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The difference between successive electron-addition chemical potentials. In a constant interaction model it is $E_{\mathrm{add}}=e^2/C_{\Sigma }+{ϵ}_{N+1}-{ϵ}_N$, separating electrostatic charging from orbital level spacing. Spin, exchange and correlation can change the orbital contribution.

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,A smooth circular lateral confinement is often better approximated by $m^{\ast }{\omega }_0^2{r}^2/2$, giving ${ϵ}_{n_x{n}_y}=\hslash {\omega }_0(n_x+n_y+1)$ 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 $2,6,12,20,\dots$. 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 ${\omega }_c=eB/m^{\ast }>0$ and $L_z=\ell \hslash$ in the usual symmetric gauge. ThenReversing 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 $Q_g=C_s{V}_s+C_d{V}_d+C_g{V}_g+Q_0$, where $Q_0$ represents a fixed offset charge in the chosen convention. The constant interaction model givesThe charging energy is conventionally $E_C=e^2/(2C_{\Sigma })$, whereas adjacent charge-state potentials differ byThis 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 $\Delta V_g=E_{\mathrm{add}}/(e{\alpha }_g)$ 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 $k_{B}T$ and tunnel broadening $\hslash \Gamma$ 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.^[ref]

An isolated nanometre-sized island has small total capacitance $C_{\Sigma }=C_S+C_D+C_G+\cdots$, so discrete charge carries appreciable charging energy. A simple constant interaction model isAway from charge degeneracy, adding or removing an electron costs more energy than a sufficiently small thermal/bias window supplies. Coulomb blockade suppresses sequential tunnelling between resolved charge states.

A single-electron transistor has source and drain reservoirs connected to an island through two tunnel junctions, plus an insulated capacitive gate. Each junction has resistance and capacitance; the gate shifts electrostatic energies without carrying the transport current. The original device schematic labels all components.

Require $k_{B}T\ll E_C$, a source–drain window below the relevant transition, and junction resistances well above $h/e^2\simeq 25.8\mathrm{k\Omega }$ to suppress strong quantum charge fluctuations. Electromagnetic noise and lifetime broadening must also be small. Gate-induced charge degeneracy permits conduction even at small bias; there is no gate-independent blockade threshold. Resolving orbital levels additionally requires temperature and broadening below their spacing between energy levels.

Assume the usual one-lead-biased measurement, $V_s=V_{sd}$ with drain grounded and the gate referred to that ground; the printed map does not separately specify the wiring. This assumption is needed to assign individual source/drain capacitances. In this Coulomb diamond capacitance extraction, the smaller zero-bias gate intervals are 5 mV wide. In the constant interaction model with no orbital gap between those additions,An upper diamond tip lies at gate 123.75 mV between the zero-bias vertices 120 and 125 mV and at bias 2.50 mV. Its positive and negative edge slopes are therefore $s_{+}=2.50/3.75=2/3$ and $s_{-}=-2.50/1.25=-2$. Reservoir alignment givesThe gate capacitance is not negligible in the second denominator. ConsequentlyThe consistency check is $e^2/C_{\Sigma }=2.50\mathrm{meV}$, matching the smaller tip's bias energy. The conventional electrostatic charging energy $E_C=e^2/(2C_{\Sigma })$ is 1.25 meV; it is half the charging energy contribution to the addition energy. A different applied-bias convention changes the slope formulas, so the convention must accompany the extraction.

The zero-bias gate period in the first plot is $\Delta V_g=20\mathrm{mV}$. In the constant interaction model, changing the gate-induced island charge by one electron givesThe tip is at $|V_{sd}|=5.0\mathrm{mV}$ above or below zero; use this half-height, not the full $10\mathrm{mV}$ vertical extent. The addition energy is $E_{\mathrm{add}}=e(5.0\mathrm{mV})=e^2/C_{\Sigma }$, giving

To distinguish the junction capacitances, explicitly adopt the circuit convention $V_s=V_{sd}$ and $V_d=0$. The diamond with zero-bias endpoints $20$ and $40\mathrm{mV}$ has its positive tip near $(V_g,V_{sd})=(35,5)\mathrm{mV}$. Its two edge slopes are therefore $+1/3$ and $-1$.

Writing the island electrochemical potential as a charge-dependent constant minus $e(C_s{V}_{sd}+C_g{V}_g)/C_{\Sigma }$, alignment with the drain gives slope $-C_g/C_s$, while alignment with the source gives $C_g/(C_{\Sigma }-C_s)$. ThusThe source/drain names are exchanged if the opposite lead is biased under the voltage convention. The figure alone establishes the pair $\{8,16\}\mathrm{aF}$ unless that wiring convention is specified. Symmetric-looking diamond heights do not imply equal junction capacitances.

The pattern is characteristic of a single-electron transistor: a small conducting island is connected to source and drain by two tunnel junctions, and to a gate through a capacitance. Inside a Coulomb diamond, changing the island's integer charge costs more energy than the leads can supply, so sequential tunnelling is blocked. Outside it, an island electrochemical potential lies in the transport window and current can flow.

For clear Coulomb blockade, the island charging scale must exceed thermal broadening, and the tunnel junctions must be sufficiently resistive to preserve well-defined charge states:in the simplest circuit. Stray capacitances should be included in $C_{\Sigma }$ if appreciable. The gate changes electrostatic energy but should not provide a conducting leakage path. Lifetime broadening and cotunnelling should be small enough not to wash out the blocked regions.

The almost identical unconfined diamonds are consistent with a constant interaction model having negligible or unresolved changes in single-particle level spacing.

An electrostatically defined quantum dot uses depletion gates to confine a small island laterally in a two-dimensional electron gas. Two narrow barriers allow weak tunnelling to source and drain; a plunger gate tunes the island's electrochemical levels, and additional gates can form a double quantum dot. Adjusting the gates resolves successive charge states and eventually reaches the single-electron regime. An electrical dot is defined by confinement and tunnel coupling, not simply by the drawn metal area.

In a constant interaction model,Here $Q_g$ is gate-induced charge and $C_{\Sigma }$ total island capacitance. Successive additions cost $E_{\mathrm{add}}=e^2/C_{\Sigma }+{ϵ}_{N+1}-{ϵ}_N$, with spin, exchange and correlations modifying the orbital term. Coulomb blockade appears when no allowed charge transition lies in the source–drain energy window. Gate sweeps give charge steps and transport peaks; finite-bias Coulomb diamonds constrain charging and orbital energies under a stated bias convention. Resolved single-electron operation requires thermal and tunnel broadening smaller than the relevant addition energies.

Quantum-dot charge sensing avoids relying on a measurable current through a nearly isolated dot. A nearby quantum point contact biased on a sensitive conductance slope, or a single-electron transistor, responds capacitively when one electron enters or leaves the dot. A time trace of sensor current can therefore distinguish $N$ from $N-1$ and resolve individual tunnelling events. The sensor measures charge; a spin-dependent step is needed to measure spin.

For one electron, Zeeman splitting gives a lower and a higher spin level separated by $E_Z=|g^{\ast }|{\mu }_B|B|$. In energy-selective spin readout, initialize an empty dot and pulse both levels below the reservoir chemical potential to load one electron. Then move to a readout configurationAn electron in the higher state can tunnel out, temporarily leaving the dot empty, after which a lower-state electron can refill it. An electron initially in the lower state remains trapped. The charge sensor records a pulse for the first outcome and no escape pulse for the second. This is spin-to-charge conversion; it measures the spin projection along the field through a state-dependent tunnelling probability. A primary single-shot experiment uses this conversion with a quantum-point-contact charge detector.^[ref]

The readout window must be long enough to detect escape but short compared with excited-spin relaxation:Detector bandwidth must also resolve the empty-state dwell time before refilling. Thermal activation, relaxation before escape, missed fast events and sensor backaction cause errors. Since the excited electron can leave and be replaced, this simple sequence does not preserve an arbitrary initial spin state.

For coupled dots, label charge occupation by $(n_L,n_R)$. Gate-controlled detuning can make $(1,1)$ and $(0,2)$ accessible. Two electrons in the lowest orbital of one dot must form a singlet state; a triplet state requires a different orbital or another higher-energy configuration. Choose a measurement window where the $(0,2)$ singlet is available but the $(0,2)$ triplet is not. Thenunder spin-conserving tunnelling. The second process is forbidden by spin symmetry and the Pauli exclusion principle, producing Pauli spin blockade. Charge sensing distinguishes transfer from a blocked $(1,1)$ state. This measures singlet versus triplet content; without a reference spin or additional control it is not a measurement of each electron's absolute spin direction separately.

The exchange interaction in a double quantum dot provides control of these states. In an effective spin description,For a symmetric weak-tunnelling two-site model with onsite charging cost $U$, $J\simeq 4t_c^2/U$. Gates change tunnel coupling $t_c$ and detuning, allowing preparation and evolution followed by a charge-conversion measurement. Spin mixing from local magnetic-field differences or spin-orbit effects can lift blockade, so the sensing interval must be chosen before unwanted relaxation obscures the initial state. Single-dot readout uses spin-dependent escape; double-dot readout can use spin-dependent charge transfer. In both cases the measurable signal is a resolved charge change.