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# Electrical conduction

Electrical conduction is the movement of charged particles through a material (current) in response to an electric field. The underlying mechanism for this movement depends on the material.

The ease with which current can be made to flow in a material is measured by conductivity σ, defined as:

J = σ E

or its reciprocal resistivity ρ:

J = E / ρ

### Scattering

Effective mass (m*) is an important concept in solid state physics. It can be shown that electrons and holes in a solid respond to an electric field almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron me (9.11×10-31 kg).

Armed with this concept, we can come up with a simple model of conduction in solids. When we apply an electric field, the electrons accelerate. No material is perfect, however, so the electrons do not accelerate forever. We assume that dotted through the crystal are scatterers. Should the electron bump into one of these scatterers, it will rebound, its velocity randomised. On average, the electron will travel for a time τ before it hits one of these sites.

Using this model, we can calculate conductivity from characteristic scattering time:

$\sigma = \frac{ne^2 \tau}{m^*}$

where n is the number of free electrons per unit volume, or electron density, and e is the electron charge. This equation suggests that perhaps conductivity is independent of electric field – when this was observed experimentally, it was called Ohm's law.

In metals, the electron density n is very large, and hence conductivity is high. In other words, metals are good conductors of electricity. Semiconductors naturally have a low electron density, but this can be boosted by adding impurities (doping). Insulators typically have a low electron density, possibly coupled with short scattering time.

### Band structure

Electrons follow the Pauli exclusion principle, meaning that two electrons cannot occupy the same state. Each state has a fixed kinetic energy associated with it. In a solid, there are many, many states, and the energy separation between the states is very small. However, the only states available are in certain energy regions, called bands. The disallowed regions are called Brillouin zones[?].

Electrons in the solid will tend to settle into the lowest available energy states. Loosely speaking, the highest occupied energy at zero temperature is called the Fermi energy. In semiconductors and insulators, the Fermi energy is inside one of the disallowed regions. This means that the electrons fill up to precisely the top of one of the bands, and no electrons enter the next highest band. The highest filled band is called the valence band, the next highest band is called the conduction band, and the energy difference between the two is called the band gap. Electrons in the valence band cannot accelerate in response to an electric field, because there are no states available where the electrons would be moving any faster. Hence there is no conduction.

Well, almost no conduction. Due to thermal energy, some of the electrons (say, one in every billion billion) from the valence band will spontaneously jump into the conduction band. That tiny number of electrons (100,000 per cubic centimetre or so) is responsible for conduction in pure semiconductors (an even smaller number for insulators). Exciting these electrons into the conduction band leaves behind holes in the valence band, which may also conduct electricity.

In semiconductors, impurities are added to the material. Donor (n-type) impurities have the effect of raising the energy of some of the valence band states up to very close to the conduction band, allowing electrons from normally filled sites easy access to the conduction band. Acceptor (p-type) impurities lower one of the conduction band states to just above the valence band, allowing the easy formation of holes. Hence, even one impurity atom in every billion billion (to use the aforementioned arbitrary figure) will have a significant effect on conductivity.

In metals, the Fermi energy is in the middle of one of the bands. This band is both a valence band and a conduction band. Electrons in this band can easily accelerate, since there are plenty of nearby states. Hence, the number of available carriers in a metal is very much higher than in an insulator.

Electric currents in electrolytes are flows of electrically charged atoms (ions). For example, if an electric field is placed on a solution of Na+ and Cl, the sodium ions will move towards the negative electrode (anode), and the chlorine ions will move towards the positive electrode (cathode). If the conditions are right, redox reactions will take place, which release electrons from the chlorine, and allow electrons to be absorbed into the sodium. In water ice and in certain solid electrolytes, flowing protons constitute the electric current.

In neutral gases, electrical conductivity is very low. They act as a dielectric or insulator, up until the electric field reaches a breakdown value, stripping the electrons from the atoms thus forming a plasma. This plasma allows the conduction of electricity, forming a spark, arc or lightning. In ordinary air below the breakdown field, the dominant source of electrical conduction is via mobile particles of water, which shuttle electric charge, forming a current.

Plasma is the state of matter where some of the electrons in a gas are stripped or "ionized" from their molecules or atoms. A plasma can be formed by high temperature, or by application of an electric field as noted above. Electrical conduction in a plasma is due to the motion of both the electrons and the positively-charged ions.

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