Question
Explain the energy band theory of solids. Compare the band structures of conductors, semiconductors, and insulators. Why does the conductivity of semiconductors increase with temperature while that of conductors decreases?
(CBSE 12 + JEE Main pattern)
Solution — Step by Step
In an isolated atom, electrons occupy discrete energy levels. When atoms come together in a solid, each energy level splits into closely-spaced levels forming a band. The two most important bands are:
- Valence band — the highest energy band that is completely or partially filled at 0 K
- Conduction band — the next higher band, empty or partially filled
- Band gap () — the energy gap between the top of the valence band and the bottom of the conduction band
| Property | Conductor | Semiconductor | Insulator |
|---|---|---|---|
| Band gap | Zero (bands overlap) | Small (Si: 1.1 eV, Ge: 0.67 eV) | Large (diamond: 5.5 eV) |
| Valence band | Partially filled or overlaps with conduction band | Completely filled at 0 K | Completely filled |
| Conduction band at 0 K | Partially filled | Empty | Empty |
| Resistivity | to ohm-m | to ohm-m | to ohm-m |
| Temp effect on conductivity | Decreases | Increases | Slightly increases |
Semiconductors: At higher temperature, more electrons gain enough thermal energy to jump across the band gap () from valence to conduction band. More charge carriers = higher conductivity. The increase in carriers dominates over increased lattice vibrations.
Conductors: Free electrons already exist in abundance. Raising temperature increases lattice vibrations (phonons), which scatter electrons more frequently. More scattering = higher resistance = lower conductivity. No new carriers are created because there is no gap to cross.
flowchart TD
A["Energy Band Theory"] --> B["Conductor"]
A --> C["Semiconductor"]
A --> D["Insulator"]
B --> E["No band gap / bands overlap"]
B --> F["Many free electrons at all T"]
C --> G["Small band gap: 0.5-3 eV"]
C --> H["Electrons jump gap at room T"]
D --> I["Large band gap: > 3 eV"]
D --> J["Almost no electrons can jump"]
F --> K["Higher T → more scattering → less conductivity"]
H --> L["Higher T → more carriers → more conductivity"]Why This Works
Band theory arises from quantum mechanics. When atoms are far apart, they have identical discrete energy levels. As they approach each other in a crystal, the Pauli exclusion principle forces the energy levels to split (no two electrons can have the same quantum state). With atoms, each level splits into sub-levels so close together that they form a continuous band.
Whether a material conducts or not depends on whether electrons can move to empty states. In a completely filled band, every state is occupied and electrons cannot accelerate (no empty state to move into). Conduction requires either a partially filled band (metals) or electrons promoted to an empty band (semiconductors).
Alternative Method — Qualitative Analogy
Think of it as a parking lot analogy:
- Conductor: Parking lot half-full — cars can move around freely
- Insulator: Parking lot completely full with a high wall to the next lot — no movement possible
- Semiconductor: Parking lot full but with a low wall to an empty lot — a few cars can jump over at room temperature
For JEE, the band gap values are important: Si = 1.1 eV, Ge = 0.67 eV, diamond = 5.5 eV. Also know that at room temperature (300 K) is about 0.026 eV — much smaller than even Ge’s band gap. This is why only a small fraction of electrons are thermally excited across the gap at room temperature.
Common Mistake
Students assume that if the band gap is small enough, ALL valence band electrons jump to the conduction band. Even in Ge (0.67 eV gap), only about 1 in 10 electrons is thermally excited at room temperature. The thermal energy eV is much smaller than the gap. The fraction excited follows , which is a very small number. Semiconductors work because even this tiny fraction provides enough carriers for practical conduction.