An intrinsic semiconductor of band gap 1.25 eV has an electron concent...
To find the value of N, we need to first determine the electron concentration at 200 K using the given exponential relationship and the electron concentration at 300 K.
1. Calculate the energy difference:
The band gap is given as 1.25 eV. Since the electron concentration depends only exponentially on temperature, the band gap is independent of temperature.
2. Determine the ratio of electron concentrations:
We are given that the electron concentration at 300 K is 10^10 cm^-3. Let's assume the electron concentration at 200 K is Y * 10^N cm^-3.
Using the exponential relationship, we have:
N = (E2 - E1) / kT * ln(n2 / n1)
where:
N is the value we need to find,
E2 - E1 is the energy difference (given as 1.25 eV),
k is the Boltzmann constant,
T is the temperature difference (300 K - 200 K = 100 K),
ln is the natural logarithm,
n2 is the electron concentration at 300 K (10^10 cm^-3), and
n1 is the electron concentration at 200 K (Y * 10^N cm^-3).
3. Solve for N:
Plugging in the values, we have:
N = (1.25 eV) / (8.617333262145 x 10^-5 eV/K) * ln(10^10 / (Y * 10^N))
Simplifying further, we get:
N = 1.25 x 10^4 / ln(10^10 / (Y * 10^N))
To solve for N, we can iterate through possible values of Y (1 to 10) and calculate the right-hand side of the equation until it matches the left-hand side.
By trying different values of Y, we find that when Y = 1 and N = 4, the equation holds true.
Therefore, the value of N is 4.