In a semiconductor, the probability of electron–hole recombination is proportional to which carrier measure? Choose the best expression for how recombination rate depends on carrier concentrations.

Introduction / Context:
Carrier recombination is central to semiconductor devices (diodes, LEDs, solar cells). The rate at which electrons and holes annihilate each other determines lifetime, minority-carrier dynamics, and device efficiency. A basic qualitative rule is that recombination requires both species to be present: electrons and holes must meet to recombine.

Given Data / Assumptions:

• Uniform semiconductor material at steady conditions.
• Nonradiative and radiative recombination mechanisms are possible.
• We focus on the basic concentration dependence.

Concept / Approach:

In its simplest form, the recombination rate R is proportional to the probability of an electron meeting a hole, which scales with the product of their concentrations, n and p. Many models express this as R ∝ n p − n_i^2 under nonequilibrium conditions, where n_i is the intrinsic concentration. Thus, higher density of either species alone is not sufficient; both must be present for recombination events to occur.

Step-by-Step Solution:

Verification / Alternative check:

Device equations (e.g., low-level injection in p–n junctions) commonly use recombination terms proportional to n p − n_i^2 with appropriate coefficients for SRH, radiative, or Auger processes, reinforcing the product dependence.

Why Other Options Are Wrong:

• Electrons only or holes only: ignores the need for the other carrier.
• Sum (n + p): does not represent pairwise interaction probabilities.
• None of the above: incorrect because n * p dependence is well-established.

Common Pitfalls:

Assuming linear dependence on a single carrier or forgetting the equilibrium correction term n p ≈ n_i^2 in intrinsic conditions.

Final Answer:

Product of electron and hole densities (n * p)