Why does the electrical resistivity of metallic conductors increase with rising temperature, whereas the resistivity of intrinsic semiconductors decreases exponentially?

Answer and explanation

Correct answer: D. In metals, lattice vibrations reduce relaxation time τ; in semiconductors, carrier density n surges

  1. Step 1: Resistivity is given by the Drude formula: .
  2. Step 2: In metals, the free electron density is already saturated and constant (~). As temperature rises, amplitude of lattice ion vibrations increases, increasing electron-ion collision frequency and drastically reducing the relaxation time , so increases (positive temperature coefficient ).
  3. Step 3: In semiconductors, the electron-hole density is small at room temperature. Heating provides thermal energy across the small band gap (), causing an exponential surge in carrier density , which completely outweighs the minor decrease in , resulting in a steep drop in resistivity (negative temperature coefficient ).
Answer
In metals, increased thermal lattice vibrations reduce electron relaxation time (τ), while in semiconductors, thermal excitation creates a massive increase in charge carrier density (n)
Why not the others: 'Metals lose all free electrons...' is false; 'Length expands tenfold...' vastly exaggerates thermal expansion; 'Metals undergo radioactive transmutation...' is scientifically absurd.

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