}$ for a small dimerization which can be typically found in the (TMTTF)₂X compounds is presented as follows:

${\displaystyle H=-t\sum _{i}\sum _{\sigma }\left(\left[1+\left(-1\right)^{i}\delta _{d}\right]c_{i,\sigma }^{\dagger }c_{i+1,\sigma }+h.c\right)+U\sum _{i}n_{i,\uparrow }n_{i,\downarrow }+V\sum _{i}n_{i},n_{i+1}}$

where t describes the transfer integral or the kinetic energy of the electron and ${\displaystyle c_{i,\sigma }^{\dagger }}$ and ${\displaystyle c_{i+1,\sigma }}$ are the creation and annihilation operator, respectively, for an electron with the spin ${\displaystyle \sigma =\uparrow ,\downarrow }$ at the ${\displaystyle i}$th or ${\displaystyle i+1}$th site. ${\displaystyle n_{i,\downarrow ,\uparrow }}$ denotes the density operator. For non-dimerized systems, ${\displaystyle \delta _{d}}$ can be set to zero Normally, the on-site Coulomb repulsion U stays unchanged only t and V can vary with pressure.

Examples

Detection of charge order

• 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 nuclei, for instance ¹³C as it is the case for TMTTF compounds, being active for NMR. The local probe nuclei are 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_Sp on the molecule. The charge order or charge disproportionation appear as a splitting or broadening of the certain feature in the spectrum.
• 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.^[18]
• 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 ν₃, ν₂ and the infrared out of phase mode ν₂₇.^[19] 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