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: Understanding Resistivity in Semiconductors 
The resistivity \( \rho \) of a semiconductor is given by: \[ \rho = \frac{m}{n e^2 \tau} \] where: - \( m \) is the electron mass, - \( n \) is the number density of charge carriers, - \( e \) is the charge of an electron, - \( \tau \) is the relaxation time. 
Step 2: Effect of Temperature on Resistivity 
In semiconductors, as temperature increases: The number density \( n \) of charge carriers increases significantly due to thermal excitation.  The relaxation time \( \tau \) decreases due to increased scattering. 
However, the increase in \( n \) dominates over the decrease in \( \tau \), leading to a net decrease in resistivity. 
Step 3: Choosing the Correct Curve 
Since resistivity decreases exponentially with increasing temperature in a semiconductor, the correct curve must show a steep downward trend. The given image confirms that Curve (b) represents this behavior. Final Answer: The correct behavior of resistivity with temperature in a semiconductor is represented by Curve (b).