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4.3: Resistance and Temperature

It is found that the resistivities of metals generally increase with increasing temperature, while the resistivities of semiconductors generally decrease with increasing temperature.

It may be worth thinking a little about how electrons in a metal or semiconductor conduct electricity. In a solid metal, most of the electrons in an atom are used to form covalent bonds between adjacent atoms and hence to hold the solid together. But about one electron per atom is not tied up in this way, and these “conduction electrons” are more or less free to move around inside the metal much like the molecules in a gas. We can estimate roughly the speed at which the electrons are moving. Thus we recall the formula $$\sqrt{3kT/m}$$ for the root-mean-square speed of molecules in a gas, and maybe we can apply that to electrons in a metal just for a rough order of magnitude for their speed. Boltzmann’s constant $$k$$ is about 1.38 $$\times$$ 10-23 J K-1 and the mass of the electron, m, is about 9.11 $$\times$$ 10-31 kg. If we assume that the temperature is about 27oC or 300 K, the root mean square electron speed would be about 1.2 $$\times$$ 105 m s-1.

Now consider a current of 1 A flowing in a copper wire of diameter 1 mm – i.e. cross-sectional area 7.85 $$\times$$ 10-7 m2. The density of copper is 8.9 g cm-3, and its “atomic weight” (molar mass) is 63.5 g per mole, which means that there are 6.02 $$\times$$ 1023 (Avogadro’s number) of atoms in 63.5 grams, or 8.44 $$\times$$ 1022 atoms per cm3 or 8.44 $$\times$$ 1028 atoms per m3. If we assume that there is one conduction electron per atom, then there are 8.44 $$\times$$ 1028 conduction electrons per m3, or, in our wire of diameter 1 mm, 6.63 $$\times$$ 1022 conduction electrons per metre.

The speed at which the electrons are carrying the current of one amp is the current divided by the charge per unit length, and with the charge on a single electron being 1.60 $$\times$$ 10-19 C, we find that the speed at which the electrons are carrying the current is about 9.4 $$\times$$ 10-5 m s-1.

Thus we have this picture of electrons moving in random directions at a speed of about 1.2 $$\times$$ 105 m s-1 (the thermal motion) and, superimposed on that, a very slow drift speed of only 9.4 $$\times$$ 10-5 m s-1 for the electron current. If you were able to see the electrons, you would see them dashing hither and thither at very high speeds, but you wouldn’t even notice the very slow drift in the direction of the current.

When you connect a long wire to a battery, however, the current (the slow electron drift) starts almost instantaneously along the entire length of the wire. If the electrons were in a complete vacuum, rather than in the interior of a metal, they would accelerate as long as they were in an electric field. The electrons inside the metal also accelerate, but they are repeatedly stopped in their tracks by collisions with the metal atoms – and then they start up again. If the temperature is increased, the vibrations of the atoms within the metal lattice increase, and this presumably somehow increases the resistance to the electron flow, or decreases the mean time or the mean path-length between collisions.

In a semiconductor, most of the electrons are required for valence bonding between the atoms – but there are a few (much fewer than one per atom) free, conduction electrons. As the temperature is increased, more electrons are shaken free from their valence duties, and they then take on the task of conducting electricity. Thus the conductivity of a semiconductor increases with increasing temperature.

The temperature coefficient of resistance, a, of a metal (or other substance) is the fractional increase in its resistivity per unit rise in temperature:

$\alpha = \frac{1}{\rho}\frac{d\rho}{dT}\label{4.3.1}$

In SI units it would be expressed in K-1. However, in many practical applications the temperature coefficient is defined in relation to the change in resistance compared with the resistivity at a temperature of 20oC, and is given by the equation

$\rho = \rho_{20} [1+\alpha (t-20)],\label{4.3.2}$

where t is the temperature in degrees Celsius.

Examples:

Silver: 3.8 $$\times$$ 10-3 C o -1

Copper: 3.9 $$\times$$ 10-3

Aluminium: 3.9 $$\times$$ 10-3

Tungsten: 4.5 $$\times$$ 10-3

Iron: 5.0 $$\times$$ 10-3

Carbon: -0.5 $$\times$$ 10-3

Germanium: -48 $$\times$$ 10-3

Silicon: -75 $$\times$$ 10-3

Some metallic alloys with commercial names such as nichrome, manganin, constantan, eureka, etc., have fairly large resistivities and very low temperature coefficients.

As a matter of style, note that the kelvin is a unit of temperature, much a the metre is a unit of length. Thus, when discussing temperatures, there is no need to use the “degree” symbol with the kelvin. When you are talking about some other temperature scale, such as Celsius, one needs to say “20 degrees on the Celsius scale” – thus 20oC. But when one is talking about a temperature interval of so many Celsius degrees, this is written Co. I have adhered to this convention above.

The resistivity of platinum as a function of temperature is used as the basis of the platinum resistance thermometer, useful under conditions and temperatures where other types of thermometers may not be useful, and it is also used for defining a practical temperature scale at high temperatures. A bolometer is an instrument used for detecting and measuring infrared radiation. The radiation is focussed on a blackened platinum disc, which consequently rises in temperature. The temperature rise is measured by measuring the increase in resistance. A thermistor is a semiconducting device whose resistance is very sensitive to temperature, and it can be used for measuring or controlling temperature.