Resistance and Temperature, Non-Ohmic Conductors

Hot metals resist more

The resistivity \(\rho = m/ne^2\tau\) holds two quantities that could change with temperature: \(n\) and \(\tau\). In a metal every atom has already released its free electrons, so heating adds none; \(n\) stays fixed. Heating makes the ions vibrate harder, and a shaking ion is a bigger target. The electrons collide more often, \(\tau\) falls, and \(\rho\) rises.

Over a modest range of temperature the rise is close to a straight line. The fractional change in resistance is proportional to the change in temperature, \(\Delta R/R_0 = \alpha\,\Delta T\). The constant \(\alpha\) is the temperature coefficient of resistance, in per °C. For copper it is about \(4 \times 10^{-3}\) per °C.

The formula measures \(\Delta T\) from the temperature that belongs to \(R_0\), not from zero. A conductor of \(1.72\,\Omega\) at \(20\,°\text{C}\), with \(\alpha = 3.93 \times 10^{-3}\) per °C, has \(1.72(1 - 3.93 \times 10^{-3} \times 20)\) \(= 1.58\,\Omega\) at \(0\,°\text{C}\). At \(100\,°\text{C}\) it has \(1.72(1 + 3.93 \times 10^{-3} \times 80)\) \(= 2.26\,\Omega\).

This steady rise is what the platinum resistance thermometer of the chapter on Temperature and Thermal Expansion reads. A pure platinum wire has a large, steady \(\alpha\), so its resistance gives its temperature.

Semiconductors and alloys go their own way

In a semiconductor such as silicon, almost every electron is bound at low temperature, so \(n\) is tiny. Heat frees more carriers, and \(n\) grows very fast with temperature. That rise far outweighs the fall in \(\tau\). So a semiconductor's resistance falls as it warms, and its \(\alpha\) is negative. The chapter on Semiconductors and the p-n Junction explains where the carriers come from.

A thermistor is a small bead of semiconductor used for exactly this. In Fig 4.1 its resistance at \(100\,°\text{C}\) is less than a tenth of its value at \(20\,°\text{C}\). A small change of temperature gives a large change of resistance, which a circuit can read easily.

Alloys behave differently again. In manganin or constantan the mixed atoms already scatter the electrons strongly, whatever the temperature. Heating adds little to that disorder, so \(\alpha\) is almost zero. Resistance boxes, standard resistors and the wires of measuring bridges are made of these alloys. Their marked value holds even when they warm under a current.

When the graph bends

A non-ohmic conductor is one whose V-I graph is not a straight line through the origin. The filament of a lamp is the everyday case. A larger current heats the thin tungsten wire more, and a hotter metal resists more. So each extra ampere needs more extra volts than the one before, and the graph bends.

For the filament above, P is at \(1\,\text{A}\) and \(2\,\text{V}\), and Q at \(2\,\text{A}\) and \(6\,\text{V}\). Its resistance is \(2/1 = 2\,\Omega\) at P and \(6/2 = 3\,\Omega\) at Q. The filament is hotter at Q. Every point on the curve has a resistance \(V/I\), but no single number describes the filament.

The tangent gives a second number, the extra volts per extra ampere, \(dV/dI\). It is \(3\,\Omega\) at P and \(5\,\Omega\) at Q. For the copper wire the two numbers are equal everywhere, because its graph is its own tangent. For an ohmic conductor, and only for one, \(V/I = dV/dI = R\).

Some devices bend in another way. A diode passes a large current one way and almost none the other way. So its graph is not even the same for both signs of \(V\). The chapter on Diodes, Transistors and Logic Gates builds circuits from that one-way behaviour.