In solid-state physics, consider a good electrical conductor (metal) at room temperature: how many “holes” are effectively present compared with free electrons?

Introduction / Context:
In semiconductors, we routinely speak of two types of mobile charge carriers: electrons in the conduction band and holes in the valence band. In metals (good conductors), the situation is different: the conduction band is partially filled, and electrical conduction is dominated by electrons. Understanding the relative presence of “holes” in a metal versus electrons clarifies why metals behave differently from semiconductors and why device physics uses different models for each material class.

Given Data / Assumptions:

• Material: a typical good conductor (e.g., copper, aluminum) at room temperature.
• Carriers in metals are overwhelmingly electrons.
• “Holes” here refers to the semiconductor concept of missing electrons in the valence band acting as positive carriers.

Concept / Approach:
A metal’s Fermi level lies within a conduction band (or overlaps bands), so many electronic states are available near that energy. Electrical conduction is well described by an electron gas model (Drude or free-electron model). The “hole” concept is not a dominant or convenient description in metals at room temperature. If one insists on describing positive effective carriers, their density is negligible compared with the enormous concentration of conduction electrons. Hence, the most accurate comparative statement is that holes are far fewer than free electrons in a metal at room temperature.

Step-by-Step Solution:
Identify material class: metal (conductor), not a doped semiconductor.Recall carrier picture: electron-dominated conduction; valence-band holes are not the primary carriers.Compare populations: hole density ≪ electron density in metals at room temperature.Select the option that states “less than the free electrons.”

Verification / Alternative check:
Contrast with intrinsic silicon: n ≈ p at a given temperature. In a metal, the free-electron density (~10^28 m^-3) dwarfs any meaningful “hole” population used in semiconductor models, validating the selection.

Why Other Options Are Wrong:
Same number as free electrons: true only for intrinsic semiconductors (n = p), not metals. “many”: too vague and misleading; does not reflect the dominance of electrons. “Only those produced by thermal energy”: suggests a semiconductor-like intrinsic generation picture, not appropriate for metals. “None of the above” is incorrect because a correct comparative statement exists.

Common Pitfalls:
Assuming the n = p relation from intrinsic semiconductors carries over to metals; overgeneralizing the “hole” concept beyond its useful semiconductor context. Metals conduct primarily through electrons in partially filled bands.

Final Answer:
less than the free electrons