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

This paper theoretically investigates Andreev reflection and shot noise spectroscopy in junctions between $s$-wave superconducting tips and topological superconductors by deriving analytical expressions and numerical simulations to establish guidelines for identifying topological superconductivity via STM/STS experiments.

The Gist

Imagine you are trying to figure out what a mysterious, invisible object looks like. You can't see it, but you can poke it with a tiny, sensitive probe. In the world of physics, this probe is called a Scanning Tunneling Microscope (STM), and the object is a Topological Superconductor—a strange material that conducts electricity without resistance and has special "surface states" that act like highways for electrons.

Usually, scientists use a metal tip to poke these materials. But this paper proposes using a superconducting tip (a tip that also conducts electricity perfectly) to get a much clearer picture. The authors, a team of physicists from Osaka and Tokyo, created a theoretical "instruction manual" for how to interpret the data from this new method.

Here is the breakdown of their work using simple analogies:

1. The Setup: Two Superconductors Meeting

Think of the experiment as a bridge between two islands.

• Island A (The Tip): A standard, well-behaved superconductor (like a calm, orderly city).
• Island B (The Sample): A Topological Superconductor (a chaotic, exotic city with secret underground tunnels).

When you bring these two islands close together, electrons try to jump across the gap. The paper focuses on a specific way they jump called Andreev Reflection.

2. The Main Event: The "Dance Partner" Swap

In a normal metal, an electron just hops across. But in this superconducting bridge, something magical happens called Andreev Reflection.

Imagine a dancer (an electron) from the Tip tries to enter the Sample. Because the Sample is a superconductor, it doesn't want a single dancer; it wants a pair (a Cooper pair).

• The electron from the Tip arrives.
• It grabs a "partner" (a hole, which is like an empty seat waiting to be filled) from the Sample.
• Together, they form a pair and cross the bridge.
• Meanwhile, the original dancer leaves behind a "ghost" (a hole) in the Tip.

The authors calculated that this "dance" is the dominant way electricity flows when the voltage is low. It's like a specialized dance club where you can only enter if you bring a partner.

3. The Measurement: Listening to the Music (dI/dV)

The scientists measure the current (how many dancers are crossing) and the noise (how chaotic the dancing is).

• The Conductance Map (dI/dV): This is like a map of the dance floor. The paper predicts that depending on the "shape" of the exotic city (the Topological Superconductor), the map will show specific peaks.
  • If the city has a smooth, flat surface, the map looks like a V-shape.
  • If the city has a flat "drumhead" of special states, the map shows a sharp spike right in the middle.
  • If the city has a "Fermi arc" (a one-way street), the map looks flat.
  • The Analogy: It's like tapping a drum. A hollow drum sounds different than a solid block. By listening to the "tapping" (the electrical signal), you can tell what the drum is made of.

4. The Secret Clue: The Fano Factor (The Noise Meter)

This is the paper's most exciting contribution. They looked at Shot Noise, which is the "static" or "crackling" sound of the current.

• Normal Tunneling: If single electrons are hopping across one by one, the noise is like raindrops hitting a roof. The "Fano factor" (a measure of noise) is 1.
• Andreev Tunneling: If electrons are hopping in pairs (the dance partners), the noise is different. It's like raindrops falling in clumps of two. The Fano factor jumps to 2.

The Big Discovery: The paper claims that if you use a superconducting tip, you can measure this noise. If you see a Fano factor of 2, you have proof that the "dance partner swap" (Andreev reflection) is happening. This confirms the material is a topological superconductor with special surface states.

5. The Catch: The Tip Must Be Clean

The authors warn that this only works if the Tip is very clean.

• The Problem: If the Tip is dirty (has "residual states"), single electrons might sneak across alone, even when they shouldn't. This is like having a few people ignoring the "dance partner" rule and just walking across.
• The Result: If too many single walkers are present, the noise looks like rain (Factor 1) instead of clumps (Factor 2), and you get the wrong answer.
• The Solution: You need a very high-quality, clean superconducting tip to ensure the "dance" is the only thing happening.

Summary

This paper provides a theoretical recipe book for scientists. It tells them:

1. How to set up the experiment: Use a superconducting tip.
2. What to look for: Specific peaks in the electrical signal that match the shape of the material's surface.
3. How to be sure: Measure the "noise" (Fano factor). If it equals 2, you've found the exotic "dance" of topological superconductivity.

They tested this recipe on several theoretical models (like the "BW state," "Chiral state," and "Polar state") and showed that each produces a unique fingerprint. This gives scientists a reliable way to identify these mysterious materials in the real world, specifically mentioning that their theory helps explain recent observations in a material called UTe2.

Technical Summary

Title: Theory of Andreev and Shot Noise Spectroscopy for Topological Superconductors Probed by s-wave Superconducting Tips

Problem Statement
Scanning tunneling microscopy (STM) and spectroscopy (STS) are established tools for investigating superconductivity, particularly for visualizing quasiparticle excitations and impurity-induced bound states. Recently, the use of s-wave superconducting tips has gained traction due to their high energy resolution (coherence peaks) and the ability to probe Josephson currents. However, the potential of superconducting-tip STM for investigating topological superconductors remains largely unexplored. While existing theoretical studies on superconducting tips focus primarily on Josephson currents, the tunneling current originating from topological surface states—specifically via Andreev reflection—has received little attention. Furthermore, the nature of electron tunneling in junctions between a conventional s-wave superconductor and a topological superconductor differs significantly from conventional superconductor-superconductor (S-S) junctions due to the presence of surface quasiparticle states at arbitrary energies and the spin-triplet nature of many topological superconductors, which can suppress Josephson tunneling.

Methodology
The authors develop a theoretical framework to investigate the differential conductance ($dI/dV$) and current noise in a junction between an s-wave superconductor (the tip) and a topological superconductor (the sample). The study employs two complementary approaches:

1. Analytical Framework (Weak Tunneling Regime):

   • Utilizing the real-time Keldysh path-integral formalism, the authors derive an effective tunneling action by expanding the tunneling Hamiltonian up to fourth order.
   • This expansion incorporates single-particle tunneling, Josephson tunneling, and Andreev reflection processes.
   • By applying a saddle-point approximation, they derive analytical expressions for the tunneling current and current noise. The resulting equations describe a resistively and capacitively shunted junction (RCSJ) model extended to include Andreev reflection currents.
   • The current noise is analyzed to derive the Fano factor ($F$), which characterizes the effective charge ($e^*$) involved in transport ($F = e^*/e$).

2. Numerical Simulations (General Regime):

   • To address regimes beyond weak tunneling, including strong coupling where multiple Andreev reflections (MAR) become significant, the authors perform numerical calculations using the Keldysh Green's function formalism.
   • They solve the Dyson equation for the junction Green's functions, treating the tunneling matrix as a perturbation to all orders. This allows for the calculation of $dI/dV$ spectra and noise across the entire range of tunneling strengths.
   • The simulations model realistic experimental conditions by including finite temperature and quasiparticle lifetime broadening (Dynes broadening) for both the tip and the sample.

The study systematically applies these methods to various representative topological superconductors, including p-wave (Balian-Werthamer, chiral, helical, polar) and d-wave ($d_{x^2-y^2}$) states, analyzing specific crystal surfaces (e.g., (100), (001), (110)) to determine the resulting surface density of states (DOS) and Andreev bound states (ABSs).

Key Contributions and Results

• Analytical Expressions for Current and Noise: The paper provides analytical formulas for the Andreev reflection current and associated shot noise. It establishes that in the low-bias regime ($eV < \Delta_L$, where $\Delta_L$ is the tip gap), the transport is dominated by Andreev reflection, leading to a Fano factor of $F=2$, indicative of $2e$ charge transfer.
• Spectral Features ($dI/dV$): The authors identify characteristic peak structures in the $dI/dV$ spectra:
  • Total Coherence Peaks: Occur at $eV = \pm(\Delta_L + \Delta_R)$ due to single-particle tunneling between coherence peaks.
  • Andreev Peaks: In the subgap regime, peaks appear at $eV = \pm\Delta_R$ and, crucially, at $eV = \pm\Delta_R/2$ if the topological superconductor possesses a finite zero-energy DOS (e.g., Majorana zero modes or drumhead states).
  • Negative Differential Conductance: The spectra can exhibit negative $dI/dV$ regions due to the resonant nature of tunneling between singularities in the DOS of both electrodes.
• Role of Surface DOS and ABSs: The study demonstrates that the $dI/dV$ characteristics are directly governed by the surface DOS, which varies by pairing symmetry and surface orientation:
  • V-shaped DOS: Found in BW p-wave and surfaces without ABSs, leading to super-Ohmic behavior ($I \propto V^3$) at low bias.
  • Flat-type DOS: Found in chiral/helical states with Fermi arcs, resulting in a finite zero-energy DOS.
  • Peak-type DOS: Found in polar p-wave and $d_{x^2-y^2}$ states with drumhead states, producing sharp zero-bias conductance peaks (ZBPs).
• Noise Spectroscopy and Tip Quality: A critical finding is the competition between single-particle tunneling (mediated by residual DOS due to Dynes broadening, scaling as $|T|^2$) and Andreev reflection (scaling as $|T|^4$). In the weak-tunneling limit, if the tip is not sufficiently clean (large Dynes parameter), single-particle tunneling can dominate, causing the Fano factor to remain near $F=1$ rather than reaching the expected $F=2$. This highlights that high-quality, clean superconducting tips are essential for observing pure Andreev signatures.
• Differentiation of Mechanisms: The paper notes that while a zero-bias peak (ZBP) in $dI/dV$ can arise from either Josephson tunneling or Andreev reflection, shot noise measurements can distinguish them. A ZBP with $F=2$ confirms an Andreev origin, whereas Josephson tunneling exhibits different noise characteristics.

Significance
The authors state that their work establishes guidelines for probing topological superconductivity using STM with s-wave superconducting tips. By providing a catalog of $dI/dV$ spectra and Fano factors for various topological states, the paper offers theoretical benchmarks for future STS experiments. Specifically, the results suggest that combining conductance spectroscopy with shot noise measurements can provide direct insights into the pairing symmetry of the superconducting order parameter and the nature of surface states. The study also offers a potential route to identify exotic surface Majorana states, referencing recent observations of zero-bias peaks in the superconductor UTe$_2$ as a possible realization of such states.