Fourier-transformed local density of states and tunneling into a $d$-wave superconductor with bosonic modes

We study the effects of the electronic coupling to bosonic modes on scanning tunneling microscopy (STM) into a $d$-wave superconductor. We propose to investigate these effects by means of a different technique: a Fourier transformed inelastic electron tunneling spectroscopy (FT-IETS). Specifically, in this technique, the Fourier spectrum of the energy derivative local density of states is addressed, which is proportional to the $(d^{2}I∕d{V}^2)(\mathbf{q},eV)$ characteristics measured in FT-IETS STM. We consider the role of the electron scattering due to a boson with the specific examples of the $B_{1g}$ phonon, the breathing mode phonon, and spin resonance at $(\pi ,\pi )$. It is found that the $B_{1g}$ mode with a highly anisotropic momentum-dependent coupling matrix element gives rise to well defined features in the Fourier spectrum, at the energy of mode plus gap, with a momentum transfer along the $\mathrm{Cu}\text{−}\mathrm{O}$ bond direction of cuprates. This result is in striking contrast to the cases of the coupling to other modes and also to the case of no mode coupling. The origin of this difference is explored in detail. A comparison with the recent STM experiments is briefly discussed.

Figure 1

Density of states (left column) and its energy derivative (right column) as a function of energy for a clean $d$-wave superconductor with the electronic coupling to $B_{1g}$-I, br-I, and $(\pi ,\pi )$ spin-resonance modes. The case of no mode coupling is also shown for comparison.

Figure 2

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

Figure 3

(Color) The imaginary and real parts of the single-particle and anomalous Green’s function, $A(\mathbf{k};E)$, $B(\mathbf{k};E)$, $J(\mathbf{k};E)$, and $K(\mathbf{k};E)$ [as defined by Eqs. (29, 30, 31, 32)], at the energy $E=-({\Delta }_0+{\Omega }_0)$ for the electronic coupling to the $B_{1g}$ phonon (first row), spin-resonance mode (second row), and the case of no mode coupling.

Figure 4

(Color) 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 $B_{1g}$ phonon mode. Here the energy has been measured by scaling ${\Delta }_0=30\mathrm{meV}$.