FR38 · Heat transport

Interface Thermal Resistance: The Temperature Jump at a Boundary

Join two solids and push heat across. Why does the temperature jump at the boundary, and when does that jump dominate? Many phonons reaching the boundary between two solids bounce back, so the temperature drops suddenly at the joint. The more the solids differ in stiffness and density, and the colder they are, the larger the jump. In a layer thinner than the Kapitza length, about 1.4 micrometres in this model for a strongly mismatched pair at room temperature, the boundaries resist heat more than the material.

Push heat through silicon and a second film. Phonons bounce off the boundary, so the temperature jumps there; compare acoustic and diffuse mismatch models.

Interface conductance $90.9\,\mathrm{MW\,m^{-2}\,K^{-1}}$

Silicon thickness with the same resistance (Kapitza length) $1.41\times10^{3}\,\mathrm{nm}$

Share of the temperature drop at the interface $44.4\,\%$

Conductance from the thermometers, over the model value $1.00 \pm 0.0089$

Set the second solid and the stack

Temperature (K)

300

T

Sound speed of the film (km/s)

2

v_2

Density of the film (g/cm³)

11

\rho_2

Stack thickness (nm)

1000

d

Drag the white interface along the stack.

Interface position, noise, packets and seed

Share of the stack that is silicon

0.5

Thermometer noise (K)

0.03

Packets in the crossing tally

1500

Seed of the noise and the packets

3803

Model, units and sources

$G=\frac{v_1}{4}\,\Gamma_{12}\,C_1(T;\,\omega<\omega_{D,2}),\qquad \Gamma_{12}=2\!\int_0^1\alpha_{12}(\mu)\,\mu\,d\mu,\qquad \alpha^{\mathrm{DMM}}_{12}=\frac{v_2^{-2}}{v_1^{-2}+v_2^{-2}}$

Each side is a gray Debye solid with one speed for its three acoustic branches. Phonons above the film's cutoff cannot enter it. The packet tally draws incidence angles and frequencies from the same weights and should match the model within its error bars. The stack holds a 10 K drop between two reservoirs; the jump uses the textbook conductance, which compares the temperatures of phonons heading into the interface, so identical solids still show the radiation-limit jump.

Wikipedia, Interfacial thermal resistance (revision of 2025-12-05) · Song and Chen (2021) · Ioffe NSM, Si mechanical properties

Course chapter: interface thermal resistance

Model assumptions

Two gray Debye solids, each with three acoustic branches of one speed, joined at a flat interface. Side 1 uses the silicon density and the gray speed of the film Lab; side 2 is an assumed solid set by its speed and density, with silicon's atom density. The acoustic mismatch model treats phonons as plane waves at a perfect, flat bond; the diffuse mismatch model lets the interface forget where each phonon came from. Both are the textbook limits, not predictions for a real interface. The conductance compares the temperatures of phonons arriving at the interface; even identical solids keep a small leftover jump in this formula, which the course explains. Bulk conductivities use the kinetic formula at 300 K and stay fixed when the temperature changes. Thermometer readings are synthetic: model temperature plus seeded Gaussian noise. The interface close-up is magnified and schematic.

Temperature through the stack

Yellow: model. Pink: synthetic thermometers (bars: two standard deviations of noise). Green dashes: line fit on each side. White line: interface.

Interface conductance against temperature

Blue: acoustic mismatch. Orange: diffuse mismatch. White dashes: radiation limit. Yellow ring: model now. Pink: thermometer fit (bars: two standard errors).

Chance that a silicon phonon crosses

Blue: acoustic mismatch. Orange: diffuse mismatch. Green: packet tally, selected model (bars: two standard errors).

More numbers and how they are checked

Debye temperatures: silicon, film: $0$
Acoustic impedances: silicon, film: $0$
Flux-weighted chance to cross: $0$
Radiation limit: every arriving phonon crosses: $0$
Same conductance computed from the film side (detailed balance): $0$
Bulk conductivities: silicon, film: $0$
Heat flux through the stack: $0$
Jump from the thermometer fits, with one standard error: $0$
Packets that crossed, with one standard error: $0$
Share the model predicts for the packets: $0$
Packets in the tally: $0$
Thermometer noise: $0$

Detailed balance requires the same conductance from either side; the two rows above check it. The packet tally samples angles and frequencies independently of the quadrature that gives the model conductance.

Physics tutorial

Why heat stumbles at a boundary

BackgroundWhen heat flows from one solid into another, the temperature does not change smoothly across the joint. It drops suddenly at the boundary, as if a thin insulating sheet were hidden there. That extra resistance is called interface thermal resistance, or Kapitza resistance after the physicist who first measured it at liquid-helium temperatures.

Why it mattersA chip stacks silicon, oxides and metals in layers tens of nanometres thick. Once a layer is thinner than its Kapitza length, the boundaries resist heat more than the material itself does.

Start with the essentials

Focus question: Join two solids and push heat across. Why does the temperature jump at the boundary, and when does that jump dominate?
One-sentence intuition: Only some phonons get through. The conductance is the ballistic flux of the arriving phonons times their chance to cross, $G=\tfrac14 v_1 C_1 \Gamma_{12}$ . The acoustic mismatch model computes that chance from the acoustic impedance $Z=\rho v$ and Snell's law; the diffuse mismatch model from detailed balance alone.

Core mathematical model

Acoustic mismatch transmission

\alpha_{12}(\theta_1)=\frac{4Z_1Z_2\cos\theta_1\cos\theta_2}{\left(Z_1\cos\theta_2+Z_2\cos\theta_1\right)^2},\qquad \frac{\sin\theta_1}{v_1}=\frac{\sin\theta_2}{v_2}

Beyond the critical angle, where $\sin\theta_1>v_1/v_2$ , no phonon crosses.

Kapitza length

\ell_K=\frac{\kappa_1}{G}

The thickness of silicon that resists heat as much as the interface does. A stack much thinner than this is dominated by its interfaces.

Common difficulties

A perfect joint still shows a jump in this formula

Typical misconceptionIf both sides are the same material, the formula gives zero resistance.

Better mental modelWith matched solids every phonon crosses, but $G$ stops at the radiation limit $\tfrac14 vC$ , not infinity. The formula compares the temperatures of phonons heading into the interface, not the temperatures thermometers read inside each layer. The course derives the correction that removes this leftover jump.

Run the experiment

01
See the jump
Start with Soft, heavy film: a clear jump.
What to observe: The stack colour changes abruptly at the white interface, most packets in the close-up bounce back in purple, and the thermometer fits miss each other by several kelvin at the interface.
02
Find the critical angle
Choose Stiff, light film: a critical angle.
What to observe: In the transmission chart the blue curve falls to zero beyond about 32 degrees, and the green packet tallies follow it.
03
Cool it down
Choose Cold interface: the jump takes over, then drag the temperature slider back up.
What to observe: At 10 K almost the whole temperature drop sits at the interface. The conductance falls roughly as the cube of temperature at low temperature, as the dot on the log-log chart shows.

Interface Thermal Resistance: The Temperature Jump at a Boundary

Why heat stumbles at a boundary

Core mathematical model

Acoustic mismatch transmission

Kapitza length

Common difficulties

A perfect joint still shows a jump in this formula

Run the experiment

See the jump

Find the critical angle

Cool it down

Related experiments and articles

From phonon flights to second sound, interfaces and thermoelectrics