Physics:Charge ordering

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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, 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 TCO=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 [math]\displaystyle{ \delta_d }[/math] for a small dimerization which can be typically found in the (TMTTF)2X compounds is presented as follows:

[math]\displaystyle{ 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]

where t describes the transfer integral or the kinetic energy of the electron and [math]\displaystyle{ c^{\dagger}_{i,\sigma} }[/math] and [math]\displaystyle{ c_{i+1,\sigma} }[/math] are the creation and annihilation operator, respectively, for an electron with the spin [math]\displaystyle{ \sigma = \uparrow , \downarrow }[/math] at the [math]\displaystyle{ i }[/math]th or [math]\displaystyle{ i+1 }[/math]th site. [math]\displaystyle{ n_{i,\downarrow, \uparrow} }[/math] denotes the density operator. For non-dimerized systems, [math]\displaystyle{ \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.

Examples

Organic conductors

Organic conductors consist of donor and 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, antiferromagnetic state, superconductivity, charge-density wave to name only some of them.[8][9]

Quasi-one-dimensional organic conductors

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 TCO 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 TCO 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 TCO 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.

Anion X TCO (K)
(TMTTF)2Br 28
(TMTTF)2PF6 70
(TMTTF)2AsF6 100.6
(TMTTF)2SbF6 154
(TMTTF)2BF4 83
(TMTTF)2ReO4 227.5
(DI-DCNQI)2Ag 220
TTM-TTPI3 120

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 TC. 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 NMR measurements on the θ-(BEDT-TTF)2RbZn(SCN)4 compound uncovered the known metal to insulator transition at TCO= 195 K as an charge order transition.[14]

Compound TCO (K)
α-(BEDT-TTF)2I3 135
θ-(BEDT-TTF)2TlCo(SCN)4 240
θ-(BEDT-TTF)2TlZn(SCN)4 165
θ-(BEDT-TTF)2RbZn(SCN)4 195
θ-(BEDT-TTF)2RbCo(SCN)4 190

Transition metal oxides

The most prominent transition metal oxide revealing a CO transition is the magnetite Fe3O4 being a mixed-valence oxide where the iron atoms have a statistical distribution of Fe3+ and Fe2+ 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.[15]

Compound[16] TCO (K)
Y0.5NiO3 582
YBaCo2O5 220
CaFeO3 290
Ba3NaRu2O9 210
TbBaFe2O5 282
Fe3O4 123
Li0.5MnO2 290
LaSrMn3O7 210
Na0.25Mn3O6 176
YBaMn2O6 498
TbBaMn2O6 473
PrCaMn2O6 230
α'-NaV2O5 34

Alkali metal oxides

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 χSp 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

  1. ↑ Wise, W.D. (2008). "Charge-density-wave origin of cuprate checkerboard visualized by scanning tunnelling microscopy". PNAS 4 (9): 696–699. doi:10.1038/nphys1021. Bibcode: 2008NatPh...4..696W. 
  2. ↑ Savitzky, B. (2017). "Bending and breaking of stripes in a charge ordered manganite". Nature Communications 8 (1): 1883. doi:10.1038/s41467-017-02156-1. PMID 29192204. Bibcode: 2017NatCo...8.1883S. 
  3. ↑ Verwey, E.J.W. (1939). "Electronic conduction of magnetite (Fe3O4) and its transition point at low temperatures". Nature 144 (3642): 327–328. doi:10.1038/144327b0. Bibcode: 1939Natur.144..327V. 
  4. ↑ Verwey, E.J.W.; Haayman, P.W. (1941). "Electronic conductivity and transition point of magnetite (Fe3O4)". Physica 8 (9): 979–987. doi:10.1016/S0031-8914(41)80005-6. Bibcode: 1941Phy.....8..979V. 
  5. ↑ Senn, M. S.; Wright, J. P.; Attfield, J. P. (2011). "Charge order and three-site distortions in the Verwey structure of magnetite". Nature 481 (7380): 173–6. doi:10.1038/nature10704. PMID 22190035. Bibcode: 2012Natur.481..173S. https://www.pure.ed.ac.uk/ws/files/10796489/Charge_order_and_three_site_distortions_in_the_Verwey_structure_of_magnetite.pdf. 
  6. ↑ El Baggari, I. (2017). "Nature and evolution of incommensurate charge order in manganites visualized with cryogenic scanning transmission electron microscopy". PNAS 115 (7): 1–6. doi:10.1073/pnas.1714901115. PMID 29382750. Bibcode: 2018PNAS..115.1445E. 
  7. ↑ Yoshioka, H.; Tsuchuuzu, M; Seo, H. (2007). "Charge-Ordered State versus Dimer-Mott Insulator at Finite Temperatures". Journal of the Physical Society of Japan 76 (10): 103701. doi:10.1143/JPSJ.76.103701. Bibcode: 2007JPSJ...76j3701Y. 
  8. ↑ Ishiguro, T. (1998). Organic Superconductors. Berlin Heidelberg New York: Springer-Verlag. ISBN 978-3-540-63025-8. https://archive.org/details/organicsupercond0000ishi. 
  9. ↑ Toyota, N. (2007). Low-dimensional Molecular Metals. Berlin Heidelberg: Springer-Verlag. ISBN 978-3-540-49574-1. 
  10. ↑ Jeromé, D. (1991). "The Physics of Organic Superconductors". Science 252 (5012): 1509–1514. doi:10.1126/science.252.5012.1509. PMID 17834876. Bibcode: 1991Sci...252.1509J. 
  11. ↑ Nad, F.; Monceau, P. (2006). "Dielectric response of the charge ordered state in quasi-one-dimensional organic conductors". Journal of the Physical Society of Japan 75 (5): 051005. doi:10.1143/JPSJ.75.051005. Bibcode: 2006JPSJ...75e1005N. 
  12. ↑ Coulon, C.; Parkin, S.S.P.; Laversanne, R. (1985). "STRUCTURELESS TRANSITION AND STRONG LOCALIZATION EFFECTS IN BIS-TETRAMETHYLTETRAHTHIAFULVALENIUM SALTS [(TMTTF)2X]". Physical Review B 31 (6): 3583–3587. doi:10.1103/PhysRevB.31.3583. PMID 9936250. Bibcode: 1985PhRvB..31.3583C. 
  13. ↑ Chow, D.S. (2000). "Charge Ordering in the TMTTF Family of Molecular Conductors". Physical Review Letters 85 (8): 1698–1701. doi:10.1103/PhysRevLett.85.1698. PMID 10970592. Bibcode: 2000PhRvL..85.1698C. 
  14. ↑ Miyagawa, K.; Kawamoto, A.; Kanoda, K. (2000). "Charge ordering in a quasi-two-dimensional organic conductor". Physical Review B 62 (12): R7679. doi:10.1103/PhysRevB.62.R7679. Bibcode: 2000PhRvB..62.7679M. 
  15. ↑ Rao, C. N. R. (1997). "Materials Science: Charge Ordering in Manganates". Science 276 (5314): 911–912. doi:10.1126/science.276.5314.911. 
  16. ↑ Attfield, J.P. (2006). "Charge ordering in transition metal oxides". Solid State Sciences 8 (8): 861–867. doi:10.1016/j.solidstatesciences.2005.02.011. Bibcode: 2006SSSci...8..861A. 
  17. ↑ Colman, Ross H.; Okur, H. Esma; Kockelmann, Winfried; Brown, Craig M.; Sans, Annette; Felser, Claudia; Jansen, Martin; Prassides, Kosmas (2019-10-21). "Elusive Valence Transition in Mixed-Valence Sesquioxide Cs4O6". Inorganic Chemistry (American Chemical Society (ACS)) 58 (21): 14532–14541. doi:10.1021/acs.inorgchem.9b02122. ISSN 0020-1669. PMID 31633914. 
  18. ↑ Coulon, C.; Lalet, G.; Pouget, J.-P.; Foury-Leylekian, P.; Moradpour, A. (2007). "Anisotropic conductivity and charge ordering in (TMTTF)(2)X salts probed by ESR". Physical Review B 76 (8): 085126. doi:10.1103/PhysRevB.76.085126. Bibcode: 2007PhRvB..76h5126C. 
  19. ↑ Dressel, M.; Drichko, N. (2004). "Optical Properties of Two-Dimensional Organic Conductors: Signatures of Charge Ordering and Correlation Effects". Chemical Reviews 104 (11): 5689–5715. doi:10.1021/cr030642f. PMID 15535665. 



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