🔬 mesoscale physics

Hall effect in viscous flows of two-dimensional electrons in samples with edges of arbitrary roughness

This paper develops a theoretical model for hydrodynamic magnetotransport in two-dimensional electron systems with arbitrary edge roughness, predicting that specific non-monotonic behaviors in Hall resistance—such as high-field saturation and magnetic-field-induced minima—serve as distinct experimental signatures of the viscous flow regime and allow for the quantification of edge roughness.

Original authors: A. V. Gert, P. S. Alekseev

Published 2026-08-28

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CC BY 4.0

Original authors: A. V. Gert, P. S. Alekseev

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

In the quiet, ultra-clean interiors of certain modern materials, electrons do not behave like the solitary, bouncing billiard balls often imagined in introductory physics. Instead, when these materials are pure enough, the electrons collide with each other so frequently that they begin to move as a collective, flowing substance. This state is known as the hydrodynamic regime, where the electron gas acts much like a viscous fluid, such as honey or water, rather than a collection of independent particles. In this fluid-like state, the electrons possess a property called viscosity, which describes their internal resistance to flow, and they can form complex patterns like whirlpools when pushed through narrow channels. Understanding how this electron fluid behaves, especially when subjected to magnetic fields, is crucial for developing next-generation electronics and for probing the fundamental nature of matter in its most pristine forms.

A team of researchers at the Ioffe Institute in St. Petersburg has now developed a detailed theory describing how this viscous electron fluid moves through long, narrow strips of material that have rough or uneven edges. While previous studies had looked at how these fluids behave in perfectly smooth containers or in specific, idealized conditions, this new work addresses the messy reality of actual laboratory samples, where the edges are rarely perfect. The researchers focused on a specific phenomenon called the Hall effect, which occurs when a magnetic field pushes a flowing electric current sideways, creating a measurable voltage. In standard, non-viscous materials, this sideways voltage is predictable and steady. However, in these viscous electron fluids, the interaction between the fluid's internal friction, the magnetic field, and the texture of the container's walls creates a much more complicated and surprising behavior.

The scientists built a mathematical model to simulate the flow of two-dimensional electrons in long samples with edges of varying roughness. They treated the roughness not as a complex geometric puzzle, but as a single, adjustable property that determines how much the fluid "sticks" to the walls versus how much it "slips" past them. In their simulations, they observed that when the edges are very smooth, the fluid glides easily, and the sideways voltage changes in a simple, predictable way as the magnetic field gets stronger. However, when the edges are rough, causing the fluid to stick and slow down significantly, the behavior changes dramatically. The researchers found that under these rough conditions, the extra voltage generated by the fluid's viscosity does not just grow or shrink steadily. Instead, it rises, dips to a distinct minimum, and then rises again as the magnetic field increases. This dip, or minimum, is a unique fingerprint of the hydrodynamic regime, appearing only when the fluid is viscous enough and the edges are rough enough to create a specific balance of forces.

The study reveals that the exact position and depth of this minimum depend on three main factors: how rough the edges are, how wide the sample is, and how many impurities exist inside the material itself. If the sample is very wide, or if the edges are smoothed out, or if there are too many defects inside the material, this special dip disappears, and the voltage returns to a simple, monotonic behavior. The researchers also discovered a precise, universal relationship connecting the resistance of the material along the direction of the flow with the sideways Hall resistance. This connection holds true regardless of the specific edge conditions, offering a powerful new tool for experimentalists. By measuring how the resistance changes with the magnetic field, scientists can now determine not only if they are observing a viscous electron fluid but also quantify exactly how rough the edges of their sample are and how clean the material is inside.

This work provides a clear set of signs for identifying the hydrodynamic regime in real-world experiments, moving beyond the need for perfectly idealized samples. The presence of a non-monotonic dip in the Hall resistance, which vanishes as the sample becomes wider or the edges smoother, serves as a reliable indicator that the electrons are flowing as a viscous fluid. Furthermore, the ability to extract the degree of edge roughness from these measurements allows researchers to characterize the quality of their ultra-pure samples with greater precision. By confirming that these theoretical predictions match the complex behaviors seen in high-quality materials like gallium arsenide quantum wells, the study bridges the gap between abstract fluid dynamics and the tangible properties of solid-state electronics, offering a new lens through which to view the collective motion of electrons.

Technical Summary: Hall Effect in Viscous Flows of Two-Dimensional Electrons in Samples with Edges of Arbitrary Roughness

Problem Statement
In ultra-clean conductors, such as high-quality graphene and GaAs quantum wells, the electron-electron (ee) collision rate can exceed the rate of scattering by impurities, leading to the formation of a viscous electron fluid and a hydrodynamic transport regime. While the hydrodynamic contribution to the Hall effect has been observed experimentally, a comprehensive analytical theory describing this effect in samples with arbitrary edge roughness and weak bulk defects has been lacking. Specifically, the influence of edge conditions—ranging from nearly specular reflection to complete sticking (no-slip)—on the Hall resistance, particularly in the presence of a magnetic field, requires further theoretical clarification. Previous studies have addressed specific boundary conditions or relied on numerical solutions of the kinetic equation; this work aims to provide a unified analytical framework.

Methodology
The authors develop an analytical theory for the stationary hydrodynamic magnetotransport of two-dimensional (2D) electrons in long samples ( $L \gg W$ ) with arbitrary edge roughness and low densities of bulk defects. The approach is based on the following theoretical constructs:

Kinetic Equation Simplification: The relaxation of the electron distribution function is modeled using a simplified collision integral. In the bulk, this includes momentum-conserving ee-collisions (relaxing shear stresses) and momentum-relaxing impurity scattering. At the edges, a local collision operator is introduced to model scattering of arbitrary strength.
Boundary Conditions: The edge scattering is characterized by a single parameter, $a$ $a$ , with dimensions of speed. This parameter interpolates between different physical regimes:
- $a \ll v_F$ : Smooth edges with nearly specular reflection (zero normal velocity gradient).
- $a \sim v_F$ : Moderately rough edges with diffuse reflection.
- $a \gg v_F$ : Extremely rough edges leading to almost complete sticking (zero velocity).
  These conditions yield a magnetic-field-dependent slip length, $l_{sl} = \eta_{xx}(\omega_c)/a$ , where $\eta_{xx}$ is the diagonal viscosity component.
Governing Equations: The authors solve the Navier-Stokes equations for the hydrodynamic velocity $V(y)$ and the Hall electric field $E_H(y)$ in the presence of a perpendicular magnetic field $B$ . The viscosity tensor includes both diagonal ( $\eta_{xx}$ ) and off-diagonal (Hall viscosity, $\eta_{xy}$ ) components.
Analytical Solution: The system is solved to derive expressions for the velocity profile, longitudinal current, and the resulting longitudinal ( $\rho_{xx}$ ) and Hall ( $\rho_{xy}$ ) resistivities.

Key Contributions and Results

Velocity Profiles and Longitudinal Resistance: The study calculates electron flow profiles for samples of varying widths and edge roughness. It demonstrates that the giant negative magnetoresistance observed in narrow, clean samples with rough edges gradually disappears as edge smoothness increases or the sample width grows, eventually transitioning to an Ohmic regime dominated by bulk defect scattering.
Hall Resistance Behavior: The primary result is the calculation of the hydrodynamic correction to the Hall resistance, $\delta\rho_{xy} = \rho_{xy} - \rho_{xy}^{st}$ $δ ρ_{x y} = ρ_{x y} - ρ_{x y}^{s t}$ , where $\rho_{xy}^{st}$ $ρ_{x y}^{s t}$ is the standard kinematic Hall resistivity.
- Non-Monotonicity: For sufficiently rough edges (where the edge relaxation parameter $s < 1$ ), $\delta\rho_{xy}$ exhibits a non-monotonic dependence on the magnetic field $B$ , characterized by a distinct minimum.
- Saturation: In the limit of high magnetic fields, this correction saturates to a constant value that is independent of the edge roughness parameter $a$ .
- Transition to Monotonicity: As edge roughness decreases (parameter $a$ decreases), the sample width increases, or bulk scattering by defects increases, the minimum in $\delta\rho_{xy}(B)$ shifts to larger magnetic fields and eventually disappears, resulting in a monotonic dependence.
Universal Relation: The authors derive a universal relation connecting the longitudinal and Hall resistances within this model: $\rho_{xx}(\beta) + \delta\rho_{xy}(\beta) / (2\beta) \equiv \rho_D$ , where $\rho_D$ is the Drude resistivity. This relation offers a potential consistency check for experimental data.
Parameter Determination: The position of the minimum in the Hall resistance curve ( $\beta_{min}$ ) and the saturated high-field value are shown to depend on the sample width, bulk scattering time, and the edge roughness parameter $a$ .

Significance and Claims
The paper claims that the derived properties of the Hall resistance—specifically the existence, position, and magnitude of the minimum in $\delta\rho_{xy}(B)$ as a function of edge roughness—serve as distinct signatures for identifying the hydrodynamic regime of electron transport in experiments.

Furthermore, the authors assert that these theoretical results provide a practical method for characterizing ultra-pure samples. By measuring the Hall resistance dependence on the magnetic field and comparing it with the derived analytical expressions, one can determine the degree of edge roughness and other transport parameters of the sample. The work bridges the gap between purely hydrodynamic theories and ballistic-hydrodynamic crossover regimes, offering a unified description that accounts for realistic boundary conditions from slip to sticking.

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