Which of the following graph correctly represents the variation of resistivity 's' with temperature 'T' for a semiconductor material ?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3)
In a semiconductor the number density of charge carriers \( n \) rises very rapidly with temperature, because more electrons are freed from the valence band. The resistivity is
\( \displaystyle \rho = \frac{m}{n e^{2}\tau} \)
The rise in \( n \) is much larger than the fall in the relaxation time \( \tau \), so the resistivity decreases with temperature (a negative temperature coefficient). It is very large at low temperatures and falls steeply, flattening out at high temperatures. Only graph (3) has this shape.
Why the other options are incorrect
Option (1) — Incorrect. This curve rises sharply with temperature; the resistivity of a semiconductor falls as \( T \) increases.
Option (2) — Incorrect. A straight line rising with \( T \) is the behaviour of a metal (conductor) over a limited range of temperature.
Option (4) — Incorrect. This curve rises with \( T \) and then levels off; a semiconductor's resistivity decreases with temperature.
NCERT (2026-27):
Temperature Dependence of Resistivity – “… Unlike metals, the resistivities of semiconductors decrease with increasing temperatures.”
Temperature Dependence of Resistivity – “… For insulators and semiconductors, however, n increases with temperature. This increase more than compensates any decrease in \( \tau \) in Eq.(3.23) so that for such materials, ρ decreases with temperature.”