Charge ordering

Charge ordering (CO) is a (first- or second-order)

This long range order phenomena was first discovered in magnetite (Fe₃O₄) by Verwey in 1939.^[3][4] He observed an increase of the

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

Organic conductors

Organic conductors consist of

The model system of one-dimensional conductors is the Bechgaard-Fabre salts family, (TMTTF)₂X and (TMTSF)₂X, 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)₂X.^[10] The transition temperature T_CO for the TMTTF subfamily was registered over two order of magnitudes for the centrosymmetric anions X = Br, PF₆, AsF₆, SbF₆ and the non-centrosymmetric anions X= BF₄ and ReO₄.^[11] In the middle of the eighties, a new "structureless transition" was discovered by Coulon et al.^[12] 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 ¹³C-NMR^[13] 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.

Quasi-two-dimensional organic conductors

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_C. 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.
At the turn of the 20th century, first

The alkali metal oxides rubidium sesquioxide (Rb₄O₆) and caesium sesquioxide (Cs₄O₆) display charge ordering.^[17]

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