Quantum Cascade Laser — Intersubband Population Inversion

Quantum-Well Design

Well width L 5.5 nm

Bias per period 260 mV

Lattice temperature 77 K

Injection current density 800 A/cm²

Laser State

BELOW THRESHOLD

Photon energy

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Emission wavelength

–

Population inversion ΔN

–

Lower-level lifetime τ₁

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How it works

A quantum cascade laser (QCL) does not use two different materials' bandgap — every photon comes from an intersubband transition between two confined electron states in the conduction band of one material, engineered by quantum-well width. For a well of width L (infinite-well approximation):

E_n = n²π²ħ² / (2 m* L²)
ħω = E₂ − E₁  →  λ = hc / (E₂ − E₁)

An electron tunnels into the excited subband (E₂), radiatively/non-radiatively decays to the lower subband (E₁) emitting a photon, then must be swept out of E₁ fast — via LO-phonon emission — before the next electron arrives, or it "backfills" and kills the gain. Steady-state rate equations for one period:

dN₂/dt = R_pump·η(V) − N₂/τ₂ = 0  →  N₂ = R_pump·η·τ₂
dN₁/dt = N₂/τ₂ − N₁/τ₁ = 0        →  N₁ = N₂·(τ₁/τ₂)
ΔN = N₂ − N₁ = N₂·(1 − τ₁/τ₂)

Well width — sets E₂−E₁ via particle-in-a-box confinement, so it directly tunes the emission wavelength.
Bias per period — aligns the injector ground state with E₂; injection efficiency η(V) peaks at the design resonance and falls off (Gaussian) when misaligned, exactly like a real chirped-superlattice cascade.
Lattice temperature — raises τ₁ (thermal backfilling / phonon reabsorption slow the lower-level emptying), which is the dominant reason QCLs need cooling or careful design to run near room temperature.
Injection current density — sets the pump rate R_pump feeding electrons into E₂ every period.

The electrons you see cascading down the staircase spend time at each level proportional to τ₂ (upper, cyan) and τ₁ (lower, amber) — watch how raising temperature visibly lengthens the amber dwell and pushes ΔN toward threshold.

Upper subband E₂ (dwell ∝ τ₂) Lower subband E₁ (dwell ∝ τ₁) Emitted photon

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