GATE 2022 EC Q.34: solution and answer

GATE 2022 · ECQ.34 · 1 mark · NAT

An ideal MOS capacitor (p-type semiconductor) is shown in the figure. The MOS capacitor is under strong inversion with $V_{G} = 2$ V. The corresponding inversion charge density ( $Q_{I N}$ ) is $2.2 μ C / cm^{2}$ . Assume oxide capacitance per unit area as $C_{O X} = 1.7 μ F / cm^{2}$ . For $V_{G} = 4$ V, the value of $Q_{I N}$ is ______ $μ C / cm^{2}$ (rounded off to one decimal place).

GATE 2022 EC question 34: Cross-section of an ideal MOS capacitor: from left to right, a thin black rectangular 'Metal' gate electrode, then a thin… — Figure from the GATE 2022 EC paper. It is part of the question.

Answer and solution

ANSWER

Q_IN at V_G = 4 V is 5.6 μC/cm².

Official answer: 5.5 to 5.7Source: Official answer key, GATE 2022 EC (IIT Kharagpur)

State the ideal MOS gate-voltage charge balance: $V_{G} = V_{F B} + ϕ_{s} + \frac{Q _{d e p} + Q _{I N}}{C _{O X}}$ , with $V_{F B} = 0$ for an ideal capacitor.
In strong inversion, $ϕ_{s}$ (pinned near $2 ϕ_{F}$ ) and $Q_{d e p}$ (at its maximum depletion value) saturate and no longer change with further increases in $V_{G}$ . Hence beyond the onset of strong inversion, $Δ V_{G} = \frac{Δ Q _{I N}}{C _{O X}}$ .
Given: at $V_{G} = 2$ V, $Q_{I N} = 2.2 μ C / cm^{2}$ ; $C_{O X} = 1.7 μ F / cm^{2}$ .
$Δ V_{G} = 4 V - 2 V = 2 V$ .
$Δ Q_{I N} = C_{O X} \times Δ V_{G} = 1.7 μ F / cm^{2} \times 2 V = 3.4 μ C / cm^{2}$ .
$Q_{I N} (4 V) = Q_{I N} (2 V) + Δ Q_{I N} = 2.2 + 3.4 = 5.6 μ C / cm^{2}$ .

KEY IDEA

In an ideal MOS capacitor, gate voltage divides among flat-band voltage (0, ideal case), surface potential, and the voltage dropped across the oxide due to all semiconductor-side charge: $V_{G} = V_{F B} + ϕ_{s} + \frac{Q _{d e p} + Q _{I N}}{C _{O X}}$ . Once strong inversion is reached, $ϕ_{s}$ and the depletion charge $Q_{d e p}$ saturate (pinned near $2 ϕ_{F}$ and its corresponding maximum value) and stop changing with further increases in $V_{G}$ . So any additional gate voltage beyond the point of strong inversion goes entirely into building up more inversion charge: $Δ V_{G} = \frac{Δ Q _{I N}}{C _{O X}}$ .

WATCH OUT FOR

Treating the MOS capacitor as a simple linear capacitor and assuming $Q_{I N}$ is proportional to the full gate voltage $V_{G}$ (i.e., computing $Q_{I N} (4 V) = 2 \times Q_{I N} (2 V) = 4.4 μ C / cm^{2}$ ) — this ignores that $ϕ_{s}$ and $Q_{d e p}$ have already saturated by the time strong inversion is reached, so only the incremental voltage $Δ V_{G}$ (not the total $V_{G}$ ) maps to incremental inversion charge.

From the GATE 2022 EC paper. Every question of this paper · Download the full paper · All 7 MOS devices questions, with notes

GATE 2022 EC Q.34: solution and answer

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