Two-channel point-contact tunneling theory of superconductors

We introduce a two-channel tunneling model to generalize the widely used BTK theory of point-contact conductance between a normal metal contact and superconductor. Tunneling of electrons can occur via localized surface states or directly, resulting in a Fano resonance in the differential conductance $G=dI/dV$. We present an analysis of $G$ within the two-channel model when applied to soft point contacts between normal metallic silver particles and prototypical heavy-fermion superconductors ${\mathrm{CeCoIn}}_5$ and ${\mathrm{CeRhIn}}_5$ at high pressures. In the normal state the Fano line shape of the measured $G$ is well described by a model with two tunneling channels and a large temperature-independent background conductance. In the superconducting state a strongly suppressed Andreev reflection signal is explained by the presence of the background conductance. We report Andreev signal in ${\mathrm{CeCoIn}}_5$ consistent with standard $d_{x^2-y^2}$-wave pairing, assuming an equal mixture of tunneling into [100] and [110] crystallographic interfaces, whereas in ${\mathrm{CeRhIn}}_5$ at 1.8 and 2.0 GPa the signal is described by a $d_{x^2-y^2}$-wave gap with reduced nodal region, i.e., increased slope of the gap opening on the Fermi surface. A possibility is that the shape of the high-pressure Andreev signal is affected by the proximity of a line of quantum critical points that extends from 1.75 to 2.3 GPa, which is not accounted for in our description of the heavy-fermion superconductor.

Figure 1

(a) Cartoon of all contributions to the standard tunneling Hamiltonian of a point contact with overlap integrals $\left(t,t_{\mathrm{loc}},v\right)$. (b) Schematics of surface defects generated by the PCS contact with atomically rough metallic tip and corresponding wave function amplitudes $|{\Psi }_{\mathrm{loc}}{|}^2$ of “sharp” and “broad” localized defect states. The tunneling conductance will be governed by the stronger overlap between the wave functions of the tip and sharp defect states compared to broad surface states. Here we suggest that the radial wave function of sharp localized states extends further into the open space, normal to the surface of the (heavy-fermion) superconductor.

Figure 2

Cartoon of the quasiparticle scattering processes at the NS interface between the metal tip and the superconductor (a). Note that the velocity of the retroreflected hole is antiparallel to that of the incoming electron, while both have the same momentum (direction of arrow). The importance of the sign change of the gap function, connecting trajectories of the scattered quasiparticle in S with momenta $p_{\mathrm{in}}$ and $p_{\mathrm{out}}$, is shown for (b) isotropic $s$, (c) $d_{xy}$, and (d) $d_{x^2-y^2}$ wave gap functions $\Delta \left(p\right)$ relative to the surface normal $n_S$.

Figure 3

Cartoon of point-contact tunneling process in the BTK regime with tunneling overlap integral $t$.

Figure 4

BTK differential conductance at $T=0$ is plotted for high $\left(t=1.0\right)$ and low $\left(0.1\right)$ transparency with $s$-wave gap ${\Delta }_0=0.2$ and in the normal state with ${\Delta }_0=0$ (dashed red line).

Figure 5

Cartoon of PCS tunneling through the localized state with energy level $E_0$ into the superconductor (SC) for the Lorentzian line shape regime with $t=0$. We show tunneling limits (a) $t_{\mathrm{loc}}\gg v$, (b) $t_{\mathrm{loc}}\sim v$, and (c) $t_{\mathrm{loc}}\ll v$.

Figure 6

Lorentzian differential conductance at $T=0$ for resonant $E_0=0$ (top) and off-resonant $E_0=3$ (bottom) localized state with $s$-wave gap ${\Delta }_0=0.2$ (black solid line) and normal state ${\Delta }_0=0$ (dashed red line). The columns depict from left to right three characteristic parameter sets, $t_{\mathrm{loc}}=1.38,v=0.32,t_{\mathrm{loc}}=v=1.0$, and $t_{\mathrm{loc}}=0.32,v=1.38$.

Figure 7

Cartoon of PCS tunneling through the localized state with energy level $E_0$ into the superconductor (SC) for the Fano line shape regime with $t\ne 0$. We show tunneling limits (a) $t_{\mathrm{loc}}\gg v$, (b) $t_{\mathrm{loc}}\sim v$, and (c) $t_{\mathrm{loc}}\ll v$.

Figure 8

Fano differential conductance at $T=0$ for resonant $E_0=0$ (top) and off-resonant $E_0=3$ (bottom) localized state with $s$-wave gap ${\Delta }_0=0.2$ (black solid line) and normal state ${\Delta }_0=0$ (dashed red line). The columns depict three characteristic parameter sets with increasing direct tunneling $t$ from left to right, $t=0.1,0.3,1.0$ and $t_{\mathrm{loc}}=v=1.0$ otherwise.

Figure 9

Fitting the Fano line shape of ${\mathrm{CeCoIn}}_5$ at ambient pressure in the normal state. The transparency $\mathcal{D}=0.210$ is in the tunneling rather than high transparency limit. The solid (black) lines are fits with constant background conductance $G_0=282.6{\left(k\Omega \right)}^{-1}$, which is shown as red-shaded background. The contact's pin code is $(t,t_{\mathrm{loc}},v,E_0)=$ (0.243, $0.066 \sqrt{\mathrm{eV}},-0.055 \sqrt{\mathrm{eV}},-0.53$ meV).

Figure 10

Fitting the Fano line shape of ${\mathrm{CeRhIn}}_5$ at 1.8 GPa in the normal state. The transparency $\mathcal{D}=0.532$ is in the intermediate transparency limit. The solid (black) lines are fits with background conductance $G_0\left(V\right)$, which is shown as red-shaded background. The contact's pin code is (0.270, $0.066 \sqrt{\mathrm{eV}},-0.065 \sqrt{\mathrm{eV}}$, 3.29 meV).

Figure 11

Fitting the Andreev signal of ${\mathrm{CeCoIn}}_5$ in the superconducting state at ambient pressure and at 1.31 K with $T_c=2.3$ K. We assumed isotropic $s$-wave gap (left) and $d$-wave gap (right). The $d$ wave is plotted for standard (black) and five times steeper (red) slope of the gap at nodes with equal weight of tunneling into [100] $\left(0^{\circ }\right)$ and [110] $\left({45}^{\circ }\right)$ surface orientations. Inset: angle dependence of gap function.

Figure 12

Fitting the Andreev signal of ${\mathrm{CeRhIn}}_5$ in the superconducting state at 1.8 GPa and at 1.16 K with $T_c\sim 2.0$ K and ${\Delta }_0=1.76$ meV. We assumed isotropic $s$-wave gap (left) and $d$-wave gap (right). The $d$ wave is plotted for standard (black) and five times steeper (red) slope of the gap at nodes with equal weight of tunneling into [100] and [110] surface orientations. Inset: angle dependence of gap function.

Figure 13

Fitting the Andreev signal of ${\mathrm{CeRhIn}}_5$ in the superconducting state at 2.0 GPa and at 1.18 K with $T_c\sim 2.0$ K and ${\Delta }_0=1.38$ meV. The contact's transparency is $\mathcal{D}=0.253$, which is in the tunneling rather than high transparency limit. We assumed isotropic $s$-wave gap (left) and $d$-wave gap (right). The $d$ wave is plotted for standard (black) and five times steeper (red) slope of the gap at nodes with equal weight of tunneling into [100] and [110] surface orientations. The contact's pin code is (0.253, $0.096 \sqrt{\mathrm{eV}},-0.075 \sqrt{\mathrm{eV}}$, 2.36 meV). Inset: angle dependence of gap function.