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Scientists from the USA, China and Japan have studied the Furrier transform scanning tunneling spectroscopy (FT-STS) of Bi-2212 crystals to find the relation between the features of r-space and k-space electronic structures of HTSCs: such as the modulation of the electronic local density of states (LDOS~g(r,ω)) observed by STM, and coherence peak, Fermi surface and d-wave gap determined by ARPES. The banana like shape of the constant quasiparticle energy near the Fermi surface of the HTSCs suggest that we should have 16 distinct ±q pairs of maximal intensity in the Fourier transform of r-space local density of states, corresponding to each vector connected the end of each banana (Fig. 1). To detect such a pattern the atomic resolution is requires for all measurements and the filed of view at the measurements should be L>450 Å (∆q=2π/L to be enough small). Fig. 2 shows such a resolution.

From the analysis of the data, it has been concluded that the Fermi surface and the amplitude and angular dependence of the gap are agreement with the ARPES results (Fig. 3b&c). This indicates the clear link between the r-space and k-space characteristics of the electronic states, because the matrix elements for tunneling and photoemission are quite different. ∆(θ_k)=∆₀[0.818 cos(2θ_k)+0.182 cos(6θ_k)] has been derived from this study with the gap amplitude of ∆₀=39.3 meV.

The authors have discussed some complementary results from this study:

• The g(r,ω) data are more relevant to the proposals that the LDOS modulation might result from the existence of a charge-density-wave order parameter with fixed q-vector, (for example stripes) rater than the quantum interference models.

• Although the g(q,ω) are not completely symmetric relative to the bias potential (for example ω=+14 and -14 in Fig. 2), the deduced ∆(k) from the same positive and negative bias energy is indistinguishable within errors. This is provides evidence that the quasiparticles of this system are particle-hole superposition, consistent with the Bogoliubov description.

• The strong response in g(q,ω) for all ω at q<0.15 may reflect long-wavelengths inhomogeneity  in the integrated LDOS. This reveals an obvious candidate for weak but ubiquitous potential scattering that could produce the LDOS modulations.

• For a given ω the Umklapp LDOS modulation signal is localized to the nanoscale regions where ω is equal to the local gap value (Fig. 4) whish is different from the pattern for usual crystals. This implies strong nanoscale spatial variation in the quasiparticle dispersion near k=(π/a,0), and therefore significant scattering.

Ref.: K. McElroy, R. W. Simmonds, J. E. Hoffman, D.-H. Lee, J. Orenstein, H. Eisaki, S. Uchidak & J. C. Davis; Nature 422 (2003) 592.