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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]

For the spin separation at a smooth quantum Hall edge, linearize the confinement energy at a crossing. Compensating a spin energy difference by a potential-energy change givesHere $V^{\prime }$ is interpreted as the specified slope of electron potential energy. If it denotes electrostatic voltage per length instead, the denominator is $e|V^{\prime }|$. The two edges of a neck and the two sides of an antidot have oppositely directed slopes; the formula is a positive local separation on any one edge. For local filling two, only the $n=0$ spin pair is occupied at the neck minimum: $E_{0,+}+U_{\min }<E_F<E_{1,-}+U_{\min }$. Raising both branches with the confinement produces two crossings on each side, as shown in the original neck sketch. This fixed-slope estimate precedes self-consistent electrostatic screening.

An antidot is a depleted potential hill. guiding centres circulate around its closed equipotentials, with chirality opposite to the outer sample edge. A phase-coherent round trip contributes magnetic phase $2\pi BS/{\Phi }_0$, with ${\Phi }_0=h/e$. A localized orbit resonant with the incident energy can transfer an electron between extended channels. If the outgoing channel returns toward the injecting reservoir, the process is resonant reflection and reduces electrical conductance. If coupling instead connects an otherwise blocked incident channel to the far reservoir, it is resonant transmission and adds electrical conductance. Transmission peaks require that forward connectivity; a closed orbit by itself does not guarantee them. Interchannel scattering can change which orbit participates.^[ref]

For the antidot resonance conductance bounds with one active spin-resolved resonance, writeHere ${\Gamma }_L,{\Gamma }_R$ are energy broadenings; the corresponding escape rates are ${\Gamma }_L/\hslash ,{\Gamma }_R/\hslash$. In a reflection-dominated configuration with two otherwise transmitted channels, $G=(e^2/h)(2-\mathcal{L})$: a $2e^2/h$ background can dip to $e^2/h$. In a transmission-dominated configuration with one spectator channel and the second otherwise blocked, $G=(e^2/h)(1+\mathcal{L})$: peaks rise from $e^2/h$ toward $2e^2/h$. These are the two ideal channel-count examples plotted below. More generally, $G=(e^2/h)(N_{\mathrm{open}}-R)$ or $(e^2/h)(N_{\mathrm{dir}}+T)$; a different spectator count changes the integer offset. The constriction filling alone does not specify every tunnel coupling or resonance height. Equal couplings permit unit resonant transfer in the isolated-level model; asymmetry, temperature and dephasing reduce the contrast. Holding the effective orbit area approximately fixed gives $\Delta B\simeq h/(eS)$ for one branch, not a universal charging-corrected period.

In quantum-dot versus antidot resonance topology, a quantum dot is a potential well holding a finite electron island. Its closed guiding-centre circulation is reversed relative to an antidot at the same field. With two barriers in series, current follows source $\to$ dot $\to$ drain; off resonance there need not be a direct conducting path. A gate shifts a level or an addition electrochemical potential into alignment with the reservoirs and produces a electrical conductance peak. For an isolated coherent spin-resolved level, the zero-temperature peak is at most $e^2/h$ and reaches it only for equal couplings; unresolved noninteracting spin degeneracy doubles this bound. In weak sequential tunnelling, peak height depends on rates and temperature rather than being quantized. Charging produces Coulomb blockade peaks with spacing set by addition energy and gate lever arm. An antidot also supports discrete orbit/charging resonances, but can be side-coupled to already open channels and produce dips. Gate voltage changes both orbit area and confinement, whereas a magnetic sweep directly changes enclosed flux.

Coulomb interactions require a self-consistent confinement potential. A gradual decrease in electron density cannot leave arbitrary uncompensated donor charge at no cost. Partly occupied Landau levels have a large charge response: electrostatic screening flattens their effective energy near the Fermi energy over compressible quantum Hall edge strips. Between them, integer-filled incompressible quantum Hall edge strips have a gap, weak density response and most of the potential variation. In an idealized cross-section the effective branch thus alternates between flat, partially occupied portions and rising portions crossing a gap; its carrier density varies oppositely. The bare levels still have Landau level and spin splittings locally, but the common electrostatic potential is no longer an imposed straight slope. exchange interaction can additionally renormalize spin gaps. Widths must follow electrostatics and the density gradient, so the unscreened $E_Z/|V^{\prime }|$ separation is not a quantitative screened-strip width.^[ref]

A gated quantum dot is an electron island confined in all spatial directions, weakly coupled by tunnel barriers to source and drain and tuned by a plunger gate. Small capacitance makes charge addition costly, while confinement resolves orbital levels. In the constant-interaction model,The charging contribution to successive addition potentials is $e^2/C_{\Sigma }$; the often used charging-energy scale $E_C=e^2/(2C_{\Sigma })$ differs by a factor of two. Orbital spacing and spin filling add further structure.

For weak coupling, $k_{B}T$ and lifetime broadening must be smaller than the relevant addition energies to see Coulomb blockade. Sequential tunnelling proceeds when an addition level is inside the source-drain bias window and the appropriate charge states are accessible. A gate sweep gives conductance peaks near charge degeneracies and low conductance between them. A bias-versus-gate map shows Coulomb diamonds, with excited-state lines when additional levels enter the window. Their slopes and sizes depend on source, drain and gate capacitances and the bias convention. Cotunnelling gives higher-order transport inside nominally blocked regions; stronger coupling and many-body effects can invalidate the simple sequential picture.

Coherent dot states as a qubit. A single-electron spin has basis $|0⟩=|\uparrow ⟩$, $|1⟩=|\downarrow ⟩$ in a magnetic field, split by $E_Z=|g^{\ast }|{\mu }_{B}B$. A qubit is a controllable coherent state $a|\uparrow ⟩+b|\downarrow ⟩$, not just a pair of classical current levels. Initialize through spin-selective loading or relaxation; resonant magnetic or spin-orbit-assisted electrical driving produces rotations. Adjacent occupied dots can couple through a gate-controlled exchange interaction. In the weak-tunnel symmetric Hubbard limit,Local spin rotations combined with a suitable exchange entangling operation provide universal control in the ideal spin-qubit proposal. Gate pulses must avoid unwanted orbital excitation and leakage, and complete coherent operations before dephasing destroys the information.^[ref]

For measurement, place the reservoir electrochemical potential between Zeeman levels so that only one spin state can tunnel out, or use Pauli spin blockade in a double dot. A nearby quantum point contact or single-electron transistor detects the resulting charge change. Such spin-to-charge conversion makes transport useful for reading a spin even when the stored information is not itself a net current.^[ref]

A double dot can alternatively encode a charge qubit in left/right occupation, with Hamiltonian $H=(\epsilon {\sigma }_z+2t_c{\sigma }_x)/2$. It allows electrical control and charge sensing, but couples strongly to charge noise. Spin qubits also suffer hyperfine, electrical and spin-orbit-related noise, as well as relaxation and readout errors. An observed Coulomb diamond alone establishes charging physics, not initialization, coherent control or a functioning qubit.