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Formation of Energy Bands

Use the energy band model (valence band, conduction band and band gap) to distinguish conductors, semiconductors and insulators, and to explain temperature effects (A Level Physics).

  • A-Level Semiconductor Physics extension
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Learning objectives

  • Explain energy bands, intrinsic conduction and how temperature and light change semiconductor resistance.

1. Definitions (Must Know)

A. Valence band

The valence band is the band of electron energies that are normally occupied by electrons bound to atoms in a solid.

B. Conduction band

The conduction band is the band of electron energies where electrons are free to move through the solid and contribute to conduction.

C. Band gap, E_g

The band gap, E_g, is the energy difference between the top of the valence band and the bottom of the conduction band.

It is a range of energies with no allowed electron states.

Energy-band comparison for metals, semiconductors and insulatorsThree labelled energy diagrams compare overlapping bands, a small forbidden gap and a large forbidden gap.Metalconduction statesvalence statesbands overlapmany mobile electronsSemiconductorconduction statesvalence statessmall gapcarriers can be excitedInsulatorconduction statesvalence stateslarge gapvery few carriersEnergy increases upwards; the drawing is qualitative and not to scale.
Scroll diagram horizontally to read all labels.
Electrical behaviour depends on the occupied states and the energy needed to reach available conducting states: overlap for a metal, a modest gap for a semiconductor and a much larger gap for an insulator.

2. Key Ideas (What Earns Marks)

  • When many atoms form a solid, discrete atomic energy levels split into many very closely spaced levels, forming bands.
  • Conduction depends mainly on whether electrons can access the conduction band:
    • metals: valence and conduction bands overlap or the conduction band is partially filled → many mobile electrons
    • insulators: large band gap → very few electrons can reach conduction band at room temperature
    • semiconductors: small band gap → some electrons can be thermally excited into conduction band
  • Temperature effect:
    • metals: resistivity increases mainly due to increased collisions
    • semiconductors: resistivity decreases mainly because the number density of charge carriers increases
Keep it syllabus-safe

For 9478 questions, you usually only need the trend statements above (especially the temperature dependence). Use band language as a supporting explanation, not the main “answer format” unless asked.

3. Detailed Explanations

A. How bands form (qualitative)

An isolated atom has discrete electron energy levels.

When a large number of atoms come close together in a solid, interactions between atoms cause each atomic level to split into many closely spaced levels. With many atoms, the spacing becomes so small that we treat them as continuous bands.

B. Why a band gap matters for conduction

For an electron to become a mobile charge carrier in this model, it must be in an allowed state in the conduction band.

  • If the conduction band is already partially filled (metal), electrons can move and conduction is easy.
  • If the conduction band is empty and separated by a large gap (insulator), thermal energy at room temperature is not enough to excite many electrons across the gap.
  • If the gap is small (semiconductor), thermal excitation can create:
    • a conduction electron
    • a vacancy in the valence band, modelled as a hole (a positive charge carrier)

For a typical semiconductor, increasing temperature increases the number density of charge carriers (more electrons reach the conduction band), so conductivity increases and resistivity decreases.

For a metal, the number density is approximately constant; increased temperature mainly increases collision frequency, reducing mobility, so resistivity increases.

Resistivity vs temperature: metal vs semiconductor (scaled)

Two contrasting trends: metals have resistivity that increases with temperature; intrinsic semiconductors have resistivity that decreases with temperature.

Scroll across the graph to read all labels.

Two contrasting trends: metals have resistivity that increases with temperature; intrinsic semiconductors have resistivity that decreases with temperature.Two contrasting trends: metals have resistivity that increases with temperature; intrinsic semiconductors have resistivity that decreases with temperature.
This is why filament lamps (metals) increase resistance when hot, while NTC thermistors (semiconductors) decrease resistance when hot.
Open full-size graph
View figure data
Values for Resistivity vs temperature: metal vs semiconductor (scaled)
Temperature (relative)Metal (trend)Intrinsic semiconductor (trend)
00.41.05
200.520.7
400.640.45
600.780.28
800.920.17
1001.050.1

4. Common Mistakes

  • Saying “insulators have no electrons” (they have electrons; they are not free to move).
  • Thinking the band gap is a “physical distance” (it is an energy gap).
  • Using band theory to replace required circuit reasoning (many questions only want trends / definitions).

5. Exam Tips

  • If asked “why an NTC thermistor resistance decreases with temperature”: say “carrier number density increases (dominant effect)”.
  • If asked “why a metal filament lamp resistance increases with temperature”: say “collision frequency increases → mobility decreases”.
  • If asked to distinguish conductor/insulator/semiconductor from a diagram: focus on whether the conduction band is accessible/partly filled.

6. Worked Examples

Modelled example 1

Identify the material from the band diagram

Core

Problem

A band diagram shows a large energy gap between a full valence band and an empty conduction band. Identify the material type and explain its poor conduction at room temperature.
Study the worked solution

  1. Read the band structure

    Method

    The conduction band is empty and separated from the valence band by a large E_g.

    Reason

    The gap is an energy interval with no allowed electron states.

    Working

    E_g large

  2. Infer carrier access

    Method

    Very few electrons reach the conduction band at room temperature.

    Reason

    Typical thermal energy is insufficient to promote many electrons across the large gap.

    Working

    kT≪ E_g ⇒ n_conduction very small

  3. Classify the material

    Method

    The material is an insulator.

    Reason

    Its very small mobile-carrier population gives poor electrical conduction.

    Working

    n_mobile ≈ 0 ⇒ σ small

Common misconception 2

Explaining semiconductor temperature trend

Find and correct the mistake

Learner claim

A learner says heating must increase a semiconductor’s resistivity because it increases collisions, just as in a metal. Diagnose the claim in 2–3 lines.

Try this before viewing the solution

Dominant semiconductor effect

View solution step by step

  1. Acknowledge collisions

    Method

    Heating can increase lattice scattering and reduce carrier mobility.

    Reason

    The learner has identified a real effect, but not necessarily the dominant one.

    Working

    T↑ ⇒ μ↓

  2. Track carrier density

    Method

    Many more electrons are thermally excited into the conduction band, creating electron–hole pairs.

    Reason

    The semiconductor has a small enough band gap for thermal excitation to change carrier number substantially.

    Working

    T↑ ⇒ nₑ↑, nₕ↑

  3. Compare effects

    Method

    The carrier-density increase dominates, so conductivity rises and resistivity falls.

    Reason

    Conductivity depends on both carrier density and mobility.

    Working

    σ ≈ nqμ↑ ⇒ ρ = 1/σ↓

7. Mind Stretchers

Mind stretcher 1: Question 1Extension

Explain why two materials can have the same band gap size but different resistivities at room temperature.

Show Answer

Resistivity depends not only on carrier number density but also on carrier mobility, which depends on scattering/collisions and material structure. So even with the same band gap, differences in mobility and impurity levels can change resistivity.

Mind stretcher 2: Question 2Extension

A metal has many free electrons. Why does it still have a non-zero resistance?

Show Answer

Even though there are many carriers, electrons collide with lattice ions/defects. These collisions transfer energy to the lattice (heating) and limit the drift velocity, producing resistance.

8. Optional (Enrichment)

A. Quantum numbers and detailed band structure (beyond syllabus)

Energy bands ultimately come from quantum mechanics, including the splitting of electron states when atoms form a periodic lattice. For most A Level questions, the qualitative band model is sufficient.