Imaging the Essential Role of Spin Fluctuations in High-$T_c$ Superconductivity

We have used scanning tunneling spectroscopy to investigate short-length electronic correlations in three-layer ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+\delta }$ (Bi-2223). We show that the superconducting gap and the energy ${\Omega }_{\mathrm{dip}}$, defined as the difference between the dip minimum and the gap, are both modulated in space following the lattice superstructure and are locally anticorrelated. Based on fits of our data to a microscopic strong-coupling model, we show that ${\Omega }_{\mathrm{dip}}$ is an accurate measure of the collective-mode energy in Bi-2223. We conclude that the collective mode responsible for the dip is a local excitation with a doping dependent energy and is most likely the ($\pi$, $\pi$) spin resonance.

Figure 1

STM topography and crystal structure of Bi-2223. (a) Surface topography of an underdoped sample ($T_c=103\pm 2\mathrm{K}$) acquired at 5 K in constant-current mode (tunneling current set to 0.2 nA and sample bias to 0.4 V). The Bi atoms exposed after cleaving the sample are observed as bright spots. The in-plane unit cell vectors of the ideal crystal structure, $\mathit{a}$ and $\mathit{b}$, and of the superstructure, ${\mathit{a}}_s$, are indicated. Lines of constant phase are depicted [see also (b)]. The so-called missing atom row [32] is observed at $\phi =(2n+1)\pi$. (b) Top panel: upper half of the ideal Bi-2223 unit cell. Bottom panel: schematic representation of the periodic modulation of atomic positions along the $\mathit{c}$ axis. The in-plane location is measured as a phase with $\phi =2n\pi$ corresponding to minima in the vertical position of the Bi atoms of the layer exposed by cleaving. (c) One-dimensional corrugation of the surface as a function of $\phi$. The profile was obtained by computing the average height within the box depicted in (a) and by displacing this box along the white line.

Figure 2

Local gap modulation and average spectra for nearly optimally doped Bi-2223. (a) Topographic image of a $17\times 17{\mathrm{nm}}^2$ region and (b) simultaneously measured local gap map ${\Delta }_p(\mathit{r})$. The white line is a spatial reference. (c) Topographic image and (d) simultaneous local gap map in a $46\times 46{\mathrm{nm}}^2$ field of view in a different region of the sample. The color scales in (b) and (d) are identical and bright areas depict larger gap values. (e) Average tunneling conductance spectra obtained from thousands of spectra with the same gap acquired over the field of view of (b). Measurements were performed at 5 K.

Figure 3

Spatial modulation of the gap and dip energy in nearly optimally doped Bi-2223. (a) Evolution of the $dI/dV$ spectra with the phase $\phi$: every curve results from averaging thousands of spectra with the same $\phi$. The green and red dots indicate the coherence peaks and the dip feature for negative bias, respectively. (b) Color-scale representation of ${\Delta }_p$ distributions for spectra with the same $\phi$: darker pixels correspond to more frequent gap values. The yellow curve depicts the average gap as a function of $\phi$, ${\overline{\Delta }}_p(\phi )$. (c) Negative-bias part of the ${\Delta }_p$-averaged spectra shown in Fig. 2c, offset by ${\Delta }_p^{-}$. Open symbols correspond to the experimental data whereas the lines are fits of the spectra with a strong-coupling model (see text). Black dots indicate the energy location of the dip in the experimental spectra and the dotted gray line is a guide to the eye. The spectra in (a) and (c) are vertically offset for clarity.

Figure 4

Spatial modulation of the collective-mode energy (CME) in nearly optimally doped Bi-2223. (a) Topographic image of a $30\times 15{\mathrm{nm}}^2$ region including the field of view of Fig. 2a. Inset: square modulus of the Fourier transform of the topography depicting the two peaks at the wave vectors of the superstructure $\pm 2\pi /a_s$. (b) Simultaneously acquired CME map. White pixels indicate spectra where the energy of the dip was not resolved. Inset: Fourier transform of the CME map. (c) Phase dependence of the CME ${\overline{\Omega }}_{\mathrm{dip}}(\phi )$ (green line) and gap ${\overline{\Delta }}_p(\phi )$ (blue line) averaged at constant $\phi$. The gap and the CME are locally anticorrelated. (d) Color-scale representation of ${\Omega }_{\mathrm{dip}}$ distributions for spectra with the same ${\Delta }_p$. The yellow curve corresponds to the average ${\Omega }_{\mathrm{dip}}$ as a function of ${\Delta }_p$, computed from the geometrical mean of each histogram. The open squares indicate the CME obtained from fitting the average spectra shown in Fig. 3c.