Theory of Andreev and shot noise spectroscopy for topological superconductors probed by $s$-wave superconducting tips

TL;DR

This work develops a comprehensive theoretical framework for Andreev and shot-noise spectroscopy of topological superconductors probed by $s$-wave STM tips. It employs a real-time Keldysh action to decompose tunneling into single-particle, Andreev, and Josephson channels, yielding analytical currents and noise expressions and a nonperturbative Dyson treatment to cover all tunneling regimes. The authors map surface Andreev bound states to distinctive features in $dI/dV$ spectra and Fano factors across representative $p$- and $d$-wave TS, providing concrete predictions for experimental STS signatures and guidelines on tip quality and tunneling transparency. The results offer a practical route to identify pairing symmetry and topological surface states in TS materials via high-resolution differential conductance and noise measurements.

Abstract

Scanning tunneling microscopy (STM) and spectroscopy (STS) with $s$-wave superconducting tips has been widely applied to probe exotic superconductors, but its potential for investigating topological superconductors remains unclear. In junctions between an $s$-wave superconductor and a topological superconductor, the dominant tunneling process is Andreev reflection, in which Cooper pairs from the $s$-wave superconductor tunnel as particle--hole excitations into the surface state of the topological superconductor. In this work, we theoretically investigate the fundamental properties of Andreev and shot noise spectroscopy on topological superconductors, focusing on the $dI/dV$ characteristics and current noise. We develop a real-time description of an effective tunneling action incorporating Andreev reflection processes in the Keldysh formalism and derive analytical expressions for the Andreev reflection current and the associated current noise. Furthermore, we perform numerical simulations for representative topological superconductors and provide a catalog of $dI/dV$ spectra and the Fano factor. Our results establish guidelines for probing topological superconductivity using STM with $s$-wave superconducting tips, and provide theoretical benchmarks for future STS experiments.

Figures (8)

• Figure 1: Schematic illustration of STM measurements with an $s$-wave superconducting tip for a topological superconductor. In the low-bias regime, the tunneling current is dominated by Andreev reflection processes, where Cooper pairs from the $s$-wave superconductor tunnel into the particles and holes within the surface states of the topological superconductor.
• Figure 2: (a) Schematic illustration of the full Hamiltonian, including electron tunneling between the left and right lattices. $x$ denotes the coordinate along the interface, and for spatially local tunneling, as in STM, the tunneling matrix element is taken as $T_{x,x'} = T \delta(x)\delta(x')$. (b) Circuit diagram of a superconducting junction.
• Figure 3: (a) The DOS of an $s$-wave superconductor with the Dynes factor $\delta = 0.01\Delta$. (b) Current-voltage ($I$-$V$) characteristics of an $s$-wave/$s$-wave superconducting junction for various effective transparencies $\alpha$. The corresponding values of $\alpha$ are indicated by the color bar in panel (d). (c) Differential conductance $dI/dV$ in the weak tunneling regime. Subharmonic gap structures due to MAR appear at $|eV| = 2\Delta/n$. (e) Single-particle tunneling process at $|eV|=2\Delta$. (f) Single Andreev reflection at $|eV| = \Delta$. (g) MAR at sub-gap bias. (h) Single particle tunneling mediated by residual states inside the gap at low bias. (i) Fano factor $F=P_N/2eI$.
• Figure 4: Probing the $p$-wave BW state (right electrode) using an $s$-wave superconducting STM tip (left electrode). (a) Surface DOS of the BW state and $s$-wave superconductor, with the Dyens parameters $\delta_L=0.01\Delta_L$ and $\delta_R = 0.025\Delta_L$. (b) The $dI/dV$ spectra for various effective transparencies $\alpha$. The corresponding values of $\alpha$ are indicated by the color bar in panel (d). (c) The $dI/dV$ spectra in the weak tunnel regime. (e) Single-particle tunneling between the coherence peak of the BW state and one of the $s$-wave superconductors. (f) Single particle tunneling at $eV = \Delta_L$. (g) Single Andreev reflection. (h) Single particle tunneling mediated by residual states inside the gap at low bias. (i) Fano factor $F=P_N/2eI$.
• Figure 5: Probing the chiral or helical state (right electrode) using an $s$-wave superconducting STM tip (left electrode). (a) Surface DOS of the chiral superconductor for the $(100)$, $(010)$, and $(001)$ planes, together with that of the $s$-wave superconductor. The Dynes parameters are set to $\delta_R = 0.025\Delta_L$ and $\delta_L = 0.01\Delta_L$. (b,c) Calculated $dI/dV$ spectra for tunneling into the $(100)$ and $(010)$ surfaces of the chiral state, for different values of the transparency $\alpha$. The values of $\alpha$ are indicated by the color bar in panel (e). (d) Corresponding Fano factor. (f,g) Calculated $dI/dV$ spectra for tunneling into the $(001)$ surfaces of the chiral state. (h) Corresponding Fano factor.