Mobility of free electrons in a semiconductor depends on temperature variation:

The mobility of free electrons in a semiconductor refers to the ease with which they can move through the crystal lattice of the material. This mobility is influenced by various factors, including temperature.

Inversely proportional relationship:
The correct answer is option 'B' - inversely proportional. This means that as the temperature increases, the mobility of free electrons decreases, and vice versa. Let's understand this relationship in more detail.

Reasoning:
The mobility of free electrons in a semiconductor is influenced by scattering mechanisms within the crystal lattice. These scattering mechanisms include impurities, defects, and lattice vibrations. Temperature affects the scattering mechanisms and, consequently, the mobility of electrons.

Relationship between temperature and scattering mechanisms:
- At low temperatures, the scattering mechanisms are less active, and the crystal lattice is relatively stable. As a result, the mobility of free electrons is high because they experience fewer obstacles in their movement.
- As the temperature increases, the scattering mechanisms become more active. The increased thermal energy causes lattice vibrations to become more pronounced, which leads to more scattering of electrons. This scattering reduces the mobility of free electrons.

Quantitative relationship:
The relationship between temperature and mobility of free electrons can be described by the Matthiessen's rule, which states that the total scattering rate is the sum of individual scattering rates due to different mechanisms. The total scattering rate is given by:

1/μ = 1/μ_impurity + 1/μ_defect + 1/μ_vibration

where μ represents the mobility of free electrons, and the subscripts indicate the scattering mechanisms.

Conclusion:
In conclusion, the mobility of free electrons in a semiconductor is inversely proportional to temperature. As the temperature increases, the scattering mechanisms become more active, leading to more obstacles for the free electrons to overcome. This reduces their mobility within the crystal lattice. Conversely, at lower temperatures, the mobility of free electrons is higher due to decreased scattering.