Strong coupling to multiple phonon modes in high-temperature superconductors

The spectra of the second derivative of tunneling current $d^{2}I∕d{V}^2$ in the high-temperature superconductors $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ show clear dip and peak features due to strong coupling to the bosonic modes mediating electron pairing. The energy positions of nearly all the peaks in $-d^{2}I∕d{V}^2$-like spectra match precisely with those in the phonon density of states obtained by inelastic neutron scattering. The results demonstrate that the bosonic modes mediating the electron pairing are phonons and that high-temperature superconductivity arises primarily from strong coupling to multiple phonon modes.

Figure 1

(Color online) (a) Normalized STM conductance $g_{NM}=(dI∕dV)∕{(dI∕dV)}_B$ versus energy $E=eV$ for a slightly overdoped $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ (YBCO) with $T_c=90\mathrm{K}$, where ${(dI∕dV)}_B$ is the background conductance at high biases. The STM conductance spectrum was taken on the (001) crystal face with a Pt-Ir tip at $4.2\mathrm{K}$ (Ref. 28). (b) The renormalized conductance $\overline{g}=g_{NM}∕N_{\mathit{BCS}}$ versus $\omega =eV-\Delta$ (the energy measured from the gap $\Delta$) for $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$. The solid line is a smoothed curve obtained using a cubic-spline interpolation.

Figure 2

(Color online) (a) $-\Delta (d\overline{g}∕d\omega )$ spectrum for the $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ crystal together with the phonon density of states $F\left(\omega \right)$ obtained from inelastic neutron scattering (Ref. 29). The left scale is for $-\Delta (d\overline{g}∕d\omega )$ and the right scale is for $F\left(\omega \right)$. The vertical dashed lines mark peak and/or shoulder features in $-\Delta (d\overline{g}∕d\omega )$. (b) $-2\Delta (d\overline{g}∕d\omega )$ spectrum together with the phonon density of states $F\left(\omega \right)$ for the conventional phonon-mediated superconductor Pb. Here, $\overline{g}=N_s∕\left(N_n{N}_{\mathit{BCS}}\right)$, $N_s$ is the superconducting density of states, and $N_n$ is the density of states in the normal state. The left scale is for $-2\Delta (d\overline{g}∕d\omega )$ and the right scale is for $F\left(\omega \right)$. The data of $N_s∕N_n$ are from Ref. 1, and the data of $F\left(\omega \right)$ are from Ref. 30. The vertical dashed lines mark peak and/or shoulder features in $-\Delta (d\overline{g}∕d\omega )$.

Figure 3

(Color online) (a) $-\Delta (d\overline{g}∕d\omega )$ spectrum for a ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ crystal with $T_c=93\mathrm{K}$ and the phonon density of states $F\left(\omega \right)$ at $296\mathrm{K}$. The high-resolution neutron data of $F\left(\omega \right)$ are reproduced from Fig. 1(a) of Ref. 31. The left scale is for $-4\Delta (d\overline{g}∕d\omega )$ and the right scale is for $F\left(\omega \right)$. The vertical dashed lines mark peak and/or shoulder features in $-4\Delta (d\overline{g}∕d\omega )$. (b) The $-d^2\left(\mathrm{Re}\Sigma \right)∕d{\omega }^2$ spectrum ($\Sigma$ is electron self-energy) (Ref. 32) and the $-d\overline{g}∕d\omega$ tunneling spectrum of BSCCO crystal. The $-d^2\left(\mathrm{Re}\Sigma \right)∕d{\omega }^2$ spectrum is reproduced from Ref. 32. Vertical dashed lines mark the peak positions in the $-d^2\left(\mathrm{Re}\Sigma \right)∕d{\omega }^2$ spectrum.