At what temp would an intrensic semiconductor behave like a perfect insulator?Why?

n an intrinsic semiconductor such as silicon at temperatures above absolute zero, there will be some electrons which are excited across the band gap into the conduction band and which can support current flow. When the electron in pure silicon crosses the gap, it leaves behind an electron vacancy or "hole" in the regular silicon lattice. Under the influence of an external voltage, both the electron and the hole can move across the material.

So the answer is , at absolute zero Kelvins.

To determine the temperature at which an intrinsic semiconductor behaves like a perfect insulator, we need to understand the basic behavior of intrinsic semiconductors and how temperature affects their electrical properties.

Intrinsic semiconductors, such as pure silicon (Si) or germanium (Ge), have their valence band completely filled with electrons and their conduction band completely empty at absolute zero temperature (-273.15 degrees Celsius or 0 Kelvin). However, as temperature increases, some electrons gain enough thermal energy to transition from the valence band to the conduction band, leaving behind holes in the valence band.

The number of thermally excited electrons (and holes) in the conduction (valence) band increases exponentially with temperature. This exponential relationship is described by the familiar equation:

n = Nc * exp(-Eg / (2kT))

where:
n is the concentration of electrons in the conduction band
Nc is the effective density of states in the conduction band
Eg is the energy band gap between the valence and conduction bands
k is Boltzmann's constant (8.6173 x 10^-5 electron-volts per Kelvin)
T is the absolute temperature

As the equation suggests, if the energy band gap (Eg) between the valence and conduction bands is larger, it requires more energy or higher temperature for electrons to bridge the gap and transition to the conduction band.

Now, to answer your question, an intrinsic semiconductor behaves like a perfect insulator when the number of thermally excited electrons in the conduction band is negligible. This happens when the concentration of electrons (n) is effectively zero.

Substituting n = 0 into the equation mentioned above, we get:

0 = Nc * exp(-Eg / (2kT))

Taking the natural logarithm of both sides:

ln(0) = ln(Nc * exp(-Eg / (2kT)))

0 = ln(Nc) - (Eg / (2kT))

At this point, we can see that ln(0) is undefined. Thus, it is not possible to have a temperature at which an intrinsic semiconductor behaves like a perfect insulator. As temperature increases, there will always be a non-zero concentration of thermally excited electrons in the conduction band, meaning the semiconductor will never truly become a perfect insulator.

However, it is important to note that at low temperatures, the concentration of thermally excited electrons can be extremely low, effectively making the intrinsic semiconductor act as an insulator for practical purposes.