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Mesoscopic physics  
A subdiscipline of condensed-matter physics that focuses on the properties of solids in a size range intermediate between bulk matter and individual atoms or molecules. The size scale of interest is determined by the appearance of novel physical phenomena absent in bulk solids and has no rigid definition; however, the systems studied are normally in the range of 100 nanometers (10-7 meter, the size of a typical virus) to 1000 nm (the size of a typical bacterium). Other branches of science, such as chemistry and molecular biology, also deal with objects in this size range, but mesoscopic physics has dealt primarily with artificial structures of metal or semiconducting material which have been fabricated by the techniques employed for producing microelectronic circuits. Thus it has a close connection to the fields of nanofabrication and nanotechnology. The boundaries of this field are not sharp; nonetheless, its emergence as a distinct area of investigation was stimulated by the discovery of three categories of new phenomena in such systems: interference effects, quantum size effects, and charging effects. See also: Artificially layered structures; Nanostructure; Quantized electronic structure (QUEST); Semiconductor heterostructures
Interference effects
Although quantum mechanics predicts that any particle should exhibit properties characteristic of waves, electrons in bulk solids may be treated for most purposes as particles. The reason is that the wavelike properties are suppressed by interactions between the electrons and the vibrations of the lattice of ions forming the solid. These interactions become weaker as the temperature of the solid is lowered and thermal vibrations are reduced. At temperatures less than 4 K (-452°F), the vibrations are so rare that electrons can traverse several thousand nanometers (the size of a typical mesoscopic device) without any significant interaction, and the wave properties of the electron affect measurable physical quantities such as the electric conductivity. See also: Quantum mechanics
The most important wave property of electrons is that of interference. Unlike a classical particle, an electron wave can split at one point in space into many wavefronts which follow different paths to another point in space. These different wavefronts may add or subtract from each other at that point, leading to constructive or destructive interference. In the former case there will be an increased probability of finding an electron at that point; in the latter case there will be zero probability. The conduction electrons in solids are split into many wavefronts by defects or impurities in the crystal lattice. The density of such defects determines the average electrical conductivity of a bulk solid (at low temperatures) since they impede the flow of the electrons. However, in the mesoscopic regime scattering from defects also induces interference effects which modulate the flow of electrons. This modulation depends not only on the density of impurities but on their precise location relative to one another; hence it differs for each specimen, even if the specimens are made of the same material and fabricated under identical conditions. Thus the concept of conductivity as a property of a given material becomes useless for mesoscopic solids. Instead, the conductance of a given specimen is studied: the ratio of the measured current to the driving voltage. See also: Conductance; Conduction (electricity); Electrical resistivity; Interference of waves
The experimental signature of mesoscopic interference effects is the appearance of reproducible fluctuations in physical quantities. Such time-independent fluctuations have been measured in the magnetization, thermopower coefficient, alternating current, direct current, and nonlinear conductance. For example, the conductance of a given specimen oscillates in an apparently random manner as a function of experimental parameters such as a magnetic field or the electron density (Fig. 1). However, the same pattern may be retraced if the experimental parameters are cycled back to their original values; in fact, the patterns observed are reproducible over a period of days. In experiments a magnetic field is most commonly used to demonstrate this effect. A magnetic field induces these fluctuations because according to quantum theory a field alters the phase of the electron waves. The general principle that a magnetic field modulates quantum-mechanical interference is known as the Aharonov-Bohm effect. See also: Aharonov-Bohm effect
Fig. 1   Conductance fluctuations in mesoscopic conductors. (a) Conductance of 2000-nm gold wire as a function of magnetic field measured at temperature 0.04 K (after R. A. Webb and S. Washburn, Quantum interference fluctuations in disordered materials, Phys. Today, 41(12): 46-55, December 1988). (b) Conductance of 600-nm GaAs/AlGaAs semiconducting heterostructure as a function of electron density (controlled by an electrostatic gate) at temperature 0.1 K (after M. W. Keller et al. Magnetotransport in a chaotic scattering cavity with a turnable electron density, Surf. Sci., 305:501-506, 1994). The patterns observed are reproducible over a period of days, and the amplitudes of both fluctuations are of order e2/h.
Quantum size effects
Another prediction of quantum mechanics is that electrons confined to a particular region of space may exist only in a certain set of allowed energy levels. The spacing between these levels increases as the confining region becomes smaller; also, in typical systems the spacing between the levels decreases with increasing overall energy. Conduction electrons in a bulk solid are confined only to the macroscopic dimensions of the solid, and the energy-level spacing becomes so small that the discreteness of the allowed energies is undetectable. In mesoscopic solids, however, the spacing is large enough that the discreteness of the levels affects measurable physical properties. Because conduction electrons in metals have much higher energies than those in semiconductors and the level spacing decreases with energy, it turns out that these quantum size effects are most important in mesoscopic semiconductors. One striking phenomenon which arises from these quantum size effects is the steplike increase of the conductance of electrons flowing through a constriction of several hundred nanometers' width. In classical physics the conductance should increase in proportion to the width of the constriction. In quantum physics the electrons become confined laterally as they enter the constriction and are allowed only certain energies. No further current can be generated as the constriction is widened until a new state appears because of the decrease in the energy-level spacing. The result is that instead of increasing linearly with width the conductance increases in a series of sharp steps. It is found that these steps are of magnitude 2e 2/h, a result which is expected from the basic quantum theory of the process. See also: Energy level (quantum mechanics)
Another mesoscopic system that shows quantum size effects consists of isolated islands of electrons that may be formed at the appropriately patterned interface between two different semiconducting materials. [Gallium arsenide (GaAs) and aluminum gallium arsenide (AlGaAs) are most common.] The electrons typically are confined to a disk-shaped region, and hence these systems have been termed quantum dots. Quantum dots may be regarded as artificial atoms with a new set of energy levels never before realized in nature. The confinement of the electrons in these systems changes their interaction with electromagnetic radiation significantly. The electronic absorption energies are shifted from their values in bulk semiconductors, and the light that is absorbed or emitted by the dot is restricted to certain frequencies because of confinement. Quantum dots can be used to make semiconductor lasers that operate at very low power, a property of great technological interest. See also: Laser
Charging effects
Isolated mesoscopic solids such as quantum dots or metallic grains on an insulating substrate also show novel effects associated with the discreteness of the charge on the electron. Normally the free-electron charge in a conductor can be regarded as a continuous fluid which is infinitely divisible. This is so both because the amount of charge involved in macroscopic systems is usually very large compared to the charge on a single electron and because quantum mechanics allows even a single electron to be spread out over a very large area. Hence the charge on a capacitor plate is tunable to a small fraction of the electron charge (e) if it is in contact with a reservoir of electron charge such as a battery. However, if the capacitor is mesoscopic, this ceases to be true. See also: Capacitor
The simplest case is that of an isolated mesoscopic grain of metal such as aluminum. The grain has a certain number of atoms and will be in its lowest energy state if it retains the correct number of electrons to neutralize the total positive charge of the atomic nuclei. The energy cost of removing an electron is inversely proportional to the capacitance of the grain, which decreases with its volume. For a mesoscopic grain this energy exceeds thermal energies at roughly 1 K; below this temperature the number of electrons on the grain will not vary because of thermal fluctuations.
If such a grain is fabricated as part of an electronic circuit, it will block the flow of current until the driving voltage is large enough to compensate for the energy needed to increase the electronic charge on the grain by e. However, if an additional electric field is applied to the grain through an auxiliary capacitor, the singly ionized state of the grain can be made energetically allowed and a current will flow through the circuit. With this principle it has been possible to fabricate circuits for which a current of 109 electrons per second (approximately 10-9 A) can be switched on and off by the addition of one-half the electron charge to the control capacitor. These devices are known as single-electron transistors (SETs) and are by far the most sensitive electrometers (instruments for measuring electrical charge) presently known. See also: Electrometer; Transistor
Building on these concepts, devices employing several single-electron transistors have been demonstrated in which a modulation of the control capacitor with a frequency f leads to an electrical current precisely equal to ef. In such devices, electrons are transferred one by one onto and off the grain. It is even possible to measure jumps in the current corresponding to the transfer of each electron (Fig. 2). Such devices are expected to have metrological applications as an absolute standard of electric current. See also: Electrical units and standards
Fig. 2   Time variation of the electric current through a single-electron transistor electrometer, measuring the charge on a mesoscopic metallic grain. Each jump corresponds to the removal or addition of a single electron from the grain. (After M. H. Devoret, D. Esteve, and C. Urbins, Single-electron transfer in metallic nanostructures, Nature, 360:547-553, 1992)

A. Douglas Stone

How to cite this article
Please cite this article as follows:
A. Douglas Stone, "Mesoscopic physics", in AccessScience@McGraw-Hill,, last modified: May 6, 2002.
For Further Study
Topic Page: arrowPhysics:  arrowSolid state physics
B. L. Altshuler, P. A. Lee, and R. A. Webb (eds.), Mesoscopic Phenomena in Solids, 1991
M. H. Devoret, D. Esteve, and C. Urbina, Single-electron transfer in metallic nanostructures, Nature, 360:547-553, 1992
A. Khurana, Ballistic electron transport through a narrow channel is quantized, Phys. Today, 41(11):21-23, 1988
M. A. Reed, Quantum dots, Sci. Amer., 268(1):118-123, January 1993
R. A. Webb and S. Washburn, Quantum interference fluctuations in disordered metals, Phys. Today, 41(12):46-55, December 1988

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