Density of states modulations from oxygen phonons in $d$-wave superconductors: Reconciling angle-resolved photoemission spectroscopy and scanning tunneling microscopy

Scanning tunneling microscopy (STM) measurements have observed modulations in the density of states (DOS) of a number of high-$T_c$ cuprates. These modulations have been interpreted in terms of electron-boson coupling analogous to the dispersion “kinks” observed by angle-resolved photoemission spectroscopy (ARPES). However, a direct a reconciliation of the energy scales and features observed by the two probes is presently lacking. In this paper we examine the general features of electron-boson coupling in a $d$-wave superconductor using Eliashberg theory, focusing on the structure of the modulations and the role of self energy contributions ${\lambda }_z$ and ${\lambda }_{\varphi }$. We identify the features in the DOS that correspond to the gap-shifted bosonic mode energies and discuss how the structure of the modulations provides information about an underlying pairing mechanism and the pairing nature of the boson. We argue that the scenario most consistent with the STM data is that of a low-energy boson mode renormalizing over a second dominant pairing interaction and we identify this low-energy mode as the out-of-phase bond buckling oxygen phonon. The influence of inelastic damping on the phonon-modulated DOS is also examined for the case of ${\text{Bi}}_2{\text{Si}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$. Using this simplified framework we are able to account for the observed isotope shift and anticorrelation between the local gap and mode energies. Combined, this work provides a direct reconciliation of the band-structure renormalizations observed by both ARPES and STM in terms of coupling to optical oxygen phonons.

Figure 1

(Color online) [(a1) and (a2)] The DOS $N\left(\omega \right)$ and corresponding real and imaginary parts of the gap function $\Delta \left(\omega \right)$, respectively, for an $s$-wave superconductor coupled to a single bosonic mode. [(b1) and (b2)] $N\left(\omega \right)$ and $\Delta \left(\omega \right)$ for a $d$-wave superconductor with ${\lambda }_z=1.6$ and ${\lambda }_{\varphi }=0.8{\lambda }_z$. The insets of panels (a1) and (b1), show $dN/d\omega$ and ${\alpha }^{2}F\left(\nu +{\Delta }_0\right)$ in order to highlight the correspondence between features in the DOS and peaks in the boson spectrum. The secondary structure in $\Delta \left(\omega \right)$ at $\omega ={\Delta }_0+2{\Omega }_0$ are due to self-energy effects due to two-phonon processes.

Figure 2

(Color online) (a) $N\left(\omega \right)$ for a $d$-wave superconductor where the $\Omega \sim 52\text{meV}$ mode (${\lambda }_{b,z}=1.6$ and ${\lambda }_{b,\varphi }=0$) renormalizes over a dominate mode associated with spin fluctuations with ${\Omega }_{sf}=260\text{meV}$ $\left({\lambda }_{sf,z}={\lambda }_{sf,\varphi }=1.6\right)$. Inset: $dN/d\omega$ and ${\alpha }^{2}F\left(\nu -{\Delta }_0\right)$ highlighting the correspondence between the low-energy mode and structure in $N\left(\omega \right)$. [(b) and (c)] $\Delta \left(\omega \right)$ and $Z\left(\omega \right)$ for this case. The vertical dashed lines indicate the energy of ${\Delta }_0+{\Omega }_0$ where ${\Omega }_0$ is the energy of the center of the low-energy spectra density [inset, panel (a)].

Figure 3

(Color online) The structures in the DOS $N\left(\omega \right)$ and gap function $\Delta \left(\omega \right)$ when the structure of the bare band has been retained. [(a1) and (a2)] $N\left(\omega \right)$ and $\Delta \left(\omega \right)$, respectively, for a $d$-wave superconductor coupled to a single low-energy mode which provides all of the pairing. [(b1) and (b2)] $N\left(\omega \right)$ and $\Delta \left(\omega \right)$ for the two mode case analogous to the case shown in Figs. 1c1 and 1c2. The insets of panels (a1) and (b1) show $dN/d\omega$ and the low-energy component of ${\alpha }^{2}F\left(\nu +{\Delta }_0\right)$ in order to show the correspondence between structure in $N\left(\omega \right)$ and peaks in ${\alpha }^{2}F\left(\nu \right)$. [(c1) and (c2)] The real and imaginary parts of $Z\left(\omega \right)$ calculated in the infinite- (solid) and finite-band (dashed) Eliashberg formalisms. Here, a single iteration of the Eliashberg equations has been assumed with coupling to an Einstein mode centered at $\Omega =52\text{meV}$ and assuming a $d$-wave gap with ${\Delta }_0=35\text{meV}$.

Figure 4

(Color online) (a) $N\left(\omega \right)$ and (b) $dN/d\omega$ for $\omega$ in the neighborhood of the low-energy boson renormalizations for various values of the low-energy mode’s spectral width ${\Gamma }_b$ (in meV).

Figure 5

(Color online) (a) The density of states calculated for coupling to the $B_{1g}$ phonon branch at $T=0$. Each spectra is calculated using a five-parameter tight-binding band structure and a $d$-wave gap with ${\Delta }_0=20$ (black), 40 (red), and 60 (blue) meV. [(b)–(d)] $dN/d\omega$ for the indicated gap size. The red dashed lines indicate the position of the roots corresponding to estimates for ${\Delta }_0$ and ${\Delta }_0+{\Omega }_0$, respectively.

Figure 6

(Color online) A waterfall plot of $N\left(\omega \right)$ calculated for doping values spanning the large gap to small gap regions. Each DOS was calculated for coupling the same mode used in Fig. 5. The gap values indicated in the legend denote the input values of ${\Delta }_0$. [(b)–(d)] $dN/d\omega$ for selected DOS presented in panel (a) for the hole side $\left(\omega >0\right)$ of the spectrum. The red arrows indicate the roots used to estimate the energies ${\Delta }_0$ and ${\Delta }_0+{\Omega }_0$. (e) The mode energy estimate obtained from the position of the local minima relative to the coherence peak. The blue data points correspond to data for ${}^{16}\text{O}$ simulations while the red data points correspond to ${}^{18}\text{O}$ simulations.