The conductivity of a semiconductor increases with increase in temperature because

Options

A. (a) number density of free current carries increases
B. (b) relaxation time increases
C. both number density of carriers and relaxation time decreases but effect of decrease in relaxation time is much less than increase in number density .
D. (d) number density of current carriers increases, relaxation time decreases but effect of decrease in relaxation time is much less than increases in number density

Explanation

To understand why the conductivity of a semiconductor increases with an increase in temperature, we can break down the explanation into a few key concepts and steps:

### Step 1: Understanding Conductivity
Conductivity (\(\sigma\)) of a material is defined as the ability of that material to conduct electric current. It is influenced by two main factors: the number density of charge carriers (electrons or holes) and the mobility of these carriers.

### Step 2: Number Density of Charge Carriers
In semiconductors, the number density (\(n\)) refers to the number of free charge carriers (electrons in the conduction band or holes in the valence band) per unit volume. As the temperature increases, more electrons gain enough energy to jump from the valence band to the conduction band, increasing the number density of free charge carriers.

### Step 3: Effect of Temperature on Number Density
When the temperature of a semiconductor is raised:
- More electrons are thermally excited from the valence band to the conduction band.
- This results in an increase in the number density of free charge carriers (\(n\)).

### Step 4: Relaxation Time
Relaxation time (\(\tau\)) is the average time between successive collisions of charge carriers. In semiconductors, as temperature increases, the atoms in the lattice vibrate more vigorously, which can lead to more frequent collisions. Thus, the relaxation time tends to decrease with increasing temperature.

### Step 5: Overall Effect on Conductivity
The conductivity of a semiconductor can be expressed as:
\[ \sigma = n \cdot e \cdot \mu \]
where:
- \(n\) = number density of charge carriers
- \(e\) = charge of an electron
- \(\mu\) = mobility of the charge carriers

Although the relaxation time decreases with increasing temperature, the increase in the number density of charge carriers has a more significant effect on conductivity. Therefore, the overall conductivity of the semiconductor increases with temperature.

### Conclusion
In summary, the conductivity of a semiconductor increases with an increase in temperature because:
1. The number density of free current carriers increases due to thermal excitation.
2. Although the relaxation time decreases, the increase in number density has a more dominant effect on conductivity.