Question

In a semiconductor, intrinsic concentration of charge carriers varies with:

• A. \( T^{1/2} \)
• B. \( T \)
• C. \( T^{3/2} \)
• D. \( T^{-1/2} \)

Hint:

While the exponential term \(\exp(-E_g / 2k_B T)\) causes the most significant change in carrier concentration with temperature, the pre-exponential \(T^{3/2}\) term is also a fundamental part of the relationship derived from the density of states. Always check if this term is among the options.

Answer 1

The Correct Option is C

Step 1: State the formula for intrinsic carrier concentration (\(n_i\)).The intrinsic carrier concentration (\(n_i\)) in a semiconductor is given by \(n_i = A T^{3/2} \exp\left(-\frac{E_g}{2k_B T}\right)\), where \(A\) is a material constant, \(T\) is temperature, \(E_g\) is the energy bandgap, and \(k_B\) is the Boltzmann constant.
Step 2: Examine the temperature dependence.The formula for \(n_i\) has two temperature-dependent components: a pre-exponential \(T^{3/2}\) term and an exponential term \(\exp(-E_g / 2k_B T)\). The exponential term is the primary driver of temperature dependence, but the question asks for the *variation with* \(T\), which includes the pre-exponential \(T^{3/2}\) factor.
Step 3: Match with provided options.Since the options are power-law dependencies on \(T\), and the pre-exponential factor in the formula for \(n_i\) is \(T^{3/2}\), this aligns with option (3).