The temperature (T) dependence of resistivity (rho) of a semiconductor is represented by :

Options

A.
B.
C.
D.

Answer

To understand the temperature dependence of resistivity (\(\rho\)) of a semiconductor, we can break down the problem step by step.

### Step-by-Step Solution:

1. **Understanding Resistivity and Temperature**:
The resistivity of a semiconductor changes with temperature. The relationship can be expressed as:
\[ \Delta \rho = \rho_0 \alpha \Delta T \]
where:
- \(\Delta \rho\) is the change in resistivity,
- \(\rho_0\) is the initial resistivity,
- \(\alpha\) is the temperature coefficient of resistivity,
- \(\Delta T\) is the change in temperature.

2. **Expressing the Change in Resistivity**:
We can express the change in resistivity in terms of initial and final temperatures:
\[ \Delta \rho = \rho(T) - \rho_0 = \rho_0 \alpha (T - T_0) \]
Rearranging gives:
\[ \rho(T) = \rho_0 + \rho_0 \alpha (T - T_0) \]

3. **Identifying the Linear Relationship**:
The equation can be rewritten as:
\[ \rho(T) = \rho_0(1 + \alpha (T - T_0)) \]
This shows that resistivity is a linear function of temperature, where:
- The slope of the line is \(\rho_0 \alpha\),
- The y-intercept is \(\rho_0\).

4. **Sign of the Temperature Coefficient**:
For semiconductors, the temperature coefficient \(\alpha\) is typically negative. This means that as the temperature increases, the resistivity decreases, leading to a negative slope in the graph of resistivity versus temperature.

5. **Graphical Representation**:
Since \(\alpha\) is negative, the graph of \(\rho\) versus \(T\) will have a negative slope. This indicates that the resistivity decreases with an increase in temperature.

6. **Conclusion**:
Based on the analysis, the correct representation of the temperature dependence of resistivity for a semiconductor will show a downward slope in the graph.

### Final Answer:
The correct option for the temperature dependence of resistivity of a semiconductor is the one that shows a negative slope.