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The temperature dependence of the electrical conductivity \((\sigma)\) of three intrinsic semiconductors A, B and C is shown in figure.

Plot of ln(sigma) against 1/T: three falling straight lines; A is the steepest, C is less steep and B is the least steep

Let EA, EB and EC be the bandgaps of A, B and C, respectively. Which one of the following relations is correct?

  • A.E\(_{C}\) \(\gt\) E\(_{A}\) \(\gt\) E\(_{B}\)
  • B.E\(_{B}\) \(\gt\) E\(_{C}\) \(\gt\) E\(_{A}\)
  • C.E\(_{A}\) \(\gt\) E\(_{B}\) \(\gt\) E\(_{C}\)
  • D.E\(_{A}\) \(\gt\) E\(_{C}\) \(\gt\) E\(_{B}\)

Answer: D

The bandgaps satisfy EA > EC > EB.

For an intrinsic semiconductor the conductivity varies as

\[ \begin{aligned} \sigma &\propto e^{-E_{g}/2k_{B}T} \\ \ln\sigma &= \text{const} - \frac{E_{g}}{2k_{B}}\cdot\frac{1}{T} \end{aligned} \]

so on a plot of ln σ against 1/T the slope is -Eg/2kB. The steeper the line, the larger the bandgap. Line A is the steepest, C next and B the least steep.

The other orders mis-rank the slopes: putting C first, B first, or B above C would all require B or C to fall faster than the drawing shows.

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