---
title: "Phonon heat transport: from mean free paths to second sound, interfaces and thermoelectrics"
description: "Learn when Fourier’s law fails for heat: phonon mean free paths, thin films, thermal gratings, second sound, interface jumps and thermoelectric efficiency."
canonical_url: "https://sandboxphysics.com/stories/phonon-heat-transport/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/phonon-heat-transport/"
last_modified: "2026-10-31"
---

[Phonon heat transport](https://sandboxphysics.com/stories/phonon-heat-transport/)

FR38 · An interactive course

# Phonon heat transport: from mean free paths to second sound, interfaces and thermoelectrics

Sandbox Physics · 2026-10-30

In an insulator or a semiconductor, heat is carried mostly by phonons, the quantized vibrations of the crystal lattice. Fourier’s law treats their motion as diffusion, which works when every distance of interest holds many collisions. This course follows what happens when it does not: films thinner than a mean free path, thermal gratings that fade too slowly or even swing negative, a temperature jump at the boundary between two materials, and the way phonon conduction limits a thermoelectric generator. Each formula is worked out with numbers and checked against a simulation you can run.

[Start with mean free paths](https://sandboxphysics.com/stories/phonon-heat-transport/heat-carriers-and-mean-free-path/) Open an experiment

A phonon flies about one mean free path $\Lambda$ between collisions. In a film much thicker than $\Lambda$ (left) heat crosses in many short steps and Fourier’s law holds. When the Knudsen number $\mathrm{Kn}=\Lambda/L$ approaches one (right), many phonons fly from the hot wall to the cold wall without scattering. Original schematic, not to scale.

## From one phonon flight to heat waves, interface jumps and thermoelectrics

The first chapter estimates how far a phonon flies in silicon from three handbook numbers, and the second writes the Boltzmann equation that tracks those flights. The third solves a film between two walls exactly and shows where Fourier’s law overshoots. The fourth paints thermal gratings with two laser pulses and reads the diffusivity from their decay; the fifth adds collisions that conserve momentum, which turn the decay into a damped heat wave called second sound. The sixth follows phonons across the boundary between two materials, and the seventh keeps the energy ledger of a thermoelectric leg. The last chapter reads the 2019 paper by Huberman and colleagues on second sound in graphite.

You need derivatives, the exponential function and a little kinetic theory of gases. The Boltzmann equation, the cosine stripes of a thermal grating and the Seebeck and Peltier effects are introduced where they first appear. [FR38](https://github.com/yangzichao/sandboxphysics/issues/68)

1. 01 · L1

   ## [What carries heat in a crystal, and how far a phonon flies](https://sandboxphysics.com/stories/phonon-heat-transport/heat-carriers-and-mean-free-path/)

   Estimate silicon’s phonon mean free path, about 38 nm, from three handbook numbers, then use the Knudsen number to tell flight from diffusion.

2. 02 · L2

   ## [The phonon Boltzmann equation: from flights and collisions to Fourier’s law](https://sandboxphysics.com/stories/phonon-heat-transport/phonon-boltzmann-equation/)

   Derive phonon conductivity and the ballistic limit from the Boltzmann equation with one relaxation time, and see why Fourier’s law overshoots in 40 nm silicon.

3. 03 · L3

   ## [Heat in thin films: between ballistic and diffusive](https://sandboxphysics.com/stories/phonon-heat-transport/ballistic-and-diffusive-films/)

   Solve the gray film between black walls exactly, find that 40 nm of silicon carries only 43% of Fourier’s flux, and see where the wall jumps come from.

4. 04 · L2

   ## [Transient thermal gratings: measuring heat diffusion with interference fringes](https://sandboxphysics.com/stories/phonon-heat-transport/transient-thermal-grating/)

   Two laser pulses paint hot and cold stripes. Their fading rate gives the diffusivity, and slower fading at short periods means phonons have stopped diffusing.

5. 05 · L3

   ## [Normal scattering and second sound: why heat can slosh back and forth like a wave](https://sandboxphysics.com/stories/phonon-heat-transport/normal-scattering-and-second-sound/)

   Separate momentum-conserving normal scattering from resistive scattering, derive the damped heat-wave equation, and compute the second-sound speed and window.

6. 06 · L3

   ## [Interface thermal resistance: why the temperature jumps where two materials meet](https://sandboxphysics.com/stories/phonon-heat-transport/interface-thermal-resistance/)

   Estimate interface conductance with acoustic and diffuse mismatch models, find a 1.4 µm Kapitza length for silicon on a soft film, and fix a textbook formula.

7. 07 · L2

   ## [The energy ledger of thermoelectric conversion: why phonon conduction limits efficiency](https://sandboxphysics.com/stories/phonon-heat-transport/thermoelectric-energy-ledger/)

   Keep the energy ledger of one thermoelectric leg: conduction, Peltier and Joule heat, the maximum-efficiency formula, and why cutting phonon conduction helps.

8. 08 · L4

   ## [Reading Huberman and colleagues on second sound in graphite (2019)](https://sandboxphysics.com/stories/phonon-heat-transport/reading-huberman-2019/)

   Read the graphite second-sound paper figure by figure, recompute the speed from the dip time, and leave with a checklist for similar papers.

## Experiments along the way

[01 **Race flight against diffusion** Heat one side of a silicon-like film and watch phonon packets cross it. Shrink the film until Fourier’s law overshoots and the ballistic limit takes over.](https://sandboxphysics.com/labs/phonon-ballistic-diffusive-film/) [02 **Make heat ring** Paint hot and cold stripes with two pump pulses and watch them fade, ring or swap sign; then test diffusion against a damped wave on held-out periods.](https://sandboxphysics.com/labs/transient-thermal-grating/) [03 **Find the jump at a boundary** Push heat through silicon and a second film, watch phonons cross or bounce at the boundary, and read the jump from synthetic thermometers.](https://sandboxphysics.com/labs/interface-thermal-resistance/)

All three experiments run models, not measured data. The film experiment solves the gray Boltzmann equation between two black walls and checks it against an independent integral-equation solution and a packet Monte Carlo. The grating experiment evolves a gray phonon distribution with normal and resistive collisions and checks it against a separate Laplace-domain solution. The interface experiment sends phonon packets at a boundary, tallies what crosses, and compares the tally with the acoustic and diffuse mismatch formulas. Each shows its checks beside the model output, and none stands in for a measurement on a real sample.

## Sources and what we recomputed

1. [Ioffe Institute, NSM Archive: Thermal properties of Silicon (Si) and Mechanical properties, elastic constants, lattice vibrations of Silicon (Si)](http://www.ioffe.ru/SVA/NSM/Semicond/Si/thermal.html)

   Both pages read in full (the mechanical page is at mechanic.html in the same folder). Values used at 300 K: specific heat 0.7 J/(g K), thermal conductivity 1.3 W/(cm K), thermal diffusivity 0.8 cm²/s, density 2.329 g/cm³, and the sound speeds 8.43 km/s (longitudinal) and 5.84 km/s (transverse) along the [100] direction. The course combines the two speeds into one Debye average; that average is our step, not a value from the pages.

2. [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](https://arxiv.org/abs/1108.3770)

   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.

3. [S. Lee, D. Broido, K. Esfarjani and G. Chen, Hydrodynamic phonon transport in suspended graphene, Nature Communications 6, 6290 (2015), doi:10.1038/ncomms7290](https://doi.org/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.

4. [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](https://arxiv.org/abs/1901.09160)

   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.

5. [Wikipedia, Interfacial thermal resistance, revision 1325816272 (5 December 2025)](https://en.wikipedia.org/w/index.php?title=Interfacial_thermal_resistance&oldid=1325816272)

   This revision read in full. Used here: the definition of the temperature jump, the acoustic and diffuse mismatch models and their assumptions, detailed balance, the statement that the acoustic model typically gives an upper limit and the diffuse model a lower limit for the resistance, and the range of 80 to 300 MW/(m² K) for most phonon-mediated interfaces. Wikipedia is a secondary source; the course uses it for definitions and typical ranges only.

6. [Q. Song and G. Chen, Evaluation of diffuse mismatch model for phonon scattering at disordered interfaces, Physical Review B 104, 085310 (2021), arXiv:2106.04745v2](https://arxiv.org/abs/2106.04745)

   Partly read: the abstract, the introduction and the opening subsection of the theory section on the assumptions of the diffuse mismatch model and detailed balance. Used here: the model assumes a phonon forgets its origin after diffuse scattering, and the paper finds that a single disordered interface does not erase that memory, which limits the model. The atomistic Green’s-function results and later sections were not read and are not quoted.

7. [H. S. Kim, W. Liu, G. Chen, C.-W. Chu and Z. Ren, Relationship between thermoelectric figure of merit and energy conversion efficiency, PNAS 112(27), 8205 (2015), doi:10.1073/pnas.1510231112](https://pmc.ncbi.nlm.nih.gov/articles/PMC4500231/)

   Main text read on PubMed Central; the SI Appendix, where the derivations sit, was not opened. Used here: the conventional maximum-efficiency formula from the constant-property model, written as the Carnot efficiency times a factor that depends on the figure of merit at the average temperature; the heat input as conduction plus Peltier heat minus half the Joule heat; the load ratio and its optimum; and the finding that the conventional formula overestimates the efficiency of SnSe more than twofold at a hot side of 700 °C because its properties change strongly with temperature. Equation numbers are not quoted.

8. [Wikipedia, Thermoelectric materials, revision 1375991214 (21 September 2026)](https://en.wikipedia.org/w/index.php?title=Thermoelectric_materials&oldid=1375991214)

   Partly read: the lead and the sections on the figure of merit, device efficiency and power factor. Used here: zT as the electrical conductivity times the Seebeck coefficient squared times temperature over the thermal conductivity; the maximum-efficiency formula with the Carnot efficiency as its first factor, exact only for temperature-independent properties; and coefficients of performance of 0.3 to 0.6 for commercial thermoelectric refrigerators. The materials sections were not read. Wikipedia is a secondary source.