---
title: "Interface Thermal Resistance: The Temperature Jump at a Boundary Interactive Lab | Sandbox Physics"
description: "Join two solids and push heat across. Why does the temperature jump at the boundary, and when does that jump dominate?"
canonical_url: "https://sandboxphysics.com/labs/interface-thermal-resistance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/interface-thermal-resistance/"
last_modified: "2026-10-31"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** When 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 matters** A 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 misconception** If both sides are the same material, the formula gives zero resistance.

**Better mental model** With 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

1. 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.

2. 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.

3. 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.

Physical conclusion

An interface is a filter for phonons. How much heat passes depends on how alike the two solids are and on how many phonons are around to carry it, and a thin stack can be ruled by its boundaries.

## Related experiments and articles

Phonon heat transport · Reading path

### From phonon flights to second sound, interfaces and thermoelectrics

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Interface thermal resistance: why the temperature jumps where two materials meet**](https://sandboxphysics.com/stories/phonon-heat-transport/interface-thermal-resistance/)
2. [Article **The energy ledger of thermoelectric conversion: why phonon conduction limits efficiency**](https://sandboxphysics.com/stories/phonon-heat-transport/thermoelectric-energy-ledger/)
3. [Lab **Ballistic and Diffusive Heat in a Thin Film** How far a film cuts phonon conduction](https://sandboxphysics.com/labs/phonon-ballistic-diffusive-film/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)