Bridging atomic and mesoscopic length scales with Replica Scanning Tunneling Microscopy: Visualizing the intra-unit cell pair density modulation of superconducting FeSe at micron length scale

This paper introduces Replica Scanning Tunneling Microscopy (R-STM), a novel technique that bridges atomic and mesoscopic length scales by utilizing replica signals to efficiently track atomic-scale electronic modulations over large areas, demonstrating that the pair density modulation in superconducting FeSe persists consistently up to hundreds of nanometers.

The Gist

The Big Problem: The "Microscope vs. Map" Dilemma

Imagine you have a super-powerful camera (a Scanning Tunneling Microscope, or STM) that can take photos of individual atoms. It's like having a camera so sharp you can see the texture of a single grain of sand.

However, there's a catch. To take a photo of a whole city block with that level of detail, you would need to take billions of photos, one for every single grain of sand. If you tried to do this, it would take years to finish the job, and your camera might break from the heat and effort before you were done.

So, scientists usually do this instead:

1. Take a quick, blurry photo of the whole city block (the "mesoscopic" scale).
2. Pick one tiny street corner and zoom in to take a super-detailed photo of the atoms there.

The Problem: This leaves a huge question unanswered. Does that specific pattern of atoms on the street corner exist everywhere else in the city, or is it just a local accident? We couldn't know for sure without spending years taking detailed photos of the whole block.

The Solution: The "Alias" Trick (R-STM)

The scientists in this paper invented a clever trick called Replica STM (R-STM). They realized they could use a mathematical quirk called "aliasing" (the same thing that makes a spinning wagon wheel look like it's moving backward in a movie) to their advantage.

The Analogy: The Strobe Light and the Fan
Imagine a ceiling fan spinning very fast.

• The Real Fan: The blades are moving at 1,000 RPM.
• The Strobe Light: You are taking photos of the fan, but your camera is slow. It only takes a picture once every second.
• The Result: In your photos, the fan doesn't look like it's spinning fast. It looks like it's spinning slowly backward.

In physics, this "slow backward spin" is a replica of the real speed. It's a fake signal, but it is mathematically locked to the real signal. If you know the math, you can look at the "fake" slow spin and calculate exactly how fast the "real" fan is spinning.

How the Scientists Used It:
Instead of trying to take a super-detailed photo of a huge area (which takes too long), they took a "low-resolution" photo of a large area (about 200 nanometers wide). Because they didn't take enough points to see the atoms clearly, the atomic patterns created "ghost images" or replicas that looked like much larger, slower waves.

By studying these "ghost waves," they could mathematically prove that the tiny atomic patterns were actually present across the entire large area. They bridged the gap between the "atomic world" and the "micron world" without waiting years for the data.

What Did They Find? (The FeSe Mystery)

They tested this new method on a material called FeSe (Iron Selenide), which is a superconductor (a material that conducts electricity with zero resistance).

Recently, scientists discovered a weird pattern in FeSe called a "pair density modulation." Imagine that the electrons in the superconductor are dancing in pairs. In this material, the "tightness" of their dance steps changes rhythmically as you move across the surface.

• The Old Question: We saw this dance rhythm in tiny, zoomed-in photos. But does this rhythm continue for hundreds of nanometers, or does it stop after a few atoms?
• The R-STM Answer: Using their "ghost wave" trick, they scanned a large area and found the rhythm. They proved that this specific electron dance pattern persists consistently over distances up to hundreds of nanometers. It's not just a local glitch; it's a property of the whole material.

Why This Matters

This paper is like giving scientists a new pair of glasses.

• Before: They had to choose between seeing the "forest" (the big picture) or the "trees" (the atoms), but not both at once.
• Now: With R-STM, they can look at the forest and instantly know exactly what the trees look like, because the "ghosts" of the trees are visible in the forest.

This technique is fast, practical, and can be used on many different types of materials and microscopes. It solves a major bottleneck in physics, allowing researchers to understand how tiny atomic secrets play out on a larger, human-visible scale.

Technical Summary

1. Problem Statement

Scanning Tunneling Microscopy (STM) is the premier technique for visualizing electronic density of states (DOS) with atomic resolution. However, a significant technical bottleneck exists when attempting to correlate atomic-scale phenomena with mesoscopic (micron-scale) behavior:

• Time Constraints: Achieving atomic resolution over large areas (e.g., $>100$ nm) requires acquiring millions of data points, each with a full $dI/dV$ (conductance) spectrum. Given typical cryogenic STM bandwidths (~10 kHz), mapping a $500 \text{ nm} \times 500 \text{ nm}$ area with atomic precision can take over 11 days.
• Tip Instability: Long acquisition times increase the likelihood of tip drift or instability, often ruining the dataset.
• The "Zoom-In" Limitation: The standard workflow involves mapping a large area at low resolution and then zooming in on small regions for high resolution. This approach fails to answer a critical question: Does a specific atomic-scale modulation persist coherently over large, mesoscopic length scales?

2. Methodology: Replica Scanning Tunneling Microscopy (R-STM)

The authors propose Replica STM (R-STM), a technique that leverages the Nyquist-Shannon sampling theorem and the phenomenon of aliasing to bridge the gap between atomic and mesoscopic scales without requiring high-density sampling.

• Principle of Aliasing: When a periodic signal with a wavenumber $k_{signal}$ (e.g., atomic lattice spacing) is sampled at a rate $k_S$ (determined by the distance between measurement points) that is insufficient to satisfy the Nyquist criterion ($k_S < 2k_{signal}$), "replica" signals appear at lower wavenumbers ($k_{replica}$).
• Mathematical Relationship: The replica wavenumber is defined as:
  $$k_{replica} = |k_{signal} - n k_S|$$
  where $n$ is an integer.
• The R-STM Protocol:
  1. Acquire a large-area map (e.g., $>200$ nm) with a fixed, relatively low point density (undersampled).
  2. Perform a Fourier Transform (FT) of the large map.
  3. Identify peaks at $k_{replica}$.
  4. "Up-fold" the Reciprocal Space: By mathematically translating the observed replica peaks by integer multiples of the sampling vector ($n \cdot k_S$), the original atomic-scale wavenumber ($k_{signal}$) can be reconstructed.
  5. Verification: If a modulation is observed at $k_{replica}$ in a large map, it proves the existence of the corresponding $k_{signal}$ modulation across the entire scanned area.

3. Key Contributions

• Novel Technique: Introduction of R-STM, which transforms the limitation of undersampling (aliasing) into a feature for studying long-range coherence.
• Efficiency: Drastically reduces acquisition time by allowing large-area mapping with standard point densities (e.g., $256 \times 256$ points over $200$ nm) while retaining information about atomic-scale features.
• Proof of Concept: Demonstrated on two distinct materials: the topographic lattice of UTe$_2$ and the electronic pair density modulation in FeSe.

4. Key Results

A. Topographic Lattice of UTe$_2$

• Observation: The authors mapped the Te atomic chains on the (011) surface of UTe$_2$.
• Small Scale: A $50$ nm map showed clear atomic Bragg peaks in the Fourier transform.
• Large Scale: Maps of $100$ nm and nearly $200$ nm were acquired with the same number of points ($256 \times 256$).
• Result: The Fourier transforms of these large maps showed "aliased" peaks. By applying the up-folding procedure, these peaks were successfully mapped back to the original atomic Bragg vectors.
• Conclusion: The Te chain periodicity persists coherently over micron-length scales.

B. Pair Density Modulation in FeSe

• Context: Recent studies identified a spatial modulation of the superconducting gap (pair density wave) in FeSe, linked to broken glide symmetry and nematicity. Previous observations were limited to small ($<10$ nm) areas.
• Experiment: The authors acquired a $200$ nm $\times$ $200$ nm map of the tunneling conductance ($dI/dV$) in FeSe at $1$ K, using $256 \times 256$ points.
• Analysis:
  • The large-scale map showed a periodic signal with a wavelength much larger than the atomic lattice.
  • Fourier analysis revealed peaks at $k_{replica}$.
  • Up-folding these peaks by the sampling vector ($k_S$) revealed the original wavevectors ($q_1$ and $q_3$) associated with the atomic-scale pair density modulation.
• Conclusion: The pair density modulation in FeSe is not a local defect or short-range phenomenon; it persists with the same characteristic wavelength over hundreds of nanometers (mesoscopic scales).

5. Significance and Impact

• Bridging Scales: R-STM provides a practical method to correlate atomic-scale quantum phenomena with mesoscopic behavior, a task previously hindered by time and stability constraints.
• Validation of Long-Range Order: The technique confirmed that the pair density wave in FeSe is a robust, long-range ordered state, resolving questions about its spatial coherence.
• Broad Applicability: While demonstrated on STM, the authors note the method is applicable to any scanning probe microscopy (SPM) technique (e.g., AFM, MFM) where periodic signals are traced. It is particularly useful for studying:
  • Superconducting vortex lattices.
  • Charge density waves (CDW).
  • Magnetic patterns.
  • Local charge density modulations around impurities.
• Comparison to Sparse Sampling: Unlike adaptive sparse sampling (which requires iterative reconstruction and prior knowledge of the signal sparsity), R-STM is a direct, post-processing technique that works specifically on periodic signals, making it computationally simpler and more robust for crystalline materials.

In summary, the paper presents a paradigm shift in how STM data is acquired and analyzed, turning the "aliasing" artifact into a powerful tool for visualizing the long-range coherence of atomic-scale electronic modulations.