Quantum Well Heterostructure — Bound-State Subband Levels

Heterostructure Parameters

Well width L 8.0 nm

Barrier height (band offset) ΔE_c 250 meV

Electron effective mass m*/m₀ 0.067

Selected Subband

Bound-State Energies

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Bound states

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Penetration depth 1/κ

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Ground-state confinement E₁

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How it works

A quantum-well heterostructure (e.g. a thin GaAs layer sandwiched between AlGaAs barriers) traps electrons in a finite square well of width L and depth V₀ set by the conduction-band offset ΔE_c between the two materials. Solving the time-independent Schrödinger equation with the electron's in-crystal effective mass m* gives discrete, quantized subband energies E_n instead of a continuum:

Inside well:   ψ ∝ cos(kx) or sin(kx),   k = √(2m*E)/ħ
Outside well:  ψ ∝ e^(−κ|x|),            κ = √(2m*(V₀−E))/ħ

Matching ψ and ψ′ at the walls gives the exact
transcendental bound-state conditions:
  even states:  k·tan(kL/2) = κ
  odd states:  −k·cot(kL/2) = κ

Each root of these equations is one confined subband. Because κ is finite (not infinite, as in an idealized "particle in a box"), the wavefunction leaks — tunnels — an exponential distance 1/κ into the barrier, which is exactly what this simulator solves numerically for every slider setting.

Well width L — a narrower well pushes every level up in energy (tighter confinement) and can push higher subbands out of the well entirely.
Barrier height ΔE_c — a shallower barrier (smaller band offset) supports fewer bound states and lets wavefunctions leak further into the barrier.
Effective mass m*/m₀ — a lighter effective mass (e.g. InAs ≈ 0.023, GaAs ≈ 0.067, GaN ≈ 0.2) raises confinement energies for the same geometry.
Subband buttons — pick which bound state's wavefunction and probability cloud are highlighted.

Real-world relevance: this exact confinement mechanism sets the emission wavelength of quantum-well laser diodes and LEDs, the intersubband transitions used in quantum cascade lasers and QWIP infrared detectors, and the two-dimensional electron gas (2DEG) that gives HEMT transistors their high mobility.

Conduction-band edge E_c(x) Selected wavefunction ψ_n(x) Electron probability cloud |ψ_n|²

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