Type-II Quantum Dot: Radial Band Diagram & Charge Separation

Dot Geometry

Core radius R_c 1.4 nm

Shell thickness t_s 1.2 nm

Band Alignment

Conduction offset ΔE_c (shell − core) -0.35 eV

Negative → electron prefers the shell

Valence offset ΔE_v (shell − core) 0.45 eV

Positive → hole prefers the core

Carrier Localization

Electron: shell · Hole: core

Live Readouts

Overlap integral J–

Radiative lifetime, ×τ₀–

e–h separation Δr–

How it works

Electron and hole envelope functions are found by solving the s-state (l=0) effective-mass radial Schrodinger equation independently for each carrier, using u(r) = r·R(r):

-(ħ²/2) d/dr[ (1/m*(r)) du/dr ] + V(r) u(r) = E u(r)
u(0) = 0,  u(r>R_dot) = 0  (hard outer wall)

The confining potential is a two-region finite well/barrier set by the band offsets: V=0 inside the core, V=ΔE inside the shell. A type-II alignment has ΔE_c and ΔE_v pushing the electron and hole to opposite regions (e.g. electron in the shell, hole in the core) — the staggered gap characteristic of pairs like CdTe/CdSe. A type-I alignment (both offsets confine both carriers to the same region) keeps electron and hole overlapped.

This 2D view plots the solved potential wells and probability densities directly along a radial line through the dot's centre — a band diagram — instead of rendering a 3D point cloud. The top strip mirrors u(r)² about the origin to show a diametric slice through the spherically-symmetric dot; the bottom strip renders the same two densities as radial cross-sections you can compare side by side.

J = ∫ u_e(r) u_h(r) dr        (each u normalized: ∫u² dr = 1)
Rate ∝ J²  →  lifetime τ ∝ 1/J²

Pushing the carriers apart shrinks J toward zero, which is exactly why type-II core/shell quantum dots show radiative lifetimes tens to hundreds of times longer than type-I dots of the same size — useful for photovoltaics and photocatalysis, where you want the separated charge to survive long enough to do work before it recombines.

Grid: 220-point finite-difference discretization, ground state found by shifted inverse-power iteration (Thomas tridiagonal solves) — identical numerics to the 3D version of this simulator.
Effective masses (in units of m₀, representative of II-VI semiconductors): electron 0.11 (core) / 0.13 (shell); hole 0.40 (core) / 0.45 (shell).
The shaded curves are the actual solved radial probability density u(r)² of each carrier's ground state — not decorative.

Electron u(r)² / potential Hole u(r)² / potential