Question

The resistivity (\( \rho \)) of a semiconductor varies with temperature. Which of the following curves represents the correct behavior?

• A. Curve (a)
• B. Curve (b)
• C. Curve (c)
• D. Curve (d)

Hint:

In semiconductors, as temperature increases, the number of charge carriers increases rapidly, leading to an exponential decrease in resistivity. This behavior is opposite to that of metals, where resistivity increases with temperature due to electron scattering.

Answer 1

The Correct Option is B

Step 1: Semiconductor Resistivity Formula
Semiconductor resistivity \( \rho \) is defined as: \[ \rho = \frac{m}{n e^2 \tau} \] \( m \) = electron mass, \( n \) = charge carrier density, \( e \) = electron charge, \( \tau \) = relaxation time. 
Step 2: Temperature Impact on Resistivity 
For semiconductors, rising temperatures: Increase charge carrier density \( n \) substantially via thermal excitation. Decrease relaxation time \( \tau \) due to heightened scattering. 
The rise in \( n \) has a greater effect than the drop in \( \tau \), causing overall resistivity reduction. 
Step 3: Curve Selection Rationale 
Semiconductor resistivity drops exponentially with temperature. Thus, the correct curve must exhibit a sharp decline. The provided image indicates Curve (b) matches this behavior. Final Answer: Curve (b) correctly illustrates the relationship between resistivity and temperature in a semiconductor.