Tunneling spectra of strongly coupled superconductors: Role of dimensionality

We investigate numerically the signatures of collective modes in the tunneling spectra of superconductors. The larger strength of the signatures observed in the high-$T_c$ superconductors, as compared to classical low-$T_c$ materials, is explained by the low dimensionality of these layered compounds. We also show that the strong-coupling structures are dips (zeros in the $d^{2}I/d{V}^2$ spectrum) in $d$-wave superconductors, rather than the steps (peaks in $d^{2}I/d{V}^2$) observed in classical $s$-wave superconductors. Finally we question the usefulness of effective density of states models for the analysis of tunneling data in $d$-wave superconductors.

Figure 1

(a) Density of states $N\left(\omega \right)$ for superconducting electrons coupled to a $\left(\pi ,\pi \right)$ spin resonance, as a function of the $c$-axis hopping $t_z$. For $t_z=0$, the system is two dimensional while for $t_z=200\text{meV}$, it is three dimensional. (b) Normal-state bare DOS $N_0\left(\omega \right)$ for the same $t_z$ values. (c) Tunneling conductance for the same $t_z$ values. The temperature is $T=2\text{K}$ and a Gaussian broadening of 4 meV has been applied. The circles show the experimental data of Ref. 8 compared to the $t_z=0$ spectrum. The curves in (a), (b), and (c) are offset vertically for clarity.

Figure 2

Scattering rate $-\text{Im}{\widehat{\Sigma }}_{11}$ as a function of energy for several values of $t_z$ at (a) the nodal point ${\mathit{k}}_{\parallel }\approx \left(0.41,0.41\right)\pi /a$ and (b) the antinodal point ${\mathit{k}}_{\parallel }\approx \left(1,0.05\right)\pi /a$ of the Fermi surface shown in the insets. Curves are offset vertically for clarity. The dashed vertical lines delimit the energy range $\left|\omega \right|<{\Omega }_{\text{sr}}$ where inelastic scattering by the spin resonance is forbidden. The dotted vertical lines indicate $\pm \left({\Omega }_{\text{sr}}+{\Delta }_0\right)$.

Figure 3

Partial BCS density of states in two (2D) and three (3D) dimensions. The thin solid lines show the contribution coming from the $\left(\pi ,0\right)$ region of the two-dimensional Brillouin zone, shaded in black in the inset while the dashed lines show the contribution of the remainder of the zone (shaded in gray). The thick line is the total BCS DOS.

Figure 4

(a) Comparison of the $d$-wave and $s$-wave DOS for $t_z=0$ (2D) and $t_z=200\text{meV}$ (3D). The thick lines show $N\left(\omega \right)$ as in Fig. 1a. The thin lines show $N\left(\omega \right)$ computed with an $s$-wave gap of the same magnitude ${\Delta }_0=46\text{meV}$, and all other parameters unchanged. (b) Second-derivative tunneling spectrum, $d^{2}I/d{V}^2$, in the region of the strong-coupling signature at negative bias. In the $s$-wave case, there is a peak in $d^{2}I/d{V}^2$ close to the energy ${\Omega }_{\text{sr}}+{\Delta }_0$ while in the $d$-wave case, there is a sign change in 2D and no clear signature in 3D.

Figure 5

Normal-state DOS. The thin lines show the $T=2\text{K}$ DOS for ${\Delta }_0=0$; the thick lines show the $T=200\text{K}$ DOS for ${\Delta }_0=0$ and ${\Gamma }_{\text{sr}}=14\text{meV}$ (see text). All other parameters are as in Fig. 1a.

Figure 6

(Left panels) Real part (solid lines) and imaginary part (dashed lines) of the pairing function defined in Eq. (6) for $t_z=0$ (2D) and $t_z=200\text{meV}$ (3D). (Right panels) Comparison of the effective tunneling DOS $N_T\left(\omega \right)$ of Eq. (7) with the actual DOS $N\left(\omega \right)$ of Eq. (5). The dashed lines show $N_T\left(\omega \right)/N_0\left(\omega \right)$. In all graphs, the dotted vertical lines mark the energy $\pm \left({\Omega }_{\text{sr}}+{\Delta }_0\right)$.