GATE 2020 EC Q.7: solution and answer

GATE 2020 · ECQ.7 · 1 mark · MCQ

Consider the recombination process via bulk traps in a forward biased pn homojunction diode. The maximum recombination rate is $U_{ma x}$ . If the electron and the hole capture cross-sections are equal, which one of the following is FALSE?

(A)
With all other parameters unchanged, $U_{ma x}$ decreases if the intrinsic carrier density is reduced.
(B)
$U_{ma x}$ occurs at the edges of the depletion region in the device.
(C)
$U_{ma x}$ depends exponentially on the applied bias.
(D)
With all other parameters unchanged, $U_{ma x}$ increases if the thermal velocity of the carriers increases.

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Answer and solution

ANSWER

The FALSE statement is (B): $U_{ma x}$ does NOT occur at the edges of the depletion region — it occurs in the interior, at the point where the electron and hole concentrations are equal ( $n \approx p$ ).

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

Setup — SRH recombination through bulk traps. For a single bulk trap level, the Shockley–Read–Hall recombination rate is $U = \frac{n p - n _{i}^{2}}{τ _{p} ( n + n _{1} ) + τ _{n} ( p + p _{1} )}$ where the capture-limited lifetimes are $τ_{n} = \frac{1}{σ _{n} v _{t h} N _{t}}$ and $τ_{p} = \frac{1}{σ _{p} v _{t h} N _{t}}$ .
Equal capture cross-sections. Given $σ_{n} = σ_{p}$ , the two lifetimes become equal: $τ_{n} = τ_{p} = τ$ . The rate simplifies to $U = \frac{n p - n _{i}^{2}}{τ ( n + p + n _{1} + p _{1} )} .$
Where the maximum occurs. Across the depletion region of a forward-biased junction, the product of carrier concentrations stays constant, $n p = n_{i}^{2} e^{q V / k T}$ , even though $n$ and $p$ individually vary from one edge to the other. For appreciable forward bias, $n_{1}, p_{1} ≪ n, p$ , so $U$ is maximized where $n + p$ is minimized subject to $n p$ fixed. By the AM–GM inequality, this minimum occurs at $n = p = n p = n_{i} e^{q V /2 k T}$ — an interior point of the space-charge region, not at either edge (the n-side edge has $n ≫ p$ ; the p-side edge has $p ≫ n$ ).
Value of $U_{ma x}$ . Substituting $n = p = n_{i} e^{q V /2 k T}$ : $U_{ma x} = \frac{n _{i}^{2} e ^{q V / k T} - n _{i}^{2}}{2 τ n _{i} e ^{q V /2 k T}} \approx \frac{n _{i}}{2 τ} e^{q V /2 k T} for q V ≫ k T .$
Check (A): $U_{ma x} \propto n_{i}$ directly, so reducing the intrinsic carrier density $n_{i}$ (all else unchanged) decreases $U_{ma x}$ — statement is TRUE.
Check (C): $U_{ma x} \propto e^{q V /2 k T}$ , an exponential dependence on the applied bias $V$ — statement is TRUE.
Check (D): Since $τ \propto 1/ v_{t h}$ , we get $U_{ma x} \propto v_{t h}$ ; increasing the thermal velocity of the carriers increases $U_{ma x}$ — statement is TRUE.
Check (B): The location of maximum recombination is the interior point where $n \approx p$ , not the edges of the depletion region — statement is FALSE.
Conclusion. The FALSE statement is (B).

KEY IDEA

In SRH recombination through bulk traps, with equal capture cross-sections ( $τ_{n} = τ_{p} = τ$ ), the recombination rate $U = \frac{n p - n _{i}^{2}}{τ ( n + n _{1} ) + τ ( p + p _{1} )}$ is maximized (for a fixed product $n p$ , which is constant across the depletion region under bias) at the point where $n \approx p$ — by AM-GM, $n + p$ is minimized when $n = p = n p$ . This point of equal carrier concentrations lies in the interior of the space-charge region, not at its edges (where one carrier type dominates), so statement (B) is the false one.

WATCH OUT FOR

Confusing "maximum recombination" with "maximum generation/depletion edge effects," or assuming the peak SRH rate tracks the majority-carrier profile (which would incorrectly place it at an edge). Students also sometimes forget that with $σ_{n} = σ_{p}$ the trap becomes an effective midgap-like center for symmetry purposes, which is what makes the $n \approx p$ condition the maximizing condition rather than some edge-biased condition.

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GATE 2020 EC Q.7: solution and answer

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