Q.How does electrical conductivity of a semiconductor change with temperature? Why?

Electrical Properties of Solids: From Atoms to Circuits

Imagine a single atom. Its electrons live in well-defined shells, each at a specific energy. Now bring two atoms close together. Their electron clouds overlap, and the sharp energy levels split into two slightly different energies — one for the bonding combination, one for the antibonding. Bring a billion atoms together in a solid, and those two levels split into a billion closely spaced levels, forming a continuous band of allowed energies.

That is the core idea: in a solid, discrete atomic energy levels broaden into energy bands, separated by gaps where no electron can exist. The highest occupied band at absolute zero is the valence band; the next empty band above it is the conduction band. The gap between them is the band gap ($E_g$).

Conductors, Insulators, and Semiconductors

Conductors (metals) have either no band gap — the valence and conduction bands overlap — or a partially filled conduction band. Electrons have a vast number of empty states right next to them in energy, so a tiny electric field sets them drifting. That is why copper and aluminium conduct so well.

Insulators have a large band gap, typically $> 3\ \text{eV}$. At room temperature, almost no electrons have enough thermal energy to jump the gap. The valence band is full, the conduction band is empty, and no current flows. Diamond ($E_g \approx 5.5\ \text{eV}$) is a classic example.

Semiconductors sit in between. Their band gap is small — about $1.1\ \text{eV}$ for silicon, $0.67\ \text{eV}$ for germanium. At absolute zero they behave like insulators, but at room temperature enough electrons are thermally excited across the gap to give a small but useful conductivity. This conductivity rises sharply with temperature, opposite to metals.

The Hole: A Missing Electron

When an electron jumps from the valence band to the conduction band, it leaves behind a vacancy — a missing negative charge. That vacancy behaves as a positive charge carrier called a hole. Under an electric field, a neighbouring electron can move into the hole, leaving a new hole behind. The hole effectively moves in the opposite direction to the electrons, carrying positive charge.

In a pure (intrinsic) semiconductor, every electron excited leaves one hole behind, so $n = p$. This is called intrinsic conduction.

Doping: Engineering Conductivity

Pure silicon is not very useful. Its conductivity is too low and too sensitive to temperature. The real power comes from doping — deliberately adding impurity atoms to control the number of charge carriers.

n-type doping: Add a group-15 element (phosphorus, arsenic) to silicon. Silicon is group 14, so the impurity has five valence electrons. Four form bonds with neighbouring silicon atoms; the fifth is loosely bound and easily donated to the conduction band. The impurity is called a donor. Now $n > p$ — electrons are the majority carriers. …