Strong-coupling analysis of scanning tunneling spectra in Bi${}_2$Sr${}_2$Ca${}_2$Cu${}_3$O${}_{10+\delta }$

We study a series of spectra measured in the superconducting state of optimally doped Bi${}_2$Sr${}_2$Ca${}_2$Cu${}_3$O${}_{10+\delta }$ (Bi-2223) by scanning tunneling spectroscopy. Each spectrum, as well as the average of spectra presenting the same gap, is fitted using a strong-coupling model taking into account the band structure, the BCS gap, and the interaction of electrons with the spin resonance. After describing our measurements and the main characteristics of the strong-coupling model, we report the whole set of parameters determined from the fits, and we discuss trends as a function of the gap magnitude. We also simulate angle-resolved photoemission spectra, and compare with recent experimental results.

Figure 1

Decrease of the tunnel current with increasing tip-sample distance $z$ for one of our junctions (circles). The solid line is an exponential fit. Inset: Same data on a log scale.

Figure 2

The complete experimental data set considered in this study. The raw Bi-2223 tunneling conductance spectra were grouped according to their peak-to-peak gap ${\Delta }_p$, and were normalized in order to have the same spectral weight in the range of the figure (black curves). The red curves are the average spectra, and the shaded blue curves show the standard deviation of the distribution of tunneling conductances at each bias. The vertical bars indicate a weak feature, possibly related to the VHS of the inner-layer band (see Sec. 3).

Figure 3

Evolution of the theoretical tunneling spectrum with varying five model parameters. The base parameters are (all energies in meV) $M=1$, $\Gamma =2$, $t_{1-5}=(-250,80,0,0,0)$, $\mu =-320$, ${\Delta }_0=40$, ${\Omega }_s=40$, ${\Gamma }_s=5$, ${\Delta }_q=1.5/a$, $\alpha =0.6$, and they correspond to the middle red curve in each series. The $\mathit{k}$-point mesh contains $1024\times 1024$ points. In (a), the chemical potential is varied to move the Van Hove singularity from $-80$ to $+80$ meV; the thin lines show the corresponding noninteracting DOS (i.e., with ${\Delta }_0=0$ and $\alpha =0$). (b), (c), and (d) show the result of varying the resonance energy ${\Omega }_s$, $\mathit{q}$-space width $\Delta q$, and the coupling $\alpha$, respectively. The effect of changing ${\Gamma }_s$ is illustrated by the thin curves in (b).

Figure 4

Same data as Fig. 3d on an expanded energy range covering the whole bandwidth. Part of the low-energy spectral weight is transferred to the band edges.

Figure 5

Series of experimental spectra with a peak-to-peak gap ${\Delta }_p=44$ meV (yellow circles), and fits to the strong-coupling model (red lines). The histograms show the distributions of fitted values for the energy of the VHS, the BCS gap, and the spin-resonance energy. Similar results are obtained for all series in Fig. 2. The standard deviations of the distributions reflect the sample inhomogeneity, and are given for all parameters in Table .

Figure 6

Average spectra of Fig. 2 (circles) and fits with the model described in Sec. 3 (red curves). The parameters are given in Table .

Figure 7

Average spectra of Fig. 2, with energies measured from one coherence peak. The dashed lines show the negative-bias spectrum, mirrored at positive energies to highlight the electron-hole asymmetry. The circles indicate the dip minimum, the triangles show the inflection point (peak in $d^{2}I/d{V}^2$), and the straight lines are guides to the eye.

Figure 8

Evolution of four properties with ${\Delta }_p$. The empty symbols show the values obtained by fitting the average spectra. The full symbols with error bars show the average and standard deviation of the distributions obtained by fitting all individual spectra. The crosses in (b) represent the peak-to-dip energy difference of the average spectra, corresponding to the circles in Fig. 7. (c) Renormalization factor at the M point, Eq. (19). (d) Nodal velocity.

Figure 9

Renormalization factor ${\lambda }_{\mathit{k}}$ along the renormalized Fermi surface. (a) Superconducting state: The parameters are those corresponding to the average spectra of Fig. 6 with the corresponding value of ${\Delta }_p$ written on each curve. Inset: Fermi surface for ${\Delta }_p=42$ meV, and definition of the Fermi-surface angle $\varphi$. (b) Normal state: The solid lines show the value of ${\lambda }_{\mathit{k}}$ obtained by setting $T=110$ K and ${\Delta }_0=0$, and keeping the other parameters unchanged. The dashed lines correspond to $T=110$ K, ${\Delta }_0=0$, and ${\Gamma }_s=25$ meV.

Figure 10

Spectral function calculated with the parameters fitted to the ${\Delta }_p=42$ meV spectrum in Fig. 6. (a) Along high-symmetry lines in the Brillouin zone. The color scale shows the variation of the spectral function from zero (white) to its maximum (yellow). The orange line is the noninteracting dispersion. The red line shows the BCS dispersion; the width of the line is proportional to the spectral weight. (b), (c), and (d) Cuts at three characteristic energies. The color scale is the same in all graphs.

Figure 11

Simulated ARPES intensity for optimally doped Bi-2223 at $T=10$ K. The spectral function of Fig. 10 was multiplied by the Fermi function, and filtered to mimic a momentum resolution of $0.05/a$ and an energy resolution of 18 meV. The color scale and the three momentum cuts correspond approximately to those of Fig. 2 in Ref. 87. The black lines show the noninteracting dispersion, and the red lines shows the quasiparticle dispersion (maximum of momentum distribution curves) for energies larger than the gap. The arrows indicate the “kink,” where the quasiparticle dispersion deviates from its low-energy linear behavior.