Phonon hydrodynamics represents a unique regime of heat transport in semiconductors where collective phonon behavior dominates over individual particle-like scattering. This phenomenon emerges when momentum-conserving normal phonon collisions outweigh resistive Umklapp processes, leading to fluid-like thermal transport analogous to viscous flow in gases. The hydrodynamic regime is characterized by two key signatures: second sound and Poiseuille flow of phonons, both observable under specific temperature and purity conditions.
Theoretical frameworks for phonon hydrodynamics originate from the solution of the linearized Boltzmann transport equation under the Callaway approximation. When normal scattering rates exceed boundary and Umklapp scattering rates, the phonon distribution function deviates from local equilibrium, exhibiting collective modes. The Guyer-Krumhansl equation extends Fourier’s law by incorporating nonlocal terms, describing heat flux q as:
∇²q - (1/Λ²)q = (1/ν) ∂q/∂t
Here, Λ is the phonon mean free path for normal processes, and ν is the phonon drift velocity. This equation predicts wave-like thermal propagation (second sound) and parabolic flow profiles (Poiseuille flow).
Second sound refers to the propagation of temperature waves at finite velocity, distinct from both diffusive heat flow and ordinary sound waves. In this mode, phonons undergo collective oscillations, creating wavefronts of thermal energy. The second sound velocity c₂ depends on the average phonon group velocity vₚ and the ratio of normal to Umklapp scattering rates:
c₂ = vₚ / √3
Experimental observation requires materials with high crystal quality and temperatures where normal scattering dominates. Graphite exhibits second sound between 85 K and 125 K, with measured velocities reaching 2500 m/s. Bismuth shows similar behavior below 3 K, with c₂ ≈ 300 m/s, confirmed via heat pulse experiments where laser-generated thermal waves propagate with minimal dispersion.
Poiseuille flow of phonons occurs when momentum-conserving collisions lead to viscous damping, creating a parabolic heat flux profile across constrained geometries. This is analogous to laminar fluid flow in pipes. The thermal conductivity κ in this regime follows:
κ = (C vₚ² τ_N) / 5
where C is the heat capacity, and τ_N is the normal scattering time. The Poiseuille regime emerges when the sample width w satisfies Λ_U ≫ w ≫ Λ_N, with Λ_U and Λ_N being the Umklapp and normal mean free paths. In bismuth nanowires of 300 nm width, thermal conductivity measurements reveal a κ ∝ w² dependence below 1.5 K, directly indicating phonon viscosity.
Material selection critically determines hydrodynamic behavior. Graphite’s strong sp² bonds and high Debye temperature (≈ 2000 K) enable long-lived normal processes. Its layered structure further suppresses Umklapp scattering along the basal planes. Bismuth’s low carrier density and anisotropic phonon dispersion promote hydrodynamic effects at cryogenic temperatures. In both materials, isotopic purity enhances momentum conservation by reducing mass-defect scattering.
Experimental techniques for probing phonon hydrodynamics include time-domain thermoreflectance (TDTR) for second sound detection and microfabricated thermal bridges for Poiseuille flow studies. TDTR measurements on graphite reveal oscillatory thermal decay profiles at 100 K, with wave periods matching theoretical c₂ values. Suspended Bi membranes exhibit thermal conductance quantization at sub-Kelvin temperatures, consistent with phonon viscosity models.
The hydrodynamic regime has practical implications for thermal management. Devices operating in this range exhibit non-Fourier heat conduction, enabling novel thermal waveguides and delay lines. However, achieving room-temperature phonon hydrodynamics remains challenging due to increased Umklapp scattering. Recent work on aligned carbon nanotubes suggests possible pathways through engineered phonon band structures.
Current research focuses on extending hydrodynamic effects to higher temperatures via material engineering. Strain tuning of phonon dispersions and superlattice structures show promise in enhancing normal scattering. Theoretical studies predict that certain topological insulators may exhibit hydrodynamic transport due to protected phonon modes, though experimental confirmation is pending.
Understanding phonon hydrodynamics provides fundamental insights into condensed matter physics. The phenomenon bridges the gap between ballistic and diffusive transport, offering a unified framework for extreme thermal regimes. Future advances in ultrafast spectroscopy and nanoscale thermometry may uncover new materials exhibiting hydrodynamic behavior, potentially revolutionizing heat control in semiconductor devices.