Assertion–Reason (Semiconductors) Assertion (A): For any semiconductor in thermal equilibrium, the carrier concentrations satisfy n p = n_i^2. Reason (R): A p-type semiconductor is obtained by adding a trivalent impurity to intrinsic material.

Introduction / Context:
The mass–action law n p = n_i^2 is a cornerstone of equilibrium semiconductor physics, relating electron and hole concentrations at a given temperature. Doping determines whether the material is p-type or n-type but does not invalidate the equilibrium relation itself. This question checks clarity on these two separate ideas.

Given Data / Assumptions:

• Thermal equilibrium (no external excitation or strong injection).
• n and p denote electron and hole concentrations; n_i is intrinsic concentration.
• p-type doping uses acceptor (trivalent) impurities in group-IV semiconductors like Si/Ge.

Concept / Approach:

Mass–action law: n p = n_i^2 arises from detailed balance between generation and recombination processes and the position of Fermi level. Doping shifts individual values of n and p but their product remains fixed at equilibrium. Meanwhile, p-type doping via trivalent atoms (e.g., B in Si) creates acceptor states leading to hole majority carriers.

Step-by-Step Solution:

Verification / Alternative check:

Textbook derivations from Fermi statistics show n = N_C exp [−(E_C − E_F)/kT], p = N_V exp [−(E_F − E_V)/kT]; multiplying yields n p = N_CN_V exp [−(E_C − E_V)/kT] = n_i^2.

Why Other Options Are Wrong:

• A true, R false: R is factually correct.
• R explains A: incorrect linkage.
• A false: contradicts equilibrium semiconductor theory.

Common Pitfalls:

Believing that heavy doping breaks the mass–action law; at equilibrium it still holds (with bandgap narrowing caveats at extreme levels, but the principle remains).

Final Answer:

Both A and R are true but R is not the correct explanation of A