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: Concept Introduction:
The Fermi-Dirac distribution, \(f(E)\), determines the probability of an electron occupying an energy level \(E\) at temperature \(T\). The Fermi level, \(E_F\), is a crucial parameter within this distribution.
Step 2: Essential Formula:
The Fermi-Dirac distribution is defined as:
\[ f(E) = \frac{1}{e^{(E - E_F) / k_B T} + 1} \]
Where:
\(E\) = energy level, \(E_F\) = Fermi level, \(k_B\) = Boltzmann constant, \(T\) = absolute temperature.
Step 3: Detailed Solution:
To find the occupation probability at the Fermi level (\(E = E_F\)), substitute \(E_F\) for \(E\) in 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 \(e^0 = 1\):
\[ f(E_F) = \frac{1}{1 + 1} = \frac{1}{2} \]
This result is valid for all temperatures \(T>0\) K.
Step 4: Conclusion:
The Fermi level \(E_F\) represents the energy level with a 1/2 probability of electron occupation at any temperature above absolute zero.