Question

The effective density of states of electrons (\(N_c\)) at the conduction band edge of the intrinsic semiconductor varies with temperature, as:

• A. \(T^{2/3}\)
• B. \(T^{3/2}\)
• C. \(T^{4/3}\)
• D. \(T^{5/2}\)

Hint:

Remember the temperature dependencies of key semiconductor parameters:

Effective density of states (\(N_c, N_v\)): \(\propto T^{3/2}\)
Intrinsic carrier concentration (\(n_i\)): \(\propto T^{3/2} \exp(-E_g / 2k_B T)\) (The exponential term dominates).
Mobility (\(\mu\)): \(\propto T^{-3/2}\) (due to lattice scattering).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The effective density of states (\(N_c\)) in the conduction band is a parameter that simplifies the calculation of the total number of free electrons. It represents the density of available states if they were all located at the conduction band edge energy, \(E_c\).
Step 2: Key Formula or Approach:
The formula for the effective density of states in the conduction band is derived by integrating the density of states function multiplied by the probability of occupation over all energies in the band. The result is:
\[ N_c = 2 \left( \frac{2\pi m_e^ k_B T}{h^2} \right)^{3/2} \]
where \(m_e^ \) is the effective mass of the electron, \(k_B\) is the Boltzmann constant, T is the absolute temperature, and h is the Planck constant.
Step 3: Detailed Explanation:
From the formula, we can see that all terms except for the temperature T are constants for a given semiconductor material. Therefore, the effective density of states \(N_c\) has a direct dependence on temperature:
\[ N_c \propto T^{3/2} \]
Similarly, the effective density of states in the valence band, \(N_v\), also has a \(T^{3/2}\) dependence.
Step 4: Final Answer:
The effective density of states of electrons (\(N_c\)) varies with temperature as \(T^{3/2}\).