The number of holes in an intrinsic semiconductor

Analysis

This question tests your understanding of the fundamental properties of intrinsic semiconductors (pure semiconductors like Silicon or Germanium) at thermal equilibrium.

Concept Explanation

An intrinsic semiconductor is a pure material with no significant dopant impurities.

1. Generation: When thermal energy is applied to an intrinsic semiconductor, electrons gain enough energy to break their covalent bonds and move into the conduction band, leaving behind a "hole" in the valence band.
2. Pair Production: This process is called electron-hole pair generation. Because every time an electron is promoted to the conduction band, exactly one hole is created in the valence band, the rate of electron generation is equal to the rate of hole generation.
3. Equilibrium: In an intrinsic semiconductor, the concentration of free electrons ($n$) is always equal to the concentration of holes ($p$). Mathematically, this is expressed as: $$n = p = n_i$$where$n_i$ is the intrinsic carrier concentration.

Step-by-Step Solution

1. Identify the material: The question specifies an intrinsic semiconductor.
2. Apply the principle: In pure (intrinsic) semiconductors, charge carriers are created only by thermal excitation.
3. Relate the carriers: Since each excitation creates one free electron and one hole simultaneously, the number of free electrons must equal the number of holes.
4. Evaluate the options: • A is incorrect because they are equal. • B is incorrect because holes do not outnumber electrons in an intrinsic material. • C is correct because $n = p$. • D is incorrect because electrons do not outnumber holes in an intrinsic material.

Conclusion

In an intrinsic semiconductor, the number of holes is exactly equal to the number of free electrons because they are generated in pairs.

Exam Tip: If the semiconductor were extrinsic (doped), this would change! In an n-type semiconductor, electrons are the majority carriers, and in a p-type semiconductor, holes are the majority carriers. Always check if the question says "intrinsic" or "doped."