A phone lies on the dashboard of a car parked in the afternoon sun. Its case warms to 45 °C. The copper tracks on its circuit board now conduct a little worse than they did in the morning. The silicon chips beside them conduct better, several times better. The same heat does opposite things to the two materials.
From one atom to many
The chapter on the Bohr atom drew the energy levels of a single atom as a ladder. An electron in that atom may have only the energies of the rungs. Now bring a second, identical atom close. The outer electrons of the two atoms begin to feel both nuclei. Each single level then splits into two levels with slightly different energies.
Bring \(N\) atoms together, and each level splits into \(N\) levels. A silicon chip 5 mm square and 1 mm thick holds about \(10^{21}\) atoms. So each level of the single atom becomes \(10^{21}\) levels, packed into a range of a few electron volts. They are far too close to tell apart. The allowed energies form a continuous energy band.
Fig 1.1 shows the idea with six atoms. The inner, tightly held electrons of each atom hardly feel the neighbours, so their levels stay sharp. The outer levels spread the most. The band that holds the outer, valence electrons is the valence band. The next band above it is the conduction band. Between them lies a range of energy with no level at all, the band gap \(E_g\).
The valence band is the highest band that is filled by the valence electrons at the absolute zero of temperature. The conduction band is the band just above it. The band gap \(E_g\) is the energy between the top of the valence band and the bottom of the conduction band. No electron in the crystal can have an energy inside it.
A full band carries no current
A band can hold only a fixed number of electrons, two for each of its levels. An electric field tries to speed electrons up, which means moving each into a level of slightly higher energy. In a completely full band there is no empty level to move into. So the electrons of a full band cannot respond to a field, and they carry no current. Conduction needs a band that is only partly full.
A crystal is made of \(N\) identical atoms. Into how many levels does one level of a single atom spread?
The valence band of a crystal is completely full at 0 K. Why can the crystal not conduct, even though it contains a great many electrons?
To carry a current, electrons must be speeded up by the field, which means moving into levels of slightly higher energy. In a full band every level is already taken, so no electron can change its state. The electrons stay, and the net current is zero.
Three kinds of solid
The size of the gap sorts solids into three kinds. In a metal such as copper, the highest band holding electrons is only partly full, or two bands overlap. Empty levels lie just above the filled ones, so the electrons move freely under the smallest field. A metal conducts at every temperature, even at 0 K.
In an insulator such as diamond, the valence band is full and the gap is large, 5.4 eV. The conduction band is empty, and almost nothing can lift an electron that far. In a semiconductor such as silicon or germanium, the gap is small, 1.1 eV or 0.7 eV. At 0 K a semiconductor is an insulator. At room temperature a few electrons gain enough energy to cross the gap.
How large is "small"? The random thermal energy of a particle at room temperature is about \(kT = 0.026\,\text{eV}\), where \(k\) is Boltzmann's constant and \(T = 300\,\text{K}\). The gap of silicon is \(1.1/0.026 \approx 42\) times this. Only a tiny fraction of electrons ever collect so much energy by chance. But a crystal has so many electrons that the few are enough to count. For diamond the gap is 208 times \(kT\), and no electron crosses.
A photon can cross the gap too
Light can lift an electron across the gap, if one photon brings at least \(E_g\). From the chapter on photons, a photon of wavelength \(\lambda\) carries \(E = hc/\lambda\), and \(hc = 1240\,\text{eV nm}\). So the longest wavelength that lifts an electron across a gap \(E_g\) is \(\lambda_{max} = hc/E_g\).
Visible light runs from \(400\,\text{nm}\) to \(700\,\text{nm}\). Find the longest wavelength that each of silicon, germanium and diamond can absorb by lifting an electron across its gap. Which of them absorb visible light?
The gaps are \(1.1\), \(0.7\) and \(5.4\,\text{eV}\). Use \(\lambda_{max} = 1240/E_g\) nm.
Silicon: \(1240/1.1\). Germanium: \(1240/0.7\). Diamond: \(1240/5.4\).
Silicon absorbs up to \(1130\,\text{nm}\), germanium up to \(1770\,\text{nm}\), and diamond only up to \(230\,\text{nm}\).
All visible light is shorter than \(1130\,\text{nm}\), so silicon and germanium absorb all of it and look dark grey. Visible light is far longer than \(230\,\text{nm}\), so diamond cannot absorb it, and the light passes through. Diamond is clear because its gap is large.
A crystal starts to absorb light at wavelengths shorter than \(620\,\text{nm}\). What is its band gap, in eV? Take \(hc = 1240\,\text{eV nm}\).
Which statement about a pure semiconductor at 0 K is correct?
Heat does opposite things
The resistance \(R\) of a wire is the potential difference across it divided by the current through it, \(R = V/I\). This is Ohm's law, which the unit on electricity treats in full. A small resistance means the wire conducts well. Two things decide it: how many carriers of charge there are, and how freely they move.
In a metal the number of free electrons is fixed, about one per atom. Heating makes the ions vibrate harder, so the electrons are scattered more often. The resistance of copper rises by about 0.4 % for every kelvin. In a semiconductor, heating also adds scattering. But it lifts many more electrons across the gap, and the number of carriers grows very fast. This growth wins easily, so the resistance falls.
Watch the readouts as you move the slider. At 300 K silicon has \(1.5 \times 10^{16}\) free electrons in each cubic metre. That sounds large, but a cubic metre of silicon holds \(5 \times 10^{28}\) atoms. Only one atom in about \(3 \times 10^{12}\) has lost an electron. So pure silicon conducts very poorly at room temperature, and much better when warm.
Now return to the phone in the sun. From 25 °C to 45 °C the copper tracks gain 20 K, and their resistance rises by about 8 %. The number of free electrons in the silicon grows about four times over the same 20 K. The chips conduct several times better, while the copper conducts a little worse. The same heat has opposite effects, because it changes different things in the two materials.
A semiconductor is not a weak metal. A metal conducts because its bands overlap, and it conducts even at 0 K. A semiconductor has a real gap, conducts not at all at 0 K, and conducts better the hotter it gets. Its resistance changes with temperature in the opposite direction to a metal's.
Pure silicon has \(5 \times 10^{28}\) atoms and \(1.5 \times 10^{16}\) free electrons in each cubic metre at 300 K. Estimate how many atoms there are for each free electron, in units of \(10^{12}\).
A copper wire and a piece of pure silicon are cooled from room temperature in liquid nitrogen. What happens to their resistances?
By 1930 the new quantum mechanics could explain why metals conduct. It could not yet explain the strange crystals such as silicon and cuprous oxide, whose conduction grew with temperature. In 1931 Alan Wilson, a young physicist at Cambridge, used band theory to explain them. A full band cannot conduct, he showed. A crystal with a small gap above a full band conducts only when heat lifts electrons across. His two papers gave the semiconductor its modern meaning.
Most phone batteries contain a thermistor, a small piece of semiconducting material whose resistance falls steeply as it warms. The phone measures that resistance many times a minute. If the battery grows too hot while charging, the resistance drops, and the phone slows or stops the charging. The same property that made the chips in the sun conduct better is used here as a thermometer.
- Thinking a semiconductor conducts a little at every temperature, like a poor metal.A pure semiconductor at 0 K is an insulator. It conducts only because heat or light lifts electrons across the gap.
- Expecting every resistance to rise with temperature.That holds for metals. A semiconductor gains carriers much faster than it gains scattering, so its resistance falls.
- Using \(\lambda_{max} = hc/E_g\) the wrong way round.Light of wavelength shorter than \(\lambda_{max}\) is absorbed. Longer wavelengths carry too little energy and pass through.