This is the 2D companion to the 3D piezoelectric quantum-dot sim: same strain and piezoelectric-field physics, but instead of an animated 3D wavefunction cloud, the electron and hole envelope wavefunctions here come from actually solving the effective-mass Schrodinger equation for a particle in a linearly tilted infinite square well of width h — the textbook "quantum-confined Stark effect in a box" problem — by diagonalizing it in a finite basis of ordinary particle-in-a-box states:
H_nm = E_n δ_nm + α ⟨n|z-h/2|m⟩
E_n = n²π²ħ² / (2m*h²) [box eigenenergies]
⟨n|z-h/2|m⟩ = −8h·nm / [π²(n²−m²)²] (n+m odd), 0 otherwise
α_e = −eF_pz , α_h = +eF_pz [opposite tilt for electron vs. hole]
Diagonalizing this real, symmetric N×N matrix (N=30, via a Jacobi rotation solver run in-browser) gives the true ground-state energies and wavefunctions nonperturbatively — valid even at fields strong enough that ordinary perturbation theory breaks down. The strain and Fpz themselves come from the identical alloy-interpolated piezoelectric/elastic formulas as the 3D sim:
a(x) = a_GaN + x(a_InN − a_GaN), ε_xx = (a_GaN−a(x))/a(x)
P_pz = 2ε_xx(e31 − e33·C13/C33), F_pz = −P_pz/(ε0εr)
Instead of a heuristic tanh-saturation formula for the electron–hole separation, the band diagram below plots the actual reconstructed probability densities |ψe(z)|² and |ψh(z)|² = |Σn cnφn(z)|², and the separation 〈z〉, overlap integral ∫ψeψhdz, and redshift ΔE = −(shifte+shifth) are all read directly off the diagonalized eigenstates.
- Indium fraction — same alloy interpolation as the 3D sim; sets the lattice mismatch and hence the tilt strength α fed into the matrix.
- Dot height — sets the box width h in the Hamiltonian; both the unperturbed level spacing and the tilt term scale with it.
- Dot radius — geometric reference only (matches the 3D sim); the 1D axial solve does not depend on it.
- Field & charge overlay — toggles the polarization sheet-charge markers and internal-field arrows drawn across the tilted well.
Verified against known results: the diagonalization reproduces the flat-well analytic E_n at zero field, its ground-state shift converges to the standard second-order Stark perturbation-theory formula in the weak-field limit, and the matrix-element formula above was checked against direct numerical integration of the box eigenfunctions.