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.

Forward flux / left supply $0$

Reverse flux / left supply $0$

Net flux / left supply $0$

Forward barrier $0$

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.

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.

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 misconceptionA visible band step means no electrons can cross until bias exactly removes it.

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

Run the experiment

01
Watch electrons cross
Choose Carriers cross and vary Interface band step.
What to observe: The transmitted stream thins rapidly as the barrier grows.
02
Raise the temperature
Choose Thermal tail and reduce Carrier temperature.
What to observe: Fewer electrons in the high-energy tail reach the collector.
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.