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MHT-CET Physics · Semiconductor Devices

Energy Bands and Doping

Solids are sorted by the gap between their valence and conduction bands; a semiconductor's small gap lets heat free a few electrons, and doping with a pentavalent or trivalent impurity adds electrons (n-type) or holes (p-type) by the million.

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Why this matters

18 PYQs, none HARD. Two things are asked: how the bands look in a conductor, an insulator and a semiconductor (and what temperature does to a semiconductor's resistance), and which impurity makes which type — with the carrier counts that follow.

Concept 1 of 2: Conductors, Insulators and Semiconductors

Electrons conduct only if they can move into empty energy states. A metal's bands overlap, so they always can; an insulator's gap is too wide to cross; a semiconductor's gap is small enough that at room temperature a few electrons jump it, leaving holes behind. Heat frees more of them, so a semiconductor conducts BETTER when hot.

Definition

  • Insulator: valence band full, conduction band empty, gap large (>3> 3 eV).
  • Semiconductor at room temperature: conduction band partly filled, valence band partly empty; gap about 1 eV (Si 1.1 eV).
  • Intrinsic (pure) semiconductor: every free electron leaves a hole, so ne=nhn_e = n_h; conduction only from broken covalent bonds.
  • Temperature up: more carriers, so a semiconductor's resistivity FALLS steeply (a metal's rises).

Intrinsic semiconductor

ne=nh=nin_e = n_h = n_i

Worked example

Copper, silicon and diamond at room temperature: which has a partly filled conduction band because of thermal excitation across a small gap?
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 10th May Shift 1 · Q29Easy

Example 1 · Semiconductor Devices · Band Theory, Doping, and Semiconductor Types

In insulators

A semiconductor's valence band is 'completely filled'

Only at absolute zero. At room temperature some electrons have left it, so the valence band is partly EMPTY and the conduction band partly FILLED. The 'completely filled' options describe an insulator.
Drill 8 more on conductors, insulators and semiconductors

Concept 2 of 2: n-type and p-type Doping

Add a pentavalent atom (P, As, Sb) and its fifth electron is nearly free: electrons become the majority carriers — n-type. Add a trivalent atom (B, Al, In) and one bond is left with a hole: holes become the majority — p-type. Either way the carrier count rises enormously, so the resistivity falls.

Definition

  • n-type: pentavalent donor; electrons majority, holes minority; donor levels sit in the gap just BELOW the conduction band.
  • p-type: trivalent acceptor; holes majority, electrons minority; acceptor levels just above the valence band.
  • The crystal stays electrically neutral either way.
  • Mass-action law: nenh=ni2n_en_h = n_i^2. With donors ND≫niN_D \gg n_i: ne≈NDn_e \approx N_D, nh=ni2NDn_h = \dfrac{n_i^2}{N_D}.
  • 1 ppm of dopant in 4×10284 \times 10^{28} atoms/m³ gives 4×10224 \times 10^{22} carriers/m³.

Mass-action law

ne nh=ni2n_e\,n_h = n_i^2

Worked example

Silicon has ni=1.5×1016 m−3n_i = 1.5 \times 10^{16}\ \text{m}^{-3}. It is doped with 102210^{22} donor atoms per m³. Electron and hole concentrations?
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 22 April Shift I · Q4Moderate

Example 2 · Semiconductor Devices · Band Theory, Doping, and Semiconductor Types

In a pure silicon crystal electron-hole concentration is 101610^{16} per m3m^{3} at 301 K . Now 102110^{21} atoms of phosphorus are added per cubic meter. The new hole concentration in silicon is (in per m3m^{3})

n-type means negatively charged

The 'n' names the majority CARRIER, not the charge of the crystal: every donor electron came with a donor atom, so the crystal is neutral.
Drill 8 more on n-type and p-type doping

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (2)

Test yourself on Semiconductor Devices

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