Temperature Dependence of Resistance

For metals, resistance increases linearly with temperature. α is the temperature coefficient of resistance (unit: K⁻¹). For semiconductors, α is negative — resistance decreases with temperature.

Metals

In metals, $\rho = m/ne^2\tau$. As temperature rises, lattice ions vibrate more vigorously, reducing the mean free path and hence $\tau$. Since $n$ changes negligibly:

$$\rho \propto \frac{1}{\tau} \implies \rho \text{ increases with } T$$

Empirically, over a moderate temperature range:

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

Since $R \propto \rho$ (same geometry):

$$\boxed{R_T = R_0[1 + \alpha(T - T_0)]}$$

$\alpha > 0$ for metals (typically $\sim 4\times10^{-3}$ K⁻¹ for copper).

Semiconductors

In semiconductors, carrier density $n$ increases strongly with temperature (more electrons excited across the band gap). This dominates over the decrease in $\tau$, so $\rho$ decreases with temperature: $\alpha < 0$.

Special cases

• NTC thermistors: semiconductors with large negative $\alpha$ — used as temperature sensors
• PTC thermistors: certain materials where $\alpha$ becomes large and positive above a critical temperature
• Superconductors: $\rho \to 0$ below a critical temperature $T_c$