The study of single-electron phenomena associated with tunnelling in semiconductor nanostructures has emerged in recent years as a major forefront of condensed matter physics, whose implications range from fundamental physics to electronic device applications. This paper presents a tutorial review of the subject, with emphasis on the role of single-electron charging in such semiconductor 'quantum dots'. The main purpose is to describe the various phenomena observed in these experiments and to present the theoretical understanding of these phenomena in an introductory fashion. The paper attempts to explain the underlying physics at the intuitive level and tries to draw, as much as possible, a unifying perspective on a relatively large body of knowledge acquired within a short time by the conjunction of many individual contributions.

Measurements are reported of the temperature dependence of the conductance through a small region of electron gas separated from its leads by tunnel junctions. The quantization of charge and energy in the small region gives rise to sharp, nearly periodic peaks in the conductance as a function of electron density, one for each electron added to the isolated region. Because the charge and energy are quantized, we call this an artificial atom. At low temperature, the conductance is limited by resonant tunneling through a single quantum level of the artificial atom, but at high temperatures several levels participate. Changes in the temperature dependence of the width and height of conductance peaks display clear evidence for this crossover from single-level to multilevel transport.

Transport measurements are presented of a small island of electrons confined within a semiconductor heterostructure. Low bias measurements are used to infer the addition spectrum of the island as a function of gate voltage and magnetic field. Nonlinear measurements are used to explore the excitation spectrum of the island. These measurements are interpreted in terms of a model that treats the Coulomb interactions between electrons in a self-consistent manner.

Transport measurements are used to study the quantized energy levels in a small electron gas, referred to as a Coulomb island. First, large-bias measurements reveal the excitation spectrum of the island. Second, small-bias measurements probe the intrinsic line shape of energy levels. Line shapes are generally fit well by thermally broadened Lorentzians. In addition, a decrease in the charging energy of the island is observed as the coupling to its leads is increased. This effect is shown to be consistent with a dramatic increase in one of the island-lead capacitances

Coulomb interactions are shown to influence the addition spectrum of a small electron gas in the quantum Hall regime in ways that cannot be described by a classical charging energy. The interaction energy between electrons is observed to depend upon Landau-level index, and the evolution of the addition spectrum with magnetic field is found to depend strongly on Coulomb interactions. A self-consistent model of the island is introduced that can account for these results

Results are reported for low temperature measurements of the conductance through small regions of a two-dimensional electron gas (2DEG). An unconventional GaAs heterostructure is used to form a 2DEG whose density can be tuned by the gate voltage applied to its conductive substrate. Electron beam lithography is used to pattern a narrow channel in the 2DEG interrupted by two constrictions, defining a small 2DEG island between them. The conductance is found to oscillate periodically with the gate voltage, namely with electron density. Calculations of the capacitance between the substrate and the island show that the period of oscillation corresponds to adding one electron to the island. The oscillatory behavior results primarily from the discreteness of charge and the Coulomb interaction between electrons. However, the observed temperature dependence of these oscillations requires a more sophisticated treatment which includes the quantized electron energy levels as well. The magnetic field dependence of the oscillations allows us to extract the discrete energy spectrum of the quantum dot in the quantum-Hall regime.

Transport measurements of a Coulomb island, a semiconductor dot small enough that Coulomb interactions dominate transport are presented. At moderate magnetic fields (B=2--4 T) the amplitude and position of the Coulomb-regulated conductance peaks show distinct periodic structure as a function of B. This structure is shown to result from the B dependence of the quantized single-particle energy states on the island. Analysis of successive peaks is used to map out the single-particle level spectrum of the island as a function of B.