Electrical Conductivity

Different materials respond very differently to an applied electric field. Metals allow current to flow freely, rubber and glass resist it almost completely, and semiconductors fall somewhere between these extremes. The property that quantifies how readily a material allows the flow of electric current is called electrical conductivity.
It is the reciprocal of resistivity and provides a direct measure of the ease of charge transport through a material. Understanding electrical conductivity at both the macroscopic and microscopic levels is essential for analysing the behaviour of conductors, semiconductors, and insulators in electric circuits.

Definition and Expression of Electrical Conductivity

Electrical conductivity σ is defined as the reciprocal of resistivity ρ:

σ = 1/ρ

Since resistivity ρ is defined through R = ρl/A, conductivity can also be expressed as:

σ = l / RA

The SI unit of electrical conductivity is siemens per metre (S m⁻¹), where:

1 S m⁻¹ = 1 Ω⁻¹ m⁻¹

A higher value of σ indicates a better conductor. Copper, one of the best metallic conductors, has σ ≈ 6 × 10⁷ S m⁻¹, while a typical insulator such as glass has σ ≈ 10⁻¹² S m⁻¹.

Relation Between Conductivity and Current Density

The relationship between current density J, conductivity σ, and electric field E is given by the microscopic form of Ohm's law:

J = σE

Where:

J is the current density (A m⁻²), i.e., current per unit cross-sectional area
E is the electric field (V m⁻¹)
σ is the electrical conductivity (S m⁻¹)

This form of Ohm's law is more fundamental than V = IR as it relates field quantities independent of the geometry of the conductor.

Microscopic Origin of Conductivity

From the electron drift model of conduction, the conductivity of a metallic conductor is:

σ = ne²τ / m

Where:

n = number density of free electrons (m⁻³)
e = charge on electron (C)
τ = average relaxation time between successive collisions (s)
m = mass of electron (kg)

This expression shows that conductivity increases with:

Higher electron number density n, as more charge carriers are available
Longer relaxation time τ as fewer collisions impede electron motion

Effect of Temperature on Conductivity

Metallic Conductors

In metals, n remains essentially constant with temperature. As the temperature rises, lattice ion vibrations increase in amplitude, reducing the relaxation time τ. From σ = ne²τ/m, decreasing τ decreases σ. The conductivity of metals, therefore, decreases with increasing temperature.

Semiconductors

In semiconductors, the number of charge carriers n increases exponentially with temperature as thermal energy promotes electrons from the valence band to the conduction band. This rise in n dominates over the decrease in τ, causing conductivity to increase with temperature, opposite to metallic behaviour.

Insulators

Insulators have an extremely small n at room temperature. Their conductivity is negligibly small and increases only slightly with temperature, remaining far below practically useful values.

Classification Based on Conductivity

Material	Conductivity (S m⁻¹)	Example
Conductors	10⁶ to 10⁸	Copper, Silver
Semiconductors	10⁻⁴ to 10⁴	Silicon, Germanium
Insulators	10⁻²⁰ to 10⁻¹⁰	Glass, Rubber

Conclusion

Electrical conductivity σ = 1/ρ quantifies the ease with which a material conducts electric current. At the microscopic level, σ = ne²τ/m, showing that conductivity depends on the density of free charge carriers and their average relaxation time. Metals have high conductivity that decreases with temperature due to reduced relaxation time.

Semiconductors have moderate conductivity that increases with temperature due to exponential growth in carrier density. Insulators have negligible conductivity under normal conditions. These distinctions form the basis for classifying and selecting materials for electrical and electronic applications.

FAQs

Q1. Why is conductivity preferred over resistance for characterising materials?

Resistance depends on geometry, length and cross-sectional area. Conductivity is an intrinsic material property independent of dimensions, making it a more meaningful parameter for comparing different materials.

Q2. Why do alloys generally have lower conductivity than pure metals?

Alloying introduces irregularities in the crystal lattice, increasing electron scattering and reducing relaxation time τ. From σ = ne²τ/m, a shorter τ directly reduces conductivity compared to the pure metal.

Q3. Why does increasing temperature increase conductivity in semiconductors but decrease it in metals?

In semiconductors, carrier density n rises exponentially with temperature, dominating over decreasing τ. In metals, n is fixed, and only τ decreases, reducing conductivity. The dominant factor differs fundamentally between the two material types.