Temperature Dependence of Resistivity

The Linear Resistivity Law

The resistivity of a material depends on temperature, and different materials respond differently. Over a temperature range that is not too large, the resistivity of a metallic conductor is given to good approximation by NCERT Eq. (3.26):

$$\rho_T = \rho_0\left[1 + \alpha\,(T - T_0)\right]$$

Here $\rho_T$ is the resistivity at temperature $T$, $\rho_0$ is the resistivity at a chosen reference temperature $T_0$, and $\alpha$ is the temperature coefficient of resistivity. Because resistance is proportional to resistivity for a fixed conductor, the same dependence carries straight over to resistance:

$$R_T = R_0\left[1 + \alpha\,\Delta T\right], \qquad \Delta T = T - T_0$$

Equation (3.26) implies that a plot of $\rho_T$ against $T$ is a straight line whose slope is set by $\alpha$. The relation is an approximation valid only over a limited range around $T_0$ where the curve is effectively straight, a point the next figure makes explicit.

Two cautions are worth fixing at the outset. First, the law is empirical and local: it describes the behaviour near a chosen reference temperature, and the coefficient $\alpha$ is itself an average over the range used, not a universal constant. Second, the resistance form $R_T = R_0[1 + \alpha\,\Delta T]$ holds only because the conductor's geometry — its length and cross-section — is assumed unchanged; what genuinely varies with temperature is the material property $\rho$, and $R$ simply inherits that variation. This is exactly the principle behind the platinum resistance thermometer, where a measured resistance is read back as a temperature.

Temperature Coefficient of Resistivity

Rearranging the linear law gives the meaning of $\alpha$ directly. In the range where resistivity increases linearly with temperature, $\alpha$ is the fractional increase in resistivity per unit increase in temperature:

$$\alpha = \frac{\rho_T - \rho_0}{\rho_0\,(T - T_0)} = \frac{1}{\rho_0}\,\frac{\Delta\rho}{\Delta T}$$

From the defining equation the dimension of $\alpha$ is $(\text{temperature})^{-1}$, so its SI unit is $\mathrm{K^{-1}}$. Since a temperature difference has the same numerical value in kelvin and in degrees Celsius, $\alpha$ may equally be quoted per °C. For metals $\alpha$ is positive; for semiconductors and insulators it is negative.

All four rows trace back to a single equation. The derivation of conductivity in NCERT gives $\sigma = ne^2\tau/m$, and inverting it yields the resistivity expression at the heart of this topic, Eq. (3.27):

$$\rho = \frac{1}{\sigma} = \frac{m}{n\,e^2\,\tau}$$

So $\rho$ depends inversely on two quantities: the number $n$ of free carriers per unit volume, and the average time $\tau$ (the relaxation time) between collisions. Temperature pulls on both, and which one wins decides the sign of $\alpha$.

Metals: Positive α

As temperature rises, the average speed of the conduction electrons increases, so they collide with the fixed lattice ions more frequently. The average time between collisions $\tau$ therefore decreases with temperature. In a metal, the carrier density $n$ does not depend on temperature to any appreciable extent. With $n$ fixed and $\tau$ falling, $\rho = m/(ne^2\tau)$ must increase — exactly the rising graph observed for copper. This is why metals carry a positive temperature coefficient of resistivity.

Semiconductors and Insulators: Negative α

For insulators and semiconductors the story changes because the carrier density $n$ is itself strongly temperature-dependent. As temperature rises, $n$ increases — more electrons gain enough thermal energy to become free to conduct. This increase more than compensates any decrease in $\tau$, so in $\rho = m/(ne^2\tau)$ the resistivity decreases with temperature. The temperature coefficient is therefore negative, and the resistivity of semiconductors such as silicon and germanium falls as they warm up — the opposite of metallic behaviour.

Alloys: Near-Zero α

Some materials behave almost indifferently to temperature. Nichrome — an alloy of nickel, iron and chromium — shows a very weak dependence of resistivity on temperature, and manganin and constantan share this property. Their temperature coefficient is vanishingly small (of the order of $10^{-6}\ \mathrm{{}^{\circ}C^{-1}}$) while their resistivity is comparatively high.

That combination is precisely what a wire-bound standard resistor needs: a stable, well-defined resistance that barely drifts as the wire warms during use. This is why these alloys, and not pure metals, are chosen for standard resistance coils.

It helps to read the alloy case through the same $\rho = m/(ne^2\tau)$ lens. An alloy is still metallic, so its carrier density $n$ stays essentially fixed with temperature; the disorder of the mixed lattice already scatters electrons so strongly that the additional scattering introduced by heating is a small fractional change. The relaxation time $\tau$ therefore varies only weakly, $\rho$ barely moves, and $\alpha$ comes out tiny. The high baseline resistivity and the near-flat response are two faces of the same heavily-disordered structure.

Superconductivity

Studying resistivity at very low temperatures led to a striking discovery: certain metals and alloys lose their resistivity completely below a characteristic transition (critical) temperature $T_c$ that is specific to each material. Below $T_c$ the resistivity becomes effectively zero ($\rho \to 0$), and a current once set up in such a material persists indefinitely without any external source to maintain it. These materials are called superconductors. Developing materials that superconduct nearer to room temperature is an active and consequential field of research, with applications such as energy-efficient electromagnets for magnetic levitation.