Parameters
Parameter A50
Parameter B25
About
Select semiconductor material and well type. Adjust well width and barrier height to compute energy levels, transition energies, and emission wavelengths.
Result 1
—
Result 2
—
Select semiconductor material and well type. Adjust well width and barrier height to compute energy levels, transition energies, and emission wavelengths.
Select semiconductor material and well type. Adjust well width and barrier height to compute energy levels, transition energies, and emission wavelengths.
The core physics is described by the time-independent Schrödinger equation for a particle in a potential $V(z)$. Inside the well, the potential is lower, creating bound states.
$$ -\frac{\hbar^2}{2m^*}\frac{d^2\psi(z)}{dz^2}+ V(z)\psi(z) = E\psi(z) $$Here, $\hbar$ is the reduced Planck constant, $m^*$ is the effective mass of the electron (or hole) in the semiconductor, $\psi(z)$ is the wavefunction, $E$ is the energy of the state, and $V(z)$ is the potential profile defining the well.
For a finite well of width $L$ and barrier height $V_0$, we solve the Schrödinger equation separately in the well and barrier regions and enforce boundary conditions (continuity of $\psi$ and $(1/m^*)(d\psi/dz)$). This leads to transcendental equations that determine the allowed energy levels $E_n$.
$$ \sqrt{\frac{m^*_w E}{\hbar^2}}\tan\left(\sqrt{\frac{m^*_w E}{\hbar^2}}\frac{L}{2}\right) = \sqrt{\frac{m^*_b (V_0 - E)}{\hbar^2}}\quad \text{(for even states)} $$Here, $m^*_w$ and $m^*_b$ are the effective masses in the well and barrier, respectively. The solutions to this equation, which you see calculated instantly in the simulator, are the discrete bound-state energies $E_1, E_2,...$.
Semiconductor Lasers (VCSELs / Edge-Emitters): Quantum wells are the active region in most modern diode lasers. By designing the well width and composition, engineers precisely control the emission wavelength. For instance, the tool can show you how changing from GaAs to GaN shifts the emission from infrared to blue/UV wavelengths.
High-Electron-Mobility Transistors (HEMTs): These high-speed transistors used in satellite communications and radar systems rely on a 2D electron gas formed at a heterojunction, which is essentially a type of quantum well. The confined electrons have very high mobility, enabling fast switching.
Quantum Cascade Lasers: These advanced mid-infrared lasers are based on a series of coupled quantum wells. Electrons "cascade" down a staircase of precisely engineered energy levels, emitting a photon at each step. Designing such a structure requires meticulous calculation of inter-subband transitions, similar to the transition energies this calculator provides.
Photodetectors and Modulators: Quantum well infrared photodetectors (QWIPs) are designed to absorb specific infrared wavelengths by tailoring the well's energy levels. Similarly, electro-absorption modulators, which encode data onto laser beams in fiber optics, use the quantum-confined Stark effect in wells to change absorption with an electric field.
When you start using this tool, there are a few common pitfalls to watch out for. First, the misconception that "effective mass is a constant that only changes with material." In reality, even within the same material, its value differs depending on which band the electron is in (the conduction band or a hole in the valence band). This tool uses the conduction band electron value by default. For example, the effective mass of a hole in GaAs is much heavier than that of an electron, so even for the same well width, the energy level spacing on the valence band side is completely different. When considering light emission, you need to look at the energy levels for both electrons and holes.
Next, errors in setting the "barrier height V₀" for the finite well. This is determined by the material's "band offset," and it's often handled in units of "meV (millielectronvolts)." For instance, in the GaAs/AlGaAs system, with an Al composition ratio of 0.3, V₀ is about 300 meV. Be careful: entering "1" in the tool is interpreted as 1 eV (=1000 meV), resulting in an unrealistically large barrier. In practice, always verify the correct value from papers or datasheets.
Finally, complacency because "it's just a 1D model, so it's simple." This calculation is for an ideal, flat well. In actual devices, the well interfaces can be rough at the atomic level, and impurities may exist. Consequently, the sharp energy levels calculated here will broaden slightly in reality (lifetime shortens). The trick is to treat simulation results as an "ideal reference," using the discrepancy with measured values to assess material quality.
The calculation of this quantum well actually underpins a much wider range of fields than you might think. First on the list is "quantum dot solar cells." Quantum dots are essentially quantum wells made small in all three dimensions. Here, the well size is varied to design efficient absorption across a broad spectrum of sunlight. The intuition you gain from this tool—that "energy changes with size"—forms the basis for that.
Next is "HEMTs (High Electron Mobility Transistors)." These are the heart of wireless communication. The "two-dimensional electron gas" formed at the AlGaAs/GaAs interface is actually a state where electrons are confined in a very shallow, triangular-shaped quantum well (a heterojunction). Understanding electron behavior here requires the finite well model, especially the concept of wavefunction penetration (tunneling) into the barriers.
Another interesting field is "quantum sensing." For example, the energy levels of a quantum well respond sensitively to external electric fields (the quantum-confined Stark effect) or strain (the piezoelectric effect). This property is leveraged in the principle design of sensors that detect extremely minute electric fields or pressures, where these fundamental calculations come into play.
If you want to dive deeper, we recommend first learning the "matrix mechanics" approach to solving it. The transcendental equation for the finite well solved by this tool is actually a special case solution that "assumes symmetry and separates into even and odd functions." More generally, you define the wavefunctions inside and outside the well piecewise and let a computer solve the smooth connection conditions at the boundaries in matrix form (e.g., using the transfer-matrix method or shooting method). Mastering this approach allows you to handle asymmetric wells or multi-stage wells (superlattices).
Regarding the mathematical background, try reframing it as an "eigenvalue problem." The Schrödinger equation $ \hat{H} \psi = E \psi $ states that the eigenvalues of the Hamiltonian operator $\hat{H}$ are $E$, and the eigenfunctions are $\psi$. The quantum well problem is a special case where the potential term $V(x)$ becomes a box shape when this operator is written in the position representation. Adopting this viewpoint makes it easier to imagine extending the concept to more complex potential shapes.
For the next practical step, moving to the "time-dependent Schrödinger equation" is recommended. By tracking how the stationary state wavefunctions obtained with this tool behave over time, you can simulate dynamic phenomena like the "stimulated emission" process during laser operation or the "tunneling time" for electrons moving between wells. This becomes the next step in discussing device operating speed and efficiency.