[Paper Review] Ballistic Heat Transfer and Energy Waves in an Electron System

Trung V. Phan, Justin C. W. Song+1 more|arXiv (Cornell University)|Jun 20, 2013

Spectroscopy and Quantum Chemical Studies22 citations

TL;DR

This paper proposes that ballistic energy transfer via collective electronic waves—termed 'electronic second sound'—can occur in graphene due to rapid electron-electron scattering, enabling heat propagation at ~0.71×10⁶ m/s, which is ~10³ times faster than phonon-based second sound. The waves are uncharged at charge neutrality but couple to plasmons upon doping, enabling all-electric excitation and detection.

ABSTRACT

Materials in which heat and entropy can be transmitted by directed ballistic pulses can trigger new approaches to energy transduction in solids. We predict that a ballistic energy transfer mode, with heat propagation governed by a wave equation rather than a diffusion equation, can be realized for a thermal electron-hole plasma in graphene. The new behavior originates from rapid exchange of energy and momentum in particle collisions leading to energy propagation as a collective weakly-damped oscillation. Due to the electronic nature of this mode, the estimated propagation velocity can be ~10^3 times larger than that for previously studied phonon mechanisms. The energy mode is uncharged at charge neutrality, but becomes coupled to charge dynamics upon doping. This coupling can be used for all-electric excitation and detection of energy transport.

Motivation & Objective

• To demonstrate that collective energy waves can propagate ballistically in a solid-state system via electron-electron interactions.
• To identify graphene as a platform for realizing electronic second sound due to its relativistic Dirac fermions and strong electron-electron scattering.
• To enable all-electric excitation and detection of energy transport by coupling energy waves to plasmons through doping.
• To establish a theoretical framework for wave-like heat transport governed by a wave equation rather than diffusion.
• To explore the potential of ballistic energy pulses for high-speed energy transduction in nanoscale devices.

Proposed method

• Derive the wave equation for energy and momentum transport using the 4×4 stress-energy tensor and conservation laws for energy and momentum in a relativistic electron gas.
• Model electron dynamics using the Boltzmann equation with rapid carrier-carrier scattering rates (γ_N) dominating over disorder and electron-lattice scattering.
• Use hydrodynamic approximation to express energy flux and momentum density in terms of distribution function perturbations, leading to coupled equations for density and velocity fluctuations.
• Perform angular averaging of distribution function perturbations to derive effective hydrodynamic equations for energy and momentum transport.
• Derive the dispersion relation for charge-coupled thermal waves by combining momentum transport with electrostatic potential via Poisson's equation.
• Use polylogarithm functions and Fermi-Dirac statistics to compute the plasmonic correction term in the dispersion relation near charge neutrality.

Experimental results

Research questions

• RQ1 Can energy in a solid propagate as a collective wave rather than by diffusion, and if so, under what conditions?
• RQ2 What is the propagation speed of such energy waves in a relativistic electron system like graphene?
• RQ3 How does doping affect the coupling between energy waves and charge dynamics in graphene?
• RQ4 Can energy waves be excited and detected electrically using plasmonic modes in graphene?
• RQ5 What is the role of electron-electron scattering in enabling wave-like energy transport instead of diffusive transport?

Key findings

• Energy waves in graphene propagate ballistically with a velocity of v′ = v/√2 ≈ 0.71×10⁶ m/s, which is ~10³ times faster than phonon-based second sound.
• At charge neutrality (μ = 0), the energy waves are uncharged and decoupled from charge dynamics, enabling pure thermal wave propagation.
• Upon doping away from neutrality, the energy waves couple to plasmons, enabling all-electric excitation and detection via gate-tunable carrier density.
• The dispersion relation for the coupled waves is ω² = (v²/2)k² + 2πe²λn₀v²|k|, with the plasmonic correction term vanishing at μ = 0.
• Near charge neutrality, the plasmonic correction scales as ∝ μ², confirming weak coupling and enabling clean observation of pure electronic second sound.
• The wave-like energy transport is stable in the hydrodynamic regime (ω < γ_N), distinguishing it from collisionless collective modes that occur at higher frequencies.