Resolve the charging energy of an island

Let the island's total capacitance to gates and leads be $C_\Sigma$. In the constant-interaction model, the electrostatic part of its energy is a parabola $E_C(N-n_g)^2$, where N is integer excess electron number and $n_g$ is the dimensionless charge induced by gates and fixed offsets. We define $E_C=e^2/(2C_\Sigma)$. Some references instead call $e^2/C_\Sigma$ the charging energy, so a quoted numerical value needs its defining formula before comparisons can be made.

Single-particle confinement energies can be added to this electrostatic term. The energy needed to add the Nth electron is a difference of total energies, not the charging coefficient itself. With constant orbital energies, consecutive addition energies differ by $2E_C$ plus the relevant single-particle spacing. Spin multiplicity can let two electrons occupy the same orbital, but it does not remove the electrostatic cost of adding the second charge. Exchange and correlation may require a more detailed model.

First neglect single-particle spacings, as appropriate for a metallic island or a charging-dominated approximation. Neighboring states N and N plus one have equal energy at $n_g=N+1/2$. Away from that crossing, one integer minimizes the energy. Near a crossing, weak tunneling can alternate occupations and carry current if both lead transitions are energetically allowed. The equality is relative to the reservoir electrochemical reference; a finite bias opens a window and shifts the relevant alignment conditions.

For a gate capacitance $C_g$ and otherwise fixed voltages, increasing gate voltage by $e/C_g$ shifts induced charge by one electron. The lever arm $\alpha=C_g/C_\Sigma$ converts a small gate-voltage change into an addition-energy shift of magnitude $e\alpha\Delta V_g$. When orbital spacings are important, peak separations can depart from the simple metallic period. A measured finite-bias blockade threshold also depends on source and drain capacitances and voltage division, so it cannot universally be identified with $E_C/e$.

A sequential-tunneling description treats charge changes as separate weak transition events and uses probabilities for the island's occupations. Large tunnel resistances suppress charge uncertainty from coupling to the reservoirs, while low temperature suppresses thermal access to unfavorable states. Neither condition guarantees exactly zero current between peaks: thermally activated transport and higher-order cotunneling can remain. Strong coupling can wash out charge quantization and bring in correlations such as Kondo physics that the classical charging parabola does not describe.

The distinction between a gate-induced continuous offset and integer electron number is central. A gate changes electrostatic preference continuously without forcing the actual isolated island to hold a fractional electron count in a particular charge eigenstate. Near degeneracy the mean occupation can be fractional because probabilities or coherent superpositions involve adjacent integer states, depending on coupling. The elementary energy diagram determines which states are favorable; transition rates and reservoir occupations are still needed to calculate a conductance peak's height and width.