L1 · Chapter 1

What carries heat in a crystal, and how far a phonon flies

Sandbox Physics · 2026-10-30

In silicon and in electrically insulating crystals, heat is carried mostly by vibrations of the lattice. Treat those vibrations as a gas of particles, phonons, that fly at the speed of sound and now and then collide, and three numbers you can look up (heat capacity, sound speed and thermal conductivity) tell you how far a phonon travels between collisions. For silicon at room temperature the answer is about 38 nm. This chapter works out that number, explains why it is only a rough scale, and uses it to decide when Fourier’s law still holds.

Heat rides on lattice vibrations

In metals free electrons carry most of the heat; in dielectrics and semiconductors phonons do, as the paper by Maznev and colleagues states in its first paragraph. A phonon is a quantum of lattice vibration: a vibrational mode of frequency $\omega$ carries energy $\hbar\omega$ per quantum and moves at its group velocity. For long-wavelength phonons the group velocity is the speed of sound.

Where the crystal is hotter the vibrations are stronger and there are more phonons. Phonons fly from hot to cold regions, colliding with other phonons, impurities and boundaries, and hand their energy on stop by stop. This picture of a particle gas is enough for most of this course.

Source: A. A. Maznev

The kinetic formula: heat capacity, speed and mean free path

Let $C$ be the heat capacity per volume, $v$ the phonon speed and $\Lambda$ the mean distance flown between collisions, the mean free path. A phonon crossing a plane brings the temperature of a point about one mean free path upstream. In a gradient $dT/dx$ the energy difference it carries is proportional to $C\Lambda\,dT/dx$ , times its speed.

Averaging over directions, the mean square of the velocity component along $x$ is $v^2/3$ , which gives the kinetic formula below. Chapter 2 derives it properly from the phonon Boltzmann equation, factor of one third included.

\kappa=\tfrac13\,C\,v\,\Lambda,\qquad D=\frac{\kappa}{C}=\tfrac13\,v\,\Lambda

Thermal conductivity

\kappa

and thermal diffusivity

D

. Know any two of the three quantities and you have the third.

Source: A. A. Maznev · Ioffe Institute

The numbers for silicon

The Ioffe Institute semiconductor tables give, for silicon at 300 K, a specific heat of 0.7 J/(g K), a density of 2.329 g/cm³ and a thermal conductivity of 1.3 W/(cm K). Specific heat times density gives the heat capacity per volume, $C=1.63\times10^{6}\,\mathrm{J\,m^{-3}\,K^{-1}}$ .

The same tables list sound speeds along [100] of 8.43 km/s (longitudinal) and 5.84 km/s (transverse). Averaging one longitudinal and two transverse branches the Debye way (formula below) gives a single speed $v=6351\,\mathrm{m/s}$ . The average is our modelling choice, not a number from the tables.

The kinetic formula then gives a mean free path $\Lambda=3\kappa/(Cv)=37.7\,\mathrm{nm}$ and a time between collisions $\tau=\Lambda/v=5.93\,\mathrm{ps}$ . As a check, $D=\kappa/C=0.797\,\mathrm{cm^2/s}$ , in line with the 0.8 cm²/s the tables list separately.

\frac{3}{v^{3}}=\frac{1}{v_L^{3}}+\frac{2}{v_T^{3}}

Debye average: the low-frequency density of states goes as the inverse cube of the speed, so the three branches add as inverse cubes.

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.

Source: Ioffe Institute · A. A. Maznev

One number is not enough: mean free paths spread widely

Maznev and colleagues call the single mean free path for all phonons, the so-called gray-body approximation, an oversimplification. Low-frequency phonons scatter weakly and can fly much farther than the average. Whatever distance you look at, some low-frequency phonons cross it without scattering.

They cite a calculation by Henry and Chen in which phonons with mean free paths longer than 1 µm carry about 40% of the room-temperature conductivity of silicon. So 37.7 nm is a kinetic scale that tells you when a gray model starts to struggle; real silicon already departs from Fourier’s law at micrometres, as Chapter 4 shows.

All three experiments in this course use a gray model anyway, because it isolates ballistic, diffusive and wave-like behaviour cleanly. Wherever the course uses gray numbers it says that they are illustrative.

Source: A. A. Maznev

The Knudsen number: flight or diffusion

Compare the mean free path with the distance $L$ the heat must cross and you get the Knudsen number, $\mathrm{Kn}=\Lambda/L$ . Far below 1, phonons collide many times on the way and heat follows the diffusion equation. Far above 1, phonons barely collide and fly straight across; this is ballistic transport. Near 1 you are in between.

The Knudsen number also fixes the ratio of two time scales. Diffusing across $L$ takes about $L^2/D$ ; flying straight across takes $L/v$ .

\frac{L^{2}/D}{L/v}=\frac{3L}{\Lambda}=\frac{3}{\mathrm{Kn}}

The smaller the Knudsen number, the slower diffusion is than straight flight. The step uses

D=v\Lambda/3

Source: A. A. Maznev · Ioffe Institute

Worked example: heat crossing 1 µm of silicon

Problem: heat has to cross $L=1\,\mu\mathrm m$ of silicon at room temperature. With the gray numbers above, find the Knudsen number, the diffusion time, the flight time and their ratio.

The Knudsen number is $\mathrm{Kn}=37.7\,\mathrm{nm}/1000\,\mathrm{nm}=0.0377$ , well inside the diffusive range. Diffusion takes $L^2/D=(10^{-6}\,\mathrm m)^2/(7.97\times10^{-5}\,\mathrm{m^2/s})=12.5\,\mathrm{ns}$ ; flight takes $L/v=157\,\mathrm{ps}$ .

The ratio is $12.5\,\mathrm{ns}/157\,\mathrm{ps}=79.6$ , exactly $3/\mathrm{Kn}$ . In the gray picture 1 µm of silicon is thick and Fourier’s law applies. As the previous section said, real silicon also holds long-path phonons that do not diffuse over this distance, so the gray verdict is optimistic for the real material.

Source: Ioffe Institute · A. A. Maznev

Try it in the Lab

Read the Knudsen number when the film is as thick as a mean free path

Choose the preset “Thickness near the mean free path”. Read the Knudsen number and watch whether packets fly across in one go or bounce around inside the film.
Predict first: at 800 nm, how close will the settled film conductivity come to the bulk value? Then choose “Thick film: Fourier returns” and check.

Cut the mean free path tenfold

Choose “Short mean free path, same film”: the film is still 40 nm but the mean free path is 4 nm, so $\mathrm{Kn}=0.1$ . Watch the packet paths turn into zigzags.
Compare its settled conductivity ratio with the first preset and say what brought a film of the same thickness back close to Fourier’s law.

Exercises with explained answers

What is the gray Knudsen number of a 100 nm silicon film at room temperature, and which range does it fall in?
Show the answer
$\mathrm{Kn}=37.7/100=0.377$ , in the transition range between diffusion and free flight. The film is only a little more than two mean free paths thick.
How long does heat take to diffuse across 10 µm of silicon, and how long to fly straight across?
Show the answer
Diffusion takes $L^2/D=10^{-10}/(7.97\times10^{-5})\,\mathrm s=1.25\,\mu\mathrm s$ ; flight takes $L/v=1.57\,\mathrm{ns}$ . The ratio is about 796, which is $3/\mathrm{Kn}$ .
A silicon alloy has the heat capacity and sound speed of pure silicon but half its conductivity. What is its gray mean free path?
Show the answer
The mean free path is proportional to the conductivity, $\Lambda=3\kappa/(Cv)$ . Half the conductivity gives half the path, $18.8\,\mathrm{nm}$ .
Show that the ratio of diffusion time to flight time is $3/\mathrm{Kn}$ .
Show the answer
Diffusion takes $L^2/D$ and flight $L/v$ ; their ratio is $Lv/D$ . With $D=v\Lambda/3$ this is $3L/\Lambda=3/\mathrm{Kn}$ .

Sources and what we recomputed

Ioffe Institute, NSM Archive: Thermal properties of Silicon (Si) and Mechanical properties, elastic constants, lattice vibrations of Silicon (Si)
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.
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.