Question Analysis: Assertion-Reason type question based on Semiconductor Physics (NCERT Class 12, Chapter 14).
Step 1 (Governing Principle): [0.25 Mark]
The electrical conductivity ($\sigma$) of a semiconductor is given by the relation:
$$\sigma = e(n_e \mu_e + n_h \mu_h)$$
where $n_e$ and $n_h$ are the number densities of electrons and holes, and $\mu_e$ and $\mu_h$ are their respective mobilities.
Step 2 (Step-by-Step Logical Deduction): [0.50 Mark]
- In a pure semiconductor (Ge), the energy gap ($E_g$) is small. As temperature ($T$) increases, thermal energy ($k_BT$) allows more electrons to jump from the valence band to the conduction band.
- This thermal excitation leads to an exponential increase in charge carrier concentration ($n_e$ and $n_h$) according to the relation: $$n_i \propto T^{3/2} e^{-E_g / 2k_BT}$$
- Although mobility ($\mu$) decreases slightly with temperature due to increased lattice scattering, the exponential increase in carrier concentration ($n_e, n_h$) dominates, leading to a net increase in conductivity ($\sigma$).
Step 3 (Final Concluded Answer): [0.25 Mark]
Correct Option: Both (A) and (R) are true and (R) is the correct explanation of (A).
Since the increase in the number of charge carriers outweighs the decrease in mobility, the conductivity of Ge increases with temperature.
💡 EL10 Concept Insight: Think of a semiconductor like a crowded room where people (electrons) are stuck in their seats (valence band); as the room gets hotter (higher temperature), people get enough energy to jump up and start moving around, which allows electricity to flow much more easily!