GATE 2024 EC Q.22: solution and answer

GATE 2024 · ECQ.22 · 1 mark · MCQ

For non-degenerately doped n-type silicon, which one of the following plots represents the temperature ( $T$ ) dependence of free electron concentration ( $n$ )?

GATE 2024 EC question 22: Four small plots, each with axes log (n) (cm^-3) on the vertical axis and 1/T (K^-1) on the horizontal axis (increasing… — Figure from the GATE 2024 EC paper. It is part of the question.

(A)
Figure — plot showing (left to right): a decreasing curve from high $lo g (n)$ , then a flat plateau, then another decreasing curve near the right end (three-region S-shaped curve)
(B)
Figure — plot showing a single flat horizontal line (constant $lo g (n)$ across the whole range)
(C)
Figure — plot showing a flat plateau on the left that bends downward only near the right end (two-region curve, no intrinsic rise on the left)
(D)
Figure — plot showing a single straight line with constant negative slope from upper-left to lower-right (simple monotonic decrease)

Answer and solution

ANSWER

The correct option is (A).

Official answer: ASource: Official answer key, GATE 2024 EC (IISc Bengaluru)

The axes are $lo g (n)$ vertically and $1/ T$ horizontally, so moving left to right on the plot means $T$ decreasing (cooling down) from very high temperature to very low temperature.
High-temperature (intrinsic) region — left part of the plot: at high $T$ , thermally generated electron-hole pairs dominate, so $n \approx n_{i} (T)$ , where $n_{i} (T) = N_{C} N_{V} e^{- E_{g} /2 k T}$ . As $T$ falls (moving right, $1/ T$ increasing), $n_{i}$ falls steeply because of the exponential dependence on $1/ T$ . So $lo g (n)$ starts high on the left and drops sharply.
Mid-temperature (extrinsic/saturation) region — middle of the plot: once $T$ has dropped enough that $n_{i} ≪ N_{D}$ , essentially all donor atoms are ionized and $n \approx N_{D}$ . Since $N_{D}$ is a fixed doping concentration, $lo g (n)$ stays constant here regardless of further changes in $T$ — this produces the flat horizontal plateau.
Low-temperature (freeze-out) region — right part of the plot: as $T$ keeps falling further (very high $1/ T$ ), donor electrons no longer have enough thermal energy to stay ionized ( $E_{C} - E_{D}$ becomes comparable to $k T$ ), so donors progressively freeze out and $n$ drops steeply again with decreasing $T$ (i.e. with increasing $1/ T$ ).
Putting the three regions together left to right: a steep drop (intrinsic), a flat plateau (extrinsic/saturation, $n \approx N_{D}$ ), then another steep drop (freeze-out). This is exactly the three-segment S-shaped curve shown in option (A).
Option (B) shows only a flat line — it misses both the intrinsic and freeze-out regions. Option (C) shows a plateau followed by only one drop — it misses the high- $T$ intrinsic decrease on the left. Option (D) is a single monotonic straight line — it shows no plateau at all, which would be true only for a degenerately or very heavily doped sample, not the non-degenerate case stated in the question. So (A) is the only plot consistent with the physics of a non-degenerately doped semiconductor across the full temperature range.

KEY IDEA

Carrier concentration in a non-degenerately doped semiconductor vs. $1/ T$ shows three distinct regions on a $lo g (n)$ vs $1/ T$ plot: a steep intrinsic-region drop at high $T$ (low $1/ T$ ), a flat extrinsic plateau at $n \approx N_{D}$ in the middle, and a steep freeze-out drop at low $T$ (high $1/ T$ ).

WATCH OUT FOR

Confusing the direction of the horizontal axis — since it is $1/ T$ (not $T$ ) increasing to the right, students sometimes place the intrinsic region on the wrong side, or forget that a non-degenerately doped sample must show a plateau (ruling out option D, which only applies to something like a fully degenerate/heavily doped sample with no distinct freeze-out/extrinsic split), or forget the freeze-out region altogether (ruling out option C).

From the GATE 2024 EC paper. Every question of this paper · Download the full paper · All 9 Semiconductors questions, with notes

GATE 2024 EC Q.22: solution and answer

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