Local modulations of the spin-fluctuation-mediated pairing interaction by impurities in $d$-wave superconductors

We present a self-consistent real space formulation of spin-fluctuation mediated $d$-wave pairing. By calculating all relevant inhomogeneous spin and charge susceptibilities in real space within the random phase approximation (RPA), we obtain the effective pairing interaction and study its spatial dependence near both local potential and hopping impurities. A remarkably large enhancement of the pairing interaction may be obtained near the impurity site. We discuss the relevance of our result to inhomogeneities observed by scanning tunneling spectroscopy on the surface of cuprate superconductors.

Figure 1

(a) Local charge density $n_i$ in the normal phase prior to calculation of the effective pairing interaction. For the results presented here, the parameters are $U=2.2$, $t^{\prime }=-0.3$ (all energies are given in units of $t$), and doping $x=0.15$. A strong pointlike nonmagnetic impurity ($V_{\mathrm{imp}}=10$) is situated at site $(x_{\mathrm{imp}},y_{\mathrm{imp}})=(13,13)$. (b) Effective pairing interaction between nearest neighbors for the same system as in (a). For each site, the average potential to the four nearest neighbors is plotted.

Figure 2

(a) Local charge density $n_i$ in a system with a strong nonmagnetic impurity ($V_{\mathrm{imp}}=10$) and a spatially homogeneous superconducting pairing potential $V_{\text{eff}}=-0.43$. Parameters are $U=2.2$, $t^{\prime }=-0.3$, and doping $x=0.15$. (b) Same as (a) but in the case of the spatially inhomogeneous spin-fluctuation mediated pairing potential $V_{\text{eff}}(i,j)$ from Fig. 1b. (c) and (d) Local superconducting $d$-wave order parameter corresponding to the two cases in (a) and (b), respectively. (e) and (f) Local density of states at energy $E=0.78$ (close to maximum gap value) corresponding to the two cases in (a) and (b), respectively.

Figure 3

Maximum of the superconducting $d$-wave gap, ${\Delta }_{\max }$, as a function of Coulomb interaction $U$ for pointlike nonmagnetic impurities and a homogeneous system. For all runs, the system displays no antiferromagnetic order, since the critical $U$ in this system is $U_{c2}>2.4$. The inset shows the profile of the $d$-wave gap as a function of site $(x,y_{\mathrm{imp}})$.

Figure 4

Average of $U{\chi }_{\uparrow \downarrow }^0\left(\mathbf{Q}\right)$ around $\mathbf{Q}=(\pi ,\pi )$ for different values of $U$ as a function of site ${\mathbf{r}}_i=(x,y_{\mathrm{imp}})$. Black lines are $U\langle {\chi }_{\uparrow \downarrow }^0\left(\mathbf{Q}\right)\rangle$ for a homogeneous system, whereas colored lines are for systems with a nonmagnetic impurity at $(x_{\mathrm{imp}},y_{\mathrm{imp}})=(13,13)$.

Figure 5

(a) and (b) Effective pairing interaction between nearest neighbors. For the results presented here the parameters are $U=2.2$, $t^{\prime }=-0.3$, doping $x=0.15$. A reduced hopping impurity of strength [$-20$ % (a) and $-100$ % (b)] is situated at site $(x_{\mathrm{imp}},y_{\mathrm{imp}})=(13,13)$. (c) and (d) Local superconducting $d$-wave order parameter for the two impurity cases. (e) and (f) Average of $U{\chi }_{\uparrow \downarrow }^0\left(\mathbf{Q}\right)$ around $\mathbf{Q}=(\pi ,\pi )$ as a function of site ${\mathbf{r}}_i=(x,y_{\mathrm{imp}})$.

Figure 6

Real-space diagrams for longitudinal spin fluctuations to fifth order in $U$.

Figure 7

Real-space diagrams for transverse spin fluctuations to third order.