Extinction of impurity resonances in large-gap regions of inhomogeneous $d$-wave superconductors

Impurity resonances observed by scanning tunneling spectroscopy in the superconducting state have been used to deduce properties of the underlying pure state. Here we study a long-standing puzzle associated with these measurements, the apparent extinction of these resonances for Ni and Zn impurities in large-gap regions of the inhomogeneous ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+x}$ superconductor. We calculate the effect of order parameter and hopping suppression near the impurity site and find that these two effects are sufficient to explain the missing resonances in the case of Ni. There are several possible scenarios for the extinction of the Zn resonances, which we discuss in turn; in addition, we propose measurements which could distinguish among them.

Figure 1

(Color online) Schematic picture of a small-gap region. Circles are ionic sites, $X$ represents an impurity, and the bonds represent the absolute values of the bond order parameter $\Delta$ on bonds in the bulk (top) and in a small-gap patch (bottom).

Figure 2

(Color online) LDOS in a $d$-wave superconductor with a Ni impurity at the center of a $5\times 5$ patch of [(a)–(b)] suppressed $\left(\delta \Delta =-0.05t\right)$, [(c)–(d)] constant $\left(\delta \Delta =0\right)$, and [(e)–(f)] enhanced $\left(\delta \Delta =+0.05t\right)$ superconducting gap. The Ni potentials used for this study are $V_3=-0.8t$ and $V_0=0.6t$. The figures in the left column [(a), (c), and (e)] show spectra on the IS itself, while right column [(b), (d), and (f)] represents NN sites. In each case, the solid black curve shows the LDOS on the site in question in the absence of the impurity, the solid color curve indicates the total LDOS with the impurity, and the dashed and dotted lines show the spin resolved LDOS for up and down spins, respectively.

Figure 3

(Color online) Same as Fig. 2 but including local impurity-induced reduced hopping $\delta t=-0.24t$.

Figure 4

LDOS real-space pattern $N\left(\mathbf{r},\pm {\Omega }_0\right)$ at the resonance frequency for Ni impurities modeled as described in the text. Left panel: no suppression of hopping by impurity assumed. Subpanels (a)–(b), (c)–(d), and (e)–(f) show the resonance pattern for gap sizes in the surrounding $5\times 5$ patch corresponding to $\delta \Delta =+0.05t$, 0, and $-0.05t$, respectively. Left column [(a), (c), and (e)] and right column [(b), (d), and (f)] correspond to negative and positive resonance frequencies $\pm {\Omega }_0$, respectively. Right panel: same as left panel but for including additional modulations of the hopping $\delta t=-0.24t$.

Figure 5

(Color online) LDOS at IS and NN sites plotted in left and right panels, respectively, for various gap modulations $\delta \Delta =-0.05t$, 0, and $+0.05t$ over a $5\times 5$ patch around the impurity described by an onsite potential $V_3=2.5t$. Black curves show the LDOS in the absence of an impurity, while colored curves correspond to the situation where an impurity is present. Hoppings are not modulated, $\delta t=0$

Figure 6

(Color online) Same as Fig. 5 but with a filter mechanism applied as described in text.

Figure 7

(Color online) LDOS for a Zn impurity represented by a potential $V_3=5t$ and calculated with a constant scattering rate $\eta =0.01t$ where the hopping is suppressed using a Gaussian distribution with the maximum $\delta t$ on the four bonds around the impurity and a decay length of one lattice spacing. (a), (c), and (e) show the LDOS at the IS site, and (b), (d), and (f) show the LDOS at the NN site. (a) and (b) show the LDOS in a region of constant gap $\left(\delta \Delta =0\right)$ for different hopping suppression where the values of $\delta t$ shown vary from 0 (top) to $-t$ (bottom) in steps of $-0.1t$. (c) and (d) show the LDOS in a small-gap region without suppression of the hopping, while (e) and (f) show it in large-gap region with maximum suppression of the hopping $\left(\delta t=-t\right)$.

Figure 8

(Color online) LDOS of a phase impurity with $\alpha =1.5$ as a function of the change in the local gap $\delta \Delta$ within the $5\times 5$ patch in which it lies. The red curve is similar to the results in Ref. 18, which reported the LDOS near a phase impurity in a homogeneous superconductor. The inset shows the LDOS without the phase impurity (same color code).