Temperature dependence of resistivity

Resistivity is not a true “material constant”: it depends appreciably on temperature. Neglecting this dependence is only valid as a first approximation; in reality the resistance of a wire changes noticeably when it heats up.

Metals: linear dependence

For metals, over an interval that is not too large around a reference temperature $T_0$, the dependence is approximately linear:

$\ev{\rho(T) = \rho_0\bigl[1 + \alpha\,(T - T_0)\bigr]}$

where $\rho_0$ is the resistivity at $T_0$ and $\alpha$ is the thermal coefficient of resistivity (units $1/\mathrm{K}$). For copper $\alpha \approx 3{,}9\cdot10^{-3}\,\mathrm{K}^{-1}$: a change of 100 K alters the resistivity by about 40%, an effect that is far from negligible.

Microscopic justification

At higher temperature the ions of the crystal lattice oscillate more about their equilibrium positions. The conduction electrons collide with these ions more frequently, their mean free path shortens and the resistivity increases. The hotter the metal, the more the lattice “vibrates” and hinders the ordered motion of the charges.

Semiconductors: opposite dependence

For semiconductors (silicon, germanium) the dependence is opposite and much more pronounced: resistivity decreases as temperature increases, usually exponentially,

$\rho \propto \exp\!\left(\frac{E_g}{2k_B T}\right)$

where $E_g$ is the material’s band gap energy. Heating a semiconductor “frees” new charge carriers in the lattice — electrons that acquire enough energy to take part in conduction — and the resistivity plummets. This is the opposite effect to that seen in metals, and underlies the operation of thermistors and many electronic devices.

Thermal coefficient of some materials

Material	$\alpha$ ($10^{-3}/$K)
Silver	+3.8
Copper	+3.9
Aluminium	+3.9
Iron	+5.0
Constantan (alloy)	~0
Carbon (graphite)	−0.5

Summary

Metal: $\rho$ increases with $T$ ($\alpha > 0$). Semiconductor: $\rho$ decreases with $T$. Constantan: $\alpha \approx 0$, which is why it is used in precision reference resistors, whose resistance stays stable as temperature varies.

Example — Incandescent light bulb

A tungsten-filament light bulb, at room temperature ($T \approx 300$ K), has a measured resistance $R_\text{off} = 10\ \Omega$. Once lit, the filament reaches about 2700 K and its resistance (calculated from $V/i$ at steady state) is $R_\text{on} \approx 150\ \Omega$. Tungsten has $\alpha \approx 4{,}5\cdot10^{-3}\,\mathrm{K}^{-1}$, so $\frac{R_\text{on}}{R_\text{off}} \approx 1 + \alpha\,(2700-300) \approx 1 + 4{,}5\cdot10^{-3}\cdot 2400 \approx 11{,}8$ The estimate gives $R_\text{on}/R_\text{off} \approx 12$, against an experimental ratio of $\sim 15$: the linear approximation is stretched to its limit (a jump of 2400 K), but it captures the qualitative effect. This is why, at switch-on, a light bulb “draws” a peak current much higher than its steady-state current, $i_\text{peak} \approx V/R_\text{off} \approx 12$ times $i_\text{steady}$ — and it is also why bulbs typically burn out right at switch-on.

Historical context — Superconductors: $\rho$ "falling" to zero

For some materials (mercury, niobium, YBaCuO alloys), below a characteristic critical temperature $T_c$ the resistivity does not simply decrease: it vanishes exactly. A current set going in a superconducting loop keeps circulating for years without dissipating anything. Heike Kamerlingh Onnes discovered the phenomenon in 1911 in mercury cooled to 4.2 K with liquid helium (Nobel Prize 1913). For “high-$T_c$ superconductors” (Bednorz and Müller, 1986; Nobel Prize 1987) the temperatures reach $\sim 90$ K, accessible with liquid nitrogen. It remains, even today, only partially understood, and one of the most active frontiers of condensed-matter physics.

Links

Topics: Circuiti elettrici Concepts: Legge di Ohm

Related exercises: Problem — resistance of a light bulb · Problem — resistance and power of an ohmic conductor · Problem — voltage drop in a copper cable