---
title: "Heterojunction Injection Interactive Lab | Sandbox Physics"
description: "How do a band step, bias and temperature change carrier injection? A higher conduction-band step leaves fewer arriving electrons with enough energy to cross. Forward bias lowers the forward barrier, and a warmer carrier distribution puts more electrons above it. At equilibrium, the reservoir populations balance the opposing transmitted fluxes."
canonical_url: "https://sandboxphysics.com/labs/heterojunction-injection/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/heterojunction-injection/"
last_modified: "2026-10-24"
---

C-X1b · Carriers meet an interface barrier

# Heterojunction Injection

**How do a band step, bias and temperature change carrier injection?** A higher conduction-band step leaves fewer arriving electrons with enough energy to cross. Forward bias lowers the forward barrier, and a warmer carrier distribution puts more electrons above it. At equilibrium, the reservoir populations balance the opposing transmitted fluxes.

Follow electrons arriving at an interface between two semiconductors. Change the band step and record the forward and reverse thermionic fluxes.

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

This classical thermionic step model assumes nondegenerate reservoirs with equal effective density of states and carrier velocities. Their equilibrium populations differ by the Boltzmann factor of the band offset. Forward bias raises the left band edge relative to the fixed right band edge while each incident supply remains prescribed. Electrons transmit above an uphill step and cross a downhill step freely; the direction of the barrier reverses when bias exceeds the offset. Flux is normalized to the left incident supply, so no material current density is inferred. The band profile is imposed, not solved with Poisson’s equation. Tunneling, traps, reflection from effective-mass mismatch, scattering, space charge and hole injection are omitted. The materials and geometry are synthetic.

Synthetic model acquisition

## Transmission through the interface

The stored sweep retains the settings used to generate it.

> Synthetic sweep data. Axis definitions and model assumptions are retained in the text.

$V/\mathrm V,\quad (\Gamma_{\rightarrow}-\Gamma_{\leftarrow})/\Gamma_0\;(\%)$

The dot marks the operating point stored with this sweep. Reacquire after changing settings.

### Conduction-band energy along the interface

The left band edge rises with bias; the right edge stays at the band offset. Numbers are energies in electronvolts relative to the unbiased left edge. Position is schematic.

Nobel · Condensed matter

## What did the original experiments establish?

Follow the apparatus, observations and interpretation in the historical evidence. Each article explains where this teaching model fits.

[Read the historical evidence](https://sandboxphysics.com/stories/condensed-matter/semiconductor-heterostructures/) [All six reading paths](https://sandboxphysics.com/stories/condensed-matter/)

Physics tutorial

## Heterojunction Injection

**Background** Joining semiconductors can create a conduction-band step. A thermal carrier reservoir contains a distribution of energies, so some electrons cross a barrier that reflects most others.

**Why it matters** A band diagram becomes useful when it predicts a flux. Compare the barrier height with the carrier thermal energy and then vary the bias.

Start with the essentials

- **Focus question**: How do a band step, bias and temperature change carrier injection?
- **One-sentence intuition**: Injection depends on how many arriving carriers lie above the interface barrier.

### Core mathematical model

#### Thermal energy scale

$$
E_T=k_B T
$$

Temperature broadens the carrier energy distribution.

#### Directional barriers

$$
\begin{aligned}\Delta E_+&=\max(\Delta E_c-qV,0)\\\Delta E_-&=\max(qV-\Delta E_c,0)\end{aligned}
$$

Only uphill travel has a classical barrier. A sufficiently large bias reverses the direction of the step.

#### Thermionic net flux

$$
\begin{aligned}\Gamma_{\rm net}/\Gamma_0&=e^{-\Delta E_+/(k_BT)}-w_R e^{-\Delta E_-/(k_BT)}\\w_R&=e^{-\Delta E_c/(k_BT)}\end{aligned}
$$

The right reservoir has a smaller equilibrium supply. Each term includes its incident population and any uphill barrier; both fluxes use the left incident supply as normalization.

### Common difficulties

#### A common inference to test

**Typical misconception** A visible band step means no electrons can cross until bias exactly removes it.

**Better mental model** The thermal distribution has a high-energy tail. Crossing is exponentially suppressed, rather than abruptly forbidden, at a positive barrier.

### Run the experiment

1. 01

   #### Watch electrons cross

   Choose Carriers cross and vary Interface band step.

   **What to observe:** The transmitted stream thins rapidly as the barrier grows.

2. 02

   #### Raise the temperature

   Choose Thermal tail and reduce Carrier temperature.

   **What to observe:** Fewer electrons in the high-energy tail reach the collector.

3. 03

   #### Restore equilibrium

   Choose Balanced counterflows and record a sweep.

   **What to observe:** At zero bias, the more populated low-band reservoir supplies more electrons, but its uphill barrier reduces transmission until the opposing fluxes balance.

Physical conclusion

Injection depends on how many arriving carriers lie above the interface barrier.

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)