Temperature Dependence of Resistivity: Overview, Questions, Preparation

In metals, as temperature increases, the resistivity typically also increases. This is because in metals, electrical conduction is primarily due to the movement of free electrons. When the temperature rises, the lattice structure of the metal vibrates more vigorously. These lattice vibrations, or phonons, scatter the free electrons, making it more difficult for them to move through the material. As a result, the resistance increases.

The relationship between resistivity (ρ) and temperature (T) for metals is often described by the empirical equation:

ρ(T) = ρ₀[1 + α(T - T₀)]

Where:

ρ(T) is the resistivity at temperature T

ρ₀ is the resistivity at a reference temperature T₀

α is the temperature coefficient of resistivity (a positive constant)

T₀ is the reference temperature

As mentioned, various materials exhibit varying degrees of sensitivity to temperature changes. Certain materials, such as Nichrome (an alloy of nickel, iron, and chromium), as well as Manganin and Constantan, show a minimal sensitivity of resistivity to changes in temperature. Because of this characteristic, these materials are commonly employed in the production of wire-wound standard resistors. This choice is made because their resistance values remain remarkably stable even when subjected to variations in temperature.

In semiconductors, the behaviour is quite different. As temperature increases, the resistivity typically decreases. This is because in semiconductors, the electrical conduction is primarily due to the movement of charge carriers (either electrons or holes) across the energy band gap.

With increasing temperature, more electrons gain enough thermal energy to jump from the valence band to the conduction band, creating more charge carriers and reducing resistivity. We can qualitatively explain this temperature-dependent resistivity based on our derivation.

ρ= 1/σ = m/ne²τ

ρ is the resistivity of the material

n is the number of free electrons per unit volume

τ is the average time between collisions

According to this equation, the resistivity of a material is determined by several factors, including the number (n) of free electrons per unit volume and the average time (τ) between collisions. As we elevate the temperature, the average velocity of the electrons, which serve as current carriers, increases, leading to more frequent collisions. Consequently, the average collision time (τ) decreases with rising temperature.

In the case of metals, the number of free electrons (n) is not significantly influenced by temperature. Therefore, the decrease in the value of t with increasing temperature leads to an observed increase in resistivity. However, for insulators and semiconductors, the number of free electrons (n) increases with temperature. This increase more than compensates for any reduction in τ. As a result, for such materials, resistivity decreases with temperature.