Momentum-resolved spectroscopy of superconductivity with the quantum twisting microscope

This paper establishes a theoretical framework for the quantum twisting microscope (QTM) as a direct momentum-resolved probe of superconductivity in two-dimensional materials, enabling the measurement of pairing symmetry, coherence factors, and rotational symmetry breaking through planar tunneling between a rotated graphene tip and a sample.

The Gist

Imagine you are trying to understand a complex dance floor where electrons are the dancers. In a normal metal, they shuffle around randomly. But in a superconductor, they pair up and move in perfect, synchronized steps. The "secret sauce" of this dance is the pairing potential—the invisible rulebook that tells the electrons how to pair up.

For decades, scientists have struggled to read this rulebook. They knew the dance was happening, but they couldn't see how the dancers were holding hands or if they were holding hands differently depending on where they were on the floor.

This paper introduces a revolutionary new tool called the Quantum Twisting Microscope (QTM) and explains how it can finally read that rulebook, step by step.

The Problem: The "Blurry" Camera

Think of traditional microscopes (like the Scanning Tunneling Microscope or STM) as a camera with a very wide-angle lens. It takes a picture of the whole dance floor and tells you, "Hey, there's a lot of dancing happening here!" But it blurs out the details. It can't tell you if the dancers in the corner are holding hands differently than the dancers in the center. It averages everything out, hiding the specific "momentum" (direction and speed) of the pairs.

The Solution: The Quantum Twisting Microscope (QTM)

The authors propose using the QTM, which is like a high-tech, rotating spotlight that can freeze-frame the dance.

Here is how it works, using a simple analogy:

1. The Setup: Imagine you have a dance floor (the sample) and a spotlight (the tip). Both are made of a special material called graphene.
2. The Twist: The magic happens when you rotate the spotlight relative to the dance floor.
3. The Momentum Lock: Because of the laws of quantum physics, when the spotlight shines on the floor, it only "sees" the dancers moving in a very specific direction that matches the angle of the light.
   • Analogy: Imagine you are looking through a long, narrow tube. You can only see what is directly in front of you. If you rotate the tube, you see a completely different slice of the room. The QTM does this with electrons. By rotating the tip, the scientists can scan the dance floor in tiny, specific slices of direction (momentum).

What They Can Now See

With this new "rotating tube," the scientists can measure three incredibly important things that were previously invisible:

1. The Strength of the Dance (Pairing Magnitude)

In some superconductors, the electrons hold hands tightly everywhere. In others, they hold hands tightly in some spots and loosely in others.

• The QTM Magic: By rotating the tip, they can measure exactly how strong the "hand-holding" is at every single point on the map. They found that in some materials, the strength changes depending on the direction, proving the dance isn't uniform.

2. The "Nodal" Holes (Where the Dance Stops)

Some superconductors have "nodes"—places where the electrons simply don't pair up at all. It's like a hole in the dance floor where no one is dancing.

• The QTM Magic: The microscope can find these holes. If the scientists rotate the tip and the signal suddenly drops to zero, they know, "Aha! There is a node right here!" This helps them identify the specific type of superconductivity.

3. Breaking the Symmetry (The "Nematic" Dance)

Imagine a hexagonal dance floor (like a honeycomb). Usually, the dance looks the same if you rotate the floor by 120 degrees. But in some exotic superconductors, the dance breaks this symmetry. Maybe the dancers in the "top" direction hold hands differently than those in the "bottom" direction.

• The QTM Magic: Because the microscope scans three specific directions at once (due to the geometry of the graphene), it can instantly spot if the dance is "lopsided." If the signal splits into two different paths, it proves the symmetry is broken.

Why This Matters for the Future

The paper applies this theory to Magic-Angle Twisted Bilayer Graphene (MATBG), a material that has been a huge mystery. Scientists have been arguing about why it becomes superconducting. Is it because of the "heavy" electrons (like a slow, heavy dance)? Or the "light" electrons (a fast, energetic dance)?

The QTM acts as a detective that can distinguish between these two suspects.

• If the pairing happens mostly on the "heavy" electrons, the microscope will see a gap (a pause in the dance) at specific angles.
• If it's the "light" electrons, the gap appears at different angles.

The Big Picture

Think of the QTM as the difference between listening to a symphony through a wall (hearing a muffled, blended sound) and sitting in the front row with a high-end microphone that can isolate every single violin, cello, and drum.

This paper provides the theoretical blueprint for that microphone. It tells us that by simply twisting a tiny graphene tip, we can finally map out the microscopic rules of superconductivity. This could be the key to designing room-temperature superconductors, which would revolutionize everything from power grids to quantum computers.

In short: They built a theoretical "rotating flashlight" that lets us see exactly how electrons pair up in 2D materials, solving mysteries that have puzzled physicists for years.

Technical Summary

1. Problem Statement

Understanding the microscopic origin of unconventional superconductivity, particularly in two-dimensional (2D) materials like magic-angle twisted bilayer graphene (MATBG), requires precise knowledge of the superconducting pairing potential $\Delta(\mathbf{k})$ in momentum space.

• Limitations of Current Techniques:
  • Scanning Tunneling Microscopy (STM): While STM provides high spatial resolution, it integrates over in-plane momentum. Consequently, it measures the local density of states (LDOS) but averages out the momentum dependence of the pairing magnitude and Bogoliubov coherence factors. It cannot directly resolve the symmetry of the order parameter or locate nodal points in momentum space.
  • ARPES (Angle-Resolved Photoemission Spectroscopy): While ARPES offers momentum resolution, it typically probes only occupied states (electron removal). In superconductors, accessing the hole branch (electron addition) at low temperatures is difficult, limiting the ability to extract coherence factors and the full pairing magnitude away from the Fermi level.
• The Gap: There is a need for a probe that combines momentum resolution with the ability to measure both electron and hole excitations to directly extract the pairing magnitude $|\Delta_{\mathbf{k}}|$ and coherence factors $|u_{\mathbf{k}}|^2, |v_{\mathbf{k}}|^2$.

2. Methodology: The Quantum Twisting Microscope (QTM)

The authors develop a theoretical framework for using the Quantum Twisting Microscope (QTM), a planar tunneling device consisting of a 2D crystalline tip (graphene) rotated by an angle $\theta$ relative to a 2D sample.

• Physical Principle:
  • Momentum Conservation: Due to the large contact area and crystalline order of the tip, tunneling conserves in-plane momentum.
  • Dirac Tip: The tip is a monolayer graphene (MLG) sheet with a linear Dirac dispersion. By rotating the tip, its Dirac point traces specific trajectories ($K_\theta$) in the sample's momentum space.
  • Tunneling Channels: In twisted bilayer graphene (TBG) systems, the tip Dirac point couples to three distinct sample momenta ($K_{\theta, n}$) related by $C_{3z}$ rotational symmetry via Umklapp processes.
• Measurement Protocols:
  1. Dirac-Point Spectroscopy (Bias Scan): The tip chemical potential is set to the Dirac point ($\mu_T = 0$). The bias voltage $V_b$ is varied to sweep the tip Dirac cone across the sample bands. The second derivative of the current ($d^2I/dV_b^2$) reveals sharp features corresponding to the sample's spectral function.
  2. Zero-Bias Conductance (Fermi Circle Scan): The tip chemical potential $\mu_T$ is varied (expanding the tip's Fermi circle) while keeping $V_b \approx 0$. This allows the tip Fermi surface to intersect nodal points in the sample, producing sharp zero-bias conductance peaks.

3. Key Contributions and Theoretical Framework

The paper derives an analytical expression for the differential conductance in the superconducting state, demonstrating how the QTM directly accesses superconducting parameters:

• Direct Extraction of Pairing Magnitude:
  The authors show that the relative intensities of the electron ($+E_k$) and hole ($-E_k$) coherence peaks in the $d^2I/dV_b^2$ spectrum encode the Bogoliubov coherence factors:
  $$R(K_\theta) = \frac{I''_+(K_\theta)}{I''_-(K_\theta)} = \frac{|u_{K_\theta}|^2}{|v_{K_\theta}|^2} = \frac{E_{K_\theta} + \xi_{K_\theta}}{E_{K_\theta} - \xi_{K_\theta}}$$
  Combining the energy separation ($2E_k$) and the intensity ratio ($R$), one can directly calculate the pairing magnitude at a specific momentum:
  $$|\Delta_{K_\theta}| = E_{K_\theta} \sqrt{1 - \left( \frac{1-R}{1+R} \right)^2}$$
  This allows mapping $|\Delta_{\mathbf{k}}|$ even away from the Fermi momentum.

• Symmetry Detection:
  Because the QTM probes three $C_{3z}$-related momenta simultaneously in a single line scan:

  • If the order parameter preserves $C_{3z}$ symmetry (e.g., s-wave), the three traces coincide.
  • If $C_{3z}$ is broken (e.g., nematic $p_x$ or $p_y$ wave), the traces split, directly revealing the symmetry breaking.

• Nodal Point Detection:

  • Gap Closing: If a trajectory passes through a node, the gap vanishes ($|\Delta|=0$), appearing as a zero-bias feature in the spectrum.
  • Geometric Triangulation: By performing zero-bias scans at two different rotation angles ($\theta_1, \theta_2$), the intersection of the tip Fermi circles with the nodal points allows for the geometric triangulation of the nodal momentum $\mathbf{k}_0$.

• Model Application:
  The framework is applied to two models:

  1. Bistritzer-MacDonald (BM) Model: A non-interacting model for MATBG.
  2. Topological Heavy-Fermion (HF) Model: An interacting model where superconductivity arises from hybridization between localized $f$-electrons and itinerant $c$-electrons.

4. Results

• Symmetry Resolution: Simulations show that for $s$-wave pairing, the three $C_{3z}$-related traces are degenerate. For $p_x$ and $p_y$ pairings, the traces split, with $p_x$ showing a nodal point where one gap vanishes.
• Coherence Factor Mapping: The paper demonstrates the successful reconstruction of the pairing magnitude $|\Delta(\mathbf{k})|$ and the variation of coherence factors across the Brillouin zone, distinguishing between isotropic and anisotropic pairing.
• Heavy-Fermion Distinction: In the HF model, the QTM can distinguish whether superconductivity originates from the flat $f$-bands or the dispersive $c$-bands.
  • Gaps appearing at specific rotation angles corresponding to the $f$-band edges indicate $f$-electron pairing.
  • Gaps near angles corresponding to the $c$-band indicate $c$-electron pairing.
  • This addresses the open question of the microscopic origin of pairing in MATBG.
• Nodal Triangulation: The zero-bias protocol successfully identifies nodal points by observing sharp peaks in conductance when the tip Fermi circle crosses the node, allowing for the geometric determination of the node's location.

5. Significance

• Direct Probe of Pairing Symmetry: The QTM offers a unique capability to measure the momentum-dependent pairing potential $|\Delta_{\mathbf{k}}|$ directly, bypassing the momentum averaging inherent in STM.
• Microscopic Origin Identification: By distinguishing between different band characters (flat vs. dispersive) and their respective superconducting gaps, the QTM can determine the specific electronic components driving superconductivity in correlated systems.
• Complementarity to ARPES: Unlike ARPES, the QTM can probe both electron and hole branches at low temperatures and is sensitive to the single-particle spectral function in a tunneling geometry, making it ideal for 2D van der Waals heterostructures.
• Future Directions: The authors suggest that extending this to stronger tunneling regimes could enable Andreev spectroscopy, and adding spin/valley selectivity (via magnetic fields or spin-orbit coupling) could resolve the internal flavor structure of non-unitary pairing.

In conclusion, this work establishes the Quantum Twisting Microscope as a powerful, momentum-resolved tool capable of solving fundamental questions regarding the symmetry, magnitude, and microscopic origin of unconventional superconductivity in 2D materials.