Extrinsic semiconductor at higher temperature: if the intrinsic carrier concentration doubles, what happens to majority and minority carrier densities?

Difficulty: Medium

The majority carrier density is doubled
The minority carrier density is doubled
The minority carrier density becomes 4 times the original value
Both majority and minority carrier densities double
Both reduce by half due to recombination

Correct Answer: The minority carrier density becomes 4 times the original value

Explanation:

Introduction / Context:
In doped semiconductors, temperature raises intrinsic carrier concentration n_i, which influences minority carriers strongly through the mass-action law. Understanding this interplay is critical when predicting leakage currents and designing devices for high-temperature environments.

Given Data / Assumptions:

Extrinsic semiconductor (assume n-type for concreteness) with donor density N_D and n_i ≪ N_D initially.
Temperature increases so that n_i → 2 n_i.
Low-level injection; complete ionization of dopants; equilibrium conditions apply.

Concept / Approach:
The mass-action law states n * p = n_i^2 at equilibrium. In a strongly n-type sample, the majority concentration n ≈ N_D (weakly affected by modest changes in n_i), while the minority concentration p ≈ n_i^2 / n ≈ n_i^2 / N_D. If n_i doubles, p scales with n_i^2 and therefore quadruples. Majority carriers remain essentially set by ionized donors until very high temperatures push the material toward intrinsic behavior.

Step-by-Step Solution:

Initial: p_1 ≈ n_i^2 / N_D.After heating: n_i → 2 n_i ⇒ p_2 ≈ (2 n_i)^2 / N_D = 4 n_i^2 / N_D.Therefore p_2 / p_1 = 4; the minority concentration quadruples.Majority concentration remains ≈ N_D for moderate temperature rise.

Verification / Alternative check:

Check limiting case: if n begins to deviate from N_D, the material tends to intrinsic and both carriers approach n_i; the stated result holds before that limit.

Why Other Options Are Wrong:

Doubling either density ignores the n_i^2 dependence of the minority carriers.Majority carriers do not double unless the material becomes intrinsic.

Common Pitfalls:

Forgetting the square dependence p ∝ n_i^2 in extrinsic regimes; assuming symmetric changes for majority and minority carriers.

Final Answer:

The minority carrier density becomes 4 times the original value

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Extrinsic semiconductor at higher temperature: if the intrinsic carrier concentration doubles, what happens to majority and minority carrier densities?

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