🔬 mesoscale physics

Characterizing electronic scattering rates with transport in multiterminal devices

This paper demonstrates that a single linearized Boltzmann simulation of a five-terminal device can distinguish between ballistic, hydrodynamic, and diffusive transport regimes and extract specific scattering rates without requiring space-resolved imaging.

Original authors: Jack H. Farrell, Andrew Lucas

Published 2026-05-06

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Jack H. Farrell, Andrew Lucas

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 a crowded dance floor inside a tiny, two-dimensional electronic device. The "dancers" are electrons. How these electrons move depends on how often they bump into each other and how often they bump into the walls or impurities in the room.

Scientists have long known that these electrons can move in three very different "styles" or regimes:

Ballistic: Like sprinters in a race, they zoom in straight lines without stopping or turning, only hitting the walls.
Hydrodynamic: Like a thick liquid (honey or water), they flow together, swirling around obstacles and sticking to the crowd's momentum.
Diffusive (Ohmic): Like a chaotic crowd in a busy hallway, they bump into everything constantly, moving in a slow, random zig-zag.

Usually, to figure out which style the electrons are using, scientists need expensive, high-tech cameras to take pictures of the electrons moving inside the device (spatial imaging). This paper argues that you don't need a camera. You just need a cleverly designed "fan-shaped" device with five doors (terminals) and a simple measurement of how much current flows out of each door.

Here is the breakdown of their discovery using everyday analogies:

The "Fan" Experiment

The researchers designed a device shaped like a fan with one entry point (the source) and five exit points (drains) arranged at different angles.

The Setup: They push electrons in from the bottom.
The Observation: They measure how the current splits up among the five exits.

How the "Split" Tells the Story

The way the current divides depends entirely on the "dance style" of the electrons:

In the Ballistic Regime (The Sprinters):
Imagine throwing a handful of marbles through a funnel. If they don't hit anything, they keep going straight. In this regime, electrons prefer to go straight to the exit directly opposite the entry. They ignore the side exits. If you see most of the current going straight and very little to the sides, you know the electrons are in "ballistic" mode.
In the Hydrodynamic Regime (The Liquid):
Imagine pouring water into that same funnel. Water flows around corners and spreads out. If the electrons are acting like a fluid, they will easily flow into the side exits, even if those exits are at an angle. The current spreads out more evenly.
In the Diffusive Regime (The Crowd):
Imagine a crowd of people shuffling through a maze. They bump into walls and each other so much that the shape of the path matters. The current distribution changes based on the width and length of the channels, following standard rules of resistance (like electricity in a wire).

The "Magic" of the Crossover

The most exciting part of the paper is what happens in the middle, where the electrons are switching from one style to another (the "crossover" regime).

The authors found that by looking at the exact ratio of current in the different exits, they could solve a mathematical puzzle to figure out two hidden numbers:

How often electrons bump into the walls/impurities (slowing them down).
How often electrons bump into each other (helping them flow like a fluid).

Think of it like a detective looking at tire tracks on a road. You don't need to see the car to know if it was speeding, drifting, or braking; you just need to measure the angle and depth of the tracks. Similarly, the "tracks" left by the current in the five exits reveal the hidden "scattering rates" (how often collisions happen) without needing to take a picture of the electrons.

The "Tomographic" Twist

The paper also mentions a more subtle, intermediate state called "tomographic" flow. This is like a dance where the electrons remember their direction for a little while longer than usual before they forget and scatter randomly.
The researchers found that their five-terminal fan setup is sensitive enough to spot this specific behavior. It's like being able to tell the difference between a fluid that flows smoothly and one that has a slight "memory" of its previous direction, just by looking at how the water splashes out of the fan's exits.

Why This Matters

Previously, to measure these specific details, scientists needed complex microscopes to watch the electrons move in real-time. This paper shows that a simple, standard electrical experiment on a multi-terminal chip is enough to get the same detailed information. It turns a complex, invisible microscopic world into a set of simple numbers you can read with a multimeter.

In short: You don't need a camera to see how electrons behave; you just need to ask them "which way are you going?" by measuring the current at five different doors. The answers tell you exactly how they are colliding and interacting.

Technical Summary: Characterizing Electronic Scattering Rates with Transport in Multiterminal Devices

Problem Statement
Strongly interacting electrons in clean two-dimensional (2D) devices are theorized to exhibit distinct transport regimes: ballistic, hydrodynamic, and diffusive (Ohmic). Realistic samples often exist in crossover regimes between these idealized limits. While bulk observables like nonlocal conductivity and the Gurzhi effect can qualitatively distinguish pairs of regimes (e.g., hydrodynamic vs. diffusive), sharply distinguishing all regimes—particularly the intermediate "tomographic" regime characterized by a large number of nearly-conserved quantities—remains challenging. Historically, quantitatively determining microscopic scattering parameters (such as momentum-relaxing and momentum-conserving rates) has required sophisticated space-resolved imaging to resolve phenomena like Poiseuille flow profiles, viscous backflow, or vorticity. The authors argue that full space-resolution is not strictly necessary to quantitatively measure these parameters if one utilizes multiterminal devices with sufficient contacts.

Methodology
The authors propose a theoretical framework utilizing a specific five-terminal "fan" geometry (Fig. 1) to diagnose transport regimes. The core methodology involves:

Linearized Boltzmann Model: They solve the linearized Boltzmann transport equation for isotropic Fermi liquids near $T=0$ . The deviation from equilibrium is expanded in Fourier modes on the Fermi surface.
Scattering Model: The collision integral is parameterized by three rates:
- $\gamma_{mr}$ : Momentum-relaxing scattering (impurities/phonons).
- $\gamma_{ee}$ : Momentum-conserving electron-electron scattering.
- $\gamma_3$ : A parameter governing the relaxation of odd harmonics, characterizing the "tomographic" regime where even-odd scattering structures dominate.
Numerical Framework: They employ an open-source numerical framework (FermiSea.jl) using a nodal discontinuous Galerkin (DG) method to solve the equation in arbitrary 2D domains with realistic boundary conditions (diffuse and specular walls).
Experimental Protocol: The study focuses on a grounded-contact experiment where a source contact is biased, and the remaining contacts act as grounded drains. The primary observables are the current branching ratios ( $I_a/I_N$ ) into the various drain arms.

Key Results

Regime Diagnosis via Current Partition: The distribution of current among the drain contacts is strongly regime-dependent.
- Ballistic Limit: Carriers follow straight trajectories, preferentially entering aligned drains. Current fractions into non-aligned or backward-facing drains are suppressed ( $I_a/I_N < 1$ ).
- Hydrodynamic Limit: Viscous flow allows current to redistribute around bends. The current fractions depend on channel width ( $w$ ) and length ( $\ell$ ) differently than in the Ohmic limit (scaling as $w^{-3}$ vs. $w^{-1}$ ).
- Crossover Regime: The three independent current fractions ( $I_{NE}/I_N$ , $I_E/I_N$ , $I_{SE}/I_N$ ) exhibit a non-degenerate dependence on $\gamma_{ee}$ and $\gamma_{mr}$ .
Inverse Problem Conditioning: The authors demonstrate that the crossover regime ( $\gamma_{ee} \sim v_F/L$ ) presents a well-conditioned inverse problem. By measuring the current fractions, one can quantitatively "triangulate" and extract both $\gamma_{ee}$ and $\gamma_{mr}$ without spatial imaging. Conversely, the strongly Ohmic regime is shown to be ill-conditioned for this extraction.
Tomographic Signatures: The framework is sensitive to the tomographic regime ( $\gamma_3 < \gamma_{ee}$ ). The authors show that varying the ratio $\gamma_3/\gamma_{ee}$ (modeled via temperature dependence) leads to distinct, sharp features in the current branching ratios, particularly a sign change in $(I_{NE}/I_N - 1)$ at intermediate $\gamma_{ee}$ . This offers a potential quantitative method to distinguish tomographic flow from viscous or ballistic transport, which is difficult to achieve using only sample size scaling.
Robustness: The results hold for both current-driven (fixed current, floating voltage probes) and voltage-driven (fixed voltage, grounded drains) configurations, with the latter being the primary focus of the main text.

Significance and Claims
The paper claims that multiterminal devices offer a simpler experimental route to characterize transport regimes in electron liquids compared to space-resolved imaging experiments. Specifically:

Quantitative Extraction: It provides a method to quantitatively determine microscopic scattering parameters ( $\gamma_{mr}$ and $\gamma_{ee}$ ) in the crossover regime using only bulk current measurements.
Tomographic Identification: It suggests that multiterminal observables may provide the most quantitative way currently available to distinguish the intermediate tomographic regime from viscous or ballistic transport, addressing a gap where scaling predictions are difficult to detect with limited sample sizes.
Geometric Sensitivity: The study illustrates that carefully designed device geometries can convert subtle information about the collision integral (specifically the even-odd structure of scattering) into robust, directly accessible bulk observables.

The authors maintain a modest stance, noting that their results rely on specific geometric assumptions (e.g., circular Fermi surfaces) and that interpreting data from non-standard geometries often requires detailed comparison with numerical models. They do not claim to have experimentally realized these results but rather provide the theoretical framework and numerical evidence that such a characterization is feasible.