Maximum Temperature for Silicon with Given Intrinsic Carrier Concentration

Introduction
Intrinsic carrier concentration in silicon is an important parameter that determines the electrical conductivity of silicon. It is a function of temperature and bandgap energy. At high temperatures, the intrinsic carrier concentration increases, which can affect the performance of silicon-based devices. Therefore, it is important to determine the maximum temperature allowed for silicon with a given intrinsic carrier concentration.

Formula
The intrinsic carrier concentration in silicon can be calculated using the following formula:

ni^2 = (Nv x Nc) x exp(-Eg/2kT)

Where,
ni is the intrinsic carrier concentration
Nv is the effective density of states in the valence band
Nc is the effective density of states in the conduction band
Eg is the bandgap energy
k is the Boltzmann constant
T is the temperature

Calculation
Given: ni = 1.0 x 10^26 per cc

Assuming Nv and Nc are constants, we can simplify the formula as:

ni^2 = A x exp(-Eg/2kT)

Where, A = Nv x Nc

Taking the logarithm of both sides, we get:

ln(ni^2) = ln(A) - Eg/2kT

Rearranging the formula, we get:

T = Eg/2k x (1/ln(ni^2) - ln(A))

Substituting the given values, we get:

T = 1.21 eV/2k x (1/ln(1.0 x 10^26) - ln(A))

Assuming A = 2.8 x 10^19 cm^-3, we get:

T = 723 K or 450°C

Conclusion
The maximum temperature allowed for silicon with a given intrinsic carrier concentration of 1.0 x 10^26 per cc is 723 K or 450°C. Beyond this temperature, the intrinsic carrier concentration will increase, which can affect the performance of silicon-based devices.