Why does the resistance of a semiconductor decrease as the temperature increases?

A semiconductor is a material that has a resistivity between that of conductors and insulators. The most common semiconductors are made from Silicon (Si) and Germanium (Ge). A semiconductor has empty energy levels, called conduction band, and a band filled with electrons, called the valence band. At absolute zero temperature, the valence band is fully occupied, and the conduction band is empty. As a result, there's no free charge carrier to conduct electricity.

As the temperature increases, thermal energy provides to the electrons in the valence band. This energy helps electrons jump from the valence band to the conduction band, which are now free to move. An increase in free electrons (charge carriers) leads to increased conductivity and decreased resistance in a semiconductor.

Semiconductors can be classified into two types: extrinsic and intrinsic. Extrinsic semiconductors are doped with impurity elements to introduce more free charge carriers (either electrons or holes). For intrinsic semiconductors, the concentration of free charge carriers solely depends on the temperature.

The intrinsic carrier concentration (ni) is a measure of how many electrons and holes are available in a semiconductor at a particular temperature. It can be expressed as: \[n_i = B \cdot T^{3/2} \cdot e^{-\frac{E_g}{2 \cdot k_B \cdot T}}\] where: - \(n_i\) is the intrinsic carrier concentration - \(B\) is a material-dependent constant - \(T\) is the temperature in kelvin - \(E_g\) is the energy bandgap between the valence and conduction bands - \(k_B\) is the Boltzmann constant As temperature \(T\) increases, the exponential term in the equation dominates, and the intrinsic carrier concentration \(n_i\) also increases.

The increase in the intrinsic carrier concentration directly affects conductivity (\(\sigma\)) of the semiconductor: \[\sigma = e \cdot (n \cdot \mu_n + p \cdot \mu_p)\] where \(e\) is the charge of an electron, \(n\) and \(p\) are the electron and hole concentrations respectively, and \(\mu_n\) and \(\mu_p\) are their respective mobilities. As the carrier concentration increases with temperature, conductivity also increases. Since resistance (\(R\)) is the reciprocal of conductivity, as the conductivity increases, resistance decreases: \[R = \frac{1}{\sigma}\] Thus, the resistance of a semiconductor decreases as the temperature increases due to the increased number of free charge carriers resulting from a higher intrinsic carrier concentration.