= Solution
A <quantum dot> confines electronic motion in all three directions. A <semiconductor> dot can be thin enough that only one transverse subband is occupied, with discrete lateral states controlled by gates. In a rectangular hard-wall model,
$$
\epsilon_{n_xn_yn_z}=\frac{\pi^2\hbar^2}{2m^*}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right).
$$
A smooth circular lateral confinement is often better approximated by $m^*\omega_0^2r^2/2$, giving $\epsilon_{n_xn_y}=\hbar\omega_0(n_x+n_y+1)$ above the transverse ground <energy>. Shell $q=n_x+n_y$ contains $q+1$ orbitals, twice that many <electrons> before interactions split the <spin degeneracy>. Hence closed-shell <electron> numbers are $2,6,12,20,\ldots$. These are single-particle <quantum numbers>; an interacting dot's states are many-electron eigenstates, not simply one orbital occupied by a whole charged island.
A perpendicular <magnetic field> changes the harmonic spectrum to the <Fock-Darwin spectrum>. For an <electron> of charge $-e$, take $\omega_c=eB/m^*>0$ and $L_z=\ell\hbar$ in the usual symmetric gauge. Then
$$
\epsilon_{n\ell s}=\hbar\Omega(2n+|\ell|+1)+\frac{\ell\hbar\omega_c}{2}+g^*\mu_BBs,\qquad \Omega=\sqrt{\omega_0^2+\frac{\omega_c^2}{4}},\quad s=\pm\frac12.
$$
Reversing the definition of the angular label changes the displayed sign of its orbital term, not the set of energies. The Zeeman term lifts <spin degeneracy>, while exchange can favor parallel-spin filling in partially filled shells. These effects are distinguished by how peaks evolve with field rather than by identifying every irregular peak spacing with one-particle confinement.
For electrostatics, let $C_\Sigma=C_s+C_d+C_g$ and $Q_g=C_sV_s+C_dV_d+C_gV_g+Q_0$, where $Q_0$ represents a fixed offset charge in the chosen convention. The <constant interaction model> gives
$$
U(N)=\sum_{i=1}^N\epsilon_i+\frac{(Ne-Q_g)^2}{2C_\Sigma},\qquad \mu_N=U(N)-U(N-1)=\epsilon_N+\frac{(N-1/2)e^2-eQ_g}{C_\Sigma}.
$$
The <charging energy> is conventionally $E_C=e^2/(2C_\Sigma)$, whereas adjacent charge-state potentials differ by
$$
\boxed{\mu_{N+1}-\mu_N=\frac{e^2}{C_\Sigma}+\epsilon_{N+1}-\epsilon_N.}
$$
This <addition energy> separates a capacitive contribution from orbital spacing. Exchange, nonuniform screening and correlations are corrections to the <constant interaction model>; a change in dot shape or effective <capacitance> can also alter spacings.
Connect the island weakly to source and drain reservoirs through tunnel barriers. <sequential tunnelling> is suppressed in a charge-stable region when neither adding nor removing an <electron> is energetically accessible. Sweeping the gate brings a charge <electrochemical potential> through the reservoir Fermi level, giving a Coulomb-blockade conductance peak. The gate lever arm is $\alpha_g=C_g/C_\Sigma$, so $\Delta V_g=E_{\rm add}/(e\alpha_g)$ locally if the <capacitances> and level spectrum do not vary appreciably. At finite source-drain bias, alignment with either reservoir defines the boundaries of a <Coulomb diamond>. Its slopes constrain <capacitance> ratios, and lines parallel to its edges can reveal excited-state transport. The conversion from a diamond's bias height to an <energy> depends on bias convention; a full-height/half-height factor must not be assumed without declaring the circuit.
A clear charging spectrum requires $k_BT$ and <tunnel broadening> $\hbar\Gamma$ smaller than the relevant <addition energy>; resolving individual orbital levels additionally requires them smaller than the orbital spacing. Tunnel resistances large compared with $h/e^2$ suppress strong charge fluctuations in the usual weak-junction regime. Blockade is not an assertion of mathematically zero current: higher-order <cotunnelling> can remain, and strong coupling or low-temperature <spin> correlations can invalidate the sequential model. An experimental check therefore combines gate sweeps with bias spectroscopy, <temperature> dependence and barrier tuning to separate level structure from broadening or uncontrolled charge traps.
In a primary vertical-dot experiment, gate-controlled current oscillations were converted into <addition energies> using finite-bias diamond spectroscopy; varying <magnetic field> then followed shell and <spin> filling. Reproducing enhanced <addition energies> at <electron> numbers 2 and 6 in several devices strengthened the shell assignment. The <electron> <temperature> was independently estimated above the refrigerator base <temperature>, an essential distinction when judging spectral resolution. Field-dependent reordering also revealed exchange effects beyond a purely spin-degenerate orbital model.<Shell Filling and Spin Effects in a Few Electron Quantum Dot by S. Tarucha and others>[pp. 3613–3616, gate control, finite-bias calibration, sample comparisons, electron-temperature estimate and field-dependent spin filling]{ref}
\Image[/past-exam-of-the-advanced-quantum-condensed-matter-physics-course-of-the-university-of-cambridge/2016/paper-1-dot-charging.png]
{title=Original constant-interaction charge parabolas and addition-energy example showing charging, spin pairing and shell closure}
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