Energy bands in solids

A wire, a window and a chip

Copper carries a current. Glass does not. A chip of silicon is somewhere in between. It carries a little current, and we can change that current with heat, light or a few foreign atoms. All the electronics of a phone depends on this middle behaviour. To see why a solid behaves in one of the three ways, we look at the energies of its electrons.

From levels to bands

An atom on its own has sharp energy levels. Bring many atoms close together to make a crystal. The outer electrons of neighbouring atoms interact, and each level splits into as many levels as there are atoms. In a crystal the number of atoms is about $10^{23}$, so these levels lie very close together. They form a band. The inner electrons stay close to their own atoms, so their levels remain narrow.

Two bands decide the electrical behaviour.

• The valence band holds the electrons that bind the atoms together. At $0$ K it is the highest band that has electrons in it.
• The conduction band is the next band above it. An electron in this band is free to move through the whole crystal.
• Between the two is the band gap $E_g$. No electron can have an energy inside the gap.

Why does a full band carry no current? An electric field pushes an electron to a slightly higher energy. If every nearby level is already taken, the electron cannot move up. So a full band cannot give a current. A band that is only partly full has empty levels next door, and its electrons can respond to the field.

Three kinds of solid

• Conductor. The conduction band is partly filled (sodium), or it overlaps the valence band (magnesium). There are plenty of free electrons at every temperature. The resistivity is $10^{-8}$ to $10^{-2}$ ohm metre.
• Insulator. The valence band is full and the conduction band is empty. The gap is large, more than about $3$ eV. Diamond has $E_g = 5.5$ eV. An electric field cannot lift electrons across. The resistivity is $10^{11}$ to $10^{19}$ ohm metre.
• Semiconductor. The picture is the same as for an insulator, but the gap is smaller, less than about $3$ eV. Silicon has $E_g = 1.12$ eV and germanium has $E_g = 0.67$ eV. At $0$ K a semiconductor is an insulator. At room temperature, some electrons have enough thermal energy to cross. The resistivity is $10^{-5}$ to $10^{6}$ ohm metre.

Standard values and NCERT. The standard values are $1.12$ eV for silicon and $0.67$ eV for germanium. NCERT prints about $1.1$ eV and $0.7$ eV. The difference is only rounding. Use the first pair in calculations.

Electrons and holes

When an electron crosses into the conduction band, it leaves an empty place in the valence band. This empty place is a hole. Neighbouring valence electrons can hop into it, and each hop moves the hole one step the other way. So a hole behaves like a positive charge $+e$ that moves in the direction of the field. A pure semiconductor has the same number of electrons and holes:

where $n_i$ is the intrinsic carrier number per cubic metre. Both kinds of carrier give a current, and the two currents add. A hole disappears when an electron falls into it. This is recombination. In balance, heat makes as many pairs as recombination removes.

Temperature and light

How many electrons cross the gap? Thermal energy at temperature $T$ is about $kT$, where $k = 8.617 \times 10^{-5}\ \text{eV/K}$. At $300$ K, $kT = 0.0259$ eV, much smaller than the gap. Only a tiny share of electrons cross. The share is governed by the factor

This is the main temperature dependence of the carrier number. A larger gap or a lower temperature makes the factor much smaller. The full formula also has a slower factor in $T$, and we leave it out in these estimates.

• A metal gets more resistive when it is heated. Its lattice vibrates more and scatters the electrons. The number of carriers stays the same.
• A semiconductor gets less resistive when it is heated. The extra carriers matter more than the extra scattering. This gives a negative temperature coefficient. A thermistor uses it.

Light can also lift an electron. A photon of wavelength $\lambda$ has energy $E = \dfrac{hc}{\lambda}$, and $hc = 1240\ \text{eV nm}$. The photon works if $E \ge E_g$, so the longest wavelength that works is $\lambda = \dfrac{hc}{E_g}$.

Set the gap to $5.5$ eV (diamond) and read the thermal factor at $300$ K. Then set it to $0.67$ eV. How many powers of ten apart are they?

A worked example. Can red light of wavelength $700$ nm create electron-hole pairs in silicon? Can it in diamond?

1. The photon energy. $E = \dfrac{1240\ \text{eV nm}}{700\ \text{nm}} = 1.77$ eV.
2. Silicon. $1.77\ \text{eV} > 1.12\ \text{eV}$. Yes, the photon lifts an electron across.
3. Diamond. $1.77\ \text{eV} < 5.5\ \text{eV}$. No. The photon passes through, and that is why a clean diamond is clear.
4. The cut-off. For silicon, $\lambda_{max} = \dfrac{1240\ \text{eV nm}}{1.12\ \text{eV}} = 1107$ nm, in the infrared. Silicon responds to all visible light.

Practise with new numbers

Try it in Python

The program lists five materials. For each, it finds the longest wavelength that lifts an electron across the gap, and the thermal factor at $300$ K.

Common mistakes

• Saying a semiconductor has no band gap. It has one. It is just small enough for heat to lift some electrons across.
• Thinking a hole is a particle that exists on its own. It is the absence of an electron in the valence band. It acts as a positive charge $+e$.
• Saying a heated semiconductor gets more resistive, like a metal. Its resistance falls, because the carrier number grows very fast.
• Using the wrong direction for the cut-off. A larger gap needs a shorter wavelength.
• Forgetting that both carriers give a current. The electron current and the hole current add.

Remember these

• Bands: the valence band is the highest full band, the conduction band is above it, and the band gap $E_g$ is between them.
• Conductor: partly full or overlapping bands. Insulator: $E_g$ above about $3$ eV. Semiconductor: $E_g$ below about $3$ eV, Si $1.12$ eV, Ge $0.67$ eV.
• A pure semiconductor has $n_e = n_h = n_i$. The number grows as $e^{-E_g/2kT}$, so the resistance falls with temperature.
• The longest wavelength that makes an electron-hole pair is $\lambda = \dfrac{1240\ \text{eV nm}}{E_g}$.

References

• NCERT Physics Class 12, Semiconductor Electronics, the sections on energy bands and intrinsic semiconductors.
• OpenStax University Physics Volume 3, the chapter Condensed Matter Physics, on the band theory of solids.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A solid has a full valence band and a gap of $0.8$ eV above it. Is it a conductor, an insulator or a semiconductor?

2. Why does the resistance of a semiconductor fall when it is heated, while the resistance of a metal rises?

3. What is the energy of a photon of wavelength $620$ nm, and can it make a pair in germanium?

4. In which band is a hole, and what does it stand for?