Coulomb Blockade: Energy Landscape & Staircase

Quantum Dot

Dot radius R 10.0 nm

Dielectric constant ε_r 10.0

Temperature T 4.0 K

Gate voltage V_g 0 mV

Live Readouts

Charging energy E_C

— meV

Total capacitance C_Σ

— aF

Electrons on dot ⟨N⟩

0.00

E_C / k_BT

—

How it works

This is the same single-electron-box model as the 3D version, drawn as two 2D-native plots instead of a rendered scene. A quantum dot's self-capacitance sets its charging energy:

C_self = 4π ε0 εr R
C_Σ    = C_self + C_g        (C_g: gate coupling, here 0.5·C_self)
E_C    = e² / (2 C_Σ)

With N excess electrons and dimensionless gate charge n_g = C_gV_g/e, the electrostatic energy is a parabola E(N) = E_C·(N − n_g)². The top-right panel plots this parabola directly against continuous N, with a marker on every integer N sized and colored by its Boltzmann weight exp(−E(N)/k_BT) — the true "energy landscape" the electron count is drawn from. The bottom panel plots the full charge-staircase curve ⟨N⟩(V_g) analytically across the whole sweep range, so you see the entire staircase shape at once, not just the current point.

At T → 0 the island always sits in the integer N that minimizes E(N), so the staircase has sharp unit steps. As k_BT approaches E_C, nearby charge states become thermally populated and the steps blur into a smooth ramp — watch both panels respond together as you move the temperature slider.

R, ε_r — set the island's capacitance and hence E_C (the parabola's curvature).
T — sets k_BT; widens the spread of colored markers around the ground state and smears the staircase.
V_g — sweeps n_g, sliding the parabola's minimum and moving the marker along the staircase curve.

Island + docked electrons E(N) landscape / population Ground state

Coulomb Blockade: Energy Landscape & Staircase (2D)