For an intrinsic semiconductor at thermal equilibrium, how do the numbers of free electrons and holes compare?

Introduction / Context:
Carrier balance in intrinsic semiconductors underpins diode and transistor operation. Intrinsic material is chemically pure, without intentional doping. Knowing the equality of electrons and holes is foundational for interpreting conductivity, Fermi level position, and temperature behavior.

Given Data / Assumptions:

• Material is intrinsic (no donor or acceptor impurities added).
• Thermal equilibrium conditions apply.
• Temperature is uniform.

Concept / Approach:
In intrinsic semiconductors, thermal generation creates electron-hole pairs. Each generation event produces one electron and one hole. Recombination removes them in pairs. At equilibrium, the intrinsic carrier concentration n_i satisfies n = p = n_i. Therefore, the number of free electrons equals the number of holes.

Step-by-Step Solution:
1) Recognize intrinsic, undoped material.2) Generation produces electron-hole pairs in equal numbers.3) At equilibrium, rates of generation and recombination are equal, keeping n and p equal.4) Conclude n = p, that is, electrons equal holes.

Verification / Alternative check:
Band theory places the Fermi level near mid-gap for intrinsic material, consistent with symmetric electron and hole populations in the conduction and valence bands at equilibrium.

Why Other Options Are Wrong:

• Less than or greater than: That occurs in doped n-type or p-type materials, not intrinsic.
• Impossible to say: It is well defined by intrinsic physics.
• None of the above: Incorrect because equality is correct.

Common Pitfalls:
Confusing intrinsic with lightly doped behavior leads to wrong conclusions. Always check whether the material is doped or pure when comparing carrier populations.

Final Answer:
Equals the number of holes.