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Terahertz s-SNOM reveals nonlocal nanoscale conductivity of graphene

This study utilizes terahertz scattering-type near-field optical microscopy to directly measure and quantify the nonlocal nanoscale conductivity of graphene, demonstrating that nonlocal effects significantly influence charge transport even at practical device dimensions and establishing a critical foundation for designing future ultracompact photonic and electronic systems.

Original authors: Henrik B. Lassen, William V. Carstensen, Leonid Iliushyn, Timothy J. Booth, Peter Bøggild, Edmund J. R. Kelleher, Peter U. Jepsen

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Henrik B. Lassen, William V. Carstensen, Leonid Iliushyn, Timothy J. Booth, Peter Bøggild, Edmund J. R. Kelleher, Peter U. Jepsen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how water flows through a pipe. In the old days, engineers thought of water as a simple, uniform fluid. If you knew the pressure and the pipe size, you could predict exactly how fast it would move. They treated the water's "slipperiness" (conductivity) as a fixed property of the liquid itself, no matter how small the pipe was.

But what happens if the pipe gets so tiny that it's only a few atoms wide? Suddenly, the water molecules start bumping into the walls, swirling in eddies, and reacting to the shape of the pipe in ways the old "uniform fluid" rules can't predict. The flow becomes messy, complex, and dependent on where you are looking inside the pipe.

This is exactly what this paper is about, but instead of water in a pipe, the scientists are studying electrons flowing through graphene (a material made of a single layer of carbon atoms, like a sheet of chicken wire) at Terahertz speeds (super-fast vibrations).

Here is the breakdown of their discovery in everyday terms:

1. The Old Rule vs. The New Reality

For a long time, scientists treated electricity in materials like a local rule: "If you push here, the electron moves right there." They assumed the material's ability to conduct electricity (conductivity) was a fixed number, like the color of a car.

However, when devices get incredibly small (nanoscale) and operate at super-fast speeds (Terahertz), this rule breaks. The electrons stop behaving like independent runners on a track. Instead, they start acting like a crowd at a concert. If you push one person in the crowd, the whole group shifts, waves, and reacts together. The movement of an electron at one spot depends on what's happening far away from it. This is called nonlocality.

2. The Problem: We Couldn't See It

The problem was that we didn't have a way to "see" this crowd behavior.

Old tools were like trying to measure a crowd from a helicopter: you only see the average movement, missing the individual interactions.
Other tools were too slow or looked at the wrong "color" of light to catch these fast, tiny interactions.

Scientists knew this "nonlocal" behavior should exist, but they couldn't prove it or measure it directly in real materials.

3. The Solution: The "Super-Microscope"

The team built a special tool called Terahertz s-SNOM. Think of this as a super-powered, ultra-sensitive needle that acts like a microscope.

The Needle: It's an incredibly sharp metal tip (like a needle on a record player, but much smaller).
The Light: They shine a beam of Terahertz light (a type of invisible light between microwaves and infrared) onto the tip.
The Magic: When the light hits the tip, it gets squeezed into a tiny, intense spot right at the very tip's point. This creates a "super-focusing" effect, allowing them to probe the electrons in a space as small as 50 nanometers (about 1/1000th the width of a human hair).

It's like using a laser pointer to read the fine print on a book, but the "laser" is so focused it can read the ink molecules themselves.

4. What They Found: The "Ghost" Effect

When they used this needle to look at graphene, they found something surprising. Even though the graphene sheets were relatively large (visible to the naked eye), the electricity flowing through them was behaving as if the material was much smaller than it actually was.

The Analogy: Imagine walking through a hallway. Normally, you walk straight. But if the hallway is filled with invisible, sticky fog (the nonlocal effect), your path curves and slows down based on how far away the walls are, even if you aren't touching them yet.
The Discovery: The electrons in the graphene were "feeling" the boundaries and the shape of the material from a distance. Their conductivity wasn't just a simple number; it was a complex dance that changed depending on how fast they were moving and how tightly they were packed.

5. Why This Matters

This is a big deal for the future of technology.

The Limit: As we make computers and phones smaller and faster, we are hitting a wall. We thought we could just keep shrinking the wires. This paper says, "Stop! You can't just shrink them forever because the electrons start behaving weirdly."
The Map: The scientists didn't just find the problem; they drew a map of it. They measured exactly how the electrons behave in these tiny spaces.
The Future: Now, engineers can use this map to design better, faster, and smaller devices. Instead of guessing, they can predict exactly how electricity will flow in the next generation of super-fast chips.

In a Nutshell

This paper is like discovering that the rules of traffic change when you drive a car the size of a marble. The scientists used a super-sharp, super-fast "needle" to watch electrons in graphene and proved that at the smallest scales, electricity doesn't just flow locally; it reacts to the whole environment around it. This changes how we design the tiny, super-fast electronics of tomorrow.

1. Problem Statement

As photonic and electronic technologies shrink to nanometer dimensions and operate at terahertz (THz) speeds, the classical assumption that electrical conductivity is a purely local material parameter ( $\sigma$ ) breaks down. In low-dimensional conductors like graphene, charge transport becomes intrinsically nonlocal, meaning conductivity depends on both frequency ( $\omega$ ) and momentum ( $q$ ), denoted as $\sigma(\omega, q)$ .

The Gap: While nonlocal effects have been theoretically predicted for decades (e.g., by Lindhard and Mermin), direct experimental measurement of $\sigma(\omega, q)$ in real materials has been elusive.
Limitations of Existing Methods:
- Contact-based transport: Measures device-integrated conductance, lacking spatial resolution.
- Conventional optical probes: Operate at frequencies where nonlocal effects are confined to a few nanometers, often suppressed or treated as minor corrections.
- Existing Near-field studies: Have visualized plasmons but often relied on local-response models, failing to capture the momentum-dependent nature of conductivity at deep sub-wavelength scales.

The authors argue that in the THz regime, nonlocal effects extend to tens or hundreds of nanometers, making them experimentally accessible and critical for designing ultracompact devices.

2. Methodology

The study employs Terahertz Scattering-type Scanning Near-field Optical Microscopy (THz-s-SNOM) combined with quantitative electrodynamic modeling.

Instrumentation:
- A commercial THz-s-SNOM system (Attocube THz-NeaSCOPE) using a tapping-mode Atomic Force Microscope (AFM).
- Tip Modification: Standard AFM tips (nominal radius <50 nm) were gently "dented" to increase their apex radius to several hundred nanometers (e.g., ~385 nm). This modification enhances the scattering signal strength.
- Signal Detection: Single-cycle THz pulses (0.5–1.5 THz) interact with the metallic tip. The scattered signal is demodulated at higher harmonics ( $m=2$ to $5$) of the tapping frequency to isolate the near-field component and suppress far-field background.
Samples: Mechanically exfoliated graphene flakes (monolayer, bilayer, and trilayer) on SiO $_2$ /Si substrates.
Modeling & Analysis:
- Finite Dipole Model (FDM): Used to simulate the tip-sample interaction and retrieve the complex conductivity from the measured amplitude and phase.
- Nonlocal Conductivity Model: The experimental data was fitted against the Lovat-BGK model (based on semiclassical Boltzmann transport with the Bhatnagar-Gross-Krook approximation). This model accounts for spatial dispersion ( $\sigma(\omega, q)$ ).
- Comparison: Results were contrasted with the conventional local Drude model to demonstrate the necessity of nonlocal theory.
- Field Confinement Analysis: Finite Element Method (FEM) simulations revealed that despite the large tip radius, the THz field is "super-focused" under the tip, confining the interaction to a radius (~50 nm) much smaller than the tip itself. This generates high in-plane momenta ( $q$ ) essential for probing nonlocality.

3. Key Contributions

Direct Measurement of $\sigma(\omega, q)$ : The paper provides the first direct experimental extraction of the complex, momentum-dependent conductivity of graphene in the THz regime with ~50 nm spatial resolution.
Validation of Nonlocal Dominance: It demonstrates that nonlocal response is not a minor correction but the dominant factor governing THz conductivity in monolayer graphene, even at length scales relevant to practical devices (~50 nm).
Superfocusing Mechanism: The authors clarify that the "superfocusing" of the THz field under the AFM tip allows the probing of high-momentum states ( $q \sim 0.02 \text{ nm}^{-1}$ ), which is the key to observing nonlocal effects at THz frequencies.
Layer-Dependent Contrast: The study reveals distinct THz contrast between mono-, bi-, and trilayer graphene, a feature previously difficult to observe in the THz regime where graphene was often considered a near-perfect reflector.

4. Key Results

Spectroscopic Evidence:
- Measured conductivity spectra showed a negative imaginary part ( $\text{Im}(\sigma) < 0$ ) within the 0.5–1.5 THz range.
- This behavior contradicts the local Drude model (which predicts $\text{Im}(\sigma) > 0$ for $\omega\tau < 1$ ) but matches the resonant-like behavior predicted by the Lovat nonlocal model.
- The nonlocal response arises from the interplay between carrier inertia and spatially dispersive charge-density dynamics.
Quantitative Parameters:
- Fitting the Lovat model yielded Fermi energies ( $E_F$ ) of ~231 meV (monolayer) and ~157 meV (bilayer), and scattering times ( $\tau$ ) of ~62 fs and ~50 fs, respectively.
- The effective in-plane momentum probed was $q \approx 0.02 \text{ nm}^{-1}$ , corresponding to a length scale of ~50 nm.
Spatial Mapping:
- Conductivity maps revealed significant heterogeneity across single flakes, resembling "eroded geological features."
- A pronounced anticorrelation was observed between Fermi energy and scattering time across the flake, likely due to variations in the transfer process or local doping.
- Clear differentiation was achieved between mono-, bi-, and trilayer regions based on conductivity magnitude.
Field Localization: Simulations confirmed that the nonlocal "reach" (spatial extent of the response) in the THz regime is hundreds of nanometers, overlapping significantly with the confined THz field under the tip, unlike in the mid-IR where nonlocality is confined to a few nanometers.

5. Significance

Paradigm Shift in Modeling: The findings establish that local-response models are insufficient for designing nanoscale THz devices. Ignoring nonlocality leads to qualitatively incorrect predictions regarding field confinement, dispersion, and loss.
Design Relevance: Nonlocal conductivity is identified as a measurable, design-relevant material property. Engineers must account for $\sigma(\omega, q)$ when scaling devices to the 50 nm regime.
Broad Applicability: The methodology is applicable to a wide class of low-dimensional conductors (e.g., topological insulators, oxide interfaces) where long carrier mean free paths coexist with deep-subwavelength confinement.
Foundation for Future Tech: This work provides a quantitative foundation for predicting performance limits in ultracompact photonic and electronic systems, bridging the gap between nanophotonics and high-speed electronics.

In summary, this paper moves nonlocal conductivity from a theoretical concept to an experimentally verified, dominant physical phenomenon in the THz regime, fundamentally changing how nanoscale electronic transport must be modeled and measured.