At what temperature would an intrinsic semiconductor behave like a perfect insulator?

Intrinsic semiconductors are materials like pure silicon or germanium, which have a balance of electrons and holes due to thermal excitation. At absolute zero temperature (0 Kelvin), these materials would behave like perfect insulators because there wouldn't be any thermally generated charge carriers (electrons and holes) available for conduction.

However, as you increase the temperature, thermal energy provides electrons with enough energy to jump from the valence band to the conduction band, creating electron-hole pairs. This increases the conductivity of the semiconductor. The temperature at which the intrinsic semiconductor behaves like a perfect insulator depends on the energy gap between the valence band and the conduction band. This energy gap is known as the bandgap (Eg).

The relationship between the conductivity (σ) and temperature (T) in intrinsic semiconductors is given by the exponential equation known as the intrinsic carrier concentration equation:

ni=AT3/2e−Eg2kTni=AT3/2e−2kTEg

Where:

• nini is the intrinsic carrier concentration.
• AA is a constant.
• TT is the temperature in Kelvin.
• EgEg is the bandgap energy.
• kk is Boltzmann's constant.

As the temperature increases, the exponential term in the equation decreases. Therefore, at higher temperatures, the intrinsic carrier concentration increases, and the material becomes more conductive. Conversely, at lower temperatures, the intrinsic carrier concentration decreases, and the material behaves more like an insulator.

However, it's important to note that "perfect insulator" is a theoretical concept. In practical terms, even at low temperatures, there can still be some level of conductivity due to impurities or defects in the material.