Doped semiconductors at room temperature: If germanium and silicon samples have the same impurity (dopant) density, how do their resistivities compare at room temperature?

Introduction / Context:
Resistivity ρ of a doped semiconductor depends on free carrier concentration and mobility: ρ = 1 / (q (n μ_n + p μ_p)). For the same dopant density, the majority carrier concentration is similar in magnitude, so material-dependent mobility and intrinsic properties determine relative resistivity at room temperature.

Given Data / Assumptions:

• Equal impurity concentration in germanium (Ge) and silicon (Si).
• Room temperature conditions.
• Standard mobility values: carriers in Ge generally have higher mobilities than in Si.

Concept / Approach:

Because Ge typically exhibits higher carrier mobility than Si, the conductivity σ = q n μ (for dominant carrier type) is greater in Ge at the same carrier concentration. Therefore, resistivity ρ = 1/σ is lower in Ge and correspondingly higher in Si.

Step-by-Step Solution:

Verification / Alternative check:

Empirical mobility data: μ_n and μ_p in Ge are larger than in Si at 300 K, matching observed lower resistivity in Ge for equal doping levels.

Why Other Options Are Wrong:

• Equal resistivity: ignores mobility differences.
• Negative resistivity: unphysical in this context.
• Ge higher than Si contradicts mobility trends.

Common Pitfalls:

• Confusing intrinsic carrier concentration effects with doped majority-carrier regimes; at typical doping, dopants dominate.

Final Answer:

resistivity of silicon will be higher than that of germanium