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.
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ABoth A and R are true and R is the correct explanation of A
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BBoth A and R are true but R is not the correct explanation of A
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CA is true but R is false
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DA is false but R is true
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EBoth A and R are false
Answer
Correct Answer: Both A and R are true but R is not the correct explanation of A
Explanation
Introduction / Context:The mass–action law n p = ni^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; ni is intrinsic concentration.
- p-type doping uses acceptor (trivalent) impurities in group-IV semiconductors like Si/Ge.
Concept / Approach:
Mass–action law: n p = ni^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:
Evaluate A: n p = ni^2 → true in thermal equilibrium.Evaluate R: p-type via trivalent dopants → true statement.Causal link: R does not explain A; doping type does not derive mass–action law.Therefore, the correct choice is “both true, but R not the explanation”.Verification / Alternative check:
Textbook derivations from Fermi statistics show n = NC exp [−(EC − EF)/kT], p = NV exp [−(EF − EV)/kT]; multiplying yields n p = NCNV exp [−(EC − EV)/kT] = ni^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