Effects of Pairing Potential Scattering on Fourier-Transformed Inelastic Tunneling Spectra of High-$T_c$ Cuprate Superconductors with Bosonic Modes

Recent scanning tunneling microscopy (STM) experimentally observed strong gap inhomogeneity in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (BSCCO). We argue that disorder in the pair potential underlies the gap inhomogeneity, and investigate its role in the Fourier-transformed inelastic tunneling spectra as revealed in the STM. We find that the random pair potential induces unique $\mathbf{q}$-space patterns in the local density of states (LDOS) of a $d$-wave superconductor. We consider the effects of electron coupling to various bosonic modes and find the pattern of LDOS modulation due to coupling to the $B_{1g}$ phonon mode to be consistent with the one observed in the inelastic electron tunnneling STM experiment in BSCCO. These results suggest strong electron-lattice coupling as an essential part of the superconducting state in high-$T_c$ materials.

Figure 1

The Fourier spectrum of the energy derivative LDOS at the energy $E=-({\Delta }_0+{\Omega }_0)$ for a $d$-wave superconductor with the electronic coupling to the $B_{1g}\mathrm{\text{−}}\mathrm{I}$, $br\mathrm{\text{−}}\mathrm{I}$, $B_{1g}\mathrm{\text{−}}\mathrm{II}$, $br\mathrm{\text{−}}\mathrm{II}$, spin resonace modes. For comparison, the spectrum is also shown for the case of no mode coupling.

Figure 2

The Fourier spectrum of the derivative of the LDOS is shown at the various values of the energy for the case of the electronic coupling to the $B_{1g}\mathrm{\text{−}}\mathrm{I}$ phonon mode. Here the energy has been measured by scaling ${\Delta }_0=30\mathrm{meV}$.