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: Concept Overview:
The effective density of states in the conduction band, denoted as \(N_c\), simplifies calculations of free electron concentration. It represents the equivalent density of available states concentrated at the conduction band edge energy, \(E_c\).
Step 2: Formula:
The effective density of states in the conduction band is calculated using the following formula:
\[ N_c = 2 \left( \frac{2\pi m_e^ k_B T}{h^2} \right)^{3/2} \]
where \(m_e^ \) represents 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: Temperature Dependence:
The formula shows that for a specific semiconductor material, all parameters except temperature (T) are constant. Consequently, the effective density of states \(N_c\) is directly related to temperature as:
\[ N_c \propto T^{3/2} \]
The effective density of states in the valence band, \(N_v\), exhibits the same \(T^{3/2}\) temperature dependence.
Step 4: Conclusion:
The effective density of states for electrons (\(N_c\)) scales with temperature as \(T^{3/2}\).