Probing superconductivity with tunneling spectroscopy in rhombohedral graphene

This paper develops a microscopic tunneling approach to demonstrate how scanning tunneling spectroscopy can distinguish between different superconducting pairing scenarios in rhombohedral tetralayer graphene, including unique signatures of broken time-reversal symmetry, spatially dependent Andreev conductance for topologically distinct states, and features of competing moiré superconductivity.

The Gist

Imagine a stack of four graphene sheets, arranged in a specific diamond-like pattern called "rhombohedral." Recently, scientists discovered that under certain conditions, this material can become a superconductor—a substance that conducts electricity with zero resistance. But here's the twist: this superconductivity doesn't start from a calm, balanced state. Instead, it emerges from a chaotic, "valley-polarized" state where electrons are forced to choose a side, much like a crowd of people all rushing toward one exit of a stadium while ignoring the other.

The authors of this paper, Denis Sedov and Mathias Scheurer, are theoretical physicists. They didn't build a new machine; they built a sophisticated mathematical "flashlight" to help experimentalists see what's happening inside this material. Their tool is a technique called Scanning Tunneling Spectroscopy (STS).

Here is a simple breakdown of their work using everyday analogies:

1. The Problem: A Hidden Symphony

When electrons in this graphene stack pair up to become superconductors (forming "Cooper pairs"), they do so in a very complex dance. Because the electrons are "valley-polarized" (they are all in one specific valley of the material's energy landscape), the usual rules of symmetry are broken. It's like a dance where the partners are spinning in a direction that breaks the usual mirror-image rules of the ballroom.

The big question is: What kind of dance are they doing? Are they spinning in a simple circle, a complex spiral, or a chaotic jumble? The paper claims that standard measurements can't easily tell the difference between these dance styles.

2. The Tool: The "Weak" vs. "Strong" Flashlight

The authors propose using their STS "flashlight" in two different ways to reveal the secret dance moves:

• The Weak Flashlight (Weak Tunneling): Imagine shining a very dim, gentle light on the dancers. This measures the density of states—essentially, how many dancers are available to move at a specific energy level.

  • What they found: In this material, because the symmetry is broken, the "dance floor" looks different than usual. Instead of a clean, hard edge where the music stops (a gap), you see sharp peaks and strange plateaus. It's like hearing a song where the silence between notes is filled with unexpected echoes. This tells you that something unusual is happening, but not exactly what kind of dance it is.

• The Strong Flashlight (Strong Tunneling): Now, imagine turning the light up bright and pushing harder. This triggers a process called Andreev reflection.

  • The Analogy: Think of an electron trying to enter a club (the superconductor). In a normal club, it just walks in. In this superconductor, the bouncer (the superconducting order) forces the electron to swap places with a "hole" (a missing electron) before letting it in. The electron leaves, and the hole enters.
  • The Discovery: The authors found that this "swap" process is extremely sensitive to the direction of the dance. If the electrons are dancing in a specific "chiral" (handed) way, the swap happens easily. If they are dancing in a different way, the swap is blocked by symmetry. By moving the tip of their microscope to different spots on the graphene (like moving from one side of the dance floor to the other), they can see which dance style is present. It's like checking if a spinning top spins clockwise or counter-clockwise by watching how it reacts to a push from different angles.

3. The Three Dance Styles (Topological Classes)

The paper identifies three distinct "classes" of superconducting states, distinguished by a mathematical property called the Chern number (think of it as the number of times the dancers twist around a central point):

• Class A (Trivial): The dancers twist zero times.
• Class E and E (Topological):* The dancers twist once clockwise or once counter-clockwise.

The authors show that by using the "Strong Flashlight" at different locations on the graphene, you can tell these three classes apart. If you move the probe and the signal changes in a specific cyclic pattern, you know you are looking at a topological superconductor.

4. The "Moiré" Superconductor (The Moving Carpet)

Finally, the paper explores a more exotic scenario. Sometimes, instead of the whole crowd dancing in unison, the dance floor itself seems to ripple. This is called a "3-q moiré superconductor."

• The Analogy: Imagine a carpet with a pattern. If you lay a second, slightly different patterned carpet on top, you see a new, larger pattern emerge (a moiré pattern). In this case, the superconductivity creates a new, larger "super-lattice" pattern across the material.
• The Result: The authors calculated that the "density of dancers" (LDOS) would vary across this new pattern. Some spots would be quiet (low density), while others would be loud (high density). This spatial variation is a unique fingerprint that distinguishes this state from the others.

Summary

In short, Sedov and Scheurer have provided a theoretical "cheat sheet" for experimentalists. They claim that by carefully measuring how electrons tunnel into rhombohedral graphene at different strengths and different locations, scientists can finally identify:

1. If the superconductivity is "chiral" (handed).
2. Which specific topological class it belongs to.
3. If the superconductivity is forming a complex, rippling "moiré" pattern across the material.

They are essentially saying: "We have the map and the compass; now, experimentalists, go look at the terrain with these specific tools, and you will finally see the true nature of this exotic superconductor."

Technical Summary

1. Problem Statement

The paper addresses the challenge of experimentally identifying the microscopic nature of superconductivity in rhombohedral tetralayer graphene (RTG). Recent experiments have revealed that RTG can host superconducting states emerging from a valley-polarized normal state (where Fermi surfaces exist only near the $K$ or $K'$ points, breaking time-reversal symmetry, TRS).

Key theoretical uncertainties include:

• Pairing Momentum: Whether Cooper pairs have commensurate momentum ($q=0$, center-of-mass momentum $2K$) or incommensurate momentum ($q \neq 0$, finite momentum pairing).
• Order Parameter Symmetry: The specific irreducible representation (IR) of the pairing state under the $C_{3z}$ rotational symmetry (Classes $A$, $E$, or $E^*$).
• Topological Distinction: Whether the states are topologically trivial ($C=0$) or non-trivial ($C=\pm 1$) based on their Chern numbers.
• Competing Phases: The potential existence of a "moiré superconductor" formed by a superposition of three momenta ($3q$ state) that breaks translational symmetry.

Standard tunneling spectroscopy (STS) often struggles to distinguish these complex, symmetry-broken states, particularly when TRS is broken in the normal state.

2. Methodology

The authors develop a microscopic tunneling approach based on the Keldysh formalism to model Scanning Tunneling Spectroscopy (STS) in RTG.

• Hamiltonian Model:

  • The target system (RTG) is modeled using an 8-band continuum model ($h_k$) incorporating the specific band structure of ABC-stacked graphene.
  • Superconductivity is introduced via a general pairing term $\Delta_{k, -k'}$ that allows for arbitrary momentum dependence ($k-k' \neq 0$) and broken TRS.
  • The tunneling is treated as a local coupling between a 1D lead (tip) and the RTG, described by tunneling matrix elements $t_k$.

• Current Decomposition:
  The tunneling current is derived and split into four terms. The authors focus on two distinct regimes:

  1. Weak Tunneling Regime: The current is dominated by the transfer of single electrons. The conductance $G(eV)$ is proportional to the Local Density of States (LDOS) of the Bogoliubov quasiparticles.
  2. Strong Tunneling Regime: The Andreev current ($I_A$) becomes significant. This current arises from Andreev reflection (electron $\to$ hole conversion) and is proportional to the superconducting order parameter $\Delta_{k, -k'}$. Crucially, $I_A$ does not vanish inside the superconducting gap, unlike the normal LDOS.

• Symmetry Analysis:
  The authors analyze how the order parameter transforms under the three distinct $C_{3z}$ rotational symmetries of the RTG lattice (rotations around sublattice sites $A_1$, $B_1$, and $B_2$). They derive selection rules for the Andreev current based on the interplay between the tunneling symmetry and the order parameter's transformation properties.

3. Key Contributions

• Generalized Tunneling Formalism: Development of a framework capable of handling arbitrary finite-momentum pairing ($q \neq 0$) and low-symmetry normal states, specifically tailored for valley-polarized systems.
• Differentiation of Pairing Regimes: Identification of unique spectral signatures in both weak and strong tunneling regimes that distinguish between:
  • Nodal vs. fully gapped superconductivity.
  • Commensurate ($q=0$) vs. incommensurate ($q \neq 0$) pairing.
  • Topologically distinct pairing classes ($A$ vs. $E/E^*$).
• Spatially Resolved Probing: Demonstration that moving the STM tip between different sublattice sites ($A_1$ vs. $B_{1,2}$) acts as a symmetry probe, cyclically permuting the visibility of Andreev peaks for different topological classes.
• Characterization of 3-q States: Prediction of spatially modulated LDOS patterns for the "moiré superconductor" (3-q state), distinguishing it from uniform 1-q states.

4. Key Results

A. Weak Tunneling Regime (LDOS Probing)

• Broken TRS Signatures: In valley-polarized RTG, the lack of TRS ($\xi_k \neq \xi_{-k}$) leads to unique LDOS features.
  • Nodal Regime: Characterized by a finite-height plateau near the Fermi energy.
  • Gapped Regime: Instead of standard coherence peaks at the gap edge, the conductance shows step-like behavior. Sharp peaks appear above the hard gap, corresponding to saddle points in the Bogoliubov spectrum.
• Finite Momentum ($q \neq 0$): In the 1-q state, the hard gap is reduced, and Van Hove singularities split into two less pronounced peaks due to particle-hole nesting effects.

B. Strong Tunneling Regime (Andreev Spectroscopy)

• Symmetry Selection Rules: The Andreev current is highly sensitive to the phase of the order parameter.
  • For commensurate pairing ($q=0$): The Andreev signal vanishes for certain symmetries due to destructive interference in the $k$-sum.
    • If tunneling occurs on the $A_1$ sublattice, the $A$ state (trivial under $C_{3z}^{A_1}$) exhibits a strong sub-gap Andreev signal.
    • The $E$ and $E^*$ states (non-trivial under $C_{3z}^{A_1}$) show a vanishing or suppressed Andreev signal at $A_1$.
  • Sublattice Dependence: Moving the tip to $B_1$ or $B_2$ changes the symmetry constraints. The state that was "dark" (no signal) at $A_1$ becomes "bright" at $B_{1,2}$. This allows experimentalists to distinguish between Chern numbers $C=0, \pm 1$ by mapping the Andreev peak intensity across the unit cell.
• Spin Polarization: Full spin polarization in the normal state suppresses the Andreev peak at zero bias ($eV=0$) due to fermionic antisymmetry, but a finite signal remains at $eV \neq 0$.

C. Finite Momentum and 3-q States

• 1-q States: Even with finite $q$ (breaking $C_{3z}$), the strong tunneling regime retains the ability to distinguish between $A$, $E$, and $E^*$ classes, although the sub-gap Andreev current becomes non-zero for all classes.
• 3-q "Moiré" Superconductor:
  • This state involves a superposition of three momenta ($q_1, q_2, q_3$) related by $C_{3z}$, breaking translational symmetry.
  • Spatial Modulation: The LDOS exhibits a distinct spatial pattern within the moiré unit cell.
    • At positions where the three order parameter phases interfere constructively (e.g., near $\Gamma_R$), the LDOS is strongly suppressed (large gap).
    • At positions of destructive interference (e.g., near $K_R, M_R$), the gap closes or is significantly reduced.
  • This spatial variation serves as a definitive fingerprint for the 3-q state, distinguishable from uniform 1-q states.

5. Significance

This work provides a comprehensive theoretical roadmap for interpreting STS experiments in valley-polarized graphene systems.

• Experimental Guidance: It offers specific protocols (varying bias voltage, tunneling strength, and tip position) to unambiguously identify the superconducting order parameter's symmetry, momentum, and topology.
• Resolution of Debates: By distinguishing between trivial and topological pairing states and identifying signatures of competing translational-symmetry breaking phases, the paper helps resolve ongoing debates about the nature of superconductivity in multilayer graphene.
• Novel Physics: It highlights how broken time-reversal symmetry in the normal state fundamentally alters tunneling spectroscopy, creating unique features (like step-like conductance and sublattice-dependent Andreev peaks) that are absent in conventional superconductors.

The authors conclude that combining weak and strong tunneling measurements at high-symmetry lattice sites is the key to unlocking the microscopic physics of these complex correlated systems.