Intrinsic semiconductor at room temperature — comparing conducting electrons to valence electrons In an intrinsic (undoped) semiconductor at room temperature, how does the number of conducting electrons compare with the total number of valence electrons present in the crystal lattice?

Introduction / Context:
This question checks conceptual understanding of intrinsic (pure, undoped) semiconductors such as silicon and germanium at room temperature. The focus is on the relative populations: the total valence electrons bound in covalent bonds versus the much smaller population of thermally excited carriers (electrons and holes) that participate in conduction.

Given Data / Assumptions:

• Material is intrinsic (no intentional doping).
• Temperature is around room temperature (approximately 300 K).
• Electrons are promoted from the valence band to the conduction band by thermal energy, creating electron–hole pairs.

Concept / Approach:
In an intrinsic semiconductor, the number of electrons in the conduction band equals the number of holes in the valence band, denoted by n_i. The vast majority of electrons remain in valence-band bonding states. Because the band gap is on the order of electron-volts, only a tiny fraction of electrons has sufficient thermal energy to cross the gap at 300 K. Thus, n_i is many orders of magnitude smaller than the total number of valence electrons per unit volume.

Step-by-Step Solution:
Recall: intrinsic carrier concentration n_i for Si at 300 K is roughly 10^10 cm^−3; for Ge it is roughly 10^13 cm^−3 (order of magnitude).Compare with atomic density ~ 5 × 10^22 cm^−3 and four valence electrons per atom for Si/Ge → total valence electrons ~ 2 × 10^23 cm^−3.Compute fraction: n_i / (valence electrons) ~ 10^10 / 10^23 = 10^−13 for Si (extremely small).Therefore, the thermally excited conducting electrons constitute a very small fraction of the total valence electrons.

Verification / Alternative check:
The exponential dependence n_i ∝ exp(−E_g / (2kT)) ensures that for band gaps of ~1 eV at 300 K, n_i is tiny compared to the density of valence electrons. This holds broadly for common semiconductors.

Why Other Options Are Wrong:
“Almost equal” or “about half” ignore the band-gap barrier. “Small as compared to” is imprecise; the correct emphasis is “a very small fraction.” “Exactly zero” is incorrect because finite temperature always generates some carriers.

Common Pitfalls:

• Confusing intrinsic with extrinsic (doped) behavior.
• Underestimating the effect of the exponential dependence on band gap.

Final Answer:
Is a very small fraction of the number of valence electrons