Real-time visualisation of the bound-state energies and wavefunctions of a semiconductor quantum well. Change the well width and barrier height with sliders and compute the quantised energies and transition wavelength of GaAs/AlGaAs-type structures at once.
Material & structure
Semiconductor system
Well width L (nm)
nm
Barrier height V₀ (eV)
eV
V0 = 0.30 eV
Presets
Results
1
Selected state n
—
Eₙ (meV)
8.0
Well width L (nm)
—
Bound states
—
E₁ (meV)
—
λ₂₁ (nm)
Well potential and wavefunction ψₙ(x,t)
State n1
Re ψₙ(x,t) (oscillating)|ψₙ|² (probability density)Other model (comparison)
What is a quantum well? I have heard the name, but what is the structure exactly, and what is so remarkable about it?
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Roughly speaking, it is an "electron sandwich" that confines electrons in a film only nanometres thick. A material such as GaAs (gallium arsenide) is sandwiched above and below by another material, AlGaAs. Electrons are confined to the thin GaAs layer and can take only discrete energy values (levels). That is the key that determines the emission wavelength of semiconductor lasers. Move the "well width L" slider above in this simulator and the effect is obvious at once.
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Really! What is the difference between an "infinite well" and a "finite well"? The tool seems to switch between them.
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The infinite well is an idealised model with infinitely high walls, and it is easy to compute. Real materials, however, have walls of finite height (the barrier height V₀). In a finite well the electron wavefunction leaks slightly into the walls (tunnelling). Choose "finite well" in the tool and lower the barrier height V₀: you will see the energy levels drop and the leakage grow. In practice the finite-well model is by far the more common.
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I see! Why does the graph change when I switch the material from GaAs to InGaAs? What is the "effective mass"?
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Good observation! An electron in a crystal moves differently from an electron in vacuum. This "resistance to motion" is expressed by the effective mass $m^*$, which differs from material to material. InGaAs has a smaller effective mass than GaAs, meaning its electrons are lighter and more mobile. Switch the material in the tool and the spacing of the energy levels changes for the same well width. This matters especially in designing emission wavelengths, and quantum cascade lasers exploit the difference skilfully.
Frequently Asked Questions
Narrowing the well strengthens quantum confinement and raises the energy levels. Raising the barrier height binds more levels inside the well and lets the higher levels be computed stably. The emission wavelength is derived from the energy difference between the lowest levels, so the narrower the well, the shorter the wavelength (blue shift).
The "semiconductor system" selector switches the effective mass between GaAs (m*=0.067m₀), In₀.₅₃Ga₀.₄₇As (0.041m₀) and GaN (0.20m₀), and the barrier height V₀ is set with the slider for the finite well. For other materials, choose the one with the closest effective mass as a substitute. Details of the band structure (non-parabolicity and so on) are not taken into account.
With a finite barrier height the electron wavefunction leaks outside the well (tunnelling). The leakage grows as the energy level approaches the barrier height, and because the effective well width increases, the energy drops slightly. The infinite-well model ignores this effect.
Usually the emission of a semiconductor quantum well is dominated by the transition between the lowest conduction-band level (n=1) and the highest valence-band level (heavy-hole n=1). The λ₂₁ shown by this tool is the wavelength of the n=2→n=1 (intersubband) transition within the conduction band, not an interband emission wavelength. To obtain the interband emission wavelength, add the electron E₁ and the hole quantisation energy to the band gap and convert that energy.
Real-World Applications
Semiconductor lasers: Electron transitions between the discrete energy levels of a quantum well generate light of sharp wavelength. Controlling the well width L in steps of a few nanometres lets the emission wavelength (colour) be designed precisely. Essential for DVD/Blu-ray pickups and optical communications.
Quantum cascade lasers: A structure in which many quantum wells are stacked in a cascade. A single electron falls from well to well, emitting a photon each time, so mid-infrared to terahertz light is generated with high efficiency. Applied to gas analysis and non-invasive medical diagnostics.
High-electron-mobility transistors (HEMT): The two-dimensional electron gas formed at the AlGaAs/GaAs interface (which can be regarded as a wide quantum well) is used as the channel. Because the electrons suffer little impurity scattering, fast, low-noise operation is possible, used for example in satellite-communication amplifiers.
Quantum-dot solar cells: These use "quantum dots", quantum wells shrunk in all three dimensions. Thanks to the discrete levels they may absorb long-wavelength light that ordinary solar cells cannot use, and research aims at a leap in conversion efficiency.
Common Misconceptions and Pitfalls
There are a few points that trip people up when they start using this tool. First, assuming that "the effective mass is a constant that depends only on the material". In fact, even in the same material the value depends on which band the carrier is in (the conduction band, or holes in the valence band). This tool uses the conduction-band electron value by default. The effective mass of holes in GaAs, for example, is much larger than that of electrons, so for the same well width the level spacing on the valence-band side is completely different. If you are thinking about emission, you need to look at both the electron and hole levels.
Next, mis-setting the "barrier height V₀" of the finite well. It is determined by the material's band offset and is often quoted in meV (millielectronvolts). In the GaAs/AlGaAs system, for example, an Al fraction of 0.3 gives V₀ of about 300 meV. Entering "1" in the tool is interpreted as 1 eV (=1000 meV), an unrealistically huge barrier, so be careful. In practice, check the correct value in papers or data sheets.
Finally, relaxing because "it is only a one-dimensional model, so it is simple". This calculation is for an ideal flat well. In real devices the interfaces of the well are rough at the atomic level and impurities are present, so the sharp levels obtained from the calculation are somewhat broadened in reality (shorter lifetimes). The trick is to treat the simulation result as the ideal picture and to use the difference from measurements to assess material quality.
Choose the semiconductor system (GaAs / InGaAs / GaN) and infinite or finite well, and set the well width L between 1 and 20 nm
For the finite well, vary the barrier height V₀ between 0.05 and 1.0 eV to mimic the band offset of the AlGaAs/GaAs interface (about 0.3 eV for an Al fraction of 0.3)
Every change recomputes immediately: E₁, Eₙ, the number of bound states, λ₂₁ (the n=2→1 transition wavelength) and the wavefunction animation of the selected state n are updated
Worked Example
GaAs quantum well (effective mass m*=0.067m₀), well width L=10 nm, infinite-well model: the ground level is E₁=56.1 meV and the first excited level E₂=224.5 meV, so the transition energy ΔE=168.4 meV gives λ₂₁=7,364 nm (7.36 μm). Widening the well to 15 nm lowers E₁ to 24.9 meV and lengthens λ₂₁ to 16,568 nm. Switching to the finite well (V₀=0.30 eV) at the same L=10 nm leaves 3 bound states with E₁=34.2 meV, E₂=132.6 meV and λ₂₁=12,602 nm – lower levels and a longer wavelength than the infinite well.
Practical Notes
As the Al fraction x of AlGaAs increases, the band gap widens and the Γ-valley effective mass also changes, so accurate material parameters are essential
For extremely narrow wells (<3 nm) band non-parabolicity and intermediate-band levels can no longer be neglected and the simulation accuracy drops
At room temperature thermal broadening widens the peak FWHM, so comparison with measurements at cooling temperatures of 80 K or below is recommended
This tool assumes a one-dimensional flat well for conduction-band electrons (parabolic band, same effective mass in the barriers). λ₂₁ is the intersubband wavelength; the interband emission wavelength additionally needs the band gap and the hole levels
🎬 Watch it in motion
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