Semiconductors

Model of thermal carriers and conduction in semiconductors. Theory, formulas, worked examples and interactive graphs.

Conceptual context

A semiconductor is not simply halfway between a metal and an insulator. Its essential feature is that the number of mobile carriers can change strongly with thermal energy. $T$ provides the microscopic agitation that promotes carriers, while $E_g$ sets the energetic door they must cross. A small door produces many thermally activated electron-hole pairs; a large door suppresses them. This is why the central question links band structure, $T$, carrier density, and electrical response. The material does not need to change its atoms to change behavior: its carrier population changes because the occupation of electronic states changes.

🟢 Essential level

In an intrinsic semiconductor, conduction appears because thermal agitation creates pairs of mobile charge carriers. An electron that crosses the $E_g$ leaves a free hole in the lower band, so that the $n$ and the $p$ increase simultaneously. The reference quantity that describes this thermodynamic equilibrium in the crystal is the $n_i$.

By raising the $T$, the available thermal energy increases, and a significantly larger number of these carriers is generated. In contrast, if the $E_g$ is wider, the energy jump becomes more difficult and the carrier population decreases. Finally, the overall $\sigma$ of the material depends both on the number of electrons and holes present and on the ease with which these charges move under the action of an applied electric field. It is like a thermal drawbridge: the more heat energy pushes upward, the more "travelers" manage to cross the gap, and the higher the bridge sits, the fewer succeed at any given temperature.

🔵 Formal level

Basic formulas

The formal model uses three linked relations. The intrinsic population is determined by $T$, $E_g$, $k_B$, $N_c$, and $N_v$.

Intrinsic carrier density

This relation combines an available-state factor with a thermal activation factor. It answers the central question because $n_i$ rises with $T$ and falls very rapidly when $E_g$ increases.

Real student questions

Why can a semiconductor conduct better when heated, while a metal often conducts worse? In the semiconductor, heating creates more carriers; in a metal, many carriers already exist and mobility loss dominates.

Is $E_g$ a physical wall in space? No. It is an energy separation between allowed electronic bands.

Does $n_i$ count electrons or holes? In an intrinsic material it represents thermally generated pairs, so it equals each mobile population.

Why include $\sigma$ if the question is about carriers? Because carrier density explains availability, but measurable current also depends on mobility and charge.

Final synthesis

A semiconductor turns thermal energy into mobile carriers. $T$ supplies opportunity, $E_g$ sets the cost, and $n_i$ summarizes the equilibrium. Conduction appears when that population combines with mobility.