Question

For temperature greater than 0 K, the fermi level is the level where the probability of occupation of electrons is:

• A. 1/2
• B. 1/6
• C. 1/8
• D. 1/4

Hint:

The Fermi level is often described as the "half-filling" point. At absolute zero (0 K), all states below \(E_F\) are 100% filled, and all states above are 0% filled. For any temperature T>0 K, the occupation probability right at the Fermi level itself is always 50%, or 1/2.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The probability of occupation of an electron energy level E in a solid at a temperature T is given by the Fermi-Dirac distribution function, \(f(E)\). The Fermi level (\(E_F\)) is a key parameter in this distribution.
Step 2: Key Formula or Approach:
The Fermi-Dirac distribution function is given by:
\[ f(E) = \frac{1}{e^{(E - E_F) / k_B T} + 1} \]
where \(E\) is the energy of the level, \(E_F\) is the Fermi level, \(k_B\) is the Boltzmann constant, and T is the absolute temperature.
Step 3: Detailed Explanation:
The question asks for the probability of occupation at the Fermi level itself. This means we need to evaluate the function at \(E = E_F\).
Substituting \(E = E_F\) into the formula:
\[ f(E_F) = \frac{1}{e^{(E_F - E_F) / k_B T} + 1} \]
\[ f(E_F) = \frac{1}{e^{0 / k_B T} + 1} \]
Since \(e^0 = 1\), the equation becomes:
\[ f(E_F) = \frac{1}{1 + 1} = \frac{1}{2} \]
This result holds for any temperature \(T>0\) K.
Step 4: Final Answer:
By definition, the Fermi level is the energy level at which the probability of occupation by an electron is exactly 1/2, for any temperature above absolute zero.