L4 · Chapter 8

Reading Huberman and colleagues on second sound in graphite (2019)

Sandbox Physics · 2026-10-30

Open arXiv:1901.09160 version 1 alongside this page: the paper by Huberman and colleagues in Science, “Observation of second sound in graphite at temperatures above 100 K”. Each of its four main figures answers one question: how a thermal grating decays at room temperature, why the signal turns negative at low temperature, how fast that negative swing travels and whether it could be some other wave, and over what range of temperatures and periods the effect appears. This chapter reads the figures in order. For each it says where to look, what it establishes and where it is easiest to misread, and it recomputes a few key numbers with the formulas of earlier chapters.

The claim and the measured quantity

The claim is in the abstract: with time-resolved optical measurements on 5 to 20 µm length scales, the authors directly observed second sound in graphite above 100 K, and first-principles calculations predict wave-like phonon hydrodynamics on a scale of about 1 µm up to almost room temperature.

First pin down what is measured. The sample is highly oriented pyrolytic graphite with grains of about 10 µm, all with their c axes normal to the surface. The 515 nm pump is absorbed within a skin depth of about 30 nm and makes the thermal grating; thermal expansion ripples the surface and the ripple diffracts the probe. Heterodyne detection gives a signal proportional to the ripple amplitude $u(t)$ . In the supplement the heterodyne phase is calibrated on tungsten, confirming that graphite gives only a phase-grating signal from the ripple.

The easy misreading: the signal is thermal expansion, not temperature itself. The authors note in their discussion that out of equilibrium different phonon modes have different Grüneisen parameters, so expansion need not track temperature exactly, while their simulation outputs temperature.

Source: S. Huberman · A. A. Maznev

Figure 1: still exponential at 300 K, but not diffusion

Look at the waveforms in Fig. 1B and the apparent diffusivity in Fig. 1C. The 37.5 µm period gives 11 cm²/s, in line with the literature. As the period shrinks to 3.7 µm, $L^2/(4\pi^2\tau)$ falls: the decay is slower than the diffusion equation predicts, although every trace is still exponential. The first-principles simulation reproduces the trend.

The easy misreading: an exponential trace does not mean diffusion. The test for diffusion is whether the decay time scales as $L^2$ . In discussing Fig. 4 the authors call this regime quasi-diffusive: exponential decay, but a time that does not scale as $L^2$ . It is the size effect Maznev and colleagues predicted in 2011.

Source: S. Huberman · A. A. Maznev

Figure 2: the signal turns negative at 85 K, and it is not a surface wave

Look at Fig. 2A. At 85 K the signal at every period shows damped oscillations and falls below zero. In heterodyne detection a sign flip means the spatial phase of the grating has shifted by half a period: former temperature maxima are now minima. The authors point out that in diffusion heat only flows from hot to cold, so maxima and minima cannot trade places, which makes the sign flip a hallmark of wave-like heat transport. The dip gets shallower at longer periods and eventually vanishes, and it moves to later times as the period grows.

The inset of Fig. 2A plots the frequency obtained from the position of the first minimum, which corresponds to half an oscillation period, against the wavenumber. From a linear fit the authors get a speed of about 3200 m/s.

The easy misreading: thermal-grating signals often contain oscillations from surface acoustic waves. The authors give two reasons to rule this out. The surface acoustic wave on the graphite basal plane travels at 1480 m/s, the fast transverse wave at 14700 m/s and the longitudinal wave faster still, none near 3200 m/s; and acoustic waves have no reason to disappear when the temperature or the period is raised.

Source: S. Huberman

The simulation: a Boltzmann equation with the full scattering matrix

The authors solve the linearized phonon Boltzmann equation with the full three-phonon scattering matrix, built from density-functional inputs, plus mass-disorder scattering from the natural isotope content (1.1% carbon-13). Following Chiloyan and colleagues they compute frequency-domain Green’s functions and Fourier-transform them into time-domain waveforms. The calculation has no fitted parameters.

In Fig. 2B the frequency-domain Green’s function shows a resonant peak, the hallmark of second sound. The speed from the position of the resonance is 3650 m/s, somewhat above the measured value. The theory correctly places the second-sound speed between the slow and fast transverse speeds. The authors note that second sound in other materials was slower than the slowest phonon; what is peculiar to graphite is its exceptionally slow transverse mode, whose large anharmonicity and density of states drive intense normal scattering.

The easy misreading: simulation and experiment agree qualitatively, in the authors’ own words. The supplement compares waveforms trace by trace. The simulation is at 80 K and the experiment at 85 K, as the caption of Fig. 3 says.

Source: S. Huberman · S. Lee

Figures 3 and 4: how wide the window is

Fig. 3 fixes the period at 10 µm and changes the temperature. The oscillation is still clear at 104 K and 125 K and gone at 150 K; it also disappears at 50 K. Below 80 K experiment and simulation drift apart: the 50 K simulation still shows a dip near 1.5 ns, though no longer negative. The authors suggest the simulation’s assumption of a thermalized initial distribution as one possible cause, since it worsens as transport turns ballistic at low temperature.

The same figure carries a ballistic-limit simulation with all scattering switched off: the dip disappears and the decay becomes slower. The authors explain that normal scattering redistributes energy into modes with higher group velocity. This differs from the gray model of Chapter 5, where all phonons share one speed and free flight alone produces a small negative dip.

Fig. 4 is the predicted second-sound window: between 50 and 250 K, with higher temperatures needing shorter periods. For a 10 µm period the window closes near 150 K, as observed; at 1.5 µm it stays open up to 250 K. The authors also say their setup lacked the time resolution for periods below about 5 µm. Fig. S6 in the supplement shows much less damping in isotopically pure graphite, so its window would be wider.

Source: S. Huberman · S. Lee

How much the pump heats the sample

A natural worry is how much the pump heats the sample, and whether the measurement still reflects graphite at 85 K. The supplement estimates a grating amplitude of about 57 K just after the pump at 85 K and an average rise of about 28 K, but by the time of the dip cross-plane conduction has lowered the surface temperature about twentyfold. For the 7.5 µm trace the dip bottoms out near 1.2 ns, when the average temperature is only about 1.4 K above 85 K and the grating amplitude during the oscillation is about 0.2 K.

They also checked linearity: changing the pump energy or probe power by a factor of two leaves the waveform unchanged, and the pump energy had to rise tenfold before a change showed. The measurement is a linear response to a small perturbation and can be compared with a linearized Boltzmann equation.

Source: S. Huberman

Worked example: second-sound speed from the time of the minimum

Problem: if the grating oscillates as an undamped standing wave at speed $v_{II}$ , the first minimum comes after half a period. Find the time of the minimum for 7.5 µm and 10 µm periods at 3200 m/s, and compare with the roughly 1.2 ns the supplement gives for the 7.5 µm trace.

The standing wave has angular frequency $\Omega=v_{II}q$ and period $2\pi/\Omega=L/v_{II}$ , so $t_{\min}=L/(2v_{II})$ . At 7.5 µm, $7.5\times10^{-6}/(2\times3200)\,\mathrm s=1.17\,\mathrm{ns}$ , consistent with about 1.2 ns; at 10 µm, $1.56\,\mathrm{ns}$ .

Damping pushes the minimum slightly later: in the telegrapher solution of Chapter 5 it falls at $\pi/\omega'$ with $\omega'<\Omega$ . A speed estimated as $L/(2t_{\min})$ therefore comes out slightly low. The theory-to-measurement ratio is $3650/3200=1.14$ : the first-principles speed is about 14% higher.

The course’s thermal-grating Lab is a gray model with one sound speed and two scattering times. It reproduces the logic of the sign flip, the window and the held-out test, but it is not graphite: it cannot give graphite’s speed, and in free flight it shows a dip that real graphite does not have in Fig. 3.

Source: S. Huberman · S. Lee

A checklist for similar papers

One: what quantity is actually measured, and which assumptions connect it to the physical quantity under discussion (here, temperature)? Two: can the sign of the signal be trusted? Here it rests on heterodyne detection and phase calibration. Three: which alternative explanations were ruled out, and with what independent evidence? Here, surface waves and the other sound speeds. Four: does the effect change with the parameters the way the explanation says? Here the dip fades with period and appears and disappears with temperature. Five: does the comparison with theory use fitted parameters? Here it does not, and the agreement is qualitative. Six: where are the setup’s blind spots? Here, periods below 5 µm and the initial distribution at low temperature.

Source: S. Huberman · A. A. Maznev

Try it in the Lab

Retrace the paper’s argument in the gray model

Choose “Second sound: stripes swap”, pause when the record reaches its lowest point and read the time, infer the wave speed with this chapter’s formula, and compare it with the gray sound speed over the square root of three.
Drag the stage to change the period. As the period grows, does the dip get shallower and later, as in Fig. 2A of the paper?
Choose “Free flight: a small dip” and say where the gray model differs from the ballistic-limit simulation in Fig. 3 of the paper.

Exercises with explained answers

A 10 µm grating reaches its first minimum at 1.56 ns. Treating it as an undamped standing wave, what is the second-sound speed?
Show the answer
$v_{II}=L/(2t_{\min})=10\times10^{-6}/(2\times1.56\times10^{-9})\,\mathrm{m/s}\approx3200\,\mathrm{m/s}$ . With damping the minimum comes later, so this estimate is slightly low.
What two reasons do the authors give for ruling out surface acoustic waves as the source of the oscillation?
Show the answer
The speed does not match: the surface acoustic wave on the graphite basal plane travels at 1480 m/s, the fast transverse wave at 14700 m/s and the longitudinal wave faster still, none at 3200 m/s. And the behaviour does not match: acoustic waves have no reason to disappear when the sample is warmed or the period is lengthened, while the observed oscillation does.
The supplement estimates a grating amplitude of about 0.2 K and an average rise of about 1.4 K at the time of the dip. Why do these numbers matter for the conclusion?
Show the answer
They show that the sample is still near 85 K and only slightly perturbed, so the measurement is a linear response. Only then can it be compared directly with a linearized Boltzmann equation and read as a property of graphite at 85 K rather than an effect of laser heating. The unchanged waveform when the pump energy is doubled supports this.
Why does the graphite simulation lose its dip when all scattering is switched off, while free flight in the gray model of Chapter 5 still dips below zero?
Show the answer
In the gray model all phonons share one speed, so free flight gives $\sin(qvt)/(qvt)$ , with all phonons in step, and the amplitude dips negative. Real graphite has a wide spread of phonon speeds whose oscillations cancel, so there is no dip in the ballistic limit. In a real material the dip needs normal scattering to make the phonons drift together.

Sources and what we recomputed

A. A. Maznev, J. A. Johnson and K. A. Nelson, Onset of nondiffusive phonon transport in transient thermal grating decay, Physical Review B 84, 195206 (2011), arXiv:1108.3770v2
Read in full (arXiv v2). Introduction: the kinetic-theory mean free path of tens of nanometres, the gray-body approximation as an oversimplification, the slab between two black walls favoured by theorists, the grating period set by the crossing angle of the pump beams, and grating decay times of about 0.3 to 30 ns in silicon for periods of 1 to 10 µm. The abstract states that the reduction of the effective conductivity in silicon is significant at periods as large as 10 µm. The course quotes no equation numbers from this paper.
S. Lee, D. Broido, K. Esfarjani and G. Chen, Hydrodynamic phonon transport in suspended graphene, Nature Communications 6, 6290 (2015), doi:10.1038/ncomms7290
Main text and Methods read; the Supplementary Information was not opened. Used here: normal scattering conserves crystal momentum while umklapp and other resistive processes do not; second sound needs a frequency window between the resistive and the normal scattering rates; in a three-dimensional Debye solid the second-sound speed is close to the sound speed divided by the square root of three; and the predicted second-sound speeds in graphene of roughly 2000 to 3000 m/s. The course quotes no equation numbers from this paper.
S. Huberman, R. A. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A. A. Maznev, G. Chen and K. A. Nelson, Observation of second sound in graphite at temperatures above 100 K, Science (2019), doi:10.1126/science.aav3548, arXiv:1901.09160v1
Read in full from arXiv v1, including the supplementary materials. Used here: the transient-grating setup with a 515 nm pump and a 532 nm probe in heterodyne detection; the diffusion time constant written as the period squared over four pi squared times the diffusivity; 11 cm²/s at a 37.5 µm period at 300 K and the slower-than-diffusive decay at short periods (Fig. 1); the sign flip at 85 K and the velocity of about 3200 m/s from the first minimum (Fig. 2); oscillations at 104 K and 125 K that vanish at 150 K and at 50 K (Fig. 3); the predicted window between 50 and 250 K (Fig. 4); the first-principles value of 3650 m/s; and, from the supplement, the dip minimum near 1.2 ns for the 7.5 µm trace and the grating amplitude of about 0.2 K during the oscillation. The published Science version was not compared line by line with arXiv v1.