Physics · Ch 2 — Current Electricity

Temperature Dependence of Resistivity

2.2.4

Temperature Dependence of Resistivity

A material's resistivity depends measurably on its temperature. For a wide range of temperatures, experiment shows that a conductor's resistivity increases linearly with temperature:

\rho_T = \rho_0\left[1+\alpha(T-T_0)\right] \qquad (2.27)

where $\rho_T$ is the resistivity at temperature T (°C), $\rho_0$ is the resistivity at a reference temperature $T_0$ (usually $20^{\circ}C$ ), and $\alpha$ , the temperature coefficient of resistivity, is defined as the ratio of the increase in resistivity per degree rise in temperature to the resistivity at $T_0$ :

\alpha = \dfrac{1}{\rho_0}\cdot\dfrac{\Delta\rho}{\Delta T} \qquad (2.29\text{, resistivity form})

Since $R = \rho l/A$ , the identical relation carries straight over to resistance itself:

R_T = R_0\left[1+\alpha(T-T_0)\right] \qquad (2.28), \qquad \alpha = \dfrac{1}{R_0}\cdot\dfrac{\Delta R}{\Delta T}

For conductors, $\alpha$ is positive (Table 2.3 lists typical values for common metals): as temperature rises, the metal ions vibrate more vigorously, so the drifting electrons collide with them more frequently, shortening the mean free time $\tau$ ; since $\rho = m/(n e^2\tau)$ (2.30), a shorter $\tau$ directly means a higher resistivity. Although this linear rise holds over a wide practical temperature range, at very low temperatures the graph bends away from a straight line and the resistivity approaches some small, finite value as $T\to0$ , rather than continuing linearly all the way down (Figure 2.13(b)).

For semiconductors, $\alpha$ is negative -- resistivity FALLS as temperature rises (Figure 2.14), the opposite of a conductor's behaviour, and germanium and silicon in Table 2.3 both show large negative $\alpha$ values. As explained by $\rho=m/(ne^2\tau)$ : raising a semiconductor's temperature both increases the number density n of free charge carriers released from their atoms AND decreases the mean free time $\tau$ through more frequent collisions, but the increase in n dominates completely over the decrease in $\tau$ , so the net effect is a falling resistivity. A semiconductor engineered specifically to have a large negative temperature coefficient of resistivity is called a thermistor. …

Figure 2.13Temperature dependence of resistivity: (a) conductor, (b) non-linear low-temperature region

What this figure shows. Graph (a) plots resistivity $\rho$ (in $\Omega\text{m}$ ) on the vertical axis against temperature T (in K) on the horizontal axis for a typical conductor, showing a straight rising line that does not pass through the origin but intercepts the vertical axis at a positive value $\rho_0$ . Graph (b) zooms in on the very-low-temperature region of the same material, showing the line bending away from straight and flattening out to approach some small finite resistivity value as $T\to0$ , rather than continuing in a straight line all the way down to zero …

Figure 2.14Temperature dependence of resistivity for a semiconductor

What this figure shows. A graph of resistivity $\rho$ (in $\Omega\text{m}$ ) against temperature T (in K) for a semiconductor, showing a curve that falls steeply as temperature increases -- the opposite trend to the rising straight line seen for a conductor in Figure 2.13(a), visually illustrating the negative temperature coefficient of resistivity that defines semiconducting behaviour …

Table 2.3Temperature coefficient of resistivity for various materials

Material	Temperature Coefficient of Resistivity α [(°C)^-1]
Silver	3.8 × 10^-3
Copper	3.9 × 10^-3
Gold	3.4 × 10^-3
Aluminum	3.9 × 10^-3
Tungsten	4.5 × 10^-3
Iron	5.0 × 10^-3
Platinum	3.92 × 10^-3
Lead	3.9 × 10^-3
Nichrome	0.4 × 10^-3

Misc Example 2.13Resistance at a new temperature given alpha

Worked out. A coil has resistance $3\ \Omega$ at $20^{\circ}C$ with $\alpha = 0.004/^{\circ}C$ , and its resistance at $100^{\circ}C$ is required. Using $R_T = R_0[1+\alpha(T-T_0)]$ : $R_{100} = 3[1+0.004\times(100-20)] = 3[1+0.004\times80] = 3\times1.32 = 3.96\ \Omega$ . …

Misc Example 2.14Finding alpha from two resistance-temperature readings

Worked out. A material's resistance is $45\ \Omega$ at $20^{\circ}C$ and $85\ \Omega$ at $40^{\circ}C$ ; its temperature coefficient of resistivity is required. Using $\alpha = \dfrac{1}{R_0}\cdot\dfrac{\Delta R}{\Delta T} = \dfrac{85-45}{45\times(40-20)} = \dfrac{40}{45\times20} = \dfrac{40}{900} \approx 0.044$ per °C. This unusually large value (compared with the pure-metal values in Table 2.3) is typical of the way such 'find alpha' problems are set, using round numbers rather than a real named material' …

Misc ~note-superconductorsSuperconductors

Worked out. A short aside on superconductivity: the resistance of certain materials drops to exactly zero below a critical (transition) temperature $T_c$ , a phenomenon first observed by Kammerlingh Onnes in 1911, who found that mercury becomes a superconductor at 4.2 K. Because R is genuinely zero (not merely very small) below $T_c$ , a current once induced in a superconducting loop persists indefinitely without needing any driving potential difference to sustain it -- unlike an ordinary conductor, where removing …

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