Fingerprinting superconductors by disentangling Andreev and quasiparticle currents across tunable tunnel junctions

Abstract

Tunneling Andreev reflection (TAR) spectroscopy offers a powerful new approach to fingerprint superconducting pairing symmetry at the atomic scale. By leveraging the exponential sensitivity of excess tunneling decay rate to Andreev reflection, TAR robustly distinguishes between s-wave, d-wave, and more complex order parameters, overcoming limitations of traditional conductance-based techniques. Here, using atomistic superconducting transport simulations, we show that the additivity of excess decay rate enables clear separation of Andreev and quasiparticle currents. In particular, we reveal how their competition as well as higher-order scattering processes shape both the decay rate spectra and their dependence on the coupling strength. We show that this phenomenology stems from the fact that Andreev reflection dominates mid-gap conductance for s-wave superconductors, it is suppressed for the d-wave, and it coexists with quasiparticle tunneling in sign-changing symmetries if the expectation value for the superconducting gap remains finite. These distinct spectral fingerprints pave the way for atomically resolved identification of unconventional superconducting states.

Figures (5)

• Figure 1: Simulated conductance and $\kappa$-spectra for s-wave superconductor at weak to intermediate coupling strength. (a) Schematic of single quasiparticle tunneling (top) and Andreev reflection (bottom) between a metallic tip and a superconducting sample. (b) Conductance spectroscopy for a range of coupling. (c) Conductance scaling as a function of the normal conductance, $G_N/G_0$ (measured outside the superconducting gap), for tunneling ($G_e$) and Andreev reflection ($G_A$). (d) $\kappa/\kappa_{N}$ calculated for energies inside the gap $(E=0)$, outside the gap (E=4$\Delta$) and separately for Andreev reflection outside the gap ($\kappa_A$). (e) Excess decay rate ($\kappa/\kappa_{N}$) spectra calculated from (b). Red highlights spectral reconstruction using Eq. \ref{['eq:kappaadd']} from the main text. Dotted line shows the scaling of only the Andreev reflection channel. (f) Partitioning of the conductance data in (b) into contributions from tunneling (dotted) and Andreev reflection (solid). Colorscales in (b), (d) and (e) follow the values of the coupling strength. $G_0$ is the conductance quantum.
• Figure 2: Near-contact conductance and $\kappa$-spectra for s-wave superconductor at strong coupling: (a) Conductance spectroscopy as a function of increasing coupling strength partitioned into contributions from tunneling (dotted) and Andreev reflection (solid). (b) Corresponding $\kappa/\kappa_{N}$ spectroscopy. Most notable, $\kappa/\kappa_N$ significantly exceeds the anticipated value of 2, and can reach as high as 4 as the mid-gap condutance approaches 2$G_0$. (c) Scaling of the conductance due to tunneling ($G_e$) and Andreev reflection ($G_A$) for several energies as a function of normal conductance (measured outside superconducting gap). Colorscales in (a) and (b) reflect the different tip-substrate coupling strengths $T=|\gamma|^2$. $G_0$ is the conductance quantum.
• Figure 3: (a) Conductance for tunneling and Andreev reflection as a function of the tip-substrate coupling strength $T=\gamma^2$. The conductance for all channels grows slower when approaching unity coupling due to an effective reduction of the height of the tunneling barrier. The calculated decay rate (b) directly shows this effect for both tunneling (orange) and Andreev reflection (brown). However, the excess decay rate for Andreev reflection is maintained for all coupling strengths. The excess ratio is exactly 2/1 for Andreev reflection beyond the superconducting gap (brown), and is larger than 2/1 for Andreev reflection in the middle of the gap (blue). Larger than 2 ratio can be rationalized by introducing higher order Andreev reflections. For example, the dashed teal line in (b) corresponds to $\kappa$ calculated from a fictitious conductance mechanism using the formula $C_1 G_e^2 + C_2 G_e^4$ for Andreev reflection. By construction, this fictitious mechanism is a weighted sum of single and multiple (higher order) Andreev reflections whose schematic trajectories are depicted in (c). The value of the coefficients $C_1$ = 5.5 and $C_2$ = 11.6 were determined from least-squares polynomial fit to the blue curve in the conductance range up to 0.8$G_0$. Although the agreement is only qualitative, the higher order contributions clearly increase the decay rate beyond the excess ratio of 2, similar to the differences between the blue and brown curves in (b)
• Figure 4: $\kappa$-spectroscopy for a d-wave superconductor, where tunneling Andreev reflection is suppressed: (a) Fermi surface for a two-Fermi-surface electron structure with the d-wave order parameter (b) Conductance spectroscopy as a function of increasing coupling strength. The Andreev contribution to this conductance is below numerical accuracy, so that calculated conductance is from single quasiparticle tunneling at any energy; (c) $\kappa/\kappa_{N}$ spectra calculated from conductance in (b). (d) The expectation value of the superconducting gap function ($|\langle \Delta \rangle| = |\int \Delta(\mathbf k) d^2 \mathbf k|$) across the Brillouin zone is zero at all energies (blue curve), while the average absolute value of the SC gap ($\langle | \Delta | \rangle = \int |\Delta(\mathbf k) | d^2 \mathbf k$) remains finite. (e) Conductance and (f) unnormalized decay rate $\kappa$ as a function of the tip-substrate coupling $T = |\gamma^2|$, at energies in the middle $(E=0)$ and outside the $(E=5\Delta)$ the superconducting gap. Mid-gap (blue) conductance saturates slightly slower than outside the gap (orange) when approaching contact, which is effectively captured by the slight differences of the unnormalized $\kappa$ (e) and $\kappa/\kappa_{N}$ spectroscopy (c). Colorscales in (b) and (c) follow the values of the tip-substrate coupling strength $T=|\gamma|^2$.
• Figure 5: $\kappa$-spectroscopy for $s_{\pm}$ superconductor, where there exists competition between quasiparticle tunneling and Andreev reflection as a function of energy: (a) Fermi surface of a two-Fermi-surface superconductor with the $s_{\pm}$ superconducting order parameter. (b) Conductance spectroscopy as a function of increasing coupling strength. Solid lines are the Andreev component ($G_A$), and dotted lines are quasiparticle tunneling ($G_e$); (c) $\kappa/\kappa_{N}$ spectra calculated from conductance in (b). (d) Expectation value of both superconducting gap ($|\langle \Delta \rangle|$) and gap magnitude $\langle |\Delta |\rangle$ remain finite at all energies. Scaling of conductance (e) and unnormalized decay rate $\kappa$ (f) as a function of the tip-substrate coupling $T = |\gamma^2|$, at several energies across the gap.