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112 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES
5.2 Checkerboard Pattern in the cuprates
The phase diagram of the hole-doped cuprates shows mainly four different electronic
phases existing at low temperature. Among them are the Mott insulator
phase, the superconducting phase, and the more usual Landau Fermi liquid phase
(or metallic phase). A fourth phase, of controversial nature, occurs at light doping,
located above the superconducting phase and was named the pseudogap
regime. This later regime shows peculiar electronic phenomena, prompting for
proposals that it might contain a hidden charge or spin order.
With an increasing number of materials and novel experimental techniques
of constantly improving resolution, novel features in the global phase diagram of
high-T c cuprate superconductors have emerged. One of the most striking is the
observation, in some systems, of a form of local electronic ordering, especially
around 1/8 hole doping.
The recent scanning tunneling microscopy/spectroscopy (STM/STS) experiments
of underdoped Bi 2 Sr 2 CaCu 2 O 8+δ (BSCO) in the pseudogap state have
shown evidence of energy-independent real-space modulations of the low-energy
density of states (DOS) [122, 123, 124] with a spatial period close to four lattice
spacings. A similar spatial variation of the electronic states has also been observed
in the pseudogap phase of Ca 2−x Na x CuO 2 Cl 2 single crystals (x =0.08 ∼ 0.12)
by similar STM/STS techniques [125].
Low-temperature (T = 100mK) electronic structure imaging studies of a
lightly hole-doped copper oxide Ca 2−x Na x CuO 2 Cl 2 using Tunneling spectroscopy
were carried on (see Fig.5.1). The doping was fixed in these studies by using samples
with Na concentrations x =0.08, 0.10, 0.12, which leads to superconducting
T c =0, 15, 20K respectively.
Indeed, for the considered samples, the spectrum exhibits a V shaped energy
gap centered at the Fermi energy. It was mainly found that the states within this
gap undergo spatial modulations with a checkerboard structure, and has 4 × 4
CuO 2 in the unit-cell. The structure changes only weakly inside the V-shaped
energy gap 1 2 .
Although it is not clear yet whether such phenomena are either generic features
or really happening in the bulk of the system, they nevertheless raise important
theoretical questions about the stability of such structures in the framework
of microscopic strongly correlated models.
Moreover, intricate atomic-scale electronic structure variations also exist within
the checkerboard larger structure. Therefore, the experimental results would be in
1 Note that the first observation was made around a vortex core in J.E. Hoffman, E.W. Hudson,K.M.Lang,V.Madhavan,H.Eisaki,S.Uchida,andJ.C.Davis,Science295,
466 (2002).
2 Note that the energy-dependent spatial modulations of the tunneling conductance of optimally
doped BSCO can be understood in terms of elastic scattering of quasiparticles. See
J.E. Hoffman, K. McElroy, D.H. Lee, K.M. Lang, H. Eisaki, S. Uchida, and J.C. Davis Science
297 1148 (2002).