GATE 2020 EC Q.6: solution and answer

GATE 2020 · ECQ.6 · 1 mark · MCQ

A single crystal intrinsic semiconductor is at a temperature of 300 K with effective density of states for holes twice that of electrons. The thermal voltage is 26 mV. The intrinsic Fermi level is shifted from mid-bandgap energy level by

(A)
18.02 meV.
(B)
9.01 meV.
(C)
13.45 meV.
(D)
26.90 meV.

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Answer and solution

ANSWER

The intrinsic Fermi level is shifted from mid-bandgap by 9.01 meV, corresponding to option B.

Official answer: BSource: Official answer key, GATE 2020 EC (IIT Delhi)

Mid-bandgap energy: $E_{mi d} = \frac{E _{c} + E _{v}}{2}$ .
The intrinsic Fermi level satisfies $N_{c} exp (- \frac{E _{c} - E _{i}}{k T}) = N_{v} exp (- \frac{E _{i} - E _{v}}{k T})$ (equal electron and hole concentrations in an intrinsic semiconductor). Solving for $E_{i}$ gives $E_{i} = \frac{E _{c} + E _{v}}{2} + \frac{k T}{2} ln (\frac{N _{v}}{N _{c}})$ , so the shift from mid-gap is $E_{i} - E_{mi d} = \frac{k T}{2} ln (\frac{N _{v}}{N _{c}})$ .
Given $N_{v} = 2 N_{c}$ , so $\frac{N _{v}}{N _{c}} = 2$ .
Shift magnitude $= \frac{k T}{2} ln (2) = \frac{26 meV}{2} \times 0.693 = 13 meV \times 0.693$ .
$= 9.01 meV$ .
Since $N_{v} > N_{c}$ , $ln (N_{v} / N_{c}) > 0$ , so $E_{i}$ is shifted above mid-gap (toward the conduction band edge $E_{c}$ ) by this amount, magnitude $9.01 meV$ .

KEY IDEA

For an intrinsic semiconductor, equating $n_{i} = N_{c} e^{- (E_{c} - E_{i}) / k T}$ with $p_{i} = N_{v} e^{- (E_{i} - E_{v}) / k T}$ gives $E_{i} = \frac{E _{c} + E _{v}}{2} + \frac{k T}{2} ln (\frac{N _{v}}{N _{c}})$ , so a larger $N_{v}$ relative to $N_{c}$ pushes $E_{i}$ above mid-gap, toward $E_{c}$ .

WATCH OUT FOR

Getting the magnitude right (using $k T ln (2) /2$ ) but reversing the direction — assuming a larger $N_{v}$ pulls $E_{i}$ toward the valence band, when the correct derivation shows it pushes $E_{i}$ above mid-gap, toward the conduction band. Since the options here ask only for magnitude, this sign error does not change which option is correct, but it must not be stated as fact.

From the GATE 2020 EC paper. Every question of this paper · Download the full paper · All 9 Semiconductors questions, with notes

GATE 2020 EC Q.6: solution and answer

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