Double Quantum Well — Matrix Diagonalisation

Well Geometry

Well Width L2.00 nm

Well Depth V₀0.300 eV

Centre-to-Centre d6.00 nm

⚠ Wells overlap — d must exceed L. Increase d or decrease L.

Effective Mass (m₀ = 9.109×10⁻³¹ kg)

m* well0.067 m₀

m* barrier0.092 m₀

Dielectric Constant εᵣ

εᵣ well12.9

εᵣ barrier10.1

Electric Field

Apply electric field V′(x) Ramp screened by εᵣ(x) — exact, no perturbation theory

Field ε (total drop)+0.000 eV

Bound States — click row to toggle ψ

BenDaniel-Duke Hamiltonian:
T = −ℏ²/2 · ∂/∂x [1/m*(x) · ∂/∂x]
Discretised on uniform grid (dx=0.05 nm).

Dielectric screening:
V′(x) = −e·F·∫dx/εᵣ(x) — field is weaker inside high-ε regions.

Eigensolver: Sturm bisection (eigenvalues) + inverse iteration (eigenvectors). Exact for any field strength — no perturbation theory.

Potential & eigenfunctions (ψ amplitude scaled)