Coupled electron-phonon hydrodynamics and viscous thermoelectric equations

This paper presents a first-principles framework that unifies electron and phonon hydrodynamics into coupled viscous thermoelectric equations, demonstrating how composite "relaxon" excitations govern non-diffusive transport and predicting distinct signatures of electron-phonon drag in complex device geometries.

The Gist

Imagine a crowded dance floor inside a solid material like graphite. Usually, when you push a crowd, people bump into each other chaotically, spreading out slowly like a drop of ink in water. This is how electricity and heat usually move in most materials: diffusively. It's messy, slow, and predictable.

But in certain special materials and conditions, the crowd behaves differently. Instead of bumping and scattering randomly, they move together in a coordinated, fluid-like wave, swirling and flowing like water in a river. This is hydrodynamic transport.

This paper is a groundbreaking guidebook on how to predict and understand what happens when two different types of dancers (electrons and phonons) try to dance together on this floor.

Here is the story of the paper, broken down into simple concepts:

1. The Two Dancers: Electrons and Phonons

To understand the paper, you need to know the two main characters:

• Electrons: The tiny, charged particles that carry electricity. Think of them as the "electric dancers."
• Phonons: These aren't particles you can hold; they are vibrations of the material's atoms, like sound waves or heat ripples. Think of them as the "heat dancers."

In most materials, these two groups dance separately. The electrons bump into the phonons, but they don't really coordinate. They just get in each other's way.

2. The "Drag" Effect: Holding Hands

The paper focuses on a phenomenon called electron-phonon drag. Imagine the heat dancers (phonons) are moving in a strong current. If they hold hands with the electric dancers (electrons), they can drag the electrons along with them. Conversely, if the electric dancers are rushing, they can drag the heat dancers.

Usually, scientists treated these two groups as separate fluids. But this paper argues that in materials like graphite, they can become a single, mixed fluid (a "bifluid") where they move together, or a coupled pair where they influence each other's flow significantly.

3. The New Rulebook: Viscous Thermoelectric Equations (VTE)

For decades, scientists used two different rulebooks:

• One for electricity (Gurzhi's equations).
• One for heat (Viscous Heat Equations).

The authors created a super-rulebook called the Viscous Thermoelectric Equations (VTE). This new book unifies the old ones. It describes how electricity and heat flow when they act like a thick, sticky fluid (like honey) rather than a thin gas.

The Key Innovation:
The authors realized that to describe this mixed dance, you can't just look at the temperature or the voltage. You have to account for viscosity (stickiness). Just like honey resists flowing through a narrow pipe, these electron-phonon fluids resist changing direction, creating swirls and eddies.

4. The "Relaxon" Concept: The Team Players

To make the math work, the authors introduced a new idea called a "Relaxon."

• Imagine a chaotic crowd where everyone is moving randomly.
• Now, imagine that crowd organizing into specific "teams" or "modes" that move together.
• A Relaxon is one of these teams. It's a collective wave of electrons and phonons moving in sync.

The paper shows that these "teams" have a specific "parity" (a mathematical symmetry). Some teams are responsible for the usual, boring flow of heat and electricity (diffusion). Others are responsible for the cool, fluid-like swirling (hydrodynamics). By studying these teams, the authors can predict exactly how the material will behave.

5. The Graphite Experiment: The "Tunnel and Chamber"

To prove their theory, the authors simulated a specific shape: a long tunnel leading into a wide circular chamber (like a funnel opening into a room).

• The Old Prediction (Diffusive): If you push heat or electricity through this shape, it should flow straight through. The temperature and voltage should change smoothly and evenly.
• The New Prediction (Hydrodynamic): Because the fluid is "sticky" (viscous), when it hits the wide chamber, it doesn't just stop. It swirls.
  • Vortices: The flow creates tiny whirlpools.
  • Backflow: In some spots, the heat or electricity actually flows backward, against the direction you pushed it!
  • Inversion: The temperature or voltage in the center of the chamber might actually be higher or lower than you'd expect, creating a "reverse" pattern compared to normal materials.

6. The "Smoking Gun": How to Spot It

The paper gives scientists a checklist to prove this is happening in real life:

1. Look for Swirls: If you can image the flow, you should see whirlpools.
2. Check the Center: Measure the temperature or voltage in the middle of the chamber. If it's "inverted" (opposite to the gradient), it's a sign of hydrodynamics.
3. The "Compressibility" Test: In normal electron flow (like in graphene), the fluid is "incompressible" (it doesn't squish). But in this mixed electron-phonon fluid, the authors predict the fluid is compressible. This means the voltage pattern won't be a perfect, smooth curve; it will have "bumps" and irregularities that standard math can't explain.

Why Does This Matter?

This isn't just about fancy math. It's about the future of electronics.

• Heat Management: Computers get hot because heat diffuses slowly. If we can make heat flow like a fluid, we might be able to direct it away from sensitive parts much faster.
• New Devices: By understanding how to control these "swirls" and "backflows," engineers could design devices that focus electricity or heat in specific spots, creating super-efficient thermoelectric generators or ultra-fast transistors.

In Summary:
This paper is like discovering that traffic in a city doesn't just jam randomly; under the right conditions, cars can flow like a river, creating whirlpools and moving against the current. The authors have written the new traffic laws (the VTE) to predict exactly how this happens when the "cars" (electrons) and the "wind" (phonons) hold hands and move together.

Technical Summary

1. Problem Statement

While hydrodynamic transport of heat (phonons) and charge (electrons) has been observed separately in various materials (e.g., graphite, graphene), the simultaneous emergence and interaction of electron-phonon bifluids remain poorly understood.

• The Gap: Existing theoretical frameworks typically treat electron-only or phonon-only hydrodynamics separately, or assume a "perfectly mixed" single-fluid regime where electrons and phonons share a single drift velocity.
• The Challenge: There is a lack of a first-principles framework to quantitatively describe the non-perfectly-mixed regime, where electron and phonon fluids coexist with distinct drift velocities, interact via drag, and exhibit complex viscous effects.
• Specific Questions: Can semimetals like graphite display multi-component transport physics? Can electron-phonon drag significantly alter phonon hydrodynamics (and vice versa)? What are the experimental signatures of this coupled regime?

2. Methodology

The authors developed a comprehensive theoretical and computational framework bridging microscopic quantum mechanics and macroscopic continuum hydrodynamics.

• Microscopic Foundation: They start from the linearized, coupled electron-phonon Boltzmann Transport Equation (epBTE). They reformulate this equation using a relaxon framework, where collective excitations (relaxons) are defined as eigenvectors of the scattering matrix.
• Symmetry and Parity Analysis: A key innovation is the classification of relaxons based on parity (odd vs. even) under time-reversal and wavevector inversion:
  • Odd Relaxons: Govern standard diffusive transport (driven by gradients of temperature and potential).
  • Even Relaxons: Govern hydrodynamic viscous flows (driven by gradients of drift velocities).
  • Drag Effects: Electron-phonon drag hybridizes these relaxons, mixing electron and phonon components into single collective excitations.
• Coarse-Graining: Using these microscopic conservation laws (energy, charge, and quasi-conserved crystal momentum), the authors coarse-grain the epBTE into a set of mesoscopic partial differential equations called the Viscous Thermoelectric Equations (VTE).
• First-Principles Implementation: The framework was implemented in the open-source Phoebe code using Density Functional Theory (DFT) to calculate scattering rates, band structures, and coupling matrix elements for graphite.
• Device Simulation: The VTE were solved numerically for complex device geometries (tunnel-chamber and mixing devices) to predict macroscopic signatures under various doping levels and boundary conditions.

3. Key Contributions

A. Theoretical Unification

The VTE formally unify three previously distinct regimes:

1. Gurzhi's Equation: Describes electron-only hydrodynamics (incompressible flow).
2. Viscous Heat Equations (VHE): Describes phonon-only hydrodynamics.
3. Coupled Bifluid Regime: The new contribution, extending these to cover systems where electrons and phonons have different drift velocities ($u_e \neq u_p$) and interact via drag.

B. Identification of Transport Coefficients

The paper derives explicit expressions for all transport coefficients in terms of relaxon properties (lifetimes $\tau_\xi$ and velocities):

• Diffusive Coefficients: Electrical conductivity ($\sigma$), thermal conductivity ($\kappa$), Seebeck ($S$), and Peltier ($\alpha$) are determined by odd-parity relaxons.
• Viscous Coefficients: Shear and bulk viscosities for electrons ($\eta_{ee}$), phonons ($\eta_{pp}$), and cross-viscosities ($\eta_{ep}, \eta_{pe}$) representing momentum exchange are determined by even-parity relaxons.
• Drag Viscosity: The cross-viscosities are non-zero only when electron-phonon drag is present, quantifying the momentum diffusion between the two fluids.

C. Prediction of Non-Harmonic Potentials

Unlike standard diffusive transport or electron-only hydrodynamics (which often assume incompressibility and harmonic potentials), the VTE predict that electron fluids in coupled regimes are compressible ($\nabla \cdot u_e \neq 0$). This leads to non-harmonic electric potential fields ($\nabla^2 V \neq 0$), a distinct signature of coupled hydrodynamics.

4. Key Results (Case Study: Graphite)

A. Doping-Dependent Transport

Simulations on graphite at 70–120 K revealed:

• Seebeck Coefficient: Electron-phonon drag significantly impacts the Seebeck coefficient, with the effect maximized at lower temperatures and varying strongly with doping (electron vs. hole).
• Viscosities:
  • Electronic Shear Viscosity: Decreases with temperature (liquid-like behavior) and is highly sensitive to doping.
  • Phonon Shear Viscosity: Increases with temperature (gas-like behavior) and is largely insensitive to doping.
  • Drag: Drag effects are significant in both strongly electron-doped and weakly hole-doped graphite, but the resulting hydrodynamic behavior differs.

B. Macroscopic Signatures in Device Geometries

Using a "tunnel-chamber" device geometry, the authors predicted distinct signatures of coupled hydrodynamics:

1. Vortices and Backflow: In strongly electron-doped graphite, the VTE predict vortices (closed streamlines) for both heat and charge fluxes. This leads to backflow, where heat and charge flow against their respective gradients in the chamber.
2. Inversion of Fields: The temperature and voltage profiles in the chamber are inverted compared to diffusive predictions (e.g., the center of the chamber becomes hotter/colder than the boundaries in a way that defies standard diffusion).
3. Decoupling of Heat and Charge:
   • In strongly electron-doped graphite: Both heat and charge exhibit hydrodynamic vortices and inversion.
   • In weakly hole-doped graphite: Heat exhibits hydrodynamic vortices (phonon-dominated), but charge flow remains diffusive (no vortices, no voltage inversion). This proves that heat and charge hydrodynamics can exist independently.
4. Non-Harmonic Potential: The electric potential in the hydrodynamic regime violates the mean-value property of harmonic functions, providing a measurable test for compressibility.

C. Mixing Geometry

In a device where electron and phonon flows merge, the authors observed that the two fluids flow with distinct length scales (Gurzhi lengths), maintaining separate Poiseuille profiles before mixing, further confirming the bifluid nature.

5. Significance and Impact

• New Paradigm for Thermoelectrics: The work establishes that electron-phonon drag is not just a correction to diffusive transport but can drive a transition to a coupled hydrodynamic regime with unique, non-diffusive signatures.
• Experimental Guidance: The paper provides "smoking-gun" signatures for experimentalists to identify coupled hydrodynamics:
  • Temperature/Voltage Inversion: Measuring non-monotonic profiles in micro-devices.
  • Non-Harmonic Potentials: Detecting deviations from Laplace's equation in voltage maps.
  • Simultaneous Backflow: Observing heat and charge flowing against gradients simultaneously.
• Technological Potential: Understanding these viscous effects could lead to novel electronic devices where heat and charge flows can be manipulated independently (e.g., focusing charge flux while dissipating heat, or vice versa) by exploiting device geometry and doping.
• Open-Source Tool: The authors released the updated Phoebe software suite, enabling the community to perform first-principles calculations of coupled electron-phonon hydrodynamics for other materials.

In summary, this paper provides the first rigorous, first-principles framework to describe the coupled electron-phonon bifluid regime, moving beyond the single-fluid approximation and revealing rich, non-intuitive transport phenomena in semimetals like graphite.