Heat flux deflection induced by hydrodynamic electron transport in a homogeneous Corbino disk under magnetic field

This study demonstrates that in a homogeneous 2D Corbino disk under a magnetic field, hydrodynamic electron transport induces a deflection of heat flux into the tangential direction, a phenomenon driven by momentum-conserving scattering and capable of reversing heat flow direction under specific electric or thermal gradients.

The Gist

The Big Picture: When Electrons Act Like a Crowd, Not Individuals

Imagine a crowded hallway.

• Normal Behavior (Diffusive): Usually, people (electrons) walk down the hall, bump into walls, bump into each other, and get distracted. They move slowly and randomly. If you push them from one end, they shuffle forward in a messy, slow line. This is how electricity usually works in most materials.
• Hydrodynamic Behavior: Now, imagine the hallway is so packed that people are holding hands and moving as a single, fluid unit. If you push the back of the line, the whole group surges forward together, like water flowing through a pipe. They don't just bump and stop; they swirl, spin, and flow around each other smoothly.

This paper studies what happens when electrons behave like that "fluid crowd" (hydrodynamic electron transport) inside a specific shape called a Corbino disk (think of a flat, round donut or a washer) while a magnetic field is applied.

The Setup: The Donut and the Magnet

The researchers set up a simulation with a few key ingredients:

1. The Donut (Corbino Disk): A flat, circular piece of material (like graphene) with a hole in the middle.
2. The Push: They either pushed electrons from the outside edge toward the center (using electricity) or heated the center to make them flow outward (using heat).
3. The Twist (Magnetic Field): They applied a magnetic field pointing straight up through the donut.

The Discovery: The "Drift" Effect

In normal physics, if you push electrons from the outside of a donut to the center, they go straight in. If you heat the center, heat flows straight out.

But in this "fluid" electron regime, something weird happened:
The flow didn't go straight. It curved.

• The Analogy: Imagine you are trying to walk straight toward the center of a spinning merry-go-round. Even if you aim straight, the spinning motion (the magnetic field) pushes you sideways.
• The Result: The electrons didn't just flow radially (in or out); they started swirling around the circle (tangentially). This is called Heat Flux Deflection. The heat and electricity took a curved path instead of a straight line.

Why Did This Happen? (The Two Types of Collisions)

The paper explains that this depends on how the electrons interact with each other:

1. The "Bouncers" (Momentum-Relaxing Scattering):

   • Imagine the electrons are like people in a chaotic mosh pit who keep bumping into walls and losing their energy. They stop and start constantly.
   • Result: The magnetic field can't make them turn because they are too busy stopping and starting. The flow stays straight. The "swirl" is suppressed.

2. The "Swarm" (Momentum-Conserving Scattering):

   • Imagine the electrons are like a school of fish. When one fish turns, the others turn with it. They conserve their collective momentum.
   • Result: Because they move together as a fluid, the magnetic field can easily push the whole "river" of electrons sideways. The flow curves dramatically. This is the Hydrodynamic Regime.

The "Reversal" Surprise

The most fascinating part of the paper is what happens when you change how you push the electrons:

• Scenario A (Electric Push): If you push electrons from the outside in with electricity, the magnetic field makes them swirl clockwise.
• Scenario B (Heat Push): If you heat the center to push them outward, the magnetic field makes them swirl counter-clockwise.

The Analogy: Think of a river.

• If you push a boat downstream (electricity), the current might spin it one way.
• If you push a boat upstream (heat flow), the same current might spin it the other way.
  The direction of the "swirl" flips depending on whether the electrons are being pushed by an electric force or a temperature difference.

Why Should We Care?

We usually think about electricity (how fast a computer chip runs) and heat (how hot it gets) as separate problems. But this paper shows that in advanced materials (like graphene), electricity and heat are deeply linked.

If we want to build super-fast, tiny electronic devices, we can't just manage the electricity; we have to manage the "fluid" flow of heat too. If we ignore this "swirling" effect, our devices might overheat or behave unpredictably.

Summary

• Normal electrons are like messy pedestrians; they go straight.
• Hydrodynamic electrons are like a fluid river; they can swirl and curve.
• Magnetic fields make this fluid electron river twist sideways.
• The Twist: Depending on whether you push with electricity or heat, the river swirls in opposite directions.

This research helps us understand how to control these "electron fluids" to build better, cooler, and faster future technology.

Technical Summary

1. Problem Statement

While hydrodynamic electron transport (where electron-electron scattering dominates, leading to fluid-like behavior) has been extensively studied regarding its electrical properties (e.g., negative nonlocal resistance, Poiseuille flow), its thermal properties remain underexplored.

• The Gap: Existing literature largely focuses on electrical conductivity and mobility. However, in modern semiconductor devices, thermal management under ultra-high heat flux is critical. Since electrical and thermal transports are coupled, understanding the thermal behavior of hydrodynamic electrons is essential.
• The Specific Challenge: How does a magnetic field affect heat transport in a hydrodynamic electron fluid within a specific geometry (Corbino disk)? Specifically, does the heat flux follow the temperature gradient (Fourier's law) or exhibit deflection due to Lorentz forces and collective electron motion?

2. Methodology

The authors employed a rigorous numerical approach to solve the coupled transport equations:

• Governing Equations:
  • Electron Boltzmann Transport Equation (eBTE): Used to describe the evolution of the electron distribution function $f$ under external electric ($E$) and magnetic ($B$) fields.
  • Poisson Equation: Coupled with eBTE to self-consistently calculate the electrostatic potential $\phi$ and charge density $\rho$.
• Scattering Models (Callaway Model):
  • Momentum-Conserving (MC) Scattering: Electron-electron interactions ($\tau_{mc}$) that drive the system toward a local equilibrium with a drift velocity $u$ (hydrodynamic regime).
  • Momentum-Relaxing (MR) Scattering: Interactions with impurities and phonons ($\tau_{mr}$) that drive the system toward a standard Fermi-Dirac distribution (diffusive regime).
• Numerical Techniques:
  • Solver: An implicit discrete ordinate method (DOM) was used to iteratively solve the stationary eBTE.
  • Nonlinearity Handling: The Newton method was applied to solve the nonlinear relationships between the Fermi-Dirac distribution, chemical potential, and temperature.
  • Geometry: A homogeneous 2D Corbino disk (annulus) with inner radius $r_{in}$ and outer radius $r_{out}$.
  • Material: Simulations were based on graphene parameters (linear dispersion, isotropic).
• Simulation Conditions:
  • Perpendicular magnetic field ($B = 0.01$ T).
  • Two driving scenarios: (1) Radial electric potential gradient ($\nabla \phi$) and (2) Radial temperature gradient ($\nabla T$).
  • Regimes compared: Diffusive ($\tau_{mr} \ll \tau_{mc}$) vs. Hydrodynamic ($\tau_{mc} \ll \tau_{mr}$).

3. Key Contributions

1. Discovery of Heat Flux Deflection: The study identifies a phenomenon where heat flux in the hydrodynamic regime deviates from the radial direction, acquiring a significant tangential component. This breaks the standard expectation of radial heat flow in a Corbino disk.
2. Mechanism Elucidation: The paper provides a physical explanation based on the competition between the driving force (electric field or temperature gradient) and the Lorentz force, mediated by the scattering mechanisms.
3. Directional Reversal: It demonstrates that the direction of the heat flux deflection (clockwise vs. counter-clockwise) reverses depending on whether the system is driven by an electric field or a temperature gradient.
4. Unified Framework: It establishes a numerical framework coupling eBTE and Poisson equations specifically for analyzing thermal transport in hydrodynamic regimes, bridging the gap between electrical and thermal hydrodynamics.

4. Key Results

A. Heat Flux Deflection Phenomenon

• Hydrodynamic Regime: When MC scattering dominates, heat flux does not flow strictly radially. Instead, it exhibits a tangential deflection, creating curved streamlines.
• Diffusive Regime: When MR scattering dominates, the deflection is negligible, and heat flow remains largely radial, adhering closer to Fourier's law.
• Quantitative Analysis: The deflection angle $\theta$ (where $\tan \theta = q_t / q_r$) is significantly larger in the hydrodynamic regime.

B. Influence of Scattering Mechanisms

• Suppression by MR: Momentum-relaxing scattering suppresses the deflection by dissipating the momentum gained from the driving force before the Lorentz force can significantly alter the trajectory.
• Promotion by MC: Momentum-conserving scattering allows electrons to maintain a collective drift velocity, making them more susceptible to the Lorentz force, thereby enhancing the deflection.

C. Driving Force Dependence (Electric vs. Thermal)

• Electric Field Driven ($\nabla \phi$):
  • Electrons move from the outer boundary to the inner boundary (against the potential gradient).
  • Lorentz force causes a clockwise deflection of electrons.
  • Result: Electric current flows counter-clockwise; Heat flux flows clockwise.
• Temperature Gradient Driven ($\nabla T$):
  • Heat flows from the hot inner boundary to the cold outer boundary.
  • The temperature gradient drives the electron distribution function to deviate away from the center.
  • Lorentz force causes a counter-clockwise deflection of electrons.
  • Result: Electric current flows clockwise; Heat flux flows counter-clockwise.
• Significance: The direction of heat flux deflection is reversed when switching between electric and thermal driving forces in the hydrodynamic regime.

D. Macroscopic Distributions

• Particle Density: In the hydrodynamic regime under an electric field, particle density accumulates significantly near the inner boundary due to reduced resistance and efficient inward flow.
• Temperature Rise: The hydrodynamic regime exhibits a larger temperature rise under the same potential difference compared to the diffusive regime, indicating stronger thermoelectric effects.

5. Significance and Implications

• Thermal Management: The findings suggest that in hydrodynamic materials (like high-quality graphene at low temperatures), heat transport is highly anisotropic and controllable via magnetic fields. This is crucial for designing thermal management strategies in nanodevices where heat flux deflection could lead to localized hot spots or unexpected cooling patterns.
• Fundamental Physics: The study confirms that the "fluid" nature of electrons extends to thermal transport, not just electrical. The reversal of heat flux direction based on the driving force type highlights the complex interplay between thermodynamics and electrodynamics in the hydrodynamic limit.
• Device Design: Understanding these deflection effects is vital for the design of Corbino disk-based sensors and thermoelectric devices, as standard diffusive models will fail to predict heat distribution accurately in the hydrodynamic regime.

In conclusion, this work provides the first comprehensive theoretical demonstration of magnetic-field-induced heat flux deflection in hydrodynamic electron transport, revealing a rich interplay between scattering mechanisms, driving forces, and geometry that fundamentally alters thermal transport behavior.