Quantum Dot Charging Energy Explorer

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

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Charging energy E_C

— meV

Total capacitance C_Σ

— aF

Electrons on dot ⟨N⟩

0.00

E_C / k_BT

—

How it works

A quantum dot is a tiny conductive island. Adding one extra electron costs electrostatic energy — the charging energy — set by the island's self-capacitance to its surroundings:

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)

A smaller, less-screened dot has less capacitance, so a single electron costs more energy — this is why nanoscale quantum dots show Coulomb blockade at liquid-helium temperatures while a macroscopic capacitor never does.

In the constant-interaction (single-electron-box) model, the electrostatic energy of the island with N excess electrons and dimensionless gate charge n_g = C_gV_g/e is a parabola:

E(N) = E_C · (N − n_g)²

At T = 0 the island always sits in the integer N that minimizes E(N), so ⟨N⟩ jumps in sharp unit steps as V_g is swept — the Coulomb staircase. At finite temperature, nearby N states are thermally populated with Boltzmann weights exp(−E(N)/k_BT), and ⟨N⟩ = Σ N·w(N) / Σ w(N). The staircase stays sharp only while E_C ≫ k_BT; once k_BT approaches E_C, thermal fluctuations wash the steps into a smooth ramp and Coulomb blockade is lost.

R, ε_r — set the island's capacitance and hence E_C; smaller/less-screened dots have a larger charging energy.
T — sets k_BT; watch the staircase on the energy ladder sharpen or blur as T changes relative to E_C.
V_g — sweeps n_g; each time n_g crosses a half-integer, the ground-state electron number N increases by one and an electron docks onto the dot.

This is the same physics behind single-electron transistors and quantum-dot qubits: the charging energy is the "energy scale" that must exceed both k_BT and the source-drain bias for single-electron effects to be observable.

Dot + docked electrons E(N) energy ladder Ground state (min E)

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