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 Definition
Semiconductor resistivity \( \rho \) is defined by the equation: \[ \rho = \frac{m}{n e^2 \tau} \] In this formula: - \( m \) represents electron mass, - \( n \) denotes the number density of charge carriers, - \( e \) signifies the electron charge, - \( \tau \) indicates the relaxation time. 
Step 2: Temperature's Impact on Resistivity 
With rising temperature in semiconductors: The charge carrier density \( n \) experiences a substantial increase due to thermal excitation.  The relaxation time \( \tau \) declines as scattering intensifies. 
The dominant factor is the increase in \( n \), which outweighs the reduction in \( \tau \), resulting in an overall decrease in resistivity. 
Step 3: Selecting the Appropriate Representation 
Given that semiconductor resistivity exhibits an exponential decrease with increasing temperature, the correct graphical representation must depict a sharp downward trend. The provided image confirms that Curve (b) accurately illustrates this relationship. Final Determination: Curve (b) correctly depicts the relationship between resistivity and temperature in a semiconductor.