Quantum confinement

Nanotechnology has yielded a number of unique structures that are not found anywhere in nature. Most demonstrate an essential quality of quantum mechanics known as quantum confinement. The idea behind confinement is all about keeping electrons trapped in a small area. The sizes we're talking about here for confinement have to be less than 30 nm for effective confinement. Quantum confinement comes in several flavors. 2-D confinement is only restricted in one dimension, and the result is a quantum well (or plane). These are what most lasers are currently built from. 1-D confinement occurs in nanowires. 0-D confinement is found only in the quantum dot.

Quantum confinement is so important for one thing, it leads to new electronic properties that are not present in today's semiconductor devices. Consider the quantum dot. The typical quantum dot is anywhere between 3-60 nm in diameter. That's still 30 to 600 times the size of a typical atom. A quantum dot exhibits 0-D confinement, meaning that electrons are confined in all three dimensions. The only things in nature that have 0-D confinement are atoms. So a quantum dot can be loosely described as an 'artificial atom'. This is vitally important because we can't readily experiment on regular atoms. They're too small and too difficult to isolate in an experiment. Quantum dots, on the other hand, are large enough to be manipulated by magnetic fields and can even be moved around with an STM or AFM. We can deduce many important atomistic characteristics from a quantum dot that would otherwise be impossible to research in an atom.


The size dependence of the bandgap is the most identifiable aspect of quantum confinement in semiconductors; the bandgap increases as the nanostructure size decreases.
Nature, Two- versus three-dimensional quantum confinement in indium phosphide wires and dots, Nature Materials 2, 517 - 520 (01 Aug 2003)



Effect of quantum confinement of surface electrons on adatom–adatom interactions

We demonstrate that the surface-state mediated interaction between adatoms can be significantly modified by the quantum confinement of surface electrons. We show that quantum corrals and quantum mirrors constructed on metal surfaces can be used to tailor the exchange interaction between magnetic adatoms at large distances. We discuss the self-organization of adatoms on metal surfaces caused by quantum confinement.http://www.iop.org/EJ/abstract/1367-2630/9/10/388










Even though electrostatically induced forces seem to be rather weak, the electrostatic force between e.g an electron and a proton, that together make up a hydrogen atom, is about 40 orders of magnitude stronger than the gravitational force acting between them.

Electrostatic phenomena include examples as simple as the attraction of plastic wrap to your hand after you remove it from a package, to the apparently spontaneous explosion of grain silos, to damage of electronic components during manufacturing, to the operation of photocopiers. Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. Although charge exchange happens whenever any two surfaces contact and separate, the effects of charge exchange are usually only noticed when at least one of the surfaces has a high resistance to electrical flow. This is because the charges that transfer to or from the highly resistive surface are more or less trapped there for a long enough time for their effects to be observed. These charges then remain on the object until they either bleed off to ground or are quickly neutralized by a discharge: e.g., the familiar phenomenon of a static 'shock' is caused by the neutralization of charge built up in the body from contact with nonconductive surfaces.


Gauss's law

Gauss' law states that "the total electric flux through a closed surface is proportional to the total electric charge enclosed within the surface". The constant of proportionality is the permittivity of free space.

Mathematically, Gauss's law takes the form of an integral equation:

ʃˢ ɛ E. dA = ʃ ρ.dV

Alternatively, in differential form, the equation becomes

∆ . ɛ E = ρ

Poisson's equation

The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential φ and the charge density ρ:

∆2 φ = - ρ / ɛ

This relationship is a form of Poisson's equation. Where ɛ is Vacuum permittivity.

Laplace's equation

In the absence of unpaired electric charge, the equation becomes

∆2 φ = O