Observation of second sound in graphite over 200 K

The study investigates heat transport beyond Fourier diffusion in graphite, focusing on the hydrodynamic regime where strong momentum-conserving normal phonon scattering leads to collective phonon drift and wave-like heat propagation (second sound). Prior observations of second sound were limited to very low temperatures or larger transport lengths; more recent theory and experiments suggested hydrodynamics and second sound could persist at much higher temperatures in layered materials. However, the wavevector dependence (dispersion) of second-sound velocity and its interplay with ballistic phonons (thermal zero sound) at micrometer length scales had not been clarified experimentally due to time-resolution limits. The purpose of this work is to directly observe second sound above 200 K in graphite with micrometer-scale thermal gratings, quantify the dispersion of thermal-wave velocity versus grating period, and disentangle contributions from ballistic, hydrodynamic, and diffusive phonons.

Classical thermal transport in dielectrics/semiconductors follows Fourier’s law, but at small scales or low temperatures phonon transport can be ballistic or hydrodynamic. Second sound was first observed in superfluid helium and later in solids (e.g., NaF) at cryogenic temperatures. Recent theoretical and experimental works predicted/observed hydrodynamic phonon transport and second sound in layered/van der Waals materials at elevated temperatures, including prior TTG-based observation in graphite around 100 K. Theoretical treatments include solutions of the Peierls-Boltzmann transport equation (PBTE) using models such as Callaway’s split into normal and resistive scattering, as well as correlation-function approaches to analyze second-sound frequency windows. Earlier heat-pulse experiments also showed early-arriving ballistic phonons, though their interpretation as mechanical first sound is limited; the concept of zero sound (ballistic regime) provides a better analogy. Despite these advances, the dispersion of second-sound velocity with wavevector and its separation from ballistic contributions in micrometer TTG geometries remained underexplored.

• Experimental: A femtosecond pump–probe transient thermal grating (TTG) setup was implemented to achieve sub-nanosecond temporal resolution and resolve micrometer-scale transport dynamics. Ultrafast pulses (~290 fs) at 515 nm (pump) and 532 nm (probe) at 25 kHz repetition rate were generated. The pump was chopped (~2 kHz) and sent through a delay stage (up to 15 ns delay). Interference of ±1 diffraction orders at the sample produced a sinusoidal temperature grating with grating periods down to ~2 µm. Heterodyned detection of the diffracted probe field measured surface displacement from thermal expansion. Acoustic oscillations were minimized by phase adjustment of the reference field. Measurements were performed on natural graphite in a cryostat over 100–250 K, varying grating period to map second-sound signatures (signal sign reversal/negative dip) and extract wave velocity and propagation length from dip timing and normalized depth.
• Theoretical/simulation: TTG responses were simulated using a Green’s-function solution of the PBTE with the full scattering matrix and first-principles inputs, without fitting parameters. Ab initio calculations (VASP with PAW and optB88 vdW functional) provided harmonic and anharmonic force constants (Phonopy, third-order IFCs, ShengBTE) on dense k/q meshes to compute phonon properties and scattering rates (normal and resistive). Additional analytical frameworks included the Callaway model to define intrinsic second-sound speed and propagation length from energy/momentum conservation, and a ballistic-limit calculation (no scattering) to define thermal zero-sound velocity based on the timing of the first dip in the TTG response. Mode-resolved contributions (ZA/LA/TA) and wavevector-dependent dispersion were analyzed. Isotope effects were evaluated by comparing natural versus isotopically enriched/pure graphite in the simulations. Derived metrics included: (1) wave velocity vp from the first negative-dip time (half-period travel), (2) propagation length from normalized dip depth, and (3) estimated fractions of ballistic, hydrodynamic, and diffusive phonons as functions of temperature and grating period based on mean free paths and travel distances.

• Direct observation of second sound in graphite at temperatures above 200 K using femtosecond TTG. The hallmark negative dip (signal sign reversal) was clearly resolved at 200–225 K for micrometer grating periods, whereas at 250 K it was obscured by acoustic contributions at the achieved SNR.
• Dispersion: The measured thermal-wave velocity increases as the grating period decreases, indicating wavevector-dependent dispersion. The measured velocities and propagation lengths versus grating period show qualitative agreement with first-principles PBTE simulations.
• Thermal zero sound: Simulations in the ballistic limit reveal a faster thermal zero-sound wave compared to intrinsic second sound. The observed thermal wave at finite scattering is a superposition of second sound and thermal zero sound, with the relative weight depending on grating period and temperature. At shorter periods, more phonons have MFPs exceeding the period, increasing ballistic (zero-sound) contributions and raising the apparent wave speed.
• Temperature trends: At fixed grating period, the measured wave velocity decreases monotonically with increasing temperature due to increased scattering (reduced ballistic participation). Propagation length increases with grating period as ballistic effects diminish.
• Three-group phonon picture: Transport is explained by concurrent ballistic, hydrodynamic, and diffusive phonons. The normalized negative dip depth (second-sound strength) is maximized at intermediate grating periods; it is reduced at large periods by diffusive (R-scattering) dominance and at small periods by mixing with ballistic waves from multiple polarizations.
• Isotope effect: Simulations predict strong isotope influence (isotopes act as resistive scattering). At 125 K, isotopically enriched graphite shows >15% higher thermal-wave speed and significantly longer propagation lengths than natural graphite across grating periods. In isotopically pure graphite, second sound is predicted to persist to room temperature for grating periods of 1.0–1.5 µm.
• Defect sensitivity: At 100 K, a vacancy density of ~0.02% in simulations reconciles discrepancies with experiment, indicating defect/impurity scattering becomes more important at lower temperatures.
• Detection threshold: Experimentally, a normalized dip depth ≈0.05 was the lower limit for direct observation of second sound under the measurement conditions.

The study demonstrates that heat can propagate as temperature waves (second sound) in graphite at much higher temperatures than traditionally expected, addressing the open question of second sound’s presence and dispersion at micrometer scales and above 200 K. The observed increase of wave velocity with decreasing grating period reflects growing ballistic (thermal zero-sound) contributions, while hydrodynamic phonons govern the negative-dip signature. The interplay of ballistic, hydrodynamic, and diffusive phonons provides a unified framework explaining the dependence of wave velocity, propagation length, and dip depth on temperature and grating period. Agreement with first-principles PBTE simulations validates the modeling of phonon scattering (normal vs resistive) and supports the interpretation of the measured dispersion. The strong isotope effect underscores the role of resistive scattering in limiting hydrodynamic phenomena and suggests routes to engineer thermal waves via isotopic purification. These findings extend phonon hydrodynamics into more accessible temperature regimes, with implications for nonlocal thermal transport, thermal signal processing, and the design of materials and devices that exploit wave-like heat propagation.

This work reports direct observation of second sound in graphite above 200 K using femtosecond TTG and establishes the dispersion of thermal-wave velocity with grating period. First-principles-based simulations explain the results via the superposition of ballistic (thermal zero sound), hydrodynamic (second sound), and diffusive phonons. A pronounced isotope effect is predicted and indicates that isotopically pure graphite could exhibit second sound at room temperature for micrometer-scale gratings. The study advances understanding of phonon hydrodynamics at elevated temperatures and provides a framework to control thermal waves through length-scale selection and isotopic engineering. Future research directions include experimental verification of room-temperature second sound in isotopically pure graphite, systematic studies of defect/isotope concentrations, exploration of nonlocal thermal conductivity’s relation to second sound, and extending measurements over broader frequency–wavevector ranges with enhanced SNR and acoustic suppression.

• Experimental SNR limited direct observation at 250 K; acoustic vibrations can obscure the sign reversal unless carefully suppressed by phase tuning.
• Discrepancies between simulations and measurements at 100 K likely arise from sample defects/impurities; modeling suggests ~0.02% vacancy density explains reduced dip depth. Defect scattering becomes increasingly important at lower temperatures.
• The detection threshold for second sound in this setup required a normalized dip depth of about 0.05, potentially missing weaker signals.
• While temporal resolution was improved with pulsed probing, finite bandwidth and grating-period range restrict the accessible frequency–wavevector space.
• The relationship between nonlocal thermal conductivity formalisms and observed second sound is noted but not resolved in this work.