Question
Why does the resistance of an intrinsic semiconductor decrease with temperature, while the resistance of a metal increases? Aren’t both just made of atoms with electrons?
Solution — Step by Step
In a metal, the number of free electrons is roughly fixed — almost every atom contributes one. Heating doesn’t create new carriers.
In an intrinsic semiconductor, free electrons exist only when thermal energy excites them across the band gap (). Heating dramatically increases the carrier count.
In metals, hotter lattice atoms vibrate more, scattering the (fixed-count) electrons. Mean free path drops, mobility drops, resistance rises.
In semiconductors, scattering also increases with temperature, reducing mobility. But the carrier count grows exponentially with via . Carrier-count growth dominates scattering, so net resistance falls sharply.
Metals: with fixed, falls with , so rises.
Semiconductors: rises exponentially, dominating , so falls.
Final answer: The temperature dependence of resistance is opposite because metals have fixed carriers while semiconductors must thermally generate theirs.
Why This Works
The single equation explains both behaviours. The trick is recognising which factor — or — dominates the temperature response.
For metals, is set by chemistry, not heat. For semiconductors, is set by Boltzmann statistics, exponentially temperature-sensitive.
Alternative Method
Look at the band-gap picture. Metals have overlapping conduction and valence bands — no gap to cross. Semiconductors have a real gap, and only thermal excitation populates the conduction band.
JEE Main 2023 asked: “If the band gap doubles, by what factor does change at fixed ?” Answer: , so the ratio is , a decrease.
Common Mistake
Saying “semiconductors conduct better when hot because electrons move faster”. Wrong reason — carrier mobility actually decreases. The real cause is the surge in carrier count.