Quantum dot

A quantum dot confines electrons and electron holes in all three spatial directions, producing discrete energy levels. A semiconductor island surrounded by a larger-band gap material is one realization. Its optical excitation and electron-hole recombination depend on confinement, Coulomb interactions and electric-dipole selection rules.

A small electron island confined laterally by depletion gates in an already vertically confined two-dimensional electron gas. Tunnel barriers couple it to reservoirs, and a plunger gate tunes charge occupation. Coulomb blockade and orbital confinement resolve individual electron additions.

A single confined electron's two spin states form a qubit when initialization, coherent rotations and measurement are controlled. Local driving rotates one spin; gate-controlled exchange can entangle neighboring dot spins. Spin-to-charge conversion permits electrical readout. Observing charge blockade alone does not demonstrate coherent qubit operation.

Two capacitively and tunnel-coupled electron islands with separately tunable occupations. Detuning controls which dot holds an electron and whether two-electron charge states such as $(1,1)$ and $(0,2)$ are accessible.

A single electron shared by two dots has a left/right charge basis. In that basis, a two-level model is $H=(\epsilon {\sigma }_z+2t_c{\sigma }_x)/2$, where detuning sets the energy imbalance and tunnelling mixes the states. Electrical control and charge sensing are direct, but electric-potential fluctuations also couple strongly to the encoded information.

An effective coupling that separates the two-electron singlet state and triplet states by energy $J$. For a symmetric two-site Hubbard model with weak tunnelling $t_c$ and charging cost $U$, $J\simeq 4t_c^2/U$. Gate-controlled detuning and tunnelling change the interaction; the simple expression does not hold at arbitrary detuning or strong charge hybridization.

A measurement method in which spin-dependent tunnelling or allowed charge transfer maps a spin outcome onto a distinguishable dot charge state. Quantum-dot charge sensing then records that charge. The measurement axis or singlet/triplet observable must be specified.

Map states with unequal escape rates ${\Gamma }_f\gg {\Gamma }_s$ onto distinct charge outcomes after an interval between their characteristic escape times. Both removal energies can lie above the reservoir chemical potential, so their splitting need not be thermally resolved. A useful rate difference must have a physical origin: a demonstrated single-dot example uses a two-electron singlet state and triplet state with different occupied orbitals. A bare single-electron Zeeman pair in the same orbital does not automatically provide the required rate contrast.^[ref]

For independent exponential escape, equal prior probabilities and no state relaxation, the mean classification error is $P_e(\tau )=[e^{-{\Gamma }_f\tau }+1-e^{-{\Gamma }_s\tau }]/2$. Setting its derivative to zero gives the displayed interval. The error increases both when the fast state is not given enough time to escape and when the slow state is given too much. Detector bandwidth, relaxation and reloading require an expanded measurement model.

Suppression of charge transfer by spin symmetry and the Pauli exclusion principle. For a double quantum dot, a $(1,1)$ singlet state can enter the lowest $(0,2)$ singlet, whereas a $(1,1)$ triplet state is blocked if the required $(0,2)$ triplet lies above the accessible energy window. Spin mixing or relaxation can lift blockade.

A resonantly driven local electron-spin rotation can give a blocked two-dot triplet a singlet component, allowing spin-conserving charge transfer to a lowest-orbital doubly occupied dot. A common rotation of two perfectly identical spins does not mix singlet and triplet sectors, so relative spin rotation or an additional mixing mechanism is essential. Leakage-current spectroscopy tests resonant lifting; pulse-duration oscillations add evidence for coherent control.

Place the reservoir chemical potential between the lower and higher Zeeman-split one-electron dot levels. The excited state can leave while the ground state remains trapped; a time-resolved charge sensor detects the temporary empty state. Thermal errors, missed fast tunnelling and relaxation before escape limit fidelity.

Detection of a dot's charge through its capacitive influence on a nearby quantum point contact or single-electron transistor. Discrete charge changes shift the sensor response, allowing individual tunnel events to be resolved even when direct current through the dot is too small to measure.

For a circular two-dimensional parabolic dot in perpendicular magnetic field, orbital levels are $\hslash \Omega (2n+|\ell |+1)\pm \ell \hslash {\omega }_c/2$, with ${\Omega }^2={\omega }_0^2+{\omega }_c^2/4$. The sign depends on the angular-momentum and signed-charge convention. At zero field the shell degeneracy is $q+1$ before spin; Zeeman and interaction terms require separate treatment.

A nanocrystal with one semiconducting composition surrounded by another. band offsets, interface quality and shell thickness govern carrier confinement and coupling to external traps. type-I band alignment favors electron and hole localization together; type-II band alignment can separate them. A larger shell band gap does not by itself determine those offsets.

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.

Filling a degenerate orbital shell produces a larger addition energy for the next electron. In a circular two-dimensional harmonic potential, shell $q=n_x+n_y$ has $q+1$ orbitals and capacity $2(q+1)$ including spin. Cumulative closed-shell occupancies are $(q+1)(q+2)=2,6,12,20,\dots$. Charging and orbital contributions must be separated, and anisotropy or exchange can split the ideal shells.

A quantum dot of InAs embedded in gallium arsenide confines injected electrons and electron holes. Strain-driven island formation supplies dots with lateral dimensions of order tens of nanometres and heights of a few nanometres. In an electrically driven implementation, a dot lies in a junction's intrinsic region. Selecting one dot spatially and one radiative line spectrally turns discrete exciton recombination into a single-photon source. Pulsed injection must avoid re-excitation during the same collection window.

The pioneering electrically driven device used an areal dot density around $5\times 10^8\mathrm{c}{\mathrm{m}}^{-2}$, a GaAs p-i-n structure and an aperture selecting individual-dot light.^[ref]