Preeminent Role of the Van Hove Singularity in the Strong-Coupling Analysis of Scanning Tunneling Spectroscopy for Two-Dimensional Cuprate Superconductors

In two dimensions the noninteracting density of states displays a van Hove singularity (VHS) which introduces an intrinsic electron-hole asymmetry, absent in three dimensions. We show that due to this VHS the strong-coupling analysis of tunneling spectra in high-$T_c$ superconductors must be reconsidered. Based on a microscopic model which reproduces the experimental data with excellent accuracy, we elucidate the peculiar role played by the VHS in shaping the tunneling spectra, and show that more conventional analysis of strong-coupling effects can lead to severe errors.

Figure 1

Typical STM conductance of Bi2223 ($T_c=111\mathrm{K}$) at $T=2\mathrm{K}$ (dots). The data are an average of 15 spectra taken at different positions on the same sample, and having the same peak-to-peak gap ${\Delta }_p=38\mathrm{meV}$. Error bars give the standard deviation of this average. The inset shows the spectrum on a larger energy scale. Also shown are three model predictions (see text): free-electrons $d$-wave BCS (shaded area), $d$-wave BCS with realistic dispersion including VHS (dashed line), and $d$-wave BCS including VHS and coupling to the collective mode (full line). The total spectral weight in the energy range of the figure is the same for all curves.

Figure 2

High-resolution calculation of the DOS in the absence [$N_0(\omega )$, dashed line, shifted vertically] and in the presence [$N(\omega )$, full line] of the coupling to the ($\pi$, $\pi$) resonance. The energy and shape of the structures induced by this coupling (arrows) is an “inverted image” of the singularities present in the BCS DOS ($a$, $b$, $b^{\prime }$, $c$, $c^{\prime }$), shifted by the mode energy ${\Omega }_s$. The parameters are as in Fig. 1, except for ${\Delta }_0=40$, ${\xi }_M=-40$, and ${\Omega }_s=30\mathrm{meV}$. The inset shows the self-energy diagram with the full line representing the BCS Green’s function and the wavy line the spin susceptibility.

Figure 3

Evolution of the theoretical tunneling conductance upon varying ${\Omega }_s$ (a) or ${\xi }_M$ (b). The other model parameters are as in Fig. 1. Energies are measured relative to the peak maximum at $-{\Delta }_p$. The values of ${\Omega }_s$ and ${\xi }_M$ span a larger range than their possible variation in the experiment (Fig. 4), but were chosen so with the aim of amplifying the trends.

Figure 4

STM conductance spectra of Bi2223 ($T_c=111\mathrm{K}$) at $T=2\mathrm{K}$. Each curve is an average of several spectra taken at different locations on the same sample, all having the indicated peak-to-peak gap ${\Delta }_p$. ${\Omega }_{\mathrm{dip}}$ is the energy difference between the dip minimum (dot) and the peak maximum at negative bias, relative to which voltages are measured.