A full lesson: a cartoon movie to start, then how sharp atomic levels spread into bands, counting the 4N and 8N states of silicon, the band gap E_g with real numbers, the metal–semiconductor–insulator verdict, why diamond is clear and silicon is grey, NEET shortcuts, problems to solve yourself, and 12 MCQs.
One atom has sharp energy levels. In a crystal, billions of atoms push those levels into wide bands. A full valence band and an empty conduction band are separated by a gap E_g. Zero gap: metal. Small gap: semiconductor. Big gap: insulator.
At the school science fair, Meera builds a tester: a battery, a bulb and two clips. Copper wire: the bulb blazes. A glass rod: nothing. A thin silicon strip: the bulb glows faintly, and when Kabir warms the strip with a hair dryer, it glows brighter. Kabir asks the obvious question: every atom has electrons, so why do some solids let them move and others do not?
The answer is not "how many electrons", but which energies they are allowed to have. In a crystal those energies form bands with a gap between them. The size of the gap, E_g, decides everything.
Try at home. Look at a diamond (or a glass marble) and at a solar cell or a computer chip. The diamond is clear, the silicon looks grey and shiny. By Stage 6 you will explain the difference with one number: the band gap.
"Electrons are like people in a building with two floors. If the lower floor is completely full and the stairs are too tall, nobody can move. That is the band picture in one line."
"Why should a completely full band carry no current, even though it is full of electrons?"
To carry current an electron must gain a little energy from the field and move into a slightly higher empty level. In a full band there is no empty level to move into, so the field cannot change the electrons' motion. Net current is zero.
In an isolated atom each electron has a sharp, fixed energy level. In a crystal, the outer orbits of neighbouring atoms come close and overlap. Each electron also sees a slightly different pattern of charges around it, so each gets a slightly different energy. N atoms turn one level into N closely spaced levels: a nearly continuous energy band. Bring the atoms together and add more of them.
Outer orbits overlap and each electron feels a slightly different surrounding. Pauli's principle forbids identical states, so the levels separate a little.
With about 10²² atoms per cm³ the spacing inside a band is tiny: the band is practically continuous.
Inner orbits do not overlap much, so their levels stay sharp. Bands that matter for conduction come from the outer (valence) electrons.
"One atom: a line. Two atoms: two lines. A crystal: a thick band. The band is just billions of lines packed so close you cannot see the gaps between them."
Each Si or Ge atom has 4 outer electrons (two in s, two in p). A crystal of N atoms therefore has 4N valence electrons. The outer shell can hold 8 electrons per atom: 8N states (2N s + 6N p). At the real interatomic spacing these 8N states split into two bands of 4N states each. The 4N electrons exactly fill the lower band at 0 K. Choose N and watch the electrons take their seats.
| Quantity (N atoms of Si/Ge) | Number | At 0 K |
|---|---|---|
| Valence electrons | 4N | all in the lower band |
| Outer-shell states | 8N = 2N (s) + 6N (p) | split into two bands |
| Valence band states | 4N | completely full |
| Conduction band states | 4N | completely empty |
"Careful with 8N: that is the total of both bands. Each band has 4N. The valence band is the band containing the valence electrons; the band just above it is the conduction band."
The top of the valence band is E_V, the bottom of the conduction band is E_C. The band gap (forbidden gap) is E_g = E_C − E_V. No electron can have an energy inside it. Heat gives electrons random kicks of about kT (0.026 eV at room temperature). Change E_g and the temperature and compare.
Measured in electron-volts: 1 eV = 1.6 × 10⁻¹⁹ J. Si: E_g = 1.1 eV = 1.76 × 10⁻¹⁹ J.
The gap of Si is about 42 kT, so only a tiny fraction of electrons crosses. Diamond's gap is about 209 kT: practically none.
Small changes in E_g or T change this enormously. Ge (0.7 eV) has about 2000 times more free electrons than Si (1.1 eV) at 300 K.
A photon with energy ≥ E_g can lift an electron across. Si: λ ≤ 1127 nm. This is used later for photodiodes, LEDs and solar cells.
"Remember the magic number 1240. Wavelength in nanometres equals 1240 divided by energy in electron-volts. It turns a band gap into a colour in one step."
A current needs electrons that can move up into empty levels just above them. Pick a solid, switch on the electric field and raise the temperature. Watch who moves.
Either the conduction band is partly filled, or the bands overlap. Empty levels sit right above filled ones: very many free electrons, high conductivity.
Valence band full, conduction band empty, gap too wide for thermal energy. No free electrons: very low conductivity.
At 0 K it is an insulator. At room temperature a few electrons cross. Each leaves a vacancy in the valence band; both electrons and vacancies conduct.
"Zero, small, big: metal, semiconductor, insulator. And the vacancy left behind in the valence band is so important it gets its own name in the next lesson: the hole."
"A semiconductor at 0 K and an insulator at 0 K. Can you tell them apart by their conductivity?"
No. Both have a full valence band and an empty conduction band, so both conduct nothing. The difference appears only when you warm them: the small-gap semiconductor starts conducting.
Group 14 shows the whole story: C (diamond) 5.4 eV, Si 1.1 eV, Ge 0.7 eV, Sn 0 eV. Same group, four different behaviours. Light reveals the gap too: a photon is absorbed only if its energy hf ≥ E_g. Pick a material and slide the wavelength of the light shining on it.
| Solid (group 14) | E_g (eV) | Type | Absorbs light shorter than |
|---|---|---|---|
| C (diamond) | 5.4 | insulator | 230 nm (UV only: clear) |
| Si | 1.1 | semiconductor | 1127 nm (all visible: grey, opaque) |
| Ge | 0.7 | semiconductor | 1771 nm |
| Sn | 0 | behaves as a metal | all wavelengths (shiny) |
"A solar cell is grey because silicon swallows every colour of sunlight, and each swallowed photon frees an electron. A diamond lets visible light straight through because its gap is too big."
Metal, semiconductor, insulator. Bands overlap or partly filled ⇒ metal.
Valence band: 4N states, full at 0 K. Conduction band: 4N states, empty at 0 K.
E_C = bottom of conduction band, E_V = top of valence band. Forbidden: no states inside.
Si 1.1 eV → 1127 nm; a 2 eV gap → 620 nm (red). Photon absorbed only if E ≥ E_g.
E_g/kT: Si ≈ 42, diamond ≈ 209. Bigger ratio, exponentially fewer carriers.
Gap shrinks down the group. Order of conductivity (pure, 300 K): Sn > Ge > Si > C.
"If a question shows a band diagram, look only at two things: is there a gap, and how big is it compared with 3 eV? Then read off the answer."
Cover the solution, write the formula, and solve on paper. Take 1 eV = 1.6 × 10⁻¹⁹ J, k = 8.62 × 10⁻⁵ eV/K, hc = 1240 eV nm.
A silicon crystal has 5 × 10²² atoms. How many valence electrons does it have? How many states are in the valence band, in the conduction band, and in the whole outer shell?
4 valence electrons per atom; 8 outer states per atom split equally into two bands.
Express the band gap of silicon (1.1 eV) and diamond (5.4 eV) in joules.
Multiply by 1.6 × 10⁻¹⁹.
Find kT at 300 K in eV. How many times kT is the gap of (a) Si, (b) diamond?
kT = 8.62 × 10⁻⁵ × 300.
Find the longest wavelength that can lift an electron across the gap in (a) Si, (b) diamond. Which of them is transparent to visible light (400–700 nm)?
λ_max = hc / E_g = 1240 / E_g nm.
A semiconductor starts absorbing light only when the wavelength is shorter than 620 nm. Find its band gap and classify it.
E_g = 1240 / λ_max.
Take hc = 1240 eV nm and kT = 0.026 eV at 300 K. Write the rule first, then click.
Score: 0 / 12
| If the question says… | Write this first | The trap |
|---|---|---|
| "why do energy bands form" | outer orbits overlap; each electron's surroundings differ | Saying "electrons collide" or "atoms ionise" |
| "number of states in the valence band of Si (N atoms)" | 4N | Answering 8N (that is both bands together) |
| "at 0 K, intrinsic Si" | VB full, CB empty → insulator | Saying "a few electrons in CB" |
| "band diagram with overlapping bands" | metal | Calling it a semiconductor with a tiny gap |
| "E_g = 5.4 eV / 1.1 eV / 0.7 eV" | insulator / semiconductor / semiconductor | Using 1 eV as the cut-off instead of 3 eV |
| "maximum wavelength to excite" | λ = 1240 / E_g nm | Forgetting nm, or using 12400 (that is for Å) |
| "forbidden energy gap" | E_C − E_V, no allowed states | Writing E_V − E_C (negative) |
| "C, Si, Ge, Sn — order of gap" | C > Si > Ge > Sn | Assuming same group ⇒ same type |
Lesson 3 looks at the same physics in real space: covalent bonds in Si and Ge, a broken bond giving a free electron and a hole, n_e = n_h = n_i, and why we dope semiconductors.
Values used: E_g(C) = 5.4 eV, E_g(Si) = 1.1 eV, E_g(Ge) = 0.7 eV, E_g(Sn) = 0, E_g(GaAs) = 1.42 eV; k = 8.62 × 10⁻⁵ eV/K; hc = 1240 eV nm; 1 eV = 1.6 × 10⁻¹⁹ J. The band-formation and carrier curves are schematic.
Follow the four steps: understand it, remember it, try it, then prove it with past papers.