Temperature Dependence of Resistivity: Explained Simply

Introduction

Resistivity is one of the most commonly mentioned properties when we consider the conductivity of materials in terms of electricity. It determines the degree of resistance of a substance to the conductivity of electric current. But not resistance in stone, but it varies with temperature. Such dependence on temperature is not shared by all materials. Sensitivity to changes in the thermometer is very different in conductors, semiconductors, and insulators since they respond either positively or negatively to changes in their resistivity. This relationship is significant, not only in theory but in practice in electrical wiring, heating coils, resistors, and sensors, among other applications. We are going to examine the temperature dependence of resistivity and its importance.

Understanding Resistivity and Temperature Relation

In physics, the resistivity of a material at temperature T can be expressed as:

ρT=ρ0[1+α(T−T₀)]

Here:

ρT = resistivity at temperature T
ρ0 = resistivity at reference temperature T₀
α = temperature coefficient of resistivity

This relation works well when the temperature range is not too large. It suggests that resistivity changes approximately linearly with temperature, at least within a moderate range around the reference point.

Temperature Coefficient of Resistivity (α)

The coefficient α(alphaα) tells us how much the resistivity changes per degree change in temperature.

Positive α (alpha): Resistivity increases with temperature (common in metals).
Negative α (alpha): Resistivity decreases with temperature (common in semiconductors and insulators).
Units: Since it measures change per degree, its dimension is (temperature)^-1.

Think of α (alpha) as the sensitivity factor: large values mean that the of resistivity is highly temperature-dependent, while small values imply stability.

Temperature Dependence in Conductors

In metallic conductors, resistivity typically increases as it gets hotter. The number of free electrons stays roughly the same, but as the temperature rises, the atoms vibrate more. Electrons collide with these vibrating atoms more often, reducing the average time between collisions, known as the relaxation time (τ). Resistivity is inversely proportional to τ, so higher temperatures lead to higher resistivity.

1. Example: Copper and Silver

Copper: Resistivity = 1.7 × 10^-8 Ω m, α = 0.0068 /°C.
Silver: Resistivity = 1.6 × 10^-8 Ω m, α = 0.0041 /°C.

At room temperature, both are excellent conductors. But as heat builds up, their resistivity rises noticeably.

2. Graphical Behavior

The relation predicts a straight line, but experiments show that at very low temperatures, the line bends slightly. The figure illustrates copper’s resistivity rising as temperature increases.

3. Special Cases: Alloys

Not all conductors behave the same. Alloys often show a much weaker dependence on temperature.

Nichrome (nickel, iron, chromium alloy): Its resistivity barely changes with temperature. This makes it ideal for heating elements and precision resistors, where stability matters more than high conductivity. (Fig. shows its nearly flat slope.)
Manganin and Constantan: Similar in behavior, widely used in wire-wound resistors. Their resistance hardly shifts with heating, making them reliable in electrical measurements.

Alloys prove that sometimes a "less perfect" conductor can be more useful.

Temperature Dependence in Semiconductors

Semiconductors like silicon and germanium behave oppositely to metals. Their resistivity decreases as temperature rises.

Why? Raising the temperature excites more electrons from the valence band into the conduction band, dramatically increasing the number of free charge carriers (n). This increase outweighs the drop in relaxation time (τ), so resistivity falls.

Silicon: Resistivity at 0°C ≈ 2300 Ω m, α = -0.07 /°C.
Germanium: Resistivity at 0°C ≈ 0.46 Ω m, α = -0.05 /°C.

The figure shows this effect clearly: resistivity drops smoothly as temperature rises.

This characteristic is the basis for thermistors—thermally sensitive devices that operate on large drops in resistance with heating.

Temperature Dependence in Insulators

Insulators are extreme cases. Their resistivity is extremely high at room temperature but decreases when heated, similar to semiconductors.

Examples: Glass, hard rubber, quartz.
At 0°C, glass can have resistivity in the range of 10¹⁰–10¹⁴ Ω m.

With increasing temperature, some bound electrons gain enough energy to hop into conduction states, reducing resistivity. Though they never become good conductors, their resistance does decrease with temperature.

Microscopic Explanation of Temperature Dependence

The general expression for resistivity is:

ρ = ^m/_ne²τ

where:

m = mass of electron
n = number of free electrons per unit volume
e = electron charge
τ = average time between collisions

From this:

In metals, n is fixed, but τ decreases as collisions rise, so resistivity increases with T.
In semiconductors and insulators, n increases rapidly with T, overcoming the decrease in τ, so resistivity decreases with T.

This duality explains why the same temperature increase can cause opposite trends in different materials.

Graphical Representation

Metals (e.g., copper): Resistivity increases almost linearly.
Alloys (e.g., nichrome): Weak dependence, almost flat slope.
Semiconductors (e.g., silicon, germanium): Resistivity falls rapidly.

These contrasting curves highlight how diverse material responses can be.

Comparative Table of Resistivity and α

The following insights come from Table:

Conductors: Extremely low resistivity (order of 10^-8 Ω m) with positive α.
Alloys: Slightly higher resistivity but much lower α, meaning stability.
Semiconductors: Resistivity varies widely (from 10^-1 to 10³ Ω m), negative α.
Insulators: Resistivity can be astronomically high (10¹⁰−10¹⁶ Ωm), sensitive to heating.

This classification helps engineers and scientists pick the right material for the right job.

Applications of the Temperature Dependence of Resistivity

1. Thermistors and Sensors

Semiconductors with negative α(alpha) are used as thermistors, detecting small temperature changes by measuring resistance shifts.

2. Heating Elements

Nichrome wires energize electric heaters, toasters, and hairdryers. Their resistivity changes minimally with temperature, maintaining constant heating.

3. Precision Resistors

Manganin and Constantan provide stable resistance, used in lab instruments where accuracy is crucial.

4. Electrical Wiring

Copper and aluminum, despite rising resistivity with temperature, remain the backbone of wiring systems because of their low base resistivity.

Conclusion

The temperature dependence of resistivity is more than a theoretical curiosity—it’s the reason we can design heaters, sensors, and stable resistors.

Metals: Resistivity rises with temperature.
Alloys: Resistivity remains almost unchanged.
Semiconductors & Insulators: Resistivity falls with temperature.

By understanding these trends, we can appreciate how the microscopic dance of electrons and atoms shapes the macroscopic world of technology around us.

Advertisement

Education

Nikka Chauhan

| Posted on September 26, 2025