"The nanoparticte has a diamond lattice structure in the shape of a cube n unit cells on a side, having a width na, where a is the unit cell dimension. Column 2 gives sizes for GaAs, which has a=0.565nm.

"The nanoparticte has a diamond lattice structure in the shape of a cube n unit cells on a side, having a width na, where a is the unit cell dimension. Column 2 gives sizes for GaAs, which has a=0.565nm.

the particle. This is expected because according to the way the calculation was carried out, only one of the two types of atoms in the GaAs structure contributes to the surface.

A charge carrier in a conductor or semiconductor has its forward motion in an applied electric field periodically interrupted by scattering off phonons and defects. An electron or hole moving with a drift velocity v will, on the average, experience a scattering event every t seconds, and travel a distance / called the mean free path between collisions, where

This is called intraband scattering because die charge carrier remains in the same band after scattering, such as the valence band in the case of holes. Mean free paths in metals depend strongly on the impurity content, and in ordinary metals typical values might be in the low nanometer range, perhaps from 2 to 50 nm. In very pure samples they will, of course, be much longer. The resistivity of a poly crystalline conductor or semiconductor composed of microcrystaliites with diameters significantly greater than the mean free path resembles that of a network of interconnected resistors, but when the microcrystallite dimensions approach or become less than /, die resistivity depends mainly on scattering off boundaries between crystallites. Both types of metallic nanostructures are common.

Various types of defects in a lattice can interrupt the forward motion of conduction electrons, and limit the mean free path. Examples of zero-dimensional defects are missing atoms called vacancies, and extra atoms called interstitial atoms located between standard lattice sites. A vacancy-interstitial pair is called a Frenkel defect. An example of a one-dimensional dislocation is a lattice defect at an edge, or a partial line of missing atoms. Common two-dimensional defects are a boundary between grains, and a stacking fault arising from a sudden change in the stacking arrangement of close-packed planes. A vacant space called a pore, a cluster of vacancies, and a precipitate of a second phase are three-dimensional defects. All of these can bring about the scattering of electrons, and thereby limit the electrical conductivity. Scone nanostructures are too small to have internal defects.

Another size effect arises from Ihe level of doping of a semiconductor. For typical doping levels of 1014 to 1018 donors/cm3 a quantum-dot cube lOOnm on a side would have, on the average, from 10-1 to 103 conduction electrons. The former figure of 10-1 electrons per cubic centimeter means that on the average only 1 quantum dot in 10 will have one of these electrons. A smaller quantum-dot cube only lOnm on a side would have, on the average 1 electron for the 1018 doping level, and be very unlikely to have any conduction electrons for the 1014 doping level. A similar analysis can be made for quantum wires and quantum wells, and the results shown in Table 9.2 demonstrate that these quantum structures are typically characterized by very small numbers or concentrations of electrons that can cany current. This results in the phenomena of single-electron tunneling ami the Coulomb blockade discussed below.

9.3.2. Conduction Electrons and Dimensionality

We are accustomed to studying electronic systems that exist in three dimensions, and are large or macroscopic in size. In this case the conduction electrons are delocalized, and move freely throughout the entire conducting medium such as a copper wire. It is clear that all the wire dimensions are very large compared to the distances between atoms. The situation changes when one or more dimensions of the copper becomes so small that it approaches several times the spacings between the atoms in the lattice. When this occurs, the delocalization is impeded, and the electrons experience confinement. For example, consider a flat plate of copper that is 10cm long, 10cm wide, and only 3.6nm thick This thickness corresponds to the length of only 10 unit cells, which means that 20% of the atoms are in unit cells at the surface of the copper. The conduction electrons would be delocalized in the plane of the plate, but confined in the narrow dimension, a configuration referred to as a quantum well. A quantum wire is a structure such as a copper wire that is long in

Table 9.2. Conduction electron content of smaller size (on left) and larger size (on right) quantum structures containing donor concentrations of 10,4-1018cm-3

Quantum Electron Electron

Structure Size Content Size Content

Bulk material — 10l4-10'8cnT3 — 1014-10"W3

Quantum well lOnm thick l-104(raT2 lOOnm thick 10-105pm-2

Quantum wire iOxlO-nm 10_2-102nm"' lOOnm x lOOnm l-104|im-1

cross section cross section

Quantum dot lOnm on a side 10~4-1 lOOnm on a side lO-'-lO3

Table 9.3. Delocallzatton and confinement dimensionalities of quantum nanostructures

Quantum Structure

Delocalization Dimensions

Confinement Dimensions

Bulk conductor

3 (x,y,z)

0 0

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