Ohm's law in atomic scale

Atomic wire made of phosphorus. It links to silicon atoms.
Credit: Physics World

Ohm's Law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

 where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wire. He presented a slightly more complex equation than the one above to explain his experimental results. The above equation is the modern form of Ohm's law.

In physics, the term Ohm's law is also used to refer to various generalizations of the law originally formulated by Ohm. The simplest example of this is:

where J is the current density at a given location in a resistive material, E is the electric field at that location, and σ is a material dependent parameter called the conductivity. This reformulation of Ohm's law is due to Gustav Kirchhoff.

Microscopic origins of Ohm's law

The dependence of the current density on the applied electric field is essentially quantum mechanical in nature; (see Classical and quantum conductivity.) A qualitative description leading to Ohm's law can be based upon classical mechanics using the Drude model developed by Paul Drude in 1900.

The Drude model treats electrons (or other charge carriers) like pinballs bouncing between the ions that make up the structure of the material. (read more in Wikipedia).

An atomic scale experiment results 

"A new technique for embedding atomic-scale wires within crystals of silicon has revealed that Ohm's law can hold true for wires just four atoms thick and one atom tall. The result comes as a surprise because conventional wisdom suggests that quantum effects should cause large deviations from Ohm's law for such tiny wires. Paradoxically, the researchers hope the finding will aid the development of quantum computers."

in Physics World (click to read entire article)

A team of Arizona State University created a channel in the silicon by removing layers of silicon atoms, by using the tip of a scanning probe microscope. The surface was then exposed to phosphorus gas, followed by the deposition of silicon atoms. The result was a chain of phosphorus atoms embedded inside a silicon crystal – an atomic wire. The team found that the resistivity of those wires was constant. That means the resistance of such a wire is proportional to its length and inversely proportional to its area, just as we seen in Ohm's law.

Implications

The discovery has several implications, including:

• For engineers it could provide a roadmap to future nanoscale computational devices where atomic sizes are at the end of Moore's law. The theory shows that a single dense row of phosphorus atoms embedded in silicon will be the ultimate limit of downscaling.

• For computer scientists, it places donor-atom based silicon quantum computing closer to realization.

• And for physicists, the results show that Ohm's Law, which demonstrates the relationship between electrical current, resistance and voltage, continues to apply all the way down to an atomic-scale wire.

in ScienceDaily