FR38 · Heat transport

Ballistic and Diffusive Heat in a Thin Film

Heat a film on one side. Does the heat diffuse across like ink, or fly straight to the far wall? In a thick film, phonons scatter many times on the way across, heat spreads slowly and Fourier's law holds. When the film is about one mean free path thick, many phonons fly straight across, so the cold side warms early, yet the steady heat flow is smaller than Fourier's law predicts. No film, however thin, carries more than the ballistic limit set by the phonon speed and heat capacity.

Heat a silicon-like film on one side and watch phonons cross it. Thin films carry less heat than Fourier predicts; thick films follow Fourier again.

Knudsen number: mean free path / thickness $0.943$

Heat leaving the cold wall now, over the ballistic limit $0.276$

Settled film conductivity over the bulk value $0.429$

Time since the hot wall switched on $20.0\,\mathrm{ps}$

Set the film

Film thickness (nm)

40

L

Phonon mean free path (nm)

37.7

\Lambda

Drag the blue cold wall to change the thickness. Sliders are logarithmic.

Solver resolution, packets and seed

Directions per hemisphere in the solver

12

Monte Carlo packets in a settled film

6000

Seed of the Monte Carlo run

3807

Model, units and sources

$\frac{\partial e}{\partial t}+\mu\frac{\partial e}{\partial x}=\frac{\theta-e}{\mathrm{Kn}},\qquad \frac{q}{q_B}=2\int_{-1}^{1}\mu\,e\,d\mu,\qquad q_B=\frac{Cv\,\Delta T}{4}$

Lengths in units of the thickness and times in units of the flight time across it. The solver uses discrete directions, a limited upwind scheme and exact relaxation; the Monte Carlo run uses the same model with packets that scatter after exponential free times. The green settled value comes from a separate integral equation and never uses either simulation.

Ioffe NSM, Si thermal properties · Ioffe NSM, Si mechanical properties · Maznev, Johnson and Nelson (2011)

Course chapter: ballistic and diffusive heat in a film

Model assumptions

Gray phonon Boltzmann equation in the relaxation-time approximation: one speed and one mean free path for every phonon, isotropic scattering, black walls that absorb every phonon and emit at their own temperature. Silicon numbers near 300 K come from the Ioffe NSM tables; the single speed (the Debye average of the [100] sound speeds) and the single mean free path from the kinetic formula are model choices. Real silicon has a broad spread of mean free paths, so a gray model underestimates how thick a film must be before Fourier returns. The temperature is a deviation from the cold wall, small enough that the equation is linear. The vertical position of each packet is drawn only to make the stage readable.

Temperature across the film, now

Yellow: solver. Pink: Monte Carlo slabs (bars: one standard error). Blue dashes: Fourier. Purple dots: free flight.

Heat leaving the cold wall

Yellow: solver. Pink: Monte Carlo. Blue dashes: Fourier. Purple dots: free flight. Green dashes: exact settled value.

Settled conductance against Knudsen number

Green: exact gray solution (dot: this film). Yellow ring: solver now. Blue dashes: Fourier. Purple dots: ballistic limit.

More numbers and how they are checked

Film thickness: $0$
Mean free path: $0$
Flight time across the film: $0$
Settled flux over the ballistic limit (integral equation): $0$
What Fourier's law would give, same ratio: $0$
Heat entering at the hot wall now, over the ballistic limit: $0$
Stored energy minus net energy in (solver check): $0$
Monte Carlo packets in the film: $0$
Solver cells across the film: $0$
Solver directions: $0$

The solver and the Monte Carlo run share one model but no code: if they agree within the error bars, neither has a bookkeeping mistake the other lacks. The settled value comes from an integral equation solved separately; the solver approaches it as the run ends.

Physics tutorial

A race between flight and diffusion

BackgroundIn an insulating crystal, heat is carried by phonons, packets of lattice vibration that fly at about the speed of sound until they scatter. Between scatterings a phonon travels one mean free path on average; in silicon near room temperature a single-speed estimate gives a few tens of nanometres.

Why it mattersFourier's law assumes every phonon scatters many times on its way across. Chips now contain silicon layers only a few mean free paths thick, where that assumption fails and the film carries less heat than Fourier's law promises.

Start with the essentials

Focus question: Heat a film on one side. Does the heat diffuse across like ink, or fly straight to the far wall?
One-sentence intuition: One number decides: the Knudsen number $\mathrm{Kn}=\Lambda/L$ . When it is small, heat arrives slowly and Fourier's law holds. When it is large, phonons cross in one flight, the far wall warms early, and the settled flux is capped at the ballistic limit $q_B=Cv\,\Delta T/4$ however thin the film is.

Core mathematical model

Gray Boltzmann equation

\frac{\partial e}{\partial t}+v\mu\frac{\partial e}{\partial x}=\frac{\theta-e}{\tau},\qquad \theta=\frac12\int_{-1}^{1}e\,d\mu

Here $e$ is the energy carried in direction cosine $\mu$ , and scattering pulls it toward the local temperature $\theta$ after a mean time $\tau=\Lambda/v$ .

Fourier against ballistic

\frac{q_F}{q_B}=\frac{\kappa\,\Delta T/L}{Cv\,\Delta T/4}=\frac{4}{3}\,\mathrm{Kn}

With $\kappa=Cv\Lambda/3$ , Fourier's law would exceed the ballistic limit once $\mathrm{Kn}>3/4$ . The exact gray solution never does.

Common difficulties

Thinner is not always better

Typical misconceptionA thinner film has a shorter path, so it should carry heat ever faster.

Better mental modelHalving the thickness doubles the Fourier flux only while phonons still scatter inside. Once they fly across in one go, the flux stops growing and saturates at the ballistic limit; the film behaves like a gap between two radiating walls.

Run the experiment

01
Watch the race
Start with Thickness near the mean free path and watch the yellow curve in the cold-wall flux chart.
What to observe: The flux rises before the dashed Fourier curve and settles well below Fourier's value; pink Monte Carlo points follow the same curve.
02
Make it thin, then thick
Choose Very thin: phonons fly across, then Thick film: Fourier returns.
What to observe: The thin film follows the free-flight curve and settles near the ballistic limit; the thick film follows the Fourier curve and its settled point sits on the Fourier line.
03
Change the material, not the size
Choose Short mean free path, same film.
What to observe: The same 40 nm film now behaves almost diffusively: what matters is the ratio of mean free path to thickness, not the thickness alone.

Ballistic and Diffusive Heat in a Thin Film

A race between flight and diffusion

Core mathematical model

Gray Boltzmann equation

Fourier against ballistic

Common difficulties

Thinner is not always better

Run the experiment

Watch the race

Make it thin, then thick

Change the material, not the size

Related experiments and articles

From phonon flights to second sound, interfaces and thermoelectrics