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: Recall the formula for intrinsic carrier concentration (\(n_i\)). The intrinsic carrier concentration in a semiconductor is a function of temperature \(T\) and the energy bandgap \(E_g\). The standard formula is: \[ n_i = A T^{3/2} \exp\left(-\frac{E_g}{2k_B T}\right) \] where \(A\) is a material-specific constant and \(k_B\) is the Boltzmann constant.
Step 2: Analyze the temperature dependence. The formula shows that \(n_i\) depends on temperature in two ways: through the \(T^{3/2}\) term and the exponential term \(\exp(-E_g / 2k_B T)\). The exponential term describes the dominant temperature dependence, but the question asks how the concentration *varies with* T, and the pre-exponential factor \(T^{3/2}\) is a key part of this variation.
Step 3: Compare with the given options. The options provided are power-law dependencies on \(T\). The pre-exponential factor in the expression for \(n_i\) is \(T^{3/2}\), which matches option (3).