The canonical example of a quantum-mechanical two-level system is spin. The simplest picture of spin is a magnetic moment pointing up or down. The full quantum properties of spin become apparent in phenomena such as superpositions of spin states, entanglement among spins, and quantum measurements. Many of these phenomena have been observed in experiments performed on ensembles of particles with spin. Only in recent years have systems been realized in which individual electrons can be trapped and their quantum properties can be studied, thus avoiding unnecessary ensemble averaging. This review describes experiments performed with quantum dots, which are nanometer-scale boxes defined in a semiconductor host material. Quantum dots can hold a precise but tunable number of electron spins starting with 0, 1, 2, etc. Electrical contacts can be made for charge transport measurements and electrostatic gates can be used for controlling the dot potential. This system provides virtually full control over individual electrons. This new, enabling technology is stimulating research on individual spins. This review describes the physics of spins in quantum dots containing one or two electrons, from an experimentalist’s viewpoint. Various methods for extracting spin properties from experiment are presented, restricted exclusively to electrical measurements. Furthermore, experimental techniques are discussed that allow for 1 the rotation of an electron spin into a superposition of up and down, 2 the measurement of the quantum state of an individual spin, and 3 the control of the interaction between two neighboring spins by the Heisenberg exchange interaction. Finally, the physics of the relevant relaxation and dephasing mechanisms is reviewed and experimental results are compared with theories for spin-orbit and hyperfine interactions. All these subjects are directly relevant for the fields of quantum information processing and spintronics with single spins i.e., single spintronics.

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Spins in few-electron quantum dots

Brownian motors: noisy transport far from equilibrium

Random-matrix theories in quantum physics : common concepts

How localized electrons interact with delocalized electrons is a central question to many problems in sold-state physics1,2,3. The simplest manifestation of this situation is the Kondo effect, which occurs when an impurity atom with an unpaired electron is placed in a metal2. At low temperatures a spin singlet state is formed between the unpaired localized electron and delocalized electrons at the Fermi energy. Theories predict4,5,6,7 that a Kondo singlet should form in a single-electron transistor (SET), which contains a confined ‘droplet’ of electrons coupled by quantum-mechanical tunnelling to the delocalized electrons in the transistor's leads. If this is so, a SET could provide a means of investigating aspects of the Kondo effect under controlled circumstances that are not accessible in conventional systems: the number of electrons can be changed from odd to even, the difference in energy between the localized state and the Fermi level can be tuned, the coupling to the leads can be adjusted, voltage differences can be applied to reveal non-equilibrium Kondo phenomena7, and a single localized state can be studied rather than a statistical distribution. But for SETs fabricated previously, the binding energy of the spin singlet has been too small to observe Kondo phenomena. Ralph and Buhrman8 have observed the Kondo singlet at a single accidental impurity in a metal point contact, but with only two electrodes and without control over the structure they were not able to observe all of the features predicted. Here we report measurements on SETs smaller than those made previously, which exhibit all of the predicted aspects of the Kondo effect in such a system.

Kondo effect in a single-electron transistor

Electron transport experiments on two lateral quantum dots coupled in series are reviewed. An introduction to the charge stability diagram is given in terms of the electrochemical potentials of both dots. Resonant tunneling experiments show that the double dot geometry allows for an accurate determination of the intrinsic lifetime of discrete energy states in quantum dots. The evolution of discrete energy levels in magnetic field is studied. The resolution allows one to resolve avoided crossings in the spectrum of a quantum dot. With microwave spectroscopy it is possible to probe the transition from ionic bonding (for weak interdot tunnel coupling) to covalent bonding (for strong interdot tunnel coupling) in a double dot artificial molecule. This review is motivated by the relevance of double quantum dot studies for realizing solid state quantum bits.

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Electron transport through double quantum dots

Coulomb blockade effects are investigated in lateral transport through a quantum dot defined in a two-dimensional electron gas. Tunneling through excited states of the quantum dot is observed for various tunneling barriers. It is shown that transport occurring via transitions between ground states with different numbers of electrons can be suppressed by the occupation of excited states. Measurements in a magnetic field parallel to the current give evidence for tunneling processes involving states with different spin.

Competing channels in single-electron tunneling through a quantum dot.

Single-electron tunneling through two coupled quantum dots is strongly influenced by the electrostatic interaction between the dots. At low temperatures, we observe the formation of a molecularlike state in the double-dot system. The dots are defined by split gates in an ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As-GaAs heterostructure. They are of different size and coupled in series, leading to two different charging energies. In the linear transport regime we find the charging diagram for the double quantum dot. This diagram exhibits the operation points of the system to be employed as an electron pump. \textcopyright{} 1996 The American Physical Society.

Single-electron tunneling through a double quantum dot: The artificial molecule.

Zero-bias anomalies in the conductance through quantum dots have recently been identified as Kondo resonances and explained in terms of the Anderson impurity model. The effect requires a degeneracy and it has been proposed that this should occur for odd electron numbers on the dot. In this paper we present data, obtained on a split-gate quantum dot with a small number of electrons, which are in disagreement with this expectation. The mapping of the Anderson model on the quantum dot is discussed in terms of an interacting $N$ electron system demonstrating why this expectation can fail.

Absence of odd-even parity behavior for Kondo resonances in quantum dots.

A low-temperature scanning force microscope sensitive to electrostatics is used to investigate the potential distribution of a two-dimensional electron system (2DES) under quantum Hall conditions at a temperature T = 1.4 K. We mapped out the Hall potential profiles for the Landau level filling factor range 1 < v < 14 with submicron resolution. At integer filling factors, the potential drop is rather nonlinear, but depends strongly on the scan position. Obviously inhomogeneities of the sample strongly affect the Hall potential profile if the bulk of the 2DES is nonconductive and cannot screen. At filling factors slightly above an even integer value, the Hall potential drops at prominent positions at both edges which are associated with the positions of the incompressible strips as confirmed by comparison with theoretical predictions. With these broad incompressible strips electrically decoupling the edge region from the compressible bulk region, a nearly vanishing Hall potential drop is observed in the compressible bulk region, suggesting that most of the current is flowing close to the edges. In addition, potential profiles near the current injecting contacts reveal the well known hot spot region in the direct vicinity of the contact, but recover to the described shape within a distance of only 6 μm.

Hall potential profiles in the quantum Hall regime measured by a scanning force microscope

Scanning force microscope measurements of the Hall potential distribution of a two-dimensional electron system (2DES) in the quantum Hall regime reveal the presence of a partial depletion along the border between the 2DES and the good ohmic metal contact: incompressible strips are formed in high magnetic fields along this border, decoupling the bulk region of the 2DES from the contact. This finding clarifies the role of potential probe contacts in Hall bar devices and might affect the interpretation of conductance measurements on Corbino devices.

J. Weis

Papers

Competing channels in single-electron tunneling through a quantum dot.

Single-electron tunneling through a double quantum dot: The artificial molecule.

Absence of odd-even parity behavior for Kondo resonances in quantum dots.

Hall potential profiles in the quantum Hall regime measured by a scanning force microscope

Hall potential distribution in the quantum Hall regime in the vicinity of a potential probe contact