Carrier generation–recombination balance in an intrinsic semiconductor at steady temperature At a fixed temperature T, which statement correctly describes the rates of thermal electron–hole pair generation and recombination in an intrinsic (undoped) semiconductor under steady conditions?

Introduction / Context:
Understanding intrinsic semiconductors requires grasping the dynamic equilibrium between thermal generation of carriers and their recombination. Steady-state conditions at a fixed temperature mean that average carrier concentrations remain constant in time, which constrains the relationship between these opposing processes.

Given Data / Assumptions:

• Material is intrinsic (no doping, no illumination).
• Temperature T is fixed and uniform.
• No external injection currents or optical generation; only thermal processes operate.

Concept / Approach:
In steady state, carrier concentrations n and p are time-invariant. Therefore, the net rate of change dn/dt and dp/dt must be zero. This implies that the thermal generation rate G_th equals the recombination rate R for both electrons and holes. Any persistent imbalance would produce a time evolution of n and p, contradicting steady state.

Step-by-Step Solution:
Let G_th be thermal generation rate and R be recombination rate.Steady state condition: dn/dt = G_th − R = 0 and dp/dt = G_th − R = 0.Thus, G_th = R; carrier concentrations remain constant at their intrinsic values n_i and p_i with n_i = p_i.

Verification / Alternative check:
Perturbation from steady state (e.g., sudden illumination) creates a transient where G − R ≠ 0, but as the system relaxes, recombination and generation re-balance and steady concentrations are re-established at the given temperature.

Why Other Options Are Wrong:
“Always lower” or “always higher” would lead to monotonic decay or growth of carrier density, violating steady-state conditions. “Any of the above” disregards the equilibrium requirement. “Generation ceases” is unphysical at finite temperature; thermal generation is continuous.

Common Pitfalls:

• Confusing steady state (time-independent) with thermal equilibrium; both enforce G = R for intrinsic conditions.
• Assuming recombination dominates at low temperature; while absolute rates change, equality holds at steady state.

Final Answer:
Generation rate equals recombination rate at steady state